0$, given streaming access to an array of length $n$ provides a $(1+\\delta)$-multiplicative approximation to the \\emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((\\log^2 n)/\\delta)$ space.","The previous best known approximation using polylogarithmic space was a multiplicative 2-factor.","Our algorithm can be used to estimate the length of the LIS to within an additive $\\delta n$ for any $\\delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$.","Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time.","It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS.","We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$.","Our algorithm works in the asymmetric setting (inspired by \\cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$.","More precisely, it gives an additive-$\\delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(\\log^2 n)/\\delta)$, where $k$ is the maximum number of times any one symbol appears in $y$."],["Introduction","Two classic optimization problems concerning subsequences (substrings) of arrays (strings) are the longest increasing subsequence (LIS) and longest common subsequence (LCS) problems.","A string of length $n$ over alphabet $\\Sigma $ is represented as a function $x:[n] \\rightarrow \\Sigma $ .","A subsequence of length $k$ is a string $x(i_1)x(i_2)\\ldots x(i_k)$ , where $1 \\le i_1 < i_2 < \\cdots < i_k \\le n$ .","In the LIS problem, the alphabet $\\Sigma $ comes equipped with a (total or partial) order $\\triangleleft $ , and we look for the longest subsequence whose terms are in increasing order.","In the LCS problem we are given two strings $x$ and $y$ and look for the longest string which is a subsequence of each of them.","Note that the LIS of $x$ is the LCS of $x$ and its sorted version.","Both of these problems can be solved by dynamic programs.","The LIS can be found on $O(n\\log n)$ time [28], [17], [2].","This is known to be optimal, even for (comparison based) algorithms that only determine the length of the LIS [27].","The LCS problem has a fairly direct $O(n^2)$ algorithm [11], which can be improved to $O(n^2/\\log ^2n)$ [24], [9].","It is a notoriously difficult open problem to improve this bound, or prove some matching lower bounds.","It is often natural to focus on the complements of the LIS and LCS lengths, which are related to some notion of distances between strings.","The distance to monotonicity of (the length $n$ string) $x$ , denoted $DM(x)$ is defined to be $n-LIS(x)$ , and is the the minimum number of values that need to be changed to make $x$ .","The (insertion-deletion) edit distance of (two length $n$ strings) $x,y$ , denoted $E(x,y)$ is defined to be $n-LCS(x,y)$ and is the minimum number of insertions and deletions needed to change one string into the other.","(Note that $E(x,y)$ is bounded between $L(x,y)$ and $2L(x,y)$ where $L(x,y)$ is the Levenshtein distance, where insertions, deletions, and substitutions are allowed.)","Of course the algorithmic problems of exactly computing $LIS(x)$ and $DM(x)$ are equivalent, but approximating them can be very different.","In recent years, there has been a lot of attention on giving approximate solutions for LIS and LCS that are much more efficient that the basic dynamic programming solutions.","Any improved results for LCS would be very interesting, since the best known quadratic time solution is infeasible for very large strings.","These problems can be studied in a variety of settings - sampling, streaming, and communication.","The streaming setting has been the focus of many results [20], [30], [18], [13].","The model for the LIS is that we are allowed one (or constant) passes over the input string $x$ , and only have access to sublinear storage.","The usual formulation of LCS in the streaming model postulates that we have only one-way access to both strings $x$ and $y$ .","We consider an alternative asymmetric model in which we have one-way access to string $x$ (called the input string) but random accesss to string $y$ (called the fixed string).","This model is more powerful that the standard one, but it is still far from clear how to obtain space efficient approximations to $E(x,y)$ in this model.","(This model was inspired by recent work of [6] concerning the time complexity of approximating edit distance in the random access model.","One part of their work introduced an asymmetric version of the random access model in which one pays only for accesses to one of the strings, and established time lower bounds for good approximations that hold even in this more powerful model.)"],["Results","Our first result is a streaming algorithm for approximating the distance to monotonicity.","There is a randomized one-pass streaming algorithm that for any $\\delta >0$ , takes as input an array (of length $n$ ), makes one-pass through the array, uses space $O(\\delta ^{-1}\\log ^2 n)$ and with error probability $n^{-\\Omega (1)}$ outputs an estimate to $DM(x)$ that is between $DM(x)$ and $(1+\\delta )DM(x)$ .","Previously there was a polylogarithmic time algorithm that gave a factor 2-approximation [13], and an algorithm that gave arbitrarily good multiplicative approximations to $LIS(x)$ (which is harder than approximating $DM(x)$ ) but required $\\Omega (\\sqrt{n})$ space [20].","The improvement in the approximation ratio from 2 to $1+\\delta $ (for polylogarithmic space algorithms) is not just “chipping away” at a constant, but provides a significant qualitative difference: previous polylogarithmic space algorithms might return an estimate of 0 when the LIS length is $n/2$ , while our algorithm can detect increasing subsequences of length a small fraction of $n$ .","More precisely, it is easy to see that if $V$ is an estimate of $DM(x)$ that is between $DM(x)$ and $(1+\\delta )DM$ then $n-V$ is within an additive $\\frac{\\delta }{1+\\delta } n$ of $LIS(x)$ , and so our algorithm can provide an estimation interval for $LIS(x)/n$ of arbitrarily small width.","The previous polylogarithmic time streaming algorithm only gave such an algorithm for $\\delta \\ge 1$ , which only guarantees an estimation interval for $LIS(x)/n$ of width 1/2.","The algorithm promised by Theorem REF is derived as a special case of a more general algorithm (Theorem REF ) that finds increasing sequences in partial orders.","This algorithm will also be applied to give a good (additive) approximation algorithm for edit distance in the asymmetric setting, whose space is polylogarithmic in the case that no symbol appears many times in the fixed string.","Let $\\delta \\in (0,1]$ .","Suppose $y$ is a fixed string of length $n$ and $x$ an input string of length $n$ to which we have streaming access.","There is a randomized algorithm that makes one pass through $x$ and, with error probability $n^{-\\Omega (1)}$ , outputs an additive $\\delta n$ -approximation to $E(x,y)$ and uses space $O(k\\log ^2 n/\\delta )$ where $k$ is the maximum number of times any symbol appears in $y$ .","There is a deterministic algorithm that runs in space $O(\\sqrt{(n\\log n)/\\delta })$ -space and outputs a $(1+\\delta )$ -multiplicative (which is also a $\\delta n$ -additive) approximation to $E(x,y)$ ."],["Techniques","A notable feature of our algorithm is its conceptual simplicity.","The pseudocode for the LIS approximation is just a few lines.","The algorithm has parameters $\\alpha (i,t)$ for $1 \\le i < t$ ) whose exact formula is a bit cumbersome to state at this point.","We set $\\alpha (i,t)$ to be 0 for $i \\ge t-O(\\log (n))$ and approximately $1/(t-i)$ otherwise.","The algorithm maintains a set of indices $R$ , and for each $i \\in R$ , we store $x(i)$ and an estimate $r(i)$ of $DM(x[1,i])$ , where $x[1,i]$ is the length $i$ prefix of $x$ .","For convenience, we add dummy elements $x(0) = x(n+1) = -\\infty $ and begin with $R = \\lbrace 0\\rbrace $ .","For each time $t \\ge 1$ , we perform the following update: Define $R^{\\prime } = \\lbrace i \\in R | x(i) \\le x(t)\\rbrace $ .","Set $r(t)=\\min _{i \\in R^{\\prime }}(r(i)+t-1-i)$ .","$R \\longleftarrow R \\cup \\lbrace t\\rbrace $ .","Remove each $i \\in R$ independently with probability $\\alpha (i,t)$ .","The final output is $r(n+1)$ .","The space used by the algorithm is (essentially) the maximum size of $|R|$ .","The update time is determined by step 1, which runs in time $O(|R|)$ .","Without the third step, the algorithm is a simple quadratic time exact algorithm for $DM(x)$ using linear space.","(The $O(n\\log n)$ time algorithms [17], [2] also work in a streaming fashion, but store data much more cleverly.)","More precisely, at step $t$ , $R=\\lbrace 0,\\ldots ,t\\rbrace $ and for each $i \\le t$ , $r(i)=DM(x[1,i])$ .","The third step reduces the set $R$ , thereby reducing the space of the algorithm.","The space used by the algorithm is (essentially) the maximum size of $R$ .","Intuitively, the algorithm “forgets” $(x(i),r(i))$ for those $i$ removed from $R$ .","The set $R$ of remembered indices is a subset of $[1,t]$ whose density decays as one goes back in time from the present time $t$ .","When we compute $r(t)$ , it may no longer be equal to the distance to monotonicity of the prefix of $x$ of length $i$ , but it will be at least this value.","This forgetting strategy is tailored to ensure that $r(t)$ is also at most a $(1+\\delta )$ -factor away from the distance to monotonicity.","We also ensure that that (with high probability) the set $R$ does not exceed size $O(\\delta ^{-1}\\log ^2 n)$ Just to give an indication of the difficulty, consider an algorithm that forgets uniformly at random.","At some time $t$ , the set of remembered indices $R$ is a uniform random set of size $O(\\delta ^{-1}\\log ^2 n)$ up to index $t$ .","These are used to compute $r(t)$ and include $t$ in $R$ .","The algorithm then forgets a uniform random index in $R$ to maintain the space bound.","Since we want to get a $(1+\\delta )$ -factor approximation, the algorithm must be able to detect an LIS of length $\\Omega (\\delta n)$ .","Of the indices in $R$ (up to time $t$ ), it is possible that around a $O(\\delta )$ -fraction of them are in the LIS.","Suppose we reach a small stretch of indices not on the LIS.","If this has size even $\\textrm {poly}(\\delta ^{-1}\\log n)$ , it is likely that all LIS indices in $R$ are forgotten.","But how do we selectively remember the LIS indices without knowing the LIS in advance?","That is the challenge of the forgetting strategy.","All past polylogarithmic space algorithms [20], [13] for LIS use combinatorial characterizations of increasing sequences based on inversion counting [14], [12], [26], [1].","While this is a very powerful technique, it does not lead to accurate approximations for the LIS, and (apparently) do not yield any generalizations to LCS.","The idea of remembering selected information about the sequence that becomes sparser as one goes back in time was first used by [20] for the inversion counting approach.","Our work seems to be the first to use this to directly mimic the dynamic program, though the idea is quite natural and has almost certainly been considered before.","The main contribution here is to analyze this algorithm, and determine the values of parameters that allow it to be both space efficient and a good approximation.","This line of thinking can be exploited to deal with asymmetric streaming LCS.","We construct a simple reduction of LCS to finding the longest chain in a specific partial order.","This reduction has a streaming implementation, so the input stream can be directly seen as just elements of this resulting partial order.","This reduction blows up the size of the input, and the size of the largest chain can become extremely small.","If each symbol occurs $k$ times in $x$ and $y$ , then the resulting partial order has $nk$ elements.","Nonetheless, the longest chain still has length at most $n$ .","We require very accurate estimates for the length of the longest chain.","This is where the power of the $(1+\\delta )$ -approximation comes in.","We can choose $\\delta $ to be much smaller to account for the input blow up, and still get a good approximation.","Note that if we only had a $1.01$ -approximation for the longest chain problem, this reduction would not be useful.","Our $\\widetilde{O}(\\sqrt{n})$ -space algorithm also works according to the basic principle of following a dynamic program, although it uses one different from the previous algorithms.","This can be thought of as generalization of the $\\widetilde{O}(\\sqrt{n})$ -space algorithm for LIS[20].","We maintain a $\\widetilde{O}(\\sqrt{n})$ -space deterministic sketch of the data structure maintained by the exact algorithm.","By breaking the stream up into the right number of chunks, we can update this sketch using $\\widetilde{O}(\\sqrt{n})$ -space.","Previous work The study of LIS and LCS in the streaming setting was initiated by Liben-Nowell et al [23], although their focus was mostly on exactly computing the LIS.","Sun and Woodruff [30] improved upon these algorithms and lower bounds and also proved bounds for the approximate version.","Most relevant for our work, they prove that randomized protocols that compute a $(1+\\varepsilon )$ -approximation of the LIS length essentially require $\\Omega (\\varepsilon ^{-1}\\log n)$ .","Gopalan et al [20] provide the first polylogarithmic space algorithm that approximates the distance to monotonicity.","This was based on inversion counting ideas in [26], [1].","Ergun and Jowhari [13] give a 2-approximation using the basic technique of inversion counting, but develop a different algorithm.","Gál and Gopalan [18] and independently Ergun and Jowhari [13] proved an $\\Omega (\\sqrt{n})$ lower bound for deterministic protocols that approximate that LIS length up to a multiplicative constant factor.","For randomized protocols, the Sun and Woodruff bound of $\\Omega (\\log n)$ is the best known.","One of the major open problems is to get a $o(\\sqrt{n})$ space randomized protocol (or an $\\Omega (\\sqrt{n})$ lower bound) for constant factor approximations for the LIS length.","Note that our work does not imply anything non-trivial for this problem.","We are unaware of any lower bounds for estimating the distance to monotonicity in the streaming setting.","A significant amount of work has been done in studying the LIS (or rather, the distance to monotonicity) in the context of property testing [14], [12], [15], [26], [1].","The property of monotonicity has been studied over a variety of domains, of which the boolean hypercube and the set $[n]$ (which is the LIS setting) have usually been of special interest [19], [12], [16], [22], [1], [26], [10].","In previous work, the authors of this paper found a $(1+\\delta )$ -multiplicative approximation algorithm for the distance to monotonicity (in the random access model) that runs in time $O(\\textrm {poly}\\log (n))$ [29].","As the present result does for the streaming model, that result also improved on the previous best factor 2 approximation for that model.","Despite the superficial similarity between the statement of results, the models considered in these two papers are quite different, and the algorithm we give here in the streaming model is completely different from the complicated algorithm we gave in the sublinear time model.","The LCS and edit distance have an extemely long and rich history, especially in the applied domain.","We point the interesting reader out to [21], [25] for more details.","Andoni et al [6] achieved a breakthrough by giving a near-linear time algorithm (in the random access model) that gives polylogarithmic time approximations for the edit distance.","This followed a long line of results well documented in [6].","They initiate the study of the asymmetric edit distance, where one string is known and we are only charged for accesses to the other string.","For the case of non-repetitive strings, there has been a body of work on studying the Ulam distance between permutations [4], [5], [3], [7].","Paths in posets We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).","We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.","Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.","This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP.","Weighted $P$ -sequences and the approximate minimum-defect path problem We use $P$ to denote a fixed set endowed with a partial order $\\triangleleft $ .","The partial order relation is given by an oracle which, given $u,v \\in P$ outputs $u \\triangleleft v$ or $\\lnot (u \\triangleleft v)$ .","For a natural number $n$ we write $[n]$ for the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ .","A sequence ${\\bf \\sigma }=(\\sigma (1),\\ldots ,\\sigma (n)) \\in P$ is called a $P$ -sequence.","The number of terms ${\\bf \\sigma }$ is called the length of ${\\bf \\sigma }$ and is denoted $|{\\bf \\sigma }|$ ; we normally use $n$ to denote $|{\\bf \\sigma }|$ .","A weighted $P$ -sequence consists of a $P$ -sequence ${\\bf \\sigma }$ together with a sequence $(w(1),\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .","In all our final applications $w(i)$ will always be 1.","Nonetheless, we solve this slightly more general weighted version.","We have the following additional definitions: For $t \\in [n]$ , ${\\bf \\sigma }_{\\le t}$ denotes the sequence $(\\sigma _1,\\ldots ,\\sigma _t)$ .","Also for $J \\subseteq [n]$ , $J_{\\le t}$ denotes the set $J \\cap \\lbrace 1,\\ldots ,t\\rbrace $ .","For $J \\subseteq [n]$ , $w(J)=\\sum _{j \\in J} w(j)$ .","The digraph $D=D({\\bf \\sigma })$ associated to the $P$ -sequence ${\\bf \\sigma }$ has vertex set $[n]$ (where $n=|{\\bf \\sigma }|$ ) and arc set $\\lbrace i \\rightarrow j: i0$ .","The output is a number $A$ such that: ${\\rm Prob}[A \\in [\\mbox{\\rm min-defect}({\\bf \\sigma },w),(1+\\delta ) \\mbox{\\rm min-defect}({\\bf \\sigma },w)]] \\ge 1-\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is said to be a one-sided error algorithm.","Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.","For the AMDP, the input consists of the parameters $\\delta $ and $\\gamma $ together with a sequence of $n$ pairs $((\\sigma (t),w(t)):t \\in [n])$ .","We think of the input as arriving in a sequence of discrete time steps, where $\\delta ,\\gamma $ arrive at time step 0 and for $t \\in [n]$ , $(\\sigma (t),w(t))$ arrives at time step $t$ .","The main complexity parameter of interest is the auxiliary memory needed.","For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.","Associated to a weighted $P$ -sequence $({\\bf \\sigma },w)$ we define the parameter: $\\rho =\\rho (w)=\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\rho =n^{O(1)}$ .","The main technical theorem about AMDP is the following.","There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\frac{\\ln (n/\\gamma )\\ln (\\rho )}{\\delta })$ .","In particular, if $\\rho =n^{O(1)}$ and $\\gamma =1/n^{O(1)}$ then the space is $O(\\frac{(\\ln (n))^2}{\\delta })$ .","The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","We first review this dynamic program.","Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .","Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.","Set $w(n+1)=0$ .","We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.","We initialize $s(0) = 0$ and $W(0) = 0$ .","For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .","It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .","In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .","The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).","The polylog space streaming algorithm We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .","Our approximation algorithm is a natural variant of the exact algorithm.","At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .","The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.","The amount of space used by the algorithm is proportional to the maximum size of $R$ .","We first define the probabilities $p(i,t)$ .","Similar quantities were defined in [20].","$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .","We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .","The following update is performed for each time step $t \\in [n+1]$ .","The final output is just $r(n+1)$ .","$W(t)=W(t-1)+w(t)$ .","$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .","The index $t$ is inserted in $R$ .","Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .","On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .","The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .","The above theorem does not exactly give what was promised in Theorem REF .","For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.","To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.","This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .","We now prove Theorem REF .","The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","The second part will be proved in the two subsection.","The final assertion of Theorem REF showing the space bound is deferred to Appendix .","Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.","Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .","Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .","We need some additional definitions.","For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .","Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .","We assume without loss of generality that $n+1 \\in C$ .","We write $R^t$ for the subset $R$ at the end of step $t$ .","Note that $R^t \\subseteq [t]$ .","We define $F^t=[t]-R^t$ .","An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.","Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .","We write $U^t$ for the set of indices that are unsafe at time $t$ .","An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.","We denote the set of unsafe indices by $U$ .","On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .","On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .","We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .","Assume $t \\ge 1$ and that the result holds for $j 0$ .","We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .","There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.","The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.","The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].","Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .","There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .","Our aim is to implement (an approximation of) this algorithm in sublinear space.","As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.","In some sense, we only maintain a small subset of the partial solutions.","Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .","We convert an LCS into finding the longest chain in a set of points $P$ .","We construct a set of anchor points in the plane, which may not be in $P$ .","Our aim is to just maintain the longest chain between pairs of anchor points.","Let $\\delta > 0$ be some fixed parameter.","We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .","For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.","$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.","Our analysis remains identical..","So we can drop the floors used in the definition of $S_i$ .","Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .","We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .","This is the set of anchor points that we discussed earlier.","Note that they are placed according to a geometric grid.","A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.","There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .","We have a basic claim about the LCS of two strings (proof deferred to Appendix ).","This gives us a simple bound on the defect that we shall exploit.","Lemma makes an important argument.","It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.","Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.","Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .","The defect $n-r$ is at least $|j_a - i|$ .","There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .","We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.","Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .","Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .","Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .","We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .","Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .","If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .","Otherwise, it is not added.","Note that $j_b \\ge \\ell _p$ , simply by construction.","The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.","It now remains to bound the defect of $L^{\\prime }$ .","Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .","Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .","This means that ${j_b} > \\ell _{p}$ .","Let $i_c$ be the smallest index larger than $p\\bar{n}$ .","So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .","Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .","We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .","Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .","By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .","The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .","The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .","$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.","The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .","Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .","Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.","The total running time is $O(n\\bar{n}^2)$ .","Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.","He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.","The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .","A union bound over all $t$ proves the third assertion of Theorem REF .","Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .","Fix $t \\in [n]$ .","Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .","For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.","Then $|R_t|=\\sum _{i \\le t} Z_i$ .","Let $\\mu =\\mathbb {E}[|R_t|]$ .","Below we show that $\\mu \\le M$ .","We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .","Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .","Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).","It remains to show that $\\mu \\le M$ .","We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.","For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .","We prove by backwards induction on $r$ .","For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.","Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .","We start with a technical statement.","$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .","This completes the proof.","$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.","Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .","(of Claim ) Assume wlog that $i_k \\ge j_k$ .","Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .","This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .","Hence the defect is at least $i_k - j_k$ .","Repeating over all $k$ , we complete the proof.","$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).","If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .","Suppose neither of these are true.","Then $j_a > i_a \\ge i$ .","Let us focus on the characters of $x$ that are not matched.","No character of $x$ with index in $[i,i_a)$ is matched.","The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).","So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .","So the number of unmatched characters in $x$ is at least $j_a - i$ .","$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .","We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .","This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .","The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .","Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","This is obtained by just maximizing over all possible $j^{\\prime }$ .","We now apply Claim .","We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .","Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.","This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .","This can be done for all $y(j)$ , $j \\in S_i$ .","The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .","$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.","Consider the index $i\\bar{n}$ .","Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","This requires space $O(|S_i|) = O(\\bar{n}))$ .","By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.","Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.","Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .","The space bound is $O(\\bar{n})$ .","The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .","$\\Box $"],["Paths in posets","We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).","We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.","Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.","This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP."],["Weighted $P$ -sequences and the approximate minimum-defect path problem","We use $P$ to denote a fixed set endowed with a partial order $\\triangleleft $ .","The partial order relation is given by an oracle which, given $u,v \\in P$ outputs $u \\triangleleft v$ or $\\lnot (u \\triangleleft v)$ .","For a natural number $n$ we write $[n]$ for the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ .","A sequence ${\\bf \\sigma }=(\\sigma (1),\\ldots ,\\sigma (n)) \\in P$ is called a $P$ -sequence.","The number of terms ${\\bf \\sigma }$ is called the length of ${\\bf \\sigma }$ and is denoted $|{\\bf \\sigma }|$ ; we normally use $n$ to denote $|{\\bf \\sigma }|$ .","A weighted $P$ -sequence consists of a $P$ -sequence ${\\bf \\sigma }$ together with a sequence $(w(1),\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .","In all our final applications $w(i)$ will always be 1.","Nonetheless, we solve this slightly more general weighted version.","We have the following additional definitions: For $t \\in [n]$ , ${\\bf \\sigma }_{\\le t}$ denotes the sequence $(\\sigma _1,\\ldots ,\\sigma _t)$ .","Also for $J \\subseteq [n]$ , $J_{\\le t}$ denotes the set $J \\cap \\lbrace 1,\\ldots ,t\\rbrace $ .","For $J \\subseteq [n]$ , $w(J)=\\sum _{j \\in J} w(j)$ .","The digraph $D=D({\\bf \\sigma })$ associated to the $P$ -sequence ${\\bf \\sigma }$ has vertex set $[n]$ (where $n=|{\\bf \\sigma }|$ ) and arc set $\\lbrace i \\rightarrow j: i0$ .","The output is a number $A$ such that: ${\\rm Prob}[A \\in [\\mbox{\\rm min-defect}({\\bf \\sigma },w),(1+\\delta ) \\mbox{\\rm min-defect}({\\bf \\sigma },w)]] \\ge 1-\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is said to be a one-sided error algorithm.","Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.","For the AMDP, the input consists of the parameters $\\delta $ and $\\gamma $ together with a sequence of $n$ pairs $((\\sigma (t),w(t)):t \\in [n])$ .","We think of the input as arriving in a sequence of discrete time steps, where $\\delta ,\\gamma $ arrive at time step 0 and for $t \\in [n]$ , $(\\sigma (t),w(t))$ arrives at time step $t$ .","The main complexity parameter of interest is the auxiliary memory needed.","For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.","Associated to a weighted $P$ -sequence $({\\bf \\sigma },w)$ we define the parameter: $\\rho =\\rho (w)=\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\rho =n^{O(1)}$ .","The main technical theorem about AMDP is the following.","There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\frac{\\ln (n/\\gamma )\\ln (\\rho )}{\\delta })$ .","In particular, if $\\rho =n^{O(1)}$ and $\\gamma =1/n^{O(1)}$ then the space is $O(\\frac{(\\ln (n))^2}{\\delta })$ .","The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","We first review this dynamic program.","Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .","Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.","Set $w(n+1)=0$ .","We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.","We initialize $s(0) = 0$ and $W(0) = 0$ .","For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .","It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .","In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .","The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).","The polylog space streaming algorithm We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .","Our approximation algorithm is a natural variant of the exact algorithm.","At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .","The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.","The amount of space used by the algorithm is proportional to the maximum size of $R$ .","We first define the probabilities $p(i,t)$ .","Similar quantities were defined in [20].","$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .","We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .","The following update is performed for each time step $t \\in [n+1]$ .","The final output is just $r(n+1)$ .","$W(t)=W(t-1)+w(t)$ .","$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .","The index $t$ is inserted in $R$ .","Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .","On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .","The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .","The above theorem does not exactly give what was promised in Theorem REF .","For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.","To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.","This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .","We now prove Theorem REF .","The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","The second part will be proved in the two subsection.","The final assertion of Theorem REF showing the space bound is deferred to Appendix .","Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.","Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .","Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .","We need some additional definitions.","For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .","Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .","We assume without loss of generality that $n+1 \\in C$ .","We write $R^t$ for the subset $R$ at the end of step $t$ .","Note that $R^t \\subseteq [t]$ .","We define $F^t=[t]-R^t$ .","An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.","Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .","We write $U^t$ for the set of indices that are unsafe at time $t$ .","An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.","We denote the set of unsafe indices by $U$ .","On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .","On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .","We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .","Assume $t \\ge 1$ and that the result holds for $j 0$ .","We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .","There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.","The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.","The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].","Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .","There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .","Our aim is to implement (an approximation of) this algorithm in sublinear space.","As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.","In some sense, we only maintain a small subset of the partial solutions.","Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .","We convert an LCS into finding the longest chain in a set of points $P$ .","We construct a set of anchor points in the plane, which may not be in $P$ .","Our aim is to just maintain the longest chain between pairs of anchor points.","Let $\\delta > 0$ be some fixed parameter.","We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .","For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.","$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.","Our analysis remains identical..","So we can drop the floors used in the definition of $S_i$ .","Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .","We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .","This is the set of anchor points that we discussed earlier.","Note that they are placed according to a geometric grid.","A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.","There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .","We have a basic claim about the LCS of two strings (proof deferred to Appendix ).","This gives us a simple bound on the defect that we shall exploit.","Lemma makes an important argument.","It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.","Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.","Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .","The defect $n-r$ is at least $|j_a - i|$ .","There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .","We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.","Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .","Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .","Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .","We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .","Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .","If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .","Otherwise, it is not added.","Note that $j_b \\ge \\ell _p$ , simply by construction.","The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.","It now remains to bound the defect of $L^{\\prime }$ .","Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .","Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .","This means that ${j_b} > \\ell _{p}$ .","Let $i_c$ be the smallest index larger than $p\\bar{n}$ .","So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .","Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .","We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .","Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .","By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .","The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .","The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .","$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.","The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .","Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .","Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.","The total running time is $O(n\\bar{n}^2)$ .","Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.","He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.","The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .","A union bound over all $t$ proves the third assertion of Theorem REF .","Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .","Fix $t \\in [n]$ .","Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .","For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.","Then $|R_t|=\\sum _{i \\le t} Z_i$ .","Let $\\mu =\\mathbb {E}[|R_t|]$ .","Below we show that $\\mu \\le M$ .","We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .","Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .","Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).","It remains to show that $\\mu \\le M$ .","We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.","For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .","We prove by backwards induction on $r$ .","For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.","Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .","We start with a technical statement.","$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .","This completes the proof.","$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.","Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .","(of Claim ) Assume wlog that $i_k \\ge j_k$ .","Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .","This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .","Hence the defect is at least $i_k - j_k$ .","Repeating over all $k$ , we complete the proof.","$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).","If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .","Suppose neither of these are true.","Then $j_a > i_a \\ge i$ .","Let us focus on the characters of $x$ that are not matched.","No character of $x$ with index in $[i,i_a)$ is matched.","The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).","So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .","So the number of unmatched characters in $x$ is at least $j_a - i$ .","$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .","We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .","This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .","The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .","Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","This is obtained by just maximizing over all possible $j^{\\prime }$ .","We now apply Claim .","We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .","Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.","This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .","This can be done for all $y(j)$ , $j \\in S_i$ .","The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .","$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.","Consider the index $i\\bar{n}$ .","Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","This requires space $O(|S_i|) = O(\\bar{n}))$ .","By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.","Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.","Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .","The space bound is $O(\\bar{n})$ .","The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .","$\\Box $"],["The algorithm","Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","We first review this dynamic program."],["Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$","It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .","Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.","Set $w(n+1)=0$ .","We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.","We initialize $s(0) = 0$ and $W(0) = 0$ .","For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .","It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .","In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .","The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ )."],["The polylog space streaming algorithm","We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .","Our approximation algorithm is a natural variant of the exact algorithm.","At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .","The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.","The amount of space used by the algorithm is proportional to the maximum size of $R$ .","We first define the probabilities $p(i,t)$ .","Similar quantities were defined in [20].","$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .","We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .","The following update is performed for each time step $t \\in [n+1]$ .","The final output is just $r(n+1)$ .","$W(t)=W(t-1)+w(t)$ .","$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .","The index $t$ is inserted in $R$ .","Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .","On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .","The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .","The above theorem does not exactly give what was promised in Theorem REF .","For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.","To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.","This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .","We now prove Theorem REF .","The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .","The second part will be proved in the two subsection.","The final assertion of Theorem REF showing the space bound is deferred to Appendix .","Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.","Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .","Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .","We need some additional definitions.","For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .","Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .","We assume without loss of generality that $n+1 \\in C$ .","We write $R^t$ for the subset $R$ at the end of step $t$ .","Note that $R^t \\subseteq [t]$ .","We define $F^t=[t]-R^t$ .","An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.","Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .","We write $U^t$ for the set of indices that are unsafe at time $t$ .","An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.","We denote the set of unsafe indices by $U$ .","On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .","On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .","We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .","Assume $t \\ge 1$ and that the result holds for $j 0$ .","We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .","There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.","The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.","The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].","Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .","There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .","Our aim is to implement (an approximation of) this algorithm in sublinear space.","As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.","In some sense, we only maintain a small subset of the partial solutions.","Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .","We convert an LCS into finding the longest chain in a set of points $P$ .","We construct a set of anchor points in the plane, which may not be in $P$ .","Our aim is to just maintain the longest chain between pairs of anchor points.","Let $\\delta > 0$ be some fixed parameter.","We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .","For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.","$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.","Our analysis remains identical..","So we can drop the floors used in the definition of $S_i$ .","Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .","We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .","This is the set of anchor points that we discussed earlier.","Note that they are placed according to a geometric grid.","A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.","There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .","We have a basic claim about the LCS of two strings (proof deferred to Appendix ).","This gives us a simple bound on the defect that we shall exploit.","Lemma makes an important argument.","It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.","Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.","Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .","The defect $n-r$ is at least $|j_a - i|$ .","There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .","We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.","Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .","Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .","Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .","We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .","Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .","If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .","Otherwise, it is not added.","Note that $j_b \\ge \\ell _p$ , simply by construction.","The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.","It now remains to bound the defect of $L^{\\prime }$ .","Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .","Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .","This means that ${j_b} > \\ell _{p}$ .","Let $i_c$ be the smallest index larger than $p\\bar{n}$ .","So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .","Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .","We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .","Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .","By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .","The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .","The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .","$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.","The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .","Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .","Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.","The total running time is $O(n\\bar{n}^2)$ .","Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.","He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.","The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .","A union bound over all $t$ proves the third assertion of Theorem REF .","Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .","Fix $t \\in [n]$ .","Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .","For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.","Then $|R_t|=\\sum _{i \\le t} Z_i$ .","Let $\\mu =\\mathbb {E}[|R_t|]$ .","Below we show that $\\mu \\le M$ .","We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .","Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .","Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).","It remains to show that $\\mu \\le M$ .","We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.","For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .","We prove by backwards induction on $r$ .","For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.","Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .","We start with a technical statement.","$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .","This completes the proof.","$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.","Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .","(of Claim ) Assume wlog that $i_k \\ge j_k$ .","Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .","This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .","Hence the defect is at least $i_k - j_k$ .","Repeating over all $k$ , we complete the proof.","$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).","If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .","Suppose neither of these are true.","Then $j_a > i_a \\ge i$ .","Let us focus on the characters of $x$ that are not matched.","No character of $x$ with index in $[i,i_a)$ is matched.","The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).","So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .","So the number of unmatched characters in $x$ is at least $j_a - i$ .","$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .","We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .","This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .","The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .","Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","This is obtained by just maximizing over all possible $j^{\\prime }$ .","We now apply Claim .","We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .","Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.","This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .","This can be done for all $y(j)$ , $j \\in S_i$ .","The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .","$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.","Consider the index $i\\bar{n}$ .","Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","This requires space $O(|S_i|) = O(\\bar{n}))$ .","By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.","Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.","Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .","The space bound is $O(\\bar{n})$ .","The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .","$\\Box $"],["Applying AMDP to LIS and LCS","We now show how to apply Theorem REF to LIS and LCS.","The application to LIS is quite obvious.","We first set some notation about points in the two-dimensional plane.","We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.","$z(2)$ ) refers to the first (resp.","second) coordinate of $z$ .","We use the standard coordinate-wise partial order on $z$ .","So $z \\triangleleft z^{\\prime }$ iff $z(1) < z^{\\prime }(1)$ and $z(2) < z^{\\prime }(2)$ .","(of Theorem REF ) The input is a stream $x(1), x(2), \\ldots , x(n)$ .","Think of the $i$ th element of the stream as the point $(i,x(i))$ .","So the input is thought of as a sequence of points.","Note that the points arrive in increasing order of first coordinate.","Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).","We set $\\gamma = n^{O(1)}$ and $\\rho = n$ in Theorem REF .","$\\Box $ The application to LCS is somewhat more subtle.","Again, we think of the input as a set of points in the two-dimensional plane.","But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\delta $ .","Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .","Then there is a $O(\\delta ^{-1}k\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\delta n$ -approximation of $E(x,y)$ .","We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.","Let $P$ be the set of pairs $\\lbrace (i,j)|x(i)=y(j)\\rbrace $ under the partial order $(i,j) < (i^{\\prime },j^{\\prime })$ if $i < i^{\\prime }$ and $j < j^{\\prime }$ .","It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.","Now we associate to the pair of strings $x,y$ the sequence $\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\prime },j^{\\prime })$ is $i < i^{\\prime }$ or if $i=i^{\\prime }$ and $j < j^{\\prime }$ .)","Note that $\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.","Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\sigma $ -paths as defined in the AMDP.","Thus the length of the LCS is equal to the size of the largest $\\sigma $ -path.","It is not true that $E(x,y)$ is equal to $min-defect(\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\sigma $ is in general longer than $n$ .","Given full access to $y$ , and a streamed $x$ .","We have a bound on $|\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .","We now argue that an additive $\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\delta /k)$ -approximation for AMDP of $P$ .","Let the length of the longest chain in $P$ be $\\ell $ and the min-defect be $m$ .","Let $d$ be a shorthand for $E(x,y)$ .","We have $\\ell + m = |P|$ and $\\ell + d = n$ .","The output of AMDP is an estimate $est$ such that $m \\le est \\le (1+\\delta /k)m$ .","We estimate $d$ by $est_d = est + n - |P|$ .","We show that $est_d \\in [d,d + \\delta n]$ .","We have $est_d = est + n - |P| \\ge m + n - |P| = n-\\ell = d$ .","We can also get an upper bound.","$est_d & = & est + n - |P| \\\\& \\le & m + n - |P| + \\delta m/k \\\\& = & d + \\delta m/k \\ \\textrm {(since |P| - m = \\ell and d = n - \\ell )} \\\\& \\le & d + \\delta n \\ \\textrm {(since m \\le |P| \\le nk)}$ Hence, we use the parameters $\\delta /k, \\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.","An application of Theorem REF completes the proof.","$\\Box $"],["Deterministic streaming algorithm for LCS","We now discuss a deterministic $\\sqrt{n}$ -space algorithm for LCS.","This can be used for large alphabets to beat the bound given in Theorem .","For any consistent sequence (CS), the size of the complement is called the defect.","For indices $i,j \\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.","The main theorem is: Let $\\delta > 0$ .","We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .","There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.","The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.","The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].","Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .","There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .","Our aim is to implement (an approximation of) this algorithm in sublinear space.","As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.","In some sense, we only maintain a small subset of the partial solutions.","Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .","We convert an LCS into finding the longest chain in a set of points $P$ .","We construct a set of anchor points in the plane, which may not be in $P$ .","Our aim is to just maintain the longest chain between pairs of anchor points.","Let $\\delta > 0$ be some fixed parameter.","We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .","For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.","$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.","Our analysis remains identical..","So we can drop the floors used in the definition of $S_i$ .","Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .","We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .","This is the set of anchor points that we discussed earlier.","Note that they are placed according to a geometric grid.","A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.","There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .","We have a basic claim about the LCS of two strings (proof deferred to Appendix ).","This gives us a simple bound on the defect that we shall exploit.","Lemma makes an important argument.","It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.","Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.","Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .","The defect $n-r$ is at least $|j_a - i|$ .","There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .","We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.","Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .","Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .","Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .","We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .","Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .","If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .","Otherwise, it is not added.","Note that $j_b \\ge \\ell _p$ , simply by construction.","The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.","It now remains to bound the defect of $L^{\\prime }$ .","Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .","Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .","This means that ${j_b} > \\ell _{p}$ .","Let $i_c$ be the smallest index larger than $p\\bar{n}$ .","So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .","Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .","We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .","Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .","By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .","The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .","The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .","$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.","The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .","Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .","Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.","The total running time is $O(n\\bar{n}^2)$ ."],["Acknowledgements","The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.","He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions."],["The space bound of Theorem ","The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .","A union bound over all $t$ proves the third assertion of Theorem REF .","Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .","Fix $t \\in [n]$ .","Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .","For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.","Then $|R_t|=\\sum _{i \\le t} Z_i$ .","Let $\\mu =\\mathbb {E}[|R_t|]$ .","Below we show that $\\mu \\le M$ .","We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .","Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .","Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).","It remains to show that $\\mu \\le M$ .","We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.","For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .","We prove by backwards induction on $r$ .","For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.","Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .","We start with a technical statement.","$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .","This completes the proof.","$\\Box $"],["Proofs from Section ","We first prove another claim from which the proof of Claim follows.","Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .","(of Claim ) Assume wlog that $i_k \\ge j_k$ .","Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .","This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .","Hence the defect is at least $i_k - j_k$ .","Repeating over all $k$ , we complete the proof.","$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).","If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .","Suppose neither of these are true.","Then $j_a > i_a \\ge i$ .","Let us focus on the characters of $x$ that are not matched.","No character of $x$ with index in $[i,i_a)$ is matched.","The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).","So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .","So the number of unmatched characters in $x$ is at least $j_a - i$ .","$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .","We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .","This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .","The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .","Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .","This is obtained by just maximizing over all possible $j^{\\prime }$ .","We now apply Claim .","We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .","Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.","This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .","This can be done for all $y(j)$ , $j \\in S_i$ .","The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .","$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.","Consider the index $i\\bar{n}$ .","Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .","This requires space $O(|S_i|) = O(\\bar{n}))$ .","By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.","Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.","Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .","The space bound is $O(\\bar{n})$ .","The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .","$\\Box $"]],"string":"[\n [\n \"Space efficient streaming algorithms for the distance to monotonicity\\n and asymmetric edit distance\"\n ],\n [\n \"Abstract Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem.\",\n \"We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used.\",\n \"We present an algorithm which, for any $\\\\delta > 0$, given streaming access to an array of length $n$ provides a $(1+\\\\delta)$-multiplicative approximation to the \\\\emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((\\\\log^2 n)/\\\\delta)$ space.\",\n \"The previous best known approximation using polylogarithmic space was a multiplicative 2-factor.\",\n \"Our algorithm can be used to estimate the length of the LIS to within an additive $\\\\delta n$ for any $\\\\delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$.\",\n \"Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time.\",\n \"It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS.\",\n \"We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$.\",\n \"Our algorithm works in the asymmetric setting (inspired by \\\\cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$.\",\n \"More precisely, it gives an additive-$\\\\delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(\\\\log^2 n)/\\\\delta)$, where $k$ is the maximum number of times any one symbol appears in $y$.\"\n ],\n [\n \"Introduction\",\n \"Two classic optimization problems concerning subsequences (substrings) of arrays (strings) are the longest increasing subsequence (LIS) and longest common subsequence (LCS) problems.\",\n \"A string of length $n$ over alphabet $\\\\Sigma $ is represented as a function $x:[n] \\\\rightarrow \\\\Sigma $ .\",\n \"A subsequence of length $k$ is a string $x(i_1)x(i_2)\\\\ldots x(i_k)$ , where $1 \\\\le i_1 < i_2 < \\\\cdots < i_k \\\\le n$ .\",\n \"In the LIS problem, the alphabet $\\\\Sigma $ comes equipped with a (total or partial) order $\\\\triangleleft $ , and we look for the longest subsequence whose terms are in increasing order.\",\n \"In the LCS problem we are given two strings $x$ and $y$ and look for the longest string which is a subsequence of each of them.\",\n \"Note that the LIS of $x$ is the LCS of $x$ and its sorted version.\",\n \"Both of these problems can be solved by dynamic programs.\",\n \"The LIS can be found on $O(n\\\\log n)$ time [28], [17], [2].\",\n \"This is known to be optimal, even for (comparison based) algorithms that only determine the length of the LIS [27].\",\n \"The LCS problem has a fairly direct $O(n^2)$ algorithm [11], which can be improved to $O(n^2/\\\\log ^2n)$ [24], [9].\",\n \"It is a notoriously difficult open problem to improve this bound, or prove some matching lower bounds.\",\n \"It is often natural to focus on the complements of the LIS and LCS lengths, which are related to some notion of distances between strings.\",\n \"The distance to monotonicity of (the length $n$ string) $x$ , denoted $DM(x)$ is defined to be $n-LIS(x)$ , and is the the minimum number of values that need to be changed to make $x$ .\",\n \"The (insertion-deletion) edit distance of (two length $n$ strings) $x,y$ , denoted $E(x,y)$ is defined to be $n-LCS(x,y)$ and is the minimum number of insertions and deletions needed to change one string into the other.\",\n \"(Note that $E(x,y)$ is bounded between $L(x,y)$ and $2L(x,y)$ where $L(x,y)$ is the Levenshtein distance, where insertions, deletions, and substitutions are allowed.)\",\n \"Of course the algorithmic problems of exactly computing $LIS(x)$ and $DM(x)$ are equivalent, but approximating them can be very different.\",\n \"In recent years, there has been a lot of attention on giving approximate solutions for LIS and LCS that are much more efficient that the basic dynamic programming solutions.\",\n \"Any improved results for LCS would be very interesting, since the best known quadratic time solution is infeasible for very large strings.\",\n \"These problems can be studied in a variety of settings - sampling, streaming, and communication.\",\n \"The streaming setting has been the focus of many results [20], [30], [18], [13].\",\n \"The model for the LIS is that we are allowed one (or constant) passes over the input string $x$ , and only have access to sublinear storage.\",\n \"The usual formulation of LCS in the streaming model postulates that we have only one-way access to both strings $x$ and $y$ .\",\n \"We consider an alternative asymmetric model in which we have one-way access to string $x$ (called the input string) but random accesss to string $y$ (called the fixed string).\",\n \"This model is more powerful that the standard one, but it is still far from clear how to obtain space efficient approximations to $E(x,y)$ in this model.\",\n \"(This model was inspired by recent work of [6] concerning the time complexity of approximating edit distance in the random access model.\",\n \"One part of their work introduced an asymmetric version of the random access model in which one pays only for accesses to one of the strings, and established time lower bounds for good approximations that hold even in this more powerful model.)\"\n ],\n [\n \"Results\",\n \"Our first result is a streaming algorithm for approximating the distance to monotonicity.\",\n \"There is a randomized one-pass streaming algorithm that for any $\\\\delta >0$ , takes as input an array (of length $n$ ), makes one-pass through the array, uses space $O(\\\\delta ^{-1}\\\\log ^2 n)$ and with error probability $n^{-\\\\Omega (1)}$ outputs an estimate to $DM(x)$ that is between $DM(x)$ and $(1+\\\\delta )DM(x)$ .\",\n \"Previously there was a polylogarithmic time algorithm that gave a factor 2-approximation [13], and an algorithm that gave arbitrarily good multiplicative approximations to $LIS(x)$ (which is harder than approximating $DM(x)$ ) but required $\\\\Omega (\\\\sqrt{n})$ space [20].\",\n \"The improvement in the approximation ratio from 2 to $1+\\\\delta $ (for polylogarithmic space algorithms) is not just “chipping away” at a constant, but provides a significant qualitative difference: previous polylogarithmic space algorithms might return an estimate of 0 when the LIS length is $n/2$ , while our algorithm can detect increasing subsequences of length a small fraction of $n$ .\",\n \"More precisely, it is easy to see that if $V$ is an estimate of $DM(x)$ that is between $DM(x)$ and $(1+\\\\delta )DM$ then $n-V$ is within an additive $\\\\frac{\\\\delta }{1+\\\\delta } n$ of $LIS(x)$ , and so our algorithm can provide an estimation interval for $LIS(x)/n$ of arbitrarily small width.\",\n \"The previous polylogarithmic time streaming algorithm only gave such an algorithm for $\\\\delta \\\\ge 1$ , which only guarantees an estimation interval for $LIS(x)/n$ of width 1/2.\",\n \"The algorithm promised by Theorem REF is derived as a special case of a more general algorithm (Theorem REF ) that finds increasing sequences in partial orders.\",\n \"This algorithm will also be applied to give a good (additive) approximation algorithm for edit distance in the asymmetric setting, whose space is polylogarithmic in the case that no symbol appears many times in the fixed string.\",\n \"Let $\\\\delta \\\\in (0,1]$ .\",\n \"Suppose $y$ is a fixed string of length $n$ and $x$ an input string of length $n$ to which we have streaming access.\",\n \"There is a randomized algorithm that makes one pass through $x$ and, with error probability $n^{-\\\\Omega (1)}$ , outputs an additive $\\\\delta n$ -approximation to $E(x,y)$ and uses space $O(k\\\\log ^2 n/\\\\delta )$ where $k$ is the maximum number of times any symbol appears in $y$ .\",\n \"There is a deterministic algorithm that runs in space $O(\\\\sqrt{(n\\\\log n)/\\\\delta })$ -space and outputs a $(1+\\\\delta )$ -multiplicative (which is also a $\\\\delta n$ -additive) approximation to $E(x,y)$ .\"\n ],\n [\n \"Techniques\",\n \"A notable feature of our algorithm is its conceptual simplicity.\",\n \"The pseudocode for the LIS approximation is just a few lines.\",\n \"The algorithm has parameters $\\\\alpha (i,t)$ for $1 \\\\le i < t$ ) whose exact formula is a bit cumbersome to state at this point.\",\n \"We set $\\\\alpha (i,t)$ to be 0 for $i \\\\ge t-O(\\\\log (n))$ and approximately $1/(t-i)$ otherwise.\",\n \"The algorithm maintains a set of indices $R$ , and for each $i \\\\in R$ , we store $x(i)$ and an estimate $r(i)$ of $DM(x[1,i])$ , where $x[1,i]$ is the length $i$ prefix of $x$ .\",\n \"For convenience, we add dummy elements $x(0) = x(n+1) = -\\\\infty $ and begin with $R = \\\\lbrace 0\\\\rbrace $ .\",\n \"For each time $t \\\\ge 1$ , we perform the following update: Define $R^{\\\\prime } = \\\\lbrace i \\\\in R | x(i) \\\\le x(t)\\\\rbrace $ .\",\n \"Set $r(t)=\\\\min _{i \\\\in R^{\\\\prime }}(r(i)+t-1-i)$ .\",\n \"$R \\\\longleftarrow R \\\\cup \\\\lbrace t\\\\rbrace $ .\",\n \"Remove each $i \\\\in R$ independently with probability $\\\\alpha (i,t)$ .\",\n \"The final output is $r(n+1)$ .\",\n \"The space used by the algorithm is (essentially) the maximum size of $|R|$ .\",\n \"The update time is determined by step 1, which runs in time $O(|R|)$ .\",\n \"Without the third step, the algorithm is a simple quadratic time exact algorithm for $DM(x)$ using linear space.\",\n \"(The $O(n\\\\log n)$ time algorithms [17], [2] also work in a streaming fashion, but store data much more cleverly.)\",\n \"More precisely, at step $t$ , $R=\\\\lbrace 0,\\\\ldots ,t\\\\rbrace $ and for each $i \\\\le t$ , $r(i)=DM(x[1,i])$ .\",\n \"The third step reduces the set $R$ , thereby reducing the space of the algorithm.\",\n \"The space used by the algorithm is (essentially) the maximum size of $R$ .\",\n \"Intuitively, the algorithm “forgets” $(x(i),r(i))$ for those $i$ removed from $R$ .\",\n \"The set $R$ of remembered indices is a subset of $[1,t]$ whose density decays as one goes back in time from the present time $t$ .\",\n \"When we compute $r(t)$ , it may no longer be equal to the distance to monotonicity of the prefix of $x$ of length $i$ , but it will be at least this value.\",\n \"This forgetting strategy is tailored to ensure that $r(t)$ is also at most a $(1+\\\\delta )$ -factor away from the distance to monotonicity.\",\n \"We also ensure that that (with high probability) the set $R$ does not exceed size $O(\\\\delta ^{-1}\\\\log ^2 n)$ Just to give an indication of the difficulty, consider an algorithm that forgets uniformly at random.\",\n \"At some time $t$ , the set of remembered indices $R$ is a uniform random set of size $O(\\\\delta ^{-1}\\\\log ^2 n)$ up to index $t$ .\",\n \"These are used to compute $r(t)$ and include $t$ in $R$ .\",\n \"The algorithm then forgets a uniform random index in $R$ to maintain the space bound.\",\n \"Since we want to get a $(1+\\\\delta )$ -factor approximation, the algorithm must be able to detect an LIS of length $\\\\Omega (\\\\delta n)$ .\",\n \"Of the indices in $R$ (up to time $t$ ), it is possible that around a $O(\\\\delta )$ -fraction of them are in the LIS.\",\n \"Suppose we reach a small stretch of indices not on the LIS.\",\n \"If this has size even $\\\\textrm {poly}(\\\\delta ^{-1}\\\\log n)$ , it is likely that all LIS indices in $R$ are forgotten.\",\n \"But how do we selectively remember the LIS indices without knowing the LIS in advance?\",\n \"That is the challenge of the forgetting strategy.\",\n \"All past polylogarithmic space algorithms [20], [13] for LIS use combinatorial characterizations of increasing sequences based on inversion counting [14], [12], [26], [1].\",\n \"While this is a very powerful technique, it does not lead to accurate approximations for the LIS, and (apparently) do not yield any generalizations to LCS.\",\n \"The idea of remembering selected information about the sequence that becomes sparser as one goes back in time was first used by [20] for the inversion counting approach.\",\n \"Our work seems to be the first to use this to directly mimic the dynamic program, though the idea is quite natural and has almost certainly been considered before.\",\n \"The main contribution here is to analyze this algorithm, and determine the values of parameters that allow it to be both space efficient and a good approximation.\",\n \"This line of thinking can be exploited to deal with asymmetric streaming LCS.\",\n \"We construct a simple reduction of LCS to finding the longest chain in a specific partial order.\",\n \"This reduction has a streaming implementation, so the input stream can be directly seen as just elements of this resulting partial order.\",\n \"This reduction blows up the size of the input, and the size of the largest chain can become extremely small.\",\n \"If each symbol occurs $k$ times in $x$ and $y$ , then the resulting partial order has $nk$ elements.\",\n \"Nonetheless, the longest chain still has length at most $n$ .\",\n \"We require very accurate estimates for the length of the longest chain.\",\n \"This is where the power of the $(1+\\\\delta )$ -approximation comes in.\",\n \"We can choose $\\\\delta $ to be much smaller to account for the input blow up, and still get a good approximation.\",\n \"Note that if we only had a $1.01$ -approximation for the longest chain problem, this reduction would not be useful.\",\n \"Our $\\\\widetilde{O}(\\\\sqrt{n})$ -space algorithm also works according to the basic principle of following a dynamic program, although it uses one different from the previous algorithms.\",\n \"This can be thought of as generalization of the $\\\\widetilde{O}(\\\\sqrt{n})$ -space algorithm for LIS[20].\",\n \"We maintain a $\\\\widetilde{O}(\\\\sqrt{n})$ -space deterministic sketch of the data structure maintained by the exact algorithm.\",\n \"By breaking the stream up into the right number of chunks, we can update this sketch using $\\\\widetilde{O}(\\\\sqrt{n})$ -space.\",\n \"Previous work The study of LIS and LCS in the streaming setting was initiated by Liben-Nowell et al [23], although their focus was mostly on exactly computing the LIS.\",\n \"Sun and Woodruff [30] improved upon these algorithms and lower bounds and also proved bounds for the approximate version.\",\n \"Most relevant for our work, they prove that randomized protocols that compute a $(1+\\\\varepsilon )$ -approximation of the LIS length essentially require $\\\\Omega (\\\\varepsilon ^{-1}\\\\log n)$ .\",\n \"Gopalan et al [20] provide the first polylogarithmic space algorithm that approximates the distance to monotonicity.\",\n \"This was based on inversion counting ideas in [26], [1].\",\n \"Ergun and Jowhari [13] give a 2-approximation using the basic technique of inversion counting, but develop a different algorithm.\",\n \"Gál and Gopalan [18] and independently Ergun and Jowhari [13] proved an $\\\\Omega (\\\\sqrt{n})$ lower bound for deterministic protocols that approximate that LIS length up to a multiplicative constant factor.\",\n \"For randomized protocols, the Sun and Woodruff bound of $\\\\Omega (\\\\log n)$ is the best known.\",\n \"One of the major open problems is to get a $o(\\\\sqrt{n})$ space randomized protocol (or an $\\\\Omega (\\\\sqrt{n})$ lower bound) for constant factor approximations for the LIS length.\",\n \"Note that our work does not imply anything non-trivial for this problem.\",\n \"We are unaware of any lower bounds for estimating the distance to monotonicity in the streaming setting.\",\n \"A significant amount of work has been done in studying the LIS (or rather, the distance to monotonicity) in the context of property testing [14], [12], [15], [26], [1].\",\n \"The property of monotonicity has been studied over a variety of domains, of which the boolean hypercube and the set $[n]$ (which is the LIS setting) have usually been of special interest [19], [12], [16], [22], [1], [26], [10].\",\n \"In previous work, the authors of this paper found a $(1+\\\\delta )$ -multiplicative approximation algorithm for the distance to monotonicity (in the random access model) that runs in time $O(\\\\textrm {poly}\\\\log (n))$ [29].\",\n \"As the present result does for the streaming model, that result also improved on the previous best factor 2 approximation for that model.\",\n \"Despite the superficial similarity between the statement of results, the models considered in these two papers are quite different, and the algorithm we give here in the streaming model is completely different from the complicated algorithm we gave in the sublinear time model.\",\n \"The LCS and edit distance have an extemely long and rich history, especially in the applied domain.\",\n \"We point the interesting reader out to [21], [25] for more details.\",\n \"Andoni et al [6] achieved a breakthrough by giving a near-linear time algorithm (in the random access model) that gives polylogarithmic time approximations for the edit distance.\",\n \"This followed a long line of results well documented in [6].\",\n \"They initiate the study of the asymmetric edit distance, where one string is known and we are only charged for accesses to the other string.\",\n \"For the case of non-repetitive strings, there has been a body of work on studying the Ulam distance between permutations [4], [5], [3], [7].\",\n \"Paths in posets We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).\",\n \"We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.\",\n \"Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.\",\n \"This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP.\",\n \"Weighted $P$ -sequences and the approximate minimum-defect path problem We use $P$ to denote a fixed set endowed with a partial order $\\\\triangleleft $ .\",\n \"The partial order relation is given by an oracle which, given $u,v \\\\in P$ outputs $u \\\\triangleleft v$ or $\\\\lnot (u \\\\triangleleft v)$ .\",\n \"For a natural number $n$ we write $[n]$ for the set $\\\\lbrace 1,2,\\\\ldots ,n\\\\rbrace $ .\",\n \"A sequence ${\\\\bf \\\\sigma }=(\\\\sigma (1),\\\\ldots ,\\\\sigma (n)) \\\\in P$ is called a $P$ -sequence.\",\n \"The number of terms ${\\\\bf \\\\sigma }$ is called the length of ${\\\\bf \\\\sigma }$ and is denoted $|{\\\\bf \\\\sigma }|$ ; we normally use $n$ to denote $|{\\\\bf \\\\sigma }|$ .\",\n \"A weighted $P$ -sequence consists of a $P$ -sequence ${\\\\bf \\\\sigma }$ together with a sequence $(w(1),\\\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .\",\n \"In all our final applications $w(i)$ will always be 1.\",\n \"Nonetheless, we solve this slightly more general weighted version.\",\n \"We have the following additional definitions: For $t \\\\in [n]$ , ${\\\\bf \\\\sigma }_{\\\\le t}$ denotes the sequence $(\\\\sigma _1,\\\\ldots ,\\\\sigma _t)$ .\",\n \"Also for $J \\\\subseteq [n]$ , $J_{\\\\le t}$ denotes the set $J \\\\cap \\\\lbrace 1,\\\\ldots ,t\\\\rbrace $ .\",\n \"For $J \\\\subseteq [n]$ , $w(J)=\\\\sum _{j \\\\in J} w(j)$ .\",\n \"The digraph $D=D({\\\\bf \\\\sigma })$ associated to the $P$ -sequence ${\\\\bf \\\\sigma }$ has vertex set $[n]$ (where $n=|{\\\\bf \\\\sigma }|$ ) and arc set $\\\\lbrace i \\\\rightarrow j: i0$ .\",\n \"The output is a number $A$ such that: ${\\\\rm Prob}[A \\\\in [\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w),(1+\\\\delta ) \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)]] \\\\ge 1-\\\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ is said to be a one-sided error algorithm.\",\n \"Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.\",\n \"For the AMDP, the input consists of the parameters $\\\\delta $ and $\\\\gamma $ together with a sequence of $n$ pairs $((\\\\sigma (t),w(t)):t \\\\in [n])$ .\",\n \"We think of the input as arriving in a sequence of discrete time steps, where $\\\\delta ,\\\\gamma $ arrive at time step 0 and for $t \\\\in [n]$ , $(\\\\sigma (t),w(t))$ arrives at time step $t$ .\",\n \"The main complexity parameter of interest is the auxiliary memory needed.\",\n \"For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.\",\n \"Associated to a weighted $P$ -sequence $({\\\\bf \\\\sigma },w)$ we define the parameter: $\\\\rho =\\\\rho (w)=\\\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\\\rho =n^{O(1)}$ .\",\n \"The main technical theorem about AMDP is the following.\",\n \"There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\\\frac{\\\\ln (n/\\\\gamma )\\\\ln (\\\\rho )}{\\\\delta })$ .\",\n \"In particular, if $\\\\rho =n^{O(1)}$ and $\\\\gamma =1/n^{O(1)}$ then the space is $O(\\\\frac{(\\\\ln (n))^2}{\\\\delta })$ .\",\n \"The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"We first review this dynamic program.\",\n \"Exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\\\sigma (n+1)$ that is greater than all other elements of $P$ .\",\n \"Thus all arcs $j \\\\longrightarrow n+1$ for $j \\\\in [n]$ are present.\",\n \"Set $w(n+1)=0$ .\",\n \"We define sequences $s(0),\\\\ldots ,s(n+1)$ and $W(0),\\\\ldots ,W(n+1)$ as follows.\",\n \"We initialize $s(0) = 0$ and $W(0) = 0$ .\",\n \"For $t \\\\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\\\\\s(t) & = & \\\\min (s(i)+W(t-1)-W(i): i < t \\\\\\\\& & \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)).$ Thus $W(t)=w([t])$ .\",\n \"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\\\pi )$ over all paths $\\\\pi $ whose maximum element is $\\\\sigma (t)$ .\",\n \"In particular, $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)=s(n+1)$ .\",\n \"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).\",\n \"The polylog space streaming algorithm We denote our streaming algorithm by $\\\\Gamma =\\\\Gamma ({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ .\",\n \"Our approximation algorithm is a natural variant of the exact algorithm.\",\n \"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .\",\n \"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.\",\n \"The amount of space used by the algorithm is proportional to the maximum size of $R$ .\",\n \"We first define the probabilities $p(i,t)$ .\",\n \"Similar quantities were defined in [20].\",\n \"$& & q(i,t) \\\\\\\\& = & \\\\min \\\\left\\\\lbrace 1,\\\\frac{1+\\\\delta }{\\\\delta }\\\\ln (4t^3/\\\\gamma )\\\\frac{w(i)}{W(t)-W(i-1)}\\\\right\\\\rbrace \\\\\\\\& & p(i,i) = 1 \\\\ \\\\ \\\\ \\\\ p(i,t) = \\\\frac{q(i,t)}{q(i,t-1)} \\\\text{ for $t>i$},$ Note that in the typical case that $\\\\delta = \\\\theta (1)$ and $\\\\gamma = \\\\log (n)^{-\\\\Theta (1)}$ , we have $q(i,t)$ is $\\\\Theta (\\\\ln (n)/(t-i))$ .\",\n \"We initialize $R=\\\\lbrace 0\\\\rbrace $ , $r(0)=0$ and $W(0)=0$ .\",\n \"The following update is performed for each time step $t \\\\in [n+1]$ .\",\n \"The final output is just $r(n+1)$ .\",\n \"$W(t)=W(t-1)+w(t)$ .\",\n \"$r(t)=\\\\min (r(i)+W(t-1)-W(i): i \\\\in R\\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)$ .\",\n \"The index $t$ is inserted in $R$ .\",\n \"Each element $i \\\\in R$ is (independently) discarded with probability $1-p(i,t)$ .\",\n \"On input $({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ , the algorithm $\\\\Gamma $ satisfies: $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"${\\\\rm Prob}[r(n+1) > (1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)] \\\\le \\\\gamma /2$ .\",\n \"The probability that $|R|$ ever exceeds $ \\\\frac{2e^2}{\\\\delta }\\\\ln (2\\\\rho )\\\\ln (4n^3/\\\\gamma )$ is at most $\\\\gamma /2$ .\",\n \"The above theorem does not exactly give what was promised in Theorem REF .\",\n \"For the algorithm $\\\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.\",\n \"To achieve the guarantee of Theorem REF we modify $\\\\Gamma $ to an algorithm $\\\\Gamma ^{\\\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.\",\n \"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\\\gamma /2$ , the probability that the output of $\\\\Gamma ^{\\\\prime }$ exceeds $(1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ is at most $\\\\gamma $ .\",\n \"We now prove Theorem REF .\",\n \"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\\\le n+1$ we have $r(j) \\\\ge s(j)$ and thus $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"The second part will be proved in the two subsection.\",\n \"The final assertion of Theorem REF showing the space bound is deferred to Appendix .\",\n \"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.\",\n \"Let $R_t$ denote the set $R$ after processing $\\\\sigma (t),w(t)$ .\",\n \"Observe that the definition of $p(i,j)$ implies: For each $i \\\\le t \\\\le n$ , ${\\\\rm Prob}[i \\\\in R_t]=\\\\prod _{j \\\\in [i,t]}p(i,j)=q(i,t)$ .\",\n \"We need some additional definitions.\",\n \"For $I \\\\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\\\bar{I}$ .\",\n \"Let $C$ be the index set of some fixed chain having minimum $\\\\mbox{\\\\rm defect}$ , so that the minimum defect is equal to $w(\\\\bar{C})$ .\",\n \"We assume without loss of generality that $n+1 \\\\in C$ .\",\n \"We write $R^t$ for the subset $R$ at the end of step $t$ .\",\n \"Note that $R^t \\\\subseteq [t]$ .\",\n \"We define $F^t=[t]-R^t$ .\",\n \"An index $i \\\\in R^t$ is said to be remembered at time $t$ and $i \\\\in F^t$ is said to be forgotten by time $t$.\",\n \"Index $i \\\\in C$ is said to be unsafe at time $t$ if every index in $C \\\\cap [i,t] \\\\subseteq F^t$ , i.e., every index of $C \\\\cap [i,t]$ is forgotten by time $t$ .\",\n \"We write $U^t$ for the set of indices that are unsafe at time $t$ .\",\n \"An index $i \\\\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.\",\n \"We denote the set of unsafe indices by $U$ .\",\n \"On any execution, the set $U$ is determined by the sequence $R^1,\\\\ldots ,R^n$ .\",\n \"On any execution of the algorithm, $r(n+1) \\\\le w(\\\\bar{C} \\\\cup U)$ .\",\n \"We prove by induction on $t$ that if $t \\\\in C$ then $r(t) \\\\le w(\\\\bar{C}_{\\\\le t-1} \\\\cup U^{t-1})$ .\",\n \"Assume $t \\\\ge 1$ and that the result holds for $j 0$ .\",\n \"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .\",\n \"There is a deterministic one-pass streaming algorithm that computes a $(1+\\\\delta )$ -approximation to $E(x,y)$ that uses $O(\\\\sqrt{(n \\\\ln n)/\\\\delta })$ space.\",\n \"The algorithm performs $O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ updates, each taking $O(n^2\\\\ln n/\\\\delta )$ time.\",\n \"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].\",\n \"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .\",\n \"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\\\in [n]$ .\",\n \"Our aim is to implement (an approximation of) this algorithm in sublinear space.\",\n \"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.\",\n \"In some sense, we only maintain a small subset of the partial solutions.\",\n \"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .\",\n \"We convert an LCS into finding the longest chain in a set of points $P$ .\",\n \"We construct a set of anchor points in the plane, which may not be in $P$ .\",\n \"Our aim is to just maintain the longest chain between pairs of anchor points.\",\n \"Let $\\\\delta > 0$ be some fixed parameter.\",\n \"We set $\\\\bar{n} = \\\\sqrt{(n\\\\ln n)/\\\\delta }$ and $\\\\mu = (\\\\ln n)/\\\\bar{n} = \\\\sqrt{(\\\\delta \\\\ln n)/n}$ .\",\n \"For each $i \\\\in [n/\\\\bar{n}]$ , the set $S_i$ of indices is defined as follows.\",\n \"$ S_i = \\\\lbrace \\\\lfloor i\\\\bar{n} + b(1+\\\\mu )^r \\\\rfloor \\\\big | r \\\\ge 0, b \\\\in \\\\lbrace -1, +1\\\\rbrace \\\\rbrace $ For convenience, we treat $\\\\bar{n}$ , $n/\\\\bar{n}$ , and $(1+\\\\mu )^r$ as integersFormally, we need to take floors of these quantities.\",\n \"Our analysis remains identical..\",\n \"So we can drop the floors used in the definition of $S_i$ .\",\n \"Note that the $|S_i| = O(\\\\mu ^{-1}\\\\ln n) = O(\\\\bar{n})$ .\",\n \"We refer to the family of sets $\\\\lbrace S_1, S_2, \\\\ldots \\\\rbrace $ by ${\\\\cal S}$ .\",\n \"This is the set of anchor points that we discussed earlier.\",\n \"Note that they are placed according to a geometric grid.\",\n \"A common subsequence of $x$ and $y$ is consistent with ${\\\\cal S}$ if the following happens.\",\n \"There exists a sequence of indices $\\\\ell _1 \\\\le \\\\ell _2 \\\\le \\\\ldots \\\\ell _m$ such that $\\\\ell _i \\\\in S_i$ and if character $x(k)$ ($k \\\\in [i\\\\bar{n}, (i+1)\\\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\\\prime })$ , then $k^{\\\\prime } \\\\in [\\\\ell _i, \\\\ell _{i+1}]$ .\",\n \"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).\",\n \"This gives us a simple bound on the defect that we shall exploit.\",\n \"Lemma makes an important argument.\",\n \"It argues that the the anchor points ${\\\\cal S}$ were chosen such that an ${\\\\cal S}$ -consistent sequence is “almost\\\" the LCS.\",\n \"Suppose that $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.\",\n \"Let $i \\\\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\\\ge i$ .\",\n \"The defect $n-r$ is at least $|j_a - i|$ .\",\n \"There exists an ${\\\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\\\delta )E(x,y)$ .\",\n \"We start with an LCS $L$ of $x$ and $y$ and “round\\\" it to be ${\\\\cal S}$ -consistent.\",\n \"Let $L$ be $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ .\",\n \"Consider some $p \\\\in [n/\\\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"Set $\\\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .\",\n \"We construct a new common sequence $L^{\\\\prime }$ by removing certain matches from $L$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .\",\n \"If $i_b \\\\in [p\\\\bar{n},(p+1)\\\\bar{n}]$ and $j_b \\\\le \\\\ell _{p+1}$ , then we add this pair to $L^{\\\\prime }$ .\",\n \"Otherwise, it is not added.\",\n \"Note that $j_b \\\\ge \\\\ell _p$ , simply by construction.\",\n \"The new common sequence $L^{\\\\prime }$ is ${\\\\cal S}$ -consistent.\",\n \"It now remains to bound the defect of $L^{\\\\prime }$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b)) \\\\in L$ that is not present in $L^{\\\\prime }$ .\",\n \"Let $i_b \\\\in [(p-1)\\\\bar{n},p\\\\bar{n}]$ .\",\n \"This means that ${j_b} > \\\\ell _{p}$ .\",\n \"Let $i_c$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"So $\\\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .\",\n \"Let $\\\\ell _{p} = p\\\\bar{n} + (1+\\\\mu )^r $ .\",\n \"We have $j_c - p\\\\bar{n} = [(1+\\\\mu )^r, (1+\\\\mu )^{r+1}]$ .\",\n \"Since $j_b \\\\in [\\\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\\\mu )^{r+1} - (1+\\\\mu )^r$ $= \\\\mu (1+\\\\mu )^r$ .\",\n \"By Claim , $E(x,y) \\\\ge j_c - p\\\\bar{n} \\\\ge (1+\\\\mu )^r$ .\",\n \"The number of characters of $x$ with indices in $[(p-1)\\\\bar{n},p\\\\bar{n}]$ that are not in $L^{\\\\prime }$ is at most $\\\\mu E(x,y)$ .\",\n \"The total number of characters of $L^{\\\\prime }$ not in $L$ is at most $\\\\mu (n/\\\\bar{n}) E(x,y)$ $\\\\le \\\\delta E(x,y)$ .\",\n \"$\\\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.\",\n \"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .\",\n \"Suppose we are given the lengths of the largest ${\\\\cal S}$ -consistent common subsequences between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"Also, suppose we have access to $x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y$ .\",\n \"Then, we can compute the lengths of the largest ${\\\\cal S}$ -consistent common sequences between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ (for all $j \\\\in S_{i+1}$ ) using $\\\\bar{n}$ space.\",\n \"The total running time is $O(n\\\\bar{n}^2)$ .\",\n \"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.\",\n \"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.\",\n \"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\\\gamma /2n$ .\",\n \"A union bound over all $t$ proves the third assertion of Theorem REF .\",\n \"Let $M=\\\\frac{2}{\\\\delta } \\\\ln (4n^3/\\\\gamma )\\\\ln (e\\\\rho )$ .\",\n \"Fix $t \\\\in [n]$ .\",\n \"Then ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le \\\\gamma /2n$ .\",\n \"For $i \\\\in [t]$ let $Z_i=1$ if $i \\\\in R_t$ and 0 otherwise.\",\n \"Then $|R_t|=\\\\sum _{i \\\\le t} Z_i$ .\",\n \"Let $\\\\mu =\\\\mathbb {E}[|R_t|]$ .\",\n \"Below we show that $\\\\mu \\\\le M$ .\",\n \"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\\\sum _i Z_i$ , and let $\\\\mu =\\\\mathbb {E}[Z]$ .\",\n \"Then for any $C \\\\ge 0$ , ${\\\\rm Prob}[Z \\\\ge C] \\\\le (e\\\\mu /C)^{C}$ .\",\n \"Applying this proposition with $C=e^2M$ gives ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le e^{-C}$ which is at most $\\\\gamma /2n$ (with a lot of room to spare).\",\n \"It remains to show that $\\\\mu \\\\le M$ .\",\n \"We have: $ \\\\mu & = & \\\\sum _{i=1}^t \\\\mathbb {E}[Z_i] = \\\\sum _{i=1}^t q(i,t) \\\\\\\\& & \\\\le \\\\frac{2}{\\\\delta }\\\\ln (4n^3/\\\\gamma ) \\\\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.\",\n \"For $r \\\\ge 1$ , $\\\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\\\le \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)})$ .\",\n \"We prove by backwards induction on $r$ .\",\n \"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.\",\n \"Assume up to $r \\\\ge 2$ , and we shall prove the statement for $r-1$ .\",\n \"We start with a technical statement.\",\n \"$& & \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& = & \\\\ln (\\\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\\\\\& = & - \\\\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\\\\\& \\\\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\\\sum _{i=r-1}^t \\\\frac{w(i)}{W(t)-W(i-1)} \\\\\\\\& = & \\\\sum _{i=r}^t \\\\frac{w(i)}{W(t)-W(i-1)} + \\\\frac{w(r-1)}{W(t) - W(r-2)} \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)}) + \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\\\Box $ Thus $\\\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\\\le \\\\ln (eW(t)/w(t)) \\\\le \\\\ln (e\\\\rho )$ , and so $\\\\mu \\\\le M$ .\",\n \"This completes the proof.\",\n \"$\\\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.\",\n \"Given a common subsequence $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ , the defect is at least $\\\\max _{k \\\\le r} (|i_k - j_k|)$ .\",\n \"(of Claim ) Assume wlog that $i_k \\\\ge j_k$ .\",\n \"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .\",\n \"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .\",\n \"Hence the defect is at least $i_k - j_k$ .\",\n \"Repeating over all $k$ , we complete the proof.\",\n \"$\\\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).\",\n \"If either $j_a \\\\in [i,i_a]$ or $i \\\\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .\",\n \"Suppose neither of these are true.\",\n \"Then $j_a > i_a \\\\ge i$ .\",\n \"Let us focus on the characters of $x$ that are not matched.\",\n \"No character of $x$ with index in $[i,i_a)$ is matched.\",\n \"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).\",\n \"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .\",\n \"So the number of unmatched characters in $x$ is at least $j_a - i$ .\",\n \"$\\\\Box $ (of Claim ) Consider some $j \\\\in S_{i+1}$ , and set $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ .\",\n \"We wish to compute the largest ${\\\\cal S}$ -consistent CS between in $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"Suppose we look at the portion of this CS in $x(1,i\\\\bar{n})$ .\",\n \"This forms a ${\\\\cal S}$ -consistent sequence between $\\\\bar{x}$ and $y(1,j^{\\\\prime })$ , for some $j^{\\\\prime } \\\\in S_i$ .\",\n \"The remaining portion of the CS is just the LCS between $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y(j^{\\\\prime },j)$ .\",\n \"Hence, given the LCS length of $\\\\bar{x}$ and $y(j^{\\\\prime },j)$ , for all $j^{\\\\prime } \\\\in S_i$ , we can compute the length of the largest ${\\\\cal S}$ -consistent CS between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"This is obtained by just maximizing over all possible $j^{\\\\prime }$ .\",\n \"We now apply Claim .\",\n \"We have $\\\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .\",\n \"Using $O(\\\\bar{n})$ space, we can compute all the LCS lengths desired.\",\n \"This gives the length of the largest ${\\\\cal S}$ -consistent CS that ends at $y(j)$ .\",\n \"This can be done for all $y(j)$ , $j \\\\in S_i$ .\",\n \"The total running time is $O(|S_{i+1}|n\\\\bar{n}) = O(n\\\\bar{n}^2)$ .\",\n \"$\\\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Consider the index $i\\\\bar{n}$ .\",\n \"Suppose we have currently stored the lengths of the largest ${\\\\cal S}$ -consistent CS between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"This requires space $O(|S_i|) = O(\\\\bar{n}))$ .\",\n \"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\\\bar{n})$ space.\",\n \"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Lemma tells us that this defect is a $(1+\\\\delta )$ - approximation to $E(x,y)$ .\",\n \"The space bound is $O(\\\\bar{n})$ .\",\n \"The number of updates is $O(n/\\\\bar{n}) = O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ , and the time for each update is $O(n{\\\\bar{n}}^2) = O((n^2\\\\ln n)/\\\\delta )$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"Paths in posets\",\n \"We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).\",\n \"We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.\",\n \"Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.\",\n \"This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP.\"\n ],\n [\n \"Weighted $P$ -sequences and the approximate minimum-defect path problem\",\n \"We use $P$ to denote a fixed set endowed with a partial order $\\\\triangleleft $ .\",\n \"The partial order relation is given by an oracle which, given $u,v \\\\in P$ outputs $u \\\\triangleleft v$ or $\\\\lnot (u \\\\triangleleft v)$ .\",\n \"For a natural number $n$ we write $[n]$ for the set $\\\\lbrace 1,2,\\\\ldots ,n\\\\rbrace $ .\",\n \"A sequence ${\\\\bf \\\\sigma }=(\\\\sigma (1),\\\\ldots ,\\\\sigma (n)) \\\\in P$ is called a $P$ -sequence.\",\n \"The number of terms ${\\\\bf \\\\sigma }$ is called the length of ${\\\\bf \\\\sigma }$ and is denoted $|{\\\\bf \\\\sigma }|$ ; we normally use $n$ to denote $|{\\\\bf \\\\sigma }|$ .\",\n \"A weighted $P$ -sequence consists of a $P$ -sequence ${\\\\bf \\\\sigma }$ together with a sequence $(w(1),\\\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .\",\n \"In all our final applications $w(i)$ will always be 1.\",\n \"Nonetheless, we solve this slightly more general weighted version.\",\n \"We have the following additional definitions: For $t \\\\in [n]$ , ${\\\\bf \\\\sigma }_{\\\\le t}$ denotes the sequence $(\\\\sigma _1,\\\\ldots ,\\\\sigma _t)$ .\",\n \"Also for $J \\\\subseteq [n]$ , $J_{\\\\le t}$ denotes the set $J \\\\cap \\\\lbrace 1,\\\\ldots ,t\\\\rbrace $ .\",\n \"For $J \\\\subseteq [n]$ , $w(J)=\\\\sum _{j \\\\in J} w(j)$ .\",\n \"The digraph $D=D({\\\\bf \\\\sigma })$ associated to the $P$ -sequence ${\\\\bf \\\\sigma }$ has vertex set $[n]$ (where $n=|{\\\\bf \\\\sigma }|$ ) and arc set $\\\\lbrace i \\\\rightarrow j: i0$ .\",\n \"The output is a number $A$ such that: ${\\\\rm Prob}[A \\\\in [\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w),(1+\\\\delta ) \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)]] \\\\ge 1-\\\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ is said to be a one-sided error algorithm.\",\n \"Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.\",\n \"For the AMDP, the input consists of the parameters $\\\\delta $ and $\\\\gamma $ together with a sequence of $n$ pairs $((\\\\sigma (t),w(t)):t \\\\in [n])$ .\",\n \"We think of the input as arriving in a sequence of discrete time steps, where $\\\\delta ,\\\\gamma $ arrive at time step 0 and for $t \\\\in [n]$ , $(\\\\sigma (t),w(t))$ arrives at time step $t$ .\",\n \"The main complexity parameter of interest is the auxiliary memory needed.\",\n \"For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.\",\n \"Associated to a weighted $P$ -sequence $({\\\\bf \\\\sigma },w)$ we define the parameter: $\\\\rho =\\\\rho (w)=\\\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\\\rho =n^{O(1)}$ .\",\n \"The main technical theorem about AMDP is the following.\",\n \"There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\\\frac{\\\\ln (n/\\\\gamma )\\\\ln (\\\\rho )}{\\\\delta })$ .\",\n \"In particular, if $\\\\rho =n^{O(1)}$ and $\\\\gamma =1/n^{O(1)}$ then the space is $O(\\\\frac{(\\\\ln (n))^2}{\\\\delta })$ .\",\n \"The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"We first review this dynamic program.\",\n \"Exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\\\sigma (n+1)$ that is greater than all other elements of $P$ .\",\n \"Thus all arcs $j \\\\longrightarrow n+1$ for $j \\\\in [n]$ are present.\",\n \"Set $w(n+1)=0$ .\",\n \"We define sequences $s(0),\\\\ldots ,s(n+1)$ and $W(0),\\\\ldots ,W(n+1)$ as follows.\",\n \"We initialize $s(0) = 0$ and $W(0) = 0$ .\",\n \"For $t \\\\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\\\\\s(t) & = & \\\\min (s(i)+W(t-1)-W(i): i < t \\\\\\\\& & \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)).$ Thus $W(t)=w([t])$ .\",\n \"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\\\pi )$ over all paths $\\\\pi $ whose maximum element is $\\\\sigma (t)$ .\",\n \"In particular, $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)=s(n+1)$ .\",\n \"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).\",\n \"The polylog space streaming algorithm We denote our streaming algorithm by $\\\\Gamma =\\\\Gamma ({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ .\",\n \"Our approximation algorithm is a natural variant of the exact algorithm.\",\n \"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .\",\n \"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.\",\n \"The amount of space used by the algorithm is proportional to the maximum size of $R$ .\",\n \"We first define the probabilities $p(i,t)$ .\",\n \"Similar quantities were defined in [20].\",\n \"$& & q(i,t) \\\\\\\\& = & \\\\min \\\\left\\\\lbrace 1,\\\\frac{1+\\\\delta }{\\\\delta }\\\\ln (4t^3/\\\\gamma )\\\\frac{w(i)}{W(t)-W(i-1)}\\\\right\\\\rbrace \\\\\\\\& & p(i,i) = 1 \\\\ \\\\ \\\\ \\\\ p(i,t) = \\\\frac{q(i,t)}{q(i,t-1)} \\\\text{ for $t>i$},$ Note that in the typical case that $\\\\delta = \\\\theta (1)$ and $\\\\gamma = \\\\log (n)^{-\\\\Theta (1)}$ , we have $q(i,t)$ is $\\\\Theta (\\\\ln (n)/(t-i))$ .\",\n \"We initialize $R=\\\\lbrace 0\\\\rbrace $ , $r(0)=0$ and $W(0)=0$ .\",\n \"The following update is performed for each time step $t \\\\in [n+1]$ .\",\n \"The final output is just $r(n+1)$ .\",\n \"$W(t)=W(t-1)+w(t)$ .\",\n \"$r(t)=\\\\min (r(i)+W(t-1)-W(i): i \\\\in R\\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)$ .\",\n \"The index $t$ is inserted in $R$ .\",\n \"Each element $i \\\\in R$ is (independently) discarded with probability $1-p(i,t)$ .\",\n \"On input $({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ , the algorithm $\\\\Gamma $ satisfies: $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"${\\\\rm Prob}[r(n+1) > (1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)] \\\\le \\\\gamma /2$ .\",\n \"The probability that $|R|$ ever exceeds $ \\\\frac{2e^2}{\\\\delta }\\\\ln (2\\\\rho )\\\\ln (4n^3/\\\\gamma )$ is at most $\\\\gamma /2$ .\",\n \"The above theorem does not exactly give what was promised in Theorem REF .\",\n \"For the algorithm $\\\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.\",\n \"To achieve the guarantee of Theorem REF we modify $\\\\Gamma $ to an algorithm $\\\\Gamma ^{\\\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.\",\n \"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\\\gamma /2$ , the probability that the output of $\\\\Gamma ^{\\\\prime }$ exceeds $(1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ is at most $\\\\gamma $ .\",\n \"We now prove Theorem REF .\",\n \"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\\\le n+1$ we have $r(j) \\\\ge s(j)$ and thus $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"The second part will be proved in the two subsection.\",\n \"The final assertion of Theorem REF showing the space bound is deferred to Appendix .\",\n \"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.\",\n \"Let $R_t$ denote the set $R$ after processing $\\\\sigma (t),w(t)$ .\",\n \"Observe that the definition of $p(i,j)$ implies: For each $i \\\\le t \\\\le n$ , ${\\\\rm Prob}[i \\\\in R_t]=\\\\prod _{j \\\\in [i,t]}p(i,j)=q(i,t)$ .\",\n \"We need some additional definitions.\",\n \"For $I \\\\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\\\bar{I}$ .\",\n \"Let $C$ be the index set of some fixed chain having minimum $\\\\mbox{\\\\rm defect}$ , so that the minimum defect is equal to $w(\\\\bar{C})$ .\",\n \"We assume without loss of generality that $n+1 \\\\in C$ .\",\n \"We write $R^t$ for the subset $R$ at the end of step $t$ .\",\n \"Note that $R^t \\\\subseteq [t]$ .\",\n \"We define $F^t=[t]-R^t$ .\",\n \"An index $i \\\\in R^t$ is said to be remembered at time $t$ and $i \\\\in F^t$ is said to be forgotten by time $t$.\",\n \"Index $i \\\\in C$ is said to be unsafe at time $t$ if every index in $C \\\\cap [i,t] \\\\subseteq F^t$ , i.e., every index of $C \\\\cap [i,t]$ is forgotten by time $t$ .\",\n \"We write $U^t$ for the set of indices that are unsafe at time $t$ .\",\n \"An index $i \\\\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.\",\n \"We denote the set of unsafe indices by $U$ .\",\n \"On any execution, the set $U$ is determined by the sequence $R^1,\\\\ldots ,R^n$ .\",\n \"On any execution of the algorithm, $r(n+1) \\\\le w(\\\\bar{C} \\\\cup U)$ .\",\n \"We prove by induction on $t$ that if $t \\\\in C$ then $r(t) \\\\le w(\\\\bar{C}_{\\\\le t-1} \\\\cup U^{t-1})$ .\",\n \"Assume $t \\\\ge 1$ and that the result holds for $j 0$ .\",\n \"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .\",\n \"There is a deterministic one-pass streaming algorithm that computes a $(1+\\\\delta )$ -approximation to $E(x,y)$ that uses $O(\\\\sqrt{(n \\\\ln n)/\\\\delta })$ space.\",\n \"The algorithm performs $O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ updates, each taking $O(n^2\\\\ln n/\\\\delta )$ time.\",\n \"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].\",\n \"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .\",\n \"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\\\in [n]$ .\",\n \"Our aim is to implement (an approximation of) this algorithm in sublinear space.\",\n \"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.\",\n \"In some sense, we only maintain a small subset of the partial solutions.\",\n \"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .\",\n \"We convert an LCS into finding the longest chain in a set of points $P$ .\",\n \"We construct a set of anchor points in the plane, which may not be in $P$ .\",\n \"Our aim is to just maintain the longest chain between pairs of anchor points.\",\n \"Let $\\\\delta > 0$ be some fixed parameter.\",\n \"We set $\\\\bar{n} = \\\\sqrt{(n\\\\ln n)/\\\\delta }$ and $\\\\mu = (\\\\ln n)/\\\\bar{n} = \\\\sqrt{(\\\\delta \\\\ln n)/n}$ .\",\n \"For each $i \\\\in [n/\\\\bar{n}]$ , the set $S_i$ of indices is defined as follows.\",\n \"$ S_i = \\\\lbrace \\\\lfloor i\\\\bar{n} + b(1+\\\\mu )^r \\\\rfloor \\\\big | r \\\\ge 0, b \\\\in \\\\lbrace -1, +1\\\\rbrace \\\\rbrace $ For convenience, we treat $\\\\bar{n}$ , $n/\\\\bar{n}$ , and $(1+\\\\mu )^r$ as integersFormally, we need to take floors of these quantities.\",\n \"Our analysis remains identical..\",\n \"So we can drop the floors used in the definition of $S_i$ .\",\n \"Note that the $|S_i| = O(\\\\mu ^{-1}\\\\ln n) = O(\\\\bar{n})$ .\",\n \"We refer to the family of sets $\\\\lbrace S_1, S_2, \\\\ldots \\\\rbrace $ by ${\\\\cal S}$ .\",\n \"This is the set of anchor points that we discussed earlier.\",\n \"Note that they are placed according to a geometric grid.\",\n \"A common subsequence of $x$ and $y$ is consistent with ${\\\\cal S}$ if the following happens.\",\n \"There exists a sequence of indices $\\\\ell _1 \\\\le \\\\ell _2 \\\\le \\\\ldots \\\\ell _m$ such that $\\\\ell _i \\\\in S_i$ and if character $x(k)$ ($k \\\\in [i\\\\bar{n}, (i+1)\\\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\\\prime })$ , then $k^{\\\\prime } \\\\in [\\\\ell _i, \\\\ell _{i+1}]$ .\",\n \"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).\",\n \"This gives us a simple bound on the defect that we shall exploit.\",\n \"Lemma makes an important argument.\",\n \"It argues that the the anchor points ${\\\\cal S}$ were chosen such that an ${\\\\cal S}$ -consistent sequence is “almost\\\" the LCS.\",\n \"Suppose that $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.\",\n \"Let $i \\\\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\\\ge i$ .\",\n \"The defect $n-r$ is at least $|j_a - i|$ .\",\n \"There exists an ${\\\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\\\delta )E(x,y)$ .\",\n \"We start with an LCS $L$ of $x$ and $y$ and “round\\\" it to be ${\\\\cal S}$ -consistent.\",\n \"Let $L$ be $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ .\",\n \"Consider some $p \\\\in [n/\\\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"Set $\\\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .\",\n \"We construct a new common sequence $L^{\\\\prime }$ by removing certain matches from $L$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .\",\n \"If $i_b \\\\in [p\\\\bar{n},(p+1)\\\\bar{n}]$ and $j_b \\\\le \\\\ell _{p+1}$ , then we add this pair to $L^{\\\\prime }$ .\",\n \"Otherwise, it is not added.\",\n \"Note that $j_b \\\\ge \\\\ell _p$ , simply by construction.\",\n \"The new common sequence $L^{\\\\prime }$ is ${\\\\cal S}$ -consistent.\",\n \"It now remains to bound the defect of $L^{\\\\prime }$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b)) \\\\in L$ that is not present in $L^{\\\\prime }$ .\",\n \"Let $i_b \\\\in [(p-1)\\\\bar{n},p\\\\bar{n}]$ .\",\n \"This means that ${j_b} > \\\\ell _{p}$ .\",\n \"Let $i_c$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"So $\\\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .\",\n \"Let $\\\\ell _{p} = p\\\\bar{n} + (1+\\\\mu )^r $ .\",\n \"We have $j_c - p\\\\bar{n} = [(1+\\\\mu )^r, (1+\\\\mu )^{r+1}]$ .\",\n \"Since $j_b \\\\in [\\\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\\\mu )^{r+1} - (1+\\\\mu )^r$ $= \\\\mu (1+\\\\mu )^r$ .\",\n \"By Claim , $E(x,y) \\\\ge j_c - p\\\\bar{n} \\\\ge (1+\\\\mu )^r$ .\",\n \"The number of characters of $x$ with indices in $[(p-1)\\\\bar{n},p\\\\bar{n}]$ that are not in $L^{\\\\prime }$ is at most $\\\\mu E(x,y)$ .\",\n \"The total number of characters of $L^{\\\\prime }$ not in $L$ is at most $\\\\mu (n/\\\\bar{n}) E(x,y)$ $\\\\le \\\\delta E(x,y)$ .\",\n \"$\\\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.\",\n \"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .\",\n \"Suppose we are given the lengths of the largest ${\\\\cal S}$ -consistent common subsequences between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"Also, suppose we have access to $x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y$ .\",\n \"Then, we can compute the lengths of the largest ${\\\\cal S}$ -consistent common sequences between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ (for all $j \\\\in S_{i+1}$ ) using $\\\\bar{n}$ space.\",\n \"The total running time is $O(n\\\\bar{n}^2)$ .\",\n \"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.\",\n \"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.\",\n \"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\\\gamma /2n$ .\",\n \"A union bound over all $t$ proves the third assertion of Theorem REF .\",\n \"Let $M=\\\\frac{2}{\\\\delta } \\\\ln (4n^3/\\\\gamma )\\\\ln (e\\\\rho )$ .\",\n \"Fix $t \\\\in [n]$ .\",\n \"Then ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le \\\\gamma /2n$ .\",\n \"For $i \\\\in [t]$ let $Z_i=1$ if $i \\\\in R_t$ and 0 otherwise.\",\n \"Then $|R_t|=\\\\sum _{i \\\\le t} Z_i$ .\",\n \"Let $\\\\mu =\\\\mathbb {E}[|R_t|]$ .\",\n \"Below we show that $\\\\mu \\\\le M$ .\",\n \"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\\\sum _i Z_i$ , and let $\\\\mu =\\\\mathbb {E}[Z]$ .\",\n \"Then for any $C \\\\ge 0$ , ${\\\\rm Prob}[Z \\\\ge C] \\\\le (e\\\\mu /C)^{C}$ .\",\n \"Applying this proposition with $C=e^2M$ gives ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le e^{-C}$ which is at most $\\\\gamma /2n$ (with a lot of room to spare).\",\n \"It remains to show that $\\\\mu \\\\le M$ .\",\n \"We have: $ \\\\mu & = & \\\\sum _{i=1}^t \\\\mathbb {E}[Z_i] = \\\\sum _{i=1}^t q(i,t) \\\\\\\\& & \\\\le \\\\frac{2}{\\\\delta }\\\\ln (4n^3/\\\\gamma ) \\\\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.\",\n \"For $r \\\\ge 1$ , $\\\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\\\le \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)})$ .\",\n \"We prove by backwards induction on $r$ .\",\n \"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.\",\n \"Assume up to $r \\\\ge 2$ , and we shall prove the statement for $r-1$ .\",\n \"We start with a technical statement.\",\n \"$& & \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& = & \\\\ln (\\\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\\\\\& = & - \\\\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\\\\\& \\\\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\\\sum _{i=r-1}^t \\\\frac{w(i)}{W(t)-W(i-1)} \\\\\\\\& = & \\\\sum _{i=r}^t \\\\frac{w(i)}{W(t)-W(i-1)} + \\\\frac{w(r-1)}{W(t) - W(r-2)} \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)}) + \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\\\Box $ Thus $\\\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\\\le \\\\ln (eW(t)/w(t)) \\\\le \\\\ln (e\\\\rho )$ , and so $\\\\mu \\\\le M$ .\",\n \"This completes the proof.\",\n \"$\\\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.\",\n \"Given a common subsequence $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ , the defect is at least $\\\\max _{k \\\\le r} (|i_k - j_k|)$ .\",\n \"(of Claim ) Assume wlog that $i_k \\\\ge j_k$ .\",\n \"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .\",\n \"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .\",\n \"Hence the defect is at least $i_k - j_k$ .\",\n \"Repeating over all $k$ , we complete the proof.\",\n \"$\\\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).\",\n \"If either $j_a \\\\in [i,i_a]$ or $i \\\\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .\",\n \"Suppose neither of these are true.\",\n \"Then $j_a > i_a \\\\ge i$ .\",\n \"Let us focus on the characters of $x$ that are not matched.\",\n \"No character of $x$ with index in $[i,i_a)$ is matched.\",\n \"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).\",\n \"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .\",\n \"So the number of unmatched characters in $x$ is at least $j_a - i$ .\",\n \"$\\\\Box $ (of Claim ) Consider some $j \\\\in S_{i+1}$ , and set $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ .\",\n \"We wish to compute the largest ${\\\\cal S}$ -consistent CS between in $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"Suppose we look at the portion of this CS in $x(1,i\\\\bar{n})$ .\",\n \"This forms a ${\\\\cal S}$ -consistent sequence between $\\\\bar{x}$ and $y(1,j^{\\\\prime })$ , for some $j^{\\\\prime } \\\\in S_i$ .\",\n \"The remaining portion of the CS is just the LCS between $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y(j^{\\\\prime },j)$ .\",\n \"Hence, given the LCS length of $\\\\bar{x}$ and $y(j^{\\\\prime },j)$ , for all $j^{\\\\prime } \\\\in S_i$ , we can compute the length of the largest ${\\\\cal S}$ -consistent CS between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"This is obtained by just maximizing over all possible $j^{\\\\prime }$ .\",\n \"We now apply Claim .\",\n \"We have $\\\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .\",\n \"Using $O(\\\\bar{n})$ space, we can compute all the LCS lengths desired.\",\n \"This gives the length of the largest ${\\\\cal S}$ -consistent CS that ends at $y(j)$ .\",\n \"This can be done for all $y(j)$ , $j \\\\in S_i$ .\",\n \"The total running time is $O(|S_{i+1}|n\\\\bar{n}) = O(n\\\\bar{n}^2)$ .\",\n \"$\\\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Consider the index $i\\\\bar{n}$ .\",\n \"Suppose we have currently stored the lengths of the largest ${\\\\cal S}$ -consistent CS between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"This requires space $O(|S_i|) = O(\\\\bar{n}))$ .\",\n \"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\\\bar{n})$ space.\",\n \"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Lemma tells us that this defect is a $(1+\\\\delta )$ - approximation to $E(x,y)$ .\",\n \"The space bound is $O(\\\\bar{n})$ .\",\n \"The number of updates is $O(n/\\\\bar{n}) = O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ , and the time for each update is $O(n{\\\\bar{n}}^2) = O((n^2\\\\ln n)/\\\\delta )$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"The algorithm\",\n \"Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"We first review this dynamic program.\"\n ],\n [\n \"Exact computation of $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$\",\n \"It will be convenient to extend the $P$ -sequence by an element $\\\\sigma (n+1)$ that is greater than all other elements of $P$ .\",\n \"Thus all arcs $j \\\\longrightarrow n+1$ for $j \\\\in [n]$ are present.\",\n \"Set $w(n+1)=0$ .\",\n \"We define sequences $s(0),\\\\ldots ,s(n+1)$ and $W(0),\\\\ldots ,W(n+1)$ as follows.\",\n \"We initialize $s(0) = 0$ and $W(0) = 0$ .\",\n \"For $t \\\\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\\\\\s(t) & = & \\\\min (s(i)+W(t-1)-W(i): i < t \\\\\\\\& & \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)).$ Thus $W(t)=w([t])$ .\",\n \"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\\\pi )$ over all paths $\\\\pi $ whose maximum element is $\\\\sigma (t)$ .\",\n \"In particular, $\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)=s(n+1)$ .\",\n \"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).\"\n ],\n [\n \"The polylog space streaming algorithm\",\n \"We denote our streaming algorithm by $\\\\Gamma =\\\\Gamma ({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ .\",\n \"Our approximation algorithm is a natural variant of the exact algorithm.\",\n \"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .\",\n \"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.\",\n \"The amount of space used by the algorithm is proportional to the maximum size of $R$ .\",\n \"We first define the probabilities $p(i,t)$ .\",\n \"Similar quantities were defined in [20].\",\n \"$& & q(i,t) \\\\\\\\& = & \\\\min \\\\left\\\\lbrace 1,\\\\frac{1+\\\\delta }{\\\\delta }\\\\ln (4t^3/\\\\gamma )\\\\frac{w(i)}{W(t)-W(i-1)}\\\\right\\\\rbrace \\\\\\\\& & p(i,i) = 1 \\\\ \\\\ \\\\ \\\\ p(i,t) = \\\\frac{q(i,t)}{q(i,t-1)} \\\\text{ for $t>i$},$ Note that in the typical case that $\\\\delta = \\\\theta (1)$ and $\\\\gamma = \\\\log (n)^{-\\\\Theta (1)}$ , we have $q(i,t)$ is $\\\\Theta (\\\\ln (n)/(t-i))$ .\",\n \"We initialize $R=\\\\lbrace 0\\\\rbrace $ , $r(0)=0$ and $W(0)=0$ .\",\n \"The following update is performed for each time step $t \\\\in [n+1]$ .\",\n \"The final output is just $r(n+1)$ .\",\n \"$W(t)=W(t-1)+w(t)$ .\",\n \"$r(t)=\\\\min (r(i)+W(t-1)-W(i): i \\\\in R\\\\mbox{ such that } \\\\sigma _i \\\\longrightarrow \\\\sigma _t)$ .\",\n \"The index $t$ is inserted in $R$ .\",\n \"Each element $i \\\\in R$ is (independently) discarded with probability $1-p(i,t)$ .\",\n \"On input $({\\\\bf \\\\sigma },w,\\\\delta ,\\\\gamma )$ , the algorithm $\\\\Gamma $ satisfies: $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"${\\\\rm Prob}[r(n+1) > (1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)] \\\\le \\\\gamma /2$ .\",\n \"The probability that $|R|$ ever exceeds $ \\\\frac{2e^2}{\\\\delta }\\\\ln (2\\\\rho )\\\\ln (4n^3/\\\\gamma )$ is at most $\\\\gamma /2$ .\",\n \"The above theorem does not exactly give what was promised in Theorem REF .\",\n \"For the algorithm $\\\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.\",\n \"To achieve the guarantee of Theorem REF we modify $\\\\Gamma $ to an algorithm $\\\\Gamma ^{\\\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.\",\n \"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\\\gamma /2$ , the probability that the output of $\\\\Gamma ^{\\\\prime }$ exceeds $(1+\\\\delta )\\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ is at most $\\\\gamma $ .\",\n \"We now prove Theorem REF .\",\n \"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\\\le n+1$ we have $r(j) \\\\ge s(j)$ and thus $r(n+1) \\\\ge \\\\mbox{\\\\rm min-defect}({\\\\bf \\\\sigma },w)$ .\",\n \"The second part will be proved in the two subsection.\",\n \"The final assertion of Theorem REF showing the space bound is deferred to Appendix .\",\n \"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.\",\n \"Let $R_t$ denote the set $R$ after processing $\\\\sigma (t),w(t)$ .\",\n \"Observe that the definition of $p(i,j)$ implies: For each $i \\\\le t \\\\le n$ , ${\\\\rm Prob}[i \\\\in R_t]=\\\\prod _{j \\\\in [i,t]}p(i,j)=q(i,t)$ .\",\n \"We need some additional definitions.\",\n \"For $I \\\\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\\\bar{I}$ .\",\n \"Let $C$ be the index set of some fixed chain having minimum $\\\\mbox{\\\\rm defect}$ , so that the minimum defect is equal to $w(\\\\bar{C})$ .\",\n \"We assume without loss of generality that $n+1 \\\\in C$ .\",\n \"We write $R^t$ for the subset $R$ at the end of step $t$ .\",\n \"Note that $R^t \\\\subseteq [t]$ .\",\n \"We define $F^t=[t]-R^t$ .\",\n \"An index $i \\\\in R^t$ is said to be remembered at time $t$ and $i \\\\in F^t$ is said to be forgotten by time $t$.\",\n \"Index $i \\\\in C$ is said to be unsafe at time $t$ if every index in $C \\\\cap [i,t] \\\\subseteq F^t$ , i.e., every index of $C \\\\cap [i,t]$ is forgotten by time $t$ .\",\n \"We write $U^t$ for the set of indices that are unsafe at time $t$ .\",\n \"An index $i \\\\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.\",\n \"We denote the set of unsafe indices by $U$ .\",\n \"On any execution, the set $U$ is determined by the sequence $R^1,\\\\ldots ,R^n$ .\",\n \"On any execution of the algorithm, $r(n+1) \\\\le w(\\\\bar{C} \\\\cup U)$ .\",\n \"We prove by induction on $t$ that if $t \\\\in C$ then $r(t) \\\\le w(\\\\bar{C}_{\\\\le t-1} \\\\cup U^{t-1})$ .\",\n \"Assume $t \\\\ge 1$ and that the result holds for $j 0$ .\",\n \"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .\",\n \"There is a deterministic one-pass streaming algorithm that computes a $(1+\\\\delta )$ -approximation to $E(x,y)$ that uses $O(\\\\sqrt{(n \\\\ln n)/\\\\delta })$ space.\",\n \"The algorithm performs $O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ updates, each taking $O(n^2\\\\ln n/\\\\delta )$ time.\",\n \"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].\",\n \"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .\",\n \"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\\\in [n]$ .\",\n \"Our aim is to implement (an approximation of) this algorithm in sublinear space.\",\n \"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.\",\n \"In some sense, we only maintain a small subset of the partial solutions.\",\n \"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .\",\n \"We convert an LCS into finding the longest chain in a set of points $P$ .\",\n \"We construct a set of anchor points in the plane, which may not be in $P$ .\",\n \"Our aim is to just maintain the longest chain between pairs of anchor points.\",\n \"Let $\\\\delta > 0$ be some fixed parameter.\",\n \"We set $\\\\bar{n} = \\\\sqrt{(n\\\\ln n)/\\\\delta }$ and $\\\\mu = (\\\\ln n)/\\\\bar{n} = \\\\sqrt{(\\\\delta \\\\ln n)/n}$ .\",\n \"For each $i \\\\in [n/\\\\bar{n}]$ , the set $S_i$ of indices is defined as follows.\",\n \"$ S_i = \\\\lbrace \\\\lfloor i\\\\bar{n} + b(1+\\\\mu )^r \\\\rfloor \\\\big | r \\\\ge 0, b \\\\in \\\\lbrace -1, +1\\\\rbrace \\\\rbrace $ For convenience, we treat $\\\\bar{n}$ , $n/\\\\bar{n}$ , and $(1+\\\\mu )^r$ as integersFormally, we need to take floors of these quantities.\",\n \"Our analysis remains identical..\",\n \"So we can drop the floors used in the definition of $S_i$ .\",\n \"Note that the $|S_i| = O(\\\\mu ^{-1}\\\\ln n) = O(\\\\bar{n})$ .\",\n \"We refer to the family of sets $\\\\lbrace S_1, S_2, \\\\ldots \\\\rbrace $ by ${\\\\cal S}$ .\",\n \"This is the set of anchor points that we discussed earlier.\",\n \"Note that they are placed according to a geometric grid.\",\n \"A common subsequence of $x$ and $y$ is consistent with ${\\\\cal S}$ if the following happens.\",\n \"There exists a sequence of indices $\\\\ell _1 \\\\le \\\\ell _2 \\\\le \\\\ldots \\\\ell _m$ such that $\\\\ell _i \\\\in S_i$ and if character $x(k)$ ($k \\\\in [i\\\\bar{n}, (i+1)\\\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\\\prime })$ , then $k^{\\\\prime } \\\\in [\\\\ell _i, \\\\ell _{i+1}]$ .\",\n \"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).\",\n \"This gives us a simple bound on the defect that we shall exploit.\",\n \"Lemma makes an important argument.\",\n \"It argues that the the anchor points ${\\\\cal S}$ were chosen such that an ${\\\\cal S}$ -consistent sequence is “almost\\\" the LCS.\",\n \"Suppose that $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.\",\n \"Let $i \\\\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\\\ge i$ .\",\n \"The defect $n-r$ is at least $|j_a - i|$ .\",\n \"There exists an ${\\\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\\\delta )E(x,y)$ .\",\n \"We start with an LCS $L$ of $x$ and $y$ and “round\\\" it to be ${\\\\cal S}$ -consistent.\",\n \"Let $L$ be $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ .\",\n \"Consider some $p \\\\in [n/\\\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"Set $\\\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .\",\n \"We construct a new common sequence $L^{\\\\prime }$ by removing certain matches from $L$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .\",\n \"If $i_b \\\\in [p\\\\bar{n},(p+1)\\\\bar{n}]$ and $j_b \\\\le \\\\ell _{p+1}$ , then we add this pair to $L^{\\\\prime }$ .\",\n \"Otherwise, it is not added.\",\n \"Note that $j_b \\\\ge \\\\ell _p$ , simply by construction.\",\n \"The new common sequence $L^{\\\\prime }$ is ${\\\\cal S}$ -consistent.\",\n \"It now remains to bound the defect of $L^{\\\\prime }$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b)) \\\\in L$ that is not present in $L^{\\\\prime }$ .\",\n \"Let $i_b \\\\in [(p-1)\\\\bar{n},p\\\\bar{n}]$ .\",\n \"This means that ${j_b} > \\\\ell _{p}$ .\",\n \"Let $i_c$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"So $\\\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .\",\n \"Let $\\\\ell _{p} = p\\\\bar{n} + (1+\\\\mu )^r $ .\",\n \"We have $j_c - p\\\\bar{n} = [(1+\\\\mu )^r, (1+\\\\mu )^{r+1}]$ .\",\n \"Since $j_b \\\\in [\\\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\\\mu )^{r+1} - (1+\\\\mu )^r$ $= \\\\mu (1+\\\\mu )^r$ .\",\n \"By Claim , $E(x,y) \\\\ge j_c - p\\\\bar{n} \\\\ge (1+\\\\mu )^r$ .\",\n \"The number of characters of $x$ with indices in $[(p-1)\\\\bar{n},p\\\\bar{n}]$ that are not in $L^{\\\\prime }$ is at most $\\\\mu E(x,y)$ .\",\n \"The total number of characters of $L^{\\\\prime }$ not in $L$ is at most $\\\\mu (n/\\\\bar{n}) E(x,y)$ $\\\\le \\\\delta E(x,y)$ .\",\n \"$\\\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.\",\n \"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .\",\n \"Suppose we are given the lengths of the largest ${\\\\cal S}$ -consistent common subsequences between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"Also, suppose we have access to $x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y$ .\",\n \"Then, we can compute the lengths of the largest ${\\\\cal S}$ -consistent common sequences between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ (for all $j \\\\in S_{i+1}$ ) using $\\\\bar{n}$ space.\",\n \"The total running time is $O(n\\\\bar{n}^2)$ .\",\n \"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.\",\n \"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.\",\n \"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\\\gamma /2n$ .\",\n \"A union bound over all $t$ proves the third assertion of Theorem REF .\",\n \"Let $M=\\\\frac{2}{\\\\delta } \\\\ln (4n^3/\\\\gamma )\\\\ln (e\\\\rho )$ .\",\n \"Fix $t \\\\in [n]$ .\",\n \"Then ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le \\\\gamma /2n$ .\",\n \"For $i \\\\in [t]$ let $Z_i=1$ if $i \\\\in R_t$ and 0 otherwise.\",\n \"Then $|R_t|=\\\\sum _{i \\\\le t} Z_i$ .\",\n \"Let $\\\\mu =\\\\mathbb {E}[|R_t|]$ .\",\n \"Below we show that $\\\\mu \\\\le M$ .\",\n \"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\\\sum _i Z_i$ , and let $\\\\mu =\\\\mathbb {E}[Z]$ .\",\n \"Then for any $C \\\\ge 0$ , ${\\\\rm Prob}[Z \\\\ge C] \\\\le (e\\\\mu /C)^{C}$ .\",\n \"Applying this proposition with $C=e^2M$ gives ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le e^{-C}$ which is at most $\\\\gamma /2n$ (with a lot of room to spare).\",\n \"It remains to show that $\\\\mu \\\\le M$ .\",\n \"We have: $ \\\\mu & = & \\\\sum _{i=1}^t \\\\mathbb {E}[Z_i] = \\\\sum _{i=1}^t q(i,t) \\\\\\\\& & \\\\le \\\\frac{2}{\\\\delta }\\\\ln (4n^3/\\\\gamma ) \\\\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.\",\n \"For $r \\\\ge 1$ , $\\\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\\\le \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)})$ .\",\n \"We prove by backwards induction on $r$ .\",\n \"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.\",\n \"Assume up to $r \\\\ge 2$ , and we shall prove the statement for $r-1$ .\",\n \"We start with a technical statement.\",\n \"$& & \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& = & \\\\ln (\\\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\\\\\& = & - \\\\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\\\\\& \\\\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\\\sum _{i=r-1}^t \\\\frac{w(i)}{W(t)-W(i-1)} \\\\\\\\& = & \\\\sum _{i=r}^t \\\\frac{w(i)}{W(t)-W(i-1)} + \\\\frac{w(r-1)}{W(t) - W(r-2)} \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)}) + \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\\\Box $ Thus $\\\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\\\le \\\\ln (eW(t)/w(t)) \\\\le \\\\ln (e\\\\rho )$ , and so $\\\\mu \\\\le M$ .\",\n \"This completes the proof.\",\n \"$\\\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.\",\n \"Given a common subsequence $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ , the defect is at least $\\\\max _{k \\\\le r} (|i_k - j_k|)$ .\",\n \"(of Claim ) Assume wlog that $i_k \\\\ge j_k$ .\",\n \"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .\",\n \"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .\",\n \"Hence the defect is at least $i_k - j_k$ .\",\n \"Repeating over all $k$ , we complete the proof.\",\n \"$\\\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).\",\n \"If either $j_a \\\\in [i,i_a]$ or $i \\\\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .\",\n \"Suppose neither of these are true.\",\n \"Then $j_a > i_a \\\\ge i$ .\",\n \"Let us focus on the characters of $x$ that are not matched.\",\n \"No character of $x$ with index in $[i,i_a)$ is matched.\",\n \"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).\",\n \"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .\",\n \"So the number of unmatched characters in $x$ is at least $j_a - i$ .\",\n \"$\\\\Box $ (of Claim ) Consider some $j \\\\in S_{i+1}$ , and set $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ .\",\n \"We wish to compute the largest ${\\\\cal S}$ -consistent CS between in $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"Suppose we look at the portion of this CS in $x(1,i\\\\bar{n})$ .\",\n \"This forms a ${\\\\cal S}$ -consistent sequence between $\\\\bar{x}$ and $y(1,j^{\\\\prime })$ , for some $j^{\\\\prime } \\\\in S_i$ .\",\n \"The remaining portion of the CS is just the LCS between $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y(j^{\\\\prime },j)$ .\",\n \"Hence, given the LCS length of $\\\\bar{x}$ and $y(j^{\\\\prime },j)$ , for all $j^{\\\\prime } \\\\in S_i$ , we can compute the length of the largest ${\\\\cal S}$ -consistent CS between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"This is obtained by just maximizing over all possible $j^{\\\\prime }$ .\",\n \"We now apply Claim .\",\n \"We have $\\\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .\",\n \"Using $O(\\\\bar{n})$ space, we can compute all the LCS lengths desired.\",\n \"This gives the length of the largest ${\\\\cal S}$ -consistent CS that ends at $y(j)$ .\",\n \"This can be done for all $y(j)$ , $j \\\\in S_i$ .\",\n \"The total running time is $O(|S_{i+1}|n\\\\bar{n}) = O(n\\\\bar{n}^2)$ .\",\n \"$\\\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Consider the index $i\\\\bar{n}$ .\",\n \"Suppose we have currently stored the lengths of the largest ${\\\\cal S}$ -consistent CS between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"This requires space $O(|S_i|) = O(\\\\bar{n}))$ .\",\n \"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\\\bar{n})$ space.\",\n \"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Lemma tells us that this defect is a $(1+\\\\delta )$ - approximation to $E(x,y)$ .\",\n \"The space bound is $O(\\\\bar{n})$ .\",\n \"The number of updates is $O(n/\\\\bar{n}) = O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ , and the time for each update is $O(n{\\\\bar{n}}^2) = O((n^2\\\\ln n)/\\\\delta )$ .\",\n \"$\\\\Box $\"\n ],\n [\n \"Applying AMDP to LIS and LCS\",\n \"We now show how to apply Theorem REF to LIS and LCS.\",\n \"The application to LIS is quite obvious.\",\n \"We first set some notation about points in the two-dimensional plane.\",\n \"We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.\",\n \"$z(2)$ ) refers to the first (resp.\",\n \"second) coordinate of $z$ .\",\n \"We use the standard coordinate-wise partial order on $z$ .\",\n \"So $z \\\\triangleleft z^{\\\\prime }$ iff $z(1) < z^{\\\\prime }(1)$ and $z(2) < z^{\\\\prime }(2)$ .\",\n \"(of Theorem REF ) The input is a stream $x(1), x(2), \\\\ldots , x(n)$ .\",\n \"Think of the $i$ th element of the stream as the point $(i,x(i))$ .\",\n \"So the input is thought of as a sequence of points.\",\n \"Note that the points arrive in increasing order of first coordinate.\",\n \"Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).\",\n \"We set $\\\\gamma = n^{O(1)}$ and $\\\\rho = n$ in Theorem REF .\",\n \"$\\\\Box $ The application to LCS is somewhat more subtle.\",\n \"Again, we think of the input as a set of points in the two-dimensional plane.\",\n \"But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\\\delta $ .\",\n \"Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .\",\n \"Then there is a $O(\\\\delta ^{-1}k\\\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\\\delta n$ -approximation of $E(x,y)$ .\",\n \"We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.\",\n \"Let $P$ be the set of pairs $\\\\lbrace (i,j)|x(i)=y(j)\\\\rbrace $ under the partial order $(i,j) < (i^{\\\\prime },j^{\\\\prime })$ if $i < i^{\\\\prime }$ and $j < j^{\\\\prime }$ .\",\n \"It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.\",\n \"Now we associate to the pair of strings $x,y$ the sequence $\\\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\\\prime },j^{\\\\prime })$ is $i < i^{\\\\prime }$ or if $i=i^{\\\\prime }$ and $j < j^{\\\\prime }$ .)\",\n \"Note that $\\\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.\",\n \"Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\\\sigma $ -paths as defined in the AMDP.\",\n \"Thus the length of the LCS is equal to the size of the largest $\\\\sigma $ -path.\",\n \"It is not true that $E(x,y)$ is equal to $min-defect(\\\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\\\sigma $ is in general longer than $n$ .\",\n \"Given full access to $y$ , and a streamed $x$ .\",\n \"We have a bound on $|\\\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .\",\n \"We now argue that an additive $\\\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\\\delta /k)$ -approximation for AMDP of $P$ .\",\n \"Let the length of the longest chain in $P$ be $\\\\ell $ and the min-defect be $m$ .\",\n \"Let $d$ be a shorthand for $E(x,y)$ .\",\n \"We have $\\\\ell + m = |P|$ and $\\\\ell + d = n$ .\",\n \"The output of AMDP is an estimate $est$ such that $m \\\\le est \\\\le (1+\\\\delta /k)m$ .\",\n \"We estimate $d$ by $est_d = est + n - |P|$ .\",\n \"We show that $est_d \\\\in [d,d + \\\\delta n]$ .\",\n \"We have $est_d = est + n - |P| \\\\ge m + n - |P| = n-\\\\ell = d$ .\",\n \"We can also get an upper bound.\",\n \"$est_d & = & est + n - |P| \\\\\\\\& \\\\le & m + n - |P| + \\\\delta m/k \\\\\\\\& = & d + \\\\delta m/k \\\\ \\\\textrm {(since |P| - m = \\\\ell and d = n - \\\\ell )} \\\\\\\\& \\\\le & d + \\\\delta n \\\\ \\\\textrm {(since m \\\\le |P| \\\\le nk)}$ Hence, we use the parameters $\\\\delta /k, \\\\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.\",\n \"An application of Theorem REF completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Deterministic streaming algorithm for LCS\",\n \"We now discuss a deterministic $\\\\sqrt{n}$ -space algorithm for LCS.\",\n \"This can be used for large alphabets to beat the bound given in Theorem .\",\n \"For any consistent sequence (CS), the size of the complement is called the defect.\",\n \"For indices $i,j \\\\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.\",\n \"The main theorem is: Let $\\\\delta > 0$ .\",\n \"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .\",\n \"There is a deterministic one-pass streaming algorithm that computes a $(1+\\\\delta )$ -approximation to $E(x,y)$ that uses $O(\\\\sqrt{(n \\\\ln n)/\\\\delta })$ space.\",\n \"The algorithm performs $O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ updates, each taking $O(n^2\\\\ln n/\\\\delta )$ time.\",\n \"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].\",\n \"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .\",\n \"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\\\in [n]$ .\",\n \"Our aim is to implement (an approximation of) this algorithm in sublinear space.\",\n \"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.\",\n \"In some sense, we only maintain a small subset of the partial solutions.\",\n \"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .\",\n \"We convert an LCS into finding the longest chain in a set of points $P$ .\",\n \"We construct a set of anchor points in the plane, which may not be in $P$ .\",\n \"Our aim is to just maintain the longest chain between pairs of anchor points.\",\n \"Let $\\\\delta > 0$ be some fixed parameter.\",\n \"We set $\\\\bar{n} = \\\\sqrt{(n\\\\ln n)/\\\\delta }$ and $\\\\mu = (\\\\ln n)/\\\\bar{n} = \\\\sqrt{(\\\\delta \\\\ln n)/n}$ .\",\n \"For each $i \\\\in [n/\\\\bar{n}]$ , the set $S_i$ of indices is defined as follows.\",\n \"$ S_i = \\\\lbrace \\\\lfloor i\\\\bar{n} + b(1+\\\\mu )^r \\\\rfloor \\\\big | r \\\\ge 0, b \\\\in \\\\lbrace -1, +1\\\\rbrace \\\\rbrace $ For convenience, we treat $\\\\bar{n}$ , $n/\\\\bar{n}$ , and $(1+\\\\mu )^r$ as integersFormally, we need to take floors of these quantities.\",\n \"Our analysis remains identical..\",\n \"So we can drop the floors used in the definition of $S_i$ .\",\n \"Note that the $|S_i| = O(\\\\mu ^{-1}\\\\ln n) = O(\\\\bar{n})$ .\",\n \"We refer to the family of sets $\\\\lbrace S_1, S_2, \\\\ldots \\\\rbrace $ by ${\\\\cal S}$ .\",\n \"This is the set of anchor points that we discussed earlier.\",\n \"Note that they are placed according to a geometric grid.\",\n \"A common subsequence of $x$ and $y$ is consistent with ${\\\\cal S}$ if the following happens.\",\n \"There exists a sequence of indices $\\\\ell _1 \\\\le \\\\ell _2 \\\\le \\\\ldots \\\\ell _m$ such that $\\\\ell _i \\\\in S_i$ and if character $x(k)$ ($k \\\\in [i\\\\bar{n}, (i+1)\\\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\\\prime })$ , then $k^{\\\\prime } \\\\in [\\\\ell _i, \\\\ell _{i+1}]$ .\",\n \"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).\",\n \"This gives us a simple bound on the defect that we shall exploit.\",\n \"Lemma makes an important argument.\",\n \"It argues that the the anchor points ${\\\\cal S}$ were chosen such that an ${\\\\cal S}$ -consistent sequence is “almost\\\" the LCS.\",\n \"Suppose that $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.\",\n \"Let $i \\\\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\\\ge i$ .\",\n \"The defect $n-r$ is at least $|j_a - i|$ .\",\n \"There exists an ${\\\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\\\delta )E(x,y)$ .\",\n \"We start with an LCS $L$ of $x$ and $y$ and “round\\\" it to be ${\\\\cal S}$ -consistent.\",\n \"Let $L$ be $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ .\",\n \"Consider some $p \\\\in [n/\\\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"Set $\\\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .\",\n \"We construct a new common sequence $L^{\\\\prime }$ by removing certain matches from $L$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .\",\n \"If $i_b \\\\in [p\\\\bar{n},(p+1)\\\\bar{n}]$ and $j_b \\\\le \\\\ell _{p+1}$ , then we add this pair to $L^{\\\\prime }$ .\",\n \"Otherwise, it is not added.\",\n \"Note that $j_b \\\\ge \\\\ell _p$ , simply by construction.\",\n \"The new common sequence $L^{\\\\prime }$ is ${\\\\cal S}$ -consistent.\",\n \"It now remains to bound the defect of $L^{\\\\prime }$ .\",\n \"Consider a matched pair $(x(i_b), y(j_b)) \\\\in L$ that is not present in $L^{\\\\prime }$ .\",\n \"Let $i_b \\\\in [(p-1)\\\\bar{n},p\\\\bar{n}]$ .\",\n \"This means that ${j_b} > \\\\ell _{p}$ .\",\n \"Let $i_c$ be the smallest index larger than $p\\\\bar{n}$ .\",\n \"So $\\\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .\",\n \"Let $\\\\ell _{p} = p\\\\bar{n} + (1+\\\\mu )^r $ .\",\n \"We have $j_c - p\\\\bar{n} = [(1+\\\\mu )^r, (1+\\\\mu )^{r+1}]$ .\",\n \"Since $j_b \\\\in [\\\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\\\mu )^{r+1} - (1+\\\\mu )^r$ $= \\\\mu (1+\\\\mu )^r$ .\",\n \"By Claim , $E(x,y) \\\\ge j_c - p\\\\bar{n} \\\\ge (1+\\\\mu )^r$ .\",\n \"The number of characters of $x$ with indices in $[(p-1)\\\\bar{n},p\\\\bar{n}]$ that are not in $L^{\\\\prime }$ is at most $\\\\mu E(x,y)$ .\",\n \"The total number of characters of $L^{\\\\prime }$ not in $L$ is at most $\\\\mu (n/\\\\bar{n}) E(x,y)$ $\\\\le \\\\delta E(x,y)$ .\",\n \"$\\\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.\",\n \"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .\",\n \"Suppose we are given the lengths of the largest ${\\\\cal S}$ -consistent common subsequences between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"Also, suppose we have access to $x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y$ .\",\n \"Then, we can compute the lengths of the largest ${\\\\cal S}$ -consistent common sequences between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ (for all $j \\\\in S_{i+1}$ ) using $\\\\bar{n}$ space.\",\n \"The total running time is $O(n\\\\bar{n}^2)$ .\"\n ],\n [\n \"Acknowledgements\",\n \"The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.\",\n \"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.\"\n ],\n [\n \"The space bound of Theorem \",\n \"The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\\\gamma /2n$ .\",\n \"A union bound over all $t$ proves the third assertion of Theorem REF .\",\n \"Let $M=\\\\frac{2}{\\\\delta } \\\\ln (4n^3/\\\\gamma )\\\\ln (e\\\\rho )$ .\",\n \"Fix $t \\\\in [n]$ .\",\n \"Then ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le \\\\gamma /2n$ .\",\n \"For $i \\\\in [t]$ let $Z_i=1$ if $i \\\\in R_t$ and 0 otherwise.\",\n \"Then $|R_t|=\\\\sum _{i \\\\le t} Z_i$ .\",\n \"Let $\\\\mu =\\\\mathbb {E}[|R_t|]$ .\",\n \"Below we show that $\\\\mu \\\\le M$ .\",\n \"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\\\sum _i Z_i$ , and let $\\\\mu =\\\\mathbb {E}[Z]$ .\",\n \"Then for any $C \\\\ge 0$ , ${\\\\rm Prob}[Z \\\\ge C] \\\\le (e\\\\mu /C)^{C}$ .\",\n \"Applying this proposition with $C=e^2M$ gives ${\\\\rm Prob}[|R_t| \\\\ge e^2M] \\\\le e^{-C}$ which is at most $\\\\gamma /2n$ (with a lot of room to spare).\",\n \"It remains to show that $\\\\mu \\\\le M$ .\",\n \"We have: $ \\\\mu & = & \\\\sum _{i=1}^t \\\\mathbb {E}[Z_i] = \\\\sum _{i=1}^t q(i,t) \\\\\\\\& & \\\\le \\\\frac{2}{\\\\delta }\\\\ln (4n^3/\\\\gamma ) \\\\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.\",\n \"For $r \\\\ge 1$ , $\\\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\\\le \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)})$ .\",\n \"We prove by backwards induction on $r$ .\",\n \"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.\",\n \"Assume up to $r \\\\ge 2$ , and we shall prove the statement for $r-1$ .\",\n \"We start with a technical statement.\",\n \"$& & \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& = & \\\\ln (\\\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\\\\\& = & - \\\\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\\\\\& \\\\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\\\sum _{i=r-1}^t \\\\frac{w(i)}{W(t)-W(i-1)} \\\\\\\\& = & \\\\sum _{i=r}^t \\\\frac{w(i)}{W(t)-W(i-1)} + \\\\frac{w(r-1)}{W(t) - W(r-2)} \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-1))}{w(t)}) + \\\\ln (\\\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\\\\\& \\\\le & \\\\ln (\\\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\\\Box $ Thus $\\\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\\\le \\\\ln (eW(t)/w(t)) \\\\le \\\\ln (e\\\\rho )$ , and so $\\\\mu \\\\le M$ .\",\n \"This completes the proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"Proofs from Section \",\n \"We first prove another claim from which the proof of Claim follows.\",\n \"Given a common subsequence $x(i_1), x(i_2), \\\\ldots , x(i_r)$ and $y(j_1), y(j_2), \\\\ldots , y(j_r)$ , the defect is at least $\\\\max _{k \\\\le r} (|i_k - j_k|)$ .\",\n \"(of Claim ) Assume wlog that $i_k \\\\ge j_k$ .\",\n \"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .\",\n \"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .\",\n \"Hence the defect is at least $i_k - j_k$ .\",\n \"Repeating over all $k$ , we complete the proof.\",\n \"$\\\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).\",\n \"If either $j_a \\\\in [i,i_a]$ or $i \\\\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .\",\n \"Suppose neither of these are true.\",\n \"Then $j_a > i_a \\\\ge i$ .\",\n \"Let us focus on the characters of $x$ that are not matched.\",\n \"No character of $x$ with index in $[i,i_a)$ is matched.\",\n \"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).\",\n \"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .\",\n \"So the number of unmatched characters in $x$ is at least $j_a - i$ .\",\n \"$\\\\Box $ (of Claim ) Consider some $j \\\\in S_{i+1}$ , and set $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ .\",\n \"We wish to compute the largest ${\\\\cal S}$ -consistent CS between in $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"Suppose we look at the portion of this CS in $x(1,i\\\\bar{n})$ .\",\n \"This forms a ${\\\\cal S}$ -consistent sequence between $\\\\bar{x}$ and $y(1,j^{\\\\prime })$ , for some $j^{\\\\prime } \\\\in S_i$ .\",\n \"The remaining portion of the CS is just the LCS between $\\\\bar{x} = x(i\\\\bar{n},(i+1)\\\\bar{n})$ and $y(j^{\\\\prime },j)$ .\",\n \"Hence, given the LCS length of $\\\\bar{x}$ and $y(j^{\\\\prime },j)$ , for all $j^{\\\\prime } \\\\in S_i$ , we can compute the length of the largest ${\\\\cal S}$ -consistent CS between $x(1,(i+1)\\\\bar{n})$ and $y(1,j)$ .\",\n \"This is obtained by just maximizing over all possible $j^{\\\\prime }$ .\",\n \"We now apply Claim .\",\n \"We have $\\\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .\",\n \"Using $O(\\\\bar{n})$ space, we can compute all the LCS lengths desired.\",\n \"This gives the length of the largest ${\\\\cal S}$ -consistent CS that ends at $y(j)$ .\",\n \"This can be done for all $y(j)$ , $j \\\\in S_i$ .\",\n \"The total running time is $O(|S_{i+1}|n\\\\bar{n}) = O(n\\\\bar{n}^2)$ .\",\n \"$\\\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Consider the index $i\\\\bar{n}$ .\",\n \"Suppose we have currently stored the lengths of the largest ${\\\\cal S}$ -consistent CS between $x(1,i\\\\bar{n})$ and $y(1,j)$ , for all $j \\\\in S_i$ .\",\n \"This requires space $O(|S_i|) = O(\\\\bar{n}))$ .\",\n \"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\\\bar{n})$ space.\",\n \"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\\\cal S}$ -consistent CS.\",\n \"Lemma tells us that this defect is a $(1+\\\\delta )$ - approximation to $E(x,y)$ .\",\n \"The space bound is $O(\\\\bar{n})$ .\",\n \"The number of updates is $O(n/\\\\bar{n}) = O(\\\\sqrt{(\\\\delta n)/\\\\ln n})$ , and the time for each update is $O(n{\\\\bar{n}}^2) = O((n^2\\\\ln n)/\\\\delta )$ .\",\n \"$\\\\Box $\"\n ]\n]"},"id":{"kind":"string","value":"1204.1098"}}},{"rowIdx":1866986,"cells":{"text":{"kind":"list like","value":[["Transport in coupled graphene-nanotube quantum devices"],["Abstract We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.","We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.","The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.","We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20% of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.","Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.","This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads."],["Fabrication of coupled graphene-nanotube quantum devices S. Engels$^{1,2,3}$ , P. Weber$^{1,2,3}$ , B. Terrés$^{1,2,3}$ , J. Dauber$^{1,2,3}$ , C. Meyer$^{2,3}$ , C. Volk$^{1,2,3}$ , S. Trellenkamp$^{2}$ , U. Wichmann$^{1}$ and C. Stampfer$^{1,2,3}$ stephan.engels@rwth-aachen.de $^1$ II.","Institute of Physics B, RWTH Aachen University, 52074 Aachen, Germany, EU $^2$ Peter Grünberg Institute (PGI-6/8/9), Forschungszentrum Jülich, 52425 Jülich, Germany, EU $^3$ JARA – Fundamentals of Future Information Technologies We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.","We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.","The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.","We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20$\\%$ of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.","Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.","This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads.","graphene, carbon nanotube, charge detector, quantum dot 71.10.Pm, 73.21.-b, 81.07.Ta, 81.05.ue Carbon nanomaterials, such as graphene and carbon nanotubes (CNTs) attract increasing interest mainly due to their promises for flexible electronics, high-frequency devices and spin-based quantum circuits [1], [2], [3].","Both materials consist of sp$^2$ -bound carbon and exhibit unique electronic properties resulting in the suppression of direct backscattering, high carrier mobilities and low intrinsic spin noise.","In particular the weak hyperfine interaction makes graphene and CNTs interesting host materials for quantum dots which promise the implementation of long-living spin qubits [4].","Up to the present state quantum dots (QDs) and double quantum dots have been demonstrated successfully in both carbon nanomaterials.","In particular, the fabrication of ultra-clean few-carrier QDs with well-defined spin states in quasi one-dimensional (1-D) carbon nanotubes attracted great interest [5], [6], [7].","A comparable quality in graphene is not yet reached, which is mainly due to its gap less band structure making it hard to controllably confine electrons and holes.","State-of-the-art graphene QDs are therefore either based on nanoribbons [8], [9] or etched islands [10], [11], [12], [13], [14] and in both cases edge roughness and disorder are dominating their properties.","However, in contrast to quasi 1-D nanotubes, the 2-D nature of graphene makes it easy to integrate lateral graphene gates and in-plane charge sensors [15], which are both important for the control and readout of QD states.","Here, we present the fabrication and characterization of quantum devices based on both graphene and carbon nanotubes, which combine the advantages of the two carbon allotropes and open the route to unprecedented quantum devices.","Figure: (color online) (a),(b) and (c) Schematic illustrations of the three main fabrication steps.","(d) Scanning force microscopy (SFM) image of a SiO 2 \\rm {SiO}_2 substrate with CVD grown CNTs (white arrows) and subsequently deposited graphene (black arrows).","(e) SFM image of a similar sample after the dry etching process.","White dashed lines indicate areas exposed to an Ar /O 2 \\rm {Ar/O}_{2} plasma which results in etching both carbon materials as indicated by the black points.","(f) Raman spectrum of the graphene flake at the position indicated by the green dot in panel (e).","The spectrum shows clear characteristics of single-layer graphene.In particular, we first discuss a carbon nanotube QD with a capacitively coupled graphene nanoribbon acting as electrostatic gate and charge detector.","Charge read-out schemes have proven to be difficult for carbon nanotubes QDs, the main challenge lying in the random orientation and position of the nanotubes on the substrate.","Two possible strategies have been put forward so far to overcome this difficulty.","The first one consists in placing a metallic single electron transistor (SET) close the the CNT QD [16], [17].","The second relies in capacitively coupling the QD to an SET realized in the same [6] or in a neighboring CNT [18] via a deposited metallic gate.","Our work adds a third detection scheme based on a graphene charge detector which is easy to fabricate and has a more than sufficient charge sensitivity to detect single charging events.","In addition to the charge sensing, we show a graphene-carbon nanotube hybrid device where a nanotube QD is contacted with both source and drain graphene leads.","Thus, a quantum dot device exclusively built of the two different carbon allotropes.","This configuration is especially interesting since it opens the route to quantum dot devices with a tunable density of states in the source and drain leads.","In particular, it may allow to experimentally investigate the pseudo-gap Kondo model which predicts that graphene undergoes a quantum phase transition [19], [20] from a phase with a screened to an unscreened impurity moment at low charge carrier densities [21].","The predicted quantum phase transition might manifest itself in the characteristic scaling of observable quantities (e.g.","the Kondo resonance) in a graphene-contacted quantum dot.","Figure: (color online) Scanning force micrograph ofa carbon nanotube (CNT) quantum dot device with a nearby structuredgraphene nanoribbon (GNR) for charge sensing.","The schematic illustrationshows the used measurement circuit and the applied voltages.","For detailedinformation see text.The fabrication process is based on chemical vapor deposition (CVD) growth of carbon nanotubes and subsequent deposition of mechanically exfoliated natural graphite.","In Figs.","REF (a) to REF (c) we show the three main fabrication steps for making all-carbon graphene-nanotube devices.","As a first step single-walled carbon nanotubes are grown on 290 nm $\\rm {SiO}_{2}$ on highly p-doped Si substrates by CVD using a Ferritin-based iron catalyst method [22].","The single-walled carbon nanotubes have a diameter of around 1.5-2 nm and are up to several micrometers in length.","In a next step, graphene is deposited on these pretreated substrates by mechanical exfoliation of natural graphite [23].","Finally, a graphene nanoribbon (GNR) or graphene leads are patterned from the deposited graphene by an electron beam lithography (EBL) step followed dry etching.","Figs.","REF (d) and REF (e) show scanning force microscopy (SFM) images of two examples of the deposition process.","Due to the stochastic nature of the method a great variety of different configurations is obtained.","For example, the graphene flake shown in Fig.","REF (d) (black arrow) lies on top or nearby of a number of individual CNTs (white arrows) whereas in Fig.","REF (e) we show an example where a nanotube (white arrow) is contacted by two graphene flakes (black arrows).","A crucial parameter for the success of this stochastic fabrication process is the density of grown CNTs which can be well controlled by the Ferritin-based CVD process [22].","In this study we used a density of approx.","1-2 CNTs per $\\rm {\\mu m}^2$ .","In Fig.","REF (f) we show a Raman spectrum taken at the position indicated by the green dot in Fig.","REF (e).","From the full width at half maximum of the 2D peak of 34 $\\rm {cm}^{-1}$ and the relative intensity of the 2D and G peak of I(G/2D)$\\approx $ 0.65, we conclude that the graphene is of single-layer nature [24], [25] (laser excitation of 532 nm).","A Raman spectrum of the bottom graphene flake at the position of the intersecting CNT reveals a bilayer flake (not shown).","The contacted CNT structure consists of two segments.","Most likely the upper segment is a small bundle of single-walled CNTs whereas the lower segment (white arrow in Fig.","REF (e)) is an individual single-walled CNT.","This conclusion is supported by a comparison of the SFM profiles of the single-layer graphene and the lower carbon nanotube segment, as well as the high accuracy of the Ferritin-based CVD growth process [22].","The third fabrication step consists of EBL followed by an $\\rm {Ar/O}_{2}$ based dry etching step for (i) structuring graphene and (ii) removing unwanted CNTs.","In Fig.","REF (e) we highlight areas (dashed lines) where graphene and CNTs have been successfully removed (see black points).","Finally, we used EBL, metal evaporation (5 nm Cr/ 50 nm Au) and lift-off for contacting the devices.","Figure: (color online) (a),(b) Back gate characteristics of the carbon nanotube QD (a) and the graphene nanoribbon (b).","Both measurements were recorded at a source-drain bias voltage of V CNT \\rm {V}_{CNT}=V GNR \\rm {V}_{GNR}=0.5 mV and V REF \\rm {V}_{REF}=0 V. The upper inset in panel (a) shows Coulomb blockade resonances of the CNT QD as a function of V BG \\rm {V}_{BG}.","The inset in panel (b) highlights the conductance resonances in the GNR in the same range of V BG \\rm {V}_{BG}.","(c) Differential conductance on the CNT QD.","The white dotted line indicates an excited state with an energy of Δ\\Delta =0.9 meV.","(d) Conductance of the CNT as function of ΔV BG \\Delta \\rm {V}_{BG}=V BG \\rm {V}_{BG}-13.63 V and V REF * \\rm {V}_{REF^{*}} at V CNT \\rm {V}_{CNT}=20 mV.","The black dashed line highlights the relative lever arm α BG , GNR \\rm {\\alpha }_{BG,GNR}.Figure: (color online) (a) QD conductance and (b) GNR conductance as a function of V REF \\rm {V}_{REF} with a simultaneous change in BG voltage following V BG \\rm {V}_{BG}=a-b·\\cdot V REF \\rm {V}_{REF} with a=48.166 V and b=-0.335.","A bias voltage of V CNT \\rm {V}_{CNT}=V GNR \\rm {V}_{GNR}=0.50.5 mV was applied to both structures.","The inset in (b) shows a schematic of the charge detecting mechanism.","(c) Derivative of (b) showing well pronounced peaks at every charging event.","(d) Dependence of the CNT QD conductance on V BG \\rm {V}_{BG} and V REF \\rm {V}_{REF} measured at V CNT \\rm {V}_{CNT}=0.5 mV on a logarithmic scale.","Periodic Coulomb blockade resonances with a positive slope of α REF , BG =\\rm {\\alpha }_{REF,BG}=1.25 are observed.","The resulting electron occupation numbers of the QD are given in white letters.","(e) Conductance of the GNR depending on V BG \\rm {V}_{BG} and V REF \\rm {V}_{REF} with V GNR \\rm {V}_{GNR}=0.5 mV.","(f) Absolute value of the GNR transconductance.","Elevated conductance traces with a positive slope (dotted line) match perfectly with the Coulomb resonances in (d).","Detection lines are even visible where the current is too small to be measured directly (see white arrows).","Measurements shown in (a)-(c) were measured parallel to line (2).","All measurements were recorded simultaneously.Fig.","REF shows a SFM image of an all-carbon device consisting of a carbon nanotube lying in the close vicinity to an etched graphene nanoribbon (GNR) which acts as a charge detector (CD).","By Raman spectroscopy we identify the bright colored area to be a graphene nanoribbon of bilayer nature (not shown) and following the above argument we conclude that the nanotube is a single-walled carbon nanotube.","Both carbon nanostructures are separated by roughly 150 nm, the nanoribbon has a width of around 100 nm and the CNT quantum dot is defined by two metal electrodes (indicated in blue) which are separated by 350 nm.","As illustrated in Fig.","REF , we apply a symmetric bias voltage ($\\rm {V}_{CNT}$ and $\\rm {V}_{GNR}$ respectively) to both structures.","The overall Fermi level can be tuned by the back gate voltage $\\rm {V}_{BG}$ applied to the highly doped Si substrate.","Additionally, we can use the CNT (GNR) as a lateral gate for the GNR (CNT) by applying a reference potential $\\rm {V}_{REF}$ ($\\rm {V}_{REF^{*}}$ ).","All presented measurements were performed in a pumped $^4\\rm {He}$ -cryostat at a base temperature of T$\\approx $ 1.5 K using low-frequency lock-in techniques.","Figs.","REF (a) and REF (b) show the back gate characteristics i.e.","the conductance as function of $\\rm {V}_{BG}$ of both, the CNT (Fig.","REF (a)) and the graphene nanoribbon (Fig.","REF (b)) for a constant bias voltage $\\rm {V}_{CNT}$ =$\\rm {V}_{GNR}$ =0.5 mV.","The CNT reveals a semiconducting behavior with a large band gap resulting in an extended BG region of suppressed current (-60 V$< \\rm {V}_{BG} < $ 40 V).","A high resolution measurement performed at the edge of this gap (see inset in Fig.","REF (a)) exhibits reproducible, well resolved and sharp Coulomb peaks.","The peak width of the sharpest resonances is given by the electron temperature [26] which can be extracted to be 2 K. In Fig.","REF (c) we show so-called Coulomb diamond measurements i.e.","the differential conductance d$\\rm {I}_{CNT}$ /d$\\rm {V}_{CNT}$ plotted as function of $\\rm {V}_{BG}$ and $\\rm {V}_{CNT}$ .","From the extent of suppressed current of the diamonds in bias ($\\rm {V}_{CNT}$ ) direction we can extract a charging energy of $\\rm {E}_C \\approx $ 6-9 meV of the CNT quantum dot and a BG lever arm, $\\rm {\\alpha }_{BG}$ =0.22.","Following Ref.","[27] we can relate the charging energy with the length of the nanotube segment forming the quantum dot ($\\rm {L}_{QD}$ ) by roughly $\\rm {E}_C$ =1.4 eV/$\\rm {L}_{QD}$ (nm).","This provides an order of magnitude estimate of $\\rm {L}_{QD}\\approx $ 150-235nm which is in reasonable agreement with the device geometry.","The change in differential conductance parallel to the diamond edges is attributed to excited states providing additional transport channels.","The observed excited states exhibit an energy of $\\Delta $$\\approx $ 0.9 meV (see e.g.","white dotted line in Fig.","REF (c)).","The independently measured back gate characteristic of the bilayer graphene nanoribbon is shown in Fig.","REF (b).","This low bias ($\\rm {V}_{GNR}$ =0.5 mV) measurement highlights that transport can be tuned from the hole regime (left inset) into the so-called transport gap [28], [29], [30], [31], [32], [33], [34], [35] starting at around $\\rm {V}_{BG}$$\\approx $ 40 V. Within the gap region transport is governed by localized states resulting in sharp resonances of the conductance as shown in the right inset of Fig.","REF (c).","Interestingly, both carbon nanostructures have different doping levels.","In contrast to the n-doped CNT we observe a significant p-doping of the GNR which is most likely due to atmospheric $\\rm {O}_{2}$ binding on the graphene edges [36].","Fig.","REF (d) shows the conductance of the CNT in dependence of the BG voltage $\\rm {V}_{BG}$ =13.63 V + $\\Delta \\rm {V}_{BG}$ and the reference voltage on the nanoribbon $\\rm {V}_{REF^{*}}$ at $\\rm {V}_{CNT}$ =20 mV, which demonstrates the gating effect of the GNR on the CNT QD.","Lines of higher conductance can be attributed to Coulomb resonances of the CNT and from their slopes we extract a relative lever arm of $\\rm {\\alpha }_{BG,GNR}$ =0.23.","If both quantum devices are now operated simultaneously, the nanoribbon device can be used to detect individual charging events on the nanotube QD device.","This phenomenon is shown in Fig.","4.","The BG voltage is put to an offset ($\\rm {V}_{BG}$ =48.166 V) such that (i) the CNT is in the Coulomb blockade regime and (ii) the conductance of the graphene nanoribbon exhibits sharp and well-reproducible resonances.","Figs.","REF (a) and REF (b) show the simultaneously measured low-bias ($\\rm {V}_{CNT}$ =0.5 meV) conductance through the nanotube $\\rm {G}_{CNT}$ and the nanoribbon $\\rm {G}_{GNR}$ as function of $\\rm {V}_{REF}$ where the BG voltage is simultaneously adjusted according to $\\rm {V}_{BG}$ =a-b$\\cdot $$\\rm {V}_{REF}$ with a=48.166 V and b=-0.335.","Similar to the inset of Fig.","REF (a), $\\rm {G}_{CNT}$ exhibits Coulomb peaks, which are indicating single charging events in the CNT QD.","In the simultaneously measured trace of the GNR, we observe distinct steps in the conductance at the exact positions of the CNT QD charging events.","The steps in conductance can measure up to 20 $\\%$ of the total resonance amplitude and are due to the capacitive coupling of both nanostructures.","Increasing the reference potential $\\rm {V}_{REF}$ and decreasing $\\rm {V}_{BG}$ both leads to a higher chemical potential in the QD and subsequently to a lower occupation number at every event.","Consequently, the GNR resonance shifts to lower values of $\\rm {V}_{REF}$ giving rise to the shape of the charge detecting resonance in Fig.","REF (b).","The unconventional shape of the detection signal is explained by the inset of Fig.","REF (b).","In this schematic, each resonance curve (in gray) would represent the conductance resonance of the GNR at a fixed charge state of the CNT QD.","However, the latter is itself a (discontinuous) function of $\\rm {V}_{REF}$ , so that the measured conductance through the GNR is not a smooth curve but show a step whenever the charge state of the CNT QD (and thus the conductance resonance) changes.","By relating the step height to the noise level of our measurement system we achieve an estimate for the charge sensitivity with an upper limit of $~10^{-3} \\rm {e}/\\sqrt{\\rm {Hz}}$ which is in agreement with previous experiments on GNR charge detectors [15].","In order to highlight the charge detection we further plot the transconductance i.e.","the derivative of $\\rm {G}_{GNR}$ with respect to $\\rm {V}_{REF}$ as shown in Fig.","REF (c).","Each individual transconductance peak (dip) is very well aligned with the directly measured Coulomb peaks.","For proving this more rigerously we performed measurements as function of both $\\rm {V}_{REF}$ and $\\rm {V}_{BG}$ and extracted relative lever arms.","Corresponding charge stability diagrams are shown in Figs.","REF (d) and REF (e).","As anticipated, the evolution of Coulomb resonances of the CNT QD (shown in Fig.","REF (d)) follows a constant positive slope with a relative lever arm of $\\rm {\\alpha }_{REF,BG}\\approx $ 1.25 and the occupation numbers (denoted in white letters) decrease with increasing $\\rm {V}_{REF}$ and decreasing $\\rm {V}_{BG}$ .","For the charge detector the observed patterns in Fig.","REF (e) show similarities to those of a multi-dot system.","Further analysis of the transconductance $\\rm {dG}_{CD}/\\rm {dV}_{REF}$ reveals at least three lines with negative slopes.","These lines show that the transport through the GNR is governed by a series of quantum dots where each of the lines can be attributed to single QDs located at different positions along the nanoribbon.","The extracted lever arms of the visible conductance resonances are $\\rm {\\alpha }_{REF,BG}^{(1)}$ =$-0.43$ , $\\rm {\\alpha }_{REF,BG}^{(2)}$ =$-0.31$ and $\\rm {\\alpha }_{ REF,BG}^{(3)}$ =$-0.12$ as indicated in Fig.","REF (f).","Additionally the plot exhibits clearly visible features (see white dotted line) with in this case positive slopes with their positions matching perfectly to those in Fig.","REF (d).","Consequently they can be associated with charging events in the CNT QD which are detected by the quantum dots situated in the graphene nanoribbon.","Detection lines are even visible where the current through the dot is too low to be measured directly (see white arrows in Figs.","REF (d) and (f)).","In order to further improve the sensitivity, the detector could potentially be included in a RF-readout setup.","The charge sensitivities reported for this type of setups are in the order of $~10^{-4} \\rm {e}/\\sqrt{\\rm {Hz}}$ to $~10^{-5} \\rm {e}/\\sqrt{\\rm {Hz}}$ [16], [17].","While a quantitative comparison of the presented dc-readout technique with other dc-charge sensors on carbon nanotube QDs [6], [18] can not be provided (no sensitivities are given), a qualitative comparison of the data shows that the graphene charge detector yields at least equal visibility of the charge states.","Finally, we discuss a carbon nanotube device where both metal leads (source and drain) are substituted by two graphene flakes.","The device structure has been discussed earlier (see fabrication section and Fig.","REF (e)) and a close up of the final device is shown as inset in Fig.","REF (a).","For measuring the conductance of the nanotube, Cr/Au metal contacts are deposited on each graphene flake by an EBL step followed by metalization and lift-off.","The metal contacts are designed in a way that they do not touch the carbon nanotube.","As a first important observation we see that current flows from one graphene flake to the other through the carbon nanotube.","Moreover, we do not observe a $\\rm {V}_{BG}$ -regime where the current is fully suppressed (not shown) which suggests the presence of a metallic nanotube.","In Fig.","REF (b) we show the current $\\rm {I}_{CNT}$ in dependence of the voltage $\\rm {V}_{SG}$ applied to a metallic side gate which was deposited next to the nanotube as indicated by the blue area in Fig.","REF (a) and allows to locally gate the central segment of the carbon nanotube.","The current $\\rm {I}_{CNT}$ exhibits periodic Coulomb oscillations.","In Fig.","REF (c) we plot the current through the nanotube $\\rm {I}_{CNT}$ on a logarithmic scale as function of the bias voltage $\\rm {V}_{BIAS}$ and $\\rm {V}_{SG}$ for a fixed value of $\\rm {V}_{BG}$ =35.15 V. The measurement exhibits clear diamond shaped patterns which can be attributed to Coulomb diamonds.","From the addition energies of 1-1.5 $\\rm {meV}$ we can estimate the quantum dot length of 0.9-1.5 $\\rm {\\mu m}$ [27], which is in good agreement with the 1.2 $\\rm {\\mu m}$ spacing between the two graphene leads.","This leads to the conclusion that the quantum dot in the nanotube extends over the full length set by the distance of the graphene leads.","Moreover, the observation of Coulomb blockade oscillations and peaks manifests the presence of tunneling barriers at the interface of both carbon allotropes.","From the maxima of the Coulomb peaks and assuming similar rates for tunneling in and out of the dot we extract tunneling rates in the low GHz regime.","The origin of the observed tunneling barriers is not fully understood at the present state.","Since graphene is a semi-metal and the investigated carbon nanotube is a metallic carbon nanotube, it is believed that the tunneling barriers form similarly to those metallic-CNT devices with metal contacts.","Here, tunneling barriers may arise from imperfect electrical contacts to the nanotube [37].","Figure: (color online)(a) SFM image of the investigated device (see also Fig. (e)).","(b) Current I CNT \\rm {I}_{CNT} through the nanotube as function of the voltage V SG \\rm {V}_{SG} applied to a metal electrode placed roughly 50 nm from the nanotube (see blue area in (a)) for a fixed back gate voltage V BG \\rm {V}_{BG}=35 V and V BIAS \\rm {V}_{BIAS}=0.3 mV.","(c) Current I CNT \\rm {I}_{CNT} through the nanotube on a logarithmic scale as function of V BIAS \\rm {V}_{BIAS} and V SG \\rm {V}_{SG}.","V BG \\rm {V}_{BG} is constant at 35.15 V.In summary, we present the fabrication and characterization of carbon nanotube-graphene hybrid devices.","We show the example of a structure where a nanotube quantum dot is capacitively coupled to a graphene nanoribbon.","Sharp resonances in the graphene nanoribbon conductance give rise to clear charge detection signals with an upper estimate for the charge sensitivity of $~10^{-3} \\rm {e}/\\sqrt{\\rm {Hz}}$ .","This kind of graphene charge detector represents a third charge detection scheme for CNT QDs in addition to the existing strategies based on metal single electron transistors [16], [17] and capacitively coupled carbon nanotube segments [6], [18].","The presented charge detector is rather easy to fabricate.","Its sensitivity can be further improved by including it into a RF-circuit.","In addition, we study a device where two carbon allotropes (graphene and CNT) are electrically coupled.","We find that the interfaces between the graphene and the carbon nanotube resembles tunneling barriers.","This device represents an important preliminary step towards experiments investigating the quantum phase transition from screened to unscreened impurity moments.","Both results open the road to more sophisticated devices which are entirely fabricated out of carbon nanostructures and exploit the different advantages of these promising materials.","Acknowledgment — The authors wish to thank L. Durrer and C. Hierold for their support and help on the growth of the CNT samples.","We thank A. Steffen, R. Lehmann and J. Mohr for the help on the sample fabrication.","Discussions with F. Haupt, J. Splettstösser, D. Schuricht and M. Wegewijs and support by the JARA Seed Fund and the DFG (SPP-1459 and FOR-912) are gratefully acknowledged."]],"string":"[\n [\n \"Transport in coupled graphene-nanotube quantum devices\"\n ],\n [\n \"Abstract We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.\",\n \"We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.\",\n \"The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.\",\n \"We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20% of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.\",\n \"Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.\",\n \"This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads.\"\n ],\n [\n \"Fabrication of coupled graphene-nanotube quantum devices S. Engels$^{1,2,3}$ , P. Weber$^{1,2,3}$ , B. Terrés$^{1,2,3}$ , J. Dauber$^{1,2,3}$ , C. Meyer$^{2,3}$ , C. Volk$^{1,2,3}$ , S. Trellenkamp$^{2}$ , U. Wichmann$^{1}$ and C. Stampfer$^{1,2,3}$ stephan.engels@rwth-aachen.de $^1$ II.\",\n \"Institute of Physics B, RWTH Aachen University, 52074 Aachen, Germany, EU $^2$ Peter Grünberg Institute (PGI-6/8/9), Forschungszentrum Jülich, 52425 Jülich, Germany, EU $^3$ JARA – Fundamentals of Future Information Technologies We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.\",\n \"We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.\",\n \"The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.\",\n \"We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20$\\\\%$ of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.\",\n \"Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.\",\n \"This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads.\",\n \"graphene, carbon nanotube, charge detector, quantum dot 71.10.Pm, 73.21.-b, 81.07.Ta, 81.05.ue Carbon nanomaterials, such as graphene and carbon nanotubes (CNTs) attract increasing interest mainly due to their promises for flexible electronics, high-frequency devices and spin-based quantum circuits [1], [2], [3].\",\n \"Both materials consist of sp$^2$ -bound carbon and exhibit unique electronic properties resulting in the suppression of direct backscattering, high carrier mobilities and low intrinsic spin noise.\",\n \"In particular the weak hyperfine interaction makes graphene and CNTs interesting host materials for quantum dots which promise the implementation of long-living spin qubits [4].\",\n \"Up to the present state quantum dots (QDs) and double quantum dots have been demonstrated successfully in both carbon nanomaterials.\",\n \"In particular, the fabrication of ultra-clean few-carrier QDs with well-defined spin states in quasi one-dimensional (1-D) carbon nanotubes attracted great interest [5], [6], [7].\",\n \"A comparable quality in graphene is not yet reached, which is mainly due to its gap less band structure making it hard to controllably confine electrons and holes.\",\n \"State-of-the-art graphene QDs are therefore either based on nanoribbons [8], [9] or etched islands [10], [11], [12], [13], [14] and in both cases edge roughness and disorder are dominating their properties.\",\n \"However, in contrast to quasi 1-D nanotubes, the 2-D nature of graphene makes it easy to integrate lateral graphene gates and in-plane charge sensors [15], which are both important for the control and readout of QD states.\",\n \"Here, we present the fabrication and characterization of quantum devices based on both graphene and carbon nanotubes, which combine the advantages of the two carbon allotropes and open the route to unprecedented quantum devices.\",\n \"Figure: (color online) (a),(b) and (c) Schematic illustrations of the three main fabrication steps.\",\n \"(d) Scanning force microscopy (SFM) image of a SiO 2 \\\\rm {SiO}_2 substrate with CVD grown CNTs (white arrows) and subsequently deposited graphene (black arrows).\",\n \"(e) SFM image of a similar sample after the dry etching process.\",\n \"White dashed lines indicate areas exposed to an Ar /O 2 \\\\rm {Ar/O}_{2} plasma which results in etching both carbon materials as indicated by the black points.\",\n \"(f) Raman spectrum of the graphene flake at the position indicated by the green dot in panel (e).\",\n \"The spectrum shows clear characteristics of single-layer graphene.In particular, we first discuss a carbon nanotube QD with a capacitively coupled graphene nanoribbon acting as electrostatic gate and charge detector.\",\n \"Charge read-out schemes have proven to be difficult for carbon nanotubes QDs, the main challenge lying in the random orientation and position of the nanotubes on the substrate.\",\n \"Two possible strategies have been put forward so far to overcome this difficulty.\",\n \"The first one consists in placing a metallic single electron transistor (SET) close the the CNT QD [16], [17].\",\n \"The second relies in capacitively coupling the QD to an SET realized in the same [6] or in a neighboring CNT [18] via a deposited metallic gate.\",\n \"Our work adds a third detection scheme based on a graphene charge detector which is easy to fabricate and has a more than sufficient charge sensitivity to detect single charging events.\",\n \"In addition to the charge sensing, we show a graphene-carbon nanotube hybrid device where a nanotube QD is contacted with both source and drain graphene leads.\",\n \"Thus, a quantum dot device exclusively built of the two different carbon allotropes.\",\n \"This configuration is especially interesting since it opens the route to quantum dot devices with a tunable density of states in the source and drain leads.\",\n \"In particular, it may allow to experimentally investigate the pseudo-gap Kondo model which predicts that graphene undergoes a quantum phase transition [19], [20] from a phase with a screened to an unscreened impurity moment at low charge carrier densities [21].\",\n \"The predicted quantum phase transition might manifest itself in the characteristic scaling of observable quantities (e.g.\",\n \"the Kondo resonance) in a graphene-contacted quantum dot.\",\n \"Figure: (color online) Scanning force micrograph ofa carbon nanotube (CNT) quantum dot device with a nearby structuredgraphene nanoribbon (GNR) for charge sensing.\",\n \"The schematic illustrationshows the used measurement circuit and the applied voltages.\",\n \"For detailedinformation see text.The fabrication process is based on chemical vapor deposition (CVD) growth of carbon nanotubes and subsequent deposition of mechanically exfoliated natural graphite.\",\n \"In Figs.\",\n \"REF (a) to REF (c) we show the three main fabrication steps for making all-carbon graphene-nanotube devices.\",\n \"As a first step single-walled carbon nanotubes are grown on 290 nm $\\\\rm {SiO}_{2}$ on highly p-doped Si substrates by CVD using a Ferritin-based iron catalyst method [22].\",\n \"The single-walled carbon nanotubes have a diameter of around 1.5-2 nm and are up to several micrometers in length.\",\n \"In a next step, graphene is deposited on these pretreated substrates by mechanical exfoliation of natural graphite [23].\",\n \"Finally, a graphene nanoribbon (GNR) or graphene leads are patterned from the deposited graphene by an electron beam lithography (EBL) step followed dry etching.\",\n \"Figs.\",\n \"REF (d) and REF (e) show scanning force microscopy (SFM) images of two examples of the deposition process.\",\n \"Due to the stochastic nature of the method a great variety of different configurations is obtained.\",\n \"For example, the graphene flake shown in Fig.\",\n \"REF (d) (black arrow) lies on top or nearby of a number of individual CNTs (white arrows) whereas in Fig.\",\n \"REF (e) we show an example where a nanotube (white arrow) is contacted by two graphene flakes (black arrows).\",\n \"A crucial parameter for the success of this stochastic fabrication process is the density of grown CNTs which can be well controlled by the Ferritin-based CVD process [22].\",\n \"In this study we used a density of approx.\",\n \"1-2 CNTs per $\\\\rm {\\\\mu m}^2$ .\",\n \"In Fig.\",\n \"REF (f) we show a Raman spectrum taken at the position indicated by the green dot in Fig.\",\n \"REF (e).\",\n \"From the full width at half maximum of the 2D peak of 34 $\\\\rm {cm}^{-1}$ and the relative intensity of the 2D and G peak of I(G/2D)$\\\\approx $ 0.65, we conclude that the graphene is of single-layer nature [24], [25] (laser excitation of 532 nm).\",\n \"A Raman spectrum of the bottom graphene flake at the position of the intersecting CNT reveals a bilayer flake (not shown).\",\n \"The contacted CNT structure consists of two segments.\",\n \"Most likely the upper segment is a small bundle of single-walled CNTs whereas the lower segment (white arrow in Fig.\",\n \"REF (e)) is an individual single-walled CNT.\",\n \"This conclusion is supported by a comparison of the SFM profiles of the single-layer graphene and the lower carbon nanotube segment, as well as the high accuracy of the Ferritin-based CVD growth process [22].\",\n \"The third fabrication step consists of EBL followed by an $\\\\rm {Ar/O}_{2}$ based dry etching step for (i) structuring graphene and (ii) removing unwanted CNTs.\",\n \"In Fig.\",\n \"REF (e) we highlight areas (dashed lines) where graphene and CNTs have been successfully removed (see black points).\",\n \"Finally, we used EBL, metal evaporation (5 nm Cr/ 50 nm Au) and lift-off for contacting the devices.\",\n \"Figure: (color online) (a),(b) Back gate characteristics of the carbon nanotube QD (a) and the graphene nanoribbon (b).\",\n \"Both measurements were recorded at a source-drain bias voltage of V CNT \\\\rm {V}_{CNT}=V GNR \\\\rm {V}_{GNR}=0.5 mV and V REF \\\\rm {V}_{REF}=0 V. The upper inset in panel (a) shows Coulomb blockade resonances of the CNT QD as a function of V BG \\\\rm {V}_{BG}.\",\n \"The inset in panel (b) highlights the conductance resonances in the GNR in the same range of V BG \\\\rm {V}_{BG}.\",\n \"(c) Differential conductance on the CNT QD.\",\n \"The white dotted line indicates an excited state with an energy of Δ\\\\Delta =0.9 meV.\",\n \"(d) Conductance of the CNT as function of ΔV BG \\\\Delta \\\\rm {V}_{BG}=V BG \\\\rm {V}_{BG}-13.63 V and V REF * \\\\rm {V}_{REF^{*}} at V CNT \\\\rm {V}_{CNT}=20 mV.\",\n \"The black dashed line highlights the relative lever arm α BG , GNR \\\\rm {\\\\alpha }_{BG,GNR}.Figure: (color online) (a) QD conductance and (b) GNR conductance as a function of V REF \\\\rm {V}_{REF} with a simultaneous change in BG voltage following V BG \\\\rm {V}_{BG}=a-b·\\\\cdot V REF \\\\rm {V}_{REF} with a=48.166 V and b=-0.335.\",\n \"A bias voltage of V CNT \\\\rm {V}_{CNT}=V GNR \\\\rm {V}_{GNR}=0.50.5 mV was applied to both structures.\",\n \"The inset in (b) shows a schematic of the charge detecting mechanism.\",\n \"(c) Derivative of (b) showing well pronounced peaks at every charging event.\",\n \"(d) Dependence of the CNT QD conductance on V BG \\\\rm {V}_{BG} and V REF \\\\rm {V}_{REF} measured at V CNT \\\\rm {V}_{CNT}=0.5 mV on a logarithmic scale.\",\n \"Periodic Coulomb blockade resonances with a positive slope of α REF , BG =\\\\rm {\\\\alpha }_{REF,BG}=1.25 are observed.\",\n \"The resulting electron occupation numbers of the QD are given in white letters.\",\n \"(e) Conductance of the GNR depending on V BG \\\\rm {V}_{BG} and V REF \\\\rm {V}_{REF} with V GNR \\\\rm {V}_{GNR}=0.5 mV.\",\n \"(f) Absolute value of the GNR transconductance.\",\n \"Elevated conductance traces with a positive slope (dotted line) match perfectly with the Coulomb resonances in (d).\",\n \"Detection lines are even visible where the current is too small to be measured directly (see white arrows).\",\n \"Measurements shown in (a)-(c) were measured parallel to line (2).\",\n \"All measurements were recorded simultaneously.Fig.\",\n \"REF shows a SFM image of an all-carbon device consisting of a carbon nanotube lying in the close vicinity to an etched graphene nanoribbon (GNR) which acts as a charge detector (CD).\",\n \"By Raman spectroscopy we identify the bright colored area to be a graphene nanoribbon of bilayer nature (not shown) and following the above argument we conclude that the nanotube is a single-walled carbon nanotube.\",\n \"Both carbon nanostructures are separated by roughly 150 nm, the nanoribbon has a width of around 100 nm and the CNT quantum dot is defined by two metal electrodes (indicated in blue) which are separated by 350 nm.\",\n \"As illustrated in Fig.\",\n \"REF , we apply a symmetric bias voltage ($\\\\rm {V}_{CNT}$ and $\\\\rm {V}_{GNR}$ respectively) to both structures.\",\n \"The overall Fermi level can be tuned by the back gate voltage $\\\\rm {V}_{BG}$ applied to the highly doped Si substrate.\",\n \"Additionally, we can use the CNT (GNR) as a lateral gate for the GNR (CNT) by applying a reference potential $\\\\rm {V}_{REF}$ ($\\\\rm {V}_{REF^{*}}$ ).\",\n \"All presented measurements were performed in a pumped $^4\\\\rm {He}$ -cryostat at a base temperature of T$\\\\approx $ 1.5 K using low-frequency lock-in techniques.\",\n \"Figs.\",\n \"REF (a) and REF (b) show the back gate characteristics i.e.\",\n \"the conductance as function of $\\\\rm {V}_{BG}$ of both, the CNT (Fig.\",\n \"REF (a)) and the graphene nanoribbon (Fig.\",\n \"REF (b)) for a constant bias voltage $\\\\rm {V}_{CNT}$ =$\\\\rm {V}_{GNR}$ =0.5 mV.\",\n \"The CNT reveals a semiconducting behavior with a large band gap resulting in an extended BG region of suppressed current (-60 V$< \\\\rm {V}_{BG} < $ 40 V).\",\n \"A high resolution measurement performed at the edge of this gap (see inset in Fig.\",\n \"REF (a)) exhibits reproducible, well resolved and sharp Coulomb peaks.\",\n \"The peak width of the sharpest resonances is given by the electron temperature [26] which can be extracted to be 2 K. In Fig.\",\n \"REF (c) we show so-called Coulomb diamond measurements i.e.\",\n \"the differential conductance d$\\\\rm {I}_{CNT}$ /d$\\\\rm {V}_{CNT}$ plotted as function of $\\\\rm {V}_{BG}$ and $\\\\rm {V}_{CNT}$ .\",\n \"From the extent of suppressed current of the diamonds in bias ($\\\\rm {V}_{CNT}$ ) direction we can extract a charging energy of $\\\\rm {E}_C \\\\approx $ 6-9 meV of the CNT quantum dot and a BG lever arm, $\\\\rm {\\\\alpha }_{BG}$ =0.22.\",\n \"Following Ref.\",\n \"[27] we can relate the charging energy with the length of the nanotube segment forming the quantum dot ($\\\\rm {L}_{QD}$ ) by roughly $\\\\rm {E}_C$ =1.4 eV/$\\\\rm {L}_{QD}$ (nm).\",\n \"This provides an order of magnitude estimate of $\\\\rm {L}_{QD}\\\\approx $ 150-235nm which is in reasonable agreement with the device geometry.\",\n \"The change in differential conductance parallel to the diamond edges is attributed to excited states providing additional transport channels.\",\n \"The observed excited states exhibit an energy of $\\\\Delta $$\\\\approx $ 0.9 meV (see e.g.\",\n \"white dotted line in Fig.\",\n \"REF (c)).\",\n \"The independently measured back gate characteristic of the bilayer graphene nanoribbon is shown in Fig.\",\n \"REF (b).\",\n \"This low bias ($\\\\rm {V}_{GNR}$ =0.5 mV) measurement highlights that transport can be tuned from the hole regime (left inset) into the so-called transport gap [28], [29], [30], [31], [32], [33], [34], [35] starting at around $\\\\rm {V}_{BG}$$\\\\approx $ 40 V. Within the gap region transport is governed by localized states resulting in sharp resonances of the conductance as shown in the right inset of Fig.\",\n \"REF (c).\",\n \"Interestingly, both carbon nanostructures have different doping levels.\",\n \"In contrast to the n-doped CNT we observe a significant p-doping of the GNR which is most likely due to atmospheric $\\\\rm {O}_{2}$ binding on the graphene edges [36].\",\n \"Fig.\",\n \"REF (d) shows the conductance of the CNT in dependence of the BG voltage $\\\\rm {V}_{BG}$ =13.63 V + $\\\\Delta \\\\rm {V}_{BG}$ and the reference voltage on the nanoribbon $\\\\rm {V}_{REF^{*}}$ at $\\\\rm {V}_{CNT}$ =20 mV, which demonstrates the gating effect of the GNR on the CNT QD.\",\n \"Lines of higher conductance can be attributed to Coulomb resonances of the CNT and from their slopes we extract a relative lever arm of $\\\\rm {\\\\alpha }_{BG,GNR}$ =0.23.\",\n \"If both quantum devices are now operated simultaneously, the nanoribbon device can be used to detect individual charging events on the nanotube QD device.\",\n \"This phenomenon is shown in Fig.\",\n \"4.\",\n \"The BG voltage is put to an offset ($\\\\rm {V}_{BG}$ =48.166 V) such that (i) the CNT is in the Coulomb blockade regime and (ii) the conductance of the graphene nanoribbon exhibits sharp and well-reproducible resonances.\",\n \"Figs.\",\n \"REF (a) and REF (b) show the simultaneously measured low-bias ($\\\\rm {V}_{CNT}$ =0.5 meV) conductance through the nanotube $\\\\rm {G}_{CNT}$ and the nanoribbon $\\\\rm {G}_{GNR}$ as function of $\\\\rm {V}_{REF}$ where the BG voltage is simultaneously adjusted according to $\\\\rm {V}_{BG}$ =a-b$\\\\cdot $$\\\\rm {V}_{REF}$ with a=48.166 V and b=-0.335.\",\n \"Similar to the inset of Fig.\",\n \"REF (a), $\\\\rm {G}_{CNT}$ exhibits Coulomb peaks, which are indicating single charging events in the CNT QD.\",\n \"In the simultaneously measured trace of the GNR, we observe distinct steps in the conductance at the exact positions of the CNT QD charging events.\",\n \"The steps in conductance can measure up to 20 $\\\\%$ of the total resonance amplitude and are due to the capacitive coupling of both nanostructures.\",\n \"Increasing the reference potential $\\\\rm {V}_{REF}$ and decreasing $\\\\rm {V}_{BG}$ both leads to a higher chemical potential in the QD and subsequently to a lower occupation number at every event.\",\n \"Consequently, the GNR resonance shifts to lower values of $\\\\rm {V}_{REF}$ giving rise to the shape of the charge detecting resonance in Fig.\",\n \"REF (b).\",\n \"The unconventional shape of the detection signal is explained by the inset of Fig.\",\n \"REF (b).\",\n \"In this schematic, each resonance curve (in gray) would represent the conductance resonance of the GNR at a fixed charge state of the CNT QD.\",\n \"However, the latter is itself a (discontinuous) function of $\\\\rm {V}_{REF}$ , so that the measured conductance through the GNR is not a smooth curve but show a step whenever the charge state of the CNT QD (and thus the conductance resonance) changes.\",\n \"By relating the step height to the noise level of our measurement system we achieve an estimate for the charge sensitivity with an upper limit of $~10^{-3} \\\\rm {e}/\\\\sqrt{\\\\rm {Hz}}$ which is in agreement with previous experiments on GNR charge detectors [15].\",\n \"In order to highlight the charge detection we further plot the transconductance i.e.\",\n \"the derivative of $\\\\rm {G}_{GNR}$ with respect to $\\\\rm {V}_{REF}$ as shown in Fig.\",\n \"REF (c).\",\n \"Each individual transconductance peak (dip) is very well aligned with the directly measured Coulomb peaks.\",\n \"For proving this more rigerously we performed measurements as function of both $\\\\rm {V}_{REF}$ and $\\\\rm {V}_{BG}$ and extracted relative lever arms.\",\n \"Corresponding charge stability diagrams are shown in Figs.\",\n \"REF (d) and REF (e).\",\n \"As anticipated, the evolution of Coulomb resonances of the CNT QD (shown in Fig.\",\n \"REF (d)) follows a constant positive slope with a relative lever arm of $\\\\rm {\\\\alpha }_{REF,BG}\\\\approx $ 1.25 and the occupation numbers (denoted in white letters) decrease with increasing $\\\\rm {V}_{REF}$ and decreasing $\\\\rm {V}_{BG}$ .\",\n \"For the charge detector the observed patterns in Fig.\",\n \"REF (e) show similarities to those of a multi-dot system.\",\n \"Further analysis of the transconductance $\\\\rm {dG}_{CD}/\\\\rm {dV}_{REF}$ reveals at least three lines with negative slopes.\",\n \"These lines show that the transport through the GNR is governed by a series of quantum dots where each of the lines can be attributed to single QDs located at different positions along the nanoribbon.\",\n \"The extracted lever arms of the visible conductance resonances are $\\\\rm {\\\\alpha }_{REF,BG}^{(1)}$ =$-0.43$ , $\\\\rm {\\\\alpha }_{REF,BG}^{(2)}$ =$-0.31$ and $\\\\rm {\\\\alpha }_{ REF,BG}^{(3)}$ =$-0.12$ as indicated in Fig.\",\n \"REF (f).\",\n \"Additionally the plot exhibits clearly visible features (see white dotted line) with in this case positive slopes with their positions matching perfectly to those in Fig.\",\n \"REF (d).\",\n \"Consequently they can be associated with charging events in the CNT QD which are detected by the quantum dots situated in the graphene nanoribbon.\",\n \"Detection lines are even visible where the current through the dot is too low to be measured directly (see white arrows in Figs.\",\n \"REF (d) and (f)).\",\n \"In order to further improve the sensitivity, the detector could potentially be included in a RF-readout setup.\",\n \"The charge sensitivities reported for this type of setups are in the order of $~10^{-4} \\\\rm {e}/\\\\sqrt{\\\\rm {Hz}}$ to $~10^{-5} \\\\rm {e}/\\\\sqrt{\\\\rm {Hz}}$ [16], [17].\",\n \"While a quantitative comparison of the presented dc-readout technique with other dc-charge sensors on carbon nanotube QDs [6], [18] can not be provided (no sensitivities are given), a qualitative comparison of the data shows that the graphene charge detector yields at least equal visibility of the charge states.\",\n \"Finally, we discuss a carbon nanotube device where both metal leads (source and drain) are substituted by two graphene flakes.\",\n \"The device structure has been discussed earlier (see fabrication section and Fig.\",\n \"REF (e)) and a close up of the final device is shown as inset in Fig.\",\n \"REF (a).\",\n \"For measuring the conductance of the nanotube, Cr/Au metal contacts are deposited on each graphene flake by an EBL step followed by metalization and lift-off.\",\n \"The metal contacts are designed in a way that they do not touch the carbon nanotube.\",\n \"As a first important observation we see that current flows from one graphene flake to the other through the carbon nanotube.\",\n \"Moreover, we do not observe a $\\\\rm {V}_{BG}$ -regime where the current is fully suppressed (not shown) which suggests the presence of a metallic nanotube.\",\n \"In Fig.\",\n \"REF (b) we show the current $\\\\rm {I}_{CNT}$ in dependence of the voltage $\\\\rm {V}_{SG}$ applied to a metallic side gate which was deposited next to the nanotube as indicated by the blue area in Fig.\",\n \"REF (a) and allows to locally gate the central segment of the carbon nanotube.\",\n \"The current $\\\\rm {I}_{CNT}$ exhibits periodic Coulomb oscillations.\",\n \"In Fig.\",\n \"REF (c) we plot the current through the nanotube $\\\\rm {I}_{CNT}$ on a logarithmic scale as function of the bias voltage $\\\\rm {V}_{BIAS}$ and $\\\\rm {V}_{SG}$ for a fixed value of $\\\\rm {V}_{BG}$ =35.15 V. The measurement exhibits clear diamond shaped patterns which can be attributed to Coulomb diamonds.\",\n \"From the addition energies of 1-1.5 $\\\\rm {meV}$ we can estimate the quantum dot length of 0.9-1.5 $\\\\rm {\\\\mu m}$ [27], which is in good agreement with the 1.2 $\\\\rm {\\\\mu m}$ spacing between the two graphene leads.\",\n \"This leads to the conclusion that the quantum dot in the nanotube extends over the full length set by the distance of the graphene leads.\",\n \"Moreover, the observation of Coulomb blockade oscillations and peaks manifests the presence of tunneling barriers at the interface of both carbon allotropes.\",\n \"From the maxima of the Coulomb peaks and assuming similar rates for tunneling in and out of the dot we extract tunneling rates in the low GHz regime.\",\n \"The origin of the observed tunneling barriers is not fully understood at the present state.\",\n \"Since graphene is a semi-metal and the investigated carbon nanotube is a metallic carbon nanotube, it is believed that the tunneling barriers form similarly to those metallic-CNT devices with metal contacts.\",\n \"Here, tunneling barriers may arise from imperfect electrical contacts to the nanotube [37].\",\n \"Figure: (color online)(a) SFM image of the investigated device (see also Fig. (e)).\",\n \"(b) Current I CNT \\\\rm {I}_{CNT} through the nanotube as function of the voltage V SG \\\\rm {V}_{SG} applied to a metal electrode placed roughly 50 nm from the nanotube (see blue area in (a)) for a fixed back gate voltage V BG \\\\rm {V}_{BG}=35 V and V BIAS \\\\rm {V}_{BIAS}=0.3 mV.\",\n \"(c) Current I CNT \\\\rm {I}_{CNT} through the nanotube on a logarithmic scale as function of V BIAS \\\\rm {V}_{BIAS} and V SG \\\\rm {V}_{SG}.\",\n \"V BG \\\\rm {V}_{BG} is constant at 35.15 V.In summary, we present the fabrication and characterization of carbon nanotube-graphene hybrid devices.\",\n \"We show the example of a structure where a nanotube quantum dot is capacitively coupled to a graphene nanoribbon.\",\n \"Sharp resonances in the graphene nanoribbon conductance give rise to clear charge detection signals with an upper estimate for the charge sensitivity of $~10^{-3} \\\\rm {e}/\\\\sqrt{\\\\rm {Hz}}$ .\",\n \"This kind of graphene charge detector represents a third charge detection scheme for CNT QDs in addition to the existing strategies based on metal single electron transistors [16], [17] and capacitively coupled carbon nanotube segments [6], [18].\",\n \"The presented charge detector is rather easy to fabricate.\",\n \"Its sensitivity can be further improved by including it into a RF-circuit.\",\n \"In addition, we study a device where two carbon allotropes (graphene and CNT) are electrically coupled.\",\n \"We find that the interfaces between the graphene and the carbon nanotube resembles tunneling barriers.\",\n \"This device represents an important preliminary step towards experiments investigating the quantum phase transition from screened to unscreened impurity moments.\",\n \"Both results open the road to more sophisticated devices which are entirely fabricated out of carbon nanostructures and exploit the different advantages of these promising materials.\",\n \"Acknowledgment — The authors wish to thank L. Durrer and C. Hierold for their support and help on the growth of the CNT samples.\",\n \"We thank A. Steffen, R. Lehmann and J. Mohr for the help on the sample fabrication.\",\n \"Discussions with F. Haupt, J. Splettstösser, D. Schuricht and M. Wegewijs and support by the JARA Seed Fund and the DFG (SPP-1459 and FOR-912) are gratefully acknowledged.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1474"}}},{"rowIdx":1866987,"cells":{"text":{"kind":"list like","value":[["Phase space reduction of the one-dimensional Fokker-Planck (Kramers)\n equation"],["Abstract A pointlike particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered.","We present a rigorous mapping of the Fokker-Planck equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x.","The result is the Smoluchowski equation, valid in the overdamped limit, m->0, with a series of corrections expanded in powers of m. They are determined unambiguously within the recurrence mapping procedure.","The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator."],["I. Introduction","The Brownian motion of a particle in a confined system represents an essential model used in description of stochastic transport through quasi one-dimensional (1D) systems, e.g.","channels in nanomaterials, pores or fibers in biological structures.","In a 1D system, the trajectory $x(t)$ of a Brownian particle is described by the Langevin equation $m\\ddot{x}+\\gamma \\dot{x}+\\partial _xU(x)=f(t)\\ .$ Here, $m$ denotes mass of the particle, $U(x)$ represents the driving potential, $\\gamma $ is an effective friction coefficient and $f(t)$ is the stochastic force, satisfying the usual conditions on averaged values, $\\left< f(t)\\right>=0$ and $\\left=2\\gamma k_BT\\delta (t-t^{\\prime })$ ; $T$ is the temperature and $k_B$ the Boltzmann constant.","The corresponding phase space density $\\rho (x,v,t)$ of the particle satisfies the Fokker-Planck (FP), or Kramers (kinetic) equation, $\\left(\\partial _t+v\\partial _x-\\frac{U^{\\prime }(x)}{m}\\partial _v\\right)\\rho (x,v,t)&& \\cr && \\hspace{-108.405pt}=\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\rho (x,v,t),\\hspace{21.68121pt}$ where $x$ is the spatial coordinate, $v$ denotes its velocity and $\\beta =1/k_BT$ is the inverse temperature.","Solutions of the Langevin equation, as well as the corresponding kinetic equation, are studied over almost a century [1], [2], [3].","Still, motion of a particle in confined geometries represents usually a complicated problem, requiring next reductions of the used description.","Due to simplicity, and often for relevance, mainly the overdamped limit is studied.","The mass dependent term in Eq.","(REF ), $m\\ddot{x}$ , is then considered negligible and the particle's spatial density $p(x,t)$ is governed by the Smoluchowski equation, $\\partial _tp(x,t)=D_0\\partial _xe^{-\\beta U(x)}\\partial _xe^{\\beta U(x)}p(x,t)\\ ,$ containing no information about the mass of the particle; $D_0$ denotes the diffusion constant.","Instead of the full phase space, one works with only the spatial coordinate $x$ .","Of course, the question is the price for this simplification, as well as possibility of also including properly inertia of the massive particles in the reduced (real space only) description of the Brownian dynamics.","Recent studies showed its importance in understanding rectification of the transport in ratchets [4], or its influence on the effective diffusion coefficient [5], [6], [7] in a narrow channel.","For our purpose to demonstrate the phase space reduction, we will deal here with only the 1D FP (Kramers) equation (REF ).","The eq.","(REF ) is reducible to the Smoluchowski eq.","(REF ) in the limit $m\\rightarrow 0$ .","The reduction procedure [8], [9], [10] is based on an instant thermalization of the particle's velocity after any move in the $x$ direction in the case of an infinitely small mass.","The situation resembles derivation of the Fick-Jacobs (FJ) equation [11], [12], $\\partial _tp(x,t)=\\partial _x A(x)\\partial _x\\frac{p(x,t)}{A(x)},$ appearing as the result of the dimensional reduction of the diffusion equation in a 2D channel with varying cross section $A(x)$ , onto the longitudinal coordinate $x$ , if equilibration in the transverse ($y$ ) direction is instant.","The function $p(x,t)$ denotes the linear (1D) density of the particle.","Of course, as in reduction of diffusion to the FJ equation, that of (REF ) to (REF ) as $m\\rightarrow 0$ is a singular limit, and must be handled with care, but from the viewpoint of the reduction of (REF ), no such caveat is needed.","Recently, an exact mapping procedure was proposed [13], [14], enabling us to also derive the corrections to the FJ equation (REF ), which are necessary, if the transverse equilibration is not instant.","The key was to introduce anisotropy of the diffusion constant in the diffusion equation, $\\partial _t\\rho (x,y,t)=\\left(\\partial _x^2+\\frac{1}{\\epsilon }\\partial _y^2\\right)\\rho (x,y,t),$ governing the 2D (spatial) density $\\rho (x,y,t)$ .","For $\\epsilon \\rightarrow 0$ , the infinitely fast transverse diffusion immediately flattens the $y$ profile of $\\rho (x,y,t)\\rightarrow \\rho (x,t)=p(x,t)/A(x)$ .","Then integration of Eq.","(REF ) over the cross section, together with the reflecting boundary conditions satisfied at the hard walls, results in the FJ eq.","(REF ).","In the case of a slower transverse diffusion, the mapping procedure generates a series of corrections to the FJ equation controlled by $\\epsilon >0$ , developing the $y$ profile of $\\rho (x,y,t)$ , which is already curved depending on the flux and geometry of the channel.","The procedure was extended to mapping of diffusion in an external field $U(x,y)$ , e.g.","for diffusion in a channel with soft walls [15], where the particle is kept near the $x$ axis by the parabolic potential, $\\beta U(x,y)=\\alpha (x)y^2$ ; $\\alpha (x)$ represents the varying stiffness of the walls.","The equation to be mapped onto the $x$ coordinate is the 2D Smoluchowski equation, $\\left(\\partial _t-\\partial _xe^{-\\alpha (x)y^2}\\partial _xe^{\\alpha (x)y^2}\\right)\\rho (x,y,t)=\\hspace{57.81621pt}\\cr =\\frac{1}{\\epsilon }\\partial _ye^{-\\alpha (x)y^2}\\partial _ye^{\\alpha (x)y^2}\\rho (x,y,t).\\hspace{14.45377pt}$ Integrating over $y$ and applying the mapping scheme gives the mapped 1D equation in an extended Smoluchowski form, $\\partial _tp(x,t)=\\partial _xe^{-V(x)}\\left[1+\\epsilon \\hat{Z}(x,\\partial _x)\\right]\\partial _xe^{V(x)}p(x,t),$ governing the 1D density $p(x,t)$ , where $V(x)$ stands for an effective potential and $\\hat{Z}$ is the correction operator sought as an expansion in the small parameter $\\epsilon $ .","The potential $V(x)$ and the coefficients of $\\hat{Z}$ depend on $\\alpha (x)$ and both are fixed unambiguously within the recurrence mapping procedure.","The central idea of this paper is a conjecture that the FP eq.","(REF ) can be reduced to an extended Smoluchowski-like 1D form, governing the spatial density $p(x,t)$ , after integration over $v$ and applying a similar mapping scheme.","The velocity $v$ thus represents a \"transverse\" coordinate instead of $y$ , with the mass $m$ playing the role of the small parameter $\\epsilon $ .","Then the Smoluchowski equation (REF ) should be obtained in the limit $m\\rightarrow 0$ [8], [9].","Performing the recurrence procedure, a series of corrections to this equation in powers of $m$ would be derived.","Then the final mapped equation would also respect inertia of the Brownian particles, although working only in real space.","Use of the mapping procedure developed for diffusion [13], [14], [15], [16] is not straightforward; the left-hand side operator of the FP equation (REF ) has a different structure than that of the diffusion (REF ) or Smoluchowski equation (REF ).","Still, there is a way to apply the general scheme of the mapping in this case and to perform reduction of the phase space onto the real space in the way described above.","Presentation of this algorithm is the primary aim of this study.","The result of the mapping of the FP equation (REF ) is again an equation of the form (REF ) with $V(x)=\\beta U(x)$ and $\\epsilon $ replaced by $m$ .","In the limit of stationary flow, this equation can be simplified by subsequent reduction of the operator $1-m\\hat{Z}$ to a function $D(x)$ , a spatially dependent effective diffusion coefficient [12], [17], $\\partial _tp(x,t)=\\partial _xe^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x,t)\\ .$ The leading term of $D(x)$ is proportional to $mU^{\\prime \\prime }(x)$ and the whole series of corrections to the Smoluchowski equation can be summed up, giving $D(x)=D_0\\frac{1-\\sqrt{1-4mU^{\\prime \\prime }(x)/\\gamma ^2}}{2mU^{\\prime \\prime }(x)/\\gamma ^2},$ with $D_0=1/\\gamma \\beta $ , if the higher derivatives of $U(x)$ are neglected.","In the following section, we analyze how to apply the mapping scheme for reduction of the phase space in the FP equation (REF ).","In Sect.","III, our considerations are verified on an exactly solvable model, the FP equation with no field, $U(x)=0$ .","Analysis of this example helps us to construct the recurrence scheme for calculation of the series of corrections to the zeroth order approximation, the Smoluchowski eq.","(REF ), in the small parameter $m$ .","Finally, the complete mapping procedure for an arbitrary potential $U(x)$ is presented in Sect.","IV.","The mapped equation of type (REF ), as well as the formula (REF ) is derived, and checked on the damped linear harmonic oscillator."],["II. Preliminary considerations","The key points of the mapping procedure, as formulated for diffusion [13], [14], are recalled in this Section.","Based on physical considerations, we adjust the general scheme of the mapping for dimensional reduction of the FP equation (REF ) to this situation.","The mapping procedure represents a consistent transition from a fine grain to a coarse grain description of some evolution process.","The process is described in details by some partial differential equation (PDE), governing the density of particles $\\rho ({\\bf r},t)$ in the full space, defined by the coordinates ${\\bf r}$ .","The dimensional reduction projects this equation onto another PDE, which governs the density $p(x,t)$ in the reduced space of the coordinate $x$ .","The coordinate $x$ is one of the coordinates of the full space, ${\\bf r}=(x,{\\bf y})$ , and the mapping accomplishes integration over the transverse coordinates ${\\bf y}$ .","In the case of the FP equation (REF ), phase space $(x,v)$ represents the full space and the dimensional reduction integrates over the \"transverse\" coordinate, the velocity $v$ .","Hence we have the defining relation between the densities $\\rho $ and $p$ : $p(x,t)=\\int _{-\\infty }^{\\infty }\\rho (x,v,t)dv\\ .$ The phase-space density $\\rho (x,v,t)$ is expected to be near the thermodynamic equilibrium and so approximately proportional to $\\exp (-\\beta mv^2/2)$ , which provides convergence of the integral in Eq.","(REF ).","Then the first step of the mapping is also integration of Eq.","(REF ) over $v$ .","If completed, we get $\\int _{-\\infty }^{\\infty }\\left(\\partial _t+v\\partial _x\\right)\\rho (x,v,t)dv=0;$ the other terms are zero due to $\\rho (x,v,t)\\rightarrow 0$ in the limit $v\\rightarrow \\pm \\infty $ .","This equation represents nothing but mass conservation in the reduced space, $\\partial _tp(x,t)+\\partial _xj(x,t)=0,$ where the 1D flux $j$ is defined by the relation $j(x,t)=\\int _{-\\infty }^{\\infty }v\\rho (x,v,t)dv\\ .$ In contrast to diffusion, where $j=-D_0\\partial _xp$ is fixed, here the flux $j$ is a function formally independent of the density $p$ , so we also need the evolution equation for this quantity.","After integration of Eq.","(REF ) multiplied by $v$ , we obtain $\\partial _tj(x,t)+\\partial _x\\int _{-\\infty }^{\\infty }v^2\\rho (x,v,t)dv+\\frac{U^{\\prime }(x)}{m}p(x,t)=\\ \\ \\ \\cr =-\\frac{\\gamma }{\\beta m^2}\\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\rho (x,v,t)dv\\ \\ $ [after some algebra and applying the definitions (REF ) and (REF ).","This step recalls the Grad's method of moments [18], [19].","For the 1D Kramers equation (REF ), taking only a couple of the zeroth ($p$ ) and the first ($j$ ) order moment of the phase space density $\\rho $ is satisfactory for generating the selfconsistent system of the mapped (real space) equations (REF ) and (REF ).]","The next key point of the mapping algorithm is that of expressing the full-space density $\\rho (x,v,t)$ using the 1D density $p(x,t)$ and also the flux $j(x,t)$ in this case.","This relation enables us to complete integrations in Eq.","(REF ) and get the evolution equation for $j(x,t)$ , together with the 1D mass conservation, in closed form.","The initial task is to find the zeroth order approximation, valid in the limit $m\\rightarrow 0$ .","Our first proposal for such a relation between $\\rho (x,v,t)$ and the reduced space quantities $p$ and $j$ is based on the following physical construction: For an infinitely small mass of the particle, the stochastic force thermalizes its velocity $v$ almost immediately after any move along the spatial coordinate $x$ .","Similar to the transverse equilibration of the 2D density of a particle diffusing in a narrow channel with biasing transverse force [15], [20], [21], [22], one could try the formula with separated Boltzmann factor in the fast relaxing \"direction\" $v$ , $\\rho (x,v,t)\\simeq \\sqrt{\\beta m/2\\pi }p(x,t)\\exp (-\\beta mv^2/2)$ .","It is easy to check that it does not work here; the flux $j$ becomes zero according to Eq.","(REF ).","To prevent this failure, let us suppose that the distribution in $v$ is shifted by the local mean (macroscopic) velocity $v_0$ , depending on the local flux, $j(x,t)=v_0(x,t)p(x,t)$ .","Then we have $\\nonumber \\rho (x,v,t)&\\simeq & \\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta m(v-v_0)^2/2}p(x,t)\\cr &\\simeq &\\sqrt{\\frac{\\beta m}{2\\pi }}\\left[1+\\beta mvv_0+ ...\\right]e^{-\\beta mv^2/2}p(x,t).$ Retaining only these two terms in the square brackets and replacing $v_0p$ by the flux $j$ , one gets $\\rho _0(x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}\\left[p(x,t)+\\beta mvj(x,t)\\right],$ which will be taken as the sought zeroth order relation between $\\rho $ and the reduced space quantities, $p$ and $j$ .","This heuristic formula will be verified later by the exact mapping algorithm.","Still, one can check immediately that the relation (REF ) represents correctly a kind of backward mapping of the 1D (spatial) functions $p$ and $j$ onto the phase space densities $\\rho $ ; if substituted for $\\rho (x,v,t)$ in the defining relations (REF ) and (REF ), we obtain identities.","Applying Eq.","(REF ) to Eq.","(REF ), the integrals over $v$ can be completed and the result, $\\partial _tj(x,t)+\\partial _x\\frac{p(x,t)}{\\beta m}+\\frac{U^{\\prime }(x)}{m}p(x,t)=-\\frac{\\gamma }{m}j(x,t),$ together with the mass conservation, Eq.","(REF ), forms a closed couple of PDE, governing the mapped quantities $p$ and $j$ .","In the limit $m\\rightarrow 0$ , the first term in Eq.","(REF ), $\\partial _tj$ , becomes negligible and the equation expresses the zeroth order relation between the flux $j$ and the density $p$ , $j(x,t)=-\\frac{1}{\\beta \\gamma }e^{-\\beta U(x)}\\partial _xe^{\\beta U(x)}p(x,t)\\ .$ If combined with the mass conservation, Eq.","(REF ), we get the Smoluchowski equation (REF ); $1/\\beta \\gamma =D_0$ represents the diffusion constant.","The calculation presented shows that the Smoluchowski equation (REF ) is related to the FP equation (REF ) in the same way as the Fick-Jacobs equation [11] to the diffusion equation valid in a narrow 2D channel.","Both mapped equations describe an asymptotic behavior of the full-space density infinitely rapidly equilibrating in the transverse direction; the velocity $v$ plays the role of the transverse coordinate for the FP equation.","Our considerations indicate that the mass of the particle, $m$ , becomes the small parameter, controlling the series of corrections to the Smoluchowski equation in the case when the transverse equilibration is not infinitely fast.","Following the scheme of the mapping procedure [13], [14], the next point is that of searching for the true relation between the phase space density $\\rho (x,v,t)$ and the 1D quantities $p(x,t)$ and $j(x,t)$ , replacing the heuristic formula (REF ), valid for nonzero $m$ .","Without losing generality, it can be written in the form $\\rho (x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}\\Big [\\hat{\\omega }(x,v)p(x,t)\\hspace{28.90755pt}\\cr \\hspace{-28.90755pt}+\\beta mv\\hat{\\eta }(x,v)j(x,t)\\Big ].$ If the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ (with $\\partial _x$ implicit) are expandable in $m$ , one can substitute for $\\rho (x,v,t)$ in the FP equation (REF ) and fix the coefficients of these operators to satisfy this equation in each order of $m$ , similar to the mapping of diffusion.","Then, using the relation of backward mapping (REF ) in Eq.","(REF ) gives the expansion of the evolution equation for $j$ and finally, in combination with mass conservation (REF ), the sought series of corrections to the Smoluchowski equation (REF ) in terms of the finite mass $m$ .","To verify whether this scheme is viable, we analyze the exactly solvable case with $U(x)=0$ in the next Section."],["III. Exactly solvable model","The exact solution of the FP equation (REF ) with no potential, $U(x)=0$ , is presented in this Section.","We demonstrate the mapping on the example of the phase space density $\\rho (x,v,t)$ evolving from the initial density $\\rho (x,v,0)$ with thermalized velocity $v$ .","The mapped equation, as well as the form of the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ in Eq.","(REF ) can be found explicitly in this case.","This analysis will direct us in construction of the recurrence mapping scheme in Sect.","IV.","The case $U(x)=0$ is exactly solvable [23], the Green's function $G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })$ of the FP equation (REF ), $&&\\hspace{-14.45377pt}\\left[\\partial _t+v\\partial _x-\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\right]G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })\\cr &&\\hspace{72.26999pt}=\\delta (x-x^{\\prime })\\delta (v-v^{\\prime })\\delta (t-t^{\\prime })\\ ,$ can be derived explicitly (see Appendix A), $G&=&\\frac{\\gamma \\beta }{4\\pi }\\frac{\\Theta (\\tau -\\tau ^{\\prime })}{\\sqrt{\\tau -\\tau ^{\\prime }-\\tanh (\\tau -\\tau ^{\\prime })}\\sqrt{1-q^2}}\\cr &&\\times \\exp \\bigg (-\\frac{\\left[2(\\xi -\\xi ^{\\prime })-(u+u^{\\prime })\\tanh (\\tau -\\tau ^{\\prime })\\right]^2}{4[\\tau -\\tau ^{\\prime }-\\tanh (\\tau -\\tau ^{\\prime })]}\\ \\ \\cr &&\\hspace{28.90755pt}-\\frac{q}{1-q^2}\\left[q(u^2+u^{\\prime 2})-2uu^{\\prime }\\right]-u^2\\bigg )\\ ,$ if expressed in the scaled coordinates, $\\tau &=&\\gamma t/2m,\\cr \\xi &=&(\\beta m/2)^{3/2}\\frac{\\gamma x}{\\beta m^2},\\cr u&=&\\sqrt{\\beta m/2}\\ v,$ and $q=\\exp [-2(\\tau -\\tau ^{\\prime })]$ .","If the thermalized particle is inserted at time $t=0$ with a spatial distribution $p_0(x)$ , $\\rho (x,v,0)=\\sqrt{\\frac{\\beta m}{2\\pi }}p_0(x)e^{-\\beta mv^2/2},$ evolution of the density $\\rho $ is given by the formula $\\rho (x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}\\int _{-\\infty }^{\\infty }dv^{\\prime }\\int dx^{\\prime } G(x,v,t;x^{\\prime },v^{\\prime },0)\\ \\ \\cr &&\\hspace{72.26999pt}\\times p_0(x^{\\prime })e^{-\\beta mv^{\\prime 2}/2}\\cr &=&\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{4\\pi D_0\\sqrt{Z}}\\exp \\bigg (-\\frac{[\\xi -\\xi ^{\\prime }]^2}{Q}\\cr &&\\hspace{14.45377pt}-\\frac{Q}{Z}\\left[u-\\frac{(1-e^{-2\\tau })}{2Q}(\\xi -\\xi ^{\\prime })\\right]^2\\bigg );$ $D_0=1/\\gamma \\beta $ , the abbreviations $Q&=&\\tau -\\frac{1}{2}\\left(1-e^{-2\\tau }\\right),\\cr Z&=&Q-\\frac{1}{4}\\left(1-e^{-2\\tau }\\right)^2$ are used and the integration over $x^{\\prime }$ runs over the whole (unspecified) 1D spatial domain.","Then the spatial (1D) density $p$ and the flux $j$ are integrated directly according to Eqs.","(REF ) and (REF ), $p(x,t)&=&\\sqrt{\\frac{2}{\\beta m}}\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{4D_0\\sqrt{\\pi Q}}e^{-(\\xi -\\xi ^{\\prime })^2/Q},\\hspace{50.58878pt}\\cr j(x,t)&=&\\frac{2}{\\beta m}\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{8D_0\\sqrt{\\pi Q^3}}\\left(1-e^{-2\\tau }\\right)\\cr &&\\hspace{65.04256pt}\\times \\left(\\xi -\\xi ^{\\prime }\\right) e^{-(\\xi -\\xi ^{\\prime })^2/Q}.$ It is easy to check that the mass conservation (REF ), $\\partial _{\\tau }p+\\sqrt{\\beta m/2}\\partial _{\\xi }j=0$ in the scaled coordinates, is satisfied.","The quantity $Q$ plays the role of a \"stretched\" time [24].","For short times, $t\\ll 2m/\\gamma $ , $Q\\simeq \\tau ^2$ and $Z\\simeq 4\\tau ^3/3$ .","The formulas (REF ) and (REF ) describe correctly behavior of the Newtonian particles in this limit.","The mapping procedure, as outlined in the previous Section, requires us to study asymptotic behavior in the opposite limit, $t\\gg 2m/\\gamma $ .","For large times, $\\tau \\rightarrow \\infty $ , the stretched time $Q$ becomes $\\tau $ and the formulas (REF ) represent the general solution of the diffusion equation, as expected according to the previous Section.","Now it is necessary to verify that the mass $m$ can serve as a small parameter controlling the series of corrections to the diffusion equation and its solution.","It may look problematic at first glance, because $Q$ contains $\\exp (-2\\tau )=\\exp (-\\gamma t/m)$ , representing essential singularity of the variable $m\\rightarrow 0$ .","Then the formulas (REF ) [and similarly Eq.","(REF )] are not expandable in $m$ .","Nevertheless, this property is still consistent with the general scheme of the mapping, as analyzed in Ref.","[14].","The dimensional reduction, as demonstrated on anisotropic diffusion in a narrow channel [14], also reduces the Hilbert space of the full-space problem.","Let us denote $\\hat{M}(\\epsilon )$ the spatial operator of the evolution equation, i.e.","$\\hat{M}(\\epsilon )=\\partial _x^2+(1/\\epsilon )\\partial _y^2$ for anisotropic 2D diffusion; the eigenvalues $\\lambda _i$ and the eigenfunctions $\\psi _i(x,y)$ are given by the equation $-\\hat{M}(\\epsilon )\\psi _i(\\epsilon ;x,y)=\\lambda _i(\\epsilon )\\psi _i(\\epsilon ;x,y),$ supplemented by proper boundary conditions at the walls of the channel.","The parameter of anisotropy $\\epsilon <1$ splits the spectrum into two parts, the low-lying states, whose eigenvalues $\\lambda _l(\\epsilon )$ remain finite for $\\epsilon \\rightarrow 0$ and the transients with $\\lambda _r(\\epsilon )$ diverging $\\sim 1/\\epsilon $ .","Then the exact 2D density $\\rho $ evolves as $\\rho (\\epsilon ;x,y,t)=\\sum _{i}c_i\\psi _i(\\epsilon ;x,y)e^{-\\lambda _i(\\epsilon ) t},$ the constants $c_i$ are given by the initial condition and the summation runs over the whole spectrum.","The transients contribute to the sum by the terms proportional to $\\exp (-\\bar{\\lambda }_r t/\\epsilon )$ , where $\\bar{\\lambda }_r=\\epsilon \\lambda _r(\\epsilon )$ are finite in the limit $\\epsilon \\rightarrow 0$ .","The result is a formula containing the essential singularity in the parameter $\\epsilon $ near zero, similar to Eq.","(REF ) with singular $\\exp (-\\gamma t/m)$ for $m\\rightarrow 0$ .","The mapping procedure reduces the full Hilbert space of all $\\psi _i$ onto the space defined only by the low-lying states $\\psi _l$ .","If the 1D density $p(x,t)$ is integrated from Eq.","(REF ) and mapped backward onto the full Hilbert space (by some operator $\\hat{\\omega }$ ), the transients will be canceled; the sum (REF ) after the mapping there and back runs only over the low-lying states $\\psi _l$ .","The terms retained involve no essential singularity in $\\epsilon $ ; the formula for $\\rho $ considered in the mapping procedure represents the regular part of the exact 2D density $\\rho $ with respect to the parameter $\\epsilon $ near zero.","This reduction is natural for the zero-th order (Fick-Jacobs) approximation, as the transients decay infinitely fast due to their infinite eigenvalues $\\lambda _r(\\epsilon \\rightarrow 0)$ .","Nevertheless, the mapping based on fixing the series of corrections expanded in $\\epsilon $ can work only with the regular part of the 2D density.","Correspondingly, the formulas (REF ) and (REF ) are exact, including the contributions of the transients, which are represented by the singular terms $\\sim \\exp (-\\gamma t/m)$ .","The mapping requires us to analyze only the regular parts, $p_{reg}(x,t)&=&\\int \\frac{p(x^{\\prime })dx^{\\prime }}{2\\sqrt{\\pi D_0(t-D_0\\beta m)}}\\cr &&\\times \\exp \\left[-(x-x^{\\prime })^2)/4D_0(t-D_0\\beta m)\\right],\\hspace{14.45377pt}\\cr j_{reg}(x,t)&=&\\int \\frac{(x-x^{\\prime })p(x^{\\prime })dx^{\\prime }}{4\\sqrt{\\pi D_0(t-D_0\\beta m)^3}}\\cr &&\\times \\exp \\left[-(x-x^{\\prime })^2/4D_0(t-D_0\\beta m)\\right]$ and $\\rho _{reg}(x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}\\int \\frac{p(x^{\\prime })dx^{\\prime }}{2\\sqrt{\\pi D_0(t-3D_0\\beta m/2)}}\\hspace{28.90755pt}\\cr &&\\hspace{-36.135pt}\\times \\exp \\left[-\\frac{(x-x^{\\prime }-D_0\\beta mv)^2}{4D_0(t-3D_0\\beta m/2)}-\\frac{1}{2}\\beta mv^2\\right],$ written in the unscaled coordinates, obtained after taking only the regular parts of $Q$ and $Z$ (REF ), $Q_{reg}=\\tau -1/2$ and $Z_{reg}=\\tau -3/4$ , in Eqs.","(REF ) and (REF ).","Of course, the formulas are applied for $t\\gg D_0\\beta m$ , when the transients vanish.","We omit writing the subscript \"reg\" in the following text.","In comparison with the overdamped limit $m\\rightarrow 0$ , evolution of the spatial (1D) density $p$ and the flux $j$ is only corrected by a time shift, $t\\rightarrow t-D_0\\beta m$ in the formulas (REF ).","The Gaussian distribution is retarded by the time $t_0=D_0\\beta m=m/\\gamma $ , corresponding to the mean time necessary for losing information about the original velocity.","The value of the shift $t_0$ is constant in $x$ and $t$ , so the density $p$ (REF ) still obeys the diffusion equation, $\\partial _tp(x,t)=D_0\\partial _x^2p(x,t),$ and $j(x,t)=-D_0\\partial _xp(x,t)$ .","There is no correction of the zero-th order mapped equation due to the finite mass of the particle.","Let us stress that the case of $U(x)=0$ is extremely simple, similar to the mapping of the 2D diffusion in a flat narrow channel, also giving no corrections to the Fick-Jacobs approximation.","Nevertheless, the relation of the backward mapping, generating the phase-space density $\\rho $ from the mapped quantities $p$ and $j$ , is not quite trivial.","In the formula for $\\rho $ , Eq.","(REF ), the time is shifted by $3t_0/2$ and the displacement $x-x^{\\prime }$ by $vt_0$ with respect to the distribution of a massless particle.","The shortest way to construct the relation of backward mapping is by applying the shift operators in $t$ and $x$ on $p(x,t)$ , compensating the different shifts of time and displacement in Eqs.","(REF ) and (REF ), $\\rho (x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}e^{-(t_0/2)\\partial _t-vt_0\\partial _x}p(x,t).$ Due to Eq.","(REF ), the operator $\\partial _t$ in the exponent can be replaced by $D_0\\partial _x^2$ .","After expanding the shift operator $\\exp (-vt_0\\partial _x)$ and using $j=-D_0\\partial _xp$ , we arrive at the relation of the form (REF ), $\\rho (x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }(\\beta mv^2)^k\\hspace{21.68121pt}\\cr &&\\hspace{-36.135pt}\\times (D_0t_0)^k\\partial _x^{2k}\\left[\\frac{1}{(2k)!","}p(x,t)+\\frac{\\beta mv}{(2k+1)!","}j(x,t)\\right];$ the explicit formulas for $\\hat{\\omega }$ and $\\hat{\\eta }$ to be substituted in Eq.","(REF ) in the case $U(x)=0$ are $\\hat{\\omega }(x,v)&=&e^{-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }\\frac{(\\beta mv^2)^k}{(2k)!","}(D_0t_0)^k\\partial _x^{2k},\\hspace{21.68121pt}\\cr \\hat{\\eta }(x,v)&=&e^{-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }\\frac{(\\beta mv^2)^k}{(2k+1)!","}(D_0t_0)^k\\partial _x^{2k}.$ Both operators are expandable in $m$ , ($t_0=m/\\gamma $ ); their zero-th order coefficients equal unity, consistent with the heuristic formula (REF ).","Also applying the relation (REF ) in the definitions (REF ) and (REF ) gives identity.","The final step is that of verifying the evolution equation for $j$ , Eq.","(REF ).","Two integrals are to be completed with $\\rho $ expressed by the backward mapping, Eq.","(REF ), $\\int _{-\\infty }^{\\infty }v^2\\rho (x,y,t)dv=\\left(1+D_0t_0\\partial _x^2\\right)p(x,t)/\\beta m,\\cr \\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\rho (x,v,t)dv=\\beta m j(x,t),\\ \\ $ which are now valid exactly.","If substituted in Eq.","(REF ), and the equation (REF ) is applied, we get $(1+t_0\\partial _t)[j(x,t)+D_0\\partial _xp(x,t)]=0$ after simple algebra.","This equation validates the relation $j=-D_0\\partial _xp$ for nonzero $m$ as well and thus, if combined with the mass conservation (REF ), also the diffusion equation (REF ) without any corrections.","If compared with the calculation of Eq.","(REF ), the 1-st order term $\\sim \\partial _tj$ , neglected in the previous Section, is compensated here by other 1-st order term coming from the exact relation (REF ), appearing in the integrals (REF ).","The FP equation (REF ) with zero potential is too simple to give nonzero corrections to the diffusion equation (REF ).","Nevertheless, it helped us to understand the structure of the mapping.","It shows that the scheme suggested in the previous section is viable.","The relation of the backward mapping has the form of Eq.","(REF ), at least for $U(x)=0$ , and the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ can be expanded in $m$ , or $t_0=m/\\gamma $ , $\\hat{\\omega }=\\sum _{k=0}^{\\infty }t_0^k\\hat{\\omega }_k(x,u),\\hspace{14.45377pt}\\hat{\\eta }=\\sum _{k=0}^{\\infty }t_0^k\\hat{\\eta }_k(x,u).$ Integration in Eqs.","(REF ) indicates that the coefficients $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ should be sought dependent up on the scaled velocity $u$ rather than $v$ (compare to Ref.","[25]); otherwise each term in Eq.","(REF ) would contribute in several succeeding orders in the integrals (REF ).","The mixing of orders would hinder us in constructing the recurrence scheme generating corrections to Eq.","(REF ).","In the notation of Eq.","(REF ), $\\hat{\\omega }_0=\\hat{\\eta }_0=1$ and $\\hat{\\omega }_1(x,u)&=&(u^2-1/2)D_0\\partial _x^2,\\hspace{21.68121pt}\\cr \\hat{\\eta }_1(x,u)&=&(u^2/3-1/2)D_0\\partial _x^2,\\cr \\hat{\\omega }_2(x,u)&=&(u^4/6-u^2/2+1/8)D_0^2\\partial _x^4,\\cr .&.&.$ valid for $U(x)=0$ according to Eqs.","(REF ), will be used for testing the results of the recurrence procedure in the next Section."],["IV. Mapping procedure","We now finish the construction of the mapping procedure, outlined in the Section II, for an arbitrary analytic potential $U(x)$ .","Supposing the backward mapping of the form (REF ) with the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ expanded in $t_0$ ($m$ ) according to Eqs.","(REF ), we find recurrence relations fixing the coefficients $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ .","Completing the integrals in Eq.","(REF ), we obtain a series of corrections to the zero-th order relation $j=-D_0\\partial _xp$ .","Combined with mass conservation, Eq.","(REF ), it gives the Smoluchowski equation corrected due to nonzero mass of the particle.","The essential relation determining the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ is the FP equation (REF ), which has to be satisfied for any solution of the reduced problem, the density $p(x,t)$ and the flux $j(x,t)$ , after their backward mapping (REF ) onto the full-dimensional Hilbert space.","If the expansion in $t_0=m/\\gamma $ of both operators, Eq.","(REF ), is supposed, we have $&&\\hspace{-21.68121pt}\\left[\\partial _t+\\sqrt{\\frac{2}{\\beta m}}u\\partial _x-\\frac{\\beta U^{\\prime }(x)}{\\sqrt{2\\beta m}}\\partial _u-\\frac{1}{2t_0}\\partial _u e^{-u^2}\\partial _u e^{u^2}\\right]\\sum _{k=0}^{\\infty }t_0^k\\cr &&\\hspace{-21.68121pt}\\times e^{-u^2}\\left[\\hat{\\omega }_k(x,u)p(x,t)+\\sqrt{2\\beta m}u\\hat{\\eta }_k(x,u)j(x,t)\\right]=0$ after introducing the scaled velocity $u=\\sqrt{\\beta m/2}v$ in Eq.","(REF ).","The factor $\\beta m$ is replaced by $t_0/D_0$ in the following calculations.","Thus half-integer powers of $t_0$ appear in Eq.","(REF ) [25].","As this equation has to be satisfied for any $t_0$ , we can split it for clarity into two relations: the first one, including only the integer powers of $t_0$ , $\\sum _{k=0}^{\\infty }t_0^k\\bigg [\\partial _t\\hat{\\omega }_kp+2u^2\\partial _x\\hat{\\eta }_kj-\\beta U^{\\prime }(x)e^{u^2}\\partial _uue^{-u^2}\\hat{\\eta }_kj\\cr -\\frac{1}{2t_0}e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_kp\\bigg ]=0,\\ \\ \\ $ and the second one, collecting the half-integer powers, $\\sum _{k=0}^{\\infty }t_0^{k-1/2}\\bigg [2t_0u\\partial _t\\hat{\\eta }_kj+D_0\\Big (2u\\partial _x-\\beta U^{\\prime }(x)e^{u^2}\\partial _ue^{-u^2}\\Big )\\cr \\times \\hat{\\omega }_kp-e^{u^2}\\partial _u e^{-u^2}\\partial _u u\\hat{\\eta }_k j\\bigg ]=0.\\hspace{21.68121pt}$ Notice that Eqs.","(REF ) and (REF ) do not violate parity of $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ in $u$ .","If used for construction of the recurrence relations between the coefficients, all they have to have the same parity as $\\hat{\\omega }_0=\\hat{\\eta }_0=1$ ; hence $\\hat{\\omega }_k(x,u)=\\hat{\\omega }_k(x,-u)$ and $\\hat{\\eta }_k(x,u)=\\hat{\\eta }_k(x,-u)$ .","This symmetry enables us to find the normalization (or identity) conditions for $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ .","The backward mapped $\\rho $ , Eq.","(REF ), with the operators $\\hat{\\omega }$ , $\\hat{\\eta }$ expanded in $t_0$ , Eq.","(REF ), substituted in the definitions (REF ) and (REF ) has to give identities for any $t_0$ , $p(x,t)$ and $j(x,t)$ .","Thus we obtain $\\frac{1}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }du e^{-u^2}\\hat{\\omega }_k(x,u)=\\delta _{0,k},$ $\\frac{1}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }2u^2du e^{-u^2}\\hat{\\eta }_k(x,u)=\\delta _{0,k}.$ The operators $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ are supposed not to depend on time, so the time derivative commutes with them and acts directly on $p(x,t)$ or $j(x,t)$ in Eqs.","(REF ), (REF ).","To derive the operators unambiguously, using only spatial derivatives, we express $\\partial _tp=-\\partial _xj$ from the mass conservation, Eq.","(REF ).","However, the time derivative of $j$ cannot be expressed in a similar way from Eq.","(REF ), because $\\partial _tj$ is not the leading term there.","If the backward mapping, Eqs.","(REF ) and (REF ), is applied to the integrals of Eq.","(REF ), we get $\\nonumber \\int _{-\\infty }^{\\infty }v^2\\rho (x,v,t) dv=\\frac{1}{\\beta m}\\left(1+\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k(x)\\right)p(x,t),$ where the operators $\\hat{I}_k$ are given by $\\hat{I}_k(x)=\\frac{2}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }u^2du e^{-u^2}\\hat{\\omega }_k(x,u);$ the right-hand side integral of Eq.","(REF ) results in $\\nonumber \\frac{\\gamma }{\\beta m^2}\\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\rho (x,v,t)dv&=&\\cr \\frac{\\gamma }{\\sqrt{\\pi }m}\\sum _{k=0}^{\\infty }t_0^k\\int _{-\\infty }^{\\infty }due^{-u^2}\\partial _uu\\hat{\\eta }_k(x,u)j(x,t)&=&\\frac{j(x,t)}{t_0}$ after integrating by parts and using the normalization relation (REF ).","Then, instead of the evolution equation for $j$ , Eq.","(REF ) has to be understood as an expression relating $j$ and $p$ , $(1+t_0\\partial _t)j(x,t)&=&-D_0\\bigg [e^{-\\beta U(x)}\\partial _x e^{\\beta U(x)}\\hspace{50.58878pt}\\cr &&\\hspace{21.68121pt}+\\partial _x\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k(x)\\bigg ]p(x,t),$ and the flux $j$ , as well as its time derivative, is expressed using $p$ according to this equation.","The term $t_0\\partial _t$ acts now on $p$ as the 1-st order correction in $(1+t_0\\partial _t)^{-1}p$ after completing inversion and commutation with the spatial operators in the square brackets of Eq.","(REF ).","Then $\\partial _tp$ is systematically replaced by $-\\partial _x j$ , contributing to the next corrections in the higher orders of $t_0$ .","The result is a formula for $j$ expressed by some purely spatial operator acting on $p$ , $j(x,t)=-D_0 e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]\\partial _x e^{\\beta U(x)}p(x,t),$ where the operators $\\hat{Z}_k$ are related to $\\hat{I}_k$ , Eq.","(REF ).","Using this form, we are able to write the final mapped equation explicitly after combination with Eq.","(REF ), $\\partial _tp&=&D_0\\partial _x e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k\\right]\\partial _x e^{\\beta U(x)}p\\cr &=&\\sum _{k=0}^{\\infty }t_0^k\\hat{Q}_kp=\\hat{Q}p,$ which is the Smoluchowski equation (REF ) in the zero-th order, extended by a series of mass dependent corrections in $t_0$ .","The operators $\\hat{Q}_k$ are introduced to simplify notation in the following calculations.","Now the operators $\\hat{Z}_k$ can be expressed explicitly using $\\hat{I}_k$ .","Expanding $(1+t_0\\partial _t)^{-1}p$ in $t_0$ and applying Eq.","(REF ), we have $\\nonumber (1+t_0\\partial _t)^{-1}p=(1-t_0\\hat{Q}+...)p=\\Big (1+\\sum _{k=0}^{\\infty }t_0^{k+1}\\hat{Q}_k\\Big )^{-1}\\hspace{-7.22743pt}p,$ which is to be used in the operator equation, $&&\\hspace{-14.45377pt}e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k\\right]\\partial _xe^{\\beta U(x)}=\\bigg [e^{-\\beta U(x)}\\partial _x e^{\\beta U(x)}\\cr &&\\hspace{43.36243pt}+\\partial _x\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k\\bigg ]\\Big (1+\\sum _{k=0}^{\\infty }t_0^{k+1}\\hat{Q}_k\\Big )^{-1},$ obtained after comparison of Eqs.","(REF ) and (REF ).","Expanding in $t_0$ and comparing the coefficients of the same powers of $t_0$ , we get a sequence of relations fixing $\\hat{Z}_k$ , $e^{-\\beta U}\\hat{Z}_1\\partial _x e^{\\beta U}&=&\\partial _x \\hat{I}_1-e^{-\\beta U}\\partial _x e^{\\beta U}\\hat{Q}_0,\\cr e^{-\\beta U}\\hat{Z}_2\\partial _x e^{\\beta U}&=&\\partial _x \\hat{I}_2-\\partial _x \\hat{I}_1\\hat{Q}_0\\cr &&+e^{-\\beta U}\\partial _x e^{\\beta U}(\\hat{Q}_0^2-\\hat{Q}_1),\\cr .&.&.$ Finally, we derive the recurrence scheme, generating the operators $\\hat{\\omega }_k$ , determining $\\hat{I}_k$ , Eq.","(REF ), and thus also $\\hat{Z}_k$ or $\\hat{Q}_k$ , Eqs.","(REF ), (REF ).","Notice that the $\\hat{\\eta }_k$ do not directly enter the mapped equation (REF ), but they are necessary in the recurrence formulas for $\\hat{\\omega }_k$ .","The recurrence scheme for $\\hat{\\omega }_k$ is defined by Eq.","(REF ).","This equation has to be satisfied for any $p$ and $j$ , solving the mapped problem, but these quantities, although considered before as formally independent, are related by Eq.","(REF ).","To obtain an equation for operators, $j$ has to be expressed by $p$ .","Applying the relation (REF ) and Eq.","(REF ) for $\\partial _tp$ , we get $&&\\hspace{-14.45377pt}\\sum _{n=0}^{\\infty }t_0^n\\bigg [D_0\\Big (\\hat{\\omega }_n\\partial _x-2u^2\\partial _x\\hat{\\eta }_n+\\beta U^{\\prime }e^{u^2}\\partial _uue^{-u^2}\\hat{\\eta }_n\\Big )e^{-\\beta U(x)}\\cr &&\\hspace{-7.22743pt}\\times \\sum _{l=0}^{\\infty }t_0^l\\hat{Z}_l\\partial _xe^{\\beta U(x)}-\\frac{1}{2t_0}e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_n\\bigg ]p=0,$ valid for any function $p(x,t)$ ; we have $\\hat{Z}_0=1$ .","To lowest order, $t_0^{-1}$ , only the term $e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_0=0$ .","It is satisfied by $\\hat{\\omega }_0=1$ , the only solution nondiverging at $u\\rightarrow \\pm \\infty $ and also satisfying the normalization, Eq.","(REF ).","In the higher orders, we derive the recurrence relation, $e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_{n+1}&=&2D_0\\sum _{k=0}^n\\Big (\\hat{\\omega }_k\\partial _x-2u^2\\partial _x\\hat{\\eta }_k+\\beta U^{\\prime }(x)\\cr &&\\hspace{-57.81621pt}\\times e^{u^2}\\partial _uu e^{-u^2}\\hat{\\eta }_k\\Big )e^{-\\beta U(x)}\\hat{Z}_{n-k}\\partial _xe^{\\beta U(x)}.$ Calculation of the $\\hat{\\omega }_{n+1}$ requires us to know the $\\hat{\\eta }_k$ up to $k=n$ .","They are generated from Eq.","(REF ).","Again, $j$ , as well as $\\partial _t j$ , have to be expressed by $p$ using the relations (REF ) and (REF ).","Then Eq.","(REF ) becomes $&&\\sum _{n=0}^{\\infty }t_0^{n-1/2}\\bigg [e^{u^2}\\partial _u e^{-u^2}\\partial _u u\\hat{\\eta }_n e^{-\\beta U}\\sum _{k=0}^{\\infty }t_0^k\\hat{Z}_k\\partial _xe^{\\beta U}\\cr &&-2D_0t_0u\\hat{\\eta }_ne^{-\\beta U}\\sum _{k=0}^{\\infty }t_0^k\\hat{Z}_k\\partial _xe^{\\beta U}\\partial _xe^{-\\beta U}\\sum _{l=0}^{\\infty }t_0^l\\hat{Z}_l\\partial _xe^{\\beta U}\\cr &&+\\big (2u\\partial _x-\\beta U^{\\prime }e^{u^2}\\partial _u e^{-u^2}\\big )\\hat{\\omega }_n\\bigg ]p=0$ valid for any function $p(x,t)$ .","In lowest order, $t_0^{-1/2}$ , $e^{u^2}\\partial _ue^{-u^2}\\partial _u u\\hat{\\eta }_0e^{-\\beta U}\\partial _xe^{\\beta U}&=&-2u\\partial _x+\\beta U^{\\prime }e^{u^2}\\partial _ue^{-u^2}\\cr =-2u(\\partial _x+\\beta U^{\\prime })&=&-2ue^{-\\beta U}\\partial _x e^{\\beta U},$ where we have used $\\hat{\\omega }_0=1$ and $\\hat{Z}_0=1$ .","After the first integration, we have $\\partial _uu\\hat{\\eta }_0=-e^{u^2}\\int 2udue^{-u^2}=1+\\hat{C}_1e^{u^2};$ the integration constant $\\hat{C}_1=0$ provides convergence as $u\\rightarrow \\pm \\infty $ .","The next integration gives $\\hat{\\eta }_0=1+(1/u)\\hat{C}_0$ ; $\\hat{C}_0=0$ .","This calculation validates our heuristic formula (REF ) in the zero-th order approximation.","In the higher orders, Eq.","(REF ) generates the relations $&&\\hspace{-14.45377pt}e^{u^2}\\partial _ue^{-u^2}\\partial _u u\\hat{\\eta }_ne^{-\\beta U}\\partial _xe^{\\beta U}=(\\beta U^{\\prime }e^{u^2}\\partial _ue^{-u^2}-2u\\partial _x)\\hat{\\omega }_n\\cr &&+2D_0\\sum _{k,l=0}^{k+l0$ depend on $u$ .","The recurrence procedure starts from $\\hat{\\omega }_0=1$ and $\\hat{Z}_0=1$ .","Calculation of the next order correction requires first expressing the $\\hat{\\eta }_n$ according to Eq.","(REF ), or (REF ) for $n=0$ , as shown above.","Then $\\hat{\\omega }_{n+1}$ is derived from Eq.","(REF ), $\\hat{I}_{n+1}$ integrated according to Eq.","(REF ) and finally $\\hat{Z}_{n+1}$ expressed from Eq.","(REF ).","To demonstrate the procedure, we derive the first order correction, $\\hat{Z}_1$ .","We use already calculated $\\hat{\\eta }_0=1$ .","For $n=0$ , Eq.","(REF ) becomes $e^{u^2}\\partial _u e^{-u^2}\\partial _u\\hat{\\omega }_1=2D_0(1-2u^2)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}.$ After the first integration, $\\partial _u\\hat{\\omega }_1=2D_0e^{u^2}\\left(ue^{-u^2}+C_1\\right)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)},$ the integration constant $C_1=0$ , to provide convergence for $u\\rightarrow \\pm \\infty $ .","The integration constant $C_0$ after the next integration is fixed to satisfy the normalization, Eq.","(REF ), $\\int _{-\\infty }^{\\infty }due^{-u^2}D_0(u^2+C_0)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}=0,$ hence $C_0=-1/2$ and $\\hat{\\omega }_1=D_0(u^2-1/2)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}.$ For $U(x)=0$ , we recover the corresponding formula in Eq.","(REF ).","Integration over $u$ in Eq.","(REF ) results in $\\hat{I}_1=D_0\\exp [-\\beta U(x)]\\partial _x^2\\exp [\\beta U(x)]$ , giving finally $\\hat{Z}_1=D_0\\left(e^{\\beta U}\\partial _x e^{-\\beta U}\\partial _x-\\partial _x e^{\\beta U}\\partial _xe^{-\\beta U}\\right)=D_0\\beta U^{\\prime \\prime }(x)$ from Eq.","(REF ).","In the higher orders, $\\hat{\\eta }_n$ , $\\hat{\\omega }_{n+1}$ are calculated according to Eqs.","(REF ) and (REF ).","The integration constants after double integration have to provide convergence for $u\\rightarrow \\pm \\infty $ , requiring the operators to be even in $u$ , and the normalization, Eq.","(REF ).","The condition (REF ) for $\\hat{\\eta }_n$ is satisfied automatically; it serves as a check on the computation.","The derivation is tedious, and we present only the results in second order, $\\hat{\\eta }_1&=&D_0(u^2/3-1/2)e^{-\\beta U(x)}\\partial _x^2e^{\\beta U(x)},\\cr \\hat{\\omega }_2&=&\\frac{D_0^2}{2}e^{-\\beta U(x)}\\Big [\\Big (\\frac{u^4}{3}-u^2+\\frac{1}{4}\\Big )\\partial _x^3\\cr &&+\\Big (u^2-\\frac{1}{2}\\Big )\\big (4\\beta U^{\\prime \\prime }(x)\\partial _x+3\\beta U^{(3)}(x)\\big )\\Big ]\\partial _x e^{\\beta U(x)},\\cr \\hat{Z}_2&=&\\frac{D_0^2}{2}\\Big [4\\big (\\beta U^{\\prime \\prime }(x)\\big )^2-\\beta ^2U^{\\prime }(x)U^{(3)}(x)+\\beta U^{(4)}(x)\\cr &&\\hspace{21.68121pt}+3\\beta U^{(3)}(x)\\partial _x\\Big ].$ Again, the formulas (REF ) for $U(x)=0$ are recovered.","There are no contributions to $\\hat{Z}_n$ in this case, too, as expected according to the analysis in the previous Section.","Also, linear potentials, $U(x)=-Fx$ , have no effect on validity of the uncorrected Smoluchowski equation (REF ).","The particle driven by a constant force $F$ move asymptotically with constant mean velocity $v_0=F/\\gamma $ and the distribution $p(y,t)$ in coordinate $y$ , shifted by the drift, $y=x-v_0t$ , is again Gaussian as in the case of no potential.","The situation becomes different if the driving force $F(x)$ is not constant.","If the mass $m$ or the time of the thermalization $t_0=m/\\gamma $ is small, but nonzero, the particle appearing at a new position $x$ has to accommodate to the new local mean velocity.","On the other hand, it carries some mean momentum from its previous position and needs some time to change it.","Meanwhile it slips to some other position than predicted by purely stochastic dynamics due to its inertia, or non-zero mass.","The effects of such slipping are indicated by the corrections $\\hat{Z}_n$ of the Smoluchowski equation and they are nonzero for potentials with nonzero $U^{\\prime \\prime }(x)$ , or higher derivatives.","As seen from Eq.","(REF ), the $\\hat{Z}_n$ are not only functions, but operators, containing $\\partial _x$ in the higher orders.","So the mapped equation (REF ) has exactly the same structure as the mapped equations for diffusion [16], or biased diffusion [15], [22], [26].","Being inspired by these works, Eq.","(REF ) can be simplified by replacing the correction operators $D_0[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k]$ by a function $D(x)$ , a spatially dependent effective diffusion coefficient, $\\partial _tp(x,t)=\\partial _xe^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x,t),$ which becomes valid in the limit of stationary flow, i.e.","when the spatial density $p$ and the flux $j$ change very slowly, $p(x,t)\\rightarrow p(x)$ .","Due to mass conservation, Eq.","(REF ), the flux $j(x,t)=j$ is constant (but nonzero) in $x$ as well.","If expressed from Eq.","(REF ), $j=-e^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x),$ the function $\\partial _x(\\exp [\\beta U(x)]p(x))=-j\\exp [\\beta U(x)]/D(x)$ is dependent only on the system; $m$ , $\\gamma $ and the potential $U(x)$ , for any stationary solution $p(x)$ .","So we can substitute for it in Eq.","(REF ), $j=D_0e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]e^{\\beta U(x)}\\frac{j}{D(x)},$ valid for stationary flow, and calculate $D(x)$ unambiguously from the expansion of the corrections $\\hat{Z}_k$ , $\\frac{D_0}{D(x)}=e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]^{-1}e^{\\beta U(x)},$ as a series in $t_0$ , $D(x)/D_0&=&1+D_0t_0\\beta U^{\\prime \\prime }+(D_0t_0)^2\\Big [2(\\beta U^{\\prime \\prime })^2\\cr &&\\hspace{14.45377pt}+\\beta ^2U^{\\prime }U^{(3)}+\\beta U^{(4)}/2\\Big ]+...\\ .$ The next simplification is that of neglecting all the derivatives but $U^{\\prime \\prime }(x)$ , i.e.","approximating the real $U(x)$ locally by a quadratic potential.","In this case, the expansion (REF ) can be summed up to infinity, $D(x)/D_0&\\simeq &\\sum _{n=0}^{\\infty }\\frac{(2n)!}{n!(n+1)!","}\\left[D_0t_0\\beta U^{\\prime \\prime }(x)\\right]^n\\cr &&=\\frac{1-\\sqrt{1-4D_0t_0\\beta U^{\\prime \\prime }(x)}}{2D_0t_0\\beta U^{\\prime \\prime }(x)},$ the proof is given in the Appendix B.","The formula works for $D_0t_0\\beta U^{\\prime \\prime }(x)=mU^{\\prime \\prime }(x)/\\gamma ^2<1/4$ , which is the condition for non-oscillatory movement of a particle in a quadratic well with friction, the damped harmonic oscillator.","If $U^{\\prime \\prime }(x)=\\kappa $ is constant, the trajectory of a particle averaged over the stochastic force is governed by $m\\langle \\ddot{x}\\rangle +\\gamma \\langle \\dot{x}\\rangle +\\kappa \\langle x\\rangle =0$ from Eq.","(REF ).","The particular solutions are $\\langle x(t)\\rangle =\\exp (\\alpha t)$ with $\\alpha =-(\\gamma \\pm \\sqrt{\\gamma ^2-4m\\kappa })/2m$ .","Requiring $\\alpha $ to be a real number gives the same condition.","This simple example demonstrates restriction of the theory presented to nonoscillatory movement of the particle in potential wells on its way along a 1D channel.","A small mass $m$ is expected, to enable the friction quickly to damp the momentum of a particle; i.e.","to have the relaxation in the velocity faster than in the real space coordinate $x$ .","Figure: Plot of the effective diffusion coefficient D(x)D(x) dependenton the mass of the particle mm and friction γ\\gamma according toEq.","(), valid if the derivatives of the potential higherthan U '' (x)U^{\\prime \\prime }(x) are neglected.A more detailed insight to the restrictions of the dimensional reduction of the phase space controlled by the mass $m$ can be obtained by comparison of the Green's function (GF) of the mapped equation (REF ) with $D(x)$ given by Eq.","(REF ) and GF of the Kramers equation (REF ) for the damped harmonic oscillator, $U(x)=\\kappa x^2/2=m\\omega _0^2x^2/2$ , which is exactly solvable.","The solution $G=G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })$ of the equation $\\Big (\\partial _t+v\\partial _x-\\omega _0^2x\\partial _v-\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\Big ) G\\cr =\\delta (x-x^{\\prime })\\delta (v-v^{\\prime })\\delta (t-t^{\\prime })\\hspace{28.45274pt}$ reads [1] $G=\\frac{(s_1-s_2)e^{\\gamma t/m}}{2\\pi \\sqrt{ab-h^2}}\\exp \\Big [-\\Big (a(\\xi -\\xi _0)^2+b(\\eta -\\eta _0)^2\\cr +2h(\\xi -\\xi _0)(\\eta -\\eta _0)\\Big )\\big /2(ab-h^2)\\Big ],\\hspace{28.45274pt}$ where $s_{1,2}=-\\frac{\\gamma }{2m}\\pm \\sqrt{\\frac{\\gamma ^2}{4m^2}-\\omega _0^2},$ $\\xi =(s_1x-v)e^{-s_2t},\\hspace{28.45274pt}\\xi _0=(s_1x^{\\prime }-v^{\\prime }),\\hspace{36.98866pt}\\cr \\eta =(s_2x-v)e^{-s_1t},\\hspace{28.45274pt}\\eta _0=(s_2x^{\\prime }-v^{\\prime }),\\hspace{36.98866pt}\\cr a=\\frac{\\gamma }{\\beta m^2s_1}\\Big (1-e^{-2s_1t}\\Big ),\\ \\ b=\\frac{\\gamma }{\\beta m^2s_2}\\Big (1-e^{-2s_2t}\\Big ),\\cr h=\\frac{2}{\\beta m}\\Big (1-e^{\\gamma t/m}\\Big ).\\hspace{56.9055pt}$ Similar to the case of no potential in the Section III, let us suppose that a thermalized particle (equilibrated in velocity) was inserted at a position $x_0$ at time $t^{\\prime }=0$ , $\\rho _0(x^{\\prime },v^{\\prime })=\\sqrt{\\frac{\\beta m}{2\\pi }}\\delta (x^{\\prime }-x_0)e^{-\\beta mv^{\\prime 2}/2}.$ After integration over $v^{\\prime }$ and $v$ , we get the corresponding spatial density $p(x,t)=\\int _{-\\infty }^{\\infty }G(x,v,t;x^{\\prime }v^{\\prime },0)\\rho _0(x^{\\prime },v^{\\prime })dx^{\\prime }dv^{\\prime }dv=\\cr \\sqrt{\\frac{\\beta m}{2\\pi Z}}\\omega _0(s_1-s_2)\\exp \\Big [-\\frac{\\beta m\\omega _0^2}{2 Z}\\hspace{42.67912pt}\\cr \\times \\Big (s_1(x-x_0e^{s_2t}) -s_2(x-x_0e^{s_1t})\\Big )^2\\Big ];\\hspace{14.22636pt}$ $\\nonumber Z=\\Big [s_1(1+e^{s_2t})-s_2(1+e^{s_1t})\\Big ]\\Big [s_1(1-e^{s_2t})-s_2(1-e^{s_1t})\\Big ].$ On the other hand, the coefficient $D(x)$ , Eq.","(REF ), becomes constant for the quadratic potential, $D(x)=\\frac{\\gamma /2m-\\sqrt{\\gamma ^2/4m^2-\\omega _0^2}}{\\beta m\\omega _0^2}=\\frac{-s_1}{\\beta m\\omega _0^2},$ and GF of the corresponding mapped equation (REF ), $\\Big (\\partial _t-\\partial _xD(x)e^{-\\beta m\\omega _0^2x^2/2}\\partial _xe^{\\beta m\\omega _0^2x^2/2}\\Big )g(x,t;x_0,t_0)\\cr =\\delta (x-x_0)\\delta (t-t_0),\\hspace{56.9055pt}$ can be easily found by a calculation similar to the derivation of Eq.","(REF ), Appendix A.","The result, $g(x,t;x_0,0)=\\sqrt{\\frac{\\beta m}{2\\pi (1-e^{2s_1t})}}\\omega _0\\exp \\Big [-\\frac{1}{2}\\beta m\\omega _0^2\\cr \\times \\big (x-x_0e^{s_1t}\\big )^2\\big /\\big (1-e^{2s_1t}\\big )\\Big ],\\hspace{14.22636pt}$ describes evolution of the real space density of a thermalized particle inserted at $x_0$ , too, and can be directly compared with the formula (REF ).","First, let us notice that in comparison with Eq.","(REF ), the exponential $e^{s_2t}$ disappeared from the formula (REF ).","The root $s_2\\simeq -\\gamma /m$ for $m\\rightarrow 0$ makes $e^{s_2t}\\simeq e^{-\\gamma t/m}$ the term essentially singular in $m$ and so invisible for the recurrence procedure, which works with the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ expanded in $m$ .","Using our argumentation from the Section III, $e^{s_2t}$ represents the \"transients\" neglected by the mapping.","On the other hand, $e^{s_1t}\\simeq e^{-m\\omega _0^2t/\\gamma }$ is regular in $m$ small, representing the contribution of the low-lying states, retained by the method.","Next, let us stress that the equation (REF ) with $D(x)$ expressed by the expansion (REF ) was derived in the limit of the stationary flow; for the net flux almost constant, which is not the case of the process described by the Eqs.","(REF ) and (REF ).","Nevertheless, using the approximations $\\nonumber s_1-s_2\\big (1\\pm e^{s_1t}\\big )&\\simeq & (s_1-s_2)\\big (1\\pm e^{s_1t}\\big ),\\cr s_1x-s_2\\big (x-x_0e^{s_1t}\\big )&\\simeq & (s_1-s_2)\\big (x-x_0e^{s_1t}\\big ),$ applicable for $e^{s_1t}\\ll 1$ , the regularized formula (REF ) (with $e^{s_2t}$ neglected) becomes finally Eq.","(REF ); i.e.","it represents correctly the asymptotic behavior of the spatial density $p(x,t)$ for large time $t$ .","If $m$ approaches $\\gamma /2\\omega _0$ , the transients contributing by $e^{s_2t}$ , neglected by the mapping, become important.","In the oscillating regime, $s_{1,2}$ are complex numbers and both are necessary for expressing the real density $p(x,t)$ in Eq.","(REF ).","The mapping which splits the Hilbert space to the retained low-lying states and the neglected transients, controlled by $m$ small, loses its justification and the method stops working.","Mapping in this region requires a different method to be applied.","It will be an object of our study in the future."],["V. Conclusion","Although modeling of transport in confined systems is often based on study of the Langevin equation, Eq.","(REF ) in the simplest 1D case, the solutions necessary in practical applications are often accessible only in two limits: either the friction $\\gamma \\rightarrow 0$ , when the particles obey Newtonian dynamics, or the mass $m\\rightarrow 0$ , which corresponds to stochastic dynamics.","Any solution in the region between these limits requires working in phase space, which makes the problem much more complicated.","The present paper shows how to describe the region of finite $m/\\gamma $ while still working in real space, as in the case of stochastic dynamics.","The equation governing evolution of the spatial density $p(x,t)$ is the Smoluchowski equation, corresponding to the limit of a massless particle, extended by a series of corrections in powers of $t_0=m/\\gamma $ ; $t_0$ can be interpreted as the typical time of thermalizing of the particle's initial velocity by the stochastic force.","In general, the extended Smoluchowski equation has the form of Eq.","(REF ), $D_0=1/\\gamma \\beta $ denotes the diffusion constant and the operators $\\hat{Z}_k$ are systematically derived within the recurrence procedure presented in the Sect.","IV.","In the limit of stationary flow, i.e.","when the flux is almost constant but nonzero in time and space, this equation can be simplified to Eq.","(REF ), where the effective diffusion coefficient $D(x)$ is calculated unambiguously from the operators $\\hat{Z}_k$ , Eq.","(REF ).","In the simplest approximation, when all the derivatives of the potential higher than $U^{\\prime \\prime }(x)$ are neglected, the series of corrections can be summed up to infinity, giving the formula for $D(x)$ in a closed form, Eq.","(REF ), described in Fig.","1.","Then the equation describes stationary flow in a quadratic potential.","The theory works while $4mU^{\\prime \\prime }(x)<\\gamma ^2$ , until the averaged trajectory of a single particle is not oscillatory in the potential wells along the 1D channel.","The mapping in the oscillatory regime requires the next analysis, which will be done in the future.","Technically, the paper demonstrates that the projection technique developed for mapping of diffusion in 2D (3D) channels with varying cross section [13], [14], [16], can be adapted for the dimensional reduction of a process described by an evolution equation of a different type than the diffusion or Smoluchowski equation.","The method has been modified significantly; $m/\\gamma $ had to be confirmed as the small parameter controlling the expansion of the corrections $\\hat{Z}_k$ , as well as the operators of the backward mapping, $\\hat{\\omega }$ and $\\hat{\\eta }$ .","In contrast to diffusion, the flux $j(x,t)$ is handled here as a quantity independent of the density $p(x,t)$ .","Thus the recurrence procedure, calculating expansions of the correction operators $\\hat{Z}_k$ and the operators $\\hat{\\omega }$ , $\\hat{\\eta }$ , is in principle the result of combination of three equations, Eqs.","(REF ), (REF ) and also (REF ), with the relation of the backward mapping, Eq.","(REF ).","It is worthwhile to notice that including the mass dependent corrections to the Smoluchowski equation results in the equations (REF ) or (REF ), which are of the same form as the comparable equations obtained from the mapping of diffusion in channels with varying cross section.","On the other hand, the effective coefficient $D(x)$ (REF ) has a different symmetry than the similar formulas extending the Fick-Jacobs equation [16], [17] for confined diffusion.","Also $D(x)$ can be greater than 1 here (see Fig.","1); i.e.","the quasi stationary flux is accelerated when passing through a shallow potential well, depending on the nonzero mass of the particles.","These interesting properties could be observed in simulations similar to that verifying $D(x)$ in the extended Fick-Jacobs equation [27], [28].","The effects of slipping of the particles diffusing under a nonconstant force $F(x)$ due to their inertia, as described in Sect.","IV, might also influence the interesting phenomena in the micro and nano world, such as Brownian pumps [29], [30], rectification of the flux in quasi 1D structures [31], stochastic resonance [20], [32], [33], [34], or the negative mobility [35].","Study of such applications of the theory presented is expected in the future."],["Acknowledgments","Support from VEGA grant No.","2/0049/12 and CE SAS QUTE project is gratefully acknowledged.","P.K.","also thanks CIMS, New York University for kind hospitality."],["Appendix A: Exact solution","The Green's function (REF ) solving the FP equation with zero potential, Eq.","(REF ), is calculated here.","First we introduce the scaled coordinates $\\xi $ , $u$ , $\\tau $ according to Eqs.","(REF ) and define the function $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })$ , $G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })=e^{-u^2/2}\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })e^{u^{\\prime 2}/2},$ satisfying the transformed equation (REF ), $\\left[\\partial _{\\tau }+u\\partial _{\\xi }-\\partial _u^2+u^2-1\\right]\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })\\hspace{36.135pt}\\cr =\\frac{\\gamma \\beta }{4}e^{(u^2-u^{\\prime 2})/2}\\delta (\\xi -\\xi ^{\\prime })\\delta (u-u^{\\prime })\\delta (\\tau -\\tau ^{\\prime }),\\ $ the exponential factor becomes unity due to $\\delta (u-u^{\\prime })$ .","After the Fourier transform in $\\xi $ and $\\tau $ , $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })=\\int \\frac{dkd\\nu }{4\\pi ^2}e^{ik(\\xi -\\xi ^{\\prime })-i\\nu (\\tau -\\tau ^{\\prime })}\\Gamma _{k,\\nu }(u;u^{\\prime }),$ and shifting the velocities by $ik/2$ , $w=u+ik/2$ and $w^{\\prime }=u^{\\prime }+ik/2$ , the equation $\\left[-i\\nu -\\partial _w^2+w^2-1+k^2/4\\right]\\Gamma _{k,\\nu }(w;w^{\\prime })=\\frac{\\gamma \\beta }{4}\\delta (u-u^{\\prime })$ becomes solvable if $\\Gamma _{k,\\nu }(w;w^{\\prime })$ is expressed in the basis set of the linear harmonic oscillator $\\psi _n(w)$ , $\\Gamma _{k,\\nu }(w;w^{\\prime })=\\sum _{n=0}^{\\infty }\\Gamma _n(k,\\nu )\\psi _n(w)\\psi _n^*(w^{\\prime }).$ The eigenfunctions $\\psi _n(w)$ satisfy $\\left(-\\partial _w^2+w^2\\right)\\psi _n(w)=\\lambda _n\\psi _n(w)=(2n+1)\\psi _n(w)$ and we use the integral representation of the Hermite polynomials $H_n(w)$ [36] $\\psi _n(w)&=&\\frac{1}{\\@root 4 \\of {\\pi }\\sqrt{2^n n!","}}H_n(w)e^{-w^2/2}\\cr &=&\\sqrt{\\frac{2^n}{n!\\pi ^{3/2}}}e^{-w^2/2}\\int _{-\\infty }^{\\infty }(w+ir)^ne^{-r^2}dr\\hspace{14.45377pt}$ in the next calculation.","Using the transformations above, we find $\\Gamma _n(k,\\nu )=\\frac{\\gamma \\beta /4}{-i\\nu +\\lambda _n-1+k^2/4}.$ Applying it in the formulas (REF ) and (REF ), we integrate the last one over $\\nu $ in the complex plane, $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })&&\\hspace{-14.45377pt}=\\frac{\\Theta (\\tau -\\tau ^{\\prime })}{8\\pi D_0}\\int _{-\\infty }^{\\infty }dke^{ik(\\xi -\\xi ^{\\prime })-k^2(\\tau -\\tau ^{\\prime })/4}\\cr &&\\times \\sum _{n=0}^{\\infty }e^{-2n(\\tau -\\tau ^{\\prime })}\\psi _n(w)\\psi _n^*(w^{\\prime }),$ $\\Theta (x)$ denotes the Heaviside unit step function and $D_0=1/\\gamma \\beta $ is the diffusion constant.","Now the integral relation (REF ) is used for $\\psi _n(w)$ and $\\psi _n^*(w^{\\prime })$ and the summation over $n$ can be readily completed.","Finally, the straightforward triple integration over $k$ , $r$ , $r^{\\prime }$ is performed and using the transformation (REF ) results in the formula (REF )."],["Appendix B: Quadratic approximation","Derivation of the formula (REF ) for the effective diffusion coefficient $D(x)$ with all the derivatives higher than $U\"(x)$ neglected is presented here.","This approximation corresponds to local replacing of the potential by a parabola, $U(x)\\simeq \\kappa (x-x_0)^2/2+U_0$ , where $\\kappa ,\\ x_0$ and $U_0$ are fitting parameters.","First we simplify Eq.","(REF ).","For quadratic potential, the right hand side can be rewritten as $\\nonumber e^{-\\beta U(x)}\\left(1+t_0\\hat{Z}\\right)^{-1}e^{\\beta U(x)}=\\left(1+t_0 e^{-\\beta U}\\hat{Z}(x)e^{\\beta U}\\right)^{-1},$ $t_0\\hat{Z}=\\sum _{n=1}^{\\infty }t_0\\hat{Z}_n$ ; the difference contains only the higher derivatives of $U(x)$ , which are zero.","Hence $D(x)/D_0=1+e^{-\\beta U(x)}\\sum _{n=1}^{\\infty }t_0^n\\hat{Z}_ne^{\\beta U(x)}.$ The formulas for $D(x)$ have been derived considering stationary flow; $j(x,t)=j$ is constant.","It simplifies the relation (REF ); $\\partial _tj=0$ .","Thus the right hand side represents stationary flux, which can be directly compared with Eq.","(REF ), giving a much simpler relation between $\\hat{Z}_n$ and $\\hat{I}_n$ than Eq.","(REF ), $e^{-\\beta U(x)}\\hat{Z}_n(x)\\partial _xe^{\\beta U(x)}p(x)=\\partial _x\\hat{I}_n(x)p(x)$ for any stationary solution $p(x)$ .","Calculation of the coefficients of $D(x)$ according to Eq.","(REF ) requires us to take $\\partial _x\\exp [\\beta U(x)]p(x)=\\exp [\\beta U(x)]$ , hence finally $e^{-\\beta U(x)}\\hat{Z}_ne^{\\beta U(x)}&=&\\frac{2}{\\sqrt{\\pi }}\\partial _x\\int _{-\\infty }^{\\infty }u^2due^{-u^2}\\hspace{57.81621pt}\\cr &&\\times \\hat{\\omega }_n(x,u)e^{-\\beta U(x)}\\int dx e^{\\beta U(x)}$ after application of Eq.","(REF ).","Before writing the explicit formulas for $\\hat{\\omega }_n$ for the quadratic potential, we define the polynomials $P_n(u)&=&\\sum _{k=0}^{n}\\frac{(-1)^{n-k}2^{2k-n}}{(2k)!(n-k)!","}u^{2k},\\cr Q_n(u)&=&\\sum _{k=0}^{n}\\frac{(-1)^{n-k}2^{2k-n}}{(2k+1)!(n-k)!","}u^{2k},$ $n=1,2, ...$ , coming from the expansions of $\\hat{\\omega }_n$ and $\\hat{\\eta }_n$ for zero potential in $t_0$ , Eqs.","(REF ).","The first few polynomials are visible in the round brackets of Eq.","(REF ).","One can check by direct integration that $\\int _{-\\infty }^{\\infty }Q_n(u)u^2e^{-u^2}du&=&\\frac{\\sqrt{\\pi }}{2^{n+1}}\\sum _{k=0}^n\\frac{(-1)^{k-n}}{k!(n-k)!","}=0,\\cr \\int _{-\\infty }^{\\infty }P_n(u)u^2e^{-u^2}du&=&\\frac{\\sqrt{\\pi }}{2}\\delta _{n,1},$ corresponding to the normalization of $\\hat{\\eta }_n$ , Eq.","(REF ), and the relations (REF ), (REF ), proving no correction to the Smoluchowski equation in the case $U(x)=0$ .","The operators $\\hat{\\omega }_n$ and $\\hat{\\eta }_n$ for the quadratic potential have the form $\\hat{\\omega }_n&=&e^{-\\beta U(x)}\\sum _{k=1}^nc_{n,k}P_k(u)\\left(\\beta U^{\\prime \\prime }\\right)^{n-k}\\partial _x^{2k}e^{\\beta U(x)},\\hspace{21.68121pt}\\cr \\hat{\\eta }_n&=&e^{-\\beta U(x)}\\sum _{k=1}^nc_{n,k}Q_k(u)\\left(\\beta U^{\\prime \\prime }\\right)^{n-k}\\partial _x^{2k}e^{\\beta U(x)},$ with the coefficients $c_{n,k}=D_0^n\\frac{2k\\ (2n-1)!}{(n-k)!(n+k)!","}.$ Due to the integrals, Eq.","(REF ), only the first terms with $P_1(u)$ in Eq.","(REF ) contribute to the expansion of $D(x)$ , Eq.","(REF ).","Then the functions become $e^{-\\beta U(x)}\\hat{Z}_ne^{\\beta U(x)}&=&c_{n,1}\\partial _xe^{-\\beta U(x)}\\left(\\beta U^{\\prime \\prime }\\right)^{n-1}\\partial _xe^{\\beta U(x)}\\cr &=&\\frac{2\\ (2n-1)!}{(n-1)!(n+1)!","}\\left(D_0\\beta U^{\\prime \\prime }\\right)^n,$ taking $U^{(3)}(x)=0$ into account.","Applied in Eq.","(REF ) it results in the expansion of $D(x)$ , Eq.","(REF ).","Finally, one has to verify that the formulas (REF ) satisfy the recurrence relations (REF ) and (REF ), acting on the function $p(x)=\\exp [-\\beta U(x)]\\int dx\\exp [\\beta U(x)]$ .","Although the equations simplify notably due to neglecting the derivatives higher than $U^{\\prime \\prime }(x)$ , we omit the details of this tedious but straightforward calculation."]],"string":"[\n [\n \"Phase space reduction of the one-dimensional Fokker-Planck (Kramers)\\n equation\"\n ],\n [\n \"Abstract A pointlike particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered.\",\n \"We present a rigorous mapping of the Fokker-Planck equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x.\",\n \"The result is the Smoluchowski equation, valid in the overdamped limit, m->0, with a series of corrections expanded in powers of m. They are determined unambiguously within the recurrence mapping procedure.\",\n \"The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.\"\n ],\n [\n \"I. Introduction\",\n \"The Brownian motion of a particle in a confined system represents an essential model used in description of stochastic transport through quasi one-dimensional (1D) systems, e.g.\",\n \"channels in nanomaterials, pores or fibers in biological structures.\",\n \"In a 1D system, the trajectory $x(t)$ of a Brownian particle is described by the Langevin equation $m\\\\ddot{x}+\\\\gamma \\\\dot{x}+\\\\partial _xU(x)=f(t)\\\\ .$ Here, $m$ denotes mass of the particle, $U(x)$ represents the driving potential, $\\\\gamma $ is an effective friction coefficient and $f(t)$ is the stochastic force, satisfying the usual conditions on averaged values, $\\\\left< f(t)\\\\right>=0$ and $\\\\left=2\\\\gamma k_BT\\\\delta (t-t^{\\\\prime })$ ; $T$ is the temperature and $k_B$ the Boltzmann constant.\",\n \"The corresponding phase space density $\\\\rho (x,v,t)$ of the particle satisfies the Fokker-Planck (FP), or Kramers (kinetic) equation, $\\\\left(\\\\partial _t+v\\\\partial _x-\\\\frac{U^{\\\\prime }(x)}{m}\\\\partial _v\\\\right)\\\\rho (x,v,t)&& \\\\cr && \\\\hspace{-108.405pt}=\\\\frac{\\\\gamma }{\\\\beta m^2}\\\\partial _ve^{-\\\\beta mv^2/2}\\\\partial _v e^{\\\\beta mv^2/2}\\\\rho (x,v,t),\\\\hspace{21.68121pt}$ where $x$ is the spatial coordinate, $v$ denotes its velocity and $\\\\beta =1/k_BT$ is the inverse temperature.\",\n \"Solutions of the Langevin equation, as well as the corresponding kinetic equation, are studied over almost a century [1], [2], [3].\",\n \"Still, motion of a particle in confined geometries represents usually a complicated problem, requiring next reductions of the used description.\",\n \"Due to simplicity, and often for relevance, mainly the overdamped limit is studied.\",\n \"The mass dependent term in Eq.\",\n \"(REF ), $m\\\\ddot{x}$ , is then considered negligible and the particle's spatial density $p(x,t)$ is governed by the Smoluchowski equation, $\\\\partial _tp(x,t)=D_0\\\\partial _xe^{-\\\\beta U(x)}\\\\partial _xe^{\\\\beta U(x)}p(x,t)\\\\ ,$ containing no information about the mass of the particle; $D_0$ denotes the diffusion constant.\",\n \"Instead of the full phase space, one works with only the spatial coordinate $x$ .\",\n \"Of course, the question is the price for this simplification, as well as possibility of also including properly inertia of the massive particles in the reduced (real space only) description of the Brownian dynamics.\",\n \"Recent studies showed its importance in understanding rectification of the transport in ratchets [4], or its influence on the effective diffusion coefficient [5], [6], [7] in a narrow channel.\",\n \"For our purpose to demonstrate the phase space reduction, we will deal here with only the 1D FP (Kramers) equation (REF ).\",\n \"The eq.\",\n \"(REF ) is reducible to the Smoluchowski eq.\",\n \"(REF ) in the limit $m\\\\rightarrow 0$ .\",\n \"The reduction procedure [8], [9], [10] is based on an instant thermalization of the particle's velocity after any move in the $x$ direction in the case of an infinitely small mass.\",\n \"The situation resembles derivation of the Fick-Jacobs (FJ) equation [11], [12], $\\\\partial _tp(x,t)=\\\\partial _x A(x)\\\\partial _x\\\\frac{p(x,t)}{A(x)},$ appearing as the result of the dimensional reduction of the diffusion equation in a 2D channel with varying cross section $A(x)$ , onto the longitudinal coordinate $x$ , if equilibration in the transverse ($y$ ) direction is instant.\",\n \"The function $p(x,t)$ denotes the linear (1D) density of the particle.\",\n \"Of course, as in reduction of diffusion to the FJ equation, that of (REF ) to (REF ) as $m\\\\rightarrow 0$ is a singular limit, and must be handled with care, but from the viewpoint of the reduction of (REF ), no such caveat is needed.\",\n \"Recently, an exact mapping procedure was proposed [13], [14], enabling us to also derive the corrections to the FJ equation (REF ), which are necessary, if the transverse equilibration is not instant.\",\n \"The key was to introduce anisotropy of the diffusion constant in the diffusion equation, $\\\\partial _t\\\\rho (x,y,t)=\\\\left(\\\\partial _x^2+\\\\frac{1}{\\\\epsilon }\\\\partial _y^2\\\\right)\\\\rho (x,y,t),$ governing the 2D (spatial) density $\\\\rho (x,y,t)$ .\",\n \"For $\\\\epsilon \\\\rightarrow 0$ , the infinitely fast transverse diffusion immediately flattens the $y$ profile of $\\\\rho (x,y,t)\\\\rightarrow \\\\rho (x,t)=p(x,t)/A(x)$ .\",\n \"Then integration of Eq.\",\n \"(REF ) over the cross section, together with the reflecting boundary conditions satisfied at the hard walls, results in the FJ eq.\",\n \"(REF ).\",\n \"In the case of a slower transverse diffusion, the mapping procedure generates a series of corrections to the FJ equation controlled by $\\\\epsilon >0$ , developing the $y$ profile of $\\\\rho (x,y,t)$ , which is already curved depending on the flux and geometry of the channel.\",\n \"The procedure was extended to mapping of diffusion in an external field $U(x,y)$ , e.g.\",\n \"for diffusion in a channel with soft walls [15], where the particle is kept near the $x$ axis by the parabolic potential, $\\\\beta U(x,y)=\\\\alpha (x)y^2$ ; $\\\\alpha (x)$ represents the varying stiffness of the walls.\",\n \"The equation to be mapped onto the $x$ coordinate is the 2D Smoluchowski equation, $\\\\left(\\\\partial _t-\\\\partial _xe^{-\\\\alpha (x)y^2}\\\\partial _xe^{\\\\alpha (x)y^2}\\\\right)\\\\rho (x,y,t)=\\\\hspace{57.81621pt}\\\\cr =\\\\frac{1}{\\\\epsilon }\\\\partial _ye^{-\\\\alpha (x)y^2}\\\\partial _ye^{\\\\alpha (x)y^2}\\\\rho (x,y,t).\\\\hspace{14.45377pt}$ Integrating over $y$ and applying the mapping scheme gives the mapped 1D equation in an extended Smoluchowski form, $\\\\partial _tp(x,t)=\\\\partial _xe^{-V(x)}\\\\left[1+\\\\epsilon \\\\hat{Z}(x,\\\\partial _x)\\\\right]\\\\partial _xe^{V(x)}p(x,t),$ governing the 1D density $p(x,t)$ , where $V(x)$ stands for an effective potential and $\\\\hat{Z}$ is the correction operator sought as an expansion in the small parameter $\\\\epsilon $ .\",\n \"The potential $V(x)$ and the coefficients of $\\\\hat{Z}$ depend on $\\\\alpha (x)$ and both are fixed unambiguously within the recurrence mapping procedure.\",\n \"The central idea of this paper is a conjecture that the FP eq.\",\n \"(REF ) can be reduced to an extended Smoluchowski-like 1D form, governing the spatial density $p(x,t)$ , after integration over $v$ and applying a similar mapping scheme.\",\n \"The velocity $v$ thus represents a \\\"transverse\\\" coordinate instead of $y$ , with the mass $m$ playing the role of the small parameter $\\\\epsilon $ .\",\n \"Then the Smoluchowski equation (REF ) should be obtained in the limit $m\\\\rightarrow 0$ [8], [9].\",\n \"Performing the recurrence procedure, a series of corrections to this equation in powers of $m$ would be derived.\",\n \"Then the final mapped equation would also respect inertia of the Brownian particles, although working only in real space.\",\n \"Use of the mapping procedure developed for diffusion [13], [14], [15], [16] is not straightforward; the left-hand side operator of the FP equation (REF ) has a different structure than that of the diffusion (REF ) or Smoluchowski equation (REF ).\",\n \"Still, there is a way to apply the general scheme of the mapping in this case and to perform reduction of the phase space onto the real space in the way described above.\",\n \"Presentation of this algorithm is the primary aim of this study.\",\n \"The result of the mapping of the FP equation (REF ) is again an equation of the form (REF ) with $V(x)=\\\\beta U(x)$ and $\\\\epsilon $ replaced by $m$ .\",\n \"In the limit of stationary flow, this equation can be simplified by subsequent reduction of the operator $1-m\\\\hat{Z}$ to a function $D(x)$ , a spatially dependent effective diffusion coefficient [12], [17], $\\\\partial _tp(x,t)=\\\\partial _xe^{-\\\\beta U(x)}D(x)\\\\partial _xe^{\\\\beta U(x)}p(x,t)\\\\ .$ The leading term of $D(x)$ is proportional to $mU^{\\\\prime \\\\prime }(x)$ and the whole series of corrections to the Smoluchowski equation can be summed up, giving $D(x)=D_0\\\\frac{1-\\\\sqrt{1-4mU^{\\\\prime \\\\prime }(x)/\\\\gamma ^2}}{2mU^{\\\\prime \\\\prime }(x)/\\\\gamma ^2},$ with $D_0=1/\\\\gamma \\\\beta $ , if the higher derivatives of $U(x)$ are neglected.\",\n \"In the following section, we analyze how to apply the mapping scheme for reduction of the phase space in the FP equation (REF ).\",\n \"In Sect.\",\n \"III, our considerations are verified on an exactly solvable model, the FP equation with no field, $U(x)=0$ .\",\n \"Analysis of this example helps us to construct the recurrence scheme for calculation of the series of corrections to the zeroth order approximation, the Smoluchowski eq.\",\n \"(REF ), in the small parameter $m$ .\",\n \"Finally, the complete mapping procedure for an arbitrary potential $U(x)$ is presented in Sect.\",\n \"IV.\",\n \"The mapped equation of type (REF ), as well as the formula (REF ) is derived, and checked on the damped linear harmonic oscillator.\"\n ],\n [\n \"II. Preliminary considerations\",\n \"The key points of the mapping procedure, as formulated for diffusion [13], [14], are recalled in this Section.\",\n \"Based on physical considerations, we adjust the general scheme of the mapping for dimensional reduction of the FP equation (REF ) to this situation.\",\n \"The mapping procedure represents a consistent transition from a fine grain to a coarse grain description of some evolution process.\",\n \"The process is described in details by some partial differential equation (PDE), governing the density of particles $\\\\rho ({\\\\bf r},t)$ in the full space, defined by the coordinates ${\\\\bf r}$ .\",\n \"The dimensional reduction projects this equation onto another PDE, which governs the density $p(x,t)$ in the reduced space of the coordinate $x$ .\",\n \"The coordinate $x$ is one of the coordinates of the full space, ${\\\\bf r}=(x,{\\\\bf y})$ , and the mapping accomplishes integration over the transverse coordinates ${\\\\bf y}$ .\",\n \"In the case of the FP equation (REF ), phase space $(x,v)$ represents the full space and the dimensional reduction integrates over the \\\"transverse\\\" coordinate, the velocity $v$ .\",\n \"Hence we have the defining relation between the densities $\\\\rho $ and $p$ : $p(x,t)=\\\\int _{-\\\\infty }^{\\\\infty }\\\\rho (x,v,t)dv\\\\ .$ The phase-space density $\\\\rho (x,v,t)$ is expected to be near the thermodynamic equilibrium and so approximately proportional to $\\\\exp (-\\\\beta mv^2/2)$ , which provides convergence of the integral in Eq.\",\n \"(REF ).\",\n \"Then the first step of the mapping is also integration of Eq.\",\n \"(REF ) over $v$ .\",\n \"If completed, we get $\\\\int _{-\\\\infty }^{\\\\infty }\\\\left(\\\\partial _t+v\\\\partial _x\\\\right)\\\\rho (x,v,t)dv=0;$ the other terms are zero due to $\\\\rho (x,v,t)\\\\rightarrow 0$ in the limit $v\\\\rightarrow \\\\pm \\\\infty $ .\",\n \"This equation represents nothing but mass conservation in the reduced space, $\\\\partial _tp(x,t)+\\\\partial _xj(x,t)=0,$ where the 1D flux $j$ is defined by the relation $j(x,t)=\\\\int _{-\\\\infty }^{\\\\infty }v\\\\rho (x,v,t)dv\\\\ .$ In contrast to diffusion, where $j=-D_0\\\\partial _xp$ is fixed, here the flux $j$ is a function formally independent of the density $p$ , so we also need the evolution equation for this quantity.\",\n \"After integration of Eq.\",\n \"(REF ) multiplied by $v$ , we obtain $\\\\partial _tj(x,t)+\\\\partial _x\\\\int _{-\\\\infty }^{\\\\infty }v^2\\\\rho (x,v,t)dv+\\\\frac{U^{\\\\prime }(x)}{m}p(x,t)=\\\\ \\\\ \\\\ \\\\cr =-\\\\frac{\\\\gamma }{\\\\beta m^2}\\\\int _{-\\\\infty }^{\\\\infty }e^{-\\\\beta mv^2/2}\\\\partial _ve^{\\\\beta mv^2/2}\\\\rho (x,v,t)dv\\\\ \\\\ $ [after some algebra and applying the definitions (REF ) and (REF ).\",\n \"This step recalls the Grad's method of moments [18], [19].\",\n \"For the 1D Kramers equation (REF ), taking only a couple of the zeroth ($p$ ) and the first ($j$ ) order moment of the phase space density $\\\\rho $ is satisfactory for generating the selfconsistent system of the mapped (real space) equations (REF ) and (REF ).]\",\n \"The next key point of the mapping algorithm is that of expressing the full-space density $\\\\rho (x,v,t)$ using the 1D density $p(x,t)$ and also the flux $j(x,t)$ in this case.\",\n \"This relation enables us to complete integrations in Eq.\",\n \"(REF ) and get the evolution equation for $j(x,t)$ , together with the 1D mass conservation, in closed form.\",\n \"The initial task is to find the zeroth order approximation, valid in the limit $m\\\\rightarrow 0$ .\",\n \"Our first proposal for such a relation between $\\\\rho (x,v,t)$ and the reduced space quantities $p$ and $j$ is based on the following physical construction: For an infinitely small mass of the particle, the stochastic force thermalizes its velocity $v$ almost immediately after any move along the spatial coordinate $x$ .\",\n \"Similar to the transverse equilibration of the 2D density of a particle diffusing in a narrow channel with biasing transverse force [15], [20], [21], [22], one could try the formula with separated Boltzmann factor in the fast relaxing \\\"direction\\\" $v$ , $\\\\rho (x,v,t)\\\\simeq \\\\sqrt{\\\\beta m/2\\\\pi }p(x,t)\\\\exp (-\\\\beta mv^2/2)$ .\",\n \"It is easy to check that it does not work here; the flux $j$ becomes zero according to Eq.\",\n \"(REF ).\",\n \"To prevent this failure, let us suppose that the distribution in $v$ is shifted by the local mean (macroscopic) velocity $v_0$ , depending on the local flux, $j(x,t)=v_0(x,t)p(x,t)$ .\",\n \"Then we have $\\\\nonumber \\\\rho (x,v,t)&\\\\simeq & \\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}e^{-\\\\beta m(v-v_0)^2/2}p(x,t)\\\\cr &\\\\simeq &\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}\\\\left[1+\\\\beta mvv_0+ ...\\\\right]e^{-\\\\beta mv^2/2}p(x,t).$ Retaining only these two terms in the square brackets and replacing $v_0p$ by the flux $j$ , one gets $\\\\rho _0(x,v,t)=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}e^{-\\\\beta mv^2/2}\\\\left[p(x,t)+\\\\beta mvj(x,t)\\\\right],$ which will be taken as the sought zeroth order relation between $\\\\rho $ and the reduced space quantities, $p$ and $j$ .\",\n \"This heuristic formula will be verified later by the exact mapping algorithm.\",\n \"Still, one can check immediately that the relation (REF ) represents correctly a kind of backward mapping of the 1D (spatial) functions $p$ and $j$ onto the phase space densities $\\\\rho $ ; if substituted for $\\\\rho (x,v,t)$ in the defining relations (REF ) and (REF ), we obtain identities.\",\n \"Applying Eq.\",\n \"(REF ) to Eq.\",\n \"(REF ), the integrals over $v$ can be completed and the result, $\\\\partial _tj(x,t)+\\\\partial _x\\\\frac{p(x,t)}{\\\\beta m}+\\\\frac{U^{\\\\prime }(x)}{m}p(x,t)=-\\\\frac{\\\\gamma }{m}j(x,t),$ together with the mass conservation, Eq.\",\n \"(REF ), forms a closed couple of PDE, governing the mapped quantities $p$ and $j$ .\",\n \"In the limit $m\\\\rightarrow 0$ , the first term in Eq.\",\n \"(REF ), $\\\\partial _tj$ , becomes negligible and the equation expresses the zeroth order relation between the flux $j$ and the density $p$ , $j(x,t)=-\\\\frac{1}{\\\\beta \\\\gamma }e^{-\\\\beta U(x)}\\\\partial _xe^{\\\\beta U(x)}p(x,t)\\\\ .$ If combined with the mass conservation, Eq.\",\n \"(REF ), we get the Smoluchowski equation (REF ); $1/\\\\beta \\\\gamma =D_0$ represents the diffusion constant.\",\n \"The calculation presented shows that the Smoluchowski equation (REF ) is related to the FP equation (REF ) in the same way as the Fick-Jacobs equation [11] to the diffusion equation valid in a narrow 2D channel.\",\n \"Both mapped equations describe an asymptotic behavior of the full-space density infinitely rapidly equilibrating in the transverse direction; the velocity $v$ plays the role of the transverse coordinate for the FP equation.\",\n \"Our considerations indicate that the mass of the particle, $m$ , becomes the small parameter, controlling the series of corrections to the Smoluchowski equation in the case when the transverse equilibration is not infinitely fast.\",\n \"Following the scheme of the mapping procedure [13], [14], the next point is that of searching for the true relation between the phase space density $\\\\rho (x,v,t)$ and the 1D quantities $p(x,t)$ and $j(x,t)$ , replacing the heuristic formula (REF ), valid for nonzero $m$ .\",\n \"Without losing generality, it can be written in the form $\\\\rho (x,v,t)=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}e^{-\\\\beta mv^2/2}\\\\Big [\\\\hat{\\\\omega }(x,v)p(x,t)\\\\hspace{28.90755pt}\\\\cr \\\\hspace{-28.90755pt}+\\\\beta mv\\\\hat{\\\\eta }(x,v)j(x,t)\\\\Big ].$ If the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ (with $\\\\partial _x$ implicit) are expandable in $m$ , one can substitute for $\\\\rho (x,v,t)$ in the FP equation (REF ) and fix the coefficients of these operators to satisfy this equation in each order of $m$ , similar to the mapping of diffusion.\",\n \"Then, using the relation of backward mapping (REF ) in Eq.\",\n \"(REF ) gives the expansion of the evolution equation for $j$ and finally, in combination with mass conservation (REF ), the sought series of corrections to the Smoluchowski equation (REF ) in terms of the finite mass $m$ .\",\n \"To verify whether this scheme is viable, we analyze the exactly solvable case with $U(x)=0$ in the next Section.\"\n ],\n [\n \"III. Exactly solvable model\",\n \"The exact solution of the FP equation (REF ) with no potential, $U(x)=0$ , is presented in this Section.\",\n \"We demonstrate the mapping on the example of the phase space density $\\\\rho (x,v,t)$ evolving from the initial density $\\\\rho (x,v,0)$ with thermalized velocity $v$ .\",\n \"The mapped equation, as well as the form of the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ in Eq.\",\n \"(REF ) can be found explicitly in this case.\",\n \"This analysis will direct us in construction of the recurrence mapping scheme in Sect.\",\n \"IV.\",\n \"The case $U(x)=0$ is exactly solvable [23], the Green's function $G(x,v,t;x^{\\\\prime },v^{\\\\prime },t^{\\\\prime })$ of the FP equation (REF ), $&&\\\\hspace{-14.45377pt}\\\\left[\\\\partial _t+v\\\\partial _x-\\\\frac{\\\\gamma }{\\\\beta m^2}\\\\partial _ve^{-\\\\beta mv^2/2}\\\\partial _ve^{\\\\beta mv^2/2}\\\\right]G(x,v,t;x^{\\\\prime },v^{\\\\prime },t^{\\\\prime })\\\\cr &&\\\\hspace{72.26999pt}=\\\\delta (x-x^{\\\\prime })\\\\delta (v-v^{\\\\prime })\\\\delta (t-t^{\\\\prime })\\\\ ,$ can be derived explicitly (see Appendix A), $G&=&\\\\frac{\\\\gamma \\\\beta }{4\\\\pi }\\\\frac{\\\\Theta (\\\\tau -\\\\tau ^{\\\\prime })}{\\\\sqrt{\\\\tau -\\\\tau ^{\\\\prime }-\\\\tanh (\\\\tau -\\\\tau ^{\\\\prime })}\\\\sqrt{1-q^2}}\\\\cr &&\\\\times \\\\exp \\\\bigg (-\\\\frac{\\\\left[2(\\\\xi -\\\\xi ^{\\\\prime })-(u+u^{\\\\prime })\\\\tanh (\\\\tau -\\\\tau ^{\\\\prime })\\\\right]^2}{4[\\\\tau -\\\\tau ^{\\\\prime }-\\\\tanh (\\\\tau -\\\\tau ^{\\\\prime })]}\\\\ \\\\ \\\\cr &&\\\\hspace{28.90755pt}-\\\\frac{q}{1-q^2}\\\\left[q(u^2+u^{\\\\prime 2})-2uu^{\\\\prime }\\\\right]-u^2\\\\bigg )\\\\ ,$ if expressed in the scaled coordinates, $\\\\tau &=&\\\\gamma t/2m,\\\\cr \\\\xi &=&(\\\\beta m/2)^{3/2}\\\\frac{\\\\gamma x}{\\\\beta m^2},\\\\cr u&=&\\\\sqrt{\\\\beta m/2}\\\\ v,$ and $q=\\\\exp [-2(\\\\tau -\\\\tau ^{\\\\prime })]$ .\",\n \"If the thermalized particle is inserted at time $t=0$ with a spatial distribution $p_0(x)$ , $\\\\rho (x,v,0)=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}p_0(x)e^{-\\\\beta mv^2/2},$ evolution of the density $\\\\rho $ is given by the formula $\\\\rho (x,v,t)&=&\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}\\\\int _{-\\\\infty }^{\\\\infty }dv^{\\\\prime }\\\\int dx^{\\\\prime } G(x,v,t;x^{\\\\prime },v^{\\\\prime },0)\\\\ \\\\ \\\\cr &&\\\\hspace{72.26999pt}\\\\times p_0(x^{\\\\prime })e^{-\\\\beta mv^{\\\\prime 2}/2}\\\\cr &=&\\\\int \\\\frac{p_0(x^{\\\\prime })dx^{\\\\prime }}{4\\\\pi D_0\\\\sqrt{Z}}\\\\exp \\\\bigg (-\\\\frac{[\\\\xi -\\\\xi ^{\\\\prime }]^2}{Q}\\\\cr &&\\\\hspace{14.45377pt}-\\\\frac{Q}{Z}\\\\left[u-\\\\frac{(1-e^{-2\\\\tau })}{2Q}(\\\\xi -\\\\xi ^{\\\\prime })\\\\right]^2\\\\bigg );$ $D_0=1/\\\\gamma \\\\beta $ , the abbreviations $Q&=&\\\\tau -\\\\frac{1}{2}\\\\left(1-e^{-2\\\\tau }\\\\right),\\\\cr Z&=&Q-\\\\frac{1}{4}\\\\left(1-e^{-2\\\\tau }\\\\right)^2$ are used and the integration over $x^{\\\\prime }$ runs over the whole (unspecified) 1D spatial domain.\",\n \"Then the spatial (1D) density $p$ and the flux $j$ are integrated directly according to Eqs.\",\n \"(REF ) and (REF ), $p(x,t)&=&\\\\sqrt{\\\\frac{2}{\\\\beta m}}\\\\int \\\\frac{p_0(x^{\\\\prime })dx^{\\\\prime }}{4D_0\\\\sqrt{\\\\pi Q}}e^{-(\\\\xi -\\\\xi ^{\\\\prime })^2/Q},\\\\hspace{50.58878pt}\\\\cr j(x,t)&=&\\\\frac{2}{\\\\beta m}\\\\int \\\\frac{p_0(x^{\\\\prime })dx^{\\\\prime }}{8D_0\\\\sqrt{\\\\pi Q^3}}\\\\left(1-e^{-2\\\\tau }\\\\right)\\\\cr &&\\\\hspace{65.04256pt}\\\\times \\\\left(\\\\xi -\\\\xi ^{\\\\prime }\\\\right) e^{-(\\\\xi -\\\\xi ^{\\\\prime })^2/Q}.$ It is easy to check that the mass conservation (REF ), $\\\\partial _{\\\\tau }p+\\\\sqrt{\\\\beta m/2}\\\\partial _{\\\\xi }j=0$ in the scaled coordinates, is satisfied.\",\n \"The quantity $Q$ plays the role of a \\\"stretched\\\" time [24].\",\n \"For short times, $t\\\\ll 2m/\\\\gamma $ , $Q\\\\simeq \\\\tau ^2$ and $Z\\\\simeq 4\\\\tau ^3/3$ .\",\n \"The formulas (REF ) and (REF ) describe correctly behavior of the Newtonian particles in this limit.\",\n \"The mapping procedure, as outlined in the previous Section, requires us to study asymptotic behavior in the opposite limit, $t\\\\gg 2m/\\\\gamma $ .\",\n \"For large times, $\\\\tau \\\\rightarrow \\\\infty $ , the stretched time $Q$ becomes $\\\\tau $ and the formulas (REF ) represent the general solution of the diffusion equation, as expected according to the previous Section.\",\n \"Now it is necessary to verify that the mass $m$ can serve as a small parameter controlling the series of corrections to the diffusion equation and its solution.\",\n \"It may look problematic at first glance, because $Q$ contains $\\\\exp (-2\\\\tau )=\\\\exp (-\\\\gamma t/m)$ , representing essential singularity of the variable $m\\\\rightarrow 0$ .\",\n \"Then the formulas (REF ) [and similarly Eq.\",\n \"(REF )] are not expandable in $m$ .\",\n \"Nevertheless, this property is still consistent with the general scheme of the mapping, as analyzed in Ref.\",\n \"[14].\",\n \"The dimensional reduction, as demonstrated on anisotropic diffusion in a narrow channel [14], also reduces the Hilbert space of the full-space problem.\",\n \"Let us denote $\\\\hat{M}(\\\\epsilon )$ the spatial operator of the evolution equation, i.e.\",\n \"$\\\\hat{M}(\\\\epsilon )=\\\\partial _x^2+(1/\\\\epsilon )\\\\partial _y^2$ for anisotropic 2D diffusion; the eigenvalues $\\\\lambda _i$ and the eigenfunctions $\\\\psi _i(x,y)$ are given by the equation $-\\\\hat{M}(\\\\epsilon )\\\\psi _i(\\\\epsilon ;x,y)=\\\\lambda _i(\\\\epsilon )\\\\psi _i(\\\\epsilon ;x,y),$ supplemented by proper boundary conditions at the walls of the channel.\",\n \"The parameter of anisotropy $\\\\epsilon <1$ splits the spectrum into two parts, the low-lying states, whose eigenvalues $\\\\lambda _l(\\\\epsilon )$ remain finite for $\\\\epsilon \\\\rightarrow 0$ and the transients with $\\\\lambda _r(\\\\epsilon )$ diverging $\\\\sim 1/\\\\epsilon $ .\",\n \"Then the exact 2D density $\\\\rho $ evolves as $\\\\rho (\\\\epsilon ;x,y,t)=\\\\sum _{i}c_i\\\\psi _i(\\\\epsilon ;x,y)e^{-\\\\lambda _i(\\\\epsilon ) t},$ the constants $c_i$ are given by the initial condition and the summation runs over the whole spectrum.\",\n \"The transients contribute to the sum by the terms proportional to $\\\\exp (-\\\\bar{\\\\lambda }_r t/\\\\epsilon )$ , where $\\\\bar{\\\\lambda }_r=\\\\epsilon \\\\lambda _r(\\\\epsilon )$ are finite in the limit $\\\\epsilon \\\\rightarrow 0$ .\",\n \"The result is a formula containing the essential singularity in the parameter $\\\\epsilon $ near zero, similar to Eq.\",\n \"(REF ) with singular $\\\\exp (-\\\\gamma t/m)$ for $m\\\\rightarrow 0$ .\",\n \"The mapping procedure reduces the full Hilbert space of all $\\\\psi _i$ onto the space defined only by the low-lying states $\\\\psi _l$ .\",\n \"If the 1D density $p(x,t)$ is integrated from Eq.\",\n \"(REF ) and mapped backward onto the full Hilbert space (by some operator $\\\\hat{\\\\omega }$ ), the transients will be canceled; the sum (REF ) after the mapping there and back runs only over the low-lying states $\\\\psi _l$ .\",\n \"The terms retained involve no essential singularity in $\\\\epsilon $ ; the formula for $\\\\rho $ considered in the mapping procedure represents the regular part of the exact 2D density $\\\\rho $ with respect to the parameter $\\\\epsilon $ near zero.\",\n \"This reduction is natural for the zero-th order (Fick-Jacobs) approximation, as the transients decay infinitely fast due to their infinite eigenvalues $\\\\lambda _r(\\\\epsilon \\\\rightarrow 0)$ .\",\n \"Nevertheless, the mapping based on fixing the series of corrections expanded in $\\\\epsilon $ can work only with the regular part of the 2D density.\",\n \"Correspondingly, the formulas (REF ) and (REF ) are exact, including the contributions of the transients, which are represented by the singular terms $\\\\sim \\\\exp (-\\\\gamma t/m)$ .\",\n \"The mapping requires us to analyze only the regular parts, $p_{reg}(x,t)&=&\\\\int \\\\frac{p(x^{\\\\prime })dx^{\\\\prime }}{2\\\\sqrt{\\\\pi D_0(t-D_0\\\\beta m)}}\\\\cr &&\\\\times \\\\exp \\\\left[-(x-x^{\\\\prime })^2)/4D_0(t-D_0\\\\beta m)\\\\right],\\\\hspace{14.45377pt}\\\\cr j_{reg}(x,t)&=&\\\\int \\\\frac{(x-x^{\\\\prime })p(x^{\\\\prime })dx^{\\\\prime }}{4\\\\sqrt{\\\\pi D_0(t-D_0\\\\beta m)^3}}\\\\cr &&\\\\times \\\\exp \\\\left[-(x-x^{\\\\prime })^2/4D_0(t-D_0\\\\beta m)\\\\right]$ and $\\\\rho _{reg}(x,v,t)&=&\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}\\\\int \\\\frac{p(x^{\\\\prime })dx^{\\\\prime }}{2\\\\sqrt{\\\\pi D_0(t-3D_0\\\\beta m/2)}}\\\\hspace{28.90755pt}\\\\cr &&\\\\hspace{-36.135pt}\\\\times \\\\exp \\\\left[-\\\\frac{(x-x^{\\\\prime }-D_0\\\\beta mv)^2}{4D_0(t-3D_0\\\\beta m/2)}-\\\\frac{1}{2}\\\\beta mv^2\\\\right],$ written in the unscaled coordinates, obtained after taking only the regular parts of $Q$ and $Z$ (REF ), $Q_{reg}=\\\\tau -1/2$ and $Z_{reg}=\\\\tau -3/4$ , in Eqs.\",\n \"(REF ) and (REF ).\",\n \"Of course, the formulas are applied for $t\\\\gg D_0\\\\beta m$ , when the transients vanish.\",\n \"We omit writing the subscript \\\"reg\\\" in the following text.\",\n \"In comparison with the overdamped limit $m\\\\rightarrow 0$ , evolution of the spatial (1D) density $p$ and the flux $j$ is only corrected by a time shift, $t\\\\rightarrow t-D_0\\\\beta m$ in the formulas (REF ).\",\n \"The Gaussian distribution is retarded by the time $t_0=D_0\\\\beta m=m/\\\\gamma $ , corresponding to the mean time necessary for losing information about the original velocity.\",\n \"The value of the shift $t_0$ is constant in $x$ and $t$ , so the density $p$ (REF ) still obeys the diffusion equation, $\\\\partial _tp(x,t)=D_0\\\\partial _x^2p(x,t),$ and $j(x,t)=-D_0\\\\partial _xp(x,t)$ .\",\n \"There is no correction of the zero-th order mapped equation due to the finite mass of the particle.\",\n \"Let us stress that the case of $U(x)=0$ is extremely simple, similar to the mapping of the 2D diffusion in a flat narrow channel, also giving no corrections to the Fick-Jacobs approximation.\",\n \"Nevertheless, the relation of the backward mapping, generating the phase-space density $\\\\rho $ from the mapped quantities $p$ and $j$ , is not quite trivial.\",\n \"In the formula for $\\\\rho $ , Eq.\",\n \"(REF ), the time is shifted by $3t_0/2$ and the displacement $x-x^{\\\\prime }$ by $vt_0$ with respect to the distribution of a massless particle.\",\n \"The shortest way to construct the relation of backward mapping is by applying the shift operators in $t$ and $x$ on $p(x,t)$ , compensating the different shifts of time and displacement in Eqs.\",\n \"(REF ) and (REF ), $\\\\rho (x,v,t)=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}e^{-\\\\beta mv^2/2}e^{-(t_0/2)\\\\partial _t-vt_0\\\\partial _x}p(x,t).$ Due to Eq.\",\n \"(REF ), the operator $\\\\partial _t$ in the exponent can be replaced by $D_0\\\\partial _x^2$ .\",\n \"After expanding the shift operator $\\\\exp (-vt_0\\\\partial _x)$ and using $j=-D_0\\\\partial _xp$ , we arrive at the relation of the form (REF ), $\\\\rho (x,v,t)&=&\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}e^{-\\\\beta mv^2/2-(D_0t_0/2)\\\\partial _x^2}\\\\sum _{k=0}^{\\\\infty }(\\\\beta mv^2)^k\\\\hspace{21.68121pt}\\\\cr &&\\\\hspace{-36.135pt}\\\\times (D_0t_0)^k\\\\partial _x^{2k}\\\\left[\\\\frac{1}{(2k)!\",\n \"}p(x,t)+\\\\frac{\\\\beta mv}{(2k+1)!\",\n \"}j(x,t)\\\\right];$ the explicit formulas for $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ to be substituted in Eq.\",\n \"(REF ) in the case $U(x)=0$ are $\\\\hat{\\\\omega }(x,v)&=&e^{-(D_0t_0/2)\\\\partial _x^2}\\\\sum _{k=0}^{\\\\infty }\\\\frac{(\\\\beta mv^2)^k}{(2k)!\",\n \"}(D_0t_0)^k\\\\partial _x^{2k},\\\\hspace{21.68121pt}\\\\cr \\\\hat{\\\\eta }(x,v)&=&e^{-(D_0t_0/2)\\\\partial _x^2}\\\\sum _{k=0}^{\\\\infty }\\\\frac{(\\\\beta mv^2)^k}{(2k+1)!\",\n \"}(D_0t_0)^k\\\\partial _x^{2k}.$ Both operators are expandable in $m$ , ($t_0=m/\\\\gamma $ ); their zero-th order coefficients equal unity, consistent with the heuristic formula (REF ).\",\n \"Also applying the relation (REF ) in the definitions (REF ) and (REF ) gives identity.\",\n \"The final step is that of verifying the evolution equation for $j$ , Eq.\",\n \"(REF ).\",\n \"Two integrals are to be completed with $\\\\rho $ expressed by the backward mapping, Eq.\",\n \"(REF ), $\\\\int _{-\\\\infty }^{\\\\infty }v^2\\\\rho (x,y,t)dv=\\\\left(1+D_0t_0\\\\partial _x^2\\\\right)p(x,t)/\\\\beta m,\\\\cr \\\\int _{-\\\\infty }^{\\\\infty }e^{-\\\\beta mv^2/2}\\\\partial _ve^{\\\\beta mv^2/2}\\\\rho (x,v,t)dv=\\\\beta m j(x,t),\\\\ \\\\ $ which are now valid exactly.\",\n \"If substituted in Eq.\",\n \"(REF ), and the equation (REF ) is applied, we get $(1+t_0\\\\partial _t)[j(x,t)+D_0\\\\partial _xp(x,t)]=0$ after simple algebra.\",\n \"This equation validates the relation $j=-D_0\\\\partial _xp$ for nonzero $m$ as well and thus, if combined with the mass conservation (REF ), also the diffusion equation (REF ) without any corrections.\",\n \"If compared with the calculation of Eq.\",\n \"(REF ), the 1-st order term $\\\\sim \\\\partial _tj$ , neglected in the previous Section, is compensated here by other 1-st order term coming from the exact relation (REF ), appearing in the integrals (REF ).\",\n \"The FP equation (REF ) with zero potential is too simple to give nonzero corrections to the diffusion equation (REF ).\",\n \"Nevertheless, it helped us to understand the structure of the mapping.\",\n \"It shows that the scheme suggested in the previous section is viable.\",\n \"The relation of the backward mapping has the form of Eq.\",\n \"(REF ), at least for $U(x)=0$ , and the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ can be expanded in $m$ , or $t_0=m/\\\\gamma $ , $\\\\hat{\\\\omega }=\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\hat{\\\\omega }_k(x,u),\\\\hspace{14.45377pt}\\\\hat{\\\\eta }=\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\hat{\\\\eta }_k(x,u).$ Integration in Eqs.\",\n \"(REF ) indicates that the coefficients $\\\\hat{\\\\omega }_k$ and $\\\\hat{\\\\eta }_k$ should be sought dependent up on the scaled velocity $u$ rather than $v$ (compare to Ref.\",\n \"[25]); otherwise each term in Eq.\",\n \"(REF ) would contribute in several succeeding orders in the integrals (REF ).\",\n \"The mixing of orders would hinder us in constructing the recurrence scheme generating corrections to Eq.\",\n \"(REF ).\",\n \"In the notation of Eq.\",\n \"(REF ), $\\\\hat{\\\\omega }_0=\\\\hat{\\\\eta }_0=1$ and $\\\\hat{\\\\omega }_1(x,u)&=&(u^2-1/2)D_0\\\\partial _x^2,\\\\hspace{21.68121pt}\\\\cr \\\\hat{\\\\eta }_1(x,u)&=&(u^2/3-1/2)D_0\\\\partial _x^2,\\\\cr \\\\hat{\\\\omega }_2(x,u)&=&(u^4/6-u^2/2+1/8)D_0^2\\\\partial _x^4,\\\\cr .&.&.$ valid for $U(x)=0$ according to Eqs.\",\n \"(REF ), will be used for testing the results of the recurrence procedure in the next Section.\"\n ],\n [\n \"IV. Mapping procedure\",\n \"We now finish the construction of the mapping procedure, outlined in the Section II, for an arbitrary analytic potential $U(x)$ .\",\n \"Supposing the backward mapping of the form (REF ) with the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ expanded in $t_0$ ($m$ ) according to Eqs.\",\n \"(REF ), we find recurrence relations fixing the coefficients $\\\\hat{\\\\omega }_k$ and $\\\\hat{\\\\eta }_k$ .\",\n \"Completing the integrals in Eq.\",\n \"(REF ), we obtain a series of corrections to the zero-th order relation $j=-D_0\\\\partial _xp$ .\",\n \"Combined with mass conservation, Eq.\",\n \"(REF ), it gives the Smoluchowski equation corrected due to nonzero mass of the particle.\",\n \"The essential relation determining the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ is the FP equation (REF ), which has to be satisfied for any solution of the reduced problem, the density $p(x,t)$ and the flux $j(x,t)$ , after their backward mapping (REF ) onto the full-dimensional Hilbert space.\",\n \"If the expansion in $t_0=m/\\\\gamma $ of both operators, Eq.\",\n \"(REF ), is supposed, we have $&&\\\\hspace{-21.68121pt}\\\\left[\\\\partial _t+\\\\sqrt{\\\\frac{2}{\\\\beta m}}u\\\\partial _x-\\\\frac{\\\\beta U^{\\\\prime }(x)}{\\\\sqrt{2\\\\beta m}}\\\\partial _u-\\\\frac{1}{2t_0}\\\\partial _u e^{-u^2}\\\\partial _u e^{u^2}\\\\right]\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\cr &&\\\\hspace{-21.68121pt}\\\\times e^{-u^2}\\\\left[\\\\hat{\\\\omega }_k(x,u)p(x,t)+\\\\sqrt{2\\\\beta m}u\\\\hat{\\\\eta }_k(x,u)j(x,t)\\\\right]=0$ after introducing the scaled velocity $u=\\\\sqrt{\\\\beta m/2}v$ in Eq.\",\n \"(REF ).\",\n \"The factor $\\\\beta m$ is replaced by $t_0/D_0$ in the following calculations.\",\n \"Thus half-integer powers of $t_0$ appear in Eq.\",\n \"(REF ) [25].\",\n \"As this equation has to be satisfied for any $t_0$ , we can split it for clarity into two relations: the first one, including only the integer powers of $t_0$ , $\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\bigg [\\\\partial _t\\\\hat{\\\\omega }_kp+2u^2\\\\partial _x\\\\hat{\\\\eta }_kj-\\\\beta U^{\\\\prime }(x)e^{u^2}\\\\partial _uue^{-u^2}\\\\hat{\\\\eta }_kj\\\\cr -\\\\frac{1}{2t_0}e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u\\\\hat{\\\\omega }_kp\\\\bigg ]=0,\\\\ \\\\ \\\\ $ and the second one, collecting the half-integer powers, $\\\\sum _{k=0}^{\\\\infty }t_0^{k-1/2}\\\\bigg [2t_0u\\\\partial _t\\\\hat{\\\\eta }_kj+D_0\\\\Big (2u\\\\partial _x-\\\\beta U^{\\\\prime }(x)e^{u^2}\\\\partial _ue^{-u^2}\\\\Big )\\\\cr \\\\times \\\\hat{\\\\omega }_kp-e^{u^2}\\\\partial _u e^{-u^2}\\\\partial _u u\\\\hat{\\\\eta }_k j\\\\bigg ]=0.\\\\hspace{21.68121pt}$ Notice that Eqs.\",\n \"(REF ) and (REF ) do not violate parity of $\\\\hat{\\\\omega }_k$ and $\\\\hat{\\\\eta }_k$ in $u$ .\",\n \"If used for construction of the recurrence relations between the coefficients, all they have to have the same parity as $\\\\hat{\\\\omega }_0=\\\\hat{\\\\eta }_0=1$ ; hence $\\\\hat{\\\\omega }_k(x,u)=\\\\hat{\\\\omega }_k(x,-u)$ and $\\\\hat{\\\\eta }_k(x,u)=\\\\hat{\\\\eta }_k(x,-u)$ .\",\n \"This symmetry enables us to find the normalization (or identity) conditions for $\\\\hat{\\\\omega }_k$ and $\\\\hat{\\\\eta }_k$ .\",\n \"The backward mapped $\\\\rho $ , Eq.\",\n \"(REF ), with the operators $\\\\hat{\\\\omega }$ , $\\\\hat{\\\\eta }$ expanded in $t_0$ , Eq.\",\n \"(REF ), substituted in the definitions (REF ) and (REF ) has to give identities for any $t_0$ , $p(x,t)$ and $j(x,t)$ .\",\n \"Thus we obtain $\\\\frac{1}{\\\\sqrt{\\\\pi }}\\\\int _{-\\\\infty }^{\\\\infty }du e^{-u^2}\\\\hat{\\\\omega }_k(x,u)=\\\\delta _{0,k},$ $\\\\frac{1}{\\\\sqrt{\\\\pi }}\\\\int _{-\\\\infty }^{\\\\infty }2u^2du e^{-u^2}\\\\hat{\\\\eta }_k(x,u)=\\\\delta _{0,k}.$ The operators $\\\\hat{\\\\omega }_k$ and $\\\\hat{\\\\eta }_k$ are supposed not to depend on time, so the time derivative commutes with them and acts directly on $p(x,t)$ or $j(x,t)$ in Eqs.\",\n \"(REF ), (REF ).\",\n \"To derive the operators unambiguously, using only spatial derivatives, we express $\\\\partial _tp=-\\\\partial _xj$ from the mass conservation, Eq.\",\n \"(REF ).\",\n \"However, the time derivative of $j$ cannot be expressed in a similar way from Eq.\",\n \"(REF ), because $\\\\partial _tj$ is not the leading term there.\",\n \"If the backward mapping, Eqs.\",\n \"(REF ) and (REF ), is applied to the integrals of Eq.\",\n \"(REF ), we get $\\\\nonumber \\\\int _{-\\\\infty }^{\\\\infty }v^2\\\\rho (x,v,t) dv=\\\\frac{1}{\\\\beta m}\\\\left(1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{I}_k(x)\\\\right)p(x,t),$ where the operators $\\\\hat{I}_k$ are given by $\\\\hat{I}_k(x)=\\\\frac{2}{\\\\sqrt{\\\\pi }}\\\\int _{-\\\\infty }^{\\\\infty }u^2du e^{-u^2}\\\\hat{\\\\omega }_k(x,u);$ the right-hand side integral of Eq.\",\n \"(REF ) results in $\\\\nonumber \\\\frac{\\\\gamma }{\\\\beta m^2}\\\\int _{-\\\\infty }^{\\\\infty }e^{-\\\\beta mv^2/2}\\\\partial _v e^{\\\\beta mv^2/2}\\\\rho (x,v,t)dv&=&\\\\cr \\\\frac{\\\\gamma }{\\\\sqrt{\\\\pi }m}\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\int _{-\\\\infty }^{\\\\infty }due^{-u^2}\\\\partial _uu\\\\hat{\\\\eta }_k(x,u)j(x,t)&=&\\\\frac{j(x,t)}{t_0}$ after integrating by parts and using the normalization relation (REF ).\",\n \"Then, instead of the evolution equation for $j$ , Eq.\",\n \"(REF ) has to be understood as an expression relating $j$ and $p$ , $(1+t_0\\\\partial _t)j(x,t)&=&-D_0\\\\bigg [e^{-\\\\beta U(x)}\\\\partial _x e^{\\\\beta U(x)}\\\\hspace{50.58878pt}\\\\cr &&\\\\hspace{21.68121pt}+\\\\partial _x\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{I}_k(x)\\\\bigg ]p(x,t),$ and the flux $j$ , as well as its time derivative, is expressed using $p$ according to this equation.\",\n \"The term $t_0\\\\partial _t$ acts now on $p$ as the 1-st order correction in $(1+t_0\\\\partial _t)^{-1}p$ after completing inversion and commutation with the spatial operators in the square brackets of Eq.\",\n \"(REF ).\",\n \"Then $\\\\partial _tp$ is systematically replaced by $-\\\\partial _x j$ , contributing to the next corrections in the higher orders of $t_0$ .\",\n \"The result is a formula for $j$ expressed by some purely spatial operator acting on $p$ , $j(x,t)=-D_0 e^{-\\\\beta U(x)}\\\\left[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k(x)\\\\right]\\\\partial _x e^{\\\\beta U(x)}p(x,t),$ where the operators $\\\\hat{Z}_k$ are related to $\\\\hat{I}_k$ , Eq.\",\n \"(REF ).\",\n \"Using this form, we are able to write the final mapped equation explicitly after combination with Eq.\",\n \"(REF ), $\\\\partial _tp&=&D_0\\\\partial _x e^{-\\\\beta U(x)}\\\\left[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k\\\\right]\\\\partial _x e^{\\\\beta U(x)}p\\\\cr &=&\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\hat{Q}_kp=\\\\hat{Q}p,$ which is the Smoluchowski equation (REF ) in the zero-th order, extended by a series of mass dependent corrections in $t_0$ .\",\n \"The operators $\\\\hat{Q}_k$ are introduced to simplify notation in the following calculations.\",\n \"Now the operators $\\\\hat{Z}_k$ can be expressed explicitly using $\\\\hat{I}_k$ .\",\n \"Expanding $(1+t_0\\\\partial _t)^{-1}p$ in $t_0$ and applying Eq.\",\n \"(REF ), we have $\\\\nonumber (1+t_0\\\\partial _t)^{-1}p=(1-t_0\\\\hat{Q}+...)p=\\\\Big (1+\\\\sum _{k=0}^{\\\\infty }t_0^{k+1}\\\\hat{Q}_k\\\\Big )^{-1}\\\\hspace{-7.22743pt}p,$ which is to be used in the operator equation, $&&\\\\hspace{-14.45377pt}e^{-\\\\beta U(x)}\\\\left[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k\\\\right]\\\\partial _xe^{\\\\beta U(x)}=\\\\bigg [e^{-\\\\beta U(x)}\\\\partial _x e^{\\\\beta U(x)}\\\\cr &&\\\\hspace{43.36243pt}+\\\\partial _x\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{I}_k\\\\bigg ]\\\\Big (1+\\\\sum _{k=0}^{\\\\infty }t_0^{k+1}\\\\hat{Q}_k\\\\Big )^{-1},$ obtained after comparison of Eqs.\",\n \"(REF ) and (REF ).\",\n \"Expanding in $t_0$ and comparing the coefficients of the same powers of $t_0$ , we get a sequence of relations fixing $\\\\hat{Z}_k$ , $e^{-\\\\beta U}\\\\hat{Z}_1\\\\partial _x e^{\\\\beta U}&=&\\\\partial _x \\\\hat{I}_1-e^{-\\\\beta U}\\\\partial _x e^{\\\\beta U}\\\\hat{Q}_0,\\\\cr e^{-\\\\beta U}\\\\hat{Z}_2\\\\partial _x e^{\\\\beta U}&=&\\\\partial _x \\\\hat{I}_2-\\\\partial _x \\\\hat{I}_1\\\\hat{Q}_0\\\\cr &&+e^{-\\\\beta U}\\\\partial _x e^{\\\\beta U}(\\\\hat{Q}_0^2-\\\\hat{Q}_1),\\\\cr .&.&.$ Finally, we derive the recurrence scheme, generating the operators $\\\\hat{\\\\omega }_k$ , determining $\\\\hat{I}_k$ , Eq.\",\n \"(REF ), and thus also $\\\\hat{Z}_k$ or $\\\\hat{Q}_k$ , Eqs.\",\n \"(REF ), (REF ).\",\n \"Notice that the $\\\\hat{\\\\eta }_k$ do not directly enter the mapped equation (REF ), but they are necessary in the recurrence formulas for $\\\\hat{\\\\omega }_k$ .\",\n \"The recurrence scheme for $\\\\hat{\\\\omega }_k$ is defined by Eq.\",\n \"(REF ).\",\n \"This equation has to be satisfied for any $p$ and $j$ , solving the mapped problem, but these quantities, although considered before as formally independent, are related by Eq.\",\n \"(REF ).\",\n \"To obtain an equation for operators, $j$ has to be expressed by $p$ .\",\n \"Applying the relation (REF ) and Eq.\",\n \"(REF ) for $\\\\partial _tp$ , we get $&&\\\\hspace{-14.45377pt}\\\\sum _{n=0}^{\\\\infty }t_0^n\\\\bigg [D_0\\\\Big (\\\\hat{\\\\omega }_n\\\\partial _x-2u^2\\\\partial _x\\\\hat{\\\\eta }_n+\\\\beta U^{\\\\prime }e^{u^2}\\\\partial _uue^{-u^2}\\\\hat{\\\\eta }_n\\\\Big )e^{-\\\\beta U(x)}\\\\cr &&\\\\hspace{-7.22743pt}\\\\times \\\\sum _{l=0}^{\\\\infty }t_0^l\\\\hat{Z}_l\\\\partial _xe^{\\\\beta U(x)}-\\\\frac{1}{2t_0}e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u\\\\hat{\\\\omega }_n\\\\bigg ]p=0,$ valid for any function $p(x,t)$ ; we have $\\\\hat{Z}_0=1$ .\",\n \"To lowest order, $t_0^{-1}$ , only the term $e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u\\\\hat{\\\\omega }_0=0$ .\",\n \"It is satisfied by $\\\\hat{\\\\omega }_0=1$ , the only solution nondiverging at $u\\\\rightarrow \\\\pm \\\\infty $ and also satisfying the normalization, Eq.\",\n \"(REF ).\",\n \"In the higher orders, we derive the recurrence relation, $e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u\\\\hat{\\\\omega }_{n+1}&=&2D_0\\\\sum _{k=0}^n\\\\Big (\\\\hat{\\\\omega }_k\\\\partial _x-2u^2\\\\partial _x\\\\hat{\\\\eta }_k+\\\\beta U^{\\\\prime }(x)\\\\cr &&\\\\hspace{-57.81621pt}\\\\times e^{u^2}\\\\partial _uu e^{-u^2}\\\\hat{\\\\eta }_k\\\\Big )e^{-\\\\beta U(x)}\\\\hat{Z}_{n-k}\\\\partial _xe^{\\\\beta U(x)}.$ Calculation of the $\\\\hat{\\\\omega }_{n+1}$ requires us to know the $\\\\hat{\\\\eta }_k$ up to $k=n$ .\",\n \"They are generated from Eq.\",\n \"(REF ).\",\n \"Again, $j$ , as well as $\\\\partial _t j$ , have to be expressed by $p$ using the relations (REF ) and (REF ).\",\n \"Then Eq.\",\n \"(REF ) becomes $&&\\\\sum _{n=0}^{\\\\infty }t_0^{n-1/2}\\\\bigg [e^{u^2}\\\\partial _u e^{-u^2}\\\\partial _u u\\\\hat{\\\\eta }_n e^{-\\\\beta U}\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\hat{Z}_k\\\\partial _xe^{\\\\beta U}\\\\cr &&-2D_0t_0u\\\\hat{\\\\eta }_ne^{-\\\\beta U}\\\\sum _{k=0}^{\\\\infty }t_0^k\\\\hat{Z}_k\\\\partial _xe^{\\\\beta U}\\\\partial _xe^{-\\\\beta U}\\\\sum _{l=0}^{\\\\infty }t_0^l\\\\hat{Z}_l\\\\partial _xe^{\\\\beta U}\\\\cr &&+\\\\big (2u\\\\partial _x-\\\\beta U^{\\\\prime }e^{u^2}\\\\partial _u e^{-u^2}\\\\big )\\\\hat{\\\\omega }_n\\\\bigg ]p=0$ valid for any function $p(x,t)$ .\",\n \"In lowest order, $t_0^{-1/2}$ , $e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u u\\\\hat{\\\\eta }_0e^{-\\\\beta U}\\\\partial _xe^{\\\\beta U}&=&-2u\\\\partial _x+\\\\beta U^{\\\\prime }e^{u^2}\\\\partial _ue^{-u^2}\\\\cr =-2u(\\\\partial _x+\\\\beta U^{\\\\prime })&=&-2ue^{-\\\\beta U}\\\\partial _x e^{\\\\beta U},$ where we have used $\\\\hat{\\\\omega }_0=1$ and $\\\\hat{Z}_0=1$ .\",\n \"After the first integration, we have $\\\\partial _uu\\\\hat{\\\\eta }_0=-e^{u^2}\\\\int 2udue^{-u^2}=1+\\\\hat{C}_1e^{u^2};$ the integration constant $\\\\hat{C}_1=0$ provides convergence as $u\\\\rightarrow \\\\pm \\\\infty $ .\",\n \"The next integration gives $\\\\hat{\\\\eta }_0=1+(1/u)\\\\hat{C}_0$ ; $\\\\hat{C}_0=0$ .\",\n \"This calculation validates our heuristic formula (REF ) in the zero-th order approximation.\",\n \"In the higher orders, Eq.\",\n \"(REF ) generates the relations $&&\\\\hspace{-14.45377pt}e^{u^2}\\\\partial _ue^{-u^2}\\\\partial _u u\\\\hat{\\\\eta }_ne^{-\\\\beta U}\\\\partial _xe^{\\\\beta U}=(\\\\beta U^{\\\\prime }e^{u^2}\\\\partial _ue^{-u^2}-2u\\\\partial _x)\\\\hat{\\\\omega }_n\\\\cr &&+2D_0\\\\sum _{k,l=0}^{k+l0$ depend on $u$ .\",\n \"The recurrence procedure starts from $\\\\hat{\\\\omega }_0=1$ and $\\\\hat{Z}_0=1$ .\",\n \"Calculation of the next order correction requires first expressing the $\\\\hat{\\\\eta }_n$ according to Eq.\",\n \"(REF ), or (REF ) for $n=0$ , as shown above.\",\n \"Then $\\\\hat{\\\\omega }_{n+1}$ is derived from Eq.\",\n \"(REF ), $\\\\hat{I}_{n+1}$ integrated according to Eq.\",\n \"(REF ) and finally $\\\\hat{Z}_{n+1}$ expressed from Eq.\",\n \"(REF ).\",\n \"To demonstrate the procedure, we derive the first order correction, $\\\\hat{Z}_1$ .\",\n \"We use already calculated $\\\\hat{\\\\eta }_0=1$ .\",\n \"For $n=0$ , Eq.\",\n \"(REF ) becomes $e^{u^2}\\\\partial _u e^{-u^2}\\\\partial _u\\\\hat{\\\\omega }_1=2D_0(1-2u^2)e^{-\\\\beta U(x)}\\\\partial _x^2 e^{\\\\beta U(x)}.$ After the first integration, $\\\\partial _u\\\\hat{\\\\omega }_1=2D_0e^{u^2}\\\\left(ue^{-u^2}+C_1\\\\right)e^{-\\\\beta U(x)}\\\\partial _x^2 e^{\\\\beta U(x)},$ the integration constant $C_1=0$ , to provide convergence for $u\\\\rightarrow \\\\pm \\\\infty $ .\",\n \"The integration constant $C_0$ after the next integration is fixed to satisfy the normalization, Eq.\",\n \"(REF ), $\\\\int _{-\\\\infty }^{\\\\infty }due^{-u^2}D_0(u^2+C_0)e^{-\\\\beta U(x)}\\\\partial _x^2 e^{\\\\beta U(x)}=0,$ hence $C_0=-1/2$ and $\\\\hat{\\\\omega }_1=D_0(u^2-1/2)e^{-\\\\beta U(x)}\\\\partial _x^2 e^{\\\\beta U(x)}.$ For $U(x)=0$ , we recover the corresponding formula in Eq.\",\n \"(REF ).\",\n \"Integration over $u$ in Eq.\",\n \"(REF ) results in $\\\\hat{I}_1=D_0\\\\exp [-\\\\beta U(x)]\\\\partial _x^2\\\\exp [\\\\beta U(x)]$ , giving finally $\\\\hat{Z}_1=D_0\\\\left(e^{\\\\beta U}\\\\partial _x e^{-\\\\beta U}\\\\partial _x-\\\\partial _x e^{\\\\beta U}\\\\partial _xe^{-\\\\beta U}\\\\right)=D_0\\\\beta U^{\\\\prime \\\\prime }(x)$ from Eq.\",\n \"(REF ).\",\n \"In the higher orders, $\\\\hat{\\\\eta }_n$ , $\\\\hat{\\\\omega }_{n+1}$ are calculated according to Eqs.\",\n \"(REF ) and (REF ).\",\n \"The integration constants after double integration have to provide convergence for $u\\\\rightarrow \\\\pm \\\\infty $ , requiring the operators to be even in $u$ , and the normalization, Eq.\",\n \"(REF ).\",\n \"The condition (REF ) for $\\\\hat{\\\\eta }_n$ is satisfied automatically; it serves as a check on the computation.\",\n \"The derivation is tedious, and we present only the results in second order, $\\\\hat{\\\\eta }_1&=&D_0(u^2/3-1/2)e^{-\\\\beta U(x)}\\\\partial _x^2e^{\\\\beta U(x)},\\\\cr \\\\hat{\\\\omega }_2&=&\\\\frac{D_0^2}{2}e^{-\\\\beta U(x)}\\\\Big [\\\\Big (\\\\frac{u^4}{3}-u^2+\\\\frac{1}{4}\\\\Big )\\\\partial _x^3\\\\cr &&+\\\\Big (u^2-\\\\frac{1}{2}\\\\Big )\\\\big (4\\\\beta U^{\\\\prime \\\\prime }(x)\\\\partial _x+3\\\\beta U^{(3)}(x)\\\\big )\\\\Big ]\\\\partial _x e^{\\\\beta U(x)},\\\\cr \\\\hat{Z}_2&=&\\\\frac{D_0^2}{2}\\\\Big [4\\\\big (\\\\beta U^{\\\\prime \\\\prime }(x)\\\\big )^2-\\\\beta ^2U^{\\\\prime }(x)U^{(3)}(x)+\\\\beta U^{(4)}(x)\\\\cr &&\\\\hspace{21.68121pt}+3\\\\beta U^{(3)}(x)\\\\partial _x\\\\Big ].$ Again, the formulas (REF ) for $U(x)=0$ are recovered.\",\n \"There are no contributions to $\\\\hat{Z}_n$ in this case, too, as expected according to the analysis in the previous Section.\",\n \"Also, linear potentials, $U(x)=-Fx$ , have no effect on validity of the uncorrected Smoluchowski equation (REF ).\",\n \"The particle driven by a constant force $F$ move asymptotically with constant mean velocity $v_0=F/\\\\gamma $ and the distribution $p(y,t)$ in coordinate $y$ , shifted by the drift, $y=x-v_0t$ , is again Gaussian as in the case of no potential.\",\n \"The situation becomes different if the driving force $F(x)$ is not constant.\",\n \"If the mass $m$ or the time of the thermalization $t_0=m/\\\\gamma $ is small, but nonzero, the particle appearing at a new position $x$ has to accommodate to the new local mean velocity.\",\n \"On the other hand, it carries some mean momentum from its previous position and needs some time to change it.\",\n \"Meanwhile it slips to some other position than predicted by purely stochastic dynamics due to its inertia, or non-zero mass.\",\n \"The effects of such slipping are indicated by the corrections $\\\\hat{Z}_n$ of the Smoluchowski equation and they are nonzero for potentials with nonzero $U^{\\\\prime \\\\prime }(x)$ , or higher derivatives.\",\n \"As seen from Eq.\",\n \"(REF ), the $\\\\hat{Z}_n$ are not only functions, but operators, containing $\\\\partial _x$ in the higher orders.\",\n \"So the mapped equation (REF ) has exactly the same structure as the mapped equations for diffusion [16], or biased diffusion [15], [22], [26].\",\n \"Being inspired by these works, Eq.\",\n \"(REF ) can be simplified by replacing the correction operators $D_0[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k]$ by a function $D(x)$ , a spatially dependent effective diffusion coefficient, $\\\\partial _tp(x,t)=\\\\partial _xe^{-\\\\beta U(x)}D(x)\\\\partial _xe^{\\\\beta U(x)}p(x,t),$ which becomes valid in the limit of stationary flow, i.e.\",\n \"when the spatial density $p$ and the flux $j$ change very slowly, $p(x,t)\\\\rightarrow p(x)$ .\",\n \"Due to mass conservation, Eq.\",\n \"(REF ), the flux $j(x,t)=j$ is constant (but nonzero) in $x$ as well.\",\n \"If expressed from Eq.\",\n \"(REF ), $j=-e^{-\\\\beta U(x)}D(x)\\\\partial _xe^{\\\\beta U(x)}p(x),$ the function $\\\\partial _x(\\\\exp [\\\\beta U(x)]p(x))=-j\\\\exp [\\\\beta U(x)]/D(x)$ is dependent only on the system; $m$ , $\\\\gamma $ and the potential $U(x)$ , for any stationary solution $p(x)$ .\",\n \"So we can substitute for it in Eq.\",\n \"(REF ), $j=D_0e^{-\\\\beta U(x)}\\\\left[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k(x)\\\\right]e^{\\\\beta U(x)}\\\\frac{j}{D(x)},$ valid for stationary flow, and calculate $D(x)$ unambiguously from the expansion of the corrections $\\\\hat{Z}_k$ , $\\\\frac{D_0}{D(x)}=e^{-\\\\beta U(x)}\\\\left[1+\\\\sum _{k=1}^{\\\\infty }t_0^k\\\\hat{Z}_k(x)\\\\right]^{-1}e^{\\\\beta U(x)},$ as a series in $t_0$ , $D(x)/D_0&=&1+D_0t_0\\\\beta U^{\\\\prime \\\\prime }+(D_0t_0)^2\\\\Big [2(\\\\beta U^{\\\\prime \\\\prime })^2\\\\cr &&\\\\hspace{14.45377pt}+\\\\beta ^2U^{\\\\prime }U^{(3)}+\\\\beta U^{(4)}/2\\\\Big ]+...\\\\ .$ The next simplification is that of neglecting all the derivatives but $U^{\\\\prime \\\\prime }(x)$ , i.e.\",\n \"approximating the real $U(x)$ locally by a quadratic potential.\",\n \"In this case, the expansion (REF ) can be summed up to infinity, $D(x)/D_0&\\\\simeq &\\\\sum _{n=0}^{\\\\infty }\\\\frac{(2n)!}{n!(n+1)!\",\n \"}\\\\left[D_0t_0\\\\beta U^{\\\\prime \\\\prime }(x)\\\\right]^n\\\\cr &&=\\\\frac{1-\\\\sqrt{1-4D_0t_0\\\\beta U^{\\\\prime \\\\prime }(x)}}{2D_0t_0\\\\beta U^{\\\\prime \\\\prime }(x)},$ the proof is given in the Appendix B.\",\n \"The formula works for $D_0t_0\\\\beta U^{\\\\prime \\\\prime }(x)=mU^{\\\\prime \\\\prime }(x)/\\\\gamma ^2<1/4$ , which is the condition for non-oscillatory movement of a particle in a quadratic well with friction, the damped harmonic oscillator.\",\n \"If $U^{\\\\prime \\\\prime }(x)=\\\\kappa $ is constant, the trajectory of a particle averaged over the stochastic force is governed by $m\\\\langle \\\\ddot{x}\\\\rangle +\\\\gamma \\\\langle \\\\dot{x}\\\\rangle +\\\\kappa \\\\langle x\\\\rangle =0$ from Eq.\",\n \"(REF ).\",\n \"The particular solutions are $\\\\langle x(t)\\\\rangle =\\\\exp (\\\\alpha t)$ with $\\\\alpha =-(\\\\gamma \\\\pm \\\\sqrt{\\\\gamma ^2-4m\\\\kappa })/2m$ .\",\n \"Requiring $\\\\alpha $ to be a real number gives the same condition.\",\n \"This simple example demonstrates restriction of the theory presented to nonoscillatory movement of the particle in potential wells on its way along a 1D channel.\",\n \"A small mass $m$ is expected, to enable the friction quickly to damp the momentum of a particle; i.e.\",\n \"to have the relaxation in the velocity faster than in the real space coordinate $x$ .\",\n \"Figure: Plot of the effective diffusion coefficient D(x)D(x) dependenton the mass of the particle mm and friction γ\\\\gamma according toEq.\",\n \"(), valid if the derivatives of the potential higherthan U '' (x)U^{\\\\prime \\\\prime }(x) are neglected.A more detailed insight to the restrictions of the dimensional reduction of the phase space controlled by the mass $m$ can be obtained by comparison of the Green's function (GF) of the mapped equation (REF ) with $D(x)$ given by Eq.\",\n \"(REF ) and GF of the Kramers equation (REF ) for the damped harmonic oscillator, $U(x)=\\\\kappa x^2/2=m\\\\omega _0^2x^2/2$ , which is exactly solvable.\",\n \"The solution $G=G(x,v,t;x^{\\\\prime },v^{\\\\prime },t^{\\\\prime })$ of the equation $\\\\Big (\\\\partial _t+v\\\\partial _x-\\\\omega _0^2x\\\\partial _v-\\\\frac{\\\\gamma }{\\\\beta m^2}\\\\partial _ve^{-\\\\beta mv^2/2}\\\\partial _v e^{\\\\beta mv^2/2}\\\\Big ) G\\\\cr =\\\\delta (x-x^{\\\\prime })\\\\delta (v-v^{\\\\prime })\\\\delta (t-t^{\\\\prime })\\\\hspace{28.45274pt}$ reads [1] $G=\\\\frac{(s_1-s_2)e^{\\\\gamma t/m}}{2\\\\pi \\\\sqrt{ab-h^2}}\\\\exp \\\\Big [-\\\\Big (a(\\\\xi -\\\\xi _0)^2+b(\\\\eta -\\\\eta _0)^2\\\\cr +2h(\\\\xi -\\\\xi _0)(\\\\eta -\\\\eta _0)\\\\Big )\\\\big /2(ab-h^2)\\\\Big ],\\\\hspace{28.45274pt}$ where $s_{1,2}=-\\\\frac{\\\\gamma }{2m}\\\\pm \\\\sqrt{\\\\frac{\\\\gamma ^2}{4m^2}-\\\\omega _0^2},$ $\\\\xi =(s_1x-v)e^{-s_2t},\\\\hspace{28.45274pt}\\\\xi _0=(s_1x^{\\\\prime }-v^{\\\\prime }),\\\\hspace{36.98866pt}\\\\cr \\\\eta =(s_2x-v)e^{-s_1t},\\\\hspace{28.45274pt}\\\\eta _0=(s_2x^{\\\\prime }-v^{\\\\prime }),\\\\hspace{36.98866pt}\\\\cr a=\\\\frac{\\\\gamma }{\\\\beta m^2s_1}\\\\Big (1-e^{-2s_1t}\\\\Big ),\\\\ \\\\ b=\\\\frac{\\\\gamma }{\\\\beta m^2s_2}\\\\Big (1-e^{-2s_2t}\\\\Big ),\\\\cr h=\\\\frac{2}{\\\\beta m}\\\\Big (1-e^{\\\\gamma t/m}\\\\Big ).\\\\hspace{56.9055pt}$ Similar to the case of no potential in the Section III, let us suppose that a thermalized particle (equilibrated in velocity) was inserted at a position $x_0$ at time $t^{\\\\prime }=0$ , $\\\\rho _0(x^{\\\\prime },v^{\\\\prime })=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi }}\\\\delta (x^{\\\\prime }-x_0)e^{-\\\\beta mv^{\\\\prime 2}/2}.$ After integration over $v^{\\\\prime }$ and $v$ , we get the corresponding spatial density $p(x,t)=\\\\int _{-\\\\infty }^{\\\\infty }G(x,v,t;x^{\\\\prime }v^{\\\\prime },0)\\\\rho _0(x^{\\\\prime },v^{\\\\prime })dx^{\\\\prime }dv^{\\\\prime }dv=\\\\cr \\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi Z}}\\\\omega _0(s_1-s_2)\\\\exp \\\\Big [-\\\\frac{\\\\beta m\\\\omega _0^2}{2 Z}\\\\hspace{42.67912pt}\\\\cr \\\\times \\\\Big (s_1(x-x_0e^{s_2t}) -s_2(x-x_0e^{s_1t})\\\\Big )^2\\\\Big ];\\\\hspace{14.22636pt}$ $\\\\nonumber Z=\\\\Big [s_1(1+e^{s_2t})-s_2(1+e^{s_1t})\\\\Big ]\\\\Big [s_1(1-e^{s_2t})-s_2(1-e^{s_1t})\\\\Big ].$ On the other hand, the coefficient $D(x)$ , Eq.\",\n \"(REF ), becomes constant for the quadratic potential, $D(x)=\\\\frac{\\\\gamma /2m-\\\\sqrt{\\\\gamma ^2/4m^2-\\\\omega _0^2}}{\\\\beta m\\\\omega _0^2}=\\\\frac{-s_1}{\\\\beta m\\\\omega _0^2},$ and GF of the corresponding mapped equation (REF ), $\\\\Big (\\\\partial _t-\\\\partial _xD(x)e^{-\\\\beta m\\\\omega _0^2x^2/2}\\\\partial _xe^{\\\\beta m\\\\omega _0^2x^2/2}\\\\Big )g(x,t;x_0,t_0)\\\\cr =\\\\delta (x-x_0)\\\\delta (t-t_0),\\\\hspace{56.9055pt}$ can be easily found by a calculation similar to the derivation of Eq.\",\n \"(REF ), Appendix A.\",\n \"The result, $g(x,t;x_0,0)=\\\\sqrt{\\\\frac{\\\\beta m}{2\\\\pi (1-e^{2s_1t})}}\\\\omega _0\\\\exp \\\\Big [-\\\\frac{1}{2}\\\\beta m\\\\omega _0^2\\\\cr \\\\times \\\\big (x-x_0e^{s_1t}\\\\big )^2\\\\big /\\\\big (1-e^{2s_1t}\\\\big )\\\\Big ],\\\\hspace{14.22636pt}$ describes evolution of the real space density of a thermalized particle inserted at $x_0$ , too, and can be directly compared with the formula (REF ).\",\n \"First, let us notice that in comparison with Eq.\",\n \"(REF ), the exponential $e^{s_2t}$ disappeared from the formula (REF ).\",\n \"The root $s_2\\\\simeq -\\\\gamma /m$ for $m\\\\rightarrow 0$ makes $e^{s_2t}\\\\simeq e^{-\\\\gamma t/m}$ the term essentially singular in $m$ and so invisible for the recurrence procedure, which works with the operators $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ expanded in $m$ .\",\n \"Using our argumentation from the Section III, $e^{s_2t}$ represents the \\\"transients\\\" neglected by the mapping.\",\n \"On the other hand, $e^{s_1t}\\\\simeq e^{-m\\\\omega _0^2t/\\\\gamma }$ is regular in $m$ small, representing the contribution of the low-lying states, retained by the method.\",\n \"Next, let us stress that the equation (REF ) with $D(x)$ expressed by the expansion (REF ) was derived in the limit of the stationary flow; for the net flux almost constant, which is not the case of the process described by the Eqs.\",\n \"(REF ) and (REF ).\",\n \"Nevertheless, using the approximations $\\\\nonumber s_1-s_2\\\\big (1\\\\pm e^{s_1t}\\\\big )&\\\\simeq & (s_1-s_2)\\\\big (1\\\\pm e^{s_1t}\\\\big ),\\\\cr s_1x-s_2\\\\big (x-x_0e^{s_1t}\\\\big )&\\\\simeq & (s_1-s_2)\\\\big (x-x_0e^{s_1t}\\\\big ),$ applicable for $e^{s_1t}\\\\ll 1$ , the regularized formula (REF ) (with $e^{s_2t}$ neglected) becomes finally Eq.\",\n \"(REF ); i.e.\",\n \"it represents correctly the asymptotic behavior of the spatial density $p(x,t)$ for large time $t$ .\",\n \"If $m$ approaches $\\\\gamma /2\\\\omega _0$ , the transients contributing by $e^{s_2t}$ , neglected by the mapping, become important.\",\n \"In the oscillating regime, $s_{1,2}$ are complex numbers and both are necessary for expressing the real density $p(x,t)$ in Eq.\",\n \"(REF ).\",\n \"The mapping which splits the Hilbert space to the retained low-lying states and the neglected transients, controlled by $m$ small, loses its justification and the method stops working.\",\n \"Mapping in this region requires a different method to be applied.\",\n \"It will be an object of our study in the future.\"\n ],\n [\n \"V. Conclusion\",\n \"Although modeling of transport in confined systems is often based on study of the Langevin equation, Eq.\",\n \"(REF ) in the simplest 1D case, the solutions necessary in practical applications are often accessible only in two limits: either the friction $\\\\gamma \\\\rightarrow 0$ , when the particles obey Newtonian dynamics, or the mass $m\\\\rightarrow 0$ , which corresponds to stochastic dynamics.\",\n \"Any solution in the region between these limits requires working in phase space, which makes the problem much more complicated.\",\n \"The present paper shows how to describe the region of finite $m/\\\\gamma $ while still working in real space, as in the case of stochastic dynamics.\",\n \"The equation governing evolution of the spatial density $p(x,t)$ is the Smoluchowski equation, corresponding to the limit of a massless particle, extended by a series of corrections in powers of $t_0=m/\\\\gamma $ ; $t_0$ can be interpreted as the typical time of thermalizing of the particle's initial velocity by the stochastic force.\",\n \"In general, the extended Smoluchowski equation has the form of Eq.\",\n \"(REF ), $D_0=1/\\\\gamma \\\\beta $ denotes the diffusion constant and the operators $\\\\hat{Z}_k$ are systematically derived within the recurrence procedure presented in the Sect.\",\n \"IV.\",\n \"In the limit of stationary flow, i.e.\",\n \"when the flux is almost constant but nonzero in time and space, this equation can be simplified to Eq.\",\n \"(REF ), where the effective diffusion coefficient $D(x)$ is calculated unambiguously from the operators $\\\\hat{Z}_k$ , Eq.\",\n \"(REF ).\",\n \"In the simplest approximation, when all the derivatives of the potential higher than $U^{\\\\prime \\\\prime }(x)$ are neglected, the series of corrections can be summed up to infinity, giving the formula for $D(x)$ in a closed form, Eq.\",\n \"(REF ), described in Fig.\",\n \"1.\",\n \"Then the equation describes stationary flow in a quadratic potential.\",\n \"The theory works while $4mU^{\\\\prime \\\\prime }(x)<\\\\gamma ^2$ , until the averaged trajectory of a single particle is not oscillatory in the potential wells along the 1D channel.\",\n \"The mapping in the oscillatory regime requires the next analysis, which will be done in the future.\",\n \"Technically, the paper demonstrates that the projection technique developed for mapping of diffusion in 2D (3D) channels with varying cross section [13], [14], [16], can be adapted for the dimensional reduction of a process described by an evolution equation of a different type than the diffusion or Smoluchowski equation.\",\n \"The method has been modified significantly; $m/\\\\gamma $ had to be confirmed as the small parameter controlling the expansion of the corrections $\\\\hat{Z}_k$ , as well as the operators of the backward mapping, $\\\\hat{\\\\omega }$ and $\\\\hat{\\\\eta }$ .\",\n \"In contrast to diffusion, the flux $j(x,t)$ is handled here as a quantity independent of the density $p(x,t)$ .\",\n \"Thus the recurrence procedure, calculating expansions of the correction operators $\\\\hat{Z}_k$ and the operators $\\\\hat{\\\\omega }$ , $\\\\hat{\\\\eta }$ , is in principle the result of combination of three equations, Eqs.\",\n \"(REF ), (REF ) and also (REF ), with the relation of the backward mapping, Eq.\",\n \"(REF ).\",\n \"It is worthwhile to notice that including the mass dependent corrections to the Smoluchowski equation results in the equations (REF ) or (REF ), which are of the same form as the comparable equations obtained from the mapping of diffusion in channels with varying cross section.\",\n \"On the other hand, the effective coefficient $D(x)$ (REF ) has a different symmetry than the similar formulas extending the Fick-Jacobs equation [16], [17] for confined diffusion.\",\n \"Also $D(x)$ can be greater than 1 here (see Fig.\",\n \"1); i.e.\",\n \"the quasi stationary flux is accelerated when passing through a shallow potential well, depending on the nonzero mass of the particles.\",\n \"These interesting properties could be observed in simulations similar to that verifying $D(x)$ in the extended Fick-Jacobs equation [27], [28].\",\n \"The effects of slipping of the particles diffusing under a nonconstant force $F(x)$ due to their inertia, as described in Sect.\",\n \"IV, might also influence the interesting phenomena in the micro and nano world, such as Brownian pumps [29], [30], rectification of the flux in quasi 1D structures [31], stochastic resonance [20], [32], [33], [34], or the negative mobility [35].\",\n \"Study of such applications of the theory presented is expected in the future.\"\n ],\n [\n \"Acknowledgments\",\n \"Support from VEGA grant No.\",\n \"2/0049/12 and CE SAS QUTE project is gratefully acknowledged.\",\n \"P.K.\",\n \"also thanks CIMS, New York University for kind hospitality.\"\n ],\n [\n \"Appendix A: Exact solution\",\n \"The Green's function (REF ) solving the FP equation with zero potential, Eq.\",\n \"(REF ), is calculated here.\",\n \"First we introduce the scaled coordinates $\\\\xi $ , $u$ , $\\\\tau $ according to Eqs.\",\n \"(REF ) and define the function $\\\\Gamma (\\\\xi ,u,\\\\tau ;\\\\xi ^{\\\\prime },u^{\\\\prime },\\\\tau ^{\\\\prime })$ , $G(x,v,t;x^{\\\\prime },v^{\\\\prime },t^{\\\\prime })=e^{-u^2/2}\\\\Gamma (\\\\xi ,u,\\\\tau ;\\\\xi ^{\\\\prime },u^{\\\\prime },\\\\tau ^{\\\\prime })e^{u^{\\\\prime 2}/2},$ satisfying the transformed equation (REF ), $\\\\left[\\\\partial _{\\\\tau }+u\\\\partial _{\\\\xi }-\\\\partial _u^2+u^2-1\\\\right]\\\\Gamma (\\\\xi ,u,\\\\tau ;\\\\xi ^{\\\\prime },u^{\\\\prime },\\\\tau ^{\\\\prime })\\\\hspace{36.135pt}\\\\cr =\\\\frac{\\\\gamma \\\\beta }{4}e^{(u^2-u^{\\\\prime 2})/2}\\\\delta (\\\\xi -\\\\xi ^{\\\\prime })\\\\delta (u-u^{\\\\prime })\\\\delta (\\\\tau -\\\\tau ^{\\\\prime }),\\\\ $ the exponential factor becomes unity due to $\\\\delta (u-u^{\\\\prime })$ .\",\n \"After the Fourier transform in $\\\\xi $ and $\\\\tau $ , $\\\\Gamma (\\\\xi ,u,\\\\tau ;\\\\xi ^{\\\\prime },u^{\\\\prime },\\\\tau ^{\\\\prime })=\\\\int \\\\frac{dkd\\\\nu }{4\\\\pi ^2}e^{ik(\\\\xi -\\\\xi ^{\\\\prime })-i\\\\nu (\\\\tau -\\\\tau ^{\\\\prime })}\\\\Gamma _{k,\\\\nu }(u;u^{\\\\prime }),$ and shifting the velocities by $ik/2$ , $w=u+ik/2$ and $w^{\\\\prime }=u^{\\\\prime }+ik/2$ , the equation $\\\\left[-i\\\\nu -\\\\partial _w^2+w^2-1+k^2/4\\\\right]\\\\Gamma _{k,\\\\nu }(w;w^{\\\\prime })=\\\\frac{\\\\gamma \\\\beta }{4}\\\\delta (u-u^{\\\\prime })$ becomes solvable if $\\\\Gamma _{k,\\\\nu }(w;w^{\\\\prime })$ is expressed in the basis set of the linear harmonic oscillator $\\\\psi _n(w)$ , $\\\\Gamma _{k,\\\\nu }(w;w^{\\\\prime })=\\\\sum _{n=0}^{\\\\infty }\\\\Gamma _n(k,\\\\nu )\\\\psi _n(w)\\\\psi _n^*(w^{\\\\prime }).$ The eigenfunctions $\\\\psi _n(w)$ satisfy $\\\\left(-\\\\partial _w^2+w^2\\\\right)\\\\psi _n(w)=\\\\lambda _n\\\\psi _n(w)=(2n+1)\\\\psi _n(w)$ and we use the integral representation of the Hermite polynomials $H_n(w)$ [36] $\\\\psi _n(w)&=&\\\\frac{1}{\\\\@root 4 \\\\of {\\\\pi }\\\\sqrt{2^n n!\",\n \"}}H_n(w)e^{-w^2/2}\\\\cr &=&\\\\sqrt{\\\\frac{2^n}{n!\\\\pi ^{3/2}}}e^{-w^2/2}\\\\int _{-\\\\infty }^{\\\\infty }(w+ir)^ne^{-r^2}dr\\\\hspace{14.45377pt}$ in the next calculation.\",\n \"Using the transformations above, we find $\\\\Gamma _n(k,\\\\nu )=\\\\frac{\\\\gamma \\\\beta /4}{-i\\\\nu +\\\\lambda _n-1+k^2/4}.$ Applying it in the formulas (REF ) and (REF ), we integrate the last one over $\\\\nu $ in the complex plane, $\\\\Gamma (\\\\xi ,u,\\\\tau ;\\\\xi ^{\\\\prime },u^{\\\\prime },\\\\tau ^{\\\\prime })&&\\\\hspace{-14.45377pt}=\\\\frac{\\\\Theta (\\\\tau -\\\\tau ^{\\\\prime })}{8\\\\pi D_0}\\\\int _{-\\\\infty }^{\\\\infty }dke^{ik(\\\\xi -\\\\xi ^{\\\\prime })-k^2(\\\\tau -\\\\tau ^{\\\\prime })/4}\\\\cr &&\\\\times \\\\sum _{n=0}^{\\\\infty }e^{-2n(\\\\tau -\\\\tau ^{\\\\prime })}\\\\psi _n(w)\\\\psi _n^*(w^{\\\\prime }),$ $\\\\Theta (x)$ denotes the Heaviside unit step function and $D_0=1/\\\\gamma \\\\beta $ is the diffusion constant.\",\n \"Now the integral relation (REF ) is used for $\\\\psi _n(w)$ and $\\\\psi _n^*(w^{\\\\prime })$ and the summation over $n$ can be readily completed.\",\n \"Finally, the straightforward triple integration over $k$ , $r$ , $r^{\\\\prime }$ is performed and using the transformation (REF ) results in the formula (REF ).\"\n ],\n [\n \"Appendix B: Quadratic approximation\",\n \"Derivation of the formula (REF ) for the effective diffusion coefficient $D(x)$ with all the derivatives higher than $U\\\"(x)$ neglected is presented here.\",\n \"This approximation corresponds to local replacing of the potential by a parabola, $U(x)\\\\simeq \\\\kappa (x-x_0)^2/2+U_0$ , where $\\\\kappa ,\\\\ x_0$ and $U_0$ are fitting parameters.\",\n \"First we simplify Eq.\",\n \"(REF ).\",\n \"For quadratic potential, the right hand side can be rewritten as $\\\\nonumber e^{-\\\\beta U(x)}\\\\left(1+t_0\\\\hat{Z}\\\\right)^{-1}e^{\\\\beta U(x)}=\\\\left(1+t_0 e^{-\\\\beta U}\\\\hat{Z}(x)e^{\\\\beta U}\\\\right)^{-1},$ $t_0\\\\hat{Z}=\\\\sum _{n=1}^{\\\\infty }t_0\\\\hat{Z}_n$ ; the difference contains only the higher derivatives of $U(x)$ , which are zero.\",\n \"Hence $D(x)/D_0=1+e^{-\\\\beta U(x)}\\\\sum _{n=1}^{\\\\infty }t_0^n\\\\hat{Z}_ne^{\\\\beta U(x)}.$ The formulas for $D(x)$ have been derived considering stationary flow; $j(x,t)=j$ is constant.\",\n \"It simplifies the relation (REF ); $\\\\partial _tj=0$ .\",\n \"Thus the right hand side represents stationary flux, which can be directly compared with Eq.\",\n \"(REF ), giving a much simpler relation between $\\\\hat{Z}_n$ and $\\\\hat{I}_n$ than Eq.\",\n \"(REF ), $e^{-\\\\beta U(x)}\\\\hat{Z}_n(x)\\\\partial _xe^{\\\\beta U(x)}p(x)=\\\\partial _x\\\\hat{I}_n(x)p(x)$ for any stationary solution $p(x)$ .\",\n \"Calculation of the coefficients of $D(x)$ according to Eq.\",\n \"(REF ) requires us to take $\\\\partial _x\\\\exp [\\\\beta U(x)]p(x)=\\\\exp [\\\\beta U(x)]$ , hence finally $e^{-\\\\beta U(x)}\\\\hat{Z}_ne^{\\\\beta U(x)}&=&\\\\frac{2}{\\\\sqrt{\\\\pi }}\\\\partial _x\\\\int _{-\\\\infty }^{\\\\infty }u^2due^{-u^2}\\\\hspace{57.81621pt}\\\\cr &&\\\\times \\\\hat{\\\\omega }_n(x,u)e^{-\\\\beta U(x)}\\\\int dx e^{\\\\beta U(x)}$ after application of Eq.\",\n \"(REF ).\",\n \"Before writing the explicit formulas for $\\\\hat{\\\\omega }_n$ for the quadratic potential, we define the polynomials $P_n(u)&=&\\\\sum _{k=0}^{n}\\\\frac{(-1)^{n-k}2^{2k-n}}{(2k)!(n-k)!\",\n \"}u^{2k},\\\\cr Q_n(u)&=&\\\\sum _{k=0}^{n}\\\\frac{(-1)^{n-k}2^{2k-n}}{(2k+1)!(n-k)!\",\n \"}u^{2k},$ $n=1,2, ...$ , coming from the expansions of $\\\\hat{\\\\omega }_n$ and $\\\\hat{\\\\eta }_n$ for zero potential in $t_0$ , Eqs.\",\n \"(REF ).\",\n \"The first few polynomials are visible in the round brackets of Eq.\",\n \"(REF ).\",\n \"One can check by direct integration that $\\\\int _{-\\\\infty }^{\\\\infty }Q_n(u)u^2e^{-u^2}du&=&\\\\frac{\\\\sqrt{\\\\pi }}{2^{n+1}}\\\\sum _{k=0}^n\\\\frac{(-1)^{k-n}}{k!(n-k)!\",\n \"}=0,\\\\cr \\\\int _{-\\\\infty }^{\\\\infty }P_n(u)u^2e^{-u^2}du&=&\\\\frac{\\\\sqrt{\\\\pi }}{2}\\\\delta _{n,1},$ corresponding to the normalization of $\\\\hat{\\\\eta }_n$ , Eq.\",\n \"(REF ), and the relations (REF ), (REF ), proving no correction to the Smoluchowski equation in the case $U(x)=0$ .\",\n \"The operators $\\\\hat{\\\\omega }_n$ and $\\\\hat{\\\\eta }_n$ for the quadratic potential have the form $\\\\hat{\\\\omega }_n&=&e^{-\\\\beta U(x)}\\\\sum _{k=1}^nc_{n,k}P_k(u)\\\\left(\\\\beta U^{\\\\prime \\\\prime }\\\\right)^{n-k}\\\\partial _x^{2k}e^{\\\\beta U(x)},\\\\hspace{21.68121pt}\\\\cr \\\\hat{\\\\eta }_n&=&e^{-\\\\beta U(x)}\\\\sum _{k=1}^nc_{n,k}Q_k(u)\\\\left(\\\\beta U^{\\\\prime \\\\prime }\\\\right)^{n-k}\\\\partial _x^{2k}e^{\\\\beta U(x)},$ with the coefficients $c_{n,k}=D_0^n\\\\frac{2k\\\\ (2n-1)!}{(n-k)!(n+k)!\",\n \"}.$ Due to the integrals, Eq.\",\n \"(REF ), only the first terms with $P_1(u)$ in Eq.\",\n \"(REF ) contribute to the expansion of $D(x)$ , Eq.\",\n \"(REF ).\",\n \"Then the functions become $e^{-\\\\beta U(x)}\\\\hat{Z}_ne^{\\\\beta U(x)}&=&c_{n,1}\\\\partial _xe^{-\\\\beta U(x)}\\\\left(\\\\beta U^{\\\\prime \\\\prime }\\\\right)^{n-1}\\\\partial _xe^{\\\\beta U(x)}\\\\cr &=&\\\\frac{2\\\\ (2n-1)!}{(n-1)!(n+1)!\",\n \"}\\\\left(D_0\\\\beta U^{\\\\prime \\\\prime }\\\\right)^n,$ taking $U^{(3)}(x)=0$ into account.\",\n \"Applied in Eq.\",\n \"(REF ) it results in the expansion of $D(x)$ , Eq.\",\n \"(REF ).\",\n \"Finally, one has to verify that the formulas (REF ) satisfy the recurrence relations (REF ) and (REF ), acting on the function $p(x)=\\\\exp [-\\\\beta U(x)]\\\\int dx\\\\exp [\\\\beta U(x)]$ .\",\n \"Although the equations simplify notably due to neglecting the derivatives higher than $U^{\\\\prime \\\\prime }(x)$ , we omit the details of this tedious but straightforward calculation.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1145"}}},{"rowIdx":1866988,"cells":{"text":{"kind":"list like","value":[["The paradox of soft singularity crossing and its resolution by\n distributional cosmological quantitities"],["Abstract A cosmological model of a flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter exhibits a future soft singularity, where the pressure of the anti-Chaplygin gas diverges (while its energy density is finite).","Despite infinite tidal forces the geodesics pass through the singularity.","Due to the dust component, the Hubble parameter has a non-zero value at the encounter with the singularity, therefore the dust implies further expansion.","With continued expansion however, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence from the point of view of the anti-Chaplygin gas only a contraction is allowed.","Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously.","This obviosly could not happen.","We solve the paradox by redefining the anti-Chaplygin gas in a distributional sense.","Then a contraction could follow the expansion phase at the singularity at the price of a jump in the Hubble parameter.","Although such an abrupt change is not common in any cosmological evolution, we explicitly show that the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity.","We also prove that the Israel junction conditions are obeyed through the singular spatial hypersurface.","In particular we enounce and prove a more general form of the Lanczos equation."],["Introduction","The problem of cosmological singularities has been attracting the attention of theoreticians working in gravity and cosmology since the early fifties [1], [2], [3].","In the sixties general theorems about the conditions for the appearance of singularities were proven [4], [5] and the oscillatory regime of approaching the singularity [6], the Mixmaster universe [7] was discovered.","Basically, until the end of nineties almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius.","However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [8]-[18], characterized by a diverging $\\ddot{a}$ whereas both the scale factor $a$ and $\\dot{a}$ are finite.","Then, the Hubble parameter $H=\\dot{a}/a~$ and the energy density $\\rho $ are also finite, while the first derivative of the Hubble parameter and the pressure $p$ diverge.","Until recent years, however, sudden future singularities attracted only a limited interest among researchers.","The interest grew due to two reasons.","The recent discovery of the cosmic acceleration [19] has stimulated the elaboration of dark energy models, responsible for such a phenomenon (see e.g.","for review [20]).","Remarkably in some of these models the sudden singularities arise quite naturally.","Another source of the interest to sudden singularities is the development of brane models [10], [11], [18], where singularities of this kind could arise naturally (sometimes these singularities, arising in brane-world models, are called “quiescent” [10]).","In the investigations devoted to sudden singularities one can distinguish three main topics.","The first of them deals with the question of the compatibility of the models possessing soft singularities with observational data [15], [21], [22].","The second direction is connected with the study of quantum effects [11], [17], [23], [24], [25].","Here one can see two subdirections: the study of quantum corrections to the effective Friedmann equation, which can eliminate classical singularities or at least, change their form [10], [17], [23]; and the study of solutions of the Wheeler-DeWitt equation for the quantum state of the universe in the presence of sudden singularities [24], [25].","The third direction is connected with the possibility of the sudden singularity crossing in classical cosmology [26], [27], [28], [29], [25].","The present paper is devoted exactly to this topic.","A particular feature of the sudden future singularities is their softness [26].","As the Christoffel symbols depend only on the first derivative of the scale factor, they are regular at these singularities.","Hence, the geodesics are well behaved and they can cross the singularity [26].","One can argue that the particles crossing the singularity will generate the geometry of the spacetime, providing in such a way a “soft rebirth” of the universe after the singularity crossing [29].","Note that the possibility of crossing of some kind of cosmological singularities was noticed already in the early paper by Tipler [30].","A close idea of integrable singularities in black holes, which can give origin to a cosmogenesis was recently put forward in [31].","As a starting point we consider an interesting example of a sudden future singularity - the Big Brake which was discovered in Ref.","[32] while studying a particular tachyon cosmological model.","The particularity of the Big Brake singularity consists in the fact that the time derivative of the scale factor is not only finite, but exactly equal to zero.","That makes the analysis of the behavior in the vicinity of singularity especially convenient.","In particular, in Ref.","[22] it was shown that the predictions of the future of the universe in this model [32] are compatible with the supernovae type Ia data, while in Refs.","[24], [25] some quantum cosmological questions were studied in the presence of the Big Brake singularity.","The simplest cosmological model allowing a Big Brake singularity was also introduced in Ref.","[32].","This model is based on the perfect fluid, dubbed “anti-Chaplygin gas”.","This fluid is characterized by the equation of state $p=\\frac{A}{\\rho }, $ where $A$ is a positive constant.","Such and equation of state arises, for example, in the theory of wiggly strings [33].","In paper [32] a fluid obeying the equation of state (REF ) was called \"anti-Chaplygin gas\" in analogy with the Chaplygin gas [34] which has the equation of state $p=-A/\\rho $ and has acquired some popularity as a candidate for a unified theory of dark energy and dark matter [35].","An explicit example of the crossing of the Big Brake singularity was described in detail in paper [29], were the tachyon model [32], [22] was investigated.","In this model the tachyon field passes through the singularity, continuing its evolution with a recollapse towards a Big Crunch.","In a simpler model, based on the anti-Chaplygin gas, such a crossing is even easier to understand.","The next natural step in the analysis of the soft singularities seems to be obvious.","One can consider a soft singularity of more general type than the big brake by adding to the tachyon matter or to the anti-Chaplygin gas some dustlike matter.","However, in this case the traversability of the singularity seems to be obstructed.","The main reason for this is that while the energy density of the tachyonic field (or of the anti-Chaplygin gas) vanishes at the singularity, the energy density of the matter component does not, leaving the Hubble parameter at the singularity with a finite value.","Then some kind of the paradox arises: if the universe continues its expansion, and if the equation of state of the component of matter, responsible for the appearance of the soft singularity (in the simplest case, the anti-Chaplygin gas) is unchanged, then the expression for the energy density of this component becomes imaginary, which is unacceptable.","The situation looks rather strange: indeed, the model, including dust should be in some sense more regular, than a single exotic fluid, the anti-Chaplygin gas.","Thus, if the model based on the pure anti-Chaplygin gas has a traversable Big Brake singularity, than the more general singularity arising in the model based on the mixture of the anti-Chaplygin gas and dust should also be traversable.","Related to that, it was recently shown that general soft singularities arising in the Friedmann model, filled with the scalar field with a negative potential, inversely proportional to this field are traversable.","So, what could be wrong with the simple two-fluid model?","One can see that what we face is some sort of a clash between the equation of state of one of these fluids and the dynamics (the Friedmann and Raychaudhuri equations) and energy conservation equations.","In this paper we shall try to resolve this paradox, insisting on the preservation of the equation of state of the anti-Chaplygin gas.","The price which one has to pay for it is the obligatory use of the generalized functions for some cosmological quantities.","Namely, the anti-Chaplygin gas remains physical if rather a recollapse follows, but then the Hubble parameter would have a sharp jump, obstructing the validity of the Raychaudhuri equation (the second Friedmann equation) in the usual sense of functions.","Thus, apparently, the evolution cannot be continued through the soft singularity, unless treating the cosmological quantities as distributions.","We claim that such a generalization is mathematically rigorous, moreover, the introduction of distributions is not so drastic, as it looks at the first sight, as the pressure of the anti-Chaplygin gas diverges anyhow at the soft singularity (as it so does for the tachyon field).","Then in the Conclusion we shall dwell on the possible physical sense of the proposed constructions and its possible alternatives.","The plan of the paper is as follows.","In Section we discuss generic Friedmann space-times, which admit $\\dot{H}=-\\infty $ type singularities, while the Hubble parameter $H$ remains finite.","Such singularities are related to corresponding divergencies in the pressure of the perfect fluid filling the Friedmann universe (while its energy density stays finite).","We investigate the kinematics, the geodesic equations, the geodesic deviation equations in the vicinity of these singularities and also prove that these singularities are weak.","In Section we discuss a mixture of the anti-Chaplygin gas and dust in a flat Friedmann universe and explain the essence of the paradox.","We explicitly derive the behavior of the energy density and pressure in the vicinity of the soft singularity and we solve the geodesic equations in this region.","While the singularity turns to be traversable by the geodesics, the explicit solution also shows that the Raychaudhuri equation is violated at the singularity.","In Section we add generalized distributional contributions to both the pressure and energy density, such that (a) the equation of state of the anti-Chaplygin gas still holds and (b) the singularity becomes traversable.","We also perform checks of the Friedmann, Raychaudhuri and continuity equations, which all hold valid across the singularity in a distributional sense.","In the process we employ a number of Propositions on distributions presented and proved in Appendix A.","For the convenience of the reader we present a related semi-heuristic discussion of two known distributional identities in Appendix B.","We stress that the distributional modifications of the energy density and pressure do not modify the cosmological evolution, but they make possible the soft singularity crossing.","In Section we revisit the junction conditions along a spacelike hypersurface in a flat Friedmann universe.","The future soft singularity represents such a spatial hypersurface, along which the energy-momentum tensor diverges.","Extending the space-time through this hypersurface is possible by obeying both Israel junction conditions.","While the first condition, requiring the continuity of the induced metric is easy to satisfy (the metric stays regular at the soft singularity), the second condition relates the jump in the extrinsic curvature to the distributional part of the energy-momentum tensor through the Lanczos equation.","We will show that in flat Friedmann space-times the Lanczos equation holds for a more general class of distributional energy-momentum tensors.","With this we give a second proof that the generalized distributional energy-momentum tensor assures the traversability of the soft singularity.","In the process we employ a simple form of the Lanczos equation valid in flat Friedmann universes, derived in Appendix .","We summarize our results and give some further outlook in the Concluding Remarks.","We chose $c=1$ and $8\\pi G/3=1$ .","A subscript $S$ denotes the value of the respective quantity at the soft singularity."],["Pressure singularities in flat Friedmann universes","The line element squared of a flat Friedmann universe can be written as $ds^{2}=-dt^{2}+a^{2}\\left( t\\right) \\sum _{\\alpha }\\left( dx^{\\alpha }\\right)^{2}\\ ,$ where $x^{\\alpha }$ ($\\alpha =1,2,3$ ) are Cartesian coordinates.","The evolution of the Friedmann universe is governed by the Raychaudhuri (second Friedmann) equation $\\dot{H}=-\\frac{3}{2}\\left( \\rho +p\\right) \\ , $ and by the continuity equation for the fluid $\\dot{\\rho }+3H\\left( \\rho +p\\right) =0\\ .","$ Here the dot denotes the derivative with respect to cosmological time $t$ .","A first integral of this system is given by the first Friedmann equation $H^{2}=\\rho \\ .","$ It is easy to see that the Raychaudhuri equation can be obtained from the first Friedmann and the continuity equations."],["Kinematics in the vicinity of sudden singularities","Sudden singularities are characterized by finite $H$ and $\\dot{H}\\rightarrow -\\infty $ (finite $\\dot{a}$ and $\\ddot{a}\\rightarrow -\\infty $ ) at some finite scale factor $a$ .","The energy density of the fluid is finite but its pressure diverges at this type of singularity, therefore the term “pressure singularity” is also in use.","Then, we would like to emphasize the fact that there is an essential difference between the sudden singularities with $H=0$ and with $H>0$ .","As has been already mentioned in the Introduction, in the first case, which is called Big Brake, the universe begin contracting and running towards the Big Crunch singularity.","Exactly this occurs in models based on tachyon field with a particular potential [32], [22], [29] or in the anti-Chaplygin gas models.","In the case of the model based on the mixture of one of this fluids and dust, we encounter the second situation when the value of the Hubble constant is positive at the moment of encounter with the sudden singularity.","That means that after crossing the singularity the universe should continue its expansion, but the anti-Chaplygin gas becomes ill-defined, as it will be shown in detail in Section , devoted to the model based on mixture of the anti-Chaplygin gas and dust.","One possible way of overcoming this obstacle is to allow the jump in the sign of the Hubble parameter, which as was mentioned in the Introduction leaves valid the first Friedmann equation, the continuity equation and the equation of state, while making invalid the Raychaudhuri equation.","This last obstacle can be cured by the acceptance the distributional Dirac $\\delta $ -function type contributions into the pressure and the energy density, which is described in detail in the section IV."],["Geodesics in the vicinity of sudden singularities","The geodesic equations in flat Friedmann space-time are $\\frac{d^{2}x^{\\alpha }}{d\\lambda ^{2}}+2\\frac{\\dot{a}}{a}\\frac{dt}{d\\lambda }\\frac{dx^{\\alpha }}{d\\lambda }=0\\ ,$ $\\frac{d^{2}t}{d\\lambda ^{2}}+a\\dot{a}\\sum _{\\alpha }\\left( \\frac{dx^{\\alpha }}{d\\lambda }\\right) ^{2}=0\\ ,$ where $\\lambda $ is an affine parameter.","Integrating these equations yields $\\frac{dx^{\\alpha }}{d\\lambda }=\\frac{P^{\\alpha }}{a^{2}}\\ ,$ $\\left( \\frac{dt}{d\\lambda }\\right) ^{2}=\\epsilon +\\frac{P^{2}}{a^{2}}\\ ,$ with $P^{\\alpha }$ , $\\epsilon $ integration constants and $P^{2}=\\sum _{\\alpha }\\left( P^{\\alpha }\\right) ^{2}$ .","The quantity $\\epsilon $ is fixed by the length of the tangent vector $u^{a}$ of the geodesic as $\\epsilon =-u_{a}u^{a}$ ; i.e.","one for timelike and zero for lightlike orbits.","In a comoving system $P^{\\alpha }=0$ and $t=\\lambda $ is affine parameter.","Eqs.","(REF ) and (REF ) are singular only for vanishing scale factor (see also Ref.","[26]).","Therefore, the existence of a solution $t\\left( \\lambda \\right) $ , $x^{\\alpha }\\left(\\lambda \\right) $ of Eqs.","(REF ) and (REF ) is assured by the Cauchy-Peano theorem for any nonzero $a$ (including the soft singularity).","Thus the functions $t\\left( \\lambda \\right) $ and $x^{\\alpha }\\left( \\lambda \\right) $ , i.e.","the geodesics can be continued through the singularity occurring at finite scale factor.","Only derivatives of higher order than two of $t\\left( \\lambda \\right) $ and $x^{\\alpha }\\left( \\lambda \\right) $ are singular (as they contain $\\ddot{a}$ ), however these do not appear in the geodesic equations.","Pointlike particles moving on geodesics do not experience any singularity.","Thus, as we argued in the preceding paper [29] one is not obliged to consider such a singularity as a final state of the universe.","Indeed, passing through this singularity the matter recreates also the spacetime in a unique way, at least for such simple models, as those based on Friedmann metrics."],["Deviation equation in the vicinity of sudden singularities","The 3-spaces with $t=$ const have vanishing Riemann curvature.","However, the 4-dimensional Riemann curvature tensor has the nonvanishing components: $R_{\\,\\,\\,\\,\\,t\\beta t}^{\\alpha } &=&-\\frac{\\ddot{a}}{a}\\delta _{\\beta }^{\\alpha }=\\left( -\\dot{H}+H^{2}\\right) \\delta _{\\beta }^{\\alpha }\\ ,\\\\R_{\\,\\,\\,\\,\\,212}^{1} &=&R_{\\,\\,\\,\\,\\,313}^{1}=R_{\\,\\,\\,\\,\\,323}^{2}=\\dot{a}^{2} $ and the corresponding components arising from symmetry.","Here $\\alpha ,\\beta =1,2,3$ .","Remarkably, all components which diverge at the singularity are of the type $R_{tata}$ [29].","Therefore, the singularity arises in the mixed spatio-temporal components.","The geodesic deviation equation along the integral curves of $u=\\partial /\\partial t$ (which are geodesics with affine parameter $t$ ) is $\\dot{u}^{a}=-R_{\\ \\ cbd}^{a}\\eta ^{b}u^{c}u^{d}\\ ,$ where $\\eta ^{b}$ is the deviation vector separating neighboring geodesics, chosen to satisfy $\\eta ^{b}u_{b}=0$ .","For a Friedmann universe it becomes $\\dot{u}^{a}=-R_{\\ \\ tbt}^{a}\\eta ^{b}\\propto \\ddot{a}\\ ,$ which at the singularity diverges as $-\\infty $ .","Therefore, when approaching the singularity, the tidal forces manifest themselves as an infinite braking force stopping the further increase of the separation of geodesics, but not the evolution along the geodesics.","With $\\ddot{a}<0$ in the vicinity of the singularity, once the geodesics have passed through, they will approach each other.","Therefore a contraction phase will follow: everything that has reached the singularity will bounce back."],["The type of the singularity","In this subsection we shall present the classification of singularities, based on the point of view of finite size objects, which approach these singularities.","In principle, finite size objects could be destroyed while passing through the singularity due to the occurring infinite tidal forces.","A strong curvature singularity is defined by the requirement that an extended finite object is crushed to zero volume by tidal forces.","We give below Tipler's [30] and Królak's [36] definitions of strong curvature singularities together with the relative necessary and sufficient conditions.","An alternative definition of the softness of a singularity, based on a Raychaudhuri averaging, was developed by Dabrowski [37].","According to Tipler's definition if every volume element defined by three linearly independent, vorticity-free, geodesic deviation vectors along every causal geodesic through a point $p$ vanishes, a strong curvature singularity is encountered at the respective point $p$ [30], [26].","The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\lambda _{s}$ ($\\lambda $ is affine parameter of the curve) [38] is that the double integral $\\int _{0}^{\\lambda }d\\lambda ^{\\prime }\\int _{0}^{\\lambda ^{\\prime }}d\\lambda ^{\\prime \\prime }\\left|R_{\\,\\,ajb}^{i}u^{a}u^{b}\\right|$ diverges as $\\lambda \\rightarrow \\lambda _{s}$ .","A similar condition is valid for lightlike geodesics, with $R_{\\,\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\,ab}u^{a}u^{b}$ in the double integral.","Królak's definition is less restrictive.","A future-endless, future-incomplete null (timelike) geodesic $\\gamma $ is said to terminate in the future at a strong curvature singularity if, for each point $p\\in \\gamma $ , the expansion of every future-directed congruence of null (timelike) geodesics emanating from $p$ and containing $\\gamma $ becomes negative somewhere on $\\gamma $ [36], [39].","The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\lambda _{s}$ [38] is that the integral $\\int _{0}^{\\lambda }d\\lambda ^{\\prime }\\left|R_{\\,\\,ajb}^{i}u^{a}u^{b}\\right|$ diverges as $\\lambda \\rightarrow \\lambda _{s}$ .","Again, a similar condition is valid for lightlike geodesics, with $R_{\\,\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\,ab}u^{a}u^{b}$ in the integral.","In flat Friedmann space-time the comoving observers move on geodesics having four velocity $u=\\partial /\\partial t$ , where $t$ is affine parameter.","The nonvanishing components of Riemann tensor are given by Eq.","(REF ).","Since $H$ is finite along the geodesics, neither of the integrals (REF ) and (REF ) diverge at the singularity.","The singularity is weak (soft) according to both Tipler's and Królak's definitions.","That means although the tidal forces become infinite, the finite objects are not necessarily crushed when reaching the singularity (see also [26])."],["The paradox of the soft singularity crossing in the cosmological\nmodel based on the anti-Chaplygin gas and dust universe","We discuss an universe filled with two components.","One is the anti-Chaplygin gas with the equation of state (REF ) and other is the pressureless dust.","The solution of the continuity equation for the anti-Chaplygin gas gives the following dependence of its energy density on the scale factor: $\\rho _{ACh}=\\sqrt{\\frac{B}{a^{6}}-A}\\ , $ where $B$ is a positive constant, determining the initial condition.","The energy density of the dust-like matter is as usual $\\rho _{m}=\\frac{\\rho _{m,0}}{a^{3}}\\ ,~ $ where $\\rho _{m,0}$ is a constant.","It is clear that when during the expansion of the universe, its scale factor approaches the value $a_{S}=\\left( \\frac{B}{A}\\right) ^{\\frac{1}{6}}\\ , $ the energy density of the anti-Chaplygin gas vanishes, and its pressure grows to infinity.","That means that the deceleration also becomes infinite.","However, the energy density of dust remains finite, hence the same is true also for the Hubble parameter.","It is here that the paradox arises: if the universe continues to expand, the expression under the sign of the square root in Eq.","(REF ) becomes negative and the energy density of the anti-Chaplygin gas becomes ill-defined.","A way out of this situation is only by assuming that at this moment the Hubble parameter changes its sign, while keeping its absolute value (such that the energy density will not have a jump, as implied by the Friedmann equation).","This possibility will be studied in detail in the following subsections."],["Evolutions in the vicinity of the singularity","Let us substitute the expressions (REF ) and (REF ) into the first Friedmann equation.","We shall find its solution for the universe approaching to the soft singularity point at the moment $t_{S}$ (the latter cannot be found analytically, but its value is not important for our analysis): $a(t)=a_{S}-\\sqrt{\\frac{\\rho _{m,0}}{a_{S}}}(t_{S}-t)-\\sqrt{\\frac{2Aa_{S}^{2}}{3H_{S}}}(t_{S}-t)^{3/2}\\ , $ where $H_{S}=\\sqrt{\\frac{\\rho _{m,0}}{a_{S}^{3}}} $ is the value of the Hubble parameter at $t=t_{S}$ .","Correspondingly the leading terms of the energy densities of the anti-Chaplygin gas and dust, also of the pressure of the anti-Chaplygin gas are $\\rho _{m}=H_{S}^{2}+3H_{S}^{3}(t_{S}-t)\\ , $ $\\rho _{ACh}=\\sqrt{6AH_{S}(t_{S}-t)}\\ , $ $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}(t_{S}-t)}}\\ .","$ One can see that the expressions (REF ), (REF ) and (REF ) cannot be continued for $t>t_{S}$ due to the emerging negative quantities under the square roots.","The assumption of a sharp transition from expansion to contraction implies the following changes in Eqs.","(REF ) – (REF ): $a(t)=a_{S}-\\sqrt{\\frac{\\rho _{m,0}}{a_{S}}}|t_{S}-t|-\\sqrt{\\frac{2Aa_{S}^{2}}{3H_{S}}}|t_{S}-t|^{3/2}\\ , $ $\\rho _{m}=H_{S}^{2}+3H_{S}^{3}|t_{S}-t|\\ , $ $\\rho _{ACh}=\\sqrt{6AH_{S}|t_{S}-t|}\\ , $ $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}\\ .","$ The quantities (REF )–(REF ) are well-defined and continuous at the moment of the singularity crossing.","The expression for the pressure (REF ) is divergent, but this divergence is integrable and this is sufficient for our purposes.","These new expressions satisfy the Friedmann equation, the continuity equations and the equation of state for the anti-Chaplygin gas.","However, the time derivatives of these quantities are not continuous and it is the reason of the failure of the Raychaudhuri equation.","We shall analyze this problem in the following section, but before we discuss the geodesics in the vicinity of the singularity."],["Singularity crossing geodesics","We can integrate explicitly the geodesics equations (REF ) and (REF ) in the vicinity of singularity, using the expression (REF ) for the cosmological factor, also taken in the vicinity of singularity.","Choosing the affine parameter in such a way that the point $\\lambda =0$ corresponds to the singularity crossing we obtain up to the second order in $\\lambda $ terms $t=t_{S}+\\sqrt{\\epsilon +\\frac{P^{2}}{a_{S}^{2}}}\\lambda +\\frac{P^{2}H_{S}}{2a_{S}^{2}}sgn(\\lambda )\\lambda ^{2}\\ , $ $x^{\\alpha }=x_{S}^{\\alpha }+\\frac{P^{\\alpha }}{a_{S}^{2}}\\lambda +\\sqrt{\\epsilon +\\frac{P^{2}}{a_{S}^{2}}}\\frac{P^{\\alpha }H_{S}}{a_{S}^{2}}sgn(\\lambda )\\lambda ^{2}\\ .","$ One can see from Eqs.","(REF ) and (REF ) that not only the time and spatial coordinates of the geodesics are continuous at the soft singularity crossing, but also their first derivatives with respect to the affine parameter $\\lambda $ ."],["Singularity crossing, the Raychaudhuri equation and distributions\n","Let us discuss the expressions for the Hubble parameter and its time derivative in the vicinity of the singularity.","Starting from the expression (REF ) we obtain $H(t) &=&H_{S}sgn(t_{S}-t) \\\\&&+\\sqrt{\\frac{3A}{2H_{S}a_{S}^{4}}}sgn(t_{S}-t)\\sqrt{|t_{S}-t|}~,$ $\\dot{H}=-2H_{S}\\delta (t_{S}-t)-\\sqrt{\\frac{3A}{8H_{S}a_{S}^{4}}}\\frac{sgn(t_{S}-t)}{\\sqrt{|t_{S}-t|}}~.","$ Naturally, the $\\delta $ -term in $\\dot{H}$ arises because of the jump in $H$ , as the expansion of the universe is followed by a contraction.","To restore the validity of the Raychaudhuri equation we shall add a singular $\\delta $ -term to the pressure of the anti-Chaplygin gas, which will acquire the form $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}+\\frac{4}{3}H_{S}\\delta (t_{S}-t)~.$ The equation of state (REF ) of the anti-Chaplygin gas is preserved, if we also modify the expression for its energy density: $\\rho _{ACh}=\\frac{A}{\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}+\\frac{4}{3}H_{S}\\delta (t_{S}-t)}~.","$ The last expression should be understood in the sense of the composition of distributions (see Appendix A and the references therein).","In order to prove that $p_{ACh}$ and $\\rho _{ACh}$ represent a self-consistent solution of the system of cosmological equations, we shall use the following distributional identities: $\\left[ sgn\\left( \\tau \\right) g\\left( \\left|\\tau \\right|\\right) \\right] \\delta \\left( \\tau \\right) &=&0\\ , \\\\\\left[ f\\left( \\tau \\right) +C\\delta \\left( \\tau \\right) \\right] ^{-1}&=&f^{-1}\\left( \\tau \\right) \\ , \\\\\\frac{d}{d\\tau }\\left[ f\\left( \\tau \\right) +C\\delta \\left( \\tau \\right) \\right] ^{-1} &=&\\frac{d}{d\\tau }f^{-1}\\left( \\tau \\right) \\ .$ Here $g\\left( \\left|\\tau \\right|\\right) $ is bounded on every finite interval, $f\\left( \\tau \\right) >0$ and $C>0$ is a constant.","These identities follow from the Propositions 1, 2 and the Corollary enounced and proved in the Appendix A.","The parameter $\\tau $ stays instead of the difference $t_{S}-t$ .","Because of Eq.","(), the energy density (REF ) behaves as a continuous function which vanishes at the singularity.","The first term in the expression for the pressure (REF ) diverges at the singularity.","Therefore the addition of a Dirac delta term, which is not changing the value of $p_{ACh}$ at any $\\tau \\ne 0$ (i.e.","$t\\ne t_{S}$ ) does not look too drastic and might be considered as a some kind of renormalization.","To prove that Friedmann, Raychaudhuri and continuity equations are satisfied we must only investigate those terms, appearing in the field equations, which contain Dirac $\\delta $ -functions, since without them, these equations can be reduced to those we have found in the previous section.","First, we check the continuity equation for the anti-Chaplygin gas.","Due to the identities ()-(), the $\\delta \\left(\\tau \\right) $ -terms occurring in $\\rho _{ACh}$ and $\\dot{\\rho }_{ACh}$ could be dropped.","We keep them however in order to have the equation of state explicitly satisfied.","Then the $\\delta \\left( \\tau \\right) $ -term appearing in $3Hp_{ACh}$ vanishes, because the Hubble parameter changes sign at the singularity [see Eq.","(REF )].","The $\\delta \\left( \\tau \\right) $ -term appearing in $\\rho _{ACh}$ does not affect the Friedmann equation due to the identity ().","Finally, the $\\delta $ -term arising in the time derivative of the Hubble parameter in the left-hand side of the Raychaudhuri equation is compensated by the conveniently chosen $\\delta $ -term in the right-hand side of Eq.","(REF )."],["The junction conditions across the singularity","In this section we discuss the singularity crossing in a slightly different way, by analyzing the junction conditions.","We have to match two space-time regions across the space-like hypersurface $\\tau =0$ .","The junction of two space-time regions has to obey the Israel matching conditions [40], namely, the induced metric should be continuous and the extrinsic curvature of the junction hypersurface could possibly have a jump, which is related to the distributional energy-momentum tensor on the hypersurface by the Lanczos equation.","The scale factor being continuous across the singularity the first Israel condition is obeyed.","We will next prove that the second Israel junction condition (the Lanczos equation [41], [40]) is also satisfied.","For this we have to check whether Eq.","$\\frac{\\partial }{\\partial t}\\left( Ha^{2}\\right) =\\left\\lbrace -H^{2}+\\frac{3}{2}\\left[ \\widetilde{\\rho }-\\widetilde{p}-\\overline{p}\\delta \\left( \\tau \\right) \\right] \\right\\rbrace a^{2}\\ , $ (see Appendix ), still implies the Lanczos equation $\\Delta H|_{t_{s}}=-\\frac{3}{2}\\overline{p}\\ , $ derived in Appendix , when $\\widetilde{p}+\\overline{p}\\delta \\left( \\tau \\right) =p_{ACh}$ , $\\widetilde{\\rho }=\\rho _{m}+\\rho _{ACh}$ and $\\rho _{ACh}$ is generalized to a distribution $\\rho _{ACh}a^{2}=\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}~.$ Here $\\omega >0$ , $R\\left( \\tau \\right) >0$ , $Q\\left( \\tau \\right) >0$ and $P\\left( \\tau \\right) $ is bounded.","When Eq.","(REF ) is applied to a test function $\\varphi \\left( \\tau \\right) $ , the terms containing $H^{2}$ and $\\rho _{m}$ give regular contributions and the limits of the respective integrals vanish, similarly as discussed in Appendix .","Also, due to Proposition 2 given in the Appendix A, the integral of the distributional term containing $\\rho _{ACh}$ becomes $\\int _{-\\varepsilon }^{\\varepsilon }\\frac{P\\left( \\tau \\right) \\varphi \\left(\\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }d\\tau ~,$ which also vanishes for $\\varepsilon \\rightarrow 0$ .","We still have to consider the contributions $\\int _{-\\varepsilon }^{\\varepsilon }\\left[ \\widetilde{p}+\\overline{p}\\delta \\left( \\tau \\right) \\right] \\varphi \\left( \\tau \\right) a^{2}d\\tau \\ .$ Although the contribution $\\widetilde{p}\\varphi \\left( \\tau \\right) a^{2}$ to the integrand is singular at $\\tau =0$ , its integral can be conveniently evaluated by the Residue Theorem.","For this we remark, that the integrand is an analytically extendible function into the complex plane in the vicinity of $\\tau =0$ and its residue is zero, therefore the integral vanishes.","Finally, the contribution containing $\\overline{p}$ leads by integration and the limiting process to the right hand side of the Lanczos equation (REF ).","Therefore we have proven that the space-time regions separated by the singular spatial hypersurface, representing the pressure singularity, can be joined.","In other words, the singularity becomes traversable."],["Concluding Remarks","It is known that certain models of cosmological fluids in Friedmann universes, like the anti-Chaplygin gas or the tachyon field with a special potential [32], evolve into a sudden future singularity, which in spite of a diverging pressure, is weak.","It was argued that singularities of this kind could be traversable despite infinite tidal forces emerging at the singularity for an infinitesimally short time [26].","In Ref.","[29] the process of crossing of the Big Brake singularity was described in some detail for the tachyon model [32].","(The particularity of the Big Brake singularity, consists in the fact that at the crossing of such a singularity the Hubble variable is not only finite, but vanishes.)","We also note recent discussions [42] on crossing the “traditional” Big Bang and Big Crunch singularities.","In the present paper we considered a simple cosmological model containing a mixture of anti-Chaplygin gas and dust.","We have shown that the geodesics equations and their solutions are still well-defined in this case, however the inclusion of dust generates a nonzero value of the Hubble parameter at the singularity encounter, generating the following paradox.","The dust would require a continued expansion, which would make the energy density and pressure of the anti-Chaplygin gas ill-defined.","A contraction in turn, would be compatible with the anti-Chaplygin gas, nevertheless implying an abrupt change of the Hubble parameter from expansion to contraction.","The jump in the Hubble parameter implies the appearance of the $\\delta $ function in the Raychaudhuri equation (which contains $\\dot{H}$ ).","We have cured this situation by redefining the pressure and energy density of the anti-Chaplygin gas as distributions.","As an equivalent interpretation, the pressure can be generalized by the addition of a distributional contribution, while the energy density left unchanged, at the price of redefining the equation of state of the anti-Chaplygin gas in a distributional sense.","Then all cosmological equations are satisfied in the same distributional sense.","We have also shown, that the Israel junction conditions are obeyed through the singular spatial hypersurface, in particular we have enounced and proved a more general form of the Lanczos equation.","The results rely on two Propositions and a Corollary proven in Appendix A.","The resolution of the paradox at the soft singularity crossing by the introduction of distributional quantities and equations may look unusual, however distributional quantities, localized on hypersurfaces are quite commonly used in general relativity and other gravitational theories.","Spacetime regions are frequently matched by the inclusion of distributional layers; also shock-waves can be modeled by Dirac $\\delta $ -functions.","Braneworld models [44], [43] arise due to the orbifold boundary conditions, the non-smoothness of the 5-dimensional metric at the brane (the jump in its extrinsic curvature) being directly related to the distributional 3+1 standard model fields embedded in the 5-dimensional spacetime.","Besides, metrics allowing distributional curvature were considered earlier for studying strings and other distributional sources in general relativity [45].","The applications of the distributional quantities to the study of Schwarzschild geometry and point massive particles in general relativity were used in Refs.","[46] and [47] respectively.","More generically the connection between singularities and the distributional treatment of the physical quantities is well-known in quantum field theory.","Indeed, the appearance of the ultraviolet divergences can be understood as the result of the indefiniteness of the product of distributions and the renormalization procedure could be interpreted as a definition of such a product [48].","We hope that the investigations presented here may turn useful in deriving similar results in connection with the traversability of other types of sudden singularities.","While mathematically self-consistent, the scenario presented in this paper may look somewhat counter-intuitive from the physical point of view.","This is because its essential ingredient is the abrupt change of the expansion into a contraction.","However, such a behavior is not more counter-intuitive that the absolutely elastic bounce of the ball from a rigid wall, as known in classical mechanics.","Indeed, in the latter case the velocity and the momentum of the ball change their direction abruptly.","That means that an infinite force acts from the wall onto the ball during an infinitely small interval of time.","The result of this action is however integrable and results in a finite change of the momentum of the ball.","In fact, the absolutely elastic bounce is an idealization of a process of finite time-span during which inelastic deformations of both the ball and the wall are likely.","It is reasonable to think that something similar occurs also in the two-fluid universe model presented in this paper, which undergoes a transition from an expanding to a contracting phase.","The smoothing of this process should involve some (temporary) geometrically induced change of the equation of state of matter.","Note, that such changes are not uncommon in cosmology.","In the tachyon model [32] which was starting point of our studies of the Big Brake singularities, there was the tachyon -pseudotachyon transformation driven by the continuity of the cosmological evolution.","In a cosmological model with the phantom field with the cusped potential [49], the transformations between phantom and standard scalar field were considered.","Thus, one can imagine that the real process of the transition from the expansion to contraction induced by passing through a soft singularity can imply some temporary change of the equation of state which makes the above processes smoother.","We hope to explore such a scenario in the future."],["ACKNOWLEDGMENTS","We are grateful to V. Gorini, M. O. Katanaev, V. N. Lukash, U. Moschella, D. Polarski and A.","A. Starobinsky for useful discussions.","The work of ZK was supported by OTKA grant no.","100216 and AK was partially supported by the RFBR grant no.","11-02-00643."],["Propositions on the product and the composition of distributions\n","To investigate how the Friedmann universe crosses a soft singularity, we must solve the field equation in distributional sense.","For this purpose we give the definitions of the product and of the composition of distributions and prove two propositions.","Fisher derived the following result: $\\left[sgn\\left( \\tau \\right) \\left|\\tau \\right|^{\\lambda }\\right] \\delta \\left( \\tau \\right) =0$ for $\\lambda >-1$ [50].","Our first proposition generalizes this equation for $\\lambda \\ge 0$ .","The second proposition generalizes Antosik's result: $\\left[ 1+\\delta \\left( \\tau \\right) \\right] ^{-1}=1$ [51].","Finally, we show a corollary.","Let $\\rho \\left( \\tau \\right) $ be any infinitely differentiable function having the following properties: $i)$ $\\rho \\left( \\tau \\right) =0$ for $\\left|\\tau \\right|\\ge 1$ ; $ii)$ $\\rho \\left( \\tau \\right) \\ge 0$ ; $iii)$ $\\rho \\left( \\tau \\right) =\\rho \\left( -\\tau \\right) $ ; $iv)$ $\\int _{-1}^{1}\\rho \\left( \\tau \\right) d\\tau =1$ .","Then $\\delta _{n}\\left(\\tau \\right) =n\\rho \\left( n\\tau \\right) $ (with $n=1$ , 2, ...) is a regular sequence of infinitely differentiable functions converging to Dirac delta function: $\\lim _{n\\rightarrow \\infty }\\left\\langle \\delta _{n},\\varphi \\right\\rangle =\\left\\langle \\delta ,\\varphi \\right\\rangle $ for any $\\varphi \\in \\mathcal {D}$ [52].","Here $\\mathcal {D}$ denotes the space of test functions having continuous derivatives of all orders and compact support.","The action of an $f\\in \\mathcal {D}^{\\prime }$ distribution on test functions $\\varphi $ is given by $\\left\\langle f,\\varphi \\right\\rangle $ , which in the case when $f$ is an ordinary locally summable function is nothing but $\\int _{-\\infty }^{\\infty }f\\left( \\tau \\right) \\varphi \\left(\\tau \\right) d\\tau $ .","We note that $\\delta _{n}\\left( \\tau \\right) $ has the compact support: $\\left( -1/n,1/n\\right) $ .","We will also use the $n$ -th derivative of $f\\in \\mathcal {D}^{\\prime }$ acts as$\\ \\left\\langle df\\left(\\tau \\right) /d\\tau ^{n},\\varphi \\right\\rangle =\\left( -1\\right)^{n}\\left\\langle f\\left( \\tau \\right) ,d\\varphi /d\\tau ^{n}\\right\\rangle $ .","For an arbitrary distribution $f$ , the function $f_{n}\\left( \\tau \\right)=f\\ast \\delta _{n}\\equiv \\left\\langle f\\left( \\tau -x\\right) ,\\delta _{n}\\left( x\\right) \\right\\rangle $ gives a sequence of infinitely differentiable functions converging to $f$ .","Definition 1 : The commutative product of $f$ and $g$ exists and is equal to $h$ on the open interval $\\left( a,b\\right) $ ($-\\infty \\le a0$ , $P\\left( \\tau \\right) $ is bounded, $R\\left( \\tau \\right) \\ne 0$ , and in some range close $\\tau =0$ the signs of $R\\left(\\tau \\right) $ and $Q\\left( \\tau \\right) $ are the same if $Q\\left( \\tau \\right) \\ne 0$ ) exists if $P\\left( \\tau \\right) /R^{\\omega }\\left( \\tau \\right) $ existsWe note that this proposition can be held even if $P\\left( \\tau \\right) =1$ and $R\\left( \\tau \\right) =\\delta \\left( \\tau \\right) $ with $\\omega =1,$ $2,$ $...$ .","Indeed, $\\delta ^{-\\omega }\\left( \\tau \\right) $ exists in neutrix limit and $\\delta ^{-\\omega }\\left( \\tau \\right) =0$ [55].","Thus in the definition 2, the usual limit must be changed for neutrix limit for this case.","and $\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}=\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }\\ .$ By the definition of composition of distributions, we should calculate $&&\\left\\langle \\frac{P_{n}\\left( \\tau \\right) }{\\left[ R_{m}\\left( \\tau \\right) +Q_{m}\\left( \\tau \\right) \\delta _{m}\\left( \\tau \\right) \\right]_{n}^{\\omega }},\\varphi \\left( \\tau \\right) \\right\\rangle \\\\&=&\\int _{-\\infty }^{\\infty }\\int _{-1/n}^{1/n}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( x\\right) dxd\\tau }{\\left[R_{m}\\left( \\tau -x\\right) +Q_{m}\\left( \\tau \\right) \\delta _{m}\\left( \\tau -x\\right) \\right] ^{\\omega }}\\ .$ Performing a change of the variables as $\\tau =\\tau $ , $y=m\\left( \\tau -x\\right) $ , we have $&=&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\\\&=&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{1}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{R_{m}^{\\omega }\\left( y/m\\right) }dyd\\tau \\\\&&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{2}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\ ,$ where $\\Omega _{2}=\\left\\lbrace y:\\left|y\\right|<1\\text{ and }\\rho \\left( y\\right) \\ne 0\\right\\rbrace $ and $\\Omega _{1}=\\mathbb {R}-\\Omega _{2}$ .","The double limit of the first term is $&&\\lim _{n\\rightarrow \\infty }\\lim _{m\\rightarrow \\rightarrow \\infty }-\\frac{1}{m}\\int _{-\\infty }^{\\infty }d\\tau \\varphi \\left( \\tau \\right) P_{n}\\left(\\tau \\right) \\\\&&\\times \\int _{\\Omega _{1}}\\frac{\\delta _{n}\\left( \\tau -y/m\\right) }{R_{m}^{\\omega }\\left( y/m\\right) }dy \\\\&=&\\lim _{n\\rightarrow \\infty }\\lim _{m\\rightarrow \\rightarrow \\infty }\\int _{-\\infty }^{\\infty }d\\tau \\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\\\&&\\times \\int _{\\begin{array}{c} \\left|x\\right|<1/n, \\\\ m\\left|\\tau -x\\right|\\in \\Omega _{1}\\end{array}}\\frac{\\delta _{n}\\left( x\\right) }{R_{m}^{\\omega }\\left( \\tau -x\\right) }dx \\\\&=&\\left\\langle \\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) },\\varphi \\left( \\tau \\right) \\right\\rangle \\ .$ We investigate the absolute value of the second integral.","According to our assumptions for $R$ and $Q$ , and since we are interested in $m\\rightarrow \\infty $ , we can choose $m$ large enough to let the signs of $R$ and $Q$ be the same, then for $\\omega >0$ : $&&\\left|\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{2}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\right|\\\\&\\le &\\left|\\frac{1}{m^{1+\\omega }}\\int _{-\\infty }^{\\infty }\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\int _{\\Omega _{2}}\\frac{\\delta _{n}\\left( \\tau -y/m\\right) }{Q_{m}\\left( y/m\\right) \\rho ^{\\omega }\\left(y\\right) }dyd\\tau \\right|\\ .$ Performing a change of the variables as $z=n\\left( \\tau -y/m\\right) $ , $y=y$ , we have $&\\le &\\frac{1}{m^{1+\\omega }}\\int _{-1}^{1}\\int _{\\Omega _{2}}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\frac{\\rho \\left( z\\right) }{\\rho ^{\\omega }\\left( y\\right) }\\right|dydz \\\\&\\le &\\frac{1}{m^{1+\\omega }}\\sup _{\\Omega _{2},\\left|z\\right|\\le 1}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\rho ^{-\\omega }\\left( y\\right) \\right|\\\\&&\\times \\int _{-1}^{1}\\rho \\left( z\\right) dz\\int _{-1}^{1}dy \\\\&=&\\frac{2}{m^{1+\\omega }}\\sup _{\\Omega _{2},\\left|z\\right|\\le 1}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\rho ^{-\\omega }\\left( y\\right) \\right|\\ ,$ that vanishes for $m\\rightarrow \\infty $ if $P$ is bounded.","Corollary 1 : The distribution $d\\left\\lbrace P\\left( \\tau \\right) \\left[ R\\left( \\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{-\\omega }\\right\\rbrace /d\\tau $ (with the same properties for $P$ , $R$ , $Q$ and $\\omega $ as in proposition 2) exists if $P\\left( \\tau \\right) /R^{\\omega }\\left( \\tau \\right) $ and its derivative exist, and $\\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}=\\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }\\ .$ Applying the derivative of a distribution at tests functions, and using the fact that $d\\varphi /d\\tau \\in \\mathcal {D}$ for any $\\varphi \\in \\mathcal {D}$ , and by the proposition 2, we have $&&\\left\\langle \\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{\\left[ R\\left(\\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right]^{\\omega }},\\varphi \\right\\rangle \\\\&=&-\\left\\langle \\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }},\\frac{d}{d\\tau }\\varphi \\right\\rangle \\\\&=&-\\left\\langle \\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right)},\\frac{d}{d\\tau }\\varphi \\right\\rangle =\\left\\langle \\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) },\\varphi \\right\\rangle \\ .$"],["Two simple examples of the product and of the decomposition of\ndistributions","The definition of the product and the composition of distributions, used in this paper and presented in the Appendix A are not often encountered in physics.","Thus, to give the reader some flavor of the corresponding considerations, using simpler means we decided to give two semi-heuristic examples of such products and compositions.","We consider first a remarkable formula $\\mathcal {P}\\left( \\frac{1}{x}\\right) \\delta (x)=-\\frac{1}{2}\\delta ^{\\prime }(x)~, $ which was first proven in [57].","Here $\\mathcal {P}$ means the principal value of the corresponding function.","We shall prove here that the regularizing succession of functions with compact support $\\rho $ , employed in the Appendix A and the references therein, can be chosen alternatively as the family of the Cauchy-Lorentz functions $f_{\\epsilon }(x)=\\frac{1}{\\pi }\\frac{\\epsilon }{x^{2}+\\epsilon ^{2}}~.$ It is well known that when the small parameter $\\epsilon \\rightarrow 0$ , the functions of this family tend in the distributional sense to the Dirac $\\delta $ function.","Obviously, the convolution of the function (REF with the Dirac $\\delta $ function gives again the same function (REF ): $f_{\\epsilon }\\ast \\delta (x)=f_{\\epsilon }(x)~.","$ The calculation of the convolution of the principal value $\\mathcal {P}\\left(\\frac{1}{x}\\right) $ with the function $f_{\\epsilon }(x)$ is slightly more complicated: $&&\\mathcal {P}\\left( \\frac{1}{x}\\right) \\ast f_{\\epsilon }(x)=\\lim _{\\varepsilon \\rightarrow 0}\\left( \\int _{-\\infty }^{x-\\varepsilon }dy\\frac{1}{x-y}\\frac{\\epsilon }{\\pi (y^{2}+\\epsilon ^{2})}\\right.","\\\\&&\\left.","+\\int _{x+\\varepsilon }^{\\infty }dy\\frac{1}{x-y}\\frac{\\epsilon }{\\pi (y^{2}+\\epsilon ^{2})}\\right) =\\frac{x}{x^{2}+\\epsilon ^{2}}~.$ The product of the expressions (REF ) and (REF ) is $\\mathcal {P}\\left( \\frac{1}{x}\\right) _{\\epsilon }\\ast \\delta _{\\epsilon }(x)=\\frac{\\epsilon x}{\\pi (x^{2}+\\epsilon ^{2})^{2}}~.","$ Let us now consider a family of functions $\\frac{df_{\\epsilon }(x)}{dx}=-\\frac{2x\\epsilon }{\\pi (x^{2}+\\epsilon ^{2})^{2}}~.","$ One can easily prove that if the family of functions (REF ) converges in the distributional sense to the Dirac $\\delta $ function, the family of their derivatives (REF ) converges to the derivative of the delta function.","Now, comparing the right-hand sides of Eqs.","(REF ) and (REF ) we see that when $\\epsilon \\rightarrow 0$ the product in the left-hand side of Eq.","(REF ) converges in the distributional sense to $-\\frac{1}{2}\\delta ^{\\prime }(x)$ and thus the correctness of the equality (REF ) is checked.","Now let us discuss the Antosik identity [51] $\\frac{1}{1+\\delta (x)}=1.","$ Here we have the composition of the distributions $F(g)$ , where $F=\\frac{1}{g}$ and $g(x)=1+\\delta (x)$ .","Calculating the convolutions of the distributions $F$ and $g$ with the Cauchy-Lorentz functions (REF ) we obtain $F_{\\sigma }(g)=F\\ast f_{\\sigma }(g)=\\frac{g}{g^{2}+\\sigma ^{2}},$ $g_{\\epsilon }(x)=1+\\frac{\\epsilon }{x^{2}+\\epsilon ^{2}}.","$ Correspondingly the composition of these functions is $F_{\\sigma }(g_{\\epsilon })=\\frac{1+\\frac{\\epsilon }{\\pi (x^{2}+\\epsilon ^{2}}}{\\sigma ^{2}+\\left( 1+\\frac{\\epsilon }{\\pi (x^{2}+\\epsilon ^{2}}\\right) ^{2}} $ and it is easy to check that $\\lim _{\\sigma \\rightarrow 0}\\lim _{\\epsilon \\rightarrow 0}F_{\\sigma }(g_{\\epsilon })=1,$ confirming the identity (REF )."],["The Lanczos equation","For a generic junction surface the Lanczos equation emerges from the Gauss-Codazzi relations [58], [43].","The projected Lie derivative of the extrinsic curvature $K_{ab}$ in the normal direction $n$ to the surface is $h_{a}^{i}h_{b}^{j}\\mathcal {L}_{\\mathbf {n}}K_{ij}=-3\\epsilon \\left(h_{a}^{i}h_{b}^{k}T_{ik}-\\frac{h_{ab}}{2}g^{ik}T_{ik}\\right) +\\mathcal {Z}_{ab}\\ $ (Eq.","(21) of [43] in the units $8\\pi G/3=1$ ), with $\\mathcal {Z}_{ab} &=&-\\epsilon \\mathcal {R}_{ab}+2K_{ac}K_{b}^{c}-g^{ik}K_{ik}K_{ab} \\\\&&+D_{b}\\alpha _{a}-\\epsilon \\alpha _{b}\\alpha _{a}\\ .$ Here $g_{ab}$ is the space-time metric, $h_{ab}=g_{ab}-\\epsilon n_{a}n_{b}$ ($\\epsilon =n^{a}n_{a}=\\left\\lbrace -1,1\\right\\rbrace $ ) is the induced metric on the junction surface, and $T_{ab}$ is the energy-momentum tensor.","The tensor $\\mathcal {Z}_{ab}$ depends only on geometrical quantities: $\\mathcal {R}_{ab}$ and $D$ are the Ricci tensor and covariant derivative induced on the hypersurface, and $\\alpha _{a}=n^{c}\\nabla _{c}n_{a}$ , with $\\nabla $ the 4-dimensional covariant derivative.","When the energy-momentum tensor is a sum $T_{ik}=\\Pi _{ik}+\\Upsilon _{ik}\\delta \\left( \\tau \\right) $ (where $\\tau $ is the coordinate adapted to $n$ , i.e.","$n=t_{S}^{-1}\\partial /\\partial \\tau $ , and $n^{a}\\Upsilon _{ab}=0$ ), with $\\Pi _{ik}$ the regular 4-dimensional part and $\\Upsilon _{ik}$ the distributional part on the hypersurface, integration of Eq.","(REF ) across $\\tau $ through an infinitesimal range containing the hypersurface keeps only the distributional part, leading to the Lanczos equation [41], [40].","$\\Delta K_{ab}=-3\\epsilon \\left( \\Upsilon _{ab}-\\frac{\\Upsilon }{2}h_{ab}\\right) \\ ,$ or equivalently $-3\\epsilon \\Upsilon _{ab}=\\Delta K_{ab}-h_{ab}\\Delta K\\ .$ Here $\\Upsilon $ is the trace of $\\Upsilon _{ab}$ .","As $\\mathcal {Z}_{ab}$ is finite, its contribution to the integral across the infinitesimal range also vanishes.","Without a distributional energy-momentum part, the extrinsic curvature should be continuous.","Let us now specialize this for a junction along a maximally symmetric $\\tau =0$ spacelike hypersurface (a hyperplane with $\\mathcal {R}_{ab}=0$ ) embedded in a flat Friedmann space-time.","The normal vector $n$ has zero acceleration $\\alpha _{a}=0$ and the extrinsic curvature becomes $K_{ab}=\\dot{a}a\\widetilde{h}_{ab}$ , with $\\widetilde{h}_{ab}$ the 3-dimensional Euclidean metric.","The curvature term is $\\mathcal {Z}_{ab}=-H^{2}a^{2}\\widetilde{h}_{ab} $ and the energy momentum tensors are $\\Pi _{ab}=\\widetilde{\\rho }n_{a}n_{b}+\\widetilde{p}a^{2}\\widetilde{h}_{ab}$ and $\\Upsilon _{ab}=\\overline{p}a^{2}\\widetilde{h}_{ab}$ .","Since the projected Lie-derivative in Eq.","(REF ) becomes a time derivative, the equation reads $\\frac{\\partial }{\\partial t}\\left( Ha^{2}\\right) =\\left\\lbrace -H^{2}+\\frac{3}{2}\\left[ \\widetilde{\\rho }-\\widetilde{p}-\\overline{p}\\delta \\left( \\tau \\right) \\right] \\right\\rbrace a^{2}\\ , $ which is a combination of the Raychaudhuri and Friedmann equations.","For finite $H$ , $\\widetilde{\\rho }$ and $\\widetilde{p}$ as before the integration of Eq.","(REF ) across an infinitesimal time range $\\tau $ leads to the Lanczos equation $\\Delta H|_{t_{s}}=-\\frac{3}{2}\\overline{p}\\ .","$"]],"string":"[\n [\n \"The paradox of soft singularity crossing and its resolution by\\n distributional cosmological quantitities\"\n ],\n [\n \"Abstract A cosmological model of a flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter exhibits a future soft singularity, where the pressure of the anti-Chaplygin gas diverges (while its energy density is finite).\",\n \"Despite infinite tidal forces the geodesics pass through the singularity.\",\n \"Due to the dust component, the Hubble parameter has a non-zero value at the encounter with the singularity, therefore the dust implies further expansion.\",\n \"With continued expansion however, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence from the point of view of the anti-Chaplygin gas only a contraction is allowed.\",\n \"Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously.\",\n \"This obviosly could not happen.\",\n \"We solve the paradox by redefining the anti-Chaplygin gas in a distributional sense.\",\n \"Then a contraction could follow the expansion phase at the singularity at the price of a jump in the Hubble parameter.\",\n \"Although such an abrupt change is not common in any cosmological evolution, we explicitly show that the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity.\",\n \"We also prove that the Israel junction conditions are obeyed through the singular spatial hypersurface.\",\n \"In particular we enounce and prove a more general form of the Lanczos equation.\"\n ],\n [\n \"Introduction\",\n \"The problem of cosmological singularities has been attracting the attention of theoreticians working in gravity and cosmology since the early fifties [1], [2], [3].\",\n \"In the sixties general theorems about the conditions for the appearance of singularities were proven [4], [5] and the oscillatory regime of approaching the singularity [6], the Mixmaster universe [7] was discovered.\",\n \"Basically, until the end of nineties almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius.\",\n \"However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [8]-[18], characterized by a diverging $\\\\ddot{a}$ whereas both the scale factor $a$ and $\\\\dot{a}$ are finite.\",\n \"Then, the Hubble parameter $H=\\\\dot{a}/a~$ and the energy density $\\\\rho $ are also finite, while the first derivative of the Hubble parameter and the pressure $p$ diverge.\",\n \"Until recent years, however, sudden future singularities attracted only a limited interest among researchers.\",\n \"The interest grew due to two reasons.\",\n \"The recent discovery of the cosmic acceleration [19] has stimulated the elaboration of dark energy models, responsible for such a phenomenon (see e.g.\",\n \"for review [20]).\",\n \"Remarkably in some of these models the sudden singularities arise quite naturally.\",\n \"Another source of the interest to sudden singularities is the development of brane models [10], [11], [18], where singularities of this kind could arise naturally (sometimes these singularities, arising in brane-world models, are called “quiescent” [10]).\",\n \"In the investigations devoted to sudden singularities one can distinguish three main topics.\",\n \"The first of them deals with the question of the compatibility of the models possessing soft singularities with observational data [15], [21], [22].\",\n \"The second direction is connected with the study of quantum effects [11], [17], [23], [24], [25].\",\n \"Here one can see two subdirections: the study of quantum corrections to the effective Friedmann equation, which can eliminate classical singularities or at least, change their form [10], [17], [23]; and the study of solutions of the Wheeler-DeWitt equation for the quantum state of the universe in the presence of sudden singularities [24], [25].\",\n \"The third direction is connected with the possibility of the sudden singularity crossing in classical cosmology [26], [27], [28], [29], [25].\",\n \"The present paper is devoted exactly to this topic.\",\n \"A particular feature of the sudden future singularities is their softness [26].\",\n \"As the Christoffel symbols depend only on the first derivative of the scale factor, they are regular at these singularities.\",\n \"Hence, the geodesics are well behaved and they can cross the singularity [26].\",\n \"One can argue that the particles crossing the singularity will generate the geometry of the spacetime, providing in such a way a “soft rebirth” of the universe after the singularity crossing [29].\",\n \"Note that the possibility of crossing of some kind of cosmological singularities was noticed already in the early paper by Tipler [30].\",\n \"A close idea of integrable singularities in black holes, which can give origin to a cosmogenesis was recently put forward in [31].\",\n \"As a starting point we consider an interesting example of a sudden future singularity - the Big Brake which was discovered in Ref.\",\n \"[32] while studying a particular tachyon cosmological model.\",\n \"The particularity of the Big Brake singularity consists in the fact that the time derivative of the scale factor is not only finite, but exactly equal to zero.\",\n \"That makes the analysis of the behavior in the vicinity of singularity especially convenient.\",\n \"In particular, in Ref.\",\n \"[22] it was shown that the predictions of the future of the universe in this model [32] are compatible with the supernovae type Ia data, while in Refs.\",\n \"[24], [25] some quantum cosmological questions were studied in the presence of the Big Brake singularity.\",\n \"The simplest cosmological model allowing a Big Brake singularity was also introduced in Ref.\",\n \"[32].\",\n \"This model is based on the perfect fluid, dubbed “anti-Chaplygin gas”.\",\n \"This fluid is characterized by the equation of state $p=\\\\frac{A}{\\\\rho }, $ where $A$ is a positive constant.\",\n \"Such and equation of state arises, for example, in the theory of wiggly strings [33].\",\n \"In paper [32] a fluid obeying the equation of state (REF ) was called \\\"anti-Chaplygin gas\\\" in analogy with the Chaplygin gas [34] which has the equation of state $p=-A/\\\\rho $ and has acquired some popularity as a candidate for a unified theory of dark energy and dark matter [35].\",\n \"An explicit example of the crossing of the Big Brake singularity was described in detail in paper [29], were the tachyon model [32], [22] was investigated.\",\n \"In this model the tachyon field passes through the singularity, continuing its evolution with a recollapse towards a Big Crunch.\",\n \"In a simpler model, based on the anti-Chaplygin gas, such a crossing is even easier to understand.\",\n \"The next natural step in the analysis of the soft singularities seems to be obvious.\",\n \"One can consider a soft singularity of more general type than the big brake by adding to the tachyon matter or to the anti-Chaplygin gas some dustlike matter.\",\n \"However, in this case the traversability of the singularity seems to be obstructed.\",\n \"The main reason for this is that while the energy density of the tachyonic field (or of the anti-Chaplygin gas) vanishes at the singularity, the energy density of the matter component does not, leaving the Hubble parameter at the singularity with a finite value.\",\n \"Then some kind of the paradox arises: if the universe continues its expansion, and if the equation of state of the component of matter, responsible for the appearance of the soft singularity (in the simplest case, the anti-Chaplygin gas) is unchanged, then the expression for the energy density of this component becomes imaginary, which is unacceptable.\",\n \"The situation looks rather strange: indeed, the model, including dust should be in some sense more regular, than a single exotic fluid, the anti-Chaplygin gas.\",\n \"Thus, if the model based on the pure anti-Chaplygin gas has a traversable Big Brake singularity, than the more general singularity arising in the model based on the mixture of the anti-Chaplygin gas and dust should also be traversable.\",\n \"Related to that, it was recently shown that general soft singularities arising in the Friedmann model, filled with the scalar field with a negative potential, inversely proportional to this field are traversable.\",\n \"So, what could be wrong with the simple two-fluid model?\",\n \"One can see that what we face is some sort of a clash between the equation of state of one of these fluids and the dynamics (the Friedmann and Raychaudhuri equations) and energy conservation equations.\",\n \"In this paper we shall try to resolve this paradox, insisting on the preservation of the equation of state of the anti-Chaplygin gas.\",\n \"The price which one has to pay for it is the obligatory use of the generalized functions for some cosmological quantities.\",\n \"Namely, the anti-Chaplygin gas remains physical if rather a recollapse follows, but then the Hubble parameter would have a sharp jump, obstructing the validity of the Raychaudhuri equation (the second Friedmann equation) in the usual sense of functions.\",\n \"Thus, apparently, the evolution cannot be continued through the soft singularity, unless treating the cosmological quantities as distributions.\",\n \"We claim that such a generalization is mathematically rigorous, moreover, the introduction of distributions is not so drastic, as it looks at the first sight, as the pressure of the anti-Chaplygin gas diverges anyhow at the soft singularity (as it so does for the tachyon field).\",\n \"Then in the Conclusion we shall dwell on the possible physical sense of the proposed constructions and its possible alternatives.\",\n \"The plan of the paper is as follows.\",\n \"In Section we discuss generic Friedmann space-times, which admit $\\\\dot{H}=-\\\\infty $ type singularities, while the Hubble parameter $H$ remains finite.\",\n \"Such singularities are related to corresponding divergencies in the pressure of the perfect fluid filling the Friedmann universe (while its energy density stays finite).\",\n \"We investigate the kinematics, the geodesic equations, the geodesic deviation equations in the vicinity of these singularities and also prove that these singularities are weak.\",\n \"In Section we discuss a mixture of the anti-Chaplygin gas and dust in a flat Friedmann universe and explain the essence of the paradox.\",\n \"We explicitly derive the behavior of the energy density and pressure in the vicinity of the soft singularity and we solve the geodesic equations in this region.\",\n \"While the singularity turns to be traversable by the geodesics, the explicit solution also shows that the Raychaudhuri equation is violated at the singularity.\",\n \"In Section we add generalized distributional contributions to both the pressure and energy density, such that (a) the equation of state of the anti-Chaplygin gas still holds and (b) the singularity becomes traversable.\",\n \"We also perform checks of the Friedmann, Raychaudhuri and continuity equations, which all hold valid across the singularity in a distributional sense.\",\n \"In the process we employ a number of Propositions on distributions presented and proved in Appendix A.\",\n \"For the convenience of the reader we present a related semi-heuristic discussion of two known distributional identities in Appendix B.\",\n \"We stress that the distributional modifications of the energy density and pressure do not modify the cosmological evolution, but they make possible the soft singularity crossing.\",\n \"In Section we revisit the junction conditions along a spacelike hypersurface in a flat Friedmann universe.\",\n \"The future soft singularity represents such a spatial hypersurface, along which the energy-momentum tensor diverges.\",\n \"Extending the space-time through this hypersurface is possible by obeying both Israel junction conditions.\",\n \"While the first condition, requiring the continuity of the induced metric is easy to satisfy (the metric stays regular at the soft singularity), the second condition relates the jump in the extrinsic curvature to the distributional part of the energy-momentum tensor through the Lanczos equation.\",\n \"We will show that in flat Friedmann space-times the Lanczos equation holds for a more general class of distributional energy-momentum tensors.\",\n \"With this we give a second proof that the generalized distributional energy-momentum tensor assures the traversability of the soft singularity.\",\n \"In the process we employ a simple form of the Lanczos equation valid in flat Friedmann universes, derived in Appendix .\",\n \"We summarize our results and give some further outlook in the Concluding Remarks.\",\n \"We chose $c=1$ and $8\\\\pi G/3=1$ .\",\n \"A subscript $S$ denotes the value of the respective quantity at the soft singularity.\"\n ],\n [\n \"Pressure singularities in flat Friedmann universes\",\n \"The line element squared of a flat Friedmann universe can be written as $ds^{2}=-dt^{2}+a^{2}\\\\left( t\\\\right) \\\\sum _{\\\\alpha }\\\\left( dx^{\\\\alpha }\\\\right)^{2}\\\\ ,$ where $x^{\\\\alpha }$ ($\\\\alpha =1,2,3$ ) are Cartesian coordinates.\",\n \"The evolution of the Friedmann universe is governed by the Raychaudhuri (second Friedmann) equation $\\\\dot{H}=-\\\\frac{3}{2}\\\\left( \\\\rho +p\\\\right) \\\\ , $ and by the continuity equation for the fluid $\\\\dot{\\\\rho }+3H\\\\left( \\\\rho +p\\\\right) =0\\\\ .\",\n \"$ Here the dot denotes the derivative with respect to cosmological time $t$ .\",\n \"A first integral of this system is given by the first Friedmann equation $H^{2}=\\\\rho \\\\ .\",\n \"$ It is easy to see that the Raychaudhuri equation can be obtained from the first Friedmann and the continuity equations.\"\n ],\n [\n \"Kinematics in the vicinity of sudden singularities\",\n \"Sudden singularities are characterized by finite $H$ and $\\\\dot{H}\\\\rightarrow -\\\\infty $ (finite $\\\\dot{a}$ and $\\\\ddot{a}\\\\rightarrow -\\\\infty $ ) at some finite scale factor $a$ .\",\n \"The energy density of the fluid is finite but its pressure diverges at this type of singularity, therefore the term “pressure singularity” is also in use.\",\n \"Then, we would like to emphasize the fact that there is an essential difference between the sudden singularities with $H=0$ and with $H>0$ .\",\n \"As has been already mentioned in the Introduction, in the first case, which is called Big Brake, the universe begin contracting and running towards the Big Crunch singularity.\",\n \"Exactly this occurs in models based on tachyon field with a particular potential [32], [22], [29] or in the anti-Chaplygin gas models.\",\n \"In the case of the model based on the mixture of one of this fluids and dust, we encounter the second situation when the value of the Hubble constant is positive at the moment of encounter with the sudden singularity.\",\n \"That means that after crossing the singularity the universe should continue its expansion, but the anti-Chaplygin gas becomes ill-defined, as it will be shown in detail in Section , devoted to the model based on mixture of the anti-Chaplygin gas and dust.\",\n \"One possible way of overcoming this obstacle is to allow the jump in the sign of the Hubble parameter, which as was mentioned in the Introduction leaves valid the first Friedmann equation, the continuity equation and the equation of state, while making invalid the Raychaudhuri equation.\",\n \"This last obstacle can be cured by the acceptance the distributional Dirac $\\\\delta $ -function type contributions into the pressure and the energy density, which is described in detail in the section IV.\"\n ],\n [\n \"Geodesics in the vicinity of sudden singularities\",\n \"The geodesic equations in flat Friedmann space-time are $\\\\frac{d^{2}x^{\\\\alpha }}{d\\\\lambda ^{2}}+2\\\\frac{\\\\dot{a}}{a}\\\\frac{dt}{d\\\\lambda }\\\\frac{dx^{\\\\alpha }}{d\\\\lambda }=0\\\\ ,$ $\\\\frac{d^{2}t}{d\\\\lambda ^{2}}+a\\\\dot{a}\\\\sum _{\\\\alpha }\\\\left( \\\\frac{dx^{\\\\alpha }}{d\\\\lambda }\\\\right) ^{2}=0\\\\ ,$ where $\\\\lambda $ is an affine parameter.\",\n \"Integrating these equations yields $\\\\frac{dx^{\\\\alpha }}{d\\\\lambda }=\\\\frac{P^{\\\\alpha }}{a^{2}}\\\\ ,$ $\\\\left( \\\\frac{dt}{d\\\\lambda }\\\\right) ^{2}=\\\\epsilon +\\\\frac{P^{2}}{a^{2}}\\\\ ,$ with $P^{\\\\alpha }$ , $\\\\epsilon $ integration constants and $P^{2}=\\\\sum _{\\\\alpha }\\\\left( P^{\\\\alpha }\\\\right) ^{2}$ .\",\n \"The quantity $\\\\epsilon $ is fixed by the length of the tangent vector $u^{a}$ of the geodesic as $\\\\epsilon =-u_{a}u^{a}$ ; i.e.\",\n \"one for timelike and zero for lightlike orbits.\",\n \"In a comoving system $P^{\\\\alpha }=0$ and $t=\\\\lambda $ is affine parameter.\",\n \"Eqs.\",\n \"(REF ) and (REF ) are singular only for vanishing scale factor (see also Ref.\",\n \"[26]).\",\n \"Therefore, the existence of a solution $t\\\\left( \\\\lambda \\\\right) $ , $x^{\\\\alpha }\\\\left(\\\\lambda \\\\right) $ of Eqs.\",\n \"(REF ) and (REF ) is assured by the Cauchy-Peano theorem for any nonzero $a$ (including the soft singularity).\",\n \"Thus the functions $t\\\\left( \\\\lambda \\\\right) $ and $x^{\\\\alpha }\\\\left( \\\\lambda \\\\right) $ , i.e.\",\n \"the geodesics can be continued through the singularity occurring at finite scale factor.\",\n \"Only derivatives of higher order than two of $t\\\\left( \\\\lambda \\\\right) $ and $x^{\\\\alpha }\\\\left( \\\\lambda \\\\right) $ are singular (as they contain $\\\\ddot{a}$ ), however these do not appear in the geodesic equations.\",\n \"Pointlike particles moving on geodesics do not experience any singularity.\",\n \"Thus, as we argued in the preceding paper [29] one is not obliged to consider such a singularity as a final state of the universe.\",\n \"Indeed, passing through this singularity the matter recreates also the spacetime in a unique way, at least for such simple models, as those based on Friedmann metrics.\"\n ],\n [\n \"Deviation equation in the vicinity of sudden singularities\",\n \"The 3-spaces with $t=$ const have vanishing Riemann curvature.\",\n \"However, the 4-dimensional Riemann curvature tensor has the nonvanishing components: $R_{\\\\,\\\\,\\\\,\\\\,\\\\,t\\\\beta t}^{\\\\alpha } &=&-\\\\frac{\\\\ddot{a}}{a}\\\\delta _{\\\\beta }^{\\\\alpha }=\\\\left( -\\\\dot{H}+H^{2}\\\\right) \\\\delta _{\\\\beta }^{\\\\alpha }\\\\ ,\\\\\\\\R_{\\\\,\\\\,\\\\,\\\\,\\\\,212}^{1} &=&R_{\\\\,\\\\,\\\\,\\\\,\\\\,313}^{1}=R_{\\\\,\\\\,\\\\,\\\\,\\\\,323}^{2}=\\\\dot{a}^{2} $ and the corresponding components arising from symmetry.\",\n \"Here $\\\\alpha ,\\\\beta =1,2,3$ .\",\n \"Remarkably, all components which diverge at the singularity are of the type $R_{tata}$ [29].\",\n \"Therefore, the singularity arises in the mixed spatio-temporal components.\",\n \"The geodesic deviation equation along the integral curves of $u=\\\\partial /\\\\partial t$ (which are geodesics with affine parameter $t$ ) is $\\\\dot{u}^{a}=-R_{\\\\ \\\\ cbd}^{a}\\\\eta ^{b}u^{c}u^{d}\\\\ ,$ where $\\\\eta ^{b}$ is the deviation vector separating neighboring geodesics, chosen to satisfy $\\\\eta ^{b}u_{b}=0$ .\",\n \"For a Friedmann universe it becomes $\\\\dot{u}^{a}=-R_{\\\\ \\\\ tbt}^{a}\\\\eta ^{b}\\\\propto \\\\ddot{a}\\\\ ,$ which at the singularity diverges as $-\\\\infty $ .\",\n \"Therefore, when approaching the singularity, the tidal forces manifest themselves as an infinite braking force stopping the further increase of the separation of geodesics, but not the evolution along the geodesics.\",\n \"With $\\\\ddot{a}<0$ in the vicinity of the singularity, once the geodesics have passed through, they will approach each other.\",\n \"Therefore a contraction phase will follow: everything that has reached the singularity will bounce back.\"\n ],\n [\n \"The type of the singularity\",\n \"In this subsection we shall present the classification of singularities, based on the point of view of finite size objects, which approach these singularities.\",\n \"In principle, finite size objects could be destroyed while passing through the singularity due to the occurring infinite tidal forces.\",\n \"A strong curvature singularity is defined by the requirement that an extended finite object is crushed to zero volume by tidal forces.\",\n \"We give below Tipler's [30] and Królak's [36] definitions of strong curvature singularities together with the relative necessary and sufficient conditions.\",\n \"An alternative definition of the softness of a singularity, based on a Raychaudhuri averaging, was developed by Dabrowski [37].\",\n \"According to Tipler's definition if every volume element defined by three linearly independent, vorticity-free, geodesic deviation vectors along every causal geodesic through a point $p$ vanishes, a strong curvature singularity is encountered at the respective point $p$ [30], [26].\",\n \"The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\\\lambda _{s}$ ($\\\\lambda $ is affine parameter of the curve) [38] is that the double integral $\\\\int _{0}^{\\\\lambda }d\\\\lambda ^{\\\\prime }\\\\int _{0}^{\\\\lambda ^{\\\\prime }}d\\\\lambda ^{\\\\prime \\\\prime }\\\\left|R_{\\\\,\\\\,ajb}^{i}u^{a}u^{b}\\\\right|$ diverges as $\\\\lambda \\\\rightarrow \\\\lambda _{s}$ .\",\n \"A similar condition is valid for lightlike geodesics, with $R_{\\\\,\\\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\\\,ab}u^{a}u^{b}$ in the double integral.\",\n \"Królak's definition is less restrictive.\",\n \"A future-endless, future-incomplete null (timelike) geodesic $\\\\gamma $ is said to terminate in the future at a strong curvature singularity if, for each point $p\\\\in \\\\gamma $ , the expansion of every future-directed congruence of null (timelike) geodesics emanating from $p$ and containing $\\\\gamma $ becomes negative somewhere on $\\\\gamma $ [36], [39].\",\n \"The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\\\lambda _{s}$ [38] is that the integral $\\\\int _{0}^{\\\\lambda }d\\\\lambda ^{\\\\prime }\\\\left|R_{\\\\,\\\\,ajb}^{i}u^{a}u^{b}\\\\right|$ diverges as $\\\\lambda \\\\rightarrow \\\\lambda _{s}$ .\",\n \"Again, a similar condition is valid for lightlike geodesics, with $R_{\\\\,\\\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\\\,ab}u^{a}u^{b}$ in the integral.\",\n \"In flat Friedmann space-time the comoving observers move on geodesics having four velocity $u=\\\\partial /\\\\partial t$ , where $t$ is affine parameter.\",\n \"The nonvanishing components of Riemann tensor are given by Eq.\",\n \"(REF ).\",\n \"Since $H$ is finite along the geodesics, neither of the integrals (REF ) and (REF ) diverge at the singularity.\",\n \"The singularity is weak (soft) according to both Tipler's and Królak's definitions.\",\n \"That means although the tidal forces become infinite, the finite objects are not necessarily crushed when reaching the singularity (see also [26]).\"\n ],\n [\n \"The paradox of the soft singularity crossing in the cosmological\\nmodel based on the anti-Chaplygin gas and dust universe\",\n \"We discuss an universe filled with two components.\",\n \"One is the anti-Chaplygin gas with the equation of state (REF ) and other is the pressureless dust.\",\n \"The solution of the continuity equation for the anti-Chaplygin gas gives the following dependence of its energy density on the scale factor: $\\\\rho _{ACh}=\\\\sqrt{\\\\frac{B}{a^{6}}-A}\\\\ , $ where $B$ is a positive constant, determining the initial condition.\",\n \"The energy density of the dust-like matter is as usual $\\\\rho _{m}=\\\\frac{\\\\rho _{m,0}}{a^{3}}\\\\ ,~ $ where $\\\\rho _{m,0}$ is a constant.\",\n \"It is clear that when during the expansion of the universe, its scale factor approaches the value $a_{S}=\\\\left( \\\\frac{B}{A}\\\\right) ^{\\\\frac{1}{6}}\\\\ , $ the energy density of the anti-Chaplygin gas vanishes, and its pressure grows to infinity.\",\n \"That means that the deceleration also becomes infinite.\",\n \"However, the energy density of dust remains finite, hence the same is true also for the Hubble parameter.\",\n \"It is here that the paradox arises: if the universe continues to expand, the expression under the sign of the square root in Eq.\",\n \"(REF ) becomes negative and the energy density of the anti-Chaplygin gas becomes ill-defined.\",\n \"A way out of this situation is only by assuming that at this moment the Hubble parameter changes its sign, while keeping its absolute value (such that the energy density will not have a jump, as implied by the Friedmann equation).\",\n \"This possibility will be studied in detail in the following subsections.\"\n ],\n [\n \"Evolutions in the vicinity of the singularity\",\n \"Let us substitute the expressions (REF ) and (REF ) into the first Friedmann equation.\",\n \"We shall find its solution for the universe approaching to the soft singularity point at the moment $t_{S}$ (the latter cannot be found analytically, but its value is not important for our analysis): $a(t)=a_{S}-\\\\sqrt{\\\\frac{\\\\rho _{m,0}}{a_{S}}}(t_{S}-t)-\\\\sqrt{\\\\frac{2Aa_{S}^{2}}{3H_{S}}}(t_{S}-t)^{3/2}\\\\ , $ where $H_{S}=\\\\sqrt{\\\\frac{\\\\rho _{m,0}}{a_{S}^{3}}} $ is the value of the Hubble parameter at $t=t_{S}$ .\",\n \"Correspondingly the leading terms of the energy densities of the anti-Chaplygin gas and dust, also of the pressure of the anti-Chaplygin gas are $\\\\rho _{m}=H_{S}^{2}+3H_{S}^{3}(t_{S}-t)\\\\ , $ $\\\\rho _{ACh}=\\\\sqrt{6AH_{S}(t_{S}-t)}\\\\ , $ $p_{ACh}=\\\\sqrt{\\\\frac{A}{6H_{S}(t_{S}-t)}}\\\\ .\",\n \"$ One can see that the expressions (REF ), (REF ) and (REF ) cannot be continued for $t>t_{S}$ due to the emerging negative quantities under the square roots.\",\n \"The assumption of a sharp transition from expansion to contraction implies the following changes in Eqs.\",\n \"(REF ) – (REF ): $a(t)=a_{S}-\\\\sqrt{\\\\frac{\\\\rho _{m,0}}{a_{S}}}|t_{S}-t|-\\\\sqrt{\\\\frac{2Aa_{S}^{2}}{3H_{S}}}|t_{S}-t|^{3/2}\\\\ , $ $\\\\rho _{m}=H_{S}^{2}+3H_{S}^{3}|t_{S}-t|\\\\ , $ $\\\\rho _{ACh}=\\\\sqrt{6AH_{S}|t_{S}-t|}\\\\ , $ $p_{ACh}=\\\\sqrt{\\\\frac{A}{6H_{S}|t_{S}-t|}}\\\\ .\",\n \"$ The quantities (REF )–(REF ) are well-defined and continuous at the moment of the singularity crossing.\",\n \"The expression for the pressure (REF ) is divergent, but this divergence is integrable and this is sufficient for our purposes.\",\n \"These new expressions satisfy the Friedmann equation, the continuity equations and the equation of state for the anti-Chaplygin gas.\",\n \"However, the time derivatives of these quantities are not continuous and it is the reason of the failure of the Raychaudhuri equation.\",\n \"We shall analyze this problem in the following section, but before we discuss the geodesics in the vicinity of the singularity.\"\n ],\n [\n \"Singularity crossing geodesics\",\n \"We can integrate explicitly the geodesics equations (REF ) and (REF ) in the vicinity of singularity, using the expression (REF ) for the cosmological factor, also taken in the vicinity of singularity.\",\n \"Choosing the affine parameter in such a way that the point $\\\\lambda =0$ corresponds to the singularity crossing we obtain up to the second order in $\\\\lambda $ terms $t=t_{S}+\\\\sqrt{\\\\epsilon +\\\\frac{P^{2}}{a_{S}^{2}}}\\\\lambda +\\\\frac{P^{2}H_{S}}{2a_{S}^{2}}sgn(\\\\lambda )\\\\lambda ^{2}\\\\ , $ $x^{\\\\alpha }=x_{S}^{\\\\alpha }+\\\\frac{P^{\\\\alpha }}{a_{S}^{2}}\\\\lambda +\\\\sqrt{\\\\epsilon +\\\\frac{P^{2}}{a_{S}^{2}}}\\\\frac{P^{\\\\alpha }H_{S}}{a_{S}^{2}}sgn(\\\\lambda )\\\\lambda ^{2}\\\\ .\",\n \"$ One can see from Eqs.\",\n \"(REF ) and (REF ) that not only the time and spatial coordinates of the geodesics are continuous at the soft singularity crossing, but also their first derivatives with respect to the affine parameter $\\\\lambda $ .\"\n ],\n [\n \"Singularity crossing, the Raychaudhuri equation and distributions\\n\",\n \"Let us discuss the expressions for the Hubble parameter and its time derivative in the vicinity of the singularity.\",\n \"Starting from the expression (REF ) we obtain $H(t) &=&H_{S}sgn(t_{S}-t) \\\\\\\\&&+\\\\sqrt{\\\\frac{3A}{2H_{S}a_{S}^{4}}}sgn(t_{S}-t)\\\\sqrt{|t_{S}-t|}~,$ $\\\\dot{H}=-2H_{S}\\\\delta (t_{S}-t)-\\\\sqrt{\\\\frac{3A}{8H_{S}a_{S}^{4}}}\\\\frac{sgn(t_{S}-t)}{\\\\sqrt{|t_{S}-t|}}~.\",\n \"$ Naturally, the $\\\\delta $ -term in $\\\\dot{H}$ arises because of the jump in $H$ , as the expansion of the universe is followed by a contraction.\",\n \"To restore the validity of the Raychaudhuri equation we shall add a singular $\\\\delta $ -term to the pressure of the anti-Chaplygin gas, which will acquire the form $p_{ACh}=\\\\sqrt{\\\\frac{A}{6H_{S}|t_{S}-t|}}+\\\\frac{4}{3}H_{S}\\\\delta (t_{S}-t)~.$ The equation of state (REF ) of the anti-Chaplygin gas is preserved, if we also modify the expression for its energy density: $\\\\rho _{ACh}=\\\\frac{A}{\\\\sqrt{\\\\frac{A}{6H_{S}|t_{S}-t|}}+\\\\frac{4}{3}H_{S}\\\\delta (t_{S}-t)}~.\",\n \"$ The last expression should be understood in the sense of the composition of distributions (see Appendix A and the references therein).\",\n \"In order to prove that $p_{ACh}$ and $\\\\rho _{ACh}$ represent a self-consistent solution of the system of cosmological equations, we shall use the following distributional identities: $\\\\left[ sgn\\\\left( \\\\tau \\\\right) g\\\\left( \\\\left|\\\\tau \\\\right|\\\\right) \\\\right] \\\\delta \\\\left( \\\\tau \\\\right) &=&0\\\\ , \\\\\\\\\\\\left[ f\\\\left( \\\\tau \\\\right) +C\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{-1}&=&f^{-1}\\\\left( \\\\tau \\\\right) \\\\ , \\\\\\\\\\\\frac{d}{d\\\\tau }\\\\left[ f\\\\left( \\\\tau \\\\right) +C\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{-1} &=&\\\\frac{d}{d\\\\tau }f^{-1}\\\\left( \\\\tau \\\\right) \\\\ .$ Here $g\\\\left( \\\\left|\\\\tau \\\\right|\\\\right) $ is bounded on every finite interval, $f\\\\left( \\\\tau \\\\right) >0$ and $C>0$ is a constant.\",\n \"These identities follow from the Propositions 1, 2 and the Corollary enounced and proved in the Appendix A.\",\n \"The parameter $\\\\tau $ stays instead of the difference $t_{S}-t$ .\",\n \"Because of Eq.\",\n \"(), the energy density (REF ) behaves as a continuous function which vanishes at the singularity.\",\n \"The first term in the expression for the pressure (REF ) diverges at the singularity.\",\n \"Therefore the addition of a Dirac delta term, which is not changing the value of $p_{ACh}$ at any $\\\\tau \\\\ne 0$ (i.e.\",\n \"$t\\\\ne t_{S}$ ) does not look too drastic and might be considered as a some kind of renormalization.\",\n \"To prove that Friedmann, Raychaudhuri and continuity equations are satisfied we must only investigate those terms, appearing in the field equations, which contain Dirac $\\\\delta $ -functions, since without them, these equations can be reduced to those we have found in the previous section.\",\n \"First, we check the continuity equation for the anti-Chaplygin gas.\",\n \"Due to the identities ()-(), the $\\\\delta \\\\left(\\\\tau \\\\right) $ -terms occurring in $\\\\rho _{ACh}$ and $\\\\dot{\\\\rho }_{ACh}$ could be dropped.\",\n \"We keep them however in order to have the equation of state explicitly satisfied.\",\n \"Then the $\\\\delta \\\\left( \\\\tau \\\\right) $ -term appearing in $3Hp_{ACh}$ vanishes, because the Hubble parameter changes sign at the singularity [see Eq.\",\n \"(REF )].\",\n \"The $\\\\delta \\\\left( \\\\tau \\\\right) $ -term appearing in $\\\\rho _{ACh}$ does not affect the Friedmann equation due to the identity ().\",\n \"Finally, the $\\\\delta $ -term arising in the time derivative of the Hubble parameter in the left-hand side of the Raychaudhuri equation is compensated by the conveniently chosen $\\\\delta $ -term in the right-hand side of Eq.\",\n \"(REF ).\"\n ],\n [\n \"The junction conditions across the singularity\",\n \"In this section we discuss the singularity crossing in a slightly different way, by analyzing the junction conditions.\",\n \"We have to match two space-time regions across the space-like hypersurface $\\\\tau =0$ .\",\n \"The junction of two space-time regions has to obey the Israel matching conditions [40], namely, the induced metric should be continuous and the extrinsic curvature of the junction hypersurface could possibly have a jump, which is related to the distributional energy-momentum tensor on the hypersurface by the Lanczos equation.\",\n \"The scale factor being continuous across the singularity the first Israel condition is obeyed.\",\n \"We will next prove that the second Israel junction condition (the Lanczos equation [41], [40]) is also satisfied.\",\n \"For this we have to check whether Eq.\",\n \"$\\\\frac{\\\\partial }{\\\\partial t}\\\\left( Ha^{2}\\\\right) =\\\\left\\\\lbrace -H^{2}+\\\\frac{3}{2}\\\\left[ \\\\widetilde{\\\\rho }-\\\\widetilde{p}-\\\\overline{p}\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] \\\\right\\\\rbrace a^{2}\\\\ , $ (see Appendix ), still implies the Lanczos equation $\\\\Delta H|_{t_{s}}=-\\\\frac{3}{2}\\\\overline{p}\\\\ , $ derived in Appendix , when $\\\\widetilde{p}+\\\\overline{p}\\\\delta \\\\left( \\\\tau \\\\right) =p_{ACh}$ , $\\\\widetilde{\\\\rho }=\\\\rho _{m}+\\\\rho _{ACh}$ and $\\\\rho _{ACh}$ is generalized to a distribution $\\\\rho _{ACh}a^{2}=\\\\frac{P\\\\left( \\\\tau \\\\right) }{\\\\left[ R\\\\left( \\\\tau \\\\right)+Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{\\\\omega }}~.$ Here $\\\\omega >0$ , $R\\\\left( \\\\tau \\\\right) >0$ , $Q\\\\left( \\\\tau \\\\right) >0$ and $P\\\\left( \\\\tau \\\\right) $ is bounded.\",\n \"When Eq.\",\n \"(REF ) is applied to a test function $\\\\varphi \\\\left( \\\\tau \\\\right) $ , the terms containing $H^{2}$ and $\\\\rho _{m}$ give regular contributions and the limits of the respective integrals vanish, similarly as discussed in Appendix .\",\n \"Also, due to Proposition 2 given in the Appendix A, the integral of the distributional term containing $\\\\rho _{ACh}$ becomes $\\\\int _{-\\\\varepsilon }^{\\\\varepsilon }\\\\frac{P\\\\left( \\\\tau \\\\right) \\\\varphi \\\\left(\\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right) }d\\\\tau ~,$ which also vanishes for $\\\\varepsilon \\\\rightarrow 0$ .\",\n \"We still have to consider the contributions $\\\\int _{-\\\\varepsilon }^{\\\\varepsilon }\\\\left[ \\\\widetilde{p}+\\\\overline{p}\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] \\\\varphi \\\\left( \\\\tau \\\\right) a^{2}d\\\\tau \\\\ .$ Although the contribution $\\\\widetilde{p}\\\\varphi \\\\left( \\\\tau \\\\right) a^{2}$ to the integrand is singular at $\\\\tau =0$ , its integral can be conveniently evaluated by the Residue Theorem.\",\n \"For this we remark, that the integrand is an analytically extendible function into the complex plane in the vicinity of $\\\\tau =0$ and its residue is zero, therefore the integral vanishes.\",\n \"Finally, the contribution containing $\\\\overline{p}$ leads by integration and the limiting process to the right hand side of the Lanczos equation (REF ).\",\n \"Therefore we have proven that the space-time regions separated by the singular spatial hypersurface, representing the pressure singularity, can be joined.\",\n \"In other words, the singularity becomes traversable.\"\n ],\n [\n \"Concluding Remarks\",\n \"It is known that certain models of cosmological fluids in Friedmann universes, like the anti-Chaplygin gas or the tachyon field with a special potential [32], evolve into a sudden future singularity, which in spite of a diverging pressure, is weak.\",\n \"It was argued that singularities of this kind could be traversable despite infinite tidal forces emerging at the singularity for an infinitesimally short time [26].\",\n \"In Ref.\",\n \"[29] the process of crossing of the Big Brake singularity was described in some detail for the tachyon model [32].\",\n \"(The particularity of the Big Brake singularity, consists in the fact that at the crossing of such a singularity the Hubble variable is not only finite, but vanishes.)\",\n \"We also note recent discussions [42] on crossing the “traditional” Big Bang and Big Crunch singularities.\",\n \"In the present paper we considered a simple cosmological model containing a mixture of anti-Chaplygin gas and dust.\",\n \"We have shown that the geodesics equations and their solutions are still well-defined in this case, however the inclusion of dust generates a nonzero value of the Hubble parameter at the singularity encounter, generating the following paradox.\",\n \"The dust would require a continued expansion, which would make the energy density and pressure of the anti-Chaplygin gas ill-defined.\",\n \"A contraction in turn, would be compatible with the anti-Chaplygin gas, nevertheless implying an abrupt change of the Hubble parameter from expansion to contraction.\",\n \"The jump in the Hubble parameter implies the appearance of the $\\\\delta $ function in the Raychaudhuri equation (which contains $\\\\dot{H}$ ).\",\n \"We have cured this situation by redefining the pressure and energy density of the anti-Chaplygin gas as distributions.\",\n \"As an equivalent interpretation, the pressure can be generalized by the addition of a distributional contribution, while the energy density left unchanged, at the price of redefining the equation of state of the anti-Chaplygin gas in a distributional sense.\",\n \"Then all cosmological equations are satisfied in the same distributional sense.\",\n \"We have also shown, that the Israel junction conditions are obeyed through the singular spatial hypersurface, in particular we have enounced and proved a more general form of the Lanczos equation.\",\n \"The results rely on two Propositions and a Corollary proven in Appendix A.\",\n \"The resolution of the paradox at the soft singularity crossing by the introduction of distributional quantities and equations may look unusual, however distributional quantities, localized on hypersurfaces are quite commonly used in general relativity and other gravitational theories.\",\n \"Spacetime regions are frequently matched by the inclusion of distributional layers; also shock-waves can be modeled by Dirac $\\\\delta $ -functions.\",\n \"Braneworld models [44], [43] arise due to the orbifold boundary conditions, the non-smoothness of the 5-dimensional metric at the brane (the jump in its extrinsic curvature) being directly related to the distributional 3+1 standard model fields embedded in the 5-dimensional spacetime.\",\n \"Besides, metrics allowing distributional curvature were considered earlier for studying strings and other distributional sources in general relativity [45].\",\n \"The applications of the distributional quantities to the study of Schwarzschild geometry and point massive particles in general relativity were used in Refs.\",\n \"[46] and [47] respectively.\",\n \"More generically the connection between singularities and the distributional treatment of the physical quantities is well-known in quantum field theory.\",\n \"Indeed, the appearance of the ultraviolet divergences can be understood as the result of the indefiniteness of the product of distributions and the renormalization procedure could be interpreted as a definition of such a product [48].\",\n \"We hope that the investigations presented here may turn useful in deriving similar results in connection with the traversability of other types of sudden singularities.\",\n \"While mathematically self-consistent, the scenario presented in this paper may look somewhat counter-intuitive from the physical point of view.\",\n \"This is because its essential ingredient is the abrupt change of the expansion into a contraction.\",\n \"However, such a behavior is not more counter-intuitive that the absolutely elastic bounce of the ball from a rigid wall, as known in classical mechanics.\",\n \"Indeed, in the latter case the velocity and the momentum of the ball change their direction abruptly.\",\n \"That means that an infinite force acts from the wall onto the ball during an infinitely small interval of time.\",\n \"The result of this action is however integrable and results in a finite change of the momentum of the ball.\",\n \"In fact, the absolutely elastic bounce is an idealization of a process of finite time-span during which inelastic deformations of both the ball and the wall are likely.\",\n \"It is reasonable to think that something similar occurs also in the two-fluid universe model presented in this paper, which undergoes a transition from an expanding to a contracting phase.\",\n \"The smoothing of this process should involve some (temporary) geometrically induced change of the equation of state of matter.\",\n \"Note, that such changes are not uncommon in cosmology.\",\n \"In the tachyon model [32] which was starting point of our studies of the Big Brake singularities, there was the tachyon -pseudotachyon transformation driven by the continuity of the cosmological evolution.\",\n \"In a cosmological model with the phantom field with the cusped potential [49], the transformations between phantom and standard scalar field were considered.\",\n \"Thus, one can imagine that the real process of the transition from the expansion to contraction induced by passing through a soft singularity can imply some temporary change of the equation of state which makes the above processes smoother.\",\n \"We hope to explore such a scenario in the future.\"\n ],\n [\n \"ACKNOWLEDGMENTS\",\n \"We are grateful to V. Gorini, M. O. Katanaev, V. N. Lukash, U. Moschella, D. Polarski and A.\",\n \"A. Starobinsky for useful discussions.\",\n \"The work of ZK was supported by OTKA grant no.\",\n \"100216 and AK was partially supported by the RFBR grant no.\",\n \"11-02-00643.\"\n ],\n [\n \"Propositions on the product and the composition of distributions\\n\",\n \"To investigate how the Friedmann universe crosses a soft singularity, we must solve the field equation in distributional sense.\",\n \"For this purpose we give the definitions of the product and of the composition of distributions and prove two propositions.\",\n \"Fisher derived the following result: $\\\\left[sgn\\\\left( \\\\tau \\\\right) \\\\left|\\\\tau \\\\right|^{\\\\lambda }\\\\right] \\\\delta \\\\left( \\\\tau \\\\right) =0$ for $\\\\lambda >-1$ [50].\",\n \"Our first proposition generalizes this equation for $\\\\lambda \\\\ge 0$ .\",\n \"The second proposition generalizes Antosik's result: $\\\\left[ 1+\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{-1}=1$ [51].\",\n \"Finally, we show a corollary.\",\n \"Let $\\\\rho \\\\left( \\\\tau \\\\right) $ be any infinitely differentiable function having the following properties: $i)$ $\\\\rho \\\\left( \\\\tau \\\\right) =0$ for $\\\\left|\\\\tau \\\\right|\\\\ge 1$ ; $ii)$ $\\\\rho \\\\left( \\\\tau \\\\right) \\\\ge 0$ ; $iii)$ $\\\\rho \\\\left( \\\\tau \\\\right) =\\\\rho \\\\left( -\\\\tau \\\\right) $ ; $iv)$ $\\\\int _{-1}^{1}\\\\rho \\\\left( \\\\tau \\\\right) d\\\\tau =1$ .\",\n \"Then $\\\\delta _{n}\\\\left(\\\\tau \\\\right) =n\\\\rho \\\\left( n\\\\tau \\\\right) $ (with $n=1$ , 2, ...) is a regular sequence of infinitely differentiable functions converging to Dirac delta function: $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\left\\\\langle \\\\delta _{n},\\\\varphi \\\\right\\\\rangle =\\\\left\\\\langle \\\\delta ,\\\\varphi \\\\right\\\\rangle $ for any $\\\\varphi \\\\in \\\\mathcal {D}$ [52].\",\n \"Here $\\\\mathcal {D}$ denotes the space of test functions having continuous derivatives of all orders and compact support.\",\n \"The action of an $f\\\\in \\\\mathcal {D}^{\\\\prime }$ distribution on test functions $\\\\varphi $ is given by $\\\\left\\\\langle f,\\\\varphi \\\\right\\\\rangle $ , which in the case when $f$ is an ordinary locally summable function is nothing but $\\\\int _{-\\\\infty }^{\\\\infty }f\\\\left( \\\\tau \\\\right) \\\\varphi \\\\left(\\\\tau \\\\right) d\\\\tau $ .\",\n \"We note that $\\\\delta _{n}\\\\left( \\\\tau \\\\right) $ has the compact support: $\\\\left( -1/n,1/n\\\\right) $ .\",\n \"We will also use the $n$ -th derivative of $f\\\\in \\\\mathcal {D}^{\\\\prime }$ acts as$\\\\ \\\\left\\\\langle df\\\\left(\\\\tau \\\\right) /d\\\\tau ^{n},\\\\varphi \\\\right\\\\rangle =\\\\left( -1\\\\right)^{n}\\\\left\\\\langle f\\\\left( \\\\tau \\\\right) ,d\\\\varphi /d\\\\tau ^{n}\\\\right\\\\rangle $ .\",\n \"For an arbitrary distribution $f$ , the function $f_{n}\\\\left( \\\\tau \\\\right)=f\\\\ast \\\\delta _{n}\\\\equiv \\\\left\\\\langle f\\\\left( \\\\tau -x\\\\right) ,\\\\delta _{n}\\\\left( x\\\\right) \\\\right\\\\rangle $ gives a sequence of infinitely differentiable functions converging to $f$ .\",\n \"Definition 1 : The commutative product of $f$ and $g$ exists and is equal to $h$ on the open interval $\\\\left( a,b\\\\right) $ ($-\\\\infty \\\\le a0$ , $P\\\\left( \\\\tau \\\\right) $ is bounded, $R\\\\left( \\\\tau \\\\right) \\\\ne 0$ , and in some range close $\\\\tau =0$ the signs of $R\\\\left(\\\\tau \\\\right) $ and $Q\\\\left( \\\\tau \\\\right) $ are the same if $Q\\\\left( \\\\tau \\\\right) \\\\ne 0$ ) exists if $P\\\\left( \\\\tau \\\\right) /R^{\\\\omega }\\\\left( \\\\tau \\\\right) $ existsWe note that this proposition can be held even if $P\\\\left( \\\\tau \\\\right) =1$ and $R\\\\left( \\\\tau \\\\right) =\\\\delta \\\\left( \\\\tau \\\\right) $ with $\\\\omega =1,$ $2,$ $...$ .\",\n \"Indeed, $\\\\delta ^{-\\\\omega }\\\\left( \\\\tau \\\\right) $ exists in neutrix limit and $\\\\delta ^{-\\\\omega }\\\\left( \\\\tau \\\\right) =0$ [55].\",\n \"Thus in the definition 2, the usual limit must be changed for neutrix limit for this case.\",\n \"and $\\\\frac{P\\\\left( \\\\tau \\\\right) }{\\\\left[ R\\\\left( \\\\tau \\\\right) +Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{\\\\omega }}=\\\\frac{P\\\\left( \\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right) }\\\\ .$ By the definition of composition of distributions, we should calculate $&&\\\\left\\\\langle \\\\frac{P_{n}\\\\left( \\\\tau \\\\right) }{\\\\left[ R_{m}\\\\left( \\\\tau \\\\right) +Q_{m}\\\\left( \\\\tau \\\\right) \\\\delta _{m}\\\\left( \\\\tau \\\\right) \\\\right]_{n}^{\\\\omega }},\\\\varphi \\\\left( \\\\tau \\\\right) \\\\right\\\\rangle \\\\\\\\&=&\\\\int _{-\\\\infty }^{\\\\infty }\\\\int _{-1/n}^{1/n}\\\\frac{\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\delta _{n}\\\\left( x\\\\right) dxd\\\\tau }{\\\\left[R_{m}\\\\left( \\\\tau -x\\\\right) +Q_{m}\\\\left( \\\\tau \\\\right) \\\\delta _{m}\\\\left( \\\\tau -x\\\\right) \\\\right] ^{\\\\omega }}\\\\ .$ Performing a change of the variables as $\\\\tau =\\\\tau $ , $y=m\\\\left( \\\\tau -x\\\\right) $ , we have $&=&-\\\\frac{1}{m}\\\\int _{-\\\\infty }^{\\\\infty }\\\\int _{-\\\\infty }^{\\\\infty }\\\\frac{\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{\\\\left[ R_{m}\\\\left( y/m\\\\right) +mQ_{m}\\\\left( y/m\\\\right) \\\\rho \\\\left( y\\\\right) \\\\right] ^{\\\\omega }}dyd\\\\tau \\\\\\\\&=&-\\\\frac{1}{m}\\\\int _{-\\\\infty }^{\\\\infty }\\\\int _{\\\\Omega _{1}}\\\\frac{\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{R_{m}^{\\\\omega }\\\\left( y/m\\\\right) }dyd\\\\tau \\\\\\\\&&-\\\\frac{1}{m}\\\\int _{-\\\\infty }^{\\\\infty }\\\\int _{\\\\Omega _{2}}\\\\frac{\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{\\\\left[ R_{m}\\\\left( y/m\\\\right) +mQ_{m}\\\\left( y/m\\\\right) \\\\rho \\\\left( y\\\\right) \\\\right] ^{\\\\omega }}dyd\\\\tau \\\\ ,$ where $\\\\Omega _{2}=\\\\left\\\\lbrace y:\\\\left|y\\\\right|<1\\\\text{ and }\\\\rho \\\\left( y\\\\right) \\\\ne 0\\\\right\\\\rbrace $ and $\\\\Omega _{1}=\\\\mathbb {R}-\\\\Omega _{2}$ .\",\n \"The double limit of the first term is $&&\\\\lim _{n\\\\rightarrow \\\\infty }\\\\lim _{m\\\\rightarrow \\\\rightarrow \\\\infty }-\\\\frac{1}{m}\\\\int _{-\\\\infty }^{\\\\infty }d\\\\tau \\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left(\\\\tau \\\\right) \\\\\\\\&&\\\\times \\\\int _{\\\\Omega _{1}}\\\\frac{\\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{R_{m}^{\\\\omega }\\\\left( y/m\\\\right) }dy \\\\\\\\&=&\\\\lim _{n\\\\rightarrow \\\\infty }\\\\lim _{m\\\\rightarrow \\\\rightarrow \\\\infty }\\\\int _{-\\\\infty }^{\\\\infty }d\\\\tau \\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\\\\\&&\\\\times \\\\int _{\\\\begin{array}{c} \\\\left|x\\\\right|<1/n, \\\\\\\\ m\\\\left|\\\\tau -x\\\\right|\\\\in \\\\Omega _{1}\\\\end{array}}\\\\frac{\\\\delta _{n}\\\\left( x\\\\right) }{R_{m}^{\\\\omega }\\\\left( \\\\tau -x\\\\right) }dx \\\\\\\\&=&\\\\left\\\\langle \\\\frac{P\\\\left( \\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right) },\\\\varphi \\\\left( \\\\tau \\\\right) \\\\right\\\\rangle \\\\ .$ We investigate the absolute value of the second integral.\",\n \"According to our assumptions for $R$ and $Q$ , and since we are interested in $m\\\\rightarrow \\\\infty $ , we can choose $m$ large enough to let the signs of $R$ and $Q$ be the same, then for $\\\\omega >0$ : $&&\\\\left|\\\\frac{1}{m}\\\\int _{-\\\\infty }^{\\\\infty }\\\\int _{\\\\Omega _{2}}\\\\frac{\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{\\\\left[ R_{m}\\\\left( y/m\\\\right) +mQ_{m}\\\\left( y/m\\\\right) \\\\rho \\\\left( y\\\\right) \\\\right] ^{\\\\omega }}dyd\\\\tau \\\\right|\\\\\\\\&\\\\le &\\\\left|\\\\frac{1}{m^{1+\\\\omega }}\\\\int _{-\\\\infty }^{\\\\infty }\\\\varphi \\\\left( \\\\tau \\\\right) P_{n}\\\\left( \\\\tau \\\\right) \\\\int _{\\\\Omega _{2}}\\\\frac{\\\\delta _{n}\\\\left( \\\\tau -y/m\\\\right) }{Q_{m}\\\\left( y/m\\\\right) \\\\rho ^{\\\\omega }\\\\left(y\\\\right) }dyd\\\\tau \\\\right|\\\\ .$ Performing a change of the variables as $z=n\\\\left( \\\\tau -y/m\\\\right) $ , $y=y$ , we have $&\\\\le &\\\\frac{1}{m^{1+\\\\omega }}\\\\int _{-1}^{1}\\\\int _{\\\\Omega _{2}}\\\\left|\\\\varphi \\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) P_{n}\\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) \\\\frac{\\\\rho \\\\left( z\\\\right) }{\\\\rho ^{\\\\omega }\\\\left( y\\\\right) }\\\\right|dydz \\\\\\\\&\\\\le &\\\\frac{1}{m^{1+\\\\omega }}\\\\sup _{\\\\Omega _{2},\\\\left|z\\\\right|\\\\le 1}\\\\left|\\\\varphi \\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) P_{n}\\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) \\\\rho ^{-\\\\omega }\\\\left( y\\\\right) \\\\right|\\\\\\\\&&\\\\times \\\\int _{-1}^{1}\\\\rho \\\\left( z\\\\right) dz\\\\int _{-1}^{1}dy \\\\\\\\&=&\\\\frac{2}{m^{1+\\\\omega }}\\\\sup _{\\\\Omega _{2},\\\\left|z\\\\right|\\\\le 1}\\\\left|\\\\varphi \\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) P_{n}\\\\left( \\\\frac{z}{n}+\\\\frac{y}{m}\\\\right) \\\\rho ^{-\\\\omega }\\\\left( y\\\\right) \\\\right|\\\\ ,$ that vanishes for $m\\\\rightarrow \\\\infty $ if $P$ is bounded.\",\n \"Corollary 1 : The distribution $d\\\\left\\\\lbrace P\\\\left( \\\\tau \\\\right) \\\\left[ R\\\\left( \\\\tau \\\\right) +Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{-\\\\omega }\\\\right\\\\rbrace /d\\\\tau $ (with the same properties for $P$ , $R$ , $Q$ and $\\\\omega $ as in proposition 2) exists if $P\\\\left( \\\\tau \\\\right) /R^{\\\\omega }\\\\left( \\\\tau \\\\right) $ and its derivative exist, and $\\\\frac{d}{d\\\\tau }\\\\frac{P\\\\left( \\\\tau \\\\right) }{\\\\left[ R\\\\left( \\\\tau \\\\right)+Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{\\\\omega }}=\\\\frac{d}{d\\\\tau }\\\\frac{P\\\\left( \\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right) }\\\\ .$ Applying the derivative of a distribution at tests functions, and using the fact that $d\\\\varphi /d\\\\tau \\\\in \\\\mathcal {D}$ for any $\\\\varphi \\\\in \\\\mathcal {D}$ , and by the proposition 2, we have $&&\\\\left\\\\langle \\\\frac{d}{d\\\\tau }\\\\frac{P\\\\left( \\\\tau \\\\right) }{\\\\left[ R\\\\left(\\\\tau \\\\right) +Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right]^{\\\\omega }},\\\\varphi \\\\right\\\\rangle \\\\\\\\&=&-\\\\left\\\\langle \\\\frac{P\\\\left( \\\\tau \\\\right) }{\\\\left[ R\\\\left( \\\\tau \\\\right)+Q\\\\left( \\\\tau \\\\right) \\\\delta \\\\left( \\\\tau \\\\right) \\\\right] ^{\\\\omega }},\\\\frac{d}{d\\\\tau }\\\\varphi \\\\right\\\\rangle \\\\\\\\&=&-\\\\left\\\\langle \\\\frac{P\\\\left( \\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right)},\\\\frac{d}{d\\\\tau }\\\\varphi \\\\right\\\\rangle =\\\\left\\\\langle \\\\frac{d}{d\\\\tau }\\\\frac{P\\\\left( \\\\tau \\\\right) }{R^{\\\\omega }\\\\left( \\\\tau \\\\right) },\\\\varphi \\\\right\\\\rangle \\\\ .$\"\n ],\n [\n \"Two simple examples of the product and of the decomposition of\\ndistributions\",\n \"The definition of the product and the composition of distributions, used in this paper and presented in the Appendix A are not often encountered in physics.\",\n \"Thus, to give the reader some flavor of the corresponding considerations, using simpler means we decided to give two semi-heuristic examples of such products and compositions.\",\n \"We consider first a remarkable formula $\\\\mathcal {P}\\\\left( \\\\frac{1}{x}\\\\right) \\\\delta (x)=-\\\\frac{1}{2}\\\\delta ^{\\\\prime }(x)~, $ which was first proven in [57].\",\n \"Here $\\\\mathcal {P}$ means the principal value of the corresponding function.\",\n \"We shall prove here that the regularizing succession of functions with compact support $\\\\rho $ , employed in the Appendix A and the references therein, can be chosen alternatively as the family of the Cauchy-Lorentz functions $f_{\\\\epsilon }(x)=\\\\frac{1}{\\\\pi }\\\\frac{\\\\epsilon }{x^{2}+\\\\epsilon ^{2}}~.$ It is well known that when the small parameter $\\\\epsilon \\\\rightarrow 0$ , the functions of this family tend in the distributional sense to the Dirac $\\\\delta $ function.\",\n \"Obviously, the convolution of the function (REF with the Dirac $\\\\delta $ function gives again the same function (REF ): $f_{\\\\epsilon }\\\\ast \\\\delta (x)=f_{\\\\epsilon }(x)~.\",\n \"$ The calculation of the convolution of the principal value $\\\\mathcal {P}\\\\left(\\\\frac{1}{x}\\\\right) $ with the function $f_{\\\\epsilon }(x)$ is slightly more complicated: $&&\\\\mathcal {P}\\\\left( \\\\frac{1}{x}\\\\right) \\\\ast f_{\\\\epsilon }(x)=\\\\lim _{\\\\varepsilon \\\\rightarrow 0}\\\\left( \\\\int _{-\\\\infty }^{x-\\\\varepsilon }dy\\\\frac{1}{x-y}\\\\frac{\\\\epsilon }{\\\\pi (y^{2}+\\\\epsilon ^{2})}\\\\right.\",\n \"\\\\\\\\&&\\\\left.\",\n \"+\\\\int _{x+\\\\varepsilon }^{\\\\infty }dy\\\\frac{1}{x-y}\\\\frac{\\\\epsilon }{\\\\pi (y^{2}+\\\\epsilon ^{2})}\\\\right) =\\\\frac{x}{x^{2}+\\\\epsilon ^{2}}~.$ The product of the expressions (REF ) and (REF ) is $\\\\mathcal {P}\\\\left( \\\\frac{1}{x}\\\\right) _{\\\\epsilon }\\\\ast \\\\delta _{\\\\epsilon }(x)=\\\\frac{\\\\epsilon x}{\\\\pi (x^{2}+\\\\epsilon ^{2})^{2}}~.\",\n \"$ Let us now consider a family of functions $\\\\frac{df_{\\\\epsilon }(x)}{dx}=-\\\\frac{2x\\\\epsilon }{\\\\pi (x^{2}+\\\\epsilon ^{2})^{2}}~.\",\n \"$ One can easily prove that if the family of functions (REF ) converges in the distributional sense to the Dirac $\\\\delta $ function, the family of their derivatives (REF ) converges to the derivative of the delta function.\",\n \"Now, comparing the right-hand sides of Eqs.\",\n \"(REF ) and (REF ) we see that when $\\\\epsilon \\\\rightarrow 0$ the product in the left-hand side of Eq.\",\n \"(REF ) converges in the distributional sense to $-\\\\frac{1}{2}\\\\delta ^{\\\\prime }(x)$ and thus the correctness of the equality (REF ) is checked.\",\n \"Now let us discuss the Antosik identity [51] $\\\\frac{1}{1+\\\\delta (x)}=1.\",\n \"$ Here we have the composition of the distributions $F(g)$ , where $F=\\\\frac{1}{g}$ and $g(x)=1+\\\\delta (x)$ .\",\n \"Calculating the convolutions of the distributions $F$ and $g$ with the Cauchy-Lorentz functions (REF ) we obtain $F_{\\\\sigma }(g)=F\\\\ast f_{\\\\sigma }(g)=\\\\frac{g}{g^{2}+\\\\sigma ^{2}},$ $g_{\\\\epsilon }(x)=1+\\\\frac{\\\\epsilon }{x^{2}+\\\\epsilon ^{2}}.\",\n \"$ Correspondingly the composition of these functions is $F_{\\\\sigma }(g_{\\\\epsilon })=\\\\frac{1+\\\\frac{\\\\epsilon }{\\\\pi (x^{2}+\\\\epsilon ^{2}}}{\\\\sigma ^{2}+\\\\left( 1+\\\\frac{\\\\epsilon }{\\\\pi (x^{2}+\\\\epsilon ^{2}}\\\\right) ^{2}} $ and it is easy to check that $\\\\lim _{\\\\sigma \\\\rightarrow 0}\\\\lim _{\\\\epsilon \\\\rightarrow 0}F_{\\\\sigma }(g_{\\\\epsilon })=1,$ confirming the identity (REF ).\"\n ],\n [\n \"The Lanczos equation\",\n \"For a generic junction surface the Lanczos equation emerges from the Gauss-Codazzi relations [58], [43].\",\n \"The projected Lie derivative of the extrinsic curvature $K_{ab}$ in the normal direction $n$ to the surface is $h_{a}^{i}h_{b}^{j}\\\\mathcal {L}_{\\\\mathbf {n}}K_{ij}=-3\\\\epsilon \\\\left(h_{a}^{i}h_{b}^{k}T_{ik}-\\\\frac{h_{ab}}{2}g^{ik}T_{ik}\\\\right) +\\\\mathcal {Z}_{ab}\\\\ $ (Eq.\",\n \"(21) of [43] in the units $8\\\\pi G/3=1$ ), with $\\\\mathcal {Z}_{ab} &=&-\\\\epsilon \\\\mathcal {R}_{ab}+2K_{ac}K_{b}^{c}-g^{ik}K_{ik}K_{ab} \\\\\\\\&&+D_{b}\\\\alpha _{a}-\\\\epsilon \\\\alpha _{b}\\\\alpha _{a}\\\\ .$ Here $g_{ab}$ is the space-time metric, $h_{ab}=g_{ab}-\\\\epsilon n_{a}n_{b}$ ($\\\\epsilon =n^{a}n_{a}=\\\\left\\\\lbrace -1,1\\\\right\\\\rbrace $ ) is the induced metric on the junction surface, and $T_{ab}$ is the energy-momentum tensor.\",\n \"The tensor $\\\\mathcal {Z}_{ab}$ depends only on geometrical quantities: $\\\\mathcal {R}_{ab}$ and $D$ are the Ricci tensor and covariant derivative induced on the hypersurface, and $\\\\alpha _{a}=n^{c}\\\\nabla _{c}n_{a}$ , with $\\\\nabla $ the 4-dimensional covariant derivative.\",\n \"When the energy-momentum tensor is a sum $T_{ik}=\\\\Pi _{ik}+\\\\Upsilon _{ik}\\\\delta \\\\left( \\\\tau \\\\right) $ (where $\\\\tau $ is the coordinate adapted to $n$ , i.e.\",\n \"$n=t_{S}^{-1}\\\\partial /\\\\partial \\\\tau $ , and $n^{a}\\\\Upsilon _{ab}=0$ ), with $\\\\Pi _{ik}$ the regular 4-dimensional part and $\\\\Upsilon _{ik}$ the distributional part on the hypersurface, integration of Eq.\",\n \"(REF ) across $\\\\tau $ through an infinitesimal range containing the hypersurface keeps only the distributional part, leading to the Lanczos equation [41], [40].\",\n \"$\\\\Delta K_{ab}=-3\\\\epsilon \\\\left( \\\\Upsilon _{ab}-\\\\frac{\\\\Upsilon }{2}h_{ab}\\\\right) \\\\ ,$ or equivalently $-3\\\\epsilon \\\\Upsilon _{ab}=\\\\Delta K_{ab}-h_{ab}\\\\Delta K\\\\ .$ Here $\\\\Upsilon $ is the trace of $\\\\Upsilon _{ab}$ .\",\n \"As $\\\\mathcal {Z}_{ab}$ is finite, its contribution to the integral across the infinitesimal range also vanishes.\",\n \"Without a distributional energy-momentum part, the extrinsic curvature should be continuous.\",\n \"Let us now specialize this for a junction along a maximally symmetric $\\\\tau =0$ spacelike hypersurface (a hyperplane with $\\\\mathcal {R}_{ab}=0$ ) embedded in a flat Friedmann space-time.\",\n \"The normal vector $n$ has zero acceleration $\\\\alpha _{a}=0$ and the extrinsic curvature becomes $K_{ab}=\\\\dot{a}a\\\\widetilde{h}_{ab}$ , with $\\\\widetilde{h}_{ab}$ the 3-dimensional Euclidean metric.\",\n \"The curvature term is $\\\\mathcal {Z}_{ab}=-H^{2}a^{2}\\\\widetilde{h}_{ab} $ and the energy momentum tensors are $\\\\Pi _{ab}=\\\\widetilde{\\\\rho }n_{a}n_{b}+\\\\widetilde{p}a^{2}\\\\widetilde{h}_{ab}$ and $\\\\Upsilon _{ab}=\\\\overline{p}a^{2}\\\\widetilde{h}_{ab}$ .\",\n \"Since the projected Lie-derivative in Eq.\",\n \"(REF ) becomes a time derivative, the equation reads $\\\\frac{\\\\partial }{\\\\partial t}\\\\left( Ha^{2}\\\\right) =\\\\left\\\\lbrace -H^{2}+\\\\frac{3}{2}\\\\left[ \\\\widetilde{\\\\rho }-\\\\widetilde{p}-\\\\overline{p}\\\\delta \\\\left( \\\\tau \\\\right) \\\\right] \\\\right\\\\rbrace a^{2}\\\\ , $ which is a combination of the Raychaudhuri and Friedmann equations.\",\n \"For finite $H$ , $\\\\widetilde{\\\\rho }$ and $\\\\widetilde{p}$ as before the integration of Eq.\",\n \"(REF ) across an infinitesimal time range $\\\\tau $ leads to the Lanczos equation $\\\\Delta H|_{t_{s}}=-\\\\frac{3}{2}\\\\overline{p}\\\\ .\",\n \"$\"\n ]\n]"},"id":{"kind":"string","value":"1204.1199"}}},{"rowIdx":1866989,"cells":{"text":{"kind":"list like","value":[["Ulta-slow relaxation in discontinuous-film based electron glasses"],["Abstract We present field effect measurements on discontinuous 2D thin films which are composed of a sub monolayer of nano-grains of Au, Ni, Ag or Al.","Like other electron glasses these systems exhibit slow conductance relaxation and memory effects.","However, unlike other systems, the discontinuous films exhibit a dramatic slowing down of the dynamics below a characteristic temperature $T^*$.","$T^*$ is typically between 10-50K and is sample dependent.","For $T}$ , of a 0.6*0.6 mm film becomes measurable for $T}$ with lowering T was observed in other systems [22], however, in those cases it was found that $\\frac{\\delta G}{} \\propto T^{ - \\beta } $ with $\\beta \\sim 2$ .","The power law dependence was attributed to the fact that the magnitude of the fluctuations are proportional to $L_{0}^{2}$ where $L_{0}$ is the spatial scale of an independent microscopic fluctuations which was ascribed to the correlation length of the percolation network.","This length is predicted to have a power law dependence on temperature [23], [24], [25] $ L_0 \\propto [r_h]^\\nu \\propto T^{-\\alpha \\nu }$ where $r_h$ is the characteristic hopping length, $\\alpha $ is the power of the temperature dependence of the resistance given by eq.","REF and $\\nu $ is between 1 and 2.","An exponential dependence of $\\frac{\\delta G}{}$ as seen in fig.","2 implies an unconventional change of the percolation network for $T\\frac{\\delta G}{} (blue circles) and the dip amplitude 20 minutes after the quench-coolling of the sample (black triangles) versus T.(color online)A similar conclusion can be derived from the temperature dependence of the current-voltage characteristics.","It is useful to study conductivity versus voltage curves, $\\sigma (V)$ , such as that shown in fig.","3 for an Au film.","These curves are characterized by a voltage $V_{0}$ above which the conductivity increases with voltage thus deviating from ohmic behavior.","Several theoretical works [23], [24], [25] predict that non-Ohmic conductivity should occur for: $ k_{B}TT^*$ the I-V characteristic yields $V_0 \\propto T^2$ as expected.","For $TT^*$ ) while applying $V_{g}$ =0 for a time $t_{1}$ after which a dip is well developed.","We then cool the sample to a different temperature, perform a gate voltage sweep to determine the size of the dip and change the gate voltage to $V_{g}=-13V$ .","We then wait for one hour and measure the amplitude of the dip at $V_{g}=-13$ .","We define the amplitude ratio between the new dip and the old dip (at $V_{g}=0$ ) as $A_{1h}$ .","This is taken as a measure for the \"slowness\" of the relaxation.","The dependance of $A_{1h}$ on temperature is shown in fig.","4.","It is seen that there is a dramatic decrease of $A_{1h}$ at $T \\sim T^*$ .","Similar behavior was obtained for all studied samples (over 12).","The relaxation times and $T^*$ did not seem to depend on resistance but the amplitude of the memory dip decreased as the sample approached the metal-insulator transition.","Figure: a-d: Conductance versus gate voltage at t=0 (black lightline) and at t=1 hour (blue heavy line) for differenttemperatures.","e: More detailed two dip experiment results for T=55K.","f: Second dip amplitude of the Au sample of panels a-e (red full squares) and a Ni sample (blue empty circles)as a function of temperature (color online).Slowing down of relaxation processes with decreasing temperature may seem natural, however it is contrary to the situation in other electron glass systems.","In amorphous and crystalline indium oxide [30], and granular aluminum [14] the dynamics were found either to be independent on temperature or to slow down upon increasing T. The latter has been suggested as evidence that disordered electronic systems are quantum glasses [30].","In contrast, our Au films exhibit a dramatic slowing of the dynamics upon cooling.","Furthermore, the temperature dependence of $\\tau $ is very peculiar.","Fig.","4f depicts $A_{1h}$ (which inversely depends on $\\tau $ ) as a function of T. It is seen that there is a sharp increase of relaxation times over a small temperature range.","The fact that the slowing down of the dynamics occurs at $T \\sim T^*$ , where mesoscopic conductance fluctuations become significant, leads us to postulate that it is related to a significant dilution of the current carrying network (CCN).","Note that, unlike in other electron glasses, the conductivity in the discontinuous films is strictly two dimensional.","In addition, the SEM image of figure 1 shows that the grains are not closely packed, but rather the film morphology is composed of fractal shaped clusters connected by thin (single grain) bottlenecks [31].","Hence this film is characterized by geometrical and not only electronic percolation which is typical of other hopping systems.","At low temperatures many of these bottlenecks may disconnect from the CCN because of energy mismatches thus leaving most of the sample electrically cut-off.","This gives rise to enhanced conductance fluctuations since the effective electronic system size is small.","This also accounts for the apparent saturation of the R(T) curve at low T as shown in fig 1.","Upon lowering T the electric current network becomes progressively dilute and the variable range hopping mechanism becomes less relevant as fewer sections of the sample dominate the conductivity.","Eventually, at the extreme case where the conductivity governed by a single weak link it is expected to be temperature independent.","Under these conditions, at low T there is a very small probability for an electron to tunnel into and out of the current carrying network and most of the charge carriers are trapped in isolated regions of the sample.","This causes a significant slowing down of the relaxation processes to equilibrium because these rely on many body hopping processes taking place in various parts of the sample.","Since most of the sample is very weakly connected, the relaxation to equilibrium becomes extremely slow, thus hindering the development of a new MD at low temperatures.","This can be realized from fig.","2b that shows the amplitude of the memory dip 20 minutes after the cool down, $A_{20m}$ , as a function of temperature.","It is seen that for $T}$ , of a 0.6*0.6 mm film becomes measurable for $T}$ with lowering T was observed in other systems [22], however, in those cases it was found that $\\\\frac{\\\\delta G}{} \\\\propto T^{ - \\\\beta } $ with $\\\\beta \\\\sim 2$ .\",\n \"The power law dependence was attributed to the fact that the magnitude of the fluctuations are proportional to $L_{0}^{2}$ where $L_{0}$ is the spatial scale of an independent microscopic fluctuations which was ascribed to the correlation length of the percolation network.\",\n \"This length is predicted to have a power law dependence on temperature [23], [24], [25] $ L_0 \\\\propto [r_h]^\\\\nu \\\\propto T^{-\\\\alpha \\\\nu }$ where $r_h$ is the characteristic hopping length, $\\\\alpha $ is the power of the temperature dependence of the resistance given by eq.\",\n \"REF and $\\\\nu $ is between 1 and 2.\",\n \"An exponential dependence of $\\\\frac{\\\\delta G}{}$ as seen in fig.\",\n \"2 implies an unconventional change of the percolation network for $T\\\\frac{\\\\delta G}{} (blue circles) and the dip amplitude 20 minutes after the quench-coolling of the sample (black triangles) versus T.(color online)A similar conclusion can be derived from the temperature dependence of the current-voltage characteristics.\",\n \"It is useful to study conductivity versus voltage curves, $\\\\sigma (V)$ , such as that shown in fig.\",\n \"3 for an Au film.\",\n \"These curves are characterized by a voltage $V_{0}$ above which the conductivity increases with voltage thus deviating from ohmic behavior.\",\n \"Several theoretical works [23], [24], [25] predict that non-Ohmic conductivity should occur for: $ k_{B}TT^*$ the I-V characteristic yields $V_0 \\\\propto T^2$ as expected.\",\n \"For $TT^*$ ) while applying $V_{g}$ =0 for a time $t_{1}$ after which a dip is well developed.\",\n \"We then cool the sample to a different temperature, perform a gate voltage sweep to determine the size of the dip and change the gate voltage to $V_{g}=-13V$ .\",\n \"We then wait for one hour and measure the amplitude of the dip at $V_{g}=-13$ .\",\n \"We define the amplitude ratio between the new dip and the old dip (at $V_{g}=0$ ) as $A_{1h}$ .\",\n \"This is taken as a measure for the \\\"slowness\\\" of the relaxation.\",\n \"The dependance of $A_{1h}$ on temperature is shown in fig.\",\n \"4.\",\n \"It is seen that there is a dramatic decrease of $A_{1h}$ at $T \\\\sim T^*$ .\",\n \"Similar behavior was obtained for all studied samples (over 12).\",\n \"The relaxation times and $T^*$ did not seem to depend on resistance but the amplitude of the memory dip decreased as the sample approached the metal-insulator transition.\",\n \"Figure: a-d: Conductance versus gate voltage at t=0 (black lightline) and at t=1 hour (blue heavy line) for differenttemperatures.\",\n \"e: More detailed two dip experiment results for T=55K.\",\n \"f: Second dip amplitude of the Au sample of panels a-e (red full squares) and a Ni sample (blue empty circles)as a function of temperature (color online).Slowing down of relaxation processes with decreasing temperature may seem natural, however it is contrary to the situation in other electron glass systems.\",\n \"In amorphous and crystalline indium oxide [30], and granular aluminum [14] the dynamics were found either to be independent on temperature or to slow down upon increasing T. The latter has been suggested as evidence that disordered electronic systems are quantum glasses [30].\",\n \"In contrast, our Au films exhibit a dramatic slowing of the dynamics upon cooling.\",\n \"Furthermore, the temperature dependence of $\\\\tau $ is very peculiar.\",\n \"Fig.\",\n \"4f depicts $A_{1h}$ (which inversely depends on $\\\\tau $ ) as a function of T. It is seen that there is a sharp increase of relaxation times over a small temperature range.\",\n \"The fact that the slowing down of the dynamics occurs at $T \\\\sim T^*$ , where mesoscopic conductance fluctuations become significant, leads us to postulate that it is related to a significant dilution of the current carrying network (CCN).\",\n \"Note that, unlike in other electron glasses, the conductivity in the discontinuous films is strictly two dimensional.\",\n \"In addition, the SEM image of figure 1 shows that the grains are not closely packed, but rather the film morphology is composed of fractal shaped clusters connected by thin (single grain) bottlenecks [31].\",\n \"Hence this film is characterized by geometrical and not only electronic percolation which is typical of other hopping systems.\",\n \"At low temperatures many of these bottlenecks may disconnect from the CCN because of energy mismatches thus leaving most of the sample electrically cut-off.\",\n \"This gives rise to enhanced conductance fluctuations since the effective electronic system size is small.\",\n \"This also accounts for the apparent saturation of the R(T) curve at low T as shown in fig 1.\",\n \"Upon lowering T the electric current network becomes progressively dilute and the variable range hopping mechanism becomes less relevant as fewer sections of the sample dominate the conductivity.\",\n \"Eventually, at the extreme case where the conductivity governed by a single weak link it is expected to be temperature independent.\",\n \"Under these conditions, at low T there is a very small probability for an electron to tunnel into and out of the current carrying network and most of the charge carriers are trapped in isolated regions of the sample.\",\n \"This causes a significant slowing down of the relaxation processes to equilibrium because these rely on many body hopping processes taking place in various parts of the sample.\",\n \"Since most of the sample is very weakly connected, the relaxation to equilibrium becomes extremely slow, thus hindering the development of a new MD at low temperatures.\",\n \"This can be realized from fig.\",\n \"2b that shows the amplitude of the memory dip 20 minutes after the cool down, $A_{20m}$ , as a function of temperature.\",\n \"It is seen that for $T\\dim H}H$ is a finite set of cardinality and Néron-Tate height effectively bounded.","Here $H$ ranges over all algebraic subgroups of codimension larger than their dimension.","In particular every non-torsion point $Y_0\\in C\\cap \\cup _{{\\rm codim\\,}H>\\dim H}H$ satisfies $\\hat{h}(Y_0) &\\ll _\\eta (h(C)+\\deg C)^{\\frac{N+1}{2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{N-1}{2}+\\eta },\\\\[k(Y_0):\\mathbb {Q}] &\\ll _\\eta (\\deg C [k(C):k])^{\\frac{N+1}{N-1}+\\eta }((h(C)+\\deg C)[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}])^{\\frac{N+1}{2}+\\eta }.$ In the following section we emphasise the implications of these theorems on the effective and quantitative Mordell-Lang Conjecture."],["Applications to the effective Mordell-Lang Conjecture","The CIT is well known to have implications on the Mordell-Lang Conjecture.","The toric case of this conjecture has been extensively studied, also in its effective form, by many authors.","A completely effective version in the toric case can be found in [7], Theorem 5.4.5, and generalisations in [8].","However an effective general result in abelian varieties is not known.","Since we prove an effective version of Theorem REF , we obtain the following effective cases of the Mordell-Lang Conjecture for curves in products of elliptic curves with CM, where effective means that we give a bound for the height of the set of points in $C$ and in a group of finite rank $\\Gamma $ ; the dependence on $C$ and $\\Gamma $ is completely explicit.","The assumption on the relative codimension gives a condition on the rank of $\\Gamma $ .","This method is completely different from the two classical effective methods known in abelian varieties.","The method by Chabauty and Coleman, surveyed, for example, in [27], gives a Mordell-Lang statement for a curve in its Jacobian and $\\Gamma $ of rank less than the genus; it involves a Selmer group calculation and smart computations with the Jacobian of the curve, and it can be made effective in some particular cases (see [23]).","The method by Manin and Demjanenko, described in Chapter 5.2 of [31], gives an effective Mordell theorem for curves with many independent morphisms to an abelian variety.","As remarked by Serre, the method remains of difficult application.","Let $E$ be an elliptic curve defined over the algebraic numbers.","We let $\\hat{h}$ be the standard Néron-Tate height on $E^N$ ; if $V$ is a subvariety of $E^N$ , we shall denote by $h(V)$ the normalized height of $V$ , as defined in [25].","The height of a set is as usual the supremum of the heights of its points.","If $E$ is defined over a field $k$ , we denote by $k_\\mathrm {tor}$ the field of definition of all torsion points of $E$ .","All the constants in the following theorems become explicit if the constant for the Lehmer type bound of Carrizosa in [13] is made explicit.","Theorem 2.1 Let $C$ be a weak-transverse curve in $E^N$ , with $E$ a CM elliptic curve and $N>2$ .","Let $k$ be a field of definition for $E$ .","Let $\\Gamma $ be a subgroup of $E^N$ such that the group generated by its coordinates is an ${\\mathrm {End}}(E)$ -module of rank one.","Then, for any positive $\\eta $ , there exists a constant $c_1$ , depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_1 (h(C)+\\deg C)^{\\frac{N-1}{N-2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{1}{N-2}+\\eta }.$ Let $g$ be a generator of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .","If a point $x=(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $0\\ne a_i,b_i \\in {\\mathrm {End}}(E)$ and torsion points $\\zeta _i\\in E$ such that $a_ix_i=b_i g+\\zeta _i, \\qquad i=1,\\cdots ,N.$ If all $b_i=0$ then $x$ is a torsion point, thus it has height zero.","We can suppose, for instance, that $b_1\\ne 0$ .","So $g$ depends on $x_1$ .","Substituting this dependence in the last $N-1$ equations, we obtain a system of $N-1$ linearly independent equations in the variables $x_1, \\dots ,x_N$ given by $ b_1 a_j x_j=a_1 b_j x_1+\\zeta ^{\\prime }_j, \\qquad j=2,\\cdots ,N$ for $\\zeta ^{\\prime }_j$ torsion points.","These equations define a torsion variety $H$ of codimension $N-1$ in $E^N$ .","Thus, $ (C\\cap \\Gamma )\\subseteq (C\\cap \\cup _{\\dim H=1}H) \\cup (C\\cap {\\mathrm {Tor}}_{E^N})=S_{N-1}(C)$ for $H$ ranging over all algebraic subgroup of dimension one.","However, if $N>2$ , any point $x$ in the intersection $C\\cap H$ is a $C$ -torsion anomalous point.","In addition, as $C$ is weak-transverse, each such $C$ -torsion anomalous point is maximal.","If $x$ is not a torsion point then $H$ is minimal for $x$ , because $H$ has dimension one.","Thus the relative codimension of $x$ in $H$ is one.","Applying Theorem REF we deduce the bound.","If $N=2$ , the intersection $C \\cap H$ is not torsion anomalous, so we must follow another line.","For this reason, we need the assumption of $C$ being transverse.","Theorem 2.2 Let $C$ be a transverse curve in $E^2$ with $E$ a CM elliptic curve defined over a number field $k$ .","Let $\\Gamma $ be a subgroup of $E^2$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank one, generated by $g$ .","Then, for any positive $\\eta $ there exists a constant $c_2$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_2[k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}]^{1+\\eta }(h(C)+(\\hat{h}(g)+1)\\deg C)^{2+\\eta }.$ Let $g$ be a generator of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .","Consider the curve $C^{\\prime }=C\\times g$ in $E^3$ .","Since $C$ is transverse then $C^{\\prime }$ is weak-transverse.","If a point $(x_1, x_2)$ is in $\\Gamma $ then there exist $0\\ne a_i,b_i \\in {\\mathrm {End}}(E)$ and torsion points $\\zeta _i$ such that $a_1x_1=b_1 g+\\zeta _1 , \\,\\,\\, \\,\\,\\, a_2x_2=b_2 g+\\zeta _2.$ Thus the point $(x_1,x_2,g) $ belongs to the intersection $C^{\\prime }\\cap H$ with $H$ the torsion variety of codimension 2 in $E^3$ defined by the equations $a_1x_1=b_1 x_3+\\zeta _1, \\quad a_2x_2=b_2 x_3+\\zeta _2.$ Thus $(x_1,x_2,g)\\in (C^{\\prime }\\cap \\cup _{\\dim H=1}H)$ for $H$ ranging over all algebraic subgroups of dimension one in $E^3$ .","Therefore $C\\cap \\Gamma $ is embedded in $C^{\\prime }\\cap \\cup _{\\dim H=1}H$ and $\\hat{h}(C\\cap \\Gamma )\\le \\hat{h}(C^{\\prime }\\cap \\cup _{\\dim H=1}H)$ .","However any point $x$ in the intersection $C^{\\prime }\\cap H$ is a maximal $C^{\\prime }$ -torsion anomalous point.","If $x$ is not a torsion point, then it has relative codimension one in $H$ .","We apply Theorem REF to $C^{\\prime }\\subseteq E^3$ , with $\\deg C^{\\prime }=\\deg C$ and $h(C^{\\prime })\\le 2( h(C)+\\hat{h}(g)\\deg C)$ by Zhang's inequality, recalled in Section REF below.","This gives the bound for the height of a point in $C\\cap \\Gamma $ .","The assumptions on the curve are necessary in both theorems.","Indeed for a weak-transverse curve $C$ in $E^2$ the above theorem is not true.","Consider the weak-transverse curve $C=E\\times g$ ; for any positive integer $m$ , let $\\Gamma _m$ be generated by the point $\\gamma _m=(mg,g)$ .","For any $m$ , the point $\\gamma _m$ belongs to $C$ and, for $m$ which goes to infinity, the height of $\\gamma _m$ tends to infinity.","Thus there cannot be general bounds independent of $\\Gamma $ .","This does not happen in higher codimension.","The analogue would be $C=E\\times g\\times g^{\\prime }$ where $g$ and $g^{\\prime }$ are linearly independent to ensure the weak-transversality.","No point of $C$ can be in $\\Gamma $ , as no point of the type $(x,g,g^{\\prime })$ has coordinates in $\\overline{\\Gamma }$ , which has rank one.","We also remark that for $C$ weak-transverse, $C\\times g$ is not necessarily weak-transverse: let $C=E\\times g$ , then $E\\times g\\times g$ is contained in the abelian subvariety $x_2=x_3$ .","With this method, the bound in the case $N=2$ depends on $g$ .","It is possible to remove such a dependence.","However one has to prove that for a transverse curve $C$ , the set $C\\cap \\cup _{\\dim H=1}H$ has bounded height.","This is proved in [32] Theorem 1.","The proof is effective, but not explicit in the dependence on $C$ and on $E$ .","The dependence on $C$ is given comparing the height function relative to a point with the height function relative to the intersections of $C\\cap \\Delta ^-$ where $ \\Delta ^-$ is the divisor defined by $x_1=-x_2$ .","We use [32] to deduce another effective case of the Mordell-Lang Conjecture.","Theorem 2.3 Let $C$ be a transverse curve in $E^N$ with $E$ an elliptic curve defined over a number field.","Let $\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank at most $N-1$ .","Then the set $C\\cap \\Gamma $ is finite of Néron-Tate height bounded by a constant $c(C,E^N)$ depending on $C$ and $E^N$ i.e.","$\\hat{h}(C\\cap \\Gamma ) \\le c(C,E^N).$ The proof follows the same idea as the proof of Theorem REF .","Let $g_1, \\dots , g_{N-1}$ be generators of $\\overline{\\Gamma }$ , the group generated by all the coordinates of any element in $\\Gamma $ .","If a point $(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $a_i\\in {\\mathrm {End}}(E)$ , an $N\\times (N-1)$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1, \\ldots , a_Nx_N)^{t}=B\\cdot (g_1,\\ldots ,g_{N-1})^{t}+\\zeta .$ Explicitating $(g_1, \\dots , g_{N-1})$ in terms of $x_1, \\dots , x_{N-1}$ from the first $N-1$ equations and substituting in the last equation, we obtain one equation in the variables $x_1, \\dots ,x_N$ $ a^{\\prime }_1x_1+ \\dots +a^{\\prime }_Nx_N=\\zeta ^{\\prime },$ where $\\zeta ^{\\prime }\\in E$ is a torsion point and $a^{\\prime }_i\\in {\\mathrm {End}}(E)$ .","This equation defines a torsion variety $H$ of codimension one in $E^N$ .","Thus, $ (C\\cap \\Gamma )\\subseteq (C\\cap \\cup _{{\\rm codim\\,}H\\ge 1}H)=S_1(C)$ for $H$ ranging over all algebraic subgroups of codimension at least one.","By [32] Theorem 1, this set has height bounded by an effective constant depending only on $E^N$ and $C$ .","At this point we want to extend our main theorem, at least in the case of curves, to see if with our method we could prove other cases of the Mordell-Lang Conjecture.","Theorem REF enables us to obtain a more general version of Theorem REF for $C$ weak-transverse and larger rank of $\\overline{\\Gamma }$ .","We also obtain a theorem similar to Theorem REF , with explicit dependence on $C$ .","Theorem 2.4 Let $C$ be a weak-transverse curve in $E^N$ with $E$ an elliptic curve with CM.","Let $k$ be a field of definition for $E$ .","Let $\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank $ t< N/2$ .","Then, for any positive $\\eta $ , there exists a constant $c_3$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_3(h(C)+\\deg C)^{\\frac{N-t}{N-2t}+\\eta }[k_\\mathrm {tor}(C):k_\\mathrm {tor}]^{\\frac{t}{N-2t}+\\eta }.$ Let $g_1, \\dots , g_t$ be generators of the free part of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .","If a point $(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $0\\ne a_i\\in {\\mathrm {End}}(E)$ , an $N\\times t$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (g_1,\\ldots ,g_t)^{t}+\\zeta .$ If the rank of $B$ is zero, then $x$ is a torsion point and so it has height zero.","If $B$ has positive rank $m$ , we can choose $m$ equations of the system corresponding to $m$ linearly independent rows of $B$ .","We use these equations to write the $g_j$ in terms of the $x_i$ and we substitute these expressions in the remaining equations.","We obtain a system of maximal rank with $N-m\\ge N-t$ linearly independent equations in the variables $x_1, \\dots ,x_N$ : ${\\left\\lbrace \\begin{array}{ll}a^{\\prime }_{11}x_1+ \\dots +a^{\\prime }_{1N}x_N=\\zeta ^{\\prime }_1\\\\\\vdots \\\\a^{\\prime }_{N-m,1}x_1+ \\dots +a^{\\prime }_{N-m,N}x_N=\\zeta ^{\\prime }_{N-m}\\end{array}\\right.","}$ where $\\zeta ^{\\prime }_i\\in E$ are torsion points and $a^{\\prime }_{ij}\\in {\\mathrm {End}}(E)$ .","These equations define a torsion variety $H$ of codimension $N-m$ in $E^N$ .","Thus $ (C\\cap \\Gamma )\\subseteq S_{ N-t}(C)$ for $H$ ranging over all algebraic subgroups of codimension at least $N-t$ .","However $\\dim H=N-{\\rm codim\\,}H=m\\le t$ .","Thus, if $N>2t$ $ S_{ N-t}(C)\\subseteq (C\\cap \\cup _{{\\rm codim\\,}H>\\dim H}H).$ Applying Corollary REF with $r=N-t$ gives the wished bound for the height of $C\\cap \\Gamma $ .","We notice that Theorems REF and REF are proved for weak-transverse curves, while previous effective results assumed transversality.","If we assume the transversality of $C$ we can relax the hypothesis on the rank of $\\overline{\\Gamma }$ .","Theorem 2.5 Let $C$ be a transverse curve in $E^N$ with $E$ a CM elliptic curve defined over $k$ .","Let $\\Gamma $ be a subgroup of $E^N$ such that the free part of the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank $t\\le N-1$ , generated by $g_1, \\dots , g_t$ .","Then, for any positive $\\eta $ there exists a constant $c_4$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_4 [k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}]^{\\frac{t}{N-t}+\\eta }(h(C)+ (\\hat{h}(g)+1)\\deg C)^{\\frac{N}{N-t}+\\eta }$ where $g=(g_1, \\dots , g_t)$ .","Consider the curve $C^{\\prime }=C\\times g$ in $E^{N+t}$ .","Since $C$ is transverse then $C^{\\prime }$ is weak-transverse.","If a point $(x_1, \\dots , x_N)$ is in $\\Gamma $ , then there exist $0\\ne a_i\\in {\\mathrm {End}}(E)$ , an $N\\times t$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (g_1,\\ldots ,g_t)^{t}+\\zeta .$ Thus the point $(x_1,\\dots x_N,g_1, \\dots , g_t) $ belongs to the intersection $C^{\\prime }\\cap H$ with $H$ the torsion variety of codimension $N$ and dimension $t$ in $E^{N+t}$ defined by the equations $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (y_{N+1},\\cdots , y_{N+t})^{t}+\\zeta .$ Thus, $C\\cap \\Gamma $ is embedded in $C^{\\prime }\\cap \\cup _{\\dim H=t}H$ , and $\\hat{h}(C\\cap \\Gamma )\\le \\hat{h}(C^{\\prime }\\cap \\cup _{\\dim H=t}H)$ for $H$ ranging over all algebraic subgroups of dimension $t$ .","If $N> t$ then ${\\rm codim\\,}H>\\dim H$ .","The bound for the height is then given by Corollary REF applied to $C^{\\prime }\\subseteq E^{N+t}$ , where $\\deg C=\\deg C^{\\prime }$ , $h(C^{\\prime })\\le 2( h(C)+\\hat{h}(g)\\deg C)$ by Zhang's inequality, recalled in Section REF , below."],["Applications to the quantitative Mordell-Lang Conjecture","In [29], Theorem 1.2 Rémond gives a bound on the cardinality of the intersection $C\\cap \\Gamma $ for a transverse curve in $E^ N$ and $\\Gamma $ a $\\mathbb {Z}$ -module of rank $r$ .","He obtains the following bound $\\#(C\\cap \\Gamma )\\le {c(E^N, \\mathcal {L})}^{r +1}(\\deg C)^{(r +1) N^{20}},$ where $c(E^N, \\mathcal {L})$ is a positive effective constant depending on $E^N$ and on the choice of an invertible, symmetric and ample sheaf $\\mathcal {L}$ on $E^N$ .","The Manin-Mumford Conjecture is a special case of the Mordell-Lang conjecture.","Explicit bounds on the number of torsion points in $C$ are given, for instance, by David and Philippon in [17] and by Hrushovski in [21].","David and Philippon in [17], Proposition 1.12 show that the number of torsion points on a non-torsion curve $C$ is at most $\\#(C\\cap \\mathrm {Tor}_{E^N})\\le (10^{2N+53}\\deg C)^{33},$ where $\\mathrm {Tor}_{E^N}$ is the set of all torsion points of $E^N$ .","Our method enables us to give a sharp bound for the number of non torsion points in $C\\cap \\Gamma $ for $C$ weak-transverse in $E^N$ , which, together with the just mentioned bounds for the torsion, improves in some cases the bounds of Rémond.","Notice that below we use the rank $t$ of $\\overline{\\Gamma }$ , the $\\mathrm {End}(E)$ -module of the coordinates of $\\Gamma $ .","To compare with Rémond's result, we can use the trivial relation $r<2Nt$ and $t2$ and $t=1$ , we have $S\\le d_1 & {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{(N-1)(4N^2-N-4)}{2(N-2)^2}+\\eta }(\\deg C)^{\\frac{2N^3-N^2+N-4}{2(N-2)}+\\eta }\\\\&\\cdot [k(C):k]^{\\frac{N(N-1)(2N+1)}{2(N-2)}+\\eta };$ if $C$ is transverse in $E^2$ and $t=1$ , we have $S\\le d_2 &{([k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}](h(C)+(\\hat{h}(g)+1)\\deg C))}^{29+\\eta }(\\deg C)^{22+\\eta }\\\\&\\cdot [k(C\\times g):k]^{21+\\eta }$ where $g$ is a generator of $\\overline{\\Gamma }$ ; if $C$ is weak-transverse in $E^N$ and $t{\\rm codim\\,}(B+\\zeta )$ .","So $Y$ is a component of $ V \\cap (B^{\\prime }+\\zeta ^{\\prime })$ .","In addition ${\\rm codim\\,}Y <{\\rm codim\\,}V+{\\rm codim\\,}(B+\\zeta )<{\\rm codim\\,}V+{\\rm codim\\,}(B^{\\prime }+\\zeta ^{\\prime }),$ which contradicts the minimality of $ (B+\\zeta )$ ."],["Torsion anomalous varieties as components of different intersections","In the next lemma we prove that every $V$ -torsion anomalous variety which is a component of $V\\cap (B+\\zeta )$ , is also a component of any intersection $V\\cap (A+\\zeta ^{\\prime })$ with $B+\\zeta \\subseteq A+\\zeta ^{\\prime }$ .","Moreover, we can choose $\\zeta =\\zeta ^{\\prime }$ , in fact if a translate $B+p$ is contained in another $ A+p^{\\prime }$ , then $A+p=A+p^{\\prime }$ .","Indeed $B$ is a subgroup and it contains 0, so $p-p^{\\prime }\\in A$ and $A+p^{\\prime }=A+(p-p^{\\prime })+p^{\\prime }=A+p$ .","Lemma 3.2 Let $Y$ be a maximal $V$ -torsion anomalous variety, and let $B+\\zeta $ be minimal for $Y$ .","Then $Y$ is a component of $V\\cap (A+\\zeta )$ for every algebraic subgroup $A\\supseteq B$ , with ${\\rm codim\\,}A\\ge \\dim V-\\dim Y$ .","Clearly $Y\\subseteq V\\cap (A+\\zeta )$ .","Let $X$ be an irreducible component of $V\\cap (A+\\zeta )$ which contains $Y$ , then ${\\rm codim\\,}X \\le {\\rm codim\\,}V+{\\rm codim\\,}A.$ If the inequality is strict then $X$ is anomalous and by the maximality of $Y$ we get $Y=X$ .","Otherwise, since by assumption ${\\rm codim\\,}A\\ge \\dim V-\\dim Y$ , we have ${\\rm codim\\,}X = {\\rm codim\\,}V+{\\rm codim\\,}A\\ge {\\rm codim\\,}V +\\dim V-\\dim Y={\\rm codim\\,}Y,$ but $Y\\subseteq X$ , so the opposite inequality also holds.","This implies ${\\rm codim\\,}X={\\rm codim\\,}Y$ .","Therefore both varieties are irreducible, so $Y=X$ and $Y$ is a component of $V\\cap (A+\\zeta )$ ."],["No torsion anomalous varieties on a weak-transverse translate","The simple choice of maximal and minimal varieties and the group structure allow us to prove Conjecture REF for weak-transverse translates.","Let $Y$ be an $(H+p)$ -torsion anomalous variety and let $B+\\zeta $ be minimal for $Y$ .","Then $Y$ is a component of $(H+p)\\cap (B+\\zeta )$ and $\\dim H+\\dim B-\\dim Y \\dim Y_i$ and $X_1$ is not $V$ -torsion anomalous due to the maximality of the $Y_i$ .","By assumption, $B+\\zeta _i$ is minimal for $Y_i$ for some $\\zeta _i\\in {\\mathrm {Tor}_{G}}$ .","Then $X_1$ is not contained in any $B+\\zeta $ , otherwise $Y_i\\subseteq X_1\\subseteq V\\cap (B+\\zeta )$ and $X_1$ would be a $V$ -torsion anomalous variety; this gives a contradiction.","Let $H+\\zeta $ be the torsion variety (not necessarily proper) of smallest dimension containing $X_1$ .","Then $X_1$ is weak-transverse in $H+\\zeta $ , and since $X_1$ is not $V$ -torsion anomalous $N-\\dim X_1=N-\\dim V+N-\\dim H.$ Recall that, the $Y_i$ are $V$ -torsion anomalous varieties and use (REF ) to obtain $N-\\dim Y_i0$ , there exists a positive constant $c_1$ depending on $E^N$ and $\\eta $ , such that $\\mu (Y) \\ge c_1 \\frac{(\\deg B)^{\\frac{1}{\\dim B-\\dim Y}-\\eta }}{(\\deg Y)^{{\\frac{1}{\\dim B-\\dim Y}}+ \\eta }}.$ This theorem is a special case of the main theorem of [14].","For our applications, the lower bound must take into account the degree of $B$ .","To do so one needs to consider several different line bundles on $E^N$ , and the proof in [14] is based on an equivalence between two line bundles, and on the lower bound for the essential minimum of a transverse variety with respect to the standard bundle ${\\mathcal {O}_{N}}$ .","Such a bound is given by Galateau in [18].","The result of Galateau is inspired by the toric version of Amoroso and David [1] Theorem 1.4.","The constant $c_1$ is effective and becomes explicit if the constants in [18] are made explicitIn a personal communication Galateau provided us explicit computations.."],["A relative Lehmer Estimate for points","In [13], Theorem 1.15, Carrizosa proves the so called relative Lehmer problem for CM abelian varieties.","Theorem 4.3 (Carrizosa) Let $A$ be an abelian defined over a number field $k$ and having CM.","Let $\\mathcal {L}_0$ be an ample symmetric bundle on $A$ , and let $\\mathcal {L}=\\mathcal {L}_0^{4}$ .","Let $H$ be an abelian subvariety of $A$ of dimension $g_0>0$ and $P$ be a point in $H$ , which is not a torsion point modulo all proper abelian subvarieties of $H$ .","Then there exists a positive constant $c(A, k, \\mathcal {L})$ depending only on $A$ , $k$ and $\\mathcal {L}$ such that $\\hat{h}_\\mathcal {L}(P)\\ge \\frac{c(A,k,\\mathcal {L})}{\\omega _{k_\\mathrm {tor}}(P,H)}\\left( \\frac{ \\log \\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}} H)^2)}{\\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}} H)^2)}\\right)^{\\kappa (g_0)}$ where $\\kappa (g_0)=2^{2g_0+1}g_0^{4g_0}(g_0+1)!^{2g_0}$ .","Here $k_\\mathrm {tor}=k(A_\\mathrm {tor})$ is the field of definition of all the torsion in $A$ and ${\\omega _{k_\\mathrm {tor}}(P,H)}$ is defined in the following way (see Definition 1.12 of [13]): $\\omega _{k_\\mathrm {tor}}(P,H)=\\min _V\\left( \\frac{\\deg _\\mathcal {L} V}{\\deg _\\mathcal {L} H} \\right)^{1/{\\rm codim\\,}_H V}$ where $V$ ranges over all proper subvarieties of $H$ defined over $k_\\mathrm {tor}$ , and containing $P$ .","This theorem generalises to abelian varieties a result of Ratazzi in [28] for one elliptic curve.","The proof of Ratazzi is inspired to the theorem of Amoroso and Zannier in [4] for algebraic numbers.","The effectivity in the relative Lehmer is not explicitly stated in the theorem of Carrizosa.In a short personal communication she claims that her constants are effective.","Using also the effectivity of other results, as for instance the result of Amoroso and Zannier in [4]Though in [4] the authors are not concerned with effectivity, their result can be made effective (see [2]).","and of David and Hindry [16], one may check that Carrizosa's constant can be made effective.","In addition, the complicated descent in her article can be replaced by the simple induction argument presented for tori in [3].","An analogous effective method for the relative Lehmer in tori is given by Delsinne, see [15].","As a straightforward corollary of Theorem REF we have the following: Theorem 4.4 Let $E$ be an elliptic curve with CM defined over a field $k$ .","Let $P$ be a point of infinite order in $E^N$ , and let $B+\\zeta $ be the torsion variety of minimal dimension containing $P$ , with $B$ an abelian subvariety and $\\zeta $ a torsion point.","Then for every $\\eta >0$ there exists a positive constant $c_2$ depending on $E^N$ and $\\eta $ , such that $\\hat{h}(P)\\ge c_2 \\frac{(\\deg B)^{\\frac{1}{\\dim B}-\\eta }}{[k_\\mathrm {tor}(P):k_\\mathrm {tor}]^{\\frac{1}{\\dim B}+ \\eta }}.$ We clarify how to obtain this theorem from Theorem REF .","We recall that we are assuming all abelian subvarieties to be defined over $k$ , We remark that $P-\\zeta \\in B$ .","In addition $\\hat{h}(P)=\\hat{h}(P-\\zeta )$ and $[k_\\mathrm {tor}(P):k_\\mathrm {tor}]=[k_\\mathrm {tor}(P-\\zeta ):k_\\mathrm {tor}]$ so we can apply Theorem REF to $P-\\zeta $ in $B$ .","Let $V$ be the set of all conjugates of $P-\\zeta $ over $k_\\mathrm {tor}$ .","Clearly the zero-dimensional variety $V$ is defined over $k_\\mathrm {tor}$ , and it is properly contained in $B$ .","Therefore we can say that $\\omega _{k_\\mathrm {tor}}(P-\\zeta ,B)\\le \\left( \\frac{[k_\\mathrm {tor}(P):k_\\mathrm {tor}]}{\\deg _{\\mathcal {L}}B} \\right)^{1/\\dim B}.$ Notice also that, if we take as $\\mathcal {L}$ the fourth power ${\\mathcal {O}_{N}}^{ 4}$ of the bundle corresponding to the canonical embedding, this only introduces a constant in the heights and the degrees.","Finally, it is clear that in the statement of Theorem REF the corrective factor can be replaced by $\\left( \\frac{ \\log \\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H)^2)}{\\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H)^2)}\\right)^{\\kappa (g_0)}\\gg _\\eta (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H))^{-\\eta }.$ Notice at last that if $B+\\zeta $ is the torsion variety of minimal dimension containing $P$ , then $P-\\zeta $ is not a torsion point modulo all proper abelian subvarieties of $B$ ."],["Torsion anomalous varieties which are not translates","We let $V$ be a weak-transverse variety in a power of elliptic curves.","In this section we prove the finiteness of the maximal $V$ -torsion anomalous varieties which are not translates and have relative codimension one.","Note that this theorem holds in any power of elliptic curves, independently if it has or not CM.","Theorem 5.1 Let $V\\subseteq E^N$ be a weak-transverse variety.","Then the maximal $V$ -torsion anomalous varieties of relative codimension one which are not translates are finitely many.","More precisely, let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.","Assume that $Y$ has relative codimension one in its minimal $B+\\zeta $ .","Then for any $\\eta >0$ there exist constants depending only on $E^N$ and $\\eta $ such that: $ \\deg B \\ll _\\eta (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V -1}+\\eta }, $ $h(Y) \\ll _\\eta (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1}+\\eta } $ and $ \\deg Y \\ll _\\eta \\deg V (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1} -1+\\eta }.$ In addition the torsion points $\\zeta $ belong to a finite set.","Let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.","Let $B+\\zeta $ be minimal for $Y$ .","Then $Y$ is a component of $V\\cap (B+\\zeta )$ with $B$ an abelian variety and $\\zeta $ a torsion point.","In addition ${\\rm codim\\,}Y<{\\rm codim\\,}V+{\\rm codim\\,}B$ .","We prove that $\\deg B$ is bounded only in terms of $V$ and $E^N$ ; we then deduce the bounds for ${h}(Y)$ and $\\deg Y$ .","By Lemma REF $Y$ is weak-transverse in $B+\\zeta $ , and by assumption ${\\rm codim\\,}_{B+\\zeta } Y=1$ ; therefore $Y$ is transverse in $B+\\zeta $ .","As points are translates and $V$ is not contained in a torsion variety, we have $1\\le \\dim Y<\\dim V.$ Applying the Bogomolov estimate Theorem REF to $Y$ in $B+\\zeta $ we get $\\frac{(\\deg B)^{1-\\eta }}{(\\deg Y)^{{1}+\\eta }}\\ll _\\eta \\mu (Y).$ We set $r={\\rm codim\\,}B$ .","Let $h_1, \\dots , h_r$ be the forms of increasing degrees $d_i$ such that $B+\\zeta $ is a component of their zero set, as recalled in Section REF .","Then $d_1 \\cdots d_r \\ll \\deg (B+\\zeta )=\\deg B \\ll d_1 \\cdots d_r.$ We denote $r_1=\\dim V-\\dim Y.$ Note that $r_1< r$ , because $Y$ is $V$ -torsion anomalous.","Let $A$ be the algebraic subgroup given by the first $h_1\\cdots h_{r_1}$ forms.","Then $\\deg A \\ll d_1 \\cdots d_{r_1} .$ Let $A_0$ be an irreducible component of $A$ containing $B+\\zeta $ .","Then by (REF ) we have $\\deg A_0 \\ll d_1 \\cdots d_{r_1}\\ll (\\deg B)^\\frac{r_1}{r} , $ and ${\\rm codim\\,}A_0=r_1=\\dim V-\\dim Y$ .","By Lemma REF , $Y$ is a component of $V\\cap A_0$ .","We apply the Arithmetic Bézout theorem to $V\\cap A_0$ and recall that $h(A_0)=0$ , because $A_0$ is a torsion variety.","Then $h(Y) \\ll (h(V)+ \\deg V) \\deg A_0\\ll (h(V)+\\deg V)(\\deg B)^{\\frac{r_1}{r}}.$ For the irreducible variety $Y$ of positive dimension, Zhang's inequality (REF ) says $\\mu (Y)\\le \\frac{h(Y)}{\\deg Y}.$ Combining this with (REF ) and (REF ) we obtain $\\frac{{(\\deg B)}^{1-\\eta }}{(\\deg Y)^{1+\\eta }}\\ll _\\eta \\mu (Y)\\ll (h(V)+ \\deg V)\\frac{ (\\deg B)^\\frac{r_1}{r}}{\\deg Y}.$ Recall that $Y$ is a component of $ V\\cap (B+\\zeta )$ .","By Bézout's theorem $\\deg Y \\le \\deg B\\deg V$ .","Thus ${(\\deg B)}^{1-\\eta }\\ll _\\eta (h(V)+\\deg V)(\\deg B)^\\frac{r_1}{r}{(\\deg B\\deg V)}^{\\eta }$ and therefore ${(\\deg B)^{\\frac{r-r_1}{r}-2\\eta }}\\ll _\\eta (h(V)+\\deg V){ (\\deg V)}^{\\eta }.$ Since $r-r_1={\\rm codim\\,}V -1$ , for $\\eta $ small enough we get $\\deg B \\ll _\\eta (h(V)+ \\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}+\\eta }(\\deg V)^{\\eta } .$ So we have proved that the degree of $B$ is bounded only in terms of $V$ and $E^N$ .","Since the abelian subvarieties of bounded degree are finitely many, applying Lemma REF we conclude that $\\zeta $ belongs to a finite set.","Finally, the bound on the height of $Y$ is given by (REF ) and (REF ) $h(Y)\\ll _\\eta (h(V)+\\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}+\\eta }(\\deg V)^{\\eta }.$ The bound on the degree is given by Bézout's theorem for the component $Y$ of $V\\cap A_0$ and (REF ) $\\deg Y\\ll _\\eta (h(V)+\\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}-1+\\eta }(\\deg V)^{1+\\eta }.$"],["Torsion anomalous points","In this and the following section we prove that, if $V$ is a weak-transverse variety in a power of elliptic curves with CM, then the $V$ -torsion anomalous varieties which are translates are non-dense in $V$ .","We now prove that the $V$ -torsion anomalous varieties of dimension zero are non-dense.","The proof relies on the Arithmetic Bézout theorem, the Zhang's inequality and on the relative Lehmer, Theorem REF .","As the last bound is proved only for CM elliptic curves we need this assumption.","Theorem 6.1 Let $V\\subseteq E^N$ be a weak-transverse variety, where $E$ has CM.","Then, the set of maximal $V$ -torsion anomalous points of relative codimension one is a finite set of explicitly bounded height and relative degree.","More precisely, let $k$ be a field of definition for $E$ and let $k_\\mathrm {tor}$ be the field of definition of all torsion points of $E^N$ .","Let $d$ be the dimension of $V$ .","Let $Y_0$ be a maximal $V$ -torsion anomalous point and let $B+\\zeta $ be minimal for $Y_0$ , with $\\dim B=1$ .","Then $\\deg B &\\ll _\\eta ((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{\\frac{N-1}{N-1-d}+\\eta } ,\\\\\\hat{h}(Y_0) &\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-1}{N-1-d}+\\eta }[k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]^{\\frac{d}{N-1-d}+\\eta },\\\\[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}] &\\ll _\\eta \\deg V[k_\\mathrm {tor}(V):k_\\mathrm {tor}]^{\\frac{N-1}{N-1-d}+\\eta }{(h(V)+\\deg V)}^{\\frac{d}{N-1-d}+\\eta }.$ In addition the torsion points $\\zeta $ have order effectively bounded in Theorem REF .","Let $Y_0$ be a maximal $V$ -torsion anomalous point, with $B+\\zeta $ minimal for $Y_0$ .","By assumption $\\dim B={\\rm codim\\,}_{B+\\zeta }Y_0=1$ .","We proceed to bound $\\deg B$ and, in turn, the height of $Y_0$ and its degree over $k_\\mathrm {tor}$ .","To this aim we shall use Theorem REF and the Arithmetic Bézout theorem.","By Section REF , the variety $B+\\zeta $ is a component of the torsion variety defined as the zero set of forms $h_1,\\dots ,h_{N-1}$ of increasing degrees $d_i$ and $d_1 \\cdots d_{N-1}\\ll \\deg B=\\deg (B+\\zeta )\\ll d_1 \\cdots d_{N-1}.$ Consider the torsion variety defined as the zero set of the first $d$ forms $h_1, \\dots , h_d$ , and take a connected component $A_0$ containing $B+\\zeta $ .","Then $\\deg A_0\\ll d_1 \\cdots d_d\\ll {(\\deg B)}^{\\frac{d}{N-1}}$ and ${\\rm codim\\,}A_0=d=\\dim V-\\dim Y_0.$ By Lemma REF , each component of $V\\cap (B+\\zeta )$ is a component of $V\\cap A_0$ .","All conjugates of $Y_0$ over $k_\\mathrm {tor}(V)$ are in $V\\cap (B+\\zeta )$ , so the number of components of $V\\cap A_0$ is at least $[k_\\mathrm {tor}(V,Y_0):k_\\mathrm {tor}(V)]\\ge \\frac{[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]}{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}.$ We then apply the Arithmetic Bézout theorem to $Y_0\\subseteq V\\cap A_0$ obtaining $[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]\\hat{h}(Y_0)\\ll {(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}(\\deg B)^{\\frac{d}{N-1}}.$ Applying Theorem REF to $Y_0$ in $B+\\zeta $ , we obtain that for every positive real $\\eta $ $\\hat{h}(Y_0)\\gg _\\eta \\frac{(\\deg B)^{1-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{1+\\eta }}.$ Combining (REF ) and (REF ) we have $\\frac{(\\deg B)^{1-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\eta }}&\\ll _\\eta [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\hat{h}(Y_0)\\ll \\\\&\\ll {(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}(\\deg B)^{\\frac{d}{N-1}}.$ For $\\eta $ small enough we obtain $\\deg B\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{N-1}{N-1-d}+\\eta }[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^\\eta .$ Apply now Bézout's theorem to $V\\cap A_0$ .","All the conjugates of $Y_0$ over $k_\\mathrm {tor}(V)$ are components of the intersection, so $\\frac{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]}{[k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]}\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d}{N-1-d}+\\eta } (\\deg V)^{1+\\eta },$ which gives the last bound in the statement.","Substituting (REF ) back into (REF ) we have the bound on $\\deg B$ .","Finally apply the Arithmetic Bézout theorem to $V\\cap A_0$ to get $\\hat{h}(Y_0)\\ll (h(V)+\\deg V)(\\deg B)^{\\frac{d}{N-1}}\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-1}{N-1-d}}+\\eta [k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]^{\\frac{d}{N-1-d}+\\eta }.$ Having bounded $\\deg B$ , in view of Lemma REF , the points $\\zeta $ belong to a finite set.","Notice that in Theorem REF we have effectively bounded the degree of the abelian variety $B$ , and we applied Lemma REF to prove the finiteness of the points $Y_0$ in a non effective way.","In the following theorem we give a completely effective result.","We explicitly bound the degree $[k(Y_0):\\mathbb {Q}]$ for $Y_0$ not a torsion point; this, together with the bound for $\\hat{h}(Y_0)$ in Theorem REF , allows to effectively find all $V$ -torsion anomalous points of relative codimension one which are not torsion points.","The effectivity of this result is relevant for the applications to the Mordell-Lang Conjecture shown in Section .","Theorem 6.2 Let $V$ be a weak-transverse variety in $E^N$ , where $E$ has CM.","Let $k$ be a field of definition for $E$ .","Let $d$ be the dimension of $V$ .","Let $Y_0$ be a maximal $V$ -torsion anomalous point which is not a torsion point, and let $B+\\zeta $ be minimal for $Y_0$ with $\\dim B=1$ .","Then $[k(Y_0):\\mathbb {Q}]\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d(N-1)}{(N-1-d)^2}+\\eta }{(\\deg V[k(V):k])}^{\\frac{N-1}{N-1-d}+\\eta }.$ In addition the torsion points $\\zeta $ can be chosen with $[k(\\zeta ):\\mathbb {Q}]\\ll [k(Y_0):\\mathbb {Q}]$ and order bounded by $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }.$ Finally let $S$ be the number of maximal $V$ -torsion anomalous points of relative codimension one.","Then $S\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{A_1+\\eta }{(\\deg V)}^{A_2+1+\\eta }[k(V):k]^{A_2+\\eta }$ where $A_1&= \\frac{(N-1)(2(N+1)(N-d-1)+dN(2N+1))}{2(N-d-1)^2} \\le (N+1)^4,\\\\A_2&= \\frac{N(N-1)(2N+1)}{2(N-d-1)} \\le N^3.$ In view of Theorem REF , we know that $\\deg B$ and $\\hat{h}(Y_0)$ are bounded.","We now proceed to bound $[k(Y_0):k]$ .","To this aim we need to construct an algebraic subgroup $G$ of codimension $d$ defined over $k$ , containing $Y_0$ and of controlled degree.","In order to do this we use Siegel's lemma in a similar way as in [32], Proposition 3, which in turn follows the work [10] of Bombieri, Masser and Zannier in tori.","Here we use Siegel's lemma directly on equations with coefficients in the endomorphism ring of $E$ .","We notice that $\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\mathcal {O}$ .","The coordinates of $Y_0=(x_1,\\ldots ,x_N)$ generate an $\\mathcal {O}$ -module $\\Gamma $ of rank one.","The torsion submodule of $\\Gamma $ is well known; for instance in [32], Proposition 2 we find a description for it.","Such a torsion module is clearly $\\mathcal {O}$ invariant.","As a $\\mathbb {Z}$ -module it is generated by two points $T,\\tau T$ of exact orders $R,R^{\\prime }$ respectively, where $R=c(\\tau )R^{\\prime }$ and $c(\\tau )$ is essentially the real part of $\\tau ^2$ , so a constant of the problem.","Therefore we can write $x_i=\\alpha _i g +\\beta _i T$ for a fixed point of infinite order $g$ in $E$ , with coefficients $\\alpha _i,\\beta _i\\in \\mathcal {O}$ such that $\\hat{h}(x_i)=|N_L(\\alpha _i)|\\hat{h}(g)$ and $N_L(\\beta _i)\\ll R^2.$ We want to find coefficients $a_i\\in \\mathcal {O}$ such that $\\sum _i^N a_i x_i=0$ .","This gives a linear system of 2 equations, obtained equating to zero the coefficients of $g$ and $T$ .","The system has coefficients in $\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.","We use a version of Siegel's lemma over $\\mathcal {O}$ as stated in [7], Section 2.9, to get $d$ equations with coefficients in $\\mathcal {O}$ ; multiplying them by a constant depending only on $E$ , we may assume that they have coefficients in $\\mathrm {End}(E)$ .","Thus they define the sought-for algebraic subgroup $G$ of degree $\\deg G\\ll \\left((\\max _i N_L(\\alpha _i))(\\max _i N_L(\\beta _i)) \\right)^{\\frac{d}{N-1}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .","Then $\\deg G_0\\ll \\left((\\max _i N_L(\\alpha _i)) R^2 \\right)^\\frac{d}{N-1}.$ By the maximality of $Y_0$ , the point $Y_0$ is a component of $V\\cap G_0$ and by Bézout's theorem we get $\\frac{[k(Y_0):k]}{[k(V):k]}\\le \\deg V\\deg G_0.$ Using also (REF ), we obtain $\\frac{[k(Y_0):k]}{[k(V):k]}&\\ll \\deg V \\left((\\max _i N_L(\\alpha _i)) R^2 \\right)^\\frac{d}{N-1}\\le \\\\&\\le \\deg V \\left(R^2\\frac{ \\max _i \\hat{h}(x_i)}{\\hat{h}({g})}\\right)^\\frac{d}{N-1}.$ Notice that $\\hat{h}(x_i)\\le \\hat{h}(Y_0)$ for all $i$ .","We can now apply Theorem REF to $g$ in $E$ , obtaining for every $\\eta >0$ $\\frac{1}{\\hat{h}(g)}\\ll _\\eta {[k_\\mathrm {tor}(g):k_\\mathrm {tor}]^{1+\\eta }}\\le [k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]^{1+\\eta }$ because $g$ is defined over $k(Y_0)$ .","The product $[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]\\hat{h}(Y_0)$ was bounded in (REF ), so using also the bounds in Theorem REF we obtain $[k(Y_0):k]\\ll _\\eta \\deg V [k(V):k]((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{\\frac{d}{N-1-d}+\\eta }R^{2\\frac{d}{N-1}}.$ By a result of Serre in [30], recalled also in [32], Corollary 3, for $R$ larger than a constant and $\\varphi $ the Euler function, we have $\\varphi (R)\\varphi (R^{\\prime })\\ll [k(Y_0):k] .$ In addition $R^{2-\\eta }\\ll _\\eta \\varphi (R)\\varphi (R^{\\prime })$ since in general $\\varphi (x)\\gg _\\eta x^{1-\\eta }$ and $R$ and $R^{\\prime }$ are related by a constant.","From this and (REF ), for $\\eta $ small enough we obtain $[k(Y_0):k]\\ll _\\eta \\deg V [k(V):k]{((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d}{N-d-1}+\\eta } [k(Y_0):k]^{\\frac{d}{N-1}+\\eta }.$ Since $Y_0$ is $V$ -torsion anomalous of relative dimension one, we have $d0$ .","In fact, if $B$ is such a abelian subvariety, consider its associated matrix, as recalled in Section REF ; it is an $(N-1)\\times N$ matrix and we call $d_i$ the square of the norm of its $i$ -th row.","We let $\\deg B={p_1}^{e_1}\\cdots {p_r}^{e_r}$ be the factorization of $\\deg B$ into distinct prime factors.","Then the number of possible choices for the elements $d_i$ is bounded by $\\delta (\\deg B)^{N-1}$ , where, for a positive integer $n$ , $\\delta (n)$ counts the number of divisors of $n$ .","We notice that, for every $\\eta >0$ we have $ \\delta (\\deg B)\\ll _\\eta (\\deg B)^{\\eta }.$ Now, for every choice of the $d_i$ , the number $D$ of $(N-1)\\times N$ matrices in which the square of the norm of the $i$ -th row is at most $d_i$ is bounded in the following way $D\\ll \\left(\\prod _{i=1}^{N-1} d_i\\right)^N\\ll ({\\deg B})^N.$ So for every $\\eta >0$ , the number of possible subgroups $B$ is $\\ll _\\eta (\\deg B)^{N+\\eta }.$ As for the point $\\zeta $ , it is well known that the number of torsion points in $E^N$ of order bounded by a constant $M$ is at most $M^{2N+1}$ .","In fact the number of points of order dividing a positive integer $i$ is $i^{2N}$ ; so a bound for the number of torsion points of order at most $M$ is given by $\\sum _{i=1}^M i^{2N}\\ll M^{2N+1}.$ Applying Bézout's theorem to every intersection $V\\cap (B+\\zeta )$ , we obtain that for every $\\eta >0$ the number $S$ of $V$ -torsion anomalous points of relative codimension one is bounded by $S\\ll _\\eta \\deg V (\\deg B)^{N+1+\\eta } {\\textrm {ord}(\\zeta )}^{2N+1}.$ This, combined with Theorem REF , (REF ) and (REF ), gives the required explicit bounds."],["Torsion anomalous translates of positive dimension","We let $V$ be a weak-transverse variety in a power of elliptic curves with CM.","In this section we study $V$ -torsion anomalous varieties which are translates of positive dimension.","We reduce this case to the zero dimensional case.","First we compare the $V$ -torsion anomalous translates with translates contained in $V$ .","Lemma 7.1 Let $V$ be a weak-transverse subvariety of an abelian variety of dimension $N$ .","Let $Y$ be a maximal $V$ -torsion anomalous translate, then $Y$ is a maximal translate contained in $V$ (i.e.","$Y$ is not strictly contained in any translate contained in $V$ ).","Let $Y$ be a maximal $V$ -torsion anomalous translate and suppose that $Y$ is contained in a maximal translate $(H+p)\\subseteq V$ with $\\dim (H+p)> \\dim Y $ .","Let $B+\\zeta $ be minimal for $Y$ .","Then $Y$ is a component of $V\\cap (B+\\zeta )$ and ${\\rm codim\\,}Y <{\\rm codim\\,}V +{\\rm codim\\,}B .$ Since $(H+p)\\cap (B+\\zeta )\\supset Y$ then $p=h+b+\\zeta $ for some $h\\in H$ and $b\\in B$ .","Therefore $(H+p)\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+b+\\zeta )=(H\\cap B)+b+\\zeta $ and $\\dim (H\\cap B)=\\dim ((H+p) \\cap (B+\\zeta ))\\ge \\dim Y.$ By (REF ), we deduce $\\begin{split}\\dim (H+B)&=\\dim H +\\dim B -\\dim (H\\cap B)\\\\& \\le \\dim H +\\dim B -\\dim Y \\dim B-\\dim X=1$ .","Since ${\\rm codim\\,}V\\ge g+1$ by the hypothesis and $\\dim V\\ge \\dim X+1=\\dim B$ , then ${\\rm codim\\,}B\\ge g+1$ .","From Section REF , $B+\\zeta $ is a component of the torsion variety given by the zero set of $r\\ge 2$ forms $h_1,\\ldots ,h_r$ of increasing degrees $d_i$ and $d_1 \\cdots d_r \\ll \\deg (B+\\zeta )\\ll d_1 \\cdots d_r .$ Notice that one single equation gives ${\\rm codim\\,}B\\le g$ , since $g$ is the maximal dimension of a simple factor of $A$ .","This implies that $r\\ge 2$ .","Denote by $A_0$ the algebraic subgroup given by $h_1,\\ldots ,h_{r-1}$ .","Since $r\\ge 2$ , we have $A_0\\subsetneq A$ .","In addition $\\deg A_0\\ll (\\deg B)^{\\frac{r-1}{r}}$ .","We now show that $X$ is a component of $V\\cap A_0$ .","Suppose, to the contrary, that $X\\subsetneq Y\\subseteq V\\cap A_0$ where $Y$ is a component of $V\\cap A_0$ strictly containing $X$ .","By maximality of $X$ , $Y$ cannot be $V$ -torsion anomalous, hence ${\\rm codim\\,}Y={\\rm codim\\,}V+{\\rm codim\\,}A_0.$ Moreover by construction $\\dim A_0\\le \\dim B+g$ .","This implies $\\dim B-1=\\dim X<\\dim Y=\\dim A_0-{\\rm codim\\,}V\\le \\dim B+g-{\\rm codim\\,}V$ and so ${\\rm codim\\,}V\\dim H} H\\right)$ is a finite set of effectively bounded Néron-Tate height.","Here $H$ ranges over all algebraic subgroups of codimension larger than the dimension.","More precisely the union can be taken over finitely many algebraic subgroups $\\mathcal {S}(C)=C\\cap \\bigcup _{i=1}^M H_i;$ the $H_i$ are algebraic subgroups with $\\dim H <{\\rm codim\\,}H$ , and for any real $\\eta >0$ , there exist constants depending only on $E^N$ and $\\eta $ such that $ \\deg H_i \\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)(N+2r-2)}{2(r-1)(2r-N)}+\\eta } ([k(C):k]\\deg C)^{\\frac{Nr}{2(r-1)}+\\eta },$ where $r={\\rm codim\\,}H_i$ .","Moreover $M$ can be effectively bounded.","Furthermore, if $Y_0\\in \\mathcal {S}(C)$ we have $\\hat{h}(Y_0) \\ll _\\eta (h(C)+\\deg C)^{\\frac{N+1}{2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{N-1}{2}+\\eta };$ if $Y_0$ is not a torsion point we also have $[k(Y_0):\\mathbb {Q}]\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{N+1}{N-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{N+1}{2}+\\eta },$ and the cardinality of the points in $\\mathcal {S}(C)$ which are not torsion points is bounded as $S\\ll _\\eta [k(C):k]^{B_1}(\\deg C)^{B_1+1+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{B_2+\\eta },$ where $B_1=\\frac{N(N+1)(2N+1)}{2(N-1)}, \\qquad B_2=\\frac{(3N^2+N-1)(N+1)}{4}.$ We notice that the set of all torsion points in $C$ is finite and has cardinality bounded by the effective Manin-Mumford Conjecture.","From now on, we will be concerned with points in $\\mathcal {S}(C)$ that are not torsion.","Clearly all the points in the intersection $C\\cap (\\cup _{{\\rm codim\\,}H>\\dim H}H) $ are $C$ -torsion anomalous.","In addition since $C$ is a weak-transverse curve each torsion anomalous point is maximal.","Let $Y_0\\in \\mathcal {S}(C)$ be a non-torsion point; then $Y_0\\in C\\cap H$ , with $H$ the minimal subgroup containing $Y_0$ with respect to the inclusion, and let $r={\\rm codim\\,}H$ .","Let also $B+\\zeta $ be a component of $H$ containing $Y_0$ .","Clearly $\\dim B=\\dim H$ and $Y_0\\in C\\cap (B+\\zeta )$ with $B+\\zeta $ minimal for $Y_0$ ; as done several times, we can also choose the torsion point $\\zeta $ in the orthogonal complement of $B$ .","Notice that $\\deg H\\le (\\deg B) \\mathrm {ord}(\\zeta ).$ In fact $B+\\langle \\zeta \\rangle $ is an algebraic subgroup of dimension equal to $\\dim H$ , it contains $Y_0$ and it is contained in $H$ ; thus $B+\\langle \\zeta \\rangle =H$ , by the minimality of $H$ .","The proof now follows the lines of the proofs of Theorems REF and REF .","We proceed to bound $\\deg B$ and, in turn, the height of $Y_0$ , using Theorem REF and the Arithmetic Bézout theorem.","Then, using Siegel's lemma, we get a bound for $[k(Y_0):k]$ and for the order and the number of torsion points $\\zeta $ , providing also a bound for $\\deg H$ .","Recall that we have $r={\\rm codim\\,}B={\\rm codim\\,}H$ .","We first exclude the case $r=1$ and show that the case $N-r=1$ is covered by Theorems REF and REF .","If $r=1$ then $\\dim B<{\\rm codim\\,}B$ implies $\\dim B=0$ and $N=\\dim B+{\\rm codim\\,}B=1$ contradicting the weak-transversality of $C$ .","The case $N-r=1$ corresponds to $\\dim B=1$ and $Y_0$ of relative codimension one, that can be treated with Theorems REF and REF .","Thus we can assume $r>N-r\\ge 2$ .","Moreover $2r>N$ , since by assumption ${\\rm codim\\,}B> \\dim B$ .","As done several times, by Section REF , the variety $B+\\zeta $ is a component of the zero set of forms $h_1,\\dots ,h_{r}$ of increasing degrees $d_i$ and $d_1 \\cdots d_{r}\\ll \\deg B=\\deg (B+\\zeta )\\ll d_1 \\cdots d_{r}.$ Consider the torsion variety defined as the zero set of $h_1$ , and let $A_0$ be one of its connected components containing $B+\\zeta $ .","Then $\\deg A_0\\ll d_1\\ll (\\deg B)^{\\frac{1}{r}}.$ From Theorem REF applied to $Y_0$ in $B+\\zeta $ , for every positive real $\\eta $ we get $\\hat{h}(Y_0)\\gg _\\eta \\frac{(\\deg B)^{\\frac{1}{N-r}-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\frac{1}{N-r}+\\eta }}.$ Notice that all conjugates of $Y_0$ over $k_\\mathrm {tor}(C)$ are components of $C\\cap A_0$ .","Applying the Arithmetic Bézout theorem to $Y_0$ in $C\\cap A_0$ and arguing as in the proof of Theorem REF , we have $\\frac{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]}{[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]}\\hat{h}(Y_0)\\ll (h(C)+\\deg C) (\\deg B)^{\\frac{1}{r}}.$ From (REF ) and (REF ) we get $(\\deg B)^{\\frac{2r-N}{r(N-r)}-\\eta }\\ll _\\eta (h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}][k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\frac{1}{N-r}-1+\\eta }.$ Since $2r>N$ , $N-r>1$ and $ [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\ge 1$ , for $\\eta $ small enough we obtain $\\deg B\\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{2r-N}+\\eta }.$ So, from (REF ) we have $ [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\hat{h}(Y_0)\\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r}{2r-N}+\\eta };$ while, using the right-hand side of (REF ) as a bound for $\\hat{h}(Y_0)$ and (REF ) we get $\\hat{h}(Y_0)\\ll _\\eta (h(C)+\\deg C)^{\\frac{r}{2r-N}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{(N-r)}{2r-N}+\\eta }$ as required.","Having bounded $\\deg B$ and $\\hat{h}(Y_0)$ , we now proceed to bound $[k(Y_0):k]$ only in terms of $C$ and $E^N$ .","We use Siegel's lemma to construct an algebraic subgroup $G$ of codimension $1=\\dim C$ defined over $k$ , containing $Y_0$ and of controlled degree.","The construction is exactly as in [32] Propositions 3 and 4.","We present the steps of the proof.","We know that $\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\mathcal {O}$ .","By minimality of $B+\\zeta $ , the coordinates of $Y_0=(x_1,\\ldots ,x_N)$ generate an $\\mathcal {O}$ -module $\\Gamma $ of rank equal to the dimension of $B+\\zeta $ which is $N-r$ .","Let $g_1, \\dots , g_{N-r}$ be generators of the free part of $\\Gamma $ which give the successive minima, and are chosen as in [32], Proposition 2.","Then $\\hat{h}(\\sum _j \\alpha _j g_j)\\gg \\sum _i |N_L(\\alpha _j)|\\hat{h}(g_j)$ for coefficients $\\alpha _j$ in $\\mathcal {O}$ .","In addition, like in the case of relative codimension one, the torsion part is generated by a torsion point $T$ of exact order $R$ .","Therefore we can write $x_i=\\sum _j\\alpha _{i,j} g_j +\\beta _i T$ with coefficients $\\alpha _i,\\beta _i\\in \\mathcal {O}$ and $N_L(\\beta _i)\\ll R^2.$ As in Proposition 2 of [32], we have $\\hat{h}(x_i)\\gg \\sum _j \\left|N_L(\\alpha _{i,j})\\right|\\hat{h}(g_j).$ We define $\\nu _j=(\\alpha _{1,j}, \\dots , \\alpha _{N,j}) \\quad {\\mathrm {and}} \\quad |\\nu _j|=\\max _i|N_L(\\alpha _{i,j})|.$ Then $|\\nu _j|\\ll \\frac{\\hat{h}(Y_0)}{\\hat{h}(g_j)}.$ We want to find coefficients $a_i\\in \\mathcal {O}$ such that $\\sum _i^N a_i x_i=0$ .","This gives a linear system of $N-r+1$ equations, obtained equating to zero the coefficients of $g_j$ and of $T$ .","The system has coefficients in $\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.","We use the Siegel's lemma over $\\mathcal {O}$ as stated in [7], Section 2.9.","We get one equation with coefficients in $\\mathcal {O}$ ; multiplying it by a constant depending only on $E$ we may assume that it has coefficients in $\\mathrm {End}(E)$ .","Thus it defines the sought-for algebraic subgroup $G$ of degree $\\deg G\\ll \\left((\\max _i N_L(\\beta _i))\\left(\\prod ^{N-r}_j|\\nu _j|\\right) \\right)^{\\frac{1}{r}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .","Then $\\deg G_0\\ll \\left(R^2\\prod ^{N-r}_j|\\nu _j| \\right)^\\frac{1}{r}.$ Since $C$ is weak-transverse, the point $Y_0$ is a component of $C\\cap G_0$ .","In addition $C$ and $G_0$ are defined over $k$ and Bézout's theorem gives $[k(Y_0):k]\\le [k(C):k]\\deg C\\deg G_0.$ Hence $[k(Y_0):k]\\ll [k(C):k]\\deg C \\left(R^2\\prod ^{N-r}_j|\\nu _j| \\right)^\\frac{1}{r}.$ Using (REF ) we get $[k(Y_0):k]\\ll [k(C):k]\\deg C \\left(R^2\\frac{\\hat{h}(Y_0)^{N-r}}{\\prod ^{N-r}_j\\hat{h}({g_j})}\\right)^\\frac{1}{r}.$ Following step by step the proof of Proposition 4 in [32], using Theorem REF in place of Theorem 1.3 of [16], we get $\\prod _{i=1}^{N-r}\\hat{h}(g_i)\\gg _\\eta \\frac{1}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{1+\\eta }}.$ By a result of Serre, recalled also in [32], Corollary 3, we know $[k(Y_0):k]\\gg _\\eta R^{2-\\eta }$ .","Moreover from (REF ), the product $\\hat{h}(Y_0)[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]$ is bounded.","Substituting these bounds in (REF ) and recalling that $r>2$ , for $\\eta $ small enough we obtain $[k(Y_0):k]\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{r}{r-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{(2r-N)(r-1)}+\\eta }.$ Moreover, as in the proof of Theorem REF , we can choose $\\zeta $ so that $[k(\\zeta ):k]\\ll [k(Y_0):\\mathbb {Q}], $ and $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }$ .","Substituting this and (REF ) in (REF ), we get the bound for $\\deg H$ .","Now, as in the last part of the proof of Theorem REF , for every $\\eta >0$ a bound for the number $S$ of non-torsion points $Y_0$ is given by $S\\ll _\\eta {\\deg C} \\phantom{.","}({\\deg B})^{N+1+\\eta } {\\mathrm {ord}(\\zeta )}^{2N+1}$ and combining this with the previous results we obtain the bounds.","Finally we notice that we have bounded the degree $\\deg H$ for $H$ of fixed codimension $r$ ; letting $r$ vary this proves that the subgroups $H$ can be taken in a finite set $\\lbrace H_1,\\cdots , H_M\\rbrace $ .","A bound for $M$ can be given effectively, as done in the proof of Theorem REF .","In Theorem REF we have stated a simplified version of the bounds for $\\hat{h}(Y_0),[k(Y_0):\\mathbb {Q}]$ and $S$ .","This has been done for clarity, but a closer inspection of the proof shows that we have proved the following, sharper bounds.","Corollary 9.2 With the same setting and notation of Theorem REF , take $Y_0\\in \\mathcal {S}(C)$ is not a torsion point, and $Y_0\\in C\\cap H$ with $H$ of minimal dimension $N-r$ .","Then for any real $\\eta >0$ , there exist constants depending only on $E^N$ and $\\eta $ such that $\\hat{h}(Y_0) &\\ll _\\eta (h(C)+\\deg C)^{\\frac{r}{2r-N}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{(N-r)}{2r-N}+\\eta },\\\\[k(Y_0):\\mathbb {Q}]&\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{r}{r-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{(2r-N)(r-1)}+\\eta }.$ Moreover $Y_0$ belongs to a finite set of cardinality $S$ bounded as $S\\ll _\\eta [k(C):k]^{B_1}(\\deg C)^{B_1+1+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{B_2+\\eta },$ where $ B_1=\\frac{rN(2N+1)}{2(r-1)},$ $B_2=\\frac{r(N-r)(2rN+2r-2+2N^2-N)}{2(2r-N)(r-1)}.$"],["Sur une question d'orthogonalité dans les puissances de courbes elliptiques","We are indebted to Patrice Philippon for writing this appendix which is essentially the proof of Lemma REF .","Let us mention that deeper properties of orthogonality in Mordell-Weil groups have been studied, see [5].","Soit $E$ une courbe elliptique définie sur le corps des nombres algébriques $\\overline{\\mathbb {Q}}$ , considéré plongé dans le corps des nombres complexes $\\mathbb {C}$ .","Soit $L={\\rm Frac}({\\rm End}(E))$ le corps des multiplications de $E$ , qu'on considérera comme un sous-corps de $\\overline{\\mathbb {Q}}\\subset {\\mathbb {C}}$ .","On a $L=\\mathbb {Q}$ ou $L=\\mathbb {Q}(\\sqrt{-D})$ pour un entier $D$ positif, sans facteur carré.","On suppose $E$ plongée dans un espace projectif par un diviseur ample et symétrique défini sur $\\overline{\\mathbb {Q}}$ et on note $\\hat{h}:E\\left(\\overline{\\mathbb {Q}}\\right)\\rightarrow {\\mathbb {R}}$ la hauteur normalisée (i.e.","de Néron-Tate) correspondante, qui satisfait $\\hat{h}(\\tau p)=N_{K/{\\mathbb {Q}}}(\\tau )\\hat{h}(p)$ pour $\\tau \\in {\\rm End}(E)$ , $p\\in E(\\overline{\\mathbb {Q}})$ , et qui ne s'annule qu'aux points de torsion de $E$ .","L'accouplement de Néron-Tate associé s'écrit $\\langle p,q\\rangle _{NT}=\\frac{1}{2}\\left( \\hat{h}(p+q)-\\hat{h}(p)-\\hat{h}(q)\\right)\\in \\mathbb {R}$ pour $p,q\\in E(\\overline{\\mathbb {Q}})$ , il induit naturellement une forme $\\mathbb {Q}$ -bilinéaire symétrique sur le $L$ -espace vectoriel $E(\\overline{\\mathbb {Q}})\\otimes _\\mathbb {Z}\\mathbb {Q}$ .","On introduit alors le produit scalaire (à valeurs dans $L\\otimes _\\mathbb {Q}\\mathbb {R}$ ) $\\langle p,q \\rangle = {\\left\\lbrace \\begin{array}{ll}\\langle p,q\\rangle _{NT} &\\text{si $L=\\mathbb {Q}$}\\\\\\langle p,q\\rangle _{NT}-\\frac{1}{\\sqrt{-D}}\\langle p,\\sqrt{-D} q\\rangle _{NT} &\\text{si $L=\\mathbb {Q}(\\sqrt{-D})$,}\\end{array}\\right.","}$ dont on vérifie qu'il est $L$ -sesqui-linéaire (relativement à la conjugaison sur $L$ ) et hermitien.","On étend ces produits scalaires à $E(\\overline{\\mathbb {Q}})^n\\otimes _\\mathbb {Z}\\mathbb {Q}$ par les formules $\\langle p,q\\rangle _{NT}&=\\sum _{i=1}^n \\langle p_i,q_i \\rangle _{NT}\\\\\\langle p,q\\rangle &=\\sum _{i=1}^n \\langle p_i,q_i \\rangle $ pour $p=(p_1,\\cdots ,p_n)$ et $q=(q_1,\\cdots ,q_n)$ dans $E(\\overline{\\mathbb {Q}})^n$ .","Et on munit $\\mathbb {C}^n$ , identifié à l'espace tangent en l'origine de $E(\\overline{\\mathbb {C}})^n$ , de son produit hermitien standard.","Lemma 1.1 Soient $H$ et $H^{\\prime }$ deux sous-groupes algébriques, connexes, de $E^n$ , alors leurs espaces tangents à l'origine $\\mathrm {T}H(\\mathbb {C})$ et $\\mathrm {T}H^{\\prime }(\\mathbb {C})$ sont orthogonaux dans $\\mathbb {C}^n$ si et seulement si $H(\\overline{\\mathbb {Q}})$ et $H^{\\prime }(\\overline{\\mathbb {Q}})$ sont orthogonaux dans $E^n$ (pour le produit scalaire $\\langle \\cdot ,\\cdot \\rangle $ ou de façon équivalente pour l'accouplement de Néron-Tate).","Soient $d$ et $d^{\\prime }$ les dimensions respectives de $H$ et $H^{\\prime }$ , il existe des homomorphismes $\\varphi :E^d\\rightarrow E^n$ et $\\varphi ^{\\prime }:E^{d^{\\prime }}\\rightarrow E^n$ dont les images sont $H$ et $H^{\\prime }$ respectivement.","On peut décrire ces homomorphismes sous la forme $(p_1,\\ldots ,p_d)&\\mapsto p=\\left(\\sum _{j=1}^d a_{1,j}p_j,\\ldots ,\\sum _{j=1}^d a_{n,j}p_j\\right)\\\\(q_1,\\ldots ,q_{d^{\\prime }})&\\mapsto q=\\left(\\sum _{k=1}^{d^{\\prime }} b_{1,k}q_k,\\ldots ,\\sum _{k=1}^{d^{\\prime }} b_{n,k}q_k\\right)$ et leurs applications tangentes $\\mathrm {T}_\\varphi :\\mathbb {C}^{d}\\rightarrow \\mathrm {T} H(\\mathbb {C})\\subseteq \\mathbb {C}^n$ et $\\mathrm {T}_\\varphi ^{\\prime }:\\mathbb {C}^{d^{\\prime }}\\rightarrow \\mathrm {T} H^{\\prime }(\\mathbb {C})\\subseteq \\mathbb {C}^n$ par $(u_1,\\ldots ,u_d)&\\mapsto u=\\left(\\sum _{j=1}^d a_{1,j}u_j,\\ldots ,\\sum _{j=1}^d a_{n,j}u_j\\right)\\\\(v_1,\\ldots ,v_{d^{\\prime }})&\\mapsto v=\\left(\\sum _{k=1}^{d^{\\prime }} b_{1,k}v_k,\\ldots ,\\sum _{k=1}^{d^{\\prime }} b_{n,k}v_k\\right)$ où $a_{i,j}, b_{i,k}\\in \\mathrm {End}(E)$ pour tous $i,j,k$ .","On calcule alors les produits scalaires en développant par sesquilinéarité $\\langle u,v \\rangle =\\sum _{i=1}^n\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }} a_{i,j}u_j \\overline{b_{i,k}v_k}=\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }}\\left(\\sum _{i=1}^n a_{i,j}\\overline{b_{i,k}}\\right)u_j\\overline{v_k}$ et $\\langle p,q \\rangle =\\sum _{i=1}^n\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }} a_{i,j} \\overline{b_{i,k}}\\langle p_j,q_k \\rangle =\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }}\\left(\\sum _{i=1}^n a_{i,j}\\overline{b_{i,k}}\\right)\\langle p_j,q_k \\rangle .$ Si les espaces tangents en l'origine à $H$ et $H^{\\prime }$ sont orthogonaux (resp.","si $H$ et $H^{\\prime }$ sont orthogonaux) on a $\\langle u,v\\rangle =0$ pour tous $u_1,\\dots ,u_d,v_1,\\dots ,v_{d^{\\prime }}\\in {\\mathbb {C}}$ (resp.","$\\langle p,q\\rangle =\\langle p,q\\rangle _{\\rm NT}=0$ pour tous $p_1,\\dots ,p_d,q_1,\\dots ,q_{d^{\\prime }}\\in E(\\overline{\\mathbb {Q}}$ ).","Appliqué avec $u_{j^{\\prime }}=v_{k^{\\prime }}=0$ pour $j^{\\prime }\\ne j$ , $k^{\\prime }\\ne k$ et $u_j=v_k=u\\ne 0$ (resp.","$p_{j^{\\prime }}=q_{k^{\\prime }}=0$ pour $j^{\\prime }\\ne j$ , $k^{\\prime }\\ne k$ et $p_j=q_k=p$ non de torsion), ceci entraîne les égalités $\\sum _{i=1}^na_{i,j}\\overline{b_{i,k}}=0$ , pour tous $j,k$ .","Réciproquement, ces dernières entraînent $\\langle u,v\\rangle =0$ et $\\langle p,q\\rangle =\\langle p,q\\rangle _{\\rm NT}=0$ pour tous $u\\in {\\rm T}H({\\mathbb {C}})$ , $v\\in {\\rm T}H^{\\prime }({\\mathbb {C}})$ , $p\\in H(\\overline{\\mathbb {Q}})$ et $q\\in H^{\\prime }(\\overline{\\mathbb {Q}})$ , ce qui établit l'équivalence énoncée."],["Acknowledgements","We wish to thank P. Philippon for useful discussions and for his kind communication of the appendix.","We kindly thank A. Galateau for his communication of the constants in [18].","We thank D. Bertrand, P. Habegger, M. Hindry and D. Masser for providing us useful references.","We warmly thank the FNS for the great support we receive in doing research in Mathematics.","E. Viada also likes to thank the AIM and the organizers J. Pila and U. Zannier for supporting her participation to a conference on unlikely intersections in Pisa, where this work was inspired.","S. Checcoli and F. Veneziano would like to address a special thank to their advisor U. Zannier for introducing them to mathematical research."]],"string":"[\n [\n \"On torsion anomalous intersections\"\n ],\n [\n \"Abstract A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$.\",\n \"We prove some cases of this conjecture.\",\n \"We show that the $V$-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety $V$ embedded in a product of elliptic curves with CM.\",\n \"We give explicit uniform bounds in the dependence on $V$.\",\n \"As an immediate consequence we prove the conjecture for $V$ of codimension two in a product of CM elliptic curves.\",\n \"We also point out some implications on the effective Mordell-Lang Conjecture.\"\n ],\n [\n \"Introduction\",\n \"In this article, by variety we mean an algebraic variety defined over the algebraic numbers.\",\n \"We denote by $G$ a semi-abelian variety defined over a number field $k$ and by $k_{\\\\mathrm {tor}}$ the field of definition of the torsion points of $G$ .\",\n \"Let $V$ be a subvariety of $G$ .\",\n \"The variety $V$ is a translate, respectively a torsion variety, if it is a finite union of translates of proper algebraic subgroups by points, respectively by torsion points.\",\n \"An irreducible variety $V$ is transverse, respectively weak-transverse, if it is not contained in any translate, respectively in any torsion variety.\",\n \"Of course a torsion variety is in particular a translate, and a transverse variety is weak-transverse.\",\n \"In addition transverse implies non-translate, and weak-transverse implies non-torsion.\",\n \"Notice that we are considering only proper subvarieties.\",\n \"It is a natural problem to investigate when a geometric assumption on $V$ is equivalent to the non-density of some special subsets of $V$ .\",\n \"Several classical statements, such as, for instance, the Manin-Mumford, the Mordell-Lang and the Bogomolov Conjectures—nowadays theorems, are all of this nature.\",\n \"More recently, new questions of similar type have been raised.\",\n \"The Zilber-Pink Conjecture asserts that, for a transverse variety $V$ , the intersection of $V$ with the union of all algebraic subgroups of codimension at least $\\\\dim V+1$ translated by points in a subgroup of finite rank, is non-dense in $V$ .\",\n \"This conjecture has been tackled from several points of view, but it has only been answered partially.\",\n \"For instance it is known for curves in some semi-abelian varieties.\",\n \"E. Bombieri, D. Masser and U. Zannier in [11] give a new approach for general dimensions.\",\n \"They introduce the notions of anomalous and torsion anomalous subvarieties.\",\n \"In their definitions they always avoid points.\",\n \"For us points can be torsion anomalous, but not anomalous.\",\n \"This gives a perfect match with the CIT conjecture, as clarified below.\",\n \"An irreducible subvariety $Y$ of $V$ is a $V$ -torsion anomalous variety if - $Y$ is an irreducible component of $V\\\\cap (B+\\\\zeta )$ with $B+\\\\zeta $ an irreducible torsion variety; - the dimension of $Y$ is larger than expected, i.e.\",\n \"${\\\\mathrm {codim}\\\\,} Y < {\\\\mathrm {codim}\\\\,} V + {\\\\mathrm {codim}\\\\,} B.$ The variety $B+\\\\zeta $ is minimal for $Y$ if it satisfies the above conditions and has minimal dimension.\",\n \"The relative codimension of $Y$ is the codimension of $Y$ in its minimal $B+\\\\zeta $ .\",\n \"We say that $Y$ is a maximal $V$ -torsion anomalous variety if it is $V$ -torsion anomalous and it is not contained in any $V$ -torsion anomalous variety of strictly larger dimension.\",\n \"The complement in $V$ of the union of all $V$ -torsion anomalous varieties is denoted by $V^{ta}$ .\",\n \"Clearly $V^{ta}$ is obtained removing from $V$ all maximal $V$ -torsion anomalous varieties.\",\n \"Again, for Bombieri, Masser and Zannier $V^{ta}$ is the complement of the union of all $V$ -torsion anomalous varieties of positive dimension.\",\n \"Furthermore, an irreducible variety $Y$ of positive dimension is $V$ -anomalous if it is a component of $V\\\\cap (B+p)$ with $B+p$ an irreducible translate and in addition $Y$ has dimension larger than expected.\",\n \"The complement in $V$ of the union of all $V$ -anomalous varieties is denoted by $V^{oa}.$ Clearly points should be excluded from the definition of $V$ -anomalous varieties because otherwise all points would be anomalous, making the notion uninteresting.\",\n \"On the other hand we allow points to be torsion anomalous varieties: they are exactly the torsion anomalous varieties which are not anomalous.\",\n \"It may be possible that only some components of $V\\\\cap (B+\\\\zeta )$ are anomalous, so each component has to be treated separately.\",\n \"This justifies the assumption of $Y$ being irreducible.\",\n \"In the Torsion Openness Conjecture [11], Bombieri, Masser and Zannier conjectured that the complement of the set of the torsion anomalous varieties of positive dimension is open.\",\n \"In addition, in the Torsion Finiteness Conjecture, they claim that there are only finitely many maximal torsion anomalous points.\",\n \"Here we state a slightly stronger conjecture, which includes both their conjectures.\",\n \"In addition, it specifies that $V^{ta}$ is empty exactly when $V$ is not weak-transverse.\",\n \"In other words we say that there are only finitely many maximal torsion anomalous varieties of any dimension.\",\n \"Conjecture 1 (Bombieri-Masser-Zannier) Let $V$ be a weak-transverse variety in a (semi-)abelian variety.\",\n \"Then $V^{ta}$ is a dense open subset of $V$ .\",\n \"For a hypersurface the conjecture is clearly true.\",\n \"Indeed the intersection of an irreducible torsion variety $B+\\\\zeta $ with a hypersurface is either the variety $B+\\\\zeta $ itself or it has the right dimension $\\\\dim B-1$ .\",\n \"So the only $V$ -torsion anomalous varieties are torsion varieties contained in $V$ ; but we know by the Manin-Mumford Conjecture that the maximal torsion varieties contained in $V$ are finitely many.\",\n \"Among other results, Bombieri, Masser and Zannier in [11], Theorem 1.7, prove the openness of $V^{ta}$ for an irreducible variety $V$ of codimension 2 in $\\\\mathbb {G}^n_m$ .\",\n \"Among the main ingredients in their proof is a result of Ax consisting in the analogue of Schanuel's Conjecture in fields of complex power series in several variables.\",\n \"In this paper, we first prove Conjecture REF for weak-transverse translates in a general abelian variety.\",\n \"Then we give a totally effective method which shows that in any weak-transverse variety $V$ in a product of elliptic curves with CM, the $V$ -torsion anomalous varieties of relative codimension one are non-dense.\",\n \"An immediate consequence is Conjecture REF for weak-transverse varieties of codimension 2, proving the analogue of Theorem 1.7 of [11] in a product of CM elliptic curves.\",\n \"Our method differs from theirs; in particular it is completely effective, and we also avoid the use of Ax's theorem.\",\n \"An intrinsic consequence of our effective bounds on the height is the effective CIT (stated below) for weak-transverse varieties of codimension 2 in a product of elliptic curves with CM.\",\n \"Finally we point out some implications on the effective Mordell-Lang Conjecture.\",\n \"We also show how the method for elliptic curves generalises to abelian varieties.\",\n \"The following first result, based only on some geometric considerations, is proved in Section REF .\",\n \"Theorem 1.1 Let $H+p$ be a weak-transverse translate in an abelian variety.\",\n \"Then the set of $(H+p)$ -torsion anomalous varieties is empty.\",\n \"This clarifies the situation and brings some simplifications in the proof of our main result: Theorem 1.2 Let $V$ be a weak-transverse variety in a product of elliptic curves with CM defined over a number field $k$ .\",\n \"Then the maximal $V$ -torsion anomalous varieties $Y$ of relative codimension one are finitely many; in addition their degree and normalised height are effectively bounded as $ h(Y) &\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{N-1}{N-1-\\\\dim V}+\\\\eta }[k_{\\\\mathrm {tor}}(V):k_{\\\\mathrm {tor}}]^{\\\\frac{\\\\dim V}{N-1-\\\\dim V}+\\\\eta },\\\\\\\\\\\\deg Y &\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{N-2}{N-1-\\\\dim V}+\\\\eta }[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}]^{\\\\frac{\\\\dim V-1}{N-1-\\\\dim V}+\\\\eta }.$ These are only some of the bounds we obtain.\",\n \"The method is totally effective, and we give explicit dependence on $V$ .\",\n \"The notations will be made precise in the next sections.\",\n \"In addition we bound the degree of the torsion varieties $B+\\\\zeta $ minimal for the maximal $V$ -torsion anomalous that we consider; these bounds provide, in principle, an algorithm to find all such anomalous varieties.\",\n \"The effective statements require some further notation and they are given in Theorem REF for maximal torsion anomalous varieties which are not translates, in Theorems REF and REF for maximal torsion anomalous points and in Theorems REF and REF for maximal torsion anomalous translates of positive dimension.\",\n \"The proof of our main theorem has four main ingredients: the deep Zhang inequality, the strong explicit Arithmetic Bézout theorem by Philippon, a sharper variant of an effective result by Galateau on the Bogomolov Conjecture, and the relative Lehmer bound in CM abelian varieties by Carrizosa.\",\n \"In the proofs we need to distinguish whether a $V$ -torsion anomalous variety $Y$ is or not itself a translate.\",\n \"If $Y$ is not a translate then the Zhang inequality, the explicit Arithmetic Bézout theorem, and a functorial version of the effective Bogomolov Conjecture are sufficient.\",\n \"This case is proved in a product of elliptic curves, as the Bogomolov type bounds do not require any assumption on complex multiplication.\",\n \"If $Y$ is a point we use the Zhang inequality, the explicit Arithmetic Bézout theorem and the relative Lehmer bound; here we need to assume that the elliptic curve has CM, as a sharp Lehmer type bound is known only under this hypothesis.\",\n \"Finally, we reduce the case of translates of positive dimension to the case of points.\",\n \"To this aim we also use a result recalled in the appendix by Patrice Philippon, which relates the essential minimum of a translate to the height of the point of translation.\",\n \"Theorem REF can be easily generalised to abelian varieties, in the following sense.\",\n \"Theorem 1.3 Let $V$ be a weak-transverse variety in an abelian variety $A$ with CM.\",\n \"Let $g$ be the maximal dimension of a simple factor of $A$ .\",\n \"If ${\\\\rm codim\\\\,}V\\\\ge g+1$ , then the maximal $V$ -torsion anomalous varieties of relative codimension 1 are finitely many and they have degree and normalised height effectively bounded.\",\n \"For clarity, we first present the proof of Theorem REF , where the technicalities are simpler to follow.\",\n \"We then explain how this generalises to abelian varieties in Section .\",\n \"We notice that the most interesting case remains the case of elliptic curves, in the sense that there is the largest number of subgroups.\",\n \"A breaking-through result would be the proof of Theorem REF for any relative codimension with an effective method.\",\n \"Such a general result would imply the effective Mordell-Lang Conjecture.\",\n \"We now give several consequences and applications of our main theorem.\",\n \"By definition all $V$ -torsion anomalous varieties satisfy a dimensional inequality.\",\n \"As a straightforward consequence, we see that in a variety $V$ of codimension 2 the relative codimension of a $V$ -torsion anomalous variety is either one or zero.\",\n \"Then, from Theorem REF , it follows immediately the following: Theorem 1.4 Let $V$ be a weak-transverse variety of codimension 2 in a product of elliptic curves with CM.\",\n \"Then $V^{ta}$ is a dense open subset of $V$ .\",\n \"It is sufficient to consider maximal $V$ -torsion anomalous varieties.\",\n \"Let $Y$ be a maximal $V$ -torsion anomalous component of $V\\\\cap (B+\\\\zeta )$ .\",\n \"Then, by definition of $V$ -torsion anomalous variety, $ {\\\\rm codim\\\\,}Y< {\\\\rm codim\\\\,}V+ {\\\\rm codim\\\\,}B.$ Equivalently $\\\\dim B-\\\\dim Y<{\\\\rm codim\\\\,}V=2.$ If $Y$ has relative codimension zero, then $Y= B+\\\\zeta $ and $Y$ is a component of the closure of the torsion contained in $V$ , which is a proper closed by the Manin-Mumford Conjecture.\",\n \"If $Y$ has relative codimension one, we apply our main theorem.\",\n \"Conjecture REF is well known to be related to the following conjecture, which in turn is equivalent to the Zilber-Pink Conjecture.\",\n \"For a natural number $r$ , define $S_{r}(V)=V \\\\cap \\\\bigcup _{{\\\\rm codim\\\\,}\\\\, H\\\\ge r}H,$ where $H$ runs over all algebraic subgroups of codimension at least $r$ .\",\n \"Conjecture 2 (CIT, Conjecture on the Intersection with Torsion varieties) Let $V$ be a weak-transverse variety in an abelian variety.\",\n \"Then $S_{\\\\dim V+1}(V)$ is non-dense in $V$ .\",\n \"By definition, for any torsion variety $B+\\\\zeta $ , the intersection $V^{ta}\\\\cap (B+\\\\zeta )$ has the right dimension.\",\n \"In particular, if a point of $V$ lies in some algebraic subgroup of codimension $\\\\ge \\\\dim V +1$ , then that point is contained in a $V$ -torsion anomalous variety (we would expect empty intersection), and so it does not belong to $V^{ta}$ .\",\n \"Then, as a consequence of the effective version of Theorems REF and REF , we obtain a completely effective version of the following theorem.\",\n \"Theorem 1.5 Let $V$ be a weak-transverse variety of codimension 2 or a weak-transverse translate in a product of elliptic curves with CM.\",\n \"Then $S_{\\\\dim V+1}(V)$ is non-dense in $V$ and its closure is $V \\\\setminus V^{ta}$ .\",\n \"If ${\\\\rm codim\\\\,}(B+\\\\zeta ) \\\\ge \\\\dim V+1$ , then all components of $V\\\\cap (B+\\\\zeta )$ are torsion anomalous so they do not intersect $V^{ta}$ .\",\n \"Therefore $V^{ta }\\\\cap \\\\bigcup _{{\\\\rm codim\\\\,}\\\\, H\\\\ge \\\\dim V+1}H=\\\\varnothing $ and $S_{\\\\dim V+1}(V) \\\\subseteq V\\\\setminus V^{ta}.$ By Theorems REF and REF , $V^{ta}$ is an open dense set in $V$ .\",\n \"That the closure of $S_{\\\\dim V+1}(V)$ is $V \\\\setminus V^{ta}$ is proven exactly as in [11], page 25, for tori.\",\n \"Recall that for them points are not torsion anomalous varieties.\",\n \"As an immediate corollary we obtain: Corollary 1.6 Let $C$ be a weak-transverse curve in $E^3$ where $E$ is an elliptic curve with CM defined over a number field $k$ .\",\n \"Then $S_2(C)=C\\\\setminus C^{ta}$ is a finite set of cardinality and Néron-Tate height effectively bounded.\",\n \"In particular every non-torsion point $Y_0\\\\in S_2(C)$ satisfies $\\\\hat{h}(Y_0) &\\\\ll _\\\\eta (h(C)+\\\\deg C)^{2+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{1+\\\\eta },\\\\\\\\[k (Y_0):\\\\mathbb {Q}] &\\\\ll _\\\\eta (\\\\deg C (h(C)+\\\\deg C)[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}][k(C):k])^{2+\\\\eta }.$ That $S_2(C)$ is a finite set is known for weak-transverse curves in tori ([22], Theorem 1.2) and in any product of elliptic curves (see [33]).\",\n \"That the height is bounded is proved using the Vojta inequality in a non-effective way.\",\n \"Effective bounds for the height of $S_2(C)$ are given for weak-transverse curves in tori in [9], using an effective Mordell-Lang Theorem, and in abelian varieties only for transverse curves in a product of elliptic curves in [32].\",\n \"So this corollary is a first example of an effective bound for the height for a weak-transverse curve in abelian varieties.\",\n \"In higher dimensions, the Bounded Height Conjecture proved by Habegger ([20], theorem at page 407), together with the Effective Bogomolov Bound by Galateau ([18], Theorem 1.1) and the non-density Theorem by Viada ([34], Theorem 1.6) imply the CIT for varieties with $V^{ta}$ non empty embedded in certain abelian varieties which include all CM abelian varieties.\",\n \"The original formulation of the Bounded Height Theorem in [19] did not contain an explicit height bound and the author did not discuss the effectivity of the result (however, in a forthcoming paper, Habegger provides an explicit version of the Bounded Height Theorem for tori).\",\n \"Even if the method is made effective, the assumption $V^{ta}\\\\ne \\\\varnothing $ is stronger than transversality.\",\n \"In this respect, ours is a new effective method in the context of the CIT for weak-transverse varieties.\",\n \"Many are the contributions on the CIT of several authors.\",\n \"For a more extensive list, we refer to the references given in the papers mentioned above.\",\n \"In the last part of the paper we generalise our method, obtaining an effective weak result for curves, related to the CIT.\",\n \"Theorem REF gives a complete effective version of the following result.\",\n \"Theorem 1.7 Let $C$ be a weak-transverse curve in $E^N$ , where $E$ is an elliptic curve with CM defined over a number field $k$ .\",\n \"Then, $C\\\\cap \\\\cup _{{\\\\rm codim\\\\,}H>\\\\dim H}H$ is a finite set of cardinality and Néron-Tate height effectively bounded.\",\n \"Here $H$ ranges over all algebraic subgroups of codimension larger than their dimension.\",\n \"In particular every non-torsion point $Y_0\\\\in C\\\\cap \\\\cup _{{\\\\rm codim\\\\,}H>\\\\dim H}H$ satisfies $\\\\hat{h}(Y_0) &\\\\ll _\\\\eta (h(C)+\\\\deg C)^{\\\\frac{N+1}{2}+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{\\\\frac{N-1}{2}+\\\\eta },\\\\\\\\[k(Y_0):\\\\mathbb {Q}] &\\\\ll _\\\\eta (\\\\deg C [k(C):k])^{\\\\frac{N+1}{N-1}+\\\\eta }((h(C)+\\\\deg C)[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}])^{\\\\frac{N+1}{2}+\\\\eta }.$ In the following section we emphasise the implications of these theorems on the effective and quantitative Mordell-Lang Conjecture.\"\n ],\n [\n \"Applications to the effective Mordell-Lang Conjecture\",\n \"The CIT is well known to have implications on the Mordell-Lang Conjecture.\",\n \"The toric case of this conjecture has been extensively studied, also in its effective form, by many authors.\",\n \"A completely effective version in the toric case can be found in [7], Theorem 5.4.5, and generalisations in [8].\",\n \"However an effective general result in abelian varieties is not known.\",\n \"Since we prove an effective version of Theorem REF , we obtain the following effective cases of the Mordell-Lang Conjecture for curves in products of elliptic curves with CM, where effective means that we give a bound for the height of the set of points in $C$ and in a group of finite rank $\\\\Gamma $ ; the dependence on $C$ and $\\\\Gamma $ is completely explicit.\",\n \"The assumption on the relative codimension gives a condition on the rank of $\\\\Gamma $ .\",\n \"This method is completely different from the two classical effective methods known in abelian varieties.\",\n \"The method by Chabauty and Coleman, surveyed, for example, in [27], gives a Mordell-Lang statement for a curve in its Jacobian and $\\\\Gamma $ of rank less than the genus; it involves a Selmer group calculation and smart computations with the Jacobian of the curve, and it can be made effective in some particular cases (see [23]).\",\n \"The method by Manin and Demjanenko, described in Chapter 5.2 of [31], gives an effective Mordell theorem for curves with many independent morphisms to an abelian variety.\",\n \"As remarked by Serre, the method remains of difficult application.\",\n \"Let $E$ be an elliptic curve defined over the algebraic numbers.\",\n \"We let $\\\\hat{h}$ be the standard Néron-Tate height on $E^N$ ; if $V$ is a subvariety of $E^N$ , we shall denote by $h(V)$ the normalized height of $V$ , as defined in [25].\",\n \"The height of a set is as usual the supremum of the heights of its points.\",\n \"If $E$ is defined over a field $k$ , we denote by $k_\\\\mathrm {tor}$ the field of definition of all torsion points of $E$ .\",\n \"All the constants in the following theorems become explicit if the constant for the Lehmer type bound of Carrizosa in [13] is made explicit.\",\n \"Theorem 2.1 Let $C$ be a weak-transverse curve in $E^N$ , with $E$ a CM elliptic curve and $N>2$ .\",\n \"Let $k$ be a field of definition for $E$ .\",\n \"Let $\\\\Gamma $ be a subgroup of $E^N$ such that the group generated by its coordinates is an ${\\\\mathrm {End}}(E)$ -module of rank one.\",\n \"Then, for any positive $\\\\eta $ , there exists a constant $c_1$ , depending only on $E^N$ and $\\\\eta $ , such that the set $C\\\\cap \\\\Gamma $ has Néron-Tate height bounded as $\\\\hat{h}(C\\\\cap \\\\Gamma ) \\\\le c_1 (h(C)+\\\\deg C)^{\\\\frac{N-1}{N-2}+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{\\\\frac{1}{N-2}+\\\\eta }.$ Let $g$ be a generator of $\\\\overline{\\\\Gamma }$ , the group generated by all coordinates of any element in $\\\\Gamma $ .\",\n \"If a point $x=(x_1,\\\\dots , x_N)$ is in $\\\\Gamma $ then there exist $0\\\\ne a_i,b_i \\\\in {\\\\mathrm {End}}(E)$ and torsion points $\\\\zeta _i\\\\in E$ such that $a_ix_i=b_i g+\\\\zeta _i, \\\\qquad i=1,\\\\cdots ,N.$ If all $b_i=0$ then $x$ is a torsion point, thus it has height zero.\",\n \"We can suppose, for instance, that $b_1\\\\ne 0$ .\",\n \"So $g$ depends on $x_1$ .\",\n \"Substituting this dependence in the last $N-1$ equations, we obtain a system of $N-1$ linearly independent equations in the variables $x_1, \\\\dots ,x_N$ given by $ b_1 a_j x_j=a_1 b_j x_1+\\\\zeta ^{\\\\prime }_j, \\\\qquad j=2,\\\\cdots ,N$ for $\\\\zeta ^{\\\\prime }_j$ torsion points.\",\n \"These equations define a torsion variety $H$ of codimension $N-1$ in $E^N$ .\",\n \"Thus, $ (C\\\\cap \\\\Gamma )\\\\subseteq (C\\\\cap \\\\cup _{\\\\dim H=1}H) \\\\cup (C\\\\cap {\\\\mathrm {Tor}}_{E^N})=S_{N-1}(C)$ for $H$ ranging over all algebraic subgroup of dimension one.\",\n \"However, if $N>2$ , any point $x$ in the intersection $C\\\\cap H$ is a $C$ -torsion anomalous point.\",\n \"In addition, as $C$ is weak-transverse, each such $C$ -torsion anomalous point is maximal.\",\n \"If $x$ is not a torsion point then $H$ is minimal for $x$ , because $H$ has dimension one.\",\n \"Thus the relative codimension of $x$ in $H$ is one.\",\n \"Applying Theorem REF we deduce the bound.\",\n \"If $N=2$ , the intersection $C \\\\cap H$ is not torsion anomalous, so we must follow another line.\",\n \"For this reason, we need the assumption of $C$ being transverse.\",\n \"Theorem 2.2 Let $C$ be a transverse curve in $E^2$ with $E$ a CM elliptic curve defined over a number field $k$ .\",\n \"Let $\\\\Gamma $ be a subgroup of $E^2$ such that the group of its coordinates is an ${\\\\mathrm {End}}(E)$ -module of rank one, generated by $g$ .\",\n \"Then, for any positive $\\\\eta $ there exists a constant $c_2$ depending only on $E^N$ and $\\\\eta $ , such that the set $C\\\\cap \\\\Gamma $ has Néron-Tate height bounded as $\\\\hat{h}(C\\\\cap \\\\Gamma ) \\\\le c_2[k_\\\\mathrm {tor}(C\\\\times g):k_\\\\mathrm {tor}]^{1+\\\\eta }(h(C)+(\\\\hat{h}(g)+1)\\\\deg C)^{2+\\\\eta }.$ Let $g$ be a generator of $\\\\overline{\\\\Gamma }$ , the group generated by all coordinates of any element in $\\\\Gamma $ .\",\n \"Consider the curve $C^{\\\\prime }=C\\\\times g$ in $E^3$ .\",\n \"Since $C$ is transverse then $C^{\\\\prime }$ is weak-transverse.\",\n \"If a point $(x_1, x_2)$ is in $\\\\Gamma $ then there exist $0\\\\ne a_i,b_i \\\\in {\\\\mathrm {End}}(E)$ and torsion points $\\\\zeta _i$ such that $a_1x_1=b_1 g+\\\\zeta _1 , \\\\,\\\\,\\\\, \\\\,\\\\,\\\\, a_2x_2=b_2 g+\\\\zeta _2.$ Thus the point $(x_1,x_2,g) $ belongs to the intersection $C^{\\\\prime }\\\\cap H$ with $H$ the torsion variety of codimension 2 in $E^3$ defined by the equations $a_1x_1=b_1 x_3+\\\\zeta _1, \\\\quad a_2x_2=b_2 x_3+\\\\zeta _2.$ Thus $(x_1,x_2,g)\\\\in (C^{\\\\prime }\\\\cap \\\\cup _{\\\\dim H=1}H)$ for $H$ ranging over all algebraic subgroups of dimension one in $E^3$ .\",\n \"Therefore $C\\\\cap \\\\Gamma $ is embedded in $C^{\\\\prime }\\\\cap \\\\cup _{\\\\dim H=1}H$ and $\\\\hat{h}(C\\\\cap \\\\Gamma )\\\\le \\\\hat{h}(C^{\\\\prime }\\\\cap \\\\cup _{\\\\dim H=1}H)$ .\",\n \"However any point $x$ in the intersection $C^{\\\\prime }\\\\cap H$ is a maximal $C^{\\\\prime }$ -torsion anomalous point.\",\n \"If $x$ is not a torsion point, then it has relative codimension one in $H$ .\",\n \"We apply Theorem REF to $C^{\\\\prime }\\\\subseteq E^3$ , with $\\\\deg C^{\\\\prime }=\\\\deg C$ and $h(C^{\\\\prime })\\\\le 2( h(C)+\\\\hat{h}(g)\\\\deg C)$ by Zhang's inequality, recalled in Section REF below.\",\n \"This gives the bound for the height of a point in $C\\\\cap \\\\Gamma $ .\",\n \"The assumptions on the curve are necessary in both theorems.\",\n \"Indeed for a weak-transverse curve $C$ in $E^2$ the above theorem is not true.\",\n \"Consider the weak-transverse curve $C=E\\\\times g$ ; for any positive integer $m$ , let $\\\\Gamma _m$ be generated by the point $\\\\gamma _m=(mg,g)$ .\",\n \"For any $m$ , the point $\\\\gamma _m$ belongs to $C$ and, for $m$ which goes to infinity, the height of $\\\\gamma _m$ tends to infinity.\",\n \"Thus there cannot be general bounds independent of $\\\\Gamma $ .\",\n \"This does not happen in higher codimension.\",\n \"The analogue would be $C=E\\\\times g\\\\times g^{\\\\prime }$ where $g$ and $g^{\\\\prime }$ are linearly independent to ensure the weak-transversality.\",\n \"No point of $C$ can be in $\\\\Gamma $ , as no point of the type $(x,g,g^{\\\\prime })$ has coordinates in $\\\\overline{\\\\Gamma }$ , which has rank one.\",\n \"We also remark that for $C$ weak-transverse, $C\\\\times g$ is not necessarily weak-transverse: let $C=E\\\\times g$ , then $E\\\\times g\\\\times g$ is contained in the abelian subvariety $x_2=x_3$ .\",\n \"With this method, the bound in the case $N=2$ depends on $g$ .\",\n \"It is possible to remove such a dependence.\",\n \"However one has to prove that for a transverse curve $C$ , the set $C\\\\cap \\\\cup _{\\\\dim H=1}H$ has bounded height.\",\n \"This is proved in [32] Theorem 1.\",\n \"The proof is effective, but not explicit in the dependence on $C$ and on $E$ .\",\n \"The dependence on $C$ is given comparing the height function relative to a point with the height function relative to the intersections of $C\\\\cap \\\\Delta ^-$ where $ \\\\Delta ^-$ is the divisor defined by $x_1=-x_2$ .\",\n \"We use [32] to deduce another effective case of the Mordell-Lang Conjecture.\",\n \"Theorem 2.3 Let $C$ be a transverse curve in $E^N$ with $E$ an elliptic curve defined over a number field.\",\n \"Let $\\\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\\\mathrm {End}}(E)$ -module of rank at most $N-1$ .\",\n \"Then the set $C\\\\cap \\\\Gamma $ is finite of Néron-Tate height bounded by a constant $c(C,E^N)$ depending on $C$ and $E^N$ i.e.\",\n \"$\\\\hat{h}(C\\\\cap \\\\Gamma ) \\\\le c(C,E^N).$ The proof follows the same idea as the proof of Theorem REF .\",\n \"Let $g_1, \\\\dots , g_{N-1}$ be generators of $\\\\overline{\\\\Gamma }$ , the group generated by all the coordinates of any element in $\\\\Gamma $ .\",\n \"If a point $(x_1,\\\\dots , x_N)$ is in $\\\\Gamma $ then there exist $a_i\\\\in {\\\\mathrm {End}}(E)$ , an $N\\\\times (N-1)$ matrix $B$ with coefficients in ${\\\\mathrm {End}}(E)$ and a torsion point $\\\\zeta \\\\in E^N$ such that $(a_1x_1, \\\\ldots , a_Nx_N)^{t}=B\\\\cdot (g_1,\\\\ldots ,g_{N-1})^{t}+\\\\zeta .$ Explicitating $(g_1, \\\\dots , g_{N-1})$ in terms of $x_1, \\\\dots , x_{N-1}$ from the first $N-1$ equations and substituting in the last equation, we obtain one equation in the variables $x_1, \\\\dots ,x_N$ $ a^{\\\\prime }_1x_1+ \\\\dots +a^{\\\\prime }_Nx_N=\\\\zeta ^{\\\\prime },$ where $\\\\zeta ^{\\\\prime }\\\\in E$ is a torsion point and $a^{\\\\prime }_i\\\\in {\\\\mathrm {End}}(E)$ .\",\n \"This equation defines a torsion variety $H$ of codimension one in $E^N$ .\",\n \"Thus, $ (C\\\\cap \\\\Gamma )\\\\subseteq (C\\\\cap \\\\cup _{{\\\\rm codim\\\\,}H\\\\ge 1}H)=S_1(C)$ for $H$ ranging over all algebraic subgroups of codimension at least one.\",\n \"By [32] Theorem 1, this set has height bounded by an effective constant depending only on $E^N$ and $C$ .\",\n \"At this point we want to extend our main theorem, at least in the case of curves, to see if with our method we could prove other cases of the Mordell-Lang Conjecture.\",\n \"Theorem REF enables us to obtain a more general version of Theorem REF for $C$ weak-transverse and larger rank of $\\\\overline{\\\\Gamma }$ .\",\n \"We also obtain a theorem similar to Theorem REF , with explicit dependence on $C$ .\",\n \"Theorem 2.4 Let $C$ be a weak-transverse curve in $E^N$ with $E$ an elliptic curve with CM.\",\n \"Let $k$ be a field of definition for $E$ .\",\n \"Let $\\\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\\\mathrm {End}}(E)$ -module of rank $ t< N/2$ .\",\n \"Then, for any positive $\\\\eta $ , there exists a constant $c_3$ depending only on $E^N$ and $\\\\eta $ , such that the set $C\\\\cap \\\\Gamma $ has Néron-Tate height bounded as $\\\\hat{h}(C\\\\cap \\\\Gamma ) \\\\le c_3(h(C)+\\\\deg C)^{\\\\frac{N-t}{N-2t}+\\\\eta }[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}]^{\\\\frac{t}{N-2t}+\\\\eta }.$ Let $g_1, \\\\dots , g_t$ be generators of the free part of $\\\\overline{\\\\Gamma }$ , the group generated by all coordinates of any element in $\\\\Gamma $ .\",\n \"If a point $(x_1,\\\\dots , x_N)$ is in $\\\\Gamma $ then there exist $0\\\\ne a_i\\\\in {\\\\mathrm {End}}(E)$ , an $N\\\\times t$ matrix $B$ with coefficients in ${\\\\mathrm {End}}(E)$ and a torsion point $\\\\zeta \\\\in E^N$ such that $(a_1x_1,\\\\ldots ,a_Nx_N)^{t}=B (g_1,\\\\ldots ,g_t)^{t}+\\\\zeta .$ If the rank of $B$ is zero, then $x$ is a torsion point and so it has height zero.\",\n \"If $B$ has positive rank $m$ , we can choose $m$ equations of the system corresponding to $m$ linearly independent rows of $B$ .\",\n \"We use these equations to write the $g_j$ in terms of the $x_i$ and we substitute these expressions in the remaining equations.\",\n \"We obtain a system of maximal rank with $N-m\\\\ge N-t$ linearly independent equations in the variables $x_1, \\\\dots ,x_N$ : ${\\\\left\\\\lbrace \\\\begin{array}{ll}a^{\\\\prime }_{11}x_1+ \\\\dots +a^{\\\\prime }_{1N}x_N=\\\\zeta ^{\\\\prime }_1\\\\\\\\\\\\vdots \\\\\\\\a^{\\\\prime }_{N-m,1}x_1+ \\\\dots +a^{\\\\prime }_{N-m,N}x_N=\\\\zeta ^{\\\\prime }_{N-m}\\\\end{array}\\\\right.\",\n \"}$ where $\\\\zeta ^{\\\\prime }_i\\\\in E$ are torsion points and $a^{\\\\prime }_{ij}\\\\in {\\\\mathrm {End}}(E)$ .\",\n \"These equations define a torsion variety $H$ of codimension $N-m$ in $E^N$ .\",\n \"Thus $ (C\\\\cap \\\\Gamma )\\\\subseteq S_{ N-t}(C)$ for $H$ ranging over all algebraic subgroups of codimension at least $N-t$ .\",\n \"However $\\\\dim H=N-{\\\\rm codim\\\\,}H=m\\\\le t$ .\",\n \"Thus, if $N>2t$ $ S_{ N-t}(C)\\\\subseteq (C\\\\cap \\\\cup _{{\\\\rm codim\\\\,}H>\\\\dim H}H).$ Applying Corollary REF with $r=N-t$ gives the wished bound for the height of $C\\\\cap \\\\Gamma $ .\",\n \"We notice that Theorems REF and REF are proved for weak-transverse curves, while previous effective results assumed transversality.\",\n \"If we assume the transversality of $C$ we can relax the hypothesis on the rank of $\\\\overline{\\\\Gamma }$ .\",\n \"Theorem 2.5 Let $C$ be a transverse curve in $E^N$ with $E$ a CM elliptic curve defined over $k$ .\",\n \"Let $\\\\Gamma $ be a subgroup of $E^N$ such that the free part of the group of its coordinates is an ${\\\\mathrm {End}}(E)$ -module of rank $t\\\\le N-1$ , generated by $g_1, \\\\dots , g_t$ .\",\n \"Then, for any positive $\\\\eta $ there exists a constant $c_4$ depending only on $E^N$ and $\\\\eta $ , such that the set $C\\\\cap \\\\Gamma $ has Néron-Tate height bounded as $\\\\hat{h}(C\\\\cap \\\\Gamma ) \\\\le c_4 [k_\\\\mathrm {tor}(C\\\\times g):k_\\\\mathrm {tor}]^{\\\\frac{t}{N-t}+\\\\eta }(h(C)+ (\\\\hat{h}(g)+1)\\\\deg C)^{\\\\frac{N}{N-t}+\\\\eta }$ where $g=(g_1, \\\\dots , g_t)$ .\",\n \"Consider the curve $C^{\\\\prime }=C\\\\times g$ in $E^{N+t}$ .\",\n \"Since $C$ is transverse then $C^{\\\\prime }$ is weak-transverse.\",\n \"If a point $(x_1, \\\\dots , x_N)$ is in $\\\\Gamma $ , then there exist $0\\\\ne a_i\\\\in {\\\\mathrm {End}}(E)$ , an $N\\\\times t$ matrix $B$ with coefficients in ${\\\\mathrm {End}}(E)$ and a torsion point $\\\\zeta \\\\in E^N$ such that $(a_1x_1,\\\\ldots ,a_Nx_N)^{t}=B (g_1,\\\\ldots ,g_t)^{t}+\\\\zeta .$ Thus the point $(x_1,\\\\dots x_N,g_1, \\\\dots , g_t) $ belongs to the intersection $C^{\\\\prime }\\\\cap H$ with $H$ the torsion variety of codimension $N$ and dimension $t$ in $E^{N+t}$ defined by the equations $(a_1x_1,\\\\ldots ,a_Nx_N)^{t}=B (y_{N+1},\\\\cdots , y_{N+t})^{t}+\\\\zeta .$ Thus, $C\\\\cap \\\\Gamma $ is embedded in $C^{\\\\prime }\\\\cap \\\\cup _{\\\\dim H=t}H$ , and $\\\\hat{h}(C\\\\cap \\\\Gamma )\\\\le \\\\hat{h}(C^{\\\\prime }\\\\cap \\\\cup _{\\\\dim H=t}H)$ for $H$ ranging over all algebraic subgroups of dimension $t$ .\",\n \"If $N> t$ then ${\\\\rm codim\\\\,}H>\\\\dim H$ .\",\n \"The bound for the height is then given by Corollary REF applied to $C^{\\\\prime }\\\\subseteq E^{N+t}$ , where $\\\\deg C=\\\\deg C^{\\\\prime }$ , $h(C^{\\\\prime })\\\\le 2( h(C)+\\\\hat{h}(g)\\\\deg C)$ by Zhang's inequality, recalled in Section REF , below.\"\n ],\n [\n \"Applications to the quantitative Mordell-Lang Conjecture\",\n \"In [29], Theorem 1.2 Rémond gives a bound on the cardinality of the intersection $C\\\\cap \\\\Gamma $ for a transverse curve in $E^ N$ and $\\\\Gamma $ a $\\\\mathbb {Z}$ -module of rank $r$ .\",\n \"He obtains the following bound $\\\\#(C\\\\cap \\\\Gamma )\\\\le {c(E^N, \\\\mathcal {L})}^{r +1}(\\\\deg C)^{(r +1) N^{20}},$ where $c(E^N, \\\\mathcal {L})$ is a positive effective constant depending on $E^N$ and on the choice of an invertible, symmetric and ample sheaf $\\\\mathcal {L}$ on $E^N$ .\",\n \"The Manin-Mumford Conjecture is a special case of the Mordell-Lang conjecture.\",\n \"Explicit bounds on the number of torsion points in $C$ are given, for instance, by David and Philippon in [17] and by Hrushovski in [21].\",\n \"David and Philippon in [17], Proposition 1.12 show that the number of torsion points on a non-torsion curve $C$ is at most $\\\\#(C\\\\cap \\\\mathrm {Tor}_{E^N})\\\\le (10^{2N+53}\\\\deg C)^{33},$ where $\\\\mathrm {Tor}_{E^N}$ is the set of all torsion points of $E^N$ .\",\n \"Our method enables us to give a sharp bound for the number of non torsion points in $C\\\\cap \\\\Gamma $ for $C$ weak-transverse in $E^N$ , which, together with the just mentioned bounds for the torsion, improves in some cases the bounds of Rémond.\",\n \"Notice that below we use the rank $t$ of $\\\\overline{\\\\Gamma }$ , the $\\\\mathrm {End}(E)$ -module of the coordinates of $\\\\Gamma $ .\",\n \"To compare with Rémond's result, we can use the trivial relation $r<2Nt$ and $t2$ and $t=1$ , we have $S\\\\le d_1 & {((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{(N-1)(4N^2-N-4)}{2(N-2)^2}+\\\\eta }(\\\\deg C)^{\\\\frac{2N^3-N^2+N-4}{2(N-2)}+\\\\eta }\\\\\\\\&\\\\cdot [k(C):k]^{\\\\frac{N(N-1)(2N+1)}{2(N-2)}+\\\\eta };$ if $C$ is transverse in $E^2$ and $t=1$ , we have $S\\\\le d_2 &{([k_\\\\mathrm {tor}(C\\\\times g):k_\\\\mathrm {tor}](h(C)+(\\\\hat{h}(g)+1)\\\\deg C))}^{29+\\\\eta }(\\\\deg C)^{22+\\\\eta }\\\\\\\\&\\\\cdot [k(C\\\\times g):k]^{21+\\\\eta }$ where $g$ is a generator of $\\\\overline{\\\\Gamma }$ ; if $C$ is weak-transverse in $E^N$ and $t{\\\\rm codim\\\\,}(B+\\\\zeta )$ .\",\n \"So $Y$ is a component of $ V \\\\cap (B^{\\\\prime }+\\\\zeta ^{\\\\prime })$ .\",\n \"In addition ${\\\\rm codim\\\\,}Y <{\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}(B+\\\\zeta )<{\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}(B^{\\\\prime }+\\\\zeta ^{\\\\prime }),$ which contradicts the minimality of $ (B+\\\\zeta )$ .\"\n ],\n [\n \"Torsion anomalous varieties as components of different intersections\",\n \"In the next lemma we prove that every $V$ -torsion anomalous variety which is a component of $V\\\\cap (B+\\\\zeta )$ , is also a component of any intersection $V\\\\cap (A+\\\\zeta ^{\\\\prime })$ with $B+\\\\zeta \\\\subseteq A+\\\\zeta ^{\\\\prime }$ .\",\n \"Moreover, we can choose $\\\\zeta =\\\\zeta ^{\\\\prime }$ , in fact if a translate $B+p$ is contained in another $ A+p^{\\\\prime }$ , then $A+p=A+p^{\\\\prime }$ .\",\n \"Indeed $B$ is a subgroup and it contains 0, so $p-p^{\\\\prime }\\\\in A$ and $A+p^{\\\\prime }=A+(p-p^{\\\\prime })+p^{\\\\prime }=A+p$ .\",\n \"Lemma 3.2 Let $Y$ be a maximal $V$ -torsion anomalous variety, and let $B+\\\\zeta $ be minimal for $Y$ .\",\n \"Then $Y$ is a component of $V\\\\cap (A+\\\\zeta )$ for every algebraic subgroup $A\\\\supseteq B$ , with ${\\\\rm codim\\\\,}A\\\\ge \\\\dim V-\\\\dim Y$ .\",\n \"Clearly $Y\\\\subseteq V\\\\cap (A+\\\\zeta )$ .\",\n \"Let $X$ be an irreducible component of $V\\\\cap (A+\\\\zeta )$ which contains $Y$ , then ${\\\\rm codim\\\\,}X \\\\le {\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}A.$ If the inequality is strict then $X$ is anomalous and by the maximality of $Y$ we get $Y=X$ .\",\n \"Otherwise, since by assumption ${\\\\rm codim\\\\,}A\\\\ge \\\\dim V-\\\\dim Y$ , we have ${\\\\rm codim\\\\,}X = {\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}A\\\\ge {\\\\rm codim\\\\,}V +\\\\dim V-\\\\dim Y={\\\\rm codim\\\\,}Y,$ but $Y\\\\subseteq X$ , so the opposite inequality also holds.\",\n \"This implies ${\\\\rm codim\\\\,}X={\\\\rm codim\\\\,}Y$ .\",\n \"Therefore both varieties are irreducible, so $Y=X$ and $Y$ is a component of $V\\\\cap (A+\\\\zeta )$ .\"\n ],\n [\n \"No torsion anomalous varieties on a weak-transverse translate\",\n \"The simple choice of maximal and minimal varieties and the group structure allow us to prove Conjecture REF for weak-transverse translates.\",\n \"Let $Y$ be an $(H+p)$ -torsion anomalous variety and let $B+\\\\zeta $ be minimal for $Y$ .\",\n \"Then $Y$ is a component of $(H+p)\\\\cap (B+\\\\zeta )$ and $\\\\dim H+\\\\dim B-\\\\dim Y \\\\dim Y_i$ and $X_1$ is not $V$ -torsion anomalous due to the maximality of the $Y_i$ .\",\n \"By assumption, $B+\\\\zeta _i$ is minimal for $Y_i$ for some $\\\\zeta _i\\\\in {\\\\mathrm {Tor}_{G}}$ .\",\n \"Then $X_1$ is not contained in any $B+\\\\zeta $ , otherwise $Y_i\\\\subseteq X_1\\\\subseteq V\\\\cap (B+\\\\zeta )$ and $X_1$ would be a $V$ -torsion anomalous variety; this gives a contradiction.\",\n \"Let $H+\\\\zeta $ be the torsion variety (not necessarily proper) of smallest dimension containing $X_1$ .\",\n \"Then $X_1$ is weak-transverse in $H+\\\\zeta $ , and since $X_1$ is not $V$ -torsion anomalous $N-\\\\dim X_1=N-\\\\dim V+N-\\\\dim H.$ Recall that, the $Y_i$ are $V$ -torsion anomalous varieties and use (REF ) to obtain $N-\\\\dim Y_i0$ , there exists a positive constant $c_1$ depending on $E^N$ and $\\\\eta $ , such that $\\\\mu (Y) \\\\ge c_1 \\\\frac{(\\\\deg B)^{\\\\frac{1}{\\\\dim B-\\\\dim Y}-\\\\eta }}{(\\\\deg Y)^{{\\\\frac{1}{\\\\dim B-\\\\dim Y}}+ \\\\eta }}.$ This theorem is a special case of the main theorem of [14].\",\n \"For our applications, the lower bound must take into account the degree of $B$ .\",\n \"To do so one needs to consider several different line bundles on $E^N$ , and the proof in [14] is based on an equivalence between two line bundles, and on the lower bound for the essential minimum of a transverse variety with respect to the standard bundle ${\\\\mathcal {O}_{N}}$ .\",\n \"Such a bound is given by Galateau in [18].\",\n \"The result of Galateau is inspired by the toric version of Amoroso and David [1] Theorem 1.4.\",\n \"The constant $c_1$ is effective and becomes explicit if the constants in [18] are made explicitIn a personal communication Galateau provided us explicit computations..\"\n ],\n [\n \"A relative Lehmer Estimate for points\",\n \"In [13], Theorem 1.15, Carrizosa proves the so called relative Lehmer problem for CM abelian varieties.\",\n \"Theorem 4.3 (Carrizosa) Let $A$ be an abelian defined over a number field $k$ and having CM.\",\n \"Let $\\\\mathcal {L}_0$ be an ample symmetric bundle on $A$ , and let $\\\\mathcal {L}=\\\\mathcal {L}_0^{4}$ .\",\n \"Let $H$ be an abelian subvariety of $A$ of dimension $g_0>0$ and $P$ be a point in $H$ , which is not a torsion point modulo all proper abelian subvarieties of $H$ .\",\n \"Then there exists a positive constant $c(A, k, \\\\mathcal {L})$ depending only on $A$ , $k$ and $\\\\mathcal {L}$ such that $\\\\hat{h}_\\\\mathcal {L}(P)\\\\ge \\\\frac{c(A,k,\\\\mathcal {L})}{\\\\omega _{k_\\\\mathrm {tor}}(P,H)}\\\\left( \\\\frac{ \\\\log \\\\log (\\\\omega _{k_\\\\mathrm {tor}}(P,H)(\\\\deg _{\\\\mathcal {L}} H)^2)}{\\\\log (\\\\omega _{k_\\\\mathrm {tor}}(P,H)(\\\\deg _{\\\\mathcal {L}} H)^2)}\\\\right)^{\\\\kappa (g_0)}$ where $\\\\kappa (g_0)=2^{2g_0+1}g_0^{4g_0}(g_0+1)!^{2g_0}$ .\",\n \"Here $k_\\\\mathrm {tor}=k(A_\\\\mathrm {tor})$ is the field of definition of all the torsion in $A$ and ${\\\\omega _{k_\\\\mathrm {tor}}(P,H)}$ is defined in the following way (see Definition 1.12 of [13]): $\\\\omega _{k_\\\\mathrm {tor}}(P,H)=\\\\min _V\\\\left( \\\\frac{\\\\deg _\\\\mathcal {L} V}{\\\\deg _\\\\mathcal {L} H} \\\\right)^{1/{\\\\rm codim\\\\,}_H V}$ where $V$ ranges over all proper subvarieties of $H$ defined over $k_\\\\mathrm {tor}$ , and containing $P$ .\",\n \"This theorem generalises to abelian varieties a result of Ratazzi in [28] for one elliptic curve.\",\n \"The proof of Ratazzi is inspired to the theorem of Amoroso and Zannier in [4] for algebraic numbers.\",\n \"The effectivity in the relative Lehmer is not explicitly stated in the theorem of Carrizosa.In a short personal communication she claims that her constants are effective.\",\n \"Using also the effectivity of other results, as for instance the result of Amoroso and Zannier in [4]Though in [4] the authors are not concerned with effectivity, their result can be made effective (see [2]).\",\n \"and of David and Hindry [16], one may check that Carrizosa's constant can be made effective.\",\n \"In addition, the complicated descent in her article can be replaced by the simple induction argument presented for tori in [3].\",\n \"An analogous effective method for the relative Lehmer in tori is given by Delsinne, see [15].\",\n \"As a straightforward corollary of Theorem REF we have the following: Theorem 4.4 Let $E$ be an elliptic curve with CM defined over a field $k$ .\",\n \"Let $P$ be a point of infinite order in $E^N$ , and let $B+\\\\zeta $ be the torsion variety of minimal dimension containing $P$ , with $B$ an abelian subvariety and $\\\\zeta $ a torsion point.\",\n \"Then for every $\\\\eta >0$ there exists a positive constant $c_2$ depending on $E^N$ and $\\\\eta $ , such that $\\\\hat{h}(P)\\\\ge c_2 \\\\frac{(\\\\deg B)^{\\\\frac{1}{\\\\dim B}-\\\\eta }}{[k_\\\\mathrm {tor}(P):k_\\\\mathrm {tor}]^{\\\\frac{1}{\\\\dim B}+ \\\\eta }}.$ We clarify how to obtain this theorem from Theorem REF .\",\n \"We recall that we are assuming all abelian subvarieties to be defined over $k$ , We remark that $P-\\\\zeta \\\\in B$ .\",\n \"In addition $\\\\hat{h}(P)=\\\\hat{h}(P-\\\\zeta )$ and $[k_\\\\mathrm {tor}(P):k_\\\\mathrm {tor}]=[k_\\\\mathrm {tor}(P-\\\\zeta ):k_\\\\mathrm {tor}]$ so we can apply Theorem REF to $P-\\\\zeta $ in $B$ .\",\n \"Let $V$ be the set of all conjugates of $P-\\\\zeta $ over $k_\\\\mathrm {tor}$ .\",\n \"Clearly the zero-dimensional variety $V$ is defined over $k_\\\\mathrm {tor}$ , and it is properly contained in $B$ .\",\n \"Therefore we can say that $\\\\omega _{k_\\\\mathrm {tor}}(P-\\\\zeta ,B)\\\\le \\\\left( \\\\frac{[k_\\\\mathrm {tor}(P):k_\\\\mathrm {tor}]}{\\\\deg _{\\\\mathcal {L}}B} \\\\right)^{1/\\\\dim B}.$ Notice also that, if we take as $\\\\mathcal {L}$ the fourth power ${\\\\mathcal {O}_{N}}^{ 4}$ of the bundle corresponding to the canonical embedding, this only introduces a constant in the heights and the degrees.\",\n \"Finally, it is clear that in the statement of Theorem REF the corrective factor can be replaced by $\\\\left( \\\\frac{ \\\\log \\\\log (\\\\omega _{k_\\\\mathrm {tor}}(P,H)(\\\\deg _{\\\\mathcal {L}}H)^2)}{\\\\log (\\\\omega _{k_\\\\mathrm {tor}}(P,H)(\\\\deg _{\\\\mathcal {L}}H)^2)}\\\\right)^{\\\\kappa (g_0)}\\\\gg _\\\\eta (\\\\omega _{k_\\\\mathrm {tor}}(P,H)(\\\\deg _{\\\\mathcal {L}}H))^{-\\\\eta }.$ Notice at last that if $B+\\\\zeta $ is the torsion variety of minimal dimension containing $P$ , then $P-\\\\zeta $ is not a torsion point modulo all proper abelian subvarieties of $B$ .\"\n ],\n [\n \"Torsion anomalous varieties which are not translates\",\n \"We let $V$ be a weak-transverse variety in a power of elliptic curves.\",\n \"In this section we prove the finiteness of the maximal $V$ -torsion anomalous varieties which are not translates and have relative codimension one.\",\n \"Note that this theorem holds in any power of elliptic curves, independently if it has or not CM.\",\n \"Theorem 5.1 Let $V\\\\subseteq E^N$ be a weak-transverse variety.\",\n \"Then the maximal $V$ -torsion anomalous varieties of relative codimension one which are not translates are finitely many.\",\n \"More precisely, let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.\",\n \"Assume that $Y$ has relative codimension one in its minimal $B+\\\\zeta $ .\",\n \"Then for any $\\\\eta >0$ there exist constants depending only on $E^N$ and $\\\\eta $ such that: $ \\\\deg B \\\\ll _\\\\eta (h(V)+ \\\\deg V)^{\\\\frac{{\\\\rm codim\\\\,}B}{{\\\\rm codim\\\\,}V -1}+\\\\eta }, $ $h(Y) \\\\ll _\\\\eta (h(V)+ \\\\deg V)^{\\\\frac{{\\\\rm codim\\\\,}B}{{\\\\rm codim\\\\,}V-1}+\\\\eta } $ and $ \\\\deg Y \\\\ll _\\\\eta \\\\deg V (h(V)+ \\\\deg V)^{\\\\frac{{\\\\rm codim\\\\,}B}{{\\\\rm codim\\\\,}V-1} -1+\\\\eta }.$ In addition the torsion points $\\\\zeta $ belong to a finite set.\",\n \"Let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.\",\n \"Let $B+\\\\zeta $ be minimal for $Y$ .\",\n \"Then $Y$ is a component of $V\\\\cap (B+\\\\zeta )$ with $B$ an abelian variety and $\\\\zeta $ a torsion point.\",\n \"In addition ${\\\\rm codim\\\\,}Y<{\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}B$ .\",\n \"We prove that $\\\\deg B$ is bounded only in terms of $V$ and $E^N$ ; we then deduce the bounds for ${h}(Y)$ and $\\\\deg Y$ .\",\n \"By Lemma REF $Y$ is weak-transverse in $B+\\\\zeta $ , and by assumption ${\\\\rm codim\\\\,}_{B+\\\\zeta } Y=1$ ; therefore $Y$ is transverse in $B+\\\\zeta $ .\",\n \"As points are translates and $V$ is not contained in a torsion variety, we have $1\\\\le \\\\dim Y<\\\\dim V.$ Applying the Bogomolov estimate Theorem REF to $Y$ in $B+\\\\zeta $ we get $\\\\frac{(\\\\deg B)^{1-\\\\eta }}{(\\\\deg Y)^{{1}+\\\\eta }}\\\\ll _\\\\eta \\\\mu (Y).$ We set $r={\\\\rm codim\\\\,}B$ .\",\n \"Let $h_1, \\\\dots , h_r$ be the forms of increasing degrees $d_i$ such that $B+\\\\zeta $ is a component of their zero set, as recalled in Section REF .\",\n \"Then $d_1 \\\\cdots d_r \\\\ll \\\\deg (B+\\\\zeta )=\\\\deg B \\\\ll d_1 \\\\cdots d_r.$ We denote $r_1=\\\\dim V-\\\\dim Y.$ Note that $r_1< r$ , because $Y$ is $V$ -torsion anomalous.\",\n \"Let $A$ be the algebraic subgroup given by the first $h_1\\\\cdots h_{r_1}$ forms.\",\n \"Then $\\\\deg A \\\\ll d_1 \\\\cdots d_{r_1} .$ Let $A_0$ be an irreducible component of $A$ containing $B+\\\\zeta $ .\",\n \"Then by (REF ) we have $\\\\deg A_0 \\\\ll d_1 \\\\cdots d_{r_1}\\\\ll (\\\\deg B)^\\\\frac{r_1}{r} , $ and ${\\\\rm codim\\\\,}A_0=r_1=\\\\dim V-\\\\dim Y$ .\",\n \"By Lemma REF , $Y$ is a component of $V\\\\cap A_0$ .\",\n \"We apply the Arithmetic Bézout theorem to $V\\\\cap A_0$ and recall that $h(A_0)=0$ , because $A_0$ is a torsion variety.\",\n \"Then $h(Y) \\\\ll (h(V)+ \\\\deg V) \\\\deg A_0\\\\ll (h(V)+\\\\deg V)(\\\\deg B)^{\\\\frac{r_1}{r}}.$ For the irreducible variety $Y$ of positive dimension, Zhang's inequality (REF ) says $\\\\mu (Y)\\\\le \\\\frac{h(Y)}{\\\\deg Y}.$ Combining this with (REF ) and (REF ) we obtain $\\\\frac{{(\\\\deg B)}^{1-\\\\eta }}{(\\\\deg Y)^{1+\\\\eta }}\\\\ll _\\\\eta \\\\mu (Y)\\\\ll (h(V)+ \\\\deg V)\\\\frac{ (\\\\deg B)^\\\\frac{r_1}{r}}{\\\\deg Y}.$ Recall that $Y$ is a component of $ V\\\\cap (B+\\\\zeta )$ .\",\n \"By Bézout's theorem $\\\\deg Y \\\\le \\\\deg B\\\\deg V$ .\",\n \"Thus ${(\\\\deg B)}^{1-\\\\eta }\\\\ll _\\\\eta (h(V)+\\\\deg V)(\\\\deg B)^\\\\frac{r_1}{r}{(\\\\deg B\\\\deg V)}^{\\\\eta }$ and therefore ${(\\\\deg B)^{\\\\frac{r-r_1}{r}-2\\\\eta }}\\\\ll _\\\\eta (h(V)+\\\\deg V){ (\\\\deg V)}^{\\\\eta }.$ Since $r-r_1={\\\\rm codim\\\\,}V -1$ , for $\\\\eta $ small enough we get $\\\\deg B \\\\ll _\\\\eta (h(V)+ \\\\deg V)^{\\\\frac{r}{{\\\\rm codim\\\\,}V -1}+\\\\eta }(\\\\deg V)^{\\\\eta } .$ So we have proved that the degree of $B$ is bounded only in terms of $V$ and $E^N$ .\",\n \"Since the abelian subvarieties of bounded degree are finitely many, applying Lemma REF we conclude that $\\\\zeta $ belongs to a finite set.\",\n \"Finally, the bound on the height of $Y$ is given by (REF ) and (REF ) $h(Y)\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{r}{{\\\\rm codim\\\\,}V -1}+\\\\eta }(\\\\deg V)^{\\\\eta }.$ The bound on the degree is given by Bézout's theorem for the component $Y$ of $V\\\\cap A_0$ and (REF ) $\\\\deg Y\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{r}{{\\\\rm codim\\\\,}V -1}-1+\\\\eta }(\\\\deg V)^{1+\\\\eta }.$\"\n ],\n [\n \"Torsion anomalous points\",\n \"In this and the following section we prove that, if $V$ is a weak-transverse variety in a power of elliptic curves with CM, then the $V$ -torsion anomalous varieties which are translates are non-dense in $V$ .\",\n \"We now prove that the $V$ -torsion anomalous varieties of dimension zero are non-dense.\",\n \"The proof relies on the Arithmetic Bézout theorem, the Zhang's inequality and on the relative Lehmer, Theorem REF .\",\n \"As the last bound is proved only for CM elliptic curves we need this assumption.\",\n \"Theorem 6.1 Let $V\\\\subseteq E^N$ be a weak-transverse variety, where $E$ has CM.\",\n \"Then, the set of maximal $V$ -torsion anomalous points of relative codimension one is a finite set of explicitly bounded height and relative degree.\",\n \"More precisely, let $k$ be a field of definition for $E$ and let $k_\\\\mathrm {tor}$ be the field of definition of all torsion points of $E^N$ .\",\n \"Let $d$ be the dimension of $V$ .\",\n \"Let $Y_0$ be a maximal $V$ -torsion anomalous point and let $B+\\\\zeta $ be minimal for $Y_0$ , with $\\\\dim B=1$ .\",\n \"Then $\\\\deg B &\\\\ll _\\\\eta ((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])^{\\\\frac{N-1}{N-1-d}+\\\\eta } ,\\\\\\\\\\\\hat{h}(Y_0) &\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{N-1}{N-1-d}+\\\\eta }[k_{\\\\mathrm {tor}}(V):k_{\\\\mathrm {tor}}]^{\\\\frac{d}{N-1-d}+\\\\eta },\\\\\\\\[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}] &\\\\ll _\\\\eta \\\\deg V[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}]^{\\\\frac{N-1}{N-1-d}+\\\\eta }{(h(V)+\\\\deg V)}^{\\\\frac{d}{N-1-d}+\\\\eta }.$ In addition the torsion points $\\\\zeta $ have order effectively bounded in Theorem REF .\",\n \"Let $Y_0$ be a maximal $V$ -torsion anomalous point, with $B+\\\\zeta $ minimal for $Y_0$ .\",\n \"By assumption $\\\\dim B={\\\\rm codim\\\\,}_{B+\\\\zeta }Y_0=1$ .\",\n \"We proceed to bound $\\\\deg B$ and, in turn, the height of $Y_0$ and its degree over $k_\\\\mathrm {tor}$ .\",\n \"To this aim we shall use Theorem REF and the Arithmetic Bézout theorem.\",\n \"By Section REF , the variety $B+\\\\zeta $ is a component of the torsion variety defined as the zero set of forms $h_1,\\\\dots ,h_{N-1}$ of increasing degrees $d_i$ and $d_1 \\\\cdots d_{N-1}\\\\ll \\\\deg B=\\\\deg (B+\\\\zeta )\\\\ll d_1 \\\\cdots d_{N-1}.$ Consider the torsion variety defined as the zero set of the first $d$ forms $h_1, \\\\dots , h_d$ , and take a connected component $A_0$ containing $B+\\\\zeta $ .\",\n \"Then $\\\\deg A_0\\\\ll d_1 \\\\cdots d_d\\\\ll {(\\\\deg B)}^{\\\\frac{d}{N-1}}$ and ${\\\\rm codim\\\\,}A_0=d=\\\\dim V-\\\\dim Y_0.$ By Lemma REF , each component of $V\\\\cap (B+\\\\zeta )$ is a component of $V\\\\cap A_0$ .\",\n \"All conjugates of $Y_0$ over $k_\\\\mathrm {tor}(V)$ are in $V\\\\cap (B+\\\\zeta )$ , so the number of components of $V\\\\cap A_0$ is at least $[k_\\\\mathrm {tor}(V,Y_0):k_\\\\mathrm {tor}(V)]\\\\ge \\\\frac{[k_\\\\mathrm {tor}(Y_0):k_\\\\mathrm {tor}]}{[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}]}.$ We then apply the Arithmetic Bézout theorem to $Y_0\\\\subseteq V\\\\cap A_0$ obtaining $[k_\\\\mathrm {tor}(Y_0):k_\\\\mathrm {tor}]\\\\hat{h}(Y_0)\\\\ll {(h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}]}(\\\\deg B)^{\\\\frac{d}{N-1}}.$ Applying Theorem REF to $Y_0$ in $B+\\\\zeta $ , we obtain that for every positive real $\\\\eta $ $\\\\hat{h}(Y_0)\\\\gg _\\\\eta \\\\frac{(\\\\deg B)^{1-\\\\eta }}{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^{1+\\\\eta }}.$ Combining (REF ) and (REF ) we have $\\\\frac{(\\\\deg B)^{1-\\\\eta }}{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^{\\\\eta }}&\\\\ll _\\\\eta [k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]\\\\hat{h}(Y_0)\\\\ll \\\\\\\\&\\\\ll {(h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}]}(\\\\deg B)^{\\\\frac{d}{N-1}}.$ For $\\\\eta $ small enough we obtain $\\\\deg B\\\\ll _\\\\eta {((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])}^{\\\\frac{N-1}{N-1-d}+\\\\eta }[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^\\\\eta .$ Apply now Bézout's theorem to $V\\\\cap A_0$ .\",\n \"All the conjugates of $Y_0$ over $k_\\\\mathrm {tor}(V)$ are components of the intersection, so $\\\\frac{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]}{[k_{\\\\mathrm {tor}}(V):k_{\\\\mathrm {tor}}]}\\\\ll _\\\\eta {((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])}^{\\\\frac{d}{N-1-d}+\\\\eta } (\\\\deg V)^{1+\\\\eta },$ which gives the last bound in the statement.\",\n \"Substituting (REF ) back into (REF ) we have the bound on $\\\\deg B$ .\",\n \"Finally apply the Arithmetic Bézout theorem to $V\\\\cap A_0$ to get $\\\\hat{h}(Y_0)\\\\ll (h(V)+\\\\deg V)(\\\\deg B)^{\\\\frac{d}{N-1}}\\\\ll _\\\\eta (h(V)+\\\\deg V)^{\\\\frac{N-1}{N-1-d}}+\\\\eta [k_{\\\\mathrm {tor}}(V):k_{\\\\mathrm {tor}}]^{\\\\frac{d}{N-1-d}+\\\\eta }.$ Having bounded $\\\\deg B$ , in view of Lemma REF , the points $\\\\zeta $ belong to a finite set.\",\n \"Notice that in Theorem REF we have effectively bounded the degree of the abelian variety $B$ , and we applied Lemma REF to prove the finiteness of the points $Y_0$ in a non effective way.\",\n \"In the following theorem we give a completely effective result.\",\n \"We explicitly bound the degree $[k(Y_0):\\\\mathbb {Q}]$ for $Y_0$ not a torsion point; this, together with the bound for $\\\\hat{h}(Y_0)$ in Theorem REF , allows to effectively find all $V$ -torsion anomalous points of relative codimension one which are not torsion points.\",\n \"The effectivity of this result is relevant for the applications to the Mordell-Lang Conjecture shown in Section .\",\n \"Theorem 6.2 Let $V$ be a weak-transverse variety in $E^N$ , where $E$ has CM.\",\n \"Let $k$ be a field of definition for $E$ .\",\n \"Let $d$ be the dimension of $V$ .\",\n \"Let $Y_0$ be a maximal $V$ -torsion anomalous point which is not a torsion point, and let $B+\\\\zeta $ be minimal for $Y_0$ with $\\\\dim B=1$ .\",\n \"Then $[k(Y_0):\\\\mathbb {Q}]\\\\ll _\\\\eta {((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])}^{\\\\frac{d(N-1)}{(N-1-d)^2}+\\\\eta }{(\\\\deg V[k(V):k])}^{\\\\frac{N-1}{N-1-d}+\\\\eta }.$ In addition the torsion points $\\\\zeta $ can be chosen with $[k(\\\\zeta ):\\\\mathbb {Q}]\\\\ll [k(Y_0):\\\\mathbb {Q}]$ and order bounded by $\\\\mathrm {ord}(\\\\zeta )\\\\ll _\\\\eta [k(Y_0):\\\\mathbb {Q}]^{\\\\frac{N}{2}+\\\\eta }.$ Finally let $S$ be the number of maximal $V$ -torsion anomalous points of relative codimension one.\",\n \"Then $S\\\\ll _\\\\eta {((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])}^{A_1+\\\\eta }{(\\\\deg V)}^{A_2+1+\\\\eta }[k(V):k]^{A_2+\\\\eta }$ where $A_1&= \\\\frac{(N-1)(2(N+1)(N-d-1)+dN(2N+1))}{2(N-d-1)^2} \\\\le (N+1)^4,\\\\\\\\A_2&= \\\\frac{N(N-1)(2N+1)}{2(N-d-1)} \\\\le N^3.$ In view of Theorem REF , we know that $\\\\deg B$ and $\\\\hat{h}(Y_0)$ are bounded.\",\n \"We now proceed to bound $[k(Y_0):k]$ .\",\n \"To this aim we need to construct an algebraic subgroup $G$ of codimension $d$ defined over $k$ , containing $Y_0$ and of controlled degree.\",\n \"In order to do this we use Siegel's lemma in a similar way as in [32], Proposition 3, which in turn follows the work [10] of Bombieri, Masser and Zannier in tori.\",\n \"Here we use Siegel's lemma directly on equations with coefficients in the endomorphism ring of $E$ .\",\n \"We notice that $\\\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\\\mathcal {O}$ .\",\n \"The coordinates of $Y_0=(x_1,\\\\ldots ,x_N)$ generate an $\\\\mathcal {O}$ -module $\\\\Gamma $ of rank one.\",\n \"The torsion submodule of $\\\\Gamma $ is well known; for instance in [32], Proposition 2 we find a description for it.\",\n \"Such a torsion module is clearly $\\\\mathcal {O}$ invariant.\",\n \"As a $\\\\mathbb {Z}$ -module it is generated by two points $T,\\\\tau T$ of exact orders $R,R^{\\\\prime }$ respectively, where $R=c(\\\\tau )R^{\\\\prime }$ and $c(\\\\tau )$ is essentially the real part of $\\\\tau ^2$ , so a constant of the problem.\",\n \"Therefore we can write $x_i=\\\\alpha _i g +\\\\beta _i T$ for a fixed point of infinite order $g$ in $E$ , with coefficients $\\\\alpha _i,\\\\beta _i\\\\in \\\\mathcal {O}$ such that $\\\\hat{h}(x_i)=|N_L(\\\\alpha _i)|\\\\hat{h}(g)$ and $N_L(\\\\beta _i)\\\\ll R^2.$ We want to find coefficients $a_i\\\\in \\\\mathcal {O}$ such that $\\\\sum _i^N a_i x_i=0$ .\",\n \"This gives a linear system of 2 equations, obtained equating to zero the coefficients of $g$ and $T$ .\",\n \"The system has coefficients in $\\\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.\",\n \"We use a version of Siegel's lemma over $\\\\mathcal {O}$ as stated in [7], Section 2.9, to get $d$ equations with coefficients in $\\\\mathcal {O}$ ; multiplying them by a constant depending only on $E$ , we may assume that they have coefficients in $\\\\mathrm {End}(E)$ .\",\n \"Thus they define the sought-for algebraic subgroup $G$ of degree $\\\\deg G\\\\ll \\\\left((\\\\max _i N_L(\\\\alpha _i))(\\\\max _i N_L(\\\\beta _i)) \\\\right)^{\\\\frac{d}{N-1}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .\",\n \"Then $\\\\deg G_0\\\\ll \\\\left((\\\\max _i N_L(\\\\alpha _i)) R^2 \\\\right)^\\\\frac{d}{N-1}.$ By the maximality of $Y_0$ , the point $Y_0$ is a component of $V\\\\cap G_0$ and by Bézout's theorem we get $\\\\frac{[k(Y_0):k]}{[k(V):k]}\\\\le \\\\deg V\\\\deg G_0.$ Using also (REF ), we obtain $\\\\frac{[k(Y_0):k]}{[k(V):k]}&\\\\ll \\\\deg V \\\\left((\\\\max _i N_L(\\\\alpha _i)) R^2 \\\\right)^\\\\frac{d}{N-1}\\\\le \\\\\\\\&\\\\le \\\\deg V \\\\left(R^2\\\\frac{ \\\\max _i \\\\hat{h}(x_i)}{\\\\hat{h}({g})}\\\\right)^\\\\frac{d}{N-1}.$ Notice that $\\\\hat{h}(x_i)\\\\le \\\\hat{h}(Y_0)$ for all $i$ .\",\n \"We can now apply Theorem REF to $g$ in $E$ , obtaining for every $\\\\eta >0$ $\\\\frac{1}{\\\\hat{h}(g)}\\\\ll _\\\\eta {[k_\\\\mathrm {tor}(g):k_\\\\mathrm {tor}]^{1+\\\\eta }}\\\\le [k_\\\\mathrm {tor}(Y_0):k_\\\\mathrm {tor}]^{1+\\\\eta }$ because $g$ is defined over $k(Y_0)$ .\",\n \"The product $[k_\\\\mathrm {tor}(Y_0):k_\\\\mathrm {tor}]\\\\hat{h}(Y_0)$ was bounded in (REF ), so using also the bounds in Theorem REF we obtain $[k(Y_0):k]\\\\ll _\\\\eta \\\\deg V [k(V):k]((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])^{\\\\frac{d}{N-1-d}+\\\\eta }R^{2\\\\frac{d}{N-1}}.$ By a result of Serre in [30], recalled also in [32], Corollary 3, for $R$ larger than a constant and $\\\\varphi $ the Euler function, we have $\\\\varphi (R)\\\\varphi (R^{\\\\prime })\\\\ll [k(Y_0):k] .$ In addition $R^{2-\\\\eta }\\\\ll _\\\\eta \\\\varphi (R)\\\\varphi (R^{\\\\prime })$ since in general $\\\\varphi (x)\\\\gg _\\\\eta x^{1-\\\\eta }$ and $R$ and $R^{\\\\prime }$ are related by a constant.\",\n \"From this and (REF ), for $\\\\eta $ small enough we obtain $[k(Y_0):k]\\\\ll _\\\\eta \\\\deg V [k(V):k]{((h(V)+\\\\deg V)[k_\\\\mathrm {tor}(V):k_\\\\mathrm {tor}])}^{\\\\frac{d}{N-d-1}+\\\\eta } [k(Y_0):k]^{\\\\frac{d}{N-1}+\\\\eta }.$ Since $Y_0$ is $V$ -torsion anomalous of relative dimension one, we have $d0$ .\",\n \"In fact, if $B$ is such a abelian subvariety, consider its associated matrix, as recalled in Section REF ; it is an $(N-1)\\\\times N$ matrix and we call $d_i$ the square of the norm of its $i$ -th row.\",\n \"We let $\\\\deg B={p_1}^{e_1}\\\\cdots {p_r}^{e_r}$ be the factorization of $\\\\deg B$ into distinct prime factors.\",\n \"Then the number of possible choices for the elements $d_i$ is bounded by $\\\\delta (\\\\deg B)^{N-1}$ , where, for a positive integer $n$ , $\\\\delta (n)$ counts the number of divisors of $n$ .\",\n \"We notice that, for every $\\\\eta >0$ we have $ \\\\delta (\\\\deg B)\\\\ll _\\\\eta (\\\\deg B)^{\\\\eta }.$ Now, for every choice of the $d_i$ , the number $D$ of $(N-1)\\\\times N$ matrices in which the square of the norm of the $i$ -th row is at most $d_i$ is bounded in the following way $D\\\\ll \\\\left(\\\\prod _{i=1}^{N-1} d_i\\\\right)^N\\\\ll ({\\\\deg B})^N.$ So for every $\\\\eta >0$ , the number of possible subgroups $B$ is $\\\\ll _\\\\eta (\\\\deg B)^{N+\\\\eta }.$ As for the point $\\\\zeta $ , it is well known that the number of torsion points in $E^N$ of order bounded by a constant $M$ is at most $M^{2N+1}$ .\",\n \"In fact the number of points of order dividing a positive integer $i$ is $i^{2N}$ ; so a bound for the number of torsion points of order at most $M$ is given by $\\\\sum _{i=1}^M i^{2N}\\\\ll M^{2N+1}.$ Applying Bézout's theorem to every intersection $V\\\\cap (B+\\\\zeta )$ , we obtain that for every $\\\\eta >0$ the number $S$ of $V$ -torsion anomalous points of relative codimension one is bounded by $S\\\\ll _\\\\eta \\\\deg V (\\\\deg B)^{N+1+\\\\eta } {\\\\textrm {ord}(\\\\zeta )}^{2N+1}.$ This, combined with Theorem REF , (REF ) and (REF ), gives the required explicit bounds.\"\n ],\n [\n \"Torsion anomalous translates of positive dimension\",\n \"We let $V$ be a weak-transverse variety in a power of elliptic curves with CM.\",\n \"In this section we study $V$ -torsion anomalous varieties which are translates of positive dimension.\",\n \"We reduce this case to the zero dimensional case.\",\n \"First we compare the $V$ -torsion anomalous translates with translates contained in $V$ .\",\n \"Lemma 7.1 Let $V$ be a weak-transverse subvariety of an abelian variety of dimension $N$ .\",\n \"Let $Y$ be a maximal $V$ -torsion anomalous translate, then $Y$ is a maximal translate contained in $V$ (i.e.\",\n \"$Y$ is not strictly contained in any translate contained in $V$ ).\",\n \"Let $Y$ be a maximal $V$ -torsion anomalous translate and suppose that $Y$ is contained in a maximal translate $(H+p)\\\\subseteq V$ with $\\\\dim (H+p)> \\\\dim Y $ .\",\n \"Let $B+\\\\zeta $ be minimal for $Y$ .\",\n \"Then $Y$ is a component of $V\\\\cap (B+\\\\zeta )$ and ${\\\\rm codim\\\\,}Y <{\\\\rm codim\\\\,}V +{\\\\rm codim\\\\,}B .$ Since $(H+p)\\\\cap (B+\\\\zeta )\\\\supset Y$ then $p=h+b+\\\\zeta $ for some $h\\\\in H$ and $b\\\\in B$ .\",\n \"Therefore $(H+p)\\\\cap (B+\\\\zeta )=(H+b+\\\\zeta )\\\\cap (B+\\\\zeta )=(H+b+\\\\zeta )\\\\cap (B+b+\\\\zeta )=(H\\\\cap B)+b+\\\\zeta $ and $\\\\dim (H\\\\cap B)=\\\\dim ((H+p) \\\\cap (B+\\\\zeta ))\\\\ge \\\\dim Y.$ By (REF ), we deduce $\\\\begin{split}\\\\dim (H+B)&=\\\\dim H +\\\\dim B -\\\\dim (H\\\\cap B)\\\\\\\\& \\\\le \\\\dim H +\\\\dim B -\\\\dim Y \\\\dim B-\\\\dim X=1$ .\",\n \"Since ${\\\\rm codim\\\\,}V\\\\ge g+1$ by the hypothesis and $\\\\dim V\\\\ge \\\\dim X+1=\\\\dim B$ , then ${\\\\rm codim\\\\,}B\\\\ge g+1$ .\",\n \"From Section REF , $B+\\\\zeta $ is a component of the torsion variety given by the zero set of $r\\\\ge 2$ forms $h_1,\\\\ldots ,h_r$ of increasing degrees $d_i$ and $d_1 \\\\cdots d_r \\\\ll \\\\deg (B+\\\\zeta )\\\\ll d_1 \\\\cdots d_r .$ Notice that one single equation gives ${\\\\rm codim\\\\,}B\\\\le g$ , since $g$ is the maximal dimension of a simple factor of $A$ .\",\n \"This implies that $r\\\\ge 2$ .\",\n \"Denote by $A_0$ the algebraic subgroup given by $h_1,\\\\ldots ,h_{r-1}$ .\",\n \"Since $r\\\\ge 2$ , we have $A_0\\\\subsetneq A$ .\",\n \"In addition $\\\\deg A_0\\\\ll (\\\\deg B)^{\\\\frac{r-1}{r}}$ .\",\n \"We now show that $X$ is a component of $V\\\\cap A_0$ .\",\n \"Suppose, to the contrary, that $X\\\\subsetneq Y\\\\subseteq V\\\\cap A_0$ where $Y$ is a component of $V\\\\cap A_0$ strictly containing $X$ .\",\n \"By maximality of $X$ , $Y$ cannot be $V$ -torsion anomalous, hence ${\\\\rm codim\\\\,}Y={\\\\rm codim\\\\,}V+{\\\\rm codim\\\\,}A_0.$ Moreover by construction $\\\\dim A_0\\\\le \\\\dim B+g$ .\",\n \"This implies $\\\\dim B-1=\\\\dim X<\\\\dim Y=\\\\dim A_0-{\\\\rm codim\\\\,}V\\\\le \\\\dim B+g-{\\\\rm codim\\\\,}V$ and so ${\\\\rm codim\\\\,}V\\\\dim H} H\\\\right)$ is a finite set of effectively bounded Néron-Tate height.\",\n \"Here $H$ ranges over all algebraic subgroups of codimension larger than the dimension.\",\n \"More precisely the union can be taken over finitely many algebraic subgroups $\\\\mathcal {S}(C)=C\\\\cap \\\\bigcup _{i=1}^M H_i;$ the $H_i$ are algebraic subgroups with $\\\\dim H <{\\\\rm codim\\\\,}H$ , and for any real $\\\\eta >0$ , there exist constants depending only on $E^N$ and $\\\\eta $ such that $ \\\\deg H_i \\\\ll _\\\\eta {((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{r(N-r)(N+2r-2)}{2(r-1)(2r-N)}+\\\\eta } ([k(C):k]\\\\deg C)^{\\\\frac{Nr}{2(r-1)}+\\\\eta },$ where $r={\\\\rm codim\\\\,}H_i$ .\",\n \"Moreover $M$ can be effectively bounded.\",\n \"Furthermore, if $Y_0\\\\in \\\\mathcal {S}(C)$ we have $\\\\hat{h}(Y_0) \\\\ll _\\\\eta (h(C)+\\\\deg C)^{\\\\frac{N+1}{2}+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{\\\\frac{N-1}{2}+\\\\eta };$ if $Y_0$ is not a torsion point we also have $[k(Y_0):\\\\mathbb {Q}]\\\\ll _\\\\eta ([k(C):k]\\\\deg C)^{\\\\frac{N+1}{N-1}+\\\\eta }{((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{N+1}{2}+\\\\eta },$ and the cardinality of the points in $\\\\mathcal {S}(C)$ which are not torsion points is bounded as $S\\\\ll _\\\\eta [k(C):k]^{B_1}(\\\\deg C)^{B_1+1+\\\\eta }{((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{B_2+\\\\eta },$ where $B_1=\\\\frac{N(N+1)(2N+1)}{2(N-1)}, \\\\qquad B_2=\\\\frac{(3N^2+N-1)(N+1)}{4}.$ We notice that the set of all torsion points in $C$ is finite and has cardinality bounded by the effective Manin-Mumford Conjecture.\",\n \"From now on, we will be concerned with points in $\\\\mathcal {S}(C)$ that are not torsion.\",\n \"Clearly all the points in the intersection $C\\\\cap (\\\\cup _{{\\\\rm codim\\\\,}H>\\\\dim H}H) $ are $C$ -torsion anomalous.\",\n \"In addition since $C$ is a weak-transverse curve each torsion anomalous point is maximal.\",\n \"Let $Y_0\\\\in \\\\mathcal {S}(C)$ be a non-torsion point; then $Y_0\\\\in C\\\\cap H$ , with $H$ the minimal subgroup containing $Y_0$ with respect to the inclusion, and let $r={\\\\rm codim\\\\,}H$ .\",\n \"Let also $B+\\\\zeta $ be a component of $H$ containing $Y_0$ .\",\n \"Clearly $\\\\dim B=\\\\dim H$ and $Y_0\\\\in C\\\\cap (B+\\\\zeta )$ with $B+\\\\zeta $ minimal for $Y_0$ ; as done several times, we can also choose the torsion point $\\\\zeta $ in the orthogonal complement of $B$ .\",\n \"Notice that $\\\\deg H\\\\le (\\\\deg B) \\\\mathrm {ord}(\\\\zeta ).$ In fact $B+\\\\langle \\\\zeta \\\\rangle $ is an algebraic subgroup of dimension equal to $\\\\dim H$ , it contains $Y_0$ and it is contained in $H$ ; thus $B+\\\\langle \\\\zeta \\\\rangle =H$ , by the minimality of $H$ .\",\n \"The proof now follows the lines of the proofs of Theorems REF and REF .\",\n \"We proceed to bound $\\\\deg B$ and, in turn, the height of $Y_0$ , using Theorem REF and the Arithmetic Bézout theorem.\",\n \"Then, using Siegel's lemma, we get a bound for $[k(Y_0):k]$ and for the order and the number of torsion points $\\\\zeta $ , providing also a bound for $\\\\deg H$ .\",\n \"Recall that we have $r={\\\\rm codim\\\\,}B={\\\\rm codim\\\\,}H$ .\",\n \"We first exclude the case $r=1$ and show that the case $N-r=1$ is covered by Theorems REF and REF .\",\n \"If $r=1$ then $\\\\dim B<{\\\\rm codim\\\\,}B$ implies $\\\\dim B=0$ and $N=\\\\dim B+{\\\\rm codim\\\\,}B=1$ contradicting the weak-transversality of $C$ .\",\n \"The case $N-r=1$ corresponds to $\\\\dim B=1$ and $Y_0$ of relative codimension one, that can be treated with Theorems REF and REF .\",\n \"Thus we can assume $r>N-r\\\\ge 2$ .\",\n \"Moreover $2r>N$ , since by assumption ${\\\\rm codim\\\\,}B> \\\\dim B$ .\",\n \"As done several times, by Section REF , the variety $B+\\\\zeta $ is a component of the zero set of forms $h_1,\\\\dots ,h_{r}$ of increasing degrees $d_i$ and $d_1 \\\\cdots d_{r}\\\\ll \\\\deg B=\\\\deg (B+\\\\zeta )\\\\ll d_1 \\\\cdots d_{r}.$ Consider the torsion variety defined as the zero set of $h_1$ , and let $A_0$ be one of its connected components containing $B+\\\\zeta $ .\",\n \"Then $\\\\deg A_0\\\\ll d_1\\\\ll (\\\\deg B)^{\\\\frac{1}{r}}.$ From Theorem REF applied to $Y_0$ in $B+\\\\zeta $ , for every positive real $\\\\eta $ we get $\\\\hat{h}(Y_0)\\\\gg _\\\\eta \\\\frac{(\\\\deg B)^{\\\\frac{1}{N-r}-\\\\eta }}{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^{\\\\frac{1}{N-r}+\\\\eta }}.$ Notice that all conjugates of $Y_0$ over $k_\\\\mathrm {tor}(C)$ are components of $C\\\\cap A_0$ .\",\n \"Applying the Arithmetic Bézout theorem to $Y_0$ in $C\\\\cap A_0$ and arguing as in the proof of Theorem REF , we have $\\\\frac{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]}{[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]}\\\\hat{h}(Y_0)\\\\ll (h(C)+\\\\deg C) (\\\\deg B)^{\\\\frac{1}{r}}.$ From (REF ) and (REF ) we get $(\\\\deg B)^{\\\\frac{2r-N}{r(N-r)}-\\\\eta }\\\\ll _\\\\eta (h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}][k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^{\\\\frac{1}{N-r}-1+\\\\eta }.$ Since $2r>N$ , $N-r>1$ and $ [k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]\\\\ge 1$ , for $\\\\eta $ small enough we obtain $\\\\deg B\\\\ll _\\\\eta {((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{r(N-r)}{2r-N}+\\\\eta }.$ So, from (REF ) we have $ [k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]\\\\hat{h}(Y_0)\\\\ll _\\\\eta {((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{r}{2r-N}+\\\\eta };$ while, using the right-hand side of (REF ) as a bound for $\\\\hat{h}(Y_0)$ and (REF ) we get $\\\\hat{h}(Y_0)\\\\ll _\\\\eta (h(C)+\\\\deg C)^{\\\\frac{r}{2r-N}+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{\\\\frac{(N-r)}{2r-N}+\\\\eta }$ as required.\",\n \"Having bounded $\\\\deg B$ and $\\\\hat{h}(Y_0)$ , we now proceed to bound $[k(Y_0):k]$ only in terms of $C$ and $E^N$ .\",\n \"We use Siegel's lemma to construct an algebraic subgroup $G$ of codimension $1=\\\\dim C$ defined over $k$ , containing $Y_0$ and of controlled degree.\",\n \"The construction is exactly as in [32] Propositions 3 and 4.\",\n \"We present the steps of the proof.\",\n \"We know that $\\\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\\\mathcal {O}$ .\",\n \"By minimality of $B+\\\\zeta $ , the coordinates of $Y_0=(x_1,\\\\ldots ,x_N)$ generate an $\\\\mathcal {O}$ -module $\\\\Gamma $ of rank equal to the dimension of $B+\\\\zeta $ which is $N-r$ .\",\n \"Let $g_1, \\\\dots , g_{N-r}$ be generators of the free part of $\\\\Gamma $ which give the successive minima, and are chosen as in [32], Proposition 2.\",\n \"Then $\\\\hat{h}(\\\\sum _j \\\\alpha _j g_j)\\\\gg \\\\sum _i |N_L(\\\\alpha _j)|\\\\hat{h}(g_j)$ for coefficients $\\\\alpha _j$ in $\\\\mathcal {O}$ .\",\n \"In addition, like in the case of relative codimension one, the torsion part is generated by a torsion point $T$ of exact order $R$ .\",\n \"Therefore we can write $x_i=\\\\sum _j\\\\alpha _{i,j} g_j +\\\\beta _i T$ with coefficients $\\\\alpha _i,\\\\beta _i\\\\in \\\\mathcal {O}$ and $N_L(\\\\beta _i)\\\\ll R^2.$ As in Proposition 2 of [32], we have $\\\\hat{h}(x_i)\\\\gg \\\\sum _j \\\\left|N_L(\\\\alpha _{i,j})\\\\right|\\\\hat{h}(g_j).$ We define $\\\\nu _j=(\\\\alpha _{1,j}, \\\\dots , \\\\alpha _{N,j}) \\\\quad {\\\\mathrm {and}} \\\\quad |\\\\nu _j|=\\\\max _i|N_L(\\\\alpha _{i,j})|.$ Then $|\\\\nu _j|\\\\ll \\\\frac{\\\\hat{h}(Y_0)}{\\\\hat{h}(g_j)}.$ We want to find coefficients $a_i\\\\in \\\\mathcal {O}$ such that $\\\\sum _i^N a_i x_i=0$ .\",\n \"This gives a linear system of $N-r+1$ equations, obtained equating to zero the coefficients of $g_j$ and of $T$ .\",\n \"The system has coefficients in $\\\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.\",\n \"We use the Siegel's lemma over $\\\\mathcal {O}$ as stated in [7], Section 2.9.\",\n \"We get one equation with coefficients in $\\\\mathcal {O}$ ; multiplying it by a constant depending only on $E$ we may assume that it has coefficients in $\\\\mathrm {End}(E)$ .\",\n \"Thus it defines the sought-for algebraic subgroup $G$ of degree $\\\\deg G\\\\ll \\\\left((\\\\max _i N_L(\\\\beta _i))\\\\left(\\\\prod ^{N-r}_j|\\\\nu _j|\\\\right) \\\\right)^{\\\\frac{1}{r}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .\",\n \"Then $\\\\deg G_0\\\\ll \\\\left(R^2\\\\prod ^{N-r}_j|\\\\nu _j| \\\\right)^\\\\frac{1}{r}.$ Since $C$ is weak-transverse, the point $Y_0$ is a component of $C\\\\cap G_0$ .\",\n \"In addition $C$ and $G_0$ are defined over $k$ and Bézout's theorem gives $[k(Y_0):k]\\\\le [k(C):k]\\\\deg C\\\\deg G_0.$ Hence $[k(Y_0):k]\\\\ll [k(C):k]\\\\deg C \\\\left(R^2\\\\prod ^{N-r}_j|\\\\nu _j| \\\\right)^\\\\frac{1}{r}.$ Using (REF ) we get $[k(Y_0):k]\\\\ll [k(C):k]\\\\deg C \\\\left(R^2\\\\frac{\\\\hat{h}(Y_0)^{N-r}}{\\\\prod ^{N-r}_j\\\\hat{h}({g_j})}\\\\right)^\\\\frac{1}{r}.$ Following step by step the proof of Proposition 4 in [32], using Theorem REF in place of Theorem 1.3 of [16], we get $\\\\prod _{i=1}^{N-r}\\\\hat{h}(g_i)\\\\gg _\\\\eta \\\\frac{1}{[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]^{1+\\\\eta }}.$ By a result of Serre, recalled also in [32], Corollary 3, we know $[k(Y_0):k]\\\\gg _\\\\eta R^{2-\\\\eta }$ .\",\n \"Moreover from (REF ), the product $\\\\hat{h}(Y_0)[k_{\\\\mathrm {tor}}(Y_0):k_{\\\\mathrm {tor}}]$ is bounded.\",\n \"Substituting these bounds in (REF ) and recalling that $r>2$ , for $\\\\eta $ small enough we obtain $[k(Y_0):k]\\\\ll _\\\\eta ([k(C):k]\\\\deg C)^{\\\\frac{r}{r-1}+\\\\eta }{((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{r(N-r)}{(2r-N)(r-1)}+\\\\eta }.$ Moreover, as in the proof of Theorem REF , we can choose $\\\\zeta $ so that $[k(\\\\zeta ):k]\\\\ll [k(Y_0):\\\\mathbb {Q}], $ and $\\\\mathrm {ord}(\\\\zeta )\\\\ll _\\\\eta [k(Y_0):\\\\mathbb {Q}]^{\\\\frac{N}{2}+\\\\eta }$ .\",\n \"Substituting this and (REF ) in (REF ), we get the bound for $\\\\deg H$ .\",\n \"Now, as in the last part of the proof of Theorem REF , for every $\\\\eta >0$ a bound for the number $S$ of non-torsion points $Y_0$ is given by $S\\\\ll _\\\\eta {\\\\deg C} \\\\phantom{.\",\n \"}({\\\\deg B})^{N+1+\\\\eta } {\\\\mathrm {ord}(\\\\zeta )}^{2N+1}$ and combining this with the previous results we obtain the bounds.\",\n \"Finally we notice that we have bounded the degree $\\\\deg H$ for $H$ of fixed codimension $r$ ; letting $r$ vary this proves that the subgroups $H$ can be taken in a finite set $\\\\lbrace H_1,\\\\cdots , H_M\\\\rbrace $ .\",\n \"A bound for $M$ can be given effectively, as done in the proof of Theorem REF .\",\n \"In Theorem REF we have stated a simplified version of the bounds for $\\\\hat{h}(Y_0),[k(Y_0):\\\\mathbb {Q}]$ and $S$ .\",\n \"This has been done for clarity, but a closer inspection of the proof shows that we have proved the following, sharper bounds.\",\n \"Corollary 9.2 With the same setting and notation of Theorem REF , take $Y_0\\\\in \\\\mathcal {S}(C)$ is not a torsion point, and $Y_0\\\\in C\\\\cap H$ with $H$ of minimal dimension $N-r$ .\",\n \"Then for any real $\\\\eta >0$ , there exist constants depending only on $E^N$ and $\\\\eta $ such that $\\\\hat{h}(Y_0) &\\\\ll _\\\\eta (h(C)+\\\\deg C)^{\\\\frac{r}{2r-N}+\\\\eta }[k_{\\\\mathrm {tor}}(C):k_{\\\\mathrm {tor}}]^{\\\\frac{(N-r)}{2r-N}+\\\\eta },\\\\\\\\[k(Y_0):\\\\mathbb {Q}]&\\\\ll _\\\\eta ([k(C):k]\\\\deg C)^{\\\\frac{r}{r-1}+\\\\eta }{((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{\\\\frac{r(N-r)}{(2r-N)(r-1)}+\\\\eta }.$ Moreover $Y_0$ belongs to a finite set of cardinality $S$ bounded as $S\\\\ll _\\\\eta [k(C):k]^{B_1}(\\\\deg C)^{B_1+1+\\\\eta }{((h(C)+\\\\deg C)[k_\\\\mathrm {tor}(C):k_\\\\mathrm {tor}])}^{B_2+\\\\eta },$ where $ B_1=\\\\frac{rN(2N+1)}{2(r-1)},$ $B_2=\\\\frac{r(N-r)(2rN+2r-2+2N^2-N)}{2(2r-N)(r-1)}.$\"\n ],\n [\n \"Sur une question d'orthogonalité dans les puissances de courbes elliptiques\",\n \"We are indebted to Patrice Philippon for writing this appendix which is essentially the proof of Lemma REF .\",\n \"Let us mention that deeper properties of orthogonality in Mordell-Weil groups have been studied, see [5].\",\n \"Soit $E$ une courbe elliptique définie sur le corps des nombres algébriques $\\\\overline{\\\\mathbb {Q}}$ , considéré plongé dans le corps des nombres complexes $\\\\mathbb {C}$ .\",\n \"Soit $L={\\\\rm Frac}({\\\\rm End}(E))$ le corps des multiplications de $E$ , qu'on considérera comme un sous-corps de $\\\\overline{\\\\mathbb {Q}}\\\\subset {\\\\mathbb {C}}$ .\",\n \"On a $L=\\\\mathbb {Q}$ ou $L=\\\\mathbb {Q}(\\\\sqrt{-D})$ pour un entier $D$ positif, sans facteur carré.\",\n \"On suppose $E$ plongée dans un espace projectif par un diviseur ample et symétrique défini sur $\\\\overline{\\\\mathbb {Q}}$ et on note $\\\\hat{h}:E\\\\left(\\\\overline{\\\\mathbb {Q}}\\\\right)\\\\rightarrow {\\\\mathbb {R}}$ la hauteur normalisée (i.e.\",\n \"de Néron-Tate) correspondante, qui satisfait $\\\\hat{h}(\\\\tau p)=N_{K/{\\\\mathbb {Q}}}(\\\\tau )\\\\hat{h}(p)$ pour $\\\\tau \\\\in {\\\\rm End}(E)$ , $p\\\\in E(\\\\overline{\\\\mathbb {Q}})$ , et qui ne s'annule qu'aux points de torsion de $E$ .\",\n \"L'accouplement de Néron-Tate associé s'écrit $\\\\langle p,q\\\\rangle _{NT}=\\\\frac{1}{2}\\\\left( \\\\hat{h}(p+q)-\\\\hat{h}(p)-\\\\hat{h}(q)\\\\right)\\\\in \\\\mathbb {R}$ pour $p,q\\\\in E(\\\\overline{\\\\mathbb {Q}})$ , il induit naturellement une forme $\\\\mathbb {Q}$ -bilinéaire symétrique sur le $L$ -espace vectoriel $E(\\\\overline{\\\\mathbb {Q}})\\\\otimes _\\\\mathbb {Z}\\\\mathbb {Q}$ .\",\n \"On introduit alors le produit scalaire (à valeurs dans $L\\\\otimes _\\\\mathbb {Q}\\\\mathbb {R}$ ) $\\\\langle p,q \\\\rangle = {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\langle p,q\\\\rangle _{NT} &\\\\text{si $L=\\\\mathbb {Q}$}\\\\\\\\\\\\langle p,q\\\\rangle _{NT}-\\\\frac{1}{\\\\sqrt{-D}}\\\\langle p,\\\\sqrt{-D} q\\\\rangle _{NT} &\\\\text{si $L=\\\\mathbb {Q}(\\\\sqrt{-D})$,}\\\\end{array}\\\\right.\",\n \"}$ dont on vérifie qu'il est $L$ -sesqui-linéaire (relativement à la conjugaison sur $L$ ) et hermitien.\",\n \"On étend ces produits scalaires à $E(\\\\overline{\\\\mathbb {Q}})^n\\\\otimes _\\\\mathbb {Z}\\\\mathbb {Q}$ par les formules $\\\\langle p,q\\\\rangle _{NT}&=\\\\sum _{i=1}^n \\\\langle p_i,q_i \\\\rangle _{NT}\\\\\\\\\\\\langle p,q\\\\rangle &=\\\\sum _{i=1}^n \\\\langle p_i,q_i \\\\rangle $ pour $p=(p_1,\\\\cdots ,p_n)$ et $q=(q_1,\\\\cdots ,q_n)$ dans $E(\\\\overline{\\\\mathbb {Q}})^n$ .\",\n \"Et on munit $\\\\mathbb {C}^n$ , identifié à l'espace tangent en l'origine de $E(\\\\overline{\\\\mathbb {C}})^n$ , de son produit hermitien standard.\",\n \"Lemma 1.1 Soient $H$ et $H^{\\\\prime }$ deux sous-groupes algébriques, connexes, de $E^n$ , alors leurs espaces tangents à l'origine $\\\\mathrm {T}H(\\\\mathbb {C})$ et $\\\\mathrm {T}H^{\\\\prime }(\\\\mathbb {C})$ sont orthogonaux dans $\\\\mathbb {C}^n$ si et seulement si $H(\\\\overline{\\\\mathbb {Q}})$ et $H^{\\\\prime }(\\\\overline{\\\\mathbb {Q}})$ sont orthogonaux dans $E^n$ (pour le produit scalaire $\\\\langle \\\\cdot ,\\\\cdot \\\\rangle $ ou de façon équivalente pour l'accouplement de Néron-Tate).\",\n \"Soient $d$ et $d^{\\\\prime }$ les dimensions respectives de $H$ et $H^{\\\\prime }$ , il existe des homomorphismes $\\\\varphi :E^d\\\\rightarrow E^n$ et $\\\\varphi ^{\\\\prime }:E^{d^{\\\\prime }}\\\\rightarrow E^n$ dont les images sont $H$ et $H^{\\\\prime }$ respectivement.\",\n \"On peut décrire ces homomorphismes sous la forme $(p_1,\\\\ldots ,p_d)&\\\\mapsto p=\\\\left(\\\\sum _{j=1}^d a_{1,j}p_j,\\\\ldots ,\\\\sum _{j=1}^d a_{n,j}p_j\\\\right)\\\\\\\\(q_1,\\\\ldots ,q_{d^{\\\\prime }})&\\\\mapsto q=\\\\left(\\\\sum _{k=1}^{d^{\\\\prime }} b_{1,k}q_k,\\\\ldots ,\\\\sum _{k=1}^{d^{\\\\prime }} b_{n,k}q_k\\\\right)$ et leurs applications tangentes $\\\\mathrm {T}_\\\\varphi :\\\\mathbb {C}^{d}\\\\rightarrow \\\\mathrm {T} H(\\\\mathbb {C})\\\\subseteq \\\\mathbb {C}^n$ et $\\\\mathrm {T}_\\\\varphi ^{\\\\prime }:\\\\mathbb {C}^{d^{\\\\prime }}\\\\rightarrow \\\\mathrm {T} H^{\\\\prime }(\\\\mathbb {C})\\\\subseteq \\\\mathbb {C}^n$ par $(u_1,\\\\ldots ,u_d)&\\\\mapsto u=\\\\left(\\\\sum _{j=1}^d a_{1,j}u_j,\\\\ldots ,\\\\sum _{j=1}^d a_{n,j}u_j\\\\right)\\\\\\\\(v_1,\\\\ldots ,v_{d^{\\\\prime }})&\\\\mapsto v=\\\\left(\\\\sum _{k=1}^{d^{\\\\prime }} b_{1,k}v_k,\\\\ldots ,\\\\sum _{k=1}^{d^{\\\\prime }} b_{n,k}v_k\\\\right)$ où $a_{i,j}, b_{i,k}\\\\in \\\\mathrm {End}(E)$ pour tous $i,j,k$ .\",\n \"On calcule alors les produits scalaires en développant par sesquilinéarité $\\\\langle u,v \\\\rangle =\\\\sum _{i=1}^n\\\\sum _{j=1}^d\\\\sum _{k=1}^{d^{\\\\prime }} a_{i,j}u_j \\\\overline{b_{i,k}v_k}=\\\\sum _{j=1}^d\\\\sum _{k=1}^{d^{\\\\prime }}\\\\left(\\\\sum _{i=1}^n a_{i,j}\\\\overline{b_{i,k}}\\\\right)u_j\\\\overline{v_k}$ et $\\\\langle p,q \\\\rangle =\\\\sum _{i=1}^n\\\\sum _{j=1}^d\\\\sum _{k=1}^{d^{\\\\prime }} a_{i,j} \\\\overline{b_{i,k}}\\\\langle p_j,q_k \\\\rangle =\\\\sum _{j=1}^d\\\\sum _{k=1}^{d^{\\\\prime }}\\\\left(\\\\sum _{i=1}^n a_{i,j}\\\\overline{b_{i,k}}\\\\right)\\\\langle p_j,q_k \\\\rangle .$ Si les espaces tangents en l'origine à $H$ et $H^{\\\\prime }$ sont orthogonaux (resp.\",\n \"si $H$ et $H^{\\\\prime }$ sont orthogonaux) on a $\\\\langle u,v\\\\rangle =0$ pour tous $u_1,\\\\dots ,u_d,v_1,\\\\dots ,v_{d^{\\\\prime }}\\\\in {\\\\mathbb {C}}$ (resp.\",\n \"$\\\\langle p,q\\\\rangle =\\\\langle p,q\\\\rangle _{\\\\rm NT}=0$ pour tous $p_1,\\\\dots ,p_d,q_1,\\\\dots ,q_{d^{\\\\prime }}\\\\in E(\\\\overline{\\\\mathbb {Q}}$ ).\",\n \"Appliqué avec $u_{j^{\\\\prime }}=v_{k^{\\\\prime }}=0$ pour $j^{\\\\prime }\\\\ne j$ , $k^{\\\\prime }\\\\ne k$ et $u_j=v_k=u\\\\ne 0$ (resp.\",\n \"$p_{j^{\\\\prime }}=q_{k^{\\\\prime }}=0$ pour $j^{\\\\prime }\\\\ne j$ , $k^{\\\\prime }\\\\ne k$ et $p_j=q_k=p$ non de torsion), ceci entraîne les égalités $\\\\sum _{i=1}^na_{i,j}\\\\overline{b_{i,k}}=0$ , pour tous $j,k$ .\",\n \"Réciproquement, ces dernières entraînent $\\\\langle u,v\\\\rangle =0$ et $\\\\langle p,q\\\\rangle =\\\\langle p,q\\\\rangle _{\\\\rm NT}=0$ pour tous $u\\\\in {\\\\rm T}H({\\\\mathbb {C}})$ , $v\\\\in {\\\\rm T}H^{\\\\prime }({\\\\mathbb {C}})$ , $p\\\\in H(\\\\overline{\\\\mathbb {Q}})$ et $q\\\\in H^{\\\\prime }(\\\\overline{\\\\mathbb {Q}})$ , ce qui établit l'équivalence énoncée.\"\n ],\n [\n \"Acknowledgements\",\n \"We wish to thank P. Philippon for useful discussions and for his kind communication of the appendix.\",\n \"We kindly thank A. Galateau for his communication of the constants in [18].\",\n \"We thank D. Bertrand, P. Habegger, M. Hindry and D. Masser for providing us useful references.\",\n \"We warmly thank the FNS for the great support we receive in doing research in Mathematics.\",\n \"E. Viada also likes to thank the AIM and the organizers J. Pila and U. Zannier for supporting her participation to a conference on unlikely intersections in Pisa, where this work was inspired.\",\n \"S. Checcoli and F. Veneziano would like to address a special thank to their advisor U. Zannier for introducing them to mathematical research.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1435"}}},{"rowIdx":1866992,"cells":{"text":{"kind":"list like","value":[["How Many Vote Operations Are Needed to Manipulate A Voting System?"],["Abstract In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations.","We prove the following theorem: if we fix the number of alternatives, generate $n$ votes i.i.d.","according to a distribution $\\pi$, and let $n$ go to infinity, then for any $\\epsilon >0$, with probability at least $1-\\epsilon$, the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) $\\Theta(\\sqrt n)$, (3) $\\Theta(n)$, and (4) $\\infty$.","This theorem holds for any set of vote operations, any individual vote distribution $\\pi$, and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.","We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.","Therefore, our main theorem naturally applies to these problems."],["Introduction","Voting is a popular method used to aggregate voters' preferences to make a joint decision.","Recently, voting has been used in many fields of artificial intelligence, for example in multi-agent systems [15], recommender systems [21], [32], and web-search engines [12].","One of the most desired properties for voting rules is strategy-proofness, that is, no voter has incentive to misreport her preferences to obtain a better outcome of the election.","Unfortunately, strategy-proofness is not compatible with some other desired properties, due to the celebrated Gibbard-Satterthwaite theorem [22], [36], which states that when there are at least three alternatives, no strategy-proof voting rule satisfies the following two natural properties: non-imposition (every alternative can win) and non-dictatorship (no voter is a dictator, whose top ranked alternative is always selected to be the winner).","Even though manipulation is inevitable, researchers have set out to investigate whether computational complexity can serve as a barrier against various types of strategic behavior, including manipulation.","The idea is, if we can prove that it is computationally too costly for a strategic individual to find a beneficial operation, she may give up doing so.","Initiated by Bartholdi, Tovey, and Trick [2], a fair amount of work has been done to characterize the computational complexity of various types of strategic behaviorSee [19], [17], [35] for recent surveys., including the following.","$\\bullet $ Manipulation: a voter or a coalition of voters cast false vote(s) to change the winner (and the new winner is more preferred).","$\\bullet $ Bribery: a strategic individual changes some votes by bribing the voters to make the winner preferable to her [16].","The bribery problem is closely related to the problem of computing the margin of victory [5], [25], [47].","$\\bullet $ Control: a strategic individual adds or deletes votes to make the winner more preferable to her [3].","Most previous results studying “using computational complexity as a barrier against strategic behavior” conduct worst-case analyses of computational complexity.","Recently, an increasing number of results show that manipulation, as a particular type of strategic behavior, is typically not hard to compute.","One direction, mainly pursued in the theoretical computer science community, is to obtain a quantitative version of the Gibbard-Satterthwaite theorem, showing that for any given voting rule that is “far” enough from any dictatorships, an instance of manipulation can be found easily with high probability.","This line of research was initiated by Friedgut, Kalai, and Nisan [20], where they proved the theorem for 3 alternatives and neutral voting rules.","The theorem was extended to an arbitrary number of alternatives by Isaksson, Kindler, and Mossel [23], and finally, the neutrality constraint was removed by Mossel and Racz [27].","Other extensions include Dobzinski and Procaccia [11] and Xia and Conitzer [49].","Another line of research is to characterize the “frequency of manipulability”, defined as the probability for a randomly generated preference-profile to be manipulable by a group of manipulators, where the non-manipulators' votes are generated i.i.d.","according to some distribution (for example, the uniform distribution over all possible types of preferences).","Peleg [31], Baharad and Neeman [1], and Slinko [37], [38] studied the asymptotic frequency of manipulability for positional scoring rules when the non-manipulators' votes are drawn i.i.d.","uniformly at random.","Procaccia and Rosenschein [34] showed that for positional scoring rules, when the non-manipulators’ votes are drawn i.i.d.","according to some distribution that satisfies some natural conditions, if the number of manipulators is $o(\\sqrt{n})$ , where $n$ is the number of non-manipulators, then the probability that the manipulators can succeed goes to 0 as $n$ goes to infinity; if the number of manipulator is $\\omega (n)$ , then the probability that the manipulators can succeed goes to 1.","This dichotomy theorem was generalized to a class of voting rules called generalized scoring rules (GSRs) by Xia and Conitzer [48].","A GSR is defined by two functions $f,g$ , where $f$ maps each vote to a vector in multidimensional space, called a generalized scoring vector (the dimensionality of the space is not necessarily the same as the number of alternatives).","Given a profile $P$ , let total generalized scoring vector be the sum of $f(V)$ for all votes $V$ in $P$ .","Then, $g$ selects the winner based on the total preorder of the components of the total generalized scoring vector.","We call a GSR an integer GSR, if the components of all generalized scoring vectors are integers.","(Integer) GSRs are a general class of voting rules.","One evidence is that many commonly studied voting rules are integer GSRs, including (but not limited to) approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.The definition of these commonly studied voting rules can be found in, e.g., [48].","In this paper, we define GSRs as voting rules where the inputs are profiles of linear orders.","GSRs can be easily generalized to include other types of voting rules where the inputs are not necessarily linear orders, for example, approval voting.","As another evidence, GSRs admit a natural axiomatic characterization [50], which also suggests that GSRs are equivalent to hyperplane rules [26].","The knife-edge case of $\\Theta (\\sqrt{n})$ was studied experimentally for STV and veto in [45], showing that the probability for the manipulators to succeed has a smooth phase transition.","More recently, [26] extends the dichotomy theorem to all anonymous voting rules for distributions that satisfy some mild conditions, and theoretically proved that for all generalized scoring rules, for the knife-edge case, the probability that the manipulators can achieve their goal is continuously differentiable, which suggests a smooth phase transition.","While most of the aforementioned results are about manipulation, in this paper, we focus the optimization variants of various types of strategic behavior, including manipulation, bribery, and control.","Despite being natural, to the best of our knowledge, such optimization variants have been investigated for only three types of strategic behavior.","The first is the unweighted coalitional optimization (UCO) problem, where we are asked to compute the minimum number of manipulators who can make a given alternative win [53].","Approximation algorithms have been proposed for UCO for specific voting systems, including positional scoring rules and maximin [53], [51], [52].","The second is the margin of victory problem, where we are asked to compute the smallest number of voters who can change their votes to change the winner [25], [5], [47].","The third is the minimum manipulation coalition size problem, which is similar to the margin of victory, except that all voters who change their votes must prefer the new winner to the old winner [33].","In this paper, we introduce a unified framework to study a class of strategic behavior for generalized scoring rules, which we call vote operations.","In our framework, a strategic individual seeks to change the winner by applying some operations, which are modeled as vectors in a multidimensional space.","We study three goals of the strategic individual: (1) making a favored alternative win, called constructive vote operation ($\\text{\\sc CVO}$), (2) making a disfavored alternative lose, called destructive vote operation ($\\text{\\sc DVO}$ ), and (3) change the winner of the election, called change-winner vote operation ($\\text{\\sc CWVO}$ ).","The framework will be formally defined in Section .","This is our main conceptual contribution.","Our main technical contribution is the following asymptotical characterization of the minimum number of operations that are needed for the strategic individual to achieve her goal.","Theorem REF (informally put) Fix the number of alternatives and the set of vote operations.","For any integer generalized scoring rule and any distribution $\\pi $ over votes, we generate $n$ votes i.i.d.","according to $\\pi $ and let $n$ go to infinity.","Then, for any $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\rbrace $ and any $\\epsilon >0$ , with probability at least $1-\\epsilon $ , the minimum number of operations that are necessary for the strategic individual to achieve $\\text{\\sc VO}$ falls into one of the following four categories: (1) 0, (2) $\\Theta (\\sqrt{n})$ , (3) $\\Theta (n)$ , and (4) $\\infty $ .","More informally, Theorem REF states that in large elections, to achieve a specific goal (one of the three goals described above), with probability that can be infinitely close to 1 the strategic individual needs to either do nothing (the goal is already achieved), apply $\\Theta (\\sqrt{n})$ vote operations, apply $\\Theta (n)$ vote operations, or the goal cannot be achieve no matter how many vote operations are applied.","This characterization holds for any integer generalized scoring rule, any set of vote operations, and any distribution $\\pi $ for individual votes.","The proof of Theorem REF is based on the Central Limit Theorem and on sensitivity analyses for the integer linear programmings (ILPs).","It works as follows.","We will formulate each of the strategic individual's three goals as a set of ILPs in Section .","By applying Central Limit Theorem, we show that with probability that goes to 1 the random generated preference-profile satisfies a desired property.","Then, for each such preference-profile we apply the sensitivity analyses in [8] to show that with high probability the number of operations that are necessary is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .","While Theorem REF may look quite abstract, we show later in the paper that many well-studied types of strategic behavior fall under our vote operation framework, including constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.We defer the definition of these types of strategic behavior to Section .","Therefore, we naturally obtain corollaries of Theorem REF for these types of strategic behavior.","The theorem also applies to other types of strategic behavior, for example the mixture of any types mentioned above, which is known as multimode control attacks [18]."],["Related Work and Discussion","Our main theorem applies to any integer generalized scoring rule for destructive manipulation, constructive and destructive bribery and control by adding/deleting votes.","To the best of our knowledge, no similar results were obtained even for specific voting rules for these types of strategic behavior.","Three previous papers obtained similar results for manipulation, margin of victory, and minimum manipulation coalition size.","The applications of our main theorem to these types of strategic behavior are slightly weaker, but we stress that our main theorem is significantly more general.","Three related papers.","First, the dichotomy theorem in [48] implies that, (informally) when the votes are drawn i.i.d.","from some distribution, with probability that goes to 1 the solution to constructive and destructive UCO is either 0 or approximately $\\sqrt{n}$ for some favored alternatives.","However, this result only works for the UCO problem and some distributions over the votes.","Second, it was proved in [47] that for any non-redundant generalized scoring rules that satisfy a continuity condition, when the votes are drawn i.i.d.","and we let the number of voters $n$ go to infinity, either with probability that can be arbitrarily close to 1 the margin of victory is $\\Theta (\\sqrt{n})$ , or with probability that can be arbitrarily close to 1 the margin of victory is $\\Theta (n)$ .","It is easy to show that for non-redundant voting rules, the margin of victory is never 0 or $\\infty $ .","Though it was shown in [47] that many commonly studied voting rules are GSRs that satisfy such continuity condition, in general it is not clear how restrictive the continuity condition is.","More importantly, the result only works for the margin of victory problem.","Third, in [33], the authors investigated the distribution over the minimum manipulation coalition size for positional scoring rules when the votes are drawn i.i.d.","from the uniform distribution.","However, it is not clear how their techniques can be extended beyond the uniform distributions and positional scoring rules, which are a very special case of generalized scoring rules.","Moreover, the paper only focused on the minimum manipulation coalition size problem.","Our results has both negative and positive implications.","On the negative side, our results provide yet another evidence that computational complexity is not a strong barrier against strategic behavior, because the strategic individual now has some information about the number of operations that are needed, without spending any computational cost or even without looking at the input instance.","Although the estimation of our theorem may not be very precise (because we do not know which of the four cases a given instance belongs to), such estimation may be explored to designing effective algorithms that facilitate strategic behavior.","On the positive side, this easiness of computation is not always a bad thing: sometimes we want to do such computation in order to test how robust a given preference-profile is.","For example, computing the margin of victory is an important component in designing novel risk-limiting audit methods [25], [5], [47], [39], [40], [42], [41], [43].","While being quite general, our results have two main limitations.","First, they are asymptotical results, where we fix the number of alternatives and let the number of voters go to infinity.","We do not know the convergence rate, or equivalently, how many voters are needed for the observation to hold.","In fact, this is a standard setting in previous work, especially in the studies of “frequency of manipulability”.","We feel that our results work well in settings where there are small number of alternatives and large number of voters, e.g., political elections.","Second, our results show that with high probability one of the four cases holds (0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , $\\infty $ ), but we do not know which case holds more often.","We will briefly discuss this issue in .","It is possible to refine our study for specific voting rules and specific types of strategic behavior that fall under our framework, which we leave as future work."],["Structure of the Paper","After recalling basic definitions of voting and generalized scoring rules, we present the framework in Section , where we define vote operations as well as the strategic individual's objectives.","Then, in Section we formulate the optimization problem for the strategic individual as a set of integer linear programmings (ILPs).","The main theorem will be presented in Section .","To show the wide application of the framework and the main theorem, we show that many commonly studied types of strategic behavior can be modeled as vote operations for generalized scoring rules in Section , which means that our main theorem naturally applies to these cases.","We add some discussions and point out some future directions in Section and .","Some proofs are relegated to ."],["Preliminaries","Let $\\mathcal {C}$ denote the set of alternatives (or candidates), $|\\mathcal {C}|=m$ .","We assume strict preference orders.","That is, a vote is a linear order over $\\mathcal {C}$ .","The set of all linear orders over $\\mathcal {C}$ is denoted by $L(\\mathcal {C})$ .","A preference-profile $P$ is a collection of $n$ votes for some $n\\in \\mathbb {N}$ , that is, $P\\in L(\\mathcal {C})^n$ .","Let $L(\\mathcal {C})^*=\\bigcup _{n=1}^\\infty L(\\mathcal {C})^n$ .","A voting rule $r$ is a mapping that assigns to each preference-profile a single winner.","That is, $r: L(\\mathcal {C})^*\\rightarrow \\mathcal {C}$ .","Throughout the paper, we let $n$ denote the number of votes and let $m$ denote the number of alternatives.","We now recall the definition of generalized scoring rules (GSRs) [48].","For any $K\\in \\mathbb {N}$ , let ${\\mathcal {O}}_K=\\lbrace o_1,\\ldots ,o_K\\rbrace $ .","A total preorder (preorder for short) is a reflexive, transitive, and total relation.","Let $\\text{Pre}({\\mathcal {O}}_K)$ denote the set of all preorders over ${\\mathcal {O}}_K$ .","For any $\\vec{p}\\in {\\mathbb {R}}^K$ , we let $\\text{Ord}(\\vec{p})$ denote the preorder $\\trianglerighteq $ over ${\\mathcal {O}}_K$ where $o_{k_1}\\trianglerighteq o_{k_2}$ if and only if $p_{k_1}\\ge p_{k_2}$ .","That is, the $k_1$ -th component of $\\vec{p}$ is as large as the $k_2$ -th component of $\\vec{p}$ .","For any preorder $\\trianglerighteq $ , if $o\\trianglerighteq o^{\\prime }$ and $o^{\\prime }\\trianglerighteq o$ , then we write $o=_\\trianglerighteq o^{\\prime }$ .","Each preorder $\\trianglerighteq $ naturally induces a (partial) strict order $\\vartriangleright $ , where $o\\vartriangleright o^{\\prime }$ if and only if $o\\trianglerighteq o^{\\prime }$ and $o^{\\prime }\\ntrianglerighteq o$ .","Definition 1 Let $K\\in \\mathbb {N}$ , $f:L(\\mathcal {C})\\rightarrow {\\mathbb {R}}^K$ and $g:\\text{Pre}({\\mathcal {O}}_K)\\rightarrow \\mathcal {C}$ .","$f$ and $g$ determine a generalized scoring rule (GSR) $GS(f,g)$ as follows.","For any preference-profile $P=(V_1,\\ldots , V_n)\\in L(\\mathcal {C})^n$ , abusing the notation we let $f(P)=\\sum _{i=1}^nf(V_i)$ , and let $GS(f,g)(P)=g(\\text{Ord}(f(P)))$ .","We say that $GS(f,g)$ is of order $K$.","If $f(V)\\in {\\mathbb {Z}}^K$ holds for all $V\\in L(\\mathcal {C})$ , then we call $\\text{GS}(f,g)$ an integer GSR.","For any $V\\in L(\\mathcal {C})$ , $f(V)$ is called a generalized scoring vector, $f(P)$ is called a total generalized scoring vector, and $\\text{Ord}(f(P))$ is called the induced preorder of $P$ .","The class of integer GSRs is equivalent to the class of rational GSRs, where the components of each generalized scoring vector is in $\\mathbb {Q}$ , because for any $l>0$ , $\\text{GS}(f,g)=\\text{GS}(l\\cdot f,g)$ .","Almost all commonly studied voting rules are generalized scoring rules, including (but not limited to) approval voting, Bucklin, Copeland, maximin, plurality with runoff, ranked pairs, and multi-stage voting rules that use GSRs in each stage to eliminate alternatives (including Nanson's and Baldwin's rule).","As an example, we recall the proof from [48] that the single transferable vote (STV) rule (a.k.a.","instant-runoff voting or alternative vote for single-winner elections) is an integer generalized scoring rule.","Example 1 STV selects the winner in $m$ rounds.","In each round, the alternative that gets the lowest plurality score (the number of times that the alternative is ranked in the top position) drops out, and is removed from all of the votes (so that votes for this alternative transfer to another alternative in the next round).","Ties are broken alphabetically.","The last-remaining alternative is the winner.","To see that STV is an integer GSR, we will use generalized scoring vectors with many components.","For every proper subset $S$ of alternatives, for every alternative $c$ outside of $S$ , there is a component in the vector that contains the number of times that $c$ is ranked first if all alternatives in $S$ are removed.","Let $\\bullet $ $K_{STV}=\\sum _{i=0}^{m-1}{m \\atopwithdelims ()i} (m-i)$ ; the components are indexed by $(S,j)$ , where $S$ is a proper subset of $\\mathcal {C}$ and $j\\le m,c_j \\notin S$ .","$\\bullet $ $(f_{STV}(V))_{(S,j)}=1$ , if after removing $S$ from $V$ , $c_j$ is at the top; otherwise, let $(f_{STV}(V))_{(S,j)}=0$ .","$\\bullet $ $g_{STV}$ selects the winners based on $\\trianglerighteq $ as follows.","In the first round, let $j_1$ be the index such that $o_{(\\emptyset ,j_1)}$ is ranked the lowest in $\\trianglerighteq $ among all $o_{(\\emptyset ,j)}$ (if there are multiple such $j$ 's, then we break ties alphabetically to select the least-preferred one).","Let $S_1=\\lbrace c_{j_1}\\rbrace $ .","Then, for any $2\\le i\\le m-1$ , define $S_i$ recursively as follows: $S_i=S_{i-1}\\cup \\lbrace c_{j_i}\\rbrace $ , where $j_i$ is the index such that $o_{(S_{i-1},j_i)}$ is ranked the lowest in $\\trianglerighteq $ among all $o_{(S_{i-1},j)}$ ; finally, the winner is the unique alternative in $(\\mathcal {C}\\setminus S_{m-1})$ .","GSRs admit a natural axiomatic characterization [50].","That is, GSRs are the class of voting rules that satisfy anonymity, homogeneity, and finite local consistency.","Anonymity says that the winner does not depend on the name of the voters, homogeneity says that if we duplicate the preference-profile multiple times, then the winner does not change, and finite local consistency is an approximation to the well-studied consistency axiom.","Not all voting rules are GSRs, for example, Dodgson's rule is not a GSR because it violates homogeneity [4], and the following skewed majority rule is also not a GSR because it also violates homogeneity.","Example 2 For any $\\frac{1}{2}<\\gamma <1$ , the $\\gamma $ -majority rule is defined for two alternatives $\\lbrace a,b\\rbrace $ as follows: $b$ is the winner if and only if the number of voters who prefer $b$ is more than the number of voters who prefer $a$ by at least $n^{\\gamma }$ .","Admittedly, these $\\gamma $ -majority rules are quite artificial.","Later in this paper we will see that the observation made for GSRs in our main theorem (Theorem REF ) does not hold for $\\gamma $ -majority rules for any $\\frac{1}{2}<\\gamma <1$ .","Notice that these rules satisfy anonymity, which means that the observation made in Theorem REF cannot be extended to all anonymous voting rules."],["Vote Operations","All types of strategic behavior mentioned in the introduction share the following common characteristics.","The strategic individual (who can be a group of manipulators, a briber, or a controller, etc.)","changes the winner by changing the votes in the preference-profile.","Therefore, for generalized scoring rules, any such an operation can be uniquely represented by changes in the total generalized scoring vector.","This is in contrast to some other types of strategic behavior where the strategic individual changes the set of alternatives or the voting rule [3], [44].","In this section, we first define the set of operations the strategic individual can apply, then define her goals.","Given a generalized scoring rule of order $K$ , we model the strategic behavior, called vote operations, as a set of vectors, each of which has $K$ elements, representing the changes made to the total generalized scoring vector if the strategic individual applies this operation.","We focus on integer vectors in this paper.","Definition 2 Given a GSR $\\text{GS}(f,g)$ of order $K$ , let $\\Delta =[\\vec{\\delta }_1 \\cdots \\vec{\\delta }_T]$ denote the vote operations, where for each $i\\le T$ , $\\vec{\\delta }_i\\in {\\mathbb {Z}}^K$ is a column vector that represents the changes made to the generalized scoring vector by applying the $i$ -th vote operation.","For each $l\\le K$ , let $\\Delta _l$ denote the $l$ -th row of $\\Delta $ .","We will show examples of these vote operations for some well-studied types of strategic behavior in Section .","Given the set of available operations $\\Delta $ , the strategic individual's behavior is characterized by a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ , where $\\vec{v}$ is a column vector and for each $i\\le T$ , $v_i$ represents the number of $i$ -th operation (corresponding to $\\vec{\\delta }_i$ ) that she applies.","Let $\\Vert \\vec{v}\\Vert _1=\\sum _{i=1}^Tv_i$ denote the total number of operations in $\\vec{v}$ , which is the L$_1$ -norm of $\\vec{v}$ .","It follows that $\\Delta \\cdot \\vec{v}$ is the change in the total generalized scoring vector introduced by the strategic individual, where for any $l\\le K$ , $\\Delta _l\\cdot \\vec{v}$ is the change in the $l$ -th component.","Next, we give definitions of the strategic individual's three goals and the corresponding computational problems studied in this paper.","Definition 3 In the constructive vote operation (CVO) problem, we are given a generalized scoring rule $\\text{GS}(f,g)$ , a preference-profile $P$ , a favored alternative $c$ , and a set of vote operations $\\Delta =[\\vec{\\delta }_1 \\cdots \\vec{\\delta }_T]$ , and we are asked to compute the smallest number $k$ , denoted by $\\text{\\sc CVO}(P,c)$ , such that there exists a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ with $\\Vert \\vec{v}\\Vert _1= k$ and $g\\left(\\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})\\right)=c$ .","If such $\\vec{v}$ does not exist, then we denote $\\text{\\sc CVO}(P,c)=\\infty $ .","The destructive vote operation (DVO) problem is defined similarly, where $c$ is the disfavored alternative, and we are asked to compute the smallest number $k$ , denoted by $\\text{\\sc DVO}(P,c)$ , such that there exists a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ with $\\Vert \\vec{v}\\Vert _1= k$ and $g\\left(\\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})\\right)\\ne c$ .","In the change-winner vote operation (CWVO) problem, we are not given $c$ and we are asked to compute $\\text{\\sc DVO}(P,\\text{GS}(f,g)(P))$ , denoted by $\\text{\\sc CWVO}(P)$ .","In CVO, the strategic individual seeks to make $c$ win; in DVO, the strategic individual seeks to make $c$ lose; and in CWVO, the strategic individual seeks to change the current winner.","For a given instance $(P,r)$ , $\\text{\\sc CWVO}$ is a special case of $\\text{\\sc DVO}$ , where $c=\\text{GS}(f,g)(P)$ .","We distinguish these two problems because in this paper, the input preference-profiles are generated randomly, so the winners of these preference-profiles might be different.","Therefore, when the preference-profiles are randomly generated, the distribution for the solution to $\\text{\\sc DVO}$ does not immediately give us a distribution for the solution to $\\text{\\sc CWVO}$ ."],["The ILP Formulation","Let us first put aside the strategic individual's goal for the moment (i.e., making a favored alternative win, making a disfavored alternative lose, or changing the winner) and focus on the following question: given a preference-profile $P$ and a preorder $\\trianglerighteq $ over the $K$ components of the generalized scoring vector, that is, $\\trianglerighteq \\in \\text{Pre}({\\mathcal {O}}_K)$ , how many vote manipulations are needed to change the order of the total generalized scoring vector to $\\trianglerighteq $ ?","Formally, given a $\\text{GS}(f,g)$ , a preference-profile $P$ and $\\trianglerighteq \\in \\text{Pre}({\\mathcal {O}}_K)$ , we are interested in $\\min \\lbrace \\Vert \\vec{v}\\Vert _1: \\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^K, \\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})=\\trianglerighteq \\rbrace $ .","This can be computed by the following integer linear programming ILP$_\\trianglerighteq $ , where $v_i$ represents the $i$ th component in $\\vec{v}$ , which must be a nonnegative integer.","We recall that $\\Delta _l$ denotes the $l$ -th row vector of $\\Delta $ .","$\\begin{tabular}{r@{\\hspace{28.45274pt}}rl}& \\min &\\Vert \\vec{v}\\Vert _1\\\\s.t.","&\\forall o_i=_\\trianglerighteq o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}= [f(P)]_j-[f(P)]_i\\\\&\\forall o_i\\vartriangleright o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}\\ge [f(P)]_j-[f(P)]_i+1\\\\&\\forall i:& v_i\\ge 0\\end{tabular}&&(\\text{LP}_{\\trianglerighteq })$ Now, we take the strategic individual's goal into account.","We immediately have the following lemma as a warmup, whose proofs are straightforward and are thus omitted.","Lemma 1 Given a GSR $\\text{GS}(f,g)$ , an alternative $c$ , and a preference-profile $P$ , $\\bullet $ $\\text{\\sc CVO}(P,c)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_\\trianglerighteq $ has an integer solution; $\\bullet $ $\\text{\\sc DVO}(P,c)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )\\ne c$ and LP$_\\trianglerighteq $ has an integer solution; $\\bullet $ $\\text{\\sc CWVO}(P)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )\\ne \\text{GS}(f,g)(P)$ and LP$_\\trianglerighteq $ has an integer solution.","(We do not need the input $c$ for this problem.)","Moreover, the solution to each of the three problems is the minimum objective value in all LPs corresponding to the problem.","For example, if $\\text{\\sc CVO}(P,c)<\\infty $ , then $\\hfill \\text{\\sc CVO}(P,c)=\\min _{\\Vert \\vec{v}\\Vert _1}\\lbrace \\vec{v}\\text{ is the solution to some LP}_\\trianglerighteq \\text{ where }g(\\trianglerighteq )=c\\rbrace \\hfill $"],["The Main Theorem","In this section we prove the main theorem, which states that for any fixed $m$ , any generalized scoring rules, and any set of vote operations $\\Delta $ , if $n$ votes are generated i.i.d., then for CVO (respectively, DVO, CWVO), with probability that can be infinitely close to 1, the solution is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .","We first present a simple example for the majority rule for two alternatives $\\lbrace a,b\\rbrace $ to show the taste of the proof for a very special case, and then comment on why this idea cannot be extended to GSRs.","After the proof of Theorem REF we will add more comments on the non-triviality of proof.","Example 3 Suppose there are $n$ voters, whose votes are drawn i.i.d.","from a distribution $\\pi $ over all possible votes (i.e., voting for $a$ with probability $\\pi (a)$ and voting for $b$ with probability $\\pi (b)$ , w.l.o.g.","$\\pi (a)\\ge \\pi (b)$ ).","Let $Y_a$ (respectively, $Y_b$ ) denote random variable that represents the total number of voters for $a$ (respectively, for $b$ ).","The number of manipulators that are needed to make $b$ to win (i.e., the solution to the UCO problem, see Section REF for formal definition) is thus a random variable $Y_a-Y_b$ .If $Y_a-Y_b<0$ then no manipulator is needed.","Let $X$ denote the random variable that takes 1 with probability $\\pi (a)$ and takes $-1$ with probability $\\pi (b)$ .","It follows that $Y_a-Y_b=\\underbrace{X+\\cdots +X}_{n}$ .","By the Central Limit Theorem, $Y_a-Y_b$ converges to a normal distribution with mean $n\\cdot E(X)$ and variance ${n\\cdot \\text{Var}(X)}$ .","We are interested in usually how large is $Y_a-Y_b$ .","Not surprisingly, the answer depends on the distribution $\\pi $ .","If $\\pi (a)=\\pi (b)=1/2$ , then the mean of $Y_a-Y_b$ is zero, and the probability that it is a few standard deviations away from the mean is small.","For example, the probability that its absolute value is larger than $4\\sqrt{n\\cdot \\text{Var}(X)}$ is less than $0.01$ , which means that with $99\\%$ probability the solution of UCO is no more than $4\\sqrt{n\\cdot \\text{Var}(X)}$ .","On the other hand, if $\\pi (a)> \\pi (b)$ , then the mean of $Y_a-Y_b$ is $n(\\pi (a)-\\pi (b))$ , which means that with high probability the solution of UCO is very close to $n(\\pi (a)-\\pi (b))=\\Theta (n)$ .","The idea behind the argument in Example REF can be easily extended to positional scoring rules [34].","However, we do not believe that it can be extended to generalized scoring rules, even for the case of manipulation, for the following two reasons.","First, for generalized scoring rules, the components of the generalized scoring vector do not correspond to the “scores” of alternatives.","Therefore, having two components tied in the total generalized scoring vector does not mean that two alternatives are tied.","Second, the conditions for an alternative to win can be much more complicated than the condition for positional scoring rules, which amounts to requiring that a corresponding component of the total generalized scoring vector is the largest.","Therefore, it is not easy to figure out whether the manipulators can achieve their goal by just knowing the asymptotic relationship between the components of the total generalized scoring vector.","Theorem 1 Let $\\text{GS}(f,g)$ be an integer generalized scoring rule, let $\\pi $ be a distribution over all linear orders, and let $\\Delta $ be a set of vote operations.","Suppose we fix the number of alternatives, generate $n$ votes i.i.d.","according to $\\pi $ , and let $P_n$ denote the preference-profile.","Then, for any alternative $c$ , $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\footnote {When \\text{\\sc VO}=\\text{\\sc CWVO}, we let \\text{\\sc VO}(P_n,c) denote \\text{\\sc CWVO}(P_n).","}\\rbrace $ , and any $\\epsilon >0$ , there exists $\\beta ^*>1$ such that as $n$ goes to infinity, the total probability for the following four events sum up to more than $1-\\epsilon $ : (1) $\\text{\\sc VO}(P_n,c)=0$ , (2) $\\frac{1}{\\beta ^*}\\sqrt{n}<\\text{\\sc VO}(P_n,c)<\\beta ^* \\sqrt{n}$ , (3) $\\frac{1}{\\beta ^*} n<\\text{\\sc VO}(P_n,c)<\\beta ^* n$ , and (4) $\\text{\\sc VO}(P_n,c)=\\infty $ .","Proof of Theorem REF : Let $f(P_\\pi )=\\sum _{V\\in L(\\mathcal {C})}\\pi (V)\\cdot f(V)$ , and $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ .","We first prove the theorem for $\\text{\\sc CVO}$ , and then show how to adjust the proof for $\\text{\\sc DVO}$ and $\\text{\\sc CWVO}$ .","The theorem is proved in the following two steps.","Step 1: we show that as $n$ goes to infinity, with probability that goes to one we have the following: in a randomly generated $P_n$ , the difference between any pair of components in $f(P_n)$ is either $\\Theta (\\sqrt{n})$ or $\\Theta (n)$ .","Step 2: we apply sensitivity analyses to ILPs that are similar to the ILP given in Section to prove that for any such preference-profile and any $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\rbrace $ , $\\text{\\sc VO}(P_n,c)$ is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .","The idea behind Step 2 is, for any preference-profile $P_n$ , if the difference between a pair of components in $f(P_n)$ is $\\Theta (\\sqrt{n})$ , then we consider this pair of components (not alternatives) to be “almost tied”; if the difference is $\\Theta (n)$ , then we consider them to be “far away”.","Take $\\text{\\sc CVO}$ as an example, we can easily identify the cases where $\\text{\\sc CVO}(P_n,c)$ is either 0 (when $\\text{GS}(f,g)=c$ ) or $\\infty $ (by Lemma REF ).","Then, we will first try to break these “almost tied” pairs by using LPs that are similar to LP$_\\trianglerighteq $ introduced in Section , and show that if there exists an integer solution $\\vec{v}$ , then the objective value $\\Vert \\vec{v}\\Vert _1$ is $\\Theta (\\sqrt{n})$ .","Otherwise, we have to change the orders between some “far away” pairs by using LP$_\\trianglerighteq $ 's, and show that if there exists an integer solution to some LP$_\\trianglerighteq $ with $g(\\trianglerighteq )=c$ , then the objective value is $\\Theta (n)$ .","Formally, given $n\\in \\mathbb {N}$ and $\\beta >1$ , let ${\\cal P}_\\beta $ denote the set of all $n$ -vote preference-profiles $P$ that satisfy the following two conditions (we recall that $f(P_\\pi )=\\sum _{V\\in L(\\mathcal {C})}\\pi (V)\\cdot f(V)$ ): for any pair $i,j\\le K$ , 1. if $[f(P_\\pi )]_i=[f(P_\\pi )]_j$ then $\\frac{1}{\\beta }\\sqrt{n}<|[f(P)]_i-[f(P)]_j|<\\beta \\sqrt{n}$ ; 2. if $[f(P_\\pi )]_i\\ne [f(P_\\pi )]_j$ then $\\frac{1}{\\beta } n<|[f(P)]_i-[f(P)]_j|<\\beta n$ .","The following lemma was proved in [47], which follows after the Central Limit Theorem.","Lemma 2 For any $\\epsilon >0$ , there exists $\\beta $ such that $\\lim _{n\\rightarrow \\infty } \\text{Pr}\\left(P_n\\in {\\cal P}_\\beta \\right)>1-\\epsilon $ .","For any given $\\epsilon $ , in the rest of the proof we fix $\\beta $ to be a constant guaranteed by Lemma REF .","The next lemma (whose proof is deferred to the appendix) will be frequently used in the rest of the proof.","Lemma 3 Fix an integer matrix $\\bf A$ .","There exists a constant $\\beta _{\\bf A}$ that only depends on $\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .","$\\min \\Vert \\vec{x}\\Vert _1, \\text{ s.t.","}{\\bf A}\\cdot \\vec{x}\\ge \\vec{b}$ To prove that with high probability $\\text{\\sc CVO}(P_n,c)$ is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ , we introduce the following notation.","A preorder $\\trianglerighteq ^{\\prime }$ is a refinement of another preorder $\\trianglerighteq $ , if $\\vartriangleright ^{\\prime }$ extends $\\vartriangleright $ .","That is, $\\vartriangleright \\subseteq \\vartriangleright ^{\\prime }$ .","We note that $\\trianglerighteq $ is a refinement of itself.","Let $\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq $ denote the strict orders that are in $\\vartriangleright ^{\\prime }$ but not in $\\vartriangleright $ .","That is, $(o_i,o_j)\\in (\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq )$ if and only if $o_i\\vartriangleright ^{\\prime } o_j$ and $o_i=_{\\trianglerighteq }o_j$ .","We define the following LP that is similar to LP$_\\trianglerighteq $ defined in Section , which will be used to check whether there is a way to break “almost tied” pairs of components to make $c$ win.","For any preorder $\\trianglerighteq $ and any of its refinement $\\trianglerighteq ^{\\prime }$ , we define LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ as follows.","$\\begin{tabular}{r@{\\hspace{28.45274pt}}rl}& \\min &\\Vert \\vec{v}\\Vert _1\\\\s.t.","&\\forall o_i=_{\\trianglerighteq ^{\\prime }} o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}= [f(P)]_j-[f(P)]_i\\\\&\\forall (o_i, o_j)\\in (\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq ): & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}\\ge [f(P)]_j-[f(P)]_i+1\\\\&\\forall i:& v_i\\ge 0\\end{tabular}&&(\\text{LP}_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq })$ LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ is defined with a little abuse of notation because some of its constraints depend on $\\trianglerighteq $ (not only the pairwise comparisons in $(\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq )$ ).","This will not cause confusion because we will always indicate $\\trianglerighteq $ in the subscript.","We note that there is a constraint in LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ for each pair of components $o_i,o_j$ with $o_i=_{\\trianglerighteq }o_j$ .","Therefore, LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ is used to find a solution that breaks ties in $\\trianglerighteq $ .","It follows that LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ has an integer solution $\\vec{v}$ if and only if the strategic individual can make the order between any pairs of $o_i,o_j$ with $o_i=_{\\trianglerighteq }o_j$ to be the one in $\\trianglerighteq ^{\\prime }$ by applying the $i$ -th operation $v_i$ times, and the total number of vote operations is $\\Vert \\vec{v}\\Vert _1$ .","The following two claims identify the preference-profiles in ${\\cal P}_\\beta $ for which $\\text{\\sc CVO}$ is $\\Theta (\\sqrt{n})$ and $\\Theta (n)$ , respectively, whose proofs are deferred to the appendix.","Claim 1 There exists $N\\in \\mathbb {N}$ and $\\beta ^{\\prime }>1$ such that for any $n\\ge N$ , any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that $g(\\trianglerighteq ^*)=c$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .","Claim 2 There exists $\\beta ^{\\prime }>1$ such that for any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, and (3) there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_{\\trianglerighteq }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .","Lastly, for any $P\\in {\\cal P}_\\beta $ such that $\\text{GS}(f,g)(P)\\ne c$ , the only case not covered by Claim REF and Claim REF is that there does not exist $\\trianglerighteq $ with $\\text{GS}(f,g)(\\trianglerighteq )=c$ such that LP$_\\trianglerighteq $ has an integer solution.","It follows from Lemma REF that in this case $\\text{\\sc CVO}(P,c)=\\infty $ .","We note that $\\beta ^{\\prime }$ in Claim REF and Claim REF does not depend on $n$ .","Let $\\beta ^*$ be an arbitrary number that is larger than the two $\\beta ^{\\prime }$ s. This proves the theorem for $\\text{\\sc CVO}$ .","For $\\text{\\sc DVO}$ , we only need to change $g(\\trianglerighteq ^*)=c$ to $g(\\trianglerighteq ^*)\\ne c$ in Claim REF , and change $g(\\trianglerighteq )=c$ to $g(\\trianglerighteq )\\ne c$ in Claim REF .","For $\\text{\\sc CWVO}$ , $\\text{\\sc CWVO}(P)$ is never 0 and we only need to change $g(\\trianglerighteq ^*)=c$ to $g(\\trianglerighteq ^*)\\ne \\text{GS}(f,g)(P)$ in Claim REF , and change $g(\\trianglerighteq )=c$ to $g(\\trianglerighteq ) \\ne \\text{GS}(f,g)(P)$ in Claim REF .","$\\blacksquare $ More comments on the non-triviality of the proof.","Lemma REF has been proved in [47], whose intuition is quite straightforward and naturally corresponds to a random walk in multidimensional space.","However, we did not find an obvious connection between random walk theory and the observation made in Theorem REF .","We believe that it is unlikely that an obvious connection exists.","One evidence is that the observation made in Theorem REF does not hold for some voting rules.","For example, consider the $\\gamma $ -majority rule defined in Example REF .","It is not hard to see that as $n$ goes to infinity, with probability that goes to 1 we have $\\text{\\sc CVO}(P_n,b)=\\text{\\sc DVO}(P_n,a)=\\text{\\sc CWVO}(P_n)=n^{\\gamma }/2$ , which is not any of the four cases described in Theorem REF if $\\frac{1}{2}<\\gamma <1$ .","(This implies that for any $\\frac{1}{2}<\\gamma <1$ , $\\gamma $ -majority is not a generalized scoring rule, which we already know because they do not satisfy homogeneity.)","Therefore, the proof of Theorem REF should involve analyses on the specific structure of GSRs.","The main difficulty in proving Theorem REF is, for generalized scoring rules we have to handle the cases where some components of the total generalized scoring vector are equivalent.","This only happens with negligible probability for the randomly generated preference-profile $P_n$ , but it is not clear how often the strategic individual can make some components equivalent in order to achieve her goal.","This is the main reason for us to convert the vote manipulation problem to multiple ILPs and apply Lemma REF to analyze them."],["Applications of the Main Theorem","In this section we show how to apply Theorem REF to some well-studied types of strategic behavior, including constructive and destructive unweighted coalitional optimization, bribery and control, and margin of victory and minimum manipulation coalition size.","In the sequel, we will use each subsection to define these problems and describe how they fit in our vote operation framework, and how Theorem REF applies.","In the end of the section we present a unified corollary for all these types of strategic behavior."],["Unweighted Coalitional Optimization","Definition 4 In a constructive (respectively, destructive) unweighted coalitional optimization (UCO) problem, we are given a voting rule $r$ , a preference-profile $P^{NM}$ of the non-manipulators, and a (dis)favored alternative $c\\in \\mathcal {C}$ .","We are asked to compute the smallest number of manipulators who can cast votes $P^{M}$ such that $c=r(P^{NM}\\cup P^{M})$ (respectively, $c\\ne r(P^{NM}\\cup P^{M})$ ).","To see how UCO fits in the vote operation model, we view the group of manipulators as the strategic individual, and each vote cast by a manipulator is a vote operation.","Therefore, the set of operations is exactly the set of all generalized scoring vectors $\\lbrace f(V): V\\in L(\\mathcal {C})\\rbrace $ .","To apply Theorem REF , for constructive UCO we let $\\text{\\sc VO}=\\text{\\sc CVO}$ and for destructive UCO we let $\\text{\\sc VO}=\\text{\\sc DVO}$ ."],["Bribery","In this paper we are interested in the optimization variant of the bribery problem [16].","Definition 5 In a constructive (respectively, destructive) opt-bribery problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\in \\mathcal {C}$ .","We are asked to compute the smallest number $k$ such that the strategic individual can change no more than $k$ votes such that $c$ is the winner (respectively, $c$ is not the winner).","To see how opt-bribery falls under the vote operation framework, we view each action of “changing a vote” as a vote operation.","Since the strategic individual can only change existing votes in the preference-profile, we define the set of operations to be the difference between the generalized scoring vectors of all votes and the generalized scoring vectors of votes in the support of $\\pi $ , that is, $\\lbrace f(V)-f(W): V,W\\in L(\\mathcal {C})\\text{ s.t.","}\\pi (W)>0\\rbrace $ .","Then, similarly the constructive variant corresponds to $\\text{\\sc CVO}$ and the destructive variant corresponds to $\\text{\\sc DVO}$ .","In both cases Theorem REF cannot be directly applied, because in the ILPs we did not limit the total number of each type of vote operations that can be used by the strategic individual.","Nevertheless, we can still prove a similar proposition by taking a closer look at the relationship between $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) and opt-bribery as follows: For any preference-profile, the solution to $\\text{\\sc CVO}$ (respectively, $\\text{\\sc DVO}$ ) is a lower bound on the solution to constructive (respectively, destructive) opt-bribery, because in $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ there are no constraints on the number of each type of vote operations.","We have the following four cases.","1.","If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is 0, then the solution to constructive (destructive) opt-bribery is also 0.","2.","If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\Theta (\\sqrt{n})$ , as $n$ become large enough, with probability that goes to 1 each type of votes in the support of $\\pi $ will appear $\\Theta (n)$ , which is $>\\Theta (\\sqrt{n})$ , times in the randomly generated preference-profile, which means that there are enough votes of each type for the strategic individual to change.","3.","If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\Theta (n)$ , then the solution to constructive (destructive) opt-bribery is either $\\Theta (n)$ (when the strategic individual can change all votes to achieve her goal), or $\\infty $ .","4.","If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\infty $ , then the solution to constructive (destructive) opt-bribery is also $\\infty $ .","It follows that the observation made in Theorem REF holds for opt-bribery."],["Margin of Victory (MoV)","Definition 6 Given a voting rule $r$ and a preference-profile $P$ , the margin of victory (MoV) of $P$ is the smallest number $k$ such that the winner can be changed by changing $k$ votes in $P$ .","In the mov problem, we are given $r$ and $P$ , and are asked to compute the margin of victory.","For a given instance $(P,r)$ , mov is equivalent to destructive opt-bribery, where $c=r(P)$ .","However, when the input preference-profiles are generated randomly, the winners in these profiles might be different.","Therefore, the corollary of Theorem REF for opt-bribery does not directly imply a similar corollary for mov.","This relationship is similar to the relationship between $\\text{\\sc DVO}$ and $\\text{\\sc CWVO}$ .","Despite this difference, the formulation of mov in the vote operation framework is very similar to that of opt-bribery: The set of all operations and the argument to apply Theorem REF are the same.","The only difference is that for mov, we obtain the corollary from the $\\text{\\sc CWVO}$ part of Theorem REF , while the corollary for opt-bribery is obtained from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF ."],["Minimum Manipulation Coalition Size (MMCS)","The minimum manipulation coalition size (MMCS) problem is similar to mov, except that in MMCS the winner must be improved for all voters who change their votes [33].","Definition 7 In an MMCS problem, we are given a voting rule $r$ and a preference-profile $P$ .","We are asked to compute the smallest number $k$ such that a coalition of $k$ voters can change their votes to change the winner, and all of them prefer the new winner to $r(P)$ .","Unlike mov, MMCS falls under the vote operation framework in the following dynamic way.","For each preference-profile, suppose $c$ is the current winner.","For each adversarial $d\\ne c$ , we use $\\lbrace f(V)-f(W):V,W\\in L(\\mathcal {C})\\text{ s.t.","}d\\succ _W c\\text{ and } \\pi (W)>0\\rbrace $ as the set of operations.","That is, we only allow voters who prefer $d$ to $c$ to participate in the manipulative coalition.","We also replace each of LP$_{\\trianglerighteq }$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ by multiple LPs, each of which is indexed by a pair of alternatives $(d,c)$ and the constraints are generated by using the corresponding set of operations.","Then, the corollary for MMCS follows after a similar argument to that of $\\text{\\sc CVO}$ in Theorem REF ."],["Control by Adding/Deleting Votes (CAV/CDV)","Definition 8 In a constructive (respectively, destructive) optimal control by adding votes (opt-CAV) problem, we are given a preference-profile $P$ , a (dis)favored alternative $c\\in \\mathcal {C}$ , and a set $N^{\\prime }$ of additional votes.","We are asked to compute the smallest number $k$ such that the strategic individual can add $k$ votes in $N^{\\prime }$ such that $c$ is the winner (respectively, $c$ is not the winner).","For simplicity, we assume that $|N^{\\prime }|=n$ and the votes in $N^{\\prime }$ are drawn i.i.d.","from a distribution $\\pi ^{\\prime }$ .","To show how opt-CAV falls under the vote operation model, we let the set of operations to be the generalized scoring vectors of all votes that are in the support of $\\pi ^{\\prime }$ , that is, $\\lbrace f(V): V\\in L(\\mathcal {C})\\text{ and }\\pi ^{\\prime }(V)>0\\rbrace $ .","Then, the corollary follows from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery.","Definition 9 In a constructive (respectively, destructive) optimal control by deleting votes (opt-CDV) problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\in \\mathcal {C}$ .","We are asked to compute the smallest number $k$ such that the strategic individual can delete $k$ votes in $P$ such that $c$ is the winner (respectively, $c$ is not the winner).","To show how opt-CDV falls under the vote operation framework, we let the set of operations to be the negation of generalized scoring vectors of votes in the support of $\\pi ^{\\prime }$ , that is, $\\lbrace -f(V): V\\in L(\\mathcal {C})\\text{ and }\\pi ^{\\prime }(V)>0\\rbrace $ .","Then, the corollary follows from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery."],["A Unified Corollary","The next corollary of Theorem REF summarizes the results obtained for all types of strategic behavior studied in this section.","Corollary 1 For any integer generalized scoring rule, any distribution $\\pi $ over votes, and any $X\\in $ $\\big ($ {constructive, destructive}$\\times $ {UCO, opt-bribery, opt-CAV, opt-CDV}$\\big )$$\\cup $ {MoV, MMCS}, suppose the input preference-profiles are generated i.i.d.","from $\\pi $ .For CAV, the distribution over the new votes can be generated i.i.d.","from a different distribution $\\pi ^{\\prime }$ .","Then, for any alternative $c$ and any $\\epsilon >0$ , there exists $\\beta ^*>1$ such that the total probability for the solution to $X$ to be one of the following four cases is more than $1-\\epsilon $ as $n$ goes to infinity: (1) 0, (2) between $\\frac{1}{\\beta ^*}\\sqrt{n}$ and $\\beta ^* \\sqrt{n}$ , (3) between $\\frac{1}{\\beta ^*} n$ and $\\beta ^* n$ , and (4) $\\infty $ ."],["Discussions and Future Work","In this paper, we proposed a general framework to study vote operations for generalized scoring rules.","Our main theorem is a characterization for the number of vote operations that are needed to achieve the strategic individual's goal.","We showed that the main theorem can be applied to many types of strategic behavior and many commonly used voting rules, for most of which no similar results were previously known.","We further discuss the generality of our framework in the next two paragraphs.","GSRs vs. integer GSRs.","Though integer GSRs are a subclass of GSRs, we feel that from a computational point of view, focusing integer generalized scoring rules does not sacrifice much generality.","Because the $g$ function only depends on the preorder among components in the total generalized scoring vector, if the $f$ function is scaled up by a constant, then the $g$ function will select the same winner.","Therefore, integer GSRs are equivalent to GSRs where components in the generalized scoring vectors are rational numbers.","When the components are irrational numbers, two computational problems arise.","First, it is not clear how these irrational numbers are represented, and second, it is hard to compare two irrational numbers computationally, thus even harder to compute the preorder of the components of the total generalized scoring vector.","On the other hand, integer GSRs do not have such computational constraints.","In fact, all commonly studied voting rules that are known to be GSRs are integer GSRs.","Therefore, we believe that our main theorem has a wide application (at least can be applied to many commonly studied voting rules).","On the generality of vote operations.","While the framework we proposed covers many types of strategic behavior, some other types of strategic behavior that have been widely studied are not covered by our framework.","These types of strategic behavior can be roughly categorized as follows: (1) controls that changes the set of alternatives, for example, control by adding/deleting alternatives [3] and control by introducing clones of alternatives [44], [13]; and (2) controls that change the procedure of voting, for example control by (runoff) partition of alternatives and control by partition of voters [3], and control by changing the agenda of voting [24].","Building a more general framework that covers more types of strategic behavior and studying their properties are interesting directions for future research.","As we discussed in the introduction, on the positive side, our main theorem suggests that computing the margin of victory is usually not hard, which helps implementing efficient post-election auditing methods.","One promising future direction is to design practical computational techniques for computing the margin of victory for generalized scoring rules, based on the ILP proposed in this paper.","On the negative side, our main theorem suggests that computational complexity might merely be a weak barrier against many types of strategic behavior.","Therefore, we should look for new ways to protect voting, for example introducing randomization [6], [14], [46], [29], [30], using multiple rounds [9], [28], [10], or limiting the strategic individuals' information about other voters [7].","Another interesting research direction is to investigate the phase transition of the probability for a coalition of strategic individuals to achieve their goal by using vote operations, as it was done for manipulation [45], [26]."],["Proofs","Lemma REF Fix an integer matrix $\\bf A$ .","There exists a constant $\\beta _{\\bf A}$ that only depends on $\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .","$\\min \\Vert \\vec{x}\\Vert _1, \\text{ s.t.","}{\\bf A}\\cdot \\vec{x}\\ge \\vec{b}$ Proof of Lemma REF : Let $\\bf A$ be a $m^*\\times n^*$ integer matrix, which includes the constraints $\\vec{x}\\ge \\vec{0}$ .","Suppose LP (REF ) has a (non-negative) integer solution.","We note that $\\vec{0}$ is an optimal integer solution to $\\min \\vec{1}\\cdot (\\vec{x})^{\\prime } \\text{ s.t.","}{\\bf A}\\cdot \\vec{x}\\ge \\vec{0}$ .","Then, it follows from Theorem 5 (ii) in [8] that LP (REF ) has a (non-negative) integer solution $\\vec{z}$ such that $\\Vert \\vec{z}-\\vec{0}\\Vert _\\infty \\le n^*\\cdot M({\\bf A})\\cdot (\\Vert \\vec{b}-\\vec{0}\\Vert _\\infty +2),$ where $M({\\bf A})$ is the maximum of the absolute values of the determinants of the square sub-matrices of $\\bf A$ .","Since $\\bf A$ is fixed, the right hand side becomes a constant, that is, $\\Vert \\vec{z}\\Vert _\\infty =O(\\Vert \\vec{b}\\Vert _\\infty )$ .","Therefore, there exists $\\beta _{\\bf A}$ such that the optimal value in the ILP (REF ) is no more than $\\vec{1}\\cdot (\\vec{z})^{\\prime }\\le n^*\\Vert \\vec{z}\\Vert _\\infty \\le \\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .","$\\blacksquare $ Claim REF There exists $N\\in \\mathbb {N}$ and $\\beta ^{\\prime }>1$ such that for any $n\\ge N$ , any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that $g(\\trianglerighteq ^*)=c$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .","Proof of Claim REF : Let $\\trianglerighteq =\\text{Ord}(f(P))$ .","Because $\\text{GS}(f,g)(P)\\ne c$ , $g(\\trianglerighteq )\\ne c$ .","Therefore, the strategic individual has to change the order of some components in the generalized scoring vector to make $c$ win.","We note that $P\\in {\\cal P}_\\beta $ , which means that the difference between any pair of components of $f(P)$ is more than $\\frac{1}{\\beta } \\sqrt{n}$ .","Let $d_{max}$ denote the maximum difference between any pair of components in generalized score vectors.","That is, $d_{max}=\\max _{t,t^{\\prime },L\\in L(\\mathcal {C})}\\lbrace (f(L))_t-(f(L))_{t^{\\prime }}\\rbrace $ .","In order for $c$ to win, the number of vote operations must be at least $\\frac{1}{\\beta } \\sqrt{n}/d_{max}$ .","Therefore, $\\text{\\sc CVO}(P,c)>\\frac{1}{\\beta d_{max}} \\sqrt{n}$ .","We next show the upper bound.","Because $P\\in {\\cal P}_\\beta $ , for any pair $o_i,o_j$ with $o_i=_{\\trianglerighteq _\\pi }o_j$ , $|[f(P)]_i-[f(P)]_j|<\\beta \\sqrt{n}$ .","Therefore, the right hand side of each (in)equality in LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ is no more than $\\beta \\sqrt{n}$ .","Applying Lemma REF to LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ , we have that there exists a constant $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ that only depends on $\\trianglerighteq ^*$ and $\\trianglerighteq _\\pi $ , and an integer solution $\\vec{v}$ with $\\Vert \\vec{v}\\Vert _1\\le \\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }\\sqrt{n}$ (the $\\bf A$ matrix in Lemma REF is fixed because we fix the number of alternatives $m$ , and the left hand side of each (in)equality in LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ does not depend on $n$ ).","Let $\\beta ^{\\prime }$ be the maximum of $d_{max}\\beta $ and all $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ (since we fix the number of alternatives, there are finite many $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ 's).","Since $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }>\\frac{1}{d_{max}\\beta }$ , $\\beta ^{\\prime }>1$ .","It follows that $\\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ .","We next show that for a sufficiently large $n$ , if the strategic individual applies $\\vec{v}$ , then the order over components of the total scoring vector will become $\\trianglerighteq ^*$ .","That is, $c$ can be made win.","The idea is, LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ ensures that by applying $\\vec{v}$ , ties between the “almost tied” components are broken as in $\\trianglerighteq ^*$ .","Since $\\Vert \\vec{v}\\Vert _1=O(\\sqrt{n})$ , when $n$ is large enough the order between any pair of “far away” components will not be affected.","Formally, let $\\vec{x}=f(P)+\\Delta \\cdot \\vec{v}$ .","That is, $\\vec{x}$ is the total generalized scoring vector after the strategic individual applied $\\vec{v}$ .","Because $\\vec{v}$ is a solution to LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ , for any pair $o_i,o_j$ with $o_i=_{\\trianglerighteq _\\pi }o_j$ , the order between $o_i$ and $o_j$ in $\\trianglerighteq ^*$ is the same as the order between $o_i$ and $o_j$ in $\\text{Ord}(\\vec{x})$ .","Since $\\trianglerighteq ^*$ is an extension of $\\trianglerighteq _\\pi $ , if $o_i\\vartriangleright _\\pi o_j$ , then we must have $o_i\\vartriangleright ^* o_j$ .","Therefore, we only need to check that for any $o_i\\vartriangleright _\\pi o_j$ , we have $[\\vec{x}]_i>[\\vec{x}]_j$ .","Because $P\\in {\\cal P}_\\beta $ , $|[f(P)]_i-[f(P)]_j|>\\frac{1}{\\beta }n$ .","We note that $\\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ , which means that by applying $\\vec{v}$ , the strategic individual can only change the difference between any pair of components by no more than $d_{max} \\beta ^{\\prime }\\sqrt{n}$ .","Let $N=(d_{max} \\beta ^{\\prime }\\beta )^2+1$ .","When $n\\ge N$ , $d_{max} \\beta ^{\\prime }\\sqrt{n}<\\frac{1}{\\beta }n$ , which means that for any $o_i\\vartriangleright _\\pi o_j$ , applying $\\vec{v}$ will not change the order between $o_i$ and $o_j$ in the total generalized scoring vector.","This means that by applying $\\vec{v}$ , the strategic individual can make $c$ win.","Therefore, $ \\text{\\sc CVO}(P,c)\\le \\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ .","It follows that for any $n\\ge N$ , $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .","$\\blacksquare $ Claim REF There exists $\\beta ^{\\prime }>1$ such that for any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, and (3) there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_{\\trianglerighteq }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .","Proof of Claim REF : Let $\\trianglerighteq =\\text{Ord}(f(P))$ .","Because the premises of Claim REF do not hold, the strategic individual has to change the order of some pair of components that are “far away” (that is, the difference between them is $\\Theta (n)$ before the strategic individual applies vote operations) to make $c$ win.","We note that one operation can only change the difference between a pair of components by at most $d_{max}$ .","Therefore, $\\text{\\sc CVO}(P,c)\\ge \\frac{1}{\\beta }n/d_{max}$ .","On the other hand, it follows from Lemma REF and condition (3) in the statement of the claim that $\\text{\\sc CVO}(P,c)<\\infty $ .","The only thing left to show is that there exists $\\beta ^{\\prime }>1$ such that $\\text{\\sc CVO}(P,c)<\\beta ^{\\prime }n$ for all $P\\in {\\cal P}_\\beta $ and for all $n$ .","Because $P\\in {\\cal P}_\\beta $ , for any pair $o_i,o_j$ , $|[f(P)]_i-[f(P)]_j|<\\beta n$ .","Applying Lemma REF to LP$_{\\trianglerighteq }$ and condition (3) in the statement of the claim, we have that for any $\\trianglerighteq $ , there exits $\\beta _{\\trianglerighteq }$ that only depends on $\\trianglerighteq $ , and an optimal integer solution $\\vec{v}$ such that $\\Vert \\vec{v}\\Vert _1\\le \\beta _\\trianglerighteq n$ .","Let $\\beta ^{\\prime }$ be the maximum of $d_{max} \\beta $ and all $\\beta _{\\trianglerighteq }$ (again, there are finite number of $\\beta _{\\trianglerighteq }$ 's).","It follows that $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .","$\\blacksquare $"],["Discussion: How often the solution is 0 or $\\infty $ ?","One important question is: how large is the probability that the solution to problems studied in this section is 0 or $\\infty $ ?","Not surprisingly the answer depends on both the voting rule and the type of vote operations.","The probability can be large for some voting rules.","For example, for any voting rule that always selects a given alternative $d$ as the winner, the solution to $\\text{\\sc CVO}$ (respectively, $\\text{\\sc DVO}$ ) is always 0 (respectively $\\infty $ ) for $c=d$ and is always $\\infty $ (respectively 0) for $c\\ne d$ .","However, for common voting rules the alternatives are treated almost equally (except for cases with ties).","Therefore, we may expect that for a preference-profile whose votes are generated i.i.d., each alternative has almost the same probability of being selected as the winner.","This is indeed the case in all commonly used voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.","Therefore, for these voting rules, when the votes are drawn i.i.d.","uniformly at random, the probability for $\\text{\\sc CVO}$ is approximately $\\frac{1}{m}$ and the probability for $\\text{\\sc DVO}$ is approximately $\\frac{m-1}{m}$ .","For $\\text{\\sc CWVO}$ , the answer is never 0 because changing 0 votes cannot change the winner.","We would also expect for common voting rules, for some types of strategic behavior studied in this section, with low probability the solution is $\\infty $ .","For UCO, the strategic individual can introduce many (but finitely many) votes such that the non-manipulators' votes are negligible.","For opt-bribery and mov, the strategic individual can change all votes to achieve her goal.","For MMCS, CAV, and CDV, it is not clear how large such probability is.","The following table summarizes folklore results for common voting rules when votes are drawn i.i.d.","uniformly at random.","Table: Probability for solutions to some problems to be 0 or ∞\\infty for common voting rules."]],"string":"[\n [\n \"How Many Vote Operations Are Needed to Manipulate A Voting System?\"\n ],\n [\n \"Abstract In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations.\",\n \"We prove the following theorem: if we fix the number of alternatives, generate $n$ votes i.i.d.\",\n \"according to a distribution $\\\\pi$, and let $n$ go to infinity, then for any $\\\\epsilon >0$, with probability at least $1-\\\\epsilon$, the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) $\\\\Theta(\\\\sqrt n)$, (3) $\\\\Theta(n)$, and (4) $\\\\infty$.\",\n \"This theorem holds for any set of vote operations, any individual vote distribution $\\\\pi$, and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.\",\n \"We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.\",\n \"Therefore, our main theorem naturally applies to these problems.\"\n ],\n [\n \"Introduction\",\n \"Voting is a popular method used to aggregate voters' preferences to make a joint decision.\",\n \"Recently, voting has been used in many fields of artificial intelligence, for example in multi-agent systems [15], recommender systems [21], [32], and web-search engines [12].\",\n \"One of the most desired properties for voting rules is strategy-proofness, that is, no voter has incentive to misreport her preferences to obtain a better outcome of the election.\",\n \"Unfortunately, strategy-proofness is not compatible with some other desired properties, due to the celebrated Gibbard-Satterthwaite theorem [22], [36], which states that when there are at least three alternatives, no strategy-proof voting rule satisfies the following two natural properties: non-imposition (every alternative can win) and non-dictatorship (no voter is a dictator, whose top ranked alternative is always selected to be the winner).\",\n \"Even though manipulation is inevitable, researchers have set out to investigate whether computational complexity can serve as a barrier against various types of strategic behavior, including manipulation.\",\n \"The idea is, if we can prove that it is computationally too costly for a strategic individual to find a beneficial operation, she may give up doing so.\",\n \"Initiated by Bartholdi, Tovey, and Trick [2], a fair amount of work has been done to characterize the computational complexity of various types of strategic behaviorSee [19], [17], [35] for recent surveys., including the following.\",\n \"$\\\\bullet $ Manipulation: a voter or a coalition of voters cast false vote(s) to change the winner (and the new winner is more preferred).\",\n \"$\\\\bullet $ Bribery: a strategic individual changes some votes by bribing the voters to make the winner preferable to her [16].\",\n \"The bribery problem is closely related to the problem of computing the margin of victory [5], [25], [47].\",\n \"$\\\\bullet $ Control: a strategic individual adds or deletes votes to make the winner more preferable to her [3].\",\n \"Most previous results studying “using computational complexity as a barrier against strategic behavior” conduct worst-case analyses of computational complexity.\",\n \"Recently, an increasing number of results show that manipulation, as a particular type of strategic behavior, is typically not hard to compute.\",\n \"One direction, mainly pursued in the theoretical computer science community, is to obtain a quantitative version of the Gibbard-Satterthwaite theorem, showing that for any given voting rule that is “far” enough from any dictatorships, an instance of manipulation can be found easily with high probability.\",\n \"This line of research was initiated by Friedgut, Kalai, and Nisan [20], where they proved the theorem for 3 alternatives and neutral voting rules.\",\n \"The theorem was extended to an arbitrary number of alternatives by Isaksson, Kindler, and Mossel [23], and finally, the neutrality constraint was removed by Mossel and Racz [27].\",\n \"Other extensions include Dobzinski and Procaccia [11] and Xia and Conitzer [49].\",\n \"Another line of research is to characterize the “frequency of manipulability”, defined as the probability for a randomly generated preference-profile to be manipulable by a group of manipulators, where the non-manipulators' votes are generated i.i.d.\",\n \"according to some distribution (for example, the uniform distribution over all possible types of preferences).\",\n \"Peleg [31], Baharad and Neeman [1], and Slinko [37], [38] studied the asymptotic frequency of manipulability for positional scoring rules when the non-manipulators' votes are drawn i.i.d.\",\n \"uniformly at random.\",\n \"Procaccia and Rosenschein [34] showed that for positional scoring rules, when the non-manipulators’ votes are drawn i.i.d.\",\n \"according to some distribution that satisfies some natural conditions, if the number of manipulators is $o(\\\\sqrt{n})$ , where $n$ is the number of non-manipulators, then the probability that the manipulators can succeed goes to 0 as $n$ goes to infinity; if the number of manipulator is $\\\\omega (n)$ , then the probability that the manipulators can succeed goes to 1.\",\n \"This dichotomy theorem was generalized to a class of voting rules called generalized scoring rules (GSRs) by Xia and Conitzer [48].\",\n \"A GSR is defined by two functions $f,g$ , where $f$ maps each vote to a vector in multidimensional space, called a generalized scoring vector (the dimensionality of the space is not necessarily the same as the number of alternatives).\",\n \"Given a profile $P$ , let total generalized scoring vector be the sum of $f(V)$ for all votes $V$ in $P$ .\",\n \"Then, $g$ selects the winner based on the total preorder of the components of the total generalized scoring vector.\",\n \"We call a GSR an integer GSR, if the components of all generalized scoring vectors are integers.\",\n \"(Integer) GSRs are a general class of voting rules.\",\n \"One evidence is that many commonly studied voting rules are integer GSRs, including (but not limited to) approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.The definition of these commonly studied voting rules can be found in, e.g., [48].\",\n \"In this paper, we define GSRs as voting rules where the inputs are profiles of linear orders.\",\n \"GSRs can be easily generalized to include other types of voting rules where the inputs are not necessarily linear orders, for example, approval voting.\",\n \"As another evidence, GSRs admit a natural axiomatic characterization [50], which also suggests that GSRs are equivalent to hyperplane rules [26].\",\n \"The knife-edge case of $\\\\Theta (\\\\sqrt{n})$ was studied experimentally for STV and veto in [45], showing that the probability for the manipulators to succeed has a smooth phase transition.\",\n \"More recently, [26] extends the dichotomy theorem to all anonymous voting rules for distributions that satisfy some mild conditions, and theoretically proved that for all generalized scoring rules, for the knife-edge case, the probability that the manipulators can achieve their goal is continuously differentiable, which suggests a smooth phase transition.\",\n \"While most of the aforementioned results are about manipulation, in this paper, we focus the optimization variants of various types of strategic behavior, including manipulation, bribery, and control.\",\n \"Despite being natural, to the best of our knowledge, such optimization variants have been investigated for only three types of strategic behavior.\",\n \"The first is the unweighted coalitional optimization (UCO) problem, where we are asked to compute the minimum number of manipulators who can make a given alternative win [53].\",\n \"Approximation algorithms have been proposed for UCO for specific voting systems, including positional scoring rules and maximin [53], [51], [52].\",\n \"The second is the margin of victory problem, where we are asked to compute the smallest number of voters who can change their votes to change the winner [25], [5], [47].\",\n \"The third is the minimum manipulation coalition size problem, which is similar to the margin of victory, except that all voters who change their votes must prefer the new winner to the old winner [33].\",\n \"In this paper, we introduce a unified framework to study a class of strategic behavior for generalized scoring rules, which we call vote operations.\",\n \"In our framework, a strategic individual seeks to change the winner by applying some operations, which are modeled as vectors in a multidimensional space.\",\n \"We study three goals of the strategic individual: (1) making a favored alternative win, called constructive vote operation ($\\\\text{\\\\sc CVO}$), (2) making a disfavored alternative lose, called destructive vote operation ($\\\\text{\\\\sc DVO}$ ), and (3) change the winner of the election, called change-winner vote operation ($\\\\text{\\\\sc CWVO}$ ).\",\n \"The framework will be formally defined in Section .\",\n \"This is our main conceptual contribution.\",\n \"Our main technical contribution is the following asymptotical characterization of the minimum number of operations that are needed for the strategic individual to achieve her goal.\",\n \"Theorem REF (informally put) Fix the number of alternatives and the set of vote operations.\",\n \"For any integer generalized scoring rule and any distribution $\\\\pi $ over votes, we generate $n$ votes i.i.d.\",\n \"according to $\\\\pi $ and let $n$ go to infinity.\",\n \"Then, for any $\\\\text{\\\\sc VO}\\\\in \\\\lbrace \\\\text{\\\\sc CVO},\\\\text{\\\\sc DVO},\\\\text{\\\\sc CWVO}\\\\rbrace $ and any $\\\\epsilon >0$ , with probability at least $1-\\\\epsilon $ , the minimum number of operations that are necessary for the strategic individual to achieve $\\\\text{\\\\sc VO}$ falls into one of the following four categories: (1) 0, (2) $\\\\Theta (\\\\sqrt{n})$ , (3) $\\\\Theta (n)$ , and (4) $\\\\infty $ .\",\n \"More informally, Theorem REF states that in large elections, to achieve a specific goal (one of the three goals described above), with probability that can be infinitely close to 1 the strategic individual needs to either do nothing (the goal is already achieved), apply $\\\\Theta (\\\\sqrt{n})$ vote operations, apply $\\\\Theta (n)$ vote operations, or the goal cannot be achieve no matter how many vote operations are applied.\",\n \"This characterization holds for any integer generalized scoring rule, any set of vote operations, and any distribution $\\\\pi $ for individual votes.\",\n \"The proof of Theorem REF is based on the Central Limit Theorem and on sensitivity analyses for the integer linear programmings (ILPs).\",\n \"It works as follows.\",\n \"We will formulate each of the strategic individual's three goals as a set of ILPs in Section .\",\n \"By applying Central Limit Theorem, we show that with probability that goes to 1 the random generated preference-profile satisfies a desired property.\",\n \"Then, for each such preference-profile we apply the sensitivity analyses in [8] to show that with high probability the number of operations that are necessary is either 0, $\\\\Theta (\\\\sqrt{n})$ , $\\\\Theta (n)$ , or $\\\\infty $ .\",\n \"While Theorem REF may look quite abstract, we show later in the paper that many well-studied types of strategic behavior fall under our vote operation framework, including constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.We defer the definition of these types of strategic behavior to Section .\",\n \"Therefore, we naturally obtain corollaries of Theorem REF for these types of strategic behavior.\",\n \"The theorem also applies to other types of strategic behavior, for example the mixture of any types mentioned above, which is known as multimode control attacks [18].\"\n ],\n [\n \"Related Work and Discussion\",\n \"Our main theorem applies to any integer generalized scoring rule for destructive manipulation, constructive and destructive bribery and control by adding/deleting votes.\",\n \"To the best of our knowledge, no similar results were obtained even for specific voting rules for these types of strategic behavior.\",\n \"Three previous papers obtained similar results for manipulation, margin of victory, and minimum manipulation coalition size.\",\n \"The applications of our main theorem to these types of strategic behavior are slightly weaker, but we stress that our main theorem is significantly more general.\",\n \"Three related papers.\",\n \"First, the dichotomy theorem in [48] implies that, (informally) when the votes are drawn i.i.d.\",\n \"from some distribution, with probability that goes to 1 the solution to constructive and destructive UCO is either 0 or approximately $\\\\sqrt{n}$ for some favored alternatives.\",\n \"However, this result only works for the UCO problem and some distributions over the votes.\",\n \"Second, it was proved in [47] that for any non-redundant generalized scoring rules that satisfy a continuity condition, when the votes are drawn i.i.d.\",\n \"and we let the number of voters $n$ go to infinity, either with probability that can be arbitrarily close to 1 the margin of victory is $\\\\Theta (\\\\sqrt{n})$ , or with probability that can be arbitrarily close to 1 the margin of victory is $\\\\Theta (n)$ .\",\n \"It is easy to show that for non-redundant voting rules, the margin of victory is never 0 or $\\\\infty $ .\",\n \"Though it was shown in [47] that many commonly studied voting rules are GSRs that satisfy such continuity condition, in general it is not clear how restrictive the continuity condition is.\",\n \"More importantly, the result only works for the margin of victory problem.\",\n \"Third, in [33], the authors investigated the distribution over the minimum manipulation coalition size for positional scoring rules when the votes are drawn i.i.d.\",\n \"from the uniform distribution.\",\n \"However, it is not clear how their techniques can be extended beyond the uniform distributions and positional scoring rules, which are a very special case of generalized scoring rules.\",\n \"Moreover, the paper only focused on the minimum manipulation coalition size problem.\",\n \"Our results has both negative and positive implications.\",\n \"On the negative side, our results provide yet another evidence that computational complexity is not a strong barrier against strategic behavior, because the strategic individual now has some information about the number of operations that are needed, without spending any computational cost or even without looking at the input instance.\",\n \"Although the estimation of our theorem may not be very precise (because we do not know which of the four cases a given instance belongs to), such estimation may be explored to designing effective algorithms that facilitate strategic behavior.\",\n \"On the positive side, this easiness of computation is not always a bad thing: sometimes we want to do such computation in order to test how robust a given preference-profile is.\",\n \"For example, computing the margin of victory is an important component in designing novel risk-limiting audit methods [25], [5], [47], [39], [40], [42], [41], [43].\",\n \"While being quite general, our results have two main limitations.\",\n \"First, they are asymptotical results, where we fix the number of alternatives and let the number of voters go to infinity.\",\n \"We do not know the convergence rate, or equivalently, how many voters are needed for the observation to hold.\",\n \"In fact, this is a standard setting in previous work, especially in the studies of “frequency of manipulability”.\",\n \"We feel that our results work well in settings where there are small number of alternatives and large number of voters, e.g., political elections.\",\n \"Second, our results show that with high probability one of the four cases holds (0, $\\\\Theta (\\\\sqrt{n})$ , $\\\\Theta (n)$ , $\\\\infty $ ), but we do not know which case holds more often.\",\n \"We will briefly discuss this issue in .\",\n \"It is possible to refine our study for specific voting rules and specific types of strategic behavior that fall under our framework, which we leave as future work.\"\n ],\n [\n \"Structure of the Paper\",\n \"After recalling basic definitions of voting and generalized scoring rules, we present the framework in Section , where we define vote operations as well as the strategic individual's objectives.\",\n \"Then, in Section we formulate the optimization problem for the strategic individual as a set of integer linear programmings (ILPs).\",\n \"The main theorem will be presented in Section .\",\n \"To show the wide application of the framework and the main theorem, we show that many commonly studied types of strategic behavior can be modeled as vote operations for generalized scoring rules in Section , which means that our main theorem naturally applies to these cases.\",\n \"We add some discussions and point out some future directions in Section and .\",\n \"Some proofs are relegated to .\"\n ],\n [\n \"Preliminaries\",\n \"Let $\\\\mathcal {C}$ denote the set of alternatives (or candidates), $|\\\\mathcal {C}|=m$ .\",\n \"We assume strict preference orders.\",\n \"That is, a vote is a linear order over $\\\\mathcal {C}$ .\",\n \"The set of all linear orders over $\\\\mathcal {C}$ is denoted by $L(\\\\mathcal {C})$ .\",\n \"A preference-profile $P$ is a collection of $n$ votes for some $n\\\\in \\\\mathbb {N}$ , that is, $P\\\\in L(\\\\mathcal {C})^n$ .\",\n \"Let $L(\\\\mathcal {C})^*=\\\\bigcup _{n=1}^\\\\infty L(\\\\mathcal {C})^n$ .\",\n \"A voting rule $r$ is a mapping that assigns to each preference-profile a single winner.\",\n \"That is, $r: L(\\\\mathcal {C})^*\\\\rightarrow \\\\mathcal {C}$ .\",\n \"Throughout the paper, we let $n$ denote the number of votes and let $m$ denote the number of alternatives.\",\n \"We now recall the definition of generalized scoring rules (GSRs) [48].\",\n \"For any $K\\\\in \\\\mathbb {N}$ , let ${\\\\mathcal {O}}_K=\\\\lbrace o_1,\\\\ldots ,o_K\\\\rbrace $ .\",\n \"A total preorder (preorder for short) is a reflexive, transitive, and total relation.\",\n \"Let $\\\\text{Pre}({\\\\mathcal {O}}_K)$ denote the set of all preorders over ${\\\\mathcal {O}}_K$ .\",\n \"For any $\\\\vec{p}\\\\in {\\\\mathbb {R}}^K$ , we let $\\\\text{Ord}(\\\\vec{p})$ denote the preorder $\\\\trianglerighteq $ over ${\\\\mathcal {O}}_K$ where $o_{k_1}\\\\trianglerighteq o_{k_2}$ if and only if $p_{k_1}\\\\ge p_{k_2}$ .\",\n \"That is, the $k_1$ -th component of $\\\\vec{p}$ is as large as the $k_2$ -th component of $\\\\vec{p}$ .\",\n \"For any preorder $\\\\trianglerighteq $ , if $o\\\\trianglerighteq o^{\\\\prime }$ and $o^{\\\\prime }\\\\trianglerighteq o$ , then we write $o=_\\\\trianglerighteq o^{\\\\prime }$ .\",\n \"Each preorder $\\\\trianglerighteq $ naturally induces a (partial) strict order $\\\\vartriangleright $ , where $o\\\\vartriangleright o^{\\\\prime }$ if and only if $o\\\\trianglerighteq o^{\\\\prime }$ and $o^{\\\\prime }\\\\ntrianglerighteq o$ .\",\n \"Definition 1 Let $K\\\\in \\\\mathbb {N}$ , $f:L(\\\\mathcal {C})\\\\rightarrow {\\\\mathbb {R}}^K$ and $g:\\\\text{Pre}({\\\\mathcal {O}}_K)\\\\rightarrow \\\\mathcal {C}$ .\",\n \"$f$ and $g$ determine a generalized scoring rule (GSR) $GS(f,g)$ as follows.\",\n \"For any preference-profile $P=(V_1,\\\\ldots , V_n)\\\\in L(\\\\mathcal {C})^n$ , abusing the notation we let $f(P)=\\\\sum _{i=1}^nf(V_i)$ , and let $GS(f,g)(P)=g(\\\\text{Ord}(f(P)))$ .\",\n \"We say that $GS(f,g)$ is of order $K$.\",\n \"If $f(V)\\\\in {\\\\mathbb {Z}}^K$ holds for all $V\\\\in L(\\\\mathcal {C})$ , then we call $\\\\text{GS}(f,g)$ an integer GSR.\",\n \"For any $V\\\\in L(\\\\mathcal {C})$ , $f(V)$ is called a generalized scoring vector, $f(P)$ is called a total generalized scoring vector, and $\\\\text{Ord}(f(P))$ is called the induced preorder of $P$ .\",\n \"The class of integer GSRs is equivalent to the class of rational GSRs, where the components of each generalized scoring vector is in $\\\\mathbb {Q}$ , because for any $l>0$ , $\\\\text{GS}(f,g)=\\\\text{GS}(l\\\\cdot f,g)$ .\",\n \"Almost all commonly studied voting rules are generalized scoring rules, including (but not limited to) approval voting, Bucklin, Copeland, maximin, plurality with runoff, ranked pairs, and multi-stage voting rules that use GSRs in each stage to eliminate alternatives (including Nanson's and Baldwin's rule).\",\n \"As an example, we recall the proof from [48] that the single transferable vote (STV) rule (a.k.a.\",\n \"instant-runoff voting or alternative vote for single-winner elections) is an integer generalized scoring rule.\",\n \"Example 1 STV selects the winner in $m$ rounds.\",\n \"In each round, the alternative that gets the lowest plurality score (the number of times that the alternative is ranked in the top position) drops out, and is removed from all of the votes (so that votes for this alternative transfer to another alternative in the next round).\",\n \"Ties are broken alphabetically.\",\n \"The last-remaining alternative is the winner.\",\n \"To see that STV is an integer GSR, we will use generalized scoring vectors with many components.\",\n \"For every proper subset $S$ of alternatives, for every alternative $c$ outside of $S$ , there is a component in the vector that contains the number of times that $c$ is ranked first if all alternatives in $S$ are removed.\",\n \"Let $\\\\bullet $ $K_{STV}=\\\\sum _{i=0}^{m-1}{m \\\\atopwithdelims ()i} (m-i)$ ; the components are indexed by $(S,j)$ , where $S$ is a proper subset of $\\\\mathcal {C}$ and $j\\\\le m,c_j \\\\notin S$ .\",\n \"$\\\\bullet $ $(f_{STV}(V))_{(S,j)}=1$ , if after removing $S$ from $V$ , $c_j$ is at the top; otherwise, let $(f_{STV}(V))_{(S,j)}=0$ .\",\n \"$\\\\bullet $ $g_{STV}$ selects the winners based on $\\\\trianglerighteq $ as follows.\",\n \"In the first round, let $j_1$ be the index such that $o_{(\\\\emptyset ,j_1)}$ is ranked the lowest in $\\\\trianglerighteq $ among all $o_{(\\\\emptyset ,j)}$ (if there are multiple such $j$ 's, then we break ties alphabetically to select the least-preferred one).\",\n \"Let $S_1=\\\\lbrace c_{j_1}\\\\rbrace $ .\",\n \"Then, for any $2\\\\le i\\\\le m-1$ , define $S_i$ recursively as follows: $S_i=S_{i-1}\\\\cup \\\\lbrace c_{j_i}\\\\rbrace $ , where $j_i$ is the index such that $o_{(S_{i-1},j_i)}$ is ranked the lowest in $\\\\trianglerighteq $ among all $o_{(S_{i-1},j)}$ ; finally, the winner is the unique alternative in $(\\\\mathcal {C}\\\\setminus S_{m-1})$ .\",\n \"GSRs admit a natural axiomatic characterization [50].\",\n \"That is, GSRs are the class of voting rules that satisfy anonymity, homogeneity, and finite local consistency.\",\n \"Anonymity says that the winner does not depend on the name of the voters, homogeneity says that if we duplicate the preference-profile multiple times, then the winner does not change, and finite local consistency is an approximation to the well-studied consistency axiom.\",\n \"Not all voting rules are GSRs, for example, Dodgson's rule is not a GSR because it violates homogeneity [4], and the following skewed majority rule is also not a GSR because it also violates homogeneity.\",\n \"Example 2 For any $\\\\frac{1}{2}<\\\\gamma <1$ , the $\\\\gamma $ -majority rule is defined for two alternatives $\\\\lbrace a,b\\\\rbrace $ as follows: $b$ is the winner if and only if the number of voters who prefer $b$ is more than the number of voters who prefer $a$ by at least $n^{\\\\gamma }$ .\",\n \"Admittedly, these $\\\\gamma $ -majority rules are quite artificial.\",\n \"Later in this paper we will see that the observation made for GSRs in our main theorem (Theorem REF ) does not hold for $\\\\gamma $ -majority rules for any $\\\\frac{1}{2}<\\\\gamma <1$ .\",\n \"Notice that these rules satisfy anonymity, which means that the observation made in Theorem REF cannot be extended to all anonymous voting rules.\"\n ],\n [\n \"Vote Operations\",\n \"All types of strategic behavior mentioned in the introduction share the following common characteristics.\",\n \"The strategic individual (who can be a group of manipulators, a briber, or a controller, etc.)\",\n \"changes the winner by changing the votes in the preference-profile.\",\n \"Therefore, for generalized scoring rules, any such an operation can be uniquely represented by changes in the total generalized scoring vector.\",\n \"This is in contrast to some other types of strategic behavior where the strategic individual changes the set of alternatives or the voting rule [3], [44].\",\n \"In this section, we first define the set of operations the strategic individual can apply, then define her goals.\",\n \"Given a generalized scoring rule of order $K$ , we model the strategic behavior, called vote operations, as a set of vectors, each of which has $K$ elements, representing the changes made to the total generalized scoring vector if the strategic individual applies this operation.\",\n \"We focus on integer vectors in this paper.\",\n \"Definition 2 Given a GSR $\\\\text{GS}(f,g)$ of order $K$ , let $\\\\Delta =[\\\\vec{\\\\delta }_1 \\\\cdots \\\\vec{\\\\delta }_T]$ denote the vote operations, where for each $i\\\\le T$ , $\\\\vec{\\\\delta }_i\\\\in {\\\\mathbb {Z}}^K$ is a column vector that represents the changes made to the generalized scoring vector by applying the $i$ -th vote operation.\",\n \"For each $l\\\\le K$ , let $\\\\Delta _l$ denote the $l$ -th row of $\\\\Delta $ .\",\n \"We will show examples of these vote operations for some well-studied types of strategic behavior in Section .\",\n \"Given the set of available operations $\\\\Delta $ , the strategic individual's behavior is characterized by a vector $\\\\vec{v}\\\\in {\\\\mathbb {N}}_{\\\\ge 0}^{T}$ , where $\\\\vec{v}$ is a column vector and for each $i\\\\le T$ , $v_i$ represents the number of $i$ -th operation (corresponding to $\\\\vec{\\\\delta }_i$ ) that she applies.\",\n \"Let $\\\\Vert \\\\vec{v}\\\\Vert _1=\\\\sum _{i=1}^Tv_i$ denote the total number of operations in $\\\\vec{v}$ , which is the L$_1$ -norm of $\\\\vec{v}$ .\",\n \"It follows that $\\\\Delta \\\\cdot \\\\vec{v}$ is the change in the total generalized scoring vector introduced by the strategic individual, where for any $l\\\\le K$ , $\\\\Delta _l\\\\cdot \\\\vec{v}$ is the change in the $l$ -th component.\",\n \"Next, we give definitions of the strategic individual's three goals and the corresponding computational problems studied in this paper.\",\n \"Definition 3 In the constructive vote operation (CVO) problem, we are given a generalized scoring rule $\\\\text{GS}(f,g)$ , a preference-profile $P$ , a favored alternative $c$ , and a set of vote operations $\\\\Delta =[\\\\vec{\\\\delta }_1 \\\\cdots \\\\vec{\\\\delta }_T]$ , and we are asked to compute the smallest number $k$ , denoted by $\\\\text{\\\\sc CVO}(P,c)$ , such that there exists a vector $\\\\vec{v}\\\\in {\\\\mathbb {N}}_{\\\\ge 0}^{T}$ with $\\\\Vert \\\\vec{v}\\\\Vert _1= k$ and $g\\\\left(\\\\text{Ord}(f(P)+\\\\Delta \\\\cdot \\\\vec{v})\\\\right)=c$ .\",\n \"If such $\\\\vec{v}$ does not exist, then we denote $\\\\text{\\\\sc CVO}(P,c)=\\\\infty $ .\",\n \"The destructive vote operation (DVO) problem is defined similarly, where $c$ is the disfavored alternative, and we are asked to compute the smallest number $k$ , denoted by $\\\\text{\\\\sc DVO}(P,c)$ , such that there exists a vector $\\\\vec{v}\\\\in {\\\\mathbb {N}}_{\\\\ge 0}^{T}$ with $\\\\Vert \\\\vec{v}\\\\Vert _1= k$ and $g\\\\left(\\\\text{Ord}(f(P)+\\\\Delta \\\\cdot \\\\vec{v})\\\\right)\\\\ne c$ .\",\n \"In the change-winner vote operation (CWVO) problem, we are not given $c$ and we are asked to compute $\\\\text{\\\\sc DVO}(P,\\\\text{GS}(f,g)(P))$ , denoted by $\\\\text{\\\\sc CWVO}(P)$ .\",\n \"In CVO, the strategic individual seeks to make $c$ win; in DVO, the strategic individual seeks to make $c$ lose; and in CWVO, the strategic individual seeks to change the current winner.\",\n \"For a given instance $(P,r)$ , $\\\\text{\\\\sc CWVO}$ is a special case of $\\\\text{\\\\sc DVO}$ , where $c=\\\\text{GS}(f,g)(P)$ .\",\n \"We distinguish these two problems because in this paper, the input preference-profiles are generated randomly, so the winners of these preference-profiles might be different.\",\n \"Therefore, when the preference-profiles are randomly generated, the distribution for the solution to $\\\\text{\\\\sc DVO}$ does not immediately give us a distribution for the solution to $\\\\text{\\\\sc CWVO}$ .\"\n ],\n [\n \"The ILP Formulation\",\n \"Let us first put aside the strategic individual's goal for the moment (i.e., making a favored alternative win, making a disfavored alternative lose, or changing the winner) and focus on the following question: given a preference-profile $P$ and a preorder $\\\\trianglerighteq $ over the $K$ components of the generalized scoring vector, that is, $\\\\trianglerighteq \\\\in \\\\text{Pre}({\\\\mathcal {O}}_K)$ , how many vote manipulations are needed to change the order of the total generalized scoring vector to $\\\\trianglerighteq $ ?\",\n \"Formally, given a $\\\\text{GS}(f,g)$ , a preference-profile $P$ and $\\\\trianglerighteq \\\\in \\\\text{Pre}({\\\\mathcal {O}}_K)$ , we are interested in $\\\\min \\\\lbrace \\\\Vert \\\\vec{v}\\\\Vert _1: \\\\vec{v}\\\\in {\\\\mathbb {N}}_{\\\\ge 0}^K, \\\\text{Ord}(f(P)+\\\\Delta \\\\cdot \\\\vec{v})=\\\\trianglerighteq \\\\rbrace $ .\",\n \"This can be computed by the following integer linear programming ILP$_\\\\trianglerighteq $ , where $v_i$ represents the $i$ th component in $\\\\vec{v}$ , which must be a nonnegative integer.\",\n \"We recall that $\\\\Delta _l$ denotes the $l$ -th row vector of $\\\\Delta $ .\",\n \"$\\\\begin{tabular}{r@{\\\\hspace{28.45274pt}}rl}& \\\\min &\\\\Vert \\\\vec{v}\\\\Vert _1\\\\\\\\s.t.\",\n \"&\\\\forall o_i=_\\\\trianglerighteq o_j: & (\\\\Delta _i-\\\\Delta _j)\\\\cdot \\\\vec{v}= [f(P)]_j-[f(P)]_i\\\\\\\\&\\\\forall o_i\\\\vartriangleright o_j: & (\\\\Delta _i-\\\\Delta _j)\\\\cdot \\\\vec{v}\\\\ge [f(P)]_j-[f(P)]_i+1\\\\\\\\&\\\\forall i:& v_i\\\\ge 0\\\\end{tabular}&&(\\\\text{LP}_{\\\\trianglerighteq })$ Now, we take the strategic individual's goal into account.\",\n \"We immediately have the following lemma as a warmup, whose proofs are straightforward and are thus omitted.\",\n \"Lemma 1 Given a GSR $\\\\text{GS}(f,g)$ , an alternative $c$ , and a preference-profile $P$ , $\\\\bullet $ $\\\\text{\\\\sc CVO}(P,c)<\\\\infty $ if and only if there exists $\\\\trianglerighteq $ such that $g(\\\\trianglerighteq )=c$ and LP$_\\\\trianglerighteq $ has an integer solution; $\\\\bullet $ $\\\\text{\\\\sc DVO}(P,c)<\\\\infty $ if and only if there exists $\\\\trianglerighteq $ such that $g(\\\\trianglerighteq )\\\\ne c$ and LP$_\\\\trianglerighteq $ has an integer solution; $\\\\bullet $ $\\\\text{\\\\sc CWVO}(P)<\\\\infty $ if and only if there exists $\\\\trianglerighteq $ such that $g(\\\\trianglerighteq )\\\\ne \\\\text{GS}(f,g)(P)$ and LP$_\\\\trianglerighteq $ has an integer solution.\",\n \"(We do not need the input $c$ for this problem.)\",\n \"Moreover, the solution to each of the three problems is the minimum objective value in all LPs corresponding to the problem.\",\n \"For example, if $\\\\text{\\\\sc CVO}(P,c)<\\\\infty $ , then $\\\\hfill \\\\text{\\\\sc CVO}(P,c)=\\\\min _{\\\\Vert \\\\vec{v}\\\\Vert _1}\\\\lbrace \\\\vec{v}\\\\text{ is the solution to some LP}_\\\\trianglerighteq \\\\text{ where }g(\\\\trianglerighteq )=c\\\\rbrace \\\\hfill $\"\n ],\n [\n \"The Main Theorem\",\n \"In this section we prove the main theorem, which states that for any fixed $m$ , any generalized scoring rules, and any set of vote operations $\\\\Delta $ , if $n$ votes are generated i.i.d., then for CVO (respectively, DVO, CWVO), with probability that can be infinitely close to 1, the solution is either 0, $\\\\Theta (\\\\sqrt{n})$ , $\\\\Theta (n)$ , or $\\\\infty $ .\",\n \"We first present a simple example for the majority rule for two alternatives $\\\\lbrace a,b\\\\rbrace $ to show the taste of the proof for a very special case, and then comment on why this idea cannot be extended to GSRs.\",\n \"After the proof of Theorem REF we will add more comments on the non-triviality of proof.\",\n \"Example 3 Suppose there are $n$ voters, whose votes are drawn i.i.d.\",\n \"from a distribution $\\\\pi $ over all possible votes (i.e., voting for $a$ with probability $\\\\pi (a)$ and voting for $b$ with probability $\\\\pi (b)$ , w.l.o.g.\",\n \"$\\\\pi (a)\\\\ge \\\\pi (b)$ ).\",\n \"Let $Y_a$ (respectively, $Y_b$ ) denote random variable that represents the total number of voters for $a$ (respectively, for $b$ ).\",\n \"The number of manipulators that are needed to make $b$ to win (i.e., the solution to the UCO problem, see Section REF for formal definition) is thus a random variable $Y_a-Y_b$ .If $Y_a-Y_b<0$ then no manipulator is needed.\",\n \"Let $X$ denote the random variable that takes 1 with probability $\\\\pi (a)$ and takes $-1$ with probability $\\\\pi (b)$ .\",\n \"It follows that $Y_a-Y_b=\\\\underbrace{X+\\\\cdots +X}_{n}$ .\",\n \"By the Central Limit Theorem, $Y_a-Y_b$ converges to a normal distribution with mean $n\\\\cdot E(X)$ and variance ${n\\\\cdot \\\\text{Var}(X)}$ .\",\n \"We are interested in usually how large is $Y_a-Y_b$ .\",\n \"Not surprisingly, the answer depends on the distribution $\\\\pi $ .\",\n \"If $\\\\pi (a)=\\\\pi (b)=1/2$ , then the mean of $Y_a-Y_b$ is zero, and the probability that it is a few standard deviations away from the mean is small.\",\n \"For example, the probability that its absolute value is larger than $4\\\\sqrt{n\\\\cdot \\\\text{Var}(X)}$ is less than $0.01$ , which means that with $99\\\\%$ probability the solution of UCO is no more than $4\\\\sqrt{n\\\\cdot \\\\text{Var}(X)}$ .\",\n \"On the other hand, if $\\\\pi (a)> \\\\pi (b)$ , then the mean of $Y_a-Y_b$ is $n(\\\\pi (a)-\\\\pi (b))$ , which means that with high probability the solution of UCO is very close to $n(\\\\pi (a)-\\\\pi (b))=\\\\Theta (n)$ .\",\n \"The idea behind the argument in Example REF can be easily extended to positional scoring rules [34].\",\n \"However, we do not believe that it can be extended to generalized scoring rules, even for the case of manipulation, for the following two reasons.\",\n \"First, for generalized scoring rules, the components of the generalized scoring vector do not correspond to the “scores” of alternatives.\",\n \"Therefore, having two components tied in the total generalized scoring vector does not mean that two alternatives are tied.\",\n \"Second, the conditions for an alternative to win can be much more complicated than the condition for positional scoring rules, which amounts to requiring that a corresponding component of the total generalized scoring vector is the largest.\",\n \"Therefore, it is not easy to figure out whether the manipulators can achieve their goal by just knowing the asymptotic relationship between the components of the total generalized scoring vector.\",\n \"Theorem 1 Let $\\\\text{GS}(f,g)$ be an integer generalized scoring rule, let $\\\\pi $ be a distribution over all linear orders, and let $\\\\Delta $ be a set of vote operations.\",\n \"Suppose we fix the number of alternatives, generate $n$ votes i.i.d.\",\n \"according to $\\\\pi $ , and let $P_n$ denote the preference-profile.\",\n \"Then, for any alternative $c$ , $\\\\text{\\\\sc VO}\\\\in \\\\lbrace \\\\text{\\\\sc CVO},\\\\text{\\\\sc DVO},\\\\text{\\\\sc CWVO}\\\\footnote {When \\\\text{\\\\sc VO}=\\\\text{\\\\sc CWVO}, we let \\\\text{\\\\sc VO}(P_n,c) denote \\\\text{\\\\sc CWVO}(P_n).\",\n \"}\\\\rbrace $ , and any $\\\\epsilon >0$ , there exists $\\\\beta ^*>1$ such that as $n$ goes to infinity, the total probability for the following four events sum up to more than $1-\\\\epsilon $ : (1) $\\\\text{\\\\sc VO}(P_n,c)=0$ , (2) $\\\\frac{1}{\\\\beta ^*}\\\\sqrt{n}<\\\\text{\\\\sc VO}(P_n,c)<\\\\beta ^* \\\\sqrt{n}$ , (3) $\\\\frac{1}{\\\\beta ^*} n<\\\\text{\\\\sc VO}(P_n,c)<\\\\beta ^* n$ , and (4) $\\\\text{\\\\sc VO}(P_n,c)=\\\\infty $ .\",\n \"Proof of Theorem REF : Let $f(P_\\\\pi )=\\\\sum _{V\\\\in L(\\\\mathcal {C})}\\\\pi (V)\\\\cdot f(V)$ , and $\\\\trianglerighteq _\\\\pi =\\\\text{Ord}(f(P_\\\\pi ))$ .\",\n \"We first prove the theorem for $\\\\text{\\\\sc CVO}$ , and then show how to adjust the proof for $\\\\text{\\\\sc DVO}$ and $\\\\text{\\\\sc CWVO}$ .\",\n \"The theorem is proved in the following two steps.\",\n \"Step 1: we show that as $n$ goes to infinity, with probability that goes to one we have the following: in a randomly generated $P_n$ , the difference between any pair of components in $f(P_n)$ is either $\\\\Theta (\\\\sqrt{n})$ or $\\\\Theta (n)$ .\",\n \"Step 2: we apply sensitivity analyses to ILPs that are similar to the ILP given in Section to prove that for any such preference-profile and any $\\\\text{\\\\sc VO}\\\\in \\\\lbrace \\\\text{\\\\sc CVO},\\\\text{\\\\sc DVO},\\\\text{\\\\sc CWVO}\\\\rbrace $ , $\\\\text{\\\\sc VO}(P_n,c)$ is either 0, $\\\\Theta (\\\\sqrt{n})$ , $\\\\Theta (n)$ , or $\\\\infty $ .\",\n \"The idea behind Step 2 is, for any preference-profile $P_n$ , if the difference between a pair of components in $f(P_n)$ is $\\\\Theta (\\\\sqrt{n})$ , then we consider this pair of components (not alternatives) to be “almost tied”; if the difference is $\\\\Theta (n)$ , then we consider them to be “far away”.\",\n \"Take $\\\\text{\\\\sc CVO}$ as an example, we can easily identify the cases where $\\\\text{\\\\sc CVO}(P_n,c)$ is either 0 (when $\\\\text{GS}(f,g)=c$ ) or $\\\\infty $ (by Lemma REF ).\",\n \"Then, we will first try to break these “almost tied” pairs by using LPs that are similar to LP$_\\\\trianglerighteq $ introduced in Section , and show that if there exists an integer solution $\\\\vec{v}$ , then the objective value $\\\\Vert \\\\vec{v}\\\\Vert _1$ is $\\\\Theta (\\\\sqrt{n})$ .\",\n \"Otherwise, we have to change the orders between some “far away” pairs by using LP$_\\\\trianglerighteq $ 's, and show that if there exists an integer solution to some LP$_\\\\trianglerighteq $ with $g(\\\\trianglerighteq )=c$ , then the objective value is $\\\\Theta (n)$ .\",\n \"Formally, given $n\\\\in \\\\mathbb {N}$ and $\\\\beta >1$ , let ${\\\\cal P}_\\\\beta $ denote the set of all $n$ -vote preference-profiles $P$ that satisfy the following two conditions (we recall that $f(P_\\\\pi )=\\\\sum _{V\\\\in L(\\\\mathcal {C})}\\\\pi (V)\\\\cdot f(V)$ ): for any pair $i,j\\\\le K$ , 1. if $[f(P_\\\\pi )]_i=[f(P_\\\\pi )]_j$ then $\\\\frac{1}{\\\\beta }\\\\sqrt{n}<|[f(P)]_i-[f(P)]_j|<\\\\beta \\\\sqrt{n}$ ; 2. if $[f(P_\\\\pi )]_i\\\\ne [f(P_\\\\pi )]_j$ then $\\\\frac{1}{\\\\beta } n<|[f(P)]_i-[f(P)]_j|<\\\\beta n$ .\",\n \"The following lemma was proved in [47], which follows after the Central Limit Theorem.\",\n \"Lemma 2 For any $\\\\epsilon >0$ , there exists $\\\\beta $ such that $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\text{Pr}\\\\left(P_n\\\\in {\\\\cal P}_\\\\beta \\\\right)>1-\\\\epsilon $ .\",\n \"For any given $\\\\epsilon $ , in the rest of the proof we fix $\\\\beta $ to be a constant guaranteed by Lemma REF .\",\n \"The next lemma (whose proof is deferred to the appendix) will be frequently used in the rest of the proof.\",\n \"Lemma 3 Fix an integer matrix $\\\\bf A$ .\",\n \"There exists a constant $\\\\beta _{\\\\bf A}$ that only depends on $\\\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\\\beta _{\\\\bf A}\\\\cdot \\\\Vert \\\\vec{b}\\\\Vert _\\\\infty $ .\",\n \"$\\\\min \\\\Vert \\\\vec{x}\\\\Vert _1, \\\\text{ s.t.\",\n \"}{\\\\bf A}\\\\cdot \\\\vec{x}\\\\ge \\\\vec{b}$ To prove that with high probability $\\\\text{\\\\sc CVO}(P_n,c)$ is either 0, $\\\\Theta (\\\\sqrt{n})$ , $\\\\Theta (n)$ , or $\\\\infty $ , we introduce the following notation.\",\n \"A preorder $\\\\trianglerighteq ^{\\\\prime }$ is a refinement of another preorder $\\\\trianglerighteq $ , if $\\\\vartriangleright ^{\\\\prime }$ extends $\\\\vartriangleright $ .\",\n \"That is, $\\\\vartriangleright \\\\subseteq \\\\vartriangleright ^{\\\\prime }$ .\",\n \"We note that $\\\\trianglerighteq $ is a refinement of itself.\",\n \"Let $\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq $ denote the strict orders that are in $\\\\vartriangleright ^{\\\\prime }$ but not in $\\\\vartriangleright $ .\",\n \"That is, $(o_i,o_j)\\\\in (\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq )$ if and only if $o_i\\\\vartriangleright ^{\\\\prime } o_j$ and $o_i=_{\\\\trianglerighteq }o_j$ .\",\n \"We define the following LP that is similar to LP$_\\\\trianglerighteq $ defined in Section , which will be used to check whether there is a way to break “almost tied” pairs of components to make $c$ win.\",\n \"For any preorder $\\\\trianglerighteq $ and any of its refinement $\\\\trianglerighteq ^{\\\\prime }$ , we define LP$_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq }$ as follows.\",\n \"$\\\\begin{tabular}{r@{\\\\hspace{28.45274pt}}rl}& \\\\min &\\\\Vert \\\\vec{v}\\\\Vert _1\\\\\\\\s.t.\",\n \"&\\\\forall o_i=_{\\\\trianglerighteq ^{\\\\prime }} o_j: & (\\\\Delta _i-\\\\Delta _j)\\\\cdot \\\\vec{v}= [f(P)]_j-[f(P)]_i\\\\\\\\&\\\\forall (o_i, o_j)\\\\in (\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq ): & (\\\\Delta _i-\\\\Delta _j)\\\\cdot \\\\vec{v}\\\\ge [f(P)]_j-[f(P)]_i+1\\\\\\\\&\\\\forall i:& v_i\\\\ge 0\\\\end{tabular}&&(\\\\text{LP}_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq })$ LP$_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq }$ is defined with a little abuse of notation because some of its constraints depend on $\\\\trianglerighteq $ (not only the pairwise comparisons in $(\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq )$ ).\",\n \"This will not cause confusion because we will always indicate $\\\\trianglerighteq $ in the subscript.\",\n \"We note that there is a constraint in LP$_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq }$ for each pair of components $o_i,o_j$ with $o_i=_{\\\\trianglerighteq }o_j$ .\",\n \"Therefore, LP$_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq }$ is used to find a solution that breaks ties in $\\\\trianglerighteq $ .\",\n \"It follows that LP$_{\\\\trianglerighteq ^{\\\\prime }\\\\ominus \\\\trianglerighteq }$ has an integer solution $\\\\vec{v}$ if and only if the strategic individual can make the order between any pairs of $o_i,o_j$ with $o_i=_{\\\\trianglerighteq }o_j$ to be the one in $\\\\trianglerighteq ^{\\\\prime }$ by applying the $i$ -th operation $v_i$ times, and the total number of vote operations is $\\\\Vert \\\\vec{v}\\\\Vert _1$ .\",\n \"The following two claims identify the preference-profiles in ${\\\\cal P}_\\\\beta $ for which $\\\\text{\\\\sc CVO}$ is $\\\\Theta (\\\\sqrt{n})$ and $\\\\Theta (n)$ , respectively, whose proofs are deferred to the appendix.\",\n \"Claim 1 There exists $N\\\\in \\\\mathbb {N}$ and $\\\\beta ^{\\\\prime }>1$ such that for any $n\\\\ge N$ , any $P\\\\in {\\\\cal P}_\\\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\\\trianglerighteq ^*$ of $\\\\trianglerighteq _\\\\pi =\\\\text{Ord}(f(P_\\\\pi ))$ such that $g(\\\\trianglerighteq ^*)=c$ and LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ has an integer solution, then $\\\\frac{1}{\\\\beta ^{\\\\prime }}\\\\sqrt{n}< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"Claim 2 There exists $\\\\beta ^{\\\\prime }>1$ such that for any $P\\\\in {\\\\cal P}_\\\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\\\trianglerighteq ^*$ of $\\\\trianglerighteq _\\\\pi =\\\\text{Ord}(f(P_\\\\pi ))$ such that LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ has an integer solution, and (3) there exists $\\\\trianglerighteq $ such that $g(\\\\trianglerighteq )=c$ and LP$_{\\\\trianglerighteq }$ has an integer solution, then $\\\\frac{1}{\\\\beta ^{\\\\prime }} n< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime } n$ .\",\n \"Lastly, for any $P\\\\in {\\\\cal P}_\\\\beta $ such that $\\\\text{GS}(f,g)(P)\\\\ne c$ , the only case not covered by Claim REF and Claim REF is that there does not exist $\\\\trianglerighteq $ with $\\\\text{GS}(f,g)(\\\\trianglerighteq )=c$ such that LP$_\\\\trianglerighteq $ has an integer solution.\",\n \"It follows from Lemma REF that in this case $\\\\text{\\\\sc CVO}(P,c)=\\\\infty $ .\",\n \"We note that $\\\\beta ^{\\\\prime }$ in Claim REF and Claim REF does not depend on $n$ .\",\n \"Let $\\\\beta ^*$ be an arbitrary number that is larger than the two $\\\\beta ^{\\\\prime }$ s. This proves the theorem for $\\\\text{\\\\sc CVO}$ .\",\n \"For $\\\\text{\\\\sc DVO}$ , we only need to change $g(\\\\trianglerighteq ^*)=c$ to $g(\\\\trianglerighteq ^*)\\\\ne c$ in Claim REF , and change $g(\\\\trianglerighteq )=c$ to $g(\\\\trianglerighteq )\\\\ne c$ in Claim REF .\",\n \"For $\\\\text{\\\\sc CWVO}$ , $\\\\text{\\\\sc CWVO}(P)$ is never 0 and we only need to change $g(\\\\trianglerighteq ^*)=c$ to $g(\\\\trianglerighteq ^*)\\\\ne \\\\text{GS}(f,g)(P)$ in Claim REF , and change $g(\\\\trianglerighteq )=c$ to $g(\\\\trianglerighteq ) \\\\ne \\\\text{GS}(f,g)(P)$ in Claim REF .\",\n \"$\\\\blacksquare $ More comments on the non-triviality of the proof.\",\n \"Lemma REF has been proved in [47], whose intuition is quite straightforward and naturally corresponds to a random walk in multidimensional space.\",\n \"However, we did not find an obvious connection between random walk theory and the observation made in Theorem REF .\",\n \"We believe that it is unlikely that an obvious connection exists.\",\n \"One evidence is that the observation made in Theorem REF does not hold for some voting rules.\",\n \"For example, consider the $\\\\gamma $ -majority rule defined in Example REF .\",\n \"It is not hard to see that as $n$ goes to infinity, with probability that goes to 1 we have $\\\\text{\\\\sc CVO}(P_n,b)=\\\\text{\\\\sc DVO}(P_n,a)=\\\\text{\\\\sc CWVO}(P_n)=n^{\\\\gamma }/2$ , which is not any of the four cases described in Theorem REF if $\\\\frac{1}{2}<\\\\gamma <1$ .\",\n \"(This implies that for any $\\\\frac{1}{2}<\\\\gamma <1$ , $\\\\gamma $ -majority is not a generalized scoring rule, which we already know because they do not satisfy homogeneity.)\",\n \"Therefore, the proof of Theorem REF should involve analyses on the specific structure of GSRs.\",\n \"The main difficulty in proving Theorem REF is, for generalized scoring rules we have to handle the cases where some components of the total generalized scoring vector are equivalent.\",\n \"This only happens with negligible probability for the randomly generated preference-profile $P_n$ , but it is not clear how often the strategic individual can make some components equivalent in order to achieve her goal.\",\n \"This is the main reason for us to convert the vote manipulation problem to multiple ILPs and apply Lemma REF to analyze them.\"\n ],\n [\n \"Applications of the Main Theorem\",\n \"In this section we show how to apply Theorem REF to some well-studied types of strategic behavior, including constructive and destructive unweighted coalitional optimization, bribery and control, and margin of victory and minimum manipulation coalition size.\",\n \"In the sequel, we will use each subsection to define these problems and describe how they fit in our vote operation framework, and how Theorem REF applies.\",\n \"In the end of the section we present a unified corollary for all these types of strategic behavior.\"\n ],\n [\n \"Unweighted Coalitional Optimization\",\n \"Definition 4 In a constructive (respectively, destructive) unweighted coalitional optimization (UCO) problem, we are given a voting rule $r$ , a preference-profile $P^{NM}$ of the non-manipulators, and a (dis)favored alternative $c\\\\in \\\\mathcal {C}$ .\",\n \"We are asked to compute the smallest number of manipulators who can cast votes $P^{M}$ such that $c=r(P^{NM}\\\\cup P^{M})$ (respectively, $c\\\\ne r(P^{NM}\\\\cup P^{M})$ ).\",\n \"To see how UCO fits in the vote operation model, we view the group of manipulators as the strategic individual, and each vote cast by a manipulator is a vote operation.\",\n \"Therefore, the set of operations is exactly the set of all generalized scoring vectors $\\\\lbrace f(V): V\\\\in L(\\\\mathcal {C})\\\\rbrace $ .\",\n \"To apply Theorem REF , for constructive UCO we let $\\\\text{\\\\sc VO}=\\\\text{\\\\sc CVO}$ and for destructive UCO we let $\\\\text{\\\\sc VO}=\\\\text{\\\\sc DVO}$ .\"\n ],\n [\n \"Bribery\",\n \"In this paper we are interested in the optimization variant of the bribery problem [16].\",\n \"Definition 5 In a constructive (respectively, destructive) opt-bribery problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\\\in \\\\mathcal {C}$ .\",\n \"We are asked to compute the smallest number $k$ such that the strategic individual can change no more than $k$ votes such that $c$ is the winner (respectively, $c$ is not the winner).\",\n \"To see how opt-bribery falls under the vote operation framework, we view each action of “changing a vote” as a vote operation.\",\n \"Since the strategic individual can only change existing votes in the preference-profile, we define the set of operations to be the difference between the generalized scoring vectors of all votes and the generalized scoring vectors of votes in the support of $\\\\pi $ , that is, $\\\\lbrace f(V)-f(W): V,W\\\\in L(\\\\mathcal {C})\\\\text{ s.t.\",\n \"}\\\\pi (W)>0\\\\rbrace $ .\",\n \"Then, similarly the constructive variant corresponds to $\\\\text{\\\\sc CVO}$ and the destructive variant corresponds to $\\\\text{\\\\sc DVO}$ .\",\n \"In both cases Theorem REF cannot be directly applied, because in the ILPs we did not limit the total number of each type of vote operations that can be used by the strategic individual.\",\n \"Nevertheless, we can still prove a similar proposition by taking a closer look at the relationship between $\\\\text{\\\\sc CVO}$ ($\\\\text{\\\\sc DVO}$ ) and opt-bribery as follows: For any preference-profile, the solution to $\\\\text{\\\\sc CVO}$ (respectively, $\\\\text{\\\\sc DVO}$ ) is a lower bound on the solution to constructive (respectively, destructive) opt-bribery, because in $\\\\text{\\\\sc CVO}$ and $\\\\text{\\\\sc DVO}$ there are no constraints on the number of each type of vote operations.\",\n \"We have the following four cases.\",\n \"1.\",\n \"If the solution to $\\\\text{\\\\sc CVO}$ ($\\\\text{\\\\sc DVO}$ ) is 0, then the solution to constructive (destructive) opt-bribery is also 0.\",\n \"2.\",\n \"If the solution to $\\\\text{\\\\sc CVO}$ ($\\\\text{\\\\sc DVO}$ ) is $\\\\Theta (\\\\sqrt{n})$ , as $n$ become large enough, with probability that goes to 1 each type of votes in the support of $\\\\pi $ will appear $\\\\Theta (n)$ , which is $>\\\\Theta (\\\\sqrt{n})$ , times in the randomly generated preference-profile, which means that there are enough votes of each type for the strategic individual to change.\",\n \"3.\",\n \"If the solution to $\\\\text{\\\\sc CVO}$ ($\\\\text{\\\\sc DVO}$ ) is $\\\\Theta (n)$ , then the solution to constructive (destructive) opt-bribery is either $\\\\Theta (n)$ (when the strategic individual can change all votes to achieve her goal), or $\\\\infty $ .\",\n \"4.\",\n \"If the solution to $\\\\text{\\\\sc CVO}$ ($\\\\text{\\\\sc DVO}$ ) is $\\\\infty $ , then the solution to constructive (destructive) opt-bribery is also $\\\\infty $ .\",\n \"It follows that the observation made in Theorem REF holds for opt-bribery.\"\n ],\n [\n \"Margin of Victory (MoV)\",\n \"Definition 6 Given a voting rule $r$ and a preference-profile $P$ , the margin of victory (MoV) of $P$ is the smallest number $k$ such that the winner can be changed by changing $k$ votes in $P$ .\",\n \"In the mov problem, we are given $r$ and $P$ , and are asked to compute the margin of victory.\",\n \"For a given instance $(P,r)$ , mov is equivalent to destructive opt-bribery, where $c=r(P)$ .\",\n \"However, when the input preference-profiles are generated randomly, the winners in these profiles might be different.\",\n \"Therefore, the corollary of Theorem REF for opt-bribery does not directly imply a similar corollary for mov.\",\n \"This relationship is similar to the relationship between $\\\\text{\\\\sc DVO}$ and $\\\\text{\\\\sc CWVO}$ .\",\n \"Despite this difference, the formulation of mov in the vote operation framework is very similar to that of opt-bribery: The set of all operations and the argument to apply Theorem REF are the same.\",\n \"The only difference is that for mov, we obtain the corollary from the $\\\\text{\\\\sc CWVO}$ part of Theorem REF , while the corollary for opt-bribery is obtained from the $\\\\text{\\\\sc CVO}$ and $\\\\text{\\\\sc DVO}$ parts of Theorem REF .\"\n ],\n [\n \"Minimum Manipulation Coalition Size (MMCS)\",\n \"The minimum manipulation coalition size (MMCS) problem is similar to mov, except that in MMCS the winner must be improved for all voters who change their votes [33].\",\n \"Definition 7 In an MMCS problem, we are given a voting rule $r$ and a preference-profile $P$ .\",\n \"We are asked to compute the smallest number $k$ such that a coalition of $k$ voters can change their votes to change the winner, and all of them prefer the new winner to $r(P)$ .\",\n \"Unlike mov, MMCS falls under the vote operation framework in the following dynamic way.\",\n \"For each preference-profile, suppose $c$ is the current winner.\",\n \"For each adversarial $d\\\\ne c$ , we use $\\\\lbrace f(V)-f(W):V,W\\\\in L(\\\\mathcal {C})\\\\text{ s.t.\",\n \"}d\\\\succ _W c\\\\text{ and } \\\\pi (W)>0\\\\rbrace $ as the set of operations.\",\n \"That is, we only allow voters who prefer $d$ to $c$ to participate in the manipulative coalition.\",\n \"We also replace each of LP$_{\\\\trianglerighteq }$ and LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ by multiple LPs, each of which is indexed by a pair of alternatives $(d,c)$ and the constraints are generated by using the corresponding set of operations.\",\n \"Then, the corollary for MMCS follows after a similar argument to that of $\\\\text{\\\\sc CVO}$ in Theorem REF .\"\n ],\n [\n \"Control by Adding/Deleting Votes (CAV/CDV)\",\n \"Definition 8 In a constructive (respectively, destructive) optimal control by adding votes (opt-CAV) problem, we are given a preference-profile $P$ , a (dis)favored alternative $c\\\\in \\\\mathcal {C}$ , and a set $N^{\\\\prime }$ of additional votes.\",\n \"We are asked to compute the smallest number $k$ such that the strategic individual can add $k$ votes in $N^{\\\\prime }$ such that $c$ is the winner (respectively, $c$ is not the winner).\",\n \"For simplicity, we assume that $|N^{\\\\prime }|=n$ and the votes in $N^{\\\\prime }$ are drawn i.i.d.\",\n \"from a distribution $\\\\pi ^{\\\\prime }$ .\",\n \"To show how opt-CAV falls under the vote operation model, we let the set of operations to be the generalized scoring vectors of all votes that are in the support of $\\\\pi ^{\\\\prime }$ , that is, $\\\\lbrace f(V): V\\\\in L(\\\\mathcal {C})\\\\text{ and }\\\\pi ^{\\\\prime }(V)>0\\\\rbrace $ .\",\n \"Then, the corollary follows from the $\\\\text{\\\\sc CVO}$ and $\\\\text{\\\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery.\",\n \"Definition 9 In a constructive (respectively, destructive) optimal control by deleting votes (opt-CDV) problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\\\in \\\\mathcal {C}$ .\",\n \"We are asked to compute the smallest number $k$ such that the strategic individual can delete $k$ votes in $P$ such that $c$ is the winner (respectively, $c$ is not the winner).\",\n \"To show how opt-CDV falls under the vote operation framework, we let the set of operations to be the negation of generalized scoring vectors of votes in the support of $\\\\pi ^{\\\\prime }$ , that is, $\\\\lbrace -f(V): V\\\\in L(\\\\mathcal {C})\\\\text{ and }\\\\pi ^{\\\\prime }(V)>0\\\\rbrace $ .\",\n \"Then, the corollary follows from the $\\\\text{\\\\sc CVO}$ and $\\\\text{\\\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery.\"\n ],\n [\n \"A Unified Corollary\",\n \"The next corollary of Theorem REF summarizes the results obtained for all types of strategic behavior studied in this section.\",\n \"Corollary 1 For any integer generalized scoring rule, any distribution $\\\\pi $ over votes, and any $X\\\\in $ $\\\\big ($ {constructive, destructive}$\\\\times $ {UCO, opt-bribery, opt-CAV, opt-CDV}$\\\\big )$$\\\\cup $ {MoV, MMCS}, suppose the input preference-profiles are generated i.i.d.\",\n \"from $\\\\pi $ .For CAV, the distribution over the new votes can be generated i.i.d.\",\n \"from a different distribution $\\\\pi ^{\\\\prime }$ .\",\n \"Then, for any alternative $c$ and any $\\\\epsilon >0$ , there exists $\\\\beta ^*>1$ such that the total probability for the solution to $X$ to be one of the following four cases is more than $1-\\\\epsilon $ as $n$ goes to infinity: (1) 0, (2) between $\\\\frac{1}{\\\\beta ^*}\\\\sqrt{n}$ and $\\\\beta ^* \\\\sqrt{n}$ , (3) between $\\\\frac{1}{\\\\beta ^*} n$ and $\\\\beta ^* n$ , and (4) $\\\\infty $ .\"\n ],\n [\n \"Discussions and Future Work\",\n \"In this paper, we proposed a general framework to study vote operations for generalized scoring rules.\",\n \"Our main theorem is a characterization for the number of vote operations that are needed to achieve the strategic individual's goal.\",\n \"We showed that the main theorem can be applied to many types of strategic behavior and many commonly used voting rules, for most of which no similar results were previously known.\",\n \"We further discuss the generality of our framework in the next two paragraphs.\",\n \"GSRs vs. integer GSRs.\",\n \"Though integer GSRs are a subclass of GSRs, we feel that from a computational point of view, focusing integer generalized scoring rules does not sacrifice much generality.\",\n \"Because the $g$ function only depends on the preorder among components in the total generalized scoring vector, if the $f$ function is scaled up by a constant, then the $g$ function will select the same winner.\",\n \"Therefore, integer GSRs are equivalent to GSRs where components in the generalized scoring vectors are rational numbers.\",\n \"When the components are irrational numbers, two computational problems arise.\",\n \"First, it is not clear how these irrational numbers are represented, and second, it is hard to compare two irrational numbers computationally, thus even harder to compute the preorder of the components of the total generalized scoring vector.\",\n \"On the other hand, integer GSRs do not have such computational constraints.\",\n \"In fact, all commonly studied voting rules that are known to be GSRs are integer GSRs.\",\n \"Therefore, we believe that our main theorem has a wide application (at least can be applied to many commonly studied voting rules).\",\n \"On the generality of vote operations.\",\n \"While the framework we proposed covers many types of strategic behavior, some other types of strategic behavior that have been widely studied are not covered by our framework.\",\n \"These types of strategic behavior can be roughly categorized as follows: (1) controls that changes the set of alternatives, for example, control by adding/deleting alternatives [3] and control by introducing clones of alternatives [44], [13]; and (2) controls that change the procedure of voting, for example control by (runoff) partition of alternatives and control by partition of voters [3], and control by changing the agenda of voting [24].\",\n \"Building a more general framework that covers more types of strategic behavior and studying their properties are interesting directions for future research.\",\n \"As we discussed in the introduction, on the positive side, our main theorem suggests that computing the margin of victory is usually not hard, which helps implementing efficient post-election auditing methods.\",\n \"One promising future direction is to design practical computational techniques for computing the margin of victory for generalized scoring rules, based on the ILP proposed in this paper.\",\n \"On the negative side, our main theorem suggests that computational complexity might merely be a weak barrier against many types of strategic behavior.\",\n \"Therefore, we should look for new ways to protect voting, for example introducing randomization [6], [14], [46], [29], [30], using multiple rounds [9], [28], [10], or limiting the strategic individuals' information about other voters [7].\",\n \"Another interesting research direction is to investigate the phase transition of the probability for a coalition of strategic individuals to achieve their goal by using vote operations, as it was done for manipulation [45], [26].\"\n ],\n [\n \"Proofs\",\n \"Lemma REF Fix an integer matrix $\\\\bf A$ .\",\n \"There exists a constant $\\\\beta _{\\\\bf A}$ that only depends on $\\\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\\\beta _{\\\\bf A}\\\\cdot \\\\Vert \\\\vec{b}\\\\Vert _\\\\infty $ .\",\n \"$\\\\min \\\\Vert \\\\vec{x}\\\\Vert _1, \\\\text{ s.t.\",\n \"}{\\\\bf A}\\\\cdot \\\\vec{x}\\\\ge \\\\vec{b}$ Proof of Lemma REF : Let $\\\\bf A$ be a $m^*\\\\times n^*$ integer matrix, which includes the constraints $\\\\vec{x}\\\\ge \\\\vec{0}$ .\",\n \"Suppose LP (REF ) has a (non-negative) integer solution.\",\n \"We note that $\\\\vec{0}$ is an optimal integer solution to $\\\\min \\\\vec{1}\\\\cdot (\\\\vec{x})^{\\\\prime } \\\\text{ s.t.\",\n \"}{\\\\bf A}\\\\cdot \\\\vec{x}\\\\ge \\\\vec{0}$ .\",\n \"Then, it follows from Theorem 5 (ii) in [8] that LP (REF ) has a (non-negative) integer solution $\\\\vec{z}$ such that $\\\\Vert \\\\vec{z}-\\\\vec{0}\\\\Vert _\\\\infty \\\\le n^*\\\\cdot M({\\\\bf A})\\\\cdot (\\\\Vert \\\\vec{b}-\\\\vec{0}\\\\Vert _\\\\infty +2),$ where $M({\\\\bf A})$ is the maximum of the absolute values of the determinants of the square sub-matrices of $\\\\bf A$ .\",\n \"Since $\\\\bf A$ is fixed, the right hand side becomes a constant, that is, $\\\\Vert \\\\vec{z}\\\\Vert _\\\\infty =O(\\\\Vert \\\\vec{b}\\\\Vert _\\\\infty )$ .\",\n \"Therefore, there exists $\\\\beta _{\\\\bf A}$ such that the optimal value in the ILP (REF ) is no more than $\\\\vec{1}\\\\cdot (\\\\vec{z})^{\\\\prime }\\\\le n^*\\\\Vert \\\\vec{z}\\\\Vert _\\\\infty \\\\le \\\\beta _{\\\\bf A}\\\\cdot \\\\Vert \\\\vec{b}\\\\Vert _\\\\infty $ .\",\n \"$\\\\blacksquare $ Claim REF There exists $N\\\\in \\\\mathbb {N}$ and $\\\\beta ^{\\\\prime }>1$ such that for any $n\\\\ge N$ , any $P\\\\in {\\\\cal P}_\\\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\\\trianglerighteq ^*$ of $\\\\trianglerighteq _\\\\pi =\\\\text{Ord}(f(P_\\\\pi ))$ such that $g(\\\\trianglerighteq ^*)=c$ and LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ has an integer solution, then $\\\\frac{1}{\\\\beta ^{\\\\prime }}\\\\sqrt{n}< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"Proof of Claim REF : Let $\\\\trianglerighteq =\\\\text{Ord}(f(P))$ .\",\n \"Because $\\\\text{GS}(f,g)(P)\\\\ne c$ , $g(\\\\trianglerighteq )\\\\ne c$ .\",\n \"Therefore, the strategic individual has to change the order of some components in the generalized scoring vector to make $c$ win.\",\n \"We note that $P\\\\in {\\\\cal P}_\\\\beta $ , which means that the difference between any pair of components of $f(P)$ is more than $\\\\frac{1}{\\\\beta } \\\\sqrt{n}$ .\",\n \"Let $d_{max}$ denote the maximum difference between any pair of components in generalized score vectors.\",\n \"That is, $d_{max}=\\\\max _{t,t^{\\\\prime },L\\\\in L(\\\\mathcal {C})}\\\\lbrace (f(L))_t-(f(L))_{t^{\\\\prime }}\\\\rbrace $ .\",\n \"In order for $c$ to win, the number of vote operations must be at least $\\\\frac{1}{\\\\beta } \\\\sqrt{n}/d_{max}$ .\",\n \"Therefore, $\\\\text{\\\\sc CVO}(P,c)>\\\\frac{1}{\\\\beta d_{max}} \\\\sqrt{n}$ .\",\n \"We next show the upper bound.\",\n \"Because $P\\\\in {\\\\cal P}_\\\\beta $ , for any pair $o_i,o_j$ with $o_i=_{\\\\trianglerighteq _\\\\pi }o_j$ , $|[f(P)]_i-[f(P)]_j|<\\\\beta \\\\sqrt{n}$ .\",\n \"Therefore, the right hand side of each (in)equality in LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ is no more than $\\\\beta \\\\sqrt{n}$ .\",\n \"Applying Lemma REF to LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ , we have that there exists a constant $\\\\beta _{\\\\trianglerighteq ^*,\\\\trianglerighteq _\\\\pi }$ that only depends on $\\\\trianglerighteq ^*$ and $\\\\trianglerighteq _\\\\pi $ , and an integer solution $\\\\vec{v}$ with $\\\\Vert \\\\vec{v}\\\\Vert _1\\\\le \\\\beta _{\\\\trianglerighteq ^*,\\\\trianglerighteq _\\\\pi }\\\\sqrt{n}$ (the $\\\\bf A$ matrix in Lemma REF is fixed because we fix the number of alternatives $m$ , and the left hand side of each (in)equality in LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ does not depend on $n$ ).\",\n \"Let $\\\\beta ^{\\\\prime }$ be the maximum of $d_{max}\\\\beta $ and all $\\\\beta _{\\\\trianglerighteq ^*,\\\\trianglerighteq _\\\\pi }$ (since we fix the number of alternatives, there are finite many $\\\\beta _{\\\\trianglerighteq ^*,\\\\trianglerighteq _\\\\pi }$ 's).\",\n \"Since $\\\\beta _{\\\\trianglerighteq ^*,\\\\trianglerighteq _\\\\pi }>\\\\frac{1}{d_{max}\\\\beta }$ , $\\\\beta ^{\\\\prime }>1$ .\",\n \"It follows that $\\\\Vert \\\\vec{v}\\\\Vert _1<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"We next show that for a sufficiently large $n$ , if the strategic individual applies $\\\\vec{v}$ , then the order over components of the total scoring vector will become $\\\\trianglerighteq ^*$ .\",\n \"That is, $c$ can be made win.\",\n \"The idea is, LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ ensures that by applying $\\\\vec{v}$ , ties between the “almost tied” components are broken as in $\\\\trianglerighteq ^*$ .\",\n \"Since $\\\\Vert \\\\vec{v}\\\\Vert _1=O(\\\\sqrt{n})$ , when $n$ is large enough the order between any pair of “far away” components will not be affected.\",\n \"Formally, let $\\\\vec{x}=f(P)+\\\\Delta \\\\cdot \\\\vec{v}$ .\",\n \"That is, $\\\\vec{x}$ is the total generalized scoring vector after the strategic individual applied $\\\\vec{v}$ .\",\n \"Because $\\\\vec{v}$ is a solution to LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ , for any pair $o_i,o_j$ with $o_i=_{\\\\trianglerighteq _\\\\pi }o_j$ , the order between $o_i$ and $o_j$ in $\\\\trianglerighteq ^*$ is the same as the order between $o_i$ and $o_j$ in $\\\\text{Ord}(\\\\vec{x})$ .\",\n \"Since $\\\\trianglerighteq ^*$ is an extension of $\\\\trianglerighteq _\\\\pi $ , if $o_i\\\\vartriangleright _\\\\pi o_j$ , then we must have $o_i\\\\vartriangleright ^* o_j$ .\",\n \"Therefore, we only need to check that for any $o_i\\\\vartriangleright _\\\\pi o_j$ , we have $[\\\\vec{x}]_i>[\\\\vec{x}]_j$ .\",\n \"Because $P\\\\in {\\\\cal P}_\\\\beta $ , $|[f(P)]_i-[f(P)]_j|>\\\\frac{1}{\\\\beta }n$ .\",\n \"We note that $\\\\Vert \\\\vec{v}\\\\Vert _1<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ , which means that by applying $\\\\vec{v}$ , the strategic individual can only change the difference between any pair of components by no more than $d_{max} \\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"Let $N=(d_{max} \\\\beta ^{\\\\prime }\\\\beta )^2+1$ .\",\n \"When $n\\\\ge N$ , $d_{max} \\\\beta ^{\\\\prime }\\\\sqrt{n}<\\\\frac{1}{\\\\beta }n$ , which means that for any $o_i\\\\vartriangleright _\\\\pi o_j$ , applying $\\\\vec{v}$ will not change the order between $o_i$ and $o_j$ in the total generalized scoring vector.\",\n \"This means that by applying $\\\\vec{v}$ , the strategic individual can make $c$ win.\",\n \"Therefore, $ \\\\text{\\\\sc CVO}(P,c)\\\\le \\\\Vert \\\\vec{v}\\\\Vert _1<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"It follows that for any $n\\\\ge N$ , $\\\\frac{1}{\\\\beta ^{\\\\prime }}\\\\sqrt{n}< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime }\\\\sqrt{n}$ .\",\n \"$\\\\blacksquare $ Claim REF There exists $\\\\beta ^{\\\\prime }>1$ such that for any $P\\\\in {\\\\cal P}_\\\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\\\trianglerighteq ^*$ of $\\\\trianglerighteq _\\\\pi =\\\\text{Ord}(f(P_\\\\pi ))$ such that LP$_{\\\\trianglerighteq ^*\\\\ominus \\\\trianglerighteq _\\\\pi }$ has an integer solution, and (3) there exists $\\\\trianglerighteq $ such that $g(\\\\trianglerighteq )=c$ and LP$_{\\\\trianglerighteq }$ has an integer solution, then $\\\\frac{1}{\\\\beta ^{\\\\prime }} n< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime } n$ .\",\n \"Proof of Claim REF : Let $\\\\trianglerighteq =\\\\text{Ord}(f(P))$ .\",\n \"Because the premises of Claim REF do not hold, the strategic individual has to change the order of some pair of components that are “far away” (that is, the difference between them is $\\\\Theta (n)$ before the strategic individual applies vote operations) to make $c$ win.\",\n \"We note that one operation can only change the difference between a pair of components by at most $d_{max}$ .\",\n \"Therefore, $\\\\text{\\\\sc CVO}(P,c)\\\\ge \\\\frac{1}{\\\\beta }n/d_{max}$ .\",\n \"On the other hand, it follows from Lemma REF and condition (3) in the statement of the claim that $\\\\text{\\\\sc CVO}(P,c)<\\\\infty $ .\",\n \"The only thing left to show is that there exists $\\\\beta ^{\\\\prime }>1$ such that $\\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime }n$ for all $P\\\\in {\\\\cal P}_\\\\beta $ and for all $n$ .\",\n \"Because $P\\\\in {\\\\cal P}_\\\\beta $ , for any pair $o_i,o_j$ , $|[f(P)]_i-[f(P)]_j|<\\\\beta n$ .\",\n \"Applying Lemma REF to LP$_{\\\\trianglerighteq }$ and condition (3) in the statement of the claim, we have that for any $\\\\trianglerighteq $ , there exits $\\\\beta _{\\\\trianglerighteq }$ that only depends on $\\\\trianglerighteq $ , and an optimal integer solution $\\\\vec{v}$ such that $\\\\Vert \\\\vec{v}\\\\Vert _1\\\\le \\\\beta _\\\\trianglerighteq n$ .\",\n \"Let $\\\\beta ^{\\\\prime }$ be the maximum of $d_{max} \\\\beta $ and all $\\\\beta _{\\\\trianglerighteq }$ (again, there are finite number of $\\\\beta _{\\\\trianglerighteq }$ 's).\",\n \"It follows that $\\\\frac{1}{\\\\beta ^{\\\\prime }} n< \\\\text{\\\\sc CVO}(P,c)<\\\\beta ^{\\\\prime } n$ .\",\n \"$\\\\blacksquare $\"\n ],\n [\n \"Discussion: How often the solution is 0 or $\\\\infty $ ?\",\n \"One important question is: how large is the probability that the solution to problems studied in this section is 0 or $\\\\infty $ ?\",\n \"Not surprisingly the answer depends on both the voting rule and the type of vote operations.\",\n \"The probability can be large for some voting rules.\",\n \"For example, for any voting rule that always selects a given alternative $d$ as the winner, the solution to $\\\\text{\\\\sc CVO}$ (respectively, $\\\\text{\\\\sc DVO}$ ) is always 0 (respectively $\\\\infty $ ) for $c=d$ and is always $\\\\infty $ (respectively 0) for $c\\\\ne d$ .\",\n \"However, for common voting rules the alternatives are treated almost equally (except for cases with ties).\",\n \"Therefore, we may expect that for a preference-profile whose votes are generated i.i.d., each alternative has almost the same probability of being selected as the winner.\",\n \"This is indeed the case in all commonly used voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.\",\n \"Therefore, for these voting rules, when the votes are drawn i.i.d.\",\n \"uniformly at random, the probability for $\\\\text{\\\\sc CVO}$ is approximately $\\\\frac{1}{m}$ and the probability for $\\\\text{\\\\sc DVO}$ is approximately $\\\\frac{m-1}{m}$ .\",\n \"For $\\\\text{\\\\sc CWVO}$ , the answer is never 0 because changing 0 votes cannot change the winner.\",\n \"We would also expect for common voting rules, for some types of strategic behavior studied in this section, with low probability the solution is $\\\\infty $ .\",\n \"For UCO, the strategic individual can introduce many (but finitely many) votes such that the non-manipulators' votes are negligible.\",\n \"For opt-bribery and mov, the strategic individual can change all votes to achieve her goal.\",\n \"For MMCS, CAV, and CDV, it is not clear how large such probability is.\",\n \"The following table summarizes folklore results for common voting rules when votes are drawn i.i.d.\",\n \"uniformly at random.\",\n \"Table: Probability for solutions to some problems to be 0 or ∞\\\\infty for common voting rules.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1231"}}},{"rowIdx":1866993,"cells":{"text":{"kind":"list like","value":[["Assessment of density functional approximations for the hemibonded\n structure of water dimer radical cation"],["Abstract Due to the severe self-interaction errors associated with some density functional approximations, conventional density functionals often fail to dissociate the hemibonded structure of water dimer radical cation (H2O)2+ into the correct fragments: H2O and H2O+.","Consequently, the binding energy of the hemibonded structure (H2O)2+ is not well-defined.","For a comprehensive comparison of different functionals for this system, we propose three criteria: (i) The binding energies, (ii) the relative energies between the conformers of the water dimer radical cation, and (iii) the dissociation curves predicted by different functionals.","The long-range corrected (LC) double-hybrid functional, omegaB97X-2(LP) [J.-D. Chai and M. Head-Gordon, J. Chem.","Phys., 2009, 131, 174105.","], is shown to perform reasonably well based on these three criteria.","Reasons that LC hybrid functionals generally work better than conventional density functionals for hemibonded systems are also explained in this work."],["Introduction","Water can be decomposed when it is exposed to high-energy flux.","The products of water radiolysis may contain various radical species, e.g.","hydrogen atoms (H), hydroxide radicals (OH), oxygen anions (O$^-$ ), and water cations (H$_2$ O$^+$ ), depending on the radiation infrastructure setup.","For example the overall decomposition scheme activated by $\\beta $ particles has been outlined by Garrett et al.","in 2005 where three main channels of decomposition were listed.","The cationic channel leads to the formation of ionized water living in about several tens femtoseconds and hydrated electron, followed by the generation of hydronium (H$_3$ O$^+$ ) and OH radicals through proton transfer process , .","The energized-neutral and anionic channels could result in the cleavage the oxygen-hydrogen chemical bonds to produce hydrogen and oxygen derivatives, i.e.","H, H$^-$ , H$_2$ , O, O$^-$ , OH$^-$ etc.","Subsequent chemical reactions can progress further up to the desorption of stable gas molecules H$_2$ and O$_2$ being driven by those reactive radical species .","The cationic channel is therefore particularly interesting due to its dominant products — OH radicals and solvated electrons.","The smallest system to understand the chemical dynamics of ionized water is the water dimer radical cation (H$_2$ O)$_2$$^+$ , and it has been approached by several experimental studies in the past.","Angel and Stace reported the predominant H$_3$ O$^+$ –OH central core by the collision-induced fragmentation experiment against the earlier theoretical assignment of charge-resonance hydrazine structure .","Dong et al.","observed a weak signal corresponding the formation of (H$_2$ O)$_2$$^+$ near the low-mass side of (H$_2$ O)$_2$ H$^+$ using 26.5 eV soft X-ray laser .","Gardenier, Johnson, and McCoy reported the argon-tagged predissociation infrared spectra of (H$_2$ O)$_2$$^+$ and assigned its structural pattern as a charge-localized H$_3$ O$^+$ –OH complex .","Recently, Fujii's group reported the infrared spectroscopic observations of larger (H$_2$ O)$_n$$^+$ clusters, $n = 3-11$ , where the OH radical vibrational signal was clearly identified for $n\\leqq 5$ clusters, but vibrational signature of OH radical become inseparable due to the overlap with H-bonded OH stretch in $n>6$ .","As being evidenced in the earlier studies , , theoretical investigations such as ab initio electronic structure theory and density functional theory (DFT) play an important role in understanding the infrared spectroscopic features of the ionized water clusters.","Because high-level ab initio calculations are computationally prohibited for larger ionized water clusters, e.g.","fully solvated cationic moiety, a reliable DFT method is necessary.","In earlier theoretical reports, two minimum structures of the water dimer radical cation have been identified: the proton transferred structure and the hemibonded structure, as shown in Fig.","REF , , , .","The previous DFT calculations have shown that many exchange-correlation (XC) functionals fail to predict reasonable results , , given rise to the presence of hemibonding interaction.","The hemibonding interaction, could be theoretically located in (H$_2$ O)$_n$$^+$ systems, is notorious for the serious self-interaction errors (SIEs) associated with some density functional approximations.","Both local density approximation (LDA) and generalized gradient approximations (GGAs) were reported to contain non-negligible amount of SIEs for describing the hemibonded structure , .","It has been suggested to adopt hybrid functionals with larger fractions of the exact Hartree-Fock (HF) exchange for more accurate results in the hemibonded structure , .","However, as the SIEs of functionals become larger at the dissociation limit, these suggested functionals can yield spurious barriers on their dissociation curves , which can lead to unphysical results in molecular dynamics simulations.","Clearly, more stringent criteria for choosing suitable functionals are needed.","In this work, we propose three different criteria for a comprehensive comparison of different functionals for this system.","Figure: (a) The proton transferred structure, (b) the hemibonded structure, and (c) the transition state between the structures of (a) and (b)."],["Computational Methods","Calculations are performed on the optimized geometries of the two structures of the water dimer cation and the transition state between them, optimized with the ab initio MP2 theory and various XC functionals involving BLYP , , PBE , and M06L , which are pure density functionals (i.e.","the fraction of HF exchange $\\alpha _{\\mathrm {HF}}=0.00$ ), B97 with $\\alpha _{\\mathrm {HF}}=0.19$ , B3LYP , , with $\\alpha _{\\mathrm {HF}}=0.20$ , PBE0 with $\\alpha _{\\mathrm {HF}}=0.25$ , M06 with $\\alpha _{\\mathrm {HF}}=0.27$ , M05 with $\\alpha _{\\mathrm {HF}}=0.28$ , BH&HLYP with $\\alpha _{\\mathrm {HF}}=0.50$ , M06-2X with $\\alpha _{\\mathrm {HF}}=0.54$ , M05-2X with $\\alpha _{\\mathrm {HF}}=0.56$ , M06HF with $\\alpha _{\\mathrm {HF}}=1.00$ , the $\\omega \\mbox{B97}$ series ($\\omega \\mbox{B97}$ , $\\omega \\mbox{B97}$ X , $\\omega \\mbox{B97}$ X-D , and $\\omega \\mbox{B97}$ X-2(LP) ), which are long-range corrected (LC) hybrid functionals (i.e.","the fraction of HF exchange depends on the interelectronic distance ), and the double-hybrid functional B2PLYP with $\\alpha _{\\mathrm {HF}}=0.53$ .","The DFT and MP2 calculations are performed with the 6-311++G(3df,3pd) basis set, where the reference values of binding energy are obtained from Ref.",".","For efficiency, the resolution-of-identity (RI) approximation is used for calculations with the MP2 correlation (using sufficiently large auxiliary basis sets).","The CCSD(T) dissociation curves are calculated on the fixed monomer geometries (using the CCSD(T) optimized geometry of Ref.","), with the aug-cc-pVTZ basis set.","Note that although the ZPE corrected energy of the proton transferred structure (or, referred to as the Ion structure in Ref.",".)","is inconsistent with Ref.",".","However, adopting the geometries obtained from Ref.","yields results that are consistent with Ref.",".","All of the calculations are performed with the development version of Q-Chem 3.2 .","As the basis set superposition error (BSSE) for the ionized water dimer has been shown to be insignificant (if the diffuse basis functions are adopted) , , we do not perform BSSE correction throughout this paper."],["Results and Discussion","The ZPE corrected binding energies and relative energies of the water dimer cation calculated by various XC functionals are shown in Table REF and Table REF , respectively.","The calculated dissociation curves for the hemibonded structure are shown in Fig.","REF .","A summary of the results based on these three different criteria is shown in Table REF .","The notation used for characterizing statistical errors is as follows: Mean signed errors (MSEs), mean absolute errors (MAEs) and root-mean-square (RMS) errors."],["Criterion I: Binding Energies","Table REF shows the binding energies of the two structures of water dimer radical cation and the transition state between them.","The reference data obtained from Ref.","are based on the CCSD(T) calculations with the aug-cc-pVQZ basis set.","Since the errors of XC functionals for the hemibonded structure are much larger than those for the proton transferred structure, we focus our discussion on the hemibonded structure.","From Table REF , the functionals with MAE less than 2.5 kcal/mol are $\\omega \\mbox{B97X-2(LP)}$ , M06HF, and BH&HLYP.","Table REF also confirms the trend that has been mentioned previously: Functionals with larger fractions of HF exchange give more accurate results for the hemibonded structure.","To give reasonable results for the hemibonded structure, the $\\alpha _{\\mathrm {HF}}$ of a global hybrid functional should be at least larger than 0.43, as observed in Ref.","for the MPW1K functional.","Although M06HF, containing a full HF exchange, gives a small error of the hemibonded structure, it yields a large error for the proton transferred structure (due to the incomplete cancelation of errors between the exact exchange and semilocal correlation), as shown in Table REF .","Although functionals with $\\alpha _{\\mathrm {HF}}$ larger than 0.43 have been suggested, functionals with $\\alpha _{\\mathrm {HF}}\\ge 0.5$ are not always reliable, which can be observed from the errors of the hemibonded structure calculated by B2PLYP, M05-2X and M06-2X.","But this trend still holds: the results of the M05-2X and M06-2X are much better than those of the M05 and M06.","This means that although the energy of the hemibonded structure is sensitive to the $\\alpha _{\\mathrm {HF}}$ values in XC functionals, it may also be affected by the associated density functional approximations (DFAs).","Also note that some GGA functionals, such as BLYP and PBE, cannot predict that the transition state between the two structures of the water dimer radical cation.","The HF exchange included in the $\\omega \\mbox{B97}$ series is given by $E^{\\omega \\mathrm {B97~series} }_{\\mathrm {HF~exchange} }=E^{\\mathrm {LR-HF} }_x+C_xE^{\\mathrm {SR-HF} }_x,$ where $E^{\\mathrm {LR-HF}}_x=&-\\frac{1}{2}\\sum _{\\sigma }\\sum ^{occu.","}_{i,j}\\int \\int \\psi ^*_{i\\sigma }(\\textbf {r}_1)\\psi ^*_{j\\sigma }(\\textbf {r}_2)\\nonumber \\\\&\\times \\frac{\\mbox{erf}(\\omega r_{12}) }{r_{12}}\\psi _{j\\sigma }(\\textbf {r}_1)\\psi _{i\\sigma }(\\textbf {r}_2)d\\textbf {r}_1d\\textbf {r}_2,$ and $E^{\\mathrm {SR-HF} }_x=&-\\frac{1}{2}\\sum _{\\sigma }\\sum ^{occu.","}_{i,j}\\int \\int \\psi ^*_{i\\sigma }(\\textbf {r}_1)\\psi ^*_{j\\sigma }(\\textbf {r}_2)\\nonumber \\\\&\\times \\frac{\\mbox{erfc}(\\omega r_{12}) }{r_{12}}\\psi _{j\\sigma }(\\textbf {r}_1)\\psi _{i\\sigma }(\\textbf {r}_2)d\\textbf {r}_1d\\textbf {r}_2.$ Here $r_{12}\\equiv \\left|{\\bf r}_{12}\\right|=\\left|{\\bf r}_{1}-{\\bf r}_{2}\\right|$ (atomic units are used throughout this paper).","The parameter $\\omega $ defines the range of the splitting operators.","The coefficients for the $\\omega \\mbox{B97}$ series are listed in Table REF .","Since the fraction of HF exchange in $\\omega \\mbox{B97}$ series depends on the interelectronic distance $r_{12}$ , the trend mentioned previously is not as obvious as the global hybrid functionals.","But it is clear that the $\\omega \\mbox{B97X-2(LP)}$ , a LC double-hybrid functional, gives the most accurate results compare to the other functionals in the $\\omega \\mbox{B97}$ series.","Table: Appendix"]],"string":"[\n [\n \"Assessment of density functional approximations for the hemibonded\\n structure of water dimer radical cation\"\n ],\n [\n \"Abstract Due to the severe self-interaction errors associated with some density functional approximations, conventional density functionals often fail to dissociate the hemibonded structure of water dimer radical cation (H2O)2+ into the correct fragments: H2O and H2O+.\",\n \"Consequently, the binding energy of the hemibonded structure (H2O)2+ is not well-defined.\",\n \"For a comprehensive comparison of different functionals for this system, we propose three criteria: (i) The binding energies, (ii) the relative energies between the conformers of the water dimer radical cation, and (iii) the dissociation curves predicted by different functionals.\",\n \"The long-range corrected (LC) double-hybrid functional, omegaB97X-2(LP) [J.-D. Chai and M. Head-Gordon, J. Chem.\",\n \"Phys., 2009, 131, 174105.\",\n \"], is shown to perform reasonably well based on these three criteria.\",\n \"Reasons that LC hybrid functionals generally work better than conventional density functionals for hemibonded systems are also explained in this work.\"\n ],\n [\n \"Introduction\",\n \"Water can be decomposed when it is exposed to high-energy flux.\",\n \"The products of water radiolysis may contain various radical species, e.g.\",\n \"hydrogen atoms (H), hydroxide radicals (OH), oxygen anions (O$^-$ ), and water cations (H$_2$ O$^+$ ), depending on the radiation infrastructure setup.\",\n \"For example the overall decomposition scheme activated by $\\\\beta $ particles has been outlined by Garrett et al.\",\n \"in 2005 where three main channels of decomposition were listed.\",\n \"The cationic channel leads to the formation of ionized water living in about several tens femtoseconds and hydrated electron, followed by the generation of hydronium (H$_3$ O$^+$ ) and OH radicals through proton transfer process , .\",\n \"The energized-neutral and anionic channels could result in the cleavage the oxygen-hydrogen chemical bonds to produce hydrogen and oxygen derivatives, i.e.\",\n \"H, H$^-$ , H$_2$ , O, O$^-$ , OH$^-$ etc.\",\n \"Subsequent chemical reactions can progress further up to the desorption of stable gas molecules H$_2$ and O$_2$ being driven by those reactive radical species .\",\n \"The cationic channel is therefore particularly interesting due to its dominant products — OH radicals and solvated electrons.\",\n \"The smallest system to understand the chemical dynamics of ionized water is the water dimer radical cation (H$_2$ O)$_2$$^+$ , and it has been approached by several experimental studies in the past.\",\n \"Angel and Stace reported the predominant H$_3$ O$^+$ –OH central core by the collision-induced fragmentation experiment against the earlier theoretical assignment of charge-resonance hydrazine structure .\",\n \"Dong et al.\",\n \"observed a weak signal corresponding the formation of (H$_2$ O)$_2$$^+$ near the low-mass side of (H$_2$ O)$_2$ H$^+$ using 26.5 eV soft X-ray laser .\",\n \"Gardenier, Johnson, and McCoy reported the argon-tagged predissociation infrared spectra of (H$_2$ O)$_2$$^+$ and assigned its structural pattern as a charge-localized H$_3$ O$^+$ –OH complex .\",\n \"Recently, Fujii's group reported the infrared spectroscopic observations of larger (H$_2$ O)$_n$$^+$ clusters, $n = 3-11$ , where the OH radical vibrational signal was clearly identified for $n\\\\leqq 5$ clusters, but vibrational signature of OH radical become inseparable due to the overlap with H-bonded OH stretch in $n>6$ .\",\n \"As being evidenced in the earlier studies , , theoretical investigations such as ab initio electronic structure theory and density functional theory (DFT) play an important role in understanding the infrared spectroscopic features of the ionized water clusters.\",\n \"Because high-level ab initio calculations are computationally prohibited for larger ionized water clusters, e.g.\",\n \"fully solvated cationic moiety, a reliable DFT method is necessary.\",\n \"In earlier theoretical reports, two minimum structures of the water dimer radical cation have been identified: the proton transferred structure and the hemibonded structure, as shown in Fig.\",\n \"REF , , , .\",\n \"The previous DFT calculations have shown that many exchange-correlation (XC) functionals fail to predict reasonable results , , given rise to the presence of hemibonding interaction.\",\n \"The hemibonding interaction, could be theoretically located in (H$_2$ O)$_n$$^+$ systems, is notorious for the serious self-interaction errors (SIEs) associated with some density functional approximations.\",\n \"Both local density approximation (LDA) and generalized gradient approximations (GGAs) were reported to contain non-negligible amount of SIEs for describing the hemibonded structure , .\",\n \"It has been suggested to adopt hybrid functionals with larger fractions of the exact Hartree-Fock (HF) exchange for more accurate results in the hemibonded structure , .\",\n \"However, as the SIEs of functionals become larger at the dissociation limit, these suggested functionals can yield spurious barriers on their dissociation curves , which can lead to unphysical results in molecular dynamics simulations.\",\n \"Clearly, more stringent criteria for choosing suitable functionals are needed.\",\n \"In this work, we propose three different criteria for a comprehensive comparison of different functionals for this system.\",\n \"Figure: (a) The proton transferred structure, (b) the hemibonded structure, and (c) the transition state between the structures of (a) and (b).\"\n ],\n [\n \"Computational Methods\",\n \"Calculations are performed on the optimized geometries of the two structures of the water dimer cation and the transition state between them, optimized with the ab initio MP2 theory and various XC functionals involving BLYP , , PBE , and M06L , which are pure density functionals (i.e.\",\n \"the fraction of HF exchange $\\\\alpha _{\\\\mathrm {HF}}=0.00$ ), B97 with $\\\\alpha _{\\\\mathrm {HF}}=0.19$ , B3LYP , , with $\\\\alpha _{\\\\mathrm {HF}}=0.20$ , PBE0 with $\\\\alpha _{\\\\mathrm {HF}}=0.25$ , M06 with $\\\\alpha _{\\\\mathrm {HF}}=0.27$ , M05 with $\\\\alpha _{\\\\mathrm {HF}}=0.28$ , BH&HLYP with $\\\\alpha _{\\\\mathrm {HF}}=0.50$ , M06-2X with $\\\\alpha _{\\\\mathrm {HF}}=0.54$ , M05-2X with $\\\\alpha _{\\\\mathrm {HF}}=0.56$ , M06HF with $\\\\alpha _{\\\\mathrm {HF}}=1.00$ , the $\\\\omega \\\\mbox{B97}$ series ($\\\\omega \\\\mbox{B97}$ , $\\\\omega \\\\mbox{B97}$ X , $\\\\omega \\\\mbox{B97}$ X-D , and $\\\\omega \\\\mbox{B97}$ X-2(LP) ), which are long-range corrected (LC) hybrid functionals (i.e.\",\n \"the fraction of HF exchange depends on the interelectronic distance ), and the double-hybrid functional B2PLYP with $\\\\alpha _{\\\\mathrm {HF}}=0.53$ .\",\n \"The DFT and MP2 calculations are performed with the 6-311++G(3df,3pd) basis set, where the reference values of binding energy are obtained from Ref.\",\n \".\",\n \"For efficiency, the resolution-of-identity (RI) approximation is used for calculations with the MP2 correlation (using sufficiently large auxiliary basis sets).\",\n \"The CCSD(T) dissociation curves are calculated on the fixed monomer geometries (using the CCSD(T) optimized geometry of Ref.\",\n \"), with the aug-cc-pVTZ basis set.\",\n \"Note that although the ZPE corrected energy of the proton transferred structure (or, referred to as the Ion structure in Ref.\",\n \".)\",\n \"is inconsistent with Ref.\",\n \".\",\n \"However, adopting the geometries obtained from Ref.\",\n \"yields results that are consistent with Ref.\",\n \".\",\n \"All of the calculations are performed with the development version of Q-Chem 3.2 .\",\n \"As the basis set superposition error (BSSE) for the ionized water dimer has been shown to be insignificant (if the diffuse basis functions are adopted) , , we do not perform BSSE correction throughout this paper.\"\n ],\n [\n \"Results and Discussion\",\n \"The ZPE corrected binding energies and relative energies of the water dimer cation calculated by various XC functionals are shown in Table REF and Table REF , respectively.\",\n \"The calculated dissociation curves for the hemibonded structure are shown in Fig.\",\n \"REF .\",\n \"A summary of the results based on these three different criteria is shown in Table REF .\",\n \"The notation used for characterizing statistical errors is as follows: Mean signed errors (MSEs), mean absolute errors (MAEs) and root-mean-square (RMS) errors.\"\n ],\n [\n \"Criterion I: Binding Energies\",\n \"Table REF shows the binding energies of the two structures of water dimer radical cation and the transition state between them.\",\n \"The reference data obtained from Ref.\",\n \"are based on the CCSD(T) calculations with the aug-cc-pVQZ basis set.\",\n \"Since the errors of XC functionals for the hemibonded structure are much larger than those for the proton transferred structure, we focus our discussion on the hemibonded structure.\",\n \"From Table REF , the functionals with MAE less than 2.5 kcal/mol are $\\\\omega \\\\mbox{B97X-2(LP)}$ , M06HF, and BH&HLYP.\",\n \"Table REF also confirms the trend that has been mentioned previously: Functionals with larger fractions of HF exchange give more accurate results for the hemibonded structure.\",\n \"To give reasonable results for the hemibonded structure, the $\\\\alpha _{\\\\mathrm {HF}}$ of a global hybrid functional should be at least larger than 0.43, as observed in Ref.\",\n \"for the MPW1K functional.\",\n \"Although M06HF, containing a full HF exchange, gives a small error of the hemibonded structure, it yields a large error for the proton transferred structure (due to the incomplete cancelation of errors between the exact exchange and semilocal correlation), as shown in Table REF .\",\n \"Although functionals with $\\\\alpha _{\\\\mathrm {HF}}$ larger than 0.43 have been suggested, functionals with $\\\\alpha _{\\\\mathrm {HF}}\\\\ge 0.5$ are not always reliable, which can be observed from the errors of the hemibonded structure calculated by B2PLYP, M05-2X and M06-2X.\",\n \"But this trend still holds: the results of the M05-2X and M06-2X are much better than those of the M05 and M06.\",\n \"This means that although the energy of the hemibonded structure is sensitive to the $\\\\alpha _{\\\\mathrm {HF}}$ values in XC functionals, it may also be affected by the associated density functional approximations (DFAs).\",\n \"Also note that some GGA functionals, such as BLYP and PBE, cannot predict that the transition state between the two structures of the water dimer radical cation.\",\n \"The HF exchange included in the $\\\\omega \\\\mbox{B97}$ series is given by $E^{\\\\omega \\\\mathrm {B97~series} }_{\\\\mathrm {HF~exchange} }=E^{\\\\mathrm {LR-HF} }_x+C_xE^{\\\\mathrm {SR-HF} }_x,$ where $E^{\\\\mathrm {LR-HF}}_x=&-\\\\frac{1}{2}\\\\sum _{\\\\sigma }\\\\sum ^{occu.\",\n \"}_{i,j}\\\\int \\\\int \\\\psi ^*_{i\\\\sigma }(\\\\textbf {r}_1)\\\\psi ^*_{j\\\\sigma }(\\\\textbf {r}_2)\\\\nonumber \\\\\\\\&\\\\times \\\\frac{\\\\mbox{erf}(\\\\omega r_{12}) }{r_{12}}\\\\psi _{j\\\\sigma }(\\\\textbf {r}_1)\\\\psi _{i\\\\sigma }(\\\\textbf {r}_2)d\\\\textbf {r}_1d\\\\textbf {r}_2,$ and $E^{\\\\mathrm {SR-HF} }_x=&-\\\\frac{1}{2}\\\\sum _{\\\\sigma }\\\\sum ^{occu.\",\n \"}_{i,j}\\\\int \\\\int \\\\psi ^*_{i\\\\sigma }(\\\\textbf {r}_1)\\\\psi ^*_{j\\\\sigma }(\\\\textbf {r}_2)\\\\nonumber \\\\\\\\&\\\\times \\\\frac{\\\\mbox{erfc}(\\\\omega r_{12}) }{r_{12}}\\\\psi _{j\\\\sigma }(\\\\textbf {r}_1)\\\\psi _{i\\\\sigma }(\\\\textbf {r}_2)d\\\\textbf {r}_1d\\\\textbf {r}_2.$ Here $r_{12}\\\\equiv \\\\left|{\\\\bf r}_{12}\\\\right|=\\\\left|{\\\\bf r}_{1}-{\\\\bf r}_{2}\\\\right|$ (atomic units are used throughout this paper).\",\n \"The parameter $\\\\omega $ defines the range of the splitting operators.\",\n \"The coefficients for the $\\\\omega \\\\mbox{B97}$ series are listed in Table REF .\",\n \"Since the fraction of HF exchange in $\\\\omega \\\\mbox{B97}$ series depends on the interelectronic distance $r_{12}$ , the trend mentioned previously is not as obvious as the global hybrid functionals.\",\n \"But it is clear that the $\\\\omega \\\\mbox{B97X-2(LP)}$ , a LC double-hybrid functional, gives the most accurate results compare to the other functionals in the $\\\\omega \\\\mbox{B97}$ series.\",\n \"Table: Appendix\"\n ]\n]"},"id":{"kind":"string","value":"1204.1522"}}},{"rowIdx":1866994,"cells":{"text":{"kind":"list like","value":[["Some limit theorems for flows of branching processes"],["Abstract We construct two kinds of stochastic flows of discrete Galton-Watson branching processes.","Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line."],["Introduction","Continuous-state branching processes (CB-processes) arose as weak limits of rescaled discrete Galton-Watson branching processes; see, e.g., Jiřina (1958) and Lamperti (1967).","Continuous-state branching processes with immigration (CBI-processes) are generalizations of them describing the situation where immigrants may come from other sources of particles.","Those processes can be obtained as the scaling limits of discrete branching processes with immigration; see, e.g., Kawazu and Watanabe (1971) and Li (2006).","A CBI-process was constructed in Dawson and Li (2006) as the strong solution of a stochastic equation driven by Brownian motions and Poisson random measures; see also Fu and Li (2010).","A similar construction was given in Li and Ma (2008) using a stochastic equation driven by time-space Gaussian white noises and Poisson random measures.","In the study of scaling limits of coalescent processes with multiple collisions, Bertoin and Le Gall (2006) constructed a flow of jump-type CB-processes as the weak solution flow of a system of stochastic equations driven by Poisson random measures; see also Bertoin and Le Gall (2003, 2005).","A more general flow of CBI-processes was constructed in Dawson and Li (2012) as strong solutions of stochastic equations driven by Gaussian white noises and Poisson random measures.","The flows in Bertoin and Le Gall (2006) and Dawson and Li (2012) were also treated as path-valued processes with independent increments.","Motivated by the works of Aldous and Pitman (1998) and Abraham and Delmas (2010) on tree-valued Markov processes, another flow of CBI-processes was introduced in Li (2012), which was identified as a path-valued branching process.","From the flows in Bertoin and Le Gall (2006), Dawson and Li (2012) and Li (2012), one can define some superprocesses or immigration superprocesses over the positive half line with local and nonlocal branching mechanisms.","To study the genealogy trees for critical branching processes conditioned on non-extinction, Bakhtin (2011) considered a flow of continuous CBI-processes driven by a time-space Gaussian white noise.","He obtained the flow as a rescaling limit of systems of discrete Galton-Watson processes and also pointed out the connection of the model with a superprocess conditioned on non-extinction.","In this paper, we consider two flows of discrete Galton-Watson branching processes and show suitable rescaled sequences of the flows converge to the flows of Dawson and Li (2012) and Li (2012), respectively.","The main motivation of the work is to understand the connection between discrete and continuum tree-valued processes.","Our results generalize those of Bakhtin (2011) to flows of discontinuous CB-processes.","To simplify the presentation, we only treat models without immigration, but the arguments given here carry over to those with immigration.","We shall first prove limit theorems for the induced superprocesses, from which we derive the convergence of the finite-dimensional distributions of the path-valued branching processes.","In Section 2, we give a brief review of the flows of Dawson and Li (2012) and Li (2012).","In Section 3, we consider flows consisting of independent branching processes and show their scaling limit gives a flow of the type of Dawson and Li (2012).","The formulation and convergence of interactive flows were discussed in Section 4, which lead to a flow in the class studied in Li (2012).","Let $\\mathbb {N}$ = $\\lbrace 0,1,2,\\cdots \\rbrace $ and $\\mathbb {N}_+$ = $\\lbrace 1,2,\\cdots \\rbrace $ .","For any $a\\ge 0$ let $M[0,a]$ be the set of finite Borel measures on $[0,a]$ endowed with the topology of weak convergence.","We identify $M[0,a]$ with the set $F[0,a]$ of positive right continuous increasing functions on $[0,a]$ .","Let $B[0,a]$ be the Banach space of bounded Borel functions on $[0,a]$ endowed with the supremum norm $\\Vert \\cdot \\Vert $ .","Let $C[0,a]$ denote its subspace of continuous functions.","We use $B[0,a]^+$ and $C[0,a]^+$ to denote the subclasses of positive elements and $C[0,a]^{++}$ to denote the subset of $C[0,a]^+$ of functions bounded away from zero.","For $\\mu \\in M[0,a]$ and $f\\in B[0,a]$ write $\\langle \\mu , f\\rangle $ = $\\int f d\\mu $ if the integral exists."],["Local and nonlocal branching flows","In this section, we recall some results on constructions and characterizations of the flow of CB-processes and the associated superprocess.","It is well-known that the law of a CB-process is determined by its branching mechanism $\\phi $ , which is a function on $[0,\\infty )$ and has the representation $\\phi (z)=bz+\\frac{1}{2}\\sigma ^2z^2 +\\int _0^\\infty (e^{-zu}-1+zu)m(du),$ where $\\sigma \\ge 0$ and $b$ are constants and $(u\\wedge {u}^2)m(du)$ is a finite measure on $(0,\\infty )$ .","Let $W(ds,du)$ be a white noise on $(0,\\infty )^2$ based on $dsdu$ and $\\tilde{N}(ds,dz,du)$ a compensated Poisson random measure on $(0,\\infty )^3$ with intensity $dsm(dz)du$ .","By Theorem 3.1 of Dawson and Li (2012), a CB-process with branching mechanism $\\phi $ can be constructed as the pathwise unique strong solution $\\lbrace Y_t:t\\ge 0\\rbrace $ to the stochastic equation: $Y_t \\!\\!&=\\!\\!&Y_0 + \\sigma \\int _0^t\\int _0^{Y_{s-}}W(ds,du) - \\int _0^tbY_{s-}ds \\cr \\!\\!&\\!\\!&+ \\int _0^t\\int _0^{\\infty }\\int _0^{Y_{s-}}z \\tilde{N}(ds,dz,du).$ Let us fix a constant $a\\ge 0$ and a function $\\mu \\in F[0,a]$ .","Let $\\lbrace Y_t(q): t\\ge 0\\rbrace $ denote the solution to (REF ) with $Y_0(q) =\\mu (q)$ .","We can consider the solution flow $\\lbrace Y_t(q): t\\ge 0, q\\in [0,a]\\rbrace $ of (REF ).","As observed in Dawson and Li (2012), there is a version of the flow which is increasing in $q\\in [0,a]$ .","Moreover, we can regard $\\lbrace (Y_t(q))_{t\\ge 0}: q\\in [0,a]\\rbrace $ as a path-valued stochastic process with independent increments.","Let $\\lbrace Y_t: t\\ge 0\\rbrace $ denote the $M[0,a]$ -valued process so that $Y_t[0,q] = Y_t(q)$ for every $t\\ge 0$ and $q\\in [0,a]$ .","Then $\\lbrace Y_t: t\\ge 0\\rbrace $ is a càdlàag superprocess with branching mechanism $\\phi $ and trivial spatial motion; see Theorems 3.9 and 3.11 in Dawson and Li (2012).","For $\\lambda \\ge 0$ let $t\\mapsto v(t,\\lambda )$ be the unique locally bounded positive solution of $v(t,\\lambda )= \\lambda -\\int _0^t\\phi (v(s,\\lambda ))ds, \\qquad t\\ge 0.$ For any $f\\in B[0,a]^+$ define $x\\mapsto v(t,f)(x)$ by $v(t,f)(x) =v(t,f(x))$ .","Then the superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ has transition semigroup $(Q_t)_{t\\ge 0}$ on $M[0,a]$ defined by $\\int _{M[0,a]} e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,\\nu )=\\exp \\left\\lbrace -\\langle \\mu ,v(t,f)\\rangle \\right\\rbrace , \\qquad f\\in B[0,a]^+.$ By Proposition 3.1 in Li (2011) one can see that $v(t,f)\\in C[0,a]^{++}$ for every $f\\in C[0,a]^{++}$ .","Then it is easy to verify that $(Q_t)_{t\\ge 0}$ is a Feller semigroup.","We can define another branching flow.","For this purpose, let us consider an admissible family of branching mechanisms $\\lbrace \\phi _q: q\\in [0,a]\\rbrace $ , where $\\phi _q$ is given by (REF ) with parameters $(b,m)=(b_q,m_q)$ depending on $q\\in [0,a]$ .","Here by an admissible family we mean for each $z\\ge 0$ , the function $q\\mapsto \\phi _q(z)$ is decreasing and continuously differentiable with the derivative $\\psi _\\theta (z) =-(\\partial /\\partial \\theta )\\phi _{\\theta }(z)$ of the form $\\psi _\\theta (z)=h_{\\theta }z+\\int _0^{\\infty }(1-e^{-zu})n_{\\theta }(du),$ where $h_{\\theta }\\ge 0$ and $n_{\\theta }(du)$ is a $\\sigma $ -finite kernel from $[0,a]$ to $(0,\\infty )$ satisfying $\\sup _{0\\le \\theta \\le a}\\Big [h_{\\theta }+\\int _0^{\\infty }un_{\\theta }(du)\\Big ]<\\infty .$ Then we have $\\phi _q(z)=\\phi _0(z)-\\int _0^q \\psi _{\\theta }(z)d\\theta , \\qquad z\\ge 0.$ Let $m(dz,d\\theta )$ be the measure on $(0,\\infty )\\times [0,a]$ defined by $m([c,d]\\times [0,q])=m_q[c,d],\\quad q\\in [0,a], d>c>0.$ Suppose that $W(ds,du)$ is a white noise on $(0,\\infty )^2$ based on $dsdu$ and $\\tilde{N}(ds,dz,d\\theta ,du)$ is a compensated Poisson random measure on $(0,\\infty )^2\\times [0,a]\\times (0,\\infty )$ with intensity $dsm(dz,d\\theta )du$ .","By the results in Li (2012), for any $\\mu \\in F[0,a]$ the stochastic equation $Y_t(q) \\!\\!&=\\!\\!&\\mu (q)-b_q\\int _0^t Y_{s-}(q)ds+\\sigma \\int _0^t\\int _0^{Y_{s-}(q)}W(ds,du)\\cr \\!\\!&\\!\\!&+\\int _0^t\\int _0^{\\infty }\\int _{[0,q]}\\int _0^{Y_{s-}(q)}z \\tilde{N}(ds,dz,d\\theta ,du)$ has a unique solution flow $\\lbrace Y_t(q): t\\ge 0,q\\in [0,a]\\rbrace $ .","For each $q\\in [0,a]$ , the one-dimensional process $\\lbrace Y_t(q): t\\ge 0\\rbrace $ is a CB-process with branching mechanism $\\phi _q$ .","It was proved in Li (2011) that there is a version of the flow which is increasing in $q\\in [0,a]$ .","Moreover, we can also regard $\\lbrace (Y_t(q))_{t\\ge 0}: q\\in [0,a]\\rbrace $ as a path-valued branching process.","The solution flow of (REF ) also induces a càdlàag superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ with state space $M[0,a]$ .","Let $f\\mapsto \\Psi (\\cdot ,f)$ be the operator on $C^+[0,a]$ defined by $\\Psi (x,f)=\\int _{[0,a]} f(x\\vee \\theta )h_{\\theta }d \\theta +\\int _{[0,a]}d\\theta \\int _0^{\\infty }(1-e^{-zf(x\\vee \\theta )})n_{\\theta }(dz).$ The superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ has local branching mechanism $\\phi _0$ and nonlocal branching mechanism given by (REF ); see Theorem 6.2 in Li (2012).","Then the transition semigroup $(Q_t)_{t\\ge 0}$ of $\\lbrace Y_t: t\\ge 0\\rbrace $ is defined by $\\int _{M[0,a]}e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,d\\nu )=\\exp \\Big \\lbrace -\\langle \\mu ,V_tf\\rangle \\Big \\rbrace ,\\qquad f\\in C^+[0,a],$ where $t\\mapsto V_tf$ is the unique locally bounded positive solution of $V_tf(x)=f(x)-\\int _0^t [\\phi _0(V_sf(x))-\\Psi (x,V_sf)]ds,\\qquad t\\ge 0,x\\in [0,a].$ To study the scaling limit theorems of the discrete branching flows, we need to introduce a metric on $M[0,a]$ .","Let $\\lbrace h_0,h_1,h_2,\\cdots \\rbrace $ be a countable dense subset of $\\lbrace h\\in C[0,a]^+:\\Vert h\\Vert \\le 1\\rbrace $ with $h_0\\equiv 1$ .","For convenience we assume each $h_i$ is bounded away from zero.","Then $\\lbrace h_0,h_1,h_2,\\cdots \\rbrace \\subset C[0,a]^{++}$ .","Now we define a metric $\\rho $ on $M[0,a]$ by $\\rho (\\mu ,\\nu ) = \\sum _{i=0}^{\\infty }\\frac{1}{2^i}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |), \\qquad \\mu , \\nu \\in M[0,a].$ It is easy to see that the metric is compatible with the weak convergence topology of $M[0,a]$ .","In other words, we have $\\mu _n\\rightarrow \\mu $ in $M[0,a]$ if and only if $\\rho (\\mu _n,\\mu )\\rightarrow 0$ .","For $\\nu \\in M[0,a]$ , set $\\mbox{\\rm e}_{h_i}(\\nu )= \\mbox{\\rm e}^{-\\langle \\nu ,h_i\\rangle }$ .","Theorem 2.1 The metric space $(M[0,a], \\rho )$ is a locally compact Polish (complete and separable) space, and $\\lbrace \\mbox{\\rm e}_{h_{i}}:i=0,1,2,\\cdots \\rbrace $ is strongly separating points of $M[0,a]$ , that is, for every $\\nu \\in M[0,a]$ and $\\delta >0$ , there exists a finite set $\\lbrace \\mbox{\\rm e}_{h_{i_1}},\\mbox{\\rm e}_{h_{i_2}},\\cdots ,\\mbox{\\rm e}_{h_{i_k}}\\rbrace \\subset \\lbrace \\mbox{\\rm e}_{h_i}:i=0,1,2,\\cdots \\rbrace $ such that $\\inf _{\\mu :\\rho (\\mu ,\\nu )\\ge \\delta }\\max _{1\\le j\\le k}|\\mbox{\\rm e}_{h_{i_j}}(\\mu )-\\mbox{\\rm e}_{h_{i_j}}(\\nu )|>0.$ Proof.","By Li (2011, p.4 and p.7) we know $M[0,a]$ is separable and locally compact, so there is a complete metric on $M[0,a]$ compatible with the weak convergence topology.","The following argument shows the metric $\\rho $ defined above is complete.","Suppose $\\lbrace \\mu _n\\rbrace _{n\\ge 1}\\subset M[0,a]$ is a Cauchy sequence under $\\rho $ .","Then for every $m\\ge 1$ , $\\lbrace \\langle \\mu _n,h_m\\rangle \\rbrace _{n\\ge 1}$ is also a Cauchy sequence.","We denote the limit by $\\Phi (h_m)$ .","For $f\\in C[0,a]^+$ satisfying $\\Vert f\\Vert \\le 1$ , let $\\lbrace h_{i_k}\\rbrace _{k\\ge 1}\\subset \\lbrace h_0,h_1,h_2,\\cdots \\rbrace $ be a sequence so that $\\Vert h_{i_k}-f\\Vert \\rightarrow 0$ as $k\\rightarrow \\infty $ .","For $n\\ge m\\ge 1$ we have $\\limsup _{m,n\\rightarrow \\infty }|\\langle \\nu _n,f\\rangle -\\langle \\nu _m,f\\rangle |\\!\\!&\\le \\!\\!&\\limsup _{m,n\\rightarrow \\infty }\\Big [|\\langle \\nu _n,f\\rangle -\\langle \\nu _n,h_{i_k}\\rangle |\\cr \\!\\!&\\!\\!&+|\\langle \\nu _n,h_{i_k}\\rangle -\\langle \\nu _m,h_{i_k}\\rangle |+|\\langle \\nu _m,h_{i_k}\\rangle -\\langle \\nu _m,f\\rangle |\\Big ]\\cr \\!\\!&\\le \\!\\!&2\\Phi (1)\\Vert f-h_{i_k}\\Vert .$ Then letting $k\\rightarrow \\infty $ we have $\\limsup _{m,n\\rightarrow \\infty }|\\langle \\nu _n,f\\rangle -\\langle \\nu _m,f\\rangle |=0.$ By linearity the above relation holds for all $f\\in C[0,a]$ , so the limit $\\Phi (f)=\\lim \\limits _{n\\rightarrow \\infty }\\langle \\mu _n,f\\rangle $ exists for each $f\\in C[0,a]$ .","Clearly, $f\\rightarrow \\Phi (f)$ is a positive linear functional on $C[0,a]$ .","By the Riesz representation theorem there exists $\\mu \\in M[0,a]$ so that $\\langle \\mu ,f\\rangle =\\Phi (f)$ for every $f\\in C[0,a]$ .","By the construction of $\\Phi $ we have $\\mu _n\\rightarrow \\mu $ , therefore, $\\rho (\\mu _n,\\mu )\\rightarrow 0$ .","That proves the first assertion of the theorem.","For any $\\nu \\in M[0,a]$ and $\\delta \\ge 0$ , there exists an $N_0\\in \\mathbb {N}_+$ such that $\\sum _{i=N_0+1}^{\\infty }1/{2^i}<{\\delta }/{2}$ .","Consider $\\lbrace h_0,h_1,\\cdots ,h_{N_0}\\rbrace $ , for any $\\mu \\in M[0,a]$ satisfying $\\rho (\\mu ,\\nu )\\ge \\delta $ , we have $\\sum _{i=0}^{N_0}\\frac{1}{2^i}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |)\\ge \\frac{\\delta }{2},$ and thus, $\\sum _{i=0}^{N_0}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |)\\ge \\frac{\\delta }{2}.$ It follows that $|\\langle \\mu ,h_j\\rangle -\\langle \\nu ,h_j\\rangle |\\ge \\frac{\\delta }{2N_0}~~\\mbox{for some}~~0\\le j\\le N_0.$ Since $|\\mbox{\\rm e}^{-x}-\\mbox{\\rm e}^{-y}|=\\mbox{\\rm e}^{-y}|\\mbox{\\rm e}^{y-x}-1|\\ge \\mbox{\\rm e}^{-y}\\Big [(\\mbox{\\rm e}^{|y-x|}-1)\\wedge (1-\\mbox{\\rm e}^{-|y-x|})\\Big ],\\qquad x,y\\in \\mathbb {R},$ we have $\\!\\!&\\!\\!&\\inf _{\\mu :\\rho (\\mu ,\\nu )\\ge \\delta }\\max _{0\\le i\\le N_0}|\\mbox{\\rm e}_{h_{i}}(\\mu )-\\mbox{\\rm e}_{h_{i}}(\\nu )|\\cr \\!\\!&\\!\\!&\\qquad \\quad \\ge \\mbox{\\rm e}^{-\\max _{0\\le i\\le N_0}\\langle \\nu ,h_i\\rangle }\\Big [(\\mbox{\\rm e}^{\\frac{\\delta }{2N_0}}-1)\\wedge (1-\\mbox{\\rm e}^{-\\frac{\\delta }{2N_0}})\\Big ]\\cr \\!\\!&\\!\\!&\\qquad \\quad > 0.$ That proves the second assertion.","$\\Box $"],["Flows of independent branching processes","In this section, we consider some flows of independent Galton-Watson branching processes.","We shall study the scaling limit in the setting of superprocesses.","Then we derive the convergence of the finite-dimensional distributions of the path-valued processes.","Let $\\lbrace g_i: i=0,1,2,\\cdots \\rbrace $ be a family of probability generating functions.","Given a family of $\\mathbb {N}$ -valued independent random variables $\\lbrace X_0(i): i=0,1,2,\\cdots \\rbrace $ , for each $i\\in \\mathbb {N}$ suppose that there are $X_0(i)$ independent Galton-Watson trees originating at time 0 and at place $i$ with offspring distribution given by $g_i$ .","Let us denote by $X_n(i)$ the numbers of vertices in the $n$ -th generation of the trees with root at $i$ .","In addition, we assume $(X_n(i))_{n\\ge 0}$ , $i=1,2,\\cdots $ are mutually independent.","It is well-known that for each $i\\in \\mathbb {N}$ , $(X_n(i))_{n\\ge 0}$ is a Galton-Watson branching process (GW-process) with parameter $g_i$ ; i.e., a discrete-time $\\mathbb {N}$ -valued Markov chain with $n$ -step transition matrix $P^n(j,k)$ defined by $\\sum _{k=0}^{\\infty }P^n(j,k)z^k=(g_i^n(z))^j,\\qquad |z|\\le 1,$ where $g_i^n(z)$ is defined by $g_i^n(z)=g_i(g_i^{n-1}(z))$ successively with $g_i^0(z)=z$ .","Suppose that for each integer $k\\ge 1$ we have a sequence of GW-processes $\\lbrace (X_n^{(k)}(i))_{n\\ge 0}:i\\ge 0\\rbrace $ with parameter $g_i^{(k)}$ .","Let $\\gamma _k$ be a positive real sequence so that $\\gamma _k\\rightarrow \\infty $ increasingly as $k\\rightarrow \\infty $ .","For $m,n\\in \\mathbb {N}$ , define $\\bar{X}_n^{(k)}(m)=\\sum _{i=0}^m X_n^{(k)}(i),$ and $\\qquad \\qquad \\qquad \\qquad Y_t^{(k)}(x)=\\displaystyle \\frac{1}{k}\\bar{X}_{[\\gamma _kt]}^{(k)}([kx]),\\qquad k=1,2,\\cdots ,$ where $[\\cdot ]$ denotes the integer part.","Then the increasing function $x\\mapsto Y_t^{(k)}(x)$ induces a random measure $Y_t^{(k)}(dx)$ on $[0,\\infty )$ so that $Y_t^{(k)}([0,x])=Y_t^{(k)}(x)$ for $x\\ge 0$ .","For convenience we fix a constant $a\\ge 0$ and consider the restriction of $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ to $[0,a]$ without changing the notation.","Clearly, $Y_0^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_0^{(k)}(i)\\delta _{\\frac{i}{k}}$ and $Y_t^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_{[\\gamma _kt]}^{(k)}(i)\\delta _{\\frac{i}{k}}.$ In view of (REF ), for each $i\\ge 0$ , given $X_0^{(k)}(i)=x_i\\in \\mathbb {N}$ , the conditional distribution $Q_{i,k}^{[\\gamma _kt]}(x_i/k,\\cdot )$ of $\\lbrace k^{-1}X_{[\\gamma _kt]}^{(k)}(i):t\\ge 0\\rbrace $ on $E_k=\\lbrace 0,1/k,2/k,\\cdots \\rbrace $ is determined by $\\int _{E_k} e^{-\\lambda y}Q_{i,k}^{[\\gamma _kt]}({x_i}/{k},dy) =\\exp \\left\\lbrace -\\frac{x_i}{k}v_i^{(k)}(t,\\lambda )\\right\\rbrace ,$ where $v_i^{(k)}(t,\\lambda )=-k\\log (g_i^{(k)})^{[\\gamma _kt]}(e^{-\\lambda /k})$ .","Let $Q_{\\mu _k}^{(k)}$ denote the conditional law given $Y_0^{(k)}=\\mu _k=k^{-1}\\sum _{i=0}^{[ka]}x_i\\delta _{i/{k}}\\in M_k[0,a]$ , where $M_k[0,a]:=\\lbrace k^{-1}\\sum _{i=0}^{[ka]}x_i\\delta _{i/{k}}: ~x_i\\in {\\mathbb {N}}, ~{k}^{-1}\\sum _{i=0}^{[ka]}{x_i}<\\infty \\rbrace $ .","For $f\\in B[0,a]^+$ , from (REF ) we have $Q_{\\mu _k}^{(k)}\\exp \\Big \\lbrace -\\langle Y_t^{(k)},f\\rangle \\Big \\rbrace \\!\\!&=\\!\\!&Q_{\\mu _k}^{(k)}\\exp \\bigg \\lbrace -\\sum _{i=0}^{[ka]}\\frac{1}{k}X_{[\\gamma _kt]}^{(k)}(i)f{({i}/{k})}\\bigg \\rbrace \\cr \\!\\!&=\\!\\!&\\prod _{i=1}^{[ka]}\\int _{E_k}e^{-f{({i}/{k})}y}Q_{i,k}^{[\\gamma _kt]}({x_i}/{k},dy)\\cr \\!\\!&=\\!\\!&\\exp \\bigg \\lbrace -\\sum _{i=0}^{[ka]}\\frac{x_i}{k}v_i^{(k)}(t,f({i}/{k}))\\bigg \\rbrace \\cr \\!\\!&=\\!\\!&\\exp \\Big \\lbrace -\\langle \\mu _k,v^{(k)}(t,f)\\rangle \\Big \\rbrace ,$ where $x\\mapsto v^{(k)}(t,f)(x)$ is defined by $v^{(k)}(t,f)(x)=v_{[kx]}^{(k)}(t,f(x))$ .","For any $x, z\\ge 0$ define $\\phi _k(x,z)=k\\gamma _k[g_{[kx]}^{(k)}(e^{-z/k})-e^{-z/k}].$ For convenience of statement of the results, we formulate the following condition: Condition (3.A) For each $a\\ge 0$ the sequence $\\lbrace \\phi _k(x,z)\\rbrace $ is Lipschitz with respect to $z$ uniformly on $[0,\\infty )\\times [0,a]$ and there is a continuous function $(x,z)\\mapsto \\phi (x, z)$ such that $\\phi _k(x, z) \\rightarrow \\phi (x, z)$ uniformly on $[0,\\infty )\\times [0,a]$ as $k\\rightarrow \\infty $ .","Before giving the limit theorem for the sequence of the rescaled processes, we first introduce the limit process.","By Proposition 4.3 in Li (2011), if Condition (3.A) is satisfied, the limit function $\\phi $ has the representation $\\phi (x,z)=b(x)z+\\frac{1}{2}c(x)z^2 +\\int _0^\\infty (e^{-zu}-1+zu)m(x,du),\\qquad x,z\\ge 0.$ where $b$ is a bounded function on $[0,\\infty )$ and $c$ is a positive bounded function on $[0,\\infty )$ .","$(u\\wedge {u}^2)m(x,du)$ is a bounded kernel from $[0,\\infty )$ to $(0,\\infty )$ .","Conversely, for any continuous function $(x,z)\\mapsto \\phi (x, z)$ given by (REF ), we can construct a family of probability generating functions $\\lbrace g_i^{(k)}:i=0,1,2,\\cdots \\rbrace $ so that the sequence (REF ) satisfies Condition (3.A); see, e.g., Li (2011, p.93).","For any $l\\ge 0$ , let $B_l[0,\\infty )^+$ be the set of positive bounded functions on $[0,\\infty )$ satisfying $\\Vert f\\Vert \\le l$ .","By a modification of the proof of Theorem 3.42 in Li (2011), it is not hard to show that for each $T\\ge 0$ and $l\\ge 0$ , $v^{(k)}(t,f)(x)$ converges uniformly on the set $[0,T]\\times [0,\\infty )\\times B_l[0,\\infty )^+$ of $(t,x,f)$ to the unique locally bounded positive solution $(t,x)\\mapsto v(t,f)(x)$ of the evolution equation $v(t,f)(x)=f(x)-\\int _0^t\\phi (x,v(s,f)(x))ds.","$ Let $\\lbrace Y_t: t\\ge 0\\rbrace $ be the superprocess with state space $M[0,a]$ and transition semigroup $(Q_t)_{t\\ge 0}$ defined by $\\int _{M[0,a]} e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,\\nu )=\\exp \\left\\lbrace -\\langle \\mu ,v(t,f)\\rangle \\right\\rbrace , \\qquad f\\in B[0,a]^+.$ Using (REF ) and Gronwall's inequality one can see $x\\mapsto v(t,f)(x)$ is continuous on $[0,a]$ for every $f\\in C[0,a]^+$ .","Then by Proposition 3.1 in Li (2011) it is easy to see that $ v(t,f)\\in C[0,a]^{++}$ for every $f\\in C[0,a]^{++}$ .","From this and (REF ) it follows that $(Q_t)_{t\\ge 0}$ is a Feller semigroup.","Note that if $\\phi (x, z)= \\phi (z)$ independent of $x\\ge 0$ , then $(Q_t)_{t\\ge 0}$ is the same transition semigroup as that defined by (REF ) and (REF ).","In this case, the corresponding superprocess can be defined by the stochastic integral equation (REF ).","Let $D([0,\\infty ),M[0,a])$ denote the space of càdlàg paths from $[0,\\infty )$ to $M[0,a]$ furnished with the Skorokhod topology.","The proof of the next theorem is a modification of that of Theorem 3.43 in Li (2011).","Theorem 3.1 Suppose that Condition (3.A) is satisfied.","Let $\\lbrace Y_t:t\\ge 0\\rbrace $ be a càdlàg superprocess with transition semigroup $(Q_t)_{t\\ge 0}$ defined by (REF ) and (REF ).","If $Y_0^{(k)}$ converges to $Y_0$ in distribution on $M[0,a]$ , then $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ converges to $\\lbrace Y_t:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .","Proof.","For $f\\in C[0,a]^{++}$ and $\\nu \\in M[0,a]$ set $e_f(\\nu )=e^{-\\langle \\nu ,f\\rangle }$ .","Clearly, the function $\\nu \\mapsto e_f(\\nu )$ is continuous in $\\rho $ .","We denote by $D_1$ the linear span of $\\lbrace \\mbox{\\rm e}_f:f\\in C[0,a]^{++}\\rbrace $ .","By Theorem REF we have $D_1$ is an algebra strong separating the points of $M[0,a]$ .","Let $C_0(M[0,a])$ be the space of continuous functions on $M[0,a]$ vanishing at infinity.","Then $D_1$ is uniformly dense in $C_0(M[0,a])$ by the Stone-Weierstrass theorem; see, e.g., Hewitt and Stromberg (1975, pp.98-99).","On the other hand, for any $f\\in C[0,a]^{++}$ , since $v(t,f)$ is bounded away from zero and $v_k(t,f)(x)\\rightarrow v(t,f)(x)$ uniformly on $[0,\\infty )$ for every $t\\ge 0$ , we have $v_k(t,f)$ is also bounded away from zero for $k$ sufficiently large.","Without loss of generality we may assume $v_k(t,f)\\ge c$ and $v(t,f)\\ge c$ for some $c>0$ .","Let $Q_t^{(k)}$ denote the transition semigroup of $Y_t^{(k)}$ .","We get from (REF ) and (REF ) that, for any $M\\ge 0$ , $\\!\\!&\\!\\!&\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} e_f(\\nu )-Q_te_f(\\nu )\\right|\\cr \\!\\!&\\!\\!&\\qquad \\quad =\\sup _{\\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le \\sup _{\\langle \\nu ,1\\rangle \\le M\\atop \\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\qquad +\\sup _{\\langle \\nu ,1\\rangle > M\\atop \\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le \\sup _{\\langle \\nu ,1\\rangle \\le M\\atop \\nu \\in M_k[0,a]}|\\langle \\nu ,v_k(t,f)\\rangle -\\langle \\nu ,v(t,f)\\rangle | + \\sup _{\\langle \\nu ,1\\rangle > M\\atop \\nu \\in M_k[0,a]}2e^{-\\langle \\nu ,c\\rangle }\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le M\\Vert v_k(t,f)-v(t,f)\\Vert +2e^{-Mc}.$ Since $M\\ge 0$ was arbitrary, we have $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} e_f(\\nu )-Q_te_f(\\nu )\\right|=0$ for every $t\\ge 0$ .","Thus $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} F(\\nu )-Q_tF(\\nu )\\right|=0$ for every $t\\ge 0$ and $F\\in C_0(M[0,a])$ .","By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ converges to $\\lbrace Y_t:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .","$\\Box $ Let $\\lbrace 0\\le a_10$ for every $\\lambda >0$ and $t\\ge 0$ .","In view of (REF ), for any $f\\in C[0,a]^{++}$ we have $V_tf\\in C[0,a]^{++}$ for every $t\\ge 0$ .","Then $D_1$ is invariant under $(Q_t)_{t\\ge 0}$ , which is a core of the strong generator of $(Q_t)_{t\\ge 0}$ ; see, e.g., Ethier and Kurtz (1986, p.17).","In other words, the closure of $L$ generates $(Q_t)_{t\\ge 0}$ uniquely; see, e.g., Ethier and Kurtz (1986, p.15 and p.17).","Based on (REF ) and (REF ) one can see $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}|L_ke^{-\\langle \\nu ,f\\rangle }-Le^{-\\langle \\nu ,f\\rangle }|=0$ for every $f\\in C[0,a]^{++}$ , which implies $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}|L_kF(\\nu )-LF(\\nu )|=0$ for every $F\\in D_1$ .","By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ converges to the immigration superprocess $\\lbrace Y_{t,a}:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .","$\\Box $ Let $\\lbrace 0\\le a_1c>0.$ Suppose that $W(ds,du)$ is a white noise on $(0,\\\\infty )^2$ based on $dsdu$ and $\\\\tilde{N}(ds,dz,d\\\\theta ,du)$ is a compensated Poisson random measure on $(0,\\\\infty )^2\\\\times [0,a]\\\\times (0,\\\\infty )$ with intensity $dsm(dz,d\\\\theta )du$ .\",\n \"By the results in Li (2012), for any $\\\\mu \\\\in F[0,a]$ the stochastic equation $Y_t(q) \\\\!\\\\!&=\\\\!\\\\!&\\\\mu (q)-b_q\\\\int _0^t Y_{s-}(q)ds+\\\\sigma \\\\int _0^t\\\\int _0^{Y_{s-}(q)}W(ds,du)\\\\cr \\\\!\\\\!&\\\\!\\\\!&+\\\\int _0^t\\\\int _0^{\\\\infty }\\\\int _{[0,q]}\\\\int _0^{Y_{s-}(q)}z \\\\tilde{N}(ds,dz,d\\\\theta ,du)$ has a unique solution flow $\\\\lbrace Y_t(q): t\\\\ge 0,q\\\\in [0,a]\\\\rbrace $ .\",\n \"For each $q\\\\in [0,a]$ , the one-dimensional process $\\\\lbrace Y_t(q): t\\\\ge 0\\\\rbrace $ is a CB-process with branching mechanism $\\\\phi _q$ .\",\n \"It was proved in Li (2011) that there is a version of the flow which is increasing in $q\\\\in [0,a]$ .\",\n \"Moreover, we can also regard $\\\\lbrace (Y_t(q))_{t\\\\ge 0}: q\\\\in [0,a]\\\\rbrace $ as a path-valued branching process.\",\n \"The solution flow of (REF ) also induces a càdlàag superprocess $\\\\lbrace Y_t: t\\\\ge 0\\\\rbrace $ with state space $M[0,a]$ .\",\n \"Let $f\\\\mapsto \\\\Psi (\\\\cdot ,f)$ be the operator on $C^+[0,a]$ defined by $\\\\Psi (x,f)=\\\\int _{[0,a]} f(x\\\\vee \\\\theta )h_{\\\\theta }d \\\\theta +\\\\int _{[0,a]}d\\\\theta \\\\int _0^{\\\\infty }(1-e^{-zf(x\\\\vee \\\\theta )})n_{\\\\theta }(dz).$ The superprocess $\\\\lbrace Y_t: t\\\\ge 0\\\\rbrace $ has local branching mechanism $\\\\phi _0$ and nonlocal branching mechanism given by (REF ); see Theorem 6.2 in Li (2012).\",\n \"Then the transition semigroup $(Q_t)_{t\\\\ge 0}$ of $\\\\lbrace Y_t: t\\\\ge 0\\\\rbrace $ is defined by $\\\\int _{M[0,a]}e^{-\\\\langle \\\\nu ,f\\\\rangle }Q_t(\\\\mu ,d\\\\nu )=\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\mu ,V_tf\\\\rangle \\\\Big \\\\rbrace ,\\\\qquad f\\\\in C^+[0,a],$ where $t\\\\mapsto V_tf$ is the unique locally bounded positive solution of $V_tf(x)=f(x)-\\\\int _0^t [\\\\phi _0(V_sf(x))-\\\\Psi (x,V_sf)]ds,\\\\qquad t\\\\ge 0,x\\\\in [0,a].$ To study the scaling limit theorems of the discrete branching flows, we need to introduce a metric on $M[0,a]$ .\",\n \"Let $\\\\lbrace h_0,h_1,h_2,\\\\cdots \\\\rbrace $ be a countable dense subset of $\\\\lbrace h\\\\in C[0,a]^+:\\\\Vert h\\\\Vert \\\\le 1\\\\rbrace $ with $h_0\\\\equiv 1$ .\",\n \"For convenience we assume each $h_i$ is bounded away from zero.\",\n \"Then $\\\\lbrace h_0,h_1,h_2,\\\\cdots \\\\rbrace \\\\subset C[0,a]^{++}$ .\",\n \"Now we define a metric $\\\\rho $ on $M[0,a]$ by $\\\\rho (\\\\mu ,\\\\nu ) = \\\\sum _{i=0}^{\\\\infty }\\\\frac{1}{2^i}(1\\\\wedge |\\\\langle \\\\mu ,h_i\\\\rangle -\\\\langle \\\\nu ,h_i\\\\rangle |), \\\\qquad \\\\mu , \\\\nu \\\\in M[0,a].$ It is easy to see that the metric is compatible with the weak convergence topology of $M[0,a]$ .\",\n \"In other words, we have $\\\\mu _n\\\\rightarrow \\\\mu $ in $M[0,a]$ if and only if $\\\\rho (\\\\mu _n,\\\\mu )\\\\rightarrow 0$ .\",\n \"For $\\\\nu \\\\in M[0,a]$ , set $\\\\mbox{\\\\rm e}_{h_i}(\\\\nu )= \\\\mbox{\\\\rm e}^{-\\\\langle \\\\nu ,h_i\\\\rangle }$ .\",\n \"Theorem 2.1 The metric space $(M[0,a], \\\\rho )$ is a locally compact Polish (complete and separable) space, and $\\\\lbrace \\\\mbox{\\\\rm e}_{h_{i}}:i=0,1,2,\\\\cdots \\\\rbrace $ is strongly separating points of $M[0,a]$ , that is, for every $\\\\nu \\\\in M[0,a]$ and $\\\\delta >0$ , there exists a finite set $\\\\lbrace \\\\mbox{\\\\rm e}_{h_{i_1}},\\\\mbox{\\\\rm e}_{h_{i_2}},\\\\cdots ,\\\\mbox{\\\\rm e}_{h_{i_k}}\\\\rbrace \\\\subset \\\\lbrace \\\\mbox{\\\\rm e}_{h_i}:i=0,1,2,\\\\cdots \\\\rbrace $ such that $\\\\inf _{\\\\mu :\\\\rho (\\\\mu ,\\\\nu )\\\\ge \\\\delta }\\\\max _{1\\\\le j\\\\le k}|\\\\mbox{\\\\rm e}_{h_{i_j}}(\\\\mu )-\\\\mbox{\\\\rm e}_{h_{i_j}}(\\\\nu )|>0.$ Proof.\",\n \"By Li (2011, p.4 and p.7) we know $M[0,a]$ is separable and locally compact, so there is a complete metric on $M[0,a]$ compatible with the weak convergence topology.\",\n \"The following argument shows the metric $\\\\rho $ defined above is complete.\",\n \"Suppose $\\\\lbrace \\\\mu _n\\\\rbrace _{n\\\\ge 1}\\\\subset M[0,a]$ is a Cauchy sequence under $\\\\rho $ .\",\n \"Then for every $m\\\\ge 1$ , $\\\\lbrace \\\\langle \\\\mu _n,h_m\\\\rangle \\\\rbrace _{n\\\\ge 1}$ is also a Cauchy sequence.\",\n \"We denote the limit by $\\\\Phi (h_m)$ .\",\n \"For $f\\\\in C[0,a]^+$ satisfying $\\\\Vert f\\\\Vert \\\\le 1$ , let $\\\\lbrace h_{i_k}\\\\rbrace _{k\\\\ge 1}\\\\subset \\\\lbrace h_0,h_1,h_2,\\\\cdots \\\\rbrace $ be a sequence so that $\\\\Vert h_{i_k}-f\\\\Vert \\\\rightarrow 0$ as $k\\\\rightarrow \\\\infty $ .\",\n \"For $n\\\\ge m\\\\ge 1$ we have $\\\\limsup _{m,n\\\\rightarrow \\\\infty }|\\\\langle \\\\nu _n,f\\\\rangle -\\\\langle \\\\nu _m,f\\\\rangle |\\\\!\\\\!&\\\\le \\\\!\\\\!&\\\\limsup _{m,n\\\\rightarrow \\\\infty }\\\\Big [|\\\\langle \\\\nu _n,f\\\\rangle -\\\\langle \\\\nu _n,h_{i_k}\\\\rangle |\\\\cr \\\\!\\\\!&\\\\!\\\\!&+|\\\\langle \\\\nu _n,h_{i_k}\\\\rangle -\\\\langle \\\\nu _m,h_{i_k}\\\\rangle |+|\\\\langle \\\\nu _m,h_{i_k}\\\\rangle -\\\\langle \\\\nu _m,f\\\\rangle |\\\\Big ]\\\\cr \\\\!\\\\!&\\\\le \\\\!\\\\!&2\\\\Phi (1)\\\\Vert f-h_{i_k}\\\\Vert .$ Then letting $k\\\\rightarrow \\\\infty $ we have $\\\\limsup _{m,n\\\\rightarrow \\\\infty }|\\\\langle \\\\nu _n,f\\\\rangle -\\\\langle \\\\nu _m,f\\\\rangle |=0.$ By linearity the above relation holds for all $f\\\\in C[0,a]$ , so the limit $\\\\Phi (f)=\\\\lim \\\\limits _{n\\\\rightarrow \\\\infty }\\\\langle \\\\mu _n,f\\\\rangle $ exists for each $f\\\\in C[0,a]$ .\",\n \"Clearly, $f\\\\rightarrow \\\\Phi (f)$ is a positive linear functional on $C[0,a]$ .\",\n \"By the Riesz representation theorem there exists $\\\\mu \\\\in M[0,a]$ so that $\\\\langle \\\\mu ,f\\\\rangle =\\\\Phi (f)$ for every $f\\\\in C[0,a]$ .\",\n \"By the construction of $\\\\Phi $ we have $\\\\mu _n\\\\rightarrow \\\\mu $ , therefore, $\\\\rho (\\\\mu _n,\\\\mu )\\\\rightarrow 0$ .\",\n \"That proves the first assertion of the theorem.\",\n \"For any $\\\\nu \\\\in M[0,a]$ and $\\\\delta \\\\ge 0$ , there exists an $N_0\\\\in \\\\mathbb {N}_+$ such that $\\\\sum _{i=N_0+1}^{\\\\infty }1/{2^i}<{\\\\delta }/{2}$ .\",\n \"Consider $\\\\lbrace h_0,h_1,\\\\cdots ,h_{N_0}\\\\rbrace $ , for any $\\\\mu \\\\in M[0,a]$ satisfying $\\\\rho (\\\\mu ,\\\\nu )\\\\ge \\\\delta $ , we have $\\\\sum _{i=0}^{N_0}\\\\frac{1}{2^i}(1\\\\wedge |\\\\langle \\\\mu ,h_i\\\\rangle -\\\\langle \\\\nu ,h_i\\\\rangle |)\\\\ge \\\\frac{\\\\delta }{2},$ and thus, $\\\\sum _{i=0}^{N_0}(1\\\\wedge |\\\\langle \\\\mu ,h_i\\\\rangle -\\\\langle \\\\nu ,h_i\\\\rangle |)\\\\ge \\\\frac{\\\\delta }{2}.$ It follows that $|\\\\langle \\\\mu ,h_j\\\\rangle -\\\\langle \\\\nu ,h_j\\\\rangle |\\\\ge \\\\frac{\\\\delta }{2N_0}~~\\\\mbox{for some}~~0\\\\le j\\\\le N_0.$ Since $|\\\\mbox{\\\\rm e}^{-x}-\\\\mbox{\\\\rm e}^{-y}|=\\\\mbox{\\\\rm e}^{-y}|\\\\mbox{\\\\rm e}^{y-x}-1|\\\\ge \\\\mbox{\\\\rm e}^{-y}\\\\Big [(\\\\mbox{\\\\rm e}^{|y-x|}-1)\\\\wedge (1-\\\\mbox{\\\\rm e}^{-|y-x|})\\\\Big ],\\\\qquad x,y\\\\in \\\\mathbb {R},$ we have $\\\\!\\\\!&\\\\!\\\\!&\\\\inf _{\\\\mu :\\\\rho (\\\\mu ,\\\\nu )\\\\ge \\\\delta }\\\\max _{0\\\\le i\\\\le N_0}|\\\\mbox{\\\\rm e}_{h_{i}}(\\\\mu )-\\\\mbox{\\\\rm e}_{h_{i}}(\\\\nu )|\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad \\\\ge \\\\mbox{\\\\rm e}^{-\\\\max _{0\\\\le i\\\\le N_0}\\\\langle \\\\nu ,h_i\\\\rangle }\\\\Big [(\\\\mbox{\\\\rm e}^{\\\\frac{\\\\delta }{2N_0}}-1)\\\\wedge (1-\\\\mbox{\\\\rm e}^{-\\\\frac{\\\\delta }{2N_0}})\\\\Big ]\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad > 0.$ That proves the second assertion.\",\n \"$\\\\Box $\"\n ],\n [\n \"Flows of independent branching processes\",\n \"In this section, we consider some flows of independent Galton-Watson branching processes.\",\n \"We shall study the scaling limit in the setting of superprocesses.\",\n \"Then we derive the convergence of the finite-dimensional distributions of the path-valued processes.\",\n \"Let $\\\\lbrace g_i: i=0,1,2,\\\\cdots \\\\rbrace $ be a family of probability generating functions.\",\n \"Given a family of $\\\\mathbb {N}$ -valued independent random variables $\\\\lbrace X_0(i): i=0,1,2,\\\\cdots \\\\rbrace $ , for each $i\\\\in \\\\mathbb {N}$ suppose that there are $X_0(i)$ independent Galton-Watson trees originating at time 0 and at place $i$ with offspring distribution given by $g_i$ .\",\n \"Let us denote by $X_n(i)$ the numbers of vertices in the $n$ -th generation of the trees with root at $i$ .\",\n \"In addition, we assume $(X_n(i))_{n\\\\ge 0}$ , $i=1,2,\\\\cdots $ are mutually independent.\",\n \"It is well-known that for each $i\\\\in \\\\mathbb {N}$ , $(X_n(i))_{n\\\\ge 0}$ is a Galton-Watson branching process (GW-process) with parameter $g_i$ ; i.e., a discrete-time $\\\\mathbb {N}$ -valued Markov chain with $n$ -step transition matrix $P^n(j,k)$ defined by $\\\\sum _{k=0}^{\\\\infty }P^n(j,k)z^k=(g_i^n(z))^j,\\\\qquad |z|\\\\le 1,$ where $g_i^n(z)$ is defined by $g_i^n(z)=g_i(g_i^{n-1}(z))$ successively with $g_i^0(z)=z$ .\",\n \"Suppose that for each integer $k\\\\ge 1$ we have a sequence of GW-processes $\\\\lbrace (X_n^{(k)}(i))_{n\\\\ge 0}:i\\\\ge 0\\\\rbrace $ with parameter $g_i^{(k)}$ .\",\n \"Let $\\\\gamma _k$ be a positive real sequence so that $\\\\gamma _k\\\\rightarrow \\\\infty $ increasingly as $k\\\\rightarrow \\\\infty $ .\",\n \"For $m,n\\\\in \\\\mathbb {N}$ , define $\\\\bar{X}_n^{(k)}(m)=\\\\sum _{i=0}^m X_n^{(k)}(i),$ and $\\\\qquad \\\\qquad \\\\qquad \\\\qquad Y_t^{(k)}(x)=\\\\displaystyle \\\\frac{1}{k}\\\\bar{X}_{[\\\\gamma _kt]}^{(k)}([kx]),\\\\qquad k=1,2,\\\\cdots ,$ where $[\\\\cdot ]$ denotes the integer part.\",\n \"Then the increasing function $x\\\\mapsto Y_t^{(k)}(x)$ induces a random measure $Y_t^{(k)}(dx)$ on $[0,\\\\infty )$ so that $Y_t^{(k)}([0,x])=Y_t^{(k)}(x)$ for $x\\\\ge 0$ .\",\n \"For convenience we fix a constant $a\\\\ge 0$ and consider the restriction of $\\\\lbrace Y_t^{(k)}:t\\\\ge 0\\\\rbrace $ to $[0,a]$ without changing the notation.\",\n \"Clearly, $Y_0^{(k)}=\\\\frac{1}{k}\\\\sum _{i=0}^{[ka]}X_0^{(k)}(i)\\\\delta _{\\\\frac{i}{k}}$ and $Y_t^{(k)}=\\\\frac{1}{k}\\\\sum _{i=0}^{[ka]}X_{[\\\\gamma _kt]}^{(k)}(i)\\\\delta _{\\\\frac{i}{k}}.$ In view of (REF ), for each $i\\\\ge 0$ , given $X_0^{(k)}(i)=x_i\\\\in \\\\mathbb {N}$ , the conditional distribution $Q_{i,k}^{[\\\\gamma _kt]}(x_i/k,\\\\cdot )$ of $\\\\lbrace k^{-1}X_{[\\\\gamma _kt]}^{(k)}(i):t\\\\ge 0\\\\rbrace $ on $E_k=\\\\lbrace 0,1/k,2/k,\\\\cdots \\\\rbrace $ is determined by $\\\\int _{E_k} e^{-\\\\lambda y}Q_{i,k}^{[\\\\gamma _kt]}({x_i}/{k},dy) =\\\\exp \\\\left\\\\lbrace -\\\\frac{x_i}{k}v_i^{(k)}(t,\\\\lambda )\\\\right\\\\rbrace ,$ where $v_i^{(k)}(t,\\\\lambda )=-k\\\\log (g_i^{(k)})^{[\\\\gamma _kt]}(e^{-\\\\lambda /k})$ .\",\n \"Let $Q_{\\\\mu _k}^{(k)}$ denote the conditional law given $Y_0^{(k)}=\\\\mu _k=k^{-1}\\\\sum _{i=0}^{[ka]}x_i\\\\delta _{i/{k}}\\\\in M_k[0,a]$ , where $M_k[0,a]:=\\\\lbrace k^{-1}\\\\sum _{i=0}^{[ka]}x_i\\\\delta _{i/{k}}: ~x_i\\\\in {\\\\mathbb {N}}, ~{k}^{-1}\\\\sum _{i=0}^{[ka]}{x_i}<\\\\infty \\\\rbrace $ .\",\n \"For $f\\\\in B[0,a]^+$ , from (REF ) we have $Q_{\\\\mu _k}^{(k)}\\\\exp \\\\Big \\\\lbrace -\\\\langle Y_t^{(k)},f\\\\rangle \\\\Big \\\\rbrace \\\\!\\\\!&=\\\\!\\\\!&Q_{\\\\mu _k}^{(k)}\\\\exp \\\\bigg \\\\lbrace -\\\\sum _{i=0}^{[ka]}\\\\frac{1}{k}X_{[\\\\gamma _kt]}^{(k)}(i)f{({i}/{k})}\\\\bigg \\\\rbrace \\\\cr \\\\!\\\\!&=\\\\!\\\\!&\\\\prod _{i=1}^{[ka]}\\\\int _{E_k}e^{-f{({i}/{k})}y}Q_{i,k}^{[\\\\gamma _kt]}({x_i}/{k},dy)\\\\cr \\\\!\\\\!&=\\\\!\\\\!&\\\\exp \\\\bigg \\\\lbrace -\\\\sum _{i=0}^{[ka]}\\\\frac{x_i}{k}v_i^{(k)}(t,f({i}/{k}))\\\\bigg \\\\rbrace \\\\cr \\\\!\\\\!&=\\\\!\\\\!&\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\mu _k,v^{(k)}(t,f)\\\\rangle \\\\Big \\\\rbrace ,$ where $x\\\\mapsto v^{(k)}(t,f)(x)$ is defined by $v^{(k)}(t,f)(x)=v_{[kx]}^{(k)}(t,f(x))$ .\",\n \"For any $x, z\\\\ge 0$ define $\\\\phi _k(x,z)=k\\\\gamma _k[g_{[kx]}^{(k)}(e^{-z/k})-e^{-z/k}].$ For convenience of statement of the results, we formulate the following condition: Condition (3.A) For each $a\\\\ge 0$ the sequence $\\\\lbrace \\\\phi _k(x,z)\\\\rbrace $ is Lipschitz with respect to $z$ uniformly on $[0,\\\\infty )\\\\times [0,a]$ and there is a continuous function $(x,z)\\\\mapsto \\\\phi (x, z)$ such that $\\\\phi _k(x, z) \\\\rightarrow \\\\phi (x, z)$ uniformly on $[0,\\\\infty )\\\\times [0,a]$ as $k\\\\rightarrow \\\\infty $ .\",\n \"Before giving the limit theorem for the sequence of the rescaled processes, we first introduce the limit process.\",\n \"By Proposition 4.3 in Li (2011), if Condition (3.A) is satisfied, the limit function $\\\\phi $ has the representation $\\\\phi (x,z)=b(x)z+\\\\frac{1}{2}c(x)z^2 +\\\\int _0^\\\\infty (e^{-zu}-1+zu)m(x,du),\\\\qquad x,z\\\\ge 0.$ where $b$ is a bounded function on $[0,\\\\infty )$ and $c$ is a positive bounded function on $[0,\\\\infty )$ .\",\n \"$(u\\\\wedge {u}^2)m(x,du)$ is a bounded kernel from $[0,\\\\infty )$ to $(0,\\\\infty )$ .\",\n \"Conversely, for any continuous function $(x,z)\\\\mapsto \\\\phi (x, z)$ given by (REF ), we can construct a family of probability generating functions $\\\\lbrace g_i^{(k)}:i=0,1,2,\\\\cdots \\\\rbrace $ so that the sequence (REF ) satisfies Condition (3.A); see, e.g., Li (2011, p.93).\",\n \"For any $l\\\\ge 0$ , let $B_l[0,\\\\infty )^+$ be the set of positive bounded functions on $[0,\\\\infty )$ satisfying $\\\\Vert f\\\\Vert \\\\le l$ .\",\n \"By a modification of the proof of Theorem 3.42 in Li (2011), it is not hard to show that for each $T\\\\ge 0$ and $l\\\\ge 0$ , $v^{(k)}(t,f)(x)$ converges uniformly on the set $[0,T]\\\\times [0,\\\\infty )\\\\times B_l[0,\\\\infty )^+$ of $(t,x,f)$ to the unique locally bounded positive solution $(t,x)\\\\mapsto v(t,f)(x)$ of the evolution equation $v(t,f)(x)=f(x)-\\\\int _0^t\\\\phi (x,v(s,f)(x))ds.\",\n \"$ Let $\\\\lbrace Y_t: t\\\\ge 0\\\\rbrace $ be the superprocess with state space $M[0,a]$ and transition semigroup $(Q_t)_{t\\\\ge 0}$ defined by $\\\\int _{M[0,a]} e^{-\\\\langle \\\\nu ,f\\\\rangle }Q_t(\\\\mu ,\\\\nu )=\\\\exp \\\\left\\\\lbrace -\\\\langle \\\\mu ,v(t,f)\\\\rangle \\\\right\\\\rbrace , \\\\qquad f\\\\in B[0,a]^+.$ Using (REF ) and Gronwall's inequality one can see $x\\\\mapsto v(t,f)(x)$ is continuous on $[0,a]$ for every $f\\\\in C[0,a]^+$ .\",\n \"Then by Proposition 3.1 in Li (2011) it is easy to see that $ v(t,f)\\\\in C[0,a]^{++}$ for every $f\\\\in C[0,a]^{++}$ .\",\n \"From this and (REF ) it follows that $(Q_t)_{t\\\\ge 0}$ is a Feller semigroup.\",\n \"Note that if $\\\\phi (x, z)= \\\\phi (z)$ independent of $x\\\\ge 0$ , then $(Q_t)_{t\\\\ge 0}$ is the same transition semigroup as that defined by (REF ) and (REF ).\",\n \"In this case, the corresponding superprocess can be defined by the stochastic integral equation (REF ).\",\n \"Let $D([0,\\\\infty ),M[0,a])$ denote the space of càdlàg paths from $[0,\\\\infty )$ to $M[0,a]$ furnished with the Skorokhod topology.\",\n \"The proof of the next theorem is a modification of that of Theorem 3.43 in Li (2011).\",\n \"Theorem 3.1 Suppose that Condition (3.A) is satisfied.\",\n \"Let $\\\\lbrace Y_t:t\\\\ge 0\\\\rbrace $ be a càdlàg superprocess with transition semigroup $(Q_t)_{t\\\\ge 0}$ defined by (REF ) and (REF ).\",\n \"If $Y_0^{(k)}$ converges to $Y_0$ in distribution on $M[0,a]$ , then $\\\\lbrace Y_t^{(k)}:t\\\\ge 0\\\\rbrace $ converges to $\\\\lbrace Y_t:t\\\\ge 0\\\\rbrace $ in distribution on $D([0,\\\\infty ),M[0,a])$ .\",\n \"Proof.\",\n \"For $f\\\\in C[0,a]^{++}$ and $\\\\nu \\\\in M[0,a]$ set $e_f(\\\\nu )=e^{-\\\\langle \\\\nu ,f\\\\rangle }$ .\",\n \"Clearly, the function $\\\\nu \\\\mapsto e_f(\\\\nu )$ is continuous in $\\\\rho $ .\",\n \"We denote by $D_1$ the linear span of $\\\\lbrace \\\\mbox{\\\\rm e}_f:f\\\\in C[0,a]^{++}\\\\rbrace $ .\",\n \"By Theorem REF we have $D_1$ is an algebra strong separating the points of $M[0,a]$ .\",\n \"Let $C_0(M[0,a])$ be the space of continuous functions on $M[0,a]$ vanishing at infinity.\",\n \"Then $D_1$ is uniformly dense in $C_0(M[0,a])$ by the Stone-Weierstrass theorem; see, e.g., Hewitt and Stromberg (1975, pp.98-99).\",\n \"On the other hand, for any $f\\\\in C[0,a]^{++}$ , since $v(t,f)$ is bounded away from zero and $v_k(t,f)(x)\\\\rightarrow v(t,f)(x)$ uniformly on $[0,\\\\infty )$ for every $t\\\\ge 0$ , we have $v_k(t,f)$ is also bounded away from zero for $k$ sufficiently large.\",\n \"Without loss of generality we may assume $v_k(t,f)\\\\ge c$ and $v(t,f)\\\\ge c$ for some $c>0$ .\",\n \"Let $Q_t^{(k)}$ denote the transition semigroup of $Y_t^{(k)}$ .\",\n \"We get from (REF ) and (REF ) that, for any $M\\\\ge 0$ , $\\\\!\\\\!&\\\\!\\\\!&\\\\sup _{\\\\nu \\\\in M_k[0,a]}\\\\left|Q_t^{(k)} e_f(\\\\nu )-Q_te_f(\\\\nu )\\\\right|\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad =\\\\sup _{\\\\nu \\\\in M_k[0,a]}\\\\Big |\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v_k(t,f)\\\\rangle \\\\Big \\\\rbrace -\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v(t,f)\\\\rangle \\\\Big \\\\rbrace \\\\Big |\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad \\\\le \\\\sup _{\\\\langle \\\\nu ,1\\\\rangle \\\\le M\\\\atop \\\\nu \\\\in M_k[0,a]}\\\\Big |\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v_k(t,f)\\\\rangle \\\\Big \\\\rbrace -\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v(t,f)\\\\rangle \\\\Big \\\\rbrace \\\\Big |\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\qquad +\\\\sup _{\\\\langle \\\\nu ,1\\\\rangle > M\\\\atop \\\\nu \\\\in M_k[0,a]}\\\\Big |\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v_k(t,f)\\\\rangle \\\\Big \\\\rbrace -\\\\exp \\\\Big \\\\lbrace -\\\\langle \\\\nu ,v(t,f)\\\\rangle \\\\Big \\\\rbrace \\\\Big |\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad \\\\le \\\\sup _{\\\\langle \\\\nu ,1\\\\rangle \\\\le M\\\\atop \\\\nu \\\\in M_k[0,a]}|\\\\langle \\\\nu ,v_k(t,f)\\\\rangle -\\\\langle \\\\nu ,v(t,f)\\\\rangle | + \\\\sup _{\\\\langle \\\\nu ,1\\\\rangle > M\\\\atop \\\\nu \\\\in M_k[0,a]}2e^{-\\\\langle \\\\nu ,c\\\\rangle }\\\\cr \\\\!\\\\!&\\\\!\\\\!&\\\\qquad \\\\quad \\\\le M\\\\Vert v_k(t,f)-v(t,f)\\\\Vert +2e^{-Mc}.$ Since $M\\\\ge 0$ was arbitrary, we have $\\\\lim _{k\\\\rightarrow \\\\infty }\\\\sup _{\\\\nu \\\\in M_k[0,a]}\\\\left|Q_t^{(k)} e_f(\\\\nu )-Q_te_f(\\\\nu )\\\\right|=0$ for every $t\\\\ge 0$ .\",\n \"Thus $\\\\lim _{k\\\\rightarrow \\\\infty }\\\\sup _{\\\\nu \\\\in M_k[0,a]}\\\\left|Q_t^{(k)} F(\\\\nu )-Q_tF(\\\\nu )\\\\right|=0$ for every $t\\\\ge 0$ and $F\\\\in C_0(M[0,a])$ .\",\n \"By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\\\lbrace Y_t^{(k)}:t\\\\ge 0\\\\rbrace $ converges to $\\\\lbrace Y_t:t\\\\ge 0\\\\rbrace $ in distribution on $D([0,\\\\infty ),M[0,a])$ .\",\n \"$\\\\Box $ Let $\\\\lbrace 0\\\\le a_10$ for every $\\\\lambda >0$ and $t\\\\ge 0$ .\",\n \"In view of (REF ), for any $f\\\\in C[0,a]^{++}$ we have $V_tf\\\\in C[0,a]^{++}$ for every $t\\\\ge 0$ .\",\n \"Then $D_1$ is invariant under $(Q_t)_{t\\\\ge 0}$ , which is a core of the strong generator of $(Q_t)_{t\\\\ge 0}$ ; see, e.g., Ethier and Kurtz (1986, p.17).\",\n \"In other words, the closure of $L$ generates $(Q_t)_{t\\\\ge 0}$ uniquely; see, e.g., Ethier and Kurtz (1986, p.15 and p.17).\",\n \"Based on (REF ) and (REF ) one can see $\\\\lim _{k\\\\rightarrow \\\\infty }\\\\sup _{\\\\nu \\\\in M_k[0,a]}|L_ke^{-\\\\langle \\\\nu ,f\\\\rangle }-Le^{-\\\\langle \\\\nu ,f\\\\rangle }|=0$ for every $f\\\\in C[0,a]^{++}$ , which implies $\\\\lim _{k\\\\rightarrow \\\\infty }\\\\sup _{\\\\nu \\\\in M_k[0,a]}|L_kF(\\\\nu )-LF(\\\\nu )|=0$ for every $F\\\\in D_1$ .\",\n \"By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\\\lbrace Y_{t,a}^{(k)}:t\\\\ge 0\\\\rbrace $ converges to the immigration superprocess $\\\\lbrace Y_{t,a}:t\\\\ge 0\\\\rbrace $ in distribution on $D([0,\\\\infty ),M[0,a])$ .\",\n \"$\\\\Box $ Let $\\\\lbrace 0\\\\le a_1-1$ .","For the even-dimensional representations [13] ($j$ half-integer), this led to an alternative model of the finite oscillator with the spectrum of $\\hat{H}$ again given by $n+\\frac{1}{2}$ ($n=0,1,\\ldots ,2j$ ), and with the spectrum of the position operator $\\hat{q}$ given by $\\pm (\\alpha +k) \\qquad (k=1,2,\\ldots ,j+\\frac{1}{2}),$ i.e.","a finite and mostly equidistant spectrum apart from a gap of size $2\\alpha +2$ in the middle, see Figure 1(b).","For the odd-dimensional representations [14] ($j$ integer), this led to a similar model with the same spectrum of $\\hat{H}$ but with the spectrum of $\\hat{q}$ given by (Figure 1(c)) $0, \\quad \\pm \\sqrt{k(2\\alpha +k+1)}, \\qquad (k=1,\\ldots ,j).$ In both deformations, the position wave functions could be constructed explicitly and turned out to be related to normalized Hahn (or dual Hahn) polynomials [13], [14].","It was also shown how these wave functions could be interpreted as the finite version of a paraboson oscillator.","Recall however that Wigner [25] was among the first to drop the canonical commutation relation $[\\hat{q}, \\hat{p}] = \\mathrm {i}$ , proposing a model with relations (REF ) with the extra condition $\\hat{H} = \\frac{1}{2}(\\hat{p}^2 + \\hat{q}^2).$ This is known as the Wigner quantum oscillator (or paraboson oscillator) [19], [20].","This paraboson oscillator still has an infinite energy spectrum of the form $n+a$ (with a positive representation parameter $a$ ), and the spectrum of the position operator is the real axis.","The canonical oscillator is recovered from the paraboson oscillator in the representation with $a=1/2$ , i.e.","one of the representations of the paraboson oscillator coincides with the canonical oscillator [17], [18], [12].","The algebraic structure equivalent with (REF ) and (REF ) is the Lie superalgebra $\\mathfrak {osp}(1|2)$ [18], [12].","In this context, the position and momentum operators $\\hat{q}$ and $\\hat{p}$ are odd elements of the Lie superalgebra, whereas $\\hat{H}$ is an even element.","This observation, and the fact that the canonical oscillator fits in one of the $\\mathfrak {osp}(1|2)$ representations, leads to the idea that it is perhaps more natural to consider the position and momentum operators of alternative oscillator models as odd elements of a Lie superalgebra rather than just (even) elements of a Lie algebra.","Following this last idea, we propose and investigate in this paper a finite oscillator model based on the Lie superalgebra $\\mathfrak {sl}(2|1)$ .","Indeed, apart from $\\mathfrak {osp}(1|2)$ , $\\mathfrak {sl}(2|1)$ is the simplest Lie superalgebra that can be considered as a superversion of the Lie algebra $\\mathfrak {su}(2)$ [21].","The Lie superalgebra $\\mathfrak {sl}(2|1)$ has moreover a class of representations $W_j$ of dimension $2j+1$ ($j$ integer) which are similar to the common $\\mathfrak {su}(2)$ representations, see Section 2.","In the finite oscillator model studied here in Section 3, the Hamiltonian $\\hat{H}$ is a diagonal operator with spectrum $n+\\frac{1}{2}$ ($n=0,1,\\ldots ,2j$ ) in $W_j$ .","The position operator $\\hat{q}$ is an arbitrary (self-adjoint) odd element from $\\mathfrak {sl}(2|1)$ , and the form of the momentum operator $\\hat{p}$ follows from (REF ); the model introduces in a natural way a real parameter $p$ with $00}} U, \\qquad U_0 = \\frac{1}{\\sqrt{2}}\\left( \\begin{array}{ccccccc}\\cdots & 0 & 0 & \\sqrt{2} & 0 & 0 & \\cdots \\\\& 0 & -1 & 0 & 1 & 0 & \\\\& 0 & 1 & 0 & 1 & 0 & \\\\& -1 & 0 & 0 & 0 & 1 & \\\\\\cdots & 1 & 0 & 0 & 0 & 1 & \\cdots \\\\& \\vdots &&&& \\vdots &\\end{array}\\right),$ where the general form of the orthogonal matrix $U_0$ (with only two nonzero elements per row, starting from the second row onwards) is clear from the above.","The left limit $p\\rightarrow 1$ yields a similar matrix form for $U$ ."],["Position and momentum wave functions and their properties","The position (resp.","momentum) wave functions of the $\\mathfrak {sl}(2|1)$ finite oscillator are the overlaps between the $\\hat{q}$ -eigenvectors (resp.","$\\hat{p}$ -eigenvectors) and the $\\hat{H}$ -eigenvectors (or equivalently, the $J_0$ -eigenvectors $|j,m\\rangle $ ).","Let us denote them by $\\phi ^{(p)}_{j+m}(q)$ (resp.","$\\psi ^{(p)}_{j+m}(\\bar{p})$ ), where $m=j,j-1,\\ldots ,-j$ , and where $q$ (resp.","$\\bar{p}$ ) assumes one of the discrete values $q_k$ (resp.","$p_k$ ) $(k=-j,-j+1,\\ldots ,+j)$ .","Observe that we have denoted the momentum variable of the wave function by $\\bar{p}$ in order not to confuse with the parameter $p$ ($0-1$ , $\\delta >-1$ ): $\\sum _{x=0}^N \\bar{w}(x;\\gamma ,\\delta ,N) R_m(\\lambda (x);\\gamma ,\\delta ,N)R_n(\\lambda (x);\\gamma ,\\delta ,N)=\\bar{h}(n;\\gamma ,\\delta ,N)\\delta _{mn},$ with $\\bar{w}(x;\\gamma ,\\delta ,N)$ and $\\bar{h}(n;\\gamma ,\\delta ,N)$ given by [15].","Let us also fix a notation for the orthonormal functions: $\\tilde{R}_n(\\lambda (x);\\gamma ,\\delta ,N)= \\sqrt{ \\frac{\\bar{w}(x;\\gamma ,\\delta ,N)}{\\bar{h}(n;\\gamma ,\\delta ,N)} }R_n(\\lambda (x);\\gamma ,\\delta ,N).$ Consider now some positive parameter $\\alpha >0$ .","Let $j$ be the representation parameter, $n=0,1,\\ldots ,j$ , and $0-1$ .\",\n \"For the even-dimensional representations [13] ($j$ half-integer), this led to an alternative model of the finite oscillator with the spectrum of $\\\\hat{H}$ again given by $n+\\\\frac{1}{2}$ ($n=0,1,\\\\ldots ,2j$ ), and with the spectrum of the position operator $\\\\hat{q}$ given by $\\\\pm (\\\\alpha +k) \\\\qquad (k=1,2,\\\\ldots ,j+\\\\frac{1}{2}),$ i.e.\",\n \"a finite and mostly equidistant spectrum apart from a gap of size $2\\\\alpha +2$ in the middle, see Figure 1(b).\",\n \"For the odd-dimensional representations [14] ($j$ integer), this led to a similar model with the same spectrum of $\\\\hat{H}$ but with the spectrum of $\\\\hat{q}$ given by (Figure 1(c)) $0, \\\\quad \\\\pm \\\\sqrt{k(2\\\\alpha +k+1)}, \\\\qquad (k=1,\\\\ldots ,j).$ In both deformations, the position wave functions could be constructed explicitly and turned out to be related to normalized Hahn (or dual Hahn) polynomials [13], [14].\",\n \"It was also shown how these wave functions could be interpreted as the finite version of a paraboson oscillator.\",\n \"Recall however that Wigner [25] was among the first to drop the canonical commutation relation $[\\\\hat{q}, \\\\hat{p}] = \\\\mathrm {i}$ , proposing a model with relations (REF ) with the extra condition $\\\\hat{H} = \\\\frac{1}{2}(\\\\hat{p}^2 + \\\\hat{q}^2).$ This is known as the Wigner quantum oscillator (or paraboson oscillator) [19], [20].\",\n \"This paraboson oscillator still has an infinite energy spectrum of the form $n+a$ (with a positive representation parameter $a$ ), and the spectrum of the position operator is the real axis.\",\n \"The canonical oscillator is recovered from the paraboson oscillator in the representation with $a=1/2$ , i.e.\",\n \"one of the representations of the paraboson oscillator coincides with the canonical oscillator [17], [18], [12].\",\n \"The algebraic structure equivalent with (REF ) and (REF ) is the Lie superalgebra $\\\\mathfrak {osp}(1|2)$ [18], [12].\",\n \"In this context, the position and momentum operators $\\\\hat{q}$ and $\\\\hat{p}$ are odd elements of the Lie superalgebra, whereas $\\\\hat{H}$ is an even element.\",\n \"This observation, and the fact that the canonical oscillator fits in one of the $\\\\mathfrak {osp}(1|2)$ representations, leads to the idea that it is perhaps more natural to consider the position and momentum operators of alternative oscillator models as odd elements of a Lie superalgebra rather than just (even) elements of a Lie algebra.\",\n \"Following this last idea, we propose and investigate in this paper a finite oscillator model based on the Lie superalgebra $\\\\mathfrak {sl}(2|1)$ .\",\n \"Indeed, apart from $\\\\mathfrak {osp}(1|2)$ , $\\\\mathfrak {sl}(2|1)$ is the simplest Lie superalgebra that can be considered as a superversion of the Lie algebra $\\\\mathfrak {su}(2)$ [21].\",\n \"The Lie superalgebra $\\\\mathfrak {sl}(2|1)$ has moreover a class of representations $W_j$ of dimension $2j+1$ ($j$ integer) which are similar to the common $\\\\mathfrak {su}(2)$ representations, see Section 2.\",\n \"In the finite oscillator model studied here in Section 3, the Hamiltonian $\\\\hat{H}$ is a diagonal operator with spectrum $n+\\\\frac{1}{2}$ ($n=0,1,\\\\ldots ,2j$ ) in $W_j$ .\",\n \"The position operator $\\\\hat{q}$ is an arbitrary (self-adjoint) odd element from $\\\\mathfrak {sl}(2|1)$ , and the form of the momentum operator $\\\\hat{p}$ follows from (REF ); the model introduces in a natural way a real parameter $p$ with $00}} U, \\\\qquad U_0 = \\\\frac{1}{\\\\sqrt{2}}\\\\left( \\\\begin{array}{ccccccc}\\\\cdots & 0 & 0 & \\\\sqrt{2} & 0 & 0 & \\\\cdots \\\\\\\\& 0 & -1 & 0 & 1 & 0 & \\\\\\\\& 0 & 1 & 0 & 1 & 0 & \\\\\\\\& -1 & 0 & 0 & 0 & 1 & \\\\\\\\\\\\cdots & 1 & 0 & 0 & 0 & 1 & \\\\cdots \\\\\\\\& \\\\vdots &&&& \\\\vdots &\\\\end{array}\\\\right),$ where the general form of the orthogonal matrix $U_0$ (with only two nonzero elements per row, starting from the second row onwards) is clear from the above.\",\n \"The left limit $p\\\\rightarrow 1$ yields a similar matrix form for $U$ .\"\n ],\n [\n \"Position and momentum wave functions and their properties\",\n \"The position (resp.\",\n \"momentum) wave functions of the $\\\\mathfrak {sl}(2|1)$ finite oscillator are the overlaps between the $\\\\hat{q}$ -eigenvectors (resp.\",\n \"$\\\\hat{p}$ -eigenvectors) and the $\\\\hat{H}$ -eigenvectors (or equivalently, the $J_0$ -eigenvectors $|j,m\\\\rangle $ ).\",\n \"Let us denote them by $\\\\phi ^{(p)}_{j+m}(q)$ (resp.\",\n \"$\\\\psi ^{(p)}_{j+m}(\\\\bar{p})$ ), where $m=j,j-1,\\\\ldots ,-j$ , and where $q$ (resp.\",\n \"$\\\\bar{p}$ ) assumes one of the discrete values $q_k$ (resp.\",\n \"$p_k$ ) $(k=-j,-j+1,\\\\ldots ,+j)$ .\",\n \"Observe that we have denoted the momentum variable of the wave function by $\\\\bar{p}$ in order not to confuse with the parameter $p$ ($0-1$ , $\\\\delta >-1$ ): $\\\\sum _{x=0}^N \\\\bar{w}(x;\\\\gamma ,\\\\delta ,N) R_m(\\\\lambda (x);\\\\gamma ,\\\\delta ,N)R_n(\\\\lambda (x);\\\\gamma ,\\\\delta ,N)=\\\\bar{h}(n;\\\\gamma ,\\\\delta ,N)\\\\delta _{mn},$ with $\\\\bar{w}(x;\\\\gamma ,\\\\delta ,N)$ and $\\\\bar{h}(n;\\\\gamma ,\\\\delta ,N)$ given by [15].\",\n \"Let us also fix a notation for the orthonormal functions: $\\\\tilde{R}_n(\\\\lambda (x);\\\\gamma ,\\\\delta ,N)= \\\\sqrt{ \\\\frac{\\\\bar{w}(x;\\\\gamma ,\\\\delta ,N)}{\\\\bar{h}(n;\\\\gamma ,\\\\delta ,N)} }R_n(\\\\lambda (x);\\\\gamma ,\\\\delta ,N).$ Consider now some positive parameter $\\\\alpha >0$ .\",\n \"Let $j$ be the representation parameter, $n=0,1,\\\\ldots ,j$ , and $0\\Delta R=R_{xx}- as a function of field BB below 35 T with tilt angles θ\\theta from 0o to 35o at 2 K. The dashed lines indicate the SdH dips at Landau filling factor v=5v=5 and 6.","The different curves are offset for clarification.","(d) Position of dips with v=5v=5 and 6, as well as the position of peaks with v=5v=5, plotted against the tilt angle θ\\theta .","The data are consistent with the 1/|cos(θ)|1/|\\cos (\\theta )| dependence (red lines).","The inset shows the Fourier transform of the SdH oscillation which gives a single frequency of F=138(7)F=138(7) T.The response of the carriers to the applied magnetic field and the magnitude of MR is determined by the mobility in the plane perpendicular to the magnetic field.","[11] For nearly isotropic three-dimensional (3D) FS, there should be no significant angle-dependent MR (AMR).","In (quasi-)2D systems, 2D states will only respond to perpendicular component of the magnetic field $B|\\cos (\\theta )|$ and consequently longitudinal AMR and AMR-oscillation were observed in some quasi-2D conductors (such as Sr$_2$ RuO$_4$ and $\\beta -$ (BEDT-TTF)$_2$ I$_3$ ) and the surface states of TIs.","[21], [22], [14], [15] For example, the MR of the bulk state in topological insulator has only $\\sim 10\\%$ angular dependence while the angular dependence of MR in the surface state is about ten times larger.","[15] Significant AMR was also observed in some materials with highly anisotropic 3D FS such as Bi and Cu, but the period of AMR is determined by the shape of the Fermi surface and is very different from the one in 2D systems.","In Bi, electrons exhibit a threefold valley degeneracy and in-plane mass anisotropy, so the AMR peaks each time when the magnetic field is oriented along the bisectrix axis and has a 60o period.","[23] In Cu the AMR is more complex due to the complex FSs and peaks about every 25o.","[24] The electronic structure calculations in SrMnBi$_2$ show that the states near the Fermi energy $E_F$ are dominated by the Bi states in the Bi square nets.","Consequently the dominant FS should be quasi-2D.","Angular dependent resistivity in $B=9$ T and $T=2$ K shows wide maximum when the field is parallel to the $c$ -axis ($\\theta =0^o, 180^o$ ), and sharper minimum around $\\theta =90^o, 270^o$ (Fig.","2(a)).","The whole curve of AMR in SrMnBi$_2$ follows the function of $|\\cos (\\theta )|$ very well with a 180o period (red line in Fig.","2(a)).","Moreover, the larger $\\rho _{c}$ than $\\rho _{ab}$ in Fig.","1(b) implies that the transfer integral and the coupling between layers along the $c$ -axis is very small.","All this implies that the FSs in SrMnBi$_2$ should be highly anisotropic and that the mobility of carriers along $k_z$ is much smaller than the value in $k_xk_y$ plane.","Angular dependent MR quantum oscillations are directly related to the cross section of FS.","In Fig.","2(c), the in-plane $\\Delta R=R-$ measured using same configuration as shown in the inset of Fig.","2(a) exhibits clear Shubnikov-de Hass (SdH) oscillations with tilt angles $\\theta $ from 0o to 35o.","In metals, SdH oscillations correspond to successive emptying of LLs by the magnetic field and the LLs index $n$ is related to the cross section of FS $S_F$ by $2\\pi (n+\\gamma )=S_F\\frac{\\hbar }{eB}$ .","[25], [14], [15] For a 2D FS (a cylinder), the cross section has $S_F(\\theta )=S_0/|\\cos (\\theta )|$ angular dependence and the LLs positions should be inversely proportional to $|cos(\\theta )|$ .","[14], [25] The peak (dip) positions in SrMnBi$_2$ rapidly shift toward higher field direction with increase in $\\theta $ (as indicated by the dashed lines in Fig.","2(c)).","In Fig.","2(d), the dip positions corresponding to LLs $n=5, 6$ and the peak position with $n=5$ were plotted against the tilt angle $\\theta $ and can be described very well by $1/|\\cos (\\theta )|$ (the red lines in Fig.","2(d)).","Similar behavior was observed in the surface states of TIs [14], [15] and some other layered structures.","[25], [26] The Fourier transform of the SdH oscillation (inset of Fig.","2(d)) revealed that the oscillation component shows a periodic behavior in $1/B$ with a single frequency $F=138(7)$ T. The small value of frequency is consistent with previous value in Ref.","[7] and demonstrates that the dominant FSs are very small since the Onsager relation is $F=(\\Phi _0/2\\pi ^2)A_k$ where $\\Phi _0$ is the flux quantum and $A_k$ is the cross sectional area of FS.","[25] This clearly shows that the dominant two-dimensional FSs found in Fig.","2(b) are indeed the small FSs between $\\Gamma $ and $M$ points, rather than the large FS at $\\Gamma $ point in the Brillouin zone.","Above SdH oscillation combined with the angular MR clearly suggests that the dominant FSs of SrMnBi$_2$ are small quasi-2D cylinders along $k_z$ originating from Bi square nets.","In addition, there are still conventional parabolic bands with three dimensional characteristic close to the Fermi level,[7] causing the small deviation from quasi-2D transport.","Figure: (a) The magnetic field (BB) dependence of the in-plane magnetoresistance (MR =(ρ(B)-ρ(0))/ρ(0)=(\\protect \\rho (B)-\\protect \\rho (0))/\\protect \\rho (0)) at differenttemperatures.","(b) The field derivative of in-plane MR at different temperaturerespectively.","The lines in high field regions were fitting results using MR =A 1 B+O(B 2 )=A_1B+O(B^2) and the lines in low field regions using MR =A 2 B 2 =A_2B^2.Figure: (a) Temperature dependence of the critical field B * B^* (blacksquares) and the high field MR linear coefficient A 1 A_1 (blue circles) up to50 K. The red solid line is the fitting results using B * =1 2eℏv F 2 (E F +k B T) 2 B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2.","(b) Temperature dependence of theeffective MR mobility μ MR \\protect \\mu _{MR} extracted from the weak-field MR.Now we turn to the linear nonsaturated in-plane magnetoresistance in SrMnBi$_2$ (Fig.","3(a)).","The MR is linear over a wide field range up to 50 K. This behavior extends to very low fields where the MR naturally reduces to a weak-field semiclassical quadratic dependence.","The cross-over from the weak-field $B^2$ dependence to the high-field linear dependence can be best seen by considering the field derivative of the MR, $d$ MR$/dB$ (Fig.","3(b)).","In the low field range ($B<$ 1 T at 2 K), $d$ MR$/dB$ is proportional to $B$ (as shown by lines in low-field regions), indicating the semiclassical MR $\\sim A_2B^2$ .","But above a characteristic field $B^*$ , $d$ MR$/dB$ deviates from the semiclassical behavior and saturates to a much reduced slope (as shown by lines in the high-field region).","This indicates that the MR for $B>B^*$ is dominated by a linear field dependence plus a very small quadratic term (MR$=A_1B+O(B^2)$ ).","The linear MR deviates from the semiclassical $B^2$ dependence of MR in the low field region and a saturating MR in high fields.","[11] The unusual nonsaturating linear magnetoresistance has been reported in gapless semiconductor Ag$_{2-\\delta }$ (Te/Se) [27], [28] with the linear energy spectrum in the quantum limit.","[10], [28] Recent first principle calculations confirmed that these materials have a gapless Dirac-type surface state.","[29] Linear magnetoresistance is also observed in topological insulators [14], [15] and BaFe$_2$ As$_2$ [16] with Dirac fermion states.","Another possible origin of the large linear magnetoresistance is the mobility fluctuations in a strongly inhomogeneous system.","[30] This does not apply in SrMnBi$_2$ since our sample is stoichiometric crystal without doping/disorder.","Below we show that the nonsaturating linear magnetoresistance and the deviation from the semiclassical transport in SrMnBi$_2$ is due to the linear energy dispersion.","The application of a strong perpendicular external magnetic field ($B$ ) would lead to a complete quantization of the orbital motion of carriers with linear energy dispersion, resulting in quantized LLs $E_n=sgn(n)v_F\\sqrt{2e\\hbar B|n|}$ where $n=0,\\pm 1,\\pm 2,\\cdots $ is the LL index and $v_F$ is the Fermi velocity.","[12], [13] Then the energy splitting between the lowest and 1st LLs is described by $\\triangle _{LL}=\\pm v_F\\sqrt{2e\\hbar B}$ .","[12], [13] In the quantum limit at specific temperature and field, $\\triangle _{LL}$ becomes larger than both the Fermi energy $E_F$ and the thermal fluctuations $k_BT$ at a finite temperature.","Consequently all carriers occupy the lowest Landau level and eventually the quantum transport with linear magnetoresistance shows up.","The critical field $B^*$ above which the quantum limit is satisfied at specific temperature $T$ is $B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2$ .","[16] The splitting of LLs in conventional parabolic bands is $\\triangle _{LL}=\\frac{e\\hbar B}{m^*}$ .","Hence the evolution of $\\triangle _{LL}$ with field for parabolic bands is much slower than that for Dirac fermion states, and it is difficult to observe quantum limit behavior in the moderate field range.","The temperature dependence of critical field $B^*$ in SrMnBi$_2$ clearly deviates from the linear relationship and can be well fitted by $B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2$ , as shown in Fig.","4(a).","The fitting gives the Fermi velocity $v_F\\sim 5.13\\times 10^5$ ms$^{-1}$ and $\\Delta _1\\sim 4.97$ meV.","This confirms the existence of Dirac fermion states in SrMnBi$_2$ .","In a multiband system with both Dirac and conventional parabolic-band carriers (including electrons and holes) the magnetoresistance in the semiclassical transport can be described as $MR=\\frac{\\sigma _e\\sigma _h(\\mu _e+\\mu _h)^2}{(\\sigma _e+\\sigma _h)^2}B^2$ where $\\sigma _e, \\sigma _h, \\mu _e, \\mu _h$ are the effective electron and hole conductivity and mobility in zero field respectively, when the Dirac carriers are dominant in transport.","[16], [17] Then the coefficient of the low-field $B^2$ quadratic term, $A_2$ , is related to the effective MR mobility $\\sqrt{A_2}=\\frac{\\sqrt{\\sigma _e\\sigma _h}}{\\sigma _e+\\sigma _h}(\\mu _e+\\mu _h)=\\mu _{MR}$ , which is smaller than the average mobility of carriers $\\mu _{ave}=\\frac{\\mu _e+\\mu _h}{2}$ and gives an estimate of the lower bound.","Fig.","4(b) shows the dependence of $\\mu _{MR}$ on the temperature.","At 2 K, the value of $\\mu _{MR}$ is about 3400 cm$^2$ /Vs.","The large effective MR mobility also implies that Dirac fermions dominate the transport behavior.","With increase in temperature, the value of $\\mu _{MR}$ and the coefficient of high-field linear term $A_1$ (Fig.","4(a)) decrease sharply.","This is due to thermal fluctuation smearing out the LL splitting.","In summary, we demonstrate quantum transport of 2D Dirac fermion states in bulk SrMnBi$_2$ single crystals.","The bands with linear energy dispersion lead to the large nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.","The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.","With increase in temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.","The effective magnetoresistant mobility $\\mu _{MR}\\sim 3400$ cm$^2$ /Vs comparable to values observed in graphene is observed.","The angle-dependence of magnetoresistance shows a $|\\cos (\\theta )|$ dependence while the LL positions in SdH oscillations are inversely proportional to $|\\cos (\\theta )|$ , indicating the dominant quasi-2D Fermi surfaces.","Our results show that the crystals with Bi square nets can host phenomena commonly observed so far in 2D structures and materials like graphene.","We than John Warren for help with SEM measurements.","Work at Brookhaven is supported by the U.S. DOE under contract No.","DE-AC02-98CH10886.","Work at the National High Magnetic Field Laboratory is supported by the DOE NNSA DEFG52-10NA29659 (S. W. T and D. G.), by the NSF Cooperative Agreement No.","DMR-0654118 and by the state of Florida."]],"string":"[\n [\n \"Quantum transport of two-dimensional Dirac fermions in SrMnBi2\"\n ],\n [\n \"Abstract We report two-dimensional quantum transport in SrMnBi$_2$ single crystals.\",\n \"The linear energy dispersion leads to the unusual nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.\",\n \"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.\",\n \"With increase in the temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.\",\n \"The effective magnetoresistant mobility $\\\\mu_{MR}\\\\sim 3400$ cm$^2$/Vs is derived.\",\n \"Angular dependent magnetoresistance and quantum oscillations suggest dominant two-dimensional (2D) Fermi surfaces.\",\n \"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ and imply that bulk crystals with Bi square nets can be used to study low dimensional electronic transport commonly found in 2D materials like graphene.\"\n ],\n [\n \"Quantum transport of two-dimensional Dirac fermions in SrMnBi$_2$ Kefeng Wang Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA D. Graf National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA Hechang Lei Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA S. W. Tozer National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA C. Petrovic Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA We report two-dimensional quantum transport in SrMnBi$_2$ single crystals.\",\n \"The linear energy dispersion leads to the unusual nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.\",\n \"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.\",\n \"With increase in the temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.\",\n \"The effective magnetoresistant mobility $\\\\mu _{MR}\\\\sim 3400$ cm$^2$ /Vs is derived.\",\n \"Angular dependent magnetoresistance and quantum oscillations suggest dominant two-dimensional (2D) Fermi surfaces.\",\n \"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ and imply that bulk crystals with Bi square nets can be used to study low dimensional electronic transport commonly found in 2D materials like graphene.\",\n \"72.20.My,72.80.-r,75.47.Np Dirac fermions have raised great interest in condensed matter physics, as seen on the example of materials such as graphene [1] and topological insulators (TIs).\",\n \"[2] The linear dispersion between momentum and energy of Dirac fermions brings forth some spectacular properties, such as zero effective mass and large transport mobility.\",\n \"[1], [2] In addition to the surface/interface states in TIs and graphene, Dirac states in bulk materials were discussed in organic conductors [3] and iron-based superconductors such as BaFe$_2$ As$_2$ .\",\n \"[4], [5] Recently, highly anisotropic Dirac states were observed in SrMnBi$_2$ ,[6], [7] where linear energy dispersion originates from the crossing of two Bi $6p_{x,y}$ bands in the double-sized Bi square nets.\",\n \"SrMnBi$_2$ has a crystal structure similar to that of the superconducting Fe pnictides and is a bad metal.\",\n \"[7], [8] The Fermi velocity along $\\\\Gamma -M$ symmetry line is $\\\\nu _{F}^{\\\\parallel }\\\\approx 1.51\\\\times 10^{6}$ m/s, whereas the Fermi velocity in the orthogonal direction experiences nearly one order of magnitude decrease.\",\n \"[7], [8] One of the interesting properties of Dirac materials is the quantum transport phenomena.\",\n \"[9], [10] Unlike the conventional electron gas with parabolic energy dispersion, where Landau levels (LLs) are equidistant,[11] the distance between the lowest and 1st LLs of Dirac fermions in magnetic field is very large and the quantum limit where all of the carriers occupy only the lowest LL is easily realized under moderate fields.\",\n \"[12], [13] Consequently some quantum transport phenomena such as quantum Hall effect and large linear magnetoresistance (MR) could be observed by conventional experimental methods in Dirac fermion system.\",\n \"[14], [15], [16], [17] Here we show two-dimensional (2D) quantum transport in bulk SrMnBi$_2$ single crystals.\",\n \"The linear energy dispersion leads to the unusual nonsaturated linear MR since all Dirac fermions occupy the lowest LL in the quantum limit.\",\n \"The transverse MR exhibits a crossover at a critical field $B^*$ from semiclassical weak-field MR $\\\\sim $ $B^2$ to the high-field MR $\\\\sim $ B dependence.\",\n \"The critical field $B^*$ increases with the increase in temperature and its temperature dependence satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.\",\n \"Angular dependent MR and oscillation indicates the quasi-2D Fermi surfaces (FS).\",\n \"We derive the effective magnetoresistant mobility $\\\\mu _{MR}\\\\sim 3400$ cm$^2$ /Vs.\",\n \"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ .\",\n \"Single crystals of SrMnBi$_2$ were grown using a self-flux method.\",\n \"[18] Stoichiometric mixtures of Sr (99.99$\\\\%$ ), Mn (99.9$\\\\%$ ) and excess Bi (99.99$\\\\%$ ) with ratio Sr:Mn:Bi=1:1:9 were sealed in a quartz tube, heated to 1050 $^{\\\\circ }C$ and cooled to 450 $^{\\\\circ }C$ where the crystals were decanted.\",\n \"X-ray diffraction (XRD) data were taken with Cu K$_{\\\\alpha }$ ($\\\\lambda =0.15418$ nm) radiation of Rigaku Miniflex powder diffractometer.\",\n \"Transport measurements were conducted in a Quantum Design PPMS-9 with conventional four-wire method.\",\n \"The crystal was cleaved to a rectangular shape with dimension 4$\\\\times $ 1 mm$^{2}$ in the ab-plane and 0.2 mm thickness along the c-axis.\",\n \"For in-plane resistivity $\\\\rho _{ab}(T)$ , the current path was in the ab-plane, whereas magnetic field was perpendicular to the current and parallel to the c-axis except in the rotator experiments.\",\n \"The $c$ -axis resistivity $\\\\rho _c(T)$ was measured by attaching current and voltage wires to the opposite sides of the plate-like crystals.\",\n \"[19] High field MR oscillation were performed at National High Magnetic Field Laboratory in the same configuration to the in-plane MR.\",\n \"Figure: (a) Powder XRD patterns and structural refinement results.\",\n \"The data were shown by (++) , and the fit is given by the red solid line.\",\n \"The difference curve (the green solid line) is offset.\",\n \"The inset shows the single crystal XRD pattern indicating the c-axis orientation of crystal.\",\n \"(b) Temperature dependence of the in-plane resistivity ρ ab (T)\\\\protect \\\\rho _{ab}(T) (open symbols) and cc-axis resistivity ρ c (T)\\\\rho _c(T) (filled symbols) in B=0B=0 T (squares) and B=9B=9 T (circles) magnetic field respectively.All powder and single crystal XRD reflections can be indexed in the I4/mmm space group by RIETICA software (Fig.\",\n \"1(a)).\",\n \"[20] The determined lattice parameters are $a=b=0.4561(8)$ nm and $c=2.309(6)$ nm in agreement with the published data.\",\n \"[6] The in-plane resistivity $\\\\rho _{ab}(T)$ shown in Fig.\",\n \"1(b) exhibits a metallic behavior.\",\n \"An external magnetic field enhances the resistivity.\",\n \"As the temperature is increased, MR is gradually suppressed and is rather small above $\\\\sim 60$ K. Resistivity along the $c$ -axis ($\\\\rho _c(T)$ ) is nearly two orders of magnitude larger than $\\\\rho _{ab}(T)$ and exhibits a weak crossover at high temperature.\",\n \"In what follows we will discuss in-plane MR. Angular dependent MR $\\\\rho (B,\\\\theta )$ at $T\\\\sim 2$ K is shown in Fig.\",\n \"2(a)-(b).\",\n \"The crystal was mounted on a rotating stage such that the tilt angle $\\\\theta $ between sample surface ($ab$ -plane) and the magnetic field can be continuously changed, with currents flowing in the $ab$ -plane perpendicular to magnetic field (inset in Fig.\",\n \"2(a)).\",\n \"The magnetoresistance of SrMnBi$_2$ exhibits significant angular dependence (Fig.\",\n \"2(a,b)).\",\n \"When $B$ is parallel to the $c$ -axis ($\\\\theta =0^{o}$ ), the MR is maximized and is linear above a characteristic field ($\\\\sim 1$ T).\",\n \"With increase in the tilt angle $\\\\theta $ , MR gradually decreases and becomes nearly negligible for $B$ in the $ab$ -plane ($\\\\theta =90^o$ ).\",\n \"Figure: (a) In-plane resistivity ρ\\\\rho vs. the tilt angle θ\\\\protect \\\\theta from 0o to 360o at BB = 3, 6 and 9 T and TT = 2 K. The red line is the fitting curve using |cos(θ)||\\\\cos (\\\\protect \\\\theta )| (see text).\",\n \"Inset shows the configuration of the measurement.\",\n \"(b) In-plane Resistivity ρ\\\\protect \\\\rho vs. magnetic field BB with different tilt angles θ\\\\protect \\\\theta at 2 K. (c) MR SdH oscillations ΔR=R xx -\\\\Delta R=R_{xx}- as a function of field BB below 35 T with tilt angles θ\\\\theta from 0o to 35o at 2 K. The dashed lines indicate the SdH dips at Landau filling factor v=5v=5 and 6.\",\n \"The different curves are offset for clarification.\",\n \"(d) Position of dips with v=5v=5 and 6, as well as the position of peaks with v=5v=5, plotted against the tilt angle θ\\\\theta .\",\n \"The data are consistent with the 1/|cos(θ)|1/|\\\\cos (\\\\theta )| dependence (red lines).\",\n \"The inset shows the Fourier transform of the SdH oscillation which gives a single frequency of F=138(7)F=138(7) T.The response of the carriers to the applied magnetic field and the magnitude of MR is determined by the mobility in the plane perpendicular to the magnetic field.\",\n \"[11] For nearly isotropic three-dimensional (3D) FS, there should be no significant angle-dependent MR (AMR).\",\n \"In (quasi-)2D systems, 2D states will only respond to perpendicular component of the magnetic field $B|\\\\cos (\\\\theta )|$ and consequently longitudinal AMR and AMR-oscillation were observed in some quasi-2D conductors (such as Sr$_2$ RuO$_4$ and $\\\\beta -$ (BEDT-TTF)$_2$ I$_3$ ) and the surface states of TIs.\",\n \"[21], [22], [14], [15] For example, the MR of the bulk state in topological insulator has only $\\\\sim 10\\\\%$ angular dependence while the angular dependence of MR in the surface state is about ten times larger.\",\n \"[15] Significant AMR was also observed in some materials with highly anisotropic 3D FS such as Bi and Cu, but the period of AMR is determined by the shape of the Fermi surface and is very different from the one in 2D systems.\",\n \"In Bi, electrons exhibit a threefold valley degeneracy and in-plane mass anisotropy, so the AMR peaks each time when the magnetic field is oriented along the bisectrix axis and has a 60o period.\",\n \"[23] In Cu the AMR is more complex due to the complex FSs and peaks about every 25o.\",\n \"[24] The electronic structure calculations in SrMnBi$_2$ show that the states near the Fermi energy $E_F$ are dominated by the Bi states in the Bi square nets.\",\n \"Consequently the dominant FS should be quasi-2D.\",\n \"Angular dependent resistivity in $B=9$ T and $T=2$ K shows wide maximum when the field is parallel to the $c$ -axis ($\\\\theta =0^o, 180^o$ ), and sharper minimum around $\\\\theta =90^o, 270^o$ (Fig.\",\n \"2(a)).\",\n \"The whole curve of AMR in SrMnBi$_2$ follows the function of $|\\\\cos (\\\\theta )|$ very well with a 180o period (red line in Fig.\",\n \"2(a)).\",\n \"Moreover, the larger $\\\\rho _{c}$ than $\\\\rho _{ab}$ in Fig.\",\n \"1(b) implies that the transfer integral and the coupling between layers along the $c$ -axis is very small.\",\n \"All this implies that the FSs in SrMnBi$_2$ should be highly anisotropic and that the mobility of carriers along $k_z$ is much smaller than the value in $k_xk_y$ plane.\",\n \"Angular dependent MR quantum oscillations are directly related to the cross section of FS.\",\n \"In Fig.\",\n \"2(c), the in-plane $\\\\Delta R=R-$ measured using same configuration as shown in the inset of Fig.\",\n \"2(a) exhibits clear Shubnikov-de Hass (SdH) oscillations with tilt angles $\\\\theta $ from 0o to 35o.\",\n \"In metals, SdH oscillations correspond to successive emptying of LLs by the magnetic field and the LLs index $n$ is related to the cross section of FS $S_F$ by $2\\\\pi (n+\\\\gamma )=S_F\\\\frac{\\\\hbar }{eB}$ .\",\n \"[25], [14], [15] For a 2D FS (a cylinder), the cross section has $S_F(\\\\theta )=S_0/|\\\\cos (\\\\theta )|$ angular dependence and the LLs positions should be inversely proportional to $|cos(\\\\theta )|$ .\",\n \"[14], [25] The peak (dip) positions in SrMnBi$_2$ rapidly shift toward higher field direction with increase in $\\\\theta $ (as indicated by the dashed lines in Fig.\",\n \"2(c)).\",\n \"In Fig.\",\n \"2(d), the dip positions corresponding to LLs $n=5, 6$ and the peak position with $n=5$ were plotted against the tilt angle $\\\\theta $ and can be described very well by $1/|\\\\cos (\\\\theta )|$ (the red lines in Fig.\",\n \"2(d)).\",\n \"Similar behavior was observed in the surface states of TIs [14], [15] and some other layered structures.\",\n \"[25], [26] The Fourier transform of the SdH oscillation (inset of Fig.\",\n \"2(d)) revealed that the oscillation component shows a periodic behavior in $1/B$ with a single frequency $F=138(7)$ T. The small value of frequency is consistent with previous value in Ref.\",\n \"[7] and demonstrates that the dominant FSs are very small since the Onsager relation is $F=(\\\\Phi _0/2\\\\pi ^2)A_k$ where $\\\\Phi _0$ is the flux quantum and $A_k$ is the cross sectional area of FS.\",\n \"[25] This clearly shows that the dominant two-dimensional FSs found in Fig.\",\n \"2(b) are indeed the small FSs between $\\\\Gamma $ and $M$ points, rather than the large FS at $\\\\Gamma $ point in the Brillouin zone.\",\n \"Above SdH oscillation combined with the angular MR clearly suggests that the dominant FSs of SrMnBi$_2$ are small quasi-2D cylinders along $k_z$ originating from Bi square nets.\",\n \"In addition, there are still conventional parabolic bands with three dimensional characteristic close to the Fermi level,[7] causing the small deviation from quasi-2D transport.\",\n \"Figure: (a) The magnetic field (BB) dependence of the in-plane magnetoresistance (MR =(ρ(B)-ρ(0))/ρ(0)=(\\\\protect \\\\rho (B)-\\\\protect \\\\rho (0))/\\\\protect \\\\rho (0)) at differenttemperatures.\",\n \"(b) The field derivative of in-plane MR at different temperaturerespectively.\",\n \"The lines in high field regions were fitting results using MR =A 1 B+O(B 2 )=A_1B+O(B^2) and the lines in low field regions using MR =A 2 B 2 =A_2B^2.Figure: (a) Temperature dependence of the critical field B * B^* (blacksquares) and the high field MR linear coefficient A 1 A_1 (blue circles) up to50 K. The red solid line is the fitting results using B * =1 2eℏv F 2 (E F +k B T) 2 B^*=\\\\frac{1}{2e\\\\hbar v_F^2}(E_F+k_BT)^2.\",\n \"(b) Temperature dependence of theeffective MR mobility μ MR \\\\protect \\\\mu _{MR} extracted from the weak-field MR.Now we turn to the linear nonsaturated in-plane magnetoresistance in SrMnBi$_2$ (Fig.\",\n \"3(a)).\",\n \"The MR is linear over a wide field range up to 50 K. This behavior extends to very low fields where the MR naturally reduces to a weak-field semiclassical quadratic dependence.\",\n \"The cross-over from the weak-field $B^2$ dependence to the high-field linear dependence can be best seen by considering the field derivative of the MR, $d$ MR$/dB$ (Fig.\",\n \"3(b)).\",\n \"In the low field range ($B<$ 1 T at 2 K), $d$ MR$/dB$ is proportional to $B$ (as shown by lines in low-field regions), indicating the semiclassical MR $\\\\sim A_2B^2$ .\",\n \"But above a characteristic field $B^*$ , $d$ MR$/dB$ deviates from the semiclassical behavior and saturates to a much reduced slope (as shown by lines in the high-field region).\",\n \"This indicates that the MR for $B>B^*$ is dominated by a linear field dependence plus a very small quadratic term (MR$=A_1B+O(B^2)$ ).\",\n \"The linear MR deviates from the semiclassical $B^2$ dependence of MR in the low field region and a saturating MR in high fields.\",\n \"[11] The unusual nonsaturating linear magnetoresistance has been reported in gapless semiconductor Ag$_{2-\\\\delta }$ (Te/Se) [27], [28] with the linear energy spectrum in the quantum limit.\",\n \"[10], [28] Recent first principle calculations confirmed that these materials have a gapless Dirac-type surface state.\",\n \"[29] Linear magnetoresistance is also observed in topological insulators [14], [15] and BaFe$_2$ As$_2$ [16] with Dirac fermion states.\",\n \"Another possible origin of the large linear magnetoresistance is the mobility fluctuations in a strongly inhomogeneous system.\",\n \"[30] This does not apply in SrMnBi$_2$ since our sample is stoichiometric crystal without doping/disorder.\",\n \"Below we show that the nonsaturating linear magnetoresistance and the deviation from the semiclassical transport in SrMnBi$_2$ is due to the linear energy dispersion.\",\n \"The application of a strong perpendicular external magnetic field ($B$ ) would lead to a complete quantization of the orbital motion of carriers with linear energy dispersion, resulting in quantized LLs $E_n=sgn(n)v_F\\\\sqrt{2e\\\\hbar B|n|}$ where $n=0,\\\\pm 1,\\\\pm 2,\\\\cdots $ is the LL index and $v_F$ is the Fermi velocity.\",\n \"[12], [13] Then the energy splitting between the lowest and 1st LLs is described by $\\\\triangle _{LL}=\\\\pm v_F\\\\sqrt{2e\\\\hbar B}$ .\",\n \"[12], [13] In the quantum limit at specific temperature and field, $\\\\triangle _{LL}$ becomes larger than both the Fermi energy $E_F$ and the thermal fluctuations $k_BT$ at a finite temperature.\",\n \"Consequently all carriers occupy the lowest Landau level and eventually the quantum transport with linear magnetoresistance shows up.\",\n \"The critical field $B^*$ above which the quantum limit is satisfied at specific temperature $T$ is $B^*=\\\\frac{1}{2e\\\\hbar v_F^2}(E_F+k_BT)^2$ .\",\n \"[16] The splitting of LLs in conventional parabolic bands is $\\\\triangle _{LL}=\\\\frac{e\\\\hbar B}{m^*}$ .\",\n \"Hence the evolution of $\\\\triangle _{LL}$ with field for parabolic bands is much slower than that for Dirac fermion states, and it is difficult to observe quantum limit behavior in the moderate field range.\",\n \"The temperature dependence of critical field $B^*$ in SrMnBi$_2$ clearly deviates from the linear relationship and can be well fitted by $B^*=\\\\frac{1}{2e\\\\hbar v_F^2}(E_F+k_BT)^2$ , as shown in Fig.\",\n \"4(a).\",\n \"The fitting gives the Fermi velocity $v_F\\\\sim 5.13\\\\times 10^5$ ms$^{-1}$ and $\\\\Delta _1\\\\sim 4.97$ meV.\",\n \"This confirms the existence of Dirac fermion states in SrMnBi$_2$ .\",\n \"In a multiband system with both Dirac and conventional parabolic-band carriers (including electrons and holes) the magnetoresistance in the semiclassical transport can be described as $MR=\\\\frac{\\\\sigma _e\\\\sigma _h(\\\\mu _e+\\\\mu _h)^2}{(\\\\sigma _e+\\\\sigma _h)^2}B^2$ where $\\\\sigma _e, \\\\sigma _h, \\\\mu _e, \\\\mu _h$ are the effective electron and hole conductivity and mobility in zero field respectively, when the Dirac carriers are dominant in transport.\",\n \"[16], [17] Then the coefficient of the low-field $B^2$ quadratic term, $A_2$ , is related to the effective MR mobility $\\\\sqrt{A_2}=\\\\frac{\\\\sqrt{\\\\sigma _e\\\\sigma _h}}{\\\\sigma _e+\\\\sigma _h}(\\\\mu _e+\\\\mu _h)=\\\\mu _{MR}$ , which is smaller than the average mobility of carriers $\\\\mu _{ave}=\\\\frac{\\\\mu _e+\\\\mu _h}{2}$ and gives an estimate of the lower bound.\",\n \"Fig.\",\n \"4(b) shows the dependence of $\\\\mu _{MR}$ on the temperature.\",\n \"At 2 K, the value of $\\\\mu _{MR}$ is about 3400 cm$^2$ /Vs.\",\n \"The large effective MR mobility also implies that Dirac fermions dominate the transport behavior.\",\n \"With increase in temperature, the value of $\\\\mu _{MR}$ and the coefficient of high-field linear term $A_1$ (Fig.\",\n \"4(a)) decrease sharply.\",\n \"This is due to thermal fluctuation smearing out the LL splitting.\",\n \"In summary, we demonstrate quantum transport of 2D Dirac fermion states in bulk SrMnBi$_2$ single crystals.\",\n \"The bands with linear energy dispersion lead to the large nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.\",\n \"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.\",\n \"With increase in temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.\",\n \"The effective magnetoresistant mobility $\\\\mu _{MR}\\\\sim 3400$ cm$^2$ /Vs comparable to values observed in graphene is observed.\",\n \"The angle-dependence of magnetoresistance shows a $|\\\\cos (\\\\theta )|$ dependence while the LL positions in SdH oscillations are inversely proportional to $|\\\\cos (\\\\theta )|$ , indicating the dominant quasi-2D Fermi surfaces.\",\n \"Our results show that the crystals with Bi square nets can host phenomena commonly observed so far in 2D structures and materials like graphene.\",\n \"We than John Warren for help with SEM measurements.\",\n \"Work at Brookhaven is supported by the U.S. DOE under contract No.\",\n \"DE-AC02-98CH10886.\",\n \"Work at the National High Magnetic Field Laboratory is supported by the DOE NNSA DEFG52-10NA29659 (S. W. T and D. G.), by the NSF Cooperative Agreement No.\",\n \"DMR-0654118 and by the state of Florida.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1049"}}},{"rowIdx":1866999,"cells":{"text":{"kind":"list like","value":[["Disordered quantum wires: microscopic origins of the DMPK theory and\n Ohm's law"],["Abstract We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire.","In the literature, the standard description of such wires is via random matrix theory (RMT).","Our objective is to firmly relate this theory to a microscopic model.","We correct and extend previous work (arXiv:0912.1574) on the same topic.","In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes \\beta=1,2.","In the \\beta=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\\ from TUE.","In the \\beta=1 class, we find a deviation from TOE.","It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime.","For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone.","This proof bypasses the explicit but intricate solution methods that underlie most previous results."],["Introduction","We start below with a brief introduction to the physics of quasi one-dimensional quantum wires.","In Section REF , we sketch the scope of this paper and its relation to previous works.","The appropriate random matrix theory is discussed in Section .","Our microscopic model, convergence results and proofs are presented in Sections and ."],["Phenomenology","Without yet introducing a concrete mathematical framework, we present the basic physics setup of quantum wires and try to elucidate the questions of charge transport and conductance fluctuations.","We rely heavily on the excellent review [2].","Disordered quantum wires are quasi one-dimensional pieces of dirty (disordered) conductor.","The wire has a physical length $L$ , which is most conveniently expressed in units of the mean free path $\\ell $ so that we shall use $s:= L/\\ell $ .","In a microscopic model where the parameter $\\lambda \\ge 0$ measures the strength of the disorder $\\ell \\sim \\lambda ^{-2}$ .","The width $W$ of the wire is expressed by an integer $N$ that corresponds to the number of different modes that `fit' in the wire.","Physically, $N \\sim W/\\lambda _F$ with $\\lambda _F$ the Fermi wavelength of the electrons sent through the wire, which is in turn determined by the energy of those incoming electrons.","For a purely one-dimensional wire, $N=1$ , it is well-known that an electron travelling through the wire gets localized with localization length of the order of the mean free path $\\ell $ , hence $s \\sim 1$ .","However, the localization length increases with $N$ (roughly as $s \\sim N$ , at least in the weak disorder limit $\\lambda \\rightarrow 0$ ) and we can ask how the system behaves for $s \\ll N$ , before localization sets in.","There, one can distinguish the ballistic regime $s \\le 1$ , where incoming electrons did not yet get scattered by the impurities, and the most interesting diffusive regime characterized by $1 \\ll s, \\qquad s/ N \\ll 1\\,.$ One of the fascinating aspects of this regime is the phenomenon of universal conductance fluctuations (UCF) first discussed in [14].","Let $g= g(s,N, \\lambda )$ be the conductance of the wire, expressed in units of the conductance quantum $2e^2/\\hbar $ .","It is a random quantity due to the disorder.","In the thick wire limit, its disorder average ${\\mathbb {E}}(g)$ , is roughly given by ${\\mathbb {E}}(g) \\sim \\left\\lbrace \\begin{array}{lllll} N/s & \\qquad & 1 \\ll s, \\quad s/ N \\ll 1 & \\qquad & \\textrm {(Ohm^{\\prime }s law)} \\\\[2mm] \\exp {\\lbrace - s /N \\rbrace } & \\qquad & s > N & \\qquad & \\textrm {(localization)} \\end{array} \\right.","$ Furthermore, universal conductance fluctuations mean that, in the diffusive regime defined by (REF ), $\\mathrm {Var}(g) = 2/(15\\beta ),$ independently of the microscopic details of the wire, or its length and width.","The only parameter that remains in this regime is the symmetry index $\\beta $ that refers to Dyson's symmetry classes.","We emphasize that these phenomena should emerge in a large $N$ limit only.","On the other hand, $N$ cannot be too large because then we enter the regime of two-dimensional localization, at least if we assume that the wire has one transverse dimension.","However, even if the transverse dimension is higher, the reasoning breaks down as soon as $W > \\ell $ .","It is therefore important to take a weak-disorder limit first, $\\lambda \\rightarrow 0$ , which also means that the wire's microscopic length $L=\\lambda ^{-2}s$ diverges.","Below, we try to distill some precise conjectures that are generally accepted.","From the mathematical perspective, they can be partially proven if one accepts RMT as a starting point (see Section ), but open if one starts from a more realistic model, as the one treated in Section of this article.","Conjecture 1 (Ohm's law) $\\lim _{N \\rightarrow \\infty } \\quad \\lim _{\\lambda \\rightarrow 0} \\quad \\frac{1}{N}{\\mathbb {E}}(g) = \\frac{1}{s} + o(1/s), \\qquad s \\rightarrow \\infty .$ Conjecture 2 (Universal conductance fluctuations) $\\quad \\lim _{N \\rightarrow \\infty } \\quad \\lim _{\\lambda \\rightarrow 0} \\quad \\mathrm {Var}(g) = \\frac{2}{15 \\beta } +o(1), \\qquad s \\rightarrow \\infty .$ We stress here that these conjectures reflect the minimum of what should be true according to the literature, and that the underlying heuristics is quite involved.","The present paper does partially settle theses conjectures starting from a microscopic model but with an additional scaling limit, as will be explained in the next section."],["Setup, goals, and results","The standard approach to disordered quantum wires is to model the transfer matrix of such a wire by an appropriate ensemble of random matrices.","The matrices under consideration belong to a subgroup of pseudo-unitary matrices.","Following [8] we shall call their ensembles TOE, TUE and TSE in analogy to the better known ensembles of Hamiltonians, the hermitian GOE, GUE and GSE, or the circular ensembles of unitaries: COE, CUE and CSE.","In fact, ensembles of transfer matrices come with a real positive parameter, called $s$ above and physically corresponding to the length of the wire.","They are therefore more complicated, but also more interesting: in particular the parameter $s$ tunes a localization-delocalization transition.","This can be observed for example in the Fokker-Planck equation describing the $s$ -dependence of the conductance, which is the equation usually referred to as the DMPK equation.","The natural question arises whether the RMT ensembles allow for a verification of the conjectures mentioned at the end of the previous section, with the proviso that the $\\lambda \\rightarrow 0$ should be omitted as the RMT assumes weak coupling from the start.","In the physics literature, there is overwhelming evidence for an affirmative answer, and the conjectures have been verified in [15], [3], [7], [21].","In that perspective, we shall here give a rigorous proof of Ohm's law based on a moment argument of [17], thereby confirming Conjecture 1 for the TOE, TUE and TSE.","The ultimate goal of our work is a derivation of the conjectures from a more realistic model of the wire, i.e.","from a `reasonable' microscopic Hamiltonian, namely the Anderson model on a tube of width $N$ with a disordered region of length $L$ and disorder strength $\\lambda $ .","First, we need to be in the weak coupling regime $\\lambda \\rightarrow 0$ , and therefore $L=\\lambda ^{-2}s\\rightarrow \\infty $ .","This first scaling limit yields a random matrix ensemble ${\\mathcal {G}}(s)$ , see Proposition REF .","For the conjectures to hold, a second scaling is certainly necessary, namely that of a broad wire, $N\\rightarrow \\infty $ .","At the time of writing, the validity of the conjectures in this scaling regime remains an open question.","However, if we consider an additional scaling limit in which the transversal hopping in the wire is small compared to the longitudinal hopping, see Theorem REF , we obtain, instead of the ensemble ${\\mathcal {G}}(s)$ , a new transfer matrix ensemble ${\\mathcal {A}}(s)$ that is very close to the ideal ensemble.","In fact, for $\\beta =2$ , the conductance calculated from that ensemble is the same as that calculated from the TUE.","Since we proved Ohm's law for the $N\\rightarrow \\infty $ limit of the random matrix ensemble in the first place, this provides a proof of Conjecture 1 in a weaker sense for $\\beta =2$ .","In Section REF , we comment on the ensemble ${\\mathcal {A}}(s)$ , pointing out to how and why it fails to satisfy all the symmetry properties of the ideal ensembles.","This article is to a large extent based on a previous paper [1] by two of us, which appeared on the arXiv shortly after and independently of [20].","Despite their similarity these two articles stressed different aspects of the resulting transfer matrix evolutions.","However, [1] contained an error, as pointed out by the second author of the present paper, and the symmetry properties of the model were not consistently treated.","In this article, which supersedes [1], we first extend the setup by constructing models for both $\\beta =1$ and $\\beta =2$ symmetry classesThe physically most natural way to discuss $\\beta =4$ as well would be to consider electrons with spin, which we chose not to do for reasons of simplicity.","Moreover, we incorporate technical improvements (among other things borrowing some terminology from [20]), mostly concerning the statement of the joint scaling limit in Theorem REF .","Finally, we study the convergence as $N\\rightarrow \\infty $ of a hierarchy of equations for the moments of the conductance introduced by [17].","We prove that the limit satisfies Ohm's law, see Theorem REF ."],["Random matrix theory: the DMPK equation","Transport properties of a quasi one-dimensional system are most conveniently approached through its scattering matrix, or equivalently its transfer matrix.","In this section we shall consider these objects as the fundamental quantities of the theory, understand what symmetries imply on their general structure and derive a stochastic differential equation describing their behavior as a function of the length of the disordered wires, based on an isotropy assumption, also called `(local) maximal entropy' Ansatz.","In particular, we do not assume that the transfer matrices here arise from some sort of microscopic Hamiltonian dynamics."],["Transfer matrices and symmetries","Heuristically speaking the transfer matrix of a quasi one-dimensional wire maps free waves on the far right of the sample to free waves on the far left of it.","Although this picture is physically meaningful, we shall only refer to it explicitly in Section and keep an abstract point of view here.","We first fix a preferred basis in ${\\mathbb {C}}^{2N}$ and make the following definition.","Definition 1 A transfer matrix for a wire of width $N$ is a $2N\\times 2N$ pseudo-unitary matrix, ${\\mathcal {M}}^{*}\\Sigma _z {\\mathcal {M}}= \\Sigma _z\\,,\\qquad \\text{where}\\qquad \\Sigma _z = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1\\end{pmatrix}\\,.$ Furthermore, a transfer matrix ${\\mathcal {M}}$ is time reversal invariant if $\\Sigma _x {\\mathcal {M}}\\Sigma _x = \\overline{{\\mathcal {M}}}\\,,\\qquad \\text{where}\\qquad \\Sigma _x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}\\,,$ and $\\overline{{\\mathcal {M}}}$ is the complex conjugate of ${\\mathcal {M}}$ .","Transfer matrices have a simple multiplicative composition rule.","If ${\\mathcal {M}}_1$ and ${\\mathcal {M}}_2$ are transfer matrices for two wires, then ${\\mathcal {M}}_2{\\mathcal {M}}_1$ is the transfer matrix for the composite system obtained from gluing the two pieces in series.","In view of (REF , REF ), it is natural to write a transfer matrix in block form ${\\mathcal {M}}= \\begin{pmatrix} {\\mathcal {M}}_{++} & {\\mathcal {M}}_{+-} \\\\ {\\mathcal {M}}_{-+} & {\\mathcal {M}}_{--} \\end{pmatrix}\\,.$ Combining (REF ) and the singular value decompositions ${\\mathcal {M}}_{++} = U_+ S_+ V_+$ and ${\\mathcal {M}}_{--} = U_- S_- V_-$ of the diagonal blocks, we obtain the following factorization ${\\mathcal {M}}= \\begin{pmatrix} U_+ & 0 \\\\ 0 & U_- \\end{pmatrix} \\begin{pmatrix} S & (S^2-1)^{1/2} \\\\ (S^2-1)^{1/2} & S \\end{pmatrix} \\begin{pmatrix} V_+ & 0 \\\\ 0 & V_- \\end{pmatrix}\\,.$ where $S=S_+=S_-$ .","If, moreover, time reversal invariance is imposed, then $U_- = \\overline{U_+}$ and $V_- = \\overline{V_+}$ .","Let $r$ be the reflection and $t$ be the transmission matrices, defined through ${\\mathcal {M}}\\begin{pmatrix} 1 \\\\ r \\end{pmatrix} = \\begin{pmatrix} t \\\\ 0 \\end{pmatrix}$ In particular, $t = {\\mathcal {M}}_{++}-{\\mathcal {M}}_{+-}{\\mathcal {M}}_{--}^{-1}{\\mathcal {M}}_{-+}= \\left({\\mathcal {M}}_{++}^{*}\\right)^{-1}\\,,$ where we used (REF ) in the second equality.","The so-called transmission eigenvalues $(T_k)_{k=1}^N$ are defined as the eigenvalues of the matrix $t^{*}t$ , i.e.","of $({\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++})^{-1}$ .","Hence, the transmission eigenvalues are also the inverses of the squares of the singular values contained in $S$ .","Let $T = S^{-2}$ be the diagonal matrix of transmission eigenvalues.","Many transport properties of the disordered wire can be expressed as functions thereof.","In particular, the conductance $g$ is given in units of $2e^2 / \\hbar $ by the Landauer-Büttiker formula [5], $g:= \\sum _{i=1}^N T_i = \\mathrm {Tr}\\, T = \\mathrm {Tr}\\, t^{*}t\\,,$ a formula that we accept here as a definition of $g$ ."],["The DMPK Theory","The DMPK theory introduced by [9] and independently by [16] is an evolution equation for the transfer matrix ${\\mathcal {M}}(r,s)$ of a wire on $[r,s]$ .","By the composition rule, for any $s_1\\le s_2$ , ${\\mathcal {M}}(0,s_2) ={\\mathcal {M}}(s_1,s_2) {\\mathcal {M}}(0,s_1)\\,,$ with ${\\mathcal {M}}(s,s)=1$ .","The first crucial idea is to take $s_2-s_1$ infinitesimal and write ${\\mathcal {M}}(s,s+\\mathrm {d}s) \\sim 1 + \\mathrm {d}{\\mathcal {L}}(s)$ such that $\\mathrm {d}{\\mathcal {L}}(s)$ is independent of ${\\mathcal {M}}(s)$ and contains only diffusive terms but no drift.","Mathematically, this translates into the assumption that ${\\mathcal {M}}(s)$ satisfies an Itô stochastic differential equation (SDE) $\\begin{split}\\mathrm {d}{\\mathcal {M}}(s) &= \\mathrm {d}{\\mathcal {L}}(s) {\\mathcal {M}}(s)\\,,\\\\{\\mathcal {M}}(0)&=1\\,,\\end{split}$ where ${\\mathcal {L}}(s)$ is a matrix valued Brownian motion and ${\\mathcal {L}}(0) = 0$ .","Lemma 1 Let ${\\mathcal {M}}(s)$ be a solution of the SDE (REF ).","Assume that $\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z + \\Sigma _z \\mathrm {d}{\\mathcal {L}}&= 0\\,, \\\\\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z \\mathrm {d}{\\mathcal {L}}&= 0\\,.","$ Then ${\\mathcal {M}}(s)$ is pseudo unitary, eq.","(REF ).","If moreover $\\Sigma _x \\mathrm {d}{\\mathcal {L}}\\Sigma _x = \\overline{\\mathrm {d}{\\mathcal {L}}}\\,,$ then ${\\mathcal {M}}(s)$ is also time reversal invariant, eq.","(REF ).","For the first part, we take the differential of (REF ), use (REF ) and Itô calculus to obtain ${\\mathcal {M}}^{*}\\left(\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z + \\Sigma _z \\mathrm {d}{\\mathcal {L}}+ \\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z \\mathrm {d}{\\mathcal {L}}\\right){\\mathcal {M}}= 0\\,,$ which holds if and only if both (REF ) and () hold as ${\\mathcal {M}}$ is nonsingular.","Similarly, the differential of (REF ) immediately yields (REF ).","Secondly, the DMPK theory prescribes a particular invariance of the distribution of $\\mathrm {d}{\\mathcal {L}}(s)$ .","The law of the increments $\\mathrm {d}{\\mathcal {L}}(s)$ shall be independent of $s$ and maximally isotropic in the sense that ${\\mathcal {W}}^{*}\\mathrm {d}{\\mathcal {L}}\\, {\\mathcal {W}}\\mathop {=}\\limits ^{d} \\mathrm {d}{\\mathcal {L}}\\qquad \\text{for any unitary}\\qquad {\\mathcal {W}}= \\begin{pmatrix} W_+ & 0 \\\\ 0 & W_- \\end{pmatrix}\\,.$ The unitary blocks $W_\\pm $ are independent of each other if ${\\mathcal {M}}$ does not exhibit any symmetry, whereas $W_- = \\overline{W_+}$ if time reversal symmetry is imposed.","For notational simplicity, we cast ${\\mathcal {L}}$ in block form, ${\\mathcal {L}}(s) = \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\ \\mathfrak {b}(s)^{*}& \\mathfrak {a}^{\\prime }(s) \\end{pmatrix}$ where $\\mathfrak {a}(s),\\mathfrak {a}^{\\prime }(s), \\mathfrak {b}(s)$ are independent local martingales, with $\\mathfrak {a}(s) = -\\mathfrak {a}(s)^{*}$ , similarly for $\\mathfrak {a}^{\\prime }(s)$ , and $\\mathrm {d}\\mathfrak {a}^{*}\\mathrm {d}\\mathfrak {a} = \\mathrm {d}\\mathfrak {b}\\mathrm {d}\\mathfrak {b}^{*}= \\mathrm {d}\\mathfrak {a}^{\\prime }{\\mathrm {d}\\mathfrak {a}^{\\prime }}^{*}\\,.$ The isotropy assumption (REF ) reduces to invariance conditions on the blocks.","First, $ \\mathfrak {a}_{ij}(s) = {\\left\\lbrace \\begin{array}{ll}1 / \\sqrt{2N} \\cdot (B^R_{ij}(s) + \\mathrm {i}B^I_{ij}(s)) & 1 \\le i < j \\le N \\\\\\mathrm {i}/ \\sqrt{N} \\cdot B^I_{ii}(s) & i=j \\\\-\\overline{\\mathfrak {a}_{ji}(s)} & \\text{otherwise}\\end{array}\\right.","}\\,,$ where $B^R$ and $B^I$ are independent real standard Brownian motions, and similarly but independently for $\\mathfrak {a}^{\\prime }(s)$ .","Secondly, $ \\mathfrak {b}_{ij}(s) = 1 / \\sqrt{2N} \\cdot (\\tilde{B}^R_{ij}(s) + \\mathrm {i}\\tilde{B}^I_{ij}(s))\\,,\\quad \\text{for all }i,j\\,.$ Note that the relative normalization of $\\mathfrak {a}(s)$ and $\\mathfrak {b}(s)$ are fixed by pseudounitarity, i.e.","(REF ).","In the time reversal invariant case, the matrix $\\mathfrak {a}(s)$ does not change, but $\\mathfrak {a}^{\\prime }(s) = \\overline{\\mathfrak {a}(s)}\\,,$ and $\\mathfrak {b}(s)$ becomes symmetric, $\\mathfrak {b}(s)^{*}= \\overline{\\mathfrak {b}(s)}$ with real and imaginary parts orthogonally invariant, namely $ \\mathfrak {b}_{ij}(s) = {\\left\\lbrace \\begin{array}{ll}1 / \\sqrt{2(N+1)} \\cdot (B^R_{ij}(s) + \\mathrm {i}B^I_{ij}(s)) & 1 \\le i < j \\le N \\\\1/\\sqrt{N+1} \\cdot (B^R_{ii}(s) + \\mathrm {i}B^I_{ii}(s)) & i=j \\\\\\mathfrak {b}_{ji}(s) & \\text{otherwise}\\end{array}\\right.","}\\,.$ Here again, the relative factor $\\sqrt{N/(N+1)}$ is imposed by (REF ).","From a physical point of view, the DMPK theory's interest lies in its predictions for the statistics of the transmission eigenvalues.","Indeed, the unitary invariance of the increments $\\mathrm {d}{\\mathcal {L}}$ implies that the set of $T_k$ satisfies an autonomous equation, which can be formally derived by Itô calculus from the matrix SDE (REF ): $\\begin{split}\\mathrm {d}T_k(s)&=v_k(T(s))\\mathrm {d}s+D_k(T(s))\\mathrm {d}B_k(s),\\\\T_k(0)&=1,\\end{split}$ for all $k=1,...,N$ .","The Brownian motions $B_k$ are independent, and the drift and diffusion coefficients are given explicitly by $\\begin{split}v_k&=-T_k+\\frac{2T_k}{\\beta N+2-\\beta }\\left(1-T_k+\\frac{\\beta }{2}\\sum _{j\\ne k}\\frac{T_k+T_j-2T_kT_j}{T_k-T_j}\\right)\\,,\\\\D_k&=\\sqrt{4\\frac{T_k^2(1-T_k)}{\\beta N+2-\\beta }}.\\end{split}$ The first term in the drift $v_k$ contracts all transmission eigenvalues towards 0 as the length $s$ of the wire increases.","However, and similarly to Dyson's Brownian motion, the drift also contains repulsion terms originating from second order perturbation theory.","As a consequence, the eigenvalues $T_k$ `try to avoid' degeneracy.","What makes a naive derivation formal is that Itô's formula is only applicable if the denominator $T_k-T_j$ never becomes singular, i.e.","${\\mathcal {M}}^*_{++}(s){\\mathcal {M}}_{++}(s)$ never has degenerate eigenvalues.","This is a nontrivial property for $s>0$ , and even more so as $s\\rightarrow 0^+$ since (REF ) starts with the completely degenerate ${\\mathcal {M}}(0)=1$ .","Both issues can however be tackled and the SDE (REF ) has a unique weak and strong solution, see [6].","Finally, let us comment on some deeper principles underlying the process ${\\mathcal {M}}$ and the resulting DMPK equation.","The maximal isotropy assumption that was used above, can be derived from a simple `maximal entropy assumption' on the set of infinitesimal transfer matrices $1+\\mathrm {d}{\\mathcal {L}}(s)$ that have a fixed 'scattering strength' $\\sum _k T_k$ .","Alternatively, as remarked by [12], one can also guess the DMPK equation from geometric considerations, since it is the radial part of the canonical Brownian motion on a certain symmetric space.","The reduction from Lie group to symmetric space is obtained by identifying certain transfer matrices that, in particular, have the same transmission eigenvalues.","This geometric approach was very fruitful.","For example, in [4] it was shown how it naturally explains the appearance of non-universal conductance properties in wires with off-diagonal disorder."],["Ohm's law","In the context of the DMPK theory, a treatment, or even proof, of the conjectures mentioned in the introduction is possible, as already indicated in Section REF .","The existing approaches rely on explicit calculations and are quite intricate.","Nevertheless, if one is solely after Ohm's law (and not the universal conductance fluctuations (UCF)), there is an appealing and compact approach by [17].","Below we present a rigorous version of this approach.","The following theorem shows that in the large $N$ limit, the rescaled moments of the conductance have an Ohmic behavior.","In particular, Conjecture 1 holds for the TOE, TUE and TSE.","We note that the symmetry index $\\beta $ drops out in that particular scaling.","Theorem 2 (Ohm's law) Let $\\left(T_k(s)\\right)_{k=1}^N$ be the solution of the DMPK process (REF ), and let $g_N(s) = \\sum _{k=1}^N T_k(s)\\,.$ Then for all $p\\ge 1$ and $T>0$ , $ \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}(g^p_N(s))}{N^p} = \\frac{1}{(1+s)^p}$ uniformly for $s\\in [0,T]$ .","The proof goes through a sequence of lemmas.","Lemma 3 Let $g_N^{(j)} = \\sum _k T_k^j$ for $j>1$ .","Then, for any $p\\ge 1$ , $ \\frac{\\mathrm {d}}{\\mathrm {d}s}{\\mathbb {E}}(g_N^p) = -p\\gamma _N(\\beta )\\left[{\\mathbb {E}}(g_N^{p+1}) - \\left(1-\\frac{2}{\\beta }\\right){\\mathbb {E}}(g_N^{p-1} g_N^{(2)}) -\\frac{2(p-1)}{\\beta }{\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)}))\\right].$ where $\\gamma _N(\\beta ) = \\frac{\\beta }{\\beta N +2 -\\beta }.$ Itô's formula yields $\\mathrm {d}(g_N^p) = \\sum _{i_1,\\ldots ,i_p = 1}^N \\left[\\sum _{j=1}^p T_{i_1}\\cdots dT_{i_j}\\cdots T_{i_p} + \\sum _{j\\ne k=1}^p T_{i_1}\\cdots dT_{i_j}\\cdots dT_{i_k}\\cdots T_{i_p}\\right]$ so that $ \\frac{\\mathrm {d}}{\\mathrm {d}s}{\\mathbb {E}}(g_N^p) = p{\\mathbb {E}}\\left(g_N^{p-1}\\sum _k v_k\\right) + \\frac{p(p-1)}{2}{\\mathbb {E}}\\left(g_N^{p-2}\\sum _k D_k^2\\right)\\,.$ where we used the DMPK equation (REF ).","In order to rewrite the right-hand side, note the simple identity $\\sum _k T_k \\sum _{j\\ne k}\\frac{T_k+T_j-2T_kT_j}{T_k-T_j} = \\sum _k\\sum _{j < k}(T_k+T_j-2T_kT_j) = (N-1) g_N - g_N^2 + \\sum _k T_k^2$ Therefore, $\\sum _k v_k &= -g_N + \\frac{2}{\\beta N +2 -\\beta }\\left[g_N - \\sum _k T_k^2 + (\\beta /2)\\left((N-1) g_N - g_N^2 + \\sum _k T_k^2\\right) \\right] \\nonumber \\\\&=-\\gamma _N(\\beta )\\left( g_N^2 - (1-2/\\beta )g_N^{(2)}\\right)\\,, $ and the lemma follows upon substituting this in (REF ).","Let us now consider $\\Psi _{N}(p,s) :=\\frac{{\\mathbb {E}}(g^p_N(s))}{N^p}\\,,$ The following properties are immediate from the definition and the differential equation (REF ) $\\Psi _{N}(p,0)=1$ .","$ |\\Psi _{N}(p,s) |\\le 1$ .","The function $s \\mapsto \\Psi _{N}(p,s)$ is continuously differentiable and $ |\\frac{\\partial \\Psi _{N}}{\\partial s}(p,s) |\\le c(p) < \\infty $ .","We consider the Banach space ${\\mathcal {L}}={\\mathcal {C}}([0,T], {\\mathbb {R}})$ for some $T>0$ , equipped with the supremum norm.","Let ${\\mathcal {L}}_p, p=1,2,\\ldots $ be copies of ${\\mathcal {L}}$ and define the cartesian product ${\\mathcal {K}}= \\mathop {\\times }\\limits _{p=1}^{\\infty } {\\mathcal {L}}_p$ , equipped with the product topology.","Lemma 4 There is an increasing sequence $N_n, n \\in {\\mathbb {N}}$ and an element $\\Psi \\in {\\mathcal {K}}$ such that for each $p$ , $\\Psi _{N_n}(p,\\cdot ) \\rightarrow \\Psi (p,\\cdot ) $ in ${\\mathcal {L}}$ , as $n \\rightarrow \\infty $ .","For $p=1,2,\\ldots $ , consider the sets $ {\\mathcal {S}}_p= \\left\\lbrace \\Psi _{N}(p,\\cdot ), N \\in {\\mathbb {N}}\\right\\rbrace \\subset {\\mathcal {L}}_p.$ By the properties $\\mathrm {i,ii,iii}$ above and Arzela-Ascoli's theorem, each of these sets is sequentially compact (s.c.).","Countable products of s.c. sets are s.c. in the product topology (sequential Tychonov's theorem), hence $\\times _p{\\mathcal {S}}_p \\subset {\\mathcal {K}}$ is s.c.","Therefore, the sequence $\\Psi _N \\subset \\times _p{\\mathcal {S}}_p$ has a convergent subsequence.","Since convergence in the product topology implies convergence for any $p$ , the lemma follows.","Lemma 5 Any limit point $\\Psi \\in {\\mathcal {K}}$ as in Lemma REF satisfies $\\Psi (p,s_2)-\\Psi (p,s_1) = - p \\int _{s_1}^{s_2} \\mathrm {d}s \\Psi (p+1,s), \\qquad \\Psi (p,0) = 1$ The equation (REF ) is rewritten as $ \\Psi _N(p,s_2)-\\Psi _N(p,s_1)= -p (1-r_1(N)) \\int _{s_1}^{s_2} \\mathrm {d}s \\left[\\Psi _N(p+1,s) - r_2(N,s,p) \\right]$ where $r_2$ is the sum of the second and third term between square brackets in (REF ) and $r_1(N) &= 1-N\\gamma _{N} = {\\mathcal {O}}(1/N), \\\\r_2(N,s,p) &= \\left(1-\\frac{2}{\\beta }\\right)\\frac{{\\mathbb {E}}(g_N^{p-1} g_N^{(2)})}{N^{p+1}} + \\frac{2(p-1)}{\\beta }\\frac{{\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)})}{N^{p+1}} = {\\mathcal {O}}(1/N)$ with the bounds ${\\mathcal {O}}(1/N)$ uniform in $s$ but not necessarily in $p$ .","The lemma follows by considering (REF ), for fixed $p$ , along the sequence ${N_n}$ .","Sloppily put, the above two lemmas show that $\\Psi _N(p,\\cdot )$ converges to a solution of the `limiting hierarchy of equations' (REF ).","It remains to prove that the limiting hierarchy has a unique solution, namely the right-hand side of (REF ).","Therefore, the proof of Theorem REF is completed by the next lemma.","Lemma 6 $\\Psi (p,s): = \\frac{1}{(1+s)^p}$ is the unique element in ${\\mathcal {K}}$ that satisfies (REF ) and $\\sup _p ||\\Psi (p,\\cdot ) ||_\\infty \\le 1$ .","We proceed by induction in the interval length $T$ .","Assume that the claim is proven for $T\\ge 0$ (for $T=0$ it is trivial).","Take then $0 \\le s_1 \\le T$ and $ s_1< s_2 0$ and $0\\le \\gamma < 2\\pi / N$ .","Note that as indicated in the definition of $\\mathcal {H}$ , periodic boundary conditions in the transverse direction are imposed on all operators.","As we have briefly discussed in Section , the DMPK equation comes in various guises, depending on the abstract symmetry group of the transmission matrix.","At the microscopic level, time reversal $T$ is naturally implemented by complex conjugation in the `position basis', namely $(T\\Psi )(x,z) = \\overline{\\Psi (x,z)}\\,.$ It is immediate to check that $H_\\mathrm {kin}$ is invariant under $T$ iff $\\gamma =0$ : the magnetic field indeed breaks time reversal invariance."],["Eigenvalues, eigenvectors and chaoticity","We consider the eigenvalue problem for the kinetic Hamiltonian $H_{\\mathrm {kin}}\\Psi =E\\Psi $ at some fixed energy $E$ .","By translation invariance and the periodic boundary conditions, the (non-normalizable) solutions are given by $\\Psi _{k,\\nu }(x,z)=\\frac{1}{\\sqrt{N}}e^{ik x}e^{\\frac{2\\pi i}{N}\\nu z}$ for $\\nu =1,\\ldots ,N$ , with $E =E(k,\\nu )= 2\\cos (k) + 2h_1\\cos \\left(\\gamma + \\frac{2\\pi }{N}\\nu \\right) + 2 h_2\\cos \\left(k - \\gamma - \\frac{2\\pi }{N}\\nu \\right)$ We now look for the condition on $E$ so that (REF ) has plane wave solutions, rather than exponentially decaying ones, i.e.","such that $E=E(k,\\nu )$ for some $k,\\nu $ .","In physical terms, this means that we do not want to study evanescent modes, also called `elliptic channels'.","In fact, we first fix the energy $E$ and consider solutions $k = k_\\nu (E)$ of (REF ) for any `transversal mode' $\\nu $ .","We shall drop the $E$ dependence in the sequel.","There are two such wavevectors $k_\\nu ^\\pm $ corresponding to a right moving and a left moving wave.","The relations $ k_\\nu ^+ = -k_{-\\nu }^-$ holds in the time reversal invariant case, but is broken if $\\gamma >0$ .","If $h_2=0$ , the residual symmetry $\\Psi (x,z) \\mapsto \\Psi (x,-z)$ induces the additional degeneracy $k_\\nu ^+ = -k_\\nu ^-$ that needs to be avoided.","It is a straightforward exercise to check Lemma 7 For an energy $E\\ne 0$ , $|E|<2$ , and a kinetic Hamiltonian $H_{\\mathrm {kin}}$ with parameters $0\\le \\gamma <\\frac{\\pi }{N}$ , and $h_1,h_2>0$ sufficiently small, in particular $|E|+2h_1+2h_2<2\\,,$ the equation $H_{\\mathrm {kin}}\\Psi =E\\Psi $ has $2N$ plane wave solutions $\\Psi _{k,\\nu }(x,z)=\\frac{1}{\\sqrt{N}}e^{ik_\\nu ^{\\sigma } x}e^{\\frac{2\\pi i}{N}\\nu z}$ with $\\nu \\in {\\mathbb {Z}}_N$ , $\\sigma \\in \\lbrace {+,-\\rbrace }$ .","For $\\gamma \\ne 0$ , the longitudinal wave numbers $k_\\nu ^{\\sigma }$ are non degenerate in the sense that $\\varsigma _1 k_{\\nu (1)}^{\\sigma _1}+\\varsigma _2 k_{\\nu (2)}^{\\sigma _2}+\\varsigma _3 k_{\\nu (3)}^{\\sigma _3}+\\varsigma _4 k_{\\nu (4)}^{\\sigma _4}=0\\mbox{ }(\\mathrm {mod}\\mbox{ }2\\pi )$ for signs $\\varsigma _1,...,\\varsigma _4$ only in the trivial case $\\big (\\varsigma _1, \\sigma _1, \\nu (1)\\big ) = \\big (-\\varsigma _2, \\sigma _2, \\nu (2)\\big )\\quad \\text{and}\\quad \\big (\\varsigma _3, \\sigma _3, \\nu (3)\\big ) = \\big (-\\varsigma _4, \\sigma _4, \\nu (4)\\big )\\,,$ and if $\\gamma =0$ also in the $T$ -symmetric situation $\\big (\\varsigma _1, \\sigma _1, \\nu (1)\\big ) = \\big (\\varsigma _2, -\\sigma _2, -\\nu (2)\\big )\\quad \\text{and}\\quad \\big (\\varsigma _3, \\sigma _3, \\nu (3)\\big ) = \\big (\\varsigma _4, -\\sigma _4, -\\nu (4)\\big )\\,,$ all of this up to relabeling of the indices.","The above lemma shows that the Hamiltonian has no other symmetry than time reversal invariance, in accordance with the macroscopic theory of the previous section.","It is important to realize that a residual symmetry of the kinetic Hamiltonian could, and in fact would, leave a trace in the scaling limit, even if that symmetry got broken by the disorder.","The DMPK theory considers a single ensemble of transfer matrices, whereas there are really two microscopic models, one without and one with disorder.","For our derivation, it is essential that both have the correct symmetry properties.","For further considerations, it is useful to define the chaoticity of the kinetic Hamiltonian in analogy to [20] by $\\mathrm {cha}\\left(\\gamma ,h_1,h_2\\right)=\\min \\left\\lbrace \\left|\\sum _{i=1}^4 \\varsigma _i k_{\\nu (i)}^{\\sigma _i}\\right|, ({\\varsigma },{\\sigma },{\\nu })\\mbox{ not solving } (\\ref {exceptionTRI}) \\right\\rbrace \\,,$ for $\\gamma >0$ and $\\mathrm {cha}\\left(0,h_1,h_2\\right)=\\min \\left\\lbrace \\left|\\sum _{i=1}^4 \\varsigma _i k_{\\nu (i)}^{\\sigma _i}\\right|,({\\varsigma },{\\sigma },{\\nu })\\mbox{ not solving } (\\ref {exceptionTRI})\\mbox{ or } (\\ref {exception})\\right\\rbrace .$ if $\\gamma =0$ ."],["The transfer matrix","We decompose the kinetic Hamiltonian $(H_\\mathrm {kin}\\Psi )_x= H_\\perp \\Psi _x + P \\Psi _{x+1} + P^* \\Psi _{x-1}$ as a strictly transverse operator $(H_\\perp \\phi )(z) = h_1 \\left( \\mathrm {e}^{\\mathrm {i}\\gamma }\\phi (z+1) + \\mathrm {e}^{-\\mathrm {i}\\gamma }\\phi (z-1) \\right)\\,,$ and the components inducing hopping in the longitudinal direction $(P\\phi )(z) = \\phi (z) + h_2 \\mathrm {e}^{\\mathrm {i}\\gamma } \\phi (z+1)\\,,\\quad \\text{and}\\quad (P^{*}\\phi )(z) = \\phi (z) + h_2 \\mathrm {e}^{-\\mathrm {i}\\gamma } \\phi (z-1)\\,,$ where $\\phi (z)\\in {\\mathbb {C}}^N$ .","Similarly, the random potential can be seen as a sequence of $N\\times N$ diagonal matrices $V_x$ .","With these notations, the eigenvalue equation with disorder, $\\lambda >0$ , reads $\\begin{pmatrix}\\Psi _{x+1}\\\\\\Psi _{x}\\end{pmatrix}=\\begin{pmatrix}\\left(P^*\\right)^{-1}(E-H_\\perp -\\lambda V_x)&-\\left(P^*\\right)^{-1}P\\\\{1}&{0}\\end{pmatrix}\\begin{pmatrix}\\Psi _{x}\\\\\\Psi _{x-1}\\end{pmatrix}=:T_x^\\lambda \\begin{pmatrix}\\Psi _{x}\\\\\\Psi _{x-1}\\end{pmatrix},$ thereby defining the transfer matrix $T_x^\\lambda $ (of dimension $2N$ ) for the layer $x$ as it usually appears in the mathematical literature.","By the multiplicative property, the transfer matrix for $L$ layers is simply given by the product of the one-layer matrices, $\\begin{pmatrix}\\Psi _{L+1}\\\\\\Psi _{L}\\end{pmatrix}=T_L^\\lambda \\cdots T_1^\\lambda \\begin{pmatrix}\\Psi _{1}\\\\\\Psi _{0}\\end{pmatrix},$ In order to allow for a comparison with the DMPK theory, it is important to write the transfer matrix in the natural basis for ${\\mathbb {C}}^{2N}$ , i.e.","the basis in which the first $N$ components correspond to left moving waves, and the other $N$ components are right moving.","Morever, in the time-reversal invariant case we need the correct identification between left and right moving channels, given by (REF ).","In other words, the time reversal of the vector $(\\phi _1, \\phi _2)^t$ must correspond to $(\\bar{\\phi }_2, \\bar{\\phi }_1)^t$ .","Only in that basis do the definitions of the transfer matrix given in Section REF apply.","We shall refer to it as the channel basis.","Let $Q$ be the $N\\times N$ matrix whose columns are the transverse eigenvectors $\\phi _\\nu (z) = 1/\\sqrt{N}\\exp (2\\pi \\mathrm {i}\\nu z/N)$ .","Furthermore, let $\\Pi $ be the permutation matrix that interchanges the $\\nu $ and $-\\nu $ channels, and let $\\Upsilon :=\\begin{pmatrix}\\frac{\\mathrm {e}^{\\mathrm {i}k^+}}{\\sqrt{\\left|v^+\\right|}}&\\frac{\\mathrm {e}^{\\mathrm {i}k^-}}{\\sqrt{\\left|v^-\\right|}}\\\\\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^-\\right|}}\\end{pmatrix}\\begin{pmatrix}1&0\\\\0&\\Pi \\end{pmatrix}$ where $k^\\pm $ and $v^\\pm $ denote diagonal matrices with elements $k^\\pm _{\\nu }$ and $v^\\pm _\\nu $ , and $v^\\pm _\\nu $ are the velocities in each channel, $v^\\pm _\\nu =\\left.\\frac{\\partial }{\\partial k}E(k,\\nu )\\right|_{k=k_\\nu ^\\pm }=-2\\sin \\left(k^\\pm _\\nu \\right)-2h_2\\sin \\left(k^\\pm _\\nu - \\frac{2\\pi }{N}\\nu - \\gamma \\right).$ Note that all $v^\\pm _\\nu $ are nonzero by (REF ).","The action of $T$ (complex conjugation) on the eigenvectors yields $T:\\,(k_\\nu ^\\sigma ,\\nu )\\mapsto (-k_\\nu ^\\sigma ,-\\nu )$ which equals $(k_{-\\nu }^{-\\sigma },-\\nu )$ if time reversal invariance holds, so that the $(\\sigma = -)$ block needs to be reordered by $\\nu \\leftrightarrow -\\nu $ .","Hence the permutation $\\Pi $ .","With these notations, Lemma REF reads $ \\Upsilon ^{-1}(Q\\otimes 1)^{*}T_x^0(Q\\otimes 1)\\Upsilon = \\begin{pmatrix}\\mathrm {e}^{\\mathrm {i}k^+}&0\\\\0&\\Pi \\mathrm {e}^{\\mathrm {i}k^-} \\Pi \\end{pmatrix} =: M_x^0\\,.$ The matrix $M_x^0$ is the transfer matrix for the purely kinetic transport in a single layer, which is diagonal the channel basis.","As expected, it simply adds a phase to the traveling plane wave.","It is now easily checked that $M_x^0$ is a bonafide transfer matrix in the sense of Definition REF : Eq.","(REF ) is always satisfied and (REF ) holds at $\\gamma =0$ , as conjugation by $\\Sigma _x$ precisely maps $k^\\sigma _\\nu $ to $k^{-\\sigma }_{-\\nu }=-k^\\sigma _{\\nu }$ .","The transfer matrix for the total system is again obtained by multiplication $M^\\lambda (L) := M^\\lambda _L \\cdots M^\\lambda _1 = K^{-1} \\, T_L^\\lambda \\cdots T_1^\\lambda \\, K\\,,$ where $K = (Q\\otimes 1)\\Upsilon $ is the change of basis from the position basis to the channel basis.","This random matrix and its relation the DMPK theory is the central object of study in the following."],["A scaling limit","The DMPK theory suggests that the microscopic transfer matrix for a disordered wire of length $L$ should converge to a solution of the DMPK equation in the correct macroscopic limit.","As discussed in the introduction, the natural scaling between the microscopic length $L$ and the macroscopic length $s$ is through the mean free path, $L = \\lambda ^{-2}s$ .","A naive interpretation of the DMPK theory would then be the convergence of $M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor )$ to ${\\mathcal {M}}(s)$ .","This cannot possibly hold as $M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor )$ contains rapidly oscillating terms as a function of $s$ , as already exemplified at the level of the unperturbed system (REF ): $M^0(\\lfloor \\lambda ^{-2}s \\rfloor )=\\begin{pmatrix}\\exp \\left(\\mathrm {i}\\lfloor \\lambda ^{-2}s \\rfloor k^+\\right)&0\\\\0&\\Pi \\exp \\left(\\mathrm {i}\\lfloor \\lambda ^{-2}s \\rfloor k^-\\right) \\Pi \\end{pmatrix}\\,.$ To obtain a reasonable limit, we therefore consider $A^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor ) := (M^0(\\lfloor \\lambda ^{-2}s \\rfloor ))^{-1}M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor ).$ As the set of matrices of Definition REF form a group, $A^\\lambda $ is a transfer matrix again.","Moreover, it is an easy check that in the polar decomposition (REF ), the matrix $S$ corresponding to $A^\\lambda $ is the same as that corresponding to $M^\\lambda $ , so that they have the same transmission eigenvalues.","In order to state the result of the scaling limit, $\\lambda \\rightarrow 0$ , we introduce the following processes.","For $\\gamma >0$ , let ${\\mathcal {Z}}_{\\gamma }(s) := \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\\\mathfrak {b}^*(s) &{ \\mathfrak {a}^{\\prime }}(s)\\end{pmatrix}$ with $\\mathfrak {a}_{\\mu \\mu }(s)=-\\mathfrak {a}^{\\prime }_{\\mu \\mu }(s)=\\frac{\\mathrm {i}}{\\sqrt{(4-E^2)N}}W(s)$ with the same standard real Brownian motion $W$ for all $\\mu =1,\\ldots ,N$ .","All diagonal elements of ${\\mathcal {Z}}$ are thus perfectly correlated.","For the off-diagonal elements, $\\mathfrak {a}_{\\mu \\nu }(s)=-\\overline{\\mathfrak {a}_{\\nu \\mu }(s)}&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{++}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu <\\nu \\le N\\\\\\mathfrak {a}^{\\prime }_{\\mu \\nu }(s)=-\\overline{\\mathfrak {a}^{\\prime }_{\\nu \\mu }(s)}&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{--}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu <\\nu \\le N\\\\\\mathfrak {b}_{\\mu \\nu }(s)&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{+-}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu ,\\nu \\le N$ with all the elements of $B^{++},B^{--}$ and $B^{+-}$ standard complex Brownian motions, mutually independent and independent of $W$ .","For $\\gamma =0$ , ${\\mathcal {Z}}_0(s) := \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\ \\overline{\\mathfrak {b}(s)} &\\overline{ \\mathfrak {a}(s)}\\end{pmatrix}$ with the definition of $\\mathfrak {a}$ unchanged, but $\\mathfrak {a}^{\\prime }=\\overline{\\mathfrak {a}}$ now, and a symmetric $\\mathfrak {b}$ : $\\mathfrak {b}_{\\mu \\nu }(s)=\\mathfrak {b}_{\\nu \\mu }&=\\frac{1}{\\sqrt{(4-E^2)N}}\\tilde{B}^{+-}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu \\le \\nu \\le N,$ with the entries of the $\\tilde{B}^{+-}$ independent standard complex Brownian motions again, independent of the elements of $B^{++}$ and $W$ .","The result of the scaling limit, $\\lambda \\rightarrow 0$ is summarized in the main theorem.","Theorem 8 If $h_1$ and $h_2$ depend on $\\lambda $ so that $h_1(\\lambda ) \\longrightarrow 0\\,,\\qquad h_2(\\lambda ) \\longrightarrow 0, \\qquad \\text{and}\\qquad \\lambda ^{-2}\\mathrm {cha}(\\gamma ,h_1(\\lambda ),h_2(\\lambda ))\\longrightarrow \\infty \\,,$ as $\\lambda \\rightarrow 0$ , then the process $\\left(A^\\lambda \\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right)_{s\\ge 0}$ converges in distribution to the process $\\left({\\mathcal {A}}(s)\\right)_{s\\ge 0}$ on the path space of ${\\mathbb {C}}^{2N\\times 2N}$ -valued processes endowed with Skorhod topology.","For $\\gamma \\ge 0$ , $\\left({\\mathcal {A}}(s)\\right)_{s\\ge 0}$ is given as the unique solution for $s\\ge 0$ to $\\begin{split}\\mathrm {d}{\\mathcal {A}}(s)&=\\mathrm {d}{\\mathcal {Z}}_\\gamma (s){\\mathcal {A}}(s)\\\\{\\mathcal {A}}(0)&=1.\\end{split}$ Remark.","Existence and uniqueness of ${\\mathcal {A}}$ is a standard result, as all entries of ${\\mathcal {Z}}$ are Brownian motions.","Note that no additional moment condition is needed on the random variables $V(x,z)$ .","The convergence of the hopping parameters $h_i$ to zero merely brings the resulting process ${\\mathcal {A}}$ into a isotropic form close to the DMPK process.","It is however not essential for the scaling limit per se, and the same techniques used in the proof of Theorem REF yield a limiting process for fixed $h_1, h_2\\ne 0$ , but a less isotropic one.","For $\\gamma >0$ , ${\\mathcal {Y}}_{\\gamma }(s) := \\begin{pmatrix} \\alpha (s) & \\beta (s) \\\\ \\beta (s)^{*}&{ \\alpha ^{\\prime }}(s)\\end{pmatrix}$ where $\\alpha (s), \\alpha ^{\\prime }(s)$ and $\\beta (s)$ differ from $\\mathfrak {a}(s),\\mathfrak {a}^{\\prime }(s)$ and $\\mathfrak {b}(s)$ only through their covariances, namely by the replacement $\\frac{1}{\\sqrt{(4-E^2)N}} \\quad \\longrightarrow \\quad \\frac{1}{\\sqrt{N \\left|v^+_{\\sigma _1\\nu _1}v^+_{\\sigma _2\\nu _2}\\right|}}\\,.$ For $\\gamma =0$ , ${\\mathcal {Y}}_0(s) := \\begin{pmatrix} \\alpha (s) & \\beta (s) \\\\[1mm] \\overline{\\beta (s)} &\\overline{\\alpha (s)}\\end{pmatrix}$ with the same substitution.","Proposition 9 Let $(\\gamma , h_1, h_2)$ be fixed (in particular, not dependent on $\\lambda $ ) and such that Lemma REF holds.","As $\\lambda \\rightarrow 0$ , the process $\\left(A_\\lambda \\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right)_{s\\ge 0}$ converges in distribution to $\\left({\\mathcal {G}}(s)\\right)_{s\\ge 0}$ on the path space of ${\\mathbb {C}}^{2N\\times 2N}$ -valued processes endowed with Skorhod topology.","$\\left({\\mathcal {G}}(s)\\right)_{s\\ge 0}$ is given as the unique solution for $s\\ge 0$ to $\\begin{split}\\mathrm {d}{\\mathcal {G}}(s)&=\\mathrm {d}{\\mathcal {Y}}(s){\\mathcal {G}}(s) \\\\{\\mathcal {G}}(0)&=1.","\\\\\\end{split}$"],["Discussion","Let us comment on the limiting process of transfer matrices ${\\mathcal {A}}(s)$ , and compare it to the ideal ensemble ${\\mathcal {M}}$ of Section .","As already discussed in [1], the overall factor $\\sqrt{4-E^2}$ only corresponds to a redefinition of the mean free path and is irrelevant here.","The major difference lies in the diagonal of the processes $\\mathfrak {a}, \\mathfrak {a}^{\\prime }$ generating ${\\mathcal {A}}$ , which have perfectly correlated diagonal elements, whereas they are independent of each other in their cousins generating ${\\mathcal {M}}$ .","In the case $\\beta =1$ , an additional deviation can be found in the variance of the diagonal elements of $\\mathfrak {b}$ , which are smaller here than in the ideal case by a factor $\\sqrt{2}\\cdot \\sqrt{N/(N+1)}$ .","Despite these differences, we have: Corollary 10 If $\\beta =2$ , the law of the process of transmission eigenvalues $(T_k)_{k=1}^N$ induced by ${\\mathcal {A}}$ is the same as that induced by ${\\mathcal {M}}$ .","Mathematically, this can be observed in the fact that the SDE for ${\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++}$ , obtained in a straightforward way by Itô calculus, $\\mathrm {d}({\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++} )= {\\mathcal {M}}_{++}^{*}\\mathrm {d}\\mathfrak {b}{\\mathcal {M}}_{-+} + {\\mathcal {M}}_{-+}^{*}\\mathrm {d}\\mathfrak {b}^{*}{\\mathcal {M}}_{++} + {\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++} \\mathrm {d}s + {\\mathcal {M}}_{-+}^{*}{\\mathcal {M}}_{-+} \\mathrm {d}s$ does not depend on $\\mathrm {d}\\mathfrak {a}^\\sharp $ , and by recalling that the transmission eigenvalues are directly related to the eigenvalues $\\lambda _k$ of this matrix.","For $\\beta =2$ , the limiting $\\mathrm {d}\\mathfrak {b}$ is exactly equal to the ideal one.","The underlying physical reason is that the $\\mathfrak {a}^\\sharp $ blocks in the infinitesimal transfer matrix merely change the basis of left-, respectively right-moving channels.","As such they do not change the magnitude of either the scattered or the reflected waves, hence they do not contribute to the transmission eigenvalues $T_{k}$ .","For the $\\beta =1$ case, the process of transmission eigenvalues induced by ${\\mathcal {A}}$ is different from that induced by ${\\mathcal {M}}$ .","Of course one may suspect that in the limit $N \\rightarrow \\infty $ , these processes have similar features, in particular, that the variance of the conductance in the diffusive regime is the same in both cases, and equal to its universal value.","In fact in the same limit, we even expect the non-isotropic processes of Proposition REF to share the same property.","This remains currently at the level of speculations, the reason being the relatively poor understanding of the properties of the DMPK equation itself in the large $N$ limit.","This should be contrasted with recent efforts in the study of hermitian random matrices, where the Gaussian ensembles are very well-known and the challenge is to show that other ensembles share some of their properties, see e.g.","[10] and references therein."],["Symmetry considerations","As already mentioned in Section , the driving process ${\\mathcal {L}}$ in the SDE for ${\\mathcal {M}}$ satisfies the invariance property ${\\mathcal {W}}{\\mathcal {L}}{\\mathcal {W}}^{-1} \\mathop {=}\\limits ^{d} {\\mathcal {L}}$ for ${\\mathcal {W}}= \\begin{pmatrix} {\\mathcal {W}}_+ &0 \\\\ 0 &{\\mathcal {W}}_- \\end{pmatrix}$ with ${\\mathcal {W}}_+, {\\mathcal {W}}_-$ arbitrary unitaries in the $\\beta =2$ case and satisfying ${\\mathcal {W}}_-=\\overline{{\\mathcal {W}}_+}$ in the $\\beta =1$ case.","In other words, all channel bases are assumed to be equivalent.","It is exactly this equivalence that is lost in our model.","This can be understood heuristically as follows.","Here, the size of the impurities is assumed to be much smaller than the wavelength of the scattered waves, a fact expressed by the $\\delta $ -correlation in space of the potential, i.e.","$V=\\sum _y V_y$ with $y=(x,z)$ and ${\\mathbb {E}}(V_yV_{y^{\\prime }})= \\delta _{y,y^{\\prime }}$ .","Therefore, the position basis (or its dual, the momentum basis) is naturally singled out.","There is no reason to expect another choice of basis to be equivalent and to allow for arbitrary unitaries ${\\mathcal {W}}_+,{\\mathcal {W}}_-$ in (REF ).","The symmetry that still survives (but in fact, only so because we performed the additional scaling limit $h_1, h_2 \\rightarrow 0$ ) is the relabeling of channels; indeed the scaling limit $h_1, h_2 \\rightarrow 0$ makes the group velocity of all channels $\\nu $ equal at a given energy $E$ .","Up to phases, this corresponds to restricting ${\\mathcal {W}}_+,{\\mathcal {W}}_-$ to be permutation matrices.","One easily checks that, with this restriction, the driving process ${\\mathcal {Z}}_\\gamma $ still satisfies (REF ).","Another consequence of the locality of the impurities is captured by the following heuristic argument.","Let us consider the kernel $S_{p,p^{\\prime }}$ of the scattering matrix in momentum basis, i.e.","$p=(k,\\nu )$ .","Since the scattering is weak (as already indicated, this is inherent to the setup and it is forced in our model by the $\\lambda \\rightarrow 0$ limit), multiple scattering can be neglected for short slabs of material, hence we can use the Born approximation for $S_{p,p^{\\prime }}$ : $S_{p,p^{\\prime }} = \\delta _{p,p^{\\prime }}-2 \\pi \\mathrm {i}\\lambda \\sum _y (\\tilde{V}_y)_{p,p^{\\prime }} \\delta (E(p)-E(p^{\\prime })) + {\\mathcal {O}}(\\lambda ^2)$ where $(\\tilde{V}_y)=V_ye^{\\mathrm {i}(p-p^{\\prime })y}$ .","For $p=p^{\\prime }$ , this expression depends on $p$ only through the factor $\\delta (E(p)-E(p^{\\prime }))$ which contributes a $p$ -dependent group velocity, a dependence which vanishes in the subsequent scaling limit $h_1,h_2 \\rightarrow 0$ .","This means that the diagonal elements of the transmission matrix all coincide and this is exactly what we find in the ensemble ${\\mathcal {A}}(s)$ , since in lowest order the $\\mathfrak {a}$ block corresponds to the transmission matrix.","Finally, we mention [18], [19] where geometric methods are used to analyze a similar weak coupling limit of the Anderson model on tubes, and contact is made with random matrix theory."],["Proof of transfer matrix limits","We prove Theorem REF .","The evolution of $\\left(A^\\lambda (x)\\right)_{x\\ge 0}$ is given by $A^\\lambda (0)=1$ and the stochastic difference equation $A^\\lambda (x)-A^\\lambda (x-1) &= \\left((M^0(x))^{-1} K^{-1}T^\\lambda _x K M^0(x-1)-1\\right) A^\\lambda (x-1) \\nonumber \\\\&=: \\lambda Z_x A^\\lambda (x-1),$ where $Z_x = (M^0(x))^{-1} R_x M^0(x)$ and we defined $\\lambda R_x := K^{-1}T^\\lambda _x K (M^0_x)^{-1} -1 = K^{-1}(T^\\lambda _x-T^0_x) K (M^0_x)^{-1}.$ Recall that $K$ , introduced in the previous section, stands for the change from the position basis to the channel basis.","It follows that $R_x = \\Upsilon ^{-1} \\begin{pmatrix}-Q^{*}(P^{*})^{-1}V_xQ & 0 \\\\ 0 & 0\\end{pmatrix}\\Upsilon (M^0_x)^{-1}$ Using the explicit forms of $\\Upsilon , Q$ and $P$ , this matrix reads $R_x=\\mathrm {i}\\begin{pmatrix}\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\Pi \\\\-\\Pi \\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&-\\Pi \\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\Pi \\end{pmatrix},$ which of course satisfies $(R_{++})^{*}= -R_{++}$ , $(R_{--})^{*}= -R_{--}$ and $(R_{-+})^{*}= R_{+-}$ .","In the time reversal invariant case, since $Q=Q^T,\\qquad Q\\Pi =Q^*,\\qquad \\Pi Q^*=Q$ this simplifies to $R_x=\\mathrm {i}\\begin{pmatrix}\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ^*\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\\\-\\frac{1}{\\sqrt{\\left|v^+\\right|}}QV_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&-\\frac{1}{\\sqrt{\\left|v^+\\right|}}QV_xQ^*\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\end{pmatrix}\\,.$ Since $Q^*=\\overline{Q}$ , we also have that $R_{--} =\\overline{ R_{++}}$ , and in a similar fashion $R_{-+} = \\overline{R_{+-}}$ .","Before we go further into the proof, let us explain the heuristics of the convergence to the DMPK equation.","The matrix $R_x$ contains the $N$ i.i.d random variables $V(x,z)$ , $z=1,\\ldots ,N$ , and $R_x,R_y$ are independent for $x\\ne y$ .","Under the appropriate technical conditions, we have convergence, as $ \\lambda \\rightarrow 0$ ; $\\lambda \\sum _{x=0}^{\\lfloor \\lambda ^{-2}s\\rfloor }R_x \\stackrel{\\mathrm {d}}{\\longrightarrow }{\\mathcal {R}}(s)\\,.$ where the $V(x,z)$ in $R_x$ are replaced by Brownian motions $B_z(s)$ in ${\\mathcal {R}}(s)$ .","Of course, this is nothing else than the convergence of a random walk to Brownian motion.","Now, the matrices $Z_x$ generating the discrete process $A(x)$ do contain the highly oscillating phases of $M^0(x)$ .","Let $Z^\\lambda (s) = \\lambda \\sum _{x=0}^{\\lfloor \\lambda ^{-2}s\\rfloor }Z_x\\,.$ The correlation of any two matrix elements reads ${\\mathbb {E}}\\left[\\left(Z^\\lambda (s)\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right] \\\\=\\lambda ^{2}\\sum _{x=1}^{\\lfloor \\lambda ^{-2}s\\rfloor }\\exp \\left(\\mathrm {i}x\\left(-k^{\\sigma _m}_{\\sigma _m\\nu _m}+k^{\\sigma _n}_{\\sigma _n\\nu _n}-k^{\\sigma _p}_{\\sigma _p\\nu _p}+k^{\\sigma _r}_{\\sigma _r\\nu _r}\\right)\\right){\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right].$ where the $\\sigma $ 's and $\\nu $ 's denote the block and position in the block of a certain element, the `physical' momentum, due to the permutation $\\Pi $ is not $\\nu $ , but $\\sigma \\nu $ .","The expectation in the r.h.s is independent of $x$ , so that this sum is highly oscillatory and formally converges to a $\\delta $ function.","Hence, the phases create a limiting process ${\\mathcal {Z}}(s)$ with almost completely uncorrelated entries, apart from the exceptional conditions of Lemma REF .","As a result, the number of independent random variables in ${\\mathcal {Z}}(s)$ is of order $N^2$ , whereas it was only $N$ in ${\\mathcal {R}}(s)$ , a phenomenon called `noise explosion' in [20].","Precisely, we prove: Lemma 11 Under the conditions of Theorem REF and in the same topology, the process $\\left(Z^\\lambda (s)\\right)_{s\\ge 0}$ converges in distribution to the process $\\left({\\mathcal {Z}}_\\gamma (s)\\right)_{s\\ge 0}$ , for $\\gamma \\ge 0$ .","We first recall that $Q$ is the matrix of transversal plane waves so that $(Q^{*}V_x Q)_{\\mu \\nu } = \\widehat{V_x}(\\mu -\\nu )\\,.$ In particular, all elements on the diagonal are perfectly correlated.","Now, we consider (REF ) for $s\\ge 0$ , and note that the wavevectors $k_\\nu ^\\sigma $ and the matrices $R_x$ all depend on $\\lambda $ implicitly through the dependence $h_i(\\lambda )$ .","However, ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right]$ is independent of $x$ and can be taken out of the sum.","For all choices $\\lbrace (\\nu _i,\\sigma _i): i=m,n,p,r\\rbrace $ for which the exponent does not vanish, we have that $\\left| \\lambda ^{2}\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\exp \\left(ix\\left(-k^{\\sigma _m}_{\\sigma _m\\nu _m}+k^{\\sigma _n}_{\\sigma _n\\nu _n}-k^{\\sigma _p}_{\\sigma _p\\nu _p}+k^{\\sigma _r}_{\\sigma _r\\nu _r}\\right)\\right)\\right|\\le \\frac{2\\lambda ^2}{\\mathrm {cha}(\\lambda )}\\longrightarrow 0,$ as $\\lambda \\rightarrow 0$ , by Assumption (REF ).","By Lemma REF , the the wavenumbers cancel out whenever $\\left(\\sigma _m,\\nu _m\\right)=\\left(\\sigma _n,\\nu _n\\right)\\mbox{ and }\\left(\\sigma _p,\\nu _p\\right)=\\left(\\sigma _r,\\nu _r\\right)\\,,$ corresponding to any two diagonal elements in $Z^\\lambda $ , or $\\left(\\sigma _m,\\nu _m\\right)=\\left(\\sigma _r,\\nu _r\\right)\\mbox{ and }\\left(\\sigma _p,\\nu _p\\right)=\\left(\\sigma _n,\\nu _n\\right)\\,,$ namely two mutually `transpose' entries of $Z^\\lambda $ .","In the case $\\gamma =0$ , a last possibility is given by $\\left(\\sigma _m,\\nu _m\\right)=\\left(-\\sigma _p,\\nu _p\\right)\\mbox{ and }\\left(\\sigma _n,\\nu _n\\right)=\\left(-\\sigma _r,\\nu _r\\right)\\,,$ corresponding to two elements in the same position, but within the opposite blocks.","In these three cases and for all $\\lambda $ , $\\lambda ^{2}\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }1=s.$ Furthermore, using the explicit form (REF ) of $R_x$ and the definition of $Q$ , ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right]=-\\frac{\\sigma _m\\sigma _p}{N^2\\sqrt{\\left|v^+_{\\sigma _m\\nu _m}v^+_{\\sigma _n\\nu _n}v^+_{\\sigma _p\\nu _p}v^+_{\\sigma _r\\nu _r}\\right|}} \\\\\\cdot \\sum _{z=0}^{N-1}\\exp \\left(\\frac{2\\pi i}{N}z\\left(-\\sigma _m\\nu _m+\\sigma _n\\nu _n-\\sigma _p\\nu _p+\\sigma _r\\nu _r\\right)\\right)$ which we only need to evaluate in the `stationary phase' situations (REF ), (REF ), and (REF ).","In all these cases, ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right] = -\\frac{\\sigma _m\\sigma _p}{N\\sqrt{\\left|v^+_{\\sigma _m\\nu _m}v^+_{\\sigma _n\\nu _n}v^+_{\\sigma _p\\nu _p}v^+_{\\sigma _r\\nu _r}\\right|}}.$ Note that the $v^+_{\\nu }$ still depend on $\\lambda $ and $\\nu $ , but by (REF ), they converge to the $\\nu $ -independent limit $|2\\sin (k)|$ with $2\\cos (k)=E$ .","Hence, in the limit $\\lambda \\rightarrow 0$ , we have ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right] \\longrightarrow -\\frac{\\sigma _m\\sigma _p}{(4-E^2)N}\\,.$ A very similar oscillatory sum appears in ${\\mathbb {E}}\\left[\\left(Z^\\lambda (s)^{*}\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right]$ , with parallel conclusions.","In summary, for all $s\\ge 0$ $\\begin{split}\\lim _{\\lambda \\rightarrow 0}{\\mathbb {E}}\\left[\\left(Z^\\lambda (s)\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right]&=\\int _0^s\\mathrm {d}\\left\\langle \\left({\\mathcal {Z}}\\right)_{mn}, \\left({\\mathcal {Z}}\\right)_{pr}\\right\\rangle _t,\\\\\\lim _{\\lambda \\rightarrow 0}{\\mathbb {E}}\\left[\\left(Z^\\lambda (s)^{*}\\right)_{mn}, \\left(Z^\\lambda (s)\\right)_{pr}\\right]&=\\int _0^s\\mathrm {d}\\left\\langle \\left({\\mathcal {Z}}^*\\right)_{mn},\\left({\\mathcal {Z}}\\right)_{pr}\\right\\rangle _t.\\end{split}$ with ${\\mathcal {Z}}$ given in the previous section, and $\\langle M,N \\rangle _t$ the bracket process of two martingales $M,N$ .","In particular, the block structure of ${\\mathcal {Z}}$ arises from the corresponding relations noted above in $R_x$ .","Moreover, perfect correlation of the diagonal elements of $Q^{*}V_x Q$ and the exceptional case (REF ) imply the perfect correlation of the diagonal elements of $\\mathrm {d}\\mathfrak {a}$ and $\\mathrm {d}\\mathfrak {a}^{\\prime }$ .","All other exceptional cases impose the correlations $\\vert \\mathrm {d}\\mathfrak {b}_{\\mu \\nu }\\vert ^2$ and $\\vert \\mathrm {d}\\mathfrak {a}_{\\mu \\nu }\\vert ^2$ .","Now the lemma follows as a simple generalization of Donsker's invariance principle, for example by using Chapter VII, Theorem 3.7 in [13].","They check the convergence of three characteristics, of which, in their notation, $\\left[\\sup -\\beta _3^{\\prime }\\right]$ is trivially fulfilled as $Z^\\lambda $ and ${\\mathcal {Z}}$ are martingales, $\\left[\\gamma _3^{\\prime }-{\\mathbb {R}}^+\\right]$ is the convergence of brackets as shown in (REF ), and $\\left[\\delta _{3,1}-{\\mathbb {R}}^+\\right]$ is a simple estimate on the jumps of $Z^\\lambda $ , namely $\\lim _{\\lambda \\rightarrow 0}\\sum _{x=1}^{\\lfloor \\lambda ^{-2}s\\rfloor }\\mathbb {E}\\left(g_a\\left(\\lambda \\left\\Vert R_x\\right\\Vert \\right)\\right)=0$ for all $g_a(y)=y^21_{\\lbrace |y|>a\\rbrace }$ , $a>0$ , which is trivial by $\\mathbb {E}\\left(v^2\\right)=1$ .","Notice that the covariances of the less isotropic ensemble of Proposition REF can be read off from this proof.","The main theorem will now follow from this lemma and the difference equation (REF ).","We simplify the notation of (REF ) and (REF ) by writing the real and imaginary parts of the matrix entries as elements of vectors in ${\\mathbb {R}}^d$ , $d=8N^2$ .","In this notation (REF ) reads $X^{\\lambda }_j(y)-X^{\\lambda }_j(y-1)=\\lambda \\sum _{k=1}^d\\xi ^{\\lambda }_{jk}(y)X^{\\lambda }_k(y-1)$ for all $y$ in ${\\mathbb {N}}$ , with $\\xi ^{\\lambda }(y)\\in {\\mathbb {R}}^{d\\times d}$ independent of $X^{\\lambda }(z), \\xi ^{\\lambda }(z)$ $z\\in \\lbrace 0,...,y-1\\rbrace $ .","Because of the phase factors, the law of $\\xi ^{\\lambda }(y)$ is not independent of $y\\in {\\mathbb {N}}$ , but ${\\mathbb {E}}(\\xi ^{\\lambda }(y))=0$ and $ \\Vert \\xi ^{\\lambda }(y)\\Vert ^2 \\le c^{\\prime } (\\sum _{z=1}^N |v(y,z) |)^2 \\le c \\sum _{z=1}^N v^2(y,z)$ where $c,c^{\\prime }< \\infty $ are $\\lambda $ -independent for sufficiently small $\\lambda $ , but they depend on $N$ .","This follows from (REF ) or (REF ) by noting that the velocity matrix $v^+$ converges to a nonsingular limit as $\\lambda \\rightarrow 0$ ; we will henceforth assume without comment that $\\lambda $ is sufficiently small.","Furthermore, we know from the proof of Lemma REF $\\lim _{\\lambda \\rightarrow 0}\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)=C_{ikjl}\\cdot s$ uniformly for $s\\ge 0$ from bounded intervals.","If we define $\\left({\\mathcal {B}}(s)\\right)_{s\\ge 0}$ as the matrix-valued Brownian motion with bracket process $\\left\\langle {\\mathcal {B}}_{ik},{\\mathcal {B}}_{jl}\\right\\rangle _s= C_{ikjl}\\cdot s,$ the equation (REF ) transforms to $\\mathrm {d}{\\mathcal {X}}_j(s)=\\sum _{k=1}^d\\mathrm {d}{\\mathcal {B}}_{jk}{\\mathcal {X}}_k(s).$ The initial values for $X^{\\lambda }$ and ${\\mathcal {X}}$ are identical and deterministic, the ${\\mathbb {R}}^d$ vector corresponding to the unit matrix $1_{2N}$ .","For notational convenience, we have chosen $X^{\\lambda }$ to still live on the microscopic, discrete space, what we really want to investigate is the cadlag process $\\overline{X}^{\\lambda }(s)=X^{\\lambda }\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right).$ For cadlag processes we define $\\overline{X}^{\\lambda }(s-)$ as the leftside limit of $\\overline{X}^{\\lambda }$ at $s$ .","With the filtration ${\\mathcal {F}}^{\\lambda }_s=\\sigma \\left\\lbrace \\overline{X}^{\\lambda }(t):t\\le s\\right\\rbrace $ , $\\left(\\overline{X}^{\\lambda }(s)\\right)_{s\\ge 0}$ is a $\\left\\lbrace {\\mathcal {F}}^{\\lambda }_s \\right\\rbrace $ -martingale.","Furthermore defining $V^{\\lambda }_{ij}(y)=\\lambda ^2\\sum _{x=1}^y\\sum _{k,l=1}^d\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)X^{\\lambda }_k(x-1)X^{\\lambda }_l(x-1)$ and the corresponding macroscopic $\\overline{V}^{\\lambda }_{ij}(s)=V^{\\lambda }\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)$ for all $i,j=1,...,d$ , the process $\\overline{X}^{\\lambda }_i\\overline{X}^{\\lambda }_j-\\overline{V}^{\\lambda }_{ij}$ is a $\\left\\lbrace {\\mathcal {F}}^{\\lambda }_s \\right\\rbrace $ -martingale as well, and we have by Theorem 7.4.1, [11], Lemma 12 If for any $T>0$ , and any stopping time $T^{\\lambda }_r=\\inf \\left\\lbrace s:\\left|\\overline{X}^{\\lambda }(s)\\right|\\ge r \\mbox{ or } \\left|\\overline{X}^{\\lambda }(s-)\\right|\\ge r\\right\\rbrace ,$ $r>0$ , for all $i,j=1,...,d$ $&\\lim _{\\lambda \\rightarrow 0}\\mathbb {E}\\left(\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{X}^{\\lambda }(s)-\\overline{X}^{\\lambda }(s-)\\right|^2\\right)=0\\\\&\\lim _{\\lambda \\rightarrow 0}\\mathbb {E}\\left(\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\overline{V}^{\\lambda }_{ij}(s-)\\right|\\right)=0$ and $\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\int _0^s\\mathrm {d}t\\sum _{k,l=1}^dC_{ikjl}\\overline{X}^{\\lambda }_k(t)\\overline{X}^{\\lambda }_l(t)\\right|\\stackrel{{\\mathbb {P}}}{\\rightarrow }0,$ then $\\left(\\overline{X}^{\\lambda }(s)\\right)_{s\\ge 0}$ converges in distribution on $D_{{\\mathbb {R}}^d}[0,\\infty )$ to $\\left({\\mathcal {X}}(s)\\right)_{s\\ge 0}$ .","So to prove Theorem REF , we only have to verify the conditions of Lemma REF .","We start with the following observation, Lemma 13 Let $Z_k$ , $k\\in {\\mathbb {N}}$ be i.i.d.","distributed, positive random variables, with $\\mathbb {E}\\left(Z_1\\right)<\\infty .$ Then $\\frac{1}{n}\\mathbb {E}\\left(\\max _{1\\le k\\le n}Z_k\\right)\\rightarrow 0$ as $n\\rightarrow \\infty $ .","With $q(x)={\\mathbb {P}}\\left(Z_1\\ge x\\right)$ for $x\\ge 0$ , we have $\\int _0^\\infty q(x)\\mathrm {d}x=\\mathbb {E}\\left(Z_1\\right),$ while ${\\mathbb {P}}\\left(\\max _{1\\le k\\le n}Z_k\\ge x\\right)=1-(1-q(x))^n\\le n q(x).$ Thus, $\\frac{1}{n}\\mathbb {E}\\left(\\max _{1\\le k\\le n}Z_k\\right)=\\int _0^\\infty \\frac{1-(1-q(x))^n}{n}\\mathrm {d}x$ with the integrand on the right side converging to zero and dominated by the integrable $q(x)$ , an the claim follows by dominated convergence.","For any $T>0, r>0$ given, we have $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\overline{X}^{\\lambda }(s)-\\overline{X}^{\\lambda }(s-)\\right|^2\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|X^{\\lambda }(y)-X^{\\lambda }(y-1)\\right|^2\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\lambda ^2\\sum _{j,k,l=1}^d\\xi ^{\\lambda }_{jk}(y)\\xi ^{\\lambda }_{jl}(y)X^{\\lambda }_{k}(y-1)X^{\\lambda }_{l}(y-1)\\\\\\le &r^2\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor }\\left\\Vert \\xi ^{\\lambda }(y)\\right\\Vert ^2\\\\\\le &c(\\lambda )r^2\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor }\\sum _{z=0}^{N-1}v(y,z)^2,\\end{split}$ where we used (REF ) in the last line.","Now the last line of (REF ) vanishes in expectation by Lemma REF and the fact that ${\\mathbb {E}}(v(y,z)^2)=1$ .","This proves (REF ).","For the proof of (), note $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\overline{V}^{\\lambda }_{ij}(s-)\\right|\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|V^{\\lambda }_{ij}(y)-V^{\\lambda }_{ij}(y-1)\\right|\\\\=&\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|\\sum _{k,l=1}^d\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right|\\\\\\le &\\lambda ^2r^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} T\\right\\rfloor }\\mathbb {E}\\left(\\left\\Vert \\xi ^{\\lambda }(y)\\right\\Vert ^2\\right),\\end{split}$ which obviously vanishes in expectation as $\\lambda \\rightarrow 0$ .","For (REF ), start with $\\begin{split}\\overline{V}^{\\lambda }_{ij}(s)&-\\int _0^s\\mathrm {d}t\\sum _{k,l=1}^dC_{i,k,j,l}\\overline{X}^{\\lambda }_k(t)\\overline{X}^{\\lambda }_l(t)\\\\=&\\sum _{k,l=1}^d\\left(\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right)\\\\&\\hspace{56.9055pt}-\\left(s-\\lambda ^2\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{k,l=1}^dC_{ikjl}X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right).\\end{split}$ For the last line, we have $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\left(s-\\lambda ^2\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{k,l=1}^dC_{ikjl}X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right|\\\\&\\le \\lambda ^2r^2d^2\\max _{k,l}\\left|C_{ikjl}\\right|\\rightarrow 0\\end{split}$ almost surely, and thus in probability, as $\\lambda \\rightarrow 0$ .","We omit the (finite) $k,l$ sum in (REF ) from our notation, and use partial summation with respect to $y$ to obtain $\\begin{split}\\lambda ^2&\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\\\&=\\lambda ^2X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)\\\\&\\hspace{5.69054pt}-\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\sum _{x=1}^{y}\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)-C_{ikjl}\\right)\\left(X^{\\lambda }_k(y)X^{\\lambda }_l(y)-X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right).\\end{split}$ We know from the convergences (REF ) and $\\lim _{\\lambda \\rightarrow 0}\\lambda ^2\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }C_{ikjl}=C_{ikjl}\\cdot s,$ which are both uniform for $s$ from compact sets, that $a^{\\lambda }_{ikjl}(y):=\\lambda ^2\\sum _{x=1}^{y}\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)-C_{ikjl}\\right)\\rightarrow 0$ as $\\lambda \\rightarrow 0$ uniformly in $y$ as long as $1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor $ for fixed positive $T$ .","Thus for the first term on the right-hand side in (REF ), $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\lambda ^2X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)\\right|\\\\&\\le r^2\\sup _{s\\le T}\\max _{i,k,j,l}\\left|a^{\\lambda }_{ikjl}\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right|\\rightarrow 0\\end{split}$ as $\\lambda \\rightarrow 0$ .","After plugging (REF ) into the second term on the right-hand side of (REF ), we are left with the sum of $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\lambda \\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }a^{\\lambda }_{ikjl}(y)\\left(\\sum _{k^{\\prime }=1}^d \\xi ^{\\lambda }_{kk^{\\prime }}(y)X^{\\lambda }_{k^{\\prime }}(y-1)X^{\\lambda }_l(y-1)\\right.\\right.\\\\+\\left.\\left.\\sum _{l^{\\prime }=1}^d \\xi ^{\\lambda }_{ll^{\\prime }}(y)X^{\\lambda }_{k}(y-1)X^{\\lambda }_{l^{\\prime }}(y-1)\\right)\\vphantom{\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }}\\right|\\end{split}$ and $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }a^{\\lambda }_{ikjl}(y)\\left(\\sum _{k^{\\prime },l^{\\prime }=1}^d \\xi ^{\\lambda }_{kk^{\\prime }}(y)\\xi ^{\\lambda }_{ll^{\\prime }}(y)X^{\\lambda }_{k^{\\prime }}(y-1)X^{\\lambda }_{l^{\\prime }}(y-1)\\right)\\right|\\\\\\end{split}$ converging to zero in $L^2({\\mathbb {P}})$ and $L^1({\\mathbb {P}})$ , respectively."],["Acknowledgements","Maximilian Butz benefited a lot from discussions with members of Antti Kupiainen's group at Helsinki University, and is grateful for financial support by the Academy of Finland during his stay there.","Sven Bachmann gratefully acknowledges the support of the National Science Foundation under Grant #DMS-0757581"]],"string":"[\n [\n \"Disordered quantum wires: microscopic origins of the DMPK theory and\\n Ohm's law\"\n ],\n [\n \"Abstract We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire.\",\n \"In the literature, the standard description of such wires is via random matrix theory (RMT).\",\n \"Our objective is to firmly relate this theory to a microscopic model.\",\n \"We correct and extend previous work (arXiv:0912.1574) on the same topic.\",\n \"In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes \\\\beta=1,2.\",\n \"In the \\\\beta=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\\\\ from TUE.\",\n \"In the \\\\beta=1 class, we find a deviation from TOE.\",\n \"It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime.\",\n \"For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone.\",\n \"This proof bypasses the explicit but intricate solution methods that underlie most previous results.\"\n ],\n [\n \"Introduction\",\n \"We start below with a brief introduction to the physics of quasi one-dimensional quantum wires.\",\n \"In Section REF , we sketch the scope of this paper and its relation to previous works.\",\n \"The appropriate random matrix theory is discussed in Section .\",\n \"Our microscopic model, convergence results and proofs are presented in Sections and .\"\n ],\n [\n \"Phenomenology\",\n \"Without yet introducing a concrete mathematical framework, we present the basic physics setup of quantum wires and try to elucidate the questions of charge transport and conductance fluctuations.\",\n \"We rely heavily on the excellent review [2].\",\n \"Disordered quantum wires are quasi one-dimensional pieces of dirty (disordered) conductor.\",\n \"The wire has a physical length $L$ , which is most conveniently expressed in units of the mean free path $\\\\ell $ so that we shall use $s:= L/\\\\ell $ .\",\n \"In a microscopic model where the parameter $\\\\lambda \\\\ge 0$ measures the strength of the disorder $\\\\ell \\\\sim \\\\lambda ^{-2}$ .\",\n \"The width $W$ of the wire is expressed by an integer $N$ that corresponds to the number of different modes that `fit' in the wire.\",\n \"Physically, $N \\\\sim W/\\\\lambda _F$ with $\\\\lambda _F$ the Fermi wavelength of the electrons sent through the wire, which is in turn determined by the energy of those incoming electrons.\",\n \"For a purely one-dimensional wire, $N=1$ , it is well-known that an electron travelling through the wire gets localized with localization length of the order of the mean free path $\\\\ell $ , hence $s \\\\sim 1$ .\",\n \"However, the localization length increases with $N$ (roughly as $s \\\\sim N$ , at least in the weak disorder limit $\\\\lambda \\\\rightarrow 0$ ) and we can ask how the system behaves for $s \\\\ll N$ , before localization sets in.\",\n \"There, one can distinguish the ballistic regime $s \\\\le 1$ , where incoming electrons did not yet get scattered by the impurities, and the most interesting diffusive regime characterized by $1 \\\\ll s, \\\\qquad s/ N \\\\ll 1\\\\,.$ One of the fascinating aspects of this regime is the phenomenon of universal conductance fluctuations (UCF) first discussed in [14].\",\n \"Let $g= g(s,N, \\\\lambda )$ be the conductance of the wire, expressed in units of the conductance quantum $2e^2/\\\\hbar $ .\",\n \"It is a random quantity due to the disorder.\",\n \"In the thick wire limit, its disorder average ${\\\\mathbb {E}}(g)$ , is roughly given by ${\\\\mathbb {E}}(g) \\\\sim \\\\left\\\\lbrace \\\\begin{array}{lllll} N/s & \\\\qquad & 1 \\\\ll s, \\\\quad s/ N \\\\ll 1 & \\\\qquad & \\\\textrm {(Ohm^{\\\\prime }s law)} \\\\\\\\[2mm] \\\\exp {\\\\lbrace - s /N \\\\rbrace } & \\\\qquad & s > N & \\\\qquad & \\\\textrm {(localization)} \\\\end{array} \\\\right.\",\n \"$ Furthermore, universal conductance fluctuations mean that, in the diffusive regime defined by (REF ), $\\\\mathrm {Var}(g) = 2/(15\\\\beta ),$ independently of the microscopic details of the wire, or its length and width.\",\n \"The only parameter that remains in this regime is the symmetry index $\\\\beta $ that refers to Dyson's symmetry classes.\",\n \"We emphasize that these phenomena should emerge in a large $N$ limit only.\",\n \"On the other hand, $N$ cannot be too large because then we enter the regime of two-dimensional localization, at least if we assume that the wire has one transverse dimension.\",\n \"However, even if the transverse dimension is higher, the reasoning breaks down as soon as $W > \\\\ell $ .\",\n \"It is therefore important to take a weak-disorder limit first, $\\\\lambda \\\\rightarrow 0$ , which also means that the wire's microscopic length $L=\\\\lambda ^{-2}s$ diverges.\",\n \"Below, we try to distill some precise conjectures that are generally accepted.\",\n \"From the mathematical perspective, they can be partially proven if one accepts RMT as a starting point (see Section ), but open if one starts from a more realistic model, as the one treated in Section of this article.\",\n \"Conjecture 1 (Ohm's law) $\\\\lim _{N \\\\rightarrow \\\\infty } \\\\quad \\\\lim _{\\\\lambda \\\\rightarrow 0} \\\\quad \\\\frac{1}{N}{\\\\mathbb {E}}(g) = \\\\frac{1}{s} + o(1/s), \\\\qquad s \\\\rightarrow \\\\infty .$ Conjecture 2 (Universal conductance fluctuations) $\\\\quad \\\\lim _{N \\\\rightarrow \\\\infty } \\\\quad \\\\lim _{\\\\lambda \\\\rightarrow 0} \\\\quad \\\\mathrm {Var}(g) = \\\\frac{2}{15 \\\\beta } +o(1), \\\\qquad s \\\\rightarrow \\\\infty .$ We stress here that these conjectures reflect the minimum of what should be true according to the literature, and that the underlying heuristics is quite involved.\",\n \"The present paper does partially settle theses conjectures starting from a microscopic model but with an additional scaling limit, as will be explained in the next section.\"\n ],\n [\n \"Setup, goals, and results\",\n \"The standard approach to disordered quantum wires is to model the transfer matrix of such a wire by an appropriate ensemble of random matrices.\",\n \"The matrices under consideration belong to a subgroup of pseudo-unitary matrices.\",\n \"Following [8] we shall call their ensembles TOE, TUE and TSE in analogy to the better known ensembles of Hamiltonians, the hermitian GOE, GUE and GSE, or the circular ensembles of unitaries: COE, CUE and CSE.\",\n \"In fact, ensembles of transfer matrices come with a real positive parameter, called $s$ above and physically corresponding to the length of the wire.\",\n \"They are therefore more complicated, but also more interesting: in particular the parameter $s$ tunes a localization-delocalization transition.\",\n \"This can be observed for example in the Fokker-Planck equation describing the $s$ -dependence of the conductance, which is the equation usually referred to as the DMPK equation.\",\n \"The natural question arises whether the RMT ensembles allow for a verification of the conjectures mentioned at the end of the previous section, with the proviso that the $\\\\lambda \\\\rightarrow 0$ should be omitted as the RMT assumes weak coupling from the start.\",\n \"In the physics literature, there is overwhelming evidence for an affirmative answer, and the conjectures have been verified in [15], [3], [7], [21].\",\n \"In that perspective, we shall here give a rigorous proof of Ohm's law based on a moment argument of [17], thereby confirming Conjecture 1 for the TOE, TUE and TSE.\",\n \"The ultimate goal of our work is a derivation of the conjectures from a more realistic model of the wire, i.e.\",\n \"from a `reasonable' microscopic Hamiltonian, namely the Anderson model on a tube of width $N$ with a disordered region of length $L$ and disorder strength $\\\\lambda $ .\",\n \"First, we need to be in the weak coupling regime $\\\\lambda \\\\rightarrow 0$ , and therefore $L=\\\\lambda ^{-2}s\\\\rightarrow \\\\infty $ .\",\n \"This first scaling limit yields a random matrix ensemble ${\\\\mathcal {G}}(s)$ , see Proposition REF .\",\n \"For the conjectures to hold, a second scaling is certainly necessary, namely that of a broad wire, $N\\\\rightarrow \\\\infty $ .\",\n \"At the time of writing, the validity of the conjectures in this scaling regime remains an open question.\",\n \"However, if we consider an additional scaling limit in which the transversal hopping in the wire is small compared to the longitudinal hopping, see Theorem REF , we obtain, instead of the ensemble ${\\\\mathcal {G}}(s)$ , a new transfer matrix ensemble ${\\\\mathcal {A}}(s)$ that is very close to the ideal ensemble.\",\n \"In fact, for $\\\\beta =2$ , the conductance calculated from that ensemble is the same as that calculated from the TUE.\",\n \"Since we proved Ohm's law for the $N\\\\rightarrow \\\\infty $ limit of the random matrix ensemble in the first place, this provides a proof of Conjecture 1 in a weaker sense for $\\\\beta =2$ .\",\n \"In Section REF , we comment on the ensemble ${\\\\mathcal {A}}(s)$ , pointing out to how and why it fails to satisfy all the symmetry properties of the ideal ensembles.\",\n \"This article is to a large extent based on a previous paper [1] by two of us, which appeared on the arXiv shortly after and independently of [20].\",\n \"Despite their similarity these two articles stressed different aspects of the resulting transfer matrix evolutions.\",\n \"However, [1] contained an error, as pointed out by the second author of the present paper, and the symmetry properties of the model were not consistently treated.\",\n \"In this article, which supersedes [1], we first extend the setup by constructing models for both $\\\\beta =1$ and $\\\\beta =2$ symmetry classesThe physically most natural way to discuss $\\\\beta =4$ as well would be to consider electrons with spin, which we chose not to do for reasons of simplicity.\",\n \"Moreover, we incorporate technical improvements (among other things borrowing some terminology from [20]), mostly concerning the statement of the joint scaling limit in Theorem REF .\",\n \"Finally, we study the convergence as $N\\\\rightarrow \\\\infty $ of a hierarchy of equations for the moments of the conductance introduced by [17].\",\n \"We prove that the limit satisfies Ohm's law, see Theorem REF .\"\n ],\n [\n \"Random matrix theory: the DMPK equation\",\n \"Transport properties of a quasi one-dimensional system are most conveniently approached through its scattering matrix, or equivalently its transfer matrix.\",\n \"In this section we shall consider these objects as the fundamental quantities of the theory, understand what symmetries imply on their general structure and derive a stochastic differential equation describing their behavior as a function of the length of the disordered wires, based on an isotropy assumption, also called `(local) maximal entropy' Ansatz.\",\n \"In particular, we do not assume that the transfer matrices here arise from some sort of microscopic Hamiltonian dynamics.\"\n ],\n [\n \"Transfer matrices and symmetries\",\n \"Heuristically speaking the transfer matrix of a quasi one-dimensional wire maps free waves on the far right of the sample to free waves on the far left of it.\",\n \"Although this picture is physically meaningful, we shall only refer to it explicitly in Section and keep an abstract point of view here.\",\n \"We first fix a preferred basis in ${\\\\mathbb {C}}^{2N}$ and make the following definition.\",\n \"Definition 1 A transfer matrix for a wire of width $N$ is a $2N\\\\times 2N$ pseudo-unitary matrix, ${\\\\mathcal {M}}^{*}\\\\Sigma _z {\\\\mathcal {M}}= \\\\Sigma _z\\\\,,\\\\qquad \\\\text{where}\\\\qquad \\\\Sigma _z = \\\\begin{pmatrix} 1 & 0 \\\\\\\\ 0 & -1\\\\end{pmatrix}\\\\,.$ Furthermore, a transfer matrix ${\\\\mathcal {M}}$ is time reversal invariant if $\\\\Sigma _x {\\\\mathcal {M}}\\\\Sigma _x = \\\\overline{{\\\\mathcal {M}}}\\\\,,\\\\qquad \\\\text{where}\\\\qquad \\\\Sigma _x = \\\\begin{pmatrix} 0 & 1 \\\\\\\\ 1 & 0 \\\\end{pmatrix}\\\\,,$ and $\\\\overline{{\\\\mathcal {M}}}$ is the complex conjugate of ${\\\\mathcal {M}}$ .\",\n \"Transfer matrices have a simple multiplicative composition rule.\",\n \"If ${\\\\mathcal {M}}_1$ and ${\\\\mathcal {M}}_2$ are transfer matrices for two wires, then ${\\\\mathcal {M}}_2{\\\\mathcal {M}}_1$ is the transfer matrix for the composite system obtained from gluing the two pieces in series.\",\n \"In view of (REF , REF ), it is natural to write a transfer matrix in block form ${\\\\mathcal {M}}= \\\\begin{pmatrix} {\\\\mathcal {M}}_{++} & {\\\\mathcal {M}}_{+-} \\\\\\\\ {\\\\mathcal {M}}_{-+} & {\\\\mathcal {M}}_{--} \\\\end{pmatrix}\\\\,.$ Combining (REF ) and the singular value decompositions ${\\\\mathcal {M}}_{++} = U_+ S_+ V_+$ and ${\\\\mathcal {M}}_{--} = U_- S_- V_-$ of the diagonal blocks, we obtain the following factorization ${\\\\mathcal {M}}= \\\\begin{pmatrix} U_+ & 0 \\\\\\\\ 0 & U_- \\\\end{pmatrix} \\\\begin{pmatrix} S & (S^2-1)^{1/2} \\\\\\\\ (S^2-1)^{1/2} & S \\\\end{pmatrix} \\\\begin{pmatrix} V_+ & 0 \\\\\\\\ 0 & V_- \\\\end{pmatrix}\\\\,.$ where $S=S_+=S_-$ .\",\n \"If, moreover, time reversal invariance is imposed, then $U_- = \\\\overline{U_+}$ and $V_- = \\\\overline{V_+}$ .\",\n \"Let $r$ be the reflection and $t$ be the transmission matrices, defined through ${\\\\mathcal {M}}\\\\begin{pmatrix} 1 \\\\\\\\ r \\\\end{pmatrix} = \\\\begin{pmatrix} t \\\\\\\\ 0 \\\\end{pmatrix}$ In particular, $t = {\\\\mathcal {M}}_{++}-{\\\\mathcal {M}}_{+-}{\\\\mathcal {M}}_{--}^{-1}{\\\\mathcal {M}}_{-+}= \\\\left({\\\\mathcal {M}}_{++}^{*}\\\\right)^{-1}\\\\,,$ where we used (REF ) in the second equality.\",\n \"The so-called transmission eigenvalues $(T_k)_{k=1}^N$ are defined as the eigenvalues of the matrix $t^{*}t$ , i.e.\",\n \"of $({\\\\mathcal {M}}_{++}^{*}{\\\\mathcal {M}}_{++})^{-1}$ .\",\n \"Hence, the transmission eigenvalues are also the inverses of the squares of the singular values contained in $S$ .\",\n \"Let $T = S^{-2}$ be the diagonal matrix of transmission eigenvalues.\",\n \"Many transport properties of the disordered wire can be expressed as functions thereof.\",\n \"In particular, the conductance $g$ is given in units of $2e^2 / \\\\hbar $ by the Landauer-Büttiker formula [5], $g:= \\\\sum _{i=1}^N T_i = \\\\mathrm {Tr}\\\\, T = \\\\mathrm {Tr}\\\\, t^{*}t\\\\,,$ a formula that we accept here as a definition of $g$ .\"\n ],\n [\n \"The DMPK Theory\",\n \"The DMPK theory introduced by [9] and independently by [16] is an evolution equation for the transfer matrix ${\\\\mathcal {M}}(r,s)$ of a wire on $[r,s]$ .\",\n \"By the composition rule, for any $s_1\\\\le s_2$ , ${\\\\mathcal {M}}(0,s_2) ={\\\\mathcal {M}}(s_1,s_2) {\\\\mathcal {M}}(0,s_1)\\\\,,$ with ${\\\\mathcal {M}}(s,s)=1$ .\",\n \"The first crucial idea is to take $s_2-s_1$ infinitesimal and write ${\\\\mathcal {M}}(s,s+\\\\mathrm {d}s) \\\\sim 1 + \\\\mathrm {d}{\\\\mathcal {L}}(s)$ such that $\\\\mathrm {d}{\\\\mathcal {L}}(s)$ is independent of ${\\\\mathcal {M}}(s)$ and contains only diffusive terms but no drift.\",\n \"Mathematically, this translates into the assumption that ${\\\\mathcal {M}}(s)$ satisfies an Itô stochastic differential equation (SDE) $\\\\begin{split}\\\\mathrm {d}{\\\\mathcal {M}}(s) &= \\\\mathrm {d}{\\\\mathcal {L}}(s) {\\\\mathcal {M}}(s)\\\\,,\\\\\\\\{\\\\mathcal {M}}(0)&=1\\\\,,\\\\end{split}$ where ${\\\\mathcal {L}}(s)$ is a matrix valued Brownian motion and ${\\\\mathcal {L}}(0) = 0$ .\",\n \"Lemma 1 Let ${\\\\mathcal {M}}(s)$ be a solution of the SDE (REF ).\",\n \"Assume that $\\\\mathrm {d}{\\\\mathcal {L}}^{*}\\\\Sigma _z + \\\\Sigma _z \\\\mathrm {d}{\\\\mathcal {L}}&= 0\\\\,, \\\\\\\\\\\\mathrm {d}{\\\\mathcal {L}}^{*}\\\\Sigma _z \\\\mathrm {d}{\\\\mathcal {L}}&= 0\\\\,.\",\n \"$ Then ${\\\\mathcal {M}}(s)$ is pseudo unitary, eq.\",\n \"(REF ).\",\n \"If moreover $\\\\Sigma _x \\\\mathrm {d}{\\\\mathcal {L}}\\\\Sigma _x = \\\\overline{\\\\mathrm {d}{\\\\mathcal {L}}}\\\\,,$ then ${\\\\mathcal {M}}(s)$ is also time reversal invariant, eq.\",\n \"(REF ).\",\n \"For the first part, we take the differential of (REF ), use (REF ) and Itô calculus to obtain ${\\\\mathcal {M}}^{*}\\\\left(\\\\mathrm {d}{\\\\mathcal {L}}^{*}\\\\Sigma _z + \\\\Sigma _z \\\\mathrm {d}{\\\\mathcal {L}}+ \\\\mathrm {d}{\\\\mathcal {L}}^{*}\\\\Sigma _z \\\\mathrm {d}{\\\\mathcal {L}}\\\\right){\\\\mathcal {M}}= 0\\\\,,$ which holds if and only if both (REF ) and () hold as ${\\\\mathcal {M}}$ is nonsingular.\",\n \"Similarly, the differential of (REF ) immediately yields (REF ).\",\n \"Secondly, the DMPK theory prescribes a particular invariance of the distribution of $\\\\mathrm {d}{\\\\mathcal {L}}(s)$ .\",\n \"The law of the increments $\\\\mathrm {d}{\\\\mathcal {L}}(s)$ shall be independent of $s$ and maximally isotropic in the sense that ${\\\\mathcal {W}}^{*}\\\\mathrm {d}{\\\\mathcal {L}}\\\\, {\\\\mathcal {W}}\\\\mathop {=}\\\\limits ^{d} \\\\mathrm {d}{\\\\mathcal {L}}\\\\qquad \\\\text{for any unitary}\\\\qquad {\\\\mathcal {W}}= \\\\begin{pmatrix} W_+ & 0 \\\\\\\\ 0 & W_- \\\\end{pmatrix}\\\\,.$ The unitary blocks $W_\\\\pm $ are independent of each other if ${\\\\mathcal {M}}$ does not exhibit any symmetry, whereas $W_- = \\\\overline{W_+}$ if time reversal symmetry is imposed.\",\n \"For notational simplicity, we cast ${\\\\mathcal {L}}$ in block form, ${\\\\mathcal {L}}(s) = \\\\begin{pmatrix} \\\\mathfrak {a}(s) & \\\\mathfrak {b}(s) \\\\\\\\ \\\\mathfrak {b}(s)^{*}& \\\\mathfrak {a}^{\\\\prime }(s) \\\\end{pmatrix}$ where $\\\\mathfrak {a}(s),\\\\mathfrak {a}^{\\\\prime }(s), \\\\mathfrak {b}(s)$ are independent local martingales, with $\\\\mathfrak {a}(s) = -\\\\mathfrak {a}(s)^{*}$ , similarly for $\\\\mathfrak {a}^{\\\\prime }(s)$ , and $\\\\mathrm {d}\\\\mathfrak {a}^{*}\\\\mathrm {d}\\\\mathfrak {a} = \\\\mathrm {d}\\\\mathfrak {b}\\\\mathrm {d}\\\\mathfrak {b}^{*}= \\\\mathrm {d}\\\\mathfrak {a}^{\\\\prime }{\\\\mathrm {d}\\\\mathfrak {a}^{\\\\prime }}^{*}\\\\,.$ The isotropy assumption (REF ) reduces to invariance conditions on the blocks.\",\n \"First, $ \\\\mathfrak {a}_{ij}(s) = {\\\\left\\\\lbrace \\\\begin{array}{ll}1 / \\\\sqrt{2N} \\\\cdot (B^R_{ij}(s) + \\\\mathrm {i}B^I_{ij}(s)) & 1 \\\\le i < j \\\\le N \\\\\\\\\\\\mathrm {i}/ \\\\sqrt{N} \\\\cdot B^I_{ii}(s) & i=j \\\\\\\\-\\\\overline{\\\\mathfrak {a}_{ji}(s)} & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"}\\\\,,$ where $B^R$ and $B^I$ are independent real standard Brownian motions, and similarly but independently for $\\\\mathfrak {a}^{\\\\prime }(s)$ .\",\n \"Secondly, $ \\\\mathfrak {b}_{ij}(s) = 1 / \\\\sqrt{2N} \\\\cdot (\\\\tilde{B}^R_{ij}(s) + \\\\mathrm {i}\\\\tilde{B}^I_{ij}(s))\\\\,,\\\\quad \\\\text{for all }i,j\\\\,.$ Note that the relative normalization of $\\\\mathfrak {a}(s)$ and $\\\\mathfrak {b}(s)$ are fixed by pseudounitarity, i.e.\",\n \"(REF ).\",\n \"In the time reversal invariant case, the matrix $\\\\mathfrak {a}(s)$ does not change, but $\\\\mathfrak {a}^{\\\\prime }(s) = \\\\overline{\\\\mathfrak {a}(s)}\\\\,,$ and $\\\\mathfrak {b}(s)$ becomes symmetric, $\\\\mathfrak {b}(s)^{*}= \\\\overline{\\\\mathfrak {b}(s)}$ with real and imaginary parts orthogonally invariant, namely $ \\\\mathfrak {b}_{ij}(s) = {\\\\left\\\\lbrace \\\\begin{array}{ll}1 / \\\\sqrt{2(N+1)} \\\\cdot (B^R_{ij}(s) + \\\\mathrm {i}B^I_{ij}(s)) & 1 \\\\le i < j \\\\le N \\\\\\\\1/\\\\sqrt{N+1} \\\\cdot (B^R_{ii}(s) + \\\\mathrm {i}B^I_{ii}(s)) & i=j \\\\\\\\\\\\mathfrak {b}_{ji}(s) & \\\\text{otherwise}\\\\end{array}\\\\right.\",\n \"}\\\\,.$ Here again, the relative factor $\\\\sqrt{N/(N+1)}$ is imposed by (REF ).\",\n \"From a physical point of view, the DMPK theory's interest lies in its predictions for the statistics of the transmission eigenvalues.\",\n \"Indeed, the unitary invariance of the increments $\\\\mathrm {d}{\\\\mathcal {L}}$ implies that the set of $T_k$ satisfies an autonomous equation, which can be formally derived by Itô calculus from the matrix SDE (REF ): $\\\\begin{split}\\\\mathrm {d}T_k(s)&=v_k(T(s))\\\\mathrm {d}s+D_k(T(s))\\\\mathrm {d}B_k(s),\\\\\\\\T_k(0)&=1,\\\\end{split}$ for all $k=1,...,N$ .\",\n \"The Brownian motions $B_k$ are independent, and the drift and diffusion coefficients are given explicitly by $\\\\begin{split}v_k&=-T_k+\\\\frac{2T_k}{\\\\beta N+2-\\\\beta }\\\\left(1-T_k+\\\\frac{\\\\beta }{2}\\\\sum _{j\\\\ne k}\\\\frac{T_k+T_j-2T_kT_j}{T_k-T_j}\\\\right)\\\\,,\\\\\\\\D_k&=\\\\sqrt{4\\\\frac{T_k^2(1-T_k)}{\\\\beta N+2-\\\\beta }}.\\\\end{split}$ The first term in the drift $v_k$ contracts all transmission eigenvalues towards 0 as the length $s$ of the wire increases.\",\n \"However, and similarly to Dyson's Brownian motion, the drift also contains repulsion terms originating from second order perturbation theory.\",\n \"As a consequence, the eigenvalues $T_k$ `try to avoid' degeneracy.\",\n \"What makes a naive derivation formal is that Itô's formula is only applicable if the denominator $T_k-T_j$ never becomes singular, i.e.\",\n \"${\\\\mathcal {M}}^*_{++}(s){\\\\mathcal {M}}_{++}(s)$ never has degenerate eigenvalues.\",\n \"This is a nontrivial property for $s>0$ , and even more so as $s\\\\rightarrow 0^+$ since (REF ) starts with the completely degenerate ${\\\\mathcal {M}}(0)=1$ .\",\n \"Both issues can however be tackled and the SDE (REF ) has a unique weak and strong solution, see [6].\",\n \"Finally, let us comment on some deeper principles underlying the process ${\\\\mathcal {M}}$ and the resulting DMPK equation.\",\n \"The maximal isotropy assumption that was used above, can be derived from a simple `maximal entropy assumption' on the set of infinitesimal transfer matrices $1+\\\\mathrm {d}{\\\\mathcal {L}}(s)$ that have a fixed 'scattering strength' $\\\\sum _k T_k$ .\",\n \"Alternatively, as remarked by [12], one can also guess the DMPK equation from geometric considerations, since it is the radial part of the canonical Brownian motion on a certain symmetric space.\",\n \"The reduction from Lie group to symmetric space is obtained by identifying certain transfer matrices that, in particular, have the same transmission eigenvalues.\",\n \"This geometric approach was very fruitful.\",\n \"For example, in [4] it was shown how it naturally explains the appearance of non-universal conductance properties in wires with off-diagonal disorder.\"\n ],\n [\n \"Ohm's law\",\n \"In the context of the DMPK theory, a treatment, or even proof, of the conjectures mentioned in the introduction is possible, as already indicated in Section REF .\",\n \"The existing approaches rely on explicit calculations and are quite intricate.\",\n \"Nevertheless, if one is solely after Ohm's law (and not the universal conductance fluctuations (UCF)), there is an appealing and compact approach by [17].\",\n \"Below we present a rigorous version of this approach.\",\n \"The following theorem shows that in the large $N$ limit, the rescaled moments of the conductance have an Ohmic behavior.\",\n \"In particular, Conjecture 1 holds for the TOE, TUE and TSE.\",\n \"We note that the symmetry index $\\\\beta $ drops out in that particular scaling.\",\n \"Theorem 2 (Ohm's law) Let $\\\\left(T_k(s)\\\\right)_{k=1}^N$ be the solution of the DMPK process (REF ), and let $g_N(s) = \\\\sum _{k=1}^N T_k(s)\\\\,.$ Then for all $p\\\\ge 1$ and $T>0$ , $ \\\\lim _{N\\\\rightarrow \\\\infty }\\\\frac{{\\\\mathbb {E}}(g^p_N(s))}{N^p} = \\\\frac{1}{(1+s)^p}$ uniformly for $s\\\\in [0,T]$ .\",\n \"The proof goes through a sequence of lemmas.\",\n \"Lemma 3 Let $g_N^{(j)} = \\\\sum _k T_k^j$ for $j>1$ .\",\n \"Then, for any $p\\\\ge 1$ , $ \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}s}{\\\\mathbb {E}}(g_N^p) = -p\\\\gamma _N(\\\\beta )\\\\left[{\\\\mathbb {E}}(g_N^{p+1}) - \\\\left(1-\\\\frac{2}{\\\\beta }\\\\right){\\\\mathbb {E}}(g_N^{p-1} g_N^{(2)}) -\\\\frac{2(p-1)}{\\\\beta }{\\\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)}))\\\\right].$ where $\\\\gamma _N(\\\\beta ) = \\\\frac{\\\\beta }{\\\\beta N +2 -\\\\beta }.$ Itô's formula yields $\\\\mathrm {d}(g_N^p) = \\\\sum _{i_1,\\\\ldots ,i_p = 1}^N \\\\left[\\\\sum _{j=1}^p T_{i_1}\\\\cdots dT_{i_j}\\\\cdots T_{i_p} + \\\\sum _{j\\\\ne k=1}^p T_{i_1}\\\\cdots dT_{i_j}\\\\cdots dT_{i_k}\\\\cdots T_{i_p}\\\\right]$ so that $ \\\\frac{\\\\mathrm {d}}{\\\\mathrm {d}s}{\\\\mathbb {E}}(g_N^p) = p{\\\\mathbb {E}}\\\\left(g_N^{p-1}\\\\sum _k v_k\\\\right) + \\\\frac{p(p-1)}{2}{\\\\mathbb {E}}\\\\left(g_N^{p-2}\\\\sum _k D_k^2\\\\right)\\\\,.$ where we used the DMPK equation (REF ).\",\n \"In order to rewrite the right-hand side, note the simple identity $\\\\sum _k T_k \\\\sum _{j\\\\ne k}\\\\frac{T_k+T_j-2T_kT_j}{T_k-T_j} = \\\\sum _k\\\\sum _{j < k}(T_k+T_j-2T_kT_j) = (N-1) g_N - g_N^2 + \\\\sum _k T_k^2$ Therefore, $\\\\sum _k v_k &= -g_N + \\\\frac{2}{\\\\beta N +2 -\\\\beta }\\\\left[g_N - \\\\sum _k T_k^2 + (\\\\beta /2)\\\\left((N-1) g_N - g_N^2 + \\\\sum _k T_k^2\\\\right) \\\\right] \\\\nonumber \\\\\\\\&=-\\\\gamma _N(\\\\beta )\\\\left( g_N^2 - (1-2/\\\\beta )g_N^{(2)}\\\\right)\\\\,, $ and the lemma follows upon substituting this in (REF ).\",\n \"Let us now consider $\\\\Psi _{N}(p,s) :=\\\\frac{{\\\\mathbb {E}}(g^p_N(s))}{N^p}\\\\,,$ The following properties are immediate from the definition and the differential equation (REF ) $\\\\Psi _{N}(p,0)=1$ .\",\n \"$ |\\\\Psi _{N}(p,s) |\\\\le 1$ .\",\n \"The function $s \\\\mapsto \\\\Psi _{N}(p,s)$ is continuously differentiable and $ |\\\\frac{\\\\partial \\\\Psi _{N}}{\\\\partial s}(p,s) |\\\\le c(p) < \\\\infty $ .\",\n \"We consider the Banach space ${\\\\mathcal {L}}={\\\\mathcal {C}}([0,T], {\\\\mathbb {R}})$ for some $T>0$ , equipped with the supremum norm.\",\n \"Let ${\\\\mathcal {L}}_p, p=1,2,\\\\ldots $ be copies of ${\\\\mathcal {L}}$ and define the cartesian product ${\\\\mathcal {K}}= \\\\mathop {\\\\times }\\\\limits _{p=1}^{\\\\infty } {\\\\mathcal {L}}_p$ , equipped with the product topology.\",\n \"Lemma 4 There is an increasing sequence $N_n, n \\\\in {\\\\mathbb {N}}$ and an element $\\\\Psi \\\\in {\\\\mathcal {K}}$ such that for each $p$ , $\\\\Psi _{N_n}(p,\\\\cdot ) \\\\rightarrow \\\\Psi (p,\\\\cdot ) $ in ${\\\\mathcal {L}}$ , as $n \\\\rightarrow \\\\infty $ .\",\n \"For $p=1,2,\\\\ldots $ , consider the sets $ {\\\\mathcal {S}}_p= \\\\left\\\\lbrace \\\\Psi _{N}(p,\\\\cdot ), N \\\\in {\\\\mathbb {N}}\\\\right\\\\rbrace \\\\subset {\\\\mathcal {L}}_p.$ By the properties $\\\\mathrm {i,ii,iii}$ above and Arzela-Ascoli's theorem, each of these sets is sequentially compact (s.c.).\",\n \"Countable products of s.c. sets are s.c. in the product topology (sequential Tychonov's theorem), hence $\\\\times _p{\\\\mathcal {S}}_p \\\\subset {\\\\mathcal {K}}$ is s.c.\",\n \"Therefore, the sequence $\\\\Psi _N \\\\subset \\\\times _p{\\\\mathcal {S}}_p$ has a convergent subsequence.\",\n \"Since convergence in the product topology implies convergence for any $p$ , the lemma follows.\",\n \"Lemma 5 Any limit point $\\\\Psi \\\\in {\\\\mathcal {K}}$ as in Lemma REF satisfies $\\\\Psi (p,s_2)-\\\\Psi (p,s_1) = - p \\\\int _{s_1}^{s_2} \\\\mathrm {d}s \\\\Psi (p+1,s), \\\\qquad \\\\Psi (p,0) = 1$ The equation (REF ) is rewritten as $ \\\\Psi _N(p,s_2)-\\\\Psi _N(p,s_1)= -p (1-r_1(N)) \\\\int _{s_1}^{s_2} \\\\mathrm {d}s \\\\left[\\\\Psi _N(p+1,s) - r_2(N,s,p) \\\\right]$ where $r_2$ is the sum of the second and third term between square brackets in (REF ) and $r_1(N) &= 1-N\\\\gamma _{N} = {\\\\mathcal {O}}(1/N), \\\\\\\\r_2(N,s,p) &= \\\\left(1-\\\\frac{2}{\\\\beta }\\\\right)\\\\frac{{\\\\mathbb {E}}(g_N^{p-1} g_N^{(2)})}{N^{p+1}} + \\\\frac{2(p-1)}{\\\\beta }\\\\frac{{\\\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)})}{N^{p+1}} = {\\\\mathcal {O}}(1/N)$ with the bounds ${\\\\mathcal {O}}(1/N)$ uniform in $s$ but not necessarily in $p$ .\",\n \"The lemma follows by considering (REF ), for fixed $p$ , along the sequence ${N_n}$ .\",\n \"Sloppily put, the above two lemmas show that $\\\\Psi _N(p,\\\\cdot )$ converges to a solution of the `limiting hierarchy of equations' (REF ).\",\n \"It remains to prove that the limiting hierarchy has a unique solution, namely the right-hand side of (REF ).\",\n \"Therefore, the proof of Theorem REF is completed by the next lemma.\",\n \"Lemma 6 $\\\\Psi (p,s): = \\\\frac{1}{(1+s)^p}$ is the unique element in ${\\\\mathcal {K}}$ that satisfies (REF ) and $\\\\sup _p ||\\\\Psi (p,\\\\cdot ) ||_\\\\infty \\\\le 1$ .\",\n \"We proceed by induction in the interval length $T$ .\",\n \"Assume that the claim is proven for $T\\\\ge 0$ (for $T=0$ it is trivial).\",\n \"Take then $0 \\\\le s_1 \\\\le T$ and $ s_1< s_2 0$ and $0\\\\le \\\\gamma < 2\\\\pi / N$ .\",\n \"Note that as indicated in the definition of $\\\\mathcal {H}$ , periodic boundary conditions in the transverse direction are imposed on all operators.\",\n \"As we have briefly discussed in Section , the DMPK equation comes in various guises, depending on the abstract symmetry group of the transmission matrix.\",\n \"At the microscopic level, time reversal $T$ is naturally implemented by complex conjugation in the `position basis', namely $(T\\\\Psi )(x,z) = \\\\overline{\\\\Psi (x,z)}\\\\,.$ It is immediate to check that $H_\\\\mathrm {kin}$ is invariant under $T$ iff $\\\\gamma =0$ : the magnetic field indeed breaks time reversal invariance.\"\n ],\n [\n \"Eigenvalues, eigenvectors and chaoticity\",\n \"We consider the eigenvalue problem for the kinetic Hamiltonian $H_{\\\\mathrm {kin}}\\\\Psi =E\\\\Psi $ at some fixed energy $E$ .\",\n \"By translation invariance and the periodic boundary conditions, the (non-normalizable) solutions are given by $\\\\Psi _{k,\\\\nu }(x,z)=\\\\frac{1}{\\\\sqrt{N}}e^{ik x}e^{\\\\frac{2\\\\pi i}{N}\\\\nu z}$ for $\\\\nu =1,\\\\ldots ,N$ , with $E =E(k,\\\\nu )= 2\\\\cos (k) + 2h_1\\\\cos \\\\left(\\\\gamma + \\\\frac{2\\\\pi }{N}\\\\nu \\\\right) + 2 h_2\\\\cos \\\\left(k - \\\\gamma - \\\\frac{2\\\\pi }{N}\\\\nu \\\\right)$ We now look for the condition on $E$ so that (REF ) has plane wave solutions, rather than exponentially decaying ones, i.e.\",\n \"such that $E=E(k,\\\\nu )$ for some $k,\\\\nu $ .\",\n \"In physical terms, this means that we do not want to study evanescent modes, also called `elliptic channels'.\",\n \"In fact, we first fix the energy $E$ and consider solutions $k = k_\\\\nu (E)$ of (REF ) for any `transversal mode' $\\\\nu $ .\",\n \"We shall drop the $E$ dependence in the sequel.\",\n \"There are two such wavevectors $k_\\\\nu ^\\\\pm $ corresponding to a right moving and a left moving wave.\",\n \"The relations $ k_\\\\nu ^+ = -k_{-\\\\nu }^-$ holds in the time reversal invariant case, but is broken if $\\\\gamma >0$ .\",\n \"If $h_2=0$ , the residual symmetry $\\\\Psi (x,z) \\\\mapsto \\\\Psi (x,-z)$ induces the additional degeneracy $k_\\\\nu ^+ = -k_\\\\nu ^-$ that needs to be avoided.\",\n \"It is a straightforward exercise to check Lemma 7 For an energy $E\\\\ne 0$ , $|E|<2$ , and a kinetic Hamiltonian $H_{\\\\mathrm {kin}}$ with parameters $0\\\\le \\\\gamma <\\\\frac{\\\\pi }{N}$ , and $h_1,h_2>0$ sufficiently small, in particular $|E|+2h_1+2h_2<2\\\\,,$ the equation $H_{\\\\mathrm {kin}}\\\\Psi =E\\\\Psi $ has $2N$ plane wave solutions $\\\\Psi _{k,\\\\nu }(x,z)=\\\\frac{1}{\\\\sqrt{N}}e^{ik_\\\\nu ^{\\\\sigma } x}e^{\\\\frac{2\\\\pi i}{N}\\\\nu z}$ with $\\\\nu \\\\in {\\\\mathbb {Z}}_N$ , $\\\\sigma \\\\in \\\\lbrace {+,-\\\\rbrace }$ .\",\n \"For $\\\\gamma \\\\ne 0$ , the longitudinal wave numbers $k_\\\\nu ^{\\\\sigma }$ are non degenerate in the sense that $\\\\varsigma _1 k_{\\\\nu (1)}^{\\\\sigma _1}+\\\\varsigma _2 k_{\\\\nu (2)}^{\\\\sigma _2}+\\\\varsigma _3 k_{\\\\nu (3)}^{\\\\sigma _3}+\\\\varsigma _4 k_{\\\\nu (4)}^{\\\\sigma _4}=0\\\\mbox{ }(\\\\mathrm {mod}\\\\mbox{ }2\\\\pi )$ for signs $\\\\varsigma _1,...,\\\\varsigma _4$ only in the trivial case $\\\\big (\\\\varsigma _1, \\\\sigma _1, \\\\nu (1)\\\\big ) = \\\\big (-\\\\varsigma _2, \\\\sigma _2, \\\\nu (2)\\\\big )\\\\quad \\\\text{and}\\\\quad \\\\big (\\\\varsigma _3, \\\\sigma _3, \\\\nu (3)\\\\big ) = \\\\big (-\\\\varsigma _4, \\\\sigma _4, \\\\nu (4)\\\\big )\\\\,,$ and if $\\\\gamma =0$ also in the $T$ -symmetric situation $\\\\big (\\\\varsigma _1, \\\\sigma _1, \\\\nu (1)\\\\big ) = \\\\big (\\\\varsigma _2, -\\\\sigma _2, -\\\\nu (2)\\\\big )\\\\quad \\\\text{and}\\\\quad \\\\big (\\\\varsigma _3, \\\\sigma _3, \\\\nu (3)\\\\big ) = \\\\big (\\\\varsigma _4, -\\\\sigma _4, -\\\\nu (4)\\\\big )\\\\,,$ all of this up to relabeling of the indices.\",\n \"The above lemma shows that the Hamiltonian has no other symmetry than time reversal invariance, in accordance with the macroscopic theory of the previous section.\",\n \"It is important to realize that a residual symmetry of the kinetic Hamiltonian could, and in fact would, leave a trace in the scaling limit, even if that symmetry got broken by the disorder.\",\n \"The DMPK theory considers a single ensemble of transfer matrices, whereas there are really two microscopic models, one without and one with disorder.\",\n \"For our derivation, it is essential that both have the correct symmetry properties.\",\n \"For further considerations, it is useful to define the chaoticity of the kinetic Hamiltonian in analogy to [20] by $\\\\mathrm {cha}\\\\left(\\\\gamma ,h_1,h_2\\\\right)=\\\\min \\\\left\\\\lbrace \\\\left|\\\\sum _{i=1}^4 \\\\varsigma _i k_{\\\\nu (i)}^{\\\\sigma _i}\\\\right|, ({\\\\varsigma },{\\\\sigma },{\\\\nu })\\\\mbox{ not solving } (\\\\ref {exceptionTRI}) \\\\right\\\\rbrace \\\\,,$ for $\\\\gamma >0$ and $\\\\mathrm {cha}\\\\left(0,h_1,h_2\\\\right)=\\\\min \\\\left\\\\lbrace \\\\left|\\\\sum _{i=1}^4 \\\\varsigma _i k_{\\\\nu (i)}^{\\\\sigma _i}\\\\right|,({\\\\varsigma },{\\\\sigma },{\\\\nu })\\\\mbox{ not solving } (\\\\ref {exceptionTRI})\\\\mbox{ or } (\\\\ref {exception})\\\\right\\\\rbrace .$ if $\\\\gamma =0$ .\"\n ],\n [\n \"The transfer matrix\",\n \"We decompose the kinetic Hamiltonian $(H_\\\\mathrm {kin}\\\\Psi )_x= H_\\\\perp \\\\Psi _x + P \\\\Psi _{x+1} + P^* \\\\Psi _{x-1}$ as a strictly transverse operator $(H_\\\\perp \\\\phi )(z) = h_1 \\\\left( \\\\mathrm {e}^{\\\\mathrm {i}\\\\gamma }\\\\phi (z+1) + \\\\mathrm {e}^{-\\\\mathrm {i}\\\\gamma }\\\\phi (z-1) \\\\right)\\\\,,$ and the components inducing hopping in the longitudinal direction $(P\\\\phi )(z) = \\\\phi (z) + h_2 \\\\mathrm {e}^{\\\\mathrm {i}\\\\gamma } \\\\phi (z+1)\\\\,,\\\\quad \\\\text{and}\\\\quad (P^{*}\\\\phi )(z) = \\\\phi (z) + h_2 \\\\mathrm {e}^{-\\\\mathrm {i}\\\\gamma } \\\\phi (z-1)\\\\,,$ where $\\\\phi (z)\\\\in {\\\\mathbb {C}}^N$ .\",\n \"Similarly, the random potential can be seen as a sequence of $N\\\\times N$ diagonal matrices $V_x$ .\",\n \"With these notations, the eigenvalue equation with disorder, $\\\\lambda >0$ , reads $\\\\begin{pmatrix}\\\\Psi _{x+1}\\\\\\\\\\\\Psi _{x}\\\\end{pmatrix}=\\\\begin{pmatrix}\\\\left(P^*\\\\right)^{-1}(E-H_\\\\perp -\\\\lambda V_x)&-\\\\left(P^*\\\\right)^{-1}P\\\\\\\\{1}&{0}\\\\end{pmatrix}\\\\begin{pmatrix}\\\\Psi _{x}\\\\\\\\\\\\Psi _{x-1}\\\\end{pmatrix}=:T_x^\\\\lambda \\\\begin{pmatrix}\\\\Psi _{x}\\\\\\\\\\\\Psi _{x-1}\\\\end{pmatrix},$ thereby defining the transfer matrix $T_x^\\\\lambda $ (of dimension $2N$ ) for the layer $x$ as it usually appears in the mathematical literature.\",\n \"By the multiplicative property, the transfer matrix for $L$ layers is simply given by the product of the one-layer matrices, $\\\\begin{pmatrix}\\\\Psi _{L+1}\\\\\\\\\\\\Psi _{L}\\\\end{pmatrix}=T_L^\\\\lambda \\\\cdots T_1^\\\\lambda \\\\begin{pmatrix}\\\\Psi _{1}\\\\\\\\\\\\Psi _{0}\\\\end{pmatrix},$ In order to allow for a comparison with the DMPK theory, it is important to write the transfer matrix in the natural basis for ${\\\\mathbb {C}}^{2N}$ , i.e.\",\n \"the basis in which the first $N$ components correspond to left moving waves, and the other $N$ components are right moving.\",\n \"Morever, in the time-reversal invariant case we need the correct identification between left and right moving channels, given by (REF ).\",\n \"In other words, the time reversal of the vector $(\\\\phi _1, \\\\phi _2)^t$ must correspond to $(\\\\bar{\\\\phi }_2, \\\\bar{\\\\phi }_1)^t$ .\",\n \"Only in that basis do the definitions of the transfer matrix given in Section REF apply.\",\n \"We shall refer to it as the channel basis.\",\n \"Let $Q$ be the $N\\\\times N$ matrix whose columns are the transverse eigenvectors $\\\\phi _\\\\nu (z) = 1/\\\\sqrt{N}\\\\exp (2\\\\pi \\\\mathrm {i}\\\\nu z/N)$ .\",\n \"Furthermore, let $\\\\Pi $ be the permutation matrix that interchanges the $\\\\nu $ and $-\\\\nu $ channels, and let $\\\\Upsilon :=\\\\begin{pmatrix}\\\\frac{\\\\mathrm {e}^{\\\\mathrm {i}k^+}}{\\\\sqrt{\\\\left|v^+\\\\right|}}&\\\\frac{\\\\mathrm {e}^{\\\\mathrm {i}k^-}}{\\\\sqrt{\\\\left|v^-\\\\right|}}\\\\\\\\\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}&\\\\frac{1}{\\\\sqrt{\\\\left|v^-\\\\right|}}\\\\end{pmatrix}\\\\begin{pmatrix}1&0\\\\\\\\0&\\\\Pi \\\\end{pmatrix}$ where $k^\\\\pm $ and $v^\\\\pm $ denote diagonal matrices with elements $k^\\\\pm _{\\\\nu }$ and $v^\\\\pm _\\\\nu $ , and $v^\\\\pm _\\\\nu $ are the velocities in each channel, $v^\\\\pm _\\\\nu =\\\\left.\\\\frac{\\\\partial }{\\\\partial k}E(k,\\\\nu )\\\\right|_{k=k_\\\\nu ^\\\\pm }=-2\\\\sin \\\\left(k^\\\\pm _\\\\nu \\\\right)-2h_2\\\\sin \\\\left(k^\\\\pm _\\\\nu - \\\\frac{2\\\\pi }{N}\\\\nu - \\\\gamma \\\\right).$ Note that all $v^\\\\pm _\\\\nu $ are nonzero by (REF ).\",\n \"The action of $T$ (complex conjugation) on the eigenvectors yields $T:\\\\,(k_\\\\nu ^\\\\sigma ,\\\\nu )\\\\mapsto (-k_\\\\nu ^\\\\sigma ,-\\\\nu )$ which equals $(k_{-\\\\nu }^{-\\\\sigma },-\\\\nu )$ if time reversal invariance holds, so that the $(\\\\sigma = -)$ block needs to be reordered by $\\\\nu \\\\leftrightarrow -\\\\nu $ .\",\n \"Hence the permutation $\\\\Pi $ .\",\n \"With these notations, Lemma REF reads $ \\\\Upsilon ^{-1}(Q\\\\otimes 1)^{*}T_x^0(Q\\\\otimes 1)\\\\Upsilon = \\\\begin{pmatrix}\\\\mathrm {e}^{\\\\mathrm {i}k^+}&0\\\\\\\\0&\\\\Pi \\\\mathrm {e}^{\\\\mathrm {i}k^-} \\\\Pi \\\\end{pmatrix} =: M_x^0\\\\,.$ The matrix $M_x^0$ is the transfer matrix for the purely kinetic transport in a single layer, which is diagonal the channel basis.\",\n \"As expected, it simply adds a phase to the traveling plane wave.\",\n \"It is now easily checked that $M_x^0$ is a bonafide transfer matrix in the sense of Definition REF : Eq.\",\n \"(REF ) is always satisfied and (REF ) holds at $\\\\gamma =0$ , as conjugation by $\\\\Sigma _x$ precisely maps $k^\\\\sigma _\\\\nu $ to $k^{-\\\\sigma }_{-\\\\nu }=-k^\\\\sigma _{\\\\nu }$ .\",\n \"The transfer matrix for the total system is again obtained by multiplication $M^\\\\lambda (L) := M^\\\\lambda _L \\\\cdots M^\\\\lambda _1 = K^{-1} \\\\, T_L^\\\\lambda \\\\cdots T_1^\\\\lambda \\\\, K\\\\,,$ where $K = (Q\\\\otimes 1)\\\\Upsilon $ is the change of basis from the position basis to the channel basis.\",\n \"This random matrix and its relation the DMPK theory is the central object of study in the following.\"\n ],\n [\n \"A scaling limit\",\n \"The DMPK theory suggests that the microscopic transfer matrix for a disordered wire of length $L$ should converge to a solution of the DMPK equation in the correct macroscopic limit.\",\n \"As discussed in the introduction, the natural scaling between the microscopic length $L$ and the macroscopic length $s$ is through the mean free path, $L = \\\\lambda ^{-2}s$ .\",\n \"A naive interpretation of the DMPK theory would then be the convergence of $M^\\\\lambda (\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor )$ to ${\\\\mathcal {M}}(s)$ .\",\n \"This cannot possibly hold as $M^\\\\lambda (\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor )$ contains rapidly oscillating terms as a function of $s$ , as already exemplified at the level of the unperturbed system (REF ): $M^0(\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor )=\\\\begin{pmatrix}\\\\exp \\\\left(\\\\mathrm {i}\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor k^+\\\\right)&0\\\\\\\\0&\\\\Pi \\\\exp \\\\left(\\\\mathrm {i}\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor k^-\\\\right) \\\\Pi \\\\end{pmatrix}\\\\,.$ To obtain a reasonable limit, we therefore consider $A^\\\\lambda (\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor ) := (M^0(\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor ))^{-1}M^\\\\lambda (\\\\lfloor \\\\lambda ^{-2}s \\\\rfloor ).$ As the set of matrices of Definition REF form a group, $A^\\\\lambda $ is a transfer matrix again.\",\n \"Moreover, it is an easy check that in the polar decomposition (REF ), the matrix $S$ corresponding to $A^\\\\lambda $ is the same as that corresponding to $M^\\\\lambda $ , so that they have the same transmission eigenvalues.\",\n \"In order to state the result of the scaling limit, $\\\\lambda \\\\rightarrow 0$ , we introduce the following processes.\",\n \"For $\\\\gamma >0$ , let ${\\\\mathcal {Z}}_{\\\\gamma }(s) := \\\\begin{pmatrix} \\\\mathfrak {a}(s) & \\\\mathfrak {b}(s) \\\\\\\\\\\\mathfrak {b}^*(s) &{ \\\\mathfrak {a}^{\\\\prime }}(s)\\\\end{pmatrix}$ with $\\\\mathfrak {a}_{\\\\mu \\\\mu }(s)=-\\\\mathfrak {a}^{\\\\prime }_{\\\\mu \\\\mu }(s)=\\\\frac{\\\\mathrm {i}}{\\\\sqrt{(4-E^2)N}}W(s)$ with the same standard real Brownian motion $W$ for all $\\\\mu =1,\\\\ldots ,N$ .\",\n \"All diagonal elements of ${\\\\mathcal {Z}}$ are thus perfectly correlated.\",\n \"For the off-diagonal elements, $\\\\mathfrak {a}_{\\\\mu \\\\nu }(s)=-\\\\overline{\\\\mathfrak {a}_{\\\\nu \\\\mu }(s)}&=\\\\frac{1}{\\\\sqrt{(4-E^2)N}}B^{++}_{\\\\mu \\\\nu }(s)&\\\\mbox{ for }&1\\\\le \\\\mu <\\\\nu \\\\le N\\\\\\\\\\\\mathfrak {a}^{\\\\prime }_{\\\\mu \\\\nu }(s)=-\\\\overline{\\\\mathfrak {a}^{\\\\prime }_{\\\\nu \\\\mu }(s)}&=\\\\frac{1}{\\\\sqrt{(4-E^2)N}}B^{--}_{\\\\mu \\\\nu }(s)&\\\\mbox{ for }&1\\\\le \\\\mu <\\\\nu \\\\le N\\\\\\\\\\\\mathfrak {b}_{\\\\mu \\\\nu }(s)&=\\\\frac{1}{\\\\sqrt{(4-E^2)N}}B^{+-}_{\\\\mu \\\\nu }(s)&\\\\mbox{ for }&1\\\\le \\\\mu ,\\\\nu \\\\le N$ with all the elements of $B^{++},B^{--}$ and $B^{+-}$ standard complex Brownian motions, mutually independent and independent of $W$ .\",\n \"For $\\\\gamma =0$ , ${\\\\mathcal {Z}}_0(s) := \\\\begin{pmatrix} \\\\mathfrak {a}(s) & \\\\mathfrak {b}(s) \\\\\\\\ \\\\overline{\\\\mathfrak {b}(s)} &\\\\overline{ \\\\mathfrak {a}(s)}\\\\end{pmatrix}$ with the definition of $\\\\mathfrak {a}$ unchanged, but $\\\\mathfrak {a}^{\\\\prime }=\\\\overline{\\\\mathfrak {a}}$ now, and a symmetric $\\\\mathfrak {b}$ : $\\\\mathfrak {b}_{\\\\mu \\\\nu }(s)=\\\\mathfrak {b}_{\\\\nu \\\\mu }&=\\\\frac{1}{\\\\sqrt{(4-E^2)N}}\\\\tilde{B}^{+-}_{\\\\mu \\\\nu }(s)&\\\\mbox{ for }&1\\\\le \\\\mu \\\\le \\\\nu \\\\le N,$ with the entries of the $\\\\tilde{B}^{+-}$ independent standard complex Brownian motions again, independent of the elements of $B^{++}$ and $W$ .\",\n \"The result of the scaling limit, $\\\\lambda \\\\rightarrow 0$ is summarized in the main theorem.\",\n \"Theorem 8 If $h_1$ and $h_2$ depend on $\\\\lambda $ so that $h_1(\\\\lambda ) \\\\longrightarrow 0\\\\,,\\\\qquad h_2(\\\\lambda ) \\\\longrightarrow 0, \\\\qquad \\\\text{and}\\\\qquad \\\\lambda ^{-2}\\\\mathrm {cha}(\\\\gamma ,h_1(\\\\lambda ),h_2(\\\\lambda ))\\\\longrightarrow \\\\infty \\\\,,$ as $\\\\lambda \\\\rightarrow 0$ , then the process $\\\\left(A^\\\\lambda \\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\right)_{s\\\\ge 0}$ converges in distribution to the process $\\\\left({\\\\mathcal {A}}(s)\\\\right)_{s\\\\ge 0}$ on the path space of ${\\\\mathbb {C}}^{2N\\\\times 2N}$ -valued processes endowed with Skorhod topology.\",\n \"For $\\\\gamma \\\\ge 0$ , $\\\\left({\\\\mathcal {A}}(s)\\\\right)_{s\\\\ge 0}$ is given as the unique solution for $s\\\\ge 0$ to $\\\\begin{split}\\\\mathrm {d}{\\\\mathcal {A}}(s)&=\\\\mathrm {d}{\\\\mathcal {Z}}_\\\\gamma (s){\\\\mathcal {A}}(s)\\\\\\\\{\\\\mathcal {A}}(0)&=1.\\\\end{split}$ Remark.\",\n \"Existence and uniqueness of ${\\\\mathcal {A}}$ is a standard result, as all entries of ${\\\\mathcal {Z}}$ are Brownian motions.\",\n \"Note that no additional moment condition is needed on the random variables $V(x,z)$ .\",\n \"The convergence of the hopping parameters $h_i$ to zero merely brings the resulting process ${\\\\mathcal {A}}$ into a isotropic form close to the DMPK process.\",\n \"It is however not essential for the scaling limit per se, and the same techniques used in the proof of Theorem REF yield a limiting process for fixed $h_1, h_2\\\\ne 0$ , but a less isotropic one.\",\n \"For $\\\\gamma >0$ , ${\\\\mathcal {Y}}_{\\\\gamma }(s) := \\\\begin{pmatrix} \\\\alpha (s) & \\\\beta (s) \\\\\\\\ \\\\beta (s)^{*}&{ \\\\alpha ^{\\\\prime }}(s)\\\\end{pmatrix}$ where $\\\\alpha (s), \\\\alpha ^{\\\\prime }(s)$ and $\\\\beta (s)$ differ from $\\\\mathfrak {a}(s),\\\\mathfrak {a}^{\\\\prime }(s)$ and $\\\\mathfrak {b}(s)$ only through their covariances, namely by the replacement $\\\\frac{1}{\\\\sqrt{(4-E^2)N}} \\\\quad \\\\longrightarrow \\\\quad \\\\frac{1}{\\\\sqrt{N \\\\left|v^+_{\\\\sigma _1\\\\nu _1}v^+_{\\\\sigma _2\\\\nu _2}\\\\right|}}\\\\,.$ For $\\\\gamma =0$ , ${\\\\mathcal {Y}}_0(s) := \\\\begin{pmatrix} \\\\alpha (s) & \\\\beta (s) \\\\\\\\[1mm] \\\\overline{\\\\beta (s)} &\\\\overline{\\\\alpha (s)}\\\\end{pmatrix}$ with the same substitution.\",\n \"Proposition 9 Let $(\\\\gamma , h_1, h_2)$ be fixed (in particular, not dependent on $\\\\lambda $ ) and such that Lemma REF holds.\",\n \"As $\\\\lambda \\\\rightarrow 0$ , the process $\\\\left(A_\\\\lambda \\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\right)_{s\\\\ge 0}$ converges in distribution to $\\\\left({\\\\mathcal {G}}(s)\\\\right)_{s\\\\ge 0}$ on the path space of ${\\\\mathbb {C}}^{2N\\\\times 2N}$ -valued processes endowed with Skorhod topology.\",\n \"$\\\\left({\\\\mathcal {G}}(s)\\\\right)_{s\\\\ge 0}$ is given as the unique solution for $s\\\\ge 0$ to $\\\\begin{split}\\\\mathrm {d}{\\\\mathcal {G}}(s)&=\\\\mathrm {d}{\\\\mathcal {Y}}(s){\\\\mathcal {G}}(s) \\\\\\\\{\\\\mathcal {G}}(0)&=1.\",\n \"\\\\\\\\\\\\end{split}$\"\n ],\n [\n \"Discussion\",\n \"Let us comment on the limiting process of transfer matrices ${\\\\mathcal {A}}(s)$ , and compare it to the ideal ensemble ${\\\\mathcal {M}}$ of Section .\",\n \"As already discussed in [1], the overall factor $\\\\sqrt{4-E^2}$ only corresponds to a redefinition of the mean free path and is irrelevant here.\",\n \"The major difference lies in the diagonal of the processes $\\\\mathfrak {a}, \\\\mathfrak {a}^{\\\\prime }$ generating ${\\\\mathcal {A}}$ , which have perfectly correlated diagonal elements, whereas they are independent of each other in their cousins generating ${\\\\mathcal {M}}$ .\",\n \"In the case $\\\\beta =1$ , an additional deviation can be found in the variance of the diagonal elements of $\\\\mathfrak {b}$ , which are smaller here than in the ideal case by a factor $\\\\sqrt{2}\\\\cdot \\\\sqrt{N/(N+1)}$ .\",\n \"Despite these differences, we have: Corollary 10 If $\\\\beta =2$ , the law of the process of transmission eigenvalues $(T_k)_{k=1}^N$ induced by ${\\\\mathcal {A}}$ is the same as that induced by ${\\\\mathcal {M}}$ .\",\n \"Mathematically, this can be observed in the fact that the SDE for ${\\\\mathcal {M}}_{++}^{*}{\\\\mathcal {M}}_{++}$ , obtained in a straightforward way by Itô calculus, $\\\\mathrm {d}({\\\\mathcal {M}}_{++}^{*}{\\\\mathcal {M}}_{++} )= {\\\\mathcal {M}}_{++}^{*}\\\\mathrm {d}\\\\mathfrak {b}{\\\\mathcal {M}}_{-+} + {\\\\mathcal {M}}_{-+}^{*}\\\\mathrm {d}\\\\mathfrak {b}^{*}{\\\\mathcal {M}}_{++} + {\\\\mathcal {M}}_{++}^{*}{\\\\mathcal {M}}_{++} \\\\mathrm {d}s + {\\\\mathcal {M}}_{-+}^{*}{\\\\mathcal {M}}_{-+} \\\\mathrm {d}s$ does not depend on $\\\\mathrm {d}\\\\mathfrak {a}^\\\\sharp $ , and by recalling that the transmission eigenvalues are directly related to the eigenvalues $\\\\lambda _k$ of this matrix.\",\n \"For $\\\\beta =2$ , the limiting $\\\\mathrm {d}\\\\mathfrak {b}$ is exactly equal to the ideal one.\",\n \"The underlying physical reason is that the $\\\\mathfrak {a}^\\\\sharp $ blocks in the infinitesimal transfer matrix merely change the basis of left-, respectively right-moving channels.\",\n \"As such they do not change the magnitude of either the scattered or the reflected waves, hence they do not contribute to the transmission eigenvalues $T_{k}$ .\",\n \"For the $\\\\beta =1$ case, the process of transmission eigenvalues induced by ${\\\\mathcal {A}}$ is different from that induced by ${\\\\mathcal {M}}$ .\",\n \"Of course one may suspect that in the limit $N \\\\rightarrow \\\\infty $ , these processes have similar features, in particular, that the variance of the conductance in the diffusive regime is the same in both cases, and equal to its universal value.\",\n \"In fact in the same limit, we even expect the non-isotropic processes of Proposition REF to share the same property.\",\n \"This remains currently at the level of speculations, the reason being the relatively poor understanding of the properties of the DMPK equation itself in the large $N$ limit.\",\n \"This should be contrasted with recent efforts in the study of hermitian random matrices, where the Gaussian ensembles are very well-known and the challenge is to show that other ensembles share some of their properties, see e.g.\",\n \"[10] and references therein.\"\n ],\n [\n \"Symmetry considerations\",\n \"As already mentioned in Section , the driving process ${\\\\mathcal {L}}$ in the SDE for ${\\\\mathcal {M}}$ satisfies the invariance property ${\\\\mathcal {W}}{\\\\mathcal {L}}{\\\\mathcal {W}}^{-1} \\\\mathop {=}\\\\limits ^{d} {\\\\mathcal {L}}$ for ${\\\\mathcal {W}}= \\\\begin{pmatrix} {\\\\mathcal {W}}_+ &0 \\\\\\\\ 0 &{\\\\mathcal {W}}_- \\\\end{pmatrix}$ with ${\\\\mathcal {W}}_+, {\\\\mathcal {W}}_-$ arbitrary unitaries in the $\\\\beta =2$ case and satisfying ${\\\\mathcal {W}}_-=\\\\overline{{\\\\mathcal {W}}_+}$ in the $\\\\beta =1$ case.\",\n \"In other words, all channel bases are assumed to be equivalent.\",\n \"It is exactly this equivalence that is lost in our model.\",\n \"This can be understood heuristically as follows.\",\n \"Here, the size of the impurities is assumed to be much smaller than the wavelength of the scattered waves, a fact expressed by the $\\\\delta $ -correlation in space of the potential, i.e.\",\n \"$V=\\\\sum _y V_y$ with $y=(x,z)$ and ${\\\\mathbb {E}}(V_yV_{y^{\\\\prime }})= \\\\delta _{y,y^{\\\\prime }}$ .\",\n \"Therefore, the position basis (or its dual, the momentum basis) is naturally singled out.\",\n \"There is no reason to expect another choice of basis to be equivalent and to allow for arbitrary unitaries ${\\\\mathcal {W}}_+,{\\\\mathcal {W}}_-$ in (REF ).\",\n \"The symmetry that still survives (but in fact, only so because we performed the additional scaling limit $h_1, h_2 \\\\rightarrow 0$ ) is the relabeling of channels; indeed the scaling limit $h_1, h_2 \\\\rightarrow 0$ makes the group velocity of all channels $\\\\nu $ equal at a given energy $E$ .\",\n \"Up to phases, this corresponds to restricting ${\\\\mathcal {W}}_+,{\\\\mathcal {W}}_-$ to be permutation matrices.\",\n \"One easily checks that, with this restriction, the driving process ${\\\\mathcal {Z}}_\\\\gamma $ still satisfies (REF ).\",\n \"Another consequence of the locality of the impurities is captured by the following heuristic argument.\",\n \"Let us consider the kernel $S_{p,p^{\\\\prime }}$ of the scattering matrix in momentum basis, i.e.\",\n \"$p=(k,\\\\nu )$ .\",\n \"Since the scattering is weak (as already indicated, this is inherent to the setup and it is forced in our model by the $\\\\lambda \\\\rightarrow 0$ limit), multiple scattering can be neglected for short slabs of material, hence we can use the Born approximation for $S_{p,p^{\\\\prime }}$ : $S_{p,p^{\\\\prime }} = \\\\delta _{p,p^{\\\\prime }}-2 \\\\pi \\\\mathrm {i}\\\\lambda \\\\sum _y (\\\\tilde{V}_y)_{p,p^{\\\\prime }} \\\\delta (E(p)-E(p^{\\\\prime })) + {\\\\mathcal {O}}(\\\\lambda ^2)$ where $(\\\\tilde{V}_y)=V_ye^{\\\\mathrm {i}(p-p^{\\\\prime })y}$ .\",\n \"For $p=p^{\\\\prime }$ , this expression depends on $p$ only through the factor $\\\\delta (E(p)-E(p^{\\\\prime }))$ which contributes a $p$ -dependent group velocity, a dependence which vanishes in the subsequent scaling limit $h_1,h_2 \\\\rightarrow 0$ .\",\n \"This means that the diagonal elements of the transmission matrix all coincide and this is exactly what we find in the ensemble ${\\\\mathcal {A}}(s)$ , since in lowest order the $\\\\mathfrak {a}$ block corresponds to the transmission matrix.\",\n \"Finally, we mention [18], [19] where geometric methods are used to analyze a similar weak coupling limit of the Anderson model on tubes, and contact is made with random matrix theory.\"\n ],\n [\n \"Proof of transfer matrix limits\",\n \"We prove Theorem REF .\",\n \"The evolution of $\\\\left(A^\\\\lambda (x)\\\\right)_{x\\\\ge 0}$ is given by $A^\\\\lambda (0)=1$ and the stochastic difference equation $A^\\\\lambda (x)-A^\\\\lambda (x-1) &= \\\\left((M^0(x))^{-1} K^{-1}T^\\\\lambda _x K M^0(x-1)-1\\\\right) A^\\\\lambda (x-1) \\\\nonumber \\\\\\\\&=: \\\\lambda Z_x A^\\\\lambda (x-1),$ where $Z_x = (M^0(x))^{-1} R_x M^0(x)$ and we defined $\\\\lambda R_x := K^{-1}T^\\\\lambda _x K (M^0_x)^{-1} -1 = K^{-1}(T^\\\\lambda _x-T^0_x) K (M^0_x)^{-1}.$ Recall that $K$ , introduced in the previous section, stands for the change from the position basis to the channel basis.\",\n \"It follows that $R_x = \\\\Upsilon ^{-1} \\\\begin{pmatrix}-Q^{*}(P^{*})^{-1}V_xQ & 0 \\\\\\\\ 0 & 0\\\\end{pmatrix}\\\\Upsilon (M^0_x)^{-1}$ Using the explicit forms of $\\\\Upsilon , Q$ and $P$ , this matrix reads $R_x=\\\\mathrm {i}\\\\begin{pmatrix}\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}&\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}\\\\Pi \\\\\\\\-\\\\Pi \\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}&-\\\\Pi \\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}\\\\Pi \\\\end{pmatrix},$ which of course satisfies $(R_{++})^{*}= -R_{++}$ , $(R_{--})^{*}= -R_{--}$ and $(R_{-+})^{*}= R_{+-}$ .\",\n \"In the time reversal invariant case, since $Q=Q^T,\\\\qquad Q\\\\Pi =Q^*,\\\\qquad \\\\Pi Q^*=Q$ this simplifies to $R_x=\\\\mathrm {i}\\\\begin{pmatrix}\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}&\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}Q^*V_xQ^*\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}\\\\\\\\-\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}QV_xQ\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}&-\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}QV_xQ^*\\\\frac{1}{\\\\sqrt{\\\\left|v^+\\\\right|}}\\\\end{pmatrix}\\\\,.$ Since $Q^*=\\\\overline{Q}$ , we also have that $R_{--} =\\\\overline{ R_{++}}$ , and in a similar fashion $R_{-+} = \\\\overline{R_{+-}}$ .\",\n \"Before we go further into the proof, let us explain the heuristics of the convergence to the DMPK equation.\",\n \"The matrix $R_x$ contains the $N$ i.i.d random variables $V(x,z)$ , $z=1,\\\\ldots ,N$ , and $R_x,R_y$ are independent for $x\\\\ne y$ .\",\n \"Under the appropriate technical conditions, we have convergence, as $ \\\\lambda \\\\rightarrow 0$ ; $\\\\lambda \\\\sum _{x=0}^{\\\\lfloor \\\\lambda ^{-2}s\\\\rfloor }R_x \\\\stackrel{\\\\mathrm {d}}{\\\\longrightarrow }{\\\\mathcal {R}}(s)\\\\,.$ where the $V(x,z)$ in $R_x$ are replaced by Brownian motions $B_z(s)$ in ${\\\\mathcal {R}}(s)$ .\",\n \"Of course, this is nothing else than the convergence of a random walk to Brownian motion.\",\n \"Now, the matrices $Z_x$ generating the discrete process $A(x)$ do contain the highly oscillating phases of $M^0(x)$ .\",\n \"Let $Z^\\\\lambda (s) = \\\\lambda \\\\sum _{x=0}^{\\\\lfloor \\\\lambda ^{-2}s\\\\rfloor }Z_x\\\\,.$ The correlation of any two matrix elements reads ${\\\\mathbb {E}}\\\\left[\\\\left(Z^\\\\lambda (s)\\\\right)_{mn}\\\\left(Z^\\\\lambda (s)\\\\right)_{pr}\\\\right] \\\\\\\\=\\\\lambda ^{2}\\\\sum _{x=1}^{\\\\lfloor \\\\lambda ^{-2}s\\\\rfloor }\\\\exp \\\\left(\\\\mathrm {i}x\\\\left(-k^{\\\\sigma _m}_{\\\\sigma _m\\\\nu _m}+k^{\\\\sigma _n}_{\\\\sigma _n\\\\nu _n}-k^{\\\\sigma _p}_{\\\\sigma _p\\\\nu _p}+k^{\\\\sigma _r}_{\\\\sigma _r\\\\nu _r}\\\\right)\\\\right){\\\\mathbb {E}}\\\\left[(R_x)_{mn}(R_x)_{pr}\\\\right].$ where the $\\\\sigma $ 's and $\\\\nu $ 's denote the block and position in the block of a certain element, the `physical' momentum, due to the permutation $\\\\Pi $ is not $\\\\nu $ , but $\\\\sigma \\\\nu $ .\",\n \"The expectation in the r.h.s is independent of $x$ , so that this sum is highly oscillatory and formally converges to a $\\\\delta $ function.\",\n \"Hence, the phases create a limiting process ${\\\\mathcal {Z}}(s)$ with almost completely uncorrelated entries, apart from the exceptional conditions of Lemma REF .\",\n \"As a result, the number of independent random variables in ${\\\\mathcal {Z}}(s)$ is of order $N^2$ , whereas it was only $N$ in ${\\\\mathcal {R}}(s)$ , a phenomenon called `noise explosion' in [20].\",\n \"Precisely, we prove: Lemma 11 Under the conditions of Theorem REF and in the same topology, the process $\\\\left(Z^\\\\lambda (s)\\\\right)_{s\\\\ge 0}$ converges in distribution to the process $\\\\left({\\\\mathcal {Z}}_\\\\gamma (s)\\\\right)_{s\\\\ge 0}$ , for $\\\\gamma \\\\ge 0$ .\",\n \"We first recall that $Q$ is the matrix of transversal plane waves so that $(Q^{*}V_x Q)_{\\\\mu \\\\nu } = \\\\widehat{V_x}(\\\\mu -\\\\nu )\\\\,.$ In particular, all elements on the diagonal are perfectly correlated.\",\n \"Now, we consider (REF ) for $s\\\\ge 0$ , and note that the wavevectors $k_\\\\nu ^\\\\sigma $ and the matrices $R_x$ all depend on $\\\\lambda $ implicitly through the dependence $h_i(\\\\lambda )$ .\",\n \"However, ${\\\\mathbb {E}}\\\\left[(R_x)_{mn}(R_x)_{pr}\\\\right]$ is independent of $x$ and can be taken out of the sum.\",\n \"For all choices $\\\\lbrace (\\\\nu _i,\\\\sigma _i): i=m,n,p,r\\\\rbrace $ for which the exponent does not vanish, we have that $\\\\left| \\\\lambda ^{2}\\\\sum _{x=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\exp \\\\left(ix\\\\left(-k^{\\\\sigma _m}_{\\\\sigma _m\\\\nu _m}+k^{\\\\sigma _n}_{\\\\sigma _n\\\\nu _n}-k^{\\\\sigma _p}_{\\\\sigma _p\\\\nu _p}+k^{\\\\sigma _r}_{\\\\sigma _r\\\\nu _r}\\\\right)\\\\right)\\\\right|\\\\le \\\\frac{2\\\\lambda ^2}{\\\\mathrm {cha}(\\\\lambda )}\\\\longrightarrow 0,$ as $\\\\lambda \\\\rightarrow 0$ , by Assumption (REF ).\",\n \"By Lemma REF , the the wavenumbers cancel out whenever $\\\\left(\\\\sigma _m,\\\\nu _m\\\\right)=\\\\left(\\\\sigma _n,\\\\nu _n\\\\right)\\\\mbox{ and }\\\\left(\\\\sigma _p,\\\\nu _p\\\\right)=\\\\left(\\\\sigma _r,\\\\nu _r\\\\right)\\\\,,$ corresponding to any two diagonal elements in $Z^\\\\lambda $ , or $\\\\left(\\\\sigma _m,\\\\nu _m\\\\right)=\\\\left(\\\\sigma _r,\\\\nu _r\\\\right)\\\\mbox{ and }\\\\left(\\\\sigma _p,\\\\nu _p\\\\right)=\\\\left(\\\\sigma _n,\\\\nu _n\\\\right)\\\\,,$ namely two mutually `transpose' entries of $Z^\\\\lambda $ .\",\n \"In the case $\\\\gamma =0$ , a last possibility is given by $\\\\left(\\\\sigma _m,\\\\nu _m\\\\right)=\\\\left(-\\\\sigma _p,\\\\nu _p\\\\right)\\\\mbox{ and }\\\\left(\\\\sigma _n,\\\\nu _n\\\\right)=\\\\left(-\\\\sigma _r,\\\\nu _r\\\\right)\\\\,,$ corresponding to two elements in the same position, but within the opposite blocks.\",\n \"In these three cases and for all $\\\\lambda $ , $\\\\lambda ^{2}\\\\sum _{x=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }1=s.$ Furthermore, using the explicit form (REF ) of $R_x$ and the definition of $Q$ , ${\\\\mathbb {E}}\\\\left[(R_x)_{mn}(R_x)_{pr}\\\\right]=-\\\\frac{\\\\sigma _m\\\\sigma _p}{N^2\\\\sqrt{\\\\left|v^+_{\\\\sigma _m\\\\nu _m}v^+_{\\\\sigma _n\\\\nu _n}v^+_{\\\\sigma _p\\\\nu _p}v^+_{\\\\sigma _r\\\\nu _r}\\\\right|}} \\\\\\\\\\\\cdot \\\\sum _{z=0}^{N-1}\\\\exp \\\\left(\\\\frac{2\\\\pi i}{N}z\\\\left(-\\\\sigma _m\\\\nu _m+\\\\sigma _n\\\\nu _n-\\\\sigma _p\\\\nu _p+\\\\sigma _r\\\\nu _r\\\\right)\\\\right)$ which we only need to evaluate in the `stationary phase' situations (REF ), (REF ), and (REF ).\",\n \"In all these cases, ${\\\\mathbb {E}}\\\\left[(R_x)_{mn}(R_x)_{pr}\\\\right] = -\\\\frac{\\\\sigma _m\\\\sigma _p}{N\\\\sqrt{\\\\left|v^+_{\\\\sigma _m\\\\nu _m}v^+_{\\\\sigma _n\\\\nu _n}v^+_{\\\\sigma _p\\\\nu _p}v^+_{\\\\sigma _r\\\\nu _r}\\\\right|}}.$ Note that the $v^+_{\\\\nu }$ still depend on $\\\\lambda $ and $\\\\nu $ , but by (REF ), they converge to the $\\\\nu $ -independent limit $|2\\\\sin (k)|$ with $2\\\\cos (k)=E$ .\",\n \"Hence, in the limit $\\\\lambda \\\\rightarrow 0$ , we have ${\\\\mathbb {E}}\\\\left[(R_x)_{mn}(R_x)_{pr}\\\\right] \\\\longrightarrow -\\\\frac{\\\\sigma _m\\\\sigma _p}{(4-E^2)N}\\\\,.$ A very similar oscillatory sum appears in ${\\\\mathbb {E}}\\\\left[\\\\left(Z^\\\\lambda (s)^{*}\\\\right)_{mn}\\\\left(Z^\\\\lambda (s)\\\\right)_{pr}\\\\right]$ , with parallel conclusions.\",\n \"In summary, for all $s\\\\ge 0$ $\\\\begin{split}\\\\lim _{\\\\lambda \\\\rightarrow 0}{\\\\mathbb {E}}\\\\left[\\\\left(Z^\\\\lambda (s)\\\\right)_{mn}\\\\left(Z^\\\\lambda (s)\\\\right)_{pr}\\\\right]&=\\\\int _0^s\\\\mathrm {d}\\\\left\\\\langle \\\\left({\\\\mathcal {Z}}\\\\right)_{mn}, \\\\left({\\\\mathcal {Z}}\\\\right)_{pr}\\\\right\\\\rangle _t,\\\\\\\\\\\\lim _{\\\\lambda \\\\rightarrow 0}{\\\\mathbb {E}}\\\\left[\\\\left(Z^\\\\lambda (s)^{*}\\\\right)_{mn}, \\\\left(Z^\\\\lambda (s)\\\\right)_{pr}\\\\right]&=\\\\int _0^s\\\\mathrm {d}\\\\left\\\\langle \\\\left({\\\\mathcal {Z}}^*\\\\right)_{mn},\\\\left({\\\\mathcal {Z}}\\\\right)_{pr}\\\\right\\\\rangle _t.\\\\end{split}$ with ${\\\\mathcal {Z}}$ given in the previous section, and $\\\\langle M,N \\\\rangle _t$ the bracket process of two martingales $M,N$ .\",\n \"In particular, the block structure of ${\\\\mathcal {Z}}$ arises from the corresponding relations noted above in $R_x$ .\",\n \"Moreover, perfect correlation of the diagonal elements of $Q^{*}V_x Q$ and the exceptional case (REF ) imply the perfect correlation of the diagonal elements of $\\\\mathrm {d}\\\\mathfrak {a}$ and $\\\\mathrm {d}\\\\mathfrak {a}^{\\\\prime }$ .\",\n \"All other exceptional cases impose the correlations $\\\\vert \\\\mathrm {d}\\\\mathfrak {b}_{\\\\mu \\\\nu }\\\\vert ^2$ and $\\\\vert \\\\mathrm {d}\\\\mathfrak {a}_{\\\\mu \\\\nu }\\\\vert ^2$ .\",\n \"Now the lemma follows as a simple generalization of Donsker's invariance principle, for example by using Chapter VII, Theorem 3.7 in [13].\",\n \"They check the convergence of three characteristics, of which, in their notation, $\\\\left[\\\\sup -\\\\beta _3^{\\\\prime }\\\\right]$ is trivially fulfilled as $Z^\\\\lambda $ and ${\\\\mathcal {Z}}$ are martingales, $\\\\left[\\\\gamma _3^{\\\\prime }-{\\\\mathbb {R}}^+\\\\right]$ is the convergence of brackets as shown in (REF ), and $\\\\left[\\\\delta _{3,1}-{\\\\mathbb {R}}^+\\\\right]$ is a simple estimate on the jumps of $Z^\\\\lambda $ , namely $\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\sum _{x=1}^{\\\\lfloor \\\\lambda ^{-2}s\\\\rfloor }\\\\mathbb {E}\\\\left(g_a\\\\left(\\\\lambda \\\\left\\\\Vert R_x\\\\right\\\\Vert \\\\right)\\\\right)=0$ for all $g_a(y)=y^21_{\\\\lbrace |y|>a\\\\rbrace }$ , $a>0$ , which is trivial by $\\\\mathbb {E}\\\\left(v^2\\\\right)=1$ .\",\n \"Notice that the covariances of the less isotropic ensemble of Proposition REF can be read off from this proof.\",\n \"The main theorem will now follow from this lemma and the difference equation (REF ).\",\n \"We simplify the notation of (REF ) and (REF ) by writing the real and imaginary parts of the matrix entries as elements of vectors in ${\\\\mathbb {R}}^d$ , $d=8N^2$ .\",\n \"In this notation (REF ) reads $X^{\\\\lambda }_j(y)-X^{\\\\lambda }_j(y-1)=\\\\lambda \\\\sum _{k=1}^d\\\\xi ^{\\\\lambda }_{jk}(y)X^{\\\\lambda }_k(y-1)$ for all $y$ in ${\\\\mathbb {N}}$ , with $\\\\xi ^{\\\\lambda }(y)\\\\in {\\\\mathbb {R}}^{d\\\\times d}$ independent of $X^{\\\\lambda }(z), \\\\xi ^{\\\\lambda }(z)$ $z\\\\in \\\\lbrace 0,...,y-1\\\\rbrace $ .\",\n \"Because of the phase factors, the law of $\\\\xi ^{\\\\lambda }(y)$ is not independent of $y\\\\in {\\\\mathbb {N}}$ , but ${\\\\mathbb {E}}(\\\\xi ^{\\\\lambda }(y))=0$ and $ \\\\Vert \\\\xi ^{\\\\lambda }(y)\\\\Vert ^2 \\\\le c^{\\\\prime } (\\\\sum _{z=1}^N |v(y,z) |)^2 \\\\le c \\\\sum _{z=1}^N v^2(y,z)$ where $c,c^{\\\\prime }< \\\\infty $ are $\\\\lambda $ -independent for sufficiently small $\\\\lambda $ , but they depend on $N$ .\",\n \"This follows from (REF ) or (REF ) by noting that the velocity matrix $v^+$ converges to a nonsingular limit as $\\\\lambda \\\\rightarrow 0$ ; we will henceforth assume without comment that $\\\\lambda $ is sufficiently small.\",\n \"Furthermore, we know from the proof of Lemma REF $\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\lambda ^2\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)=C_{ikjl}\\\\cdot s$ uniformly for $s\\\\ge 0$ from bounded intervals.\",\n \"If we define $\\\\left({\\\\mathcal {B}}(s)\\\\right)_{s\\\\ge 0}$ as the matrix-valued Brownian motion with bracket process $\\\\left\\\\langle {\\\\mathcal {B}}_{ik},{\\\\mathcal {B}}_{jl}\\\\right\\\\rangle _s= C_{ikjl}\\\\cdot s,$ the equation (REF ) transforms to $\\\\mathrm {d}{\\\\mathcal {X}}_j(s)=\\\\sum _{k=1}^d\\\\mathrm {d}{\\\\mathcal {B}}_{jk}{\\\\mathcal {X}}_k(s).$ The initial values for $X^{\\\\lambda }$ and ${\\\\mathcal {X}}$ are identical and deterministic, the ${\\\\mathbb {R}}^d$ vector corresponding to the unit matrix $1_{2N}$ .\",\n \"For notational convenience, we have chosen $X^{\\\\lambda }$ to still live on the microscopic, discrete space, what we really want to investigate is the cadlag process $\\\\overline{X}^{\\\\lambda }(s)=X^{\\\\lambda }\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right).$ For cadlag processes we define $\\\\overline{X}^{\\\\lambda }(s-)$ as the leftside limit of $\\\\overline{X}^{\\\\lambda }$ at $s$ .\",\n \"With the filtration ${\\\\mathcal {F}}^{\\\\lambda }_s=\\\\sigma \\\\left\\\\lbrace \\\\overline{X}^{\\\\lambda }(t):t\\\\le s\\\\right\\\\rbrace $ , $\\\\left(\\\\overline{X}^{\\\\lambda }(s)\\\\right)_{s\\\\ge 0}$ is a $\\\\left\\\\lbrace {\\\\mathcal {F}}^{\\\\lambda }_s \\\\right\\\\rbrace $ -martingale.\",\n \"Furthermore defining $V^{\\\\lambda }_{ij}(y)=\\\\lambda ^2\\\\sum _{x=1}^y\\\\sum _{k,l=1}^d\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(x)\\\\xi _{jl}^{\\\\lambda }(x)\\\\right)X^{\\\\lambda }_k(x-1)X^{\\\\lambda }_l(x-1)$ and the corresponding macroscopic $\\\\overline{V}^{\\\\lambda }_{ij}(s)=V^{\\\\lambda }\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)$ for all $i,j=1,...,d$ , the process $\\\\overline{X}^{\\\\lambda }_i\\\\overline{X}^{\\\\lambda }_j-\\\\overline{V}^{\\\\lambda }_{ij}$ is a $\\\\left\\\\lbrace {\\\\mathcal {F}}^{\\\\lambda }_s \\\\right\\\\rbrace $ -martingale as well, and we have by Theorem 7.4.1, [11], Lemma 12 If for any $T>0$ , and any stopping time $T^{\\\\lambda }_r=\\\\inf \\\\left\\\\lbrace s:\\\\left|\\\\overline{X}^{\\\\lambda }(s)\\\\right|\\\\ge r \\\\mbox{ or } \\\\left|\\\\overline{X}^{\\\\lambda }(s-)\\\\right|\\\\ge r\\\\right\\\\rbrace ,$ $r>0$ , for all $i,j=1,...,d$ $&\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\mathbb {E}\\\\left(\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}\\\\left|\\\\overline{X}^{\\\\lambda }(s)-\\\\overline{X}^{\\\\lambda }(s-)\\\\right|^2\\\\right)=0\\\\\\\\&\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\mathbb {E}\\\\left(\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}\\\\left|\\\\overline{V}^{\\\\lambda }_{ij}(s)-\\\\overline{V}^{\\\\lambda }_{ij}(s-)\\\\right|\\\\right)=0$ and $\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}\\\\left|\\\\overline{V}^{\\\\lambda }_{ij}(s)-\\\\int _0^s\\\\mathrm {d}t\\\\sum _{k,l=1}^dC_{ikjl}\\\\overline{X}^{\\\\lambda }_k(t)\\\\overline{X}^{\\\\lambda }_l(t)\\\\right|\\\\stackrel{{\\\\mathbb {P}}}{\\\\rightarrow }0,$ then $\\\\left(\\\\overline{X}^{\\\\lambda }(s)\\\\right)_{s\\\\ge 0}$ converges in distribution on $D_{{\\\\mathbb {R}}^d}[0,\\\\infty )$ to $\\\\left({\\\\mathcal {X}}(s)\\\\right)_{s\\\\ge 0}$ .\",\n \"So to prove Theorem REF , we only have to verify the conditions of Lemma REF .\",\n \"We start with the following observation, Lemma 13 Let $Z_k$ , $k\\\\in {\\\\mathbb {N}}$ be i.i.d.\",\n \"distributed, positive random variables, with $\\\\mathbb {E}\\\\left(Z_1\\\\right)<\\\\infty .$ Then $\\\\frac{1}{n}\\\\mathbb {E}\\\\left(\\\\max _{1\\\\le k\\\\le n}Z_k\\\\right)\\\\rightarrow 0$ as $n\\\\rightarrow \\\\infty $ .\",\n \"With $q(x)={\\\\mathbb {P}}\\\\left(Z_1\\\\ge x\\\\right)$ for $x\\\\ge 0$ , we have $\\\\int _0^\\\\infty q(x)\\\\mathrm {d}x=\\\\mathbb {E}\\\\left(Z_1\\\\right),$ while ${\\\\mathbb {P}}\\\\left(\\\\max _{1\\\\le k\\\\le n}Z_k\\\\ge x\\\\right)=1-(1-q(x))^n\\\\le n q(x).$ Thus, $\\\\frac{1}{n}\\\\mathbb {E}\\\\left(\\\\max _{1\\\\le k\\\\le n}Z_k\\\\right)=\\\\int _0^\\\\infty \\\\frac{1-(1-q(x))^n}{n}\\\\mathrm {d}x$ with the integrand on the right side converging to zero and dominated by the integrable $q(x)$ , an the claim follows by dominated convergence.\",\n \"For any $T>0, r>0$ given, we have $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}&\\\\left|\\\\overline{X}^{\\\\lambda }(s)-\\\\overline{X}^{\\\\lambda }(s-)\\\\right|^2\\\\\\\\=&\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2} \\\\left(T\\\\wedge T^{\\\\lambda }_r\\\\right)\\\\right\\\\rfloor }\\\\left|X^{\\\\lambda }(y)-X^{\\\\lambda }(y-1)\\\\right|^2\\\\\\\\=&\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2} \\\\left(T\\\\wedge T^{\\\\lambda }_r\\\\right)\\\\right\\\\rfloor }\\\\lambda ^2\\\\sum _{j,k,l=1}^d\\\\xi ^{\\\\lambda }_{jk}(y)\\\\xi ^{\\\\lambda }_{jl}(y)X^{\\\\lambda }_{k}(y-1)X^{\\\\lambda }_{l}(y-1)\\\\\\\\\\\\le &r^2\\\\lambda ^2\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2}T\\\\right\\\\rfloor }\\\\left\\\\Vert \\\\xi ^{\\\\lambda }(y)\\\\right\\\\Vert ^2\\\\\\\\\\\\le &c(\\\\lambda )r^2\\\\lambda ^2\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2}T\\\\right\\\\rfloor }\\\\sum _{z=0}^{N-1}v(y,z)^2,\\\\end{split}$ where we used (REF ) in the last line.\",\n \"Now the last line of (REF ) vanishes in expectation by Lemma REF and the fact that ${\\\\mathbb {E}}(v(y,z)^2)=1$ .\",\n \"This proves (REF ).\",\n \"For the proof of (), note $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}&\\\\left|\\\\overline{V}^{\\\\lambda }_{ij}(s)-\\\\overline{V}^{\\\\lambda }_{ij}(s-)\\\\right|\\\\\\\\=&\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2} \\\\left(T\\\\wedge T^{\\\\lambda }_r\\\\right)\\\\right\\\\rfloor }\\\\left|V^{\\\\lambda }_{ij}(y)-V^{\\\\lambda }_{ij}(y-1)\\\\right|\\\\\\\\=&\\\\lambda ^2\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2} \\\\left(T\\\\wedge T^{\\\\lambda }_r\\\\right)\\\\right\\\\rfloor }\\\\left|\\\\sum _{k,l=1}^d\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)X^{\\\\lambda }_k(y-1)X^{\\\\lambda }_l(y-1)\\\\right|\\\\\\\\\\\\le &\\\\lambda ^2r^2\\\\max _{1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2} T\\\\right\\\\rfloor }\\\\mathbb {E}\\\\left(\\\\left\\\\Vert \\\\xi ^{\\\\lambda }(y)\\\\right\\\\Vert ^2\\\\right),\\\\end{split}$ which obviously vanishes in expectation as $\\\\lambda \\\\rightarrow 0$ .\",\n \"For (REF ), start with $\\\\begin{split}\\\\overline{V}^{\\\\lambda }_{ij}(s)&-\\\\int _0^s\\\\mathrm {d}t\\\\sum _{k,l=1}^dC_{i,k,j,l}\\\\overline{X}^{\\\\lambda }_k(t)\\\\overline{X}^{\\\\lambda }_l(t)\\\\\\\\=&\\\\sum _{k,l=1}^d\\\\left(\\\\lambda ^2\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)-C_{ikjl}\\\\right)X^{\\\\lambda }_k(y-1)X^{\\\\lambda }_l(y-1)\\\\right)\\\\\\\\&\\\\hspace{56.9055pt}-\\\\left(s-\\\\lambda ^2\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\sum _{k,l=1}^dC_{ikjl}X^{\\\\lambda }_k\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)X^{\\\\lambda }_l\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right).\\\\end{split}$ For the last line, we have $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}&\\\\left|\\\\left(s-\\\\lambda ^2\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\sum _{k,l=1}^dC_{ikjl}X^{\\\\lambda }_k\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)X^{\\\\lambda }_l\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\right|\\\\\\\\&\\\\le \\\\lambda ^2r^2d^2\\\\max _{k,l}\\\\left|C_{ikjl}\\\\right|\\\\rightarrow 0\\\\end{split}$ almost surely, and thus in probability, as $\\\\lambda \\\\rightarrow 0$ .\",\n \"We omit the (finite) $k,l$ sum in (REF ) from our notation, and use partial summation with respect to $y$ to obtain $\\\\begin{split}\\\\lambda ^2&\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)-C_{ikjl}\\\\right)X^{\\\\lambda }_k(y-1)X^{\\\\lambda }_l(y-1)\\\\\\\\&=\\\\lambda ^2X^{\\\\lambda }_k\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)X^{\\\\lambda }_l\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)-C_{ikjl}\\\\right)\\\\\\\\&\\\\hspace{5.69054pt}-\\\\lambda ^2\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\sum _{x=1}^{y}\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(x)\\\\xi _{jl}^{\\\\lambda }(x)\\\\right)-C_{ikjl}\\\\right)\\\\left(X^{\\\\lambda }_k(y)X^{\\\\lambda }_l(y)-X^{\\\\lambda }_k(y-1)X^{\\\\lambda }_l(y-1)\\\\right).\\\\end{split}$ We know from the convergences (REF ) and $\\\\lim _{\\\\lambda \\\\rightarrow 0}\\\\lambda ^2\\\\sum _{x=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }C_{ikjl}=C_{ikjl}\\\\cdot s,$ which are both uniform for $s$ from compact sets, that $a^{\\\\lambda }_{ikjl}(y):=\\\\lambda ^2\\\\sum _{x=1}^{y}\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(x)\\\\xi _{jl}^{\\\\lambda }(x)\\\\right)-C_{ikjl}\\\\right)\\\\rightarrow 0$ as $\\\\lambda \\\\rightarrow 0$ uniformly in $y$ as long as $1\\\\le y\\\\le \\\\left\\\\lfloor \\\\lambda ^{-2}T\\\\right\\\\rfloor $ for fixed positive $T$ .\",\n \"Thus for the first term on the right-hand side in (REF ), $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}&\\\\left|\\\\lambda ^2X^{\\\\lambda }_k\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)X^{\\\\lambda }_l\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }\\\\left(\\\\mathbb {E}\\\\left(\\\\xi _{ik}^{\\\\lambda }(y)\\\\xi _{jl}^{\\\\lambda }(y)\\\\right)-C_{ikjl}\\\\right)\\\\right|\\\\\\\\&\\\\le r^2\\\\sup _{s\\\\le T}\\\\max _{i,k,j,l}\\\\left|a^{\\\\lambda }_{ikjl}\\\\left(\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor \\\\right)\\\\right|\\\\rightarrow 0\\\\end{split}$ as $\\\\lambda \\\\rightarrow 0$ .\",\n \"After plugging (REF ) into the second term on the right-hand side of (REF ), we are left with the sum of $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}\\\\left|\\\\lambda \\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }a^{\\\\lambda }_{ikjl}(y)\\\\left(\\\\sum _{k^{\\\\prime }=1}^d \\\\xi ^{\\\\lambda }_{kk^{\\\\prime }}(y)X^{\\\\lambda }_{k^{\\\\prime }}(y-1)X^{\\\\lambda }_l(y-1)\\\\right.\\\\right.\\\\\\\\+\\\\left.\\\\left.\\\\sum _{l^{\\\\prime }=1}^d \\\\xi ^{\\\\lambda }_{ll^{\\\\prime }}(y)X^{\\\\lambda }_{k}(y-1)X^{\\\\lambda }_{l^{\\\\prime }}(y-1)\\\\right)\\\\vphantom{\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }}\\\\right|\\\\end{split}$ and $\\\\begin{split}\\\\sup _{s\\\\le T\\\\wedge T^{\\\\lambda }_r}\\\\left|\\\\lambda ^2\\\\sum _{y=1}^{\\\\left\\\\lfloor \\\\lambda ^{-2}s\\\\right\\\\rfloor }a^{\\\\lambda }_{ikjl}(y)\\\\left(\\\\sum _{k^{\\\\prime },l^{\\\\prime }=1}^d \\\\xi ^{\\\\lambda }_{kk^{\\\\prime }}(y)\\\\xi ^{\\\\lambda }_{ll^{\\\\prime }}(y)X^{\\\\lambda }_{k^{\\\\prime }}(y-1)X^{\\\\lambda }_{l^{\\\\prime }}(y-1)\\\\right)\\\\right|\\\\\\\\\\\\end{split}$ converging to zero in $L^2({\\\\mathbb {P}})$ and $L^1({\\\\mathbb {P}})$ , respectively.\"\n ],\n [\n \"Acknowledgements\",\n \"Maximilian Butz benefited a lot from discussions with members of Antti Kupiainen's group at Helsinki University, and is grateful for financial support by the Academy of Finland during his stay there.\",\n \"Sven Bachmann gratefully acknowledges the support of the National Science Foundation under Grant #DMS-0757581\"\n ]\n]"},"id":{"kind":"string","value":"1204.0823"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":18669,"numItemsPerPage":100,"numTotalItems":1867602,"offset":1866900,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc0MTg1NTU1OCwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzQxODU5MTU4LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.7B0kjiBJr6QaYDKIS8lbgps4uswosxvlpy_iayrms7GvJEA2pO_wd5mK7qaKgVmoUVv-iamVjTQD7ckCC-7sAw","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false}">
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[
[
"Quarkonium at T>0"
],
[
"Abstract We report recent progress on theoretical investigations of quarkonia at finite temperature.",
"We discuss medium modification of charmonia and bottomonia from a viewpoint of local operators and point out that while charmonia are sensitive to the deconfinement transition bottomonia will be modified at much higher temperatures."
],
[
"Introduction",
"Properties of heavy quarkonia have been extensively studied since it was pointed out that they provide information on deconfinement transition of QCD.",
"[1], [2] While expected suppressions of the quarkonia have been measured in heavy ion experiments [3], interpretation of these data is not so straightforward, because of not only the complexity of the collision processes but also the fact that spectral properties of the heavy quarkonia are not well understood yet.",
"In this report, we focus on recent development of theoretical understanding of the quarkonium states at finite temperature."
],
[
"Theoretical development",
"Lattice QCD provides a unique non-perturbative first principle approach.",
"A direct study of the quarkonium spectral function $\\rho (\\omega ,T)$ is possible with the help of the maximum entropy method (MEM) which enables us to invert a current correlation function at imaginary time $G(\\tau ,T)$ via a dispersion relation.",
"[4] $G(\\tau ,T)=\\int d\\omega \\frac{\\cosh [\\omega (\\tau -1/(2T))]}{\\sinh [\\omega /(2T)]} \\rho (\\omega ,T).$ Even in recent calculations with bigger lattice sizes [5], however, quantitative information seems hard to be extracted, presumably due to limited temporal lattice size at high temperature and a small number of data points.",
"The existence of the spectral peak does not necessary mean survival of a quarkonium[6] and substantial spectral modification does not contradict with the behavior of $G(\\tau ,T)$ [7].",
"A rather promising direction seems to determine an interquark potential $V(r,T)$ containing both real and imaginary parts for a Schödinger equation by lattice QCD.",
"[8] Indeed, existence of the imaginary part in the potential was pointed out by Laine et al.",
"within a resummed perturbation theory.",
"[9] Recently significant progress has been made with an effective field theory framework for heavy quark bound states.",
"[10] Though this approach assumes a hierarchy of the energy scale which becomes complicated at finite temperature due to newly introduced medium energy scales, analytically results have been presented in some cases to give an insight of possible mechanisms of the in-medium modification of a quarkonium.",
"[11] These analyses indicate that a dominant in-medium effect on quarkonium at experimentally accessible temperatures could be a singlet to octet breakup process by gluons.",
"On the other hand, an estimation of the energy density achieved in heavy ion collisions at RHIC energies and subsequent hydrodynamic evolution lead to a lifetime of the deconfined phase long enough to melt quarkonia if $\\Gamma > 50$ MeV.",
"[12] This means that we need a theoretical estimation of the width at those temperatures, likely in the strongly interacting regime.",
"[13] One of the promising approach for this purpose is relating local operators to a medium modification of a quarkonium via an operator product expansion (OPE).",
"We will give a basic concept and a recent result below."
],
[
"Local operator approach for quarkonia",
"The interaction between a heavy quarkonium and soft gluons was first formulated by Peskin [14].",
"The key concept is a separation scale, which is binding energy $\\epsilon $ in the case of a heavy quarkonium.",
"Regarding an exchange momentum $k$ larger than $\\varepsilon $ as hard scale while the other case as soft one, we may express a matrix element via OPE as $\\sum _{i} C_i \\langle \\mathcal {O}_i \\rangle $ where $C_i$ and $\\langle \\mathcal {O}_i \\rangle $ stand for the Wilson coefficients responsible for the hard scale $k > \\epsilon $ and expectation value of local operators for the soft scale, respectively (see Fig.",
"REF ).",
"The leading order contribution is given by dimension four gluon condensate.",
"At lower temperature than the seperation scale, one may impose all the medium effect on the change of the expectation value of the operators, which can be extracted from lattice calculations.",
"Taking the real part of the matrix element immediately leads to a formula of a mass shift, which is the second order Stark effect in QCD, as $ \\Delta m_{\\bar{Q}Q} = -\\frac{7\\pi ^2}{18}\\frac{a_0^2}{\\epsilon }\\left\\langle \\frac{\\alpha _s}{\\pi }\\Delta E^2 \\right\\rangle $ where $a_0$ is the Bohr radius of the Coulombic bound state and $\\left\\langle \\frac{\\alpha _s}{\\pi }\\Delta E^2 \\right\\rangle $ is the temperature dependent part of the chromoelectric condensate.",
"The expectation value of the dimension four gluon operators can be extracted from pressure $p$ , energy density $\\varepsilon $ and effective coupling constant $\\alpha _s^{\\text{eff}}(T)$ obtained in lattice gauge theory as $\\left\\langle \\frac{\\alpha _s}{\\pi }\\Delta E^2 \\right\\rangle &=\\frac{2}{11-\\frac{2}{3}N_f}M_0(T)+\\frac{3}{4}\\frac{\\alpha _s^{\\text{eff}}(T)}{\\pi }M_2(T)$ where $M_0$ and $M_2$ correspond to the gluonic part of the QCD trace anomaly, $\\varepsilon -3p$ and that of enthalpy density, $\\varepsilon +p$ , respectively and shown in Fig.",
"REF .",
"The rapid change of the energy density results in an abrupt increase of $\\left\\langle \\frac{\\alpha _s}{\\pi }E^2\\right\\rangle $ thus downward mass shift of a heavy quarkonium in the vicnity of the phase transition.",
"Figure: Left: Schematic diagram for OPE.",
"Right: Temperature dependence of M 0 /T 4 M_0/T^4 and M 2 /T 4 M_2/T^4extracted from pure SU(3) lattice gauge theory.",
"The above framework with separation scale $\\varepsilon $ is applicable for a deeply bounded heavy quark-antiquark system.",
"In reality, this condition might be questionable for charmonium at high temperature.",
"One can turn to the current correlation function $\\Pi (q^2)= \\int d^4 x e^{iq\\cdot x}\\left\\langle T j(x)j(0)\\right\\rangle $ in terms of OPE by going to deep Euclidean region in momentum space, $q^2 =-Q^2$ .",
"Large negative $q^2$ enables us to compute $\\Pi (q^2)$ in a perturbative manner with a pointlike current such as $j^\\mu (x) = \\bar{c}(x)\\gamma ^\\mu c(x)$ and makes convergence property better than the above case.",
"The relation to the physical quarkonia, $q^2 = m_{\\bar{Q}{Q}}^2$ , is kept through the dispersion relation.",
"After the Borel transformation, which optimizes the dispersion relation such that the integral over the energy is dominated by the lowest resonance, the dispersion relation for the transformed correlator $M(\\nu )$ reads $\\mathcal {M}(\\nu ) = \\int dx^2 e^{-\\nu x^2}\\rho (2m_Qx, T)$ where $\\nu =4m_Q^2/M^2$ and $M$ is the so-called Borel mass parameter.",
"QCD sum rules [17] gives a systematic framework to extract spectral properties from the current correlation function and dispersion relation also at finite temperature by introducing medium dependent condensates unless the typical energy scale of the medium exceeds the separation scale.",
"[18] With a Breit-Wigner type pole ansatz for the physical spectral function, one can derive a constraint on the spectral change of a quarkonium at finite temperature [19], [7].",
"Recently, Gubler and Oka proposed to apply MEM to QCD sum rules.",
"[20] In this method, we do not have to assume a specfic functional form on the spectral function.",
"Futhermore, compared to lattice calculations based on Eq.",
"(REF ), we can take as many points as possible and the dispersion relation does not have a temperature dependence other than the spectral function itself.",
"Figure: Spectral function obtained from QCD sum rules with maximumentropy method.",
"Left: J/ψJ/\\psi , Right: Υ\\Upsilon .Figure REF shows spectral functions of $J/\\psi $ (left)[21] and $\\Upsilon $ (right)[22] obtained by the QCD sum rule+MEM approach.",
"Although resolution of the width in the lowest peak is not sufficient, one sees how the peak dissolves as temperature increases.",
"The drastic change around $T_c$ seen in the case of $J/\\psi $ is consistent with previous sum rule calculations [19], [7], while $\\Upsilon $ hardly reflects the phase transition but exhibits sizable modification above $2T_c$ ."
],
[
"Summary and outlook",
"Several approaches have revealed that a heavy quarkonium has a width at finite temperature induced by the medium effect, especially due to gluonic dissociations.",
"Utilizing OPE, we have shown that charmonia are sensitive to the phase transition while bottomonium are modified at much higher temperatures.",
"Since bottomonia will be more appropriate for treatment with effective field theories, we expect theoretical attempts could meet together to unveil the interaction of the heavy quarkonium in the deconfined medium, which can be explored in heavy ion colllisions at the LHC energies."
],
[
"Acknowledgements",
"The author is much indebt to S. H. Lee for a fruitful collaboration.",
"He is also grateful to P. Gubler, M. Oka and K. Suzuki for results shown in Fig.",
"REF .",
"This work is supported by YIPQS at Kyoto University."
]
] | 1204.1125 |
[
[
"Non-Hermitian approach for modeling of noise-assisted quantum electron\n transfer in photosynthetic complexes"
],
[
"Abstract We model the quantum electron transfer (ET) in the photosynthetic reaction center (RC), using a non-Hermitian Hamiltonian approach.",
"Our model includes (i) two protein cofactors, donor and acceptor, with discrete energy levels and (ii) a third protein pigment (sink) which has a continuous energy spectrum.",
"Interactions are introduced between the donor and acceptor, and between the acceptor and the sink, with noise acting between the donor and acceptor.",
"The noise is considered classically (as an external random force), and it is described by an ensemble of two-level systems (random fluctuators).",
"Each fluctuator has two independent parameters, an amplitude and a switching rate.",
"We represent the noise by a set of fluctuators with fitting parameters (boundaries of switching rates), which allows us to build a desired spectral density of noise in a wide range of frequencies.",
"We analyze the quantum dynamics and the efficiency of the ET as a function of (i) the energy gap between the donor and acceptor, (ii) the strength of the interaction with the continuum, and (iii) noise parameters.",
"As an example, numerical results are presented for the ET through the active pathway in a quinone-type photosystem II RC."
],
[
"Introduction",
"Nature has evolved photosynthetic organisms to be extremely complex bio-engines that capture visible light in their peripheral light-harvesting complexes (LHCs) and transfer excited-state energy (as excitons) through the proximal LHC of photosystem II (PSII) and photosystem I (PSI) to the RCs.",
"The primary charge separation occurs in the RC (which works as a battery), leading to the formation of an electrochemical gradient [1], [2], [3], [4].",
"During the past two decades, crystallographic structures for many photosynthetic complexes (PCs), including the LHCs and RCs, have been determined to a resolution of 2.5-3 Å [5], [6], [7].",
"(See also references therein.)",
"Like all engines, PCs operate in a thermal environment at ambient temperature and in the presence of external “classical\" sources of noise [8], [9], [10], [11], [12], [13], [14].",
"In spite of this, recent experiments based on two-dimensional laser-pulse femtosecond photon echo spectroscopy revealed a long-lived exciton-electron quantum coherence in PCs such as the Fenna-Matthews-Oslov (FMO) and marine algae [15], [7], [16].",
"Mainly, this occurs because the dynamics of the ET is so rapid (some picoseconds) that the thermal fluctuations and external noise are unable to significantly destroy quantum coherence.",
"Consequently, the exciton/electron dynamics in LHCs-RCs must be described using quantum-mechanical methods [17], [18], [19], [20], [21].",
"(See also references therein.)",
"An important consequence of this is the high ET efficiency of the peripheral antennae complexes (close to 100 %).",
"As an example, Fleming and colleagues [17] have modeled quantum coherence effects in the bacterial FMO LHC by (i) using a tight-binding model (TBM) for exciton dynamics and (ii) introducing an empirical thermal relaxation function having an exponential form, in order to describe the high efficiency of exciton energy transport.",
"Usually, in the TBM the exciton/electron ET dynamics in LHCs-RCs is described in the single exciton/electron approximation (due to limited sunlight intensities), with $N$ ( $N=7$ for the FMO in [17]) being the total number of discrete pigments/sites.",
"(Note that more complicated models which account for exciton and charged states can also be used [21].)",
"In this case, each pigment, $n \\,(n=1,\\dots ,N)$ , is represented by a two-level system with states $|0_n\\rangle $ (unoccupied) and $|1_n\\rangle $ (occupied).",
"The total Hamiltonian is $H_{tot} = H_e + H_{ph} + H_{el-ph} $ [17].",
"The first term is the Hamiltonian of exciton/electron states of the pigments in the site representation: $H_e= \\sum _1^N E_n |n\\rangle \\langle n| + \\sum ^N_{m\\ne n} V_{mn}|m\\rangle \\langle n | $ , where $E_n$ is the site energy, and $V_{mn}$ denotes the coupling between the $n$ -th and $m$ -th pigments.",
"The term $H_{ph}$ describes the thermal phonons provided by the protein environment, and the third term describes the interaction between pigments and the thermal phonons.",
"It was numerically demonstrated in [17], that in the FMO complex, quantum coherent ET is an adequate way to describe the energy transport dynamics.",
"Usually, there are two different approaches which are used to describe the influence of the protein environment on the ET.",
"One is based on the thermal environment [9].",
"In this case, the environment acts self-consistently on the electron system and, in combination with the transition amplitudes between sites/pigments, provides the ET rates between the sites and the Gibbs equilibrium state for the LHC-RC subsystem.",
"The other approach is based on considering an external “classical\" noise [20] provided by the protein vibrations.",
"This approach results in a transfer rate for the electron, but does not lead to Gibbs equilibrium states.",
"The choice of approach depends on the concrete experimental situation which the theoretical model is intended to describe.",
"In this paper we use the second approach, modeling the noise by an ensemble of fluctuators [14].",
"To simplify our description, we introduce a set of fluctuators with fitting parameters (boundaries of switching rates between relatively slow and fast fluctuators), which allows us to build a desired spectral density of noise in a wide range of frequencies.",
"In particular, the spectral density of noise, used in this paper, includes the components of white noise, $1/f$ noise, and high-frequency noise.",
"We demonstrated in [14] that this approach successfully described the experiments [22] on the quantum dynamics of superconducting qubits.",
"Here we consider the simplest model of ET in a quinone-type active pathway of the PSII RC.",
"Our model includes two protein cofactors (donor and acceptor) with discrete energy levels, with the acceptor being embedded in a third protein pigment (sink) that has a continuous energy spectrum.",
"In [19] an additional sink reservoir was empirically introduced in order to describe the high ET efficiency in the FMO complex.",
"A sink reservoir was also introduced phenomenologically in [20] to describe the asymmetry of two branches of the ET in the photosynthetic PSII RC, and in [23] to describe the dynamics of excitons in photosynthetic systems.",
"In our model, the influence of the sink is described self-consistently, using a non-Hermitian Hamiltonian approach.",
"We include the interactions between the donor and acceptor, and between the acceptor and the sink.",
"The classical noise acts only between the donor and acceptor.",
"We analyze the dynamics and the efficiency of the ET as a function of the energy gap between donor and acceptor, the strength of interaction with continuum, and the noise parameters.",
"We calculate explicitly the ET rate and efficiency as a function of parameters.",
"We demonstrate the regimes in which noise assists the ET efficiency (in particular, in which the influence of noise significantly increases the efficiency of the ET from the “donor-acceptor\" subsystem to the sink).",
"Our paper is organized as follows.",
"In Section II, using the Feshbach projection method, we introduce an effective non-Hermitian Hamiltonian to describe the RC consisting of the donor and acceptor coupled to the sink.",
"In Section III, we study the dynamics of the electron transfer without noise.",
"In Section IV, we study the decoherence effects caused by the classical noise on the ET efficiency.",
"In the Section V, we discuss the obtained results.",
"In the Appendices some important formulae are presented."
],
[
"Model description",
"We consider a model (“building block\" of the LHCs-RCs) of the RC with three sites (protein pigments): the first site, $|d\\rangle $ , is the electron donor (with the energy $E_d$ ), the second site, $|a\\rangle $ , is the electron acceptor (with the energy $E_a$ ), and the third site is a “sink\", with a continuous spectrum.",
"We assume that the acceptor is coupled to the sink, which we first model by a large number of discrete and nearly degenerate energy levels, $N \\gg 1$ (Fig.",
"REF ).",
"Figure: A reaction center consisting of the donor and acceptor discrete energy levels, with the acceptor coupled to a sink reservoir with a continuous spectrum.The Hamiltonian of this system can be written as, $H_t = E_d|d\\rangle \\langle d|+E_a|a\\rangle \\langle a| + \\frac{V}{2}(|d\\rangle \\langle a|+ |a\\rangle \\langle d|) \\nonumber \\\\+\\sum ^N_{n=1}E_n |n\\rangle \\langle n | + \\sum ^N_{m=1} \\big ( V_{am}|a\\rangle \\langle m | + V_{ma}|m\\rangle \\langle a |\\big ).$ The total Hilbert space can be divided into two orthogonal subspaces generated by two projection operators, $P= |d\\rangle \\langle d|+ |a\\rangle \\langle a|$ and $Q= \\sum _1^N(|n\\rangle \\langle n|)$ , where the $P$ -space is associated with the donor-acceptor levels and the $Q$ -space is associated with the sink.",
"These projection operators have the following properties: $P+Q=1$ , $P^{2} =P$ , $Q^{2} =Q$ and $PQ=QP=0$ .",
"Then, using the Feshbach projection method [24], [25], [26], [27], we obtain the effective non-Hermitian Hamiltonian that describes only the “donor-acceptor\" subsystem, $\\tilde{\\mathcal {H}}= E_d|d\\rangle \\langle d|+ (E_a+ \\Delta (E) - \\frac{i}{2}\\Gamma _a(E))|a\\rangle \\langle a|+\\frac{V}{2}(|d\\rangle \\langle a|+ |a\\rangle \\langle d|),$ where $\\Delta (E) - \\frac{i}{2}\\Gamma _a(E) = \\sum _n \\frac{|V_{an}|^2}{E - E_n + i\\delta }.$ To proceed further, we assume that the sink is sufficiently dense, so that one can perform an integration instead of a summation.",
"Then we have, $\\Delta (E) - \\frac{i}{2}\\Gamma _a(E) = \\int \\frac{|V_{an}|^2 g(E_n)dE_n}{E - E_n + i\\delta },$ where $g(E_n)$ is the density of states of the sink.",
"One can show that [28] $\\Delta (E) = {\\mathcal {P}} \\int \\frac{|V_{an}|^2g(E_n)dE_n}{E - E_n}, \\\\\\Gamma _a(E) = 2\\pi \\int |V_{an}|^2g(E_n)\\delta (E - E_n) dE_n,$ where $\\mathcal {P}$ denotes the principal value of the integral.",
"The exact dynamical evolution of the whole quantum system (RC) is described by the Schrödinger equation (we set $\\hbar =1$ ), $i\\frac{\\partial \\psi (t)}{\\partial t}= H_t\\psi (t).$ We assume that at $t=0$ the system is populated in the $P$ -space.",
"If the Q-space represents a smooth continuum (which is assumed below) one can neglect the dependence of $\\Delta (E)$ and $\\Gamma _a(E)$ on $E$ .",
"Denoting these functions as $\\Delta $ and $\\Gamma _a$ , one can find that the dynamics of the donor-acceptor (intrinsic) states can be described by the following Schrödinger equation with the effective non-Hermitian Hamiltonian, $\\tilde{\\mathcal {H}}$ : $i\\frac{\\partial \\psi _{p} (t)}{\\partial t}= \\tilde{\\mathcal {H}} \\psi _{p} (t),$ where $\\psi _{p} (t)=P\\psi (t)$ .",
"Further, it is convenient to rewrite $\\tilde{\\mathcal {H}}$ as $ \\tilde{\\mathcal {H}}= {\\mathcal {H}}- i \\mathcal {W}$ , where ${\\mathcal {H}} = \\varepsilon _d|d\\rangle \\langle d|+ \\varepsilon _a|a\\rangle \\langle a|+ \\frac{V}{2}(|d\\rangle \\langle a|+ |a\\rangle \\langle d|)$ is the dressed donor-acceptor Hamiltonian, $\\mathcal {W} = ({1}/{2})\\Gamma _a|a\\rangle \\langle a|$ , with $\\varepsilon _d=E_d$ and $ \\varepsilon _a= E_a+ \\Delta $ .",
"We define $\\rho _t(t)$ to be the density matrix that satisfies the conventional equation of motion with the total Hamiltonian, $H_t$ : $i\\dot{ \\rho }_t = [H_t,\\rho ]$ .",
"Next, we introduce the projected density matrix as $\\rho (t) = P\\rho _t(t) P$ .",
"Then, one can show that $\\rho (t)$ satisfies the Liouville equation, $i \\dot{ \\rho } = [\\mathcal {H},\\rho ] - i\\lbrace \\mathcal {W},\\rho \\rbrace ,$ where $\\lbrace \\mathcal {W},\\rho \\rbrace = \\mathcal {W}\\rho +\\rho \\mathcal {W}$ .",
"Assume now that the quantum system under consideration interacts with the environment.",
"We use the reduced density matrix approach to describe this interaction.",
"To include into the description of the system both processes of decoherence and tunneling to the continuum, we introduce the following generalized master equation, $ i\\dot{ \\rho } = [\\mathcal {H},\\rho ] + {\\mathcal {L}}\\rho - i\\lbrace \\mathcal {W},\\rho \\rbrace $ , where $\\mathcal {H}$ is the dressed Hamiltonian, and the Lindblad operator, $\\mathcal {L}$ , describes the coupling to the environment.",
"The commutator of the density operator, $\\rho $ , with the Hamiltonian, $\\mathcal {H}$ , is the coherent part of evolution, and the remaining part corresponds to the decoherence process causes by the interaction with the environment."
],
[
"Tunneling to the sink",
"We consider here the quantum dynamics of the ET from the donor $|d\\rangle $ ($|1\\rangle $ ) to the acceptor $|a\\rangle $ ($|2\\rangle $ ) coupled to the sink.",
"We assume that the acceptor is coupled to the $N$ -level sink reservoir and that the corresponding Hilbert subspace is dense and smooth.",
"For description of the tunneling from the acceptor to the sink we use the Feshbach projection method described above.",
"This yields the following effective non-Hermitian Hamiltonian: $\\tilde{ {\\mathcal {H}}}= \\frac{\\tilde{\\lambda }_0}{2} \\left(\\begin{array}{cc} 1& 0\\\\ 0& 1 \\end{array}\\right) + \\frac{1}{2} \\left(\\begin{array}{cc} {\\varepsilon + i\\Gamma }&V\\\\V& {-\\varepsilon -i\\Gamma } \\end{array}\\right),$ where $\\tilde{\\lambda }_0= \\varepsilon _{1} +\\varepsilon _{2} -i \\Gamma $ , $\\varepsilon = \\varepsilon _{1} -\\varepsilon _{2}$ ($\\varepsilon _n$ is the renormalized energy), $\\Gamma = \\Gamma _a/2 $ , with $\\Gamma _a$ being the relaxation rate from the acceptor to the sink.",
"Region of parameters.",
"The model involves various parameters, which are only partially known.",
"For concreteness of the numerical simulations, our choice of the parameters is based on the data taken for the ET through the active pathway in the quinone-type of the photosystem II RC [29] (in the units $\\hbar =1$ ): $\\varepsilon = 60 \\rm ps^{-1}$ and $ 10 \\rm ps^{-1} < V < 40 \\rm ps^{-1}$ .",
"The parameter $\\Gamma $ is varied in the interval: $ 1 \\rm ps^{-1}< \\Gamma < 5 \\rm ps^{-1}$ .",
"But also other values of parameters, $\\varepsilon $ and $V$ , are used in our numerical simulations.",
"(Note, that the values of parameters in energy units can be obtained by multiplying our values by $\\hbar \\approx 6.58\\times 10^{-13}\\rm meVs$ .",
"For example, $\\varepsilon = 60\\; \\rm ps^{-1}\\approx 40\\rm meV$ .)",
"In what follows we assume that initially the quantum system occupies the upper level (donor), $\\rho _{11}(0)=1$ ($\\rho _{22}(0)=0$ ).",
"Then, for the diagonal component of the density matrix the solution of the Liouville equation (REF ) is given by (for details see Appendix A), $\\rho _{11}(t) = {e^{-\\Gamma t}} \\bigg |\\Big (\\cos \\frac{\\Omega t}{2} - i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big )\\bigg |^2, \\quad \\rho _{22}(t) = {e^{-\\Gamma t}} \\bigg |\\sin \\theta \\sin \\frac{\\Omega t}{2}\\bigg |^2,$ where $\\Omega = \\sqrt{V^2 +(\\varepsilon + i\\Gamma )^2}$ is the complex Rabi frequency, $\\cos \\theta = (\\varepsilon + i\\Gamma )/\\Omega $ , and $\\sin \\theta = V/\\Omega $ .",
"Figure: Left panel: The time dependence of the population of the donor site.Right panel: ET efficiency.",
"The parameters are the following: blue line (Γ=1 ps -1 \\Gamma =1\\; \\rm ps^{-1}, V=10 ps -1 V = 10 \\rm ps^{-1} ), black line (Γ=1 ps -1 \\Gamma =1 \\rm ps^{-1}, V=20 ps -1 V = 20 \\rm ps^{-1} ), green line (Γ=5 ps -1 \\Gamma =5 \\rm ps^{-1}, V=10 ps -1 V = 10 \\rm ps^{-1} ), red line (Γ=5 ps -1 \\Gamma =5 \\rm ps^{-1}, V=20 ps -1 V = 20 \\rm ps^{-1} ).",
"In all cases ε=60 ps -1 \\varepsilon = 60 \\rm ps^{-1} .The ET efficiency can be defined as the integrated probability of trapping the electron in the sink [18], [30], $\\eta (t) = 2\\Gamma \\int _0^t \\rho _{22}(\\tau )d \\tau .$ Setting $\\Omega = \\Omega _1 +i \\Omega _2$ and performing the integration, we obtain for the ET efficiency, $\\eta (t) = 1- \\frac{ e^{-\\Gamma t}}{\\Gamma (\\Omega ^2_1 +\\Omega ^2_2)}\\big ((\\Gamma ^2 + \\Omega _1^2) (\\Gamma \\cosh {\\Omega _2 t} + \\Omega _2\\sinh {\\Omega _2 t}) \\nonumber \\\\- (\\Gamma ^2 - \\Omega _2^2) (\\Gamma \\cos {\\Omega _1 t} -\\Omega _1\\sin {\\Omega _1 t} )\\big ).$ This yields the following large-time asymptotic behavior: $\\eta (t) \\sim 1- \\frac{(\\Gamma - \\Omega _2 )(\\Gamma ^2 + \\Omega _1^2)}{2\\Gamma (\\Omega ^2_1 +\\Omega ^2_2)} e^{-(\\Gamma +\\Omega _2)t} .$ The numerical results are presented in Fig.",
"REF .",
"As one can see, for these values of parameters, and without the action of noise, the ET efficiency approaches a value close to 1 for relatively large times, $t>150$ ps.",
"Let us consider now the flat redox potential, $\\varepsilon =0$ .",
"From the relation $\\Omega _1\\Omega _2 = \\varepsilon \\Gamma $ , it follows that for $\\varepsilon = 0$ there are two possibilities: (i) $\\Omega _1=0$ , $\\Omega _2 = \\sqrt{\\Gamma ^2 - V^2}$ ($V < \\Gamma )$ ; and (ii) $\\Omega _2 = 0$ , $\\Omega _1 = \\sqrt{ V^2 - \\Gamma ^2}$ ($V > \\Gamma )$ .",
"Using these results we obtain, $\\eta (t) =\\left\\lbrace \\begin{array}{ll}1- \\displaystyle \\frac{ e^{-\\Gamma t}}{\\Omega ^2_1 }\\big ((\\Gamma ^2(1- \\cos {\\Omega _1 t}) + \\Omega _1 ( \\Omega _1- \\Gamma \\sin {\\Omega _1 t} )\\big ), & V > \\Gamma \\\\1- \\bigg (1+ \\Gamma t + \\bigg (\\displaystyle \\frac{\\Gamma t}{2}\\bigg )^2\\bigg ) e^{-\\Gamma t}, & V = \\Gamma \\\\1- \\displaystyle \\frac{ e^{-\\Gamma t}}{\\Omega ^2_2 }\\big ((\\Gamma ^2(\\cosh {\\Omega _2t}-1) + \\Omega _2 ( \\Omega _2 +\\Gamma \\sinh {\\Omega _1 t} )\\big ), & V < \\Gamma \\end{array}\\right.$ This yields the following asymptotic behavior for the ET efficiency, $\\eta (t)$ ($\\Gamma t \\gg 1$ ): $\\eta (t) \\sim \\left\\lbrace \\begin{array}{ll}1- \\displaystyle \\frac{\\Gamma ^2}{2\\Omega _1^2 } e^{-\\Gamma t}, & V > \\Gamma \\\\1- \\bigg (\\displaystyle \\frac{\\Gamma t}{2}\\bigg )^2 e^{-\\Gamma t}, & V = \\Gamma \\\\1- \\displaystyle \\frac{\\Gamma ^2 }{2\\Omega _2^2 }e^{-(\\Gamma - \\Omega _2 )t} , & V < \\Gamma \\end{array}\\right.$ Comparing Eqs.",
"(REF ) - (REF ), we conclude that the highest ET efficiency is obtained for the flat redox potential ($\\varepsilon =0$ ), and $V > \\Gamma $ ."
],
[
"Quantum evolution in the vicinity of the exceptional point",
"For the Hermitian Hamiltonian, the coalescence of eigenvalues results in different eigenvectors and the related degeneracy, referred to as a “conical intersection\", is known also as a “diabolic point\" [31].",
"However, in a quantum mechanical system governed by a non-Hermitian Hamiltonian merging not only of eigenvalues of the Hamiltonian but also of the associated eigenvectors can occur.",
"In this case, the point of coalescence is called an “exceptional point\" (EP).",
"At the EP, the eigenvectors merge, forming a Jordan block.",
"(For a review and references, see, e.g., [32].)",
"In the effective two-level system under consideration, the EP is defined by equation $\\Omega =0$ .",
"This yields $\\varepsilon =0$ and $V^2 - \\Gamma ^2 =0$ .",
"To study tunneling to the sink near a degeneracy, we assume the flat dressed redox potential, $\\varepsilon = 0$ .",
"Then, there are two different regimes of the ET depending on the relative values of $V$ and $\\Gamma $ .",
"For $V > \\Gamma $ , we have a coherent tunneling process (with oscillating probabilities, see Fig.",
"REF ), Figure: Time dependence of site populations in the vicinity of the EP (red line) for the flat redox potential (ε=0\\varepsilon =0).",
"The parameters are chosen as the following: blue line (Γ=1 ps -1 \\Gamma =1\\; \\rm ps^{-1}, V=5 ps -1 V = 5 \\rm ps^{-1} ), black line (Γ=1 ps -1 \\Gamma =1 \\rm ps^{-1}, V=10 ps -1 V = 10 \\rm ps^{-1} ), green line (Γ=5 ps -1 \\Gamma =5 \\rm ps^{-1}, V=10 ps -1 V = 10 \\rm ps^{-1} ), red line (Γ=5 ps -1 \\Gamma =5 \\rm ps^{-1}, V=5 ps -1 V = 5 \\rm ps^{-1} ) corresponds to the exceptional point.Figure: The ET efficiency in the vicinity of the EP (Γ=5 ps -1 \\Gamma = 5 \\rm ps^{-1} ).",
"Black line (V=20 ps -1 V = 20 \\rm ps^{-1} , ε=0 ps -1 \\varepsilon = 0 \\rm ps^{-1} ), blue line (V=10 ps -1 V = 10 \\rm ps^{-1} , ε=0 \\varepsilon = 0 ).",
"Red line: tunneling at the EP ( V=5 ps -1 V = 5 \\rm ps^{-1} , ε=0 \\varepsilon = 0 ).",
"Black dashed line ( V=2.5 ps -1 V = 2.5 \\rm ps^{-1} , ε=20 ps -1 \\varepsilon = 20 \\rm ps^{-1} ).",
"Green dashed line ( V=2.5 ps -1 V = 2.5 \\rm ps^{-1} , ε=0 \\varepsilon = 0 ).",
"Orange dashed line (V=2.5 ps -1 V = 2.5 \\rm ps^{-1} , ε=10 ps -1 \\varepsilon = 10 \\rm ps^{-1} ).$\\rho _{11} = e^{-\\Gamma t}\\Big (\\cos \\frac{\\Omega _0 t}{2}+ \\frac{\\Gamma }{\\Omega _0}\\sin \\frac{\\Omega _0 t}{2}\\Big )^2 , \\quad \\rho _{22}= e^{-\\Gamma t} \\frac{V^2}{\\Omega _0^2}\\sin ^2\\frac{\\Omega _0 t}{2},$ where $\\Omega _0= |V^2 - \\Gamma ^2|^{1/2}$ denotes the Rabi frequency.",
"On the other hand, for $V < \\Gamma $ , the tunneling becomes incoherent, without probability oscillations, $\\rho _{11} = e^{-\\Gamma t}\\Big (\\cosh \\frac{\\Omega _0 t}{2}+ \\frac{\\Gamma }{\\Omega _0}\\sinh \\frac{\\Omega _0 t}{2}\\Big )^2 , \\quad \\rho _{22}= e^{-\\Gamma t} \\frac{V^2}{\\Omega _0^2}\\sinh ^2\\frac{\\Omega _0 t}{2}.$ At the EP, $\\Omega _0 = 0$ , and both regimes coincide.",
"(See Fig.",
"REF , red curve.)",
"In this case, we have the following solutions for the probabilities, $\\rho _{11}(t) = {e^{-\\Gamma t}} \\bigg (1 +\\frac{\\Gamma t}{2}\\bigg )^2, \\quad \\rho _{22}(t) = {e^{-\\Gamma t}} \\bigg (\\frac{\\Gamma t}{2}\\bigg )^2.$ The results of numerical simulations of the ET efficiency in the vicinity of the EP are shown in Fig.",
"REF .",
"One can see that, for the chosen parameters, the ET efficiency can approach a value close to 1 for short times, $\\sim 2$ ps.",
"Note that the coherent tunneling regime ($V>\\Gamma $ ) is more effective for approaching a high ET efficiency for short times.",
"(See Fig.",
"REF , black and blue curves.)"
],
[
"Noise-assisted electron transfer",
"In this section, we consider ET from the donor, $|1\\rangle $ , to the acceptor, $|2\\rangle $ , coupled to the sink, in the presence of classical noise.",
"Then, the effective non-Hermitian Hamiltonian (REF ) takes the form $ \\tilde{\\mathcal {H}}= \\sum _n \\varepsilon _n |n\\rangle \\langle n |+ \\sum _{m,n} \\lambda _{mn}(t))|m\\rangle \\langle n | + \\frac{V}{2}\\sum _{m \\ne n} |m\\rangle \\langle n | - i\\Gamma |2\\rangle \\langle 2|, \\quad m,n = 1,2,$ where $\\lambda _{mn}(t))$ describes the noise.",
"In our approach, we use a spin-fluctuator model of noise with the number of fluctuators, ${{\\mathcal {N}}} \\gg 1$ [33], [34], [14].",
"The diagonal matrix elements of noise, $\\lambda _{nn}$ , are responsible for decoherence, and the off-diagonal matrix elements, $\\lambda _{mn}$ ($ m \\ne n$ ), lead to the relaxation processes.",
"The approximate equations of motion for the average diagonal components of the density matrix are given by (for details see Appendix B) $ \\frac{d}{dt}{\\langle {\\rho }}_{11}(t)\\rangle =-{\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) + {\\mathcal {O}}(|V|^4), \\\\\\frac{d}{dt}{\\langle {\\rho }}_{22}(t)\\rangle ={\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) - 2\\Gamma \\langle {\\tilde{\\rho }}_{22}(t)\\rangle + {\\mathcal {O}}(|V|^4),$ where the average $\\langle \\; \\rangle $ is taken over the random process describing noise, and ${\\mathfrak {R}}(t) =\\frac{1}{4}\\int _0^t e^{-\\Gamma (t- t^{\\prime })}\\big (\\big \\langle {\\tilde{V}}(t){\\tilde{V}}(t^{\\prime }) \\big \\rangle + \\big \\langle {\\tilde{V}}(t^{\\prime }){\\tilde{V}}(t)\\big \\rangle \\big )dt^{\\prime }.$ The model of noise.",
"In the following, we restrict ourselves to consider only diagonal noise effects, assuming that the noisy environment is the same for the donor and acceptor sites (collective noise).",
"Then, one can write $\\lambda _1(t) = g_1 \\xi (t) $ and $\\lambda _2(t) = g_2 \\xi (t) $ , where $\\xi (t)$ is a random variable describing the stationary noise with the correlation function, $\\chi (t-t^{\\prime })=\\langle \\xi (t)\\xi (t^{\\prime })\\rangle $ , and $g_{1,2}$ are the interaction constants.",
"We describe the noise by a spin-fluctuator model with the number of fluctuators, ${\\mathcal {N}} \\gg 1$ , with the correlation function, $\\chi (\\tau )$ , given by [14] $\\chi (\\tau ) = \\sigma ^2 A\\Big (E_1(2\\gamma _m \\tau ) - E_1(2\\gamma _c \\tau ) \\Big ), \\quad \\tau = |t-t^{\\prime }|,$ where $E_n(z)$ denotes the Exponential integral [35], $A= 1/\\ln (\\gamma _c/\\gamma _m)$ and $\\chi (0)= \\sigma ^2$ .",
"The spectral density of the noise, defined as $S(\\omega ) = \\frac{1}{\\pi }\\int \\limits _{0}^{\\infty }\\chi (\\tau ) \\cos (\\omega \\tau ) d\\tau ,$ is given in [14], $S(\\omega ) = \\frac{\\sigma ^2 }{\\pi \\omega \\ln (\\gamma _c/\\gamma _m)}\\bigg ( \\arctan \\Big (\\frac{\\omega }{2\\gamma _m}\\Big )- \\arctan \\Big (\\frac{\\omega }{2\\gamma _c}\\Big )\\bigg ).$ This yields the following asymptotic behavior of $S(\\omega )$ : $S(\\omega ) \\approx \\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{\\sigma ^2 }{2\\pi \\gamma _m \\ln (\\gamma _c/\\gamma _m) }\\bigg (1- \\frac{\\gamma _m}{\\gamma _c}\\bigg ),& \\omega \\ll 2\\gamma _m ,\\\\\\\\\\displaystyle \\frac{\\sigma ^2 }{2 \\omega \\ln (\\gamma _c/\\gamma _m)},& 2\\gamma _m \\ll \\omega \\ll 2\\gamma _c, \\\\\\\\\\displaystyle \\frac{2\\sigma ^2 \\gamma _c (1-\\gamma _m/\\gamma _c)}{\\pi \\omega ^2 \\ln (\\gamma _c/\\gamma _m)},& \\omega \\gg 2\\gamma _c,\\end{array}\\right.$ where $\\gamma _m$ and $\\gamma _c$ ($\\gamma _m\\ll \\gamma _c$ ) indicate the boundaries of the switching rates in the ensemble of random fluctuators.",
"As one can see from Eqs.",
"(REF ), for $\\omega \\ll 2\\gamma _m $ the spectral density of noise, $S(\\omega )$ , describes the white noise.",
"In the interval of frequencies, $2\\gamma _m \\ll \\omega \\ll 2\\gamma _c$ , one has the $1/f$ noise: $S\\sim 1/f$ ($f=\\omega /2\\pi $ ).",
"And for $ \\omega \\gg 2\\gamma _c$ , we obtain the Lorentzian spectrum.",
"Choice of parameters.",
"The correlation function includes, besides the amplitude, $\\sigma $ , two fitting parameters: $\\gamma _m$ and $\\gamma _c$ .",
"Taking into account available theoretical and experimental data [36], [37], [38], we have chosen in our numerical simulations the following parameters: $ 2\\gamma _m = 10^{-4}\\rm ps^{-1} $ , $ 2\\gamma _c = 1\\rm ps^{-1} $ .",
"Note that as our results demonstrate, a decrease of the left boundary, $\\gamma _m$ , even up to $\\gamma _m \\approx 1 \\rm s^{-1}$ practically does not change ET rates.",
"We also introduce the notation: $D=|g_1-g_2|$ .",
"The spectral density of noise corresponding to Eq.",
"(REF ) and its asymptotic behavior given by Eq.",
"(REF ) is presented in Fig.",
"REF .",
"Figure: The spectral density of noise σ=1\\sigma = 1 .",
"Left panel: S(f)S(f) given by exact formula, Eq.",
"() (blue line), and the asymptotic formula (): S(f)∼1/fS(f) \\sim 1/f (red line).",
"Right panel: lnS(f)\\ln S(f) as a function of frequency."
],
[
"Influence of noise on the electron transfer rates ($\\Gamma = 0$ )",
"For $\\Gamma = 0$ , we obtain the exact solution of the system (REF ) and (), $\\big \\langle {\\rho }_{11}(t)\\big \\rangle = \\frac{1}{2} + \\bigg (\\rho _{11}(0)-\\frac{1}{2}\\bigg )\\displaystyle e^{-2\\int _0^t {\\mathfrak {R}}(t^{\\prime })dt^{\\prime }}, \\\\\\big \\langle {\\rho }_{22}(t)\\big \\rangle = \\frac{1}{2} + \\bigg (\\rho _{22}(0) -\\frac{1}{2}\\bigg )\\displaystyle e^{-2\\int _0^t {\\mathfrak {R}}(t^{\\prime })dt^{\\prime }}.",
"$ It follows that, independent of the initial conditions and the nature of noise (producing decoherence or relaxation), in the limit, $t\\rightarrow \\infty $ , the presence of noise produces equal populations in the two levels.",
"(See Fig.",
"REF , left panel.)",
"The computation of the ET rate, ${\\mathfrak {R}}(t)$ , yields, ${\\mathfrak {R}}(t)= \\frac{V^2}{4} \\int _{-t}^{t}e^{i\\varepsilon \\tau }\\big \\langle e^{i\\kappa (\\tau )} \\big \\rangle d\\tau , \\quad \\kappa (\\tau ) = -D\\int _0^\\tau \\xi (t^{\\prime })dt^{\\prime }.$ To proceed further, we use the first order cumulant expansion (the Gaussian approximation) to evaluate the generating functional $\\big \\langle e^{i\\kappa (\\tau )} \\big \\rangle $ .",
"The computation gives $\\big \\langle e^{i\\kappa } \\big \\rangle = e^{-\\langle \\kappa ^2\\rangle /2} =\\exp \\bigg (- D^2\\int ^t_0 dt^{\\prime }\\int _{0}^{t^{\\prime }}dt^{\\prime \\prime }\\chi (t^{\\prime } -t^{\\prime \\prime }) \\bigg ).$ Let us assume that initially the system occupies only the upper level (donor), $\\rho _{11}(0)= 1$ .",
"Then, if $g_1 =g_2$ the solution for the diagonal components of the density matrix takes the form, $\\big \\langle {\\rho }_{11}(t)\\big \\rangle = \\frac{1}{2} + \\frac{1}{2} \\exp \\Big (-2\\frac{V^2 }{\\varepsilon ^2} \\sin ^2\\frac{\\varepsilon t}{2}\\Big ), \\quad \\big \\langle {\\rho }_{22}(t)\\big \\rangle = \\frac{1}{2} -\\frac{1}{2}\\exp \\Big ( -2\\frac{V^2 }{\\varepsilon ^2} \\sin ^2\\frac{\\varepsilon t}{2}\\Big ).$ One can see that up to the first order in the dimensionless parameter, $V^2/\\varepsilon ^2$ , the approximate solution () coincides with the exact solution given by Eq.",
"(REF ) (with $\\Gamma =0$ ).",
"In this case, the effect of the collective noise vanishes.",
"Figure: Left panel: Effects of the diagonal noise on the time dependence of the site populations: ρ 11 (t)\\rho _{11}(t) (blue line) and ρ 22 (t)\\rho _{22}(t) (red line).",
"Right panel: Blue line describes the time dependence of the ET rate, ℜ(t) {\\mathfrak {R}}(t), given by Eq.",
"(), 2γ m =10 -4 ps -1 2\\gamma _m = 10^{-4} \\rm ps^{-1} .",
"Green dashed line corresponds to 2γ m =1s -1 2\\gamma _m = 1 \\rm s^{-1}.Red line corresponds to the asymptotic formula ().",
"The parameters are: Dσ=ε=60 ps -1 D\\sigma = \\varepsilon = 60 \\rm ps^{-1} , 2γ c =1 ps -1 2\\gamma _c = 1 \\rm ps^{-1} and V=20 ps -1 V = 20 \\rm ps^{-1}.Relation to the Marcus' theory.",
"The asymptotic ET rate for $\\Gamma =0$ is defined as $ {\\mathcal {R}}= \\lim _{t\\rightarrow \\infty } {\\mathfrak {R}}(t)$ .",
"Using Eq.",
"(REF ), we obtain, $ {\\mathcal {R}}= \\frac{V^2}{4}\\int _{-\\infty }^{\\infty }dt \\exp \\big (i\\varepsilon t - \\Theta (t) \\big ),$ where $\\Theta (t) = D^2\\int ^t_0 dt^{\\prime }\\int _{0}^{t^{\\prime }}dt^{\\prime \\prime }\\chi (t^{\\prime } -t^{\\prime \\prime })$ .",
"To evaluate $\\Theta (t)$ , we use the approximation, $\\chi (t) \\approx \\chi (0)$ .",
"(Note that $\\chi (0)= \\sigma ^2$ .)",
"This yields $\\Theta (t) \\backsimeq ( \\sigma D t)^2/2$ .",
"Performing the integration over $t$ in (REF ), we obtain ${\\mathcal {R}} =\\frac{V^2}{4}\\sqrt{\\frac{2\\pi }{D^2 \\sigma ^2}}\\exp \\Bigg (-\\frac{\\varepsilon ^2}{2D^2 \\sigma ^2}\\Bigg ).$ In Fig.",
"REF , we compare the results of numerical calculations (blue line) of relaxation rate, ${\\mathfrak {R}}(t)$ described by by Eq.",
"(REF ) with the asymptotic formula (REF ).",
"One can see a good agreement of the asymptotic rate defined by Eq.",
"(REF ) with the formula (REF ).",
"In the case in which the number of thermally excited fluctuators, ${\\mathcal {N}}_T \\gg 1 $ , the dispersion, $\\sigma ^2$ , is a linear function of the temperature, so that $\\sigma ^2= P_0kT$ [34].",
"Inserting this expression into Eq.",
"(REF ), we obtain ${\\mathcal {R}}= |V_{12}|^2\\sqrt{\\frac{\\pi }{\\lambda kT}}\\exp \\Bigg (-\\frac{(E_1 -E_2)^2}{4\\lambda kT}\\Bigg ),$ where $\\lambda = {D^2 P_0}/{2 }$ and $|V_{12}|= V/2$ .",
"Comparing our result with the Marcus formula [8], one can see that the classical noise results in a large-time asymptotic of the ET rate which can be expressed in the form of the Marcus-type formula."
],
[
"Noise-assisted electron transfer in the reaction center ($\\Gamma \\ne 0$ )",
"We consider here noise-assisted ET to the sink described by the equations of motion (REF ) and () for the averaged components of the density matrix: $ \\frac{d}{dt}{\\langle {\\rho }}_{11}(t)\\rangle =- {\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) , \\\\\\frac{d}{dt}{\\langle {\\rho }}_{22}(t)\\rangle ={\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) - 2\\Gamma \\langle {\\tilde{\\rho }}_{22}(t)\\rangle ,$ where $ {\\mathfrak {R}}(t)= (1/2){V^2} \\int _{0}^{t} e^{-\\Gamma t}\\cos \\varepsilon \\tau \\big \\langle e^{i\\kappa (\\tau )} \\big \\rangle d\\tau $ .",
"Further, we use the Gaussian approximation to evaluate the generating functional: $\\big \\langle e^{i\\kappa (t)} \\big \\rangle = \\exp \\bigg (- (g_1 -g_2)^2\\int ^t_0 dt^{\\prime }\\int _{0}^{t^{\\prime }}dt^{\\prime \\prime }\\chi (t^{\\prime } -t^{\\prime \\prime }) \\bigg ).$ Performing the integration over $t$ in Eq.",
"(REF ), we obtain for $\\Gamma \\ne 0$ a generalization of asymptotic expression for $\\mathcal {R}$ given by Eq.",
"(REF ) for finite $\\Gamma $ , ${\\mathcal {R}}_\\Gamma = \\lim _{t\\rightarrow \\infty }{\\mathfrak {R}}(t)$ , as ${\\mathcal {R}}_{\\Gamma } =\\frac{V^2\\sqrt{2\\pi }}{8D\\sigma } \\Bigg ( \\exp \\bigg (\\frac{(\\Gamma + i\\varepsilon )^2}{2D^2 \\sigma ^2}\\bigg ){\\rm erfc}\\bigg (\\frac{\\Gamma + i\\varepsilon }{\\sqrt{2}D \\sigma }\\bigg ) + \\exp \\bigg (\\frac{(\\Gamma - i\\varepsilon )^2}{2D^2 \\sigma ^2}\\bigg ){\\rm erfc}\\bigg (\\frac{\\Gamma - i\\varepsilon }{\\sqrt{2}D \\sigma }\\bigg )\\Bigg ),$ where ${\\rm erf}c(z)$ denotes the complementary error function, ${\\rm erfc}(z)= 1 -{\\rm erf}(z)$ [35].",
"The dependence of ${\\mathcal {R}_\\Gamma }$ as function of the amplitude of noise, $D\\sigma $ , is presented in Fig.",
"REF .",
"Figure: The function ℛ Γ {\\mathcal {R}_\\Gamma } vs. the amplitude of noise, DσD\\sigma (ε=60 ps -1 \\varepsilon = 60 \\rm ps^{-1}, V=20 ps -1 V = 20 \\rm ps^{-1} ).",
"Black line (Γ=10 ps -1 \\Gamma = 10 \\rm ps^{-1} ), green line (Γ=5 ps -1 \\Gamma = 5 \\rm ps^{-1} ), blue line (Γ=1 ps -1 \\Gamma = 1 \\rm ps^{-1} ), red line (Γ=0 ps -1 \\Gamma = 0 \\,\\rm ps^{-1} ).Using these results and taking the initial conditions as ${\\rho }_{11}(0)=1$ , we obtain the approximate solution of Eqs.",
"(REF ) and () ${\\langle {\\rho }}_{11}(t)\\rangle \\approx \\bigg (\\frac{1}{2} - \\frac{\\Gamma }{2\\sqrt{ {\\mathcal {R}}_\\Gamma ^2 + \\Gamma ^2}}\\bigg )e^{-{\\mathcal {R}}_1t}+ \\bigg (\\frac{1}{2} + \\frac{\\Gamma }{2\\sqrt{ {\\mathcal {R}}_\\Gamma ^2 + \\Gamma ^2}}\\bigg )e^{-{\\mathcal {R}}_2t}, \\\\{\\langle {\\rho }}_{22}(t)\\rangle \\approx \\frac{ {\\mathcal {R}}_\\Gamma }{2\\sqrt{ {\\mathcal {R}}_\\Gamma ^2 + \\Gamma ^2}}\\bigg (e^{-{\\mathcal {R}}_2t} -e^{-{\\mathcal {R}}_1t} \\bigg ),$ where ${\\mathcal {R}}_{1,2} = {\\mathcal {R}}_\\Gamma + \\Gamma \\pm \\sqrt{ {\\mathcal {R}}_\\Gamma ^2 + \\Gamma ^2} $ .",
"Inserting () into Eq.",
"(REF ), to we obtain for the ET efficiency $\\eta (t) = 1- e^{-\\frac{({\\mathcal {R}}_1 + {\\mathcal {R}}_2)t}{2}}\\bigg ( \\cosh \\frac{({\\mathcal {R}}_1- {\\mathcal {R}}_2)t}{2} + \\frac{{\\mathcal {R}}_1 + {\\mathcal {R}}_2}{{\\mathcal {R}}_1 - {\\mathcal {R}}_2} \\sinh \\frac{({\\mathcal {R}}_1- {\\mathcal {R}}_2)t}{2} \\bigg ).$ This yields the following asymptotic behavior of the ET efficiency $\\eta (t) \\approx 1- \\frac{{\\mathcal {R}}_1 }{{\\mathcal {R}}_1 - {\\mathcal {R}}_2} e^{- {\\mathcal {R}}_2 t}.$ As can be seen from Eq.",
"(REF ) there are two ET rates, ${\\mathcal {R}}_1$ and ${\\mathcal {R}}_2$ .",
"The asymptotic behavior of ET efficiency, $\\eta (t)$ , is defined by the lowest ET rate, ${\\mathcal {R}}_2$ .",
"Figure: The time dependence of the site population (left panel) and the ET efficiency (right panel) in the presence of noise.",
"Left panel: (i) V=20 ps -1 V= 20 \\rm ps^{-1}, blue and black lines correspond to, ρ 11 (t)\\rho _{11}(t) and ρ 22 (t)\\rho _{22}(t) , respectively; (ii) V=40 ps -1 V= 40 \\rm ps^{-1}, red line corresponds to ρ 11 (t)\\rho _{11}(t) , and green line corresponds to ρ 22 (t)\\rho _{22}(t) .",
"Right panel: The ET efficiency, η(t)\\eta (t), in the presence of noise.",
"Green line: (V=20 ps -1 V=20 \\rm ps^{-1}, Dσ=30 ps -1 D\\sigma = 30 \\rm ps^{-1}); red line: (V=20 ps -1 V=20 \\rm ps^{-1}, Dσ≈60 ps -1 D\\sigma \\approx 60 \\rm ps^{-1}); black line: (V=20 ps -1 V=20 \\rm ps^{-1}, Dσ=120 ps -1 D\\sigma = 120 \\rm ps^{-1}); blue line: (V=10 ps -1 V=10 \\rm ps^{-1}, Dσ=60 ps -1 D\\sigma = 60 \\rm ps^{-1}).",
"In all cases: ε=60 ps -1 ,Γ=1 ps -1 \\varepsilon = 60 \\rm ps^{-1}, \\Gamma = 1\\rm ps^{-1} , 2γ m =10 -4 ps -1 2\\gamma _m = 10^{-4} \\rm ps^{-1} and 2γ c =1 ps -1 2\\gamma _c = 1 \\rm ps^{-1} .In Fig.",
"REF , we present the results of numerical simulations for $\\Gamma =1 \\; \\rm ps^{-1}$ , and for a given sharp redox potential, $\\varepsilon =60 \\; \\rm ps^{-1}$ .",
"Red line corresponds to the amplitude of noise which was optimized by using the modified Marcus-type formula given by Eq.",
"(REF ).",
"(See Fig.",
"REF .)",
"As one can see from a comparison of the results presented in Fig.",
"REF (for a sharp redox potential) and Fig.",
"REF , influence of noise with amplitudes near to optimal value significantly accelerates the ET to the sink."
],
[
"Conclusions",
"In this paper, we model quantum electron transfer dynamics in a photosynthetic reaction center consisting of three elementary pigment units.",
"Two of them, a donor and an acceptor, are represented by localized sites of protein pigments with discrete energy levels.",
"The donor interacts with the acceptor through the corresponding matrix element.",
"The third protein pigment (sink) has a continuous energy spectrum, and is described by two parameters: its density of states and its strength of interaction with the acceptor.",
"The sink is described self-consistently, by using the Feshbach projection method on the “donor-acceptor\" intrinsic states, within a non-Hermitian Hamiltonian approach.",
"We apply our results to the quantum dynamics of the electron transfer in the active branch of the quinone-type PSII reaction center.",
"The collective external noise produced by the environment of the proteins acts on the “donor-acceptor\" sub-system.",
"Usually, the presence of noise acts as an incoherent pump in the system under consideration.",
"But, as our results demonstrate, the simultaneous influence of both noise and the sink, significantly assist, under appropriate conditions, the quantum efficiency of the electron transfer.",
"We derived the expression for electron transfer rate which describes the tunneling to the sink, in the presence of noise.",
"We calculate explicitly the corresponding region of parameters of the noise assisted quantum electron transfer for sharp and flat redox potentials, and for noise described by an ensemble of two-level fluctuators.",
"Our results show that even in this simplified model, the quantum dynamics of the electron transfer to the sink can be rather complicated, and depends on many parameters.",
"Further analytical research and numerical simulations are required to extend our approach for (i) complicated dependencies of the density of states on energy in the sinks for flat and sharp redox potentials, in the presence of noise and thermal environments, and (ii) more complicated LHCs-RCs complexes.",
"The problem of the electron transfer optimization also requires further analysis."
],
[
"Acknowledgements",
"We are thankful to B.H.",
"McMahon for useful discussions.",
"This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No.",
"DE-AC52-06NA25396.",
"A.I.",
"Nesterov acknowledges the support from the CONACyT, Grant No.",
"118930."
],
[
"Solution of the Liouville equation",
"The solution of the the Liouville equation (REF ), $i\\dot{ \\rho } = [\\mathcal {H},\\rho ] - i \\lbrace \\mathcal {W},\\rho \\rbrace $ , is given by[39], [40], $\\rho _{11}(t) = {e^{-\\Gamma t}} \\bigg |\\Big (\\cos \\frac{\\Omega t}{2} - i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_1-i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_2\\bigg |^2,\\\\\\rho _{22}(t) = {e^{-\\Gamma t}} \\bigg |\\Big (\\cos \\frac{\\Omega t}{2} +i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_2-i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_1\\bigg |^2, $ $\\rho _{12}(t) = {e^{-\\Gamma t}} \\bigg (\\Big (\\cos \\frac{\\Omega t}{2} - i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_1-i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_2\\bigg )^\\ast \\cdot \\nonumber \\\\\\bigg (\\Big (\\cos \\frac{\\Omega t}{2} + i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_2-i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_1\\bigg ) , \\\\\\rho _{21}(t) = {e^{-\\Gamma t}} \\bigg (\\Big (\\cos \\frac{\\Omega t}{2} - i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_1-i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_2\\bigg )\\cdot \\nonumber \\\\\\bigg (\\Big (\\cos \\frac{\\Omega t}{2} + i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big ) C_2- i \\sin \\theta \\sin \\frac{\\Omega t}{2} C_1\\bigg )^\\ast ,$ where $\\cos \\theta = (\\varepsilon + i\\Gamma )/\\Omega $ , $\\sin \\theta = V/\\Omega $ , and $\\Omega = \\sqrt{V^2 +(\\varepsilon + i\\Gamma )^2}$ is the complex Rabi frequency.",
"The constants, $C_1$ and $C_2$ , are defined from the initial conditions as follows: $\\rho _{11}(0)=|C_1|^2$ , $\\rho _{22}(0)=|C_2|^2$ , $\\rho _{12}(0)=C^\\ast _1 C_2$ and $\\rho _{21}(0)=C^\\ast _2 C_1$ .",
"Let us assume that initially only the acceptor site is occupied, so that $\\rho _{11}(0)=0$ and $\\rho _{22}(0)= 1$ .",
"This yields, $\\rho _{11}(t) = {e^{-\\Gamma t}} \\bigg |\\sin \\theta \\sin \\frac{\\Omega t}{2}\\bigg |^2, \\quad \\rho _{22}(t) = {e^{-\\Gamma t}} \\bigg |\\Big (\\cos \\frac{\\Omega t}{2} + i\\cos \\theta \\sin \\frac{\\Omega t}{2}\\Big )\\bigg |^2, \\\\\\rho _{12}(t) = i{e^{-\\Gamma t}} \\bigg (\\sin \\theta \\sin \\frac{\\Omega t}{2} \\bigg )^\\ast \\cdot \\bigg (\\cos \\frac{\\Omega t}{2} + i\\cos \\theta \\sin \\frac{\\Omega t}{2} \\bigg ) , \\\\\\rho _{21}(t) = -i{e^{-\\Gamma t}} \\bigg (\\sin \\theta \\sin \\frac{\\Omega t}{2} \\bigg )\\cdot \\bigg (\\cos \\frac{\\Omega t}{2} + i\\cos \\theta \\sin \\frac{\\Omega t}{2} \\bigg )^\\ast .$ Taking $V=0$ , we find $\\rho _{11}(t)=\\rho _{12}(t)= \\rho _{21}(t)=0$ , and for $\\rho _{22}(t) $ we obtain the Weisskopf-Wigner formula for an irreversible decay, $\\rho _{22}(t) = {e^{-2\\Gamma t}} $ [41], [28].",
"Presenting $\\Omega = \\Omega _1 +i \\Omega _2 = \\sqrt{p +i q}$ , where $p= V^2+ \\varepsilon ^2- \\Gamma ^2$ and $q= 2\\varepsilon \\Gamma $ , we obtain $\\Omega _1^2 - \\Omega _2^2 = p$ and $\\Omega _1\\Omega _2 = q$ .",
"This yields, $\\Omega _1= \\pm \\frac{1}{\\sqrt{2}} \\sqrt{p + \\sqrt{p^2 + q^2}}, \\quad \\Omega _2= \\pm \\frac{1}{\\sqrt{2}} \\sqrt{-p + \\sqrt{p^2 + q^2}}$ , where the upper sign corresponds to $p>0$ , and the lower sign corresponds to $p<0$ .",
"Using these results, we obtain for $\\rho _{22}(t) $ the simple analytical expression, $\\rho _{22}(t) = \\frac{ V^2 e^{-\\Gamma t}}{2(\\Omega ^2_1 +\\Omega ^2_2)} \\big (\\cosh {\\Omega _2 t} - \\cos {\\Omega _1 t} \\big ).$"
],
[
"Equation of motion for the average density matrix",
"In the interaction representation, considering the off-diagonal elements as perturbations, so that $\\tilde{\\mathcal {H}}={\\mathcal {H}}_0 + V(t)- i\\mathcal {W}$ , where ${\\mathcal {H}}_0= \\sum _{n} \\varepsilon _n |n\\rangle \\langle n | + \\sum _{n} \\lambda _{nn} (t) |n\\rangle \\langle n | , \\\\V(t)= \\sum _{m \\ne n} ( V_{mn} +\\lambda _{mn}(t))|m\\rangle \\langle n |, \\quad \\mathcal {W} = \\Gamma |2\\rangle \\langle 2|,$ we obtain the following equations of motion, $ {\\dot{\\tilde{\\rho }}}_{11} = i({\\tilde{\\rho }}_{12}{\\tilde{V}}_{21}- {\\tilde{V}}_{12} {\\tilde{\\rho }}_{21}), \\quad {\\dot{\\tilde{\\rho }}}_{22} = i({\\tilde{\\rho }}_{21}{\\tilde{V}}_{12}- {\\tilde{V}}_{21} {\\tilde{\\rho }}_{12})-2\\Gamma {\\tilde{\\rho }}_{22}, \\\\{\\dot{\\tilde{\\rho }}}_{12} = i{\\tilde{V}}_{12}({\\tilde{\\rho }}_{11}- {\\tilde{\\rho }}_{22})- \\Gamma {\\tilde{\\rho }}_{12}, \\quad {\\dot{\\tilde{\\rho }}}_{21} = i{\\tilde{V}}_{21}({\\tilde{\\rho }}_{11}- {\\tilde{\\rho }}_{22}) - \\Gamma {\\tilde{\\rho }}_{21},$ where $ \\tilde{\\rho }= T(e^{i\\int _0^t H_0(\\tau ) d \\tau })\\rho T(e^{-i\\int _0^t H_0(\\tau ) d\\tau })$ and $ \\tilde{V}= T(e^{i\\int _0^t H_0(\\tau ) d \\tau })V T(e^{-i\\int _0^t H_0(\\tau ) d\\tau })$ .",
"Using Eqs.",
"(REF )- (), we obtain $ {\\tilde{\\rho }}_{11}(t) = { {\\tilde{\\rho }}}_{11}(0) + i\\int _0^t({\\tilde{\\rho }}_{12}(t^{\\prime }){\\tilde{V}}_{21}(t^{\\prime })- {\\tilde{V}}_{12}(t^{\\prime }) {\\tilde{\\rho }}_{21}(t^{\\prime }))dt^{\\prime }, \\\\{\\tilde{\\rho }}_{22}(t) = {\\tilde{\\rho }}_{22}(0)+ i\\int _0^t e^{-2\\Gamma (t- t^{\\prime })}({\\tilde{\\rho }}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t^{\\prime })- {\\tilde{V}}_{21}(t^{\\prime }) {\\tilde{\\rho }}_{12}(t^{\\prime })), \\\\{\\tilde{\\rho }}_{12}(t) = {\\tilde{\\rho }}_{12}(0) + i \\int _0^t e^{-\\Gamma (t- t^{\\prime })}{\\tilde{V}}_{12}(t^{\\prime }) ({\\tilde{\\rho }}_{11}(t^{\\prime })- {\\tilde{\\rho }}_{22}(t^{\\prime }))dt^{\\prime }, \\\\{\\tilde{\\rho }}_{21}(t) = {\\tilde{\\rho }}_{21}(0) + i \\int _0^t e^{-\\Gamma (t- t^{\\prime })}{\\tilde{V}}_{21}(t^{\\prime })({\\tilde{\\rho }}_{11}(t^{\\prime })- {\\tilde{\\rho }}_{22}(t^{\\prime }))dt^{\\prime }.$ We assume that initially ${\\tilde{\\rho }}_{12}(0)={\\tilde{\\rho }}_{21}(0)=0$ .",
"Now inserting (REF ) - () into Eqs.",
"(REF ) - (), and taking into account that ${\\tilde{\\rho }}_{11} = \\rho _{11}$ and ${\\tilde{\\rho }}_{22} = \\rho _{22}$ , we obtain the following system of integro-differential equations, $ &{\\dot{\\rho }}_{11} = - \\int _0^t e^{-\\Gamma (t- t^{\\prime })}\\Big ({\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime })+ {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\Big )\\Big ({ \\rho }_{11}(t^{\\prime }) -{ \\rho }_{22}(t^{\\prime })\\Big ) dt^{\\prime }, \\\\&{\\dot{\\rho }}_{22} = \\int _0^te^{-\\Gamma (t- t^{\\prime })}\\Big ({\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime })+ {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\Big )\\Big ({ \\rho }_{11}(t^{\\prime }) -{\\rho }_{22}(t^{\\prime })\\Big ) dt^{\\prime } - 2\\Gamma {\\rho }_{22}(t) , \\\\&\\dot{\\tilde{\\rho }}_{12}(t) = - \\int _0^t\\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big ) \\Big ({\\tilde{V}}_{21}(t^{\\prime }){\\tilde{\\rho }}_{12}(t^{\\prime })- {\\tilde{V}}_{12}(t^{\\prime }){\\tilde{\\rho }}_{21}(t^{\\prime })\\Big ){\\tilde{V}}_{12}(t)dt^{\\prime }- \\Gamma {\\rho }_{12}(t) \\nonumber \\\\&+ i{\\tilde{V}}_{12}(t)({\\tilde{\\rho }}_{11}(0)- {\\tilde{\\rho }}_{22}(0)) , \\\\&\\dot{\\tilde{\\rho }}_{21}(t) = - \\int _0^t\\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big ) \\Big ({\\tilde{V}}_{21}(t^{\\prime }){\\tilde{\\rho }}_{12}(t^{\\prime })- {\\tilde{V}}_{12}(t^{\\prime }){\\tilde{\\rho }}_{21}(t^{\\prime })\\Big ){\\tilde{V}}_{21}(t)dt^{\\prime }- \\Gamma {\\rho }_{21}(t) \\nonumber \\\\&+ i{\\tilde{V}}_{21}(t)({\\tilde{\\rho }}_{11}(0)- {\\tilde{\\rho }}_{22}(0)) .$ For the average components of the density matrix this yields, $ &\\frac{d}{dt}{\\langle {\\rho }}_{11}(t)\\rangle = - \\int _0^te^{-\\Gamma (t- t^{\\prime })}\\Big \\langle \\Big ({\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime })+ {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\Big )\\Big ({ \\rho }_{11}(t^{\\prime }) -{ \\rho }_{22}(t^{\\prime })\\Big ) \\Big \\rangle dt^{\\prime }, \\\\&\\frac{d}{dt}{\\langle {\\rho }}_{22}(t)\\rangle = \\int _0^te^{-\\Gamma (t- t^{\\prime })}\\Big \\langle \\Big ({\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime })+ {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\Big )\\Big ({ \\rho }_{11}(t^{\\prime }) -{\\rho }_{22}(t^{\\prime })\\Big )\\Big \\rangle dt^{\\prime } \\nonumber \\\\&- 2\\Gamma \\langle {\\rho }_{22}(t)\\rangle , \\\\&\\frac{d}{dt}{\\langle {\\rho }}_{12}(t)\\rangle = - \\int _0^t\\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big )\\Big \\langle \\Big ({\\tilde{V}}_{21}(t^{\\prime }){\\tilde{\\rho }}_{12}(t^{\\prime })- {\\tilde{V}}_{12}(t^{\\prime }){\\tilde{\\rho }}_{21}(t^{\\prime })\\Big ){\\tilde{V}}_{12}(t)\\Big \\rangle dt^{\\prime }- \\Gamma \\langle {\\rho }_{12}(t)\\rangle \\nonumber \\\\&+ i\\langle {\\tilde{V}}_{12}(t)\\rangle ({ \\rho }_{11}(0)- { \\rho }_{22}(0)) , \\\\&\\frac{d}{dt}{\\langle {\\rho }}_{21}(t)\\rangle = - \\int _0^t\\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big )\\Big \\langle \\Big ({\\tilde{V}}_{21}(t^{\\prime }){\\tilde{\\rho }}_{12}(t^{\\prime })- {\\tilde{V}}_{12}(t^{\\prime }){\\tilde{\\rho }}_{21}(t^{\\prime })\\Big ){\\tilde{V}}_{21}(t)\\Big \\rangle dt^{\\prime }- \\Gamma \\langle {\\rho }_{21}(t)\\rangle \\nonumber \\\\&+ i\\langle {\\tilde{V}}_{21}(t)\\rangle ({\\rho }_{11}(0)- {\\rho }_{22}(0)) ,$ where the average $\\langle \\; \\rangle $ is taken over the random process describing noise.",
"Generalization of the obtained results for the case ${\\tilde{\\rho }}_{12}(0) \\ne 0$ and ${\\tilde{\\rho }}_{21}(0)\\ne 0$ is straightforward.",
"In the spin-fluctuator model of noise with the number of fluctuators ${\\mathcal {N}} \\gg 1$ one has the following relations for the splitting of correlations [14], $\\big \\langle \\big ({\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime })+ {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\big )\\big ({\\tilde{\\rho }}_{11}(t^{\\prime }) -{\\tilde{\\rho }}_{22}(t^{\\prime })\\big )\\big \\rangle = \\nonumber \\\\\\big (\\big \\langle {\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime }) \\big \\rangle + \\big \\langle {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\big \\rangle \\big )\\big (\\big \\langle {\\tilde{\\rho }}_{11}(t^{\\prime })\\big \\rangle -\\big \\langle {\\tilde{\\rho }}_{22}(t^{\\prime })\\big \\rangle \\big ),$ and so on.",
"Next, using the second-order cumulant expansion, we obtain the following system of differential equations for the average components of the density matrix, $ &\\frac{d}{dt}{\\langle {\\rho }}_{11}(t)\\rangle =-\\int _0^t e^{-\\Gamma (t- t^{\\prime })}\\big (\\big \\langle {\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime }) \\big \\rangle + \\big \\langle {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\big \\rangle \\big )dt^{\\prime }\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) \\nonumber \\\\& + {\\mathcal {O}}(V^4), \\\\&\\frac{d}{dt}{\\langle {\\rho }}_{22}(t)\\rangle =\\int _0^t e^{-\\Gamma (t- t^{\\prime })}\\big (\\big \\langle {\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime }) \\big \\rangle + \\big \\langle {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\big \\rangle \\big )dt^{\\prime }\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) \\nonumber \\\\&- 2\\Gamma \\langle {\\tilde{\\rho }}_{22}(t)\\rangle + {\\mathcal {O}}(V^4),$ $&\\frac{d}{dt}\\langle {\\tilde{\\rho }}_{12}(t)\\rangle = i\\langle {\\tilde{V}}_{12}(t)\\rangle ({ \\rho }_{11}(0)- { \\rho }_{22}(0)) - \\int _0^t \\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big ) \\langle {\\tilde{V}}_{12}(t) {\\tilde{V}}_{21}(t^{\\prime })\\rangle dt^{\\prime }\\langle {\\tilde{\\rho }}_{12}(t)\\rangle \\nonumber \\\\&+ \\int _0^t \\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big ) \\langle {\\tilde{V}}_{12}(t) {\\tilde{V}}_{12}(t^{\\prime })\\rangle dt^{\\prime }\\langle {\\tilde{\\rho }}_{21}(t)\\rangle - \\Gamma \\langle {\\rho }_{12}(t)\\rangle + {\\mathcal {O}}(V^4),\\\\&\\frac{d}{dt}\\langle {\\tilde{\\rho }}_{21}(t)\\rangle = i\\langle {\\tilde{V}}_{21}(t)\\rangle ({ \\rho }_{11}(0)- { \\rho }_{22}(0))- \\int _0^t \\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big ) \\langle {\\tilde{V}}_{21}(t) {\\tilde{V}}_{21}(t^{\\prime })\\rangle dt^{\\prime }\\langle {\\tilde{\\rho }}_{12}(t)\\rangle \\nonumber \\\\& + \\int _0^t \\Big (1+ e^{-2\\Gamma (t- t^{\\prime })}\\Big )\\langle {\\tilde{V}}_{21}(t) {\\tilde{V}}_{12}(t^{\\prime })\\rangle dt^{\\prime }\\langle {\\tilde{\\rho }}_{21}(t)\\rangle - \\Gamma \\langle {\\rho }_{21}(t)\\rangle + {\\mathcal {O}}(V^4).$ We rewrite the equation of motion for the diagonal components of the density matrix as, $ \\frac{d}{dt}{\\langle {\\rho }}_{11}(t)\\rangle =- {\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ), \\\\\\frac{d}{dt}{\\langle {\\rho }}_{22}(t)\\rangle ={\\mathfrak {R}}(t)\\big (\\big \\langle {\\rho }_{11}(t)\\big \\rangle -\\big \\langle {\\rho }_{22}(t)\\big \\rangle \\big ) - 2\\Gamma \\langle {\\tilde{\\rho }}_{22}(t)\\rangle ,$ where ${\\mathfrak {R}}(t) =\\int _0^t e^{-\\Gamma (t- t^{\\prime })}\\big (\\big \\langle {\\tilde{V}}_{21}(t){\\tilde{V}}_{12}(t^{\\prime }) \\big \\rangle + \\big \\langle {\\tilde{V}}_{21}(t^{\\prime }){\\tilde{V}}_{12}(t)\\big \\rangle \\big )dt^{\\prime }$ ."
]
] | 1204.0805 |
[
[
"Extended quantum conditional entropy and quantum uncertainty\n inequalities"
],
[
"Abstract Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies.",
"There is a long history of such entropy inequalities between position and momentum.",
"Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized.",
"All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique of the original derivation of SSA."
],
[
"Extended quantum conditional entropy and quantum uncertainty inequalities © 2012 by the authors.",
"This paper may be reproduced, in its entirety, for non-commercial purposes.",
"Rupert L. Frank Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Elliott H. Lieb Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Department of Physics, Princeton University, P. O.",
"Box 708, Princeton, NJ 08542, USA Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies.",
"There is a long history of such entropy inequalities between position and momentum.",
"Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized.",
"All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique of the original derivation of SSA.",
"03.67.-a, 03.67.Mn, 03.67.Hk, 03.65.Ta A celebrated inequality of Maassen–Uffink [1], based on earlier work in [3], [2], [4], relates the `classical' entropy of a quantum density matrix in two different bases and shows that although one of them could be very small, the sum of the two is bounded below by a positive constant.",
"The word `classical' refers to $H=-\\sum _j p_j \\ln p_j$ , where $p_j$ is the expectation of a density matrix in some orthonormal basis.",
"The inequality is $H(A)+H(B)\\ge -2\\ln \\sup _{j,k} |\\langle a_j,b_k\\rangle | \\,,$ where $A$ and $B$ represent two orthonormal bases $(a_j)$ and $(b_k)$ .",
"This can be generalized to continuous bases, like position $\\mathbf {x}$ and momentum $\\mathbf {p}$ , in which case the inequality becomes (with $\\mathbf {p}=2\\pi \\mathbf {k}$ ) $-\\int \\!d^d\\mathbf {x}\\, \\rho (\\mathbf {x},\\mathbf {x})\\ln \\rho (\\mathbf {x},\\mathbf {x}) - \\int \\!d^d\\mathbf {k}\\, \\widehat{\\rho }(\\mathbf {k},\\mathbf {k})\\ln \\widehat{\\rho }(\\mathbf {k},\\mathbf {k}) \\ge 0 \\,.$ These inequalities have subsequently been improved; see the review [5] and the recent papers [6], [7], [12].",
"Only purely classical analysis needs to be utilized to prove these inequalities and only one Hilbert space is involved.",
"Another direction was opened up by the conjectures of Renes and Boileau [8], in which more than one Hilbert space appears.",
"The analogous inequalities become more difficult mathematically because of the well-known entanglement problems in quantum mechanics.",
"Indeed, as noted in [8], the Lieb–Ruskai strong subadditivity (SSA) theorem [9], or one of its equivalents, would ultimately be needed to prove the conjectures.",
"Berta et al.",
"[10] then proved a special 2–space version of the conjecture, that can be called the rank-one version, and used [9] again (this time the concavity of conditional entropy, which is equivalent to SSA) to deduce a 3–space version of the uncertainty principle.",
"This quantum version of the uncertainty principle has attracted a great deal of attention.",
"Subsequently, Coles et al.",
"[6] and Tomamichel and Renner [11] were able to eliminate the rank one condition for the 3–space version and partially removed it for the 2–space version.",
"The significant developments in [10], [6] came at the cost of rather lengthy calculations and multi-page proofs.",
"It is evidently desirable to shorten these proofs and to clarify the essential mathematical underpinning.",
"It is shown here that if one uses the original proof structure of SSA in [9] these proofs can be significantly shortened.",
"A second thing we do in this paper is eliminate the aforementioned restriction of [6] in the 2–space case.",
"Those authors point out that the obvious extension is necessarily false; we find a more general formulation that makes the extension possible.",
"Finally, we go back to the Maassen–Uffink inequality above and relate the $\\mathbf {x}$ -space entropy to the $\\mathbf {p}$ -space entropy, but this time with an auxiliary quantum system, thereby promoting the Maassen–Uffink inequality to a truly quantum one.",
"The notation in this field is not uniform and we begin by defining our terms.",
"We have three Hilbert spaces (degrees of freedom) $\\mathcal {H}_1$ , $\\mathcal {H}_2$ and $\\mathcal {H}_3$ and their tensor products $\\mathcal {H}_{123}= \\mathcal {H}_1 \\otimes \\mathcal {H}_2 \\otimes \\mathcal {H}_3$ , $\\mathcal {H}_{12}= \\mathcal {H}_1 \\otimes \\mathcal {H}_2$ etc.",
"We have density matrices $\\rho _{123}$ , $\\rho _{12}$ , etc.",
"(i.e., non-negative operators of trace one).",
"If $\\rho _{123}$ is defined on $\\mathcal {H}_{123}$ then there is a natural $\\rho _{12}$ on $\\mathcal {H}_{12}$ given by $\\rho _{12} = \\operatorname{Tr}_3 \\rho _{123}$ , where $\\operatorname{Tr}_3$ is the partial trace over $\\mathcal {H}_3$ .",
"In general, entropy is defined by $S(\\rho ) = -\\operatorname{Tr}_{\\mathcal {H}} \\rho \\ln \\rho $ , where $\\mathcal {H}$ is one of the above spaces.",
"If $\\rho $ and $\\mathcal {H}$ are understood, then $S_{12} = S(\\rho _{12})$ and $S_1=S(\\rho _1)$ .",
"Conditional entropy is defined by $S(1|2) = S_{12}-S_2 = S(\\rho _{12}) - S(\\rho _2)$ and, as shown in [9], this is a concave function of $\\rho _{12}$ .",
"This concavity is mathematically equivalent to SSA, and we use it in the Lemma below.",
"A classical measurement on $\\mathcal {H}_1$ is defined by a sequence of operators $A_1$ , $A_2$ , etc.",
"on $\\mathcal {H}_1$ such that $\\sum _j A_j^* A_j=1$ .",
"We will need two measurements in our discussion, so we will need a second sequence $B_1$ , $B_2$ , etc.",
"on $\\mathcal {H}_1$ such that $\\sum _k B_k^* B_k =1$ .",
"At the end of this paper we will let the indices $j$ and $k$ be continuous and the sums become integrals, but for simplicity we stay with sums for now.",
"A measurement $A$ is called `rank-one' if every $A_j$ is a rank-one matrix, namely, $A_j= |a_j\\rangle \\langle a_j|$ .",
"The classical/quantum conditional entropy corresponding to $A$ and $\\rho $ is H(1A|2) = -j Tr2 (Tr1 Aj12 Aj*) (Tr1 Aj12 Aj*) -S(2) .",
"Note where the summation sits.",
"(This is not the conditional entropy of $\\sum _j A_j\\rho A_j^*$ .)",
"The quantity $H(1^B|2)$ is defined similarly.",
"2–space theorem.— For any $\\rho _{12}$ , $H(1^A|2) + H(1^B|2) \\ge S(1|2) -2\\ln c_1 \\,,$ where $c_1=\\sup _{j,k} \\sqrt{ \\operatorname{Tr}_1 B_k A_j^*A_j B_k^*}$ .",
"In the special case that $\\rho _{12}=\\rho _1\\otimes \\rho _2$ is a product state one easily sees that (REF ) reduces to the 1–space theorems in [6], [7], [12], which extend the Maassen–Uffink theorem.",
"Coles et al.",
"[6] prove (REF ) if either $A$ or $B$ is rank-one and if $c_1$ is replaced by the smaller, better, number $c_\\infty = \\sup _{j,k} \\sqrt{ \\Vert B_k A_j^*A_j B_k^*\\Vert _\\infty }$ They state correctly that the theorem cannot hold for general $A$ and $B$ in the $c_\\infty $ version.",
"However, if either $A$ or $B$ is rank-one, then $c_1=c_\\infty $ .",
"So our improvement of Coles et al.",
"consists of eliminating the rank-one condition and using $c_1$ .",
"The norm $ \\Vert B_k A_j^*A_j B_k^*\\Vert _\\infty $ is the largest eigenvalue of $B_k A_j^*A_j B_k^*$ , which is also the largest eigenvalue of $\\sqrt{B_k^* B_k} A_j^*A_j \\sqrt{B_k^* B_k}$ , and thus coincides with $\\Vert \\sqrt{A_j^*A_j} \\sqrt{B_k^* B_k} \\Vert _\\infty ^2$ .",
"This shows that our $c_\\infty ^2$ is, indeed, the same as that in [6].",
"3–space theorem.",
"[6]— For any $\\rho _{123}$ , $H(1^A|2) + H(1^B|3) \\ge -2\\ln c_\\infty \\,.$ We shall prove the 3–space theorem by first proving the following theorem which is not conveniently expressed in the $H(\\cdot |\\cdot )$ notation.",
"2$\\tfrac{1}{2}$ –space theorem.— For any $\\rho _{12}$ , $H(1^A|2) - \\sum _k \\operatorname{Tr}_{12} B_k\\rho B_k^* \\ln B_k\\rho B_k^* - S(\\rho _{12})\\ge -2\\ln c_\\infty \\,.$ This looks like another 2–space theorem but, as we explain below, it is really the 3-space theorem applied to a pure state (i.e., rank-one) $\\rho _{123}$ .",
"Thus our name.",
"The key inequality behind the proofs of all our three theorems is the three operator generalization of the Golden–Thompson inequality from [13], which was also the key ingredient in the proof of SSA in [9].",
"It states that for non-negative operators $X,Y,Z$ , $\\operatorname{Tr}e^{\\ln X-\\ln Y +\\ln Z} \\le \\int _0^\\infty \\!\\!",
"dt\\, \\operatorname{Tr}X (Y+t)^{-1} Z(Y+t)^{-1}.$ Note that in the special case $Z=1$ this reduces to the classical Golden–Thompson inequality.",
"This inequality was a byproduct of the proof of concavity of the generalized Wigner–Yanase skew information.",
"We shall also need the Gibbs variational principle (equivalent to the Peierls–Bogolubov inequality), $\\operatorname{Tr}\\rho h - S(\\rho ) \\ge - \\ln \\operatorname{Tr}e^{-h}$ for any density matrix $\\rho $ and any self-adjoint $h$ , and the Davis operator Jensen inequality [14], $\\sum _j A_j^* \\left(\\ln K_j\\right) A_j \\le \\ln \\left( \\sum _j A_j^* K_j A_j \\right)$ for any positive operators $K_j$ and any $A_j$ with $\\sum _j A_j^* A_j =1$ .",
"Proof of the 2–space theorem.",
"We use (REF ) for both $A$ and $B$ to bound $H(1^A|2) + H(1^B|2) - S(1|2) \\ge \\operatorname{Tr}_{12} \\rho _{12}h - S(\\rho _{12}) \\,,$ with the operator h = 2 - j Aj*Aj (Tr1 Aj12 Aj*) - k Bk*Bk (Tr1 Bk12 Bk*).",
"We do not need to invoke (REF ) when $A$ and $B$ are rank-one; in that case (REF ) is an equality and the proof simplifies further.",
"Thus, by (REF ) and (REF ), $H(1^A|2) + H(1^B|2) - S(1|2) \\ge -\\ln \\operatorname{Tr}_{12} e^{-h} \\,,$ and it remains to show that $\\operatorname{Tr}_{12} e^{-h} \\le c_1^2$ .",
"Now comes the crucial step!",
"We use (REF ) to bound $\\operatorname{Tr}_{12} e^{-h} \\le \\int _0^\\infty dt\\, \\sum _{j,k} \\operatorname{Tr}_{12} C_{j,k}(t)$ with Cj,k(t) = Aj*Aj (Tr1 Aj12 Aj* ) (2+t)-1 Bk*Bk (Tr1 Bk12 Bk* ) (2+t)-1 = Aj*Aj Bk*Bk Dj,k(t) .",
"Here, $D_{j,k}(t)$ is the operator on $\\mathcal {H}_2$ given by $\\left( \\operatorname{Tr}_1 A_j\\rho _{12} A_j^* \\right) (\\rho _2+t)^{-1} \\left( \\operatorname{Tr}_1 B_k\\rho _{12} B_k^* \\right) (\\rho _2+t)^{-1} .$ Thus, Tr12 Cj,k(t) = ( Tr1 Aj*Aj Bk*Bk ) Tr2 Dj,k(t) c12 Tr2 Dj,k(t) .",
"We next note that $\\sum _j \\operatorname{Tr}_1 A_j\\rho _{12} A_j^* = \\sum _j \\operatorname{Tr}_1 A_j^*A_j\\rho _{12} = \\rho _2$ , and similarly for the $k$ sum, and obtain $\\sum _{j,k} \\operatorname{Tr}_2 D_{j,k}(t) = \\operatorname{Tr}_2 \\rho _2^2 (\\rho _2+t)^{-2} \\,.$ Thus, since $\\int _0^\\infty dt\\, (\\rho _2+t)^{-2} = \\rho _2^{-1}$ , $\\int _0^\\infty dt\\, \\sum _{j,k} \\operatorname{Tr}_2 D_{j,k}(t) = \\operatorname{Tr}_2 \\rho _2 = 1 \\,.$ This completes the proof of the 2–space theorem.",
"QED We shall now show that the 3–space theorem is a corollary of the 2$\\frac{1}{2}$ –space theorem, so that the proof of the 2$\\frac{1}{2}$ –space theorem will finish everything.",
"We use the following: Lemma.— $H(1^A|2)$ is a concave function on the set of non-negative operators $\\rho _{12}$ on $\\mathcal {H}_{12}$ .",
"Proof of the Lemma.— The idea is to view the sum over $j$ as the trace over an auxiliary space $\\mathcal {K}$ of a matrix that happens to be diagonal in this space, and to apply concavity of the conditional entropy [9] in $\\mathcal {H}_2\\otimes \\mathcal {K}$ .",
"The details are as follows: Let $(e_j)$ be an orthonormal basis of $\\mathcal {K}$ and consider the operator $\\Gamma = \\sum _j \\left(\\operatorname{Tr}_1 A_j\\rho _{12} A_j^*\\right) \\otimes |e_j\\rangle \\langle e_j|$ on $\\mathcal {H}_2\\otimes \\mathcal {K}$ .",
"As in the proof of the 2–space theorem we have $\\Gamma _2=\\operatorname{Tr}_{\\mathcal {K}}\\Gamma =\\rho _2$ and therefore $H(1^A|2) = S(\\Gamma ) - S(\\Gamma _2) \\,.$ This is the conditional entropy of $\\Gamma $ with respect to $\\mathcal {H}_2$ .",
"Since $\\rho _{12}\\mapsto \\Gamma $ is linear, the asserted concavity follows from the fact that conditional entropy is concave, as shown in [9].",
"QED Proof of the 3–space theorem.— It follows from the Lemma that $H(1^A|2) + H(1^B|3)$ is a concave function of $\\rho _{123}$ .",
"Thus, for the proof we may assume that $\\rho _{123}$ is a pure state (rank one).",
"In that case $S(\\rho _3)= S(\\rho _{12})$ and, since $B_k\\rho _{123} B_k^*$ is pure as well, $H(1^B|3) = - \\sum _k \\operatorname{Tr}_{12} B_k\\rho _{12} B_k^* \\ln B_k\\rho B_k^* -S(\\rho _{12}) \\,.$ This reduces the inequality of the 3–space theorem to that of the 2$\\frac{1}{2}$ –space theorem.",
"QED Proof of the 2$\\frac{1}{2}$ –space theorem.— The proof runs very parallel to that of the 2–space theorem.",
"Namely, the right side in the desired inequality is bounded from below by $\\operatorname{Tr}_{12} \\rho _{12}\\tilde{h} - S(\\rho _{12})$ , where now h = 2 - j Aj*Aj (Tr1 Aj12 Aj*) - k Bk*Bk 12 Bk* Bk .",
"Here we used (REF ).",
"After applying (REF ) as before, everything is reduced to showing $\\operatorname{Tr}_{12} e^{-\\tilde{h}} \\le c_\\infty ^2$ .",
"The crucial ingredient is again (REF ) which now leads to $\\operatorname{Tr}_{12} e^{-\\tilde{h}} \\le \\int _0^\\infty dt\\, \\sum _{j,k} \\operatorname{Tr}_{12} \\tilde{C}_{j,k}(t)$ with Cj,k(t) = Aj*Aj (Tr1 Aj12 Aj* ) (2+t)-1 Bk*Bk 12 Bk* Bk (2+t)-1 .",
"At this point the proof diverges somewhat from that of the 2–space theorem.",
"Namely, by cyclicity of the trace we write $\\operatorname{Tr}_{12} \\tilde{C}_{j,k}(t) = \\operatorname{Tr}_{12} \\left( B_k A_j^*A_j B_k^* \\tilde{D}_{j,k}(t) \\right)$ with $\\tilde{D}_{j,k}(t)$ given by $\\left(\\operatorname{Tr}_1 A_j\\rho _{12} A_j^* \\right) (\\rho _2+t)^{-1} B_k \\rho _{12} B_k^* (\\rho _2+t)^{-1} \\,.$ Since $B_k A_j^*A_j B_k^* \\left(\\operatorname{Tr}_1 A_j\\rho _{12} A_j^* \\right) \\le c_\\infty ^2 \\left(\\operatorname{Tr}_1 A_j\\rho _{12} A_j^* \\right)$ and since $(\\rho _2+t)^{-1} B_k \\rho _{12} B_k^* (\\rho _2+t)^{-1} \\ge 0$ , we have $\\operatorname{Tr}_{12} \\tilde{C}_{j,k}(t) \\le c_\\infty ^2 \\operatorname{Tr}_{12} \\tilde{D}_{j,k}(t) \\,.$ From here, everything is as before: $\\sum _{j,k} \\operatorname{Tr}_{12} \\tilde{D}_{j,k}(t) = \\operatorname{Tr}_2 \\rho _2^2 (\\rho _2+t)^{-2}$ and $\\int _0^\\infty dt\\, \\sum _{j,k} \\operatorname{Tr}_{12} \\tilde{D}_{j,k}(t) = \\operatorname{Tr}_2 \\rho _2 = 1 \\,.\\qquad \\qquad \\mathrm {QED}$ At last, we turn to the continuous version, the most important application being the position-momentum uncertainty.",
"We start with this case.",
"Take $\\mathcal {H}_1$ to be $L^2(\\mathbb {R}^d)$ , the square-integrable functions on $\\mathbb {R}^d$ .",
"The spaces $\\mathcal {H}_2$ and $\\mathcal {H}_3$ can be anything.",
"The measurement $A_j^*A_j$ is $|a_j\\rangle \\langle a_j|$ where $a_j$ is the delta-function $\\delta (\\mathbf {x}-\\mathbf {x}^{\\prime })$ for some $\\mathbf {x}^{\\prime }$ in $\\mathbb {R}^n$ .",
"The $j$ becomes $\\mathbf {x}^{\\prime }$ , which is a continuous variable, the sum $\\sum _j$ becomes the integral $\\int d^d\\mathbf {x}^{\\prime }$ , and the normalization condition $\\sum _j A_j^*A_j =1$ becomes $\\int d^d\\mathbf {x}^{\\prime }\\, \\delta (\\mathbf {x}-\\mathbf {x}^{\\prime })\\delta (\\mathbf {x}^{\\prime \\prime }-\\mathbf {x}^{\\prime })=\\delta (\\mathbf {x}-\\mathbf {x}^{\\prime \\prime })$ .",
"(We realize that the delta-function is not a function, but all of this can be made rigorous.)",
"Similarly, the $B_k^*B_k$ 's are $|b_k\\rangle \\langle b_k|$ where the $b_k$ will also not be square-integrable functions.",
"They will be plane waves $e^{i 2\\pi \\mathbf {k}\\cdot \\mathbf {x}}$ for some $\\mathbf {k}$ in $\\mathbb {R}^d$ .",
"Again, $\\mathbf {k}$ is a continuous index and sums are integrals.",
"The normalization condition $\\sum _kB_k^* B_k =1$ becomes $\\int d^d\\mathbf {k}\\,e^{-2\\pi i \\mathbf {k}\\cdot \\mathbf {x}} e^{2\\pi i \\mathbf {k}\\cdot \\mathbf {x}^{\\prime }} =\\delta (\\mathbf {x}-\\mathbf {x}^{\\prime })$ .",
"Let $\\rho _{12}$ be a density matrix on the Hilbert space $L^2(\\mathbb {R}^d)\\otimes \\mathcal {H}_2$ .",
"For every fixed $\\mathbf {x}\\in \\mathbb {R}^n$ we can define $\\langle \\mathbf {x}|\\rho _{12}|\\mathbf {x}\\rangle $ as an operator on $\\mathcal {H}_2$ .",
"This really means the partial trace $\\operatorname{Tr}_1$ of $A_j^* \\rho _{12} A_j$ .",
"Likewise we define $\\langle \\mathbf {k}|\\rho _{12}|\\mathbf {k}\\rangle $ to be $\\operatorname{Tr}_1 B_k^* \\rho _{12} B_k$ .",
"We see that $\\langle \\mathbf {x}|\\rho _{12}|\\mathbf {x}\\rangle $ is an operator-valued density in position space and $\\langle \\mathbf {k}|\\rho _{12}|\\mathbf {k}\\rangle $ is the corresponding density in momentum space.",
"Now we apply the 2–space theorem and infer that $H(1^A|2) + H(1^B|2) \\ge S(1|2) \\,,$ where $H(1^A|2) = - \\int _{\\mathbb {R}^d} d^d\\mathbf {x}\\operatorname{Tr}_2 \\langle \\mathbf {x}|\\rho _{12}|\\mathbf {x}\\rangle \\ln \\langle \\mathbf {x}|\\rho _{12}|\\mathbf {x}\\rangle -S(\\rho _2)$ and $H(1^B|2) = - \\int _{\\mathbb {R}^d} d^d\\mathbf {k}\\operatorname{Tr}_2 \\langle \\mathbf {k}|\\rho _{12}|\\mathbf {k}\\rangle \\ln \\langle \\mathbf {k}|\\rho _{12}|\\mathbf {k}\\rangle -S(\\rho _2) \\,.$ Here $c_1 = c_\\infty = \\sup _{\\mathbf {x},\\mathbf {k}} |e^{i2\\pi \\mathbf {k}\\cdot \\mathbf {x}}| =1$ and $\\ln c_1=\\ln c_\\infty =0$ .",
"This is a generalization of the uncertainty principle in [7], which is what (REF ) reduces to when $\\rho _{12}=\\rho _1\\otimes \\rho _2$ is a product state.",
"The 3–space theorem and the 2$\\frac{1}{2}$ –space theorem obviously generalize in a similar way for the Fourier transform.",
"This example of a continuum version of our theorems has the obvious generalization to positive operator-valued measures (POVMs).",
"The interested reader can work this out for him/herself, but we mention here one further specific generalization.",
"Let us take $\\mathcal {H}_1=L^2(X,d\\mathbf {x})$ , where $X$ is some configuration space (e.g., $X=\\mathbb {R}^d$ or $X=$ a torus or $X=$ a lattice) and $L^2(X,d\\mathbf {x})$ are the square integrable functions on $X$ with respect to some measure $d\\mathbf {x}$ .",
"The measurements $A_j^*A_j$ are again given by rank one projections corresponding to the functions $\\delta _{\\mathbf {x}^{\\prime }}(\\mathbf {x})$ for some $\\mathbf {x}^{\\prime }$ in $X$ .",
"Now let $\\mathcal {H}_1^{\\prime }$ be a second Hilbert space of the form $L^2(K,d\\mathbf {k})$ and let $\\mathcal {U}$ be a unitary from $\\mathcal {H}_1$ to $\\mathcal {H}_1^{\\prime }$ , which is given by a kernel $\\mathcal {U}(\\mathbf {k},\\mathbf {x})$ .",
"For each $\\mathbf {k}\\in K$ , we can think of $\\mathcal {U}(\\mathbf {k},\\mathbf {x})$ as a function of $\\mathbf {x}$ and we define the $B_k^*B_k$ 's to be the rank-one projections onto these functions.",
"In this way we obtain, as before, operators $\\langle \\mathbf {x}|\\rho _{12}|\\mathbf {x}\\rangle $ and $\\langle \\mathbf {k}|\\rho _{12}|\\mathbf {k}\\rangle $ on $\\mathcal {H}_2$ .",
"Now $H(1^A|2)$ and $H(1^B|2)$ are defined as in the Fourier transform case, except that the integration is over the sets $X$ and $K$ , respectively.",
"Generalized Fourier transform theorem.— (REF ) is valid with $c_1 = \\sup _{\\mathbf {k},\\mathbf {x}} |\\mathcal {U}(\\mathbf {k},\\mathbf {x})| \\,.$ An examples in which the classical entropies are simultaneously discrete and continuous is the following.",
"Suppose $X=\\mathbb {Z}$ , i.e., the integers, like the sites in a tight binding model.",
"An $L^2$ function is the wave function of an itinerant electron.",
"The second space $\\mathcal {H}_1^{\\prime }$ is the square integrable functions on $K=(-1/2,1/2)$ , the Brillouin zone.",
"In this case the delta-functions, $\\delta _{x x^{\\prime }}$ , on $X$ are legitimate Kronecker deltas, and the (normalized) plane waves on $X$ , parametrized by $k\\in K$ , are $e^{2\\pi i kx}$ .",
"The unitary here is $\\mathcal {U}(k,x)= e^{2\\pi i kx}$ and therefore $c_1=1$ .",
"Since $dx$ is counting measure on $\\mathbb {Z}$ the expression of $H(1^A|2)$ is a sum $\\sum _{x\\in \\mathbb {Z}}$ , whereas $dk$ is ordinary Lebesgue measure on $(-1/2,1/2)$ and $H(1^B|2)$ is an integral $\\int _K dk$ .",
"In conclusion, we have shown several things.",
"(1) The entropy inequalities in [10], [6] can be proved in a few lines essentially by imitating the original proof of strong subadditivity of entropy.",
"(2) We have carried at least one of these inequalities forward by utilizing the trace norm $c_1$ instead of the operator norm $c_\\infty $ .",
"(3) We have shown how these inequalities, suitably interpreted, extend the 2–space entropy uncertainty principle to continuous bases such as position and momentum.",
"We thank Zhihao Ma for helpful correspondence.",
"U.S. National Science Foundation grants PHY-1068285 (R.F.)",
"and PHY-0965859 (E.L.) are acknowledged."
]
] | 1204.0825 |
[
[
"Finite generators for countable group actions in the Borel and Baire\n category settings"
],
[
"Abstract For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \\cdot P := \\{ gC : g \\in G, C \\in P \\}$ generates the Borel $\\sigma$-algebra of $X$.",
"For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite entropy.",
"It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator.",
"We show that the answer is positive (in fact, there is a 32-generator) for an arbitrary countable group $G$ and $\\sigma$-compact $X$ (in particular, for locally compact $X$).",
"We also show that any continuous aperiodic action of $G$ on an arbitrary Polish space admits a 4-generator on a comeager set, thus giving a positive answer to a question of Alexander Kechris asked in the mid-90s.",
"Furthermore, assuming a positive answer to Weiss's question for arbitrary Polish spaces and $G = Z$, we prove the following dichotomy: every aperiodic Borel action of $Z$ on a Polish space $X$ admits either an invariant probability measure of infinite entropy or a finite generator.",
"As an auxiliary lemma, we prove the following statement, which may be of independent interest: every aperiodic Borel action of a countable group $G$ on a Polish space $X$ admits a $G$-equivariant Borel map to the aperiodic part of the shift action of $G$ on $2^G$.",
"We also obtain a number of other related results, among which is a criterion for the nonexistence of non-meager weakly wandering sets for continuous actions of $Z$.",
"A consequence of this is a negative answer to a question asked by Eigen--Hajian--Nadkarni, which was also independently answered by Benjamin Miller."
],
[
"Introduction",
"§1 .",
"Throughout the paper let $G$ denote a countably infinite discrete group.",
"Let $X$ be a Borel $G$ -space, i.e.",
"a standard Borel space equipped with a Borel action of $G$ .",
"Consider the following game: Player I chooses a finite or countable Borel partition $I= \\lbrace A_n\\rbrace _{n<k}$ of $X$ , $k \\le \\infty $ , then Player II chooses $x \\in X$ and Player I tries to guess $x$ by asking questions to Player II regarding which piece of the partition $x$ lands in when moved by a certain group element.",
"More precisely, for every $g \\in G$ , Player I asks to which $A_n$ does $gx$ belong and Player II gives $n_g < k$ as an answer.",
"Whether or not Player I can uniquely determine $x$ from the sequence $\\lbrace n_g\\rbrace _{g \\in G}$ of responses depends on how cleverly he chose the partition $I$ .",
"A partition is called a generator if it guarantees that Player I will determine $x$ correctly no matter which $x$ Player II chooses.",
"Here is the precise definition, which also explains the terminology.",
"Definition 1.1 (Generator) Let $k \\le \\infty $ and $I= \\lbrace A_n\\rbrace _{n<k}$ be a Borel partition of $X$ (i.e.",
"each $A_n$ is Borel).",
"$I$ is called a generator if $G I:= \\lbrace gA_n : g \\in G, n<k\\rbrace $ generates the Borel $\\sigma $ -algebra of $X$ .",
"We also call $I$ a $k$ -generator, and, if $k$ is finite, a finite generator.",
"For each $k \\le \\infty $ , we give $k^G$ the product topology and let $G$ act by shift on $k^G$ .",
"For a Borel partition $I= \\lbrace A_n\\rbrace _{n < k}$ of $X$ , let $ X \\rightarrow k^G$ be defined by $x \\mapsto (n_g)_{g \\in G}$ , where $n_g$ is such that $gx \\in A_{n_g}$ .",
"This is often called the symbolic representation map for the process $(X, G, I)$ .",
"Clearly $ is a Borel $ G$-map and, for every $ x X$, $ x)$ is the sequence of responses of Player I in the above game.",
"Based on this we have the following.$ Observation 1.2 Let $k \\le \\infty $ and $I= \\lbrace A_n\\rbrace _{n<k}$ be a Borel partition of $X$ .",
"The following are equivalent: $I$ is a generator.",
"$G I$ separates points, i.e.",
"for all distinct $x,y \\in X$ there is $A \\in G I$ such that $x \\in A \\nLeftrightarrow y \\in A$ .",
"$ is one-to-one.$ In all of the arguments below, we use these equivalent descriptions of a finite generator without comment.",
"Given a Borel $G$ -map $f : X \\rightarrow k^G$ for some $k \\le \\infty $ , define a partition $I_f = \\lbrace A_n\\rbrace _{n<k}$ by $A_n = f^{-1}(V_n)$ , where $V_n = \\lbrace \\alpha \\in k^G : \\alpha (1_G) = n\\rbrace $ .",
"Note that $f_{I_f} = f$ .",
"This and the above observation imply the following.",
"Observation 1.3 For $k \\le \\infty $ , $X$ admits a $k$ -generator if and only if there is a Borel $G$ -embedding of $X$ into $k^G$ .",
"§2 .",
"Generators arose in the study of entropy in ergodic theory.",
"Let $(X, \\mu , T)$ be a dynamical system, i.e.",
"$(X, \\mu )$ is a standard probability space and $T$ is a Borel measure preserving automorphism of $X$ .",
"We can interpret the above game as follows: $X$ is the set of possible pictures of the world, $I$ is an experiment that we are conducting and $T$ is the unit of time.",
"Assume that $I$ is finite (indeed, we want our experiment to have finitely many possible outcomes).",
"Player I repeats the experiment every day and Player II tells its outcome.",
"The goal is to find the true picture of the world (i.e.",
"$x \\in X$ that Player II has in mind) with probability 1.",
"The entropy of the experiment $I$ is defined by $h_{\\mu }(I) = - \\sum _{n<k} \\mu (A_n) \\log \\mu (A_n),$ and intuitively, it measures our probabilistic uncertainty about the outcome of the experiment.",
"For example, if for some $n < k$ , $A_n$ had probability 1, then we would be probabilistically certain that the outcome is going to be in $A_n$ .",
"Conversely, if all of $A_n$ had probability ${1 \\over k}$ , then our uncertainty would be the highest.",
"Equivalently, according to Shannon's interpretation, $h_{\\mu }(I)$ measures how much information we gain from learning the outcome of the experiment.",
"We now define the time average of the entropy of $I$ by $h_{\\mu }(I, T) = \\lim _{n \\rightarrow \\infty } {1 \\over n} h_{\\mu }(\\bigvee _{i<n} T^i I),$ where $\\bigvee $ denotes the joint of the partitions (the least common refinement).",
"The sequence in the limit is decreasing and hence the limit always exists and is finite (see or ).",
"Finally the entropy of the dynamical system $(X,\\mu ,T)$ is defined as the supremum over all (finite) experiments: $h_{\\mu }(T) = \\sup _{I} h_{\\mu }(I, T),$ and it could be finite or infinite.",
"Now it is plausible that if $I$ is a finite generator (and hence Player I wins the above game), then $h_{\\mu }(I, T)$ should be all the information there is to obtain about $X$ and hence $I$ achieves the supremum above.",
"This is indeed the case as the following theorem (Theorem 14.33 in ) shows.",
"Theorem 1.4 (Kolmogorov-Sinai, '58-59) If $I$ is a finite generator modulo $\\mu $ -NULL, then $h_{\\mu }(T) = h_{\\mu }(I, T)$ .",
"In particular, $h_{\\mu }(T) \\le \\log (|I|) < \\infty $ Here $\\mu $ -NULL denotes the $\\sigma $ -ideal of $\\mu $ -null sets and, by definition, a statement holds modulo a $\\sigma $ -ideal $\\mathfrak {I}$ if it holds on $X \\setminus Z$ , for some $Z \\in \\mathfrak {I}$ .",
"We will also use this for MEAGER, the $\\sigma $ -ideal of meager sets in a Polish space.",
"In case of ergodic systems, i.e.",
"dynamical systems where every (measurable) invariant set is either null or co-null, the converse of Kolmogorov-Sinai theorem is true (see ): Theorem 1.5 (Krieger, '70) Suppose $(X, \\mu , T)$ is ergodic.",
"If $h_{\\mu }(T) < \\log k$ , for some $k \\ge 2$ , then there is a $k$ -generator modulo $\\mu $ -NULL.",
"§3 .",
"Now let $X$ be just a Borel $\\mathbb {Z}$ -space with no measure specified.",
"Then by the Kolmogorov-Sinai theorem, if there exists an invariant Borel probability measure on $X$ with infinite entropy, then $X$ does not admit a finite generator.",
"What happens if we remove this obstruction?",
"The following question was first stated in and restated in for an arbitrary countable group.",
"Question 1.6 (Weiss, '87).",
"Let $G$ be a countable group and $X$ be a Borel $G$ -space.",
"If $X$ does not admit any invariant Borel probability measure, does it have a finite generator?",
"Assuming that the answer to this question is positive for $G = \\mathbb {Z}$ , we prove the following dichotomy: Theorem REF .",
"Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"We remark that the nonexistence of an invariant ergodic probability measure of infinite entropy does not guarantee the existence of a finite generator.",
"For example, let $X$ be a direct sum of uniquely ergodic actions $\\mathbb {Z}^{\\curvearrowright } \\!",
"X_n$ such that the entropy $h_n$ of each $X_n$ is finite but $h_n \\rightarrow \\infty $ .",
"Then $X$ does not admit an invariant ergodic probability measure with infinite entropy since otherwise it would have to be supported on one of the $X_n$ , contradicting unique ergodicity.",
"Neither does $X$ admit a finite generator since that would contradict Krieger's theorem applied to $X_n$ , for large enough $n$ .",
"However, assuming again that the answer to REF is positive for $G = \\mathbb {Z}$ , we prove the following dichotomy suggested by Kechris: Theorem REF .",
"Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy, there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"The proofs of these dichotomies presented in Section use the Ergodic Decomposition Theorem and a version of Krieger's theorem together with Theorem REF about separating the equivalence classes of a smooth equivalence relation.",
"Definition 1.7 Let $X$ be a Borel $G$ -space and denote its Borel $\\sigma $ -algebra by $\\mathfrak {B}(X)$ .",
"For a topological property $P$ (e.g.",
"Polish, $\\sigma $ -compact, etc.",
"), we say that $X$ admits a $P$ topological realization, if there exists a Hausdorff second countable topology on $X$ satisfying $P$ such that it makes the $G$ -action continuous and its induced Borel $\\sigma $ -algebra is equal to $\\mathfrak {B}(X)$ .",
"We remark that every Borel $G$ -space admits a Polish topological realization (this is actually true for an arbitrary Polish group, but it is a highly non-trivial result of Becker and Kechris, see 5.2 in ).",
"The main result of this paper is a positive answer to Question REF in case $X$ has a $\\sigma $ -compact realization: Theorem REF .",
"Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"If there is no $G$ -invariant Borel probability measure on $X$ , then $X$ admits a Borel 32-generator.",
"For example, REF has a positive answer when $G$ acts continuously on a locally compact or even $\\sigma $ -compact Polish space.",
"Before explaining the idea of the proof of the above theorem, we present previously known results as well as other related results obtained in this paper.",
"§4 .",
"In it was shown that every aperiodic (i.e.",
"having no finite orbits) $\\mathbb {Z}$ -space admits a countable generator.",
"This has been generalized to any countable group in .",
"Theorem 1.8 (Jackson-Kechris-Louveau, '02) Every aperiodic Borel $G$ -space $X$ admits a countable generator.",
"In particular, there is a Borel $G$ -embedding of $X$ into $\\mathbb {N}^G$ .",
"Recall that this is sharp in the sense that we could not hope to obtain a finite generator solely from the aperiodicity assumption.",
"Indeed, the Kolmogorov-Sinai theorem implies that dynamical systems with infinite entropy cannot have a finite generator, and there do exist continuous aperiodic actions of $\\mathbb {Z}$ with infinite entropy (e.g.",
"the action of $\\mathbb {Z}$ on $[0,1]^{\\mathbb {Z}} \\setminus A$ by shift, where $A$ is the set of periodic points and the measure is the product of the Lebesgue measure).",
"§5 .",
"The following result gives a positive answer to a version of Question REF in the measure-theoretic context (see for $G=\\mathbb {Z}$ and for arbitrary $G$ ).",
"Theorem 1.9 (Krengel, Kuntz, '74) Let $X$ be a Borel $G$ -space and let $\\mu $ be a quasi-invariant Borel probability measure on $X$ (i.e.",
"$G$ preserves the $\\mu $ -null sets).",
"If there is no invariant Borel probability measure absolutely continuous with respect to $\\mu $ , then $X$ admits a 2-generator modulo $\\mu $ -NULL.",
"The proof uses a version of the Hajian-Kakutani-Itô theorem (see and ), which states that the hypothesis of the Krengel-Kuntz theorem is equivalent to the existence of a weakly wandering set (see Definition REF ) of positive measure.",
"We show in Section that having a weakly wandering (or even just locally weakly wandering) set of full saturation implies the existence of finite generators in the Borel context (Theorem REF ).",
"However, it was shown by Eigen-Hajian-Nadkarni in that the analogue of the Hajian-Kakutani-Itô theorem fails in the Borel context.",
"In Section , we strengthen this result by showing that it fails even in the context of Baire category (Corollary REF ).",
"This result is a consequence of a criterion for non-existence of non-meager weakly wandering sets (Theorem REF ), and it implies a negative answer to the following question asked in (question (ii) on page 9): Question 1.10 (Eigen-Hajian-Nadkarni, '93).",
"Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ does not admit an invariant probability measure, is there a countably generated (by Borel sets) partition of $X$ into invariant sets, each of which admits a weakly wandering set of full saturation?",
"Ben Miller pointed out to us that a negative answer to this question could also be inferred from Propositions 3.6 and 3.7 of his PhD thesis (see ).",
"However, the implication is indirect and these propositions do not provide a criterion for non-existence of weakly wandering sets.",
"§6 .",
"In the mid-'90s, Kechris asked whether an analogue of the Krengel-Kuntz theorem holds in the context of Baire category (see 6.6.",
"(B) in ), more precisely: Question 1.11 (Kechris, mid-'90s).",
"If a countable group $G$ acts continuously on a perfect Polish space $X$ and the action is generically ergodic (i.e.",
"every invariant Borel set is meager or comeager), does it follow that there is a finite generator on an invariant comeager set?",
"Note that a positive answer to Question REF for an arbitrary group $G$ would imply a positive answer to this question because of the following theorem (cf.",
"Theorem 13.1 in ): Theorem 1.12 (Kechris-Miller, '04) Let $X$ be a Polish $G$ -space.",
"If the action is aperiodic, then there is an invariant dense $G_{\\delta }$ set $X^{\\prime } \\subseteq X$ that does not admit an invariant Borel probability measure.",
"We give an affirmative answer to Question REF in Section ; in fact, we prove the following slightly stronger result: Theorem REF .",
"Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then the action admits a 4-generator on an invariant comeager set.",
"The proof of this uses the Kuratowski-Ulam method introduced in the proofs of Theorems 12.1 and 13.1 in .",
"This method was inspired by product forcing and its idea is as follows.",
"Suppose we want to prove the existence of an object that satisfies a certain condition on a comeager set (in our case a finite partition).",
"We give a parametrized construction of such objects $A_{\\alpha }$ , where the parameter $\\alpha $ ranges over $2^{\\mathbb {N}}$ or $\\mathbb {N}^{\\mathbb {N}}$ (or any other Polish space), and then try to show that for comeager many values of $\\alpha $ , $A_{\\alpha }$ has the desired property $\\Phi $ on a comeager set.",
"In other words, we want to prove $\\forall ^* \\alpha \\forall ^* x \\Phi (\\alpha , x)$ , where $\\forall ^*$ means “for comeager many”.",
"Now the key point is that the Kuratowski-Ulam theorem allows us to switch the order of the quantifiers and prove $\\forall ^* x \\forall ^* \\alpha \\Phi (\\alpha , x)$ instead.",
"The latter is often an easier task since it allows one to work locally with fixed $x$ .",
"§7 .",
"We now briefly outline the idea of the proof of the main result (Theorem REF ).",
"First we present an equivalent condition to the nonexistence of invariant measures that is proved by Nadkarni in and is the analogue of Tarski's theorem about paradoxical decompositions (see ) for countably additive measures.",
"Let $X$ be a Borel $G$ -space and denote the set of invariant Borel probability measures on $X$ by $\\mathcal {M}_G(X)$ .",
"Also, for $S \\subseteq X$ , let $[S]_G$ denote the saturation of $S$ , i.e.",
"$[S]_G = \\bigcup _{g \\in G} gS$ .",
"The following definition makes no reference to any invariant measure on $X$ , yet provides a sufficient condition for the measure of two sets to be equal (resp.",
"$\\le $ or $<$ ).",
"Definition 1.13 Two Borel sets $A,B \\subseteq X$ are said to be equidecomposable (denoted by $A \\sim B$ ) if there are Borel partitions $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $A$ and $B$ , respectively, and $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq G$ such that $g_n A_n = B_n$ .",
"We write $A \\preceq B$ if $A \\sim B^{\\prime } \\subseteq B$ , and we write $A \\prec B$ if moreover $[B \\setminus B^{\\prime }]_G = [B]_G$ .",
"The following explains the above definition.",
"Observation 1.14 Let $A,B \\subseteq X$ be Borel sets.",
"If $A \\sim B$ , then $\\mu (A) = \\mu (B)$ for any $\\mu \\in \\mathcal {M}_G(X)$ .",
"If $A \\preceq B$ , then $\\mu (A) \\le \\mu (B)$ for any $\\mu \\in \\mathcal {M}_G(X)$ .",
"If $A \\prec B$ , then either $\\mu (A) = \\mu (B) = 0$ or $\\mu (A) < \\mu (B)$ for any $\\mu \\in \\mathcal {M}_G(X)$ .",
"Definition 1.15 A Borel set $A \\subseteq X$ is called compressible if $A \\prec A$ .",
"It is clear from the observation above that if a Borel set $A \\subseteq X$ is compressible, then $\\mu (A) = 0$ for all $\\mu \\in \\mathcal {M}_G(X)$ .",
"In particular, if $X$ itself is compressible then $\\mathcal {M}_G(X) = \\mathbb {\\emptyset }$ .",
"Thus compressibility is an apparent obstruction to having an invariant probability measure.",
"It turns out that it is the only one: Theorem 1.16 (Nadkarni, '91) Let $X$ be a Borel $G$ -space.",
"There is an invariant Borel probability measure on $X$ if and only if $X$ is not compressible.",
"The proof of this first appeared in for $G = \\mathbb {Z}$ and is also presented in Chapter 4 of for an arbitrary countable group $G$ .",
"This theorem is what makes it “possible” to work with the hypothesis of Question REF since it equates the nonexistence of a certain kind of object (an invariant probability measure) with the existence of another (the sets witnessing the compressibility of $X$ ).",
"For a finite Borel partition $I$ of $X$ , we define the notion of $I$ -compressibility, which basically means the compressibility of the image set under $ (recall that $ is the symbolic representation of $X$ with respect to $I$ ).",
"Furthermore, for $i \\in \\mathbb {N}$ , we define the notion of $i$ -compressibility (Definition REF ) and show that if $X$ is $i$ -compressible, then it admits a $2^{i+1}$ -generator (Proposition REF ).",
"In fact, under the assumption that $X$ is compressible, having a finite generator turns out to be equivalent to $X$ being $i$ -compressible for some $i \\ge 1$ (Corollary REF ).",
"This shows that $i$ -compressibility is the right notion to look at when studying Question REF .",
"Now ideally we would like to prove that the notions of compressibility and $i$ -compressibility coincide, at least for sufficiently large $i$ , since this would imply a positive answer to Question REF .",
"This turns out to be true when $X$ has a $\\sigma $ -compact realization and we show this as follows.",
"In the proof of Nadkarni's theorem, one assumes that $X$ is not compressible and constructs an invariant probability measure.",
"We give a similar construction for $i$ -compressibility instead of compressibility, but unfortunately the proof yields only a finitely additive invariant probability measure (Corollary REF ).",
"However, with the additional assumption that $X$ is $\\sigma $ -compact, we are able to concoct a countably additive invariant measure out of it (Corollary REF ), and thus obtain Theorem REF .",
"§8 .",
"Lastly, we give a positive answer to a version of Question REF with slightly stronger hypothesis.",
"It is not hard to prove (see REF ) that for a Borel $G$ -space $X$ , the nonexistence of invariant probability measures on $X$ is equivalent to the existence of so-called traveling sets of full saturation (Definition REF ).",
"We define a slightly stronger notion of a locally finitely traveling set (Definition REF ), and show in REF that if there exists such a set of full saturation, then $X$ admits a 32-generator.",
"The proof uses the machinery discussed in $§7$ .",
"§9 Organization of the paper.",
"In Section , we develop the theory of $i$ -compressibility and establish its connection with the existence of finite generators.",
"More particularly, in Subsection we give the definition of $I$ -equidecomposability and prove the important property of orbit-disjoint countable additivity (see REF ), which is what makes $i$ (defined below) a $\\sigma $ -ideal.",
"In Subsections and REF we define the notions of $i$ -compressibility and $i$ -traveling sets and establish their connection.",
"Finally, in Subsection REF , we show how to construct a finite generator using an $i$ -traveling complete section (by definition, a complete section is a set that meets every orbit, equivalently, has full saturation).",
"In Section , we prove the main theorem, which provides means of constructing finitely additive invariant measures that are non-zero on a given non-$i$ -compressible set.",
"In the following two sections we establish two corollaries of this theorem, namely REF and REF , where the former is the main result of the paper stated above and the latter is the result discussed in $§8$ .",
"In Section , we show that given a smooth equivalence relation $E$ on $X$ with $E \\supseteq E_G$ , there exists a finite partition $\\mathcal {P}$ such that $G \\mathcal {P}$ separates points in different classes of $E$ ; in fact, we give an explicit construction of such $\\mathcal {P}$ .",
"This result is then used in the following section, where we establish the potential dichotomy theorems mentioned above (REF and REF ).",
"Section establishes the existence of a 4-generator on an invariant comeager set, and Section provides various examples of $i$ -compressible actions involving locally weakly wandering sets.",
"Finally, in Section we develop a criterion for non-existence of non-meager weakly wandering sets and derive a negative answer to Question REF .",
"§10 Open questions.",
"Here are some open questions that arose in this research.",
"Let $X$ denote a Borel $G$ -space.",
"Is $X$ being compressible equivalent to $X$ being $i$ -compressible for some $i \\ge 1$ ?",
"Does the existence of a traveling complete section imply the existence of a locally finitely traveling complete section?",
"A positive answer to any of these questions would imply a positive answer to Question REF since (A) is just a rephrasing of Question REF because of REF and for (B), it follows from REF and REF .",
"In the original version of the paper, it was also asked whether $X$ always admits a $\\sigma $ -compact realization.",
"However, this was answered negatively by Conley, Kechris and Miller.",
"§11 Acknowledgements.",
"I thank my advisor Alexander Kechris for his help, support and encouragement, in particular, for suggesting the problems and guiding me throughout the research.",
"I also thank the UCLA logic group for positive feedback and the Caltech logic group for running a series of seminars in which I presented my work.",
"Finally, I thank Ben Miller, Patrick Allen and Justin Palumbo for useful conversations and comments.",
"Finite generators and $i$ -compressibility Throughout this section let $X$ be a Borel $G$ -space and $E_G$ be the orbit equivalence relation on $X$ .",
"For $A \\subseteq X$ and $G$ -invariant $P \\subseteq X$ , let $A^P := A \\cap P$ .",
"For an equivalence relation $E$ on $X$ and $A \\subseteq X$ , let $[A]_E$ denote the saturation of $A$ with respect to $E$ , i.e.",
"$[A]_E = \\lbrace x \\in X : \\exists y \\in A (x E y)\\rbrace $ .",
"In case $E = E_G$ , we use $[A]_G$ instead of $[A]_{E_G}$ .",
"Let $\\mathfrak {B}$ denote the class of all Borel sets in standard Borel spaces and let $\\Gamma $ be a $\\sigma $ -algebra of subsets of standard Borel spaces containing $\\mathfrak {B}$ and closed under Borel preimages.",
"For example, $\\Gamma = \\mathfrak {B}$ , $\\sigma (\\mathbf {\\Sigma }_1^1)$ , universally measurable sets.",
"For $A \\subseteq X$ , let $\\Gamma (A)$ denote the set of $\\Gamma $ sets relative to $A$ , i.e.",
"$\\Gamma (A) = \\lbrace B \\cap A : B \\subseteq X, B \\in \\Gamma \\rbrace $ .",
"subsection2-2 ex5 pt The notion of $I$ -equidecomposability A countable partition of $X$ is called Borel if all the sets in it are Borel.",
"For a finite Borel partition $I= \\lbrace A_i : i < k\\rbrace $ of $X$ , let $F_{I}$ denote the equivalence relation of not being separated by $G I:= \\lbrace g A_i : g \\in G, i < k\\rbrace $ , more precisely, $\\forall x,y\\in X$ , $x F_{I}y \\Leftrightarrow x) = y),$ where $ is the symbolic representation map for $ (X, G, I)$ defined above.",
"Note that if $ I$ is a generator, then $ FI$ is just the equality relation.$ For an equivalence relation $E$ on $X$ and $A,B \\subseteq X$ , $A$ is said to be $E$ -invariant relative to $B$ or just $E \\!",
"\\!",
"\\downharpoonright _{B}$ -invariant if $[A]_{E} \\cap B = A \\cap B$ .",
"Definition 2.1 ($I$ -equidecomposability) Let $A,B \\subseteq X$ , and $I$ be a finite Borel partition of $X$ .",
"$A$ and $B$ are said to be equidecomposable with $\\Gamma $ pieces (denote by $A \\sim ^{\\Gamma } B$ ) if there are $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq G$ and partitions $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $A$ and $B$ , respectively, such that for all $n \\in \\mathbb {N}$ $g_n A_n = B_n$ , $A_n \\in \\Gamma (A)$ and $B_n \\in \\Gamma (B)$ .",
"If moreover, $A_n$ and $B_n$ are $F_{I}$ -invariant relative to $A$ and $B$ , respectively, then we will say that $A$ and $B$ are $I$ -equidecomposable with $\\Gamma $ pieces and denote it by $A \\sim _{I}^{\\Gamma } B$ .",
"If $\\Gamma = \\mathfrak {B}$ , we will not mention $\\Gamma $ and will just write $\\sim $ and $\\sim _{I}$ .",
"Note that for any $I$ , $A$ , $B$ as above, $A$ and $B$ are $I$ -equidecomposable if and only if $A)$ and $B)$ are equidecomposable (although the images of Borel sets under $ are analytic, they are Borel relative to $ X)$ due to the Lusin Separation Theorem for analytic sets).",
"Also note that if $ I$ is a generator, then $ I$ coincides with $$.$ Observation 2.2 Below let $I,I_0,I_1$ denote finite Borel partitions of $X$ , and $A,B,C \\in \\Gamma (X)$ .",
"(Quasi-transitivity) If $A \\sim _{I_0}^{\\Gamma } B \\sim _{I_1}^{\\Gamma } C$ , then $A \\sim _{I}^{\\Gamma } C$ with $I= I_0 \\vee I_1$ (the least common refinement of $I_0$ and $I_1$ ).",
"($F_{I}$ -disjoint countable additivity) Let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ be partitions of $A$ and $B$ , respectively, into $\\Gamma $ sets such that $\\forall n \\ne m$ , $[A_n]_{F_{I}} \\cap [A_m]_{F_{I}} = [B_n]_{F_{I}} \\cap [B_m]_{F_{I}} = \\mathbb {\\emptyset }$ .",
"If $\\forall n\\in \\mathbb {N}$ , $A_n \\sim _{I}^{\\Gamma } B_n$ , then $A \\sim _{I}^{\\Gamma } B$ .",
"If $A \\sim B$ , then there is a Borel isomorphism $\\phi $ of $A$ onto $B$ with $\\phi (x) E_G x$ for all $x \\in A$ ; namely $\\phi (x) = g_n x$ for all $x \\in A_n$ , where $A_n, g_n$ are as in Definition REF .",
"It is easy to see that the converse is also true, i.e.",
"if such $\\phi $ exists, then $A \\sim B$ .",
"In Proposition REF we prove the analogue of this for $\\sim _{I}^{\\Gamma }$ , but first we need the following lemma and definition that take care of definability and $F_{I}$ -invariance, respectively.",
"For a Polish space $Y$ , $f : X \\rightarrow Y$ is said to be $\\Gamma $ -measurable if the preimages of open sets under $f$ are in $\\Gamma $ .",
"For $A \\in \\Gamma (X)$ and $h : A \\rightarrow G$ , define $\\hat{h} : A \\rightarrow X$ by $x \\mapsto h(x)x$ .",
"Lemma 2.3 If $h : A \\rightarrow G$ is $\\Gamma $ -measurable, then the images and preimages of sets in $\\Gamma $ under $\\hat{h}$ are in $\\Gamma $ .",
"Let $B \\subseteq A$ , $C \\subseteq X$ be in $\\Gamma $ .",
"For $g \\in G$ , set $A_g = h^{-1}(g)$ and note that $\\hat{h}(B) = \\bigcup _{g \\in G} g(A_g \\cap B)$ and $\\hat{h}^{-1}(C) = \\bigcup _{g \\in G} g^{-1}(gA_g \\cap C)$ .",
"Thus $\\hat{h}(B)$ and $\\hat{h}^{-1}(C)$ are in $\\Gamma $ by the assumptions on $\\Gamma $ .",
"The following technical definition is needed in the proofs of REF and REF .",
"Definition 2.4 For $A \\subseteq X$ and a finite Borel partition $I$ of $X$ , we say that $I$ is $A$ -sensitive or that $A$ respects $I$ if $A$ is $F_{I}$ -invariant relative to $[A]_G$ , i.e.",
"$[A]_{F_{I}}^{[A]_G} = A$ .",
"For example, if $I$ is finer than $\\lbrace A, A^c\\rbrace $ , then $I$ is $A$ -sensitive.",
"Note that if $A \\sim _{I} B$ and $A$ respects $I$ , then so does $B$ .",
"Proposition 2.5 Let $A,B \\in \\Gamma (X)$ and let $I$ be a Borel partition of $X$ that is $A$ -sensitive.",
"Then, $A \\sim _{I}^{\\Gamma } B$ if and only if there is an $F_{I}$ -invariant $\\Gamma $ -measurable map $\\gamma : A \\rightarrow G$ such that $\\hat{\\gamma }$ is a bijection between $A$ and $B$ .",
"We refer to such $\\gamma $ as a witnessing map for $A \\sim _{I}^{\\Gamma } B$ .",
"The same holds if we delete “$F_{I}$ -invariant” and “$I$ ” from the statement.",
"$\\Rightarrow $ : If $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ , $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ are as in Definition REF , then define $\\gamma : A \\rightarrow G$ by setting $\\gamma \\!",
"\\!",
"\\downharpoonright _{A_n} \\equiv g_n$ .",
"$\\Leftarrow $ : Let $\\gamma $ be as in the lemma.",
"Fixing an enumeration $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ of $G$ with no repetitions, put $A_n = \\gamma ^{-1}(g_n)$ and $B_n = g_n A_n$ .",
"It is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ are partitions of $A$ and $B$ , respectively, into $\\Gamma $ sets.",
"Since $\\gamma $ is $F_{I}$ -invariant, each $A_n$ is $F_{I}$ -invariant relative to $A$ and hence relative to $P := [A]_G = [B]_G$ because $A$ respects $I$ .",
"It remains to show that each $B_n$ is $F_{I}$ -invariant relative to $B$ .",
"To this end, let $y \\in [B_n]_{F_{I}} \\cap B$ and thus there is $x \\in A_n$ such that $y F_{I}g_n x$ .",
"Hence $z := g_n^{-1} y \\ F_{I}\\ g_n^{-1} g_n x = x$ and therefore $z \\in A_n$ because $A_n$ is $F_{I}$ -invariant relative to $P$ .",
"Thus $y = g_n z \\in B_n$ .",
"In the rest of the subsection we work with $\\Gamma = \\mathfrak {B}$ .",
"Next we prove that $I$ -equidecomposability can be extended to $F_{I}$ -invariant Borel sets.",
"First we need the following reflection principle.",
"Lemma 2.6 (A reflection principle) Let $E$ be a Borel equivalence relation on $X$ and $B \\subseteq X$ be Borel.",
"Define the predicate $\\Phi \\subseteq Pow(X)$ as follows: $\\Phi (A) \\Leftrightarrow A \\subseteq B \\wedge A \\text{ is $E$-invariant}.$ If $A$ is analytic and $\\Phi (A)$ then there exists a Borel set $A^{\\prime } \\supseteq A$ with $\\Phi (A^{\\prime })$ .",
"Define the predicate $\\Psi \\subseteq Pow(X)$ as follows: $\\Psi (D) \\Leftrightarrow B^c \\subseteq D \\wedge D \\text{ is $E$-invariant}.$ It is clear that $\\Phi (D) \\Leftrightarrow \\Psi (D^c)$ , so it is enough to show that if $D$ is co-analytic and $\\Psi (D)$ , then there is a Borel set $D^{\\prime } \\subseteq D$ with $\\Psi (D^{\\prime })$ .",
"Note that $\\Psi (D) \\Leftrightarrow \\forall x \\in X \\forall y \\in X (x \\notin D \\wedge y \\in D \\Rightarrow x \\notin B \\wedge \\lnot xEy).$ Thus, setting $R(x,y) \\Leftrightarrow x \\notin B \\wedge \\lnot xEy$ , we apply The Burgess Reflection Theorem (see 35.18 in ) to $\\Psi $ with $\\Gamma = \\mathbf {\\Pi }_1^1$ and $A = D$ , and get a Borel $D^{\\prime } \\subseteq D$ with $\\Psi (D^{\\prime })$ .",
"Proposition 2.7 ($F_{I}$ -invariant extensions) If for some Borel partition $I$ of $X$ and Borel sets $A,B \\subseteq X$ , $A \\sim _{I} B$ , then there exists Borel sets $A^{\\prime } \\supseteq A$ and $B^{\\prime } \\supseteq B$ such that $A^{\\prime },B^{\\prime }$ are $F_{I}$ -invariant and $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"In fact, if $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ witness $A \\sim _{I} B$ , then there are $F_{I}$ -invariant Borel partitions $\\lbrace A^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}$ of $A^{\\prime }$ and $B^{\\prime }$ respectively, such that $g_n A^{\\prime }_n = B^{\\prime }_n$ and $A^{\\prime }_n \\supseteq A_n$ (and hence $B^{\\prime }_n \\supseteq B_n$ ).",
"Let $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF and put $\\bar{A}_n = [A_n]_{F_{I}}$ .",
"It is easy to see that for $n \\ne m \\in \\mathbb {N}$ , $\\bar{A}_n \\cap \\bar{A}_m = \\mathbb {\\emptyset }$ ; $g_n \\bar{A}_n \\cap g_m \\bar{A}_m = \\mathbb {\\emptyset }$ .",
"Put $\\bar{A} = [A]_{F_{I}}$ and note that $\\lbrace \\bar{A}_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $\\bar{A}$ .",
"Although $\\bar{A}_n$ and $\\bar{A}$ are $F_{I}$ -invariant, they are analytic and in general not Borel.",
"We obtain Borel analogues using $\\mathbf {\\Pi }_1^1$ -reflection theorems.",
"Set $U = \\bigcup _{n \\in \\mathbb {N}}(n \\times \\bar{A}_n)$ and define a predicate $\\Phi \\subseteq Pow(\\mathbb {N}\\times X)$ as follows: $\\Phi (W) \\Leftrightarrow \\forall n (W_n \\text{ is $F_{I}$-invariant}) \\wedge \\forall n \\ne m (W_n \\cap W_m = \\mathbb {\\emptyset } \\wedge g_n W_n \\cap g_m W_m = \\mathbb {\\emptyset }),$ where $W_n = \\lbrace x \\in X : (n,x) \\in W\\rbrace $ , the section of $W$ at $n$ .",
"Note that $\\Phi (U)$ .",
"Claim There is a Borel set $U^{\\prime } \\supseteq U$ with $\\Phi (U^{\\prime })$ .",
"Proof of Claim.",
"For $W \\subseteq \\mathbb {N}\\times X$ , let $\\Lambda (W) \\Leftrightarrow \\forall n \\ne m (W_n \\cap W_m = \\mathbb {\\emptyset } \\wedge g_n W_n \\cap g_m W_m = \\mathbb {\\emptyset })$ Note that $\\Lambda (W) \\Leftrightarrow \\forall n \\ne m \\forall x \\in X [(x \\notin W_n \\vee x \\notin W_m) \\wedge (x \\notin g_n W_n \\vee x \\notin g_m W_m)].$ Thus $\\Lambda $ is $\\mathbf {\\Pi }_1^1$ on $\\mathbf {\\Sigma }_1^1$ , and hence, by the dual form of the First Reflection Theorem for $\\mathbf {\\Pi }_1^1$ (see the discussion following 35.10 in ), there is a Borel set $V \\supseteq U$ with $\\Lambda (V)$ , since $\\Lambda (U)$ .",
"Now applying Lemma REF to $E = F_{I}$ , $B = V_n$ and $A = U_n$ , we get $F_{I}$ -invariant Borel sets $U^{\\prime }_n \\subseteq X$ with $U_n \\subseteq U^{\\prime }_n \\subseteq V_n$ .",
"Thus $U^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}(n \\times U^{\\prime }_n)$ is what we wanted.",
"$\\dashv $ Put $A^{\\prime }_n = U^{\\prime }_n$ and $A^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}A^{\\prime }_n$ .",
"Thus $\\lbrace A^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $A^{\\prime }$ into $F_{I}$ -invariant Borel sets.",
"Also, $A_n \\subseteq \\bar{A_n} \\subseteq A^{\\prime }_n$ and hence $A \\subseteq \\bar{A} \\subseteq A^{\\prime }$ .",
"Put $B_n^{\\prime } = g_n A^{\\prime }_n$ and $B^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}B_n^{\\prime }$ ; thus $\\lbrace B_n^{\\prime }\\rbrace _{n \\in \\mathbb {N}}$ is a Borel partition of $B^{\\prime }$ .",
"Also note that $B_n^{\\prime }$ are $F_{I}$ -invariant and $B^{\\prime } \\supseteq B$ since $A^{\\prime }_n$ are $F_{I}$ -invariant and $A_n \\subseteq A^{\\prime }_n$ .",
"Thus $A^{\\prime } \\sim _{I} B^{\\prime }$ and we are done.",
"Lemma 2.8 (Orbit-disjoint unions) Let $A_k,B_k \\in \\mathfrak {B}(X)$ , $k=0,1$ , be such that $[A_0]_G$ and $[A_1]_G$ are disjoint and put $A = A_0 \\cup A_1$ and $B = B_0 \\cup B_1$ .",
"If $I$ is an $A,B$ -sensitive finite Borel partition of $X$ such that $A_k \\sim _{I} B_k$ for $k=0,1$ , then $A \\sim _{I} B$ .",
"Moreover, if $\\gamma _0 : A_0 \\rightarrow G$ is a Borel map witnessing $A_0 \\sim _{I} B_0$ , then there exists a Borel map $\\gamma : A \\rightarrow G$ extending $\\gamma _0$ that witnesses $A \\sim _{I} B$ .",
"First assume without loss of generality that $X = [A]_G$ ($= [B]_G$ ) since the statement of the lemma is relative to $[A]_G$ .",
"Thus $A,B$ are $F_{I}$ -invariant.",
"Applying REF to $A_0 \\sim _{I} B_0$ , we get $F_{I}$ -invariant $A^{\\prime }_0 \\supseteq A_0, B^{\\prime }_0 \\supseteq B_0$ such that $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"Moreover, by the second part of the same lemma, if $\\gamma _0 : A_0 \\rightarrow G$ is a witnessing map for $A_0 \\sim _{I} B_0$ , then there is a witnessing map $\\delta : A^{\\prime }_0 \\rightarrow G$ for $A^{\\prime } \\sim _{I} B^{\\prime }$ extending $\\gamma _0$ .",
"Put $C = A^{\\prime }_0 \\cap A$ and note that $C$ is $F_{I}$ -invariant since so are $A^{\\prime }_0$ and $A$ .",
"Finally, put $\\bar{A}_0 = \\lbrace x \\in C : C^{[x]_G} = A^{[x]_G} \\wedge \\hat{\\delta }(C^{[x]_G}) = B^{[x]_G}\\rbrace $ and note that $\\bar{A}_0 \\supseteq A_0$ since $\\delta \\supseteq \\gamma _0$ and $[A_0]_G \\cap [A_1]_G = \\mathbb {\\emptyset }$ .",
"Claim $\\bar{A}_0$ is $F_{I}$ -invariant.",
"Proof of Claim.",
"First note that for any $F_{I}$ -invariant $D \\subseteq X$ and $z \\in X$ , $[D^{[z]_G}]_{F_{I}} = D^{[[z]_{F_{I}}]_G}$ .",
"Furthermore, if $D \\subseteq C$ , then $[\\hat{\\delta }(D)]_{F_{I}} = \\hat{\\delta }([D]_{F_{I}})$ since $\\hat{\\delta }$ and its inverse map $F_{I}$ -invariant sets to $F_{I}$ -invariant sets.",
"Now take $x \\in \\bar{A}_0$ and let $Q = [[x]_{F_{I}}]_G$ .",
"Since $A, B, C$ are $F_{I}$ -invariant, $C^Q = [C^{[x]_G}]_{F_{I}} = [A^{[x]_G}]_{F_{I}} = A^Q$ .",
"Furthermore, $\\hat{\\delta }(C^Q) = \\hat{\\delta }([C^{[x]_G}]_{F_{I}}) = [\\hat{\\delta }(C^{[x]_G})]_{F_{I}} = [B^{[x]_G}]_{F_{I}} = B^Q$ .",
"Thus, $\\forall y \\in [x]_{F_{I}}$ , $C^{[y]_G} = A^{[y]_G}$ and $\\hat{\\delta }(C^{[y]_G}) = B^{[y]_G}$ ; hence $[x]_{F_{I}} \\subseteq \\bar{A}_0$ .",
"$\\dashv $ Put $\\bar{A}_1 = A \\setminus \\bar{A}_0$ , $\\alpha _0 = \\delta \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_0}$ , $\\alpha _1 = \\gamma _1 \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_1}$ , where $\\gamma _1$ is a witnessing map for $A_1 \\sim _{I} B_1$ .",
"It is clear from the definition of $\\bar{A}_0$ that $\\bar{A}_0$ is $E_G$ -invariant relative to $A$ and hence $[\\bar{A}_0]_G \\cap [\\bar{A}_1]_G = \\mathbb {\\emptyset }$ .",
"Thus, for $k=0,1$ , it follows that $\\alpha _k$ witnesses $\\bar{A}_k \\sim _{I} \\bar{B}_k$ , where $\\bar{B}_k = \\hat{\\alpha _k}(\\bar{A}_k)$ .",
"Furthermore, it is clear that $B^{[\\bar{A}_k]_G} = \\bar{B}_k$ and, since $[\\bar{A}_0]_G \\cup [\\bar{A}_1]_G = X$ , $\\bar{B}_0 \\cup \\bar{B}_1 = B$ .",
"Now since $\\bar{A}_k$ are $F_{I}$ -invariant, $\\gamma = \\alpha _0 \\cup \\alpha _1$ is $F_{I}$ -invariant and hence witnesses $A \\sim _{I} B$ .",
"Finally, $\\alpha _0 \\!",
"\\!",
"\\downharpoonright _{A_0} = \\delta \\!",
"\\!",
"\\downharpoonright _{A_0} = \\gamma _0$ and hence $\\alpha _0 \\supseteq \\gamma _0$ .",
"Proposition 2.9 (Orbit-disjoint countable unions) For $k \\in \\mathbb {N}$ , let $A_k,B_k \\in \\mathfrak {B}(X)$ be such that $[A_k]_G$ are disjoint and put $A = \\bigcup _{k \\in \\mathbb {N}} A_k$ , $B = \\bigcup _{k \\in \\mathbb {N}} B_k$ .",
"Suppose that $I$ is an $A,B$ -sensitive finite Borel partition of $X$ such that $A_k \\sim _{I} B_k$ for all $k$ .",
"Then $A \\sim _{I} B$ .",
"We recursively apply Lemma REF as follows.",
"Put $\\bar{A}_n = \\bigcup _{k \\le n} A_k$ and $\\bar{B}_n = \\bigcup _{k \\le n} B_k$ .",
"Inductively define Borel maps $\\gamma _n : \\bigcup _{k \\le n} A_k \\rightarrow G$ such that $\\gamma _n$ is a witnessing map for $\\bar{A}_n \\sim _{I} \\bar{B}_n$ and $\\gamma _n \\sqsubseteq \\gamma _{n+1}$ .",
"Let $\\gamma _0$ be a witnessing map for $A_0 \\sim _{I} B_0$ .",
"Assume $\\gamma _n$ is defined.",
"Then $\\gamma _{n+1}$ is provided by Lemma REF applied to $\\bar{A}_n$ and $A_{n+1}$ with $\\gamma _n$ as a witness for $\\bar{A}_n \\sim _{I} \\bar{B}_n$ .",
"Thus $\\gamma _n \\sqsubseteq \\gamma _{n+1}$ and $\\gamma _{n+1}$ witnesses $\\bar{A}_{n+1} \\sim _{I} \\bar{B}_{n+1}$ .",
"Now it just remains to show that $\\gamma := \\bigcup _{n \\in \\mathbb {N}}\\gamma _n$ is $F_{I}$ -invariant since then it follows that $\\gamma $ witnesses $A \\sim _{I} B$ .",
"Let $x,y \\in A$ be $F_{I}$ -equivalent.",
"Then there is $n$ such that $x,y \\in \\bar{A}_n$ .",
"By induction on $n$ , $\\gamma _n$ is $F_{I}$ -invariant and, since $\\gamma \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_n} = \\gamma _n$ , $\\gamma (x) = \\gamma (y)$ .",
"Corollary 2.10 (Finite quasi-additivity) For $k=0,1$ , let $A_k,B_k \\in \\mathfrak {B}(X)$ be such that $A_0 \\cap A_1 = B_0 \\cap B_1 = \\mathbb {\\emptyset }$ and put $A = A_0 \\cup A_1$ , $B = B_0 \\cup B_1$ .",
"Let $I_k$ be an $A_k,B_k$ -sensitive finite Borel partition of $X$ .",
"If $A_0 \\sim _{I_0} B_0$ and $A_1 \\sim _{I_1} B_1$ , then $A \\sim _{I_0 \\vee I_1} B$ .",
"Put $I= I_0 \\vee I_1$ , $P = [A_0]_G \\cap [A_1]_G$ , $Q = [A_0]_G \\setminus [A_1]_G$ and $R = [A_1]_G \\setminus [A_0]_G$ .",
"Then $A_k^P, B_k^P$ respect $I$ , and thus $[A_0]_{F_{I}}^P \\cap [A_1]_{F_{I}}^P = \\mathbb {\\emptyset }$ , $[B_0]_{F_{I}}^P \\cap [B_1]_{F_{I}}^P = \\mathbb {\\emptyset }$ .",
"Hence $A^P \\sim _{I} B^P$ since the sets that are $F_{I}$ -invariant relative to $A_k^P$ are also $F_{I}$ -invariant relative to $A^P$ , and the same is true for $B_k^P$ and $B^P$ .",
"Also, $A^Q \\sim _{I} B^Q$ and $A^R \\sim _{I} B^R$ because $A^Q = A_0$ , $B^Q = B_0$ , $A^R = A_1$ , $B^R = B_1$ .",
"Now since $P,Q,R$ are pairwise disjoint, it follows from Proposition REF that $A \\sim _{I} B$ .",
"subsection2-2 ex5 pt The notion of $i$ -compressibility For a finite collection $\\mathcal {F}$ of subsets of $X$ , let $< \\!\\!",
"\\mathcal {F} \\!\\!",
">$ denote the partition of $X$ generated by $\\mathcal {F}$ .",
"Definition 2.11 ($i$ -equidecomposibility) For $i \\ge 1$ , $A,B \\subseteq X$ , we say that $A$ and $B$ are $i$ -equidecomposable with $\\Gamma $ pieces (write $A \\sim _i^{\\Gamma } B$ ) if there is an $A$ -sensitive partition $I$ of $X$ generated by $i$ Borel sets such that $A \\sim _{I}^{\\Gamma } B$ .",
"For a collection $\\mathcal {F}$ of Borel sets, we say that $\\mathcal {F}$ witnesses $A \\sim _i^{\\Gamma } B$ if $|\\mathcal {F}| = i$ , $I:= < \\!\\!",
"\\mathcal {F} \\!\\!",
">$ is $A$ -sensitive and $A \\sim _{I}^{\\Gamma } B$ .",
"Remark.",
"In the above definition, it might seem more natural to have $i$ be the cardinality of the partition $I$ instead of the cardinality of the collection $\\mathcal {F}$ generating $I$ .",
"However, our definition above of $i$ -equidecomposability is needed in order to show that the collection $i$ defined below forms a $\\sigma $ -ideal.",
"More precisely, the presence of $\\mathcal {F}$ is needed in the definition of $i^*$ -compressibility, which ensures that the partition $I$ in the proof of REF is $B$ -sensitive.",
"For $i \\ge 1$ , $A,B \\subseteq X$ , we write $A \\preceq _i^{\\Gamma } B$ if there is a $\\Gamma $ set $B^{\\prime } \\subseteq B$ such that $A \\sim _i^{\\Gamma } B^{\\prime }$ .",
"If moreover $[A \\setminus B]_G = [A]_G$ , then we write $A \\prec _i^{\\Gamma } B$ .",
"If $\\Gamma = \\mathfrak {B}$ , we simply write $\\sim _i, \\preceq _i, \\prec _i$ .",
"Definition 2.12 ($i$ -compressibility) For $i \\in \\mathbb {N}$ , $A \\subseteq X$ , we say that $A$ is $i$ -compressible with $\\Gamma $ pieces if $A \\prec _i^{\\Gamma } A$ .",
"Unless specified otherwise, we will be working with $\\Gamma = \\mathfrak {B}$ , in which case we simply say $i$ -compressible.",
"For a collection of sets $\\mathcal {F}$ and a $G$ -invariant set $P$ , set $\\mathcal {F}^P = \\lbrace A^P : A \\in \\mathcal {F}\\rbrace $ .",
"We will use the following observations without mentioning.",
"Observation 2.13 Let $i,j \\ge 2$ , $A,A^{\\prime },B,B^{\\prime },C \\in \\mathfrak {B}$ .",
"Let $P \\subseteq [A]_G$ denote a $G$ -invariant Borel set and $\\mathcal {F}, \\mathcal {F}_0, \\mathcal {F}_1$ denote finite collections of Borel sets.",
"If $A \\sim _i B$ then $A^P \\sim _i B^P$ .",
"If $\\mathcal {F}$ witnesses $A \\sim _i B$ , then so does $\\mathcal {F}^{[A]_G}$ .",
"If $A \\sim _i B \\sim _j C$ , then $A \\sim _{(i+j)} C$ .",
"In fact, $\\mathcal {F}_0$ and $\\mathcal {F}_1$ witness $A \\sim _i B$ and $B \\sim _j C$ , respectively, then $\\mathcal {F}= \\mathcal {F}_0 \\cup \\mathcal {F}_1$ witnesses $A \\sim _{(i+j)} C$ .",
"If $A \\preceq _i B \\preceq _j C$ , then $A \\preceq _{(i+j)} C$ .",
"If one of the first two $\\preceq $ is $\\prec $ then $A \\prec _{(i+j)} C$ .",
"If $A \\sim _i B$ and $A^{\\prime } \\sim _j B^{\\prime }$ with $A \\cap A^{\\prime } = B \\cap B^{\\prime } = \\mathbb {\\emptyset }$ , then $A \\cup A^{\\prime } \\sim _{(i+j)} B \\cup B^{\\prime }$ .",
"Part (e) follows from REF , and the rest follows directly from the definition of $i$ -equidecomposability and REF .",
"Lemma 2.14 If a Borel set $A \\subseteq X$ is $i$ -compressible, then so is $[A]_G$ .",
"In fact, if $\\mathcal {F}$ is a finite collection of Borel sets witnessing the $i$ -compressibility of $A$ , then it also witnesses that of $[A]_G$ .",
"Let $B \\subseteq A$ be a Borel set such that $[A \\setminus B]_G = [A]_G$ and $A \\sim _i B$ .",
"Furthermore, let $I$ be an $A,B$ -sensitive partition generated by a collection $\\mathcal {F}$ of $i$ Borel sets such that $A \\sim _{I} B$ .",
"Let $\\gamma : A \\rightarrow G$ be a witnessing map for $A \\sim _{I} B$ .",
"Put $A^{\\prime } = [A]_G$ , $B^{\\prime } = B \\cup (A^{\\prime } \\setminus A)$ and note that $A^{\\prime },B^{\\prime }$ respect $I$ .",
"Define $\\gamma ^{\\prime } : A^{\\prime } \\rightarrow G$ by setting $\\gamma ^{\\prime } \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A} = id \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A}$ and $\\gamma ^{\\prime } \\!",
"\\!",
"\\downharpoonright _{A} = \\gamma $ .",
"Since $A^{\\prime }$ respects $I$ and $id \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A}, \\gamma $ are $F_{I}$ -invariant, $\\gamma ^{\\prime }$ is $F_{I}$ -invariant and thus clearly witnesses $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"The following is a technical refinement of the definition of $i$ -compressibility that is (again) necessary for $i$ , defined below, to be a $\\sigma $ -ideal.",
"Definition 2.15 ($i^*$ -compressibility) For $i \\ge 1$ , we say that a Borel set $A$ is $i^*$ -compressible if there is a Borel set $B \\subseteq A$ such that $[A \\setminus B]_G = [A]_G =: P$ , $A \\sim _i B$ , and the latter is witnessed by a collection $\\mathcal {F}$ of Borel sets such that $B \\in \\mathcal {F}^{P}$ .",
"Finally, for $i \\ge 1$ , put $i = \\lbrace A \\subseteq X : \\text{there is a $G$-invariant Borel set $P \\supseteq A$ such that $P$ is $i^*$-compressible}\\rbrace .$ Lemma 2.16 Let $i \\ge 1$ and $A \\subseteq X$ be Borel.",
"If $A \\prec _i A$ , then $A \\in {i+1}$ .",
"Setting $P = [A]_G$ and applying REF , we get that $P \\prec _i P$ , i.e.",
"there is $B \\subseteq P$ such that $[P \\setminus B]_G = P$ and $P \\sim _i B$ .",
"Let $\\mathcal {F}$ be a collection of Borel sets witnessing the latter fact.",
"Then $\\mathcal {F}^{\\prime } = \\mathcal {F}\\cup \\lbrace B\\rbrace $ witnesses $P \\sim _{(i+1)} B$ and contains $B$ .",
"Proposition 2.17 For all $i \\ge 1$ , $i$ is a $\\sigma $ -ideal.",
"We only need to show that $i$ is closed under countable unions.",
"For this it is enough to show that if $A_n \\in \\mathfrak {B}(X)$ are $i^*$ -compressible $G$ -invariant Borel sets, then so is $A := \\bigcup _{n \\in \\mathbb {N}}A_n$ .",
"We may assume that $A_n$ are pairwise disjoint since we could replace each $A_n$ by $A_n \\setminus (\\bigcup _{k<n} A_k)$ .",
"Let $B_n \\subseteq A_n$ be a Borel set and $\\mathcal {F}_n = \\lbrace F^n_k\\rbrace _{k<i}$ be a collection of Borel sets with $(F^n_0)^{A_n} = B_n$ such that $\\mathcal {F}_n$ witnesses $A_n \\sim _i B_n$ and $[A_n \\setminus B_n]_G = A_n$ .",
"Using part (b) of REF , we may assume that $\\mathcal {F}_n^{A_n} = \\mathcal {F}_n$ ; in particular, $F^n_0 = B_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}}B_n$ and $F_k = \\bigcup _{n \\in \\mathbb {N}}F^n_k$ , $\\forall k<i$ ; note that $F_0 = B$ .",
"Set $\\mathcal {F}= \\lbrace F_k\\rbrace _{k<i}$ and $I= < \\!\\!",
"\\mathcal {F} \\!\\!",
">$ .",
"Since $B \\in \\mathcal {F}$ and $A$ is $G$ -invariant, $I$ is $A,B$ -sensitive.",
"Furthermore, since $\\mathcal {F}^{A_n} = \\mathcal {F}_n$ , $A_n \\sim _{I} B_n$ for all $n \\in \\mathbb {N}$ .",
"Thus, by REF , $A \\sim _{I} B$ and hence $A$ is $i^*$ -compressible.",
"subsection2-2 ex5 pt Traveling sets Definition 2.18 Let $A \\in \\Gamma (X)$ .",
"We call $A$ a traveling set with $\\Gamma $ pieces if there exists pairwise disjoint sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ in $\\Gamma (X)$ such that $A_0 = A$ and $A \\sim ^{\\Gamma } A_n$ , $\\forall n \\in \\mathbb {N}$ .",
"For a finite Borel partition $I$ , we say that $A$ is $I$ -traveling with $\\Gamma $ pieces if $A$ respects $I$ and the above condition holds with $\\sim ^{\\Gamma }$ replaced by $\\sim _{I}^{\\Gamma }$ .",
"For $i \\ge 1$ , we say that $A$ is $i$ -traveling if it is $I$ -traveling for some $A$ -sensitive partition $I$ generated by a collection of $i$ Borel sets.",
"Definition 2.19 For a set $A \\subseteq X$ , a function $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ is called a travel guide for $A$ if $\\forall x \\in A, \\gamma (x)(0) = 1_G$ and $\\forall (x,n) \\ne (y,m) \\in A \\times \\mathbb {N}$ , $\\gamma (x)(n)x \\ne \\gamma (y)(m)y$ .",
"For $A \\in \\Gamma (X)$ , a $\\Gamma $ -measurable map $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ and $n \\in \\mathbb {N}$ , set $\\gamma _n := \\gamma (\\cdot )(n) : A \\rightarrow G$ and note that $\\gamma _n$ is also $\\Gamma $ -measurable.",
"Observation 2.20 Suppose $A \\in \\Gamma (X)$ and $I$ is an $A$ -sensitive finite Borel partition of $X$ .",
"Then $A$ is $I$ -traveling with $\\Gamma $ pieces if and only if it has a $\\Gamma $ -measurable $F_{I}$ -invariant travel guide.",
"Follows from definitions and Proposition REF .",
"Now we establish the connection between compressibility and traveling sets.",
"Lemma 2.21 Let $I$ be a finite Borel partition of $X$ , $P \\in \\Gamma (X)$ be a Borel $G$ -invariant set and let $A,B$ be $\\Gamma $ subsets of $P$ .",
"If $P \\sim _{I}^{\\Gamma } B$ , then $P \\setminus B$ is $I$ -traveling with $\\Gamma $ pieces.",
"Conversely, if $A$ is $I$ -traveling with $\\Gamma $ pieces, then $P \\sim _{I}^{\\Gamma } (P \\setminus A)$ .",
"The same is true if we replace $\\sim _{I}^{\\Gamma }$ and “$I$ -traveling” with $\\sim ^{\\Gamma }$ and “traveling”, respectively.",
"For the first statement, let $\\gamma : X \\rightarrow G$ be a witnessing map for $X \\sim _{I}^{\\Gamma } B$ .",
"Put $A^{\\prime } = X \\setminus B$ and note that $A^{\\prime }$ respects $I$ since so does $P$ and hence $B$ .",
"We show that $A^{\\prime }$ is $I$ -traveling.",
"Put $A_n = (\\hat{\\gamma })^n(A^{\\prime })$ , for each $n \\ge 0$ .",
"It follows from injectivity of $\\hat{\\gamma }$ that $A_n$ are pairwise disjoint.",
"For all $n$ , recursively define $\\delta _n : A^{\\prime } \\rightarrow G$ as follows $\\left\\lbrace \\begin{array}{l}\\delta _0 = \\gamma \\!",
"\\!",
"\\downharpoonright _{A^{\\prime }} \\\\\\delta _{n+1} = \\gamma \\circ \\hat{\\delta }_n\\end{array}\\right..$ It follows from $F_{I}$ -invariance of $\\gamma $ that each $\\delta _n$ is $F_{I}$ -invariant.",
"It is also clear that $\\hat{\\delta }_n = (\\hat{\\gamma })^n$ and hence $\\delta _n$ is a witnessing map for $A^{\\prime } \\sim _{I}^{\\Gamma } A_n$ .",
"Thus $A^{\\prime }$ is $i$ -traveling with $\\Gamma $ pieces.",
"For the converse, assume that $A$ is $I$ -traveling and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"In particular, each $A_n$ respects $I$ and $A_n \\sim _{I}^{\\Gamma } A_m$ , for all $n,m \\in \\mathbb {N}$ .",
"Let $P^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}A_n$ and $B^{\\prime } = \\bigcup _{n \\ge 1} A_n$ .",
"Since $A_n \\sim _{I}^{\\Gamma } A_{n+1}$ , part (b) of REF implies that $P^{\\prime } \\sim _{I}^{\\Gamma } B^{\\prime }$ .",
"Moreover, since $P \\setminus P^{\\prime } \\sim _{I}^{\\Gamma } P \\setminus P^{\\prime }$ , we get $P \\sim _{I}^{\\Gamma } (B^{\\prime } \\cup (P \\setminus P^{\\prime })) = P \\setminus A$ .",
"For a $G$ -invariant set $P$ and $A \\subseteq P$ , we say that $A$ is a complete section for $P$ if $[A]_G = P$ .",
"The above lemma immediately implies the following.",
"Proposition 2.22 Let $P \\in \\Gamma (X)$ be $G$ -invariant and $i \\ge 1$ .",
"$P$ is $i$ -compressible with $\\Gamma $ pieces if and only if there exists a complete section for $P$ that is $i$ -traveling with $\\Gamma $ pieces.",
"The same is true with “$i$ -compressible” and “$i$ -traveling” replaced by “compressible” and “traveling”.",
"We need the following lemma in the proofs of REF and REF .",
"Lemma 2.23 Suppose $A \\subseteq X$ is an invariant analytic set that does not admit an invariant Borel probability measure.",
"Then there is an invariant Borel set $A^{\\prime } \\supseteq A$ that still does not admit an invariant Borel probability measure.",
"Let $\\mathcal {M}$ denote the standard Borel space of $G$ -invariant Borel probability measures on $X$ (see Section 17 in ).",
"Let $\\Phi \\subseteq Pow(X)$ be the following predicate: $\\Phi (W) \\Leftrightarrow \\forall \\mu \\in \\mathcal {M}(\\mu (W) = 0).$ Claim There is a Borel set $B \\supseteq A$ with $\\Phi (B)$ .",
"Proof of Claim.",
"By the dual form of the First Reflection Theorem for $\\mathbf {\\Pi }_1^1$ (see the discussion following 35.10 in ), it is enough to show that $\\Phi $ is $\\mathbf {\\Pi }_1^1$ on $\\mathbf {\\Sigma }_1^1$ .",
"To this end, let $Y$ be a Polish space and $D \\subseteq Y \\times X$ be analytic.",
"Then, for any $n \\in \\mathbb {N}$ , the set $H_n = \\lbrace (\\mu , y) \\in \\mathcal {M}\\times Y : \\mu (D_y) > {1 \\over n}\\rbrace ,$ is analytic by a theorem of Kondô-Tugué (see 29.26 of ), and hence so are the sets $H^{\\prime }_n := \\text{proj}_{Y}(H_n)$ and $H := \\bigcup _{n \\in \\mathbb {N}}H^{\\prime }_n$ .",
"Finally, note that $\\lbrace y \\in Y : \\Phi (A_y)\\rbrace = \\lbrace y \\in Y : \\exists \\mu \\in \\mathcal {M}\\exists n \\in \\mathbb {N}(\\mu (A_y) > {1 \\over n})\\rbrace ^c = H^c,$ and so $\\lbrace y \\in Y : \\Phi (A_y)\\rbrace $ is $\\mathbf {\\Pi }_1^1$ .",
"$\\dashv $ Now put $A^{\\prime } = (B)_G$ , where $(B)_G = \\lbrace x \\in B : [x]_G \\subseteq B\\rbrace $ .",
"Clearly, $A^{\\prime }$ is an invariant Borel set, $A^{\\prime } \\supseteq A$ , and $\\Phi (A^{\\prime })$ since $A^{\\prime } \\subseteq B$ and $\\Phi (B)$ .",
"Proposition 2.24 Let $X$ be a Borel $G$ -space.",
"The following are equivalent: $X$ is compressible with universally measurable pieces; There is a universally measurable complete section that is a traveling set with universally measurable pieces; There is no $G$ -invariant Borel probability measure on $X$ ; $X$ is compressible with Borel pieces; There is a Borel complete section that is a traveling set with Borel pieces.",
"Equivalence of (1) and (2) as well as (4) and (5) is asserted in REF , (4)$\\Rightarrow $ (1) is trivial, and (3)$\\Rightarrow $ (4) follows from Nadkarni's theorem (see REF ).",
"It remains to show (1)$\\Rightarrow $ (3).",
"To this end, suppose $X \\sim ^{\\Gamma } B$ , where $B^c = X \\setminus B$ is a complete section and $\\Gamma $ is the class of universally measurable sets.",
"If there was a $G$ -invariant Borel probability measure $\\mu $ on $X$ , then $\\mu (X) = \\mu (B)$ and hence $\\mu (B^c) = 0$ .",
"But since $B^c$ is a complete section, $X = \\bigcup _{g \\in G} gB^c$ , and thus $\\mu (X) = 0$ , a contradiction.",
"Now we prove an analogue of this for $i$ -compressibility.",
"Proposition 2.25 Let $X$ be a Borel $G$ -space.",
"For $i \\ge 1$ , the following are equivalent: $X$ is $i$ -compressible with universally measurable pieces; There is a universally measurable complete section that is an $i$ -traveling set with universally measurable pieces; There is a partition $I$ of $X$ generated by $i$ Borel sets such that $Y = X) \\subseteq |I|^G$ does not admit a $G$ -invariant Borel probability measure; $X$ is $i$ -compressible with Borel pieces; There is a Borel complete section that is a $i$ -traveling set with Borel pieces.",
"Equivalence of (1) and (2) as well as (4) and (5) is asserted in REF and (4)$\\Rightarrow $ (1) is trivial.",
"It remains to show (1)$\\Rightarrow $ (3)$\\Rightarrow $ (5).",
"(1)$\\Rightarrow $ (3): Suppose $X \\sim _{I}^{\\Gamma } B$ , where $B^c = X \\setminus B$ is a complete section, $I$ is a partition of $X$ generated by $i$ Borel sets, and $\\Gamma $ denotes the class of universally measurable sets.",
"Let $\\gamma : X \\rightarrow G$ be a witnessing map for $X \\sim _i^{\\Gamma } B$ .",
"By the Jankov-von Neumann uniformization theorem (see 18.1 in ), $ has a $ (11)$-measurable (hence universally measurable) right inverse $ h : Y X$.",
"Define $ : Y G$ by $ (y) = (h(y))$ and note that $$ is universally measurable being a composition of such functions.",
"Letting $ B' = (Y)$, it is straightforward to check that $ $ and thus $ B' = (X)) = B)$.",
"Now it follows that $$ is a witnessing map for $ Y B'$ and hence $ Y$ is compressible with universally measurable pieces.",
"Finally, (1)$$(3) of \\ref {equivalences to compressibility} implies that $ Y$ does not admit an invariant Borel probability measure.$ (3)$\\Rightarrow $ (5): Assume $Y$ is as in (3).",
"Then by Lemma REF , there is a Borel $G$ -invariant $Y^{\\prime } \\supseteq Y$ that does not admit a $G$ -invariant Borel probability measure.",
"Viewing $Y^{\\prime }$ as a Borel $G$ -space, we apply (3)$\\Rightarrow $ (4) of REF and get that $Y^{\\prime }$ is compressible with Borel pieces; thus there is a Borel $B^{\\prime } \\subseteq Y^{\\prime }$ with $[Y^{\\prime } \\setminus B^{\\prime }]_G = Y^{\\prime }$ such that $Y^{\\prime } \\sim B^{\\prime }$ .",
"Let $\\delta : Y^{\\prime } \\rightarrow G$ be a witnessing map for $Y^{\\prime } \\sim B^{\\prime }$ .",
"Put $B = {-1}(B^{\\prime })$ and $\\gamma = \\delta \\circ .",
"By definition, $$ is $ FI$-invariant.",
"In fact, it is straightforward to check that $$ is a witnessing map for $ X I B$ and $ [X B]G = [-1(Y B')]G = -1([Y B']G) = -1(Y) = X$.",
"Hence $ X$ is $ I$-compressible.$ We now give an example of a 1-traveling set.",
"First we need some definitions.",
"Definition 2.26 Let $X$ be a Borel $G$ -space and $A \\subseteq X$ be Borel.",
"$A$ is called aperiodic if it intersects every orbit in either 0 or infinitely many points; a partial transversal if it intersects every orbit in at most one point; smooth if there is a Borel partial transversal $T \\subseteq A$ such that $[T]_G = [A]_G$ .",
"Proposition 2.27 Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal, then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"let $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ .",
"For each $n \\in \\mathbb {N}$ , define $\\bar{n} : X \\rightarrow \\mathbb {N}$ and $\\gamma _n : T \\rightarrow G$ recursively in $n$ as follows: $\\left\\lbrace \\begin{array}{l}\\bar{n}(x) = \\text{the least $k$ such that } g_k x \\notin \\lbrace \\hat{\\gamma }_i(x) : i<n\\rbrace \\\\\\gamma _n(x) = g_{\\bar{n}(x)}\\end{array}\\right..$ Clearly, $\\bar{n}$ and $\\gamma _n$ are well-defined and Borel.",
"Define $\\gamma : T \\rightarrow G^{\\mathbb {N}}$ by setting $\\gamma (\\cdot )(n) = \\gamma _n$ .",
"It follows from the definitions that $\\gamma $ is a Borel travel guide for $T$ and hence, $T$ is a traveling set.",
"It remains to show that $\\gamma $ is $F_{I}$ -invariant, where $I= < \\!\\!",
"T \\!\\!",
">$ .",
"For this it is enough to show that $\\bar{n}$ is $F_{I}$ -invariant, which we do by induction on $n$ .",
"Since it trivially holds for $n = 0$ , we assume it is true for all $0 \\le k < n$ and show it for $n$ .",
"To this end, suppose $x,y \\in T$ with $x F_{I}y$ , and assume for contradiction that $m := \\bar{n}(x) < \\bar{n}(y)$ .",
"Thus it follows that $g_m y = \\hat{\\gamma }_k(y) \\in \\hat{\\gamma }_k(T)$ , for some $k < n$ .",
"By the induction hypothesis, $\\hat{\\gamma }_k(T)$ is $F_{I}$ -invariant and hence, $g_m x \\in \\hat{\\gamma }_k(T)$ , contradicting the definition of $\\bar{n}(x)$ .",
"Corollary 2.28 Let $X$ be an aperiodic Borel $G$ -space.",
"If a Borel set $A \\subseteq X$ is smooth, then $A \\in 1$ .",
"Let $P = [A]_G$ and let $T$ be a Borel partial transversal with $[T]_G = P$ .",
"By REF , $T$ is $I$ -traveling, where $I= < \\!\\!",
"T \\!\\!",
">$ .",
"Hence, $P \\sim _{I} P \\setminus T$ , by Lemma REF .",
"This implies that $P$ is $1^*$ -compressible since $I= < \\!\\!",
"T^c \\!\\!",
">$ and $P \\setminus T \\in \\lbrace T^c\\rbrace ^P$ .",
"subsection2-2 ex5 pt Constructing finite generators using $i$ -traveling sets Lemma 2.29 Let $A \\in \\mathfrak {B}(X)$ be a complete section and $I$ be an $A$ -sensitive finite Borel partition of $X$ .",
"If $A$ is $I$ -traveling (with Borel pieces), then there is a Borel $2|I|$ -generator.",
"If moreover $A \\in I$ , then there is a Borel $(2|I|-1)$ -generator.",
"Let $\\gamma $ be an $F_{I}$ -invariant Borel travel guide for $A$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel structure of $X$ and let $B = \\bigcup _{n \\ge 1}\\hat{\\gamma }_n(A \\cap U_n)$ .",
"By Lemma REF , each $\\hat{\\gamma }_n$ maps Borel sets to Borel sets and hence $B$ is Borel.",
"Set $J= < \\!\\!",
"B \\!\\!",
">$ , $\\mathcal {P}= I\\vee J$ and note that $|\\mathcal {P}| \\le 2 |I|$ .",
"$A$ and $B$ are disjoint since $\\lbrace \\hat{\\gamma }_n(A)\\rbrace _{n \\in \\mathbb {N}}$ is a collection of pairwise disjoint sets and $\\hat{\\gamma }_0(A) = A$ ; thus if $A \\in I$ , $|\\mathcal {P}| \\le 1 + 2(|I|-1) = 2|I|-1$ .",
"We show that $\\mathcal {P}$ is a generator, that is $G \\mathcal {P}$ separates points in $X$ .",
"Let $x \\ne y \\in X$ and assume they are not separated by $G I$ , thus $x F_{I}y$ .",
"We show that $G J$ separates $x$ and $y$ .",
"Because $A$ is a complete section, multiplying $x$ by an appropriate group element, we may assume that $x \\in A$ .",
"Since $A$ respects $I$ , $A$ is $F_{I}$ -invariant and thus $y \\in A$ .",
"Also, because $\\gamma $ is $F_{I}$ -invariant, $\\gamma _n(x) = \\gamma _n(y)$ , $\\forall n \\in \\mathbb {N}$ .",
"Let $n \\ge 1$ be such that $x \\in U_n$ but $y \\notin U_n$ .",
"Put $g = \\gamma _n(x)$ ($= \\gamma _n(y)$ ).",
"Then $gx = \\hat{\\gamma }_n(x) \\in \\hat{\\gamma }_n(A \\cap U_n)$ while $gy = \\hat{\\gamma }_n(y) \\notin \\hat{\\gamma }_n(A \\cap U_n)$ .",
"Hence, $gx \\in B$ and $gy \\notin B$ because $\\gamma _m(A) \\cap \\gamma _n(A) = \\mathbb {\\emptyset }$ for all $m \\ne n$ and $gy = \\hat{\\gamma }_n(y) \\in \\hat{\\gamma }_n(A)$ .",
"Thus $G J$ separates $x$ and $y$ .",
"Now REF and REF together imply the following.",
"Proposition 2.30 Let $X$ be a Borel $G$ -space and $i \\ge 1$ .",
"If $X$ is $i$ -compressible then there is a Borel $2^{i+1}$ -generator.",
"By REF , there exists a Borel $i$ -traveling complete section $A$ .",
"Let $I$ witness $A$ being $i$ -traveling and thus, by Lemma REF , there is a $2|I| \\le 2 \\cdot 2^i = 2^{i+1}$ -generator.",
"Example 2.31.",
"For $2 \\le n \\le \\infty $ , let $\\mathbb {F}_n$ denote the free group on $n$ generators and let $X$ be the boundary of $\\mathbb {F}_n$ , i.e.",
"the set of infinite reduced words.",
"Clearly, the product topology makes $X$ a Polish space and $\\mathbb {F}_n$ acts continuously on $X$ by left concatenation and cancellation.",
"We show that $X$ is 1-compressible and thus admits a Borel $2^2=4$ -generator by Proposition REF .",
"To this end, let $a,b$ be two of the $n$ generators of $\\mathbb {F}_n$ and let $X_a$ be the set of all words in $X$ that start with $a$ .",
"Then $X = (X_{a^{-1}} \\cup X_{a^{-1}}^c) \\sim _{I} Y$ , where $Y = bX_{a^{-1}} \\cup aX_{a^{-1}}^c$ and $I< \\!\\!",
"X_{a^{-1}} \\!\\!",
">$ .",
"Hence $X \\sim _1 Y$ .",
"Since $X \\setminus Y \\supseteq X_{a^{-1}}$ , $[X \\setminus Y]_{\\mathbb {F}_n} = X$ and thus $X$ is 1-compressible.",
"Now we obtain a sufficient condition for the existence of an embedding into a finite Bernoulli shift.",
"Corollary 2.32 Let $X$ be a Borel $G$ -space and $k \\in \\mathbb {N}$ .",
"If there exists a Borel $G$ -map $f : X \\rightarrow k^G$ such that $Y = f(X)$ does not admit a $G$ -invariant Borel probability measure, then there is a Borel $G$ -embedding of $X$ into $(2k)^G$ .",
"Let $I= I_{f}$ and hence $f = .",
"By (3)$$(5) of \\ref {equivalences to i-compressibility} (or rather the proof of it), $ X$ admits a Borel $ I$-traveling complete section.",
"Thus by Lemma \\ref {I-traveling implies finite generator}, $ X$ admits a $ 2|I| = 2k$-generator and hence, there is a Borel $ G$-embedding of $ X$ into $ (2k)G$.$ Lemma 2.33 Let $I$ be a partition of $X$ into $n$ Borel sets.",
"Then $I$ is generated by $k = \\lceil \\log _2(n) \\rceil $ Borel sets.",
"Since $2^k \\ge n$ , we can index $I$ by the set $\\mathbf {2^k}$ of all $k$ -tuples of $\\lbrace 0,1\\rbrace $ , i.e.",
"$I= \\lbrace A_{\\sigma }\\rbrace _{\\sigma \\in \\mathbf {2^k}}$ .",
"For all $i < k$ , put $B_i = \\bigcup _{\\sigma \\in \\mathbf {2^k} \\wedge \\sigma (i) = 1} A_{\\sigma }.$ Now it is clear that for all $\\sigma \\in \\mathbf {2^k}$ , $A_{\\sigma } = \\bigcap _{i<k} B_i^{\\sigma (i)}$ , where $B_i^{\\sigma (i)}$ is equal to $B_i$ if $\\sigma (i) = 1$ , and equal to $B_i^c$ , otherwise.",
"Thus $I= < \\!\\!",
"B_i : i<k \\!\\!",
">$ .",
"Proposition 2.34 If $X$ is compressible and there is a Borel $n$ -generator, then $X$ is $\\lceil \\log _2(n) \\rceil $ -compressible.",
"Let $I$ be an $n$ -generator and hence, by Lemma REF , $I$ is generated by $i$ Borel sets.",
"Since $G I$ separates points in $X$ , each $F_{I}$ -class is a singleton and hence $X \\prec X$ implies $X \\prec _{I} X$ .",
"From REF and REF we immediately get the following corollary, which justifies the use of $i$ -compressibility in studying Question REF .",
"Corollary 2.35 Let $X$ be a Borel $G$ -space that is compressible (equivalently, does not admit an invariant Borel probability measure).",
"$X$ admits a finite generator if and only if $X$ is $i$ -compressible for some $i \\ge 1$ .",
"Invariant measures and $i$ -compressibility This section is mainly devoted to proving the following theorem.",
"Theorem 3.1 Let $X$ be a Borel $G$ -space.",
"If $X$ is aperiodic, then there exists a function $m : \\mathfrak {B}(X) \\times X \\rightarrow [0,1]$ satisfying the following properties for all $A,B \\in \\mathfrak {B}(X)$ : $m(A, \\cdot )$ is Borel; $m(X, x) = 1$ , $\\forall x \\in X$ ; If $A \\subseteq B$ , then $m(A, x) \\le m(B,x)$ , $\\forall x \\in X$ ; $m(A, x) = 0$ off $[A]_G$ ; $m(A, x) > 0$ on $[A]_G$ modulo 4; $m(A,x) = m(gA, x)$ , for all $g \\in G$ , $x \\in X$ modulo 3; If $A \\cap B = \\mathbb {\\emptyset }$ , then $m(A \\cup B,x) = m(A,x) + m(B,x)$ , $\\forall x \\in X$ modulo 4.",
"Remark.",
"A version of this theorem is what lies at the heart of the proof of Nadkarni's theorem.",
"The conclusions of our theorem are modulo 4, which is potentially a smaller $\\sigma $ -ideal than the $\\sigma $ -ideal of sets contained in compressible Borel sets used in Nadkarni's version.",
"However, the price we paid for this is that part (g) asserts only finite additivity instead of countable additivity asserted by Nadkarni's version.",
"Proof of Theorem REF.",
"Our proof follows the general outline of Nadkarni's proof.",
"The construction of $m(A,x)$ is somewhat similar to that of Haar measure.",
"First, for sets $A,B$ , we define a Borel function $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ that basically gives the number of copies of $B^{[x]_G}$ that fit in $A^{[x]_G}$ when moved by group elements (piecewise).",
"Then we define a decreasing sequence of complete sections (called a fundamental sequence below), which serves as a gauge to measure the size of a given set.",
"Assume throughout that $X$ is an aperiodic Borel $G$ -space (although we only use the aperiodicity assumption in REF to assert that smooth sets are in 1).",
"Lemma 3.2 (Comparability) $\\forall A,B \\in \\mathfrak {B}(X)$ , there is a partition $X = P \\cup Q$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ , $A^P \\prec _{I} B^P$ and $B^Q \\preceq _{I} A^Q$ .",
"It is enough to prove the lemma assuming $X = [A]_G \\cap [B]_G$ since we can always include $[B]_G \\setminus [A]_G$ in $P$ and $X \\setminus [B]_G$ in $Q$ .",
"Fix an enumeration $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ for $G$ .",
"We recursively construct Borel sets $A_n,B_n,A_n^{\\prime },B_n^{\\prime }$ as follows.",
"Set $A_0^{\\prime } = A$ and $B_0^{\\prime } = B$ .",
"Assuming $A_n^{\\prime }, B_n^{\\prime }$ are defined, set $B_n = B_n^{\\prime } \\cap g_n A_n^{\\prime }$ , $A_n = g_n^{-1} B_n$ , $A_{n+1}^{\\prime } = A_n^{\\prime } \\setminus A_n$ and $B_{n+1}^{\\prime } = B_n^{\\prime } \\setminus B_n$ .",
"It is easy to see by induction on $n$ that for any $A,B$ -sensitive $I$ , $A_n,B_n$ are $F_{I}$ -invariant since so are $A,B$ .",
"Thus, setting $A^* = \\bigcup _{n \\in \\mathbb {N}}A_n$ and $B^* = \\bigcup _{n \\in \\mathbb {N}}B_n$ , we get that $A^* \\sim _{I} B^*$ since $B_n = g_n A_n$ .",
"Let $A^{\\prime } = A \\setminus A^*$ , $B^{\\prime } = B \\setminus B^*$ and set $P = [B^{\\prime }]_G$ , $Q = X \\setminus P$ .",
"Claim $[A^{\\prime }]_G \\cap [B^{\\prime }]_G = \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $\\exists x \\in A^{\\prime }$ and $n \\in \\mathbb {N}$ such that $g_n x \\in B^{\\prime }$ .",
"It is clear that $A^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} A_k^{\\prime }$ , $B^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} B_k^{\\prime }$ ; in particular, $x \\in A_n^{\\prime }$ and $g_n x \\in B_n^{\\prime }$ .",
"But then $g_n x \\in B_n$ and $x \\in A_n$ , contradicting $x \\in A^{\\prime }$ .",
"$\\dashv $ Let $I$ be an $A,B$ -sensitive partition.",
"Then $A^P = (A^*)^P$ and hence $A^P \\prec _{I} B^P$ since $(A^*)^P \\sim _{I} (B^*)^P \\subseteq B^P$ and $[B^P \\setminus (B^*)^P]_G = [B^{\\prime }]_G = P = [B^P]_G$ .",
"Similarly, $B^Q = (B^*)^Q$ and hence $B^Q \\preceq _{I} A^Q$ since $(B^*)^Q \\sim _{I} (A^*)^Q \\subseteq A^Q$ .",
"Definition 3.3 (Divisibility) Let $n \\le \\infty $ , $A,B,C \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"Write $A \\sim _{I} nB \\oplus C$ if there are Borel sets $A_k \\subseteq A$ , $k<n$ , such that $\\lbrace A_k\\rbrace _{k<n} \\cup \\lbrace C\\rbrace $ is a partition of $A$ , each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\sim _{I} B$ .",
"Write $n B \\preceq _{I} A$ if there is $C \\subseteq A$ with $A \\sim _{I} n B \\oplus C$ , and write $n B \\prec _{I} A$ if moreover $[C]_G = [A]_G$ .",
"Write $A \\preceq _{I} nB$ if there is a Borel partition $\\lbrace A_k\\rbrace _{k<n}$ of $A$ such that each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\preceq _{I} B$ .",
"If moreover, $A_k \\prec _{I} B$ for at least one $k < n$ , we write $A \\prec _{I} nB$ .",
"For $i \\ge 1$ , we use the above notation with $I$ replaced by $i$ if there is an $A,B$ -sensitive partition $I$ generated by $i$ sets for which the above conditions hold.",
"Proposition 3.4 (Euclidean decomposition) Let $A,B \\in \\mathfrak {B}(X)$ and put $R = [A]_G \\cap [B]_G$ .",
"There exists a partition $\\lbrace P_n\\rbrace _{n \\le \\infty }$ of $R$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ and $n \\le \\infty $ , $A^{P_n} \\sim _{I} nB^{P_n} \\oplus C_n$ for some $C_n$ such that $C_n \\prec _{I} B^{P_n}$ , if $n < \\infty $ .",
"We repeatedly apply Lemma REF .",
"For $n < \\infty $ , recursively define $R_n, P_n, A_n, C_n$ satisfying the following: $R_n$ are invariant decreasing Borel sets such that $n B^{R_n} \\preceq _{I} A^{R_n}$ for any $A,B$ -sensitive $I$ ; $P_n = R_n \\setminus R_{n+1}$ ; $A_n \\subseteq R_{n+1}$ are pairwise disjoint Borel sets such that for any $A,B$ -sensitive $I$ , every $A_n$ respects $I$ and $A_n \\sim _{I} B^{R_{n+1}}$ ; $C_n \\subseteq P_n$ are Borel sets such that for any $A,B$ -sensitive $I$ , every $C_n$ respects $I$ and $C_n \\prec _{I} B^{P_n}$ .",
"Set $R_0 = R$ .",
"Given $R_n$ , $\\lbrace A_k\\rbrace _{k<n}$ satisfying the above properties, let $A^{\\prime } = A^{R_n} \\setminus \\bigcup _{k<n} A_k$ .",
"We apply Lemma REF to $A^{\\prime }$ and $B^{R_n}$ , and get a partition $R_n = P_n \\cup R_{n+1}$ such that $(A^{\\prime })^{P_n} \\prec _{I} B^{P_n}$ and $B^{R_{n+1}} \\preceq _{I} (A^{\\prime })^{R_{n+1}}$ .",
"Set $C_n = (A^{\\prime })^{P_n}$ .",
"Let $A_n \\subseteq (A^{\\prime })^{R_{n+1}}$ be such that $B^{R_{n+1}} \\sim _{I} A_n$ .",
"It is straightforward to check (i)-(iv) are satisfied.",
"Now let $R_{\\infty } = \\bigcap _{n \\in \\mathbb {N}} R_n$ and $C_{\\infty } = (A \\setminus \\bigcup _{n \\in \\mathbb {N}}A_n)^{R_{\\omega }}$ .",
"Now it follows from (i)-(iv) that for all $n \\le \\infty $ , $\\lbrace A_k^{P_n}\\rbrace _{k<n} \\cup \\lbrace C_n\\rbrace $ is a partition of $A^{P_n}$ witnessing $A^{P_n} \\sim _{I} nB \\oplus C_n$ , and for all $n < \\infty $ , $C_n \\prec B^{P_n}$ .",
"For $A,B \\in \\mathfrak {B}(X)$ , let $\\lbrace P_n\\rbrace _{n \\le \\infty }$ be as in the above proposition.",
"Define $[A / B](x) = \\left\\lbrace \\begin{array}{ll}n & \\text{if } x \\in P_n, n < \\infty \\\\\\infty & \\text{if } x \\in P_{\\infty } \\text{ or } x \\in [A]_G \\setminus [B]_G \\\\0 & \\text{if } x \\in [B]_G \\setminus [A]_G \\\\-1 & \\text{otherwise}\\end{array}\\right..$ Note that $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ is a Borel function by definition.",
"Lemma 3.5 (Infinite divisibility $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ with $[A]_G = [B]_G$ , and let $I$ be a finite Borel partition of $X$ .",
"If $\\infty B \\preceq _{I} A$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} \\infty B \\oplus C$ and let $\\lbrace A_k\\rbrace _{k < \\infty }$ be as in Definition REF .",
"$A_k \\sim _{I} B \\sim _{I} A_{k+1}$ and hence $A_k \\sim _{I} A_{k+1}$ .",
"Also trivially $C \\sim _{I} C$ .",
"Thus, letting $A^{\\prime } = \\bigcup _{k < \\infty } A_{k+1} \\cup C$ , we apply (b) of REF to $A$ and $A^{\\prime }$ , and get that $A \\sim _{I} A^{\\prime }$ .",
"Because $[A \\setminus A^{\\prime }]_G = [A_0]_G = [B]_G = [A]_G$ , we have $A \\prec _{I} A$ .",
"Lemma 3.6 (Ambiguity $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"If $nB \\preceq _{I} A \\prec _{I} nB$ for some $n \\ge 1$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} nB \\oplus C$ and let $\\lbrace A_k\\rbrace _{k<n}$ be a partitions of $A \\setminus C$ witnessing $A \\sim _{I} nB \\oplus C$ .",
"Also let $\\lbrace A^{\\prime }_k\\rbrace _{k<n}$ be witnessing $A \\prec _{I} nB$ with $A^{\\prime }_0 \\prec _{I} B$ .",
"Since $A^{\\prime }_k \\preceq _{I} B \\sim _{I} A_k$ , $A^{\\prime }_k \\preceq _{I} A_k$ , for all $k<n$ and $A^{\\prime }_0 \\prec _{I} A_0$ .",
"Note that it follows from the hypothesis that $[A]_G = [B]_G$ and hence $[A_0]_G = [A]_G$ since $[A_0]_G = [B]_G$ .",
"Thus it follows from (b) of REF that $A = \\bigcup _{k<n} A^{\\prime }_k \\prec _{I} \\bigcup _{k<n} A_k \\subseteq A$ .",
"Proposition 3.7 Let $n \\in \\mathbb {N}$ and $A,A^{\\prime },B,P \\in \\mathfrak {B}(X)$ , where $P$ is invariant.",
"$[A/B] \\in \\mathbb {N}$ on $[B]_G$ modulo 3.",
"If $A \\subseteq A^{\\prime }$ , then $[A / B] \\le [A^{\\prime } / B]$ .",
"If $[A/B] = n$ on $P$ then $n B^P \\preceq _{I} A^P \\prec _{I} (n+1) B^P$ , for any finite Borel partition $I$ that is $A,B$ -sensitive.",
"In particular, $n B^P \\preceq _2 A^P \\prec _2 (n+1) B^P$ by taking $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"For $n \\ge 1$ , if $A^P \\prec _i nB^P$ , then $[A/B] < n$ on $P$ modulo ${i+1}$ ; If $A^P \\subseteq [B]_G$ and $nB^P \\preceq _i A^P$ , then $[A/B] \\ge n$ on $P$ modulo ${i+1}$ .",
"For (a), notice that REF and REF imply that $P_{\\infty } \\in 3$ .",
"(b) and (c) follow from the definition of $[A/B]$ .",
"For (d), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $A^P \\prec _{I} nB^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) \\ge n\\rbrace $ .",
"By (c), $nB^Q \\preceq _{I} A^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"For (e), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $nB^P \\preceq _{I} A^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) < n\\rbrace $ .",
"By (c), $A^Q \\prec _{I} nB^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"Definition 3.8 (Fundamental sequence) A sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections with $F_0 = X$ and $[F_n/F_{n+1}] \\ge 2$ modulo 3 is called fundamental.",
"Proposition 3.9 There exists a fundamental sequence.",
"Take $F_0 = X$ .",
"Given any complete Borel section $F$ , its intersection with every orbit is infinite modulo a smooth set (if the intersection of an orbit with a set is finite, then we can choose an element from each such nonempty intersection in a Borel way and get a Borel transversal).",
"Thus, by REF , $F$ is aperiodic modulo 1.",
"Now use Lemma REF to write $F = A \\cup B, A \\cap B = \\mathbb {\\emptyset }$ , where $A,B$ are also complete sections.",
"Let now $P,Q$ be as in Lemma REF for $A,B$ , and hence $A^P \\prec _2 B^P, B^Q \\preceq _2 A^Q$ because we can take $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"Let $A^{\\prime } = A^P \\cup B^Q, B^{\\prime } = B^P \\cup A^Q$ .",
"Then $F = A^{\\prime } \\cup B^{\\prime }, A^{\\prime } \\cap B^{\\prime } = \\mathbb {\\emptyset }$ , $A^{\\prime } \\preceq B^{\\prime }$ and $A^{\\prime }$ is also a complete Borel section.",
"By (e) of REF , $[F/A^{\\prime }] \\ge 2$ modulo 3.",
"Iterate this process to inductively define $F_n$ .",
"Fix a fundamental sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ and for any $A \\in \\mathfrak {B}(X), x \\in X$ , define $m(A,x) = \\lim _{n \\rightarrow \\infty } \\frac{[A/F_n](x)}{[X/F_n](x)}, \\qquad \\mathrm {(\\dagger )}$ if the limit exists, and 0 otherwise.",
"In the above fraction we define ${\\infty \\over \\infty } = 1$ .",
"We will prove in Proposition REF that this limit exists modulo 4.",
"But first we need the following two lemmas.",
"Lemma 3.10 (Almost cancelation) For any $A,B,C \\in X$ , $[A/B][B/C] \\le [A/C] < ([A/B] + 1)([B/C] + 1)$ on $R := [B]_G \\cap [C]_G$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,C \\!\\!",
">$ .",
"$[A/B][B/C] \\le [A/C]$ : Fix integers $i,j > 0$ and let $P = \\lbrace x \\in X : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"Since $i,j > 0$ , $P \\subseteq [A]_G \\cap [B]_G \\cap [C]_G$ and we work in $P$ .",
"By (c) of REF , $i B \\preceq _{I} A$ and $j C \\preceq _{I} B$ .",
"Thus it follows that $ij C \\preceq _{I} A$ and hence $[A / C] \\ge ij$ modulo 4 by (e) of REF .",
"$[A/C] < ([A/B] + 1)([B/C] + 1)$ : By (a) of REF , $[A/C], [A/B], [B/C] \\in \\mathbb {N}$ on $R$ modulo 3.",
"Fix $i,j \\in \\mathbb {N}$ and let $Q = \\lbrace x \\in R : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"We work in $Q$ .",
"By (c) of REF , $A \\prec _{I} (i+1) B$ and $B \\prec _{I} (j+1) C$ .",
"Thus $A \\prec _{I} (i+1)(j+1) C$ and hence $[A/C] < (i+1)(j+1)$ modulo 4 by (d) of REF .",
"Lemma 3.11 For any $A \\in \\mathfrak {B}(A)$ , $\\lim _{n \\rightarrow \\infty } [A / F_n] = \\left\\lbrace \\begin{array}{ll} \\infty & \\text{on } [A]_G \\\\ 0 & \\text{on } X \\setminus [A]_G \\end{array}\\right., \\text{ modulo } 4.$ The part about $X \\setminus [A]_E$ is clear, so work in $[A]_E$ , i.e.",
"assume $X = [A]_G$ .",
"By (a) of REF and REF , we have $\\infty > [F_1 /A] \\ge [F_1 /F_n ] [F_n / A ] \\ge 2^{n-1} [F_n /A], \\text{ modulo } 4,$ which holds for all $n$ at once since 4 is a $\\sigma $ -ideal.",
"Thus $[F_n /A] \\rightarrow 0$ modulo 4 and hence, as $[F_n /A] \\in \\mathbb {N}$ , $[F_n /A]$ is eventually 0, modulo 4.",
"So if $B_k := \\lbrace x \\in [A]_G : [F / A](x) = 0\\rbrace ,$ then $B_k \\nearrow X$ , modulo 4.",
"Now it follows from Lemma REF that $[A / F_k] > 0$ on $B_k$ modulo 4.",
"But $[A/F_{k+n}] \\ge [A / F_k ] [F_k / F_{k+n} ] \\ge 2^n [A / F_k ], \\text{ modulo } 4,$ so for every $k$ , $[A/F_n] \\rightarrow \\infty $ on $B_k$ modulo 4.",
"Since $B_k \\nearrow X$ modulo 4, we have $[A/F_n] \\rightarrow \\infty $ on $X$ , modulo 4.",
"Proposition 3.12 For any Borel set $A \\subseteq X$ , the limit in ($\\dagger $ ) exists and is positive on $[A]_G$ , modulo 4.",
"Claim Suppose $B,C \\in \\mathfrak {B}(X)$ , $i \\in \\mathbb {N}$ and $D_i = \\lbrace x \\in X : [C / F_i](x) > 0\\rbrace $ .",
"Then $\\overline{\\lim } {[B/F_n ] \\over [C/F_n]} \\le {[B/F_i] + 1 \\over [C/F_i]}$ on $D_i$ , modulo 4.",
"Proof of Claim.",
"Working in $D_i$ and using Lemma REF , $\\forall j$ we have (modulo 4) $[B/F_{i+j}] & \\le ([B/F_i ]+1) ([F_i / F_{i+j}] + 1) \\\\[C/F_{i+j}] & \\ge [C/F_i ] [ F_i /F_{i+j}] > 0,$ so ${[B/F_{i+j}]\\over [C/F_{i+j}]} & \\le {[B/F_i ]+1 \\over [C/F_i ]}\\cdot {[F_i /F_{i+j}]+1\\over [F_i /F_{i+j}]} \\\\& \\le {[B/F_i ]+1\\over [C/F_i ]} \\cdot (1 + {1\\over 2^j}),$ from which the claim follows.",
"$\\dashv $ Applying the claim to $B = A$ and $C = X$ (hence $D_i = X$ ), we get that for all $i \\in \\mathbb {N}$ $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ](x)\\over [X/F_n ](x)} \\le {[A/F_i ](x)+1 \\over [X/F_i ](x)} (\\text{modulo } 4).$ Thus $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ]\\over [X/F_n ]} \\le \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i]+1\\over [X/F_i ]} = \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i ]\\over [X/F_i ]}$ since $\\lim _{i \\rightarrow \\infty } {1 \\over [X/F_i]} = 0$ .",
"To see that $m(A,x)$ is positive on $[A]_E$ modulo 4 we argue as follows.",
"We work in $[A]_G$ .",
"Applying the above claim to $B = X$ and $C = A$ , we get ${1 \\over m(A,x)} = \\lim _{n \\rightarrow \\infty } {[X/F_n] \\over [A/F_n]} \\le {[X/F_i] + 1 \\over [A/F_i]} < \\infty \\text{ on } D_i \\text{ (modulo $4$)}.$ Thus $m(A,x) > 0$ on $\\cup _{i \\in \\mathbb {N}} D_i$ , modulo 4.",
"But $D_i \\nearrow [A]_G$ because $[A/F_i] \\rightarrow \\infty $ as $i \\rightarrow \\infty $ , and hence $m(A,x) > 0$ on $[A]_G$ modulo 4.",
"Lemma 3.13 (Invariance) For $A,F \\in \\mathfrak {B}(X)$ , $\\forall g \\in G, [A / F] = [gA / F]$ , modulo 3.",
"We may assume that $X = [A]_G \\cap [F]_G$ .",
"Fix $g \\in G$ , $n \\in \\mathbb {N}$ , and put $Q = \\lbrace x \\in X : [gA / F](x) = n\\rbrace $ .",
"We work in $Q$ .",
"Let $I= < \\!\\!",
"A,F \\!\\!",
">$ and hence $A,gA,F$ respect $I$ .",
"By (c) of REF , $nF \\preceq _{I} gA$ .",
"But clearly $gA \\sim _{I} A$ and hence $nF \\preceq _{I} A$ .",
"Thus, by (e) of REF , $[A / F] \\ge n = [gA / F]$ , modulo 3.",
"By symmetry, $[gA / F] \\ge [A / F]$ (modulo 3) and the lemma follows.",
"Lemma 3.14 (Almost additivity) For any $A,B,F \\in X$ with $A \\cap B = \\mathbb {\\emptyset }$ , $[A/F] + [B/F] \\le [A \\cup B / F] \\le [A/F] + [B/F] + 1$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,F \\!\\!",
">$ .",
"$[A/F] + [B/F] \\le [A \\cap B / F]$ : Fix $i,j \\in \\mathbb {N}$ not both 0, say $i>0$ , and let $S = \\lbrace x \\in X : [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"Since $i>0$ , $S \\subseteq [A]_G \\cap [F]_G$ and we work in $S$ .",
"By (c) of REF , $iF^S \\preceq _{I} A^S$ and $jF^S \\preceq _{I} B^S$ .",
"Hence $(i+j)F^S \\preceq _{I} (A \\cup B)^S$ and thus, by (e) of REF , $[A \\cup B / F] \\ge i + j$ , modulo 4.",
"$[A \\cap B / F] \\le [A/F] + [B/F] + 1$ : Outside $[F]_G$ , the inequality clearly holds.",
"Fix $i,j \\in \\mathbb {N}$ and let $M = \\lbrace x \\in [F]_G: [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"We work in $M$ .",
"By (c) of REF , $A \\prec _{I} (i+1)F$ and $B \\prec _{I} (j+1)F$ .",
"Thus it is clear that $A \\cup B \\prec _{I} (i + j + 2)F$ and hence $[A \\cup B / F] < i+j+2$ , modulo 4, by (d) of REF .",
"Now we are ready to finish the proof of Theorem REF .",
"Fix $A,B \\in \\mathfrak {B}(X)$ .",
"The fact that $m(A,x) \\in [0,1]$ and parts (b) and (d) follow directly from the definition of $m(A,x)$ .",
"Part (a) follows from the fact that $[A / F_n]$ is Borel for all $n \\in \\mathbb {N}$ .",
"(c) follows from (b) of Lemma REF , and (e) and (f) are asserted by REF and REF , respectively.",
"To show (g), we argue as follows.",
"By Lemma REF , $[A/F_n] + [B/F_n] \\le [A \\cup B / F_n] \\le [A/F_n] + [B/F_n] + 1$ , modulo 4, and thus $\\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} \\le \\frac{[A \\cup B / F_n]}{[X/F_n]} \\le \\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} + \\frac{1}{[X/F_n]},$ for all $n$ at once, modulo 4 (using the fact that 4 is a $\\sigma $ -ideal).",
"Since $[X/F_n] \\ge 2^n$ , passing to the limit in the inequalities above, we get $m(A,x) + m(B,x) \\le m(A \\cup B,x) \\le m(A,x) + m(B,x)$ .",
"QED (Thm REF ) Theorem REF will only be used via Corollary REF and to state it we need the following.",
"Definition 3.15 Let $X$ be a Borel $G$ -space.",
"$\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ is called a Boolean $G$ -algebra, if it is a Boolean algebra, i.e.",
"is closed under finite unions and complements, and is closed under the $G$ -action, i.e.",
"$G \\mathcal {B}= \\mathcal {B}$ .",
"Corollary 3.16 Let $X$ be a Borel $G$ -space and let $\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ be a countable Boolean $G$ -algebra.",
"For any $A \\in \\mathcal {B}$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, $\\mu $ can be taken such that there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B)=0$ .",
"Let $A \\in \\mathcal {B}$ be such that $A \\notin 4$ .",
"We may assume that $X = [A]_G$ by setting the (to be constructed) measure to be 0 outside $[A]_G$ .",
"If $X$ is not aperiodic, then by assigning equal point masses to the points of a finite orbit, we will have a probability measure on all of $\\mathfrak {B}(X)$ , so assume $X$ is aperiodic.",
"Since 4 is a $\\sigma $ -ideal and $\\mathcal {B}$ is countable, Theorem REF implies that there is a $P \\in 4$ such that (a)-(g) of the same theorem hold on $X \\setminus P$ for all $A,B \\in \\mathcal {B}$ .",
"Since $A \\notin 4$ , there exists $x_A \\in A \\setminus P$ .",
"Hence, letting $\\mu (B) = m(B,x_A)$ for all $B \\in \\mathcal {B}$ , conditions (b),(f) and (g) imply that $\\mu $ is a $G$ -invariant finitely additive probability measure on $\\mathcal {B}$ .",
"Moreover, since $x_A \\in [A]_G \\setminus P$ , $\\mu (A) = m(A, x_A) > 0$ .",
"Finally, the last assertion follows from condition (d).",
"Corollary 3.17 Let $X$ be a Borel $G$ -space.",
"For every Borel set $A \\subseteq X$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive Borel probability measure $\\mu $ (defined on all Borel sets) with $\\mu (A)>0$ .",
"The statement follows from REF and a standard application of the Compactness Theorem of propositional logic.",
"Here are the details.",
"We fix the following set of propositional variables $\\mathcal {P}= \\lbrace P_{A,r} : A \\in \\mathfrak {B}(X), r \\in [0,1]\\rbrace ,$ with the following interpretation in mind: $P_{A,r} \\Leftrightarrow \\text{``the measure of $A$ is $\\ge r$''}.$ Define the theory $T$ as the following set of sentences: for each $A,B \\in \\mathfrak {B}(X)$ , $r,s \\in [0,1]$ and $g \\in G$ , “$P_{A,0}$ ”$\\in T$ ; if $r > 0$ , then “$\\lnot P_{\\mathbb {\\emptyset }, r}$ ”$\\in T$ ; if $s \\ge r$ , then “$P_{A,s} \\rightarrow P_{A,r}$ ”$\\in T$ ; if $A \\cap B = \\mathbb {\\emptyset }$ , then “$(P_{A,r} \\wedge P_{B,s}) \\rightarrow P_{A \\cup B, r+s}$ ”, “$(\\lnot P_{A,r} \\wedge \\lnot P_{B,s}) \\rightarrow \\lnot P_{A \\cup B, r+s}$ ”$\\in T$ ; “$P_{X,1}$ ”$\\in T$ ; “$P_{A,r} \\rightarrow P_{gA,r}$ ”$\\in T$ .",
"If there is an assignment of the variables in $\\mathcal {P}$ satisfying $T$ , then for each $A \\in \\mathfrak {B}(X)$ , we can define $\\mu (A) = \\sup \\lbrace r \\in [0,1] : P_{A,r}\\rbrace .$ Note that due to (i), $\\mu $ is well defined for all $A \\in \\mathfrak {B}(X)$ .",
"In fact, it is straightforward to check that $\\mu $ is a finitely additive $G$ -invariant probability measure.",
"Thus, we only need to show that $T$ is satisfiable, for which it is enough to check that $T$ is finitely satisfiable, by the Compactness Theorem of propositional logic (or by Tychonoff's theorem).",
"Let $T_0 \\subseteq T$ be finite and let $\\mathcal {P}_0$ be the set of propositional variables that appear in the sentences in $T_0$ .",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by the sets that appear in the indices of the variables in $\\mathcal {P}_0$ .",
"By REF , there is a finitely additive $G$ -invariant probability measure $\\mu $ defined on $\\mathcal {B}$ .",
"Consider the following assignment of the variables in $\\mathcal {P}_0$ : for all $P_{A,r} \\in \\mathcal {P}_0$ , $P_{A,r} :\\Leftrightarrow \\mu (A) \\ge r.$ It is straightforward to check that this assignment satisfies $T_0$ , and hence, $T$ is finitely satisfiable.",
"Finite generators in the case of $\\sigma $ -compact spaces In this section we prove that the answer to Question REF is positive in case $X$ has a $\\sigma $ -compact realization.",
"To do this, we first prove Proposition REF , which shows how to construct a countably additive invariant probability measure on $X$ using a finitely additive one.",
"We then use REF to conclude the result.",
"For the next two statements, let $X$ be a second countable Hausdorff topological space equipped with a continuous action of $G$ .",
"Lemma 4.1 Let $\\mathcal {U}\\subseteq Pow(X)$ be a countable base for $X$ closed under the $G$ -action and finite unions/intersections.",
"Let $\\rho $ be a $G$ -invariant finitely additive probability measure on the $G$ -algebra generated by $\\mathcal {U}$ .",
"For every $A \\subseteq X$ , define $\\mu ^* (A) = \\inf \\lbrace \\sum _{n \\in \\mathbb {N}} \\rho (U_n ) : U_n \\in \\mathcal {U}\\; \\wedge \\; A \\subseteq \\bigcup _{n \\in \\mathbb {N}} U_n\\rbrace .$ Then: $\\mu ^*$ is a $G$ -invariant outer measure.",
"If $K \\subseteq X$ is compact, then $K$ is metrizable and $\\mu ^*$ is a metric outer measure on $K$ (with respect to any compatible metric).",
"It is a standard fact from measure theory that $\\mu ^*$ is an outer measure.",
"That $\\mu ^*$ is $G$ -invariant follows immediately from $G$ -invariance of $\\rho $ and the fact that $\\mathcal {U}$ is closed under the action of $G$ .",
"For (b), first note that by Urysohn metrization theorem, $K$ is metrizable, and fix a metric on $K$ .",
"If $E, F \\subseteq K$ are a positive distance apart, then so are $\\bar{E}$ and $\\bar{F}$ .",
"Hence there exist disjoint open sets $U,V$ such that $\\bar{E} \\subseteq U$ , $\\bar{F} \\subseteq V$ .",
"Because $\\bar{E}$ and $\\bar{F}$ are compact, $U,V$ can be taken to be finite unions of sets in $\\mathcal {U}$ and therefore $U,V \\in \\mathcal {U}$ .",
"Now fix $\\epsilon >0$ and let $W_n \\in \\mathcal {U}$ , be such that $E \\cup F \\subseteq \\bigcup _n W_n$ and $\\sum _n \\rho (W_n) \\le \\mu ^*(E \\cup F) + \\epsilon \\le \\mu ^*(E) + \\mu ^*(F) +\\epsilon .\\qquad \\mathrm {{(*)}}$ Note that $\\lbrace W_n \\cap U\\rbrace _{n \\in \\mathbb {N}}$ covers $E$ , $\\lbrace W_n \\cap V\\rbrace _{n \\in \\mathbb {N}}$ covers $F$ and $W_n \\cap U, W_n \\cap V \\in \\mathcal {U}$ .",
"Also, by finite additivity of $\\rho $ , $\\rho (W_n \\cap U) + \\rho (W_n \\cap V) = \\rho (W_n \\cap (U \\cup V)) \\le \\rho (W_n).$ Thus $\\mu ^*(E) + \\mu ^*(F) \\le \\sum _n \\rho (W_n \\cap U) + \\sum _n \\rho (W_n \\cap V) \\le \\sum _n \\rho (W_n),$ which, together with ($*$ ), implies that $\\mu ^*(E \\cup F) = \\mu ^*(E) + \\mu ^*(F)$ since $\\epsilon $ is arbitrary.",
"Proposition 4.2 Suppose there exist a countable base $\\mathcal {U}\\subseteq Pow(X)$ for $X$ and a compact set $K \\subseteq X$ such that the $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ admits a finitely additive $G$ -invariant probability measure $\\rho $ with $\\rho (K)>0$ .",
"Then there exists a countably additive $G$ -invariant Borel probability measure on $X$ .",
"Let $K, \\mathcal {U}$ and $\\rho $ be as in the hypothesis.",
"We may assume that $\\mathcal {U}$ is closed under the $G$ -action and finite unions/intersections.",
"Let $\\mu ^*$ be the outer measure provided by Lemma REF applied to $\\mathcal {U}$ , $\\rho $ .",
"Thus $\\mu ^*$ is a metric outer measure on $K$ and hence all Borel subsets of $K$ are $\\mu ^*$ -measurable (see 13.2 in ).",
"This implies that all Borel subsets of $Y = [K]_G = \\bigcup _{g \\in G} gK$ are $\\mu ^*$ -measurable because $\\mu ^*$ is $G$ -invariant.",
"By Carathéodory's theorem, the restriction of $\\mu ^*$ to the Borel subsets of $Y$ is a countably additive Borel measure on $Y$ , and we extend it to a Borel measure $\\mu $ on $X$ by setting $\\mu (Y^c) = 0$ .",
"Note that $\\mu $ is $G$ -invariant and $\\mu (Y) \\le 1$ .",
"It remains to show that $\\mu $ is nontrivial, which we do by showing that $\\mu (K) \\ge \\rho (K)$ and hence $\\mu (K)>0$ .",
"To this end, let $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq \\mathcal {U}$ cover $K$ .",
"Since $K$ is compact, there is a finite subcover $\\lbrace U_n\\rbrace _{n < N}$ .",
"Thus $U := \\bigcup _{n < N} U_n \\in \\mathcal {U}$ and $K \\subseteq U$ .",
"By finite additivity of $\\rho $ , we have $\\sum _{n \\in \\mathbb {N}} \\rho (U_n) \\ge \\sum _{n < N} \\rho (U_n) \\ge \\rho (U) \\ge \\rho (K),$ and hence, it follows from the definition of $\\mu ^*$ that $\\mu ^*(K) \\ge \\rho (K)$ .",
"Thus $\\mu (K) = \\mu ^*(K) > 0$ .",
"Corollary 4.3 Let $X$ be a second countable Hausdorff topological $G$ -space whose Borel structure is standard.",
"For every compact set $K \\subseteq X$ not in 4, there is a $G$ -invariant countably additive Borel probability measure $\\mu $ on $X$ with $\\mu (K) > 0$ .",
"Fix any countable base $\\mathcal {U}$ for $X$ and let $\\mathcal {B}$ be the Boolean $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ .",
"By Corollary REF , there exists a $G$ -invariant finitely additive probability measure $\\rho $ on $\\mathcal {B}$ such that $\\rho (K) > 0$ .",
"Now apply REF .",
"As a corollary, we derive the analogue of Nadkarni's theorem for 4 in case of $\\sigma $ -compact spaces.",
"Corollary 4.4 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"$X \\notin 4$ if and only if there exists a $G$ -invariant countably additive Borel probability measure on $X$ .",
"$\\Leftarrow $ : If $X \\in 4$ , then it is compressible in the usual sense and hence does not admit a $G$ -invariant Borel probability measure.",
"$\\Rightarrow $ : Suppose that $X$ is a $\\sigma $ -compact topological $G$ -space and $X \\notin 4$ .",
"Then, since $X$ is $\\sigma $ -compact and 4 is a $\\sigma $ -ideal, there is a compact set $K$ not in 4.",
"Now apply REF .",
"Remark.",
"For a Borel $G$ -space $X$ , let $\\mathcal {K}$ denote the collection of all subsets of invariant Borel sets that admit a $\\sigma $ -compact realization (when viewed as Borel $G$ -spaces).",
"Also, let $ denote the collection of all subsets of invariant compressible Borel sets.",
"It is clear that $ K$ and $ are $\\sigma $ -ideals, and what REF implies is that $\\mathcal {K}\\subseteq 4$ .",
"The question of whether $\\mathcal {K}= Pow(X)$ is just a rephrasing of $§10$ .",
"(B) of Introduction.",
"Theorem 4.5 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"If there is no $G$ -invariant Borel probability measure on $X$ , then $X$ admits a Borel 32-generator.",
"By REF , $X \\in 4$ and hence, $X$ is 4-compressible.",
"Thus, by Proposition REF , $X$ admits a Borel $2^5$ -generator.",
"Example 4.6.",
"Let $LO \\subseteq 2^{\\mathbb {N}^2}$ denote the Polish space of all countable linear orderings and let $G$ be the group of finite permutations of elements of $\\mathbb {N}$ .",
"$G$ is countable and acts continuously on $LO$ in the natural way.",
"Put $X = LO \\setminus DLO$ , where $DLO$ denotes the set of all dense linear orderings without endpoints (copies of $\\mathbb {Q}$ ).",
"It is straightforward to see that $DLO$ is a $G_{\\delta }$ subset of $LO$ and hence, $X$ is $F_{\\sigma }$ .",
"Therefore, $X$ is in fact $\\sigma $ -compact since $LO$ is compact being a closed subset of $2^{\\mathbb {N}^2}$ .",
"Also note that $X$ is $G$ -invariant.",
"Let $\\mu $ be the unique measure on $LO$ defined by $\\mu (V_{(F,<)}) = {1 \\over n!",
"}$ , where $(F,<)$ is a finite linearly ordered subset of $\\mathbb {N}$ of cardinality $n$ and $V_{(F,<)}$ is the set of all linear orderings of $\\mathbb {N}$ extending the order $<$ on $F$ .",
"As shown in , $\\mu $ is the unique invariant measure for the action of $G$ on $LO$ and $\\mu (X) = 0$ .",
"Thus there is no $G$ -invariant Borel probability measure on $X$ and hence, by the above theorem, $X$ admits a Borel 32-generator.",
"Finitely traveling sets Let $X$ be a Borel $G$ -space.",
"Definition 5.1 Let $A,B \\in \\mathfrak {B}(X)$ be equidecomposable, i.e.",
"there are $N \\le \\infty $ , $\\lbrace g_n\\rbrace _{n < N} \\subseteq G$ and Borel partitions $\\lbrace A_n\\rbrace _{n < N}$ and $\\lbrace B_n\\rbrace _{n < N}$ of $A$ and $B$ , respectively, such that $g_n A_n = B_n$ for all $n < N$ .",
"$A,B$ are said to be locally finitely equidecomposable (denote by $A \\sim _{\\text{lfin}} B$ ), if $\\lbrace A_n\\rbrace _{n < N},\\lbrace B_n\\rbrace _{n < N},\\lbrace g_n\\rbrace _{n < N}$ can be taken so that for every $x \\in A$ , $A_n \\cap [x]_G = \\mathbb {\\emptyset }$ for all but finitely many $n<N$ ; finitely equidecomposable (denote by $A \\sim _{\\text{fin}} B$ ), if $N$ can be taken to be finite.",
"The notation $\\prec _{\\text{fin}}$ , $\\prec _{\\text{lfin}}$ and the notions of finite and locally finite compressibility are defined analogous to Definitions REF and REF .",
"Definition 5.2 A Borel set $A \\subseteq X$ is called (locally) finitely traveling if there exists pairwise disjoint Borel sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ such that $A_0 = A$ and $A \\sim _{\\text{fin}} A_n$ ($A \\sim _{\\text{lfin}} A_n$ ), $\\forall n \\in \\mathbb {N}$ .",
"Proposition 5.3 If $X$ is (locally) finitely compressible then $X$ admits a (locally) finitely traveling Borel complete section.",
"We prove for finitely compressible $X$ , but note that everything below is also locally valid (i.e.",
"restricted to every orbit) for a locally compressible $X$ .",
"Run the proof of the first part of Lemma REF noting that a witnessing map $\\gamma : X \\rightarrow G$ of finite compressibility of $X$ has finite image and hence the image of each $\\delta _n$ (in the notation of the proof) is finite, which implies that the obtained traveling set $A$ is actually finitely traveling.",
"Proposition 5.4 If $X$ admits a locally finitely traveling Borel complete section, then $X \\in 4$ .",
"Let $A$ be a locally finitely traveling Borel complete section and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"Let $I_n = \\lbrace C_k^n\\rbrace _{k \\in \\mathbb {N}}$ , $J_n = \\lbrace D_k^n\\rbrace _{k \\in \\mathbb {N}}$ be Borel partitions of $A$ and $A_n$ , respectively, that together with $\\lbrace g_k^n\\rbrace _{k \\in \\mathbb {N}} \\subseteq G$ witness $A \\sim _{\\text{lfin}} A_n$ (as in Definition REF ).",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by $\\lbrace X\\rbrace \\cup \\bigcup _{n \\in \\mathbb {N}} (I_n \\cup J_n \\cup \\lbrace A_n\\rbrace )$ .",
"Now assume for contradiction that $X \\notin 4$ and hence, $A \\notin 4$ .",
"Thus, applying Corollary REF to $A$ and $\\mathcal {B}$ , we get a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B) = 0$ .",
"Claim $\\mu (A_n) = \\mu (A)$ , for all $n \\in \\mathbb {N}$ .",
"Proof of Claim.",
"For each $n$ , let $\\lbrace C_{k_i}^n\\rbrace _{i < K_n}$ be the list of those $C_k^n$ such that $C_k^n \\cap [x]_G \\ne \\mathbb {\\emptyset }$ ($K_n < \\infty $ by the definition of locally finitely traveling).",
"Set $B = A \\setminus (\\bigcup _{i < K_n} C_{k_i}^n)$ and note that by finite additivity of $\\mu $ , $\\mu (A) = \\mu (B) + \\sum _{i < K_n} \\mu (C_{k_i}^n).$ Similarly, set $B^{\\prime } = A_n \\setminus (\\bigcup _{i < K_n} D_{k_i}^n)$ and hence $\\mu (A_n) = \\mu (B^{\\prime }) + \\sum _{i < K_n} \\mu (D_{k_i}^n).$ But $B \\cap [x]_G = \\mathbb {\\emptyset }$ and $B^{\\prime } \\cap [x]_G = \\mathbb {\\emptyset }$ , and thus $\\mu (B) = \\mu (B^{\\prime }) = 0$ .",
"Also, since $g_{k_i}^n C_{k_i}^n = D_{k_i}^n$ and $\\mu $ is $G$ -invariant, $\\mu (C_{k_i}^n) = \\mu (D_{k_i}^n)$ .",
"Therefore $\\mu (A) = \\sum _{i < K_n} \\mu (C_{k_i}^n) = \\sum _{i < K_n} \\mu (D_{k_i}^n) = \\mu (A_n).$ $\\dashv $ This claim contradicts $\\mu $ being a probability measure since for large enough $N$ , $\\mu (\\bigcup _{n < N} A_n) = N \\mu (A) > 1$ , contradicting $\\mu (X) = 1$ .",
"This, together with REF , implies the following.",
"Corollary 5.5 Let $X$ be a Borel $G$ -space.",
"If $X$ admits a locally finitely traveling Borel complete section, then there is a Borel 32-generator.",
"Separating smooth-many invariant sets Assume throughout that $X$ is a Borel $G$ -space.",
"Lemma 6.1 If $X$ is aperiodic then it admits a countably infinite partition into Borel complete sections.",
"The following argument is also given in the proof of Theorem 13.1 in .",
"By the marker lemma (see 6.7 in ), there exists a vanishing sequence $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections, i.e.",
"$\\bigcap _{n \\in \\mathbb {N}} B_n = \\mathbb {\\emptyset }$ .",
"For each $n \\in \\mathbb {N}$ , define $k_n : X \\rightarrow \\mathbb {N}$ recursively as follows: $\\left\\lbrace \\begin{array}{rcl}k_0(x) & = & 0 \\\\k_{n+1}(x) &= & min \\lbrace k \\in \\mathbb {N}: B_{k_n(x)} \\cap [x]_G \\nsubseteq B_k\\rbrace \\end{array}\\right.,$ and define $A_n \\subseteq X$ by $x \\in A_n \\Leftrightarrow x \\in A_{k_n(x)} \\setminus A_{k_{n+1}(x)}.$ It is straightforward to check that $A_n$ are pairwise disjoint Borel complete sections.",
"For $A \\in \\mathfrak {B}(X)$ , if $I= < \\!\\!",
"A \\!\\!",
">$ then we use the notation $F_A$ and $f_A$ instead of $F_{I}$ and $, respectively.$ We now work towards strengthening the above lemma to yield a countably infinite partition into $F_A$ -invariant Borel complete sections.",
"Definition 6.2 (Aperiodic separation) For Borel sets $A, Y \\subseteq X$ , we say that $A$ aperiodically separates $Y$ if $f_A([Y]_G)$ is aperiodic (as an invariant subset of the shift $2^G$ ).",
"If such $A$ exists, we say that $Y$ is aperiodically separable.",
"Proposition 6.3 For $A \\in \\mathfrak {B}(X)$ , if $A$ aperiodically separates $X$ , then $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Let $Y = \\lbrace y \\in 2^G : |[y]_G| = \\infty \\rbrace $ and hence $f_A(X)$ is a $G$ -invariant subset of $Y$ .",
"By Lemma REF applied to $Y$ , there is a partition $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $Y$ into Borel complete sections.",
"Thus $A_n = f_{I}^{-1}(B_n)$ is a Borel $F_A$ -invariant complete section for $X$ and $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $X$ .",
"Let $\\mathfrak {A}$ denote the collection of all subsets of aperiodically separable Borel sets.",
"Lemma 6.4 $\\mathfrak {A}$ is a $\\sigma $ -ideal.",
"We only have to show that if $Y_n$ are aperiodically separable Borel sets, then $Y = \\bigcup _{n \\in \\mathbb {N}} Y_n \\in \\mathfrak {A}$ .",
"Let $A_n$ be a Borel set aperiodically separating $Y_n$ .",
"Since $A_n$ also aperiodically separates $[Y_n]_G$ (by definition), we can assume that $Y_n$ is $G$ -invariant.",
"Furthermore, by taking $Y_n^{\\prime } = Y_n \\setminus \\bigcup _{k<n} Y_k$ , we can assume that $Y_n$ are pairwise disjoint.",
"Now letting $A = \\bigcup _{n \\in \\mathbb {N}} (A_n \\cap Y_n)$ , it is easy to check that $A$ aperiodically separates $Y$ .",
"Let $\\mathfrak {S}$ denote the collection of all subsets of smooth sets.",
"By a similar argument as the one above, $\\mathfrak {S}$ is a $\\sigma $ -ideal.",
"Lemma 6.5 If $X$ is aperiodic, then $\\mathfrak {S}\\subseteq \\mathfrak {A}$ .",
"Let $S \\in \\mathfrak {S}$ and hence there is a Borel transversal $T$ for $[S]_G$ .",
"Fix $x \\in S$ and let $y \\ne z \\in [x]_G$ .",
"Since $T$ is a transversal, there is $g \\in G$ such that $gy \\in T$ , and hence $gz \\notin T$ .",
"Thus $f_T(y) \\ne f_T(z)$ , and so $f_T([x]_G)$ is infinite.",
"Therefore $T$ aperiodically separates $[S]_G$ .",
"For the rest of the section, fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and let $F_A^n$ be following equivalence relation: $y F_A^n z \\Leftrightarrow \\forall k < n (g_k y \\in A \\leftrightarrow g_k z \\in A).$ Note that $F_A^n$ has no more than $2^n$ equivalence classes and that $y F_A z$ if and only if $\\forall n (y F_A^n z)$ .",
"Lemma 6.6 For $A,Y \\in \\mathfrak {B}(X)$ , $A$ aperiodically separates $Y$ if and only if $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in Y^{[x]_G}) [y F_A^n z \\wedge \\lnot (y F_A z)]$ .",
"$\\Rightarrow $ : Assume that for all $x \\in Y$ , $f_A([x]_G)$ is infinite and thus $F_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ has infinitely many equivalence classes.",
"Fix $n \\in \\mathbb {N}$ and recall that $F_A^n$ has only finitely many equivalence classes.",
"Thus, by the Pigeon Hole Principle, there are $y,z \\in Y^{[x]_G}$ such that $y F_A^n z$ yet $\\lnot (y F_A z)$ .",
"$\\Leftarrow $ : Assume for contradiction that $f_A(Y^{[x]_G})$ is finite for some $x \\in Y$ .",
"Then it follows that $F_A = F_A^n$ , for some $n$ , and hence for any $y,z \\in Y^{[x]_G}$ , $y F_A^n z$ implies $y F_A z$ , contradicting the hypothesis.",
"Theorem 6.7 If $X$ is an aperiodic Borel $G$ -space, then $X \\in \\mathfrak {A}$ .",
"By Lemma REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into Borel complete sections.",
"We will inductively construct Borel sets $B_n \\subseteq C_n$ , where $C_n$ should be thought of as the set of points colored (black or white) at the $n^{th}$ step, and $B_n$ as the set of points colored black (thus $C_n \\setminus B_n$ is colored white).",
"Define a function $\\# : X \\rightarrow \\mathbb {N}$ by $x \\mapsto m$ , where $m$ is such that $x \\in A_m$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ of sets generating the Borel $\\sigma $ -algebra of $X$ .",
"Assuming that for all $k < n$ , $C_k, B_k$ are defined, let $\\bar{C}_n = \\bigcup _{k<n} C_k$ and $\\bar{B}_n = \\bigcup _{k<n} B_k$ .",
"Put $P_n = \\lbrace x \\in A_0 : \\forall k < n (g_k x \\in \\bar{C}_n) \\wedge g_n x \\notin \\bar{C}_n\\rbrace $ and set $F_n = F_{\\bar{B}_n}^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ , that is for all $x,y \\in P_n$ , $y F_n z \\Leftrightarrow \\forall k < n (g_k y \\in \\bar{B}_n \\leftrightarrow g_k z \\in \\bar{B}_n).$ Now put $C^{\\prime }_n = \\lbrace x \\in P_n : \\#(g_n x) = \\min \\#((g_nP_n)^{[x]_G})\\rbrace $ , $C^{\\prime \\prime }_n = \\lbrace x \\in C^{\\prime }_n : \\exists y, z \\in (C^{\\prime }_n)^{[x]_G} (y \\ne z \\wedge y F_n z)\\rbrace $ and $C_n = g_n C^{\\prime \\prime }_n$ .",
"Note that it follows from the definition of $P_n$ that $C_n$ is disjoint from $\\bar{C}_n$ .",
"Now in order to define $B_n$ , first define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{ the smallest $m$ such that there are } y,z \\in C^{\\prime \\prime }_n \\cap [x]_G \\text{ with } y F_n z, y \\in U_m \\text{ and } z \\notin U_m.$ Note that $\\bar{n}$ is Borel and $G$ -invariant.",
"Lastly, let $B^{\\prime }_n = \\lbrace x \\in C^{\\prime \\prime }_n : x \\in U_{\\bar{n}(x)}\\rbrace $ and $B_n = g_n B^{\\prime }_n$ .",
"Clearly $B_n \\subseteq C_n$ .",
"Now let $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $D = \\left[\\bigcup _{n \\in \\mathbb {N}} (C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)\\right]_G$ .",
"We show that $B$ aperiodically separates $Y := X \\setminus D$ and $D \\in \\mathfrak {S}$ .",
"Since $\\mathfrak {S}\\subseteq \\mathfrak {A}$ and $\\mathfrak {A}$ is an ideal, this will imply that $X \\in \\mathfrak {A}$ .",
"Claim 1 $D \\in \\mathfrak {S}$ .",
"Proof of Claim.",
"Since $\\mathfrak {S}$ is a $\\sigma $ -ideal, it is enough to show that for each $n$ , $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G \\in \\mathfrak {S}$ , so fix $n \\in \\mathbb {N}$ .",
"Clearly $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ is finite, for all $x \\in X$ , since there can be at most $2^n$ pairwise $F_n$ -nonequivalent points.",
"Thus, fixing some Borel linear ordering of $X$ and taking the smallest element from $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ for each $x \\in C^{\\prime }_n \\setminus C^{\\prime \\prime }_n$ , we can define a Borel transversal for $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G$ .",
"$\\dashv $ By Lemma REF , to show that $B$ aperiodically separates $Y$ , it is enough to show that $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in [x]_G) [y F_B^n z \\wedge \\lnot (y F_B z)]$ .",
"Fix $x \\in Y$ .",
"Claim 2 $(\\exists ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $(\\forall ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $x \\notin D$ , it follows that $(\\forall ^{\\infty } n) P_n^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $A_0$ is a complete section and $\\bar{C}_0 = \\mathbb {\\emptyset }$ , $P_0^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Let $N$ be the largest number such that $P_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus for all $n > N$ , $C_n^{[x]_G} = \\mathbb {\\emptyset }$ and hence for all $n > N$ , $\\bar{C}_n^{[x]_G} = \\bar{C}_{N+1}^{[x]_G}$ .",
"Because $C_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ , there is $y \\in A_0^{[x]_G}$ such that $\\forall k \\le N (g_k y \\in \\bar{C}_{N+1})$ ; but because $P_{N+1}^{[x]_G} = \\mathbb {\\emptyset }$ , $g_{N+1} y$ must also fall into $\\bar{C}_{N+1}$ .",
"By induction on $n > N$ , we get that for all $n>N$ , $g_n y \\in \\bar{C}_n$ and thus $g_n y \\in \\bar{C}_{N+1}$ .",
"On the other hand, it follows from the definition of $C^{\\prime }_n$ that for each $n$ , $(C^{\\prime }_n)^{[x]_G}$ intersects exactly one of $A_k$ .",
"Thus $\\bar{C}_{N+1}^{[x]_G}$ intersects at most $N+1$ of $A_k$ and hence there exists $K \\in \\mathbb {N}$ such that for all $k \\ge K$ , $\\bar{C}_{N+1}^{[x]_G} \\cap A_k = \\mathbb {\\emptyset }$ .",
"Since $\\exists ^{\\infty } n (g_n y \\in \\bigcup _{k \\ge K} A_k)$ , $\\exists ^{\\infty } n (g_n y \\notin \\bar{C}_{N+1})$ , a contradiction.",
"$\\dashv $ Now it remains to show that for all $n \\in \\mathbb {N}$ , $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ implies that $\\exists y,z \\in [x]_G$ such that $y F_B^n z$ but $\\lnot (y F_B z)$ .",
"To this end, fix $n \\in \\mathbb {N}$ and assume $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus there are $y,z \\in (C^{\\prime \\prime }_n)^{[x]_G}$ such that $y F_n z$ , $y \\in U_{\\bar{n}(x)}$ and $z \\notin U_{\\bar{n}(x)}$ ; hence, $g_n y \\in B_n$ and $g_n z \\notin B_n$ , by the definition of $B_n$ .",
"Since $C_k$ are pairwise disjoint, $B_n \\subseteq C_n$ and $g_n y, g_n z \\in C_n$ , it follows that $g_n y \\in B$ and $g_n z \\notin B$ , and therefore $\\lnot (y F_B z)$ .",
"Finally, note that $F_n = F_B^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ and hence $y F_B^n z$ .",
"Corollary 6.8 Suppose all of the nontrivial subgroups of $G$ have finite index (e.g.",
"$G = \\mathbb {Z}$ ), and let $X$ be an aperiodic Borel $G$ -space.",
"Then there exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit, i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in X$ .",
"Let $A$ be a Borel set aperiodically separating $X$ (exists by Theorem REF ) and put $Y = f_A(X)$ .",
"Then $Y \\subseteq 2^G$ is aperiodic and hence the action of $G$ on $Y$ is free since the stabilizer subgroup of every element must have infinite index and thus is trivial.",
"But this implies that for all $y \\in Y$ , $f_A^{-1}(y)$ intersects every orbit in $X$ at no more than one point, and hence $f_A$ is one-to-one on every orbit.",
"From REF and REF we immediately get the following strengthening of Lemma REF .",
"Corollary 6.9 If $X$ is aperiodic, then for some $A \\in \\mathfrak {B}(X)$ , $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Theorem 6.10 Let $X$ be an aperiodic $G$ -space and let $E$ be a smooth equivalence relation on $X$ with $E_G \\subseteq E$ .",
"There exists a partition $\\mathcal {P}$ of $X$ into 4 Borel sets such that $G \\mathcal {P}$ separates any two $E$ -nonequivalent points in $X$ , i.e.",
"$\\forall x,y \\in X (\\lnot (x E y) \\rightarrow f_{\\mathcal {P}}(x) \\ne f_{\\mathcal {P}}(y))$ .",
"By Corollary REF , there is $A \\in \\mathfrak {B}(X)$ and a Borel partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant complete sections.",
"For each $n \\in \\mathbb {N}$ , define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{the smallest $m$ such that } \\exists x^{\\prime } \\in A_0^{[x]_G} \\text{ with } g_m x^{\\prime } \\in A_n.$ Clearly $\\bar{n}$ is Borel, and because all of $A_k$ are $F_A$ -invariant, $\\bar{n}$ is also $F_A$ -invariant, i.e.",
"for all $x,y \\in X$ , $x F_A y \\rightarrow \\bar{n}(x) = \\bar{n}(y)$ .",
"Also, $\\bar{n}$ is $G$ -invariant by definition.",
"Put $A^{\\prime }_n = \\lbrace x \\in A_0 : g_{\\bar{n}(x)} x \\in A_n\\rbrace $ and note that $A^{\\prime }_n$ is $F_A$ -invariant Borel since so are $\\bar{n}$ , $A_0$ and $A_n$ .",
"Moreover, $A^{\\prime }_n$ is clearly a complete section.",
"Define $\\gamma _n : A^{\\prime }_n \\rightarrow A_n$ by $x \\mapsto g_{\\bar{n}(x)} x$ .",
"Clearly, $\\gamma _n$ is Borel and one-to-one.",
"Since $E$ is smooth, there is a Borel $h : X \\rightarrow \\mathbb {R}$ such that for all $x,y \\in X$ , $x E y \\leftrightarrow h(x) = h(y)$ .",
"Let $\\lbrace V_n\\rbrace _{n \\in \\mathbb {N}}$ be a countable family of subsets of $\\mathbb {R}$ generating the Borel $\\sigma $ -algebra of $\\mathbb {R}$ and put $U_n = h^{-1}(V_n)$ .",
"Because each equivalence class of $E$ is $G$ -invariant, so is $h$ and hence so is $U_n$ .",
"Now let $B_n = \\gamma _n(A^{\\prime }_n \\cap U_n)$ and note that $B_n$ is Borel being a one-to-one Borel image of a Borel set.",
"It follows from the definition of $\\gamma _n$ that $B_n \\subseteq A_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $\\mathcal {P}= < \\!\\!",
"A,B \\!\\!",
">$ ; in particular, $|\\mathcal {P}| \\le 4$ .",
"We show that $\\mathcal {P}$ is what we want.",
"To this end, fix $x,y \\in X$ with $\\lnot (x E y)$ .",
"If $\\lnot (x F_A y)$ , then $G < \\!\\!",
"A \\!\\!",
">$ (and hence $G \\mathcal {P}$ ) separates $x$ and $y$ .",
"Thus assume that $x F_A y$ .",
"Since $h(x) \\ne h(y)$ , there is $n$ such that $h(x) \\in V_n$ and $h(y) \\notin V_n$ .",
"Hence, by invariance of $U_n$ , $gx \\in U_n \\wedge gy \\notin U_n$ , for all $g \\in G$ .",
"Because $A^{\\prime }_n$ is a complete section, there is $g \\in G$ such that $gx \\in A^{\\prime }_n$ and hence $gy \\in A^{\\prime }_n$ since $A^{\\prime }_n$ is $F_A$ -invariant.",
"Let $m = \\bar{n}(gx)$ ($= \\bar{n}(gy)$ ).",
"Then $g_m gx \\in B_n$ while $g_m gy \\notin B_n$ although $g_m gy \\in \\gamma _n(A^{\\prime }_n) \\subseteq A_n$ .",
"Thus $g_m gx \\in B$ but $g_m gy \\notin B$ and therefore $G \\mathcal {P}$ separates $x$ and $y$ .",
"Potential dichotomy theorems In this section we prove dichotomy theorems assuming Weiss's question has a positive answer for $G = \\mathbb {Z}$ .",
"In the proofs we use the Ergodic Decomposition Theorem (see , ) and a Borel/uniform version of Krieger's finite generator theorem, so we first state both of the theorems and sketch the proof of the latter.",
"For a Borel $G$ -space $X$ , let $\\mathcal {M}_G(X)$ denote the set of $G$ -invariant Borel probability measures on $X$ and let $\\mathcal {E}_G(X)$ denote the set of ergodic ones among those.",
"Clearly both are Borel subsets of $P(X)$ (the standard Borel space of Borel probability measures on $X$ ) and thus are themselves standard Borel spaces.",
"Ergodic Decomposition Theorem 7.1 (Farrell, Varadarajan) Let $X$ be a Borel $G$ -space.",
"If $\\mathcal {M}_G(X) \\ne \\mathbb {\\emptyset }$ (and hence $\\mathcal {E}_G(X) \\ne \\mathbb {\\emptyset }$ ), then there is a Borel surjection $x \\mapsto e_x$ from $X$ onto $\\mathcal {E}_G(X)$ such that: $x E_G y \\Rightarrow e_x = e_y$ ; For each $e \\in \\mathcal {E}_G(X)$ , if $X_e = \\lbrace x \\in X : e_x = e\\rbrace $ (hence $X_e$ is invariant Borel), then $e(X_e) = 1$ and $e \\!",
"\\!",
"\\downharpoonright _{X_e}$ is the unique ergodic invariant Borel probability measure on $X_e$ ; For each $\\mu \\in \\mathcal {M}_G(X)$ and $A \\in \\mathfrak {B}(X)$ , we have $\\mu (A) = \\int e_x(A) d\\mu (x).$ For the rest of the section, let $X$ be a Borel $\\mathbb {Z}$ -space.",
"For $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , if we let $h_e$ denote the entropy of $(X, \\mathbb {Z}, e)$ , then the map $e \\mapsto h_e$ is Borel.",
"Indeed, if $\\lbrace \\mathcal {P}_k\\rbrace _{k \\in \\mathbb {N}}$ is a refining sequence of partitions of $X$ that generates the Borel $\\sigma $ -algebra of $X$ , then by 4.1.2 of , $h_e = \\lim _{k \\rightarrow \\infty } h_e(\\mathcal {P}_k, \\mathbb {Z})$ , where $h_e(\\mathcal {P}_k, \\mathbb {Z})$ denotes the entropy of $\\mathcal {P}_k$ .",
"By 17.21 of , the function $e \\mapsto h_e(\\mathcal {P}_k)$ is Borel and thus so is the map $e \\mapsto h_e$ .",
"For all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ with $h_e < \\infty $ , let $N_e$ be the smallest integer such that $\\log N_e > h_e$ .",
"The map $e \\mapsto N_e$ is Borel because so is $e \\mapsto h_e$ .",
"Krieger's Finite Generator Theorem 7.2 (Uniform version) Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"Suppose $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Assume also that all measures in $\\mathcal {E}_{\\mathbb {Z}}(X)$ have finite entropy and let $e \\mapsto N_e$ be the map defined above.",
"Then there is a partition $\\lbrace A_n\\rbrace _{n \\le \\infty }$ of $X$ into Borel sets such that $A_{\\infty }$ is invariant and does not admit an invariant Borel probability measure; For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e \\setminus A_{\\infty }$ , where $X_e = \\rho ^{-1}(e)$ .",
"Sketch of Proof.",
"Note that it is enough to find a Borel invariant set $X^{\\prime } \\subseteq X$ and a Borel $\\mathbb {Z}$ -map $\\phi : X^{\\prime } \\rightarrow \\mathbb {N}^{\\mathbb {Z}}$ , such that for each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , we have $e(X \\setminus X^{\\prime }) = 0$ ; $\\phi \\!",
"\\!",
"\\downharpoonright _{X_e \\cap X^{\\prime }}$ is one-to-one and $\\phi (X_e \\cap X^{\\prime }) \\subseteq (N_e)^{\\mathbb {Z}}$ , where $(N_e)^{\\mathbb {Z}}$ is naturally viewed as a subset of $\\mathbb {N}^{\\mathbb {Z}}$ .",
"Indeed, assume we had such $X^{\\prime }$ and $\\phi $ , and let $A_{\\infty } = X \\setminus X^{\\prime }$ and $A_n = \\phi ^{-1}(V_n)$ for all $n \\in \\mathbb {N}$ , where $V_n = \\lbrace y \\in \\mathbb {N}^{\\mathbb {Z}} : y(0) = n\\rbrace $ .",
"Then it is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ satisfies (ii).",
"Also, (I) and part (ii) of the Ergodic Decomposition Theorem imply that (i) holds for $A_{\\infty }$ .",
"To construct such a $\\phi $ , we use the proof of Krieger's theorem presented in , Theorem 4.2.3, and we refer to it as Downarowicz's proof.",
"For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , the proof constructs a Borel $\\mathbb {Z}$ -embedding $\\phi _e : X^{\\prime } \\rightarrow N_e^{\\mathbb {Z}}$ on an $e$ -measure 1 set $X^{\\prime }$ .",
"We claim that this construction is uniform in $e$ in a Borel way and hence would yield $X^{\\prime }$ and $\\phi $ as above.",
"Our claim can be verified by inspection of Downarowicz's proof.",
"The proof uses the existence of sets with certain properties and one has to check that such sets exist with the properties satisfied for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ at once.",
"For example, the set $C$ used in the proof of Lemma 4.2.5 in can be chosen so that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $C \\cap X_e$ has the required properties for $e$ (using the Shannon-McMillan-Brieman theorem).",
"Another example is the set $B$ used in the proof of the same lemma, which is provided by Rohlin's lemma.",
"By inspection of the proof of Rohlin's lemma (see 2.1 in ), one can verify that we can get a Borel $B$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $B \\cap X_e$ has the required properties for $e$ .",
"The sets in these two examples are the only kind of sets whose existence is used in the whole proof; the rest of the proof constructs the required $\\phi $ “by hand”.",
"$\\Box $ Theorem 7.3 (Dichotomy I) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy; there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"We first show that the conditions above are mutually exclusive.",
"Indeed, assume there exist an invariant ergodic Borel probability measure $e$ with infinite entropy and a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"By ergodicity, $e$ would have to be supported on one of the $Y_n$ .",
"But $Y_n$ has a finite generator and hence the dynamical system $(Y_n, \\mathbb {Z}, e)$ has finite entropy by the Kolmogorov-Sinai theorem (see REF ).",
"Thus so does $(X, \\mathbb {Z}, e)$ since these two systems are isomorphic (modulo $e$ -NULL), contradicting the assumption on $e$ .",
"Now we prove that at least one of the conditions holds.",
"Assume that there is no invariant ergodic measure with infinite entropy.",
"Now, if there was no invariant Borel probability measure at all, then, since the answer to Question REF is assumed to be positive, $X$ would admit a finite generator, and we would be done.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ , $Y_{\\infty } = A_{\\infty }$ , and for all $n \\in \\mathbb {N}$ , $Y_n = \\lbrace x \\in X^{\\prime } : N_{e_x} = n\\rbrace ,$ where the map $e \\mapsto N_e$ is as above.",
"Note that the sets $Y_n$ are invariant since $\\rho $ is invariant, so $\\lbrace Y_n\\rbrace _{n \\le \\infty }$ is a countable partition of $X$ into invariant Borel sets.",
"Since $Y_{\\infty }$ does not admit an invariant Borel probability measure, by our assumption, it has a finite generator.",
"Let $E$ be the equivalence relation on $X^{\\prime }$ defined by $\\rho $ , i.e.",
"$\\forall x,y \\in X^{\\prime }$ , $x E y \\Leftrightarrow \\rho (x) = \\rho (y).$ By definition, $E$ is a smooth Borel equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action.",
"Thus, by Theorem REF , there exists a partition $\\mathcal {P}$ of $X^{\\prime }$ into 4 Borel sets such that $\\mathbb {Z}\\mathcal {P}$ separates any two points in different $E$ -classes.",
"Now fix $n \\in \\mathbb {N}$ and we will show that $I= \\mathcal {P}\\vee \\lbrace A_i\\rbrace _{i < n}$ is a generator for $Y_n$ .",
"Indeed, take distinct $x,y \\in Y_n$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}\\mathcal {P}$ separates them and hence so does $\\mathbb {Z}I$ .",
"Thus we can assume that $x E y$ .",
"Then $e := \\rho (x) = \\rho (y)$ , i.e.",
"$x,y \\in X_e = \\rho ^{-1}(e)$ .",
"By the choice of $\\lbrace A_i\\rbrace _{i \\in \\mathbb {N}}$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e$ and hence $\\mathbb {Z}\\lbrace A_i\\rbrace _{i < N_e}$ separates $x$ and $y$ .",
"But $n = N_e$ by the definition of $Y_n$ , so $\\mathbb {Z}I$ separates $x$ and $y$ .",
"Proposition 7.4 Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ admits invariant ergodic probability measures of arbitrarily large entropy, then it admits an invariant probability measure of infinite entropy.",
"For each $n \\ge 1$ , let $\\mu _n$ be an invariant ergodic probability measure of entropy $h_{\\mu _n} > n 2^n$ such that $\\mu _n \\ne \\mu _m$ for $n \\ne m$ , and put $\\mu = \\sum _{n \\ge 1} {1 \\over 2^n} \\mu _n.$ It is clear that $\\mu $ is an invariant probability measure, and we show that its entropy $h_{\\mu }$ is infinite.",
"Fix $n \\ge 1$ .",
"Let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem and put $X_n = \\rho ^{-1}(\\mu _n)$ .",
"It is clear that $\\mu _m(X_n) = 1$ if $m = n$ and 0 otherwise.",
"For any finite Borel partition $\\mathcal {P}= \\lbrace A_i\\rbrace _{i=1}^k$ of $X_n$ , put $A_0 = X \\setminus X_n$ and $\\bar{\\mathcal {P}} = \\mathcal {P}\\cup \\lbrace A_0\\rbrace $ .",
"Let $T$ be the Borel automorphism of $X$ corresponding to the action of $1_{\\mathbb {Z}}$ , and let $h_{\\nu }(I)$ and $h_{\\nu }(I, T)$ denote, respectively, the static and dynamic entropies of a finite Borel partition $I$ of $X$ with respect to an invariant probability measure $\\nu $ .",
"Then, with the convention that $\\log (0) \\cdot 0 = 0$ , we have $h_{\\mu }(\\bar{\\mathcal {P}}) &= - \\sum _{i=0}^k \\log (\\mu (A_i)) \\mu (A_i) \\ge - \\sum _{i = 1}^k \\log (\\mu (A_i)) \\mu (A_i)= - \\sum _{i = 1}^k \\log ({1 \\over 2^n}\\mu _n(A_i)) {1 \\over 2^n} \\mu _n(A_i) \\\\&\\ge - {1 \\over 2^n} \\sum _{i = 1}^k \\log (\\mu _n(A_i)) \\mu _n(A_i) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}).$ Since $\\mathcal {P}$ is arbitrary and $X_n$ is invariant, it follows that $h_{\\mu }(\\bar{\\mathcal {P}}, T) = \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu }(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) \\ge {1 \\over 2^n} \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu _n}(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}, T).$ Now for any finite Borel partition $I$ of $X$ , it is clear that $h_{\\mu _n}(I) = h_{\\mu _n}(\\bar{\\mathcal {P}})$ (and hence $h_{\\mu _n}(I, T) = h_{\\mu _n}(\\bar{\\mathcal {P}}, T)$ ), for some $\\mathcal {P}$ as above.",
"This implies that $h_{\\mu } \\ge \\sup _{\\mathcal {P}} h_{\\mu }(\\bar{\\mathcal {P}}, T) \\ge {1 \\over 2^n} \\sup _{\\mathcal {P}} h_{\\mu _n}(\\bar{\\mathcal {P}}, T) = {1 \\over 2^n} \\sup _{I} h_{\\mu _n}(I, T) = {1 \\over 2^n} h_{\\mu _n} > n,$ where $\\mathcal {P}$ and $I$ range over finite Borel partitions of $X_n$ and $X$ , respectively.",
"Thus $h_{\\mu }\\!",
"= \\infty $ .",
"Theorem 7.5 (Dichotomy II) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"The Kolmogorov-Sinai theorem implies that the conditions are mutually exclusive, and we prove that at least one of them holds.",
"Assume that there is no invariant measure with infinite entropy.",
"If there was no invariant Borel probability measure at all, then, by our assumption, $X$ would admit a finite generator.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ and $X_e = \\rho ^{-1}(e)$ , for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ .",
"By our assumption, $A_{\\infty }$ admits a finite generator $\\mathcal {P}$ .",
"Also, by REF , there is $N \\ge 1$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $N_e \\le N$ and hence $\\mathcal {Q}:= \\lbrace A_n\\rbrace _{n < N}$ is a finite generator for $X_e$ ; in particular, $\\mathcal {Q}$ is a partition of $X^{\\prime }$ .",
"Let $E$ be the following equivalence relation on $X$ : $x E y \\Leftrightarrow (x, y \\in A_{\\infty }) \\vee (x,y \\in X^{\\prime } \\wedge \\rho (x) = \\rho (y)).$ By definition, $E$ is a smooth equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action and $A_{\\infty }$ is $\\mathbb {Z}$ -invariant.",
"Thus, by Theorem REF , there exists a partition $J$ of $X$ into 4 Borel sets such that $\\mathbb {Z}J$ separates any two points in different $E$ -classes.",
"We now show that $I:= < \\!\\!",
"J\\cup \\mathcal {P}\\cup \\mathcal {Q} \\!\\!",
">$ is a generator.",
"Indeed, fix distinct $x,y \\in X$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}J$ separates them.",
"So we can assume that $x E y$ .",
"If $x,y \\in A_{\\infty }$ , then $\\mathbb {Z}\\mathcal {P}$ separates $x$ and $y$ .",
"Finally, if $x,y \\in X^{\\prime }$ , then $x,y \\in X_e$ , where $e = \\rho (x)$ ($= \\rho (y)$ ), and hence $\\mathbb {Z}\\mathcal {Q}$ separates $x$ and $y$ .",
"Remark.",
"It is likely that the above dichotomies are also true for any amenable group using a uniform version of Krieger's theorem for amenable groups, cf.",
", but I have not checked the details.",
"Finite generators on comeager sets Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER.",
"Locally weakly wandering sets and other special cases Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
"BKbook author = Becker, H. author = Kechris, A. S. title = The Descriptive Set Theory of Polish Group Actions date = 1996 publisher = Cambridge Univ.",
"Press series = London Math.",
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"volume = 149 pages = 453-464 Krengelarticle author = Krengel, U. title = Transformations without finite invariant measure have finite strong generators conference = title = First Midwest Conference, Ergodic Theory and Probability book = series = Springer Lecture Notes volume = 160 date = 1970 pages = 133-157 Kuntzarticle author = Kuntz, A. J. title = Groups of transformations without finite invariant measures have strong generators of size 2 date = 1974 journal = Annals of Probability volume = 2 number = 1 pages = 143-146 Millerthesisbook author = Miller, B. D. title = PhD Thesis: Full groups, classification, and equivalence relations date = 2004 publisher = University of California at Los Angeles Millerarticle author = Miller, B. D. title = The existence of measures of a given cocycle, II: Probability measures date = 2008 journal = Ergodic Theory and Dynamical Systems volume = 28 number = 5 pages = 1615-1633 Munroebook author = Munroe, M. E. title = Introduction to Measure and Integration date = 1953 publisher = Addison-Wesley Nadkarniarticle author = Nadkarni, M. G. title = On the existence of a finite invariant measure date = 1991 journal = Proc.",
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],
[
"Finite generators and $i$ -compressibility",
"Throughout this section let $X$ be a Borel $G$ -space and $E_G$ be the orbit equivalence relation on $X$ .",
"For $A \\subseteq X$ and $G$ -invariant $P \\subseteq X$ , let $A^P := A \\cap P$ .",
"For an equivalence relation $E$ on $X$ and $A \\subseteq X$ , let $[A]_E$ denote the saturation of $A$ with respect to $E$ , i.e.",
"$[A]_E = \\lbrace x \\in X : \\exists y \\in A (x E y)\\rbrace $ .",
"In case $E = E_G$ , we use $[A]_G$ instead of $[A]_{E_G}$ .",
"Let $\\mathfrak {B}$ denote the class of all Borel sets in standard Borel spaces and let $\\Gamma $ be a $\\sigma $ -algebra of subsets of standard Borel spaces containing $\\mathfrak {B}$ and closed under Borel preimages.",
"For example, $\\Gamma = \\mathfrak {B}$ , $\\sigma (\\mathbf {\\Sigma }_1^1)$ , universally measurable sets.",
"For $A \\subseteq X$ , let $\\Gamma (A)$ denote the set of $\\Gamma $ sets relative to $A$ , i.e.",
"$\\Gamma (A) = \\lbrace B \\cap A : B \\subseteq X, B \\in \\Gamma \\rbrace $ .",
"subsection2-2 ex5 pt The notion of $I$ -equidecomposability A countable partition of $X$ is called Borel if all the sets in it are Borel.",
"For a finite Borel partition $I= \\lbrace A_i : i < k\\rbrace $ of $X$ , let $F_{I}$ denote the equivalence relation of not being separated by $G I:= \\lbrace g A_i : g \\in G, i < k\\rbrace $ , more precisely, $\\forall x,y\\in X$ , $x F_{I}y \\Leftrightarrow x) = y),$ where $ is the symbolic representation map for $ (X, G, I)$ defined above.",
"Note that if $ I$ is a generator, then $ FI$ is just the equality relation.$ For an equivalence relation $E$ on $X$ and $A,B \\subseteq X$ , $A$ is said to be $E$ -invariant relative to $B$ or just $E \\!",
"\\!",
"\\downharpoonright _{B}$ -invariant if $[A]_{E} \\cap B = A \\cap B$ .",
"Definition 2.1 ($I$ -equidecomposability) Let $A,B \\subseteq X$ , and $I$ be a finite Borel partition of $X$ .",
"$A$ and $B$ are said to be equidecomposable with $\\Gamma $ pieces (denote by $A \\sim ^{\\Gamma } B$ ) if there are $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq G$ and partitions $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $A$ and $B$ , respectively, such that for all $n \\in \\mathbb {N}$ $g_n A_n = B_n$ , $A_n \\in \\Gamma (A)$ and $B_n \\in \\Gamma (B)$ .",
"If moreover, $A_n$ and $B_n$ are $F_{I}$ -invariant relative to $A$ and $B$ , respectively, then we will say that $A$ and $B$ are $I$ -equidecomposable with $\\Gamma $ pieces and denote it by $A \\sim _{I}^{\\Gamma } B$ .",
"If $\\Gamma = \\mathfrak {B}$ , we will not mention $\\Gamma $ and will just write $\\sim $ and $\\sim _{I}$ .",
"Note that for any $I$ , $A$ , $B$ as above, $A$ and $B$ are $I$ -equidecomposable if and only if $A)$ and $B)$ are equidecomposable (although the images of Borel sets under $ are analytic, they are Borel relative to $ X)$ due to the Lusin Separation Theorem for analytic sets).",
"Also note that if $ I$ is a generator, then $ I$ coincides with $$.$ Observation 2.2 Below let $I,I_0,I_1$ denote finite Borel partitions of $X$ , and $A,B,C \\in \\Gamma (X)$ .",
"(Quasi-transitivity) If $A \\sim _{I_0}^{\\Gamma } B \\sim _{I_1}^{\\Gamma } C$ , then $A \\sim _{I}^{\\Gamma } C$ with $I= I_0 \\vee I_1$ (the least common refinement of $I_0$ and $I_1$ ).",
"($F_{I}$ -disjoint countable additivity) Let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ be partitions of $A$ and $B$ , respectively, into $\\Gamma $ sets such that $\\forall n \\ne m$ , $[A_n]_{F_{I}} \\cap [A_m]_{F_{I}} = [B_n]_{F_{I}} \\cap [B_m]_{F_{I}} = \\mathbb {\\emptyset }$ .",
"If $\\forall n\\in \\mathbb {N}$ , $A_n \\sim _{I}^{\\Gamma } B_n$ , then $A \\sim _{I}^{\\Gamma } B$ .",
"If $A \\sim B$ , then there is a Borel isomorphism $\\phi $ of $A$ onto $B$ with $\\phi (x) E_G x$ for all $x \\in A$ ; namely $\\phi (x) = g_n x$ for all $x \\in A_n$ , where $A_n, g_n$ are as in Definition REF .",
"It is easy to see that the converse is also true, i.e.",
"if such $\\phi $ exists, then $A \\sim B$ .",
"In Proposition REF we prove the analogue of this for $\\sim _{I}^{\\Gamma }$ , but first we need the following lemma and definition that take care of definability and $F_{I}$ -invariance, respectively.",
"For a Polish space $Y$ , $f : X \\rightarrow Y$ is said to be $\\Gamma $ -measurable if the preimages of open sets under $f$ are in $\\Gamma $ .",
"For $A \\in \\Gamma (X)$ and $h : A \\rightarrow G$ , define $\\hat{h} : A \\rightarrow X$ by $x \\mapsto h(x)x$ .",
"Lemma 2.3 If $h : A \\rightarrow G$ is $\\Gamma $ -measurable, then the images and preimages of sets in $\\Gamma $ under $\\hat{h}$ are in $\\Gamma $ .",
"Let $B \\subseteq A$ , $C \\subseteq X$ be in $\\Gamma $ .",
"For $g \\in G$ , set $A_g = h^{-1}(g)$ and note that $\\hat{h}(B) = \\bigcup _{g \\in G} g(A_g \\cap B)$ and $\\hat{h}^{-1}(C) = \\bigcup _{g \\in G} g^{-1}(gA_g \\cap C)$ .",
"Thus $\\hat{h}(B)$ and $\\hat{h}^{-1}(C)$ are in $\\Gamma $ by the assumptions on $\\Gamma $ .",
"The following technical definition is needed in the proofs of REF and REF .",
"Definition 2.4 For $A \\subseteq X$ and a finite Borel partition $I$ of $X$ , we say that $I$ is $A$ -sensitive or that $A$ respects $I$ if $A$ is $F_{I}$ -invariant relative to $[A]_G$ , i.e.",
"$[A]_{F_{I}}^{[A]_G} = A$ .",
"For example, if $I$ is finer than $\\lbrace A, A^c\\rbrace $ , then $I$ is $A$ -sensitive.",
"Note that if $A \\sim _{I} B$ and $A$ respects $I$ , then so does $B$ .",
"Proposition 2.5 Let $A,B \\in \\Gamma (X)$ and let $I$ be a Borel partition of $X$ that is $A$ -sensitive.",
"Then, $A \\sim _{I}^{\\Gamma } B$ if and only if there is an $F_{I}$ -invariant $\\Gamma $ -measurable map $\\gamma : A \\rightarrow G$ such that $\\hat{\\gamma }$ is a bijection between $A$ and $B$ .",
"We refer to such $\\gamma $ as a witnessing map for $A \\sim _{I}^{\\Gamma } B$ .",
"The same holds if we delete “$F_{I}$ -invariant” and “$I$ ” from the statement.",
"$\\Rightarrow $ : If $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ , $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ and $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ are as in Definition REF , then define $\\gamma : A \\rightarrow G$ by setting $\\gamma \\!",
"\\!",
"\\downharpoonright _{A_n} \\equiv g_n$ .",
"$\\Leftarrow $ : Let $\\gamma $ be as in the lemma.",
"Fixing an enumeration $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ of $G$ with no repetitions, put $A_n = \\gamma ^{-1}(g_n)$ and $B_n = g_n A_n$ .",
"It is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ are partitions of $A$ and $B$ , respectively, into $\\Gamma $ sets.",
"Since $\\gamma $ is $F_{I}$ -invariant, each $A_n$ is $F_{I}$ -invariant relative to $A$ and hence relative to $P := [A]_G = [B]_G$ because $A$ respects $I$ .",
"It remains to show that each $B_n$ is $F_{I}$ -invariant relative to $B$ .",
"To this end, let $y \\in [B_n]_{F_{I}} \\cap B$ and thus there is $x \\in A_n$ such that $y F_{I}g_n x$ .",
"Hence $z := g_n^{-1} y \\ F_{I}\\ g_n^{-1} g_n x = x$ and therefore $z \\in A_n$ because $A_n$ is $F_{I}$ -invariant relative to $P$ .",
"Thus $y = g_n z \\in B_n$ .",
"In the rest of the subsection we work with $\\Gamma = \\mathfrak {B}$ .",
"Next we prove that $I$ -equidecomposability can be extended to $F_{I}$ -invariant Borel sets.",
"First we need the following reflection principle.",
"Lemma 2.6 (A reflection principle) Let $E$ be a Borel equivalence relation on $X$ and $B \\subseteq X$ be Borel.",
"Define the predicate $\\Phi \\subseteq Pow(X)$ as follows: $\\Phi (A) \\Leftrightarrow A \\subseteq B \\wedge A \\text{ is $E$-invariant}.$ If $A$ is analytic and $\\Phi (A)$ then there exists a Borel set $A^{\\prime } \\supseteq A$ with $\\Phi (A^{\\prime })$ .",
"Define the predicate $\\Psi \\subseteq Pow(X)$ as follows: $\\Psi (D) \\Leftrightarrow B^c \\subseteq D \\wedge D \\text{ is $E$-invariant}.$ It is clear that $\\Phi (D) \\Leftrightarrow \\Psi (D^c)$ , so it is enough to show that if $D$ is co-analytic and $\\Psi (D)$ , then there is a Borel set $D^{\\prime } \\subseteq D$ with $\\Psi (D^{\\prime })$ .",
"Note that $\\Psi (D) \\Leftrightarrow \\forall x \\in X \\forall y \\in X (x \\notin D \\wedge y \\in D \\Rightarrow x \\notin B \\wedge \\lnot xEy).$ Thus, setting $R(x,y) \\Leftrightarrow x \\notin B \\wedge \\lnot xEy$ , we apply The Burgess Reflection Theorem (see 35.18 in ) to $\\Psi $ with $\\Gamma = \\mathbf {\\Pi }_1^1$ and $A = D$ , and get a Borel $D^{\\prime } \\subseteq D$ with $\\Psi (D^{\\prime })$ .",
"Proposition 2.7 ($F_{I}$ -invariant extensions) If for some Borel partition $I$ of $X$ and Borel sets $A,B \\subseteq X$ , $A \\sim _{I} B$ , then there exists Borel sets $A^{\\prime } \\supseteq A$ and $B^{\\prime } \\supseteq B$ such that $A^{\\prime },B^{\\prime }$ are $F_{I}$ -invariant and $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"In fact, if $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ witness $A \\sim _{I} B$ , then there are $F_{I}$ -invariant Borel partitions $\\lbrace A^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}$ of $A^{\\prime }$ and $B^{\\prime }$ respectively, such that $g_n A^{\\prime }_n = B^{\\prime }_n$ and $A^{\\prime }_n \\supseteq A_n$ (and hence $B^{\\prime }_n \\supseteq B_n$ ).",
"Let $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}, \\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF and put $\\bar{A}_n = [A_n]_{F_{I}}$ .",
"It is easy to see that for $n \\ne m \\in \\mathbb {N}$ , $\\bar{A}_n \\cap \\bar{A}_m = \\mathbb {\\emptyset }$ ; $g_n \\bar{A}_n \\cap g_m \\bar{A}_m = \\mathbb {\\emptyset }$ .",
"Put $\\bar{A} = [A]_{F_{I}}$ and note that $\\lbrace \\bar{A}_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $\\bar{A}$ .",
"Although $\\bar{A}_n$ and $\\bar{A}$ are $F_{I}$ -invariant, they are analytic and in general not Borel.",
"We obtain Borel analogues using $\\mathbf {\\Pi }_1^1$ -reflection theorems.",
"Set $U = \\bigcup _{n \\in \\mathbb {N}}(n \\times \\bar{A}_n)$ and define a predicate $\\Phi \\subseteq Pow(\\mathbb {N}\\times X)$ as follows: $\\Phi (W) \\Leftrightarrow \\forall n (W_n \\text{ is $F_{I}$-invariant}) \\wedge \\forall n \\ne m (W_n \\cap W_m = \\mathbb {\\emptyset } \\wedge g_n W_n \\cap g_m W_m = \\mathbb {\\emptyset }),$ where $W_n = \\lbrace x \\in X : (n,x) \\in W\\rbrace $ , the section of $W$ at $n$ .",
"Note that $\\Phi (U)$ .",
"Claim There is a Borel set $U^{\\prime } \\supseteq U$ with $\\Phi (U^{\\prime })$ .",
"Proof of Claim.",
"For $W \\subseteq \\mathbb {N}\\times X$ , let $\\Lambda (W) \\Leftrightarrow \\forall n \\ne m (W_n \\cap W_m = \\mathbb {\\emptyset } \\wedge g_n W_n \\cap g_m W_m = \\mathbb {\\emptyset })$ Note that $\\Lambda (W) \\Leftrightarrow \\forall n \\ne m \\forall x \\in X [(x \\notin W_n \\vee x \\notin W_m) \\wedge (x \\notin g_n W_n \\vee x \\notin g_m W_m)].$ Thus $\\Lambda $ is $\\mathbf {\\Pi }_1^1$ on $\\mathbf {\\Sigma }_1^1$ , and hence, by the dual form of the First Reflection Theorem for $\\mathbf {\\Pi }_1^1$ (see the discussion following 35.10 in ), there is a Borel set $V \\supseteq U$ with $\\Lambda (V)$ , since $\\Lambda (U)$ .",
"Now applying Lemma REF to $E = F_{I}$ , $B = V_n$ and $A = U_n$ , we get $F_{I}$ -invariant Borel sets $U^{\\prime }_n \\subseteq X$ with $U_n \\subseteq U^{\\prime }_n \\subseteq V_n$ .",
"Thus $U^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}(n \\times U^{\\prime }_n)$ is what we wanted.",
"$\\dashv $ Put $A^{\\prime }_n = U^{\\prime }_n$ and $A^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}A^{\\prime }_n$ .",
"Thus $\\lbrace A^{\\prime }_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $A^{\\prime }$ into $F_{I}$ -invariant Borel sets.",
"Also, $A_n \\subseteq \\bar{A_n} \\subseteq A^{\\prime }_n$ and hence $A \\subseteq \\bar{A} \\subseteq A^{\\prime }$ .",
"Put $B_n^{\\prime } = g_n A^{\\prime }_n$ and $B^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}B_n^{\\prime }$ ; thus $\\lbrace B_n^{\\prime }\\rbrace _{n \\in \\mathbb {N}}$ is a Borel partition of $B^{\\prime }$ .",
"Also note that $B_n^{\\prime }$ are $F_{I}$ -invariant and $B^{\\prime } \\supseteq B$ since $A^{\\prime }_n$ are $F_{I}$ -invariant and $A_n \\subseteq A^{\\prime }_n$ .",
"Thus $A^{\\prime } \\sim _{I} B^{\\prime }$ and we are done.",
"Lemma 2.8 (Orbit-disjoint unions) Let $A_k,B_k \\in \\mathfrak {B}(X)$ , $k=0,1$ , be such that $[A_0]_G$ and $[A_1]_G$ are disjoint and put $A = A_0 \\cup A_1$ and $B = B_0 \\cup B_1$ .",
"If $I$ is an $A,B$ -sensitive finite Borel partition of $X$ such that $A_k \\sim _{I} B_k$ for $k=0,1$ , then $A \\sim _{I} B$ .",
"Moreover, if $\\gamma _0 : A_0 \\rightarrow G$ is a Borel map witnessing $A_0 \\sim _{I} B_0$ , then there exists a Borel map $\\gamma : A \\rightarrow G$ extending $\\gamma _0$ that witnesses $A \\sim _{I} B$ .",
"First assume without loss of generality that $X = [A]_G$ ($= [B]_G$ ) since the statement of the lemma is relative to $[A]_G$ .",
"Thus $A,B$ are $F_{I}$ -invariant.",
"Applying REF to $A_0 \\sim _{I} B_0$ , we get $F_{I}$ -invariant $A^{\\prime }_0 \\supseteq A_0, B^{\\prime }_0 \\supseteq B_0$ such that $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"Moreover, by the second part of the same lemma, if $\\gamma _0 : A_0 \\rightarrow G$ is a witnessing map for $A_0 \\sim _{I} B_0$ , then there is a witnessing map $\\delta : A^{\\prime }_0 \\rightarrow G$ for $A^{\\prime } \\sim _{I} B^{\\prime }$ extending $\\gamma _0$ .",
"Put $C = A^{\\prime }_0 \\cap A$ and note that $C$ is $F_{I}$ -invariant since so are $A^{\\prime }_0$ and $A$ .",
"Finally, put $\\bar{A}_0 = \\lbrace x \\in C : C^{[x]_G} = A^{[x]_G} \\wedge \\hat{\\delta }(C^{[x]_G}) = B^{[x]_G}\\rbrace $ and note that $\\bar{A}_0 \\supseteq A_0$ since $\\delta \\supseteq \\gamma _0$ and $[A_0]_G \\cap [A_1]_G = \\mathbb {\\emptyset }$ .",
"Claim $\\bar{A}_0$ is $F_{I}$ -invariant.",
"Proof of Claim.",
"First note that for any $F_{I}$ -invariant $D \\subseteq X$ and $z \\in X$ , $[D^{[z]_G}]_{F_{I}} = D^{[[z]_{F_{I}}]_G}$ .",
"Furthermore, if $D \\subseteq C$ , then $[\\hat{\\delta }(D)]_{F_{I}} = \\hat{\\delta }([D]_{F_{I}})$ since $\\hat{\\delta }$ and its inverse map $F_{I}$ -invariant sets to $F_{I}$ -invariant sets.",
"Now take $x \\in \\bar{A}_0$ and let $Q = [[x]_{F_{I}}]_G$ .",
"Since $A, B, C$ are $F_{I}$ -invariant, $C^Q = [C^{[x]_G}]_{F_{I}} = [A^{[x]_G}]_{F_{I}} = A^Q$ .",
"Furthermore, $\\hat{\\delta }(C^Q) = \\hat{\\delta }([C^{[x]_G}]_{F_{I}}) = [\\hat{\\delta }(C^{[x]_G})]_{F_{I}} = [B^{[x]_G}]_{F_{I}} = B^Q$ .",
"Thus, $\\forall y \\in [x]_{F_{I}}$ , $C^{[y]_G} = A^{[y]_G}$ and $\\hat{\\delta }(C^{[y]_G}) = B^{[y]_G}$ ; hence $[x]_{F_{I}} \\subseteq \\bar{A}_0$ .",
"$\\dashv $ Put $\\bar{A}_1 = A \\setminus \\bar{A}_0$ , $\\alpha _0 = \\delta \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_0}$ , $\\alpha _1 = \\gamma _1 \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_1}$ , where $\\gamma _1$ is a witnessing map for $A_1 \\sim _{I} B_1$ .",
"It is clear from the definition of $\\bar{A}_0$ that $\\bar{A}_0$ is $E_G$ -invariant relative to $A$ and hence $[\\bar{A}_0]_G \\cap [\\bar{A}_1]_G = \\mathbb {\\emptyset }$ .",
"Thus, for $k=0,1$ , it follows that $\\alpha _k$ witnesses $\\bar{A}_k \\sim _{I} \\bar{B}_k$ , where $\\bar{B}_k = \\hat{\\alpha _k}(\\bar{A}_k)$ .",
"Furthermore, it is clear that $B^{[\\bar{A}_k]_G} = \\bar{B}_k$ and, since $[\\bar{A}_0]_G \\cup [\\bar{A}_1]_G = X$ , $\\bar{B}_0 \\cup \\bar{B}_1 = B$ .",
"Now since $\\bar{A}_k$ are $F_{I}$ -invariant, $\\gamma = \\alpha _0 \\cup \\alpha _1$ is $F_{I}$ -invariant and hence witnesses $A \\sim _{I} B$ .",
"Finally, $\\alpha _0 \\!",
"\\!",
"\\downharpoonright _{A_0} = \\delta \\!",
"\\!",
"\\downharpoonright _{A_0} = \\gamma _0$ and hence $\\alpha _0 \\supseteq \\gamma _0$ .",
"Proposition 2.9 (Orbit-disjoint countable unions) For $k \\in \\mathbb {N}$ , let $A_k,B_k \\in \\mathfrak {B}(X)$ be such that $[A_k]_G$ are disjoint and put $A = \\bigcup _{k \\in \\mathbb {N}} A_k$ , $B = \\bigcup _{k \\in \\mathbb {N}} B_k$ .",
"Suppose that $I$ is an $A,B$ -sensitive finite Borel partition of $X$ such that $A_k \\sim _{I} B_k$ for all $k$ .",
"Then $A \\sim _{I} B$ .",
"We recursively apply Lemma REF as follows.",
"Put $\\bar{A}_n = \\bigcup _{k \\le n} A_k$ and $\\bar{B}_n = \\bigcup _{k \\le n} B_k$ .",
"Inductively define Borel maps $\\gamma _n : \\bigcup _{k \\le n} A_k \\rightarrow G$ such that $\\gamma _n$ is a witnessing map for $\\bar{A}_n \\sim _{I} \\bar{B}_n$ and $\\gamma _n \\sqsubseteq \\gamma _{n+1}$ .",
"Let $\\gamma _0$ be a witnessing map for $A_0 \\sim _{I} B_0$ .",
"Assume $\\gamma _n$ is defined.",
"Then $\\gamma _{n+1}$ is provided by Lemma REF applied to $\\bar{A}_n$ and $A_{n+1}$ with $\\gamma _n$ as a witness for $\\bar{A}_n \\sim _{I} \\bar{B}_n$ .",
"Thus $\\gamma _n \\sqsubseteq \\gamma _{n+1}$ and $\\gamma _{n+1}$ witnesses $\\bar{A}_{n+1} \\sim _{I} \\bar{B}_{n+1}$ .",
"Now it just remains to show that $\\gamma := \\bigcup _{n \\in \\mathbb {N}}\\gamma _n$ is $F_{I}$ -invariant since then it follows that $\\gamma $ witnesses $A \\sim _{I} B$ .",
"Let $x,y \\in A$ be $F_{I}$ -equivalent.",
"Then there is $n$ such that $x,y \\in \\bar{A}_n$ .",
"By induction on $n$ , $\\gamma _n$ is $F_{I}$ -invariant and, since $\\gamma \\!",
"\\!",
"\\downharpoonright _{\\bar{A}_n} = \\gamma _n$ , $\\gamma (x) = \\gamma (y)$ .",
"Corollary 2.10 (Finite quasi-additivity) For $k=0,1$ , let $A_k,B_k \\in \\mathfrak {B}(X)$ be such that $A_0 \\cap A_1 = B_0 \\cap B_1 = \\mathbb {\\emptyset }$ and put $A = A_0 \\cup A_1$ , $B = B_0 \\cup B_1$ .",
"Let $I_k$ be an $A_k,B_k$ -sensitive finite Borel partition of $X$ .",
"If $A_0 \\sim _{I_0} B_0$ and $A_1 \\sim _{I_1} B_1$ , then $A \\sim _{I_0 \\vee I_1} B$ .",
"Put $I= I_0 \\vee I_1$ , $P = [A_0]_G \\cap [A_1]_G$ , $Q = [A_0]_G \\setminus [A_1]_G$ and $R = [A_1]_G \\setminus [A_0]_G$ .",
"Then $A_k^P, B_k^P$ respect $I$ , and thus $[A_0]_{F_{I}}^P \\cap [A_1]_{F_{I}}^P = \\mathbb {\\emptyset }$ , $[B_0]_{F_{I}}^P \\cap [B_1]_{F_{I}}^P = \\mathbb {\\emptyset }$ .",
"Hence $A^P \\sim _{I} B^P$ since the sets that are $F_{I}$ -invariant relative to $A_k^P$ are also $F_{I}$ -invariant relative to $A^P$ , and the same is true for $B_k^P$ and $B^P$ .",
"Also, $A^Q \\sim _{I} B^Q$ and $A^R \\sim _{I} B^R$ because $A^Q = A_0$ , $B^Q = B_0$ , $A^R = A_1$ , $B^R = B_1$ .",
"Now since $P,Q,R$ are pairwise disjoint, it follows from Proposition REF that $A \\sim _{I} B$ .",
"subsection2-2 ex5 pt The notion of $i$ -compressibility For a finite collection $\\mathcal {F}$ of subsets of $X$ , let $< \\!\\!",
"\\mathcal {F} \\!\\!",
">$ denote the partition of $X$ generated by $\\mathcal {F}$ .",
"Definition 2.11 ($i$ -equidecomposibility) For $i \\ge 1$ , $A,B \\subseteq X$ , we say that $A$ and $B$ are $i$ -equidecomposable with $\\Gamma $ pieces (write $A \\sim _i^{\\Gamma } B$ ) if there is an $A$ -sensitive partition $I$ of $X$ generated by $i$ Borel sets such that $A \\sim _{I}^{\\Gamma } B$ .",
"For a collection $\\mathcal {F}$ of Borel sets, we say that $\\mathcal {F}$ witnesses $A \\sim _i^{\\Gamma } B$ if $|\\mathcal {F}| = i$ , $I:= < \\!\\!",
"\\mathcal {F} \\!\\!",
">$ is $A$ -sensitive and $A \\sim _{I}^{\\Gamma } B$ .",
"Remark.",
"In the above definition, it might seem more natural to have $i$ be the cardinality of the partition $I$ instead of the cardinality of the collection $\\mathcal {F}$ generating $I$ .",
"However, our definition above of $i$ -equidecomposability is needed in order to show that the collection $i$ defined below forms a $\\sigma $ -ideal.",
"More precisely, the presence of $\\mathcal {F}$ is needed in the definition of $i^*$ -compressibility, which ensures that the partition $I$ in the proof of REF is $B$ -sensitive.",
"For $i \\ge 1$ , $A,B \\subseteq X$ , we write $A \\preceq _i^{\\Gamma } B$ if there is a $\\Gamma $ set $B^{\\prime } \\subseteq B$ such that $A \\sim _i^{\\Gamma } B^{\\prime }$ .",
"If moreover $[A \\setminus B]_G = [A]_G$ , then we write $A \\prec _i^{\\Gamma } B$ .",
"If $\\Gamma = \\mathfrak {B}$ , we simply write $\\sim _i, \\preceq _i, \\prec _i$ .",
"Definition 2.12 ($i$ -compressibility) For $i \\in \\mathbb {N}$ , $A \\subseteq X$ , we say that $A$ is $i$ -compressible with $\\Gamma $ pieces if $A \\prec _i^{\\Gamma } A$ .",
"Unless specified otherwise, we will be working with $\\Gamma = \\mathfrak {B}$ , in which case we simply say $i$ -compressible.",
"For a collection of sets $\\mathcal {F}$ and a $G$ -invariant set $P$ , set $\\mathcal {F}^P = \\lbrace A^P : A \\in \\mathcal {F}\\rbrace $ .",
"We will use the following observations without mentioning.",
"Observation 2.13 Let $i,j \\ge 2$ , $A,A^{\\prime },B,B^{\\prime },C \\in \\mathfrak {B}$ .",
"Let $P \\subseteq [A]_G$ denote a $G$ -invariant Borel set and $\\mathcal {F}, \\mathcal {F}_0, \\mathcal {F}_1$ denote finite collections of Borel sets.",
"If $A \\sim _i B$ then $A^P \\sim _i B^P$ .",
"If $\\mathcal {F}$ witnesses $A \\sim _i B$ , then so does $\\mathcal {F}^{[A]_G}$ .",
"If $A \\sim _i B \\sim _j C$ , then $A \\sim _{(i+j)} C$ .",
"In fact, $\\mathcal {F}_0$ and $\\mathcal {F}_1$ witness $A \\sim _i B$ and $B \\sim _j C$ , respectively, then $\\mathcal {F}= \\mathcal {F}_0 \\cup \\mathcal {F}_1$ witnesses $A \\sim _{(i+j)} C$ .",
"If $A \\preceq _i B \\preceq _j C$ , then $A \\preceq _{(i+j)} C$ .",
"If one of the first two $\\preceq $ is $\\prec $ then $A \\prec _{(i+j)} C$ .",
"If $A \\sim _i B$ and $A^{\\prime } \\sim _j B^{\\prime }$ with $A \\cap A^{\\prime } = B \\cap B^{\\prime } = \\mathbb {\\emptyset }$ , then $A \\cup A^{\\prime } \\sim _{(i+j)} B \\cup B^{\\prime }$ .",
"Part (e) follows from REF , and the rest follows directly from the definition of $i$ -equidecomposability and REF .",
"Lemma 2.14 If a Borel set $A \\subseteq X$ is $i$ -compressible, then so is $[A]_G$ .",
"In fact, if $\\mathcal {F}$ is a finite collection of Borel sets witnessing the $i$ -compressibility of $A$ , then it also witnesses that of $[A]_G$ .",
"Let $B \\subseteq A$ be a Borel set such that $[A \\setminus B]_G = [A]_G$ and $A \\sim _i B$ .",
"Furthermore, let $I$ be an $A,B$ -sensitive partition generated by a collection $\\mathcal {F}$ of $i$ Borel sets such that $A \\sim _{I} B$ .",
"Let $\\gamma : A \\rightarrow G$ be a witnessing map for $A \\sim _{I} B$ .",
"Put $A^{\\prime } = [A]_G$ , $B^{\\prime } = B \\cup (A^{\\prime } \\setminus A)$ and note that $A^{\\prime },B^{\\prime }$ respect $I$ .",
"Define $\\gamma ^{\\prime } : A^{\\prime } \\rightarrow G$ by setting $\\gamma ^{\\prime } \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A} = id \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A}$ and $\\gamma ^{\\prime } \\!",
"\\!",
"\\downharpoonright _{A} = \\gamma $ .",
"Since $A^{\\prime }$ respects $I$ and $id \\!",
"\\!",
"\\downharpoonright _{A^{\\prime } \\setminus A}, \\gamma $ are $F_{I}$ -invariant, $\\gamma ^{\\prime }$ is $F_{I}$ -invariant and thus clearly witnesses $A^{\\prime } \\sim _{I} B^{\\prime }$ .",
"The following is a technical refinement of the definition of $i$ -compressibility that is (again) necessary for $i$ , defined below, to be a $\\sigma $ -ideal.",
"Definition 2.15 ($i^*$ -compressibility) For $i \\ge 1$ , we say that a Borel set $A$ is $i^*$ -compressible if there is a Borel set $B \\subseteq A$ such that $[A \\setminus B]_G = [A]_G =: P$ , $A \\sim _i B$ , and the latter is witnessed by a collection $\\mathcal {F}$ of Borel sets such that $B \\in \\mathcal {F}^{P}$ .",
"Finally, for $i \\ge 1$ , put $i = \\lbrace A \\subseteq X : \\text{there is a $G$-invariant Borel set $P \\supseteq A$ such that $P$ is $i^*$-compressible}\\rbrace .$ Lemma 2.16 Let $i \\ge 1$ and $A \\subseteq X$ be Borel.",
"If $A \\prec _i A$ , then $A \\in {i+1}$ .",
"Setting $P = [A]_G$ and applying REF , we get that $P \\prec _i P$ , i.e.",
"there is $B \\subseteq P$ such that $[P \\setminus B]_G = P$ and $P \\sim _i B$ .",
"Let $\\mathcal {F}$ be a collection of Borel sets witnessing the latter fact.",
"Then $\\mathcal {F}^{\\prime } = \\mathcal {F}\\cup \\lbrace B\\rbrace $ witnesses $P \\sim _{(i+1)} B$ and contains $B$ .",
"Proposition 2.17 For all $i \\ge 1$ , $i$ is a $\\sigma $ -ideal.",
"We only need to show that $i$ is closed under countable unions.",
"For this it is enough to show that if $A_n \\in \\mathfrak {B}(X)$ are $i^*$ -compressible $G$ -invariant Borel sets, then so is $A := \\bigcup _{n \\in \\mathbb {N}}A_n$ .",
"We may assume that $A_n$ are pairwise disjoint since we could replace each $A_n$ by $A_n \\setminus (\\bigcup _{k<n} A_k)$ .",
"Let $B_n \\subseteq A_n$ be a Borel set and $\\mathcal {F}_n = \\lbrace F^n_k\\rbrace _{k<i}$ be a collection of Borel sets with $(F^n_0)^{A_n} = B_n$ such that $\\mathcal {F}_n$ witnesses $A_n \\sim _i B_n$ and $[A_n \\setminus B_n]_G = A_n$ .",
"Using part (b) of REF , we may assume that $\\mathcal {F}_n^{A_n} = \\mathcal {F}_n$ ; in particular, $F^n_0 = B_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}}B_n$ and $F_k = \\bigcup _{n \\in \\mathbb {N}}F^n_k$ , $\\forall k<i$ ; note that $F_0 = B$ .",
"Set $\\mathcal {F}= \\lbrace F_k\\rbrace _{k<i}$ and $I= < \\!\\!",
"\\mathcal {F} \\!\\!",
">$ .",
"Since $B \\in \\mathcal {F}$ and $A$ is $G$ -invariant, $I$ is $A,B$ -sensitive.",
"Furthermore, since $\\mathcal {F}^{A_n} = \\mathcal {F}_n$ , $A_n \\sim _{I} B_n$ for all $n \\in \\mathbb {N}$ .",
"Thus, by REF , $A \\sim _{I} B$ and hence $A$ is $i^*$ -compressible.",
"subsection2-2 ex5 pt Traveling sets Definition 2.18 Let $A \\in \\Gamma (X)$ .",
"We call $A$ a traveling set with $\\Gamma $ pieces if there exists pairwise disjoint sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ in $\\Gamma (X)$ such that $A_0 = A$ and $A \\sim ^{\\Gamma } A_n$ , $\\forall n \\in \\mathbb {N}$ .",
"For a finite Borel partition $I$ , we say that $A$ is $I$ -traveling with $\\Gamma $ pieces if $A$ respects $I$ and the above condition holds with $\\sim ^{\\Gamma }$ replaced by $\\sim _{I}^{\\Gamma }$ .",
"For $i \\ge 1$ , we say that $A$ is $i$ -traveling if it is $I$ -traveling for some $A$ -sensitive partition $I$ generated by a collection of $i$ Borel sets.",
"Definition 2.19 For a set $A \\subseteq X$ , a function $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ is called a travel guide for $A$ if $\\forall x \\in A, \\gamma (x)(0) = 1_G$ and $\\forall (x,n) \\ne (y,m) \\in A \\times \\mathbb {N}$ , $\\gamma (x)(n)x \\ne \\gamma (y)(m)y$ .",
"For $A \\in \\Gamma (X)$ , a $\\Gamma $ -measurable map $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ and $n \\in \\mathbb {N}$ , set $\\gamma _n := \\gamma (\\cdot )(n) : A \\rightarrow G$ and note that $\\gamma _n$ is also $\\Gamma $ -measurable.",
"Observation 2.20 Suppose $A \\in \\Gamma (X)$ and $I$ is an $A$ -sensitive finite Borel partition of $X$ .",
"Then $A$ is $I$ -traveling with $\\Gamma $ pieces if and only if it has a $\\Gamma $ -measurable $F_{I}$ -invariant travel guide.",
"Follows from definitions and Proposition REF .",
"Now we establish the connection between compressibility and traveling sets.",
"Lemma 2.21 Let $I$ be a finite Borel partition of $X$ , $P \\in \\Gamma (X)$ be a Borel $G$ -invariant set and let $A,B$ be $\\Gamma $ subsets of $P$ .",
"If $P \\sim _{I}^{\\Gamma } B$ , then $P \\setminus B$ is $I$ -traveling with $\\Gamma $ pieces.",
"Conversely, if $A$ is $I$ -traveling with $\\Gamma $ pieces, then $P \\sim _{I}^{\\Gamma } (P \\setminus A)$ .",
"The same is true if we replace $\\sim _{I}^{\\Gamma }$ and “$I$ -traveling” with $\\sim ^{\\Gamma }$ and “traveling”, respectively.",
"For the first statement, let $\\gamma : X \\rightarrow G$ be a witnessing map for $X \\sim _{I}^{\\Gamma } B$ .",
"Put $A^{\\prime } = X \\setminus B$ and note that $A^{\\prime }$ respects $I$ since so does $P$ and hence $B$ .",
"We show that $A^{\\prime }$ is $I$ -traveling.",
"Put $A_n = (\\hat{\\gamma })^n(A^{\\prime })$ , for each $n \\ge 0$ .",
"It follows from injectivity of $\\hat{\\gamma }$ that $A_n$ are pairwise disjoint.",
"For all $n$ , recursively define $\\delta _n : A^{\\prime } \\rightarrow G$ as follows $\\left\\lbrace \\begin{array}{l}\\delta _0 = \\gamma \\!",
"\\!",
"\\downharpoonright _{A^{\\prime }} \\\\\\delta _{n+1} = \\gamma \\circ \\hat{\\delta }_n\\end{array}\\right..$ It follows from $F_{I}$ -invariance of $\\gamma $ that each $\\delta _n$ is $F_{I}$ -invariant.",
"It is also clear that $\\hat{\\delta }_n = (\\hat{\\gamma })^n$ and hence $\\delta _n$ is a witnessing map for $A^{\\prime } \\sim _{I}^{\\Gamma } A_n$ .",
"Thus $A^{\\prime }$ is $i$ -traveling with $\\Gamma $ pieces.",
"For the converse, assume that $A$ is $I$ -traveling and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"In particular, each $A_n$ respects $I$ and $A_n \\sim _{I}^{\\Gamma } A_m$ , for all $n,m \\in \\mathbb {N}$ .",
"Let $P^{\\prime } = \\bigcup _{n \\in \\mathbb {N}}A_n$ and $B^{\\prime } = \\bigcup _{n \\ge 1} A_n$ .",
"Since $A_n \\sim _{I}^{\\Gamma } A_{n+1}$ , part (b) of REF implies that $P^{\\prime } \\sim _{I}^{\\Gamma } B^{\\prime }$ .",
"Moreover, since $P \\setminus P^{\\prime } \\sim _{I}^{\\Gamma } P \\setminus P^{\\prime }$ , we get $P \\sim _{I}^{\\Gamma } (B^{\\prime } \\cup (P \\setminus P^{\\prime })) = P \\setminus A$ .",
"For a $G$ -invariant set $P$ and $A \\subseteq P$ , we say that $A$ is a complete section for $P$ if $[A]_G = P$ .",
"The above lemma immediately implies the following.",
"Proposition 2.22 Let $P \\in \\Gamma (X)$ be $G$ -invariant and $i \\ge 1$ .",
"$P$ is $i$ -compressible with $\\Gamma $ pieces if and only if there exists a complete section for $P$ that is $i$ -traveling with $\\Gamma $ pieces.",
"The same is true with “$i$ -compressible” and “$i$ -traveling” replaced by “compressible” and “traveling”.",
"We need the following lemma in the proofs of REF and REF .",
"Lemma 2.23 Suppose $A \\subseteq X$ is an invariant analytic set that does not admit an invariant Borel probability measure.",
"Then there is an invariant Borel set $A^{\\prime } \\supseteq A$ that still does not admit an invariant Borel probability measure.",
"Let $\\mathcal {M}$ denote the standard Borel space of $G$ -invariant Borel probability measures on $X$ (see Section 17 in ).",
"Let $\\Phi \\subseteq Pow(X)$ be the following predicate: $\\Phi (W) \\Leftrightarrow \\forall \\mu \\in \\mathcal {M}(\\mu (W) = 0).$ Claim There is a Borel set $B \\supseteq A$ with $\\Phi (B)$ .",
"Proof of Claim.",
"By the dual form of the First Reflection Theorem for $\\mathbf {\\Pi }_1^1$ (see the discussion following 35.10 in ), it is enough to show that $\\Phi $ is $\\mathbf {\\Pi }_1^1$ on $\\mathbf {\\Sigma }_1^1$ .",
"To this end, let $Y$ be a Polish space and $D \\subseteq Y \\times X$ be analytic.",
"Then, for any $n \\in \\mathbb {N}$ , the set $H_n = \\lbrace (\\mu , y) \\in \\mathcal {M}\\times Y : \\mu (D_y) > {1 \\over n}\\rbrace ,$ is analytic by a theorem of Kondô-Tugué (see 29.26 of ), and hence so are the sets $H^{\\prime }_n := \\text{proj}_{Y}(H_n)$ and $H := \\bigcup _{n \\in \\mathbb {N}}H^{\\prime }_n$ .",
"Finally, note that $\\lbrace y \\in Y : \\Phi (A_y)\\rbrace = \\lbrace y \\in Y : \\exists \\mu \\in \\mathcal {M}\\exists n \\in \\mathbb {N}(\\mu (A_y) > {1 \\over n})\\rbrace ^c = H^c,$ and so $\\lbrace y \\in Y : \\Phi (A_y)\\rbrace $ is $\\mathbf {\\Pi }_1^1$ .",
"$\\dashv $ Now put $A^{\\prime } = (B)_G$ , where $(B)_G = \\lbrace x \\in B : [x]_G \\subseteq B\\rbrace $ .",
"Clearly, $A^{\\prime }$ is an invariant Borel set, $A^{\\prime } \\supseteq A$ , and $\\Phi (A^{\\prime })$ since $A^{\\prime } \\subseteq B$ and $\\Phi (B)$ .",
"Proposition 2.24 Let $X$ be a Borel $G$ -space.",
"The following are equivalent: $X$ is compressible with universally measurable pieces; There is a universally measurable complete section that is a traveling set with universally measurable pieces; There is no $G$ -invariant Borel probability measure on $X$ ; $X$ is compressible with Borel pieces; There is a Borel complete section that is a traveling set with Borel pieces.",
"Equivalence of (1) and (2) as well as (4) and (5) is asserted in REF , (4)$\\Rightarrow $ (1) is trivial, and (3)$\\Rightarrow $ (4) follows from Nadkarni's theorem (see REF ).",
"It remains to show (1)$\\Rightarrow $ (3).",
"To this end, suppose $X \\sim ^{\\Gamma } B$ , where $B^c = X \\setminus B$ is a complete section and $\\Gamma $ is the class of universally measurable sets.",
"If there was a $G$ -invariant Borel probability measure $\\mu $ on $X$ , then $\\mu (X) = \\mu (B)$ and hence $\\mu (B^c) = 0$ .",
"But since $B^c$ is a complete section, $X = \\bigcup _{g \\in G} gB^c$ , and thus $\\mu (X) = 0$ , a contradiction.",
"Now we prove an analogue of this for $i$ -compressibility.",
"Proposition 2.25 Let $X$ be a Borel $G$ -space.",
"For $i \\ge 1$ , the following are equivalent: $X$ is $i$ -compressible with universally measurable pieces; There is a universally measurable complete section that is an $i$ -traveling set with universally measurable pieces; There is a partition $I$ of $X$ generated by $i$ Borel sets such that $Y = X) \\subseteq |I|^G$ does not admit a $G$ -invariant Borel probability measure; $X$ is $i$ -compressible with Borel pieces; There is a Borel complete section that is a $i$ -traveling set with Borel pieces.",
"Equivalence of (1) and (2) as well as (4) and (5) is asserted in REF and (4)$\\Rightarrow $ (1) is trivial.",
"It remains to show (1)$\\Rightarrow $ (3)$\\Rightarrow $ (5).",
"(1)$\\Rightarrow $ (3): Suppose $X \\sim _{I}^{\\Gamma } B$ , where $B^c = X \\setminus B$ is a complete section, $I$ is a partition of $X$ generated by $i$ Borel sets, and $\\Gamma $ denotes the class of universally measurable sets.",
"Let $\\gamma : X \\rightarrow G$ be a witnessing map for $X \\sim _i^{\\Gamma } B$ .",
"By the Jankov-von Neumann uniformization theorem (see 18.1 in ), $ has a $ (11)$-measurable (hence universally measurable) right inverse $ h : Y X$.",
"Define $ : Y G$ by $ (y) = (h(y))$ and note that $$ is universally measurable being a composition of such functions.",
"Letting $ B' = (Y)$, it is straightforward to check that $ $ and thus $ B' = (X)) = B)$.",
"Now it follows that $$ is a witnessing map for $ Y B'$ and hence $ Y$ is compressible with universally measurable pieces.",
"Finally, (1)$$(3) of \\ref {equivalences to compressibility} implies that $ Y$ does not admit an invariant Borel probability measure.$ (3)$\\Rightarrow $ (5): Assume $Y$ is as in (3).",
"Then by Lemma REF , there is a Borel $G$ -invariant $Y^{\\prime } \\supseteq Y$ that does not admit a $G$ -invariant Borel probability measure.",
"Viewing $Y^{\\prime }$ as a Borel $G$ -space, we apply (3)$\\Rightarrow $ (4) of REF and get that $Y^{\\prime }$ is compressible with Borel pieces; thus there is a Borel $B^{\\prime } \\subseteq Y^{\\prime }$ with $[Y^{\\prime } \\setminus B^{\\prime }]_G = Y^{\\prime }$ such that $Y^{\\prime } \\sim B^{\\prime }$ .",
"Let $\\delta : Y^{\\prime } \\rightarrow G$ be a witnessing map for $Y^{\\prime } \\sim B^{\\prime }$ .",
"Put $B = {-1}(B^{\\prime })$ and $\\gamma = \\delta \\circ .",
"By definition, $$ is $ FI$-invariant.",
"In fact, it is straightforward to check that $$ is a witnessing map for $ X I B$ and $ [X B]G = [-1(Y B')]G = -1([Y B']G) = -1(Y) = X$.",
"Hence $ X$ is $ I$-compressible.$ We now give an example of a 1-traveling set.",
"First we need some definitions.",
"Definition 2.26 Let $X$ be a Borel $G$ -space and $A \\subseteq X$ be Borel.",
"$A$ is called aperiodic if it intersects every orbit in either 0 or infinitely many points; a partial transversal if it intersects every orbit in at most one point; smooth if there is a Borel partial transversal $T \\subseteq A$ such that $[T]_G = [A]_G$ .",
"Proposition 2.27 Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal, then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"let $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ .",
"For each $n \\in \\mathbb {N}$ , define $\\bar{n} : X \\rightarrow \\mathbb {N}$ and $\\gamma _n : T \\rightarrow G$ recursively in $n$ as follows: $\\left\\lbrace \\begin{array}{l}\\bar{n}(x) = \\text{the least $k$ such that } g_k x \\notin \\lbrace \\hat{\\gamma }_i(x) : i<n\\rbrace \\\\\\gamma _n(x) = g_{\\bar{n}(x)}\\end{array}\\right..$ Clearly, $\\bar{n}$ and $\\gamma _n$ are well-defined and Borel.",
"Define $\\gamma : T \\rightarrow G^{\\mathbb {N}}$ by setting $\\gamma (\\cdot )(n) = \\gamma _n$ .",
"It follows from the definitions that $\\gamma $ is a Borel travel guide for $T$ and hence, $T$ is a traveling set.",
"It remains to show that $\\gamma $ is $F_{I}$ -invariant, where $I= < \\!\\!",
"T \\!\\!",
">$ .",
"For this it is enough to show that $\\bar{n}$ is $F_{I}$ -invariant, which we do by induction on $n$ .",
"Since it trivially holds for $n = 0$ , we assume it is true for all $0 \\le k < n$ and show it for $n$ .",
"To this end, suppose $x,y \\in T$ with $x F_{I}y$ , and assume for contradiction that $m := \\bar{n}(x) < \\bar{n}(y)$ .",
"Thus it follows that $g_m y = \\hat{\\gamma }_k(y) \\in \\hat{\\gamma }_k(T)$ , for some $k < n$ .",
"By the induction hypothesis, $\\hat{\\gamma }_k(T)$ is $F_{I}$ -invariant and hence, $g_m x \\in \\hat{\\gamma }_k(T)$ , contradicting the definition of $\\bar{n}(x)$ .",
"Corollary 2.28 Let $X$ be an aperiodic Borel $G$ -space.",
"If a Borel set $A \\subseteq X$ is smooth, then $A \\in 1$ .",
"Let $P = [A]_G$ and let $T$ be a Borel partial transversal with $[T]_G = P$ .",
"By REF , $T$ is $I$ -traveling, where $I= < \\!\\!",
"T \\!\\!",
">$ .",
"Hence, $P \\sim _{I} P \\setminus T$ , by Lemma REF .",
"This implies that $P$ is $1^*$ -compressible since $I= < \\!\\!",
"T^c \\!\\!",
">$ and $P \\setminus T \\in \\lbrace T^c\\rbrace ^P$ .",
"subsection2-2 ex5 pt Constructing finite generators using $i$ -traveling sets Lemma 2.29 Let $A \\in \\mathfrak {B}(X)$ be a complete section and $I$ be an $A$ -sensitive finite Borel partition of $X$ .",
"If $A$ is $I$ -traveling (with Borel pieces), then there is a Borel $2|I|$ -generator.",
"If moreover $A \\in I$ , then there is a Borel $(2|I|-1)$ -generator.",
"Let $\\gamma $ be an $F_{I}$ -invariant Borel travel guide for $A$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel structure of $X$ and let $B = \\bigcup _{n \\ge 1}\\hat{\\gamma }_n(A \\cap U_n)$ .",
"By Lemma REF , each $\\hat{\\gamma }_n$ maps Borel sets to Borel sets and hence $B$ is Borel.",
"Set $J= < \\!\\!",
"B \\!\\!",
">$ , $\\mathcal {P}= I\\vee J$ and note that $|\\mathcal {P}| \\le 2 |I|$ .",
"$A$ and $B$ are disjoint since $\\lbrace \\hat{\\gamma }_n(A)\\rbrace _{n \\in \\mathbb {N}}$ is a collection of pairwise disjoint sets and $\\hat{\\gamma }_0(A) = A$ ; thus if $A \\in I$ , $|\\mathcal {P}| \\le 1 + 2(|I|-1) = 2|I|-1$ .",
"We show that $\\mathcal {P}$ is a generator, that is $G \\mathcal {P}$ separates points in $X$ .",
"Let $x \\ne y \\in X$ and assume they are not separated by $G I$ , thus $x F_{I}y$ .",
"We show that $G J$ separates $x$ and $y$ .",
"Because $A$ is a complete section, multiplying $x$ by an appropriate group element, we may assume that $x \\in A$ .",
"Since $A$ respects $I$ , $A$ is $F_{I}$ -invariant and thus $y \\in A$ .",
"Also, because $\\gamma $ is $F_{I}$ -invariant, $\\gamma _n(x) = \\gamma _n(y)$ , $\\forall n \\in \\mathbb {N}$ .",
"Let $n \\ge 1$ be such that $x \\in U_n$ but $y \\notin U_n$ .",
"Put $g = \\gamma _n(x)$ ($= \\gamma _n(y)$ ).",
"Then $gx = \\hat{\\gamma }_n(x) \\in \\hat{\\gamma }_n(A \\cap U_n)$ while $gy = \\hat{\\gamma }_n(y) \\notin \\hat{\\gamma }_n(A \\cap U_n)$ .",
"Hence, $gx \\in B$ and $gy \\notin B$ because $\\gamma _m(A) \\cap \\gamma _n(A) = \\mathbb {\\emptyset }$ for all $m \\ne n$ and $gy = \\hat{\\gamma }_n(y) \\in \\hat{\\gamma }_n(A)$ .",
"Thus $G J$ separates $x$ and $y$ .",
"Now REF and REF together imply the following.",
"Proposition 2.30 Let $X$ be a Borel $G$ -space and $i \\ge 1$ .",
"If $X$ is $i$ -compressible then there is a Borel $2^{i+1}$ -generator.",
"By REF , there exists a Borel $i$ -traveling complete section $A$ .",
"Let $I$ witness $A$ being $i$ -traveling and thus, by Lemma REF , there is a $2|I| \\le 2 \\cdot 2^i = 2^{i+1}$ -generator.",
"Example 2.31.",
"For $2 \\le n \\le \\infty $ , let $\\mathbb {F}_n$ denote the free group on $n$ generators and let $X$ be the boundary of $\\mathbb {F}_n$ , i.e.",
"the set of infinite reduced words.",
"Clearly, the product topology makes $X$ a Polish space and $\\mathbb {F}_n$ acts continuously on $X$ by left concatenation and cancellation.",
"We show that $X$ is 1-compressible and thus admits a Borel $2^2=4$ -generator by Proposition REF .",
"To this end, let $a,b$ be two of the $n$ generators of $\\mathbb {F}_n$ and let $X_a$ be the set of all words in $X$ that start with $a$ .",
"Then $X = (X_{a^{-1}} \\cup X_{a^{-1}}^c) \\sim _{I} Y$ , where $Y = bX_{a^{-1}} \\cup aX_{a^{-1}}^c$ and $I< \\!\\!",
"X_{a^{-1}} \\!\\!",
">$ .",
"Hence $X \\sim _1 Y$ .",
"Since $X \\setminus Y \\supseteq X_{a^{-1}}$ , $[X \\setminus Y]_{\\mathbb {F}_n} = X$ and thus $X$ is 1-compressible.",
"Now we obtain a sufficient condition for the existence of an embedding into a finite Bernoulli shift.",
"Corollary 2.32 Let $X$ be a Borel $G$ -space and $k \\in \\mathbb {N}$ .",
"If there exists a Borel $G$ -map $f : X \\rightarrow k^G$ such that $Y = f(X)$ does not admit a $G$ -invariant Borel probability measure, then there is a Borel $G$ -embedding of $X$ into $(2k)^G$ .",
"Let $I= I_{f}$ and hence $f = .",
"By (3)$$(5) of \\ref {equivalences to i-compressibility} (or rather the proof of it), $ X$ admits a Borel $ I$-traveling complete section.",
"Thus by Lemma \\ref {I-traveling implies finite generator}, $ X$ admits a $ 2|I| = 2k$-generator and hence, there is a Borel $ G$-embedding of $ X$ into $ (2k)G$.$ Lemma 2.33 Let $I$ be a partition of $X$ into $n$ Borel sets.",
"Then $I$ is generated by $k = \\lceil \\log _2(n) \\rceil $ Borel sets.",
"Since $2^k \\ge n$ , we can index $I$ by the set $\\mathbf {2^k}$ of all $k$ -tuples of $\\lbrace 0,1\\rbrace $ , i.e.",
"$I= \\lbrace A_{\\sigma }\\rbrace _{\\sigma \\in \\mathbf {2^k}}$ .",
"For all $i < k$ , put $B_i = \\bigcup _{\\sigma \\in \\mathbf {2^k} \\wedge \\sigma (i) = 1} A_{\\sigma }.$ Now it is clear that for all $\\sigma \\in \\mathbf {2^k}$ , $A_{\\sigma } = \\bigcap _{i<k} B_i^{\\sigma (i)}$ , where $B_i^{\\sigma (i)}$ is equal to $B_i$ if $\\sigma (i) = 1$ , and equal to $B_i^c$ , otherwise.",
"Thus $I= < \\!\\!",
"B_i : i<k \\!\\!",
">$ .",
"Proposition 2.34 If $X$ is compressible and there is a Borel $n$ -generator, then $X$ is $\\lceil \\log _2(n) \\rceil $ -compressible.",
"Let $I$ be an $n$ -generator and hence, by Lemma REF , $I$ is generated by $i$ Borel sets.",
"Since $G I$ separates points in $X$ , each $F_{I}$ -class is a singleton and hence $X \\prec X$ implies $X \\prec _{I} X$ .",
"From REF and REF we immediately get the following corollary, which justifies the use of $i$ -compressibility in studying Question REF .",
"Corollary 2.35 Let $X$ be a Borel $G$ -space that is compressible (equivalently, does not admit an invariant Borel probability measure).",
"$X$ admits a finite generator if and only if $X$ is $i$ -compressible for some $i \\ge 1$ .",
"Invariant measures and $i$ -compressibility This section is mainly devoted to proving the following theorem.",
"Theorem 3.1 Let $X$ be a Borel $G$ -space.",
"If $X$ is aperiodic, then there exists a function $m : \\mathfrak {B}(X) \\times X \\rightarrow [0,1]$ satisfying the following properties for all $A,B \\in \\mathfrak {B}(X)$ : $m(A, \\cdot )$ is Borel; $m(X, x) = 1$ , $\\forall x \\in X$ ; If $A \\subseteq B$ , then $m(A, x) \\le m(B,x)$ , $\\forall x \\in X$ ; $m(A, x) = 0$ off $[A]_G$ ; $m(A, x) > 0$ on $[A]_G$ modulo 4; $m(A,x) = m(gA, x)$ , for all $g \\in G$ , $x \\in X$ modulo 3; If $A \\cap B = \\mathbb {\\emptyset }$ , then $m(A \\cup B,x) = m(A,x) + m(B,x)$ , $\\forall x \\in X$ modulo 4.",
"Remark.",
"A version of this theorem is what lies at the heart of the proof of Nadkarni's theorem.",
"The conclusions of our theorem are modulo 4, which is potentially a smaller $\\sigma $ -ideal than the $\\sigma $ -ideal of sets contained in compressible Borel sets used in Nadkarni's version.",
"However, the price we paid for this is that part (g) asserts only finite additivity instead of countable additivity asserted by Nadkarni's version.",
"Proof of Theorem REF.",
"Our proof follows the general outline of Nadkarni's proof.",
"The construction of $m(A,x)$ is somewhat similar to that of Haar measure.",
"First, for sets $A,B$ , we define a Borel function $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ that basically gives the number of copies of $B^{[x]_G}$ that fit in $A^{[x]_G}$ when moved by group elements (piecewise).",
"Then we define a decreasing sequence of complete sections (called a fundamental sequence below), which serves as a gauge to measure the size of a given set.",
"Assume throughout that $X$ is an aperiodic Borel $G$ -space (although we only use the aperiodicity assumption in REF to assert that smooth sets are in 1).",
"Lemma 3.2 (Comparability) $\\forall A,B \\in \\mathfrak {B}(X)$ , there is a partition $X = P \\cup Q$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ , $A^P \\prec _{I} B^P$ and $B^Q \\preceq _{I} A^Q$ .",
"It is enough to prove the lemma assuming $X = [A]_G \\cap [B]_G$ since we can always include $[B]_G \\setminus [A]_G$ in $P$ and $X \\setminus [B]_G$ in $Q$ .",
"Fix an enumeration $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ for $G$ .",
"We recursively construct Borel sets $A_n,B_n,A_n^{\\prime },B_n^{\\prime }$ as follows.",
"Set $A_0^{\\prime } = A$ and $B_0^{\\prime } = B$ .",
"Assuming $A_n^{\\prime }, B_n^{\\prime }$ are defined, set $B_n = B_n^{\\prime } \\cap g_n A_n^{\\prime }$ , $A_n = g_n^{-1} B_n$ , $A_{n+1}^{\\prime } = A_n^{\\prime } \\setminus A_n$ and $B_{n+1}^{\\prime } = B_n^{\\prime } \\setminus B_n$ .",
"It is easy to see by induction on $n$ that for any $A,B$ -sensitive $I$ , $A_n,B_n$ are $F_{I}$ -invariant since so are $A,B$ .",
"Thus, setting $A^* = \\bigcup _{n \\in \\mathbb {N}}A_n$ and $B^* = \\bigcup _{n \\in \\mathbb {N}}B_n$ , we get that $A^* \\sim _{I} B^*$ since $B_n = g_n A_n$ .",
"Let $A^{\\prime } = A \\setminus A^*$ , $B^{\\prime } = B \\setminus B^*$ and set $P = [B^{\\prime }]_G$ , $Q = X \\setminus P$ .",
"Claim $[A^{\\prime }]_G \\cap [B^{\\prime }]_G = \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $\\exists x \\in A^{\\prime }$ and $n \\in \\mathbb {N}$ such that $g_n x \\in B^{\\prime }$ .",
"It is clear that $A^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} A_k^{\\prime }$ , $B^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} B_k^{\\prime }$ ; in particular, $x \\in A_n^{\\prime }$ and $g_n x \\in B_n^{\\prime }$ .",
"But then $g_n x \\in B_n$ and $x \\in A_n$ , contradicting $x \\in A^{\\prime }$ .",
"$\\dashv $ Let $I$ be an $A,B$ -sensitive partition.",
"Then $A^P = (A^*)^P$ and hence $A^P \\prec _{I} B^P$ since $(A^*)^P \\sim _{I} (B^*)^P \\subseteq B^P$ and $[B^P \\setminus (B^*)^P]_G = [B^{\\prime }]_G = P = [B^P]_G$ .",
"Similarly, $B^Q = (B^*)^Q$ and hence $B^Q \\preceq _{I} A^Q$ since $(B^*)^Q \\sim _{I} (A^*)^Q \\subseteq A^Q$ .",
"Definition 3.3 (Divisibility) Let $n \\le \\infty $ , $A,B,C \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"Write $A \\sim _{I} nB \\oplus C$ if there are Borel sets $A_k \\subseteq A$ , $k<n$ , such that $\\lbrace A_k\\rbrace _{k<n} \\cup \\lbrace C\\rbrace $ is a partition of $A$ , each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\sim _{I} B$ .",
"Write $n B \\preceq _{I} A$ if there is $C \\subseteq A$ with $A \\sim _{I} n B \\oplus C$ , and write $n B \\prec _{I} A$ if moreover $[C]_G = [A]_G$ .",
"Write $A \\preceq _{I} nB$ if there is a Borel partition $\\lbrace A_k\\rbrace _{k<n}$ of $A$ such that each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\preceq _{I} B$ .",
"If moreover, $A_k \\prec _{I} B$ for at least one $k < n$ , we write $A \\prec _{I} nB$ .",
"For $i \\ge 1$ , we use the above notation with $I$ replaced by $i$ if there is an $A,B$ -sensitive partition $I$ generated by $i$ sets for which the above conditions hold.",
"Proposition 3.4 (Euclidean decomposition) Let $A,B \\in \\mathfrak {B}(X)$ and put $R = [A]_G \\cap [B]_G$ .",
"There exists a partition $\\lbrace P_n\\rbrace _{n \\le \\infty }$ of $R$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ and $n \\le \\infty $ , $A^{P_n} \\sim _{I} nB^{P_n} \\oplus C_n$ for some $C_n$ such that $C_n \\prec _{I} B^{P_n}$ , if $n < \\infty $ .",
"We repeatedly apply Lemma REF .",
"For $n < \\infty $ , recursively define $R_n, P_n, A_n, C_n$ satisfying the following: $R_n$ are invariant decreasing Borel sets such that $n B^{R_n} \\preceq _{I} A^{R_n}$ for any $A,B$ -sensitive $I$ ; $P_n = R_n \\setminus R_{n+1}$ ; $A_n \\subseteq R_{n+1}$ are pairwise disjoint Borel sets such that for any $A,B$ -sensitive $I$ , every $A_n$ respects $I$ and $A_n \\sim _{I} B^{R_{n+1}}$ ; $C_n \\subseteq P_n$ are Borel sets such that for any $A,B$ -sensitive $I$ , every $C_n$ respects $I$ and $C_n \\prec _{I} B^{P_n}$ .",
"Set $R_0 = R$ .",
"Given $R_n$ , $\\lbrace A_k\\rbrace _{k<n}$ satisfying the above properties, let $A^{\\prime } = A^{R_n} \\setminus \\bigcup _{k<n} A_k$ .",
"We apply Lemma REF to $A^{\\prime }$ and $B^{R_n}$ , and get a partition $R_n = P_n \\cup R_{n+1}$ such that $(A^{\\prime })^{P_n} \\prec _{I} B^{P_n}$ and $B^{R_{n+1}} \\preceq _{I} (A^{\\prime })^{R_{n+1}}$ .",
"Set $C_n = (A^{\\prime })^{P_n}$ .",
"Let $A_n \\subseteq (A^{\\prime })^{R_{n+1}}$ be such that $B^{R_{n+1}} \\sim _{I} A_n$ .",
"It is straightforward to check (i)-(iv) are satisfied.",
"Now let $R_{\\infty } = \\bigcap _{n \\in \\mathbb {N}} R_n$ and $C_{\\infty } = (A \\setminus \\bigcup _{n \\in \\mathbb {N}}A_n)^{R_{\\omega }}$ .",
"Now it follows from (i)-(iv) that for all $n \\le \\infty $ , $\\lbrace A_k^{P_n}\\rbrace _{k<n} \\cup \\lbrace C_n\\rbrace $ is a partition of $A^{P_n}$ witnessing $A^{P_n} \\sim _{I} nB \\oplus C_n$ , and for all $n < \\infty $ , $C_n \\prec B^{P_n}$ .",
"For $A,B \\in \\mathfrak {B}(X)$ , let $\\lbrace P_n\\rbrace _{n \\le \\infty }$ be as in the above proposition.",
"Define $[A / B](x) = \\left\\lbrace \\begin{array}{ll}n & \\text{if } x \\in P_n, n < \\infty \\\\\\infty & \\text{if } x \\in P_{\\infty } \\text{ or } x \\in [A]_G \\setminus [B]_G \\\\0 & \\text{if } x \\in [B]_G \\setminus [A]_G \\\\-1 & \\text{otherwise}\\end{array}\\right..$ Note that $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ is a Borel function by definition.",
"Lemma 3.5 (Infinite divisibility $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ with $[A]_G = [B]_G$ , and let $I$ be a finite Borel partition of $X$ .",
"If $\\infty B \\preceq _{I} A$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} \\infty B \\oplus C$ and let $\\lbrace A_k\\rbrace _{k < \\infty }$ be as in Definition REF .",
"$A_k \\sim _{I} B \\sim _{I} A_{k+1}$ and hence $A_k \\sim _{I} A_{k+1}$ .",
"Also trivially $C \\sim _{I} C$ .",
"Thus, letting $A^{\\prime } = \\bigcup _{k < \\infty } A_{k+1} \\cup C$ , we apply (b) of REF to $A$ and $A^{\\prime }$ , and get that $A \\sim _{I} A^{\\prime }$ .",
"Because $[A \\setminus A^{\\prime }]_G = [A_0]_G = [B]_G = [A]_G$ , we have $A \\prec _{I} A$ .",
"Lemma 3.6 (Ambiguity $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"If $nB \\preceq _{I} A \\prec _{I} nB$ for some $n \\ge 1$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} nB \\oplus C$ and let $\\lbrace A_k\\rbrace _{k<n}$ be a partitions of $A \\setminus C$ witnessing $A \\sim _{I} nB \\oplus C$ .",
"Also let $\\lbrace A^{\\prime }_k\\rbrace _{k<n}$ be witnessing $A \\prec _{I} nB$ with $A^{\\prime }_0 \\prec _{I} B$ .",
"Since $A^{\\prime }_k \\preceq _{I} B \\sim _{I} A_k$ , $A^{\\prime }_k \\preceq _{I} A_k$ , for all $k<n$ and $A^{\\prime }_0 \\prec _{I} A_0$ .",
"Note that it follows from the hypothesis that $[A]_G = [B]_G$ and hence $[A_0]_G = [A]_G$ since $[A_0]_G = [B]_G$ .",
"Thus it follows from (b) of REF that $A = \\bigcup _{k<n} A^{\\prime }_k \\prec _{I} \\bigcup _{k<n} A_k \\subseteq A$ .",
"Proposition 3.7 Let $n \\in \\mathbb {N}$ and $A,A^{\\prime },B,P \\in \\mathfrak {B}(X)$ , where $P$ is invariant.",
"$[A/B] \\in \\mathbb {N}$ on $[B]_G$ modulo 3.",
"If $A \\subseteq A^{\\prime }$ , then $[A / B] \\le [A^{\\prime } / B]$ .",
"If $[A/B] = n$ on $P$ then $n B^P \\preceq _{I} A^P \\prec _{I} (n+1) B^P$ , for any finite Borel partition $I$ that is $A,B$ -sensitive.",
"In particular, $n B^P \\preceq _2 A^P \\prec _2 (n+1) B^P$ by taking $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"For $n \\ge 1$ , if $A^P \\prec _i nB^P$ , then $[A/B] < n$ on $P$ modulo ${i+1}$ ; If $A^P \\subseteq [B]_G$ and $nB^P \\preceq _i A^P$ , then $[A/B] \\ge n$ on $P$ modulo ${i+1}$ .",
"For (a), notice that REF and REF imply that $P_{\\infty } \\in 3$ .",
"(b) and (c) follow from the definition of $[A/B]$ .",
"For (d), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $A^P \\prec _{I} nB^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) \\ge n\\rbrace $ .",
"By (c), $nB^Q \\preceq _{I} A^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"For (e), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $nB^P \\preceq _{I} A^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) < n\\rbrace $ .",
"By (c), $A^Q \\prec _{I} nB^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"Definition 3.8 (Fundamental sequence) A sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections with $F_0 = X$ and $[F_n/F_{n+1}] \\ge 2$ modulo 3 is called fundamental.",
"Proposition 3.9 There exists a fundamental sequence.",
"Take $F_0 = X$ .",
"Given any complete Borel section $F$ , its intersection with every orbit is infinite modulo a smooth set (if the intersection of an orbit with a set is finite, then we can choose an element from each such nonempty intersection in a Borel way and get a Borel transversal).",
"Thus, by REF , $F$ is aperiodic modulo 1.",
"Now use Lemma REF to write $F = A \\cup B, A \\cap B = \\mathbb {\\emptyset }$ , where $A,B$ are also complete sections.",
"Let now $P,Q$ be as in Lemma REF for $A,B$ , and hence $A^P \\prec _2 B^P, B^Q \\preceq _2 A^Q$ because we can take $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"Let $A^{\\prime } = A^P \\cup B^Q, B^{\\prime } = B^P \\cup A^Q$ .",
"Then $F = A^{\\prime } \\cup B^{\\prime }, A^{\\prime } \\cap B^{\\prime } = \\mathbb {\\emptyset }$ , $A^{\\prime } \\preceq B^{\\prime }$ and $A^{\\prime }$ is also a complete Borel section.",
"By (e) of REF , $[F/A^{\\prime }] \\ge 2$ modulo 3.",
"Iterate this process to inductively define $F_n$ .",
"Fix a fundamental sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ and for any $A \\in \\mathfrak {B}(X), x \\in X$ , define $m(A,x) = \\lim _{n \\rightarrow \\infty } \\frac{[A/F_n](x)}{[X/F_n](x)}, \\qquad \\mathrm {(\\dagger )}$ if the limit exists, and 0 otherwise.",
"In the above fraction we define ${\\infty \\over \\infty } = 1$ .",
"We will prove in Proposition REF that this limit exists modulo 4.",
"But first we need the following two lemmas.",
"Lemma 3.10 (Almost cancelation) For any $A,B,C \\in X$ , $[A/B][B/C] \\le [A/C] < ([A/B] + 1)([B/C] + 1)$ on $R := [B]_G \\cap [C]_G$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,C \\!\\!",
">$ .",
"$[A/B][B/C] \\le [A/C]$ : Fix integers $i,j > 0$ and let $P = \\lbrace x \\in X : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"Since $i,j > 0$ , $P \\subseteq [A]_G \\cap [B]_G \\cap [C]_G$ and we work in $P$ .",
"By (c) of REF , $i B \\preceq _{I} A$ and $j C \\preceq _{I} B$ .",
"Thus it follows that $ij C \\preceq _{I} A$ and hence $[A / C] \\ge ij$ modulo 4 by (e) of REF .",
"$[A/C] < ([A/B] + 1)([B/C] + 1)$ : By (a) of REF , $[A/C], [A/B], [B/C] \\in \\mathbb {N}$ on $R$ modulo 3.",
"Fix $i,j \\in \\mathbb {N}$ and let $Q = \\lbrace x \\in R : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"We work in $Q$ .",
"By (c) of REF , $A \\prec _{I} (i+1) B$ and $B \\prec _{I} (j+1) C$ .",
"Thus $A \\prec _{I} (i+1)(j+1) C$ and hence $[A/C] < (i+1)(j+1)$ modulo 4 by (d) of REF .",
"Lemma 3.11 For any $A \\in \\mathfrak {B}(A)$ , $\\lim _{n \\rightarrow \\infty } [A / F_n] = \\left\\lbrace \\begin{array}{ll} \\infty & \\text{on } [A]_G \\\\ 0 & \\text{on } X \\setminus [A]_G \\end{array}\\right., \\text{ modulo } 4.$ The part about $X \\setminus [A]_E$ is clear, so work in $[A]_E$ , i.e.",
"assume $X = [A]_G$ .",
"By (a) of REF and REF , we have $\\infty > [F_1 /A] \\ge [F_1 /F_n ] [F_n / A ] \\ge 2^{n-1} [F_n /A], \\text{ modulo } 4,$ which holds for all $n$ at once since 4 is a $\\sigma $ -ideal.",
"Thus $[F_n /A] \\rightarrow 0$ modulo 4 and hence, as $[F_n /A] \\in \\mathbb {N}$ , $[F_n /A]$ is eventually 0, modulo 4.",
"So if $B_k := \\lbrace x \\in [A]_G : [F / A](x) = 0\\rbrace ,$ then $B_k \\nearrow X$ , modulo 4.",
"Now it follows from Lemma REF that $[A / F_k] > 0$ on $B_k$ modulo 4.",
"But $[A/F_{k+n}] \\ge [A / F_k ] [F_k / F_{k+n} ] \\ge 2^n [A / F_k ], \\text{ modulo } 4,$ so for every $k$ , $[A/F_n] \\rightarrow \\infty $ on $B_k$ modulo 4.",
"Since $B_k \\nearrow X$ modulo 4, we have $[A/F_n] \\rightarrow \\infty $ on $X$ , modulo 4.",
"Proposition 3.12 For any Borel set $A \\subseteq X$ , the limit in ($\\dagger $ ) exists and is positive on $[A]_G$ , modulo 4.",
"Claim Suppose $B,C \\in \\mathfrak {B}(X)$ , $i \\in \\mathbb {N}$ and $D_i = \\lbrace x \\in X : [C / F_i](x) > 0\\rbrace $ .",
"Then $\\overline{\\lim } {[B/F_n ] \\over [C/F_n]} \\le {[B/F_i] + 1 \\over [C/F_i]}$ on $D_i$ , modulo 4.",
"Proof of Claim.",
"Working in $D_i$ and using Lemma REF , $\\forall j$ we have (modulo 4) $[B/F_{i+j}] & \\le ([B/F_i ]+1) ([F_i / F_{i+j}] + 1) \\\\[C/F_{i+j}] & \\ge [C/F_i ] [ F_i /F_{i+j}] > 0,$ so ${[B/F_{i+j}]\\over [C/F_{i+j}]} & \\le {[B/F_i ]+1 \\over [C/F_i ]}\\cdot {[F_i /F_{i+j}]+1\\over [F_i /F_{i+j}]} \\\\& \\le {[B/F_i ]+1\\over [C/F_i ]} \\cdot (1 + {1\\over 2^j}),$ from which the claim follows.",
"$\\dashv $ Applying the claim to $B = A$ and $C = X$ (hence $D_i = X$ ), we get that for all $i \\in \\mathbb {N}$ $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ](x)\\over [X/F_n ](x)} \\le {[A/F_i ](x)+1 \\over [X/F_i ](x)} (\\text{modulo } 4).$ Thus $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ]\\over [X/F_n ]} \\le \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i]+1\\over [X/F_i ]} = \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i ]\\over [X/F_i ]}$ since $\\lim _{i \\rightarrow \\infty } {1 \\over [X/F_i]} = 0$ .",
"To see that $m(A,x)$ is positive on $[A]_E$ modulo 4 we argue as follows.",
"We work in $[A]_G$ .",
"Applying the above claim to $B = X$ and $C = A$ , we get ${1 \\over m(A,x)} = \\lim _{n \\rightarrow \\infty } {[X/F_n] \\over [A/F_n]} \\le {[X/F_i] + 1 \\over [A/F_i]} < \\infty \\text{ on } D_i \\text{ (modulo $4$)}.$ Thus $m(A,x) > 0$ on $\\cup _{i \\in \\mathbb {N}} D_i$ , modulo 4.",
"But $D_i \\nearrow [A]_G$ because $[A/F_i] \\rightarrow \\infty $ as $i \\rightarrow \\infty $ , and hence $m(A,x) > 0$ on $[A]_G$ modulo 4.",
"Lemma 3.13 (Invariance) For $A,F \\in \\mathfrak {B}(X)$ , $\\forall g \\in G, [A / F] = [gA / F]$ , modulo 3.",
"We may assume that $X = [A]_G \\cap [F]_G$ .",
"Fix $g \\in G$ , $n \\in \\mathbb {N}$ , and put $Q = \\lbrace x \\in X : [gA / F](x) = n\\rbrace $ .",
"We work in $Q$ .",
"Let $I= < \\!\\!",
"A,F \\!\\!",
">$ and hence $A,gA,F$ respect $I$ .",
"By (c) of REF , $nF \\preceq _{I} gA$ .",
"But clearly $gA \\sim _{I} A$ and hence $nF \\preceq _{I} A$ .",
"Thus, by (e) of REF , $[A / F] \\ge n = [gA / F]$ , modulo 3.",
"By symmetry, $[gA / F] \\ge [A / F]$ (modulo 3) and the lemma follows.",
"Lemma 3.14 (Almost additivity) For any $A,B,F \\in X$ with $A \\cap B = \\mathbb {\\emptyset }$ , $[A/F] + [B/F] \\le [A \\cup B / F] \\le [A/F] + [B/F] + 1$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,F \\!\\!",
">$ .",
"$[A/F] + [B/F] \\le [A \\cap B / F]$ : Fix $i,j \\in \\mathbb {N}$ not both 0, say $i>0$ , and let $S = \\lbrace x \\in X : [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"Since $i>0$ , $S \\subseteq [A]_G \\cap [F]_G$ and we work in $S$ .",
"By (c) of REF , $iF^S \\preceq _{I} A^S$ and $jF^S \\preceq _{I} B^S$ .",
"Hence $(i+j)F^S \\preceq _{I} (A \\cup B)^S$ and thus, by (e) of REF , $[A \\cup B / F] \\ge i + j$ , modulo 4.",
"$[A \\cap B / F] \\le [A/F] + [B/F] + 1$ : Outside $[F]_G$ , the inequality clearly holds.",
"Fix $i,j \\in \\mathbb {N}$ and let $M = \\lbrace x \\in [F]_G: [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"We work in $M$ .",
"By (c) of REF , $A \\prec _{I} (i+1)F$ and $B \\prec _{I} (j+1)F$ .",
"Thus it is clear that $A \\cup B \\prec _{I} (i + j + 2)F$ and hence $[A \\cup B / F] < i+j+2$ , modulo 4, by (d) of REF .",
"Now we are ready to finish the proof of Theorem REF .",
"Fix $A,B \\in \\mathfrak {B}(X)$ .",
"The fact that $m(A,x) \\in [0,1]$ and parts (b) and (d) follow directly from the definition of $m(A,x)$ .",
"Part (a) follows from the fact that $[A / F_n]$ is Borel for all $n \\in \\mathbb {N}$ .",
"(c) follows from (b) of Lemma REF , and (e) and (f) are asserted by REF and REF , respectively.",
"To show (g), we argue as follows.",
"By Lemma REF , $[A/F_n] + [B/F_n] \\le [A \\cup B / F_n] \\le [A/F_n] + [B/F_n] + 1$ , modulo 4, and thus $\\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} \\le \\frac{[A \\cup B / F_n]}{[X/F_n]} \\le \\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} + \\frac{1}{[X/F_n]},$ for all $n$ at once, modulo 4 (using the fact that 4 is a $\\sigma $ -ideal).",
"Since $[X/F_n] \\ge 2^n$ , passing to the limit in the inequalities above, we get $m(A,x) + m(B,x) \\le m(A \\cup B,x) \\le m(A,x) + m(B,x)$ .",
"QED (Thm REF ) Theorem REF will only be used via Corollary REF and to state it we need the following.",
"Definition 3.15 Let $X$ be a Borel $G$ -space.",
"$\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ is called a Boolean $G$ -algebra, if it is a Boolean algebra, i.e.",
"is closed under finite unions and complements, and is closed under the $G$ -action, i.e.",
"$G \\mathcal {B}= \\mathcal {B}$ .",
"Corollary 3.16 Let $X$ be a Borel $G$ -space and let $\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ be a countable Boolean $G$ -algebra.",
"For any $A \\in \\mathcal {B}$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, $\\mu $ can be taken such that there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B)=0$ .",
"Let $A \\in \\mathcal {B}$ be such that $A \\notin 4$ .",
"We may assume that $X = [A]_G$ by setting the (to be constructed) measure to be 0 outside $[A]_G$ .",
"If $X$ is not aperiodic, then by assigning equal point masses to the points of a finite orbit, we will have a probability measure on all of $\\mathfrak {B}(X)$ , so assume $X$ is aperiodic.",
"Since 4 is a $\\sigma $ -ideal and $\\mathcal {B}$ is countable, Theorem REF implies that there is a $P \\in 4$ such that (a)-(g) of the same theorem hold on $X \\setminus P$ for all $A,B \\in \\mathcal {B}$ .",
"Since $A \\notin 4$ , there exists $x_A \\in A \\setminus P$ .",
"Hence, letting $\\mu (B) = m(B,x_A)$ for all $B \\in \\mathcal {B}$ , conditions (b),(f) and (g) imply that $\\mu $ is a $G$ -invariant finitely additive probability measure on $\\mathcal {B}$ .",
"Moreover, since $x_A \\in [A]_G \\setminus P$ , $\\mu (A) = m(A, x_A) > 0$ .",
"Finally, the last assertion follows from condition (d).",
"Corollary 3.17 Let $X$ be a Borel $G$ -space.",
"For every Borel set $A \\subseteq X$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive Borel probability measure $\\mu $ (defined on all Borel sets) with $\\mu (A)>0$ .",
"The statement follows from REF and a standard application of the Compactness Theorem of propositional logic.",
"Here are the details.",
"We fix the following set of propositional variables $\\mathcal {P}= \\lbrace P_{A,r} : A \\in \\mathfrak {B}(X), r \\in [0,1]\\rbrace ,$ with the following interpretation in mind: $P_{A,r} \\Leftrightarrow \\text{``the measure of $A$ is $\\ge r$''}.$ Define the theory $T$ as the following set of sentences: for each $A,B \\in \\mathfrak {B}(X)$ , $r,s \\in [0,1]$ and $g \\in G$ , “$P_{A,0}$ ”$\\in T$ ; if $r > 0$ , then “$\\lnot P_{\\mathbb {\\emptyset }, r}$ ”$\\in T$ ; if $s \\ge r$ , then “$P_{A,s} \\rightarrow P_{A,r}$ ”$\\in T$ ; if $A \\cap B = \\mathbb {\\emptyset }$ , then “$(P_{A,r} \\wedge P_{B,s}) \\rightarrow P_{A \\cup B, r+s}$ ”, “$(\\lnot P_{A,r} \\wedge \\lnot P_{B,s}) \\rightarrow \\lnot P_{A \\cup B, r+s}$ ”$\\in T$ ; “$P_{X,1}$ ”$\\in T$ ; “$P_{A,r} \\rightarrow P_{gA,r}$ ”$\\in T$ .",
"If there is an assignment of the variables in $\\mathcal {P}$ satisfying $T$ , then for each $A \\in \\mathfrak {B}(X)$ , we can define $\\mu (A) = \\sup \\lbrace r \\in [0,1] : P_{A,r}\\rbrace .$ Note that due to (i), $\\mu $ is well defined for all $A \\in \\mathfrak {B}(X)$ .",
"In fact, it is straightforward to check that $\\mu $ is a finitely additive $G$ -invariant probability measure.",
"Thus, we only need to show that $T$ is satisfiable, for which it is enough to check that $T$ is finitely satisfiable, by the Compactness Theorem of propositional logic (or by Tychonoff's theorem).",
"Let $T_0 \\subseteq T$ be finite and let $\\mathcal {P}_0$ be the set of propositional variables that appear in the sentences in $T_0$ .",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by the sets that appear in the indices of the variables in $\\mathcal {P}_0$ .",
"By REF , there is a finitely additive $G$ -invariant probability measure $\\mu $ defined on $\\mathcal {B}$ .",
"Consider the following assignment of the variables in $\\mathcal {P}_0$ : for all $P_{A,r} \\in \\mathcal {P}_0$ , $P_{A,r} :\\Leftrightarrow \\mu (A) \\ge r.$ It is straightforward to check that this assignment satisfies $T_0$ , and hence, $T$ is finitely satisfiable.",
"Finite generators in the case of $\\sigma $ -compact spaces In this section we prove that the answer to Question REF is positive in case $X$ has a $\\sigma $ -compact realization.",
"To do this, we first prove Proposition REF , which shows how to construct a countably additive invariant probability measure on $X$ using a finitely additive one.",
"We then use REF to conclude the result.",
"For the next two statements, let $X$ be a second countable Hausdorff topological space equipped with a continuous action of $G$ .",
"Lemma 4.1 Let $\\mathcal {U}\\subseteq Pow(X)$ be a countable base for $X$ closed under the $G$ -action and finite unions/intersections.",
"Let $\\rho $ be a $G$ -invariant finitely additive probability measure on the $G$ -algebra generated by $\\mathcal {U}$ .",
"For every $A \\subseteq X$ , define $\\mu ^* (A) = \\inf \\lbrace \\sum _{n \\in \\mathbb {N}} \\rho (U_n ) : U_n \\in \\mathcal {U}\\; \\wedge \\; A \\subseteq \\bigcup _{n \\in \\mathbb {N}} U_n\\rbrace .$ Then: $\\mu ^*$ is a $G$ -invariant outer measure.",
"If $K \\subseteq X$ is compact, then $K$ is metrizable and $\\mu ^*$ is a metric outer measure on $K$ (with respect to any compatible metric).",
"It is a standard fact from measure theory that $\\mu ^*$ is an outer measure.",
"That $\\mu ^*$ is $G$ -invariant follows immediately from $G$ -invariance of $\\rho $ and the fact that $\\mathcal {U}$ is closed under the action of $G$ .",
"For (b), first note that by Urysohn metrization theorem, $K$ is metrizable, and fix a metric on $K$ .",
"If $E, F \\subseteq K$ are a positive distance apart, then so are $\\bar{E}$ and $\\bar{F}$ .",
"Hence there exist disjoint open sets $U,V$ such that $\\bar{E} \\subseteq U$ , $\\bar{F} \\subseteq V$ .",
"Because $\\bar{E}$ and $\\bar{F}$ are compact, $U,V$ can be taken to be finite unions of sets in $\\mathcal {U}$ and therefore $U,V \\in \\mathcal {U}$ .",
"Now fix $\\epsilon >0$ and let $W_n \\in \\mathcal {U}$ , be such that $E \\cup F \\subseteq \\bigcup _n W_n$ and $\\sum _n \\rho (W_n) \\le \\mu ^*(E \\cup F) + \\epsilon \\le \\mu ^*(E) + \\mu ^*(F) +\\epsilon .\\qquad \\mathrm {{(*)}}$ Note that $\\lbrace W_n \\cap U\\rbrace _{n \\in \\mathbb {N}}$ covers $E$ , $\\lbrace W_n \\cap V\\rbrace _{n \\in \\mathbb {N}}$ covers $F$ and $W_n \\cap U, W_n \\cap V \\in \\mathcal {U}$ .",
"Also, by finite additivity of $\\rho $ , $\\rho (W_n \\cap U) + \\rho (W_n \\cap V) = \\rho (W_n \\cap (U \\cup V)) \\le \\rho (W_n).$ Thus $\\mu ^*(E) + \\mu ^*(F) \\le \\sum _n \\rho (W_n \\cap U) + \\sum _n \\rho (W_n \\cap V) \\le \\sum _n \\rho (W_n),$ which, together with ($*$ ), implies that $\\mu ^*(E \\cup F) = \\mu ^*(E) + \\mu ^*(F)$ since $\\epsilon $ is arbitrary.",
"Proposition 4.2 Suppose there exist a countable base $\\mathcal {U}\\subseteq Pow(X)$ for $X$ and a compact set $K \\subseteq X$ such that the $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ admits a finitely additive $G$ -invariant probability measure $\\rho $ with $\\rho (K)>0$ .",
"Then there exists a countably additive $G$ -invariant Borel probability measure on $X$ .",
"Let $K, \\mathcal {U}$ and $\\rho $ be as in the hypothesis.",
"We may assume that $\\mathcal {U}$ is closed under the $G$ -action and finite unions/intersections.",
"Let $\\mu ^*$ be the outer measure provided by Lemma REF applied to $\\mathcal {U}$ , $\\rho $ .",
"Thus $\\mu ^*$ is a metric outer measure on $K$ and hence all Borel subsets of $K$ are $\\mu ^*$ -measurable (see 13.2 in ).",
"This implies that all Borel subsets of $Y = [K]_G = \\bigcup _{g \\in G} gK$ are $\\mu ^*$ -measurable because $\\mu ^*$ is $G$ -invariant.",
"By Carathéodory's theorem, the restriction of $\\mu ^*$ to the Borel subsets of $Y$ is a countably additive Borel measure on $Y$ , and we extend it to a Borel measure $\\mu $ on $X$ by setting $\\mu (Y^c) = 0$ .",
"Note that $\\mu $ is $G$ -invariant and $\\mu (Y) \\le 1$ .",
"It remains to show that $\\mu $ is nontrivial, which we do by showing that $\\mu (K) \\ge \\rho (K)$ and hence $\\mu (K)>0$ .",
"To this end, let $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq \\mathcal {U}$ cover $K$ .",
"Since $K$ is compact, there is a finite subcover $\\lbrace U_n\\rbrace _{n < N}$ .",
"Thus $U := \\bigcup _{n < N} U_n \\in \\mathcal {U}$ and $K \\subseteq U$ .",
"By finite additivity of $\\rho $ , we have $\\sum _{n \\in \\mathbb {N}} \\rho (U_n) \\ge \\sum _{n < N} \\rho (U_n) \\ge \\rho (U) \\ge \\rho (K),$ and hence, it follows from the definition of $\\mu ^*$ that $\\mu ^*(K) \\ge \\rho (K)$ .",
"Thus $\\mu (K) = \\mu ^*(K) > 0$ .",
"Corollary 4.3 Let $X$ be a second countable Hausdorff topological $G$ -space whose Borel structure is standard.",
"For every compact set $K \\subseteq X$ not in 4, there is a $G$ -invariant countably additive Borel probability measure $\\mu $ on $X$ with $\\mu (K) > 0$ .",
"Fix any countable base $\\mathcal {U}$ for $X$ and let $\\mathcal {B}$ be the Boolean $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ .",
"By Corollary REF , there exists a $G$ -invariant finitely additive probability measure $\\rho $ on $\\mathcal {B}$ such that $\\rho (K) > 0$ .",
"Now apply REF .",
"As a corollary, we derive the analogue of Nadkarni's theorem for 4 in case of $\\sigma $ -compact spaces.",
"Corollary 4.4 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"$X \\notin 4$ if and only if there exists a $G$ -invariant countably additive Borel probability measure on $X$ .",
"$\\Leftarrow $ : If $X \\in 4$ , then it is compressible in the usual sense and hence does not admit a $G$ -invariant Borel probability measure.",
"$\\Rightarrow $ : Suppose that $X$ is a $\\sigma $ -compact topological $G$ -space and $X \\notin 4$ .",
"Then, since $X$ is $\\sigma $ -compact and 4 is a $\\sigma $ -ideal, there is a compact set $K$ not in 4.",
"Now apply REF .",
"Remark.",
"For a Borel $G$ -space $X$ , let $\\mathcal {K}$ denote the collection of all subsets of invariant Borel sets that admit a $\\sigma $ -compact realization (when viewed as Borel $G$ -spaces).",
"Also, let $ denote the collection of all subsets of invariant compressible Borel sets.",
"It is clear that $ K$ and $ are $\\sigma $ -ideals, and what REF implies is that $\\mathcal {K}\\subseteq 4$ .",
"The question of whether $\\mathcal {K}= Pow(X)$ is just a rephrasing of $§10$ .",
"(B) of Introduction.",
"Theorem 4.5 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"If there is no $G$ -invariant Borel probability measure on $X$ , then $X$ admits a Borel 32-generator.",
"By REF , $X \\in 4$ and hence, $X$ is 4-compressible.",
"Thus, by Proposition REF , $X$ admits a Borel $2^5$ -generator.",
"Example 4.6.",
"Let $LO \\subseteq 2^{\\mathbb {N}^2}$ denote the Polish space of all countable linear orderings and let $G$ be the group of finite permutations of elements of $\\mathbb {N}$ .",
"$G$ is countable and acts continuously on $LO$ in the natural way.",
"Put $X = LO \\setminus DLO$ , where $DLO$ denotes the set of all dense linear orderings without endpoints (copies of $\\mathbb {Q}$ ).",
"It is straightforward to see that $DLO$ is a $G_{\\delta }$ subset of $LO$ and hence, $X$ is $F_{\\sigma }$ .",
"Therefore, $X$ is in fact $\\sigma $ -compact since $LO$ is compact being a closed subset of $2^{\\mathbb {N}^2}$ .",
"Also note that $X$ is $G$ -invariant.",
"Let $\\mu $ be the unique measure on $LO$ defined by $\\mu (V_{(F,<)}) = {1 \\over n!",
"}$ , where $(F,<)$ is a finite linearly ordered subset of $\\mathbb {N}$ of cardinality $n$ and $V_{(F,<)}$ is the set of all linear orderings of $\\mathbb {N}$ extending the order $<$ on $F$ .",
"As shown in , $\\mu $ is the unique invariant measure for the action of $G$ on $LO$ and $\\mu (X) = 0$ .",
"Thus there is no $G$ -invariant Borel probability measure on $X$ and hence, by the above theorem, $X$ admits a Borel 32-generator.",
"Finitely traveling sets Let $X$ be a Borel $G$ -space.",
"Definition 5.1 Let $A,B \\in \\mathfrak {B}(X)$ be equidecomposable, i.e.",
"there are $N \\le \\infty $ , $\\lbrace g_n\\rbrace _{n < N} \\subseteq G$ and Borel partitions $\\lbrace A_n\\rbrace _{n < N}$ and $\\lbrace B_n\\rbrace _{n < N}$ of $A$ and $B$ , respectively, such that $g_n A_n = B_n$ for all $n < N$ .",
"$A,B$ are said to be locally finitely equidecomposable (denote by $A \\sim _{\\text{lfin}} B$ ), if $\\lbrace A_n\\rbrace _{n < N},\\lbrace B_n\\rbrace _{n < N},\\lbrace g_n\\rbrace _{n < N}$ can be taken so that for every $x \\in A$ , $A_n \\cap [x]_G = \\mathbb {\\emptyset }$ for all but finitely many $n<N$ ; finitely equidecomposable (denote by $A \\sim _{\\text{fin}} B$ ), if $N$ can be taken to be finite.",
"The notation $\\prec _{\\text{fin}}$ , $\\prec _{\\text{lfin}}$ and the notions of finite and locally finite compressibility are defined analogous to Definitions REF and REF .",
"Definition 5.2 A Borel set $A \\subseteq X$ is called (locally) finitely traveling if there exists pairwise disjoint Borel sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ such that $A_0 = A$ and $A \\sim _{\\text{fin}} A_n$ ($A \\sim _{\\text{lfin}} A_n$ ), $\\forall n \\in \\mathbb {N}$ .",
"Proposition 5.3 If $X$ is (locally) finitely compressible then $X$ admits a (locally) finitely traveling Borel complete section.",
"We prove for finitely compressible $X$ , but note that everything below is also locally valid (i.e.",
"restricted to every orbit) for a locally compressible $X$ .",
"Run the proof of the first part of Lemma REF noting that a witnessing map $\\gamma : X \\rightarrow G$ of finite compressibility of $X$ has finite image and hence the image of each $\\delta _n$ (in the notation of the proof) is finite, which implies that the obtained traveling set $A$ is actually finitely traveling.",
"Proposition 5.4 If $X$ admits a locally finitely traveling Borel complete section, then $X \\in 4$ .",
"Let $A$ be a locally finitely traveling Borel complete section and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"Let $I_n = \\lbrace C_k^n\\rbrace _{k \\in \\mathbb {N}}$ , $J_n = \\lbrace D_k^n\\rbrace _{k \\in \\mathbb {N}}$ be Borel partitions of $A$ and $A_n$ , respectively, that together with $\\lbrace g_k^n\\rbrace _{k \\in \\mathbb {N}} \\subseteq G$ witness $A \\sim _{\\text{lfin}} A_n$ (as in Definition REF ).",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by $\\lbrace X\\rbrace \\cup \\bigcup _{n \\in \\mathbb {N}} (I_n \\cup J_n \\cup \\lbrace A_n\\rbrace )$ .",
"Now assume for contradiction that $X \\notin 4$ and hence, $A \\notin 4$ .",
"Thus, applying Corollary REF to $A$ and $\\mathcal {B}$ , we get a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B) = 0$ .",
"Claim $\\mu (A_n) = \\mu (A)$ , for all $n \\in \\mathbb {N}$ .",
"Proof of Claim.",
"For each $n$ , let $\\lbrace C_{k_i}^n\\rbrace _{i < K_n}$ be the list of those $C_k^n$ such that $C_k^n \\cap [x]_G \\ne \\mathbb {\\emptyset }$ ($K_n < \\infty $ by the definition of locally finitely traveling).",
"Set $B = A \\setminus (\\bigcup _{i < K_n} C_{k_i}^n)$ and note that by finite additivity of $\\mu $ , $\\mu (A) = \\mu (B) + \\sum _{i < K_n} \\mu (C_{k_i}^n).$ Similarly, set $B^{\\prime } = A_n \\setminus (\\bigcup _{i < K_n} D_{k_i}^n)$ and hence $\\mu (A_n) = \\mu (B^{\\prime }) + \\sum _{i < K_n} \\mu (D_{k_i}^n).$ But $B \\cap [x]_G = \\mathbb {\\emptyset }$ and $B^{\\prime } \\cap [x]_G = \\mathbb {\\emptyset }$ , and thus $\\mu (B) = \\mu (B^{\\prime }) = 0$ .",
"Also, since $g_{k_i}^n C_{k_i}^n = D_{k_i}^n$ and $\\mu $ is $G$ -invariant, $\\mu (C_{k_i}^n) = \\mu (D_{k_i}^n)$ .",
"Therefore $\\mu (A) = \\sum _{i < K_n} \\mu (C_{k_i}^n) = \\sum _{i < K_n} \\mu (D_{k_i}^n) = \\mu (A_n).$ $\\dashv $ This claim contradicts $\\mu $ being a probability measure since for large enough $N$ , $\\mu (\\bigcup _{n < N} A_n) = N \\mu (A) > 1$ , contradicting $\\mu (X) = 1$ .",
"This, together with REF , implies the following.",
"Corollary 5.5 Let $X$ be a Borel $G$ -space.",
"If $X$ admits a locally finitely traveling Borel complete section, then there is a Borel 32-generator.",
"Separating smooth-many invariant sets Assume throughout that $X$ is a Borel $G$ -space.",
"Lemma 6.1 If $X$ is aperiodic then it admits a countably infinite partition into Borel complete sections.",
"The following argument is also given in the proof of Theorem 13.1 in .",
"By the marker lemma (see 6.7 in ), there exists a vanishing sequence $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections, i.e.",
"$\\bigcap _{n \\in \\mathbb {N}} B_n = \\mathbb {\\emptyset }$ .",
"For each $n \\in \\mathbb {N}$ , define $k_n : X \\rightarrow \\mathbb {N}$ recursively as follows: $\\left\\lbrace \\begin{array}{rcl}k_0(x) & = & 0 \\\\k_{n+1}(x) &= & min \\lbrace k \\in \\mathbb {N}: B_{k_n(x)} \\cap [x]_G \\nsubseteq B_k\\rbrace \\end{array}\\right.,$ and define $A_n \\subseteq X$ by $x \\in A_n \\Leftrightarrow x \\in A_{k_n(x)} \\setminus A_{k_{n+1}(x)}.$ It is straightforward to check that $A_n$ are pairwise disjoint Borel complete sections.",
"For $A \\in \\mathfrak {B}(X)$ , if $I= < \\!\\!",
"A \\!\\!",
">$ then we use the notation $F_A$ and $f_A$ instead of $F_{I}$ and $, respectively.$ We now work towards strengthening the above lemma to yield a countably infinite partition into $F_A$ -invariant Borel complete sections.",
"Definition 6.2 (Aperiodic separation) For Borel sets $A, Y \\subseteq X$ , we say that $A$ aperiodically separates $Y$ if $f_A([Y]_G)$ is aperiodic (as an invariant subset of the shift $2^G$ ).",
"If such $A$ exists, we say that $Y$ is aperiodically separable.",
"Proposition 6.3 For $A \\in \\mathfrak {B}(X)$ , if $A$ aperiodically separates $X$ , then $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Let $Y = \\lbrace y \\in 2^G : |[y]_G| = \\infty \\rbrace $ and hence $f_A(X)$ is a $G$ -invariant subset of $Y$ .",
"By Lemma REF applied to $Y$ , there is a partition $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $Y$ into Borel complete sections.",
"Thus $A_n = f_{I}^{-1}(B_n)$ is a Borel $F_A$ -invariant complete section for $X$ and $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $X$ .",
"Let $\\mathfrak {A}$ denote the collection of all subsets of aperiodically separable Borel sets.",
"Lemma 6.4 $\\mathfrak {A}$ is a $\\sigma $ -ideal.",
"We only have to show that if $Y_n$ are aperiodically separable Borel sets, then $Y = \\bigcup _{n \\in \\mathbb {N}} Y_n \\in \\mathfrak {A}$ .",
"Let $A_n$ be a Borel set aperiodically separating $Y_n$ .",
"Since $A_n$ also aperiodically separates $[Y_n]_G$ (by definition), we can assume that $Y_n$ is $G$ -invariant.",
"Furthermore, by taking $Y_n^{\\prime } = Y_n \\setminus \\bigcup _{k<n} Y_k$ , we can assume that $Y_n$ are pairwise disjoint.",
"Now letting $A = \\bigcup _{n \\in \\mathbb {N}} (A_n \\cap Y_n)$ , it is easy to check that $A$ aperiodically separates $Y$ .",
"Let $\\mathfrak {S}$ denote the collection of all subsets of smooth sets.",
"By a similar argument as the one above, $\\mathfrak {S}$ is a $\\sigma $ -ideal.",
"Lemma 6.5 If $X$ is aperiodic, then $\\mathfrak {S}\\subseteq \\mathfrak {A}$ .",
"Let $S \\in \\mathfrak {S}$ and hence there is a Borel transversal $T$ for $[S]_G$ .",
"Fix $x \\in S$ and let $y \\ne z \\in [x]_G$ .",
"Since $T$ is a transversal, there is $g \\in G$ such that $gy \\in T$ , and hence $gz \\notin T$ .",
"Thus $f_T(y) \\ne f_T(z)$ , and so $f_T([x]_G)$ is infinite.",
"Therefore $T$ aperiodically separates $[S]_G$ .",
"For the rest of the section, fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and let $F_A^n$ be following equivalence relation: $y F_A^n z \\Leftrightarrow \\forall k < n (g_k y \\in A \\leftrightarrow g_k z \\in A).$ Note that $F_A^n$ has no more than $2^n$ equivalence classes and that $y F_A z$ if and only if $\\forall n (y F_A^n z)$ .",
"Lemma 6.6 For $A,Y \\in \\mathfrak {B}(X)$ , $A$ aperiodically separates $Y$ if and only if $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in Y^{[x]_G}) [y F_A^n z \\wedge \\lnot (y F_A z)]$ .",
"$\\Rightarrow $ : Assume that for all $x \\in Y$ , $f_A([x]_G)$ is infinite and thus $F_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ has infinitely many equivalence classes.",
"Fix $n \\in \\mathbb {N}$ and recall that $F_A^n$ has only finitely many equivalence classes.",
"Thus, by the Pigeon Hole Principle, there are $y,z \\in Y^{[x]_G}$ such that $y F_A^n z$ yet $\\lnot (y F_A z)$ .",
"$\\Leftarrow $ : Assume for contradiction that $f_A(Y^{[x]_G})$ is finite for some $x \\in Y$ .",
"Then it follows that $F_A = F_A^n$ , for some $n$ , and hence for any $y,z \\in Y^{[x]_G}$ , $y F_A^n z$ implies $y F_A z$ , contradicting the hypothesis.",
"Theorem 6.7 If $X$ is an aperiodic Borel $G$ -space, then $X \\in \\mathfrak {A}$ .",
"By Lemma REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into Borel complete sections.",
"We will inductively construct Borel sets $B_n \\subseteq C_n$ , where $C_n$ should be thought of as the set of points colored (black or white) at the $n^{th}$ step, and $B_n$ as the set of points colored black (thus $C_n \\setminus B_n$ is colored white).",
"Define a function $\\# : X \\rightarrow \\mathbb {N}$ by $x \\mapsto m$ , where $m$ is such that $x \\in A_m$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ of sets generating the Borel $\\sigma $ -algebra of $X$ .",
"Assuming that for all $k < n$ , $C_k, B_k$ are defined, let $\\bar{C}_n = \\bigcup _{k<n} C_k$ and $\\bar{B}_n = \\bigcup _{k<n} B_k$ .",
"Put $P_n = \\lbrace x \\in A_0 : \\forall k < n (g_k x \\in \\bar{C}_n) \\wedge g_n x \\notin \\bar{C}_n\\rbrace $ and set $F_n = F_{\\bar{B}_n}^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ , that is for all $x,y \\in P_n$ , $y F_n z \\Leftrightarrow \\forall k < n (g_k y \\in \\bar{B}_n \\leftrightarrow g_k z \\in \\bar{B}_n).$ Now put $C^{\\prime }_n = \\lbrace x \\in P_n : \\#(g_n x) = \\min \\#((g_nP_n)^{[x]_G})\\rbrace $ , $C^{\\prime \\prime }_n = \\lbrace x \\in C^{\\prime }_n : \\exists y, z \\in (C^{\\prime }_n)^{[x]_G} (y \\ne z \\wedge y F_n z)\\rbrace $ and $C_n = g_n C^{\\prime \\prime }_n$ .",
"Note that it follows from the definition of $P_n$ that $C_n$ is disjoint from $\\bar{C}_n$ .",
"Now in order to define $B_n$ , first define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{ the smallest $m$ such that there are } y,z \\in C^{\\prime \\prime }_n \\cap [x]_G \\text{ with } y F_n z, y \\in U_m \\text{ and } z \\notin U_m.$ Note that $\\bar{n}$ is Borel and $G$ -invariant.",
"Lastly, let $B^{\\prime }_n = \\lbrace x \\in C^{\\prime \\prime }_n : x \\in U_{\\bar{n}(x)}\\rbrace $ and $B_n = g_n B^{\\prime }_n$ .",
"Clearly $B_n \\subseteq C_n$ .",
"Now let $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $D = \\left[\\bigcup _{n \\in \\mathbb {N}} (C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)\\right]_G$ .",
"We show that $B$ aperiodically separates $Y := X \\setminus D$ and $D \\in \\mathfrak {S}$ .",
"Since $\\mathfrak {S}\\subseteq \\mathfrak {A}$ and $\\mathfrak {A}$ is an ideal, this will imply that $X \\in \\mathfrak {A}$ .",
"Claim 1 $D \\in \\mathfrak {S}$ .",
"Proof of Claim.",
"Since $\\mathfrak {S}$ is a $\\sigma $ -ideal, it is enough to show that for each $n$ , $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G \\in \\mathfrak {S}$ , so fix $n \\in \\mathbb {N}$ .",
"Clearly $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ is finite, for all $x \\in X$ , since there can be at most $2^n$ pairwise $F_n$ -nonequivalent points.",
"Thus, fixing some Borel linear ordering of $X$ and taking the smallest element from $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ for each $x \\in C^{\\prime }_n \\setminus C^{\\prime \\prime }_n$ , we can define a Borel transversal for $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G$ .",
"$\\dashv $ By Lemma REF , to show that $B$ aperiodically separates $Y$ , it is enough to show that $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in [x]_G) [y F_B^n z \\wedge \\lnot (y F_B z)]$ .",
"Fix $x \\in Y$ .",
"Claim 2 $(\\exists ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $(\\forall ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $x \\notin D$ , it follows that $(\\forall ^{\\infty } n) P_n^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $A_0$ is a complete section and $\\bar{C}_0 = \\mathbb {\\emptyset }$ , $P_0^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Let $N$ be the largest number such that $P_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus for all $n > N$ , $C_n^{[x]_G} = \\mathbb {\\emptyset }$ and hence for all $n > N$ , $\\bar{C}_n^{[x]_G} = \\bar{C}_{N+1}^{[x]_G}$ .",
"Because $C_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ , there is $y \\in A_0^{[x]_G}$ such that $\\forall k \\le N (g_k y \\in \\bar{C}_{N+1})$ ; but because $P_{N+1}^{[x]_G} = \\mathbb {\\emptyset }$ , $g_{N+1} y$ must also fall into $\\bar{C}_{N+1}$ .",
"By induction on $n > N$ , we get that for all $n>N$ , $g_n y \\in \\bar{C}_n$ and thus $g_n y \\in \\bar{C}_{N+1}$ .",
"On the other hand, it follows from the definition of $C^{\\prime }_n$ that for each $n$ , $(C^{\\prime }_n)^{[x]_G}$ intersects exactly one of $A_k$ .",
"Thus $\\bar{C}_{N+1}^{[x]_G}$ intersects at most $N+1$ of $A_k$ and hence there exists $K \\in \\mathbb {N}$ such that for all $k \\ge K$ , $\\bar{C}_{N+1}^{[x]_G} \\cap A_k = \\mathbb {\\emptyset }$ .",
"Since $\\exists ^{\\infty } n (g_n y \\in \\bigcup _{k \\ge K} A_k)$ , $\\exists ^{\\infty } n (g_n y \\notin \\bar{C}_{N+1})$ , a contradiction.",
"$\\dashv $ Now it remains to show that for all $n \\in \\mathbb {N}$ , $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ implies that $\\exists y,z \\in [x]_G$ such that $y F_B^n z$ but $\\lnot (y F_B z)$ .",
"To this end, fix $n \\in \\mathbb {N}$ and assume $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus there are $y,z \\in (C^{\\prime \\prime }_n)^{[x]_G}$ such that $y F_n z$ , $y \\in U_{\\bar{n}(x)}$ and $z \\notin U_{\\bar{n}(x)}$ ; hence, $g_n y \\in B_n$ and $g_n z \\notin B_n$ , by the definition of $B_n$ .",
"Since $C_k$ are pairwise disjoint, $B_n \\subseteq C_n$ and $g_n y, g_n z \\in C_n$ , it follows that $g_n y \\in B$ and $g_n z \\notin B$ , and therefore $\\lnot (y F_B z)$ .",
"Finally, note that $F_n = F_B^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ and hence $y F_B^n z$ .",
"Corollary 6.8 Suppose all of the nontrivial subgroups of $G$ have finite index (e.g.",
"$G = \\mathbb {Z}$ ), and let $X$ be an aperiodic Borel $G$ -space.",
"Then there exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit, i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in X$ .",
"Let $A$ be a Borel set aperiodically separating $X$ (exists by Theorem REF ) and put $Y = f_A(X)$ .",
"Then $Y \\subseteq 2^G$ is aperiodic and hence the action of $G$ on $Y$ is free since the stabilizer subgroup of every element must have infinite index and thus is trivial.",
"But this implies that for all $y \\in Y$ , $f_A^{-1}(y)$ intersects every orbit in $X$ at no more than one point, and hence $f_A$ is one-to-one on every orbit.",
"From REF and REF we immediately get the following strengthening of Lemma REF .",
"Corollary 6.9 If $X$ is aperiodic, then for some $A \\in \\mathfrak {B}(X)$ , $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Theorem 6.10 Let $X$ be an aperiodic $G$ -space and let $E$ be a smooth equivalence relation on $X$ with $E_G \\subseteq E$ .",
"There exists a partition $\\mathcal {P}$ of $X$ into 4 Borel sets such that $G \\mathcal {P}$ separates any two $E$ -nonequivalent points in $X$ , i.e.",
"$\\forall x,y \\in X (\\lnot (x E y) \\rightarrow f_{\\mathcal {P}}(x) \\ne f_{\\mathcal {P}}(y))$ .",
"By Corollary REF , there is $A \\in \\mathfrak {B}(X)$ and a Borel partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant complete sections.",
"For each $n \\in \\mathbb {N}$ , define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{the smallest $m$ such that } \\exists x^{\\prime } \\in A_0^{[x]_G} \\text{ with } g_m x^{\\prime } \\in A_n.$ Clearly $\\bar{n}$ is Borel, and because all of $A_k$ are $F_A$ -invariant, $\\bar{n}$ is also $F_A$ -invariant, i.e.",
"for all $x,y \\in X$ , $x F_A y \\rightarrow \\bar{n}(x) = \\bar{n}(y)$ .",
"Also, $\\bar{n}$ is $G$ -invariant by definition.",
"Put $A^{\\prime }_n = \\lbrace x \\in A_0 : g_{\\bar{n}(x)} x \\in A_n\\rbrace $ and note that $A^{\\prime }_n$ is $F_A$ -invariant Borel since so are $\\bar{n}$ , $A_0$ and $A_n$ .",
"Moreover, $A^{\\prime }_n$ is clearly a complete section.",
"Define $\\gamma _n : A^{\\prime }_n \\rightarrow A_n$ by $x \\mapsto g_{\\bar{n}(x)} x$ .",
"Clearly, $\\gamma _n$ is Borel and one-to-one.",
"Since $E$ is smooth, there is a Borel $h : X \\rightarrow \\mathbb {R}$ such that for all $x,y \\in X$ , $x E y \\leftrightarrow h(x) = h(y)$ .",
"Let $\\lbrace V_n\\rbrace _{n \\in \\mathbb {N}}$ be a countable family of subsets of $\\mathbb {R}$ generating the Borel $\\sigma $ -algebra of $\\mathbb {R}$ and put $U_n = h^{-1}(V_n)$ .",
"Because each equivalence class of $E$ is $G$ -invariant, so is $h$ and hence so is $U_n$ .",
"Now let $B_n = \\gamma _n(A^{\\prime }_n \\cap U_n)$ and note that $B_n$ is Borel being a one-to-one Borel image of a Borel set.",
"It follows from the definition of $\\gamma _n$ that $B_n \\subseteq A_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $\\mathcal {P}= < \\!\\!",
"A,B \\!\\!",
">$ ; in particular, $|\\mathcal {P}| \\le 4$ .",
"We show that $\\mathcal {P}$ is what we want.",
"To this end, fix $x,y \\in X$ with $\\lnot (x E y)$ .",
"If $\\lnot (x F_A y)$ , then $G < \\!\\!",
"A \\!\\!",
">$ (and hence $G \\mathcal {P}$ ) separates $x$ and $y$ .",
"Thus assume that $x F_A y$ .",
"Since $h(x) \\ne h(y)$ , there is $n$ such that $h(x) \\in V_n$ and $h(y) \\notin V_n$ .",
"Hence, by invariance of $U_n$ , $gx \\in U_n \\wedge gy \\notin U_n$ , for all $g \\in G$ .",
"Because $A^{\\prime }_n$ is a complete section, there is $g \\in G$ such that $gx \\in A^{\\prime }_n$ and hence $gy \\in A^{\\prime }_n$ since $A^{\\prime }_n$ is $F_A$ -invariant.",
"Let $m = \\bar{n}(gx)$ ($= \\bar{n}(gy)$ ).",
"Then $g_m gx \\in B_n$ while $g_m gy \\notin B_n$ although $g_m gy \\in \\gamma _n(A^{\\prime }_n) \\subseteq A_n$ .",
"Thus $g_m gx \\in B$ but $g_m gy \\notin B$ and therefore $G \\mathcal {P}$ separates $x$ and $y$ .",
"Potential dichotomy theorems In this section we prove dichotomy theorems assuming Weiss's question has a positive answer for $G = \\mathbb {Z}$ .",
"In the proofs we use the Ergodic Decomposition Theorem (see , ) and a Borel/uniform version of Krieger's finite generator theorem, so we first state both of the theorems and sketch the proof of the latter.",
"For a Borel $G$ -space $X$ , let $\\mathcal {M}_G(X)$ denote the set of $G$ -invariant Borel probability measures on $X$ and let $\\mathcal {E}_G(X)$ denote the set of ergodic ones among those.",
"Clearly both are Borel subsets of $P(X)$ (the standard Borel space of Borel probability measures on $X$ ) and thus are themselves standard Borel spaces.",
"Ergodic Decomposition Theorem 7.1 (Farrell, Varadarajan) Let $X$ be a Borel $G$ -space.",
"If $\\mathcal {M}_G(X) \\ne \\mathbb {\\emptyset }$ (and hence $\\mathcal {E}_G(X) \\ne \\mathbb {\\emptyset }$ ), then there is a Borel surjection $x \\mapsto e_x$ from $X$ onto $\\mathcal {E}_G(X)$ such that: $x E_G y \\Rightarrow e_x = e_y$ ; For each $e \\in \\mathcal {E}_G(X)$ , if $X_e = \\lbrace x \\in X : e_x = e\\rbrace $ (hence $X_e$ is invariant Borel), then $e(X_e) = 1$ and $e \\!",
"\\!",
"\\downharpoonright _{X_e}$ is the unique ergodic invariant Borel probability measure on $X_e$ ; For each $\\mu \\in \\mathcal {M}_G(X)$ and $A \\in \\mathfrak {B}(X)$ , we have $\\mu (A) = \\int e_x(A) d\\mu (x).$ For the rest of the section, let $X$ be a Borel $\\mathbb {Z}$ -space.",
"For $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , if we let $h_e$ denote the entropy of $(X, \\mathbb {Z}, e)$ , then the map $e \\mapsto h_e$ is Borel.",
"Indeed, if $\\lbrace \\mathcal {P}_k\\rbrace _{k \\in \\mathbb {N}}$ is a refining sequence of partitions of $X$ that generates the Borel $\\sigma $ -algebra of $X$ , then by 4.1.2 of , $h_e = \\lim _{k \\rightarrow \\infty } h_e(\\mathcal {P}_k, \\mathbb {Z})$ , where $h_e(\\mathcal {P}_k, \\mathbb {Z})$ denotes the entropy of $\\mathcal {P}_k$ .",
"By 17.21 of , the function $e \\mapsto h_e(\\mathcal {P}_k)$ is Borel and thus so is the map $e \\mapsto h_e$ .",
"For all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ with $h_e < \\infty $ , let $N_e$ be the smallest integer such that $\\log N_e > h_e$ .",
"The map $e \\mapsto N_e$ is Borel because so is $e \\mapsto h_e$ .",
"Krieger's Finite Generator Theorem 7.2 (Uniform version) Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"Suppose $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Assume also that all measures in $\\mathcal {E}_{\\mathbb {Z}}(X)$ have finite entropy and let $e \\mapsto N_e$ be the map defined above.",
"Then there is a partition $\\lbrace A_n\\rbrace _{n \\le \\infty }$ of $X$ into Borel sets such that $A_{\\infty }$ is invariant and does not admit an invariant Borel probability measure; For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e \\setminus A_{\\infty }$ , where $X_e = \\rho ^{-1}(e)$ .",
"Sketch of Proof.",
"Note that it is enough to find a Borel invariant set $X^{\\prime } \\subseteq X$ and a Borel $\\mathbb {Z}$ -map $\\phi : X^{\\prime } \\rightarrow \\mathbb {N}^{\\mathbb {Z}}$ , such that for each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , we have $e(X \\setminus X^{\\prime }) = 0$ ; $\\phi \\!",
"\\!",
"\\downharpoonright _{X_e \\cap X^{\\prime }}$ is one-to-one and $\\phi (X_e \\cap X^{\\prime }) \\subseteq (N_e)^{\\mathbb {Z}}$ , where $(N_e)^{\\mathbb {Z}}$ is naturally viewed as a subset of $\\mathbb {N}^{\\mathbb {Z}}$ .",
"Indeed, assume we had such $X^{\\prime }$ and $\\phi $ , and let $A_{\\infty } = X \\setminus X^{\\prime }$ and $A_n = \\phi ^{-1}(V_n)$ for all $n \\in \\mathbb {N}$ , where $V_n = \\lbrace y \\in \\mathbb {N}^{\\mathbb {Z}} : y(0) = n\\rbrace $ .",
"Then it is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ satisfies (ii).",
"Also, (I) and part (ii) of the Ergodic Decomposition Theorem imply that (i) holds for $A_{\\infty }$ .",
"To construct such a $\\phi $ , we use the proof of Krieger's theorem presented in , Theorem 4.2.3, and we refer to it as Downarowicz's proof.",
"For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , the proof constructs a Borel $\\mathbb {Z}$ -embedding $\\phi _e : X^{\\prime } \\rightarrow N_e^{\\mathbb {Z}}$ on an $e$ -measure 1 set $X^{\\prime }$ .",
"We claim that this construction is uniform in $e$ in a Borel way and hence would yield $X^{\\prime }$ and $\\phi $ as above.",
"Our claim can be verified by inspection of Downarowicz's proof.",
"The proof uses the existence of sets with certain properties and one has to check that such sets exist with the properties satisfied for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ at once.",
"For example, the set $C$ used in the proof of Lemma 4.2.5 in can be chosen so that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $C \\cap X_e$ has the required properties for $e$ (using the Shannon-McMillan-Brieman theorem).",
"Another example is the set $B$ used in the proof of the same lemma, which is provided by Rohlin's lemma.",
"By inspection of the proof of Rohlin's lemma (see 2.1 in ), one can verify that we can get a Borel $B$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $B \\cap X_e$ has the required properties for $e$ .",
"The sets in these two examples are the only kind of sets whose existence is used in the whole proof; the rest of the proof constructs the required $\\phi $ “by hand”.",
"$\\Box $ Theorem 7.3 (Dichotomy I) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy; there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"We first show that the conditions above are mutually exclusive.",
"Indeed, assume there exist an invariant ergodic Borel probability measure $e$ with infinite entropy and a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"By ergodicity, $e$ would have to be supported on one of the $Y_n$ .",
"But $Y_n$ has a finite generator and hence the dynamical system $(Y_n, \\mathbb {Z}, e)$ has finite entropy by the Kolmogorov-Sinai theorem (see REF ).",
"Thus so does $(X, \\mathbb {Z}, e)$ since these two systems are isomorphic (modulo $e$ -NULL), contradicting the assumption on $e$ .",
"Now we prove that at least one of the conditions holds.",
"Assume that there is no invariant ergodic measure with infinite entropy.",
"Now, if there was no invariant Borel probability measure at all, then, since the answer to Question REF is assumed to be positive, $X$ would admit a finite generator, and we would be done.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ , $Y_{\\infty } = A_{\\infty }$ , and for all $n \\in \\mathbb {N}$ , $Y_n = \\lbrace x \\in X^{\\prime } : N_{e_x} = n\\rbrace ,$ where the map $e \\mapsto N_e$ is as above.",
"Note that the sets $Y_n$ are invariant since $\\rho $ is invariant, so $\\lbrace Y_n\\rbrace _{n \\le \\infty }$ is a countable partition of $X$ into invariant Borel sets.",
"Since $Y_{\\infty }$ does not admit an invariant Borel probability measure, by our assumption, it has a finite generator.",
"Let $E$ be the equivalence relation on $X^{\\prime }$ defined by $\\rho $ , i.e.",
"$\\forall x,y \\in X^{\\prime }$ , $x E y \\Leftrightarrow \\rho (x) = \\rho (y).$ By definition, $E$ is a smooth Borel equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action.",
"Thus, by Theorem REF , there exists a partition $\\mathcal {P}$ of $X^{\\prime }$ into 4 Borel sets such that $\\mathbb {Z}\\mathcal {P}$ separates any two points in different $E$ -classes.",
"Now fix $n \\in \\mathbb {N}$ and we will show that $I= \\mathcal {P}\\vee \\lbrace A_i\\rbrace _{i < n}$ is a generator for $Y_n$ .",
"Indeed, take distinct $x,y \\in Y_n$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}\\mathcal {P}$ separates them and hence so does $\\mathbb {Z}I$ .",
"Thus we can assume that $x E y$ .",
"Then $e := \\rho (x) = \\rho (y)$ , i.e.",
"$x,y \\in X_e = \\rho ^{-1}(e)$ .",
"By the choice of $\\lbrace A_i\\rbrace _{i \\in \\mathbb {N}}$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e$ and hence $\\mathbb {Z}\\lbrace A_i\\rbrace _{i < N_e}$ separates $x$ and $y$ .",
"But $n = N_e$ by the definition of $Y_n$ , so $\\mathbb {Z}I$ separates $x$ and $y$ .",
"Proposition 7.4 Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ admits invariant ergodic probability measures of arbitrarily large entropy, then it admits an invariant probability measure of infinite entropy.",
"For each $n \\ge 1$ , let $\\mu _n$ be an invariant ergodic probability measure of entropy $h_{\\mu _n} > n 2^n$ such that $\\mu _n \\ne \\mu _m$ for $n \\ne m$ , and put $\\mu = \\sum _{n \\ge 1} {1 \\over 2^n} \\mu _n.$ It is clear that $\\mu $ is an invariant probability measure, and we show that its entropy $h_{\\mu }$ is infinite.",
"Fix $n \\ge 1$ .",
"Let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem and put $X_n = \\rho ^{-1}(\\mu _n)$ .",
"It is clear that $\\mu _m(X_n) = 1$ if $m = n$ and 0 otherwise.",
"For any finite Borel partition $\\mathcal {P}= \\lbrace A_i\\rbrace _{i=1}^k$ of $X_n$ , put $A_0 = X \\setminus X_n$ and $\\bar{\\mathcal {P}} = \\mathcal {P}\\cup \\lbrace A_0\\rbrace $ .",
"Let $T$ be the Borel automorphism of $X$ corresponding to the action of $1_{\\mathbb {Z}}$ , and let $h_{\\nu }(I)$ and $h_{\\nu }(I, T)$ denote, respectively, the static and dynamic entropies of a finite Borel partition $I$ of $X$ with respect to an invariant probability measure $\\nu $ .",
"Then, with the convention that $\\log (0) \\cdot 0 = 0$ , we have $h_{\\mu }(\\bar{\\mathcal {P}}) &= - \\sum _{i=0}^k \\log (\\mu (A_i)) \\mu (A_i) \\ge - \\sum _{i = 1}^k \\log (\\mu (A_i)) \\mu (A_i)= - \\sum _{i = 1}^k \\log ({1 \\over 2^n}\\mu _n(A_i)) {1 \\over 2^n} \\mu _n(A_i) \\\\&\\ge - {1 \\over 2^n} \\sum _{i = 1}^k \\log (\\mu _n(A_i)) \\mu _n(A_i) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}).$ Since $\\mathcal {P}$ is arbitrary and $X_n$ is invariant, it follows that $h_{\\mu }(\\bar{\\mathcal {P}}, T) = \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu }(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) \\ge {1 \\over 2^n} \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu _n}(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}, T).$ Now for any finite Borel partition $I$ of $X$ , it is clear that $h_{\\mu _n}(I) = h_{\\mu _n}(\\bar{\\mathcal {P}})$ (and hence $h_{\\mu _n}(I, T) = h_{\\mu _n}(\\bar{\\mathcal {P}}, T)$ ), for some $\\mathcal {P}$ as above.",
"This implies that $h_{\\mu } \\ge \\sup _{\\mathcal {P}} h_{\\mu }(\\bar{\\mathcal {P}}, T) \\ge {1 \\over 2^n} \\sup _{\\mathcal {P}} h_{\\mu _n}(\\bar{\\mathcal {P}}, T) = {1 \\over 2^n} \\sup _{I} h_{\\mu _n}(I, T) = {1 \\over 2^n} h_{\\mu _n} > n,$ where $\\mathcal {P}$ and $I$ range over finite Borel partitions of $X_n$ and $X$ , respectively.",
"Thus $h_{\\mu }\\!",
"= \\infty $ .",
"Theorem 7.5 (Dichotomy II) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"The Kolmogorov-Sinai theorem implies that the conditions are mutually exclusive, and we prove that at least one of them holds.",
"Assume that there is no invariant measure with infinite entropy.",
"If there was no invariant Borel probability measure at all, then, by our assumption, $X$ would admit a finite generator.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ and $X_e = \\rho ^{-1}(e)$ , for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ .",
"By our assumption, $A_{\\infty }$ admits a finite generator $\\mathcal {P}$ .",
"Also, by REF , there is $N \\ge 1$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $N_e \\le N$ and hence $\\mathcal {Q}:= \\lbrace A_n\\rbrace _{n < N}$ is a finite generator for $X_e$ ; in particular, $\\mathcal {Q}$ is a partition of $X^{\\prime }$ .",
"Let $E$ be the following equivalence relation on $X$ : $x E y \\Leftrightarrow (x, y \\in A_{\\infty }) \\vee (x,y \\in X^{\\prime } \\wedge \\rho (x) = \\rho (y)).$ By definition, $E$ is a smooth equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action and $A_{\\infty }$ is $\\mathbb {Z}$ -invariant.",
"Thus, by Theorem REF , there exists a partition $J$ of $X$ into 4 Borel sets such that $\\mathbb {Z}J$ separates any two points in different $E$ -classes.",
"We now show that $I:= < \\!\\!",
"J\\cup \\mathcal {P}\\cup \\mathcal {Q} \\!\\!",
">$ is a generator.",
"Indeed, fix distinct $x,y \\in X$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}J$ separates them.",
"So we can assume that $x E y$ .",
"If $x,y \\in A_{\\infty }$ , then $\\mathbb {Z}\\mathcal {P}$ separates $x$ and $y$ .",
"Finally, if $x,y \\in X^{\\prime }$ , then $x,y \\in X_e$ , where $e = \\rho (x)$ ($= \\rho (y)$ ), and hence $\\mathbb {Z}\\mathcal {Q}$ separates $x$ and $y$ .",
"Remark.",
"It is likely that the above dichotomies are also true for any amenable group using a uniform version of Krieger's theorem for amenable groups, cf.",
", but I have not checked the details.",
"Finite generators on comeager sets Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER.",
"Locally weakly wandering sets and other special cases Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
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[
"Invariant measures and $i$ -compressibility",
"This section is mainly devoted to proving the following theorem.",
"Theorem 3.1 Let $X$ be a Borel $G$ -space.",
"If $X$ is aperiodic, then there exists a function $m : \\mathfrak {B}(X) \\times X \\rightarrow [0,1]$ satisfying the following properties for all $A,B \\in \\mathfrak {B}(X)$ : $m(A, \\cdot )$ is Borel; $m(X, x) = 1$ , $\\forall x \\in X$ ; If $A \\subseteq B$ , then $m(A, x) \\le m(B,x)$ , $\\forall x \\in X$ ; $m(A, x) = 0$ off $[A]_G$ ; $m(A, x) > 0$ on $[A]_G$ modulo 4; $m(A,x) = m(gA, x)$ , for all $g \\in G$ , $x \\in X$ modulo 3; If $A \\cap B = \\mathbb {\\emptyset }$ , then $m(A \\cup B,x) = m(A,x) + m(B,x)$ , $\\forall x \\in X$ modulo 4.",
"Remark.",
"A version of this theorem is what lies at the heart of the proof of Nadkarni's theorem.",
"The conclusions of our theorem are modulo 4, which is potentially a smaller $\\sigma $ -ideal than the $\\sigma $ -ideal of sets contained in compressible Borel sets used in Nadkarni's version.",
"However, the price we paid for this is that part (g) asserts only finite additivity instead of countable additivity asserted by Nadkarni's version.",
"Proof of Theorem REF.",
"Our proof follows the general outline of Nadkarni's proof.",
"The construction of $m(A,x)$ is somewhat similar to that of Haar measure.",
"First, for sets $A,B$ , we define a Borel function $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ that basically gives the number of copies of $B^{[x]_G}$ that fit in $A^{[x]_G}$ when moved by group elements (piecewise).",
"Then we define a decreasing sequence of complete sections (called a fundamental sequence below), which serves as a gauge to measure the size of a given set.",
"Assume throughout that $X$ is an aperiodic Borel $G$ -space (although we only use the aperiodicity assumption in REF to assert that smooth sets are in 1).",
"Lemma 3.2 (Comparability) $\\forall A,B \\in \\mathfrak {B}(X)$ , there is a partition $X = P \\cup Q$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ , $A^P \\prec _{I} B^P$ and $B^Q \\preceq _{I} A^Q$ .",
"It is enough to prove the lemma assuming $X = [A]_G \\cap [B]_G$ since we can always include $[B]_G \\setminus [A]_G$ in $P$ and $X \\setminus [B]_G$ in $Q$ .",
"Fix an enumeration $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ for $G$ .",
"We recursively construct Borel sets $A_n,B_n,A_n^{\\prime },B_n^{\\prime }$ as follows.",
"Set $A_0^{\\prime } = A$ and $B_0^{\\prime } = B$ .",
"Assuming $A_n^{\\prime }, B_n^{\\prime }$ are defined, set $B_n = B_n^{\\prime } \\cap g_n A_n^{\\prime }$ , $A_n = g_n^{-1} B_n$ , $A_{n+1}^{\\prime } = A_n^{\\prime } \\setminus A_n$ and $B_{n+1}^{\\prime } = B_n^{\\prime } \\setminus B_n$ .",
"It is easy to see by induction on $n$ that for any $A,B$ -sensitive $I$ , $A_n,B_n$ are $F_{I}$ -invariant since so are $A,B$ .",
"Thus, setting $A^* = \\bigcup _{n \\in \\mathbb {N}}A_n$ and $B^* = \\bigcup _{n \\in \\mathbb {N}}B_n$ , we get that $A^* \\sim _{I} B^*$ since $B_n = g_n A_n$ .",
"Let $A^{\\prime } = A \\setminus A^*$ , $B^{\\prime } = B \\setminus B^*$ and set $P = [B^{\\prime }]_G$ , $Q = X \\setminus P$ .",
"Claim $[A^{\\prime }]_G \\cap [B^{\\prime }]_G = \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $\\exists x \\in A^{\\prime }$ and $n \\in \\mathbb {N}$ such that $g_n x \\in B^{\\prime }$ .",
"It is clear that $A^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} A_k^{\\prime }$ , $B^{\\prime } = \\bigcap _{k \\in \\mathbb {N}} B_k^{\\prime }$ ; in particular, $x \\in A_n^{\\prime }$ and $g_n x \\in B_n^{\\prime }$ .",
"But then $g_n x \\in B_n$ and $x \\in A_n$ , contradicting $x \\in A^{\\prime }$ .",
"$\\dashv $ Let $I$ be an $A,B$ -sensitive partition.",
"Then $A^P = (A^*)^P$ and hence $A^P \\prec _{I} B^P$ since $(A^*)^P \\sim _{I} (B^*)^P \\subseteq B^P$ and $[B^P \\setminus (B^*)^P]_G = [B^{\\prime }]_G = P = [B^P]_G$ .",
"Similarly, $B^Q = (B^*)^Q$ and hence $B^Q \\preceq _{I} A^Q$ since $(B^*)^Q \\sim _{I} (A^*)^Q \\subseteq A^Q$ .",
"Definition 3.3 (Divisibility) Let $n \\le \\infty $ , $A,B,C \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"Write $A \\sim _{I} nB \\oplus C$ if there are Borel sets $A_k \\subseteq A$ , $k<n$ , such that $\\lbrace A_k\\rbrace _{k<n} \\cup \\lbrace C\\rbrace $ is a partition of $A$ , each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\sim _{I} B$ .",
"Write $n B \\preceq _{I} A$ if there is $C \\subseteq A$ with $A \\sim _{I} n B \\oplus C$ , and write $n B \\prec _{I} A$ if moreover $[C]_G = [A]_G$ .",
"Write $A \\preceq _{I} nB$ if there is a Borel partition $\\lbrace A_k\\rbrace _{k<n}$ of $A$ such that each $A_k$ is $F_{I}$ -invariant relative to $A$ and $A_k \\preceq _{I} B$ .",
"If moreover, $A_k \\prec _{I} B$ for at least one $k < n$ , we write $A \\prec _{I} nB$ .",
"For $i \\ge 1$ , we use the above notation with $I$ replaced by $i$ if there is an $A,B$ -sensitive partition $I$ generated by $i$ sets for which the above conditions hold.",
"Proposition 3.4 (Euclidean decomposition) Let $A,B \\in \\mathfrak {B}(X)$ and put $R = [A]_G \\cap [B]_G$ .",
"There exists a partition $\\lbrace P_n\\rbrace _{n \\le \\infty }$ of $R$ into $G$ -invariant Borel sets such that for any $A,B$ -sensitive finite Borel partition $I$ of $X$ and $n \\le \\infty $ , $A^{P_n} \\sim _{I} nB^{P_n} \\oplus C_n$ for some $C_n$ such that $C_n \\prec _{I} B^{P_n}$ , if $n < \\infty $ .",
"We repeatedly apply Lemma REF .",
"For $n < \\infty $ , recursively define $R_n, P_n, A_n, C_n$ satisfying the following: $R_n$ are invariant decreasing Borel sets such that $n B^{R_n} \\preceq _{I} A^{R_n}$ for any $A,B$ -sensitive $I$ ; $P_n = R_n \\setminus R_{n+1}$ ; $A_n \\subseteq R_{n+1}$ are pairwise disjoint Borel sets such that for any $A,B$ -sensitive $I$ , every $A_n$ respects $I$ and $A_n \\sim _{I} B^{R_{n+1}}$ ; $C_n \\subseteq P_n$ are Borel sets such that for any $A,B$ -sensitive $I$ , every $C_n$ respects $I$ and $C_n \\prec _{I} B^{P_n}$ .",
"Set $R_0 = R$ .",
"Given $R_n$ , $\\lbrace A_k\\rbrace _{k<n}$ satisfying the above properties, let $A^{\\prime } = A^{R_n} \\setminus \\bigcup _{k<n} A_k$ .",
"We apply Lemma REF to $A^{\\prime }$ and $B^{R_n}$ , and get a partition $R_n = P_n \\cup R_{n+1}$ such that $(A^{\\prime })^{P_n} \\prec _{I} B^{P_n}$ and $B^{R_{n+1}} \\preceq _{I} (A^{\\prime })^{R_{n+1}}$ .",
"Set $C_n = (A^{\\prime })^{P_n}$ .",
"Let $A_n \\subseteq (A^{\\prime })^{R_{n+1}}$ be such that $B^{R_{n+1}} \\sim _{I} A_n$ .",
"It is straightforward to check (i)-(iv) are satisfied.",
"Now let $R_{\\infty } = \\bigcap _{n \\in \\mathbb {N}} R_n$ and $C_{\\infty } = (A \\setminus \\bigcup _{n \\in \\mathbb {N}}A_n)^{R_{\\omega }}$ .",
"Now it follows from (i)-(iv) that for all $n \\le \\infty $ , $\\lbrace A_k^{P_n}\\rbrace _{k<n} \\cup \\lbrace C_n\\rbrace $ is a partition of $A^{P_n}$ witnessing $A^{P_n} \\sim _{I} nB \\oplus C_n$ , and for all $n < \\infty $ , $C_n \\prec B^{P_n}$ .",
"For $A,B \\in \\mathfrak {B}(X)$ , let $\\lbrace P_n\\rbrace _{n \\le \\infty }$ be as in the above proposition.",
"Define $[A / B](x) = \\left\\lbrace \\begin{array}{ll}n & \\text{if } x \\in P_n, n < \\infty \\\\\\infty & \\text{if } x \\in P_{\\infty } \\text{ or } x \\in [A]_G \\setminus [B]_G \\\\0 & \\text{if } x \\in [B]_G \\setminus [A]_G \\\\-1 & \\text{otherwise}\\end{array}\\right..$ Note that $[A/B] : X \\rightarrow \\mathbb {N}\\cup \\lbrace -1, \\infty \\rbrace $ is a Borel function by definition.",
"Lemma 3.5 (Infinite divisibility $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ with $[A]_G = [B]_G$ , and let $I$ be a finite Borel partition of $X$ .",
"If $\\infty B \\preceq _{I} A$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} \\infty B \\oplus C$ and let $\\lbrace A_k\\rbrace _{k < \\infty }$ be as in Definition REF .",
"$A_k \\sim _{I} B \\sim _{I} A_{k+1}$ and hence $A_k \\sim _{I} A_{k+1}$ .",
"Also trivially $C \\sim _{I} C$ .",
"Thus, letting $A^{\\prime } = \\bigcup _{k < \\infty } A_{k+1} \\cup C$ , we apply (b) of REF to $A$ and $A^{\\prime }$ , and get that $A \\sim _{I} A^{\\prime }$ .",
"Because $[A \\setminus A^{\\prime }]_G = [A_0]_G = [B]_G = [A]_G$ , we have $A \\prec _{I} A$ .",
"Lemma 3.6 (Ambiguity $\\Rightarrow $ compressibility) Let $A,B \\in \\mathfrak {B}(X)$ and $I$ be a finite Borel partition of $X$ .",
"If $nB \\preceq _{I} A \\prec _{I} nB$ for some $n \\ge 1$ , then $A \\prec _{I} A$ .",
"Let $C \\subseteq A$ be such that $A \\sim _{I} nB \\oplus C$ and let $\\lbrace A_k\\rbrace _{k<n}$ be a partitions of $A \\setminus C$ witnessing $A \\sim _{I} nB \\oplus C$ .",
"Also let $\\lbrace A^{\\prime }_k\\rbrace _{k<n}$ be witnessing $A \\prec _{I} nB$ with $A^{\\prime }_0 \\prec _{I} B$ .",
"Since $A^{\\prime }_k \\preceq _{I} B \\sim _{I} A_k$ , $A^{\\prime }_k \\preceq _{I} A_k$ , for all $k<n$ and $A^{\\prime }_0 \\prec _{I} A_0$ .",
"Note that it follows from the hypothesis that $[A]_G = [B]_G$ and hence $[A_0]_G = [A]_G$ since $[A_0]_G = [B]_G$ .",
"Thus it follows from (b) of REF that $A = \\bigcup _{k<n} A^{\\prime }_k \\prec _{I} \\bigcup _{k<n} A_k \\subseteq A$ .",
"Proposition 3.7 Let $n \\in \\mathbb {N}$ and $A,A^{\\prime },B,P \\in \\mathfrak {B}(X)$ , where $P$ is invariant.",
"$[A/B] \\in \\mathbb {N}$ on $[B]_G$ modulo 3.",
"If $A \\subseteq A^{\\prime }$ , then $[A / B] \\le [A^{\\prime } / B]$ .",
"If $[A/B] = n$ on $P$ then $n B^P \\preceq _{I} A^P \\prec _{I} (n+1) B^P$ , for any finite Borel partition $I$ that is $A,B$ -sensitive.",
"In particular, $n B^P \\preceq _2 A^P \\prec _2 (n+1) B^P$ by taking $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"For $n \\ge 1$ , if $A^P \\prec _i nB^P$ , then $[A/B] < n$ on $P$ modulo ${i+1}$ ; If $A^P \\subseteq [B]_G$ and $nB^P \\preceq _i A^P$ , then $[A/B] \\ge n$ on $P$ modulo ${i+1}$ .",
"For (a), notice that REF and REF imply that $P_{\\infty } \\in 3$ .",
"(b) and (c) follow from the definition of $[A/B]$ .",
"For (d), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $A^P \\prec _{I} nB^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) \\ge n\\rbrace $ .",
"By (c), $nB^Q \\preceq _{I} A^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"For (e), let $I$ be an $A,B$ -sensitive partition of $X$ generated by $i$ Borel sets such that $nB^P \\preceq _{I} A^P$ , and put $Q = \\lbrace x \\in P : [A/B](x) < n\\rbrace $ .",
"By (c), $A^Q \\prec _{I} nB^Q$ .",
"Thus, by Lemma REF , $A^Q \\prec _{I} A^Q$ and hence, by Lemma REF , $[A^Q]_G = Q \\in C_{i+1}$ .",
"Definition 3.8 (Fundamental sequence) A sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections with $F_0 = X$ and $[F_n/F_{n+1}] \\ge 2$ modulo 3 is called fundamental.",
"Proposition 3.9 There exists a fundamental sequence.",
"Take $F_0 = X$ .",
"Given any complete Borel section $F$ , its intersection with every orbit is infinite modulo a smooth set (if the intersection of an orbit with a set is finite, then we can choose an element from each such nonempty intersection in a Borel way and get a Borel transversal).",
"Thus, by REF , $F$ is aperiodic modulo 1.",
"Now use Lemma REF to write $F = A \\cup B, A \\cap B = \\mathbb {\\emptyset }$ , where $A,B$ are also complete sections.",
"Let now $P,Q$ be as in Lemma REF for $A,B$ , and hence $A^P \\prec _2 B^P, B^Q \\preceq _2 A^Q$ because we can take $I= < \\!\\!",
"A,B \\!\\!",
">$ .",
"Let $A^{\\prime } = A^P \\cup B^Q, B^{\\prime } = B^P \\cup A^Q$ .",
"Then $F = A^{\\prime } \\cup B^{\\prime }, A^{\\prime } \\cap B^{\\prime } = \\mathbb {\\emptyset }$ , $A^{\\prime } \\preceq B^{\\prime }$ and $A^{\\prime }$ is also a complete Borel section.",
"By (e) of REF , $[F/A^{\\prime }] \\ge 2$ modulo 3.",
"Iterate this process to inductively define $F_n$ .",
"Fix a fundamental sequence $\\lbrace F_n\\rbrace _{n \\in \\mathbb {N}}$ and for any $A \\in \\mathfrak {B}(X), x \\in X$ , define $m(A,x) = \\lim _{n \\rightarrow \\infty } \\frac{[A/F_n](x)}{[X/F_n](x)}, \\qquad \\mathrm {(\\dagger )}$ if the limit exists, and 0 otherwise.",
"In the above fraction we define ${\\infty \\over \\infty } = 1$ .",
"We will prove in Proposition REF that this limit exists modulo 4.",
"But first we need the following two lemmas.",
"Lemma 3.10 (Almost cancelation) For any $A,B,C \\in X$ , $[A/B][B/C] \\le [A/C] < ([A/B] + 1)([B/C] + 1)$ on $R := [B]_G \\cap [C]_G$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,C \\!\\!",
">$ .",
"$[A/B][B/C] \\le [A/C]$ : Fix integers $i,j > 0$ and let $P = \\lbrace x \\in X : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"Since $i,j > 0$ , $P \\subseteq [A]_G \\cap [B]_G \\cap [C]_G$ and we work in $P$ .",
"By (c) of REF , $i B \\preceq _{I} A$ and $j C \\preceq _{I} B$ .",
"Thus it follows that $ij C \\preceq _{I} A$ and hence $[A / C] \\ge ij$ modulo 4 by (e) of REF .",
"$[A/C] < ([A/B] + 1)([B/C] + 1)$ : By (a) of REF , $[A/C], [A/B], [B/C] \\in \\mathbb {N}$ on $R$ modulo 3.",
"Fix $i,j \\in \\mathbb {N}$ and let $Q = \\lbrace x \\in R : [A/B](x) = i \\wedge [B/C](x) = j\\rbrace $ .",
"We work in $Q$ .",
"By (c) of REF , $A \\prec _{I} (i+1) B$ and $B \\prec _{I} (j+1) C$ .",
"Thus $A \\prec _{I} (i+1)(j+1) C$ and hence $[A/C] < (i+1)(j+1)$ modulo 4 by (d) of REF .",
"Lemma 3.11 For any $A \\in \\mathfrak {B}(A)$ , $\\lim _{n \\rightarrow \\infty } [A / F_n] = \\left\\lbrace \\begin{array}{ll} \\infty & \\text{on } [A]_G \\\\ 0 & \\text{on } X \\setminus [A]_G \\end{array}\\right., \\text{ modulo } 4.$ The part about $X \\setminus [A]_E$ is clear, so work in $[A]_E$ , i.e.",
"assume $X = [A]_G$ .",
"By (a) of REF and REF , we have $\\infty > [F_1 /A] \\ge [F_1 /F_n ] [F_n / A ] \\ge 2^{n-1} [F_n /A], \\text{ modulo } 4,$ which holds for all $n$ at once since 4 is a $\\sigma $ -ideal.",
"Thus $[F_n /A] \\rightarrow 0$ modulo 4 and hence, as $[F_n /A] \\in \\mathbb {N}$ , $[F_n /A]$ is eventually 0, modulo 4.",
"So if $B_k := \\lbrace x \\in [A]_G : [F / A](x) = 0\\rbrace ,$ then $B_k \\nearrow X$ , modulo 4.",
"Now it follows from Lemma REF that $[A / F_k] > 0$ on $B_k$ modulo 4.",
"But $[A/F_{k+n}] \\ge [A / F_k ] [F_k / F_{k+n} ] \\ge 2^n [A / F_k ], \\text{ modulo } 4,$ so for every $k$ , $[A/F_n] \\rightarrow \\infty $ on $B_k$ modulo 4.",
"Since $B_k \\nearrow X$ modulo 4, we have $[A/F_n] \\rightarrow \\infty $ on $X$ , modulo 4.",
"Proposition 3.12 For any Borel set $A \\subseteq X$ , the limit in ($\\dagger $ ) exists and is positive on $[A]_G$ , modulo 4.",
"Claim Suppose $B,C \\in \\mathfrak {B}(X)$ , $i \\in \\mathbb {N}$ and $D_i = \\lbrace x \\in X : [C / F_i](x) > 0\\rbrace $ .",
"Then $\\overline{\\lim } {[B/F_n ] \\over [C/F_n]} \\le {[B/F_i] + 1 \\over [C/F_i]}$ on $D_i$ , modulo 4.",
"Proof of Claim.",
"Working in $D_i$ and using Lemma REF , $\\forall j$ we have (modulo 4) $[B/F_{i+j}] & \\le ([B/F_i ]+1) ([F_i / F_{i+j}] + 1) \\\\[C/F_{i+j}] & \\ge [C/F_i ] [ F_i /F_{i+j}] > 0,$ so ${[B/F_{i+j}]\\over [C/F_{i+j}]} & \\le {[B/F_i ]+1 \\over [C/F_i ]}\\cdot {[F_i /F_{i+j}]+1\\over [F_i /F_{i+j}]} \\\\& \\le {[B/F_i ]+1\\over [C/F_i ]} \\cdot (1 + {1\\over 2^j}),$ from which the claim follows.",
"$\\dashv $ Applying the claim to $B = A$ and $C = X$ (hence $D_i = X$ ), we get that for all $i \\in \\mathbb {N}$ $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ](x)\\over [X/F_n ](x)} \\le {[A/F_i ](x)+1 \\over [X/F_i ](x)} (\\text{modulo } 4).$ Thus $\\overline{\\lim _{n \\rightarrow \\infty }} {[A/F_n ]\\over [X/F_n ]} \\le \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i]+1\\over [X/F_i ]} = \\underline{\\lim _{i \\rightarrow \\infty }} {[A/F_i ]\\over [X/F_i ]}$ since $\\lim _{i \\rightarrow \\infty } {1 \\over [X/F_i]} = 0$ .",
"To see that $m(A,x)$ is positive on $[A]_E$ modulo 4 we argue as follows.",
"We work in $[A]_G$ .",
"Applying the above claim to $B = X$ and $C = A$ , we get ${1 \\over m(A,x)} = \\lim _{n \\rightarrow \\infty } {[X/F_n] \\over [A/F_n]} \\le {[X/F_i] + 1 \\over [A/F_i]} < \\infty \\text{ on } D_i \\text{ (modulo $4$)}.$ Thus $m(A,x) > 0$ on $\\cup _{i \\in \\mathbb {N}} D_i$ , modulo 4.",
"But $D_i \\nearrow [A]_G$ because $[A/F_i] \\rightarrow \\infty $ as $i \\rightarrow \\infty $ , and hence $m(A,x) > 0$ on $[A]_G$ modulo 4.",
"Lemma 3.13 (Invariance) For $A,F \\in \\mathfrak {B}(X)$ , $\\forall g \\in G, [A / F] = [gA / F]$ , modulo 3.",
"We may assume that $X = [A]_G \\cap [F]_G$ .",
"Fix $g \\in G$ , $n \\in \\mathbb {N}$ , and put $Q = \\lbrace x \\in X : [gA / F](x) = n\\rbrace $ .",
"We work in $Q$ .",
"Let $I= < \\!\\!",
"A,F \\!\\!",
">$ and hence $A,gA,F$ respect $I$ .",
"By (c) of REF , $nF \\preceq _{I} gA$ .",
"But clearly $gA \\sim _{I} A$ and hence $nF \\preceq _{I} A$ .",
"Thus, by (e) of REF , $[A / F] \\ge n = [gA / F]$ , modulo 3.",
"By symmetry, $[gA / F] \\ge [A / F]$ (modulo 3) and the lemma follows.",
"Lemma 3.14 (Almost additivity) For any $A,B,F \\in X$ with $A \\cap B = \\mathbb {\\emptyset }$ , $[A/F] + [B/F] \\le [A \\cup B / F] \\le [A/F] + [B/F] + 1$ modulo 4.",
"Let $I= < \\!\\!",
"A,B,F \\!\\!",
">$ .",
"$[A/F] + [B/F] \\le [A \\cap B / F]$ : Fix $i,j \\in \\mathbb {N}$ not both 0, say $i>0$ , and let $S = \\lbrace x \\in X : [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"Since $i>0$ , $S \\subseteq [A]_G \\cap [F]_G$ and we work in $S$ .",
"By (c) of REF , $iF^S \\preceq _{I} A^S$ and $jF^S \\preceq _{I} B^S$ .",
"Hence $(i+j)F^S \\preceq _{I} (A \\cup B)^S$ and thus, by (e) of REF , $[A \\cup B / F] \\ge i + j$ , modulo 4.",
"$[A \\cap B / F] \\le [A/F] + [B/F] + 1$ : Outside $[F]_G$ , the inequality clearly holds.",
"Fix $i,j \\in \\mathbb {N}$ and let $M = \\lbrace x \\in [F]_G: [A/F](x) = i \\wedge [B/F](x) = j\\rbrace $ .",
"We work in $M$ .",
"By (c) of REF , $A \\prec _{I} (i+1)F$ and $B \\prec _{I} (j+1)F$ .",
"Thus it is clear that $A \\cup B \\prec _{I} (i + j + 2)F$ and hence $[A \\cup B / F] < i+j+2$ , modulo 4, by (d) of REF .",
"Now we are ready to finish the proof of Theorem REF .",
"Fix $A,B \\in \\mathfrak {B}(X)$ .",
"The fact that $m(A,x) \\in [0,1]$ and parts (b) and (d) follow directly from the definition of $m(A,x)$ .",
"Part (a) follows from the fact that $[A / F_n]$ is Borel for all $n \\in \\mathbb {N}$ .",
"(c) follows from (b) of Lemma REF , and (e) and (f) are asserted by REF and REF , respectively.",
"To show (g), we argue as follows.",
"By Lemma REF , $[A/F_n] + [B/F_n] \\le [A \\cup B / F_n] \\le [A/F_n] + [B/F_n] + 1$ , modulo 4, and thus $\\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} \\le \\frac{[A \\cup B / F_n]}{[X/F_n]} \\le \\frac{[A/F_n]}{[X/F_n]} + \\frac{[B/F_n]}{[X/F_n]} + \\frac{1}{[X/F_n]},$ for all $n$ at once, modulo 4 (using the fact that 4 is a $\\sigma $ -ideal).",
"Since $[X/F_n] \\ge 2^n$ , passing to the limit in the inequalities above, we get $m(A,x) + m(B,x) \\le m(A \\cup B,x) \\le m(A,x) + m(B,x)$ .",
"QED (Thm REF ) Theorem REF will only be used via Corollary REF and to state it we need the following.",
"Definition 3.15 Let $X$ be a Borel $G$ -space.",
"$\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ is called a Boolean $G$ -algebra, if it is a Boolean algebra, i.e.",
"is closed under finite unions and complements, and is closed under the $G$ -action, i.e.",
"$G \\mathcal {B}= \\mathcal {B}$ .",
"Corollary 3.16 Let $X$ be a Borel $G$ -space and let $\\mathcal {B}\\subseteq \\mathfrak {B}(X)$ be a countable Boolean $G$ -algebra.",
"For any $A \\in \\mathcal {B}$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, $\\mu $ can be taken such that there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B)=0$ .",
"Let $A \\in \\mathcal {B}$ be such that $A \\notin 4$ .",
"We may assume that $X = [A]_G$ by setting the (to be constructed) measure to be 0 outside $[A]_G$ .",
"If $X$ is not aperiodic, then by assigning equal point masses to the points of a finite orbit, we will have a probability measure on all of $\\mathfrak {B}(X)$ , so assume $X$ is aperiodic.",
"Since 4 is a $\\sigma $ -ideal and $\\mathcal {B}$ is countable, Theorem REF implies that there is a $P \\in 4$ such that (a)-(g) of the same theorem hold on $X \\setminus P$ for all $A,B \\in \\mathcal {B}$ .",
"Since $A \\notin 4$ , there exists $x_A \\in A \\setminus P$ .",
"Hence, letting $\\mu (B) = m(B,x_A)$ for all $B \\in \\mathcal {B}$ , conditions (b),(f) and (g) imply that $\\mu $ is a $G$ -invariant finitely additive probability measure on $\\mathcal {B}$ .",
"Moreover, since $x_A \\in [A]_G \\setminus P$ , $\\mu (A) = m(A, x_A) > 0$ .",
"Finally, the last assertion follows from condition (d).",
"Corollary 3.17 Let $X$ be a Borel $G$ -space.",
"For every Borel set $A \\subseteq X$ with $A \\notin 4$ , there exists a $G$ -invariant finitely additive Borel probability measure $\\mu $ (defined on all Borel sets) with $\\mu (A)>0$ .",
"The statement follows from REF and a standard application of the Compactness Theorem of propositional logic.",
"Here are the details.",
"We fix the following set of propositional variables $\\mathcal {P}= \\lbrace P_{A,r} : A \\in \\mathfrak {B}(X), r \\in [0,1]\\rbrace ,$ with the following interpretation in mind: $P_{A,r} \\Leftrightarrow \\text{``the measure of $A$ is $\\ge r$''}.$ Define the theory $T$ as the following set of sentences: for each $A,B \\in \\mathfrak {B}(X)$ , $r,s \\in [0,1]$ and $g \\in G$ , “$P_{A,0}$ ”$\\in T$ ; if $r > 0$ , then “$\\lnot P_{\\mathbb {\\emptyset }, r}$ ”$\\in T$ ; if $s \\ge r$ , then “$P_{A,s} \\rightarrow P_{A,r}$ ”$\\in T$ ; if $A \\cap B = \\mathbb {\\emptyset }$ , then “$(P_{A,r} \\wedge P_{B,s}) \\rightarrow P_{A \\cup B, r+s}$ ”, “$(\\lnot P_{A,r} \\wedge \\lnot P_{B,s}) \\rightarrow \\lnot P_{A \\cup B, r+s}$ ”$\\in T$ ; “$P_{X,1}$ ”$\\in T$ ; “$P_{A,r} \\rightarrow P_{gA,r}$ ”$\\in T$ .",
"If there is an assignment of the variables in $\\mathcal {P}$ satisfying $T$ , then for each $A \\in \\mathfrak {B}(X)$ , we can define $\\mu (A) = \\sup \\lbrace r \\in [0,1] : P_{A,r}\\rbrace .$ Note that due to (i), $\\mu $ is well defined for all $A \\in \\mathfrak {B}(X)$ .",
"In fact, it is straightforward to check that $\\mu $ is a finitely additive $G$ -invariant probability measure.",
"Thus, we only need to show that $T$ is satisfiable, for which it is enough to check that $T$ is finitely satisfiable, by the Compactness Theorem of propositional logic (or by Tychonoff's theorem).",
"Let $T_0 \\subseteq T$ be finite and let $\\mathcal {P}_0$ be the set of propositional variables that appear in the sentences in $T_0$ .",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by the sets that appear in the indices of the variables in $\\mathcal {P}_0$ .",
"By REF , there is a finitely additive $G$ -invariant probability measure $\\mu $ defined on $\\mathcal {B}$ .",
"Consider the following assignment of the variables in $\\mathcal {P}_0$ : for all $P_{A,r} \\in \\mathcal {P}_0$ , $P_{A,r} :\\Leftrightarrow \\mu (A) \\ge r.$ It is straightforward to check that this assignment satisfies $T_0$ , and hence, $T$ is finitely satisfiable.",
"Finite generators in the case of $\\sigma $ -compact spaces In this section we prove that the answer to Question REF is positive in case $X$ has a $\\sigma $ -compact realization.",
"To do this, we first prove Proposition REF , which shows how to construct a countably additive invariant probability measure on $X$ using a finitely additive one.",
"We then use REF to conclude the result.",
"For the next two statements, let $X$ be a second countable Hausdorff topological space equipped with a continuous action of $G$ .",
"Lemma 4.1 Let $\\mathcal {U}\\subseteq Pow(X)$ be a countable base for $X$ closed under the $G$ -action and finite unions/intersections.",
"Let $\\rho $ be a $G$ -invariant finitely additive probability measure on the $G$ -algebra generated by $\\mathcal {U}$ .",
"For every $A \\subseteq X$ , define $\\mu ^* (A) = \\inf \\lbrace \\sum _{n \\in \\mathbb {N}} \\rho (U_n ) : U_n \\in \\mathcal {U}\\; \\wedge \\; A \\subseteq \\bigcup _{n \\in \\mathbb {N}} U_n\\rbrace .$ Then: $\\mu ^*$ is a $G$ -invariant outer measure.",
"If $K \\subseteq X$ is compact, then $K$ is metrizable and $\\mu ^*$ is a metric outer measure on $K$ (with respect to any compatible metric).",
"It is a standard fact from measure theory that $\\mu ^*$ is an outer measure.",
"That $\\mu ^*$ is $G$ -invariant follows immediately from $G$ -invariance of $\\rho $ and the fact that $\\mathcal {U}$ is closed under the action of $G$ .",
"For (b), first note that by Urysohn metrization theorem, $K$ is metrizable, and fix a metric on $K$ .",
"If $E, F \\subseteq K$ are a positive distance apart, then so are $\\bar{E}$ and $\\bar{F}$ .",
"Hence there exist disjoint open sets $U,V$ such that $\\bar{E} \\subseteq U$ , $\\bar{F} \\subseteq V$ .",
"Because $\\bar{E}$ and $\\bar{F}$ are compact, $U,V$ can be taken to be finite unions of sets in $\\mathcal {U}$ and therefore $U,V \\in \\mathcal {U}$ .",
"Now fix $\\epsilon >0$ and let $W_n \\in \\mathcal {U}$ , be such that $E \\cup F \\subseteq \\bigcup _n W_n$ and $\\sum _n \\rho (W_n) \\le \\mu ^*(E \\cup F) + \\epsilon \\le \\mu ^*(E) + \\mu ^*(F) +\\epsilon .\\qquad \\mathrm {{(*)}}$ Note that $\\lbrace W_n \\cap U\\rbrace _{n \\in \\mathbb {N}}$ covers $E$ , $\\lbrace W_n \\cap V\\rbrace _{n \\in \\mathbb {N}}$ covers $F$ and $W_n \\cap U, W_n \\cap V \\in \\mathcal {U}$ .",
"Also, by finite additivity of $\\rho $ , $\\rho (W_n \\cap U) + \\rho (W_n \\cap V) = \\rho (W_n \\cap (U \\cup V)) \\le \\rho (W_n).$ Thus $\\mu ^*(E) + \\mu ^*(F) \\le \\sum _n \\rho (W_n \\cap U) + \\sum _n \\rho (W_n \\cap V) \\le \\sum _n \\rho (W_n),$ which, together with ($*$ ), implies that $\\mu ^*(E \\cup F) = \\mu ^*(E) + \\mu ^*(F)$ since $\\epsilon $ is arbitrary.",
"Proposition 4.2 Suppose there exist a countable base $\\mathcal {U}\\subseteq Pow(X)$ for $X$ and a compact set $K \\subseteq X$ such that the $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ admits a finitely additive $G$ -invariant probability measure $\\rho $ with $\\rho (K)>0$ .",
"Then there exists a countably additive $G$ -invariant Borel probability measure on $X$ .",
"Let $K, \\mathcal {U}$ and $\\rho $ be as in the hypothesis.",
"We may assume that $\\mathcal {U}$ is closed under the $G$ -action and finite unions/intersections.",
"Let $\\mu ^*$ be the outer measure provided by Lemma REF applied to $\\mathcal {U}$ , $\\rho $ .",
"Thus $\\mu ^*$ is a metric outer measure on $K$ and hence all Borel subsets of $K$ are $\\mu ^*$ -measurable (see 13.2 in ).",
"This implies that all Borel subsets of $Y = [K]_G = \\bigcup _{g \\in G} gK$ are $\\mu ^*$ -measurable because $\\mu ^*$ is $G$ -invariant.",
"By Carathéodory's theorem, the restriction of $\\mu ^*$ to the Borel subsets of $Y$ is a countably additive Borel measure on $Y$ , and we extend it to a Borel measure $\\mu $ on $X$ by setting $\\mu (Y^c) = 0$ .",
"Note that $\\mu $ is $G$ -invariant and $\\mu (Y) \\le 1$ .",
"It remains to show that $\\mu $ is nontrivial, which we do by showing that $\\mu (K) \\ge \\rho (K)$ and hence $\\mu (K)>0$ .",
"To this end, let $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq \\mathcal {U}$ cover $K$ .",
"Since $K$ is compact, there is a finite subcover $\\lbrace U_n\\rbrace _{n < N}$ .",
"Thus $U := \\bigcup _{n < N} U_n \\in \\mathcal {U}$ and $K \\subseteq U$ .",
"By finite additivity of $\\rho $ , we have $\\sum _{n \\in \\mathbb {N}} \\rho (U_n) \\ge \\sum _{n < N} \\rho (U_n) \\ge \\rho (U) \\ge \\rho (K),$ and hence, it follows from the definition of $\\mu ^*$ that $\\mu ^*(K) \\ge \\rho (K)$ .",
"Thus $\\mu (K) = \\mu ^*(K) > 0$ .",
"Corollary 4.3 Let $X$ be a second countable Hausdorff topological $G$ -space whose Borel structure is standard.",
"For every compact set $K \\subseteq X$ not in 4, there is a $G$ -invariant countably additive Borel probability measure $\\mu $ on $X$ with $\\mu (K) > 0$ .",
"Fix any countable base $\\mathcal {U}$ for $X$ and let $\\mathcal {B}$ be the Boolean $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ .",
"By Corollary REF , there exists a $G$ -invariant finitely additive probability measure $\\rho $ on $\\mathcal {B}$ such that $\\rho (K) > 0$ .",
"Now apply REF .",
"As a corollary, we derive the analogue of Nadkarni's theorem for 4 in case of $\\sigma $ -compact spaces.",
"Corollary 4.4 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"$X \\notin 4$ if and only if there exists a $G$ -invariant countably additive Borel probability measure on $X$ .",
"$\\Leftarrow $ : If $X \\in 4$ , then it is compressible in the usual sense and hence does not admit a $G$ -invariant Borel probability measure.",
"$\\Rightarrow $ : Suppose that $X$ is a $\\sigma $ -compact topological $G$ -space and $X \\notin 4$ .",
"Then, since $X$ is $\\sigma $ -compact and 4 is a $\\sigma $ -ideal, there is a compact set $K$ not in 4.",
"Now apply REF .",
"Remark.",
"For a Borel $G$ -space $X$ , let $\\mathcal {K}$ denote the collection of all subsets of invariant Borel sets that admit a $\\sigma $ -compact realization (when viewed as Borel $G$ -spaces).",
"Also, let $ denote the collection of all subsets of invariant compressible Borel sets.",
"It is clear that $ K$ and $ are $\\sigma $ -ideals, and what REF implies is that $\\mathcal {K}\\subseteq 4$ .",
"The question of whether $\\mathcal {K}= Pow(X)$ is just a rephrasing of $§10$ .",
"(B) of Introduction.",
"Theorem 4.5 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"If there is no $G$ -invariant Borel probability measure on $X$ , then $X$ admits a Borel 32-generator.",
"By REF , $X \\in 4$ and hence, $X$ is 4-compressible.",
"Thus, by Proposition REF , $X$ admits a Borel $2^5$ -generator.",
"Example 4.6.",
"Let $LO \\subseteq 2^{\\mathbb {N}^2}$ denote the Polish space of all countable linear orderings and let $G$ be the group of finite permutations of elements of $\\mathbb {N}$ .",
"$G$ is countable and acts continuously on $LO$ in the natural way.",
"Put $X = LO \\setminus DLO$ , where $DLO$ denotes the set of all dense linear orderings without endpoints (copies of $\\mathbb {Q}$ ).",
"It is straightforward to see that $DLO$ is a $G_{\\delta }$ subset of $LO$ and hence, $X$ is $F_{\\sigma }$ .",
"Therefore, $X$ is in fact $\\sigma $ -compact since $LO$ is compact being a closed subset of $2^{\\mathbb {N}^2}$ .",
"Also note that $X$ is $G$ -invariant.",
"Let $\\mu $ be the unique measure on $LO$ defined by $\\mu (V_{(F,<)}) = {1 \\over n!",
"}$ , where $(F,<)$ is a finite linearly ordered subset of $\\mathbb {N}$ of cardinality $n$ and $V_{(F,<)}$ is the set of all linear orderings of $\\mathbb {N}$ extending the order $<$ on $F$ .",
"As shown in , $\\mu $ is the unique invariant measure for the action of $G$ on $LO$ and $\\mu (X) = 0$ .",
"Thus there is no $G$ -invariant Borel probability measure on $X$ and hence, by the above theorem, $X$ admits a Borel 32-generator.",
"Finitely traveling sets Let $X$ be a Borel $G$ -space.",
"Definition 5.1 Let $A,B \\in \\mathfrak {B}(X)$ be equidecomposable, i.e.",
"there are $N \\le \\infty $ , $\\lbrace g_n\\rbrace _{n < N} \\subseteq G$ and Borel partitions $\\lbrace A_n\\rbrace _{n < N}$ and $\\lbrace B_n\\rbrace _{n < N}$ of $A$ and $B$ , respectively, such that $g_n A_n = B_n$ for all $n < N$ .",
"$A,B$ are said to be locally finitely equidecomposable (denote by $A \\sim _{\\text{lfin}} B$ ), if $\\lbrace A_n\\rbrace _{n < N},\\lbrace B_n\\rbrace _{n < N},\\lbrace g_n\\rbrace _{n < N}$ can be taken so that for every $x \\in A$ , $A_n \\cap [x]_G = \\mathbb {\\emptyset }$ for all but finitely many $n<N$ ; finitely equidecomposable (denote by $A \\sim _{\\text{fin}} B$ ), if $N$ can be taken to be finite.",
"The notation $\\prec _{\\text{fin}}$ , $\\prec _{\\text{lfin}}$ and the notions of finite and locally finite compressibility are defined analogous to Definitions REF and REF .",
"Definition 5.2 A Borel set $A \\subseteq X$ is called (locally) finitely traveling if there exists pairwise disjoint Borel sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ such that $A_0 = A$ and $A \\sim _{\\text{fin}} A_n$ ($A \\sim _{\\text{lfin}} A_n$ ), $\\forall n \\in \\mathbb {N}$ .",
"Proposition 5.3 If $X$ is (locally) finitely compressible then $X$ admits a (locally) finitely traveling Borel complete section.",
"We prove for finitely compressible $X$ , but note that everything below is also locally valid (i.e.",
"restricted to every orbit) for a locally compressible $X$ .",
"Run the proof of the first part of Lemma REF noting that a witnessing map $\\gamma : X \\rightarrow G$ of finite compressibility of $X$ has finite image and hence the image of each $\\delta _n$ (in the notation of the proof) is finite, which implies that the obtained traveling set $A$ is actually finitely traveling.",
"Proposition 5.4 If $X$ admits a locally finitely traveling Borel complete section, then $X \\in 4$ .",
"Let $A$ be a locally finitely traveling Borel complete section and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"Let $I_n = \\lbrace C_k^n\\rbrace _{k \\in \\mathbb {N}}$ , $J_n = \\lbrace D_k^n\\rbrace _{k \\in \\mathbb {N}}$ be Borel partitions of $A$ and $A_n$ , respectively, that together with $\\lbrace g_k^n\\rbrace _{k \\in \\mathbb {N}} \\subseteq G$ witness $A \\sim _{\\text{lfin}} A_n$ (as in Definition REF ).",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by $\\lbrace X\\rbrace \\cup \\bigcup _{n \\in \\mathbb {N}} (I_n \\cup J_n \\cup \\lbrace A_n\\rbrace )$ .",
"Now assume for contradiction that $X \\notin 4$ and hence, $A \\notin 4$ .",
"Thus, applying Corollary REF to $A$ and $\\mathcal {B}$ , we get a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B) = 0$ .",
"Claim $\\mu (A_n) = \\mu (A)$ , for all $n \\in \\mathbb {N}$ .",
"Proof of Claim.",
"For each $n$ , let $\\lbrace C_{k_i}^n\\rbrace _{i < K_n}$ be the list of those $C_k^n$ such that $C_k^n \\cap [x]_G \\ne \\mathbb {\\emptyset }$ ($K_n < \\infty $ by the definition of locally finitely traveling).",
"Set $B = A \\setminus (\\bigcup _{i < K_n} C_{k_i}^n)$ and note that by finite additivity of $\\mu $ , $\\mu (A) = \\mu (B) + \\sum _{i < K_n} \\mu (C_{k_i}^n).$ Similarly, set $B^{\\prime } = A_n \\setminus (\\bigcup _{i < K_n} D_{k_i}^n)$ and hence $\\mu (A_n) = \\mu (B^{\\prime }) + \\sum _{i < K_n} \\mu (D_{k_i}^n).$ But $B \\cap [x]_G = \\mathbb {\\emptyset }$ and $B^{\\prime } \\cap [x]_G = \\mathbb {\\emptyset }$ , and thus $\\mu (B) = \\mu (B^{\\prime }) = 0$ .",
"Also, since $g_{k_i}^n C_{k_i}^n = D_{k_i}^n$ and $\\mu $ is $G$ -invariant, $\\mu (C_{k_i}^n) = \\mu (D_{k_i}^n)$ .",
"Therefore $\\mu (A) = \\sum _{i < K_n} \\mu (C_{k_i}^n) = \\sum _{i < K_n} \\mu (D_{k_i}^n) = \\mu (A_n).$ $\\dashv $ This claim contradicts $\\mu $ being a probability measure since for large enough $N$ , $\\mu (\\bigcup _{n < N} A_n) = N \\mu (A) > 1$ , contradicting $\\mu (X) = 1$ .",
"This, together with REF , implies the following.",
"Corollary 5.5 Let $X$ be a Borel $G$ -space.",
"If $X$ admits a locally finitely traveling Borel complete section, then there is a Borel 32-generator.",
"Separating smooth-many invariant sets Assume throughout that $X$ is a Borel $G$ -space.",
"Lemma 6.1 If $X$ is aperiodic then it admits a countably infinite partition into Borel complete sections.",
"The following argument is also given in the proof of Theorem 13.1 in .",
"By the marker lemma (see 6.7 in ), there exists a vanishing sequence $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections, i.e.",
"$\\bigcap _{n \\in \\mathbb {N}} B_n = \\mathbb {\\emptyset }$ .",
"For each $n \\in \\mathbb {N}$ , define $k_n : X \\rightarrow \\mathbb {N}$ recursively as follows: $\\left\\lbrace \\begin{array}{rcl}k_0(x) & = & 0 \\\\k_{n+1}(x) &= & min \\lbrace k \\in \\mathbb {N}: B_{k_n(x)} \\cap [x]_G \\nsubseteq B_k\\rbrace \\end{array}\\right.,$ and define $A_n \\subseteq X$ by $x \\in A_n \\Leftrightarrow x \\in A_{k_n(x)} \\setminus A_{k_{n+1}(x)}.$ It is straightforward to check that $A_n$ are pairwise disjoint Borel complete sections.",
"For $A \\in \\mathfrak {B}(X)$ , if $I= < \\!\\!",
"A \\!\\!",
">$ then we use the notation $F_A$ and $f_A$ instead of $F_{I}$ and $, respectively.$ We now work towards strengthening the above lemma to yield a countably infinite partition into $F_A$ -invariant Borel complete sections.",
"Definition 6.2 (Aperiodic separation) For Borel sets $A, Y \\subseteq X$ , we say that $A$ aperiodically separates $Y$ if $f_A([Y]_G)$ is aperiodic (as an invariant subset of the shift $2^G$ ).",
"If such $A$ exists, we say that $Y$ is aperiodically separable.",
"Proposition 6.3 For $A \\in \\mathfrak {B}(X)$ , if $A$ aperiodically separates $X$ , then $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Let $Y = \\lbrace y \\in 2^G : |[y]_G| = \\infty \\rbrace $ and hence $f_A(X)$ is a $G$ -invariant subset of $Y$ .",
"By Lemma REF applied to $Y$ , there is a partition $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $Y$ into Borel complete sections.",
"Thus $A_n = f_{I}^{-1}(B_n)$ is a Borel $F_A$ -invariant complete section for $X$ and $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $X$ .",
"Let $\\mathfrak {A}$ denote the collection of all subsets of aperiodically separable Borel sets.",
"Lemma 6.4 $\\mathfrak {A}$ is a $\\sigma $ -ideal.",
"We only have to show that if $Y_n$ are aperiodically separable Borel sets, then $Y = \\bigcup _{n \\in \\mathbb {N}} Y_n \\in \\mathfrak {A}$ .",
"Let $A_n$ be a Borel set aperiodically separating $Y_n$ .",
"Since $A_n$ also aperiodically separates $[Y_n]_G$ (by definition), we can assume that $Y_n$ is $G$ -invariant.",
"Furthermore, by taking $Y_n^{\\prime } = Y_n \\setminus \\bigcup _{k<n} Y_k$ , we can assume that $Y_n$ are pairwise disjoint.",
"Now letting $A = \\bigcup _{n \\in \\mathbb {N}} (A_n \\cap Y_n)$ , it is easy to check that $A$ aperiodically separates $Y$ .",
"Let $\\mathfrak {S}$ denote the collection of all subsets of smooth sets.",
"By a similar argument as the one above, $\\mathfrak {S}$ is a $\\sigma $ -ideal.",
"Lemma 6.5 If $X$ is aperiodic, then $\\mathfrak {S}\\subseteq \\mathfrak {A}$ .",
"Let $S \\in \\mathfrak {S}$ and hence there is a Borel transversal $T$ for $[S]_G$ .",
"Fix $x \\in S$ and let $y \\ne z \\in [x]_G$ .",
"Since $T$ is a transversal, there is $g \\in G$ such that $gy \\in T$ , and hence $gz \\notin T$ .",
"Thus $f_T(y) \\ne f_T(z)$ , and so $f_T([x]_G)$ is infinite.",
"Therefore $T$ aperiodically separates $[S]_G$ .",
"For the rest of the section, fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and let $F_A^n$ be following equivalence relation: $y F_A^n z \\Leftrightarrow \\forall k < n (g_k y \\in A \\leftrightarrow g_k z \\in A).$ Note that $F_A^n$ has no more than $2^n$ equivalence classes and that $y F_A z$ if and only if $\\forall n (y F_A^n z)$ .",
"Lemma 6.6 For $A,Y \\in \\mathfrak {B}(X)$ , $A$ aperiodically separates $Y$ if and only if $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in Y^{[x]_G}) [y F_A^n z \\wedge \\lnot (y F_A z)]$ .",
"$\\Rightarrow $ : Assume that for all $x \\in Y$ , $f_A([x]_G)$ is infinite and thus $F_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ has infinitely many equivalence classes.",
"Fix $n \\in \\mathbb {N}$ and recall that $F_A^n$ has only finitely many equivalence classes.",
"Thus, by the Pigeon Hole Principle, there are $y,z \\in Y^{[x]_G}$ such that $y F_A^n z$ yet $\\lnot (y F_A z)$ .",
"$\\Leftarrow $ : Assume for contradiction that $f_A(Y^{[x]_G})$ is finite for some $x \\in Y$ .",
"Then it follows that $F_A = F_A^n$ , for some $n$ , and hence for any $y,z \\in Y^{[x]_G}$ , $y F_A^n z$ implies $y F_A z$ , contradicting the hypothesis.",
"Theorem 6.7 If $X$ is an aperiodic Borel $G$ -space, then $X \\in \\mathfrak {A}$ .",
"By Lemma REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into Borel complete sections.",
"We will inductively construct Borel sets $B_n \\subseteq C_n$ , where $C_n$ should be thought of as the set of points colored (black or white) at the $n^{th}$ step, and $B_n$ as the set of points colored black (thus $C_n \\setminus B_n$ is colored white).",
"Define a function $\\# : X \\rightarrow \\mathbb {N}$ by $x \\mapsto m$ , where $m$ is such that $x \\in A_m$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ of sets generating the Borel $\\sigma $ -algebra of $X$ .",
"Assuming that for all $k < n$ , $C_k, B_k$ are defined, let $\\bar{C}_n = \\bigcup _{k<n} C_k$ and $\\bar{B}_n = \\bigcup _{k<n} B_k$ .",
"Put $P_n = \\lbrace x \\in A_0 : \\forall k < n (g_k x \\in \\bar{C}_n) \\wedge g_n x \\notin \\bar{C}_n\\rbrace $ and set $F_n = F_{\\bar{B}_n}^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ , that is for all $x,y \\in P_n$ , $y F_n z \\Leftrightarrow \\forall k < n (g_k y \\in \\bar{B}_n \\leftrightarrow g_k z \\in \\bar{B}_n).$ Now put $C^{\\prime }_n = \\lbrace x \\in P_n : \\#(g_n x) = \\min \\#((g_nP_n)^{[x]_G})\\rbrace $ , $C^{\\prime \\prime }_n = \\lbrace x \\in C^{\\prime }_n : \\exists y, z \\in (C^{\\prime }_n)^{[x]_G} (y \\ne z \\wedge y F_n z)\\rbrace $ and $C_n = g_n C^{\\prime \\prime }_n$ .",
"Note that it follows from the definition of $P_n$ that $C_n$ is disjoint from $\\bar{C}_n$ .",
"Now in order to define $B_n$ , first define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{ the smallest $m$ such that there are } y,z \\in C^{\\prime \\prime }_n \\cap [x]_G \\text{ with } y F_n z, y \\in U_m \\text{ and } z \\notin U_m.$ Note that $\\bar{n}$ is Borel and $G$ -invariant.",
"Lastly, let $B^{\\prime }_n = \\lbrace x \\in C^{\\prime \\prime }_n : x \\in U_{\\bar{n}(x)}\\rbrace $ and $B_n = g_n B^{\\prime }_n$ .",
"Clearly $B_n \\subseteq C_n$ .",
"Now let $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $D = \\left[\\bigcup _{n \\in \\mathbb {N}} (C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)\\right]_G$ .",
"We show that $B$ aperiodically separates $Y := X \\setminus D$ and $D \\in \\mathfrak {S}$ .",
"Since $\\mathfrak {S}\\subseteq \\mathfrak {A}$ and $\\mathfrak {A}$ is an ideal, this will imply that $X \\in \\mathfrak {A}$ .",
"Claim 1 $D \\in \\mathfrak {S}$ .",
"Proof of Claim.",
"Since $\\mathfrak {S}$ is a $\\sigma $ -ideal, it is enough to show that for each $n$ , $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G \\in \\mathfrak {S}$ , so fix $n \\in \\mathbb {N}$ .",
"Clearly $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ is finite, for all $x \\in X$ , since there can be at most $2^n$ pairwise $F_n$ -nonequivalent points.",
"Thus, fixing some Borel linear ordering of $X$ and taking the smallest element from $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ for each $x \\in C^{\\prime }_n \\setminus C^{\\prime \\prime }_n$ , we can define a Borel transversal for $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G$ .",
"$\\dashv $ By Lemma REF , to show that $B$ aperiodically separates $Y$ , it is enough to show that $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in [x]_G) [y F_B^n z \\wedge \\lnot (y F_B z)]$ .",
"Fix $x \\in Y$ .",
"Claim 2 $(\\exists ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $(\\forall ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $x \\notin D$ , it follows that $(\\forall ^{\\infty } n) P_n^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $A_0$ is a complete section and $\\bar{C}_0 = \\mathbb {\\emptyset }$ , $P_0^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Let $N$ be the largest number such that $P_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus for all $n > N$ , $C_n^{[x]_G} = \\mathbb {\\emptyset }$ and hence for all $n > N$ , $\\bar{C}_n^{[x]_G} = \\bar{C}_{N+1}^{[x]_G}$ .",
"Because $C_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ , there is $y \\in A_0^{[x]_G}$ such that $\\forall k \\le N (g_k y \\in \\bar{C}_{N+1})$ ; but because $P_{N+1}^{[x]_G} = \\mathbb {\\emptyset }$ , $g_{N+1} y$ must also fall into $\\bar{C}_{N+1}$ .",
"By induction on $n > N$ , we get that for all $n>N$ , $g_n y \\in \\bar{C}_n$ and thus $g_n y \\in \\bar{C}_{N+1}$ .",
"On the other hand, it follows from the definition of $C^{\\prime }_n$ that for each $n$ , $(C^{\\prime }_n)^{[x]_G}$ intersects exactly one of $A_k$ .",
"Thus $\\bar{C}_{N+1}^{[x]_G}$ intersects at most $N+1$ of $A_k$ and hence there exists $K \\in \\mathbb {N}$ such that for all $k \\ge K$ , $\\bar{C}_{N+1}^{[x]_G} \\cap A_k = \\mathbb {\\emptyset }$ .",
"Since $\\exists ^{\\infty } n (g_n y \\in \\bigcup _{k \\ge K} A_k)$ , $\\exists ^{\\infty } n (g_n y \\notin \\bar{C}_{N+1})$ , a contradiction.",
"$\\dashv $ Now it remains to show that for all $n \\in \\mathbb {N}$ , $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ implies that $\\exists y,z \\in [x]_G$ such that $y F_B^n z$ but $\\lnot (y F_B z)$ .",
"To this end, fix $n \\in \\mathbb {N}$ and assume $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus there are $y,z \\in (C^{\\prime \\prime }_n)^{[x]_G}$ such that $y F_n z$ , $y \\in U_{\\bar{n}(x)}$ and $z \\notin U_{\\bar{n}(x)}$ ; hence, $g_n y \\in B_n$ and $g_n z \\notin B_n$ , by the definition of $B_n$ .",
"Since $C_k$ are pairwise disjoint, $B_n \\subseteq C_n$ and $g_n y, g_n z \\in C_n$ , it follows that $g_n y \\in B$ and $g_n z \\notin B$ , and therefore $\\lnot (y F_B z)$ .",
"Finally, note that $F_n = F_B^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ and hence $y F_B^n z$ .",
"Corollary 6.8 Suppose all of the nontrivial subgroups of $G$ have finite index (e.g.",
"$G = \\mathbb {Z}$ ), and let $X$ be an aperiodic Borel $G$ -space.",
"Then there exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit, i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in X$ .",
"Let $A$ be a Borel set aperiodically separating $X$ (exists by Theorem REF ) and put $Y = f_A(X)$ .",
"Then $Y \\subseteq 2^G$ is aperiodic and hence the action of $G$ on $Y$ is free since the stabilizer subgroup of every element must have infinite index and thus is trivial.",
"But this implies that for all $y \\in Y$ , $f_A^{-1}(y)$ intersects every orbit in $X$ at no more than one point, and hence $f_A$ is one-to-one on every orbit.",
"From REF and REF we immediately get the following strengthening of Lemma REF .",
"Corollary 6.9 If $X$ is aperiodic, then for some $A \\in \\mathfrak {B}(X)$ , $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Theorem 6.10 Let $X$ be an aperiodic $G$ -space and let $E$ be a smooth equivalence relation on $X$ with $E_G \\subseteq E$ .",
"There exists a partition $\\mathcal {P}$ of $X$ into 4 Borel sets such that $G \\mathcal {P}$ separates any two $E$ -nonequivalent points in $X$ , i.e.",
"$\\forall x,y \\in X (\\lnot (x E y) \\rightarrow f_{\\mathcal {P}}(x) \\ne f_{\\mathcal {P}}(y))$ .",
"By Corollary REF , there is $A \\in \\mathfrak {B}(X)$ and a Borel partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant complete sections.",
"For each $n \\in \\mathbb {N}$ , define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{the smallest $m$ such that } \\exists x^{\\prime } \\in A_0^{[x]_G} \\text{ with } g_m x^{\\prime } \\in A_n.$ Clearly $\\bar{n}$ is Borel, and because all of $A_k$ are $F_A$ -invariant, $\\bar{n}$ is also $F_A$ -invariant, i.e.",
"for all $x,y \\in X$ , $x F_A y \\rightarrow \\bar{n}(x) = \\bar{n}(y)$ .",
"Also, $\\bar{n}$ is $G$ -invariant by definition.",
"Put $A^{\\prime }_n = \\lbrace x \\in A_0 : g_{\\bar{n}(x)} x \\in A_n\\rbrace $ and note that $A^{\\prime }_n$ is $F_A$ -invariant Borel since so are $\\bar{n}$ , $A_0$ and $A_n$ .",
"Moreover, $A^{\\prime }_n$ is clearly a complete section.",
"Define $\\gamma _n : A^{\\prime }_n \\rightarrow A_n$ by $x \\mapsto g_{\\bar{n}(x)} x$ .",
"Clearly, $\\gamma _n$ is Borel and one-to-one.",
"Since $E$ is smooth, there is a Borel $h : X \\rightarrow \\mathbb {R}$ such that for all $x,y \\in X$ , $x E y \\leftrightarrow h(x) = h(y)$ .",
"Let $\\lbrace V_n\\rbrace _{n \\in \\mathbb {N}}$ be a countable family of subsets of $\\mathbb {R}$ generating the Borel $\\sigma $ -algebra of $\\mathbb {R}$ and put $U_n = h^{-1}(V_n)$ .",
"Because each equivalence class of $E$ is $G$ -invariant, so is $h$ and hence so is $U_n$ .",
"Now let $B_n = \\gamma _n(A^{\\prime }_n \\cap U_n)$ and note that $B_n$ is Borel being a one-to-one Borel image of a Borel set.",
"It follows from the definition of $\\gamma _n$ that $B_n \\subseteq A_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $\\mathcal {P}= < \\!\\!",
"A,B \\!\\!",
">$ ; in particular, $|\\mathcal {P}| \\le 4$ .",
"We show that $\\mathcal {P}$ is what we want.",
"To this end, fix $x,y \\in X$ with $\\lnot (x E y)$ .",
"If $\\lnot (x F_A y)$ , then $G < \\!\\!",
"A \\!\\!",
">$ (and hence $G \\mathcal {P}$ ) separates $x$ and $y$ .",
"Thus assume that $x F_A y$ .",
"Since $h(x) \\ne h(y)$ , there is $n$ such that $h(x) \\in V_n$ and $h(y) \\notin V_n$ .",
"Hence, by invariance of $U_n$ , $gx \\in U_n \\wedge gy \\notin U_n$ , for all $g \\in G$ .",
"Because $A^{\\prime }_n$ is a complete section, there is $g \\in G$ such that $gx \\in A^{\\prime }_n$ and hence $gy \\in A^{\\prime }_n$ since $A^{\\prime }_n$ is $F_A$ -invariant.",
"Let $m = \\bar{n}(gx)$ ($= \\bar{n}(gy)$ ).",
"Then $g_m gx \\in B_n$ while $g_m gy \\notin B_n$ although $g_m gy \\in \\gamma _n(A^{\\prime }_n) \\subseteq A_n$ .",
"Thus $g_m gx \\in B$ but $g_m gy \\notin B$ and therefore $G \\mathcal {P}$ separates $x$ and $y$ .",
"Potential dichotomy theorems In this section we prove dichotomy theorems assuming Weiss's question has a positive answer for $G = \\mathbb {Z}$ .",
"In the proofs we use the Ergodic Decomposition Theorem (see , ) and a Borel/uniform version of Krieger's finite generator theorem, so we first state both of the theorems and sketch the proof of the latter.",
"For a Borel $G$ -space $X$ , let $\\mathcal {M}_G(X)$ denote the set of $G$ -invariant Borel probability measures on $X$ and let $\\mathcal {E}_G(X)$ denote the set of ergodic ones among those.",
"Clearly both are Borel subsets of $P(X)$ (the standard Borel space of Borel probability measures on $X$ ) and thus are themselves standard Borel spaces.",
"Ergodic Decomposition Theorem 7.1 (Farrell, Varadarajan) Let $X$ be a Borel $G$ -space.",
"If $\\mathcal {M}_G(X) \\ne \\mathbb {\\emptyset }$ (and hence $\\mathcal {E}_G(X) \\ne \\mathbb {\\emptyset }$ ), then there is a Borel surjection $x \\mapsto e_x$ from $X$ onto $\\mathcal {E}_G(X)$ such that: $x E_G y \\Rightarrow e_x = e_y$ ; For each $e \\in \\mathcal {E}_G(X)$ , if $X_e = \\lbrace x \\in X : e_x = e\\rbrace $ (hence $X_e$ is invariant Borel), then $e(X_e) = 1$ and $e \\!",
"\\!",
"\\downharpoonright _{X_e}$ is the unique ergodic invariant Borel probability measure on $X_e$ ; For each $\\mu \\in \\mathcal {M}_G(X)$ and $A \\in \\mathfrak {B}(X)$ , we have $\\mu (A) = \\int e_x(A) d\\mu (x).$ For the rest of the section, let $X$ be a Borel $\\mathbb {Z}$ -space.",
"For $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , if we let $h_e$ denote the entropy of $(X, \\mathbb {Z}, e)$ , then the map $e \\mapsto h_e$ is Borel.",
"Indeed, if $\\lbrace \\mathcal {P}_k\\rbrace _{k \\in \\mathbb {N}}$ is a refining sequence of partitions of $X$ that generates the Borel $\\sigma $ -algebra of $X$ , then by 4.1.2 of , $h_e = \\lim _{k \\rightarrow \\infty } h_e(\\mathcal {P}_k, \\mathbb {Z})$ , where $h_e(\\mathcal {P}_k, \\mathbb {Z})$ denotes the entropy of $\\mathcal {P}_k$ .",
"By 17.21 of , the function $e \\mapsto h_e(\\mathcal {P}_k)$ is Borel and thus so is the map $e \\mapsto h_e$ .",
"For all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ with $h_e < \\infty $ , let $N_e$ be the smallest integer such that $\\log N_e > h_e$ .",
"The map $e \\mapsto N_e$ is Borel because so is $e \\mapsto h_e$ .",
"Krieger's Finite Generator Theorem 7.2 (Uniform version) Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"Suppose $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Assume also that all measures in $\\mathcal {E}_{\\mathbb {Z}}(X)$ have finite entropy and let $e \\mapsto N_e$ be the map defined above.",
"Then there is a partition $\\lbrace A_n\\rbrace _{n \\le \\infty }$ of $X$ into Borel sets such that $A_{\\infty }$ is invariant and does not admit an invariant Borel probability measure; For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e \\setminus A_{\\infty }$ , where $X_e = \\rho ^{-1}(e)$ .",
"Sketch of Proof.",
"Note that it is enough to find a Borel invariant set $X^{\\prime } \\subseteq X$ and a Borel $\\mathbb {Z}$ -map $\\phi : X^{\\prime } \\rightarrow \\mathbb {N}^{\\mathbb {Z}}$ , such that for each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , we have $e(X \\setminus X^{\\prime }) = 0$ ; $\\phi \\!",
"\\!",
"\\downharpoonright _{X_e \\cap X^{\\prime }}$ is one-to-one and $\\phi (X_e \\cap X^{\\prime }) \\subseteq (N_e)^{\\mathbb {Z}}$ , where $(N_e)^{\\mathbb {Z}}$ is naturally viewed as a subset of $\\mathbb {N}^{\\mathbb {Z}}$ .",
"Indeed, assume we had such $X^{\\prime }$ and $\\phi $ , and let $A_{\\infty } = X \\setminus X^{\\prime }$ and $A_n = \\phi ^{-1}(V_n)$ for all $n \\in \\mathbb {N}$ , where $V_n = \\lbrace y \\in \\mathbb {N}^{\\mathbb {Z}} : y(0) = n\\rbrace $ .",
"Then it is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ satisfies (ii).",
"Also, (I) and part (ii) of the Ergodic Decomposition Theorem imply that (i) holds for $A_{\\infty }$ .",
"To construct such a $\\phi $ , we use the proof of Krieger's theorem presented in , Theorem 4.2.3, and we refer to it as Downarowicz's proof.",
"For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , the proof constructs a Borel $\\mathbb {Z}$ -embedding $\\phi _e : X^{\\prime } \\rightarrow N_e^{\\mathbb {Z}}$ on an $e$ -measure 1 set $X^{\\prime }$ .",
"We claim that this construction is uniform in $e$ in a Borel way and hence would yield $X^{\\prime }$ and $\\phi $ as above.",
"Our claim can be verified by inspection of Downarowicz's proof.",
"The proof uses the existence of sets with certain properties and one has to check that such sets exist with the properties satisfied for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ at once.",
"For example, the set $C$ used in the proof of Lemma 4.2.5 in can be chosen so that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $C \\cap X_e$ has the required properties for $e$ (using the Shannon-McMillan-Brieman theorem).",
"Another example is the set $B$ used in the proof of the same lemma, which is provided by Rohlin's lemma.",
"By inspection of the proof of Rohlin's lemma (see 2.1 in ), one can verify that we can get a Borel $B$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $B \\cap X_e$ has the required properties for $e$ .",
"The sets in these two examples are the only kind of sets whose existence is used in the whole proof; the rest of the proof constructs the required $\\phi $ “by hand”.",
"$\\Box $ Theorem 7.3 (Dichotomy I) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy; there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"We first show that the conditions above are mutually exclusive.",
"Indeed, assume there exist an invariant ergodic Borel probability measure $e$ with infinite entropy and a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"By ergodicity, $e$ would have to be supported on one of the $Y_n$ .",
"But $Y_n$ has a finite generator and hence the dynamical system $(Y_n, \\mathbb {Z}, e)$ has finite entropy by the Kolmogorov-Sinai theorem (see REF ).",
"Thus so does $(X, \\mathbb {Z}, e)$ since these two systems are isomorphic (modulo $e$ -NULL), contradicting the assumption on $e$ .",
"Now we prove that at least one of the conditions holds.",
"Assume that there is no invariant ergodic measure with infinite entropy.",
"Now, if there was no invariant Borel probability measure at all, then, since the answer to Question REF is assumed to be positive, $X$ would admit a finite generator, and we would be done.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ , $Y_{\\infty } = A_{\\infty }$ , and for all $n \\in \\mathbb {N}$ , $Y_n = \\lbrace x \\in X^{\\prime } : N_{e_x} = n\\rbrace ,$ where the map $e \\mapsto N_e$ is as above.",
"Note that the sets $Y_n$ are invariant since $\\rho $ is invariant, so $\\lbrace Y_n\\rbrace _{n \\le \\infty }$ is a countable partition of $X$ into invariant Borel sets.",
"Since $Y_{\\infty }$ does not admit an invariant Borel probability measure, by our assumption, it has a finite generator.",
"Let $E$ be the equivalence relation on $X^{\\prime }$ defined by $\\rho $ , i.e.",
"$\\forall x,y \\in X^{\\prime }$ , $x E y \\Leftrightarrow \\rho (x) = \\rho (y).$ By definition, $E$ is a smooth Borel equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action.",
"Thus, by Theorem REF , there exists a partition $\\mathcal {P}$ of $X^{\\prime }$ into 4 Borel sets such that $\\mathbb {Z}\\mathcal {P}$ separates any two points in different $E$ -classes.",
"Now fix $n \\in \\mathbb {N}$ and we will show that $I= \\mathcal {P}\\vee \\lbrace A_i\\rbrace _{i < n}$ is a generator for $Y_n$ .",
"Indeed, take distinct $x,y \\in Y_n$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}\\mathcal {P}$ separates them and hence so does $\\mathbb {Z}I$ .",
"Thus we can assume that $x E y$ .",
"Then $e := \\rho (x) = \\rho (y)$ , i.e.",
"$x,y \\in X_e = \\rho ^{-1}(e)$ .",
"By the choice of $\\lbrace A_i\\rbrace _{i \\in \\mathbb {N}}$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e$ and hence $\\mathbb {Z}\\lbrace A_i\\rbrace _{i < N_e}$ separates $x$ and $y$ .",
"But $n = N_e$ by the definition of $Y_n$ , so $\\mathbb {Z}I$ separates $x$ and $y$ .",
"Proposition 7.4 Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ admits invariant ergodic probability measures of arbitrarily large entropy, then it admits an invariant probability measure of infinite entropy.",
"For each $n \\ge 1$ , let $\\mu _n$ be an invariant ergodic probability measure of entropy $h_{\\mu _n} > n 2^n$ such that $\\mu _n \\ne \\mu _m$ for $n \\ne m$ , and put $\\mu = \\sum _{n \\ge 1} {1 \\over 2^n} \\mu _n.$ It is clear that $\\mu $ is an invariant probability measure, and we show that its entropy $h_{\\mu }$ is infinite.",
"Fix $n \\ge 1$ .",
"Let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem and put $X_n = \\rho ^{-1}(\\mu _n)$ .",
"It is clear that $\\mu _m(X_n) = 1$ if $m = n$ and 0 otherwise.",
"For any finite Borel partition $\\mathcal {P}= \\lbrace A_i\\rbrace _{i=1}^k$ of $X_n$ , put $A_0 = X \\setminus X_n$ and $\\bar{\\mathcal {P}} = \\mathcal {P}\\cup \\lbrace A_0\\rbrace $ .",
"Let $T$ be the Borel automorphism of $X$ corresponding to the action of $1_{\\mathbb {Z}}$ , and let $h_{\\nu }(I)$ and $h_{\\nu }(I, T)$ denote, respectively, the static and dynamic entropies of a finite Borel partition $I$ of $X$ with respect to an invariant probability measure $\\nu $ .",
"Then, with the convention that $\\log (0) \\cdot 0 = 0$ , we have $h_{\\mu }(\\bar{\\mathcal {P}}) &= - \\sum _{i=0}^k \\log (\\mu (A_i)) \\mu (A_i) \\ge - \\sum _{i = 1}^k \\log (\\mu (A_i)) \\mu (A_i)= - \\sum _{i = 1}^k \\log ({1 \\over 2^n}\\mu _n(A_i)) {1 \\over 2^n} \\mu _n(A_i) \\\\&\\ge - {1 \\over 2^n} \\sum _{i = 1}^k \\log (\\mu _n(A_i)) \\mu _n(A_i) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}).$ Since $\\mathcal {P}$ is arbitrary and $X_n$ is invariant, it follows that $h_{\\mu }(\\bar{\\mathcal {P}}, T) = \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu }(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) \\ge {1 \\over 2^n} \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu _n}(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}, T).$ Now for any finite Borel partition $I$ of $X$ , it is clear that $h_{\\mu _n}(I) = h_{\\mu _n}(\\bar{\\mathcal {P}})$ (and hence $h_{\\mu _n}(I, T) = h_{\\mu _n}(\\bar{\\mathcal {P}}, T)$ ), for some $\\mathcal {P}$ as above.",
"This implies that $h_{\\mu } \\ge \\sup _{\\mathcal {P}} h_{\\mu }(\\bar{\\mathcal {P}}, T) \\ge {1 \\over 2^n} \\sup _{\\mathcal {P}} h_{\\mu _n}(\\bar{\\mathcal {P}}, T) = {1 \\over 2^n} \\sup _{I} h_{\\mu _n}(I, T) = {1 \\over 2^n} h_{\\mu _n} > n,$ where $\\mathcal {P}$ and $I$ range over finite Borel partitions of $X_n$ and $X$ , respectively.",
"Thus $h_{\\mu }\\!",
"= \\infty $ .",
"Theorem 7.5 (Dichotomy II) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"The Kolmogorov-Sinai theorem implies that the conditions are mutually exclusive, and we prove that at least one of them holds.",
"Assume that there is no invariant measure with infinite entropy.",
"If there was no invariant Borel probability measure at all, then, by our assumption, $X$ would admit a finite generator.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ and $X_e = \\rho ^{-1}(e)$ , for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ .",
"By our assumption, $A_{\\infty }$ admits a finite generator $\\mathcal {P}$ .",
"Also, by REF , there is $N \\ge 1$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $N_e \\le N$ and hence $\\mathcal {Q}:= \\lbrace A_n\\rbrace _{n < N}$ is a finite generator for $X_e$ ; in particular, $\\mathcal {Q}$ is a partition of $X^{\\prime }$ .",
"Let $E$ be the following equivalence relation on $X$ : $x E y \\Leftrightarrow (x, y \\in A_{\\infty }) \\vee (x,y \\in X^{\\prime } \\wedge \\rho (x) = \\rho (y)).$ By definition, $E$ is a smooth equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action and $A_{\\infty }$ is $\\mathbb {Z}$ -invariant.",
"Thus, by Theorem REF , there exists a partition $J$ of $X$ into 4 Borel sets such that $\\mathbb {Z}J$ separates any two points in different $E$ -classes.",
"We now show that $I:= < \\!\\!",
"J\\cup \\mathcal {P}\\cup \\mathcal {Q} \\!\\!",
">$ is a generator.",
"Indeed, fix distinct $x,y \\in X$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}J$ separates them.",
"So we can assume that $x E y$ .",
"If $x,y \\in A_{\\infty }$ , then $\\mathbb {Z}\\mathcal {P}$ separates $x$ and $y$ .",
"Finally, if $x,y \\in X^{\\prime }$ , then $x,y \\in X_e$ , where $e = \\rho (x)$ ($= \\rho (y)$ ), and hence $\\mathbb {Z}\\mathcal {Q}$ separates $x$ and $y$ .",
"Remark.",
"It is likely that the above dichotomies are also true for any amenable group using a uniform version of Krieger's theorem for amenable groups, cf.",
", but I have not checked the details.",
"Finite generators on comeager sets Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER.",
"Locally weakly wandering sets and other special cases Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
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[
"Finite generators in the case of $\\sigma $ -compact spaces",
"In this section we prove that the answer to Question REF is positive in case $X$ has a $\\sigma $ -compact realization.",
"To do this, we first prove Proposition REF , which shows how to construct a countably additive invariant probability measure on $X$ using a finitely additive one.",
"We then use REF to conclude the result.",
"For the next two statements, let $X$ be a second countable Hausdorff topological space equipped with a continuous action of $G$ .",
"Lemma 4.1 Let $\\mathcal {U}\\subseteq Pow(X)$ be a countable base for $X$ closed under the $G$ -action and finite unions/intersections.",
"Let $\\rho $ be a $G$ -invariant finitely additive probability measure on the $G$ -algebra generated by $\\mathcal {U}$ .",
"For every $A \\subseteq X$ , define $\\mu ^* (A) = \\inf \\lbrace \\sum _{n \\in \\mathbb {N}} \\rho (U_n ) : U_n \\in \\mathcal {U}\\; \\wedge \\; A \\subseteq \\bigcup _{n \\in \\mathbb {N}} U_n\\rbrace .$ Then: $\\mu ^*$ is a $G$ -invariant outer measure.",
"If $K \\subseteq X$ is compact, then $K$ is metrizable and $\\mu ^*$ is a metric outer measure on $K$ (with respect to any compatible metric).",
"It is a standard fact from measure theory that $\\mu ^*$ is an outer measure.",
"That $\\mu ^*$ is $G$ -invariant follows immediately from $G$ -invariance of $\\rho $ and the fact that $\\mathcal {U}$ is closed under the action of $G$ .",
"For (b), first note that by Urysohn metrization theorem, $K$ is metrizable, and fix a metric on $K$ .",
"If $E, F \\subseteq K$ are a positive distance apart, then so are $\\bar{E}$ and $\\bar{F}$ .",
"Hence there exist disjoint open sets $U,V$ such that $\\bar{E} \\subseteq U$ , $\\bar{F} \\subseteq V$ .",
"Because $\\bar{E}$ and $\\bar{F}$ are compact, $U,V$ can be taken to be finite unions of sets in $\\mathcal {U}$ and therefore $U,V \\in \\mathcal {U}$ .",
"Now fix $\\epsilon >0$ and let $W_n \\in \\mathcal {U}$ , be such that $E \\cup F \\subseteq \\bigcup _n W_n$ and $\\sum _n \\rho (W_n) \\le \\mu ^*(E \\cup F) + \\epsilon \\le \\mu ^*(E) + \\mu ^*(F) +\\epsilon .\\qquad \\mathrm {{(*)}}$ Note that $\\lbrace W_n \\cap U\\rbrace _{n \\in \\mathbb {N}}$ covers $E$ , $\\lbrace W_n \\cap V\\rbrace _{n \\in \\mathbb {N}}$ covers $F$ and $W_n \\cap U, W_n \\cap V \\in \\mathcal {U}$ .",
"Also, by finite additivity of $\\rho $ , $\\rho (W_n \\cap U) + \\rho (W_n \\cap V) = \\rho (W_n \\cap (U \\cup V)) \\le \\rho (W_n).$ Thus $\\mu ^*(E) + \\mu ^*(F) \\le \\sum _n \\rho (W_n \\cap U) + \\sum _n \\rho (W_n \\cap V) \\le \\sum _n \\rho (W_n),$ which, together with ($*$ ), implies that $\\mu ^*(E \\cup F) = \\mu ^*(E) + \\mu ^*(F)$ since $\\epsilon $ is arbitrary.",
"Proposition 4.2 Suppose there exist a countable base $\\mathcal {U}\\subseteq Pow(X)$ for $X$ and a compact set $K \\subseteq X$ such that the $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ admits a finitely additive $G$ -invariant probability measure $\\rho $ with $\\rho (K)>0$ .",
"Then there exists a countably additive $G$ -invariant Borel probability measure on $X$ .",
"Let $K, \\mathcal {U}$ and $\\rho $ be as in the hypothesis.",
"We may assume that $\\mathcal {U}$ is closed under the $G$ -action and finite unions/intersections.",
"Let $\\mu ^*$ be the outer measure provided by Lemma REF applied to $\\mathcal {U}$ , $\\rho $ .",
"Thus $\\mu ^*$ is a metric outer measure on $K$ and hence all Borel subsets of $K$ are $\\mu ^*$ -measurable (see 13.2 in ).",
"This implies that all Borel subsets of $Y = [K]_G = \\bigcup _{g \\in G} gK$ are $\\mu ^*$ -measurable because $\\mu ^*$ is $G$ -invariant.",
"By Carathéodory's theorem, the restriction of $\\mu ^*$ to the Borel subsets of $Y$ is a countably additive Borel measure on $Y$ , and we extend it to a Borel measure $\\mu $ on $X$ by setting $\\mu (Y^c) = 0$ .",
"Note that $\\mu $ is $G$ -invariant and $\\mu (Y) \\le 1$ .",
"It remains to show that $\\mu $ is nontrivial, which we do by showing that $\\mu (K) \\ge \\rho (K)$ and hence $\\mu (K)>0$ .",
"To this end, let $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}} \\subseteq \\mathcal {U}$ cover $K$ .",
"Since $K$ is compact, there is a finite subcover $\\lbrace U_n\\rbrace _{n < N}$ .",
"Thus $U := \\bigcup _{n < N} U_n \\in \\mathcal {U}$ and $K \\subseteq U$ .",
"By finite additivity of $\\rho $ , we have $\\sum _{n \\in \\mathbb {N}} \\rho (U_n) \\ge \\sum _{n < N} \\rho (U_n) \\ge \\rho (U) \\ge \\rho (K),$ and hence, it follows from the definition of $\\mu ^*$ that $\\mu ^*(K) \\ge \\rho (K)$ .",
"Thus $\\mu (K) = \\mu ^*(K) > 0$ .",
"Corollary 4.3 Let $X$ be a second countable Hausdorff topological $G$ -space whose Borel structure is standard.",
"For every compact set $K \\subseteq X$ not in 4, there is a $G$ -invariant countably additive Borel probability measure $\\mu $ on $X$ with $\\mu (K) > 0$ .",
"Fix any countable base $\\mathcal {U}$ for $X$ and let $\\mathcal {B}$ be the Boolean $G$ -algebra generated by $\\mathcal {U}\\cup \\lbrace K\\rbrace $ .",
"By Corollary REF , there exists a $G$ -invariant finitely additive probability measure $\\rho $ on $\\mathcal {B}$ such that $\\rho (K) > 0$ .",
"Now apply REF .",
"As a corollary, we derive the analogue of Nadkarni's theorem for 4 in case of $\\sigma $ -compact spaces.",
"Corollary 4.4 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"$X \\notin 4$ if and only if there exists a $G$ -invariant countably additive Borel probability measure on $X$ .",
"$\\Leftarrow $ : If $X \\in 4$ , then it is compressible in the usual sense and hence does not admit a $G$ -invariant Borel probability measure.",
"$\\Rightarrow $ : Suppose that $X$ is a $\\sigma $ -compact topological $G$ -space and $X \\notin 4$ .",
"Then, since $X$ is $\\sigma $ -compact and 4 is a $\\sigma $ -ideal, there is a compact set $K$ not in 4.",
"Now apply REF .",
"Remark.",
"For a Borel $G$ -space $X$ , let $\\mathcal {K}$ denote the collection of all subsets of invariant Borel sets that admit a $\\sigma $ -compact realization (when viewed as Borel $G$ -spaces).",
"Also, let $ denote the collection of all subsets of invariant compressible Borel sets.",
"It is clear that $ K$ and $ are $\\sigma $ -ideals, and what REF implies is that $\\mathcal {K}\\subseteq 4$ .",
"The question of whether $\\mathcal {K}= Pow(X)$ is just a rephrasing of $§10$ .",
"(B) of Introduction.",
"Theorem 4.5 Let $X$ be a Borel $G$ -space that admits a $\\sigma $ -compact realization.",
"If there is no $G$ -invariant Borel probability measure on $X$ , then $X$ admits a Borel 32-generator.",
"By REF , $X \\in 4$ and hence, $X$ is 4-compressible.",
"Thus, by Proposition REF , $X$ admits a Borel $2^5$ -generator.",
"Example 4.6.",
"Let $LO \\subseteq 2^{\\mathbb {N}^2}$ denote the Polish space of all countable linear orderings and let $G$ be the group of finite permutations of elements of $\\mathbb {N}$ .",
"$G$ is countable and acts continuously on $LO$ in the natural way.",
"Put $X = LO \\setminus DLO$ , where $DLO$ denotes the set of all dense linear orderings without endpoints (copies of $\\mathbb {Q}$ ).",
"It is straightforward to see that $DLO$ is a $G_{\\delta }$ subset of $LO$ and hence, $X$ is $F_{\\sigma }$ .",
"Therefore, $X$ is in fact $\\sigma $ -compact since $LO$ is compact being a closed subset of $2^{\\mathbb {N}^2}$ .",
"Also note that $X$ is $G$ -invariant.",
"Let $\\mu $ be the unique measure on $LO$ defined by $\\mu (V_{(F,<)}) = {1 \\over n!",
"}$ , where $(F,<)$ is a finite linearly ordered subset of $\\mathbb {N}$ of cardinality $n$ and $V_{(F,<)}$ is the set of all linear orderings of $\\mathbb {N}$ extending the order $<$ on $F$ .",
"As shown in , $\\mu $ is the unique invariant measure for the action of $G$ on $LO$ and $\\mu (X) = 0$ .",
"Thus there is no $G$ -invariant Borel probability measure on $X$ and hence, by the above theorem, $X$ admits a Borel 32-generator.",
"Finitely traveling sets Let $X$ be a Borel $G$ -space.",
"Definition 5.1 Let $A,B \\in \\mathfrak {B}(X)$ be equidecomposable, i.e.",
"there are $N \\le \\infty $ , $\\lbrace g_n\\rbrace _{n < N} \\subseteq G$ and Borel partitions $\\lbrace A_n\\rbrace _{n < N}$ and $\\lbrace B_n\\rbrace _{n < N}$ of $A$ and $B$ , respectively, such that $g_n A_n = B_n$ for all $n < N$ .",
"$A,B$ are said to be locally finitely equidecomposable (denote by $A \\sim _{\\text{lfin}} B$ ), if $\\lbrace A_n\\rbrace _{n < N},\\lbrace B_n\\rbrace _{n < N},\\lbrace g_n\\rbrace _{n < N}$ can be taken so that for every $x \\in A$ , $A_n \\cap [x]_G = \\mathbb {\\emptyset }$ for all but finitely many $n<N$ ; finitely equidecomposable (denote by $A \\sim _{\\text{fin}} B$ ), if $N$ can be taken to be finite.",
"The notation $\\prec _{\\text{fin}}$ , $\\prec _{\\text{lfin}}$ and the notions of finite and locally finite compressibility are defined analogous to Definitions REF and REF .",
"Definition 5.2 A Borel set $A \\subseteq X$ is called (locally) finitely traveling if there exists pairwise disjoint Borel sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ such that $A_0 = A$ and $A \\sim _{\\text{fin}} A_n$ ($A \\sim _{\\text{lfin}} A_n$ ), $\\forall n \\in \\mathbb {N}$ .",
"Proposition 5.3 If $X$ is (locally) finitely compressible then $X$ admits a (locally) finitely traveling Borel complete section.",
"We prove for finitely compressible $X$ , but note that everything below is also locally valid (i.e.",
"restricted to every orbit) for a locally compressible $X$ .",
"Run the proof of the first part of Lemma REF noting that a witnessing map $\\gamma : X \\rightarrow G$ of finite compressibility of $X$ has finite image and hence the image of each $\\delta _n$ (in the notation of the proof) is finite, which implies that the obtained traveling set $A$ is actually finitely traveling.",
"Proposition 5.4 If $X$ admits a locally finitely traveling Borel complete section, then $X \\in 4$ .",
"Let $A$ be a locally finitely traveling Borel complete section and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"Let $I_n = \\lbrace C_k^n\\rbrace _{k \\in \\mathbb {N}}$ , $J_n = \\lbrace D_k^n\\rbrace _{k \\in \\mathbb {N}}$ be Borel partitions of $A$ and $A_n$ , respectively, that together with $\\lbrace g_k^n\\rbrace _{k \\in \\mathbb {N}} \\subseteq G$ witness $A \\sim _{\\text{lfin}} A_n$ (as in Definition REF ).",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by $\\lbrace X\\rbrace \\cup \\bigcup _{n \\in \\mathbb {N}} (I_n \\cup J_n \\cup \\lbrace A_n\\rbrace )$ .",
"Now assume for contradiction that $X \\notin 4$ and hence, $A \\notin 4$ .",
"Thus, applying Corollary REF to $A$ and $\\mathcal {B}$ , we get a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B) = 0$ .",
"Claim $\\mu (A_n) = \\mu (A)$ , for all $n \\in \\mathbb {N}$ .",
"Proof of Claim.",
"For each $n$ , let $\\lbrace C_{k_i}^n\\rbrace _{i < K_n}$ be the list of those $C_k^n$ such that $C_k^n \\cap [x]_G \\ne \\mathbb {\\emptyset }$ ($K_n < \\infty $ by the definition of locally finitely traveling).",
"Set $B = A \\setminus (\\bigcup _{i < K_n} C_{k_i}^n)$ and note that by finite additivity of $\\mu $ , $\\mu (A) = \\mu (B) + \\sum _{i < K_n} \\mu (C_{k_i}^n).$ Similarly, set $B^{\\prime } = A_n \\setminus (\\bigcup _{i < K_n} D_{k_i}^n)$ and hence $\\mu (A_n) = \\mu (B^{\\prime }) + \\sum _{i < K_n} \\mu (D_{k_i}^n).$ But $B \\cap [x]_G = \\mathbb {\\emptyset }$ and $B^{\\prime } \\cap [x]_G = \\mathbb {\\emptyset }$ , and thus $\\mu (B) = \\mu (B^{\\prime }) = 0$ .",
"Also, since $g_{k_i}^n C_{k_i}^n = D_{k_i}^n$ and $\\mu $ is $G$ -invariant, $\\mu (C_{k_i}^n) = \\mu (D_{k_i}^n)$ .",
"Therefore $\\mu (A) = \\sum _{i < K_n} \\mu (C_{k_i}^n) = \\sum _{i < K_n} \\mu (D_{k_i}^n) = \\mu (A_n).$ $\\dashv $ This claim contradicts $\\mu $ being a probability measure since for large enough $N$ , $\\mu (\\bigcup _{n < N} A_n) = N \\mu (A) > 1$ , contradicting $\\mu (X) = 1$ .",
"This, together with REF , implies the following.",
"Corollary 5.5 Let $X$ be a Borel $G$ -space.",
"If $X$ admits a locally finitely traveling Borel complete section, then there is a Borel 32-generator.",
"Separating smooth-many invariant sets Assume throughout that $X$ is a Borel $G$ -space.",
"Lemma 6.1 If $X$ is aperiodic then it admits a countably infinite partition into Borel complete sections.",
"The following argument is also given in the proof of Theorem 13.1 in .",
"By the marker lemma (see 6.7 in ), there exists a vanishing sequence $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections, i.e.",
"$\\bigcap _{n \\in \\mathbb {N}} B_n = \\mathbb {\\emptyset }$ .",
"For each $n \\in \\mathbb {N}$ , define $k_n : X \\rightarrow \\mathbb {N}$ recursively as follows: $\\left\\lbrace \\begin{array}{rcl}k_0(x) & = & 0 \\\\k_{n+1}(x) &= & min \\lbrace k \\in \\mathbb {N}: B_{k_n(x)} \\cap [x]_G \\nsubseteq B_k\\rbrace \\end{array}\\right.,$ and define $A_n \\subseteq X$ by $x \\in A_n \\Leftrightarrow x \\in A_{k_n(x)} \\setminus A_{k_{n+1}(x)}.$ It is straightforward to check that $A_n$ are pairwise disjoint Borel complete sections.",
"For $A \\in \\mathfrak {B}(X)$ , if $I= < \\!\\!",
"A \\!\\!",
">$ then we use the notation $F_A$ and $f_A$ instead of $F_{I}$ and $, respectively.$ We now work towards strengthening the above lemma to yield a countably infinite partition into $F_A$ -invariant Borel complete sections.",
"Definition 6.2 (Aperiodic separation) For Borel sets $A, Y \\subseteq X$ , we say that $A$ aperiodically separates $Y$ if $f_A([Y]_G)$ is aperiodic (as an invariant subset of the shift $2^G$ ).",
"If such $A$ exists, we say that $Y$ is aperiodically separable.",
"Proposition 6.3 For $A \\in \\mathfrak {B}(X)$ , if $A$ aperiodically separates $X$ , then $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Let $Y = \\lbrace y \\in 2^G : |[y]_G| = \\infty \\rbrace $ and hence $f_A(X)$ is a $G$ -invariant subset of $Y$ .",
"By Lemma REF applied to $Y$ , there is a partition $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $Y$ into Borel complete sections.",
"Thus $A_n = f_{I}^{-1}(B_n)$ is a Borel $F_A$ -invariant complete section for $X$ and $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $X$ .",
"Let $\\mathfrak {A}$ denote the collection of all subsets of aperiodically separable Borel sets.",
"Lemma 6.4 $\\mathfrak {A}$ is a $\\sigma $ -ideal.",
"We only have to show that if $Y_n$ are aperiodically separable Borel sets, then $Y = \\bigcup _{n \\in \\mathbb {N}} Y_n \\in \\mathfrak {A}$ .",
"Let $A_n$ be a Borel set aperiodically separating $Y_n$ .",
"Since $A_n$ also aperiodically separates $[Y_n]_G$ (by definition), we can assume that $Y_n$ is $G$ -invariant.",
"Furthermore, by taking $Y_n^{\\prime } = Y_n \\setminus \\bigcup _{k<n} Y_k$ , we can assume that $Y_n$ are pairwise disjoint.",
"Now letting $A = \\bigcup _{n \\in \\mathbb {N}} (A_n \\cap Y_n)$ , it is easy to check that $A$ aperiodically separates $Y$ .",
"Let $\\mathfrak {S}$ denote the collection of all subsets of smooth sets.",
"By a similar argument as the one above, $\\mathfrak {S}$ is a $\\sigma $ -ideal.",
"Lemma 6.5 If $X$ is aperiodic, then $\\mathfrak {S}\\subseteq \\mathfrak {A}$ .",
"Let $S \\in \\mathfrak {S}$ and hence there is a Borel transversal $T$ for $[S]_G$ .",
"Fix $x \\in S$ and let $y \\ne z \\in [x]_G$ .",
"Since $T$ is a transversal, there is $g \\in G$ such that $gy \\in T$ , and hence $gz \\notin T$ .",
"Thus $f_T(y) \\ne f_T(z)$ , and so $f_T([x]_G)$ is infinite.",
"Therefore $T$ aperiodically separates $[S]_G$ .",
"For the rest of the section, fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and let $F_A^n$ be following equivalence relation: $y F_A^n z \\Leftrightarrow \\forall k < n (g_k y \\in A \\leftrightarrow g_k z \\in A).$ Note that $F_A^n$ has no more than $2^n$ equivalence classes and that $y F_A z$ if and only if $\\forall n (y F_A^n z)$ .",
"Lemma 6.6 For $A,Y \\in \\mathfrak {B}(X)$ , $A$ aperiodically separates $Y$ if and only if $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in Y^{[x]_G}) [y F_A^n z \\wedge \\lnot (y F_A z)]$ .",
"$\\Rightarrow $ : Assume that for all $x \\in Y$ , $f_A([x]_G)$ is infinite and thus $F_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ has infinitely many equivalence classes.",
"Fix $n \\in \\mathbb {N}$ and recall that $F_A^n$ has only finitely many equivalence classes.",
"Thus, by the Pigeon Hole Principle, there are $y,z \\in Y^{[x]_G}$ such that $y F_A^n z$ yet $\\lnot (y F_A z)$ .",
"$\\Leftarrow $ : Assume for contradiction that $f_A(Y^{[x]_G})$ is finite for some $x \\in Y$ .",
"Then it follows that $F_A = F_A^n$ , for some $n$ , and hence for any $y,z \\in Y^{[x]_G}$ , $y F_A^n z$ implies $y F_A z$ , contradicting the hypothesis.",
"Theorem 6.7 If $X$ is an aperiodic Borel $G$ -space, then $X \\in \\mathfrak {A}$ .",
"By Lemma REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into Borel complete sections.",
"We will inductively construct Borel sets $B_n \\subseteq C_n$ , where $C_n$ should be thought of as the set of points colored (black or white) at the $n^{th}$ step, and $B_n$ as the set of points colored black (thus $C_n \\setminus B_n$ is colored white).",
"Define a function $\\# : X \\rightarrow \\mathbb {N}$ by $x \\mapsto m$ , where $m$ is such that $x \\in A_m$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ of sets generating the Borel $\\sigma $ -algebra of $X$ .",
"Assuming that for all $k < n$ , $C_k, B_k$ are defined, let $\\bar{C}_n = \\bigcup _{k<n} C_k$ and $\\bar{B}_n = \\bigcup _{k<n} B_k$ .",
"Put $P_n = \\lbrace x \\in A_0 : \\forall k < n (g_k x \\in \\bar{C}_n) \\wedge g_n x \\notin \\bar{C}_n\\rbrace $ and set $F_n = F_{\\bar{B}_n}^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ , that is for all $x,y \\in P_n$ , $y F_n z \\Leftrightarrow \\forall k < n (g_k y \\in \\bar{B}_n \\leftrightarrow g_k z \\in \\bar{B}_n).$ Now put $C^{\\prime }_n = \\lbrace x \\in P_n : \\#(g_n x) = \\min \\#((g_nP_n)^{[x]_G})\\rbrace $ , $C^{\\prime \\prime }_n = \\lbrace x \\in C^{\\prime }_n : \\exists y, z \\in (C^{\\prime }_n)^{[x]_G} (y \\ne z \\wedge y F_n z)\\rbrace $ and $C_n = g_n C^{\\prime \\prime }_n$ .",
"Note that it follows from the definition of $P_n$ that $C_n$ is disjoint from $\\bar{C}_n$ .",
"Now in order to define $B_n$ , first define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{ the smallest $m$ such that there are } y,z \\in C^{\\prime \\prime }_n \\cap [x]_G \\text{ with } y F_n z, y \\in U_m \\text{ and } z \\notin U_m.$ Note that $\\bar{n}$ is Borel and $G$ -invariant.",
"Lastly, let $B^{\\prime }_n = \\lbrace x \\in C^{\\prime \\prime }_n : x \\in U_{\\bar{n}(x)}\\rbrace $ and $B_n = g_n B^{\\prime }_n$ .",
"Clearly $B_n \\subseteq C_n$ .",
"Now let $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $D = \\left[\\bigcup _{n \\in \\mathbb {N}} (C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)\\right]_G$ .",
"We show that $B$ aperiodically separates $Y := X \\setminus D$ and $D \\in \\mathfrak {S}$ .",
"Since $\\mathfrak {S}\\subseteq \\mathfrak {A}$ and $\\mathfrak {A}$ is an ideal, this will imply that $X \\in \\mathfrak {A}$ .",
"Claim 1 $D \\in \\mathfrak {S}$ .",
"Proof of Claim.",
"Since $\\mathfrak {S}$ is a $\\sigma $ -ideal, it is enough to show that for each $n$ , $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G \\in \\mathfrak {S}$ , so fix $n \\in \\mathbb {N}$ .",
"Clearly $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ is finite, for all $x \\in X$ , since there can be at most $2^n$ pairwise $F_n$ -nonequivalent points.",
"Thus, fixing some Borel linear ordering of $X$ and taking the smallest element from $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ for each $x \\in C^{\\prime }_n \\setminus C^{\\prime \\prime }_n$ , we can define a Borel transversal for $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G$ .",
"$\\dashv $ By Lemma REF , to show that $B$ aperiodically separates $Y$ , it is enough to show that $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in [x]_G) [y F_B^n z \\wedge \\lnot (y F_B z)]$ .",
"Fix $x \\in Y$ .",
"Claim 2 $(\\exists ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $(\\forall ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $x \\notin D$ , it follows that $(\\forall ^{\\infty } n) P_n^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $A_0$ is a complete section and $\\bar{C}_0 = \\mathbb {\\emptyset }$ , $P_0^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Let $N$ be the largest number such that $P_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus for all $n > N$ , $C_n^{[x]_G} = \\mathbb {\\emptyset }$ and hence for all $n > N$ , $\\bar{C}_n^{[x]_G} = \\bar{C}_{N+1}^{[x]_G}$ .",
"Because $C_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ , there is $y \\in A_0^{[x]_G}$ such that $\\forall k \\le N (g_k y \\in \\bar{C}_{N+1})$ ; but because $P_{N+1}^{[x]_G} = \\mathbb {\\emptyset }$ , $g_{N+1} y$ must also fall into $\\bar{C}_{N+1}$ .",
"By induction on $n > N$ , we get that for all $n>N$ , $g_n y \\in \\bar{C}_n$ and thus $g_n y \\in \\bar{C}_{N+1}$ .",
"On the other hand, it follows from the definition of $C^{\\prime }_n$ that for each $n$ , $(C^{\\prime }_n)^{[x]_G}$ intersects exactly one of $A_k$ .",
"Thus $\\bar{C}_{N+1}^{[x]_G}$ intersects at most $N+1$ of $A_k$ and hence there exists $K \\in \\mathbb {N}$ such that for all $k \\ge K$ , $\\bar{C}_{N+1}^{[x]_G} \\cap A_k = \\mathbb {\\emptyset }$ .",
"Since $\\exists ^{\\infty } n (g_n y \\in \\bigcup _{k \\ge K} A_k)$ , $\\exists ^{\\infty } n (g_n y \\notin \\bar{C}_{N+1})$ , a contradiction.",
"$\\dashv $ Now it remains to show that for all $n \\in \\mathbb {N}$ , $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ implies that $\\exists y,z \\in [x]_G$ such that $y F_B^n z$ but $\\lnot (y F_B z)$ .",
"To this end, fix $n \\in \\mathbb {N}$ and assume $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus there are $y,z \\in (C^{\\prime \\prime }_n)^{[x]_G}$ such that $y F_n z$ , $y \\in U_{\\bar{n}(x)}$ and $z \\notin U_{\\bar{n}(x)}$ ; hence, $g_n y \\in B_n$ and $g_n z \\notin B_n$ , by the definition of $B_n$ .",
"Since $C_k$ are pairwise disjoint, $B_n \\subseteq C_n$ and $g_n y, g_n z \\in C_n$ , it follows that $g_n y \\in B$ and $g_n z \\notin B$ , and therefore $\\lnot (y F_B z)$ .",
"Finally, note that $F_n = F_B^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ and hence $y F_B^n z$ .",
"Corollary 6.8 Suppose all of the nontrivial subgroups of $G$ have finite index (e.g.",
"$G = \\mathbb {Z}$ ), and let $X$ be an aperiodic Borel $G$ -space.",
"Then there exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit, i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in X$ .",
"Let $A$ be a Borel set aperiodically separating $X$ (exists by Theorem REF ) and put $Y = f_A(X)$ .",
"Then $Y \\subseteq 2^G$ is aperiodic and hence the action of $G$ on $Y$ is free since the stabilizer subgroup of every element must have infinite index and thus is trivial.",
"But this implies that for all $y \\in Y$ , $f_A^{-1}(y)$ intersects every orbit in $X$ at no more than one point, and hence $f_A$ is one-to-one on every orbit.",
"From REF and REF we immediately get the following strengthening of Lemma REF .",
"Corollary 6.9 If $X$ is aperiodic, then for some $A \\in \\mathfrak {B}(X)$ , $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Theorem 6.10 Let $X$ be an aperiodic $G$ -space and let $E$ be a smooth equivalence relation on $X$ with $E_G \\subseteq E$ .",
"There exists a partition $\\mathcal {P}$ of $X$ into 4 Borel sets such that $G \\mathcal {P}$ separates any two $E$ -nonequivalent points in $X$ , i.e.",
"$\\forall x,y \\in X (\\lnot (x E y) \\rightarrow f_{\\mathcal {P}}(x) \\ne f_{\\mathcal {P}}(y))$ .",
"By Corollary REF , there is $A \\in \\mathfrak {B}(X)$ and a Borel partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant complete sections.",
"For each $n \\in \\mathbb {N}$ , define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{the smallest $m$ such that } \\exists x^{\\prime } \\in A_0^{[x]_G} \\text{ with } g_m x^{\\prime } \\in A_n.$ Clearly $\\bar{n}$ is Borel, and because all of $A_k$ are $F_A$ -invariant, $\\bar{n}$ is also $F_A$ -invariant, i.e.",
"for all $x,y \\in X$ , $x F_A y \\rightarrow \\bar{n}(x) = \\bar{n}(y)$ .",
"Also, $\\bar{n}$ is $G$ -invariant by definition.",
"Put $A^{\\prime }_n = \\lbrace x \\in A_0 : g_{\\bar{n}(x)} x \\in A_n\\rbrace $ and note that $A^{\\prime }_n$ is $F_A$ -invariant Borel since so are $\\bar{n}$ , $A_0$ and $A_n$ .",
"Moreover, $A^{\\prime }_n$ is clearly a complete section.",
"Define $\\gamma _n : A^{\\prime }_n \\rightarrow A_n$ by $x \\mapsto g_{\\bar{n}(x)} x$ .",
"Clearly, $\\gamma _n$ is Borel and one-to-one.",
"Since $E$ is smooth, there is a Borel $h : X \\rightarrow \\mathbb {R}$ such that for all $x,y \\in X$ , $x E y \\leftrightarrow h(x) = h(y)$ .",
"Let $\\lbrace V_n\\rbrace _{n \\in \\mathbb {N}}$ be a countable family of subsets of $\\mathbb {R}$ generating the Borel $\\sigma $ -algebra of $\\mathbb {R}$ and put $U_n = h^{-1}(V_n)$ .",
"Because each equivalence class of $E$ is $G$ -invariant, so is $h$ and hence so is $U_n$ .",
"Now let $B_n = \\gamma _n(A^{\\prime }_n \\cap U_n)$ and note that $B_n$ is Borel being a one-to-one Borel image of a Borel set.",
"It follows from the definition of $\\gamma _n$ that $B_n \\subseteq A_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $\\mathcal {P}= < \\!\\!",
"A,B \\!\\!",
">$ ; in particular, $|\\mathcal {P}| \\le 4$ .",
"We show that $\\mathcal {P}$ is what we want.",
"To this end, fix $x,y \\in X$ with $\\lnot (x E y)$ .",
"If $\\lnot (x F_A y)$ , then $G < \\!\\!",
"A \\!\\!",
">$ (and hence $G \\mathcal {P}$ ) separates $x$ and $y$ .",
"Thus assume that $x F_A y$ .",
"Since $h(x) \\ne h(y)$ , there is $n$ such that $h(x) \\in V_n$ and $h(y) \\notin V_n$ .",
"Hence, by invariance of $U_n$ , $gx \\in U_n \\wedge gy \\notin U_n$ , for all $g \\in G$ .",
"Because $A^{\\prime }_n$ is a complete section, there is $g \\in G$ such that $gx \\in A^{\\prime }_n$ and hence $gy \\in A^{\\prime }_n$ since $A^{\\prime }_n$ is $F_A$ -invariant.",
"Let $m = \\bar{n}(gx)$ ($= \\bar{n}(gy)$ ).",
"Then $g_m gx \\in B_n$ while $g_m gy \\notin B_n$ although $g_m gy \\in \\gamma _n(A^{\\prime }_n) \\subseteq A_n$ .",
"Thus $g_m gx \\in B$ but $g_m gy \\notin B$ and therefore $G \\mathcal {P}$ separates $x$ and $y$ .",
"Potential dichotomy theorems In this section we prove dichotomy theorems assuming Weiss's question has a positive answer for $G = \\mathbb {Z}$ .",
"In the proofs we use the Ergodic Decomposition Theorem (see , ) and a Borel/uniform version of Krieger's finite generator theorem, so we first state both of the theorems and sketch the proof of the latter.",
"For a Borel $G$ -space $X$ , let $\\mathcal {M}_G(X)$ denote the set of $G$ -invariant Borel probability measures on $X$ and let $\\mathcal {E}_G(X)$ denote the set of ergodic ones among those.",
"Clearly both are Borel subsets of $P(X)$ (the standard Borel space of Borel probability measures on $X$ ) and thus are themselves standard Borel spaces.",
"Ergodic Decomposition Theorem 7.1 (Farrell, Varadarajan) Let $X$ be a Borel $G$ -space.",
"If $\\mathcal {M}_G(X) \\ne \\mathbb {\\emptyset }$ (and hence $\\mathcal {E}_G(X) \\ne \\mathbb {\\emptyset }$ ), then there is a Borel surjection $x \\mapsto e_x$ from $X$ onto $\\mathcal {E}_G(X)$ such that: $x E_G y \\Rightarrow e_x = e_y$ ; For each $e \\in \\mathcal {E}_G(X)$ , if $X_e = \\lbrace x \\in X : e_x = e\\rbrace $ (hence $X_e$ is invariant Borel), then $e(X_e) = 1$ and $e \\!",
"\\!",
"\\downharpoonright _{X_e}$ is the unique ergodic invariant Borel probability measure on $X_e$ ; For each $\\mu \\in \\mathcal {M}_G(X)$ and $A \\in \\mathfrak {B}(X)$ , we have $\\mu (A) = \\int e_x(A) d\\mu (x).$ For the rest of the section, let $X$ be a Borel $\\mathbb {Z}$ -space.",
"For $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , if we let $h_e$ denote the entropy of $(X, \\mathbb {Z}, e)$ , then the map $e \\mapsto h_e$ is Borel.",
"Indeed, if $\\lbrace \\mathcal {P}_k\\rbrace _{k \\in \\mathbb {N}}$ is a refining sequence of partitions of $X$ that generates the Borel $\\sigma $ -algebra of $X$ , then by 4.1.2 of , $h_e = \\lim _{k \\rightarrow \\infty } h_e(\\mathcal {P}_k, \\mathbb {Z})$ , where $h_e(\\mathcal {P}_k, \\mathbb {Z})$ denotes the entropy of $\\mathcal {P}_k$ .",
"By 17.21 of , the function $e \\mapsto h_e(\\mathcal {P}_k)$ is Borel and thus so is the map $e \\mapsto h_e$ .",
"For all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ with $h_e < \\infty $ , let $N_e$ be the smallest integer such that $\\log N_e > h_e$ .",
"The map $e \\mapsto N_e$ is Borel because so is $e \\mapsto h_e$ .",
"Krieger's Finite Generator Theorem 7.2 (Uniform version) Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"Suppose $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Assume also that all measures in $\\mathcal {E}_{\\mathbb {Z}}(X)$ have finite entropy and let $e \\mapsto N_e$ be the map defined above.",
"Then there is a partition $\\lbrace A_n\\rbrace _{n \\le \\infty }$ of $X$ into Borel sets such that $A_{\\infty }$ is invariant and does not admit an invariant Borel probability measure; For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e \\setminus A_{\\infty }$ , where $X_e = \\rho ^{-1}(e)$ .",
"Sketch of Proof.",
"Note that it is enough to find a Borel invariant set $X^{\\prime } \\subseteq X$ and a Borel $\\mathbb {Z}$ -map $\\phi : X^{\\prime } \\rightarrow \\mathbb {N}^{\\mathbb {Z}}$ , such that for each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , we have $e(X \\setminus X^{\\prime }) = 0$ ; $\\phi \\!",
"\\!",
"\\downharpoonright _{X_e \\cap X^{\\prime }}$ is one-to-one and $\\phi (X_e \\cap X^{\\prime }) \\subseteq (N_e)^{\\mathbb {Z}}$ , where $(N_e)^{\\mathbb {Z}}$ is naturally viewed as a subset of $\\mathbb {N}^{\\mathbb {Z}}$ .",
"Indeed, assume we had such $X^{\\prime }$ and $\\phi $ , and let $A_{\\infty } = X \\setminus X^{\\prime }$ and $A_n = \\phi ^{-1}(V_n)$ for all $n \\in \\mathbb {N}$ , where $V_n = \\lbrace y \\in \\mathbb {N}^{\\mathbb {Z}} : y(0) = n\\rbrace $ .",
"Then it is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ satisfies (ii).",
"Also, (I) and part (ii) of the Ergodic Decomposition Theorem imply that (i) holds for $A_{\\infty }$ .",
"To construct such a $\\phi $ , we use the proof of Krieger's theorem presented in , Theorem 4.2.3, and we refer to it as Downarowicz's proof.",
"For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , the proof constructs a Borel $\\mathbb {Z}$ -embedding $\\phi _e : X^{\\prime } \\rightarrow N_e^{\\mathbb {Z}}$ on an $e$ -measure 1 set $X^{\\prime }$ .",
"We claim that this construction is uniform in $e$ in a Borel way and hence would yield $X^{\\prime }$ and $\\phi $ as above.",
"Our claim can be verified by inspection of Downarowicz's proof.",
"The proof uses the existence of sets with certain properties and one has to check that such sets exist with the properties satisfied for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ at once.",
"For example, the set $C$ used in the proof of Lemma 4.2.5 in can be chosen so that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $C \\cap X_e$ has the required properties for $e$ (using the Shannon-McMillan-Brieman theorem).",
"Another example is the set $B$ used in the proof of the same lemma, which is provided by Rohlin's lemma.",
"By inspection of the proof of Rohlin's lemma (see 2.1 in ), one can verify that we can get a Borel $B$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $B \\cap X_e$ has the required properties for $e$ .",
"The sets in these two examples are the only kind of sets whose existence is used in the whole proof; the rest of the proof constructs the required $\\phi $ “by hand”.",
"$\\Box $ Theorem 7.3 (Dichotomy I) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy; there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"We first show that the conditions above are mutually exclusive.",
"Indeed, assume there exist an invariant ergodic Borel probability measure $e$ with infinite entropy and a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"By ergodicity, $e$ would have to be supported on one of the $Y_n$ .",
"But $Y_n$ has a finite generator and hence the dynamical system $(Y_n, \\mathbb {Z}, e)$ has finite entropy by the Kolmogorov-Sinai theorem (see REF ).",
"Thus so does $(X, \\mathbb {Z}, e)$ since these two systems are isomorphic (modulo $e$ -NULL), contradicting the assumption on $e$ .",
"Now we prove that at least one of the conditions holds.",
"Assume that there is no invariant ergodic measure with infinite entropy.",
"Now, if there was no invariant Borel probability measure at all, then, since the answer to Question REF is assumed to be positive, $X$ would admit a finite generator, and we would be done.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ , $Y_{\\infty } = A_{\\infty }$ , and for all $n \\in \\mathbb {N}$ , $Y_n = \\lbrace x \\in X^{\\prime } : N_{e_x} = n\\rbrace ,$ where the map $e \\mapsto N_e$ is as above.",
"Note that the sets $Y_n$ are invariant since $\\rho $ is invariant, so $\\lbrace Y_n\\rbrace _{n \\le \\infty }$ is a countable partition of $X$ into invariant Borel sets.",
"Since $Y_{\\infty }$ does not admit an invariant Borel probability measure, by our assumption, it has a finite generator.",
"Let $E$ be the equivalence relation on $X^{\\prime }$ defined by $\\rho $ , i.e.",
"$\\forall x,y \\in X^{\\prime }$ , $x E y \\Leftrightarrow \\rho (x) = \\rho (y).$ By definition, $E$ is a smooth Borel equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action.",
"Thus, by Theorem REF , there exists a partition $\\mathcal {P}$ of $X^{\\prime }$ into 4 Borel sets such that $\\mathbb {Z}\\mathcal {P}$ separates any two points in different $E$ -classes.",
"Now fix $n \\in \\mathbb {N}$ and we will show that $I= \\mathcal {P}\\vee \\lbrace A_i\\rbrace _{i < n}$ is a generator for $Y_n$ .",
"Indeed, take distinct $x,y \\in Y_n$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}\\mathcal {P}$ separates them and hence so does $\\mathbb {Z}I$ .",
"Thus we can assume that $x E y$ .",
"Then $e := \\rho (x) = \\rho (y)$ , i.e.",
"$x,y \\in X_e = \\rho ^{-1}(e)$ .",
"By the choice of $\\lbrace A_i\\rbrace _{i \\in \\mathbb {N}}$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e$ and hence $\\mathbb {Z}\\lbrace A_i\\rbrace _{i < N_e}$ separates $x$ and $y$ .",
"But $n = N_e$ by the definition of $Y_n$ , so $\\mathbb {Z}I$ separates $x$ and $y$ .",
"Proposition 7.4 Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ admits invariant ergodic probability measures of arbitrarily large entropy, then it admits an invariant probability measure of infinite entropy.",
"For each $n \\ge 1$ , let $\\mu _n$ be an invariant ergodic probability measure of entropy $h_{\\mu _n} > n 2^n$ such that $\\mu _n \\ne \\mu _m$ for $n \\ne m$ , and put $\\mu = \\sum _{n \\ge 1} {1 \\over 2^n} \\mu _n.$ It is clear that $\\mu $ is an invariant probability measure, and we show that its entropy $h_{\\mu }$ is infinite.",
"Fix $n \\ge 1$ .",
"Let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem and put $X_n = \\rho ^{-1}(\\mu _n)$ .",
"It is clear that $\\mu _m(X_n) = 1$ if $m = n$ and 0 otherwise.",
"For any finite Borel partition $\\mathcal {P}= \\lbrace A_i\\rbrace _{i=1}^k$ of $X_n$ , put $A_0 = X \\setminus X_n$ and $\\bar{\\mathcal {P}} = \\mathcal {P}\\cup \\lbrace A_0\\rbrace $ .",
"Let $T$ be the Borel automorphism of $X$ corresponding to the action of $1_{\\mathbb {Z}}$ , and let $h_{\\nu }(I)$ and $h_{\\nu }(I, T)$ denote, respectively, the static and dynamic entropies of a finite Borel partition $I$ of $X$ with respect to an invariant probability measure $\\nu $ .",
"Then, with the convention that $\\log (0) \\cdot 0 = 0$ , we have $h_{\\mu }(\\bar{\\mathcal {P}}) &= - \\sum _{i=0}^k \\log (\\mu (A_i)) \\mu (A_i) \\ge - \\sum _{i = 1}^k \\log (\\mu (A_i)) \\mu (A_i)= - \\sum _{i = 1}^k \\log ({1 \\over 2^n}\\mu _n(A_i)) {1 \\over 2^n} \\mu _n(A_i) \\\\&\\ge - {1 \\over 2^n} \\sum _{i = 1}^k \\log (\\mu _n(A_i)) \\mu _n(A_i) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}).$ Since $\\mathcal {P}$ is arbitrary and $X_n$ is invariant, it follows that $h_{\\mu }(\\bar{\\mathcal {P}}, T) = \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu }(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) \\ge {1 \\over 2^n} \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu _n}(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}, T).$ Now for any finite Borel partition $I$ of $X$ , it is clear that $h_{\\mu _n}(I) = h_{\\mu _n}(\\bar{\\mathcal {P}})$ (and hence $h_{\\mu _n}(I, T) = h_{\\mu _n}(\\bar{\\mathcal {P}}, T)$ ), for some $\\mathcal {P}$ as above.",
"This implies that $h_{\\mu } \\ge \\sup _{\\mathcal {P}} h_{\\mu }(\\bar{\\mathcal {P}}, T) \\ge {1 \\over 2^n} \\sup _{\\mathcal {P}} h_{\\mu _n}(\\bar{\\mathcal {P}}, T) = {1 \\over 2^n} \\sup _{I} h_{\\mu _n}(I, T) = {1 \\over 2^n} h_{\\mu _n} > n,$ where $\\mathcal {P}$ and $I$ range over finite Borel partitions of $X_n$ and $X$ , respectively.",
"Thus $h_{\\mu }\\!",
"= \\infty $ .",
"Theorem 7.5 (Dichotomy II) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"The Kolmogorov-Sinai theorem implies that the conditions are mutually exclusive, and we prove that at least one of them holds.",
"Assume that there is no invariant measure with infinite entropy.",
"If there was no invariant Borel probability measure at all, then, by our assumption, $X$ would admit a finite generator.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ and $X_e = \\rho ^{-1}(e)$ , for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ .",
"By our assumption, $A_{\\infty }$ admits a finite generator $\\mathcal {P}$ .",
"Also, by REF , there is $N \\ge 1$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $N_e \\le N$ and hence $\\mathcal {Q}:= \\lbrace A_n\\rbrace _{n < N}$ is a finite generator for $X_e$ ; in particular, $\\mathcal {Q}$ is a partition of $X^{\\prime }$ .",
"Let $E$ be the following equivalence relation on $X$ : $x E y \\Leftrightarrow (x, y \\in A_{\\infty }) \\vee (x,y \\in X^{\\prime } \\wedge \\rho (x) = \\rho (y)).$ By definition, $E$ is a smooth equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action and $A_{\\infty }$ is $\\mathbb {Z}$ -invariant.",
"Thus, by Theorem REF , there exists a partition $J$ of $X$ into 4 Borel sets such that $\\mathbb {Z}J$ separates any two points in different $E$ -classes.",
"We now show that $I:= < \\!\\!",
"J\\cup \\mathcal {P}\\cup \\mathcal {Q} \\!\\!",
">$ is a generator.",
"Indeed, fix distinct $x,y \\in X$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}J$ separates them.",
"So we can assume that $x E y$ .",
"If $x,y \\in A_{\\infty }$ , then $\\mathbb {Z}\\mathcal {P}$ separates $x$ and $y$ .",
"Finally, if $x,y \\in X^{\\prime }$ , then $x,y \\in X_e$ , where $e = \\rho (x)$ ($= \\rho (y)$ ), and hence $\\mathbb {Z}\\mathcal {Q}$ separates $x$ and $y$ .",
"Remark.",
"It is likely that the above dichotomies are also true for any amenable group using a uniform version of Krieger's theorem for amenable groups, cf.",
", but I have not checked the details.",
"Finite generators on comeager sets Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER.",
"Locally weakly wandering sets and other special cases Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
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],
[
"Finitely traveling sets",
"Let $X$ be a Borel $G$ -space.",
"Definition 5.1 Let $A,B \\in \\mathfrak {B}(X)$ be equidecomposable, i.e.",
"there are $N \\le \\infty $ , $\\lbrace g_n\\rbrace _{n < N} \\subseteq G$ and Borel partitions $\\lbrace A_n\\rbrace _{n < N}$ and $\\lbrace B_n\\rbrace _{n < N}$ of $A$ and $B$ , respectively, such that $g_n A_n = B_n$ for all $n < N$ .",
"$A,B$ are said to be locally finitely equidecomposable (denote by $A \\sim _{\\text{lfin}} B$ ), if $\\lbrace A_n\\rbrace _{n < N},\\lbrace B_n\\rbrace _{n < N},\\lbrace g_n\\rbrace _{n < N}$ can be taken so that for every $x \\in A$ , $A_n \\cap [x]_G = \\mathbb {\\emptyset }$ for all but finitely many $n<N$ ; finitely equidecomposable (denote by $A \\sim _{\\text{fin}} B$ ), if $N$ can be taken to be finite.",
"The notation $\\prec _{\\text{fin}}$ , $\\prec _{\\text{lfin}}$ and the notions of finite and locally finite compressibility are defined analogous to Definitions REF and REF .",
"Definition 5.2 A Borel set $A \\subseteq X$ is called (locally) finitely traveling if there exists pairwise disjoint Borel sets $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ such that $A_0 = A$ and $A \\sim _{\\text{fin}} A_n$ ($A \\sim _{\\text{lfin}} A_n$ ), $\\forall n \\in \\mathbb {N}$ .",
"Proposition 5.3 If $X$ is (locally) finitely compressible then $X$ admits a (locally) finitely traveling Borel complete section.",
"We prove for finitely compressible $X$ , but note that everything below is also locally valid (i.e.",
"restricted to every orbit) for a locally compressible $X$ .",
"Run the proof of the first part of Lemma REF noting that a witnessing map $\\gamma : X \\rightarrow G$ of finite compressibility of $X$ has finite image and hence the image of each $\\delta _n$ (in the notation of the proof) is finite, which implies that the obtained traveling set $A$ is actually finitely traveling.",
"Proposition 5.4 If $X$ admits a locally finitely traveling Borel complete section, then $X \\in 4$ .",
"Let $A$ be a locally finitely traveling Borel complete section and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be as in Definition REF .",
"Let $I_n = \\lbrace C_k^n\\rbrace _{k \\in \\mathbb {N}}$ , $J_n = \\lbrace D_k^n\\rbrace _{k \\in \\mathbb {N}}$ be Borel partitions of $A$ and $A_n$ , respectively, that together with $\\lbrace g_k^n\\rbrace _{k \\in \\mathbb {N}} \\subseteq G$ witness $A \\sim _{\\text{lfin}} A_n$ (as in Definition REF ).",
"Let $\\mathcal {B}$ denote the Boolean $G$ -algebra generated by $\\lbrace X\\rbrace \\cup \\bigcup _{n \\in \\mathbb {N}} (I_n \\cup J_n \\cup \\lbrace A_n\\rbrace )$ .",
"Now assume for contradiction that $X \\notin 4$ and hence, $A \\notin 4$ .",
"Thus, applying Corollary REF to $A$ and $\\mathcal {B}$ , we get a $G$ -invariant finitely additive probability measure $\\mu $ on $\\mathcal {B}$ with $\\mu (A)>0$ .",
"Moreover, there is $x \\in A$ such that $\\forall B \\in \\mathcal {B}$ with $B \\cap [x]_G = \\mathbb {\\emptyset }$ , $\\mu (B) = 0$ .",
"Claim $\\mu (A_n) = \\mu (A)$ , for all $n \\in \\mathbb {N}$ .",
"Proof of Claim.",
"For each $n$ , let $\\lbrace C_{k_i}^n\\rbrace _{i < K_n}$ be the list of those $C_k^n$ such that $C_k^n \\cap [x]_G \\ne \\mathbb {\\emptyset }$ ($K_n < \\infty $ by the definition of locally finitely traveling).",
"Set $B = A \\setminus (\\bigcup _{i < K_n} C_{k_i}^n)$ and note that by finite additivity of $\\mu $ , $\\mu (A) = \\mu (B) + \\sum _{i < K_n} \\mu (C_{k_i}^n).$ Similarly, set $B^{\\prime } = A_n \\setminus (\\bigcup _{i < K_n} D_{k_i}^n)$ and hence $\\mu (A_n) = \\mu (B^{\\prime }) + \\sum _{i < K_n} \\mu (D_{k_i}^n).$ But $B \\cap [x]_G = \\mathbb {\\emptyset }$ and $B^{\\prime } \\cap [x]_G = \\mathbb {\\emptyset }$ , and thus $\\mu (B) = \\mu (B^{\\prime }) = 0$ .",
"Also, since $g_{k_i}^n C_{k_i}^n = D_{k_i}^n$ and $\\mu $ is $G$ -invariant, $\\mu (C_{k_i}^n) = \\mu (D_{k_i}^n)$ .",
"Therefore $\\mu (A) = \\sum _{i < K_n} \\mu (C_{k_i}^n) = \\sum _{i < K_n} \\mu (D_{k_i}^n) = \\mu (A_n).$ $\\dashv $ This claim contradicts $\\mu $ being a probability measure since for large enough $N$ , $\\mu (\\bigcup _{n < N} A_n) = N \\mu (A) > 1$ , contradicting $\\mu (X) = 1$ .",
"This, together with REF , implies the following.",
"Corollary 5.5 Let $X$ be a Borel $G$ -space.",
"If $X$ admits a locally finitely traveling Borel complete section, then there is a Borel 32-generator."
],
[
"Separating smooth-many invariant sets",
"Assume throughout that $X$ is a Borel $G$ -space.",
"Lemma 6.1 If $X$ is aperiodic then it admits a countably infinite partition into Borel complete sections.",
"The following argument is also given in the proof of Theorem 13.1 in .",
"By the marker lemma (see 6.7 in ), there exists a vanishing sequence $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of decreasing Borel complete sections, i.e.",
"$\\bigcap _{n \\in \\mathbb {N}} B_n = \\mathbb {\\emptyset }$ .",
"For each $n \\in \\mathbb {N}$ , define $k_n : X \\rightarrow \\mathbb {N}$ recursively as follows: $\\left\\lbrace \\begin{array}{rcl}k_0(x) & = & 0 \\\\k_{n+1}(x) &= & min \\lbrace k \\in \\mathbb {N}: B_{k_n(x)} \\cap [x]_G \\nsubseteq B_k\\rbrace \\end{array}\\right.,$ and define $A_n \\subseteq X$ by $x \\in A_n \\Leftrightarrow x \\in A_{k_n(x)} \\setminus A_{k_{n+1}(x)}.$ It is straightforward to check that $A_n$ are pairwise disjoint Borel complete sections.",
"For $A \\in \\mathfrak {B}(X)$ , if $I= < \\!\\!",
"A \\!\\!",
">$ then we use the notation $F_A$ and $f_A$ instead of $F_{I}$ and $, respectively.$ We now work towards strengthening the above lemma to yield a countably infinite partition into $F_A$ -invariant Borel complete sections.",
"Definition 6.2 (Aperiodic separation) For Borel sets $A, Y \\subseteq X$ , we say that $A$ aperiodically separates $Y$ if $f_A([Y]_G)$ is aperiodic (as an invariant subset of the shift $2^G$ ).",
"If such $A$ exists, we say that $Y$ is aperiodically separable.",
"Proposition 6.3 For $A \\in \\mathfrak {B}(X)$ , if $A$ aperiodically separates $X$ , then $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Let $Y = \\lbrace y \\in 2^G : |[y]_G| = \\infty \\rbrace $ and hence $f_A(X)$ is a $G$ -invariant subset of $Y$ .",
"By Lemma REF applied to $Y$ , there is a partition $\\lbrace B_n\\rbrace _{n \\in \\mathbb {N}}$ of $Y$ into Borel complete sections.",
"Thus $A_n = f_{I}^{-1}(B_n)$ is a Borel $F_A$ -invariant complete section for $X$ and $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ is a partition of $X$ .",
"Let $\\mathfrak {A}$ denote the collection of all subsets of aperiodically separable Borel sets.",
"Lemma 6.4 $\\mathfrak {A}$ is a $\\sigma $ -ideal.",
"We only have to show that if $Y_n$ are aperiodically separable Borel sets, then $Y = \\bigcup _{n \\in \\mathbb {N}} Y_n \\in \\mathfrak {A}$ .",
"Let $A_n$ be a Borel set aperiodically separating $Y_n$ .",
"Since $A_n$ also aperiodically separates $[Y_n]_G$ (by definition), we can assume that $Y_n$ is $G$ -invariant.",
"Furthermore, by taking $Y_n^{\\prime } = Y_n \\setminus \\bigcup _{k<n} Y_k$ , we can assume that $Y_n$ are pairwise disjoint.",
"Now letting $A = \\bigcup _{n \\in \\mathbb {N}} (A_n \\cap Y_n)$ , it is easy to check that $A$ aperiodically separates $Y$ .",
"Let $\\mathfrak {S}$ denote the collection of all subsets of smooth sets.",
"By a similar argument as the one above, $\\mathfrak {S}$ is a $\\sigma $ -ideal.",
"Lemma 6.5 If $X$ is aperiodic, then $\\mathfrak {S}\\subseteq \\mathfrak {A}$ .",
"Let $S \\in \\mathfrak {S}$ and hence there is a Borel transversal $T$ for $[S]_G$ .",
"Fix $x \\in S$ and let $y \\ne z \\in [x]_G$ .",
"Since $T$ is a transversal, there is $g \\in G$ such that $gy \\in T$ , and hence $gz \\notin T$ .",
"Thus $f_T(y) \\ne f_T(z)$ , and so $f_T([x]_G)$ is infinite.",
"Therefore $T$ aperiodically separates $[S]_G$ .",
"For the rest of the section, fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and let $F_A^n$ be following equivalence relation: $y F_A^n z \\Leftrightarrow \\forall k < n (g_k y \\in A \\leftrightarrow g_k z \\in A).$ Note that $F_A^n$ has no more than $2^n$ equivalence classes and that $y F_A z$ if and only if $\\forall n (y F_A^n z)$ .",
"Lemma 6.6 For $A,Y \\in \\mathfrak {B}(X)$ , $A$ aperiodically separates $Y$ if and only if $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in Y^{[x]_G}) [y F_A^n z \\wedge \\lnot (y F_A z)]$ .",
"$\\Rightarrow $ : Assume that for all $x \\in Y$ , $f_A([x]_G)$ is infinite and thus $F_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ has infinitely many equivalence classes.",
"Fix $n \\in \\mathbb {N}$ and recall that $F_A^n$ has only finitely many equivalence classes.",
"Thus, by the Pigeon Hole Principle, there are $y,z \\in Y^{[x]_G}$ such that $y F_A^n z$ yet $\\lnot (y F_A z)$ .",
"$\\Leftarrow $ : Assume for contradiction that $f_A(Y^{[x]_G})$ is finite for some $x \\in Y$ .",
"Then it follows that $F_A = F_A^n$ , for some $n$ , and hence for any $y,z \\in Y^{[x]_G}$ , $y F_A^n z$ implies $y F_A z$ , contradicting the hypothesis.",
"Theorem 6.7 If $X$ is an aperiodic Borel $G$ -space, then $X \\in \\mathfrak {A}$ .",
"By Lemma REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into Borel complete sections.",
"We will inductively construct Borel sets $B_n \\subseteq C_n$ , where $C_n$ should be thought of as the set of points colored (black or white) at the $n^{th}$ step, and $B_n$ as the set of points colored black (thus $C_n \\setminus B_n$ is colored white).",
"Define a function $\\# : X \\rightarrow \\mathbb {N}$ by $x \\mapsto m$ , where $m$ is such that $x \\in A_m$ .",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ of sets generating the Borel $\\sigma $ -algebra of $X$ .",
"Assuming that for all $k < n$ , $C_k, B_k$ are defined, let $\\bar{C}_n = \\bigcup _{k<n} C_k$ and $\\bar{B}_n = \\bigcup _{k<n} B_k$ .",
"Put $P_n = \\lbrace x \\in A_0 : \\forall k < n (g_k x \\in \\bar{C}_n) \\wedge g_n x \\notin \\bar{C}_n\\rbrace $ and set $F_n = F_{\\bar{B}_n}^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ , that is for all $x,y \\in P_n$ , $y F_n z \\Leftrightarrow \\forall k < n (g_k y \\in \\bar{B}_n \\leftrightarrow g_k z \\in \\bar{B}_n).$ Now put $C^{\\prime }_n = \\lbrace x \\in P_n : \\#(g_n x) = \\min \\#((g_nP_n)^{[x]_G})\\rbrace $ , $C^{\\prime \\prime }_n = \\lbrace x \\in C^{\\prime }_n : \\exists y, z \\in (C^{\\prime }_n)^{[x]_G} (y \\ne z \\wedge y F_n z)\\rbrace $ and $C_n = g_n C^{\\prime \\prime }_n$ .",
"Note that it follows from the definition of $P_n$ that $C_n$ is disjoint from $\\bar{C}_n$ .",
"Now in order to define $B_n$ , first define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{ the smallest $m$ such that there are } y,z \\in C^{\\prime \\prime }_n \\cap [x]_G \\text{ with } y F_n z, y \\in U_m \\text{ and } z \\notin U_m.$ Note that $\\bar{n}$ is Borel and $G$ -invariant.",
"Lastly, let $B^{\\prime }_n = \\lbrace x \\in C^{\\prime \\prime }_n : x \\in U_{\\bar{n}(x)}\\rbrace $ and $B_n = g_n B^{\\prime }_n$ .",
"Clearly $B_n \\subseteq C_n$ .",
"Now let $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $D = \\left[\\bigcup _{n \\in \\mathbb {N}} (C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)\\right]_G$ .",
"We show that $B$ aperiodically separates $Y := X \\setminus D$ and $D \\in \\mathfrak {S}$ .",
"Since $\\mathfrak {S}\\subseteq \\mathfrak {A}$ and $\\mathfrak {A}$ is an ideal, this will imply that $X \\in \\mathfrak {A}$ .",
"Claim 1 $D \\in \\mathfrak {S}$ .",
"Proof of Claim.",
"Since $\\mathfrak {S}$ is a $\\sigma $ -ideal, it is enough to show that for each $n$ , $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G \\in \\mathfrak {S}$ , so fix $n \\in \\mathbb {N}$ .",
"Clearly $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ is finite, for all $x \\in X$ , since there can be at most $2^n$ pairwise $F_n$ -nonequivalent points.",
"Thus, fixing some Borel linear ordering of $X$ and taking the smallest element from $(C^{\\prime }_n \\setminus C^{\\prime \\prime }_n)^{[x]_G}$ for each $x \\in C^{\\prime }_n \\setminus C^{\\prime \\prime }_n$ , we can define a Borel transversal for $[C^{\\prime }_n \\setminus C^{\\prime \\prime }_n]_G$ .",
"$\\dashv $ By Lemma REF , to show that $B$ aperiodically separates $Y$ , it is enough to show that $(\\forall x \\in Y) (\\forall n) (\\exists y,z \\in [x]_G) [y F_B^n z \\wedge \\lnot (y F_B z)]$ .",
"Fix $x \\in Y$ .",
"Claim 2 $(\\exists ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Assume for contradiction that $(\\forall ^{\\infty } n) (C^{\\prime \\prime }_n)^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $x \\notin D$ , it follows that $(\\forall ^{\\infty } n) P_n^{[x]_G} = \\mathbb {\\emptyset }$ .",
"Since $A_0$ is a complete section and $\\bar{C}_0 = \\mathbb {\\emptyset }$ , $P_0^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Let $N$ be the largest number such that $P_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus for all $n > N$ , $C_n^{[x]_G} = \\mathbb {\\emptyset }$ and hence for all $n > N$ , $\\bar{C}_n^{[x]_G} = \\bar{C}_{N+1}^{[x]_G}$ .",
"Because $C_N^{[x]_G} \\ne \\mathbb {\\emptyset }$ , there is $y \\in A_0^{[x]_G}$ such that $\\forall k \\le N (g_k y \\in \\bar{C}_{N+1})$ ; but because $P_{N+1}^{[x]_G} = \\mathbb {\\emptyset }$ , $g_{N+1} y$ must also fall into $\\bar{C}_{N+1}$ .",
"By induction on $n > N$ , we get that for all $n>N$ , $g_n y \\in \\bar{C}_n$ and thus $g_n y \\in \\bar{C}_{N+1}$ .",
"On the other hand, it follows from the definition of $C^{\\prime }_n$ that for each $n$ , $(C^{\\prime }_n)^{[x]_G}$ intersects exactly one of $A_k$ .",
"Thus $\\bar{C}_{N+1}^{[x]_G}$ intersects at most $N+1$ of $A_k$ and hence there exists $K \\in \\mathbb {N}$ such that for all $k \\ge K$ , $\\bar{C}_{N+1}^{[x]_G} \\cap A_k = \\mathbb {\\emptyset }$ .",
"Since $\\exists ^{\\infty } n (g_n y \\in \\bigcup _{k \\ge K} A_k)$ , $\\exists ^{\\infty } n (g_n y \\notin \\bar{C}_{N+1})$ , a contradiction.",
"$\\dashv $ Now it remains to show that for all $n \\in \\mathbb {N}$ , $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ implies that $\\exists y,z \\in [x]_G$ such that $y F_B^n z$ but $\\lnot (y F_B z)$ .",
"To this end, fix $n \\in \\mathbb {N}$ and assume $(C^{\\prime \\prime }_n)^{[x]_G} \\ne \\mathbb {\\emptyset }$ .",
"Thus there are $y,z \\in (C^{\\prime \\prime }_n)^{[x]_G}$ such that $y F_n z$ , $y \\in U_{\\bar{n}(x)}$ and $z \\notin U_{\\bar{n}(x)}$ ; hence, $g_n y \\in B_n$ and $g_n z \\notin B_n$ , by the definition of $B_n$ .",
"Since $C_k$ are pairwise disjoint, $B_n \\subseteq C_n$ and $g_n y, g_n z \\in C_n$ , it follows that $g_n y \\in B$ and $g_n z \\notin B$ , and therefore $\\lnot (y F_B z)$ .",
"Finally, note that $F_n = F_B^n \\!",
"\\!",
"\\downharpoonright _{P_n}$ and hence $y F_B^n z$ .",
"Corollary 6.8 Suppose all of the nontrivial subgroups of $G$ have finite index (e.g.",
"$G = \\mathbb {Z}$ ), and let $X$ be an aperiodic Borel $G$ -space.",
"Then there exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit, i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in X$ .",
"Let $A$ be a Borel set aperiodically separating $X$ (exists by Theorem REF ) and put $Y = f_A(X)$ .",
"Then $Y \\subseteq 2^G$ is aperiodic and hence the action of $G$ on $Y$ is free since the stabilizer subgroup of every element must have infinite index and thus is trivial.",
"But this implies that for all $y \\in Y$ , $f_A^{-1}(y)$ intersects every orbit in $X$ at no more than one point, and hence $f_A$ is one-to-one on every orbit.",
"From REF and REF we immediately get the following strengthening of Lemma REF .",
"Corollary 6.9 If $X$ is aperiodic, then for some $A \\in \\mathfrak {B}(X)$ , $X$ admits a countably infinite partition into Borel $F_A$ -invariant complete sections.",
"Theorem 6.10 Let $X$ be an aperiodic $G$ -space and let $E$ be a smooth equivalence relation on $X$ with $E_G \\subseteq E$ .",
"There exists a partition $\\mathcal {P}$ of $X$ into 4 Borel sets such that $G \\mathcal {P}$ separates any two $E$ -nonequivalent points in $X$ , i.e.",
"$\\forall x,y \\in X (\\lnot (x E y) \\rightarrow f_{\\mathcal {P}}(x) \\ne f_{\\mathcal {P}}(y))$ .",
"By Corollary REF , there is $A \\in \\mathfrak {B}(X)$ and a Borel partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant complete sections.",
"For each $n \\in \\mathbb {N}$ , define a function $\\bar{n}: X \\rightarrow \\mathbb {N}$ by $x \\mapsto \\text{the smallest $m$ such that } \\exists x^{\\prime } \\in A_0^{[x]_G} \\text{ with } g_m x^{\\prime } \\in A_n.$ Clearly $\\bar{n}$ is Borel, and because all of $A_k$ are $F_A$ -invariant, $\\bar{n}$ is also $F_A$ -invariant, i.e.",
"for all $x,y \\in X$ , $x F_A y \\rightarrow \\bar{n}(x) = \\bar{n}(y)$ .",
"Also, $\\bar{n}$ is $G$ -invariant by definition.",
"Put $A^{\\prime }_n = \\lbrace x \\in A_0 : g_{\\bar{n}(x)} x \\in A_n\\rbrace $ and note that $A^{\\prime }_n$ is $F_A$ -invariant Borel since so are $\\bar{n}$ , $A_0$ and $A_n$ .",
"Moreover, $A^{\\prime }_n$ is clearly a complete section.",
"Define $\\gamma _n : A^{\\prime }_n \\rightarrow A_n$ by $x \\mapsto g_{\\bar{n}(x)} x$ .",
"Clearly, $\\gamma _n$ is Borel and one-to-one.",
"Since $E$ is smooth, there is a Borel $h : X \\rightarrow \\mathbb {R}$ such that for all $x,y \\in X$ , $x E y \\leftrightarrow h(x) = h(y)$ .",
"Let $\\lbrace V_n\\rbrace _{n \\in \\mathbb {N}}$ be a countable family of subsets of $\\mathbb {R}$ generating the Borel $\\sigma $ -algebra of $\\mathbb {R}$ and put $U_n = h^{-1}(V_n)$ .",
"Because each equivalence class of $E$ is $G$ -invariant, so is $h$ and hence so is $U_n$ .",
"Now let $B_n = \\gamma _n(A^{\\prime }_n \\cap U_n)$ and note that $B_n$ is Borel being a one-to-one Borel image of a Borel set.",
"It follows from the definition of $\\gamma _n$ that $B_n \\subseteq A_n$ .",
"Put $B = \\bigcup _{n \\in \\mathbb {N}} B_n$ and $\\mathcal {P}= < \\!\\!",
"A,B \\!\\!",
">$ ; in particular, $|\\mathcal {P}| \\le 4$ .",
"We show that $\\mathcal {P}$ is what we want.",
"To this end, fix $x,y \\in X$ with $\\lnot (x E y)$ .",
"If $\\lnot (x F_A y)$ , then $G < \\!\\!",
"A \\!\\!",
">$ (and hence $G \\mathcal {P}$ ) separates $x$ and $y$ .",
"Thus assume that $x F_A y$ .",
"Since $h(x) \\ne h(y)$ , there is $n$ such that $h(x) \\in V_n$ and $h(y) \\notin V_n$ .",
"Hence, by invariance of $U_n$ , $gx \\in U_n \\wedge gy \\notin U_n$ , for all $g \\in G$ .",
"Because $A^{\\prime }_n$ is a complete section, there is $g \\in G$ such that $gx \\in A^{\\prime }_n$ and hence $gy \\in A^{\\prime }_n$ since $A^{\\prime }_n$ is $F_A$ -invariant.",
"Let $m = \\bar{n}(gx)$ ($= \\bar{n}(gy)$ ).",
"Then $g_m gx \\in B_n$ while $g_m gy \\notin B_n$ although $g_m gy \\in \\gamma _n(A^{\\prime }_n) \\subseteq A_n$ .",
"Thus $g_m gx \\in B$ but $g_m gy \\notin B$ and therefore $G \\mathcal {P}$ separates $x$ and $y$ ."
],
[
"Potential dichotomy theorems",
"In this section we prove dichotomy theorems assuming Weiss's question has a positive answer for $G = \\mathbb {Z}$ .",
"In the proofs we use the Ergodic Decomposition Theorem (see , ) and a Borel/uniform version of Krieger's finite generator theorem, so we first state both of the theorems and sketch the proof of the latter.",
"For a Borel $G$ -space $X$ , let $\\mathcal {M}_G(X)$ denote the set of $G$ -invariant Borel probability measures on $X$ and let $\\mathcal {E}_G(X)$ denote the set of ergodic ones among those.",
"Clearly both are Borel subsets of $P(X)$ (the standard Borel space of Borel probability measures on $X$ ) and thus are themselves standard Borel spaces.",
"Ergodic Decomposition Theorem 7.1 (Farrell, Varadarajan) Let $X$ be a Borel $G$ -space.",
"If $\\mathcal {M}_G(X) \\ne \\mathbb {\\emptyset }$ (and hence $\\mathcal {E}_G(X) \\ne \\mathbb {\\emptyset }$ ), then there is a Borel surjection $x \\mapsto e_x$ from $X$ onto $\\mathcal {E}_G(X)$ such that: $x E_G y \\Rightarrow e_x = e_y$ ; For each $e \\in \\mathcal {E}_G(X)$ , if $X_e = \\lbrace x \\in X : e_x = e\\rbrace $ (hence $X_e$ is invariant Borel), then $e(X_e) = 1$ and $e \\!",
"\\!",
"\\downharpoonright _{X_e}$ is the unique ergodic invariant Borel probability measure on $X_e$ ; For each $\\mu \\in \\mathcal {M}_G(X)$ and $A \\in \\mathfrak {B}(X)$ , we have $\\mu (A) = \\int e_x(A) d\\mu (x).$ For the rest of the section, let $X$ be a Borel $\\mathbb {Z}$ -space.",
"For $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , if we let $h_e$ denote the entropy of $(X, \\mathbb {Z}, e)$ , then the map $e \\mapsto h_e$ is Borel.",
"Indeed, if $\\lbrace \\mathcal {P}_k\\rbrace _{k \\in \\mathbb {N}}$ is a refining sequence of partitions of $X$ that generates the Borel $\\sigma $ -algebra of $X$ , then by 4.1.2 of , $h_e = \\lim _{k \\rightarrow \\infty } h_e(\\mathcal {P}_k, \\mathbb {Z})$ , where $h_e(\\mathcal {P}_k, \\mathbb {Z})$ denotes the entropy of $\\mathcal {P}_k$ .",
"By 17.21 of , the function $e \\mapsto h_e(\\mathcal {P}_k)$ is Borel and thus so is the map $e \\mapsto h_e$ .",
"For all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ with $h_e < \\infty $ , let $N_e$ be the smallest integer such that $\\log N_e > h_e$ .",
"The map $e \\mapsto N_e$ is Borel because so is $e \\mapsto h_e$ .",
"Krieger's Finite Generator Theorem 7.2 (Uniform version) Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"Suppose $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Assume also that all measures in $\\mathcal {E}_{\\mathbb {Z}}(X)$ have finite entropy and let $e \\mapsto N_e$ be the map defined above.",
"Then there is a partition $\\lbrace A_n\\rbrace _{n \\le \\infty }$ of $X$ into Borel sets such that $A_{\\infty }$ is invariant and does not admit an invariant Borel probability measure; For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e \\setminus A_{\\infty }$ , where $X_e = \\rho ^{-1}(e)$ .",
"Sketch of Proof.",
"Note that it is enough to find a Borel invariant set $X^{\\prime } \\subseteq X$ and a Borel $\\mathbb {Z}$ -map $\\phi : X^{\\prime } \\rightarrow \\mathbb {N}^{\\mathbb {Z}}$ , such that for each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , we have $e(X \\setminus X^{\\prime }) = 0$ ; $\\phi \\!",
"\\!",
"\\downharpoonright _{X_e \\cap X^{\\prime }}$ is one-to-one and $\\phi (X_e \\cap X^{\\prime }) \\subseteq (N_e)^{\\mathbb {Z}}$ , where $(N_e)^{\\mathbb {Z}}$ is naturally viewed as a subset of $\\mathbb {N}^{\\mathbb {Z}}$ .",
"Indeed, assume we had such $X^{\\prime }$ and $\\phi $ , and let $A_{\\infty } = X \\setminus X^{\\prime }$ and $A_n = \\phi ^{-1}(V_n)$ for all $n \\in \\mathbb {N}$ , where $V_n = \\lbrace y \\in \\mathbb {N}^{\\mathbb {Z}} : y(0) = n\\rbrace $ .",
"Then it is clear that $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ satisfies (ii).",
"Also, (I) and part (ii) of the Ergodic Decomposition Theorem imply that (i) holds for $A_{\\infty }$ .",
"To construct such a $\\phi $ , we use the proof of Krieger's theorem presented in , Theorem 4.2.3, and we refer to it as Downarowicz's proof.",
"For each $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , the proof constructs a Borel $\\mathbb {Z}$ -embedding $\\phi _e : X^{\\prime } \\rightarrow N_e^{\\mathbb {Z}}$ on an $e$ -measure 1 set $X^{\\prime }$ .",
"We claim that this construction is uniform in $e$ in a Borel way and hence would yield $X^{\\prime }$ and $\\phi $ as above.",
"Our claim can be verified by inspection of Downarowicz's proof.",
"The proof uses the existence of sets with certain properties and one has to check that such sets exist with the properties satisfied for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ at once.",
"For example, the set $C$ used in the proof of Lemma 4.2.5 in can be chosen so that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $C \\cap X_e$ has the required properties for $e$ (using the Shannon-McMillan-Brieman theorem).",
"Another example is the set $B$ used in the proof of the same lemma, which is provided by Rohlin's lemma.",
"By inspection of the proof of Rohlin's lemma (see 2.1 in ), one can verify that we can get a Borel $B$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $B \\cap X_e$ has the required properties for $e$ .",
"The sets in these two examples are the only kind of sets whose existence is used in the whole proof; the rest of the proof constructs the required $\\phi $ “by hand”.",
"$\\Box $ Theorem 7.3 (Dichotomy I) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant ergodic Borel probability measure with infinite entropy; there exists a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"We first show that the conditions above are mutually exclusive.",
"Indeed, assume there exist an invariant ergodic Borel probability measure $e$ with infinite entropy and a partition $\\lbrace Y_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into invariant Borel sets such that each $Y_n$ has a finite generator.",
"By ergodicity, $e$ would have to be supported on one of the $Y_n$ .",
"But $Y_n$ has a finite generator and hence the dynamical system $(Y_n, \\mathbb {Z}, e)$ has finite entropy by the Kolmogorov-Sinai theorem (see REF ).",
"Thus so does $(X, \\mathbb {Z}, e)$ since these two systems are isomorphic (modulo $e$ -NULL), contradicting the assumption on $e$ .",
"Now we prove that at least one of the conditions holds.",
"Assume that there is no invariant ergodic measure with infinite entropy.",
"Now, if there was no invariant Borel probability measure at all, then, since the answer to Question REF is assumed to be positive, $X$ would admit a finite generator, and we would be done.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ , $Y_{\\infty } = A_{\\infty }$ , and for all $n \\in \\mathbb {N}$ , $Y_n = \\lbrace x \\in X^{\\prime } : N_{e_x} = n\\rbrace ,$ where the map $e \\mapsto N_e$ is as above.",
"Note that the sets $Y_n$ are invariant since $\\rho $ is invariant, so $\\lbrace Y_n\\rbrace _{n \\le \\infty }$ is a countable partition of $X$ into invariant Borel sets.",
"Since $Y_{\\infty }$ does not admit an invariant Borel probability measure, by our assumption, it has a finite generator.",
"Let $E$ be the equivalence relation on $X^{\\prime }$ defined by $\\rho $ , i.e.",
"$\\forall x,y \\in X^{\\prime }$ , $x E y \\Leftrightarrow \\rho (x) = \\rho (y).$ By definition, $E$ is a smooth Borel equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action.",
"Thus, by Theorem REF , there exists a partition $\\mathcal {P}$ of $X^{\\prime }$ into 4 Borel sets such that $\\mathbb {Z}\\mathcal {P}$ separates any two points in different $E$ -classes.",
"Now fix $n \\in \\mathbb {N}$ and we will show that $I= \\mathcal {P}\\vee \\lbrace A_i\\rbrace _{i < n}$ is a generator for $Y_n$ .",
"Indeed, take distinct $x,y \\in Y_n$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}\\mathcal {P}$ separates them and hence so does $\\mathbb {Z}I$ .",
"Thus we can assume that $x E y$ .",
"Then $e := \\rho (x) = \\rho (y)$ , i.e.",
"$x,y \\in X_e = \\rho ^{-1}(e)$ .",
"By the choice of $\\lbrace A_i\\rbrace _{i \\in \\mathbb {N}}$ , $\\lbrace A_n \\cap X_e\\rbrace _{n < N_e}$ is a generator for $X_e$ and hence $\\mathbb {Z}\\lbrace A_i\\rbrace _{i < N_e}$ separates $x$ and $y$ .",
"But $n = N_e$ by the definition of $Y_n$ , so $\\mathbb {Z}I$ separates $x$ and $y$ .",
"Proposition 7.4 Let $X$ be a Borel $\\mathbb {Z}$ -space.",
"If $X$ admits invariant ergodic probability measures of arbitrarily large entropy, then it admits an invariant probability measure of infinite entropy.",
"For each $n \\ge 1$ , let $\\mu _n$ be an invariant ergodic probability measure of entropy $h_{\\mu _n} > n 2^n$ such that $\\mu _n \\ne \\mu _m$ for $n \\ne m$ , and put $\\mu = \\sum _{n \\ge 1} {1 \\over 2^n} \\mu _n.$ It is clear that $\\mu $ is an invariant probability measure, and we show that its entropy $h_{\\mu }$ is infinite.",
"Fix $n \\ge 1$ .",
"Let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem and put $X_n = \\rho ^{-1}(\\mu _n)$ .",
"It is clear that $\\mu _m(X_n) = 1$ if $m = n$ and 0 otherwise.",
"For any finite Borel partition $\\mathcal {P}= \\lbrace A_i\\rbrace _{i=1}^k$ of $X_n$ , put $A_0 = X \\setminus X_n$ and $\\bar{\\mathcal {P}} = \\mathcal {P}\\cup \\lbrace A_0\\rbrace $ .",
"Let $T$ be the Borel automorphism of $X$ corresponding to the action of $1_{\\mathbb {Z}}$ , and let $h_{\\nu }(I)$ and $h_{\\nu }(I, T)$ denote, respectively, the static and dynamic entropies of a finite Borel partition $I$ of $X$ with respect to an invariant probability measure $\\nu $ .",
"Then, with the convention that $\\log (0) \\cdot 0 = 0$ , we have $h_{\\mu }(\\bar{\\mathcal {P}}) &= - \\sum _{i=0}^k \\log (\\mu (A_i)) \\mu (A_i) \\ge - \\sum _{i = 1}^k \\log (\\mu (A_i)) \\mu (A_i)= - \\sum _{i = 1}^k \\log ({1 \\over 2^n}\\mu _n(A_i)) {1 \\over 2^n} \\mu _n(A_i) \\\\&\\ge - {1 \\over 2^n} \\sum _{i = 1}^k \\log (\\mu _n(A_i)) \\mu _n(A_i) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}).$ Since $\\mathcal {P}$ is arbitrary and $X_n$ is invariant, it follows that $h_{\\mu }(\\bar{\\mathcal {P}}, T) = \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu }(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) \\ge {1 \\over 2^n} \\lim _{m \\rightarrow \\infty } {1 \\over m} h_{\\mu _n}(\\bigvee _{j<m} T^j \\bar{\\mathcal {P}}) = {1 \\over 2^n} h_{\\mu _n}(\\bar{\\mathcal {P}}, T).$ Now for any finite Borel partition $I$ of $X$ , it is clear that $h_{\\mu _n}(I) = h_{\\mu _n}(\\bar{\\mathcal {P}})$ (and hence $h_{\\mu _n}(I, T) = h_{\\mu _n}(\\bar{\\mathcal {P}}, T)$ ), for some $\\mathcal {P}$ as above.",
"This implies that $h_{\\mu } \\ge \\sup _{\\mathcal {P}} h_{\\mu }(\\bar{\\mathcal {P}}, T) \\ge {1 \\over 2^n} \\sup _{\\mathcal {P}} h_{\\mu _n}(\\bar{\\mathcal {P}}, T) = {1 \\over 2^n} \\sup _{I} h_{\\mu _n}(I, T) = {1 \\over 2^n} h_{\\mu _n} > n,$ where $\\mathcal {P}$ and $I$ range over finite Borel partitions of $X_n$ and $X$ , respectively.",
"Thus $h_{\\mu }\\!",
"= \\infty $ .",
"Theorem 7.5 (Dichotomy II) Suppose the answer to Question REF is positive and let $X$ be an aperiodic Borel $\\mathbb {Z}$ -space.",
"Then exactly one of the following holds: there exists an invariant Borel probability measure with infinite entropy; $X$ admits a finite generator.",
"The Kolmogorov-Sinai theorem implies that the conditions are mutually exclusive, and we prove that at least one of them holds.",
"Assume that there is no invariant measure with infinite entropy.",
"If there was no invariant Borel probability measure at all, then, by our assumption, $X$ would admit a finite generator.",
"So assume that $\\mathcal {M}_{\\mathbb {Z}}(X) \\ne \\mathbb {\\emptyset }$ and let $\\lbrace A_n\\rbrace _{n \\le \\infty }$ be as in Theorem REF .",
"Furthermore, let $\\rho $ be the map $x \\mapsto e_x$ as in the Ergodic Decomposition Theorem.",
"Set $X^{\\prime } = X \\setminus A_{\\infty }$ and $X_e = \\rho ^{-1}(e)$ , for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ .",
"By our assumption, $A_{\\infty }$ admits a finite generator $\\mathcal {P}$ .",
"Also, by REF , there is $N \\ge 1$ such that for all $e \\in \\mathcal {E}_{\\mathbb {Z}}(X)$ , $N_e \\le N$ and hence $\\mathcal {Q}:= \\lbrace A_n\\rbrace _{n < N}$ is a finite generator for $X_e$ ; in particular, $\\mathcal {Q}$ is a partition of $X^{\\prime }$ .",
"Let $E$ be the following equivalence relation on $X$ : $x E y \\Leftrightarrow (x, y \\in A_{\\infty }) \\vee (x,y \\in X^{\\prime } \\wedge \\rho (x) = \\rho (y)).$ By definition, $E$ is a smooth equivalence relation with $E \\supseteq E_{\\mathbb {Z}}$ since $\\rho $ respects the $\\mathbb {Z}$ -action and $A_{\\infty }$ is $\\mathbb {Z}$ -invariant.",
"Thus, by Theorem REF , there exists a partition $J$ of $X$ into 4 Borel sets such that $\\mathbb {Z}J$ separates any two points in different $E$ -classes.",
"We now show that $I:= < \\!\\!",
"J\\cup \\mathcal {P}\\cup \\mathcal {Q} \\!\\!",
">$ is a generator.",
"Indeed, fix distinct $x,y \\in X$ .",
"If $x$ and $y$ are in different $E$ -classes, then $\\mathbb {Z}J$ separates them.",
"So we can assume that $x E y$ .",
"If $x,y \\in A_{\\infty }$ , then $\\mathbb {Z}\\mathcal {P}$ separates $x$ and $y$ .",
"Finally, if $x,y \\in X^{\\prime }$ , then $x,y \\in X_e$ , where $e = \\rho (x)$ ($= \\rho (y)$ ), and hence $\\mathbb {Z}\\mathcal {Q}$ separates $x$ and $y$ .",
"Remark.",
"It is likely that the above dichotomies are also true for any amenable group using a uniform version of Krieger's theorem for amenable groups, cf.",
", but I have not checked the details.",
"Finite generators on comeager sets Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER.",
"Locally weakly wandering sets and other special cases Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
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[
"Finite generators on comeager sets",
"Throughout this section let $X$ be an aperiodic Polish $G$ -space.",
"We use the notation $\\forall ^*$ to mean “for comeager many $x$ ”.",
"The following lemma proves the conclusion of Lemma REF for any group on a comeager set.",
"Below, we use this lemma only to conclude that there is an aperiodically separable comeager set, while we already know from REF that $X$ itself is aperiodically separable.",
"However, the proof of the latter is more involved, so we present this lemma to keep this section essentially self-contained.",
"Lemma 8.1 There exists $A \\in \\mathfrak {B}(X)$ such that $G < \\!\\!",
"A \\!\\!",
">$ separates points in each orbit of a comeager $G$ -invariant set $D$ , i.e.",
"$f_A \\!",
"\\!",
"\\downharpoonright _{[x]_G}$ is one-to-one, for all $x \\in D$ .",
"Fix a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ with $U_0 = \\emptyset $ and let $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ be a partition of $X$ provided by Lemma REF .",
"For each $\\alpha \\in \\mathcal {N}$ (the Baire space), define $B_{\\alpha } = \\bigcup _{n \\in \\mathbb {N}}(A_n \\cap U_{\\alpha (n)}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}\\forall ^* z \\in X \\forall x,y \\in [z]_G (x \\ne y \\Rightarrow \\exists g \\in G (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }))$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show the statement with places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall ^* z \\in X$ switched.",
"Also, since orbits are countable and countable intersection of comeager sets is comeager, we can also switch the places of quantifiers $\\forall ^* \\alpha \\in \\mathcal {N}$ and $\\forall x,y \\in [z]_G$ .",
"Thus we fix $z \\in X$ and $x,y \\in [z]_G$ with $x \\ne y$ and show that $C = \\lbrace \\alpha \\in \\mathcal {N}: \\exists g \\in G \\ (gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha })\\rbrace $ is dense open.",
"To see that $C$ is open, take $\\alpha \\in C$ and let $g \\in G$ be such that $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ .",
"Let $n,m \\in \\mathbb {N}$ be such that $gx \\in A_n$ and $gy \\in A_m$ .",
"Then for all $\\beta \\in \\mathcal {N}$ with $\\beta (n) = \\alpha (n)$ and $\\beta (m) = \\alpha (m)$ , we have $gx \\in B_{\\beta } \\nLeftrightarrow gy \\in B_{\\beta }$ .",
"But the set of such $\\beta $ is open in $\\mathcal {N}$ and contained in $C$ .",
"For the density of $C$ , let $s \\in \\mathbb {N}^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists g \\in G$ with $gx \\in A_n$ .",
"Let $m \\in \\mathbb {N}$ be such that $gy \\in A_m$ .",
"Take any $t \\in \\mathbb {N}^{\\max \\lbrace n,m\\rbrace +1}$ with $t \\sqsupseteq s$ satisfying the following condition: Case 1: $n > m$ .",
"If $gy \\in U_{s(m)}$ then set $t(n) = 0$ .",
"If $gy \\notin U_{s(m)}$ , then let $k$ be such that $gx \\in U_k$ and set $t(n) = k$ .",
"Case 2: $n \\le m$ .",
"Let $k$ be such that $gx \\in U_k$ but $gy \\notin U_k$ and set $t(n) = t(m) = k$ .",
"Now it is easy to check that in any case $gx \\in B_{\\alpha } \\nLeftrightarrow gy \\in B_{\\alpha }$ , for any $\\alpha \\in \\mathcal {N}$ with $\\alpha \\sqsupseteq t$ , and so $\\alpha \\in C$ and $\\alpha \\sqsupseteq s$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, $\\exists \\alpha \\in \\mathcal {N}$ such that $D = \\lbrace z \\in X : \\forall x,y \\in [z]_G \\text{ with } x \\ne y, \\ G < \\!\\!",
"B_{\\alpha } \\!\\!",
"> \\text{separates $x$ and $y$} \\rbrace $ is comeager and clearly invariant, which completes the proof.",
"Theorem 8.2 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then there exists an invariant dense $G_{\\delta }$ set that admits a Borel 4-generator.",
"Let $A$ and $D$ be provided by Lemma REF .",
"Throwing away an invariant meager set from $D$ , we may assume that $D$ is dense $G_{\\delta }$ and hence Polish in the relative topology.",
"Therefore, we may assume without loss of generality that $X = D$ .",
"Thus $A$ aperiodically separates $X$ and hence, by REF , there is a partition $\\lbrace A_n\\rbrace _{n \\in \\mathbb {N}}$ of $X$ into $F_A$ -invariant Borel complete sections (the latter could be inferred directly from Corollary REF without using Lemma REF ).",
"Fix an enumeration $G = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ and a countable basis $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ for $X$ .",
"Denote $\\mathcal {N}_2= (\\mathbb {N}^2)^{\\mathbb {N}}$ and for each $\\alpha \\in \\mathcal {N}_2$ , define $B_{\\alpha } = \\bigcup _{n \\ge 1}(A_n \\cap g_{(\\alpha (n))_0}U_{(\\alpha (n))_1}).$ Claim $\\forall ^* \\alpha \\in \\mathcal {N}_2\\forall ^* x \\in X \\forall l \\in \\mathbb {N}\\exists n,k \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)$ .",
"Proof of Claim.",
"By Kuratowski-Ulam, it is enough to show that $\\forall x \\in X$ and $\\forall l \\in \\mathbb {N}$ , $C = \\lbrace \\alpha \\in \\mathcal {N}_2: \\exists k,n \\in \\mathbb {N}(\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is dense open.",
"To see that $C$ is open, note that for fixed $n,k,l \\in N$ , $\\alpha (n) = (k,l)$ is an open condition in $\\mathcal {N}_2$ .",
"For the density of $C$ , let $s \\in (\\mathbb {N}^2)^{<\\mathbb {N}}$ and set $n = |s|$ .",
"Since $A_n$ is a complete section, $\\exists k \\in \\mathbb {N}$ with $g_k x \\in A_n$ .",
"Any $\\alpha \\in \\mathcal {N}_2$ with $\\alpha \\sqsupseteq s$ and $\\alpha (n) = (k,l)$ belongs to $C$ .",
"Hence $C$ is dense.",
"$\\dashv $ By the claim, there exists $\\alpha \\in \\mathcal {N}_2$ such that $Y = \\lbrace x \\in X : \\forall l \\in \\mathbb {N}\\ \\exists k,n \\in \\mathbb {N}\\ (\\alpha (n) = (k,l) \\wedge g_k x \\in A_n)\\rbrace $ is comeager.",
"Throwing away an invariant meager set from $Y$ , we can assume that $Y$ is $G$ -invariant dense $G_{\\delta }$ .",
"Let $I= < \\!\\!",
"A, B_{\\alpha } \\!\\!",
">$ , and so $|I| \\le 4$ .",
"We show that $I$ is a generator on $Y$ .",
"Fix distinct $x,y \\in Y$ .",
"If $x$ and $y$ are separated by $G < \\!\\!",
"A \\!\\!",
">$ then we are done, so assume otherwise, that is $x F_A y$ .",
"Let $l \\in \\mathbb {N}$ be such that $x \\in U_l$ but $y \\notin U_l$ .",
"Then there exists $k,n \\in \\mathbb {N}$ such that $\\alpha (n) = (k,l)$ and $g_k x \\in A_n$ .",
"Since $g_k x F_A g_k y$ and $A_n$ is $F_A$ -invariant, $g_k y \\in A_n$ .",
"Furthermore, since $g_k x \\in A_n \\cap g_k U_l$ and $g_k y \\notin A_n \\cap g_k U_l$ , $g_k x \\in B_{\\alpha }$ while $g_k y \\notin B_{\\alpha }$ .",
"Hence $G < \\!\\!",
"B_{\\alpha } \\!\\!",
">$ separates $x$ and $y$ , and thus so does $GI$ .",
"Therefore $I$ is a generator.",
"Corollary 8.3 Let $X$ be a Polish $G$ -space.",
"If $X$ is aperiodic, then it is 2-compressible modulo MEAGER.",
"By Theorem 13.1 in , $X$ is compressible modulo MEAGER.",
"Also, by the above theorem, $X$ admits a 4-generator modulo MEAGER.",
"Thus REF implies that $X$ is 2-compressible modulo MEAGER."
],
[
"Locally weakly wandering sets and other special cases",
"Assume throughout the section that $X$ is a Borel $G$ -space.",
"Definition 9.1 We say that $A \\subseteq X$ is weakly wandering with respect to $H \\subseteq G$ if $(h A) \\cap (h^{\\prime } A) = \\mathbb {\\emptyset }$ , for all distinct $h, h^{\\prime } \\in H$ ; weakly wandering, if it is weakly wandering with respect to an infinite subset $H \\subseteq G$ (by shifting $H$ , we can always assume $1_G \\in H$ ); locally weakly wandering if for every $x \\in X$ , $A^{[x]_G}$ is weakly wandering.",
"For $A \\subseteq X$ and $x \\in A$ , put $\\Delta _A(x) = \\lbrace (g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}: g_0 = 1_G \\wedge \\forall n \\ne m (g_n A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset }) \\rbrace ,$ and let $F(G^{\\mathbb {N}})$ denote the Effros space of $G^{\\mathbb {N}}$ , i.e.",
"the standard Borel space of closed subsets of $G^{\\mathbb {N}}$ (see 12.C in ).",
"Proposition 9.2 Let $A \\in \\mathfrak {B}(X)$ .",
"$\\forall x \\in X$ , $\\Delta _A(x)$ is a closed set in $G^{\\mathbb {N}}$ .",
"$\\Delta _A : A \\rightarrow F(G^{\\mathbb {N}})$ is $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable and hence universally measurable.",
"$\\Delta _A$ is $F_A$ -invariant, i.e.",
"$\\forall x,y \\in A$ , if $x F_A y$ then $\\Delta _A(x) = \\Delta _A(y)$ .",
"If $s : F(G^{\\mathbb {N}}) \\rightarrow G^{\\mathbb {N}}$ is a Borel selector (i.e.",
"$s(F) \\in F$ , $\\forall F \\in F(G^{\\mathbb {N}})$ ), then $\\gamma := s \\circ \\Delta _A$ is a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable $F_A$ - and $G$ -invariant travel guide.",
"In particular, $A$ is a 1-traveling set with $\\sigma (\\mathbf {\\Sigma }_1^1)$ -pieces.",
"$\\Delta _A(x)^c$ is open since being in it is witnessed by two coordinates.",
"For $s \\in G^{<\\mathbb {N}}$ , let $B_s = \\lbrace F \\in F(G^{\\mathbb {N}}) : F \\cap V_s \\ne \\mathbb {\\emptyset }\\rbrace $ , where $V_s = \\lbrace \\alpha \\in G^{\\mathbb {N}}: \\alpha \\sqsupseteq s\\rbrace $ .",
"Since $\\lbrace B_s\\rbrace _{s \\in G^{<\\mathbb {N}}}$ generates the Borel structure of $F(G^{\\mathbb {N}})$ , it is enough to show that $\\Delta _A^{-1}(B_s)$ is analytic, for every $s \\in G^{<\\mathbb {N}}$ .",
"But $\\Delta _A^{-1}(B_s) = \\lbrace x \\in X : \\exists (g_n)_{n \\in \\mathbb {N}} \\in V_s [g_0 = 1_G \\wedge \\forall n \\ne m g_n (A^{[x]_G} \\cap g_m A^{[x]_G} = \\mathbb {\\emptyset })]\\rbrace $ is clearly analytic.",
"Assume for contradiction that $x F_A y$ , but $\\Delta _A(x) \\ne \\Delta _A(y)$ for some $x,y \\in A$ .",
"We may assume that there is $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x) \\setminus \\Delta _A(y)$ and thus $\\exists n \\ne m$ such that $g_n A^{[y]_G} \\cap g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ .",
"Hence $A^{[y]_G} \\cap g_n^{-1}g_m A^{[y]_G} \\ne \\mathbb {\\emptyset }$ and let $y^{\\prime },y^{\\prime \\prime } \\in A^{[y]_G}$ be such that $y^{\\prime \\prime } = g_n^{-1}g_m y^{\\prime }$ .",
"Let $g \\in G$ be such that $y^{\\prime } = gy$ .",
"Since $y^{\\prime } = gy$ , $y^{\\prime \\prime } = g_n^{-1}g_m g y$ are in $A$ , $x F_A y$ , and $A$ is $F_A$ -invariant, $gx, g_n^{-1}g_m g x$ are in $A$ as well.",
"Thus $A^{[x]_G} \\cap g_n^{-1}g_m A^{[x]_G} \\ne \\mathbb {\\emptyset }$ , contradicting $g_n A^{[y]_G} \\cap g_m A^{[y]_G} = \\mathbb {\\emptyset }$ (this holds since $(g_n)_{n \\in \\mathbb {N}} \\in \\Delta _A(x)$ ).",
"Follows from parts (b) and (c), and the definition of $\\Delta _A$ .",
"Theorem 9.3 Let $X$ be a Borel $G$ -space.",
"If there is a locally weakly wandering Borel complete section for $X$ , then $X$ admits a Borel 4-generator.",
"By part (d) of REF and REF , $X$ is 1-compressible.",
"Thus, by REF , $X$ admits a Borel $2^2$ -finite generator.",
"Observation 9.4 Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering and put $W_n^{\\prime } = W_n \\setminus \\bigcup _{i<n} [W_i]_G$ .",
"Then $A^{\\prime } := \\bigcup _{n \\in \\mathbb {N}}W_n^{\\prime }$ is locally weakly wandering and $[A]_G = [A^{\\prime }]_G$ .",
"Corollary 9.5 Let $X$ be a Borel $G$ -space.",
"If $X$ is the saturation of a countable union of weakly wandering Borel sets, $X$ admits a Borel 3-generator.",
"Let $A = \\bigcup _{n \\in \\mathbb {N}}W_n$ , where each $W_n$ is weakly wandering.",
"By REF , we may assume that $[W_n]_G$ are pairwise disjoint and hence $A$ is locally weakly wandering.",
"Using countable choice, take a function $p : \\mathbb {N}\\rightarrow G^{\\mathbb {N}}$ such that $\\forall n \\in \\mathbb {N}$ , $p(n) \\in \\bigcap _{x \\in W_n} \\Delta _{W_n}(x)$ (we know that $\\bigcap _{x \\in W_n} \\Delta _{W_n}(x) \\ne \\mathbb {\\emptyset }$ since $W_n$ is weakly wandering).",
"Define $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ by $x \\mapsto \\text{the smallest $k$ such that } p(k) \\in \\Delta _A(x).$ The condition $p(k) \\in \\Delta _A(x)$ is Borel because it is equivalent to $\\forall n,m \\in \\mathbb {N}, y,z \\in A \\cap [x]_G, p(k)(n)y = p(k)(m)z \\Rightarrow n=m \\wedge x=y$ ; thus $\\gamma $ is a Borel function.",
"Note that $\\gamma $ is a travel guide for $A$ by definition.",
"Moreover, it is $F_A$ -invariant because if $\\Delta _A(x) = \\Delta _A(y)$ for some $x,y \\in A$ , then conditions $p(k) \\in \\Delta _A(x)$ and $p(k) \\in \\Delta _A(y)$ hold or fail together.",
"Since $\\Delta _A$ is $F_A$ -invariant, so is $\\gamma $ .",
"Hence, Lemma REF applied to $I= < \\!\\!",
"A \\!\\!",
">$ gives a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Remark.",
"The above corollary in particular implies the existence of a 3-generator in the presence of a weakly wandering Borel complete section.",
"(For a direct proof of this, note that if $W$ is a complete section that is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ with $g_0 = 1_G$ and $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ is a family generating the Borel sets, then $I= <W, \\bigcup _{n \\ge 1}g_n (W \\cap U_n)>$ is a generator and $|I| = 3$ .)",
"This can be viewed as a Borel version of the Krengel-Kuntz theorem (see REF ) in the sense that it implies a version of the latter (our result gives a 3-generator instead of a 2-generator).",
"To see this, let $X$ be a Borel $G$ -space and $\\mu $ be a quasi-invariant measure on $X$ such that there is no invariant measure absolutely continuous with respect to $\\mu $ .",
"Assume first that the action is ergodic.",
"Then by the Hajian-Kakutani-Itô theorem, there exists a weakly wandering set $W$ with $\\mu (W)>0$ .",
"Thus $X^{\\prime } = [W]_G$ is conull and admits a 3-generator by the above, so $X$ admits a 3-generator modulo $\\mu $ -NULL.",
"For the general case, one can use Ditzen's Ergodic Decomposition Theorem for quasi-invariant measures (Theorem 5.2 in ), apply the previous result to $\\mu $ -a.e.",
"ergodic piece, combine the generators obtained for each piece into a partition of $X$ (modulo $\\mu $ -NULL) and finally apply Theorem REF to obtain a finite generator for $X$ .",
"Each of these steps requires a certain amount of work, but we will not go into the details.",
"Example 9.6.",
"Let $X = \\mathcal {N}$ (the Baire space) and $\\tilde{E}_0$ be the equivalence relation of eventual agrement of sequences of natural numbers.",
"We find a countable group $G$ of homeomorphisms of $X$ such that $E_G = \\tilde{E}_0$ .",
"For all $s,t \\in \\mathbb {N}^{<\\mathbb {N}}$ with $|s| = |t|$ , let $\\phi _{s,t} : X \\rightarrow X$ be defined as follows: $\\phi _{s,t}(x) = \\left\\lbrace \\begin{array}{ll} t \\!\\!",
"y & \\text{if } x = s \\!\\!",
"y \\\\s \\!\\!",
"y & \\text{if } x = t \\!\\!",
"y \\\\x & \\text{otherwise}\\end{array}\\right.,$ and let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in \\mathbb {N}^{<\\mathbb {N}}, |s|=|t|\\rbrace $ .",
"It is clear that each $\\phi _{s,t}$ is a homeomorphism of $X$ and $E_G = \\tilde{E}_0$ .",
"Now for $n \\in \\mathbb {N}$ , let $X_n = \\lbrace x \\in X : x(0) = n\\rbrace $ and let $g_n = \\phi _{0,n}$ .",
"Then $X_n$ are pairwise disjoint and $g_n X_0 = X_n$ .",
"Hence $X_0$ is a weakly wandering set and thus $X$ admits a Borel 3-generator by Corollary REF .",
"Example 9.7.",
"Let $X = 2^{\\mathbb {N}}$ (the Cantor space) and $E_t$ be the tail equivalence relation on $X$ , that is $x E_t y \\Leftrightarrow (\\exists n,m \\in \\mathbb {N}) (\\forall k \\in \\mathbb {N}) x(n+k) = y(m+k)$ .",
"Let $G$ be the group generated by $\\lbrace \\phi _{s,t} : s,t \\in 2^{<\\mathbb {N}}, s \\perp t\\rbrace $ , where $\\phi _{s,t}$ are defined as above.",
"To see that $E_G = E_t$ fix $x,y \\in X$ with $x E_t y$ .",
"Thus there are nonempty $s,t \\in 2^{<\\mathbb {N}}$ and $z \\in X$ such that $x = s \\!\\!",
"z$ and $y = t \\!\\!",
"z$ .",
"If $s \\perp t$ , then $y = \\phi _{s,t}(x)$ .",
"Otherwise, assume say $s \\sqsubseteq t$ and let $s^{\\prime } \\in 2^{<\\mathbb {N}}$ be such that $s \\perp s^{\\prime }$ (exists since $s \\ne \\mathbb {\\emptyset }$ ).",
"Then $s^{\\prime } \\perp t$ and $y = \\phi _{s^{\\prime },t} \\circ \\phi _{s,s^{\\prime }}(x)$ .",
"Now for $n \\in \\mathbb {N}$ , let $s_n = \\underbrace{11...1}_n 0$ and $X_n = \\lbrace x \\in X : x = s_n \\!\\!",
"y, \\text{ for some } y \\in X\\rbrace $ .",
"Note that $s_n$ are pairwise incompatible and hence $X_n$ are pairwise disjoint.",
"Letting $g_n = \\phi _{s_0,s_n}$ , we see that $g_n X_0 = X_n$ .",
"Thus $X_0$ is a weakly wandering set and hence $X$ admits a Borel 3-generator.",
"Using the function $\\Delta $ defined above, we give another proof of Proposition REF .",
"Proposition REF .",
"Let $X$ be an aperiodic Borel $G$ -space and $T \\subseteq X$ be Borel.",
"If $T$ is a partial transversal then $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"By definition, $T$ is locally weakly wandering.",
"Claim $\\Delta _T$ is Borel.",
"Proof of Claim.",
"Using the notation of the proof of part (b) of REF , it is enough to show that $\\Delta _T^{-1}(B_s)$ is Borel for every $s \\in G^{<\\mathbb {N}}$ .",
"But since $\\forall x \\in T$ , $T \\cap [x]_G$ is a singleton, $\\Delta _T(x) \\in B_s$ is equivalent to $s(0) = 1_G \\wedge (\\forall n < m < |s|)$ $s(m)x \\ne s(n)x$ .",
"The latter condition is Borel, hence so is $\\Delta _T^{-1}(B_s)$ .",
"$\\dashv $ By part (d) of REF , $\\gamma = s \\circ \\Delta _T$ is a Borel $F_T$ -invariant travel guide for $T$ .",
"Corollary 9.8 Let $X$ be a Borel $G$ -space.",
"If $X$ is smooth and aperiodic, then it admits a Borel 3-generator.",
"Since the $G$ -action is smooth, there exists a Borel transversal $T \\subseteq X$ .",
"By REF , $T$ is $< \\!\\!",
"T \\!\\!",
">$ -traveling.",
"Thus, by REF , there is a Borel $(2 \\cdot 2 - 1)$ -generator.",
"Lastly, in case of smooth free actions, a direct construction gives the optimal result as the following proposition shows.",
"Proposition 9.9 Let $X$ be a Borel $G$ -space.",
"If the $G$ -action is free and smooth, then $X$ admits a Borel 2-generator.",
"Let $T \\subseteq X$ be a Borel transversal.",
"Also let $G \\setminus \\lbrace 1_G\\rbrace = \\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ be such that $g_n \\ne g_m$ for $n \\ne m$ .",
"Because the action is free, $g_n T \\cap g_m T = \\mathbb {\\emptyset }$ for $n \\ne m$ .",
"Define $\\pi : \\mathbb {N}\\rightarrow \\mathbb {N}$ recursively as follows: $\\pi (n) = \\left\\lbrace \\begin{array}{ll} \\min \\lbrace m : g_m \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k \\\\\\min \\lbrace m : g_m, g_m g_k \\notin \\lbrace g_{\\pi (i)} : i < n\\rbrace \\rbrace & \\text{if } n=3k+1 \\\\\\text{the unique $l$ s.t. }",
"g_l = g_{\\pi (3k+1)}g_k & \\text{if }n=3k+2\\end{array}\\right..$ Note that $\\pi $ is a bijection.",
"Fix a countable family $\\lbrace U_n\\rbrace _{n \\in \\mathbb {N}}$ generating the Borel sets and put $A = \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k)}(T \\cap U_k) \\cup \\bigcup _{k \\in \\mathbb {N}} g_{\\pi (3k+1)}T$ .",
"Clearly $A$ is Borel and we show that $I= < \\!\\!",
"A \\!\\!",
">$ is a generator.",
"Fix distinct $x, y \\in X$ .",
"Note that since $T$ is a complete section, we can assume that $x \\in T$ .",
"First assume $y \\in T$ .",
"Take $k$ with $x \\in U_k$ and $y \\notin U_k$ .",
"Then $g_{\\pi (3k)} x \\in g_{\\pi (3k)}(T \\cap U_k) \\subseteq A$ and $g_{\\pi (3k)} y \\in g_{\\pi (3k)}(T \\setminus U_k)$ .",
"However $g_{\\pi (3k)}(T \\setminus U_k) \\cap A = \\emptyset $ and hence $g_{\\pi (3k)} y \\notin A$ .",
"Now suppose $y \\notin T$ .",
"Then there exists $y^{\\prime } \\in T^{[y]_G}$ and $k$ such that $g_ky^{\\prime } = y$ .",
"Now $g_{\\pi (3k+1)}x \\in g_{\\pi (3k+1)} T \\subseteq A$ and $g_{\\pi (3k+1)} y = g_{\\pi (3k+1)}g_k y^{\\prime } = g_{\\pi (3k+2)} y^{\\prime } \\in g_{\\pi (3k+2)} T$ .",
"But $g_{\\pi (3k+2)} T \\cap A = \\emptyset $ , hence $g_{\\pi (3k+1)} y \\notin A$ .",
"Corollary 9.10 Let $H$ be a Polish group and $G$ be a countable subgroup of $H$ .",
"If $G$ admits an infinite discrete subgroup, then the translation action of $G$ on $H$ admits a 2-generator.",
"Let $G^{\\prime }$ be an infinite discrete subgroup of $G$ .",
"Clearly, it is enough to show that the translation action of $G^{\\prime }$ on $H$ admits a 2-generator.",
"Since $G^{\\prime }$ is discrete, it is closed.",
"Indeed, if $d$ is a left-invariant compatible metric on $H$ , then $B_d(1_H, \\epsilon ) \\cap G^{\\prime } = \\lbrace 1_H\\rbrace $ , for some $\\epsilon >0$ .",
"Thus every $d$ -Cauchy sequence in $G^{\\prime }$ is eventually constant and hence $G^{\\prime }$ is closed.",
"This implies that the translation action of $G^{\\prime }$ on $H$ is smooth and free (see 12.17 in ), and hence REF applies.",
"A condition for non-existence of non-meager weakly wandering sets Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
"BKbook author = Becker, H. author = Kechris, A. S. title = The Descriptive Set Theory of Polish Group Actions date = 1996 publisher = Cambridge Univ.",
"Press series = London Math.",
"Soc.",
"Lecture Note Series volume = 232 DParticle author = Danilenko, A. I. author = Park, K. K. title = Generators and Bernoullian factors for amenable actions and cocycles on their orbits date = 2002 journal = Ergod.",
"Th.",
"& Dynam.",
"Sys.",
"volume = 22 pages = 1715-1745 Downarowiczbook author = Downarowicz, T. title = Entropy in Dynamical Systems date = 2011 publisher = Cambridge Univ.",
"Press series = New Mathematical Monographs Series volume = 18 EHNarticle author = Eigen, S. author = Hajian, A. author = Nadkarni, M. title = Weakly wandering sets and compressibility in a descriptive setting date = 1993 journal = Proc.",
"Indian Acad.",
"Sci.",
"volume = 103 number = 3 pages = 321-327 Farrellarticle author = Farrell, R. H. title = Representation of invariant measures date = 1962 journal = Illinois J.",
"Math.",
"volume = 6 pages = 447-467 Glasnerbook author = Glasner, E. title = Ergodic Theory via Joinings date = 2003 publisher = American Mathematical Society series = Mathematical Surveys and Monographs volume = 101 GWarticle author = Glasner, E. author = Weiss, B. title = Minimal actions of the group $S(\\mathbb {Z})$ of permutations of the integers date = 2002 journal = Geom.",
"Funct.",
"Anal.",
"volume = 12 pages = 964-988 HIarticle author = Hajian, A.",
"B. author = Itô, Y. title = Weakly wandering sets and invariant measures for a group of transformations date = 1969 journal = Journal of Math.",
"Mech.",
"volume = 18 pages = 1203-1216 HKarticle author = Hajian, A.",
"B. author = Kakutani, S. title = Weakly wandering sets and invariant measures date = 1964 journal = Trans.",
"Amer.",
"Math.",
"Soc.",
"volume = 110 pages = 136-151 JKLarticle author = Jackson, S. author = Kechris, A. S. author = Louveau, A. title = Countable Borel equivalence relations date = 2002 journal = Journal of Math.",
"Logic volume = 2 number = 1 pages = 1-80 biblebook author = Kechris, A. S. title = Classical Descriptive Set Theory date = 1995 publisher = Springer series = Graduate Texts in Mathematics volume = 156 KMbook author = Kechris, A. S. author = Miller, B. title = Topics in Orbit Equivalence date = 2004 publisher = Springer series = Lecture Notes in Math.",
"volume = 1852 Kriegerarticle author = Krieger, W. title = On entropy and generators of measure-preserving transformations date = 1970 journal = Trans.",
"of the Amer.",
"Math.",
"Soc.",
"volume = 149 pages = 453-464 Krengelarticle author = Krengel, U. title = Transformations without finite invariant measure have finite strong generators conference = title = First Midwest Conference, Ergodic Theory and Probability book = series = Springer Lecture Notes volume = 160 date = 1970 pages = 133-157 Kuntzarticle author = Kuntz, A. J. title = Groups of transformations without finite invariant measures have strong generators of size 2 date = 1974 journal = Annals of Probability volume = 2 number = 1 pages = 143-146 Millerthesisbook author = Miller, B. D. title = PhD Thesis: Full groups, classification, and equivalence relations date = 2004 publisher = University of California at Los Angeles Millerarticle author = Miller, B. D. title = The existence of measures of a given cocycle, II: Probability measures date = 2008 journal = Ergodic Theory and Dynamical Systems volume = 28 number = 5 pages = 1615-1633 Munroebook author = Munroe, M. E. title = Introduction to Measure and Integration date = 1953 publisher = Addison-Wesley Nadkarniarticle author = Nadkarni, M. G. title = On the existence of a finite invariant measure date = 1991 journal = Proc.",
"Indian Acad.",
"Sci.",
"Math.",
"Sci.",
"volume = 100 pages = 203-220 Rudolphbook author = Rudolph, D. title = Fundamentals of Measurable Dynamics date = 1990 publisher = Oxford Univ.",
"Press Varadarajanarticle author = Varadarajan, V. S. title = Groups of automorphisms of Borel spaces date = 1963 journal = Trans.",
"Amer.",
"Math.",
"Soc.",
"volume = 109 pages = 191-220 Wagonbook author = Wagon, S. title = The Banach-Tarski Paradox date = 1993 publisher = Cambridge Univ.",
"Press Weissarticle author = Weiss, B. title = Countable generators in dynamics-universal minimal models date = 1987 journal = Measure and Measurable Dynamics, Contemp.",
"Math.",
"volume = 94 pages = 321-326"
],
[
"A condition for non-existence of non-meager weakly wandering sets",
"Throughout this section let $X$ be a Polish $\\mathbb {Z}$ -space and $T$ be the homeomorphism corresponding to the action of $1 \\in \\mathbb {Z}$ .",
"Observation 10.1 Let $A \\subseteq X$ be weakly wandering with respect to $H \\subseteq \\mathbb {Z}$ .",
"Then $A$ is weakly wandering with respect to any subset of $H$ ; $r+H$ , $\\forall r \\in \\mathbb {Z}$ ; $-H$ .",
"Definition 10.2 Let $d \\ge 1$ and $F = \\lbrace n_i\\rbrace _{i<k} \\subseteq \\mathbb {Z}$ , where $n_0 < n_1 < ... < n_{k-1}$ are increasing.",
"$F$ is called $d$ -syndetic if $n_{i+1} - n_i \\le d$ for all $i < k-1$ .",
"In this case we say that the length of $F$ is $n_{k-1}-n_0$ and denote it by $||F||$ .",
"Lemma 10.3 Let $d \\ge 1$ and $F \\subseteq \\mathbb {Z}$ be a $d$ -syndetic set.",
"For any $H \\subseteq \\mathbb {Z}$ , if $|H| = d+1$ and $\\max (H) - \\min (H) < ||F|| + d$ , then $F$ is not weakly wandering with respect to $H$ (viewing $\\mathbb {Z}$ as a $\\mathbb {Z}$ -space).",
"Using (b) and (c) of REF , we may assume that $H$ is a set of non-negative numbers containing 0.",
"Let $F = \\lbrace n_i\\rbrace _{i<k}$ with $n_i$ increasing.",
"Claim $\\forall h \\in H$ , $(h + F) \\cap [n_{k-1}, n_{k-1} + d) \\ne \\mathbb {\\emptyset }$ .",
"Proof of Claim.",
"Fix $h \\in H$ .",
"Since $0 \\le h < ||F|| + d$ , $n_0 + h < n_0 + (||F|| + d) = n_{k-1} + d.$ We prove that there is $0 \\le i \\le k-1$ such that $n_i + h \\in [n_{k-1}, n_{k-1} + d)$ .",
"Otherwise, because $n_{i+1} - n_i \\le d$ , one can show by induction on $i$ that $n_i + h < n_{k-1}, \\forall i < k$ , contradicting $n_{k-1} + h \\ge n_{k-1}$ .",
"$\\dashv $ Now $|H| = d+1 > d = |\\mathbb {Z}\\cap [n_{k-1}, n_{k-1} + d)|$ , so by the Pigeon Hole Principle there exists $h \\ne h^{\\prime } \\in H$ such that $(h + F) \\cap (h^{\\prime } + F) \\ne \\mathbb {\\emptyset }$ and hence $F$ is not weakly wandering with respect to $H$ .",
"Definition 10.4 Let $d,l \\ge 1$ and $A \\subseteq X$ .",
"We say that $A$ contains a $d$ -syndetic set of length $l$ if there exists $x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set of length $\\ge l$ .",
"This is equivalent to $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ , for some $d$ -syndetic set $F \\subseteq \\mathbb {Z}$ of length $\\ge l$ .",
"For $A \\subseteq X$ , define $s_A : \\mathbb {N}\\rightarrow \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ by $d \\mapsto \\sup \\lbrace l \\in \\mathbb {N}: A \\text{ contains a } d\\text{-syndetic set of length } l\\rbrace .$ Also, for infinite $H \\subseteq \\mathbb {Z}$ , define a width function $w_H : \\mathbb {N}\\rightarrow \\mathbb {N}$ by $d \\mapsto \\min \\lbrace \\max (H^{\\prime }) - \\min (H^{\\prime }) : H^{\\prime } \\subseteq H \\wedge |H^{\\prime }| = d+1\\rbrace .$ Proposition 10.5 If $A \\subseteq X$ is weakly wandering with respect to an infinite $H \\subseteq \\mathbb {Z}$ then $\\forall d \\in \\mathbb {N}, s_A(d) + d \\le w_H(d)$ .",
"Let $H$ be an infinite subset of $\\mathbb {Z}$ and $A \\subseteq X$ , and assume that $s_A(d) + d > w_H(d)$ for some $d \\in \\mathbb {N}$ .",
"Thus $\\exists x \\in X$ such that $\\lbrace n \\in \\mathbb {Z}: T^n(x) \\in A\\rbrace $ contains a $d$ -syndetic set $F$ of length $l$ with $l + d > w_H(d)$ and $\\exists H^{\\prime } \\subseteq H$ such that $|H^{\\prime }| = d+1$ and $\\max (H^{\\prime }) - \\min (H^{\\prime }) = w_H(d)$ .",
"By Lemma REF applied to $F$ and $H^{\\prime }$ , $F$ is not weakly wandering with respect to $H^{\\prime }$ and hence neither is $A$ .",
"Thus $A$ is not weakly wandering with respect to $H$ .",
"Corollary 10.6 If $A \\subseteq X$ contains arbitrarily long $d$ -syndetic sets for some $d \\ge 1$ , then it is not weakly wandering.",
"If $A$ and $d$ are as in the hypothesis, then $s_A(d) = \\infty $ and hence, by Proposition REF , $A$ is not weakly wandering with respect to any infinite $H \\subseteq \\mathbb {Z}$ .",
"Theorem 10.7 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $V$ contains arbitrarily long $d$ -syndetic sets, i.e.",
"$\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ for arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ .",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable subset of $X$ .",
"By the Baire property, there exists a nonempty open $V \\subseteq X$ such that $A$ is comeager in $V$ .",
"By the hypothesis, there exists arbitrarily long $d$ -syndetic sets $F \\subseteq \\mathbb {Z}$ such that $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Since $A$ is comeager in $V$ and $T$ is a homeomorphism, $\\bigcap _{n \\in F} T^n(A)$ is comeager in $\\bigcap _{n \\in F} T^n(V)$ , and hence $\\bigcap _{n \\in F} T^n(A) \\ne \\mathbb {\\emptyset }$ for any $F$ for which $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus $A$ also contains arbitrarily long $d$ -syndetic sets and hence, by Corollary REF , $A$ is not weakly wandering.",
"Corollary 10.8 Let $X$ be a Polish $G$ -space.",
"Suppose for every nonempty open $V \\subseteq X$ there exists $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then $X$ does not admit a non-meager Baire measurable weakly wandering subset.",
"Fix nonempty open $V \\subseteq X$ and let $d \\ge 1$ such that $\\lbrace T^{nd}(V)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property.",
"Then for every $N$ , $F = \\lbrace kd : k \\le N\\rbrace $ is a $d$ -syndetic set of length $Nd$ and $\\bigcap _{n \\in F} T^n(V) \\ne \\mathbb {\\emptyset }$ .",
"Thus Theorem REF applies.",
"Lemma 10.9 Let $X$ be a generically ergodic Polish $G$ -space.",
"If there is a non-meager Baire measurable locally weakly wandering subset then there is a non-meager Baire measurable weakly wandering subset.",
"Let $A$ be a non-meager Baire measurable locally weakly wandering subset.",
"By generic ergodicity, we may assume that $X = [A]_G$ .",
"Throwing away a meager set from $A$ we can assume that $A$ is $G_{\\delta }$ .",
"Then, by (d) of REF , there exists a $\\sigma (\\mathbf {\\Sigma }_1^1)$ -measurable (and hence Baire measurable) $G$ -invariant travel guide $\\gamma : A \\rightarrow G^{\\mathbb {N}}$ .",
"By generic ergodicity, $\\gamma $ must be constant on a comeager set, i.e.",
"there is $(g_n)_{n \\in \\mathbb {N}} \\in G^{\\mathbb {N}}$ such that $Y := \\gamma ^{-1}((g_n)_{n \\in \\mathbb {N}})$ is comeager.",
"But then $W := A \\cap Y$ is non-meager and is weakly wandering with respect to $\\lbrace g_n\\rbrace _{n \\in \\mathbb {N}}$ .",
"Let $X = \\lbrace \\alpha \\in 2^{\\mathbb {N}} : \\alpha \\text{ has infinitely many 0-s and 1-s}\\rbrace $ and $T$ be the odometer transformation on $X$ .",
"We will refer to this $\\mathbb {Z}$ -space as the odometer space.",
"Corollary 10.10 The odometer space does not admit a non-meager Baire measurable locally weakly wandering subset.",
"Let $\\lbrace U_s\\rbrace _{s \\in 2^{<\\mathbb {N}}}$ be the standard basis.",
"Then for any $s \\in 2^{<\\mathbb {N}}$ , $T^{d}(U_s) = U_s$ for $d = |s|$ .",
"Thus $\\lbrace T^{nd}(U_s)\\rbrace _{n \\in \\mathbb {N}}$ has the finite intersection property, in fact $\\bigcap _{n \\in \\mathbb {N}} T^{nd}(U_s) = U_s$ .",
"Hence, we are done by REF and REF .",
"The following corollary shows the failure of the analogue of the Hajian-Kakutani-Itô theorem in the context of Baire category as well as gives a negative answer to Question REF .",
"Corollary 10.11 There exists a generically ergodic Polish $\\mathbb {Z}$ -space $Y$ (namely an invariant dense $G_{\\delta }$ subset of the odometer space) with the following properties: there does not exist an invariant Borel probability measure on $Y$ ; there does not exist a non-meager Baire measurable locally weakly wandering set; there does not exist a Baire measurable countably generated partition of $Y$ into invariant sets, each of which admits a Baire measurable weakly wandering complete section.",
"By the Kechris-Miller theorem (see REF ), there exists an invariant dense $G_{\\delta }$ subset $Y$ of the odometer space that does not admit an invariant Borel probability measure.",
"Now (ii) is asserted by Corollary REF .",
"By generic ergodicity of $Y$ , for any Baire measurable countably generated partition of $Y$ into invariant sets, one of the pieces of the partition has to be comeager.",
"But then that piece does not admit a Baire measurable weakly wandering complete section since otherwise it would be non-meager, contradicting (ii).",
"BKbook author = Becker, H. author = Kechris, A. S. title = The Descriptive Set Theory of Polish Group Actions date = 1996 publisher = Cambridge Univ.",
"Press series = London Math.",
"Soc.",
"Lecture Note Series volume = 232 DParticle author = Danilenko, A. I. author = Park, K. K. title = Generators and Bernoullian factors for amenable actions and cocycles on their orbits date = 2002 journal = Ergod.",
"Th.",
"& Dynam.",
"Sys.",
"volume = 22 pages = 1715-1745 Downarowiczbook author = Downarowicz, T. title = Entropy in Dynamical Systems date = 2011 publisher = Cambridge Univ.",
"Press series = New Mathematical Monographs Series volume = 18 EHNarticle author = Eigen, S. author = Hajian, A. author = Nadkarni, M. title = Weakly wandering sets and compressibility in a descriptive setting date = 1993 journal = Proc.",
"Indian Acad.",
"Sci.",
"volume = 103 number = 3 pages = 321-327 Farrellarticle author = Farrell, R. H. title = Representation of invariant measures date = 1962 journal = Illinois J.",
"Math.",
"volume = 6 pages = 447-467 Glasnerbook author = Glasner, E. title = Ergodic Theory via Joinings date = 2003 publisher = American Mathematical Society series = Mathematical Surveys and Monographs volume = 101 GWarticle author = Glasner, E. author = Weiss, B. title = Minimal actions of the group $S(\\mathbb {Z})$ of permutations of the integers date = 2002 journal = Geom.",
"Funct.",
"Anal.",
"volume = 12 pages = 964-988 HIarticle author = Hajian, A.",
"B. author = Itô, Y. title = Weakly wandering sets and invariant measures for a group of transformations date = 1969 journal = Journal of Math.",
"Mech.",
"volume = 18 pages = 1203-1216 HKarticle author = Hajian, A.",
"B. author = Kakutani, S. title = Weakly wandering sets and invariant measures date = 1964 journal = Trans.",
"Amer.",
"Math.",
"Soc.",
"volume = 110 pages = 136-151 JKLarticle author = Jackson, S. author = Kechris, A. S. author = Louveau, A. title = Countable Borel equivalence relations date = 2002 journal = Journal of Math.",
"Logic volume = 2 number = 1 pages = 1-80 biblebook author = Kechris, A. S. title = Classical Descriptive Set Theory date = 1995 publisher = Springer series = Graduate Texts in Mathematics volume = 156 KMbook author = Kechris, A. S. author = Miller, B. title = Topics in Orbit Equivalence date = 2004 publisher = Springer series = Lecture Notes in Math.",
"volume = 1852 Kriegerarticle author = Krieger, W. title = On entropy and generators of measure-preserving transformations date = 1970 journal = Trans.",
"of the Amer.",
"Math.",
"Soc.",
"volume = 149 pages = 453-464 Krengelarticle author = Krengel, U. title = Transformations without finite invariant measure have finite strong generators conference = title = First Midwest Conference, Ergodic Theory and Probability book = series = Springer Lecture Notes volume = 160 date = 1970 pages = 133-157 Kuntzarticle author = Kuntz, A. J. title = Groups of transformations without finite invariant measures have strong generators of size 2 date = 1974 journal = Annals of Probability volume = 2 number = 1 pages = 143-146 Millerthesisbook author = Miller, B. D. title = PhD Thesis: Full groups, classification, and equivalence relations date = 2004 publisher = University of California at Los Angeles Millerarticle author = Miller, B. D. title = The existence of measures of a given cocycle, II: Probability measures date = 2008 journal = Ergodic Theory and Dynamical Systems volume = 28 number = 5 pages = 1615-1633 Munroebook author = Munroe, M. E. title = Introduction to Measure and Integration date = 1953 publisher = Addison-Wesley Nadkarniarticle author = Nadkarni, M. G. title = On the existence of a finite invariant measure date = 1991 journal = Proc.",
"Indian Acad.",
"Sci.",
"Math.",
"Sci.",
"volume = 100 pages = 203-220 Rudolphbook author = Rudolph, D. title = Fundamentals of Measurable Dynamics date = 1990 publisher = Oxford Univ.",
"Press Varadarajanarticle author = Varadarajan, V. S. title = Groups of automorphisms of Borel spaces date = 1963 journal = Trans.",
"Amer.",
"Math.",
"Soc.",
"volume = 109 pages = 191-220 Wagonbook author = Wagon, S. title = The Banach-Tarski Paradox date = 1993 publisher = Cambridge Univ.",
"Press Weissarticle author = Weiss, B. title = Countable generators in dynamics-universal minimal models date = 1987 journal = Measure and Measurable Dynamics, Contemp.",
"Math.",
"volume = 94 pages = 321-326"
]
] | 1204.0829 |
[
[
"Nonlinear elliptic problems with dynamical boundary conditions of\n reactive and reactive-diffusive type"
],
[
"Abstract We investigate classical solutions of nonlinear elliptic equations with two classes of dynamical boundary conditions, of reactive and reactive-diffusive type.",
"In the latter case it is shown that well-posedness is to a large extent independent of the coupling with the elliptic equation.",
"For both types of boundary conditions we consider blow-up, global existence, global attractors and convergence to single equilibria."
],
[
"Introduction",
"The prototype of the elliptic-parabolic initial-boundary value problems that we consider in this article is $\\left\\lbrace \\begin{array}{ll}\\lambda u-d\\Delta u=f(u) & \\text{in }(0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }-\\delta \\Delta _{\\Gamma }u_{\\Gamma }+d\\partial _{\\nu }u=g(u_{\\Gamma }) & \\text{on }(0,T)\\times \\Gamma , \\\\u|_{\\Gamma }=u_{\\Gamma } & \\text{on }(0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma .\\end{array}\\right.",
"$ We assume that $ \\Omega \\subset \\mathbb {R}^{n}$ is a bounded domain with smooth boundary $ \\Gamma =\\partial \\Omega $ , that $d>0$ , $\\delta \\ge 0$ and $ f,g\\in C^{\\infty }(\\mathbb {R})$ .",
"Further, $\\Delta _{\\Gamma }$ is the Laplace-Beltrami operator and $ \\partial _{\\nu }$ is the outer normal derivative on $ \\Gamma $ .",
"It is throughout assumed that $f$ is globally Lipschitz continuous and that $ \\lambda $ is sufficiently large, in dependence on $ f$ .",
"Depending on $\\delta $ , two classes of boundary conditions are modelled by (REF ).",
"For $ \\delta >0$ we have boundary conditions of reactive-diffusive type, and for $ \\delta =0$ the boundary conditions are purely reactive.",
"The motivation to consider (REF ) comes from physics.",
"The function $ u$ represents the steady state temperature in a body $\\Omega $ such that the rate at which $ u$ evolves through the boundary $\\Gamma $ is proportional to the flux on the boundary, up to some correction $ \\delta \\Delta _{\\Gamma }u_\\Gamma ,$ $ \\delta \\ge 0$ , which from a modelling viewpoint, accounts for small diffusive effects along $ \\Gamma $ .",
"Moreover, the heat source on $ \\Gamma $ acts nonlinearly through the function $g$ .",
"Problem (REF ) is also important in conductivity (see, e.g., [18]) and harmonic analysis due to its connection to the following eigenvalue problem $\\Delta u=0\\quad \\text{ in }\\Omega ,\\qquad -\\delta \\Delta _{\\Gamma }u_\\Gamma +\\partial _{\\nu }u=\\xi u_\\Gamma \\quad \\text{ on }\\Gamma , $ which was introduced by Stekloff [34] (initially) in the case $ \\delta =0 $ .",
"This connection arises because the linear problem associated with (REF ) (i.e., by letting $ \\lambda =0$ , $f\\equiv 0$ and $g\\equiv 0$ ) can be solved by the Fourier method in terms of the eigenfunctions of (REF ) (see [38], which also includes the case $ \\delta >0$ ; cf.",
"also [39] for $ \\delta =0$ ).",
"The solvability of the linear problem (assuming $ \\delta =0$ ) was also investigated by Hintermann [20] by means of the theory of pseudo-differential operators, and by Gröger [18] and Showalter [33], by applying the theory of maximal monotone operators in the Hilbert-space setting (see, also, [13]).",
"It turns out that this connection is also essential for solvability of the nonlinear problem (REF ).",
"The mathematical study of the prototype (REF ) has a long-standing history.",
"In [23] J.-L. Lions considered the special case $ \\delta =\\lambda =0$ , $f\\equiv 0$ and $g\\left( s\\right) =-\\left|s\\right|^{p}s,$ $ p>0$ .",
"By standard compactness methods, he proved existence and uniqueness of global solutions for initial datum $ u_{0}\\in H^{1/2}\\left( \\Gamma \\right) $ in this special case.",
"Problem (REF ) was investigated in the general case by Escher [8], [9] for nontrivial functions $ f,g,$ by also treating systems of elliptic equations, but always in the case $ \\delta =0$ .",
"His papers deal with classical solvability and global existence for smooth initial data.",
"In particular, global existence of classical solutions was shown assuming $ f$ is globally Lipschitz and that $g\\left( s\\right) s\\le 0$ , for all $ s\\in \\mathbb {R}$ .",
"Constantin, Escher and Yin [5], [44] established, in the case $ \\delta =\\lambda =0$ and $f\\equiv 0,$ some natural structural conditions for the function $ g$ so that global existence of classical solutions holds.",
"Their approach is based on global existence criteria for ODEs.",
"Boundedness of the global solutions for (REF ) was shown by Fila and Quittner [12] in the case when $ \\delta =\\lambda =0$ , $f\\equiv 0$ and $g$ is a superlinear subcritical nonlinearity.",
"They have also proved that blow-up in finite time occurs for (REF ) if $ g\\left( s\\right) =\\left|s\\right|^{p-1}s-as,$ $ p>1,$ $a\\ge 0$ and if the initial datum $u_{0}$ is \"large\" enough [12].",
"Blow-up phenomena for smooth solutions of (REF ), when $ \\delta =0$ and $f\\equiv 0,$ was also observed by Kirane [21] under some general assumptions on $ g$ , i.e., when $g(s)>0$ , for all $ s\\ge s_{0}$ , and $\\int _{s_{0}}^{\\infty }\\frac{d\\xi }{g(\\xi )}<\\infty .$ A version of the problem (REF ) for which the dynamic boundary condition is replaced by $\\left|\\partial _{t}u_{\\Gamma }\\right|^{m-1}\\partial _{t}u_{\\Gamma }+d\\partial _{\\nu }u=\\left|u_{\\Gamma }\\right|^{p-1}u_{\\Gamma }\\qquad \\text{on }\\Gamma \\times \\left( 0,T\\right) ,$ for some $ m\\ge 1$ and $p\\ge 1$ was investigated by Vitillaro [39] for initial data $ u_{0}\\in H^{1/2}\\left( \\Gamma \\right) $ and $f\\equiv 0$ .",
"He mainly devotes his attention to proving the local and the global existence as well as blow-up of solutions for $ m\\ge 1$ , especially, in the nonlinear case when $ m\\ne 1$ .",
"Finally, it is interesting to note that, in the case when $ f\\ne 0$ but $f$ is not globally Lipschitz, global non-existence without blow-up and non-uniqueness phenomena for (REF ) can occur (see [11]).",
"All the papers quoted so far deal only with classical issues, such as global existence, uniqueness and blow-up phenomena for (REF ) when $ \\delta =0$ .",
"Concerning further regularity and longtime behavior of solutions, as time goes to infinity, not much seems to be known.",
"This seems to be due to the fact that the gradient structure of (REF ) has not been exploited before.",
"This issue is intimately connected with a key result on smoothness in $ \\mathbb {R}_{+}\\times \\overline{\\Omega }$ of solutions for (REF ) even when $ f\\ne 0$ (see Proposition REF ), which is essential to the study of the asymptotic behavior of the system, in terms of global attractors and $ \\omega $ -limit sets.",
"The main novelties of the present paper with respect to previous results on (REF ) are the following: (i) The local well-posedness results are extended to the case $ \\delta >0$ .",
"In fact, we will consider a more general class of elliptic problems with quasilinear, nondegenerate dynamic boundary conditions of reactive-diffusive type.",
"More precisely, we consider the following generalization of the prototype model (REF ), $\\left\\lbrace \\begin{array}{ll}\\lambda u+\\mathcal {A}u=f(u) & \\text{in }(0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }+\\mathcal {C}(u_{\\Gamma })u_{\\Gamma }+\\mathcal {B}(u)=g(u_{\\Gamma }) & \\text{on }(0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma ,\\end{array}\\right.",
"$ where $\\mathcal {A}u=-\\text{div}\\big (d\\nabla u\\big ),\\qquad \\mathcal {C}(u_{\\Gamma })u_{\\Gamma }=-\\text{div}_{\\Gamma }\\big (\\delta (\\cdot ,u_{\\Gamma })\\nabla _{\\Gamma }u_{\\Gamma }\\big ),$ such that $ d\\in C^{\\infty }(\\overline{\\Omega }),$ $\\delta \\in C^{\\infty }(\\Gamma \\times \\mathbb {R})$ with $ d\\ge d_{\\ast }>0$ and $\\delta \\ge \\delta _{\\ast }>0$ .",
"Moreover, $ \\nabla _\\Gamma $ is the surface gradient and $ \\text{div}_\\Gamma $ is the surface divergence.",
"Here and in the sequel we always assume that $ u|_\\Gamma = u_\\Gamma $ .",
"The nonlinear map $\\mathcal {B}$ in (REF ) couples the equations in the domain $ \\Omega $ and on the boundary $ \\Gamma $ in a (possibly) nontrivial way.",
"We do not impose any further structural conditions for $ \\mathcal {B}$ and $g$ other than they must be of order strictly lower than two and satisfy a local Lipschitz condition.",
"One example for $ \\mathcal {B}$ we have in mind is $\\mathcal {B}(u)=b \\nu \\cdot (\\nabla u)|_{\\Gamma }$ , with no sign restriction on $ b\\in C^{\\infty }(\\Gamma )$ .",
"We prove that for sufficiently large $\\lambda $ and a globally Lipschitz function $ f$ the problem (REF ) generates a (compact) local semiflow of solutions for $ u_{0}\\in \\mathcal {X}_{\\delta }:=W^{2-2/p,p}(\\Gamma )$ , $ p\\in \\left( n+1,\\infty \\right) $ , $ \\delta >0$ , and establish some further regularity properties for the local solution $ u=u(\\cdot ;u_{0})$ .",
"For the notion of local semiflow, we refer the reader to Section REF .",
"The independence of the well-posedness of the coupling was first observed by Vazquez and Vitillaro [38] for a linear model problem with $ \\mathcal {C }= -\\Delta _\\Gamma $ and $ \\mathcal {B}=-\\partial _{\\nu }$ in a Hilbert space setting.",
"Our approach to the quasilinear problem is based on maximal $ L^p$ -regularity properties of the corresponding linearized dynamic equation on the boundary.",
"In Section these will be verified for a general class of elliptic boundary differential operators using localization techniques.",
"The global Lipschitz condition on $ f$ allows to solve the elliptic equation on $ \\Omega $ and to rewrite (REF ) as an initial-value problem for $ u_\\Gamma $ on $\\Gamma $ , which can be treated with the general theory of [22].",
"The fact that the concrete form of the coupling $ \\mathcal {B}$ is inessential is a consequence of the fact that maximal regularity is invariant under lower order perturbations.",
"For the precise statements of these results we refer the reader to Section .",
"The corresponding result for boundary conditions of purely reactive type, i.e., $ \\mathcal {C}\\equiv 0$ and $\\mathcal {B}=d\\partial _{\\nu }$ in (REF ), was shown in [8].",
"There the result is based on the generation properties of the Dirichlet-Neumann operator and thus, the solutions enjoy worse regularity properties up to $ t=0$ .",
"In addition to this we establish the compactness of the solution semiflow on $ \\mathcal {X}_0 :=W^{1-1/p,p}(\\Gamma )$ , $ p\\in (n,\\infty )$ , in this case (see Section REF ).",
"(ii) The blow-up results for problem (REF ), from [21] and [40], are also extended to the case when $ \\delta >0$ and $ f\\ne 0$ .",
"Our approach is based on the method of subsolutions and a comparison lemma, and is inspired by [3] and [32] (see Section REF ).",
"We further show global existence of solutions of (REF ) under the natural assumption that $ g(\\xi )\\xi \\le c_{g}(\\left|\\xi \\right|^{2}+1)$ for all $ \\xi \\in \\mathbb {R}$ by performing a Moser-Alikakos iteration procedure as in [14], [27].",
"Here an inequality of Poincaré-Young type allows to connect the structure of the elliptic equation with that of the dynamic equation on $ \\Gamma $ (see Section REF ).",
"(iii) We prove the smoothness of solutions of (REF ) in both space and time exploiting a variation of parameters formula for the trace $ u_{\\Gamma }$ and the implicit function theorem, which is entirely new (see Section REF ).",
"Consequently, taking advantage of this smoothness, we can show that (REF ) has a gradient structure, and as a result establish the existence of a finite-dimensional global attractor in the phase space $ \\mathcal {X}_{\\delta }$ for both types of boundary conditions.",
"Here the main assumption is that the first eigenvalue of a Stekloff-like eigenvalue problem (similar to (REF )) is positive (see Section REF ).",
"(iv) The $ \\omega $ -limit sets of (REF ) can exhibit a complicated structure if the functions $ f,g$ are non-monotone and, a fortiori, the same is true for the global attractor.",
"Indeed, when $ f,g$ are non-monotone (i.e., the related potentials $ F\\left( s\\right)=\\int _{0}^{s}f\\left( y\\right) dy,$ $ G\\left( s\\right) =\\int _{0}^{s}g\\left(y\\right) dy$ are non-convex) this can happen if the stationary problem associated with (REF ) possesses a continuum of nonconstant solutions.",
"Some examples which show that the $ \\omega $ -limit set can be a continuum are provided in [29].",
"However, assuming the nonlinearities $ f,g$ to be real analytic, we prove the convergence of a given trajectory $ u=u(t;u_{0}),$ $u_{0}\\in \\mathcal {X}_{\\delta },$ as time goes to infinity, to a single equilibrium of (REF ).",
"This shows, in a strong form, the asymptotic stability of $ u(t;u_{0})$ for an arbitrary (but given) initial datum $ u_{0}\\in \\mathcal {X}_{\\delta }$ .",
"This type of result exploits a technique which is based on the so-called Łojasiewicz-Simon inequality (see Section REF ; cf.",
"also [35], [43]).",
"Finally, it is worth mentioning that most of our results can be also extended to systems of nonlinear elliptic equations subject to both types of boundary conditions.",
"The plan of the paper goes as follows.",
"In Section , we introduce the functional analytic framework associated with (REF ) and (REF ), respectively.",
"In Section , maximal $ L^{p}$ -regularity theory is developed for elliptic boundary differential operators of second order.",
"Then, in Section (and corresponding subsections) we prove (local) well-posedness results for (REF ) and establish the existence of a compact (local) semiflow on the corresponding phase spaces.",
"The final Section is further divided into five parts: the first part provides the key result which shows the smoothness of solutions in both space and time, while the second and third parts deal with blow-up phenomena and global existence, respectively.",
"Finally, the last two subsections deal with the asymptotic behavior as time goes to infinity, in terms of global attractors and convergence of solutions to single equilbria."
],
[
"Function spaces",
"We briefly describe the function spaces that are used in the paper.",
"Details and proofs can be found in [25], [37].",
"Throughout, all function spaces under consideration are real.",
"Let $ p\\in [1,\\infty ]$ .",
"If $ \\Omega \\subseteq \\mathbb {R}^{n}$ is open, we denote by $ L^{p}(\\Omega )$ the usual Lebesgue spaces.",
"Now let $\\Omega $ have a (sufficiently) smooth boundary.",
"Then for $ s\\ge 0$ and $p\\in [1,\\infty )$ we denote by $ H^{s,p}(\\Omega )$ the Bessel-potential spaces and by $ W^{s,p}(\\Omega )$ the Slobodetskij spaces.",
"One has $H^{s,2}(\\Omega )=W^{s,2}(\\Omega )$ for all $ s$ , but for $p\\ne 2$ the identity $ H^{s,p}(\\Omega )=W^{s,p}(\\Omega )$ is only true if $s\\in \\mathbb {N}_{0}$ .",
"If $ s\\in \\mathbb {N}_{0}$ , then $H^{s,p}(\\Omega )$ and $W^{s,p}(\\Omega )$ coincide with the usual Sobolev spaces.",
"In the case of noninteger differentiability, for our purposes it suffices to consider these spaces as interpolation spaces.",
"If $ s=[s]+s_{\\ast }\\notin \\mathbb {N}_{0}$ with $[s]\\in \\mathbb {N}_{0}$ and $ s_{\\ast }\\in (0,1)$ , then $H^{s,p}=[H^{[s],p},H^{[s]+1,p}]_{s_{\\ast }},\\qquad W^{s,p}=(W^{[s],p},W^{[s]+1,p})_{s_{\\ast },p}, $ where $ [\\cdot ,\\cdot ]_{s_{\\ast }}$ and $(\\cdot ,\\cdot )_{s_{\\ast },p}$ denote complex and real interpolation, respectively.",
"Moreover, $ H^{s,p}=[L^{p},H^{2,p}]_{s/2}$ for $s\\in (0,2)$ and $W^{s,p}(\\Omega )=(L^{p},W^{2,p})_{s/2,p}$ for $ s\\in (0,2)$ , $s\\ne 1$ .",
"A useful tool are interpolation inequalities.",
"We shall make particular use of $\\Vert u\\Vert _{H^{s,p}}\\le \\Vert u\\Vert _{L^{p}}^{1-s/2}\\Vert u\\Vert _{H^{2,p}}^{s/2},\\qquad \\Vert u\\Vert _{W^{s,p}}\\le C\\,\\Vert u\\Vert _{L^{p}}^{1-s/2}\\Vert u\\Vert _{W^{2,p}}^{s/2}, $ which is valid for all $ u\\in H^{2,p}=W^{2,p}$ .",
"The corresponding function spaces over the boundary $ \\Gamma = \\partial \\Omega $ of a bounded smooth domain $ \\Omega \\subset \\mathbb {R}^n$ are defined via local charts.",
"Let $ \\text{g}_i:U_i\\subset \\mathbb {R}^{n-1} \\rightarrow \\Gamma $ be a finite family of parametrizations such that $ \\bigcup _i \\text{g}_i(U_i)$ covers $ \\Gamma $ , and let $\\lbrace \\psi _i\\rbrace $ be a partition of unity for $\\Gamma $ subordinate to this cover.",
"Then for $ s\\ge 0$ we have $H^{s,p}(\\Gamma ) = \\big \\lbrace u \\in L^p(\\Gamma )\\;:\\; (\\psi _i u)\\circ \\text{g}_i\\in H^{s,p}(\\mathbb {R}^{n-1}) \\text{ for all $ i$}\\big \\rbrace ,$ and an equivalent norm is given by $ \\Vert u\\Vert _{H^{s,p}(\\Gamma )} = \\sum _i\\Vert (\\psi _i u)\\circ \\text{g}_i\\Vert _{H^{s,p}(\\mathbb {R}^{n-1})}.",
"$ The spaces $ W^{s,p}(\\Gamma )$ are defined in the same way, replacing $H$ by $W $ .",
"In this way the properties of the spaces over $ \\Omega $ described above carry over to the spaces over $ \\Gamma $ .",
"For $ p\\in (1,\\infty )$ and $s > 1/p$ the trace $\\text{tr} \\,u = u|_\\Gamma $ extends to a continuous operator $\\text{tr}: H^{s,p}(\\Omega ) \\rightarrow W^{s-1/p,p}(\\Gamma ).$ Here we exclude the case $ s-1/p \\in \\mathbb {N}$ for $p\\ne 2$ ."
],
[
"Semiflows",
"Let $ \\mathcal {X}$ be a Banach space and let $t^+: \\mathcal {X }\\rightarrow (0,\\infty ]$ be lower semicontinuous.",
"Then we call a map $S: \\bigcup _{x\\in \\mathcal {X}} [0,t^+(x)) \\times \\lbrace x\\rbrace \\rightarrow \\mathcal {X}$ a local semiflow on $ \\mathcal {X}$ if for all $x\\in \\mathcal {X}$ it holds that $ S(\\cdot ;x):[0,t^+(x)) \\rightarrow \\mathcal {X}$ is continuous, if $S(t,\\cdot ):B_r(x) \\subset \\mathcal {X }\\rightarrow \\mathcal {X}$ is continuous for $ t < t^+(x)$ and sufficiently small $ r >0$ , if $S(0;\\cdot ) = \\text{id}_{\\mathcal {X}}$ , $ S(t+s;x) = S(t; S(s;x))$ and if $t^+(x) < \\infty $ implies that $\\Vert S(t;x)\\Vert _{\\mathcal {X}} \\rightarrow \\infty $ as $ t\\rightarrow t^+$ .",
"In addition we call $S$ compact, if for all bounded sets $ M\\subset \\mathcal {X}$ with $t^+(M) \\ge T > 0$ and all $ t\\in (0,T)$ it holds that $S(t;M)$ is relatively compact in $\\mathcal {X}$ .",
"If $ t^+(x) = \\infty $ for all $x\\in \\mathcal {X}$ , then we call $S$ a global semiflow.",
"In this case our notion of a semiflow coincides with the one in [4].",
"Note that, in contrast to parts of the literature, we include the condition for global existence (i.e., $ t^+ = \\infty $ ) already in the definition of a local semiflow."
],
[
"Maximal $ L^{p}$ -regularity for boundary differential operators",
"In this section we show maximal $ L^p$ -regularity for elliptic boundary differential operators of second order."
],
[
"Boundary differential operators",
"Throughout, let $ \\Omega \\subset \\mathbb {R}^{n}$ be a bounded domain with smooth boundary $ \\Gamma =\\partial \\Omega $ .",
"We describe our notion of a differential operator on $ \\Gamma $ with possibly nonsmooth coefficients.",
"Let $ (0,T)$ be a finite or infinite time interval.",
"We call a globally defined, linear map $ \\mathcal {C}:(0,T)\\times C^{\\infty }(\\Gamma )\\rightarrow L^{1}(\\Gamma )$ a (non-autonomous) boundary differential operator of order $ k\\in \\mathbb {N}$ , if for all $t\\in (0,T)$ and all parametrizations $ \\text{g}:U\\subset \\mathbb {R}^{n-1}\\rightarrow \\Gamma $ it holds $(\\mathcal {C}(t,\\cdot )u)\\circ \\text{g}(x)=\\sum _{|\\gamma |\\le k}c_{\\gamma }^{\\text{g}}(t,x)D_{n-1}^{\\gamma }(u\\circ \\text{g})(x),\\qquad x\\in U,$ with local coefficients $ c_{\\gamma }^{\\text{g}}(t,\\cdot )\\in L^{1}(U)$ and $ D_{n-1}=-\\text{i}\\nabla _{n-1}$ .",
"The coefficients do not have to be globally defined and may in fact depend on the parametrization $ \\text{g}$ .",
"The examples we have in mind are the Laplace-Beltrami operator $ \\Delta _{\\Gamma }=\\text{div}_{\\Gamma }\\nabla _{\\Gamma }$ , which is in coordinates given by $(\\Delta _{\\Gamma }u)\\circ \\text{g}=\\frac{1}{\\sqrt{\\left|\\text{G}\\right|}}\\sum _{i,j=1}^{n-1}\\partial _{i}(\\sqrt{|\\text{G}|}\\text{g}^{ij}\\partial _{j}(u\\circ \\text{g})),$ and, for a tangential vector field $ \\mathcal {V}$ on $\\Gamma $ , a surface convection term $ \\mathcal {V}\\nabla _{\\Gamma }$ , i.e., $(\\mathcal {V}\\nabla _{\\Gamma }u)\\circ \\text{g}=\\sum _{i,j=1}^{n-1}\\text{g}^{ij}(\\mathcal {V}\\cdot \\partial _{i}\\text{g})\\partial _{j}(u\\circ \\text{g}).$ Here $ \\text{G}^{-1}=(\\text{g}^{ij})_{i,j}$ is the inverse of the fundamental form $ \\text{G}$ corresponding to $\\text{g}$ .",
"As in the euclidian case, the regularity of the local coefficients $ c_\\gamma ^{\\text{g}}$ decides on which scale of function spaces over $\\Gamma $ the operator $ \\mathcal {C}(t,\\cdot )$ acts.",
"For instance, if $c_{\\gamma }^{\\text{g}}(t,\\cdot )\\in L^{\\infty }(U)$ for all parametrizations $ g$ and all $ \\gamma $ , then we obtain for all $p\\in [1,\\infty ]$ an estimate $\\Vert \\mathcal {C}(t,\\cdot )u\\Vert _{L^{p}(\\Gamma )}\\le C\\,\\Vert u\\Vert _{W^{k,p}(\\Gamma )},\\qquad u\\in C^{\\infty }(\\Gamma ).$ In this case $ \\mathcal {C}(t,\\cdot )$ extends uniquely to a bounded linear map $ W^{k,p}(\\Gamma )\\rightarrow L^{p}(\\Gamma )$ , or to a closed operator on $ L^{p}(\\Gamma )$ with domain $W^{k,p}(\\Gamma )$ .",
"Of course, in view of Sobolev embeddings, for such an extension the regularity of the coefficients can be lowered in many cases.",
"Finally, structural conditions like ellipticity of a boundary differential operator $ \\mathcal {C}$ can also be imposed to hold locally with respect to all parametrizations, see e.g.",
"condition (E) below."
],
[
"Maximal $ L^p$ -regularity",
"Let $ \\mathcal {C}$ be a boundary differential operator of order $k=2$ .",
"Consider for a finite time interval $ (0,T)$ the inhomogeneous Cauchy problem $\\partial _{t}u + \\mathcal {C}(t,x) u=g(t,x)\\quad \\text{on }(0,T)\\times \\Gamma ,\\qquad u|_{t=0}=u_{0}\\quad \\text{on }\\Gamma .$ For $ p\\in (1,\\infty )$ we take $g\\in L^p ((0,T)\\times \\Gamma )$ and are thus looking for solutions $ u$ that belong to the space $\\mathbb {E}(\\Gamma ) := W^{1,p}(0,T; L^p(\\Gamma )) \\cap L^p(0,T;W^{2,p}(\\Gamma )).$ We want that for all parametrizations $ \\text{g}:U\\rightarrow \\Gamma $ and all $ |\\gamma |\\le 2$ the terms $c_\\gamma ^\\text{g} D_{n-1}^\\gamma $ arising in the local representation of $ \\mathcal {C}$ are continuous from $\\mathbb {E}(U)$ to $ L^p((0,T)\\times U)$ .",
"Then in particular $\\mathcal {C}:\\mathbb {E}(\\Gamma ) \\rightarrow L^p((0,T)\\times \\Gamma )$ is continuous.",
"Using Sobolev embeddings and Hölder's inequality, it can be shown as in [27] that the following assumptions are sufficient for this purpose.",
"(R) Let $ \\text{g}:U\\rightarrow \\Gamma $ be any parametrization of $ \\Gamma $ .",
"Then for $|\\gamma |=2$ it holds $c_{\\gamma }^{\\text{g}}\\in BU\\!C([0,T]\\times U)$ , and in case $ |\\gamma |<2$ one of the following conditions is valid: either $ p > n+1$ and $c_{\\gamma }^{\\text{g}}\\in L^{p}((0,T)\\times U)$ , or there are $ r_{\\gamma },s_{\\gamma }\\in [p,\\infty )$ with $ \\frac{1}{s_{\\gamma }}+\\frac{n-1}{2r_{\\gamma }}<1-\\frac{|\\gamma |}{2}$ such that $ c_{\\gamma }^{\\text{g}}\\in L^{s_{\\gamma }}(0,T;L^{r_{\\gamma }}(U))$ .",
"As structural conditions for $ \\mathcal {C}$ we assume local parameter-ellipticity (cf.",
"[6] for the euclidian case).",
"Observe that this is a condition only for the highest order coefficients.",
"(E) For all parametrizations $ g:U\\rightarrow \\Gamma $ , all $t\\in [0,T]$ , $ x\\in U$ and $\\xi \\in \\mathbb {R}^{n-1}$ with $|\\xi |=1$ it holds that $ \\sum _{|\\gamma |=2}c_{\\gamma }^{\\text{g}}(t,x)\\xi ^{\\gamma } > 0$ .",
"Example 3.1 Let $ \\mathcal {C }u = - \\text{div}_\\Gamma (\\delta \\nabla _\\Gamma u)$ for a $ C^1 $ -function $\\delta : (0,T)\\times \\Gamma \\rightarrow \\mathbb {R}$ with $\\delta \\ge \\delta _* > 0$ .",
"Then $ \\mathcal {C}$ satisfies (E).",
"Our maximal $ L^p$ -regularity result is now as follows.",
"Theorem 3.2 Let $ p\\in (1,\\infty )$ and $T\\in (0,\\infty )$ .",
"Assume that $ \\Omega \\subset \\mathbb {R}^n$ is a bounded domain with smooth boundary $\\Gamma = \\partial \\Omega $ , and that $ \\mathcal {C}$ is a differential operator on $ \\Gamma $ of second order satisfying (R) and (E).",
"Then there is a unique solution $u\\in \\mathbb {E}(\\Gamma )=W^{1,p} (0,T;L^{p}(\\Gamma ))\\cap L^{p}(0,T;W^{2,p}(\\Gamma ))$ of the problem $\\left\\lbrace \\begin{array}{ll}\\partial _{t}u+\\mathcal {C}(t,x)u=g(t,x) & \\emph {\\text{on }} (0,T)\\times \\Gamma , \\\\u|_{t=0}=u_{0} & \\emph {\\text{on }}\\Gamma ,\\end{array}\\right.",
"$ if and only if the data is subject to $g\\in L^{p}((0,T)\\times \\Gamma ), \\qquad u_{0}\\in W^{2-2/p,p}(\\Gamma ).$ Given $ T_{0}>0$ , there is a constant $C$ , which is independent of the data and $ T\\in (0,T_{0})$ , such that $\\Vert u\\Vert _{\\mathbb {E}(\\Gamma )}\\le C\\left( \\Vert g\\Vert _{L^{p}((0,T)\\times \\Gamma )}+\\Vert u_{0}\\Vert _{W^{2-2/p,p}(\\Gamma )}\\right).",
"$ Moreover, in the autonomous case, i.e., if $ \\mathcal {C}$ is independent of $ t $ , then $- \\mathcal {C}$ generates an analytic $C_0$ -semigroup on $ L^p(\\Gamma ) $ .",
"Step 1.",
"If $ u\\in \\mathbb {E}(\\Gamma )$ solves (REF ), then $g\\in L^{p}((0,T)\\times \\Gamma )$ follows from (R), and further $ u_{0}\\in W^{2-2/p,p}(\\Gamma )$ is a consequence of e.g.",
"[28].",
"Next, assume that a unique solution of (REF ) exists for all given data.",
"Then the corresponding solution operator is continuous $ L^{p}((0,T)\\times \\Gamma )\\times W^{2 -2/p,p}(\\Gamma )\\rightarrow \\mathbb {E}(\\Gamma )$ due to the open mapping theorem.",
"This gives (REF ).",
"The uniformity of the constant with respect to $ T\\in (0,T_0)$ follows from an extension-restriction argument and the uniqueness of solutions.",
"Further, in this case the generator property of $ - \\mathcal {C}$ follows from [7], and the strong continuity of the semigroup is a consequence of the density of $ W^{2,p}(\\Gamma )$ in $L^p(\\Gamma )$ .",
"We thus have to show the unique solvability of (REF ) in $ \\mathbb {E}(\\Gamma )$ for all data $ g\\in L^p((0,T)\\times \\Gamma )$ and $u_0\\in W^{2-2/p,p}(\\Gamma ) $ .",
"A compactness argument shows that it suffices to do this for one (possibly small) $ T>0$ , which is independent of the data.",
"Step 2.",
"Choose a finite number of parametrizations $ \\text{g}_{i}$ with domains $ U_{i}$ such that $\\bigcup _i \\text{g}_i(U_i)$ covers $\\Gamma $ , and a partition of unity $ \\lbrace \\psi _{i}\\rbrace $ for $\\Gamma $ subordinate to this cover.",
"Then $ u\\in \\mathbb {E}(\\Gamma )$ solves (REF ) if and only if for all $ i,$ the function $v_{i}:=(\\psi _{i}u)\\circ \\text{g}_{i}$ solves $\\partial _{t}v_{i}+\\mathcal {C}^{\\text{g}_{i}}v_{i}=g_{i}, \\quad \\text{on }(0,T)\\times U_{i}, \\qquad v_{i}|_{t=0}=v_{i}^{0}, \\quad \\text{on }U_{i}.$ Here the local operator $ \\mathcal {C}^{\\text{g}_{i}}$ is given by $\\mathcal {C}^{\\text{g}_{i}}(t,x)=\\sum _{|\\gamma |\\le 2}c_{\\gamma }^{\\text{g}_{i}}(t,x)D^{\\gamma },\\qquad (t,x)\\in (0,T)\\times U_{i},$ and the transformed data is given by $g_{i}:=\\left( \\psi _{i}g+[\\mathcal {C},\\psi _{i}]u\\right) \\circ \\text{g}_{i},\\qquad v_{i}^{0}:=(\\psi _{i}u^{0})\\circ \\text{g}_{i},$ where $ [\\cdot ,\\cdot ]$ denotes the commutator bracket, i.e., $[\\mathcal {C},\\psi _{i}]u = \\mathcal {C}(\\psi _i u) - \\psi _i \\mathcal {C }u.$ Identifying $ v_{i}$ , $g_{i}$ and $v_{i}^{0}$ with their trivial extensions to $ \\mathbb {R}^{n-1}$ , we obtain $v_{i}\\in \\mathbb {E}(\\mathbb {R}^{n-1}), \\qquad g_{i}\\in L ^{p}((0,T)\\times \\mathbb {R}^{n-1}), \\qquad v_{i}^{0}\\in W^{2 -2/p,p}(\\mathbb {R}^{n-1}).$ We extend the top order coefficients $ c_{\\gamma }^{\\text{g}_{i}} $ , $|\\gamma |=2$ , to coefficients $ c_{\\gamma }^{i}\\in BU\\!C([0,T]\\times \\mathbb {R}^{n-1}) $ , using only values from the image of $ c_{\\gamma }^{\\text{g}_{i}}$ (assuming e.g.",
"$ U_{i}$ to be ball and reflecting on $\\partial U_i$ ).",
"By continuity of the $ c_{\\gamma }^{\\text{g}_{i}} $ , the oscillation of the extend top order coefficients becomes small if the diameter of the $ U_{i}$ is small.",
"The lower order coefficients $ c_{\\gamma }^{\\text{g}_{i}}$ , $ |\\gamma |<2$ , are trivially extended to $c_{\\gamma }^{i}$ on $(0,T)\\times \\mathbb {R}^{n-1}$ .",
"The extended coefficients $ c_\\gamma ^i$ , $|\\gamma |\\le 2$ , induce a differential operator $ \\mathcal {C}^{i}(t,x)$ acting on functions over $ (0,T)\\times \\mathbb {R}^{n-1}$ , which satisfies (R) and (E) for $\\Gamma = \\mathbb {R}^{n-1}$ .",
"Note in particular that (E) is a pointwise condition.",
"The (trivial extension of) $ v_{i}$ solves the full space problem $\\partial _{t}w+\\mathcal {C}^{i}(t,x)w=g_{i}\\quad \\text{on }(0,T)\\times \\mathbb {R}^{n-1}, \\qquad w|_{t=0}=v_{i}^{0} \\quad \\text{on }\\mathbb {R}^{n-1}.$ It is well-known that there is a solution operator $ \\mathcal {S}_i(g_i,v_0^i) $ for (REF ), which is continuous $\\mathcal {S}_{i}:L^{p}((0,T)\\times \\mathbb {R}^{n-1})\\times W^{2 -2/p,p}(\\mathbb {R}^{n-1})\\rightarrow \\mathbb {E}(\\mathbb {R}^{n-1}).$ We refer to [27] for the case of top order coefficients with small oscillation, which applies to the present case.",
"Hence $v_{i}=\\mathcal {S}_{i}(g_{i},v_{i}^{0})|_{U_{i}}.$ Step 3.",
"Take $ \\phi _{i}\\in C^\\infty (\\Gamma )$ with $\\phi _{i}\\equiv 1$ on $ \\text{supp}\\,\\psi _{i}$ and $\\text{supp}\\,\\phi _{i}\\subset \\text{g}_i(U_{i})$ .",
"On the complete metric space $\\mathbb {Y}_{u_0}:=\\left\\lbrace u\\in \\mathbb {E}(\\Gamma )\\;:\\;u|_{t=0}=u_{0}\\right\\rbrace ,$ which is nonempty by [28], we define the map $ \\mathcal {S}_{g,u_{0}}$ by $\\mathcal {S}_{g,u_{0}}(u):=\\sum _{i}\\phi _{i}\\mathcal {S}_{i}\\big ( (\\psi _{i}g+[\\mathcal {C},\\psi _{i}]u)\\circ \\text{g}_{i},(\\psi _{i}u_{0})\\circ \\text{g}_{i}\\big ) |_{U_{i}}\\circ \\text{g}_{i}^{-1}.$ Since $ \\mathcal {S}_i(g_i,v_i^0)|_{t=0}= v_i^0$ , we have that $\\mathcal {S}_{g,u^{0}}$ is a self-mapping on $ \\mathbb {Y}_{u_0}$ .",
"We show that it is a strict contraction if $ T$ is sufficiently small.",
"For $u,v\\in \\mathbb {Y}_{u_0} $ we have $\\Vert \\mathcal {S}_{g,u^{0}}(u)-\\mathcal {S}_{g,u^{0}}(v)\\Vert _{\\mathbb {E}(\\Gamma )}& \\,\\le C\\sum _i \\Vert \\mathcal {S}_{i}\\left( [\\mathcal {C},\\psi _{i}](u-v)\\circ g_{i},0\\right) \\Vert _{\\mathbb {E}(\\mathbb {R}^{n-1})} \\\\& \\,\\le C \\sum _{i} \\Vert [\\mathcal {C},\\psi _{i}](u-v)\\circ g_{i}\\Vert _{L^{p}((0,T)\\times \\mathbb {R}^{n-1})}.",
"$ Since $ [\\mathcal {C},\\psi _{i}](u-v)\\circ \\text{g}_{i}$ involves at most the first derivatives of $ (u-v)\\circ \\text{g}_i$ , we have for arbitrary $\\eta >0$ that $\\Vert [\\mathcal {C},\\psi _{i}](u-v)\\circ \\text{g}_{i}\\Vert _{L^{p}((0,T)\\times \\mathbb {R}^{n-1})}& \\,\\le C\\,\\Vert (u-v)\\circ \\text{g}_i\\Vert _{L^{p}(0,T;W^{1,p}(\\mathbb {R}^{n-1}))} \\\\& \\,\\le \\eta \\Vert u-v\\Vert _{L^{p}(0,T;W^{2,p}(\\Gamma ))} \\\\& \\quad +C_{\\eta }\\Vert u-v\\Vert _{L^{p}((0,T)\\times \\mathbb {R}^{n-1})}\\\\& \\,\\le (\\eta +C_{\\eta }T)\\Vert u-v\\Vert _{\\mathbb {E}(\\Gamma )}, $ using the interpolation inequality for $ W^{1,p}(\\Gamma )$ , Young's inequality and Poincare's inequality for $ L^p (0,T; L^p(\\Gamma ))$ (see [28]).",
"Thus if $ \\eta $ and $T$ are sufficiently small, then $\\mathcal {S}_{g,u_{0}}$ has a unique fixed point on $ \\mathbb {Y}_{u_0}$ .",
"Observe that this is true for all $ g$ and $u_0$ , and that the choice of $T$ is independent of $g$ and $ u_0$ .",
"The considerations in Step 2 show that every solution of (REF ) is necessarily a fixed point of $ \\mathcal {S}_{g,u_{0}}$ .",
"We have thus already shown that solutions of (REF ) are unique.",
"However, due to the nonempty intersections of the $ \\text{g}_i(U_{i})$ , the fixed point does in general not solve (REF ).",
"Step 4.",
"To find $ g^{\\ast }\\in L^{p}((0,T)\\times \\Gamma )$ for which $ \\mathcal {S}_{g^{\\ast },u_{0}}$ solves (REF ) for given $g$ and $u_0 $ we consider the fixed point map $ \\mathcal {F}$ , defined by $\\mathcal {F}:L^{p}((0,T)\\times \\Gamma )\\times W^{2 -2/p,p}(\\Gamma )\\rightarrow \\mathbb {E}(\\Gamma ),\\qquad \\mathcal {S}_{h,v_{0}}\\left( \\mathcal {F}(h,v_0)\\right) =\\mathcal {F}(h,v_0).$ For $ h\\in L^{p}((0,T)\\times \\Gamma )$ and $u_0 \\in W^{2 -2/p,p}(\\Gamma )$ we have $ \\mathcal {F}(h,u_0)|_{t=0} = u_0$ and $\\left( \\partial _{t}+\\mathcal {C}\\right) \\mathcal {F}(h,u_{0})=h+\\mathcal {K}h,$ with the error term $\\mathcal {K}h:=\\sum _{i}[\\mathcal {C},\\phi _{i}]\\mathcal {S}_{i}\\big ( (\\psi _{i}h+[\\mathcal {C},\\psi _{i}]\\mathcal {F}(h,u_{0}))\\circ \\text{g}_{i},(\\psi _{i}u_{0})\\circ \\text{g}_{i}\\big )|_{U_i} \\circ \\text{g}_{i}^{-1}.$ We use again the contraction principle to show that the map $ h\\mapsto g-\\mathcal {K}h$ has a fixed point $ g^{\\ast }$ on $L^{p}((0,T)\\times \\Gamma )$ .",
"Then $ \\mathcal {F}(g^{\\ast },u_{0})$ solves (REF ) for given $g\\in L^{p}((0,T)\\times \\Gamma )$ by (REF ).",
"First note that $ \\mathcal {K}$ maps $L^{p}((0,T)\\times \\Gamma )$ into itself by construction.",
"For $ h_{1},h_{2}\\in L^{p}((0,T)\\times \\Gamma )$ we have that $ \\mathcal {F}(h_1,v_0)-\\mathcal {F}(h_2,v_0) = \\mathcal {F}(h_1-h_2,0)$ , since this difference is the unique solution of (REF ) with inhomogeneity $ h_1-h_2$ and trivial initial value.",
"We thus obtain as above that $\\Vert \\mathcal {K}h_{1}-&\\, \\mathcal {K}h_{2}\\Vert _{L^{p}((0,T)\\times \\Gamma )}\\\\&\\,\\le C\\sum _{i}\\Vert \\mathcal {S}_{i}\\big ( (\\psi _{i}(h_1-h_2)+[\\mathcal {C},\\psi _{i}]\\mathcal {F}(h_{1}-h_{2},0))\\circ \\text{g}_{i},0\\big ) \\Vert _{L^{p}(0,T;W^{1,p}(\\mathbb {R}^{n-1} ))} \\\\& \\,\\le (\\eta +C_{\\eta }T)\\sum _{i}\\Vert \\mathcal {S}_{i}\\big ( (\\psi _{i}(h_{1}-h_{2})+[\\mathcal {C},\\psi _{i}]\\mathcal {F}(h_{1}-h_{2},0))\\circ \\text{g}_{i},0\\big ) \\Vert _{\\mathbb {E}(\\Gamma )} \\\\& \\,\\le C(\\eta +C_{\\eta }T)\\left( \\Vert h_{1}-h_{2}\\Vert _{L^{p}((0,T)\\times \\Gamma )}+\\Vert [\\mathcal {C},\\psi _{i}]\\mathcal {F}(h_{1}-h_{2},0)\\Vert _{L^{p}((0,T)\\times \\Gamma )}\\right), $ where $ \\eta >0$ is arbitrary.",
"Therefore $\\mathcal {K}$ is a strict contraction for sufficiently small $ \\eta $ and $T$ if the second summand above satisfies $ \\Vert [\\mathcal {C},\\psi _{i}]\\mathcal {F}(h_{1}-h_{2},0)\\Vert _{L^{p}((0,T)\\times \\Gamma )} \\le C\\, \\Vert h_{1}-h_{2}\\Vert _{L^{p}((0,T)\\times \\Gamma )},$ with a constant $ C$ independent of $T$ .",
"To see this we estimate for $h\\in L^{p}((0,T)\\times \\Gamma )$ $\\Vert \\mathcal {F}(h,0)\\Vert _{\\mathbb {E}(\\Gamma )} &\\, = \\Vert \\mathcal {S}_{h,0}(\\mathcal {F}(h,0))\\Vert _{\\mathbb {E}(\\Gamma )} \\\\& \\,\\le \\Vert \\mathcal {S}_{h,0}(\\mathcal {F}(h,0))-\\mathcal {S}_{h,0}(0)\\Vert _{\\mathbb {E}(\\Gamma )}+\\Vert \\mathcal {S}_{h,0}(0)\\Vert _{\\mathbb {E}(\\Gamma )}\\\\& \\,\\le (\\varepsilon +C_{\\varepsilon }T)\\Vert \\mathcal {F}(h,0)\\Vert _{\\mathbb {E}(\\Gamma )}+C\\Vert h\\Vert _{L^{p}((0,T)\\times \\Gamma )}$ for given $ \\varepsilon >0$ by (REF ) and (REF ).",
"In this inequality, if $ \\varepsilon $ and $T$ are sufficiently small, then we may absorb $ (\\varepsilon +C_{\\varepsilon }T)\\Vert \\mathcal {F}(h,0)\\Vert _{\\mathbb {E}(\\Gamma )}$ into the left-hand side to obtain $\\Vert \\mathcal {F}(h,0)\\Vert _{\\mathbb {E}(\\Gamma )}\\le C\\Vert h\\Vert _{L^{p}((0,T)\\times \\Gamma )}.$ Now (REF ) follows from $\\Vert [\\mathcal {C},\\psi _{i}]\\mathcal {F}(h_{1}-h_{2},0)\\Vert _{L^{p}((0,T)\\times \\Gamma )} \\le C\\, \\Vert \\mathcal {F}(h_{1}-h_{2},0)\\Vert _{\\mathbb {E}(\\Gamma )},$ which finishes the proof."
],
[
"Dirichlet problems",
"Let $ \\Omega \\subset \\mathbb {R}^{n}$ be a bounded domain with smooth boundary $ \\Gamma $ .",
"We assume that the operator $\\mathcal {A}$ is given by $ \\mathcal {A }u = - \\text{div} \\big ( d \\nabla u\\big ), \\quad d\\in C^\\infty (\\overline{\\Omega },\\mathbb {R}), \\quad d\\ge d_* >0,$ where $ d_*$ is a constant.",
"We first consider the linear inhomogeneous Dirichlet problem $\\left\\lbrace \\begin{array}{ll}\\lambda v + \\mathcal {A }v= f & \\text{in }\\Omega , \\\\v|_\\Gamma = g & \\text{on }\\Gamma .\\end{array}\\right.",
"$ We denote $ \\text{tr}\\, u = u|_\\Gamma $ for the trace on $\\Gamma $ .",
"It follows from classical Agmon-Douglis-Nirenberg theory that there is $ \\lambda _D\\ge 0$ such that for all $ \\lambda \\ge \\lambda _D$ it holds that $(\\lambda +\\mathcal {A}, \\text{tr}): H^{2,p}(\\Omega )\\rightarrow L^p(\\Omega )\\times W^{2-1/p,p}(\\Gamma )$ is a continuous isomorphism.",
"For instance, if $ \\mathcal {A}=-d\\Delta $ for a constant $ d>0$ , then one can take $\\lambda _{D}=0$ .",
"The corresponding inverse, which is denoted by $\\mathcal {R}_\\lambda := (\\lambda +\\mathcal {A}, \\text{tr})^{-1},$ enjoys the following properties.",
"Lemma 4.1 Let $ \\mathcal {A}$ be given by (REF ) and $p\\in (1,\\infty ).$ Then for all $ \\theta \\in (1/2p,1]$ , $\\theta \\ne 1/2 + 1/2p,$ and $ \\lambda \\ge \\lambda _D$ the operator $\\mathcal {R}_\\lambda $ extends to a continuous map $\\mathcal {R}_\\lambda :L^{p}(\\Omega )\\times W^{2\\theta -1/p,p}(\\Gamma ) \\rightarrow H^{2\\theta ,p}(\\Omega ).$ There are constants $ C_{D}$ (independent of $\\lambda $ ) and $C_{\\lambda }$ such that $\\Vert \\mathcal {R}_{\\lambda }(f,g)\\Vert _{H^{2\\theta ,p}(\\Omega )}\\le C_{D}\\lambda ^{-(1-\\theta )}\\Vert f\\Vert _{L^{p}(\\Omega )}+C_{\\lambda }\\Vert g \\Vert _{W^{2\\theta -1/p,p}(\\Gamma )}.$ Here we exclude $ \\theta =1/2+1/2p$ to avoid the terminology of Besov spaces.",
"The properties of the extension of $ \\mathcal {R}_\\lambda $ are proved in [9] and are based on the Lions-Magenes extension of $ (\\lambda +\\mathcal {A}, \\text{tr})$ , see [24].",
"Moreover, it is shown in [2] that $\\Vert \\mathcal {R}_\\lambda (f,0)\\Vert _{H^{2,p}(\\Omega )} \\le C \\Vert f\\Vert _{L^p(\\Omega )},\\qquad \\Vert \\mathcal {R}_\\lambda (f,0)\\Vert _{L^p(\\Omega )} \\le C\\lambda ^{-1}\\Vert f\\Vert _{L^p(\\Omega )},$ for all $ \\lambda \\ge \\lambda _D$ .",
"Now the second statement follows from complex interpolation.",
"Let us now consider the nonlinear Dirichlet problem $\\left\\lbrace \\begin{array}{ll}\\lambda u + \\mathcal {A }u= F(u) & \\text{in }\\Omega , \\\\u|_\\Gamma = u_\\Gamma & \\text{on }\\Gamma ,\\end{array}\\right.",
"$ where the boundary data $ u_\\Gamma $ is given.",
"For $p\\in (1,\\infty )$ and $ \\theta \\in (0,1)$ we assume that $F:H^{2\\theta ,p}(\\Omega )\\rightarrow L^{p}(\\Omega )\\text{ is globallyLipschitzian with constant $ c_F \\ge 0$.}",
"$ Example 4.2 Let $ F$ be the superposition operator induced by a globally Lipschitzian $f:\\mathbb {R}\\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ , i.e., $ F(u)(x) =f(u(x),\\nabla u(x))$ .",
"Then $ F$ satisfies (REF ) for all $p$ and $ \\theta \\ge \\frac{1}{2}$ .",
"If $F$ is induced by a function that is not globally Lipschitzian, then (REF ) cannot hold.",
"Definition 4.3 Let $ p\\in (1,\\infty )$ and $\\theta \\in (1/2p,1]$ with $\\theta \\ne 1/2 + 1/2p$ .",
"For $ u_{\\Gamma }\\in W^{2\\theta -1/p,p}(\\Gamma ),$ we call $ u\\in H^{2\\theta ,p}(\\Omega )$ a solution of (REF ) if $u=\\mathcal {R}_{\\lambda }(F(u),u_{\\Gamma }).",
"$ If $ \\theta =1$ , then such $u$ is a strong solution, meaning that it satisfies (REF ) almost everywhere in $ \\Omega $ in the sense of weak derivatives.",
"Lemma 4.4 Let $ p\\in (1,\\infty )$ , $\\theta \\in (1/2p,1)$ with $ \\theta \\ne 1/2 + 1/2p$ and assume (REF ) and (REF ).",
"Then there is $ \\lambda _*\\ge \\lambda _D$ such that for all $\\lambda \\ge \\lambda _*$ the following holds.",
"For all $ u_\\Gamma \\in W^{2\\theta -1/p,p}(\\Gamma )$ there is a unique solution $ u = \\mathcal {D}_\\lambda (u_\\Gamma )$ of (REF ) in the sense of (REF ), and the solution operator $\\mathcal {D}_\\lambda : W^{2\\theta -1/p,p}(\\Gamma )\\rightarrow H^{2\\theta ,p}(\\Omega )$ is globally Lipschitz continuous.",
"If $ F\\in C^k(H^{2\\theta ,p}(\\Omega ),L^p(\\Omega ))$ for $ k\\in \\mathbb {N}\\cup \\lbrace \\infty \\rbrace $ , then $\\mathcal {D}_\\lambda \\in C^k(W^{2\\theta -1/p,p}(\\Gamma ), H^{2\\theta ,p}(\\Omega ))$ .",
"For $ u_\\Gamma \\in W^{2\\theta -1/p,p}(\\Gamma )$ the map $u\\mapsto \\mathcal {R}_\\lambda (u,u_\\Gamma )$ is a strict contraction on $ H^{2\\theta ,p}(\\Omega )$ if $ \\lambda _*$ is sufficiently large, since $\\Vert \\mathcal {R}_\\lambda (u,u_\\Gamma ) - \\mathcal {R}_\\lambda (v,u_\\Gamma )\\Vert _{H^{2\\theta ,p}(\\Omega )} & = \\Vert \\mathcal {R}_\\lambda (F(u)-F(v),0)\\Vert _{H^{2\\theta ,p}(\\Omega )} \\\\& \\le C_D \\lambda _*^{-(1-\\theta )} c_F \\Vert u-v \\Vert _{H^{2\\theta ,p}(\\Omega )}$ for $ u,v\\in H^{2\\theta ,p}(\\Omega )$ by Lemma REF and (REF ).",
"The resulting unique fixed point is the unique solution of (REF ).",
"The global Lipschitz continuity of the solution operator $ \\mathcal {D}_\\lambda $ follows from $\\Vert \\mathcal {D}_\\lambda (u_\\Gamma )& - \\mathcal {D}_\\lambda (v_\\Gamma )\\Vert _{H^{2\\theta ,p}(\\Omega )} \\\\&\\, = \\Vert \\mathcal {R}_\\lambda (F(\\mathcal {D}_\\lambda (u_\\Gamma )) - F(\\mathcal {D}_\\lambda (v_\\Gamma )), u_\\Gamma - v_\\Gamma )\\Vert _{H^{2\\theta ,p}(\\Omega )} \\\\&\\, \\le C_D \\lambda _*^{-(1-\\theta )} c_F \\Vert \\mathcal {D}_\\lambda (u_\\Gamma ) -\\mathcal {D}_\\lambda (v_\\Gamma )\\Vert _{H^{2\\theta ,p}(\\Omega )} + C_\\lambda \\Vert u_\\Gamma - v_\\Gamma \\Vert _{W^{2\\theta -1/p,p}(\\Gamma )},$ and from $ C_D \\lambda _*^{-(1-\\theta )} c_F < 1$ .",
"Finally, suppose that $F\\in C^k$ and consider the map $\\mathcal {F }: H^{2\\theta ,p}(\\Omega )\\times W^{2\\theta -1/p,p}(\\Gamma ) \\rightarrow H^{2\\theta ,p}(\\Omega ), \\quad \\mathcal {F}(u,u_\\Gamma ) = u - \\mathcal {R}_\\lambda (F(u), u_\\Gamma ).$ The unique zero of $ \\mathcal {F}(\\cdot , u_\\Gamma )$ is $\\mathcal {D}_\\lambda (u_\\Gamma )$ .",
"We have $ \\mathcal {F }\\in C^k$ , and for $h\\in H^{2\\theta ,p}(\\Omega )$ it holds $ D_1 \\mathcal {F}(u,u_\\Gamma ) h = h -\\mathcal {R}_\\lambda (F^{\\prime }(u) h, 0).$ As above we can estimate $\\Vert \\mathcal {R}_\\lambda (F^{\\prime }(u) h, 0)\\Vert _{H^{2\\theta ,p}(\\Omega )} \\le C_D\\lambda _*^{-(1-\\theta )} c_F \\Vert h\\Vert _{H^{2\\theta ,p}(\\Omega )},$ which yields that $ D_1 \\mathcal {F}(u,u_\\Gamma )$ is invertible for all $ (u,u_\\Gamma )$ if $\\lambda _*$ is large.",
"Thus $\\mathcal {D}_\\lambda \\in C^k$ by the implicit function theorem.",
"Remark 4.5 The number $ \\lambda _*$ can be chosen such that $\\lambda _* > \\max \\lbrace \\lambda _D,(C_D c_F)^{\\frac{1}{1-\\theta }}\\rbrace $ .",
"It tends to infinity as the global Lipschitz constant of $ F$ tends to infinity."
],
[
"Quasilinear boundary conditions of reactive-diffusive type",
"We consider the following class of elliptic problems with quasilinear, nondegenerate dynamic boundary conditions of reactive-diffusive type, $\\left\\lbrace \\begin{array}{ll}\\lambda u + \\mathcal {A }u=F(u) & \\text{in }(0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }+ \\mathcal {C }(u_\\Gamma )u_{\\Gamma }+ \\mathcal {B }(u)=G(u_{\\Gamma }) & \\text{on } (0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma .\\end{array}\\right.",
"$ This is a generalization of the prototype model (REF ) from the introduction.",
"We assume that $ \\mathcal {A}$ is as above and that $F$ satisfies (REF ) with $ \\theta = 1-1/(2p)$ .",
"The nonlinear boundary differential operator $ \\mathcal {C}$ is given by $ \\mathcal {C }(u_\\Gamma )v_\\Gamma = - \\text{div}_\\Gamma \\big (\\delta (\\cdot ,u_\\Gamma ) \\nabla _\\Gamma v_\\Gamma \\big ), \\quad \\delta \\in C^\\infty (\\Gamma \\times \\mathbb {R},\\mathbb {R}), \\quad \\delta \\ge \\delta _* > 0,$ where $ \\delta _*$ is a constant.",
"The nonlinear map $\\mathcal {B}$ couples the equations in the domain and on the boundary in a nontrivial way.",
"Given $ p\\in (1,\\infty )$ , we assume that $ \\mathcal {B }: H^{2-1/p,p}(\\Omega )\\rightarrow L^p(\\Gamma ) \\text{ is locallyLipschitzian}.$ We do not impose any further structural condition for $ \\mathcal {B}$ .",
"In fact, it could vanish identically.",
"Example 4.6 The prototype for $ \\mathcal {B}$ is $\\mathcal {B}(u) = B\\nu \\cdot (\\nabla u)|_\\Gamma $ for some $ B\\in C^\\infty (\\Gamma , \\mathbb {R}^{n\\times n})$ , which satisfies (REF ) if $ p > 2$ .",
"For $B = \\pm \\text{id}$ one obtains $ \\mathcal {B }= \\pm \\partial _\\nu $ .",
"In the semilinear case one can also allow $ p\\le 2$ for such $\\mathcal {B}$ , see Proposition REF below.",
"Next, for the boundary nonlinearity $ G$ we assume $G:W^{2-2/p,p}(\\Gamma )\\rightarrow L^{p}(\\Gamma )\\text{ is locallyLipschitzian.}",
"$ Example 4.7 If $ G(u_\\Gamma )(x) = g(u_\\Gamma (x))$ and $g:\\mathbb {R}\\rightarrow \\mathbb {R}$ is locally Lipschitzian, then $ G$ satisfies (REF ) if $p > \\frac{n+1}{2}$ .",
"If $ g $ depends in addition on $\\nabla _\\Gamma u_\\Gamma $ , then $p > n+1$ is required.",
"These assertions are easily verified using Sobolev's embeddings.",
"If $ g$ is polynomial, then the values of $p$ can be lowered.",
"We are now ready to precisely state what we mean by a strong solution to problem (REF ).",
"Definition 4.8 A function $ u$ is said to be a strong solution of (REF ) if it is a strong solution of the elliptic equation almost everywhere in $ (0,T)$ and if its trace $u_{\\Gamma }$ is a strong solution of the parabolic equation on $ (0,T)\\times \\Gamma $ .",
"We have the following local well-posedness result for (REF ).",
"Theorem 4.9 Let $ p\\in (n+1,\\infty )$ and assume (REF ), (REF ) with $ \\theta = 1-1/(2p)$ , (REF ), (REF ) and (REF ).",
"Then there is $ \\lambda _{\\ast }$ such that for all $\\lambda \\ge \\lambda _{\\ast }$ the following holds true.",
"The problem (REF ) generates a compact local semiflow of strong solutions on $ W^{2-2/p,p}(\\Gamma )$ .",
"For all $ T\\in (0,t^{+}(u_{0}))$ a solution $u=u(\\cdot ,u_{0})$ enjoys the regularity $u\\in C([0,T];H^{2-1/p,p}(\\Omega ))\\cap L^{p}(0,T;W^{2,p}(\\Omega )),$ $u_{\\Gamma }\\in W^{1,p}(0,T;L^{p}(\\Gamma ))\\cap L^{p}(0,T;W^{2,p}(\\Gamma )).$ Remark 4.10 The corresponding result for boundary conditions of purely reactive type, i.e., $ \\mathcal {C}\\equiv 0$ and $\\mathcal {B}=d\\partial _{\\nu }$ , was shown in [8].",
"There the result is based on the generation properties of the Dirichlet-to-Neumann operator and thus requires a good sign of the normal derivative.",
"Moreover, the solutions enjoy worse regularity properties up to $ t=0$ .",
"In presence of the surface diffusion operator $ \\mathcal {C}$ , local well-posedness becomes essentially independent of the lower order coupling $ \\mathcal {B}$ .",
"The latter was already observed in [38] for a linear problem in the special case $ \\mathcal {B}=-\\partial _{\\nu }$ .",
"The proof shows that one can take $ \\lambda _{\\ast }=\\max \\lbrace \\lambda _{D},(C_{D}c_{F})^{p}\\rbrace .$ If $ F$ is not globally Lipschitzian, then nonexistence, nonuniqueness and noncontinuation phenomena can occur (see [11]).",
"Step 1.",
"Let $ \\mathcal {D}_{\\lambda }:W^{2-2/p,p}(\\Gamma )\\rightarrow H^{2-1/p,p}(\\Omega )$ be the solution operator from Lemma REF for the nonlinear Dirichlet problem (REF ).",
"Given $ u_0\\in W^{2-2/p,p}(\\Gamma )$ , we consider the quasilinear evolution equation $\\left\\lbrace \\begin{array}{ll}\\partial _{t}u_{\\Gamma }+\\mathcal {C}(u_{\\Gamma })u_{\\Gamma }=G(u_{\\Gamma })-\\mathcal {B}(\\mathcal {D}_{\\lambda }(u_{\\Gamma })) & \\text{on }(0,T)\\times \\Gamma , \\\\u_{\\Gamma }|_{t=0}=u_{0} & \\text{on }\\Gamma .\\end{array}\\right.",
"$ for $ u_\\Gamma $ .",
"We verify the conditions of [22] to apply the abstract results on local well-posedness provided there.",
"By assumption and the Lipschitz continuity of $ \\mathcal {D}_\\lambda $ , the map $ u_\\Gamma \\mapsto G(u_\\Gamma ) - \\mathcal {B}(\\mathcal {D}_\\lambda (u_\\Gamma ))$ is locally Lipschitzian $ W^{2-2/p,p}(\\Gamma )\\rightarrow L^p(\\Gamma )$ .",
"Next we rewrite the leading term $ \\mathcal {C}(u_\\Gamma ) u_\\Gamma $ into $\\mathcal {C}(u_\\Gamma ) u_\\Gamma = - \\delta (\\cdot , u_\\Gamma ) \\Delta _\\Gamma u_\\Gamma - \\nabla _\\Gamma (\\delta (\\cdot , u_\\Gamma )) \\nabla _\\Gamma u.$ For the first term we have $\\Vert \\delta (\\cdot , u_\\Gamma )\\Delta _\\Gamma w_\\Gamma -\\delta (\\cdot ,v_\\Gamma )\\Delta _\\Gamma w_\\Gamma \\Vert _{L^p(\\Gamma )} \\le \\Vert \\delta (\\cdot , u_\\Gamma ) - \\delta (\\cdot , v_\\Gamma )\\Vert _{L^\\infty (\\Gamma )}\\Vert w_\\Gamma \\Vert _{W^{2,p}(\\Gamma )}.$ The superposition operator induced by $ \\delta $ is locally Lipschitzian as a map $ C(\\Gamma ) \\rightarrow C(\\Gamma )$ , and since $p > \\frac{n+1}{2}$ we have that $ W^{2-2/p,p}(\\Gamma ) \\hookrightarrow C(\\Gamma )$ .",
"Therefore $u_\\Gamma \\mapsto - \\delta (\\cdot , u_\\Gamma ) \\Delta _\\Gamma $ is locally Lipschitzian as a map $ W^{2-2/p,p}(\\Gamma ) \\rightarrow \\mathcal {L}(W^{2,p}(\\Gamma ), L^p(\\Gamma ))$ .",
"Further, for each $ u_\\Gamma \\in W^{2-2/p,p}(\\Gamma )$ the function $ -\\delta (\\cdot ,u_\\Gamma )$ belongs to $C(\\Gamma )$ .",
"Hence by Theorem REF , the operator $ -\\delta (\\cdot , u_\\Gamma ) \\Delta _\\Gamma $ with domain $ W^{2,p}(\\Gamma )$ on $L^p(\\Gamma )$ enjoys the property of maximal $L^p$ -regularity.",
"Finally, if $ p> n+1$ then $W^{2-2/p,p}(\\Gamma ) \\hookrightarrow C^1(\\Gamma )$ , and this implies that $ u_\\Gamma \\mapsto \\nabla _\\Gamma (\\delta (\\cdot , u_\\Gamma )) \\nabla _\\Gamma u$ is locally Lipschitzian $ W^{2-2/p,p}(\\Gamma ) \\rightarrow L^p(\\Gamma )$ .",
"It thus follows from the results of [22] that (REF ) generates a local solution semiflow on $ W^{2-2/p,p}(\\Gamma )$ , such that $u_{\\Gamma }\\in W^{1,p}(0,T;L^{p}(\\Gamma ))\\cap L^{p}(0,T;W^{2,p}(\\Gamma ))\\cap C([0,T];W^{2-2/p,p}(\\Gamma ))$ for each $ T\\in (0,t^{+}(u_{0}))$ .",
"Hence $u:=\\mathcal {D}_{\\lambda }(u_{\\Gamma })$ solves (REF ), as described in Definition REF , by the Lemmas REF and REF .",
"In this way the semiflow for (REF ) becomes a semiflow for (REF ).",
"Step 2.",
"It remains to show the compactness of the semiflow generated by (REF ).",
"To this end we modify the arguments of [22] appropriately.",
"We will use the notion and properties of the weighted spaces $ L_{\\mu }^{p}$ used in [22], which are given by $L_{\\mu }^{p}(0,T;E):=\\bigg \\lbrace v:(0,T)\\rightarrow E\\,:\\,\\text{ }\\Vert v\\Vert _{L_{\\mu }^{p}(0,T;E)}^{p}:=\\int _{0}^{T}t^{p(1-\\mu )}|v(t)|_{E}^{p}\\,dt<+\\infty \\bigg \\rbrace ,$ for some $ p\\in (1,\\infty )$ , $\\mu \\in (1/p,1],$ and a Banach space $E$ with norm $ |\\cdot |_{E}$ .",
"The corresponding Sobolev spaces $W_{\\mu }^{1,p}(0,T;E)$ are defined by $W_{\\mu }^{1,p}(0,T;E):=\\lbrace v\\in L_{\\mu }^{p}(0,T;E)\\,:\\, \\exists \\,v^{\\prime }\\in L_{\\mu }^{p}(0,T;E)\\rbrace .$ Let us now return to the proof.",
"By assumption there exists a number $ \\mu \\in (1/p,1)$ with $ 2\\mu -2/p>1+\\frac{n-1}{p}$ .",
"The same arguments as above show that $ u_{\\Gamma }\\mapsto -\\delta (\\cdot ,u_{\\Gamma })\\Delta _{\\Gamma }$ is locally Lipschitzian $ W^{2\\mu -2/p,p}(\\Gamma )\\rightarrow \\mathcal {L}(W^{2,p}(\\Gamma ),L^{p}(\\Gamma ))$ , and the lower order nonlinearities are locally Lipschitzian $ W^{2\\mu -2/p,p}(\\Gamma )\\rightarrow L^{p}(\\Gamma )$ .",
"Thus by [22], for each $ v_{0}\\in W^{2\\mu -2/p,p}(\\Gamma )$ there are $ r,T>0$ and a continuous map $\\Phi :B_{r}(v_{0})\\subset W^{2\\mu -2/p,p}(\\Gamma )\\rightarrow W_{\\mu }^{1,p}(0,T;L^{p}(\\Gamma ))\\cap L_{\\mu }^{p}(0,T;W^{2,p}(\\Gamma ))$ such that $ u_{\\Gamma }=\\Phi (u_{0})$ solves (REF ) on $(0,T)$ .",
"Now let $ M$ be a bounded subset of $W^{2-2/p,p}(\\Gamma )$ such that $ t^+(M)\\ge T > 0$ .",
"Then $M$ is relatively compact in $W^{2\\mu -2/p,p}(\\Gamma )$ .",
"Hence finitely many balls $ B_{r_i} \\subset W^{2\\mu -2/p,p}(\\Gamma )$ suffice to cover $ M$ , with corresponding solution maps $\\Phi _i$ and times $T_i$ as above.",
"Let $ T_0 = \\min T_i$ , and take $0 < t\\le T_0$ .",
"Then we have $u(t;M)= \\bigcup _{i} \\text{tr}_{t} \\Phi _i(B_i \\cap M)$ .",
"By the continuity of $ \\Phi _i $ , the set $\\Phi _i(B_i \\cap M)$ is relatively compact in $ W^{1,p}_\\mu (0,T_0;L^p(\\Gamma )) \\cap L^p_\\mu (0,T_0; W^{2,p}(\\Gamma ))$ .",
"Moreover, the trace $ \\text{tr}_{t}$ at time $t$ is continuous from the latter space into the higher regularity space $ W^{2-2/p,p}(\\Gamma )$ , due to the fact that the weight $ t^{p(1-\\mu )}$ only has an effect at $t=0$ (see [30] and [28]).",
"Thus $ u(t;M)$ is relatively compact in $ W^{2-2/p,p}(\\Gamma )$ .",
"Finally, in case $T_0 < t\\le T $ we obtain the relative compactness of $ u(t;M)$ from $u(t;M) =u(t-T_0; u(T_0;M))$ and the continuity of $ u(t-T_0; \\cdot )$ on $ W^{2-2/p,p}(\\Gamma )$ .",
"Things are simpler in the semilinear case.",
"Proposition 4.11 Let $ p\\in (1,\\infty )$ and assume (REF ) and (REF ), where $ \\delta $ is independent of $u_\\Gamma $ .",
"Suppose that there is $ \\theta \\in (1/2p,1)$ such that $F: H^{2\\theta ,p}(\\Omega ) \\rightarrow L^p(\\Omega ), \\quad G: W^{2\\theta -1/p,p}(\\Gamma )\\rightarrow L^p(\\Gamma ), \\quad \\mathcal {B }: H^{2\\theta ,p}(\\Omega ) \\rightarrow L^p(\\Gamma ),$ where $ F$ is globally Lipschitzian and $G,\\mathcal {B}$ are Lipschitzian on bounded sets.",
"Then there is $ \\lambda _*$ such that for all $\\lambda \\ge \\lambda _*$ the following holds.",
"For all $ \\sigma \\in (\\theta , 1)$ the problem (REF ) generates a compact local semiflow of strong solutions on $ W^{2\\sigma -1/p,p}(\\Gamma )$ .",
"A solution $u = u(\\cdot ;u_0)$ enjoys the regularity $u\\in C([0,t^+); H^{2 \\sigma ,p}(\\Omega )) \\cap C(0,t^+; W^{2,p}(\\Omega )),$ $u_\\Gamma \\in C([0,t^+); W^{2 \\sigma - 1/p,p}(\\Gamma )) \\cap C^1(0,t^+;L^p(\\Gamma )) \\cap C(0,t^+; W^{2,p}(\\Gamma )).$ The reformulation (REF ) of (REF ) is now an abstract semilinear problem.",
"Since $ W^{2,p}(\\Gamma ) \\hookrightarrow L^p(\\Gamma )$ is compact, the assertions follow from Theorem REF , the Lemmas REF and REF , and e.g.",
"[4]."
],
[
"Compactness in the purely reactive case",
"We complement the results in [8], [9] concerning compactness of the solution semiflow.",
"We consider problems of type $\\left\\lbrace \\begin{array}{ll}\\lambda u - \\text{div}(d \\nabla u)=f(u) & \\text{in } (0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }+ d\\partial _\\nu u = g(u_\\Gamma ) & \\text{on }(0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma .\\end{array}\\right.",
"$ Throughout this subsection we assume that $ d\\in C^\\infty (\\overline{\\Omega }), \\qquad d\\ge d_* > 0, \\qquad f,g \\in C^\\infty (\\mathbb {R}), \\qquad |f^{\\prime }|\\le c_f.$ The results of [8] and [9] can be summarized as follows.",
"Proposition 4.12 Assume (REF ).",
"Then for $ p \\in (n,\\infty )$ there is $ \\lambda _*$ such that for all $\\lambda \\ge \\lambda _*$ the problem (REF ) generates a local semiflow of classical solutions on $ W^{1-1/p,p}(\\Gamma )$ .",
"A solution $u$ enjoys the regularity $u \\in C([0,t^+); W^{1,p}(\\Omega )) \\cap C^1(0,t^+; C^\\infty (\\Gamma )) \\cap C(0,t^+; C^\\infty (\\overline{\\Omega })).$ For sufficiently large $ \\lambda $ we define the Dirichlet-Neumann operator $ \\mathcal {N}_\\lambda $ by $\\mathcal {N}_\\lambda u_\\Gamma := d\\partial _\\nu \\mathcal {R}_\\lambda (0,u_\\Gamma ),$ where $ \\mathcal {R}_\\lambda $ is from Lemma REF .",
"The following generator properties of $ \\mathcal {N}_\\lambda $ are shown in [8] (see also [9] and [10]).",
"Proposition 4.13 Let $ d\\in C^\\infty (\\overline{\\Omega })$ with $d\\ge d_* > 0$ , and let $ \\lambda $ be sufficiently large.",
"Then for all $p\\in (1,\\infty )$ and $ \\theta \\ge 0$ the operator $-\\mathcal {N}_\\lambda $ with domain $ W^{\\theta +1,p}(\\Gamma )$ generates an analytic $C_0$ -semigroup on $ W^{\\theta ,p}(\\Gamma )$ .",
"Now we prove the compactness of the semiflow generated by (REF ).",
"Proposition 4.14 For each $ p\\in (n,\\infty )$ , the local solution semiflow from Proposition REF is compact.",
"Using the solution operator $ \\mathcal {D}_\\lambda $ from Lemma REF for the nonlinear Dirichlet problem (REF ), the regularity of the solutions allows rewrite (REF ) into $\\left\\lbrace \\begin{array}{ll} \\partial _t u_\\Gamma + \\mathcal {N}_\\lambda u_\\Gamma =g(u_\\Gamma ) - d\\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma )),0) & \\text{on } (0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma ,\\end{array}\\right.$ which is a semilinear problem for $ u_\\Gamma $ .",
"Let $M\\subset W^{1-1/p,p}(\\Gamma )$ be bounded with $ t^+(M)\\ge T > 0$ .",
"Fix $t\\in (0,T)$ .",
"Note that $ D(\\mathcal {N}_\\lambda ^{\\alpha _2}) \\hookrightarrow W^{s,p}(\\Gamma )\\hookrightarrow D(\\mathcal {N}_\\lambda ^{\\alpha _1})$ for $ \\alpha _2 > s>\\alpha _1\\ge 0$ .",
"If $ \\alpha $ is sufficiently close to $1-1/p$ , then Sobolev's embedding implies that $ u_\\Gamma \\mapsto g(u_\\Gamma )$ is Lipschitzian on bounded sets as a map $ D(\\mathcal {N}_\\lambda ^\\alpha ) \\rightarrow L^p(\\Gamma )$ .",
"Using the Lemmas REF and REF and the Lipschitz properties of $ f$ and $\\mathcal {D}_\\lambda $ , for $ u_\\Gamma , v_\\Gamma \\in D(\\mathcal {N}_\\lambda ^\\alpha )$ and $ \\eta \\in (0,\\alpha )$ we estimate $\\Vert d\\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma )) - f(\\mathcal {D}_\\lambda (v_\\Gamma )),0)\\Vert _{L^p(\\Gamma )}&\\, \\le C\\Vert f(\\mathcal {D}_\\lambda (u_\\Gamma )) - f(\\mathcal {D}_\\lambda (v_\\Gamma ))\\Vert _{L^p(\\Omega )} \\\\&\\, \\le C \\Vert u_\\Gamma - v_\\Gamma \\Vert _{W^{\\eta ,p}(\\Gamma )} \\\\&\\, \\le C \\Vert u_\\Gamma - v_\\Gamma \\Vert _{D(\\mathcal {N}_\\lambda ^\\alpha )}.$ Hence $ u_\\Gamma \\mapsto d\\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma )),0)$ is globally Lipschitzian as a map $ D(\\mathcal {N}_\\lambda ^\\alpha )\\rightarrow L^p(\\Gamma )$ .",
"Therefore [4] applies to (REF ), and we obtain that $ u_\\Gamma (t;M)$ is bounded in $ D(\\mathcal {N}_\\lambda ^\\alpha )$ for all $\\alpha \\in (0,1)$ .",
"Since $ W^{1,p}(\\Gamma ) \\hookrightarrow L^p(\\Gamma )$ is compact, we conclude that $ u_\\Gamma (t;M)$ is relatively compact in $W^{1-1/p,p}(\\Gamma )$ ."
],
[
"Qualitative properties of classical solutions",
"In this section we study the qualitative properties of solutions of the semilinear problem $\\left\\lbrace \\begin{array}{ll}\\lambda u-\\text{div}(d\\nabla u)=f(u) & \\text{in }(0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }-\\text{div}_{\\Gamma }(\\delta \\nabla _{\\Gamma }u_{\\Gamma })+d\\partial _{\\nu }u=g(u_{\\Gamma }) & \\text{on }(0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma ,\\end{array}\\right.",
"$ where we assume throughout that $ \\lambda \\ge \\lambda _{\\ast }$ is sufficiently large (in dependence on the other parameters).",
"We treat the two types of boundary conditions simultaneously and assume that $\\left\\lbrace \\begin{array}{c}d\\in C^{\\infty }(\\overline{\\Omega }),\\quad d\\ge d_{\\ast }>0,\\quad f,g\\in C^{\\infty }(\\mathbb {R}),\\quad |f^{\\prime }|\\le c_{f},\\quad p\\in (n,\\infty ),\\\\\\delta \\in C^{\\infty }(\\Gamma ),\\quad \\text{and either }\\delta \\ge \\delta _{\\ast }>0\\text{ or }\\delta \\equiv 0.\\end{array}\\right.",
"$ The local well-posedness of (REF ) is provided by the Propositions REF and REF .",
"To simplify the notation we set $\\mathcal {X}_{\\delta }:=W^{2-2/p,p}(\\Gamma )\\quad \\text{if }\\delta \\ge \\delta _{\\ast },\\qquad \\mathcal {X}_{\\delta }:=W^{1-1/p,p}(\\Gamma )\\quad \\text{if }\\delta \\equiv 0,$ for the corresponding phase spaces.",
"We will make essential use of the fact that by the Propositions REF and REF , for both types of boundary conditions the trace $ u_\\Gamma $ of a strong resp.",
"classical solution of (REF ) satisfies $\\left\\lbrace \\begin{array}{ll} \\partial _t u_\\Gamma + \\mathcal {C }u_\\Gamma + \\mathcal {N}_\\lambda u_\\Gamma = g(u_\\Gamma ) - d\\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma )),0) & \\text{on } (0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\text{on }\\Gamma ,\\end{array}\\right.$ where $ \\mathcal {C }u_\\Gamma = - \\text{div}_\\Gamma (\\delta \\nabla _\\Gamma u_\\Gamma )$ , $ \\mathcal {N}_\\lambda $ is the Dirichlet-Neumann operator, $ \\mathcal {R}_\\lambda $ is from Lemma REF and $\\mathcal {D}_\\lambda $ is from Lemma REF .",
"By Theorem REF and Proposition REF , the operators $ \\mathcal {C}$ and $\\mathcal {N}_\\lambda $ are both the negative generators of an analytic $ C_0$ -semigroup on $L^p(\\Gamma )$ .",
"Therefore we may represent $ u_\\Gamma $ by the variation of constants formula with an inhomogeneity as above."
],
[
"Classical solutions",
"We show the smoothness of solutions in space and time.",
"Besides its own interest, this will become important to apply the comparison result Lemma REF below and to show that (REF ) is of gradient structure (see Section REF ).",
"The key to smoothness in time is the following.",
"Lemma 5.1 Suppose that (REF ) holds, and that $ \\varphi \\in C^\\infty (0,T; W^{1-1/p,p}(\\Gamma ))$ .",
"For each $t\\in (0,T)$ , denote by $ u= u(t,\\cdot )$ the unique solution of $\\left\\lbrace \\begin{array}{ll}\\lambda u - \\emph {\\text{div}}(d \\nabla u)= f(u) & \\emph {\\text{in }}\\Omega ,\\\\u|_\\Gamma = \\varphi (t) & \\emph {\\text{on }}\\Gamma ,\\end{array}\\right.",
"$ i.e., $ u = \\mathcal {R}_\\lambda (f(u), \\varphi (t))$ for $t\\in (0,T)$ .",
"Then $ u\\in C^\\infty (0,T; H^{1,p}(\\Omega ))$ .",
"Define $ \\mathcal {F}:(0,T)\\times H^{1,p}(\\Omega )\\rightarrow H^{1,p}(\\Omega )$ by $\\mathcal {F}(t,v):=v-\\mathcal {R}_{\\lambda }\\big (f(v),\\varphi (t)\\big ).$ By Lemma REF , for each $ t$ the unique zero of $ \\mathcal {F}$ is $u(t,\\cdot )$ .",
"The assumption on $p$ guarantees that the superposition operator $ v\\mapsto f(v)$ belongs to $C^{\\infty }(H^{1,p}(\\Omega ),L^{p}(\\Omega ))$ , with derivative $ h\\mapsto f^{\\prime }(v)h$ .",
"The regularity of $ \\varphi $ and the continuity of $\\mathcal {R}_{\\lambda }$ thus show that $ \\mathcal {F}\\in C^{\\infty }$ .",
"At $v\\in H^{1,p}(\\Omega )$ the derivative $ D_{2}\\mathcal {F}(t,v)$ is given by $ h\\mapsto h-\\mathcal {R}_{\\lambda }(f^{\\prime }(v)h,0),$ and by Lemma REF it holds $\\Vert \\mathcal {R}_{\\lambda }(f^{\\prime }(v)h,0)\\Vert _{H^{1,p}(\\Omega )}\\le C_{D}\\lambda _{\\ast }^{-1/2}c_{f}\\Vert h\\Vert _{H^{1,p}(\\Omega )}.$ Therefore $ D_{2}\\mathcal {F}(t,v)$ is invertible for all $t$ and all $v$ .",
"We obtain that for every $ t_{0}\\in (0,T)$ there are $\\varepsilon >0$ and a function $ \\Phi \\in C^{\\infty }\\big (t_{0}-\\varepsilon ,t_{0}+\\varepsilon ;H^{1,p}(\\Omega )\\big )$ such that $ \\mathcal {F}(t,\\Phi (t)) = 0$ .",
"Uniqueness implies that $ \\Phi (t)=u(t,\\cdot )$ for all $t\\in (t_{0}-\\varepsilon ,t_{0}+\\varepsilon )$ .",
"Hence $ u\\in C^{\\infty }\\big (0,T;H^{1,p}(\\Omega )\\big )$ as asserted.",
"After this preparation we can show the smoothness of solutions.",
"Proposition 5.2 Let (REF ) hold.",
"Then for all $ u_0\\in \\mathcal {X}_\\delta $ the solution $ u$ of (REF ) satisfies $ u\\in C^\\infty ((0,t^+) \\times \\overline{\\Omega }).$ Throughout we fix $ T < t^+$ .",
"Step 1.",
"First let $ \\delta \\ge \\delta _*$ .",
"For sufficiently large $ \\rho $ the operator $\\rho + \\mathcal {C}$ is invertible and commutes with $-\\mathcal {C}$ .",
"Employing local arguments as in the proof of Theorem REF and interpolation, we obtain that $ \\rho +\\mathcal {C}$ is an isomorphism $ W^{2+\\theta ,p}(\\Gamma )\\rightarrow W^{\\theta ,p}(\\Gamma )$ for all $\\theta \\ge 0$ .",
"Thus $ -\\mathcal {C}$ with domain $W^{2+\\theta ,p}(\\Gamma )$ generates an analytic $ C_0$ -semigroup on $W^{\\theta ,p}(\\Gamma )$ for all $\\theta $ .",
"The trace $ u_\\Gamma $ may be represented by $ u_\\Gamma (t,\\cdot ) = e^{- \\mathcal {C }t }u_0 + e^{- \\mathcal {C}\\cdot } *\\big (g(u_\\Gamma ) - d \\partial _\\nu u\\big )(t), \\quad t\\in (0,T].$ Since $ u \\in C([0,T]; H^{2-1/p}(\\Omega ))$ we have $g(u_\\Gamma ) - d\\partial _\\nu u\\in C([0,T]; W^{1-2/p,p}(\\Gamma ))$ .",
"We may thus consider (REF ) as an identity on $ W^{1-2/p,p}(\\Gamma )$ , and obtain from [26] that $u_\\Gamma \\in C^1 ((0,T]; W^{1-2/p,p}(\\Gamma )) \\cap C ((0,T];W^{3-2/p,p}(\\Gamma )).$ Since $ u = \\mathcal {R}_\\lambda (f(u), u_\\Gamma )$ and $f(u(t,\\cdot )) \\in H^{1-1/p}(\\Omega )$ , we further obtain from [2] that $ u(t,\\cdot )\\in H^{3-1/p,p}(\\Omega )$ for all $t$ , and that $\\Vert u(t_1,\\cdot ) - u(t_2,\\cdot )\\Vert _{H^{3-1/p,p}(\\Omega )} &\\, \\le C\\big ( \\Vert f(u(t_1,\\cdot )) - f(u(t_2,\\cdot ))\\Vert _{H^{1-1/p}(\\Omega )} \\\\&\\, \\qquad \\quad + \\Vert u_\\Gamma (t_1,\\cdot )- u_\\Gamma (t_2,\\cdot )\\Vert _{W^{3-2/p,p}(\\Gamma )}\\big ) $ with a constant $ C$ independent of $t_1,t_2\\in (0,T]$ .",
"Thus $u \\in C((0,T];W^{3-1/p,p}(\\Omega ))$ .",
"An iteration of these arguments together with Sobolev's embeddings gives $u \\in C^1((0,T]; C^\\infty (\\Gamma ))\\cap C((0,T]; C^\\infty (\\overline{\\Omega })).$ Now it follows from (REF ) that $ u_\\Gamma \\in C^\\infty ((0,T];C^\\infty (\\Gamma ))$ .",
"Moreover, Lemma REF implies that $ u\\in C^\\infty ((0,T]; H^{1,p}(\\Omega ))$ .",
"Step 2.",
"Let now $ \\delta \\equiv 0$ .",
"By Proposition REF we have $ u\\in C^1((0,T]; C^\\infty (\\Gamma )) \\cap C((0,T]; C^\\infty (\\overline{\\Omega }))$ , and further $u_\\Gamma (t,\\cdot ) = e^{-\\mathcal {N}_\\lambda t} u_0 + e^{-\\mathcal {N}_\\lambda \\cdot } * \\big (g(u_\\Gamma ) - d \\partial _\\nu \\mathcal {R}_\\lambda (f(u),0)\\big )(t), \\quad t\\in (0,T].$ As above, this formula yields $ u_\\Gamma \\in C^\\infty ((0,T]\\times \\Gamma )$ and then $ u\\in C^\\infty ((0,T]; H^{1,p}(\\Omega ))$ by Lemma REF .",
"Step 3.",
"For both types of boundary conditions it now follows from the linearity and the continuity of $ \\mathcal {R}_\\lambda $ that $\\partial _t^k u = \\mathcal {R}_\\lambda (\\partial _t^k (f(u)), \\partial _t^ku_\\Gamma )$ for all $ k\\in \\mathbb {N}$ .",
"Now argue by induction and suppose that $ \\partial _t^{k-1} u \\in C((0,T]; C^\\infty (\\overline{\\Omega }))$ .",
"Note that $ \\partial _t^k(f(u))$ is of the form $f^{\\prime }(u) \\partial _t^k u + \\psi $ , where $ \\psi \\in C((0,T]; C^\\infty (\\overline{\\Omega }))$ is a polynomial in the derivatives of $ u$ up to the order $k-1$ and derivatives of $f$ with $u$ inserted.",
"Since $ |f^{\\prime }(u)| \\le c_f$ we may apply [2] to $ \\mathcal {A }- f^{\\prime }(u)$ for all $\\lambda \\ge \\lambda _D +c_f$ and estimate as in (REF ) to obtain $ \\partial _t^k u \\in C((0,T]; C^\\infty (\\overline{\\Omega }))$ ."
],
[
"Blow-up",
"In this subsection we assume that $ d\\equiv d_* >0$ and $\\delta \\equiv \\delta _*\\ge 0$ are constants.",
"Our blow-up results are based on the method of subsolutions and the following comparison lemma.",
"Its proof is inspired by [32].",
"Lemma 5.3 Assume $ f,g\\in C^1(\\mathbb {R})$ with $|f^{\\prime }|\\le c_f $ , $ \\lambda \\ge c_f$ , $d > 0$ and $\\delta \\ge 0$ .",
"If $u,v\\in C([0,T]\\times \\overline{\\Omega }) \\cap C^1((0,T]; C(\\Gamma )) \\cap C((0,T]; C^2(\\overline{\\Omega }))$ satisfy $\\left\\lbrace \\begin{array}{ll}\\lambda v-d\\Delta v-f(v) \\ge \\lambda u - d \\Delta u - f(u) & \\emph {\\text{in}}(0,T] \\times \\Omega , \\\\\\partial _{t}v_{\\Gamma }-\\delta \\Delta _{\\Gamma }v_{\\Gamma }+d\\partial _{\\nu }v- g(v_{\\Gamma }) \\ge \\partial _{t}u_{\\Gamma }-\\delta \\Delta _{\\Gamma }u_{\\Gamma }+d\\partial _{\\nu }u- g(u_{\\Gamma }) & \\emph {\\text{on }}(0,T]\\times \\Gamma , \\\\v_\\Gamma |_{t=0}\\ge u_\\Gamma |_{t=0} & \\emph {\\text{on }}\\Gamma ,\\end{array}\\right.$ then $ v \\ge u$ on $[0,T]\\times \\overline{\\Omega }$ .",
"The assumptions on $ f$ and $\\lambda $ imply that the function $a(t,x) = \\frac{\\lambda v(t,x) - f(v(t,x)) - (\\lambda u(t,x) -f(u(t,x)))}{v(t,x) - u(t,x)}$ is continuous and nonnegative on $ [0,T]\\times \\overline{\\Omega }$ .",
"Moreover, for all $ (t,x)\\in [0,T]\\times \\overline{\\Omega }$ we can write $g(v(t,x)) - g(u(t,x)) = (L- b(t,x))(v(t,x) -u(t,x)),$ where $ L>0$ is a constant and $b$ is continuous and nonnegative on $ [0,T]\\times \\Gamma $ .",
"Define $w(t,x):= e^{Lt}(v(t,x) -u(t,x)).$ We suppose that $ m:= \\min _{ [0,T]\\times \\overline{\\Omega }} w < 0 $ and derive a contradiction.",
"Let $ (t_0,x_0)\\in (0,T]\\times \\overline{\\Omega }$ be such that $ m = w(t_0,x_0)$ .",
"The function $w$ satisfies $\\lambda w - d\\Delta w \\ge e^{Lt_0}(f(v)-f(u)) \\qquad \\text{in }\\lbrace t_0\\rbrace \\times \\Omega ,$ and is thus a classical solution of $d\\Delta w - e^{Lt_0} (\\lambda v -f(v) - (\\lambda u - f(u))) = d \\Delta w - aw \\le 0 \\qquad \\text{in }\\lbrace t_0\\rbrace \\times \\Omega .$ Since $ - a \\le 0$ we deduce from the strong maximum principle [17] that $ x_0\\in \\Gamma $ .",
"Now the Hopf lemma [17] implies $ \\partial _\\nu w(t_0,x_0) < 0$ .",
"Therefore $ \\partial _t w(t_0,x_0) - \\delta \\Delta _\\Gamma w(t_0,x_0) + b(t_0,x_0)w(t_0,x_0) > 0.$ As $ b \\ge 0$ we have $b(t_0,x_0) w(t_0,x_0) \\le 0$ , and further $ \\partial _t w(t_0,x_0) \\le 0$ since $t\\mapsto w(t,x_0)$ attains its minimum in $ t_0$ .",
"Moreover, in case $\\delta > 0$ , take orthogonal coordinates $\\text{g}:U\\subset \\mathbb {R}^{n-1} \\rightarrow \\Gamma $ for $ x_0\\in \\Gamma $ , with $\\text{g}(y_0) = x_0$ for some $ y_0 \\in U$ .",
"Then $y \\mapsto w(t_0,\\text{g}(y))$ has a local minimum in $ y_0$ , which implies that $\\nabla _y w(t_0,\\text{g}(y_0)) =0 $ and $ \\Delta _y w(t_0,\\text{g}(y_0)) \\ge 0$ .",
"Hence the formula for $ \\Delta _\\Gamma $ in coordinates yields $\\Delta _\\Gamma w(t_0,x_0) = \\Delta _\\Gamma w(t_0, \\text{g}(y_0)) = \\Delta _yw(t_0,\\text{g}(y_0)) \\ge 0.$ The signs of the terms on the left-hand side of (REF ) lead to a contradiction.",
"To obtain appropriate subsolutions we modify the ones from [3].",
"Proposition 5.4 Let (REF ) hold and assume $ d \\equiv d_*$ and $ \\delta \\equiv \\delta _*$ .",
"Let further $\\frac{g(\\xi )}{\\xi ^q} \\rightarrow + \\infty \\qquad \\text{as }\\xi \\rightarrow +\\infty ,$ for some $ q>1$ .",
"Then there is $C>0$ such that if $u_0\\in \\mathcal {X}_\\delta $ satisfies $ u_0 \\ge C$ , then the solution of (REF ) blows up in finite time.",
"Remark 5.5 For $ \\delta =0$ , blow-up results for (REF ) with $f \\ne 0$ were obtained in [40] by the so-called concavity method.",
"For $ 1<r\\le q$ , let $\\varphi (s) := ( c- (r-1)s)^{-1/(r-1)}$ , where $c:=(r-1) (\\max _{y\\in \\overline{\\Omega }} \\sum _i y_i + 1)$ , such that $ \\varphi ^{\\prime }= \\varphi ^r$ and $\\varphi ^{\\prime \\prime }= r \\varphi ^{2r-1}$ .",
"Define $ \\underline{u}$ by $\\underline{u}(t,x) : = \\varphi \\Big (\\sum _i x_i + t\\Big ) = \\Big ((r-1) \\Big [ \\max _{y\\in \\overline{\\Omega }} \\sum _i y_i - \\sum _i x_i + 1 - t\\Big ] \\Big )^{-1/(r-1)},$ which is well-defined on $ \\overline{\\Omega }$ as long as $t < 1.$ Observe that $ \\underline{u}$ is positive and that $ \\text{for all $ K>0$ there is $r>1$ such that} \\qquad \\underline{u}(t,x) \\ge K \\quad \\text{on } [0,1)\\times \\overline{\\Omega }.$ We check that $ \\underline{u}$ is a subsolution of (REF ) on $ (0,1)\\times \\overline{\\Omega }$ for a suitable choice of $r$ .",
"First consider the elliptic equation.",
"The assumption on $ \\lambda $ and $f$ yields $\\lambda \\underline{u} - f(\\underline{u}) \\le (\\lambda +c_f) \\underline{u}-f(0),$ and we have $ \\Delta \\underline{u} = n r \\underline{u}^{2r-1}$ .",
"By (REF ) (with $ \\underline{u}^{2(r-1)}$ instead of $\\underline{u}$ ) we can achieve the inequality $(\\lambda +c_f) \\le d n r \\underline{u}^{2(r-1)} + f(0)/\\underline{u} \\qquad \\text{on } (0,1)\\times \\Omega $ if $ r$ is sufficiently close to 1.",
"For the boundary equation we have $ \\partial _t \\underline{u} = \\underline{u}^r $ and $ \\partial _\\nu \\underline{u} = (\\nu \\cdot \\mathbf {1}) \\underline{u}^r $ , where $ \\mathbf {1} = (1,...,1)\\in \\mathbb {R}^n$ .",
"To treat the Laplace-Beltrami term in case $ \\delta >0$ , fix $x_0\\in \\Gamma $ and take orthogonal coordinates $ \\text{g}:U\\subset \\mathbb {R}^{n-1}\\rightarrow \\Gamma $ for $ x_0$ , such that $x_0 = \\text{g}(y_0)$ for some $y_0\\in U$ .",
"Let $|\\text{G}|$ be the Gramian and let $ \\text{G}^{-1} = (\\text{g}^{ij})_{i,j=1,...,n-1}$ be the inverse fundamental form with respect to $ \\text{g}$ .",
"We write $a(x) =\\sum _i x_i$ for simplicity.",
"Since $ (\\text{g}^{ij})_{y=y_0}$ equals the Kronecker symbol, we have $(\\Delta _\\Gamma \\underline{u})(t,x_0) &\\, = \\sum _{i,j=1}^{n-1} \\partial _i \\big [ \\sqrt{|\\text{G}|} \\text{g}^{ij} \\partial _j(\\varphi ( a\\circ \\text{g}(y_0) + t))\\big ] \\\\&\\, = \\underline{u}^r(t,x_0) \\Delta _\\Gamma a(x_0) + r \\underline{u}^{2r-1}\\sum _{i=1}^{n-1} |\\partial _i (a\\circ \\text{g})(y_0)|^2 \\\\&\\, \\ge m \\underline{u}^r(t,x_0),$ where $ m = \\min _{x\\in \\Gamma } \\Delta _\\Gamma a(x)$ .",
"Therefore on $(0,1)\\times \\Gamma $ we have $\\partial _t \\underline{u} - \\delta \\Delta _\\Gamma \\underline{u} + d\\partial _\\nu \\underline{u} \\le (1 - \\delta m + \\nu \\cdot \\mathbf {1})\\underline{u}^r \\le g(\\underline{u})$ when choosing $ r$ such that $(1 - \\delta m + \\nu \\cdot \\mathbf {1}) \\le g(\\underline{u})/\\underline{u}^r$ on $ (0,1)\\times \\Gamma $ , which is possible by assumption on $ g$ and (REF ).",
"Hence $\\underline{u}$ is a subsolution of (REF ) if $ r$ is appropriate.",
"Now take $ u_0\\in \\mathcal {X}_\\delta $ with $u_0 \\ge \\underline{u}|_{t=0}$ on $ \\overline{\\Omega }$ .",
"Let $u$ be the corresponding classical solution of (REF ).",
"Then $ u \\ge \\underline{u}$ on $\\overline{\\Omega }$ by Lemma REF , as long as $ u$ exists.",
"Thus $u$ blows up at $t=1$ .",
"In case $ f \\equiv 0$ we can refine the blow-up condition for $g$ .",
"Proposition 5.6 Let $ d> 0$ and $\\delta \\ge 0$ .",
"Suppose that there is $ \\xi _0$ such that $g(\\xi )>0$ for $\\xi \\ge \\xi _0$ , and that $\\int _{\\xi _0}^\\infty \\frac{d\\xi }{g(\\xi )} < \\infty .$ Then there is $ C>0$ such that for all $u_0\\in \\mathcal {X}_\\delta $ the solution of $\\left\\lbrace \\begin{array}{ll}\\Delta u=0 & \\emph {\\text{in }}(0,T)\\times \\Omega , \\\\\\partial _{t}u_{\\Gamma }-\\delta \\Delta _{\\Gamma }u_{\\Gamma }+d\\partial _{\\nu }u= g(u_\\Gamma ) & \\emph {\\text{on }} (0,T)\\times \\Gamma , \\\\u_\\Gamma |_{t=0}=u_{0} & \\emph {\\text{on }}\\Gamma ,\\end{array}\\right.",
"$ blows up in finite time.",
"Remark 5.7 Under the additional assumption that $ g$ is entirely positive, the above result was shown in [21] for $ \\delta =0$ .",
"For a constant initial value $ \\underline{u}_0 > 0$ the solution of (REF ) is given by the solution $ \\underline{u}$ of $u^{\\prime }=g(u)$ with $ u|_{t=0} = \\underline{u}_0$ .",
"If $\\underline{u}_0$ is sufficiently large, then it is well-known that the condition on $ g$ implies that $ \\underline{u}$ blows up in finite time.",
"By Lemma REF , any solution of (REF ) with initial value $ u_0 \\ge \\underline{u}_0$ blows up as well."
],
[
"Global existence",
"We now return to the slightly more general assumptions (REF ) with variable diffusion coefficients.",
"First we refine the blow-up conditions and show that for both types of boundary conditions an $ L^\\infty $ -bound for $u_\\Gamma $ suffices for global existence.",
"Lemma 5.8 Let (REF ) hold, and assume that for $ u_0 \\in \\mathcal {X}_\\delta $ the solution $ u$ of (REF ) satisfies $u_\\Gamma \\in L^\\infty ((0,t^+)\\times \\Gamma ).$ Then $ t^+ = \\infty $ .",
"Suppose $ t^+ < \\infty $ .",
"We show $u_\\Gamma \\in L^\\infty (0,t^+; \\mathcal {X}_\\delta )$ to derive a contradiction.",
"In both cases $ \\delta \\ge \\delta _*$ and $ \\delta \\equiv 0$ , for $T< t^+$ we may use the variation of constants formula to estimate as in the proof of [4], $\\sup _{t\\in [0,T]} \\Vert u_\\Gamma (t)\\Vert _{\\mathcal {X}_\\delta } &\\, \\le C_{t^+}\\big (1 + \\sup _{t\\in [0,T]} \\Vert g(u_\\Gamma (t))\\Vert _{L^p(\\Gamma )} \\\\&\\,\\qquad \\qquad \\qquad \\quad + \\sup _{t\\in [0,T]} \\Vert d \\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma (t))),0)\\Vert _{L^p(\\Gamma )}\\big ).$ By assumption, the second summand is bounded independent of $ T < t^+$ .",
"For the third summand we have by Lemma REF that $\\Vert d \\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma (t))),0)\\Vert _{L^p(\\Gamma )} &\\, \\le C\\Vert f(\\mathcal {D}_\\lambda (u_\\Gamma (t)))\\Vert _{L^p(\\Omega )} \\le C_\\eta \\Vert u_\\Gamma (t)\\Vert _{W^{\\eta ,p}(\\Gamma )},$ where $ \\eta > 0$ is small.",
"Given $\\varepsilon >0$ , it follows from the interpolation inequality and Young's inequality that $\\Vert u_\\Gamma (t)\\Vert _{W^{\\eta ,p}(\\Gamma )} \\le \\varepsilon \\Vert u_\\Gamma (t)\\Vert _{\\mathcal {X}_\\delta } + C_\\varepsilon \\Vert u_\\Gamma (t)\\Vert _{L^p(\\Gamma )} \\le \\varepsilon \\sup _{t\\in [0,T]} \\Vert u_\\Gamma (t)\\Vert _{\\mathcal {X}_\\delta } + C_\\varepsilon .$ For sufficiently small $ \\varepsilon $ we may absorb $\\varepsilon \\sup _{t\\in [0,T]} \\Vert u_\\Gamma (t)\\Vert _{\\mathcal {X}_\\delta }$ into the left-hand side of (REF ).",
"We thus find a bound for $ \\sup _{t\\in [0,T]} \\Vert u_\\Gamma (t)\\Vert _{\\mathcal {X}_\\delta }$ that is independent of $ T < t^+$ .",
"Hence $u_\\Gamma \\in L^\\infty (0,t^+; \\mathcal {X} _\\delta )$ .",
"Remark 5.9 It follows from the proof above that if $ g$ grows asymptotically at most polynomial, then $ u_{\\Gamma }\\in L^{\\infty }( 0,t^{+}; L^q(\\Gamma ))$ for sufficiently large $ q<\\infty $ is already sufficient for global existence.",
"Before continuing we need the following inequality of Poincaré-Young type.",
"Lemma 5.10 For all $ p\\in (1,\\infty )$ and $\\varepsilon \\in (0,1)$ there is $ \\tau >0$ such that $\\Vert u\\Vert _{L^{p}(\\Gamma )}\\le \\varepsilon \\Vert \\nabla u\\Vert _{L^{p}(\\Omega )}+\\varepsilon ^{-\\tau }\\Vert u\\Vert _{L^{1}(\\Gamma )},\\qquad \\text{for all }u\\in W^{1,p}(\\Omega ).",
"$ Step 1.",
"We use the Poincaré inequality proved in [31] to estimate $\\Vert u\\Vert _{L^p(\\Omega )} \\le \\Vert u - \\frac{1}{|\\Gamma |} \\int _\\Gamma u\\Vert _{L^p(\\Omega )} + C\\Vert u\\Vert _{L^p(\\Gamma )} \\le C\\big (\\Vert \\nabla u\\Vert _{L^p(\\Omega )} + \\Vert u\\Vert _{L^p(\\Gamma )}\\big ).$ Thus $ \\Vert \\nabla u\\Vert _{L^p(\\Omega )} + \\Vert u\\Vert _{L^p(\\Gamma )}$ is an equivalent norm on $ W^{1,p}(\\Omega )$ .",
"Step 2.",
"By a scaling argument it suffices to prove the inequality for $ \\Vert u\\Vert _{L^p(\\Gamma )} = 1$ .",
"Suppose that there is no $\\tau >0$ such that the inequality holds for a given $ \\varepsilon \\in (0,1)$ .",
"Then for any $k\\in \\mathbb {N}$ there is $ u_k \\in W^{1,p}(\\Omega )$ such that $\\Vert u_k\\Vert _{L^p(\\Gamma )} = 1 \\ge \\varepsilon \\Vert \\nabla u_k\\Vert _{L^p(\\Omega )} +\\varepsilon ^{-k} \\Vert u_k\\Vert _{L^1(\\Gamma )}.$ It follows from this inequality and Step 1 that the resulting sequence $ (u_k) $ is bounded in $W^{1,p}(\\Omega )$ .",
"Since the trace operator is a compact map from $ W^{1,p}(\\Omega )$ into $L^p(\\Gamma )$ and into $L^1(\\Gamma )$ , we find a subsequence, again denoted by $ (u_k)$ , that converges in $ L^p(\\Gamma )$ and in $L^1(\\Gamma )$ to some limit $u$ .",
"By assumption we have $ \\Vert u\\Vert _{L^p(\\Gamma )} = 1$ .",
"On the other hand, the inequality shows that $ \\Vert u_k\\Vert _{L^1(\\Gamma )} \\le \\varepsilon ^k$ for all $k$ , such that $ \\Vert u\\Vert _{L^1(\\Gamma )} = 0$ and thus $u|_\\Gamma = 0$ .",
"This is a contradiction.",
"We verify an $ L^\\infty (\\Gamma )$ -bound for solutions of (REF ) under the assumption that $g(\\xi )\\xi \\le c_{g}( \\xi ^{2}+1 ) \\quad \\text{ for all }\\xi \\in \\mathbb {R},$ where $ c_{g}$ is a nonnegative constant.",
"Observe that this sign condition complements the sufficient condition from Proposition REF for blow-up.",
"Proposition 5.11 Let (REF ) hold, and assume (REF ).",
"Then for all $ u_0 \\in \\mathcal {X}_\\delta $ the classical solution of (REF ) exists globally in time, i.e., $ t^{+}=\\infty $ .",
"We suppose that $ t^+ <\\infty $ and show $u_\\Gamma \\in L^\\infty ((0,t^+)\\times \\Gamma )$ to derive a contradiction to Lemma REF .",
"Step 1.",
"Let $ T < t^+$ .",
"By an iteration argument we will first show that $\\Vert u_\\Gamma \\Vert _{L^{\\infty }((0,T)\\times \\Gamma )}\\le C\\,\\max \\left(\\Vert u_{0}\\Vert _{L^{\\infty }(\\Gamma )},\\Vert u_\\Gamma \\Vert _{L^{\\infty }(0,T;L^{2}(\\Gamma ))}\\right), $ where $ C$ is independent of $u_\\Gamma $ and $T$ .",
"Let $k\\in \\mathbb {N}$ , fix $ t\\in (0,T)$ and write $u=u_\\Gamma = u(t,\\cdot )$ .",
"We multiply the equation on $ \\Gamma $ by $u^{2^{k}-1}$ and integrate by parts on $\\Gamma $ to obtain $\\frac{d}{dt}\\int _{\\Gamma }u^{2^{k}}dS& \\,=-(2^{k}-1)2^{2-k} \\int _{\\Gamma }\\delta |\\nabla _{\\Gamma }\\left( u^{2^{k-1}}\\right) |^{2}dS \\\\& \\qquad + 2^k \\int _{\\Gamma }g(u)u^{2^{k}-1}dS-2^{k}\\int _{\\Gamma }d \\partial _{\\nu }u u^{2^{k}-1}dS.$ Multiplying the equation on $ \\Omega $ by $u^{2^{k}-1}$ gives $-2^{k}d\\int _{\\Gamma }\\partial _{\\nu }u u^{2^{k}-1}dS & \\, =-(2^{k}-1)2^{2-k}\\int _{\\Omega }d |\\nabla \\left( u^{2^{k-1}}\\right) |^{2}dx \\\\&\\, \\qquad + 2^{k}\\int _{\\Omega }\\left( f(u)u^{2^{k}-1}-\\lambda u^{2^{k}}\\right) dx.$ Using $ -(2^{k}-1)2^{2-k}\\le -2$ , that $f$ is globally Lipschitzian and that $ \\lambda \\ge c_f$ , we obtain $\\frac{d}{dt}\\int _{\\Gamma }u^{2^{k}}dS& \\,\\le -2d_* \\int _{\\Omega } |\\nabla \\left( u^{2^{k-1}}\\right) |^{2}dx \\\\& \\,\\qquad \\quad +2^{k}\\int _{\\Omega }\\left( f(u)u^{2^{k}-1}-\\lambda u^{2^{k}}\\right) dx+2^{k}\\int _{\\Gamma }g(u)u^{2^{k}-1}dS \\\\&\\, \\le - 2 d_*\\int _{\\Omega }|\\nabla \\left( u^{2^{k-1}}\\right) |^{2}dx +C\\,2^{k} \\int _{\\Gamma }u^{2^{k}}dS + C\\, 2^k.",
"$ Given $ \\varepsilon >0$ , it follows from Lemma REF that there is $ \\tau >1$ such that $- \\int _\\Omega |\\nabla v|^2\\, dx \\le - \\varepsilon ^{-1} \\int _\\Gamma v^2 \\,dS + \\varepsilon ^{-\\tau } \\left( \\int _\\Gamma |v| \\, d S\\right)^2.$ Choosing $ \\varepsilon = \\delta 2^{-k}$ with sufficiently small $\\delta >0$ , we obtain that $ \\frac{d}{dt}\\int _{\\Gamma }u^{2^{k}}\\, dS \\le - 2^k \\int _{\\Gamma }u^{2^{k}}\\, d S + C 2^{k\\tau }\\int _{\\Gamma }u^{2^{k-1}} \\, d S + C 2^{k}, \\qquad k\\in \\mathbb {N}.$ Now (REF ) follows from a standard Moser-Alikakos iteration procedure as presented e.g.",
"in [4] (see also [27]).",
"Step 2.",
"Set $ \\varphi =\\Vert u_\\Gamma \\Vert _{L^2(\\Gamma )}^2$ .",
"Employing (REF ) with $ k=1$ , we get $\\varphi ^{\\prime }\\le C_{1}\\varphi +C_{2}$ , which we can integrate to $\\varphi \\left( t\\right) \\le C_{1}\\int _{0}^{t}\\varphi (s)ds+\\left(tC_{2}+\\varphi (0)\\right) ,\\qquad t\\in (0,T).$ Thus, by Gronwall's inequality, $\\Vert u_\\Gamma \\Vert _{L^2(\\Gamma )}^2 \\le \\left( tC_{2}+\\int _{\\Gamma }u_{0}^{2}dS\\right) e^{C_{1}t},\\qquad t\\in (0,T).$ Hence $ u_\\Gamma \\in L^\\infty (0,t^+; L^2(\\Gamma ))$ , and therefore $u_\\Gamma \\in L^\\infty ((0,t^+) \\times \\Gamma )$ by (REF ).",
"Combining the Propositions REF and REF gives the following.",
"Theorem 5.12 Let (REF ) hold, assume $ d \\equiv d_*$ and $\\delta \\equiv \\delta _*\\ge 0$ , and that $g(\\xi ) \\sim \\rho |\\xi |^{q-1} \\xi \\qquad \\text{as }|\\xi |\\rightarrow \\infty ,$ for some $ \\rho \\in \\mathbb {R}$ and $q>0$ .",
"Then for all $u_0 \\in \\mathcal {X}_\\delta $ the problem (REF ) has a unique global classical solution if and only if either $ \\rho \\le 0$ or $q\\le 1$ .",
"As for blow-up, we refine the sufficient conditions on $ g$ for global existence in case of the Laplace equation.",
"We argue as in [5], where the case $ \\delta \\equiv 0$ was considered.",
"The condition below complements the one of Proposition REF .",
"Proposition 5.13 Let (REF ) hold, assume $ d \\equiv d_*$ and $\\delta \\equiv \\delta _*\\ge 0$ , and that $ |g|\\le \\gamma $ , where $\\gamma \\in C(\\mathbb {R},(0,\\infty ))$ is such that $\\int _0^\\infty \\frac{ds}{\\gamma (s)} = \\int _{-\\infty }^0 \\frac{ds}{\\gamma (s)} =\\infty .$ Then for all $ u_0 \\in \\mathcal {X}_\\delta $ the problem (REF ) has a unique global classical solution.",
"Suppose that $ t^+ < \\infty $ , and let $\\xi (t)$ , $\\zeta (t)\\in \\overline{\\Omega }$ be such that $m(t):= \\min _{x\\in \\overline{\\Omega }} u(t,x) = u(t,\\xi (t)), \\qquad M(t):=\\max _{x\\in \\overline{\\Omega }} u(t,x) = u(t,\\zeta (t)).$ It follows from [17] that for each $ t\\in (0,t^+)$ we have $ \\xi (t), \\zeta (t) \\in \\Gamma $ .",
"Thus $\\partial _\\nu u(t,\\xi (t)) < 0$ and $ \\partial _\\nu u(t,\\zeta (t)) > 0$ by [17].",
"By [5], the function $ m$ is almost everywhere differentiable on $(0,t^+)$ with $ \\partial _t m = (\\partial _t u)(t,\\xi (t)).$ Using that $\\Delta _\\Gamma u(t,\\xi (t)) \\ge 0$ , which can be seen as in the proof of Lemma REF , we get $\\partial _t m(t) = \\delta \\Delta _\\Gamma u(t,\\xi (t)) - \\partial _\\nu u(t,\\xi (t)) + g( u(t,\\xi (t))) \\ge - \\gamma ( m(t))$ for a.e.",
"$ t\\in (0,t^+)$ .",
"In the same way we obtain $\\partial _t M(t) \\le \\gamma ( M(t))$ for a.e.",
"$ t\\in (0,t^+).$ Now the very same arguments as in the proof of [5] provide a contradiction to the assumption $ t^+ < \\infty $ ."
],
[
"Global attractors",
"Suppose that (REF ) and (REF ) hold true.",
"Then by the above results, (REF ) generates a compact global solution semiflow $S_{\\delta }(t;u_{0}):=u(t;u_{0})$ of smooth solutions in the phase space $ \\mathcal {X}_{\\delta }$ .",
"Let $ F^{\\prime }=f$ and $G^{\\prime }=g$ .",
"Then we may differentiate $\\mathcal {E}(u):=\\frac{1}{2}\\int _{\\Omega }d |\\nabla u|^{2}\\,dx+\\frac{1}{2}\\int _{\\Gamma }\\delta |\\nabla _{\\Gamma }u_{\\Gamma }|^{2}\\,dS-\\int _{\\Omega }(F(u)-\\frac{\\lambda }{2}u^{2})\\,dx-\\int _{\\Gamma }G(u_{\\Gamma })\\,dS$ with respect to time, to obtain $\\partial _{t}\\mathcal {E}(u)=-\\Vert \\partial _{t}u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}.",
"$ Thus $ \\mathcal {E}$ is a strict Lyapunov function for (REF ), and the problem is of gradient structure.",
"By [4], for the existence of a global attractor it is left to show the boundedness of the set of equilibria $ E$ of (REF ).",
"To formulate a sufficient condition for this, we note that by the global Lipschitz continuity of $ f$ there is a constant $\\tilde{c}_{f}\\in \\mathbb {R}$ such that $f(\\xi )\\xi \\le \\tilde{c}_{f}(\\xi ^{2}+1),\\qquad \\xi \\in \\mathbb {R}.$ For the boundedness of the equilibria the parameters of the problem should satisfy $\\frac{\\Vert \\sqrt{d}\\nabla \\psi \\Vert _{L^{2}(\\Omega )}^{2}+\\Vert \\sqrt{\\delta }\\nabla _{\\Gamma }\\psi \\Vert _{L^{2}(\\Gamma )}^{2}-(\\tilde{c}_{f}-\\lambda )\\Vert \\psi \\Vert _{L^{2}(\\Omega )}^{2}-c_{g}\\Vert \\psi \\Vert _{L^{2}(\\Gamma )}^{2}}{\\Vert \\psi \\Vert _{L^{2}(\\Gamma )}^{2}}\\ge \\eta >0,$ for all $ \\Theta _{\\delta }:=\\left\\lbrace \\psi \\in W^{1,2}(\\Omega )\\,:\\, \\psi |_{\\Gamma }\\in W^{1,2}(\\Gamma )\\text{ if }\\delta \\ge \\delta _{\\ast }\\right\\rbrace $ .",
"Lemma 5.14 Assume (REF ), (REF ), (REF ) and (REF ).",
"Then the set of equilibria $ E\\subset C^\\infty (\\overline{\\Omega })$ of (REF ) is bounded in $ W^{2,p}(\\Gamma )$ for $\\delta \\ge \\delta _*$ and it is bounded in $ W^{1,p}(\\Gamma )$ for $\\delta \\equiv 0$ .",
"Step 1.",
"Note that indeed $ E\\subset C^{\\infty }(\\overline{\\Omega })$ by Proposition REF .",
"Thus an equilibrium $ u$ satisfies $\\lambda u-\\text{div}(d\\nabla u)=f(u)\\quad \\text{in }\\Omega ,\\qquad -\\text{div}_{\\Gamma }(\\delta \\nabla _{\\Gamma }u)+d\\partial _{\\nu }u=g(u_{\\Gamma })\\quad \\text{on }\\Gamma .",
"$ Multiplying by $ u$ , integrating by parts and using (REF ), (REF ) and (REF ), we get $C& \\,\\ge \\Vert \\sqrt{d}\\nabla u\\Vert _{L^{2}(\\Omega )}^{2}+\\Vert \\sqrt{\\delta }\\nabla _{\\Gamma }u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}-(\\tilde{c}_{f}-\\lambda )\\Vert u\\Vert _{L^{2}(\\Omega )}^{2}-c_{g}\\Vert u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2} \\\\& \\,\\ge \\eta \\Vert u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2},$ with a constant $ C$ independent of $u$ .",
"Hence $\\sup _{u\\in E}\\Vert u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}<\\infty $ .",
"Next we obtain from (REF ) that there are $ C,\\tau >0$ such that $\\int _{\\Gamma }u_{\\Gamma }^{2^{k}}\\,dS\\le 2^{k\\tau }\\int _{\\Gamma }u_{\\Gamma }^{2^{k-1}}\\,dS+C$ for all $ u\\in E$ and $k\\in \\mathbb {N}$ .",
"Hence, by an iteration argument, $\\Vert u_{\\Gamma }\\Vert _{L^{\\infty }(\\Gamma )}\\le C\\big (1+\\Vert u_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}\\big ).$ Therefore $\\sup _{u\\in E}\\Vert u_{\\Gamma }\\Vert _{L^{\\infty }(\\Gamma )}<\\infty .$ Step 2.",
"Suppose that $ \\delta \\ge \\delta _{\\ast }$ .",
"Then for $u\\in E$ , (REF ) gives $\\Vert u_{\\Gamma }\\Vert _{W^{2,p}(\\Gamma )}& \\,\\le C\\big (\\Vert u_{\\Gamma }\\Vert _{L^{p}(\\Gamma )}+\\Vert \\Delta u_{\\Gamma }\\Vert _{L^{p}(\\Gamma )}\\big )\\\\& \\,\\le C\\big (1+\\Vert g(u_{\\Gamma })\\Vert _{L^{p}(\\Gamma )}+\\Vert \\partial _{\\nu }u\\Vert _{L^{p}(\\Gamma )}\\big )\\le C(1+\\Vert u\\Vert _{H^{2-1/p,p}(\\Omega )}\\big ).$ Recall that $ u=\\mathcal {D}_{\\lambda }(u_{\\Gamma })$ , where $\\mathcal {D}_{\\lambda }:W^{2-2/p,p}(\\Gamma )\\rightarrow H^{2-1/p,p}(\\Omega )$ is globally Lipschitzian by Lemma REF .",
"Using the interpolation inequality, Young's inequality and (REF ), for arbitrary $ \\varepsilon >0$ we get $\\Vert u\\Vert _{H^{2-1/p,p}(\\Omega )}\\le C\\big (1+\\Vert u_{\\Gamma }\\Vert _{W^{2-2/p,p}(\\Gamma )}\\big )\\le \\varepsilon \\Vert u_{\\Gamma }\\Vert _{W^{2,p}(\\Gamma )}+C_{\\varepsilon },$ where $ C_{\\varepsilon }$ does not depend on $u\\in E$ .",
"For small $\\varepsilon $ we can thus absorb $ \\varepsilon \\Vert u_{\\Gamma }\\Vert _{W^{2,p}(\\Gamma )}$ into the left-hand side of the previous inequality to obtain $ \\sup _{u\\in E}\\Vert u_{\\Gamma }\\Vert _{W^{2,p}(\\Gamma )}<\\infty $ .",
"Step 3.",
"Now let $ \\delta \\equiv 0$ .",
"Then for $u\\in E$ we have by Lemma REF that $\\Vert u_\\Gamma \\Vert _{W^{1,p}(\\Gamma )} &\\, \\le C\\big ( \\Vert u_\\Gamma \\Vert _{L^p(\\Gamma )} +\\Vert \\mathcal {N}_\\lambda u_\\Gamma \\Vert _{L^p(\\Gamma )}\\big ) \\\\&\\, \\le C\\big (1 + \\Vert \\partial _\\nu \\mathcal {R}_\\lambda (f(u),0)\\Vert _{L^p(\\Gamma )}\\big ) \\le C(1 + \\Vert u\\Vert _{L^p(\\Omega )}\\big ).$ Since $ \\Vert u\\Vert _{L^p(\\Omega )} \\le C\\big (1+ \\Vert u_\\Gamma \\Vert _{W^{1-1/p,p}(\\Gamma )}\\big )$ by Lemma REF , we may argue as above to obtain $ \\sup _{u\\in E} \\Vert u_\\Gamma \\Vert _{W^{1,p}(\\Gamma )} < \\infty $ .",
"Under the above assumptions it now follows from [4] that the semiflow $ S_{\\delta }$ generated by (REF ) has a global attractor $ \\mathcal {A}_{\\delta }.$ To verify that $\\mathcal {A}_{\\delta }$ has finite Hausdorff dimension, we need the following.",
"Lemma 5.15 Assume (REF ) and (REF ).",
"Then for each $ t >0$ the time $t$ map $S_\\delta (t;\\cdot )$ belongs to $ C^\\infty (\\mathcal {X }_\\delta )$ , and the derivative $D_2S_\\delta (t;\\cdot )$ is compact on $ \\mathcal {X }_\\delta $ .",
"Recall that (REF ) may be rewritten into the form (REF ).",
"The superposition operator induced by $ g$ belongs to $ C^\\infty (W^{s,p}(\\Gamma ), L^p(\\Gamma ))$ for all $s > \\frac{n-1}{p}$ .",
"By Lemma REF , the same is true for $ u_\\Gamma \\mapsto d\\partial _\\nu \\mathcal {R}_\\lambda (f(\\mathcal {D}_\\lambda (u_\\Gamma )),0).$ Therefore $ S_\\delta (t; \\cdot )$ is smooth on $\\mathcal {X}_\\delta $ by e.g.",
"[19].",
"Since $ S_\\delta (t;\\cdot )$ is a compact map by Theorem REF and Proposition REF , also $ D_2S_\\delta (t;\\cdot )$ is compact.",
"Since the global attractor $ \\mathcal {A}_{\\delta }$ is by definition invariant under $ S_{\\delta }(1;\\cdot )$ , it is a consequence of [36] that $ \\mathcal {A}_{\\delta }$ has finite Hausdorff dimension.",
"We summarize the the results of this subsection as follows.",
"Theorem 5.16 Assume (REF ), (REF ), (REF ) and (REF ).",
"Then the solution semiflow $ S_{\\delta }$ on $ \\mathcal {X}_{\\delta }$ for (REF ) has a global attractor $ \\mathcal {A}_{\\delta }$ , which is of finite Hausdorff dimension and coincides with the unstable set of equilibria.",
"Remark 5.17 Another (more indirect) way to prove that $ \\mathcal {A}_{\\delta }$ has finite fractal dimension is to establish the existence of a more refined object called exponential attractor $ \\mathcal {E}_{\\delta }$ , whose existence proof is based on the so-called smoothing property for the differences of any two solutions.",
"This can be easily carried out in light of the assumptions for $ f,g$ , and the smoothness both in space and time for the solutions of (REF ) (see Proposition REF , and Lemma REF below).",
"It is also worth mentioning that the above result also holds for less regular functions in (REF ).",
"We conclude with a result that states a necessary and sufficient condition such that (REF ) is satisfied.",
"To this purpose, consider the (self-adjoint) eigenvalue problem (see, e.g., [38]) $\\left( \\lambda -\\widetilde{c}_{f}\\right) \\varphi -\\text{div}\\left( d\\nabla \\varphi \\right) =0\\qquad \\text{ in }\\Omega , $ with a boundary condition that depends on the eigenvalue $ \\xi $ explicitly, $-\\text{div}_{\\Gamma }\\left( \\delta \\nabla _{\\Gamma }\\varphi \\right)+d\\partial _{\\nu }\\varphi -c_{g}\\varphi =\\xi \\varphi \\qquad \\text{ on }\\Gamma .",
"$ Proposition 5.18 Let $ \\delta \\ge 0$ .",
"Then inequality (REF ) is satisfied if and only if the first eigenvalue $ \\xi _{1}^{\\delta }=\\xi _{1}^{\\delta }\\left( \\Omega ,\\widetilde{c}_{f},c_{g}\\right) $ of (REF )-(REF ) is positive.",
"We have that (see [38]) $\\xi _{1}^{\\delta }=\\inf _{\\psi \\in \\Theta _{\\delta },\\psi \\ne 0}\\frac{\\Vert \\sqrt{d}\\nabla \\psi \\Vert _{L^{2}(\\Omega )}^{2}+\\Vert \\sqrt{\\delta }\\nabla _{\\Gamma }\\psi \\Vert _{L^{2}(\\Gamma )}^{2}-(\\tilde{c}_{f}-\\lambda )\\Vert \\psi \\Vert _{L^{2}(\\Omega )}^{2}-c_{g}\\Vert \\psi \\Vert _{L^{2}(\\Gamma )}^{2}}{\\Vert \\psi \\Vert _{L^{2}(\\Gamma )}^{2}}\\;,$ from which the assertion immediately follows.",
"Remark 5.19 It is easy to see that if $ \\lambda >\\widetilde{c}_{f}$ and $ c_{g}<C_{P}=C_{P}\\left( \\Omega ,d,\\lambda ,\\widetilde{c}_{f}\\right) $ , where $ C_{P}>0$ is the best constant in the following Poincaré-Sobolev type inequality $C_{P}\\Vert \\psi \\Vert _{L^{2}(\\Gamma )}^{2}\\le \\Vert \\sqrt{d}\\nabla \\psi \\Vert _{L^{2}(\\Omega )}^{2}+(\\lambda -\\widetilde{c}_{f})\\Vert \\psi \\Vert _{L^{2}(\\Omega )}^{2},$ then we always have $ \\xi _{1}^{\\delta }>0$ ."
],
[
"Convergence to single equilibria",
"We shall finally be concerned with the asymptotic behavior of single trajectories.",
"We first give sufficient conditions where a single homogeneous equilibrium is approached exponentially fast by every solution with respect to the $ L^{2}(\\Gamma )$ -norm.",
"Proposition 5.20 Assume (REF ) and that $ f^{\\prime }\\le \\tilde{c}_{f}$ , $ g^{\\prime }\\le c_{g}$ for $\\tilde{c}_{f},c_{g}\\in \\mathbb {R}$ , such that (REF ) is valid.",
"If (REF ) has a homogeneous equilibrium $ u_{\\ast }\\in \\mathbb {R}$ , then for all $u_{0}\\in \\mathcal {X}_{\\delta }$ we have $\\Vert u(t;u_{0})-u_{\\ast }\\Vert _{L^{2}(\\Gamma )}\\le e^{-2\\eta t}\\Vert u_{0}-u_{\\ast }\\Vert _{L^{2}(\\Gamma )},\\qquad t>0.$ We first note that if (REF ) holds true, then at most one homogeneous equilibrium can exist, since it is necessary that either $ \\tilde{c}_f - \\lambda <0$ or $ c_g < 0$ .",
"It is straightforward to see that $ g(\\xi )\\xi \\le (c_{g}+1)\\xi ^{2}+C$ , such that every solution exists globally in time by Proposition REF .",
"Let $ w=u(\\cdot ,u_{0})-u_{\\ast }$ .",
"Testing the equations for $w$ with $w$ itself, we get $\\frac{1}{2}\\partial _{t}\\Vert w_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}& \\le -\\Vert \\sqrt{d}\\nabla w\\Vert _{L^{2}(\\Omega )}^{2}-\\Vert \\sqrt{\\delta }\\nabla _{\\Gamma }w_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2} \\\\&\\qquad +(\\tilde{c}_{f}-\\lambda )\\Vert w\\Vert _{L^{2}(\\Omega )}^{2}+c_{g}\\Vert w_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}.$ Therefore $ \\partial _{t}\\Vert w_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}\\le -2\\eta \\Vert w_{\\Gamma }\\Vert _{L^{2}(\\Gamma )}^{2}$ by (REF ), and the result follows from Gronwall's inequality.",
"We follow the approach of [35] to show the convergence of solutions to single equilibria also in nontrivial situations, under the assumption that $ f,g$ are real analytic.",
"Thanks to Proposition REF we can work with smooth solutions $ u\\in C^{\\infty }\\left( (0,t^{+})\\times \\overline{\\Omega }\\right) $ .",
"In the situation of Theorem REF , the trajectory of any solution is bounded in $ \\mathcal {X}_\\delta $ , and thus relatively compact.",
"Combining this with the gradient structure of (REF ), which is due to (REF ), we obtain the following properties of the limit sets $\\omega (u_0) = \\lbrace u_* \\in \\mathcal {X}_\\delta \\,:\\, \\exists \\, t_k \\nearrow \\infty \\text{ such that }\\; u(t_k;u_0) \\rightarrow u_* \\text{ as }k\\rightarrow \\infty \\rbrace $ of trajectories (see, e.g., [36]).",
"Lemma 5.21 Assume (REF ), (REF ), (REF ) and (REF ).",
"Then for any $ u_{0}\\in \\mathcal {X}_{\\delta }$ , the set $ \\omega (u_{0})$ is a nonempty, compact and connected subset of $ \\mathcal {X}_{\\delta }$ .",
"Furthermore, we have: (i) $ \\omega (u_{0})$ is fully invariant for the corresponding semiflow $ S_{\\delta }\\left( t\\right) $ on $\\mathcal {X}_{\\delta }$ ; (ii) $ \\mathcal {E}$ is constant on $\\omega (u_{0})$ ; (iii) $ \\emph {\\text{dist}}_{\\mathcal {X}_{\\delta }}( S_{\\delta }(t)u_{0},\\omega (u_{0})) \\rightarrow 0$ as $ t\\rightarrow +\\infty $ ; (iv) $ \\omega (u_{0})$ consists of equilibria only.",
"The key to prove that each solution converges to a single equilibrium in case when $ f$ and $g$ are analytic is the following inequality of Łojasiewicz-Simon type.",
"Proposition 5.22 Let $ d\\equiv d_*$ and $\\delta \\in \\lbrace 0,1\\rbrace $ .",
"Assume that $f,g$ are real analytic, $ |f^{\\prime }|\\le c_{f}$ , $\\lambda >c_{f}$ and that (REF ) and (REF ) are satisfied.",
"Let $ u_{\\ast }\\in C^{\\infty }(\\overline{\\Omega })$ be an equilibrium of (REF ).",
"Then there are constants $ \\theta \\in (0,1/2)$ and $r>0$ , depending on $ u_{\\ast }$ , such that for any $u\\in C^{2}\\left( \\overline{\\Omega }\\right) $ with $ \\Vert u-u_{\\ast }\\Vert _{H^{2}\\left( \\Omega \\right) }+\\delta \\Vert u_{\\Gamma }-u_{\\ast }\\Vert _{H^{2}\\left( \\Gamma \\right) }\\le r,$ we have $& \\left\\Vert \\lambda u-d\\Delta u-f(u)\\right\\Vert _{L^{2}(\\Omega )}+\\left\\Vert -\\delta \\Delta _{\\Gamma }u_{\\Gamma }+d\\partial _{\\nu }u-g(u_{\\Gamma })\\right\\Vert _{L^{2}\\left( \\Gamma \\right) } \\\\& \\qquad \\ge |\\mathcal {E}(u)-\\mathcal {E}(u_{\\ast })|^{1-\\theta }.",
"$ Our proof follows closely that of [35] which only includes the case $ \\delta =1$ (cf.",
"also [43] for $g\\equiv 0$ and $\\delta =0$ ).",
"We shall briefly mention the details below in the case when $ \\delta =0$ and $ g$ is nontrivial.",
"To this end, let us first set $V_{k}:=\\left\\lbrace \\left( u,u_{\\Gamma }\\right) \\in W^{k,2}\\left( \\Omega \\right)\\times W^{k-1/2,2}\\left( \\Gamma \\right) :u_{\\Gamma }=u|_{ \\Gamma }\\right\\rbrace ,$ for $ k\\ge 1$ (by convention, we also let $V_{0}:=L^{2}\\left( \\Omega \\right)\\times L^{2}\\left( \\Gamma \\right) $ ).",
"Here and below, for the sake of simplicity of notation we will identify any function $ u$ that belongs to $ W^{k,2}\\left( \\Omega \\right) $ with $\\left( u,u_{\\Gamma }\\right) \\in V_{k}$ such that $ u_{\\Gamma }\\in W^{k-1/2,2}\\left( \\Gamma \\right)$ .",
"Next, consider the so-called Wentzell Laplacian, given by $A_{W}:=\\left(\\begin{array}{cc}\\lambda I-d\\Delta & 0 \\\\d\\partial _{\\nu } & 0\\end{array}\\right) ,$ with domain $ D\\left( A_{W}\\right) =\\left\\lbrace u\\in V_{1}:-\\Delta u\\in L^{2}\\left( \\Omega \\right) ,\\text{ }\\partial _{\\nu }u\\in L^{2}\\left( \\Gamma \\right) \\right\\rbrace $ , which we endow with its natural graph norm $ \\left\\Vert A_{W}\\cdot \\right\\Vert _{V_{0}}$ .",
"In particular, $ D\\left( A_{W}\\right)=V_{2} $ provided that the boundary $ \\Gamma $ is sufficiently regular (see [15], [16]).",
"It is also well-known that $ \\left( A_{W},D\\left(A_{W}\\right) \\right) $ is self-adjoint and positive on $ V_{0}$ .",
"By [14], we also infer that there exists a complete orthonormal family $ \\left\\lbrace \\phi _{j}\\right\\rbrace \\subset V_{0}$ , with $\\phi _{j}\\in D\\left(A_{W}\\right) $ , as well as a sequence of eigenvalues $ 0<\\lambda _{1}\\le \\lambda _{2}\\le ... \\le \\lambda _{j}\\rightarrow \\infty $ , as $ j\\rightarrow \\infty $ , such that $ A_{W}\\phi _{j}=\\lambda _{j}\\phi _{j},$ $j\\in \\mathbb {N}_{+}$ .",
"Moreover, by standard elliptic theory and bootstrap arguments we have $ \\phi _{j}\\in C^{\\infty },$ for every $j\\in \\mathbb {N}_{+}$ provided that $ \\Gamma $ is sufficiently regular (see [14]).",
"Let now $P_{m}$ be the orthogonal projector from $ V_{0}$ onto $K_{m}:=span\\left\\lbrace \\phi _{1},...,\\phi _{m}\\right\\rbrace $ .",
"Following a similar strategy to [35], it is easy to show that $\\left( A_{W}u+\\lambda _{m}P_{m}u,u\\right) _{V_{1}^{\\ast },V_{1}}\\ge C_{\\lambda ,d}\\left\\Vert u\\right\\Vert _{V_{1}}^{2}+\\frac{1}{4}\\lambda _{m}\\left\\Vert u\\right\\Vert _{V_{0}}^{2} $ holds for any $ u\\in V_{1}$ (for some positive constant $C_{\\lambda ,d}>0$ ).",
"Let $ \\psi $ be a critical point of $\\mathcal {E}(u)$ .",
"For any $\\psi \\in C^{2}\\left( \\overline{\\Omega }\\right) $ , we consider the following linearized operator $ v\\in V_{2}\\longmapsto L\\left( v\\right) $ , analogous to [35], given by $L\\left( v\\right) :=\\left(\\begin{array}{cc}\\lambda I-d\\Delta & 0 \\\\d\\partial _{\\nu } & 0\\end{array}\\right) +\\left(\\begin{array}{cc}-f^{^{\\prime }}\\left( v+\\psi \\right) & 0 \\\\0 & -g^{^{\\prime }}\\left( v+\\psi \\right)\\end{array}\\right) ,$ with domain $ \\mathcal {D}:=D\\left( A_{W}\\right) =V_{2}$ .",
"We note that one can associate with $ L\\left( 0\\right) $ the following bilinear form $b\\left(u_{1},u_{2}\\right) $ on $ V_{1}\\times V_{1}$ , as follows: $b\\left( u_{1},u_{2}\\right) =\\int _{\\Omega }\\left( \\lambda u_{1}u_{2}+d\\nabla u_{1}\\cdot \\nabla u_{2}-f^{^{\\prime }}\\left( \\psi \\right) u_{1}u_{2}\\right)dx+\\int _{\\Gamma }\\left( -g^{^{\\prime }}\\left( \\psi \\right) u_{1}u_{2}\\right)dS,$ for any $ u_{1},u_{2}\\in V_{1}$ .",
"As in [16], it can be easily shown that $ \\left( L\\left( v\\right) ,\\mathcal {D}\\right) $ is self-adjoint on $ V_{0} $ .",
"Moreover, by (REF ) it is readily seen that the operator $ L\\left( 0\\right) +\\lambda _{m}P_{m}$ is coercive with respect to the (equivalent) inner product of $ H^{1}\\left( \\Omega \\right) ,$ provided that $\\lambda _{m}>4\\max \\left\\lbrace \\left\\Vert f^{^{\\prime }}\\left( \\psi \\right)\\right\\Vert _{L^{\\infty }\\left( \\Omega \\right) },\\left\\Vert g^{^{\\prime }}\\left( \\psi \\right) \\right\\Vert _{L^{\\infty }\\left( \\Gamma \\right)}\\right\\rbrace .",
"$ Recalling that $ \\psi $ is sufficiently smooth, we note that condition (REF ) can always be achieved by choosing a sufficiently large $ m$ .",
"Next, consider the following operators: $\\Pi :=\\lambda _{m}P_{m}:V_{0}\\rightarrow V_{0},\\text{ }\\mathcal {L}\\left(v\\right) :\\mathcal {D}\\rightarrow V_{0},\\text{ }\\mathcal {L}\\left( v\\right)h=\\Pi h+L\\left( v\\right) h,$ for any $ v\\in \\mathcal {D}$ .",
"Clearly, $\\mathcal {L}\\left( 0\\right) :\\mathcal {D}\\rightarrow V_{0}$ is bijective on account of (REF ).",
"The following lemma is concerned with some regularity properties for the (unique) solution of $ \\mathcal {L}\\left( 0\\right) h=w$ , for some given $w=\\left(w_{1},w_{2}\\right) \\in V_{0}.$ Lemma 5.23 Let $ w=\\left( w_{1},w_{2}\\right) \\in W^{k-1,2}\\left( \\Omega \\right) \\times W^{k-1/2,2}\\left( \\Gamma \\right) $ , for any $ k\\in \\mathbb {N}$ , $k\\ge 1$ .",
"Then the following estimate holds: $\\left\\Vert h\\right\\Vert _{V_{k+1}}\\le C\\left( \\left\\Vert w_{1}\\right\\Vert _{W^{k-1,2}\\left( \\Omega \\right) }+\\left\\Vert w_{2}\\right\\Vert _{W^{k-1/2,2}\\left( \\Gamma \\right) }\\right) .",
"$ Moreover, it holds $\\left\\Vert h\\right\\Vert _{C^{0,\\gamma }\\left( \\overline{\\Omega }\\right)}\\le C\\left( \\left\\Vert w_{1}\\right\\Vert _{L^{p}\\left( \\Omega \\right)}+\\left\\Vert w_{2}\\right\\Vert _{L^{q}\\left( \\Gamma \\right) }\\right) ,$ as long as $ w_{1}\\in L^{p}\\left( \\Omega \\right) $ with $p>n$ and $w_{2}\\in L^{q}\\left( \\Gamma \\right) $ with $ q>n-1.$ The constant $C>0$ is independent of $ k.$ Indeed, writing the equation $ \\mathcal {L}\\left( 0\\right) h=w$ in the form $\\left\\lbrace \\begin{array}{l}\\lambda h-d\\Delta h=q_{1}:=f^{^{\\prime }}\\left( \\psi \\right) h-\\Pi h+w_{1},\\quad \\text{ a.e.",
"in }\\Omega , \\\\d\\partial _{\\nu }h=q_{2}:=g^{^{\\prime }}\\left( \\psi \\right) h-\\left( \\Pi h\\right) _{\\mid \\Gamma }+w_{2},\\quad \\text{ a.e.",
"on }\\Gamma ,\\end{array}\\right.",
"$ we have $ \\left\\Vert h\\right\\Vert _{V_{1}\\cap \\mathcal {D}}\\le C\\left\\Vert w\\right\\Vert _{V_{0}}$ since $ \\mathcal {L}\\left( 0\\right) :\\mathcal {D}\\rightarrow V_{0}$ is a bijection.",
"Recalling the standard trace-regularity theory and the fact that $ \\psi \\in C^{\\infty }$ , and applying the $W^{l,2}$ regularity theorem (see, e.g., [24]) to (REF ) with $ l=2,3,...,$ we have $\\left\\Vert h\\right\\Vert _{V_{l}}\\le C\\left\\Vert h\\right\\Vert _{W^{l,2}\\left( \\Omega \\right) }\\le C\\left( \\left\\Vert q_{1}\\right\\Vert _{W^{l-2,2}\\left( \\Omega \\right) }+\\left\\Vert h\\right\\Vert _{W^{l-1,2}\\left(\\Omega \\right) }+\\left\\Vert q_{2}\\right\\Vert _{W^{l-3/2,2}\\left( \\Gamma \\right) }\\right).$ From this (REF ) immediately follows by exploiting a standard iteration argument for $ l\\ge 2$ .",
"The proof of (REF ) is contained in [41] (see, also, [42] for the proof of the same bound in $ L^{\\infty }\\left( \\Omega \\right) \\times L^{\\infty }\\left( \\Gamma \\right) $ -norm).",
"Exploiting now the results of the preceding lemma, the proof of the (extended) Łojasiewicz-Simon inequality (REF ) can be reproduced from that of [35] with no essential modifications.",
"To apply the proposition we need that the solutions converge to the limit set in a norm that is stronger than that of $ \\mathcal {X}_\\delta $ .",
"This will be a consequence of the next lemma.",
"Let $ B_{\\mathcal {X}_{\\delta }}(R)$ be the ball in $ \\mathcal {X}_{\\delta }$ centered at the origin with radius $R>0$ .",
"Lemma 5.24 Assume (REF ), (REF ), (REF ) and (REF ).",
"Then for $ k\\in \\mathbb {N}_0$ and $R >0$ there is a constant $ C= C(k,R) > 0$ such that $\\sup _{u_0 \\in B_{\\mathcal {X}_\\delta }(R) } \\sup _{t \\ge 1}\\Vert S_\\delta (t;u_0)\\Vert _{C^k(\\overline{\\Omega })} \\le C.$ By Theorem REF , the semiflow has global attractor in $ \\mathcal {X}_\\delta $ .",
"Therefore, using Lemma REF , $\\Vert u\\Vert _{L^\\infty (\\mathbb {R}_+; H^{2-1/p,p}(\\Omega ))} \\le C(R) \\qquad \\text{if } \\delta \\ge \\delta _*,$ $\\Vert u\\Vert _{L^\\infty (\\mathbb {R}_+; W^{1,p}(\\Omega ))} \\le C(R)\\qquad \\text{if }\\delta \\equiv 0,$ for all $ u= u(\\cdot ; u_0)$ with $u_0 \\in B_{\\mathcal {X}_\\delta }(R)$ .",
"Let us consider the case $ \\delta \\equiv 0$ .",
"We use the representation $ u_\\Gamma (t) = e^{-t\\mathcal {N}_\\lambda }u_0 + e^{- \\cdot \\mathcal {N}_\\lambda } *\\big (g(u_\\Gamma ) - d \\partial _\\nu \\mathcal {R}_\\lambda (f(u),0)\\big )(t),\\qquad t > 0,$ and assume, inductively, that $\\Vert u\\Vert _{L^{\\infty }(1,\\infty ;W^{k,p}(\\Omega ))}\\le C(k,R)$ for some $ k\\in \\mathbb {N}$ .",
"Then $\\Vert g(u_{\\Gamma })\\Vert _{L^{\\infty }(1,\\infty ;W^{k-1/p,p}(\\Gamma ))}\\le C(k,R)$ , and further, using [2], $\\Vert d\\partial _{\\nu }\\mathcal {R}_{\\lambda }(f(u),0)\\Vert _{L^{\\infty }(1,\\infty ;W^{k-1/p,p}(\\Gamma ))}\\le C(k)\\Vert f(u)\\Vert _{L^{\\infty }(1,\\infty ;W^{k-1,p}(\\Omega ))}\\le C(k,R).$ Considering (REF ) as an identity on $ W^{k-1/p,p}(\\Gamma )$ , it follows from [26] that $ \\Vert u_{\\Gamma }\\Vert _{L^{\\infty }(1,\\infty ;W^{k+1-1/p,p}(\\Gamma ))}\\le C(k+1,R).$ Moreover, by [2], we have $\\Vert u\\Vert _{L^{\\infty }(1,\\infty ;W^{k+1,p}(\\Omega ))}& \\,\\le C\\big (\\Vert f(u)\\Vert _{L^{\\infty }(1,\\infty ;W^{k-1,p}(\\Omega ))}+\\Vert u_{\\Gamma }\\Vert _{L^{\\infty }(1,\\infty ;W^{k+1-1/p,p}(\\Gamma ))}\\big ) \\\\& \\,\\le C(k+1,R).$ The asserted estimate in case $ \\delta \\equiv 0$ now follows from Sobolev's embeddings.",
"The arguments in the case $ \\delta \\ge \\delta _{\\ast }$ are similar.",
"We can now prove the main result of this subsection.",
"Theorem 5.25 Let $ p\\in (n,\\infty )$ , $d >0$ and $\\delta \\in \\lbrace 0,1\\rbrace $ .",
"Assume that $ f,g$ are real analytic, $|f^{\\prime }|\\le c_{f}$ , $\\lambda >c_{f}$ and that $ g$ satisfies (REF ) and (REF ).",
"Then for any given initial datum $ u_{0}\\in \\mathcal {X}_{\\delta }$ the corresponding solution $ u\\left( t;u_{0}\\right) =S_{\\delta }(t;u_{0})$ of (REF ) exists globally in time and converges to a single equilibrium $ u_{\\ast }$ in the topology of $\\mathcal {X}_{\\delta } $ .",
"More precisely, ${\\lim _{t\\rightarrow +\\infty }}\\left( \\Vert u(t;u_{0})-u_{\\ast }\\Vert _{\\mathcal {X}_{\\delta }}+\\Vert \\partial _{t}u(t;u_{0})\\Vert _{L^{2}\\left(\\Gamma \\right) }\\right) =0.",
"$ Step 1.",
"From Theorem REF and Lemma REF we know that the solution $ u= u(\\cdot ;u_0)$ is smooth in space and time, exists globally and that the corresponding trajectory converges to the set of equilibria in $ \\mathcal {X}_\\delta $ .",
"By Lemma REF , the trajectory is also bounded in, say, $ W^{3,p}(\\Gamma )$ .",
"Since $\\omega (u_{0})\\subset C^{\\infty }(\\overline{\\Omega }),$ we can apply the interpolation inequality (REF ) with suitable $ \\theta \\in (0,1)$ to obtain that $\\text{dist}_{\\mathcal {V}_{\\delta }}(u(t;u_{0}),\\omega (u_{0}))\\rightarrow 0\\qquad \\text{as }t\\rightarrow +\\infty , $ where we have set $ \\mathcal {V}_{0}:=H^{2}\\left( \\Omega \\right) $ if $\\delta =0$ , and $ \\mathcal {V}_{1}:=H^{2}\\left( \\Omega \\right) \\oplus H^{2}\\left(\\Gamma \\right) $ if $ \\delta =1,$ respectively.",
"Step 2.",
"The function $ t \\mapsto \\mathcal {E}(u(t))$ is decreasing and bounded from below.",
"Thus $\\mathcal {E}_{\\infty }:=\\lim _{t\\rightarrow \\infty }\\mathcal {E} ( u( t))$ exists.",
"If there is $ t^\\sharp $ with $\\mathcal {E}( u( t^{\\sharp })) =\\mathcal {E}_\\infty $ , then $ u$ is an equilibrium and there is nothing to prove.",
"Hence we may suppose that for all $ t\\ge t_{0}>0,$ we have $\\mathcal {E}\\left( u\\left( t\\right) \\right) >\\mathcal {E}_{\\infty }.$ We first observe that, by Lemma REF , the functional $ \\mathcal {E}$ satisfies the Łojasiewicz-Simon inequality (REF ) near every $ u_{\\ast }\\in \\omega (u_{0}).$ Since $ \\omega (u_{0})$ is compact in $\\mathcal {X}_{\\delta }$ , we can cover it by the union of finitely many balls $ B_{j}$ with centers $ u_{\\ast }^{j}$ and radii $r_{j},$ where each radius is such that (REF ) holds in $ B_{j}$ .",
"It follows from Proposition REF that there exist uniform constants $ \\xi \\in (0,1/2)$ , $C_{L}>0$ and a neighborhood $U$ of $ \\omega (u_{0})$ in $\\mathcal {X}_\\delta $ such that (REF ) holds in $U$ .",
"Thus, recalling (REF ), we can find a time $ t_{0}\\ge 1$ such that $ u ( t; u_0) $ belongs to $U$ for all $t\\ge t_{0}.$ On account of (REF ) and (REF ) we obtain $& -\\frac{d}{dt} ( \\mathcal {E} ( u ( t)) -\\mathcal {E}_{\\infty }) ^{\\xi }\\\\& =-\\xi \\partial _{t}\\mathcal {E}( u( t)) ( \\mathcal {E}( u( t)) -\\mathcal {E}_{\\infty }) ^{\\xi -1} \\\\& \\ge \\xi C_{L}\\frac{\\left\\Vert \\partial _{t}u_\\Gamma \\right\\Vert _{L^{2}\\left( \\Gamma \\right) }^{2}}{\\left\\Vert \\lambda u-d\\Delta u-f(u)\\right\\Vert _{L^{2}\\left( \\Omega \\right) }+\\left\\Vert -\\delta \\Delta _{\\Gamma }u_{\\Gamma }+d\\partial _{\\nu }u-g(u_\\Gamma )\\right\\Vert _{L^{2}\\left( \\Gamma \\right) }}.",
"$ Recalling (REF ), we get $-\\frac{d}{dt}\\left( \\mathcal {E}\\left( u\\left( t\\right) \\right) -\\mathcal {E}_{\\infty }\\right) ^{\\xi }\\ge C\\left\\Vert \\partial _{t}u_\\Gamma (t)\\right\\Vert _{L^{2}\\left( \\Gamma \\right) }.",
"$ Integrating over $ ( t_0,\\infty ) $ and using that $\\mathcal {E}( u ( t ))\\rightarrow \\mathcal {E}_{\\infty }$ as $ t \\rightarrow \\infty $ , we infer that $\\partial _{t}u_\\Gamma \\in L^{1}\\left( t_0,\\infty ;L^{2}\\left( \\Gamma \\right)\\right).$ Step 3.",
"Since $ \\partial _t u_\\Gamma $ is uniformly continuous with values in $ L^2(\\Gamma )$ , it follows that $\\Vert \\partial _tu_\\Gamma (t)\\Vert _{L^2(\\Gamma )} \\rightarrow 0$ as $ t\\rightarrow \\infty $ .",
"Moreover, since by Lemma REF (iii) there are $ t_k \\nearrow \\infty $ and $u_{\\ast }\\in \\omega \\left( u_{0}\\right) $ such that $ u(t_{k})\\rightarrow u_{\\ast }$ in $ \\mathcal {X}_{\\delta }$ as $k \\rightarrow \\infty $ , the integrability of $\\partial _tu_\\Gamma $ implies that $ u_{\\Gamma } ( t) \\rightarrow u_*$ in $L^{2}( \\Gamma )$ as $ t \\rightarrow \\infty ,$ and then in $\\mathcal {X}_{\\delta }$ as well.",
"Hence $ \\omega ( u_{0}) =\\lbrace u_{\\ast }\\rbrace $ , and (REF ) follows.",
"Remark 5.26 One can also exploit (REF ) and (REF ) to deduce a convergence rate estimate in (REF ) of the form $\\left\\Vert u( t;u_{0}) -u_{\\ast }\\right\\Vert _{L^{2}\\left( \\Gamma \\right)}\\le C(1+t)^{-1/(1-2\\xi )},\\qquad t >0,$ for some constants $ C>0$ , $\\xi \\in \\left( 0,\\frac{1}{2}\\right) $ depending on $ u_{\\ast }$ .",
"Taking advantage of the above (lower-order) convergence estimate and the results of the previous subsections, one can also prove the corresponding estimate in higher-order norms $ W^{k,2}\\left( \\Omega \\right) $ , arguing, for instance, as in [35], [43]."
]
] | 1204.1251 |
[
[
"CAT(0) spaces with boundary the join of two Cantor sets"
],
[
"Abstract We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets.",
"If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees."
],
[
"Introduction",
"CAT(0) spaces with homeomorphic visual boundaries can have very different Tits boundaries.",
"However, if $X$ admits a proper and cocompact group action by isometries, or a geometric group action in short, then this places a restriction on the possible Tits boundaries for a given visual boundary.",
"(We follow the definition of a proper group action in Chapter I.8 of [3]; some use the term “properly discontinuous” for this.)",
"Kim Ruane has showed in [12] that for a CAT(0) space $X$ with boundary $\\partial X$ homeomorphic to the suspension of a Cantor set, if it admits a geometric group action, then the Tits boundary $\\partial _{\\mathrm {T}}X$ is isometric to the suspension of an uncountable discrete set.",
"In this paper we will show the following.",
"Theorem 1.1 If a CAT(0) space $X$ has a boundary $\\partial X$ homeomorphic to the join of two Cantor sets, $C_1$ and $C_2$ , and if $X$ admits a geometric group action by a group containing a subgroup isomorphic to $\\mathbb {Z}^2$ , then its Tits boundary $\\partial _{\\mathrm {T}}X$ is isometric to the spherical join of two uncountable discrete sets.",
"So if $X$ is geodesically complete, then $X = X_1 \\times X_2$ with $\\partial X_i$ homeomorphic to $C_i$ , $i=1,2$ .",
"As for the group acting on $X$ , we will prove the following.",
"Theorem 1.2 Let $X$ be a geodesically complete CAT(0) space such that $\\partial X$ is homeomorphic to the join of two Cantor sets.",
"Then for a group $G < \\mathrm {Isom}(X)$ acting geometrically on $X$ and containing a subgroup isomorphic to $\\mathbb {Z}^2$ , either $G$ or a subgroup of $G$ of index 2 is a uniform lattice in $\\mathrm {Isom}(X_1) \\times \\mathrm {Isom}(X_2)$ .",
"Furthermore, a finite index subgroup of $G$ is a lattice in $\\mathrm {Isom}(T_1)\\times \\mathrm {Isom}(T_2)$ , where $T_i$ is a bounded valence bushy tree quasi-isometric to $X_i$ , $i=1,2$ .",
"Remark 1.3 The assumption that $G$ contains a subgroup isomorphic to $\\mathbb {Z}^2$ is only used to obtain a hyperbolic element in $G$ with endpoints in $\\partial X \\setminus (C_1 \\cup C_2)$ , which we use in Section 4 to prove Theorem REF .",
"It is conjectured that a CAT(0) group is either Gromov hyperbolic or it contains a subgroup isomorphic to $\\mathbb {Z}^2$ .",
"Without using the assumption on $G$ , we can show that $G$ cannot be hyperbolic, which follows from Lemma REF and the Flat Plane Theorem.",
"([3], Theorem III.H.1.5) Thus if the conjecture is shown to be true for general CAT(0) groups, the assumption on $G$ will not be necessary.",
"The conjecture has been proved for some classes of CAT(0) groups, see [7] and [5] for examples.",
"If $X_i$ are proper geodesically complete, one might hope that they are trees, so $G$ will be a uniform lattice in the product of two isometry groups of trees.",
"Surprisingly, this may not be the case.",
"Ontaneda constructed a 2-complex $Z$ which is non-positively curved and geodesically complete with free group $F_n$ as its fundamental group.",
"(See proof of proposition 1 in [9]) Its universal cover is quasi-isometric to $F_n$ , so it is a Gromov hyperbolic space with Cantor set boundary, while being also a CAT(0) space.",
"Under an additional condition that the isotropy subgroup of $\\mathrm {Isom}(X_i)$ of every boundary point of $X_i$ acts cocompactly on $X_i$ , then $X_i$ is a tree.",
"(See proof of Theorem 1.3 in [6].)",
"There are irreducible lattice in a product of two trees, so $G$ may not have a finite index subgroup which splits as a product.",
"See [4] for a detailed investigation.",
"Acknowledgement 1.4 I would like to thank my advisor Chris Connell for suggesting this problem to me and providing me with a lot of valuable discussions, assistance and encouragements while I was on this project."
],
[
"Preliminaries",
"First we fix the notations.",
"For a CAT(0) space $X$ , its (visual) boundary with the cone topology is $\\partial X$ .",
"For a subset $H\\subset X$ , we denote by $\\partial H := \\overline{H} \\cap \\partial X$ , where the closure $\\overline{H}$ is taken in $\\overline{X} := X \\cup \\partial X$ .",
"The angular and the Tits metrics on the boundary are denoted as $\\angle (\\cdot ,\\cdot )$ and $\\mathrm {d}_{\\mathrm {T}}(\\cdot ,\\cdot )$ respectively.",
"We denote the boundary with the Tits metric by $\\partial _{\\mathrm {T}}X$ .",
"If $g$ is a group element acting on $X$ by isometry, we denote by $\\overline{g}$ the action of $g$ extended to $\\partial X$ by homeomorphism.",
"If $g$ acts on $X$ by a hyperbolic isometry, the two endpoints of its axes on $\\partial X$ are denoted by $g^{\\pm \\infty }$ .",
"We refer to [3] for details on basic facts about CAT(0) spaces.",
"Let $X$ be a complete CAT(0) space with $\\partial X$ homeomorphic to the join of two Cantor sets $C_1$ and $C_2$ , and $G < \\mathrm {Isom}(X)$ be a group acting on $X$ geometrically.",
"We will not assume that $G$ contains a subgroup isomorphic to $\\mathbb {Z}^2$ until Section 4.",
"By the following lemma, we can assume that $G$ stabilizes $C_1$ and $C_2$ .",
"Lemma 2.1 Either $G$ or a subgroup of $G$ of index 2 stabilizes each of $C_1$ and $C_2$ .",
"Consider $\\partial X$ as a complete bipartite graph with $C_1, C_2$ as the two sets of vertices.",
"For any $g\\in G$ , if $\\overline{g}\\cdot x_1 \\in C_1$ for some $x_1\\in C_1$ , then $\\overline{g} \\cdot C_i =C_i$ , $i=1,2$ ; otherwise $\\overline{g} \\cdot C_1 = C_2$ and $\\overline{g} \\cdot C_2 = C_1$ .",
"So the homomorphism from $G$ to symmetric group on two elements is well-defined and its kernel is the subgroup of $G$ which stabilizes each of $C_1$ and $C_2$ .",
"By an arc we specifically mean a segment from a point in $C_1$ to a point in $C_2$ which does not pass through any other point of $C_1$ or $C_2$ , and by open (closed) segment a segment on the boundary excluding (including) its two endpoints.",
"We will investigate the positions of the endpoints of hyperbolic elements in $G$ .",
"We quote a basic result on dynamics on CAT(0) space boundary by Ruane: Lemma 2.2 (Ruane, [11] Lemma 4.1) Let $g$ be a hyperbolic isometry of a CAT(0) space $X$ and let $c$ be an axis of $g$ .",
"Let $z \\in \\partial X$ , $z \\ne g^{-\\infty }$ and let $z_i = \\overline{g}^i \\cdot z$ .",
"If $w \\in \\partial X $ is an accumulation point of the sequence $(z_i)$ in the cone topology, then $\\angle (g^{-\\infty },w)+\\angle (w,g^{\\infty })=\\pi $ , and $\\angle (g^{-\\infty },z)=\\angle (g^{-\\infty },w)$ .",
"If $w\\ne g^\\infty $ , then $\\mathrm {d}_{\\mathrm {T}}(g^{-\\infty }, w)+\\mathrm {d}_{\\mathrm {T}}(w,g^{\\infty } )=\\pi $ .",
"In this case $c$ and a ray from $c(0)$ to $w$ span a flat half plane, and $\\mathrm {d}_{\\mathrm {T}}(g^{-\\infty },z)=\\mathrm {d}_{\\mathrm {T}}(g^{-\\infty },w)$ .",
"Recall that a hyperbolic isometry is of rank one if none of its axes bounds a flat half plane, and it is of higher rank otherwise.",
"Lemma 2.3 There is no rank one isometry in $G$ .",
"Take any $g \\in G$ .",
"Assume without loss of generality that $g^\\infty \\in \\partial X \\setminus C_2$ .",
"Then for any point $y\\in C_2$ , $\\overline{g}^n\\cdot y$ cannot accumulate at $g^\\infty $ since $C_2$ is closed in $\\partial X$ .",
"Any accumulation point of $\\overline{g}^n\\cdot y$ will form a boundary of a half plane with $g^{\\pm \\infty }$ by Lemma REF .",
"So $g$ is not rank one.",
"We note also that no finite subset of points on the boundary is stabilized by $G$ , which readily follows from a result by Ruane, quoted in a paper by Papasoglu and Swenson, and the fact that our $\\partial X$ is not a suspension.",
"Lemma 2.4 (Ruane, [10] Lemma 26) If $G$ virtually stabilizes a finite subset $A$ of $\\partial X$ , then $G$ virtually has $\\mathbb {Z}$ as a direct factor.",
"In this case $\\partial X$ is a suspension."
],
[
"Endpoints of a hyperbolic element",
"We will show that there is no hyperbolic element of $G$ with one of its endpoints in $C_1$ but not the other one.",
"We will proceed by contradiction, using as a key result the following theorem by Papasoglu and Swenson to $\\partial X$ , itself a strengthening of a previous result by Ballmann and Buyalo [2].",
"This theorem is applicable to our $\\partial X$ in light of the previous lemmas.",
"Theorem 3.1 (Papasoglu and Swenson, [10] Theorem 22) If the Tits diameter of $\\partial X$ is bigger than $\\frac{3\\pi }{2}$ then $G$ contains a rank 1 hyperbolic element.",
"In particular: If $G$ does not fix a point of $\\partial X$ and does not have rank 1, and $I$ is a (minimal) closed invariant set for the action of $G$ on $\\partial X$ , then for any $x\\in \\partial X$ , $\\mathrm {d}_{\\mathrm {T}}(x, I)\\le \\frac{\\pi }{2}$ .",
"We put the word minimal in parentheses as it is not a necessary condition, for if $I \\subset \\partial X$ is a closed invariant set, then it contains a minimal closed invariant set $I^{\\prime }$ , and so for any $x\\in \\partial X$ , $\\mathrm {d}_{\\mathrm {T}}(x,I) \\le \\mathrm {d}_{\\mathrm {T}}(x, I^{\\prime })\\le \\frac{\\pi }{2}$ .",
"Note that the above theorem implies that $\\partial X$ has finite Tits diameter, and hence the CAT(1) space $\\partial _{\\mathrm {T}}X$ is connected.",
"Now assume that $g\\in G$ is hyperbolic such that $g^\\infty \\in C_1$ and $g^{-\\infty } \\in \\partial X \\setminus C_1$ .",
"Lemma 3.2 $\\mathrm {Fix}(\\overline{g})$ contains boundary of a 2-flat.",
"By Lemma REF , $g^{\\pm \\infty }$ bound a half plane, so there is a segment joining $g^{\\pm \\infty }$ fixed by $g$ , then it is contained in $\\partial \\mathrm {Min}(g)$ .",
"Then by Theorems 3.2 and 3.3 of [11], $\\mathrm {Min}(g) = Y \\times \\mathbb {R}$ with $\\partial Y \\ne \\varnothing $ , and $C(g)/\\langle g \\rangle $ acts on the CAT(0) space $Y$ geometrically.",
"Since $Y$ has nonempty boundary, so by Theorem 11 of [13] there is a hyperbolic element in $C(g)/\\langle g \\rangle $ which has an axis in $Y$ with two endpoints on $\\partial Y$ .",
"Thus there is a 2-flat in $\\mathrm {Min}(g)$ .",
"Denote this 2-flat by $F$ , and let $z$ be a point in $\\partial F \\cap C_1$ other than $g^\\infty $ .",
"Figure: Boundary of a 2-flat in Min (h)\\mathrm {Min}(h)Lemma 3.3 If $F_0$ is a 2-flat whose boundary is contained in $\\mathrm {Fix}(\\overline{h})=\\partial \\mathrm {Min}(h)$ for some hyperbolic $h\\in G$ , then $\\partial F_0$ intersects each of $C_1$ and $C_2$ at exactly 2 points.",
"Suppose not, then denote the points at which $\\partial F_0$ alternatively intersects $C_1$ , $C_2$ by $x_1, y_1, x_2, y_2, \\ldots , x_n, y_n$ .",
"Consider the segment joining $x_1$ and $y_2$ .",
"We may assume that not both of $x_1$ , $y_2$ are endpoints of $h$ .",
"(If not, choose $y_1$ and $x_3$ instead.)",
"From the assumption on $\\partial F_0$ , this segment is not part of $\\partial F_0$ .",
"Its two endpoints are fixed, but the arc joining them is not in $\\mathrm {Fix}(\\overline{h})$ because $\\mathrm {Fix}(\\overline{h})$ is a suspension with suspension points $h^{\\pm \\infty }$ .",
"However, this arc is stabilized by $h$ because of the cone topology of $\\partial X$ .",
"Action of $G$ on $\\partial _{\\mathrm {T}}X$ is by isometries.",
"Take a point $p$ in the open arc between $x_1$ and $y_2$ .",
"Since $\\partial _{\\mathrm {T}}X$ is connected there exists a Tits segment in this arc from $p$ to one of $x_1$ and $y_2$ , say $x_1$ .",
"Choose a new point on this segment as $p$ if necessary, we can assume $\\mathrm {d}_{\\mathrm {T}}(p,x_1) < \\mathrm {d}_{\\mathrm {T}}(y_2, x_1)$ .",
"Now $\\mathrm {d}_{\\mathrm {T}}(\\overline{h}\\cdot p, \\overline{h} \\cdot x_1) = \\mathrm {d}_{\\mathrm {T}}(\\overline{h}\\cdot p, x_1)$ and $\\overline{h}\\cdot p$ is also on the arc.",
"$\\overline{h}\\cdot p$ cannot be on the open segment between $p$ and $x_1$ .",
"If $\\overline{h}\\cdot p$ were on the open segment between $p$ and $y_2$ , the Tits geodesic from $\\overline{h}\\cdot p$ to $x_1$ would go through $p$ or $y_2$ , both would contradict $\\mathrm {d}_{\\mathrm {T}}(\\overline{h}\\cdot p,x_1)=\\mathrm {d}_{\\mathrm {T}}(p, x_1)$ .",
"So $\\overline{h}\\cdot p = p$ .",
"Then $p \\in \\partial \\mathrm {Min}(h)$ and lies on a path in $\\partial \\mathrm {Min}(h)$ joining $h^{\\pm \\infty }$ , forcing the arc to be in $\\partial \\mathrm {Min}(h)$ , which contradicts the previous assertion.",
"Denote the segment in $\\partial X$ from $g^\\infty $ to $z$ passing through $g^{-\\infty }$ by $\\beta $ .",
"Let $y$ be the point where $\\beta $ intersects $C_2$ .",
"The essense of the following arguments is to look for a point in $\\partial _{\\mathrm {T}}X$ that is over $\\pi /2$ away from $C_1$ or $C_2$ , which are closed $G$ -invariant subsets, so obtaining a contradiction to Theorem REF .",
"Lemma 3.4 $g^{-\\infty }$ cannot be on the closed segment in $\\beta $ from $g^\\infty $ to $y$ .",
"Suppose not.",
"The Tits length of this segment from $g^\\infty $ to $y$ is at least $\\pi $ .",
"Let $0<\\delta <\\pi /2$ be such that $2\\delta \\le \\mathrm {d}_{\\mathrm {T}}(y, C_1)$ .",
"Take a point $p$ on this segment so that $\\mathrm {d}_{\\mathrm {T}}(p,g^\\infty )=\\pi /2+ \\delta $ .",
"Then $\\mathrm {d}_{\\mathrm {T}}(p,y) \\ge \\pi /2 - \\delta $ .",
"Now for any point $x\\in C_1$ other than $g^\\infty $ , if the Tits geodesic segment from $p$ to $x$ passes through $y$ , then $\\mathrm {d}_{\\mathrm {T}}(p, x)\\ge \\mathrm {d}_{\\mathrm {T}}(p,y)+\\mathrm {d}_{\\mathrm {T}}(y,C_1) \\ge (\\pi /2 - \\delta ) + 2\\delta = \\pi /2 + \\delta ;$ while if it passes through $g^\\infty $ , then obviously $\\mathrm {d}_{\\mathrm {T}}(p, x) > \\mathrm {d}_{\\mathrm {T}}(p, g^\\infty ) = \\pi /2 +\\delta $ .",
"So $\\mathrm {d}_{\\mathrm {T}}(p, C_1) \\ge \\pi /2 + \\delta $ , which contradicts Theorem REF .",
"Now we deal with the case that $g^{-\\infty }$ is in the open segment in $\\beta $ from $y$ to $z$ .",
"We state a lemma first which will also be used in later arguments.",
"Lemma 3.5 Suppose $h \\in G$ is a hyperbolic element such that $F_0 \\subset \\mathrm {Min}(h)$ whose boundary intersects $C_1$ and $C_2$ alternatively at $x_1, y_1, x_2, y_2$ .",
"Assume that the endpoint $h^{-\\infty }$ is on some open arc, say the open arc between $x_i$ and $y_j$ , while another endpoint $h^\\infty $ is not contained in the closed arc between $x_i$ and $y_j$ .",
"Then for any point $x\\in C_1$ other than $x_1$ and $x_2$ , the sequence $\\overline{h}^n \\cdot x$ can only accumulate at $x_1$ or $x_2$ .",
"Similarly, for any point $y\\in C_2$ other than $y_1$ and $y_2$ , the sequence $\\overline{h}^n \\cdot x$ can only accumulate at $y_1$ or $y_2$ .",
"Suppose not, then the sequence has an accumulation point $x^{\\prime } \\in C_1\\setminus \\lbrace x_1,x_2\\rbrace $ .",
"By Lemma REF , $x^{\\prime }$ forms boundary of a half flat plane with $h^{\\pm \\infty }$ .",
"This boundary goes from $h^\\infty $ to $x^{\\prime }$ , and then passes through $x_i$ or $y_j$ before ending at $h^{-\\infty }$ .",
"If it passes through $x_i$ , then the Tits length of segment on this boundary joining $h^\\infty $ to $x_i$ is the total length of the half-plane boundary $\\pi $ minus the length of the segment from $x_i$ to $h^{-\\infty }$ , thus it is equal to the length of the Tits geodesic segment on $\\partial F_0$ joining these two points, so there are two geodesics for these two points.",
"But this contradicts the uniqueness of Tits geodesic between two points less than $\\pi $ apart.",
"If the boundary of the half flat plane goes through $y_j$ , apply the same argument to the points $h^\\infty $ and $y_j$ and we have the same contradiction.",
"For the case $y\\in C_2\\setminus \\lbrace y_1,y_2\\rbrace $ use the same argument.",
"Lemma 3.6 $g^{-\\infty }$ cannot be in the open segment from $y$ to $z$ .",
"Suppose not.",
"For any point $z^{\\prime }\\in C_1$ other than $g^\\infty $ and $z$ , the sequence $\\overline{g}^{-n}\\cdot z^{\\prime }$ converges to $z$ by Lemma REF and Lemma REF which says that $\\overline{g}^{-n}\\cdot z^{\\prime }$ cannot accumulate at $g^\\infty $ .",
"The segment $\\beta $ has Tits length larger than $\\pi $ , so there is a point $w\\in \\beta $ which is more than $\\pi /2$ away from $g^\\infty $ and from $z$ .",
"By lower semi-continuity of the Tits metric, $\\mathrm {d}_{\\mathrm {T}}(w,z^{\\prime }) & =\\lim _{n\\rightarrow \\infty }\\mathrm {d}_{\\mathrm {T}}(\\overline{g}^{-n}\\cdot w, \\overline{g}^{-n}\\cdot z^{\\prime }) \\\\& \\ge \\mathrm {d}_{\\mathrm {T}}(\\lim _{n\\rightarrow \\infty } \\overline{g}^{-n}\\cdot w,\\lim _{n\\rightarrow \\infty }\\overline{g}^{-n}\\cdot z^{\\prime }) = \\mathrm {d}_{\\mathrm {T}}(w,z).$ So $\\mathrm {d}_{\\mathrm {T}}(w,C_1) > \\pi /2$ , a contradiction to Theorem REF .",
"Figure: ∂F\\partial F in Lemma We see from these lemmas that the endpoints of a hyperbolic element must be both in $C_1$ , or both in $C_2$ , or none is in $C_1\\cup C_2$ .",
"If $g$ is a hyperbolic element of $G$ with endpoints not in $C_1\\cup C_2$ , we have the following results.",
"Lemma 3.7 $\\partial \\mathrm {Min}(g)$ is the boundary of a 2-flat.",
"Since $\\partial \\mathrm {Min}(g)$ is a suspension, so it can only be a circle or a set of two points.",
"However, as $g$ acts on $\\partial _{\\mathrm {T}}X$ by isometry, we see that $g$ must fix the arc on which $g^\\infty $ lies.",
"So $\\partial \\mathrm {Min}(g) = \\mathrm {Fix}(\\overline{g})$ can only be a circle.",
"Then by the same reason as in Lemma REF $\\mathrm {Min}(g)$ contains a 2-flat, whose boundary is the circle.",
"Suppose for convenience that $g^\\infty $ is on the open arc from $x_1\\in C_1$ to $y_1\\in C_2$ , and $x_2\\in C_1$ , $y_2\\in C_2$ are the two other points on the boundary $\\partial F$ .",
"Lemma 3.8 For $g$ as above, $g^{-\\infty }$ can only be on the open arc from $x_2$ to $y_2$ .",
"Suppose $g^{-\\infty }$ were not on this arc.",
"Without loss of generality let $g^{-\\infty }$ be on the arc joining $y_1$ and $x_2$ .",
"Now the segment from $x_1$ to $x_2$ through $y_1$ has Tits length larger than $\\pi $ , so we can choose a point $p$ on this segment so that $p$ is at distance more than $\\pi /2$ away from $x_1$ and $x_2$ .",
"By Lemma REF , for any other point $x^{\\prime }\\in C_1$ , $\\overline{g}^n\\cdot x^{\\prime }$ cannot have an accumulation point other than $x_1$ and $x_2$ .",
"Passing to a subsequence $\\overline{g}^{n_k}\\cdot x^{\\prime } \\rightarrow x_i$ , $i=1$ or 2, we have $\\mathrm {d}_{\\mathrm {T}}(p, x^{\\prime }) & = \\lim _{n_k\\rightarrow \\infty } \\mathrm {d}_{\\mathrm {T}}(\\overline{g}^{n_k}\\cdot p,\\overline{g}^{n_k}\\cdot x^{\\prime }) \\\\& \\ge \\mathrm {d}_{\\mathrm {T}}(\\lim _{n_k\\rightarrow \\infty } \\overline{g}^{n_k}\\cdot p,\\lim _{n_k\\rightarrow \\infty } \\overline{g}^{n_k}\\cdot x^{\\prime })= \\mathrm {d}_{\\mathrm {T}}(p, x_i),$ then $\\mathrm {d}_{\\mathrm {T}}(p, C_1)> \\pi /2$ , contradicting Theorem REF ."
],
[
"Main result",
"Now we add the assumption that $G$ contains a subgroup isomorphic to $\\mathbb {Z}^2$ , then the Flat Torus Theorem ([3], Theorem II.7.1) implies that there exists two commuting hyperbolic elements $g_1, g_2\\in G$ , such that $\\mathrm {Min}(g_1)$ , formed by the axes of $g_1$ , contains axes of $g_2$ not parallel to those of $g_1$ .",
"Then an axis of $g_1$ and an axis of $g_2$ span a 2-flat in $\\mathrm {Min}(g_1)$ , and elements $g_1^n g_2^m$ are also hyperbolic and have axes in this 2-flat with endpoints dense on the boundary of this 2-flat.",
"So we can choose some hyperbolic element $g$ so that its endpoints are not in $C_1 \\cup C_2$ .",
"We start with a lemma about the orbits of the group action, then we will prove Theorem REF .",
"Lemma 4.1 For any two distinct points $w_1,w_2 \\in \\partial X$ , there exists a sequence $(g_i)_{i=0}^\\infty \\subset G$ such that the points $\\overline{g}_i \\cdot w_j$ , where $0\\le i < \\infty $ and $j\\in \\lbrace 1,2\\rbrace $ , are distinct.",
"From Lemma REF we know that every $w\\in \\partial X$ has an infinite orbit $G \\cdot w$ .",
"So let $(h_i)_{i=0}^\\infty \\subset G$ be a sequence such that $\\overline{h}_i \\cdot w_1$ are distinct.",
"We will construct the sequence $(g_i)$ inductively.",
"First set $g_0 = e$ .",
"Suppose that for $n \\ge 0$ we have $g_0, \\ldots , g_n$ such that $\\overline{g}_i \\cdot w_j$ , where $0\\le i \\le n$ , $j\\in \\lbrace 1,2\\rbrace $ , are distinct.",
"Let $S_n := \\lbrace \\overline{g}_m \\cdot w_1, \\overline{g}_m \\cdot w_2: 0\\le m \\le n \\rbrace $ .",
"Pass to a subsequence of $(h_i)$ so that $\\overline{h}_i \\cdot w_1 \\notin S_n$ .",
"(We will keep denoting any subsequence by $(h_i)$ .)",
"If there exists some $h_j$ such that $\\overline{h}_j \\cdot w_2 \\notin S_n$ , then set $g_{n+1} = h_j$ .",
"Otherwise, there exists some $\\overline{g}_m \\cdot w_k \\in S_n$ such that $\\overline{h}_i \\cdot w_2 = \\overline{g}_m \\cdot w_k$ for infinitely many $h_i$ .",
"Pass to this subsequence.",
"Since the orbit of $\\overline{g}_m \\cdot w_k$ is infinite, there exists $h^{\\prime } \\in G$ such that $\\overline{h^{\\prime }} \\cdot (\\overline{g}_m \\cdot w_k) \\notin S_n$ , so $\\overline{h^{\\prime } h_i} \\cdot w_2 \\notin S_n$ .",
"Now $\\overline{h^{\\prime } h_i} \\cdot w_1 \\notin S_n$ for infinitely many $h_i$ .",
"Set $g_{n+1} = h^{\\prime } h_i$ for one such $h_i$ .",
"Hence we get the desired sequence $(g_i)$ .",
"Remark 4.2 The only condition required on the group action is that every orbit is infinite.",
"This proof can be used to show a similar result for any finite set $\\lbrace w_1, \\ldots w_n\\rbrace $ .",
"Lemma 4.3 For any $x\\in C_1$ , $y\\in C_2$ we have $\\mathrm {d}_{\\mathrm {T}}(x,y) = \\pi / 2$ .",
"Hence $\\partial _{\\mathrm {T}}X$ is metrically a spherical join of $C_1$ and $C_2$ .",
"Consider some $g\\in G$ which is hyperbolic with endpoints not on $C_1\\cup C_2$ .",
"Let $\\partial \\mathrm {Min}(g)=\\partial F$ .",
"We will first prove that for $x_1,x_2\\in C_1 \\cap \\partial F$ , $y_1,y_2\\in C_2 \\cap \\partial F$ , we have $\\mathrm {d}_{\\mathrm {T}}(x_i,y_j) = \\pi / 2$ , where $i,j=1,2$ .",
"Take any of the four arcs making up $\\partial F$ , say the arc joining $x_1$ and $y_1$ .",
"The endpoints of hyperbolic elements in $Z_g$ are dense on $\\partial F$ , so we can pick a $g^{\\prime } \\in Z_g$ so that $g^{\\prime -\\infty }$ is as close to the midpoint of arc $x_2$ and $y_2$ as we want.",
"Let $0<\\delta <\\min (\\mathrm {d}_{\\mathrm {T}}(x_2, C_2),\\mathrm {d}_{\\mathrm {T}}(y_2, C_1))$ .",
"Pick $g^{\\prime }$ so that $\\left|\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty }, x_2) - \\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty }, y_2) \\right| < \\delta $ .",
"For any point $x\\in C_1$ other than $x_2$ , if the Tits geodesic segment from $g^{\\prime -\\infty }$ to $x$ passes through $y_2$ , then $\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x)& \\ge \\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },y_2)+\\mathrm {d}_{\\mathrm {T}}(y_2,C_1) \\\\& > \\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x_2) - \\delta + \\mathrm {d}_{\\mathrm {T}}(y_2,C_1) >\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x_2);$ while if it passes through $x_2$ then obviously $\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x) >\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x_2)$ .",
"For any $y\\in C_2$ other than $y_2$ , by similar reasoning on the Tits geodesic segment from $g^{\\prime -\\infty }$ to $y$ , we have $\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },y) > \\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },y_2)$ .",
"For any arc joining $x\\ne x_2 \\in C_1$ and $y\\ne y_2 \\in C_2$ , since $\\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x) > \\mathrm {d}_{\\mathrm {T}}(g^{\\prime -\\infty },x_2)$ , the point $x_2$ cannot be an accumulation point of $\\overline{g^{\\prime }}^n \\cdot x$ by Lemma REF , then by Lemma REF , $\\overline{g^{\\prime }}^n \\cdot x \\rightarrow x_1$ .",
"Likewise, $\\overline{g^{\\prime }}^n \\cdot y\\rightarrow y_1$ .",
"So $\\mathrm {d}_{\\mathrm {T}}(x, y) &= \\lim _{n\\rightarrow \\infty } \\mathrm {d}_{\\mathrm {T}}(\\overline{g^{\\prime }}^n \\cdot x, \\overline{g^{\\prime }}^n \\cdot y) \\\\&\\ge \\mathrm {d}_{\\mathrm {T}}(\\lim _{n\\rightarrow \\infty }\\overline{g^{\\prime }}^n \\cdot x, \\lim _{n\\rightarrow \\infty }\\overline{g^{\\prime }}^n \\cdot y) = \\mathrm {d}_{\\mathrm {T}}(x_1, y_1).",
"$ For any other arc joining $x_i$ to $y_j$ in $\\partial F$ , by lemma REF there exists $h \\in G$ such that $\\overline{h} \\cdot x_i \\ne x_2$ and $\\overline{h} \\cdot y_j \\ne y_2$ , so from the inequality (REF ) we get $\\mathrm {d}_{\\mathrm {T}}(x_i, y_j) = \\mathrm {d}_{\\mathrm {T}}(\\overline{h} \\cdot x_i, \\overline{h} \\cdot y_j) \\ge \\mathrm {d}_{\\mathrm {T}}(x_1,y_1).$ Thus all arcs have equal length $\\pi /2$ .",
"Now for any $x \\in C_1$ , $y\\in C_2$ , by Lemma REF the sequence $\\overline{g}^n\\cdot x$ can accumulate at $x_1$ or $x_2$ , and $\\overline{g}^n\\cdot y$ can accumulate at $y_1$ or $y_2$ , so passing to some subsequence $(\\overline{g}^{n_k})$ , we have convergence sequences $\\overline{g}^{n_k}\\cdot x \\rightarrow x_i$ and $\\overline{g}^{n_k}\\cdot y \\rightarrow y_j$ .",
"Then we have inequality $\\mathrm {d}_{\\mathrm {T}}(x, y) = \\lim _{n_k\\rightarrow \\infty } \\mathrm {d}_{\\mathrm {T}}(\\overline{g}^{n_k}\\cdot x, \\overline{g}^{n_k}\\cdot y) \\ge \\mathrm {d}_{\\mathrm {T}}(x_i,y_j) = \\pi /2.$ Take a point $p$ on the open arc joining $x$ and $y$ .",
"Without loss of generality assume that $p$ and $x$ are connected in $\\partial _{\\mathrm {T}}X$ by a segment in the arc.",
"For any $\\epsilon > 0$ , we may choose a new point on the segment from $p$ to $x$ to replace $p$ so that $0<\\mathrm {d}_{\\mathrm {T}}(x,p) <\\epsilon $ .",
"Consider the Tits geodesic from $p$ to some point in $C_2$ .",
"If it passes through $x$ , then it consists of the segment from $p$ to $x$ and an arc from $x$ to some point in $C_2$ , so by the inequality (REF ) its Tits length is at least $\\pi /2+\\mathrm {d}_{\\mathrm {T}}(x,p)$ .",
"By Theorem REF $\\mathrm {d}_{\\mathrm {T}}(p,C_2) \\le \\pi /2$ , so there must be a Tits geodesic from $p$ to some point in $C_2$ that does not pass through $x$ , hence it passes through $y$ .",
"Its length is at least $\\mathrm {d}_{\\mathrm {T}}(p,y)$ , so $y$ is the closest point in $C_2$ to $p$ , so $\\mathrm {d}_{\\mathrm {T}}(p,y) = \\mathrm {d}_{\\mathrm {T}}(p,C_2) \\le \\pi /2$ .",
"Then $\\mathrm {d}_{\\mathrm {T}}(x, y) \\le \\mathrm {d}_{\\mathrm {T}}(x,p) + \\mathrm {d}_{\\mathrm {T}}(p,y) < \\pi /2 +\\epsilon $ .",
"Letting $\\epsilon \\rightarrow 0$ we have $\\mathrm {d}_{\\mathrm {T}}(x, y)\\le \\pi /2$ .",
"Combining with the inequality (REF ), $\\mathrm {d}_{\\mathrm {T}}(x,y) =\\pi /2$ .",
"Theorem 4.4 If $X$ is a CAT(0) space which admits a geometric group action by a group containing a subgroup isomorphic to $\\mathbb {Z}^2$ , and $\\partial X$ is homeomorphic to the join of two Cantor sets, then $\\partial _{\\mathrm {T}}X$ is the spherical join of two uncountable discrete sets.",
"If $X$ is geodesically complete, i.e.",
"every geodesic segment in $X$ can be extended to a geodesic line, then $X$ is a product of two CAT(0) space $X_1,X_2$ with $\\partial X_i$ homeomorphic to a Cantor set.",
"We have shown that for any $x\\in C_1$ , $y\\in C_2$ , $\\mathrm {d}_{\\mathrm {T}}(x,y) = \\pi /2$ in Lemma REF , so every two distinct points in $C_i$ has Tits distance $\\pi $ for $i=1,2$ , i.e.",
"$C_i$ with the Tits metric is an uncountable discrete set.",
"Then $\\partial _{\\mathrm {T}}X$ is isomorphic to the spherical join of $C_1$ and $C_2$ , giving the first result.",
"So with the additional assumption that $X$ is geodesically complete, it follows by Theorem II.9.24 of [3] that $X$ splits as a product $X_1 \\times X_2$ , with $\\partial X_i = C_i$ for $i=1,2$ ."
],
[
"Some properties of the group",
"We will show Theorem REF in this section.",
"Assuming that $X$ is geodesically complete, and hence reducible by Theorem REF , we have the following result for the group $G$ .",
"We do not require that $G$ stabilizes each of $C_1$ and $C_2$ in this section.",
"Theorem 5.1 Let $X$ be a CAT(0) space such that $\\partial X$ is homeomorphic to the join of two Cantor sets and suppose $X$ is geodesically complete.",
"For a group $G < \\mathrm {Isom}(X)$ containing $\\mathbb {Z}^2$ and acting geometrically on $X$ , either $G$ or a subgroup of it of index 2 is a uniform lattice in $\\mathrm {Isom}(X_1) \\times \\mathrm {Isom}(X_2)$ , where $X_1, X_2$ are given by Theorem .",
"We know from Theorem REF that $X=X_1 \\times X_2$ , so we only need to show that $G$ or a subgroup of it of index 2 preserves this decomposition.",
"By Lemma REF , either $G$ or a subgroup of it of index 2 stabilizes $C_1$ and $C_2$ .",
"Replacing $G$ by its subgroup if necessary, we assume $G$ stabilizes $C_1$ and $C_2$ .",
"Denote by $\\pi _i$ the projection of $X$ to $X_i$ , $i=1,2$ .",
"Take any $p_1, p_2\\in X$ such that $\\pi _2(p_1) = \\pi _2(p_2)$ .",
"Extend $[p_1,p_2]$ to a geodesic line $\\gamma $ , its projection to each of $X_i$ is the image of a geodesic line.",
"Since $X_1$ is totally geodesic, the geodesic segment $[p_1,p_2]$ projects to a single point $\\pi _2(p_1)$ on $X_2$ , i.e.",
"a degenerated geodesic segment, so $\\pi _2(\\gamma )$ is also a degenerated geodesic line.",
"Thus the endpoints $\\gamma (\\pm \\infty )$ are in $C_1$ .",
"Now $\\overline{g} \\cdot \\gamma $ is a geodesics line passing through $\\overline{g} \\cdot p_1$ , $\\overline{g} \\cdot p_2$ , and its endpoints $\\overline{g} \\cdot \\gamma (\\pm \\infty ) \\in C_1$ , so $\\pi _2(\\overline{g} \\cdot p_1) = \\pi _2(\\overline{g} \\cdot p_2)$ .",
"Similarly, for any $q_1, q_2\\in X$ such that $\\pi _1(q_1) = \\pi _1(q_2)$ we have $\\pi _1(\\overline{g} \\cdot q_1) =\\pi _1(\\overline{g} \\cdot q_2)$ .",
"So $G$ preserves the decomposition $X =X_1 \\times X_2$ , hence the result.",
"We will show that $\\mathrm {Isom}(X_i)$ is isomorphic to a subgroup of $\\mathrm {Homeo}(C_i)$ by the following lemma.",
"Lemma 5.2 Suppose $X^{\\prime }$ is a proper complete CAT(0) space, and $G^{\\prime }<\\mathrm {Isom}(X^{\\prime })$ acts properly on $X^{\\prime }$ by isometries.",
"If $S\\subset \\partial X^{\\prime }$ is a set of points on the boundary such that the intersection $\\bigcap _{w\\in S} \\overline{\\mathrm {B}_{\\mathrm {T}}(w,\\pi /2)}$ is empty, then there exists a point $q\\in X$ such that any non-hyperbolic $g\\in \\mathrm {Isom}(X^{\\prime })$ that fixes $S$ pointwise will fix $q$ .",
"In particular, such $g$ is elliptic.",
"If $\\partial X^{\\prime }$ is not a suspension and the radius of $\\partial _{\\mathrm {T}}X^{\\prime }$ is larger than $\\pi /2$ , then the map $G^{\\prime }\\rightarrow \\mathrm {Homeo}(\\partial X^{\\prime })$ , defined by extending the action of $G^{\\prime }$ to the boundary $\\partial X^{\\prime }$ , has a finite kernel, i.e.",
"the subgroup of $G^{\\prime }$ that acts trivially on the boundary is finite.",
"Moreover, assume the action of $G^{\\prime }$ is cocompact, then the kernel fixes a subspace of $X^{\\prime }$ with boundary $\\partial X^{\\prime }$ .",
"To prove (1), observe that any such $g$ stabilizes all horospheres and thus all horoballs centered at every $w\\in S$ .",
"Take an arbitrary point $q^{\\prime }\\in X$ and choose for each $w$ a closed horoball $H_w$ centered at $w$ that contains $q^{\\prime }$ .",
"Their intersection $\\bigcap _{w\\in S} H_w$ is non-empty since it contains $q^{\\prime }$ .",
"By Lemma 3.5 of [6] $\\partial H_w = \\overline{\\mathrm {B_T}(w,\\pi /2)}$ , then $\\partial (\\bigcap _{w\\in S} H_w) \\subset \\bigcap _{w\\in S}(\\partial H_w)=\\varnothing $ .",
"So $\\bigcap _{w\\in S} H_w$ is bounded.",
"Also as every $H_w$ is stabilized by $g$ , so is $\\bigcap _{w\\in S} H_w$ .",
"As $\\bigcap _{w\\in S} H_w$ is convex and compact, it contains a unique center $q$ , where the function $\\sup \\lbrace \\mathrm {d}_X(\\cdot ,z):z\\in \\bigcap _{w\\in S} H_w \\rbrace $ is minimized.",
"Then $g$ fixes $q$ .",
"For (2), if $g\\in G^{\\prime }$ acts by hyperbolic isometry, then $\\partial \\mathrm {Min}(g)= \\mathrm {Fix}(\\overline{g})$ is a suspension.",
"Then any $g$ acting trivially on the whole boundary $\\partial X^{\\prime }$ is not hyperbolic.",
"As $\\partial _{\\mathrm {T}}X^{\\prime }$ has radius larger than $\\pi /2$ , for every $x \\in \\partial X^{\\prime }$ there is some $w\\in \\partial X^{\\prime }$ such that $\\mathrm {d}_{\\mathrm {T}}(x,w) > \\pi /2$ , so $x \\notin \\overline{\\mathrm {B_T}(w,\\pi /2)}$ , hence $S=\\partial X^{\\prime }$ satisfies the condition in (1).",
"Now (1) implies that the kernel of $G^{\\prime } \\rightarrow \\mathrm {Homeo}(\\partial X^{\\prime })$ is a subgroup of the stabilizer of some point $q \\in X^{\\prime }$ .",
"As the action of $G^{\\prime }$ is proper, the kernel is finite.",
"Let $K$ be the kernel.",
"The set fixed by $K$ is closed and convex.",
"For any point $q$ fixed by the kernel, as $g \\cdot q$ is fixed by $g K g^{-1} = K$ , then $G^{\\prime } \\cdot q$ is fixed by $K$ .",
"If the action of $G^{\\prime }$ is cocompact, then the set fixed by $K$ is quasi-dense, hence it is a subspace with boundary $\\partial X^{\\prime }$ .",
"Corollary 5.3 Let $X$ be a geodesically complete CAT(0) space such that $\\partial X$ is homeomorphic to the join of two Cantor sets.",
"Then for a group $G < \\mathrm {Isom}(X)$ containing $\\mathbb {Z}^2$ and acting geometrically on $X$ , either $G$ or a subgroup of it of index 2 is isomorphic to a subgroup of $\\mathrm {Homeo}(C_1) \\times \\mathrm {Homeo}(C_2)$ .",
"This follows from Theorem REF and Lemma REF .",
"We can still show this without the geodesic completeness assumption.",
"Theorem 5.4 Let $X$ be a CAT(0) space such that $\\partial X$ is homeomorphic to the join of two Cantor sets.",
"Then for a group $G < \\mathrm {Isom}(X)$ containing $\\mathbb {Z}^2$ and acting geometrically on $X$ , a finite quotient of either $G$ or a subgroup of $G$ of index 2 is isomorphic to a subgroup in $\\mathrm {Homeo}(C_1) \\times \\mathrm {Homeo}(C_2)$ .",
"Assume $G$ stabilizes each of $C_1$ and $C_2$ as in the proof of Theorem REF .",
"Each $g\\in G$ acts on $\\partial X$ as a homeomorphism, so it acts on $C_i \\subset \\partial X$ also as a homeomorphism.",
"Suppose $g$ acts trivially on $C_1$ and $C_2$ , i.e.",
"$g$ is in the kernel of $G\\rightarrow \\mathrm {Homeo}(C_1) \\times \\mathrm {Homeo}(C_2)$ .",
"Then for any point $x\\in \\partial X$ outside $C_1 \\cup C_2$ , the arc on which $x$ lies is a Tits geodesic segment of length $\\pi /2$ in $\\partial _{\\mathrm {T}}X$ .",
"Since $g$ acts on $\\partial _{\\mathrm {T}}X$ by isometry and both endpoints of this Tits geodesic segment are fixed by $g$ , so $g$ fixes the whole arc, thus $\\overline{g} \\cdot x = x$ .",
"Hence $g$ acts trivially on $\\partial X$ .",
"One can check that $\\partial _{\\mathrm {T}}X$ has radius larger than $\\pi /2$ , so by Lemma REF $G\\rightarrow \\mathrm {Homeo}(\\partial X)$ has finite kernel.",
"Hence the result.",
"In the case when $X$ is geodesically complete, actually we can prove a stronger result, expressed in the last statement of Theorem REF .",
"Observe that $X_i$ is a Gromov hyperbolic space by the Flat Plane Theorem, which states that a proper cocompact CAT(0) space $Y$ is hyperbolic if and only if it does not contain a subspace isometric to $\\mathbb {E}^2$ .",
"Recall that a cocompact space is defined as a space $Y$ which has a compact subset whose images under the action by $\\mathrm {Isom}(Y)$ cover $Y$ .",
"The (projected) action of $G$ on $X_i$ is cocompact, even though the image in $\\mathrm {Isom}(X_i)$ may not be discrete.",
"As $\\partial X_i$ does not contain $S^1$ , the result follows.",
"We will show $X_i$ is quasi-isometric to a tree.",
"This is equivalent to having the Bottleneck Property by a theorem of Manning, which he proved with an explicit construction: Theorem 5.5 ([8], Theorem 4.6) Let $Y$ be a geodesic metric space.",
"The following are equivalent: $Y$ is quasi-isometric to some simplicial tree $\\Gamma $ .",
"(Bottleneck Property) There is some $\\Delta > 0$ so that for all $x, y \\in Y$ there is a midpoint $m = m(x, y)$ with $d(x, m) = d(y, m) = \\frac{1}{2} d(x,y)$ and the property that any path from $x$ to $y$ must pass within less than $\\Delta $ of the point $m$ .",
"Pick a base point $p$ in $X_i$ .",
"There exists some $r>0$ such that $G \\cdot B(p,r)$ covers $X_i$ .",
"Lemma 5.6 There exists $R >0$ such that for any $x,y$ in the same connected component of $X_i \\setminus B(p,R)$ , the geodesic segment $[x,y]$ does not intersect $B(p,r)$ .",
"Suppose on the contrary that for $R_n$ increasing to infinity, we can find $x_n, y_n$ in the same connected component of $X_i\\setminus B(p,R_n)$ and $[x_n,y_n]$ intersects $B(p,r)$ .",
"Since $\\partial X_i$ is compact in the cone topology, passing to a subsequence we have $x_n \\rightarrow \\overline{x}$ , $y_n \\rightarrow \\overline{y}$ for some $\\overline{x}, \\overline{y}\\in \\partial X_i$ .",
"By [3] Lemma II.9.22, there is a geodesic line from $\\overline{x}$ to $\\overline{y}$ intersecting $B(p,r)$ .",
"In particular, $\\overline{x}\\ne \\overline{y}$ .",
"Since different connected components in the boundary of a hyperbolic space correspond to different ends of the space ([3] Exercise III.H.3.8), and $\\partial X_i$ is a Cantor set, so $\\overline{x}$ and $\\overline{y}$ are in different ends of $X_i$ , which are separated by $B(p,R_n)$ for $R_n$ large enough.",
"But then $x_n$ , $y_n$ will be in different connected components of $X_i \\setminus B(p,R_n)$ , contradicting the assumption.",
"Hence the result.",
"Lemma 5.7 $X_i$ has the Bottleneck Property.",
"For any $x,y\\in X_i$ , we may translate by some $g\\in G$ so that the midpoint $m$ of $[x,y]$ is in $B(p,r)$ .",
"We may assume that $d(x,y)> 2(R+r)$ , then $x,y \\in X_i \\setminus B(p,R)$ .",
"By Lemma , $x,y$ are in different connected components of $X_i \\setminus B(p,R)$ , hence any path connecting $x$ to $y$ must intersect $B(p,R)$ , so some point on this path is at a distance at most $R+r$ from $m$ .",
"Thus the Bottleneck Property is satisfied.",
"Lemma 5.8 $X_i$ is quasi-isometric to a bounded valence tree with no terminal vertex.",
"First we describe briefly Manning's construction in his proof of Theorem REF .",
"Let $R^{\\prime }=20\\Delta $ .",
"Start with a single point $\\star $ in $Y$ .",
"Call the vertex set containing this point $V_0$ , and let $\\Gamma _0$ be a tree with only one vertex and no edge, and $\\beta _0:\\Gamma _0 \\rightarrow Y$ be the map sending the vertex to $\\star $ .",
"Then for each $k \\ge 1$ , Let $N_{k-1}$ be the open $R$ -neighborhood of $V_{k-1}$ .",
"Let $\\mathcal {C}_k$ be the set consists of path components of $Y\\setminus N_{k-1}$ .",
"For each $C\\in \\mathcal {C}_k$ pick some point $v$ at $C \\cap \\overline{N}_k$ .",
"There is a unique path component in $\\mathcal {C}_{k-1}$ containing $C$ , corresponding to a terminal vertex $w \\in V_{k-1}$ .",
"Connect $v$ to $w$ by a geodesic segment.",
"Let $V_k$ be the union of $V_{k-1}$ and the set of new points from each of the path components in $\\mathcal {C}_k$ .",
"Add new vertices and edges to the tree $\\Gamma _{k-1}$ accordingly to get the tree $\\Gamma _k$ .",
"Extend $\\beta _{k-1}$ to $\\beta _k$ by mapping new vertices of $\\Gamma _k$ to corresponding new vertices in $V_k$ , and new edges to corresponding geodesic segments.",
"The tree $\\Gamma =\\cup _{k\\ge 0}\\Gamma _k$ , and $\\beta : \\Gamma \\rightarrow Y$ is defined to be $\\beta _k$ on $\\Gamma _k$ .",
"Apply the construction above to $X_i$ .",
"Since $X_i$ is geodesically complete, each terminal vertex in $V_{k-1}$ will be connected by at least one vertex in $V_k \\setminus V_{k-1}$ , and similarly so for terminal vertices of $\\Gamma _{k-1}$ .",
"So the tree $\\Gamma $ has no terminal vertex.",
"Manning proved that the length of each geodesic segment added in the construction is bounded above by $R^{\\prime }+6\\Delta $ .",
"Consider $w \\in V_{k-1}$ with corresponding path component $C_w \\in \\mathcal {C}_{k-1}$ .",
"Every path component $C \\in \\mathcal {C}_k$ such that $C\\subset C_w$ gives a new segment joining $w$ .",
"Together with geodesic completeness of $X_i$ , this implies that such $C$ will contain at least one path component of $X_i \\setminus B(w,R^{\\prime }+6\\Delta )$ , and every path component of $X_i \\setminus B(w,R^{\\prime }+6\\Delta )$ is contained in at most one such $C$ .",
"(Geodesic completeness is used to ensure that no such $C$ will disappear when passing to $X_i \\setminus B(w,R^{\\prime }+6\\Delta )$ .)",
"Thus the number of new vertices in $V_k$ joining $w$ is bounded by the number of path components of $X_i\\setminus B(w,R^{\\prime }+6\\Delta )$ .",
"Call the vertex in $\\Gamma $ corresponding to $w$ as $p_w$ .",
"Since no more new segments will join $w$ in subsequent steps, the degree of $p_w$ in $\\Gamma $ equals one plus the number of new vertices in $V_k$ joining $w$ .",
"Translate $X_i$ by some $g$ so that $g \\cdot w\\in B(p,r)$ .",
"The number of path components in $X_i \\setminus B(w, R^{\\prime }+6\\Delta )$ equals that in $X_i\\setminus B(g\\cdot w, R^{\\prime }+6\\Delta )$ , which is at most the number of path components in $X_i \\setminus B(p, r+ R^{\\prime }+6\\Delta )$ , as $ B(g\\cdot w, R^{\\prime }+6\\Delta ) \\subset B(p, r+ R^{\\prime }+6\\Delta )$ .",
"Hence we obtain a universal bound of the degree of $p_w$ in $\\Gamma $ , which means $\\Gamma $ has bounded valence.",
"A tree of bounded valence with no terminal vertex is quasi-isometric to the trivalent tree.",
"Such tree is called a bounded valence bushy tree.",
"Therefore we have shown the following: Theorem 5.9 If $X_i$ is a proper cocompact and geodesically complete CAT(0) space whose boundary $\\partial X_i$ is homeomorphic to a Cantor set, then $X_i$ is quasi-isometric to a bounded valence bushy tree.",
"Now each of $X_1$ , $X_2$ is quasi-isometric to a bushy tree, thus $X$ is quasi-isometric to the product of two bounded valence bushy trees, and so is $G$ .",
"Therefore we can apply a theorem by Ahlin ([1] Theorem 1) on quasi-isometric rigidity of lattices in products of trees to show that a finite index subgroup of $G$ is a lattice in $\\mathrm {Isom}(T_1 \\times T_2)$ where $T_i$ is a bounded valence bushy tree quasi-isometric to $X_i$ , $i=1,2$ .",
"Notice that $\\mathrm {Isom}(T_1) \\times \\mathrm {Isom}(T_2)$ is isomorphic to a subgroup of $\\mathrm {Isom}(T_1 \\times T_2)$ of index 1 or 2 (which can be proved similarly as Lemma REF ), we finally proved the last statement of Theorem REF ."
]
] | 1204.1047 |
[
[
"The nature of orbits in a prolate elliptical galaxy model with a bulge\n and a dense nucleus"
],
[
"Abstract We study the transition from regular to chaotic motion in a prolate elliptical galaxy dynamical model with a bulge and a dense nucleus.",
"Our numerical investigation shows that stars with angular momentum Lz less than or equal to a critical value Lzc, moving near the galactic plane, are scattered to higher z, when reaching the central region of the galaxy, thus displaying chaotic motion.",
"An inverse square law relationship was found to exist between the radius of the bulge and the critical value Lzc of the angular momentum.",
"On the other hand, a linear relationship exists between the mass of the nucleus and Lzc.",
"The numerically obtained results are explained using theoretical arguments.",
"Our study shows that there are connections between regular or chaotic motion and the physical parameters of the system, such as the star's angular momentum and mass, the scale length of the nucleus and the radius of the bulge.",
"The results are compared with the outcomes of previous work."
],
[
"Introduction",
"Today astronomers believe that the intrinsic shapes of most elliptical galaxies are either oblate or prolate.",
"It is also true that spherical galaxies are very rare but triaxial elliptical galaxies do exist (see [11], [10], [1], [15]).",
"On the other hand, there are observational data indicating the presence of a black hole or a dense massive nucleus in the central parts of elliptical galaxies ([12]).",
"All of the above gives us the opportunity to construct a dynamical model for an elliptical galaxy hosting a dense nucleus, in order to use it for the study of the global properties of motion in these stellar systems.",
"In order to describe the motion in a prolate elliptical galaxy we use the potential $V_{\\rm t} = \\frac{\\upsilon _0^2}{2}\\ln \\left(r^2 + \\alpha z^2 +c_{\\rm b}^2 \\right) - \\frac{M_{\\rm n}}{\\sqrt{r^2 + z^2 +c_{\\rm n}^2}}= V_{\\rm g} + V_{\\rm n}.",
"\\ \\ \\ $ Our model consists of two components.",
"The first component describes a prolate elliptical galaxy while the second is the potential of a dense spherical nucleus.",
"Here $(r, z)$ are the usual cylindrical coordinates, $\\upsilon _0$ is used for consistency of galactic units, $0.2 \\le \\alpha <1$ is the flattening parameter, and $c_{\\rm b}$ is the radius of the bulge component.",
"Furthermore, $M_{\\rm n}$ is the mass and $c_{\\rm n}$ is the scale length of the nucleus.",
"In an earlier paper ([4], hereafter Paper I) we have studied the transition from regular to chaotic motion in a disk galaxy model with a dense nucleus.",
"There we found that stars, moving in the $(r,z)$ plane with values of angular momentum $L_z$ less than or equal to a critical value $L_{zc}$ , are scattered to higher $z$ upon encountering the dense nucleus, thus displaying chaotic motion.",
"We also found relationships connecting the physical parameters of the system with chaos.",
"In the present paper, we shall focus our study on the transition from regular to chaotic motion in model (1), which describes a prolate elliptical galaxy hosting a dense nucleus.",
"Our aim is: (i) to look for relationships between the physical parameters and chaos, (ii) to explain the numerically found relationships using a semi-theoretical approach and (iii) to compare the present results with the outcomes found in Paper I.",
"In this research we use the well known system of galactic units, where the unit of length is 1 kpc, the unit of mass is $2.325\\times 10^7\\,M_{\\odot }$ and the unit of time is $0.97748 \\times 10^8$ yr.",
"The velocity and the angular velocity units are 10 km s$^{-1}$ and 100 km s$^{-1}$ kpc$^{-1}$ respectively, while $G$ is equal to unity.",
"The energy unit (per unit mass) is 100 km$^2$ s$^{-2}$ .",
"In the above units we use the values: $\\upsilon _0=10$ while $0 \\le M_{\\rm n} \\le 150$ , $0.75 \\le c_{\\rm b} \\le 1.25$ and $0.1 \\le c_{\\rm n} \\le 0.25$ .",
"Since the total potential $V_{\\rm t} = V_{\\rm t}\\left(r, z\\right)$ is axially symmetric and the $L_z$ component of the angular momentum is conserved, we use the effective potential $V_{\\rm eff}=\\frac{L_z^2}{2r^2}+V_{\\rm t}\\left(r,z\\right),\\ \\ \\ $ in order to study the motion in the meridian $(r-z)$ plane.",
"The equations of motion are $\\dot{r} = p_r, \\ \\ \\ \\dot{z} = p_z, \\nonumber \\\\\\dot{p_r} = -\\frac{\\partial \\ V_{\\rm eff}}{\\partial r}, \\ \\ \\ \\dot{p_z} = -\\frac{\\partial \\ V_{\\rm eff}}{\\partial z}, \\ \\ \\ $ and the corresponding Hamiltonian is written as $H=\\frac{1}{2}\\left(p_r^2 + p_z^2 \\right) + V_{\\rm eff}\\left(r,z\\right) = E, \\ \\ \\ $ where $p_r$ and $p_z$ are the momenta per unit mass conjugate to $r$ and $z$ respectively, while $E$ is the numerical value of the Hamiltonian.",
"Equation (4) is an integral of motion, which indicates that the total energy of the particle is conserved.",
"Orbit calculations are based on the numerical integration of the equations of motion (3), which were made using a Bulirsh-Stöer routine, to double precision.",
"The accuracy of the calculations was checked by the constancy of the energy integral, which was conserved up to the twelfth significant figure.",
"The paper is organized as follows.",
"In Section 2 we present numerical results for the potential of a non-active galaxy, that is when $M_{\\rm n}=0$ .",
"Furthermore, the numerical relationship is explained using some semi-theoretical arguments.",
"In Section 3, we investigate numerically the case when the dense nucleus is present.",
"Some semi-theoretical arguments are also presented, in order to explain the numerical outcomes.",
"In Section 4 a comparison with earlier work is given and discussion and conclusions of this research are presented."
],
[
"Results when the massive dense nucleus is not present",
"In this Section we shall study the behavior of orbits when the dense massive nucleus is not present, that is when $M_{\\rm n}=0$ .",
"Figure REF shows the numerical relationship between the critical value $L_{zc}$ of the angular momentum and the radius of the bulge $c_{\\rm b}$ .",
"Orbits were started near $r_0=r_{\\rm max}$ , with $z_0 = p_{r0}=0$ , while the value of $p_{z0}$ is always found from the energy integral (4).",
"The value of $r_{\\rm max}$ is the maximal root of equation $\\frac{L_z^2}{2r^2}+\\frac{1}{2}\\ln \\left(r^2 + c_{\\rm b}^2 \\right)=E,\\ \\ \\ $ which was found numerically.",
"Dots represent the numerical values while the solid line is the best fit which is an inverse square law, represented by the equation $L_{zc}=\\frac{10.5906}{c_{\\rm b}^2}.",
"\\ \\ \\ $ The value of $\\alpha $ is 0.2, while the value of $E$ is 231.",
"Orbits with values of the parameters on and below the line are chaotic, while orbits on the upper side of the line are regular.",
"Figure: Plot of L zc L_{zc} vs. c b c_{\\rm b}.",
"The values of the parameters aregiven in the text.Figure REF shows the Poincaré $(r,p_r), z=0, p_z>0$ phase plane when $\\alpha =0.2, L_z=1, E=213$ , while the value of $c_b$ is 0.75 in Figure REF (a) and 1.25 in Figure REF (b).",
"As one can see, the pattern is similar in both figures with regions of regular and chaotic motion.",
"A more detailed inspection shows that the extent of the chaotic zone is larger in the case when $c_b=0.75$ , which is when we have a galaxy with a denser bulge.",
"Figure REF shows two orbits when $\\alpha =0.2, c_b=0.9,E=231$ .",
"The value of the angular momentum in the orbit shown in Figure REF (a) is $L_z=40$ and the initial conditions are: $r_0=9.1, z_0 = p_{r0}=0$ , while the value of $p_{z0}$ is always found from the energy integral (4).",
"Note that the orbit is regular and stays very close to the galactic plane.",
"On the contrary, the orbit shown in Figure REF (b) has a value of $L_z=8$ and initial conditions: $r_0=10.0, z_0 = p_{r0}=0$ .",
"The orbit is chaotic and it is scattered off the galactic plane, displaying high values of $z$ .",
"Both orbits were calculated for a time period of 200 time units.",
"Figure: (r,p r )(r,p_r) phase planewhen the dense nucleus is absent.",
"(a) c b =0.75c_{\\rm b}=0.75 and (b) c b =1.25c_{\\rm b}=1.25.",
"The values ofall other parameters are given in the text.Figure: Orbits when the dense nucleus is not present.",
"(a) A regular orbit which stays near the galactic plane.",
"(b) A chaotic orbit which is scattered into high values ofzz.",
"See the text for details.Let us now begin to use semi-analytical arguments in order to explain the numerically found relationship of Figure REF .",
"The lines of arguments are similar to those used in Paper I.",
"As the test particle approaches very close to the center of a galaxy, there is a change in its momentum in the $z$ direction given by the equation $m\\Delta p_z = \\langle F_z \\rangle \\Delta t. \\ \\ \\ $ Here $m$ is the mass of the test particle, $\\langle F_z \\rangle $ is the total average force acting in the $z$ direction and $\\Delta t$ is the duration of the encounter.",
"Empirical evidence shows that the test particle's rise proceeds cumulatively in each case, and increases a little more with each successive pass from the center rather than with a single “violent\" encounter.",
"It is observed that the test particle gains considerable height after $n(n>1)$ crossings, when the total change in the momentum in the $z$ direction is on the order of $m \\upsilon _\\phi $ , where $\\upsilon _\\phi $ is the tangential velocity of the test particle near the center at a distance $r=\\langle r_0 \\rangle \\simeq \\langle z_0 \\rangle \\ll 1$ .",
"Therefore we write $m\\sum \\limits _{i=1}^{n}{\\Delta {{p}_{zi}}}\\approx \\langle {{F}_{z}}\\rangle \\sum \\limits _{i=1}^{n}{\\Delta {{t}_{i}}}.",
"\\ \\ \\ $ Setting $m\\sum \\limits _{i=1}^{n}{\\Delta {{p}_{zi}}}&=& m{{\\upsilon }_{\\phi }}=\\frac{m{{L}_{zc}}}{\\langle {{r}_{0}}\\rangle }\\,, \\nonumber \\\\\\sum \\limits _{i=1}^{n}{\\Delta {{t}_{i}}} &=& {{T}_{c}}, \\ \\ \\ $ in Equation (8) we find $\\frac{m{{L}_{zc}}}{\\langle {{r}_{0}}\\rangle }\\,\\approx \\,\\langle {{F}_{z}}\\rangle {{T}_{c}}.",
"\\ \\ \\ $ The force acting in the $z$ direction for a test particle of unit mass $(m=1)$ is ${{F}_{z}}=\\frac{-\\upsilon _{0}^{2} \\alpha z}{{{r}^{2}}+ \\alpha {{z}^{2}}+c_{\\rm b}^{2}}.",
"\\ \\ \\ $ Remember that $r=\\langle r_0 \\rangle \\simeq \\langle z_0 \\rangle \\ll 1$ , therefore $\\langle r_0^2 \\rangle \\simeq \\langle z_0^2\\rangle \\ll c_{\\rm b}^2$ .",
"Keeping only the linear terms in $\\langle r \\rangle $ and $\\langle z \\rangle $ in Equation (11) and taking the absolute value of the $F_z$ force, we find from relationship (10) that ${{L}_{zc}}\\,\\approx \\,\\frac{\\upsilon _{0}^{2} \\alpha {{\\left(\\langle {{r}_{0}}\\rangle \\right)}^{2}}}{c_{\\rm b}^{2}}{{T}_{c}}.",
"\\ \\ \\ $ Since $T_c$ was observed to be the same when $0.75\\le c_{\\rm b} \\le 1.25$ , we can set $k_1=\\upsilon _0^2 \\alpha \\left(\\langle r_0 \\rangle \\right)^2 T_c$ and obtain $L_{zc} \\approx \\frac{k_1}{c_{\\rm b}^2}, \\ \\ \\ $ which explains the numerical relationship of Figure REF .",
"The authors would like to make it clear that relation (13) does not reproduce the numerical results shown in Figure REF , but only shows the form of the relationship."
],
[
"Results when the massive nucleus is present",
"We investigate the case when we have an active galaxy, that is when the dense massive nucleus is present.",
"Figure REF shows a numerical relationship between the critical value $L_{zc}$ of the angular momentum and the mass of the nucleus $M_{\\rm n}$ for two values of $c_{\\rm n}$ .",
"The procedure to obtain the results shown in Figure REF is similar to that followed in Figure REF .",
"The value of $\\alpha $ is 0.5, $c_{\\rm b}=1.2$ , while the value of $E$ is 227.",
"Here we see a straight line.",
"Orbits with values of the parameters on the left side of the line, including the line, are chaotic, while orbits on the right side of the line are regular.",
"It is interesting to note that the extent of the chaotic region is larger when the value of $c_{\\rm n}$ is smaller, which is when we have a denser nucleus.",
"Figure: (r,p r )(r,p_r) phase plane whenthe dense nucleus is present.",
"(a) M n =20M_{\\rm n}=20 and (b) M n =100M_{\\rm n}=100.",
"The values of all other parameters are given inthe text.Figure: Plot of |F z ||F_z| and |F zn ||F_{zn}| vs. zz.",
"The values of all theother parameters are given in the text.Figure REF (a)–(b) shows the Poincaré $(r,p_r), z=0, p_z>0$ phase plane when: $\\alpha =0.2,L_z=10, E=230, c_{\\rm b} =1.2$ ; the value of $M_{\\rm n}$ is 20 in Figure REF (a) and 100 in Figure REF (b).",
"In both cases, we observe regular regions together with large chaotic regions.",
"Some small islands are also present indicating secondary resonances.",
"It is evident that the area covered by chaotic orbits is larger in the case of $M_{\\rm n}=100$ , which is when we have a more massive nucleus.",
"In Figure REF (a)–(b) we can see two orbits when $\\alpha =0.5, c_{\\rm b}=1.2, E=227$ .",
"The value of the angular momentum in the orbit shown in Figure REF (a) is $L_z=60$ , while the value of $M_{\\rm n}$ is 25 and $c_{\\rm n}=0.25$ .",
"The initial conditions are $r_0=6.0, z_0 = p_{r0}=0$ , while the value of $p_{z0}$ is always found from the energy integral (4).",
"Note that the orbit is regular, and stays very close to the galactic plane.",
"On the other hand, the orbit shown in Figure REF (b) has a value of $L_z=45$ , while the value of $M_{\\rm n}$ is 120 and $c_{\\rm n}=0.25$ .",
"Initial conditions are: $r_0=9.768, z_0 = p_{r0}=0$ .",
"The orbit is chaotic and it is scattered off the galactic plane, displaying high values of $z$ .",
"Both orbits were calculated for a time period of 200 time units.",
"The linear relationship of Figure REF can be obtained using semi-analytical arguments.",
"As the test particle approaches the nucleus it experiences a strong vertical force, due to the presence of the dense nucleus.",
"Figure REF shows a plot of the $F_z$ force as well as the nuclear $F_{zn}$ force as a function of $z$ , near the nucleus when $r_0=0.1$ .",
"The values of the parameters are $\\alpha =0.5, \\upsilon _0=10, c_{\\rm b}=0.8, M_{\\rm n}=100, c_{\\rm n}=0.25$ .",
"We see that the nuclear force is about 25 times as strong as $F_z$ .",
"Because of this strong force, there is a change in its momentum in the $z$ direction given by the equation $m\\Delta p_z = \\langle F_{zn} \\rangle \\Delta t, \\ \\ \\ $ where $m$ is the mass of the test particle, $\\langle F_{zn} \\rangle $ is the average nuclear force acting in the $z$ direction, while $\\Delta t$ is the duration of the encounter.",
"Here again, our numerical results show that the test particle goes to higher $z$ after $n (n>1)$ crossings, when the total change in the momentum in the $z$ direction is on the order of $m \\upsilon _\\phi $ , where $\\upsilon _\\phi $ is the tangential velocity of the test particle near the center, at a distance $r_0 \\simeq z_0 \\simeq c_{\\rm n}$ .",
"Therefore we write $m\\sum \\limits _{i=1}^{n}{\\Delta {{p}_{zi}}} \\approx \\langle {{F}_{zn}}\\rangle \\sum \\limits _{i=1}^{n}{\\Delta {{t}_{i}}}.",
"\\ \\ \\ $ If we set $m\\sum \\limits _{i=1}^{n}{\\Delta {{p}_{zi}}} &=& m{{\\upsilon }_{\\phi }}=\\frac{m{{L}_{zc}}}{\\langle {{r}_{0}}\\rangle }, \\nonumber \\\\\\sum \\limits _{i=1}^{n}{\\Delta {{t}_{i}}} &=& {{T}_{e}}, \\nonumber \\\\m &=& 1, \\nonumber \\\\\\langle F_{zn} \\rangle &=& \\frac{c_{\\rm n} M_{\\rm n}}{\\left(3c_{\\rm n}^2 \\right)^{3/2}}, \\ \\ \\ $ in Equation (15) we obtain $M_{\\rm n} \\approx k_2 L_{zc} c_{\\rm n}, \\ \\ \\ $ where $k_2=\\sqrt{27}/T_e$ .",
"Note that the values of $T_e$ were observed to be about the same when $0 < M_n \\le 150$ .",
"Equation (17) explains the numerically found relationship given in Figure REF ."
],
[
"Discussion",
"Today it is well-known that there are luminous elliptical galaxies hosting dense massive nuclei (see [13]).",
"During the past few years, a large amount of observational data has given a better and more detailed picture of these active galaxies ([2], [5], [7], [14], [6], [8], [3], [9], [16]).",
"In this article we have studied the transition from regular to chaotic motion in a prolate elliptical galaxy model.",
"Two cases were studied, which are the case where the nucleus was absent and the case where we have an active galaxy hosting a dense massive nucleus.",
"In the first case we found that an inverse square law relationship exists between the radius of the bulge of the galaxy and the critical angular momentum when all other parameters are kept fixed.",
"This relationship was also reproduced, using some semi-analytical arguments.",
"On the other hand, it was observed that for larger values of the radius of the bulge, for $c_{\\rm b} \\ge 1.5$ , the chaotic regions when the dense nucleus is absent, if any, are negligible when $0.2 \\le \\alpha < 1$ .",
"This result is in agreement with the result found in Paper I, where no chaos was observed when $c_{\\rm b}> 1.3$ and this result was independent of the mass of the bulge.",
"Thus, our results suggest that chaos is observed in disk and prolate elliptical galaxies when a dense bulge is present.",
"Interesting results are obtained when a dense nucleus is present.",
"In this case the numerically found relationship connecting $L_{zc}$ and $M_{\\rm n}$ is linear when all other parameters are kept constant.",
"Furthermore, the linear relationship depends on the value of $c_{\\rm n}$ , in such a way that the extent of the chaotic region is larger when $c_{\\rm n}$ is smaller, which is when the nucleus has a higher density.",
"The linear relationship obtained by the numerical integration of the equations of motion was also found using some semi-theoretical arguments, together with numerical evidence.",
"The present investigation shows that the results obtained are very similar to those obtained for disk galaxies in Paper I.",
"Therefore, we can say that low angular momentum stars approaching a dense massive nucleus are deflected to higher $z$ , thus displaying chaotic motion.",
"The similarity of the results obtained for different galactic models suggests that it is the strong vertical force near the dense nucleus that is responsible for this scattering combined with the star's low angular momentum, which allows the star to approach the dense nucleus.",
"Some of the latest discoveries obtained from observational astronomy show that supermassive black holes, up to a billion solar masses, inhabit the centers of all massive spheroidal galaxies, independent of their visible activity; the supermassive black holes are often quiescent with regard to their own radiation but always show dynamical behavior.",
"We believe that, with new data from active galaxies, astronomers will be able to construct better dynamical models in the near future, in order to study the properties of motion in galaxies and to find interesting relationships connecting chaos with the physical parameters of these stellar systems.",
"Useful suggestions and comments from an anonymous referee are gratefully acknowledged."
]
] | 1204.1203 |
[
[
"Variation of the gas and radiation content in the sub-Keplerian\n accretion disk around black holes and its impact to the solutions"
],
[
"Abstract We investigate the variation of the gas and the radiation pressure in accretion disks during the infall of matter to the black hole and its effect to the flow.",
"While the flow far away from the black hole might be non-relativistic, in the vicinity of the black hole it is expected to be relativistic behaving more like radiation.",
"Therefore, the ratio of gas pressure to total pressure (beta) and the underlying polytropic index (gamma) should not be constant throughout the flow.",
"We obtain that accretion flows exhibit significant variation of beta and then gamma, which affects solutions described in the standard literature based on constant beta.",
"Certain solutions for a particular set of initial parameters with a constant beta do not exist when the variation of beta is incorporated appropriately.",
"We model the viscous sub-Keplerian accretion disk with a nonzero component of advection and pressure gradient around black holes by preserving the conservations of mass, momentum, energy, supplemented by the evolution of beta.",
"By solving the set of five coupled differential equations, we obtain the thermo-hydrodynamical properties of the flow.",
"We show that during infall, beta of the flow could vary upto ~300%, while gamma upto ~20%.",
"This might have a significant impact to the disk solutions in explaining observed data, e.g.",
"super-luminal jets from disks, luminosity, and then extracting fundamental properties from them.",
"Hence any conclusion based on constant gamma and beta should be taken with caution and corrected."
],
[
"Introduction",
"Gas flows in the vicinity of a compact object, particularly black hole and neutron star, are expected to be highly relativistic.",
"However, far away of it, when infalling matter comes off, e.g., a companion star, flows are rather non-relativistic.",
"Therefore, during accretion of matter from a large distance, flows should transit from the non-relativistic to relativistic regime while reaching the vicinity of the black hole horizon or the neutron star surface, when flow is expected to be sub-Keplerian in nature.",
"Indeed, it is generally believed that the velocity and temperature at the inner edge of accretion disk are very high.",
"Not only the cases in accretion, highly relativistic flows are involved in astrophysical jets from galactic, extragalactic black holes, gamma-ray bursts etc.",
"(e.g.",
"Zensus 1997; Mirabel & Rodríguez 1999; Mészáros 2002), with the velocity $0.9-0.98$ times the speed of light.",
"The collimated dipolar outflows emerging from deep inside collapsars, according to the collapsar model of gamma-ray burst (Woosley 1993), are expected to achieve a Lorentz factor more than 100.",
"As the accretion and outflow/jet are expected to be coupled (influencing each other; Bhattacharya, Ghosh & Mukhopadhyay 2010), any change/transition (e.g.",
"as mentioned above) in the accretion flow would influence the jet and hence the underlying inferences.",
"When the velocity varies from the non-relativistic to relativistic (ultra-relativistic) regime, the corresponding polytropic index ($\\gamma $ ) in the equation of state (EOS), i.e.",
"the ratio of gas to total pressure ($\\beta $ ) in the present context, should not remain constant.",
"Note that $\\gamma $ and $\\beta $ can be approximated as constant only if the flow remains non-relativistic or ultra-relativistic throughout.",
"Therefore, the existing solutions for such flows with a fixed $\\gamma $ need to be verified and corrected if necessary with a relativistically correct EOS (see e.g.",
"Chandrasekhar 1938).",
"Most of the studies of accretion disks around compact objects have approximated $\\gamma $ to be constant throughout the flow, which cannot be the correct description for the reasons described above.",
"Blumenthal & Mathews (1976) introduced, for the first time to best of our knowledge, variable $\\gamma $ and proposed an appropriate EOS in obtaining solutions for spherical accretion and wind around Schwarzschild black holes.",
"Recently, Meliani et al.",
"(2004) used their EOS and obtained the general solutions for the relativistic Parker winds incorporating varying $\\gamma $ according to the nature of the flow.",
"Ryu et al.",
"(2006) proposed a new EOS suitable for non-relativistic and relativistic regimes with varying $\\gamma $ , to implement in the simulations of relativistic hydrodynamics.",
"Very recently, Chattopadhyay & Ryu (2009) showed that composition of the fluid plays an important role to determine the value of $\\gamma $ and then solutions of inviscid spherical flows around black holes.",
"This would have impact on a hot accretion disk as well which was shown to exhibit significant nuclear burning, producing the variation of compositions in the disk (Chakrabarti & Mukhopadhyay 1999, Mukhopadhyay & Chakrabarti 2000).",
"Earlier, while Chen & Taam (1993) discussed structure and stability of the transonic optically thick accretion disks without restricting the gas and radiation content therein, did not show the actual variation of the gas/radiation content as a function of radial coordinate and then its impact to the solutions.",
"The present work introduces a self-consistent variation of $\\gamma $ in solving viscous flows in the accretion disk around black holes.",
"We essentially concentrate on the region of the flows which is expected to be sub-Keplerian in nature.",
"Therefore, following previous work (Muchotrzeb & Paczynski 1982; Abramowicz et al.",
"1988; Chakrabarti 1996; Mukhopadhyay & Ghosh 2003; Rajesh & Mukhopadhyay 2010a) we model the flow from the Keplerian to sub-Keplerian transition radius ($x_t$ ) to the black hole event horizon ($x_+$ ).",
"We plan to concentrate, in particular, the class of flows which is not geometrically very thick, yet sub-Keplerian in nature including a nonzero component of advection and gradient of pressure in general.",
"Hence, the underlying equations can be integrated/averaged vertically without losing important physics, in obtaining the solutions.",
"Therefore, following previous work (Muchotrzeb & Paczynski 1982; Abramowicz et al.",
"1988; Chakrabarti 1996; Mukhopadhyay & Ghosh 2003; Rajesh & Mukhopadhyay 2010a, 2010b), we consider vertically integrated infall consisting of gas and radiation.",
"While the flow around $x_t$ might be non-relativistic with a high $\\gamma $ (depending on the size of the disk), as it advances towards $x_+$ , $\\gamma $ must be varying.",
"We plan to investigate, how the self-consistent variation of $\\gamma $ over the disk radii affects the established solutions with a fixed $\\gamma $ .",
"In the next section we establish the model equations describing flows.",
"Subsequently, we discuss solutions in §3 and finally summarize the results in §4."
],
[
"Model Equations",
"The basic equations describing conservations of mass and momentum in the vertically integrated disk remain same as of earlier works which assumed $\\gamma $ and then $\\beta $ to be constant throughout (e.g.",
"Abramowicz et al.",
"1988; Chakrabarti 1996; Rajesh & Mukhopadhyay 2010b).",
"Abramowicz et al.",
"(1988) also showed that the geometrically slimmer flow is stable for $\\beta >0.4$ .",
"For simplicity, following e.g.",
"Narayan & Yi (1994) and Chakrabarti (1996), we assume that the heat radiated out to be proportional to the heat generated by viscous dissipation.",
"Hence, following the same authors, we define a parameter $f$ determining the fraction of matter advected in.",
"Therefore, we recall the equation describing conservation of energy in the vertically integrated disk from Chakrabarti (1996) and Mukhopadhyay & Ghosh (2003).",
"However, in reality, $f$ should be determined, rather fixing, selfconsistently as done by Rajesh & Mukhopadhyay (2010a) very recently for hot optically thin flows around black holes.",
"For the present purpose, all the above mentioned equations of conservation are supplemented by an equation describing the variation of $\\beta $ as a function of radial coordinate, assuming a single temperature fluid.",
"Earlier authors (e.g.",
"Chen, Abramowicz & Lasota 1997) already formulated single temperature optically thin disk flows including bremsstrahlung cooling for constant $\\beta $ .",
"Therefore, without repeating the description of equations of conservation, assuming the flow to be a mixture of perfect gas and radiation (Abramowicz et al.",
"1988; Chakrabarti 1996; Rajesh & Mukhopadhyay 2010a,b), we recall straight away $&&\\beta =\\frac{P_{\\rm gas}}{P}\\,\\,=\\frac{P_{\\rm gas}}{P_{\\rm rad}+P_{\\rm gas}},\\,\\,P_{\\rm gas}=\\frac{\\rho k T}{\\mu _i m_p},$ where $k$ is the Boltzmann constant, $m_p$ the mass of a proton, $\\mu _i$ the mean molecular weight of ions, and $P$ , $\\rho $ , $T$ are respectively the total pressure, density, temperature in $K$ of the flow.",
"Now for an optically thick flow, as was initiated in the limit of geometrically slim disk by Muchotrzeb & Paczynski (1982), Abramowicz et al.",
"(1988), $P_{\\rm rad}=P_{\\rm bb}={\\rm pressure\\,\\,from\\,\\,blackbody\\,\\,radiation}=\\frac{a T^4}{3},$ where $a$ is the radiation constant.",
"On the other hand, for the optically thin hot flows (e.g.",
"Narayan & Yi 1995; Mandal & Chakrabarti 2005; Rajesh & Mukhopadhyay 2010a,b) in the presence of bremsstrahlung as radiation mechanism, which are presumably geometrically thicker than that of Muchotrzeb & Paczynski (1982), Abramowicz et al.",
"(1988), $\\nonumber P_{\\rm rad}=P_{\\rm br}&=&{\\rm pressure\\,\\,from\\,\\,bremsstrahlung\\,\\,radiation}\\\\&\\sim &2\\times 10^{-18}\\,\\rho ^2\\,h\\,T^{1/2}\\,(1+4.4\\times 10^{-10} T)\\,M$ in CGS unit, where $h$ is the half-thickness of the flow in units of $GM/c^2$ , $G$ the Newton's gravitation constant, $M$ the mass of the black hole, $c$ the speed of light.",
"Note that, in the first approximation (Rajesh & Mukhopadhyay 2010b), we do not consider other radiation mechanisms which could be effective in optically thin flows.",
"In a future work, such effects e.g.",
"synchrotron radiation, inverse-Comptonization could be included, once the present results confirm such a study worth pursuing.",
"With the choice of $\\gamma P=\\rho c_s^2$ (Mukhopadhyay & Ghosh 2003), we further write $c_s^2=\\frac{8-3\\beta }{6-3\\beta }\\frac{k T}{\\mu m_p \\beta },$ where $c_s$ is the sound speed of the flow."
],
[
"Optically thick flows",
"First we set up the model equations for optical thick and geometrically slim flows.",
"Eliminating $T$ from eqns.",
"(REF ), (REF ) and (REF ) we obtain $R\\frac{d\\beta }{dx}+\\frac{6(1-\\beta )}{c_s}\\frac{dc_s}{dx}+(\\beta -1)\\frac{1}{\\rho }\\frac{d\\rho }{dx}=0,$ where $R=\\frac{1}{\\beta }+3(1-\\beta )\\left(\\frac{1}{\\beta }-\\frac{6}{(8-3\\beta )(6-3\\beta )}\\right)$ and $x$ is the radial coordinate in units of $GM/c^2$ .",
"As $\\beta $ and $\\gamma $ are related by (Narayan & Yi 1995; Ghosh & Mukhopadhyay 2009) $\\beta =\\frac{6\\gamma -8}{3(\\gamma -1)},$ from eqn.",
"(REF ) one can easily obtain $d\\gamma /dx$ as well.",
"Now replacing $P$ , using eqns.",
"(REF ) and (REF ), from the momentum and energy conservation equations and then eliminating $\\rho $ using the mass conservation equation (see Mukhopadhyay & Ghosh 2003, for details), we obtain $\\left(\\frac{\\gamma \\vartheta }{K c_s^2}-\\frac{1}{\\vartheta }\\right)\\frac{d\\vartheta }{dx}+\\frac{1}{c_s}\\left(\\frac{L}{K}-1\\right)\\frac{dc_s}{dx}=\\frac{3}{2x}+\\frac{\\gamma }{K c_s^2}\\left(\\frac{\\lambda ^2}{x^3}-F\\right)-\\frac{1}{2F}\\frac{dF}{dx},$ $A\\frac{d\\vartheta }{dx}+\\vartheta B\\frac{dc_s}{dx}+\\frac{f I_n Z}{x}\\frac{d\\lambda }{dx}=\\frac{2 f I_n Z \\lambda }{x^2}-\\frac{3}{2}\\frac{A\\vartheta }{x}+\\frac{A\\vartheta }{2F}\\frac{dF}{dx},$ $\\nonumber &&\\frac{x}{\\vartheta }\\left[K\\,Z-(1+K)\\alpha \\vartheta ^2\\right]\\frac{d\\vartheta }{dx}-\\frac{x}{c_s}(L-K+1)(Z-\\alpha \\vartheta ^2)\\frac{dc_s}{dx}+\\vartheta \\frac{d\\lambda }{dx}\\\\&&=\\left(\\frac{5}{2}-\\frac{3}{2}K\\right)Z+(K-1)\\left(\\frac{3}{2}\\alpha \\vartheta ^2+\\alpha x\\frac{I_{n+1}}{I_n}\\frac{c_s^2}{\\gamma }\\frac{1}{2F}\\frac{dF}{dx}\\right),$ where $\\vartheta $ and $\\lambda $ are the radial velocity and specific angular momentum of the disk respectively, $f$ is kept constant for a particular flow (Mukhopadhyay & Ghosh 2003; Rajesh & Mukhopadhyay 2010b); $f\\rightarrow 1$ corresponds to the advection of entire energy in and $f\\rightarrow 0$ to no advection, $\\alpha $ the viscosity parameter, $F$ the pseudo-Newtonian gravitational force due to a rotating black hole (Mukhopadhyay 2002), the functions $A,B,Z,K,L$ are given by $\\nonumber A=\\frac{c_s^2}{\\gamma }\\frac{\\Gamma _1-K}{\\Gamma _3-1},\\,\\,\\,B=\\frac{c_s}{\\gamma }\\frac{L-K+\\Gamma _1}{\\Gamma _3-1},\\,\\,\\,Z=\\alpha \\left(\\frac{I_{n+1}}{I_n}\\frac{c_s^2}{\\gamma }+\\vartheta ^2\\right),\\\\K=1+\\frac{6(\\beta -1)}{(6-3\\beta )(8-3\\beta )R},\\,\\,\\, L=2+\\frac{36(1-\\beta )}{R(6-3\\beta )(8-3\\beta )},$ where $I_n, I_{n+1}, \\Gamma _1, \\Gamma _3$ have their usual meaning, discussed by the previous authors in detail (e.g.",
"Mukhopadhyay & Ghosh 2003; Rajesh & Mukhopadhyay 2010a,b).",
"Now combining eqns.",
"(REF ), (REF ), (REF ) we obtain $\\frac{d\\vartheta }{dx}=\\frac{f_1}{f_2},\\,\\,\\,{\\rm where}$ $\\nonumber f_1=\\left[\\frac{3}{2x}+\\frac{\\gamma }{K\\,c_s^2}\\left(\\frac{\\lambda ^2}{x^3}-F\\right)-\\frac{1}{2F}\\frac{dF}{dx}\\right]\\left[\\vartheta ^2 B+\\frac{f I_n Z}{c_s}(L-K+1)\\right.\\\\\\nonumber \\left.",
"(Z-\\alpha ^2\\vartheta ^2)\\right]-\\frac{1}{c_s}\\left(\\frac{L}{K}-1\\right)\\left[\\vartheta \\left(\\frac{2f I_nZ\\lambda }{x^2}-\\frac{3}{2}\\frac{A\\vartheta }{x}+\\frac{A\\vartheta }{2F}\\frac{dF}{dx}\\right)\\right.\\\\\\left.",
"-\\frac{f I_n Z}{x}\\left(\\left(\\frac{5}{2}-\\frac{3}{2} K\\right)Z+(K-1)\\left(\\frac{3}{2}\\alpha \\vartheta ^2+x\\alpha \\frac{I_{n+1}}{I_n}\\frac{c_s^2}{\\gamma }\\frac{1}{2F}\\frac{dF}{dx}\\right)\\right)\\right],$ $\\nonumber f_2=\\left(\\frac{\\gamma \\vartheta }{K\\,c_s^2}-\\frac{1}{\\vartheta }\\right)\\left(\\vartheta ^2B+(L-K+1)(Z-\\alpha \\vartheta ^2)\\frac{fI_nZ}{c_s}\\right)\\\\-\\frac{1}{c_s}\\left(\\frac{L}{K}-1\\right)\\left(A\\vartheta -\\frac{KZ-(1+K)\\alpha \\vartheta ^2}{\\vartheta }fI_nZ\\right).$ We strictly follow the methodology adopted by Rajesh & Mukhopadhyay (2010a) in solving eqn.",
"(REF ).",
"Then following Mukhopadhyay & Ghosh (2003) and Rajesh & Mukhopadhyay (2010a,b) the outer boundary $x_t$ , where the flow becomes purely sub-Keplerian, is specified with the condition $\\lambda /\\lambda _K=1$ ($\\lambda _K$ being value of Keplerian $\\lambda $ ).",
"As matter advances from $x_t$ to $x_+$ , it reaches the critical radius $x=x_c$ when $f_1=f_2=0$ .",
"Hence from $f_2=0$ , the algebraic equation of Mach number ($M_{ac}$ ) at $x_c$ is given by ${\\cal A}\\,M_{ac}^4+{\\cal B}\\,M_{ac}^2+{\\cal C}=0,\\,\\,\\,{\\rm where}$ $\\nonumber &&{\\cal A}=\\frac{L-K+\\Gamma _1}{K(\\Gamma _3-1)}+\\frac{L-K+1}{K}\\alpha ^2f I_{n+1}-\\left(\\frac{L}{K}-1\\right)\\alpha ^2fI_n,\\\\\\nonumber &&{\\cal B}=-\\frac{1}{\\gamma (\\Gamma _3-1)}\\left[\\left(\\frac{L}{K}-1\\right)(\\Gamma _1-K)+(L-K+\\Gamma _1)\\right]\\\\\\nonumber &&+\\frac{(L-K+1)}{K}\\frac{\\alpha ^2fI_{n+1}}{\\gamma }\\frac{I_{n+1}}{I_n}-(L-K+1)\\alpha ^2fI_{n+1}\\\\\\nonumber &&+\\left(\\frac{L}{K}-1\\right)\\alpha ^2fI_n\\frac{K-1}{\\gamma }\\frac{I_{n+1}}{I_n},\\\\&&{\\cal C}=-\\frac{(L-K+1)}{\\gamma ^2}\\alpha ^2fI_{n+1}\\frac{I_{n+1}}{I_n}+\\left(\\frac{L}{K}-1\\right)\\alpha ^2fI_nK\\left(\\frac{I_{n+1}}{I_n\\gamma }\\right)^2.$ Then $M_{ac}$ can be easily computed from eqn.",
"(REF ) for given $\\alpha $ and $f$ , once $\\gamma $ and therefore $\\beta $ is known at $x_c$ , say $\\gamma _c$ and therefore $\\beta _c$ .",
"Subsequently, $c_s$ at $x_c$ , say $c_{sc}$ , can be obtained from $f_1=0$ , once $x_c$ and the specific angular momentum of the disk at this radius $\\lambda _c$ are obtained.",
"Now using the conservation of mass, eliminating $d\\rho /dx$ from eqn.",
"(REF ), we obtain $\\frac{d\\beta }{dx}=\\frac{1}{R}\\left[-\\frac{7(1-\\beta )}{c_s}\\frac{dc_s}{dx}-\\frac{(1-\\beta )}{\\vartheta }\\frac{d\\vartheta }{dx}+(1-\\beta )\\left(-\\frac{3}{2x}+\\frac{1}{2F}\\frac{dF}{dx}\\right)\\right].$ Hence, by solving eqns.",
"(REF ) and (REF ), along with any two of eqns.",
"(REF ), (REF ), (REF ) simultaneously, we obtain $\\vartheta , c_s, \\lambda , \\beta $ (and then $\\gamma )$ as functions of $x$ from $x_t$ to $x_+$ ."
],
[
"Optically thin flows",
"In presence of bremsstrahlung radiation for optically thin flows, from eqns.",
"(REF ), (REF ) and (REF ) we obtain $R\\frac{d\\beta }{dx}+\\frac{(1-\\beta )}{c_s}\\frac{dc_s}{dx}+\\frac{(1-\\beta )}{\\rho }\\frac{d\\rho }{dx}=W,$ where $R=\\frac{1}{\\beta }-\\frac{1-\\beta }{2}\\left(\\frac{1}{\\beta }-\\frac{6}{(8-3\\beta )(6-3\\beta )}\\right),\\,\\,\\,\\,W=\\frac{1-\\beta }{2}\\left(\\frac{1}{F}\\frac{dF}{dx}- \\frac{1}{x}\\right).$ The remaining equations are same as in an optically thick flow, with $R$ given by eqn.",
"(REF ).",
"Subsequently, as in §2.1, eliminating $d\\rho /dx$ from eqn.",
"(REF ), we can obtain $d\\beta /dx$ in terms of $d\\vartheta /dx$ and $dc_s/dx$ and hence solve for $\\vartheta , c_s, \\lambda , \\beta $ (and then $\\gamma )$ ."
],
[
"Solution",
"First we discuss the solution for the optically thick flows which are geometrically slimmer.",
"Subsequently, we address the solution for the optically thin and geometrically thicker flows."
],
[
"Optically thick flows",
"We consider typical cases of accretion flow with $\\beta $ at $x_t$ being $0\\raisebox {-.4ex}{\\stackrel{<}{\\scriptstyle \\sim }}\\beta _t\\raisebox {-.4ex}{\\stackrel{<}{\\scriptstyle \\sim }}1$ ; extreme radiation dominated to extreme gas dominated flows at $x_t$ .",
"Note that the general criterion for thermal stability of the flow is $\\beta >0.4$ (Abramowicz et al.",
"1988).",
"We also assume that a part ($f$ ) of the energy dissipated in the infalling matter advects in.$Q_{\\rm avd}=Q^+-Q^-=f\\,Q^+$ , when $Q_{\\rm avd}$ , $Q^+$ and $Q^-$ respectively the amount of energy advected in, dissipated and radiated out..",
"However, note importantly that the value of $f$ is not arbitrary, rather should be chosen according to the nature of the flow; whether it is gas dominated or radiation dominated, optically thin or thick.",
"The flow consisting of significant radiation component should be radiatively more efficient and hence should have a small $f$ compared to a flow with a lesser content of radiation.",
"However, the flow with significant radiation around a rotating black hole might have a larger $f$ compared to its non-rotating counter part, as the radial velocity of infall matter is higher for a rotating black hole with a shorter infall time scale.",
"Figures REF a,b show how $\\gamma $ and $\\beta $ decrease as the flow advances from $x_t$ to $x_+$ .",
"When $\\beta _t\\sim 1$ , the flow is extremely gas dominated at $x_t$ and $\\gamma _t\\sim 5/3$ .",
"However, as the flow advances in, the gas flow starts becoming relativistic, rendering the increase of apparent content of radiation which decreases $\\beta $ .",
"At $x_+$ , $\\beta $ decreases by $\\sim 30\\%$ , while $\\gamma $ by $\\sim 9\\%$ .",
"However, as $\\beta _t$ and then $\\gamma _t$ decreases, their variation decreases as well.",
"For $\\beta _t\\sim 0.33$ , its variation during infall is only $\\sim 7\\%$ , while for a smaller $\\beta _t$ variation is negligible.",
"Therefore, for a gas dominated flow, the solutions of accretion flows, particularly for vertically integrated geometrically thin/slim disks, with a constant $\\beta $ and then $\\gamma $ as discussed in the existing literature, appear to be incorrect which need to be revised.",
"Figures REF c,d show how the variation of $\\beta $ affects the disk thermo-hydrodynamics, particularly for gas dominated flows.",
"In general, as $\\beta $ decreases at a large distance from the black hole, the content of radiation in the flow increases which leads to a smaller $\\vartheta $ therein.",
"This is because, the increase of radiation makes the flow to be radiatively more efficient, rendering the disk to be geometrically thinner and then more centrifugally dominated.",
"Hence, ignoring the variation of $\\beta $ , when $\\beta _t$ and $\\beta _c$ are high, would make the disk to be gas pressure dominated geometrically thick and quasi-spherical throughout which could reflect a wrong picture.",
"However, at the inner region of the disk, $\\vartheta $ profiles for all $\\beta $ merge.",
"This is because the flow therein is practically controlled by the black hole's gravity.",
"Interestingly, while the flow with a constant $\\beta $ ($=0.75$ ) would have a physical solution (dot-dashed line) for $\\lambda _c=3.1$ , the variation of $\\beta $ (with $\\beta _c=0.75$ ), when other initial parameters remain unchanged, leads to an unphysical solution (solid line) with an O-type critical point which does not continue from $x_t$ to $x_+$ .",
"However, if $\\lambda _c$ decreases slightly (to , e.g., $3.05$ ), then the flow with varying $\\beta $ again exhibits a physical solution (long-short-dashed line).",
"Figure REF e shows how the relation between $P$ , $\\rho $ and $c_s$ , i.e.",
"EOS, changes with the variation of $\\beta $ in the flow.",
"Clearly $dP/d\\rho \\sim P/\\rho $ (which is practically true for the present purposes) does not vary linearly with $c_s^2$ , unlike the cases modeled in the existing literature based on constant $\\gamma $ .",
"In Fig.",
"REF we compare the change of variation of $\\beta $ , $\\gamma $ and then corresponding $\\vartheta $ , with the change of $\\alpha $ .",
"It is found that the effect of variation of $\\beta $ on disk thermo-hydrodynamics is stronger for a lower viscosity.",
"If $\\beta _c$ is fixed, then $\\beta $ is higher and its variation is steeper for a lower $\\alpha $ at a large distance.",
"Higher the $\\beta $ , higher the $\\gamma $ in the flow is, inducing a quasi-spherical gas flow with a faster radial infall which may lead to an unstable/unphysical flow solution at a low $\\alpha $ .",
"In other words, a lower $\\alpha $ corresponds to a lower rate of energy-momentum transfer in the disk.",
"Hence, to describe a steady infall, $\\beta $ has to be larger, particularly at a large $x$ , which implies a quasi-spherical faster inflow of gas.",
"Therefore, the assumption of a constant $\\beta $ and then $\\gamma $ ($\\sim 1.533$ ) would reflect an incorrect solution of the disk flow, particularly for a gas dominated flow.",
"Figures REF c,d show that physically interesting solutions, exhibiting flows coming from $x_t$ to $x_+$ , require a smaller $\\lambda _c$ than that considered in the literature based on a constant $\\beta $ .",
"This supports quasi-spherical flows.",
"Figure REF e compares solutions with different sets of $\\beta _c$ and $\\alpha $ , and shows that the physical solution is achieved for a smaller $\\beta _c$ , when $\\alpha $ is lower.",
"From Fig.",
"REF we understand how the variation of Kerr parameter affects the solutions with varying $\\beta $ .",
"First we consider the maximum possible $\\lambda _c$ obtained with constant $\\beta $ (Rajesh & Mukhopadhyay 2010a) for the respective cases of $a$ .",
"Interestingly, while the physical solution around a co-rotating (prograde) black hole is available (solid line) with varying $\\beta $ for the same set of parameters as of constant $\\beta $ , $\\lambda _c$ has to be smaller for a counter-rotating (retrograde) black hole (dotted and dashed lines).",
"This is because a high $\\beta _t$ renders a quasi-spherical gas flow, which requires a smaller $\\lambda $ allowing the matter to fall inwards.",
"Note that while the maximum possible $\\lambda _c$ for a co-rotating black hole itself is smaller, for a counter-rotating black hole it is much larger.",
"Hence, the allowed range of $\\lambda _c$ is shrunk for a counter-rotating black hole compared to that inferred from the analysis based on a constant $\\beta $ .",
"For clarity, in Figs.",
"REF c,d we also show the $\\vartheta $ and $\\lambda $ profiles for $a=0.998$ and $-0.998$ (respectively dot-long-dashed and dashed-long-dashed lines) with a constant $\\beta $ for maximum possible values of $\\lambda _c$ .",
"Figure REF e shows how the EOS changes with the variation of $a$ , when $\\beta $ varies in the flow.",
"Clearly $P/\\rho $ does not vary linearly with $c_s^2$ , unlike the cases with constant $\\gamma $ .",
"Note that dashed and dashed-long-dashed lines practically overlap each other, implying that the solution with a constant $\\beta $ is recovered by decreasing $\\lambda _c$ in the case of varying $\\beta $ .",
"We have already seen in Fig.",
"REF that the variation of $\\beta $ is negligible for the radiation dominated flows.",
"This is because a small $\\beta $ corresponds to a radiatively efficient geometrically thinner flow, which effectively corresponds to a relativistic flow of radiation throughout.",
"Therefore, as the matter advances, the black hole's gravity has nothing to affect especially.",
"However, close to the black hole, $\\vartheta $ increases noticeably and the corresponding infall time scale of the matter decreases significantly, rendering a quasi-spherical flow which increases $\\beta $ by $4\\%$ , as shown in Fig.",
"REF .",
"Figure REF shows that the variation of $f$ keeping other parameters intact does not affect hydrodynamic properties, e.g.",
"Mach number, significantly.",
"However, as $f$ increases, the flow tends to become radiatively inefficient, rendering an increase in $\\beta $ significantly close to the black hole.",
"Therefore, any (observed) property related to the inner accretion flow must be influenced by $f$ and hence the variation of $\\beta $ ."
],
[
"Optically thin flows",
"There is a stronger variation of $\\beta $ and then $\\gamma $ in optically thin flows, compared to its optically thick counter parts.",
"As seen in Fig.",
"REF , while for a high $\\beta _t$ , the variation of $\\beta $ could be $\\sim 100\\%$ and of $\\gamma $ be $\\sim 15\\%$ , for a low $\\beta _t$ , the respective variations are much higher, $\\sim 300\\%$ and $\\sim 20\\%$ .",
"In all the cases depicted here, as $\\lambda $ decreases around $x_t$ (see Fig.",
"REF d), rendering the flow to be quasi-spherical, $\\beta $ increases as the flow advances in.",
"Subsequently, centrifugal force starts increasing, which renders the increase of $\\lambda $ and rate of cooling, which decreases $\\beta $ and $\\gamma $ .",
"However, close to the black hole, due to the dominance of gravitational force, matter falls very fast, rendering the flows, irrespective of $\\beta _t$ , to be quasi-spherical with high $\\beta $ .",
"Therefore, unlike our conventional thought that highly relativistic flow in the vicinity of a black hole may exhibit low $\\beta $ and then $\\gamma $ as seen in optically thick flows, in optically thin cases with bremsstrahlung radiation the situation is different.",
"This is because the inefficient/incomplete cooling process (Rajesh & Mukhopadhyay 2010a) keeps the flow to be very hot and then quasi-spherical until inner edge when the infall time is very short.",
"It is very clear from EOSs shown in Fig.",
"REF e that $P/\\rho $ does not vary linearly with $c_s^2$ , unlike the conventional cases.",
"Figure REF shows that the self-consistent inclusion of variation of $\\beta $ and then $\\gamma $ , for both the gas and radiation dominated flows, renders the unphysical solutions (solid lines), when the corresponding solutions with constant $\\beta $ and then $\\gamma $ (dotted lines) appear physical.",
"It is very clear that at around $x=50$ , the value of $\\gamma $ varies sharply with radius, rendering the flow to be gas dominated quasi-spherical with an unphysical O-type critical radius.",
"However, the flow with a smaller $\\lambda _c$ (dashed lines), keeping other parameters unchanged, exhibits the physical solution, even with varying $\\beta $ .",
"This is because the decrease of $\\lambda $ favors the quasi-spherical nature of gas dominated flow at around $x\\raisebox {-.4ex}{\\stackrel{>}{\\scriptstyle \\sim }}50$ , giving rise to a physical solution.",
"In general, the specific angular momentum of the flow, more specifically $\\lambda _c$ , is smaller in a realistic flow with varying $\\beta $ and $\\gamma $ than that with the idealistic choice of constant $\\beta $ and $\\gamma $ .",
"Figure REF shows that the variation of $f$ keeping other parameters intact neither affects hydrodynamic properties nor $\\beta $ profiles.",
"However, at a very high $f$ ($\\rightarrow 1$ , advection dominated flow), other parameters should not be kept same and $\\beta $ must be larger.",
"Such a flow is represented by long-dashed lines.",
"However, this still does not alter the Mach number profile significantly.",
"Note also that the flow with $\\beta =f=1$ strictly at a large distance does not exhibit any variation of $\\beta $ even at a smaller radius, which renders the corresponding $\\gamma =5/3$ , to be ratio of specific heats throughout.",
"Therefore a strict ADAF solution remains unaltered.",
"However, Narayan & Yi (1994) in their ADAF chose $\\gamma =1.5$ , while $\\gamma =5/3$ corresponds to a Bondi flow.",
"Moreover, for all the practical flows $\\beta , f <1$ and hence a variation of $\\beta $ should be accounted in the model, in particular to explain luminosity, as the present work intends to do.",
"Table: Parameter sets used for optically thick solutions: M=10M=10, m ˙\\dot{m} is the accretion ratein Eddington unitsTable: Parameter sets used for optically thin solutions: x c =5.8x_c=5.8, a=0a=0, α=0.01\\alpha =0.01(except for the case with β c =0.95\\beta _c=0.95, when α=0.001\\alpha =0.001), M=10M=10,m ˙\\dot{m} is the accretion rate in Eddington unitsFigure: Variation of (a) polytropic index,(b) ratio of gas pressure to total pressure,(c) Mach number, (d) ratio of specific angular momentum tothe corresponding Keplerian value.",
"In (a), (b), (c), (d) the solid, dotted, dashed, dot-long-dashed linescorresponding to the different β\\beta at transition radius are for β c =0.75,0.5,0.33,0.05\\beta _c=0.75,0.5,0.33,0.05 respectively.In (c) and (d) long-short-dashed and dot-dashed linescorrespond to β c =0.75\\beta _c=0.75, but respectively with a smaller λ c \\lambda _c (=3.05=3.05) than that of solid line(λ c =3.1\\lambda _c=3.1) and with a constant γ\\gamma with λ c =3.1\\lambda _c=3.1.",
"(e) The density-pressure-sound speed relation (EOS), whenthe profiles shown by solid and dotted lines are the corresponding EOS of the solutions shownin (c) and (d) by solid and dotted linesrespectively, while dashed line corresponds to a case with smaller λ c \\lambda _c than that of solid line.Other parameters are α=0.01\\alpha =0.01, f=0.5f=0.5, x c =5.8x_c=5.8, a=0a=0, M=10M=10.",
"See Table 1 for detail.Figure: Variation of (a) polytropic index,(b) ratio of gas pressure to total pressure,(c) Mach number, (d) ratio of specific angular momentum tothe corresponding Keplerian value, as functionsof radial coordinate.",
"In (a), (b), (c), (d) the solid and dotted lines are forα=0.01\\alpha =0.01 and 0.00010.0001 respectively when λ c =3.1\\lambda _c=3.1, and dashed and dot-long-dashed linesfor α=0.01\\alpha =0.01 and 0.00010.0001 respectively when λ c =3.05\\lambda _c=3.05.In (c) and (d) long-short-dashed and dot-dashed linesare corresponding solutions with constant γ\\gamma (∼1.533\\sim 1.533) for the cases of α=0.01\\alpha =0.01 andα=0.0001\\alpha =0.0001 respectively when λ c =3.1\\lambda _c=3.1.",
"(e) Mach number as a function of radial coordinate, when solid and dotted linesare for α=0.01\\alpha =0.01 and 0.00010.0001 respectively where β c =0.745\\beta _c=0.745, and dashedline is for α=0.0001\\alpha =0.0001 where β c =0.7\\beta _c=0.7.Other parameters are f=0.5f=0.5, x c =5.8x_c=5.8, a=0a=0, M=10M=10.",
"See Table 1 for detail.Figure: Variation of (a) polytropic index,(b) ratio of gas pressure to total pressure,(c) Mach number, (d) ratio of specific angular momentum tothe corresponding Keplerian value.",
"In (a), (b) the solid, dotted, dashed, dot-long-dashed linesrespectively correspond to parameter sets a=0.998,λ c =1.7a=0.998, \\lambda _c=1.7; a=-0.998,λ c =4a=-0.998, \\lambda _c=4;a=-0.998,λ c =3.95a=-0.998, \\lambda _c=3.95; a=0,λ c =3.05a=0, \\lambda _c=3.05.",
"In (c), (d) the solid, dotted, dashed,dot-long-dashed, dashed-long-dashed lines respectively correspond to parameter setsa=0.998,λ c =1.7a=0.998, \\lambda _c=1.7; a=-0.998,λ c =4a=-0.998, \\lambda _c=4; a=-0.998,λ c =3.95a=-0.998, \\lambda _c=3.95;a=0.998,λ c =1.7a=0.998, \\lambda _c=1.7, with constant β=0.75\\beta =0.75; a=-0.998,λ c =4a=-0.998, \\lambda _c=4, with constantβ=0.75\\beta =0.75.",
"The profiles shown by solid, dotted, dashed, dashed-long-dashedlines in (e) arethe corresponding density-pressure-sound speed relation (EOS)of the solutions shown by the same lines in (c) and (d).Other parameters are α=0.01\\alpha =0.01, f=0.5f=0.5, M=10M=10.",
"See Table 1 for detail.Figure: Variation of (a) polytropic index,(b) ratio of gas pressure to total pressure, as functions ofof radial coordinate.",
"The solid and dotted lines correspond to a=0.998,λ c =1.8a=0.998, \\lambda _c=1.8and a=0,λ c =3.2a=0, \\lambda _c=3.2 respectively.Other parameters are α=0.01\\alpha =0.01, f=0.5f=0.5, M=10M=10.",
"See Table 1 for detail.Figure: Variation of (a) Mach number, (b) ratio of gas pressure to total pressure, as functions ofof radial coordinate.",
"The solid, dotted, dashed, long-dashed lines correspond to flowswith f=0.1,0.3,0.5,0.9f=0.1,0.3,0.5,0.9 respectively, when β c =0.33\\beta _c=0.33.See Table 1 for detail.Figure: Variation of (a) polytropic index,(b) ratio of gas pressure to total pressure,(c) Mach number, (d) ratio of specific angular momentum tothe corresponding Keplerian value, (e) the density-pressure-sound speed relation (EOS).The solid, dotted, dashed, dot-long-dashed linescorresponding to the different β\\beta at transition radius are for β c =0.75,0.5,0.33,0.05\\beta _c=0.75,0.5,0.33,0.05 respectively.Other parameters are α=0.01\\alpha =0.01, f=0.5f=0.5, x c =5.8x_c=5.8, a=0a=0, M=10M=10.",
"See Table 2 for detail.Figure: Variation of (a) polytropic index, (b) Mach number, as functions of radial coordinate.The solid, dotted and dashed lines correspond to the flow with β c =0.5\\beta _c=0.5, λ c =3.22\\lambda _c=3.22;constant β\\beta , λ c =3.22\\lambda _c=3.22; and β c =0.5\\beta _c=0.5, λ c =3.2\\lambda _c=3.2 respectively.",
"(c) and (d) are same as in (a) and (b) respectively, when the solid, dotted and dashed linescorrespond to the flow with β c =0.05\\beta _c=0.05, λ c =3.27\\lambda _c=3.27;constant β\\beta , λ c =3.27\\lambda _c=3.27; and β c =0.05\\beta _c=0.05, λ c =3.25\\lambda _c=3.25 respectively.",
"Otherparameters are α=0.01\\alpha =0.01, f=0.5f=0.5, x c =5.8x_c=5.8, a=0a=0, M=10M=10.",
"See Table 2 for details.Figure: Variation of (a) Mach number, (b) ratio of gas pressure to total pressure, as functions ofof radial coordinate.",
"The solid, dotted, dashed lines correspond to flowswith f=0.1,0.5,0.8f=0.1,0.5,0.8 respectively when β c =0.75\\beta _c=0.75, and the long-dashed linecorresponds to the flow with f=0.98f=0.98 when β c =0.95\\beta _c=0.95.See Table 2 for detail."
],
[
"Summary",
"We have investigated the sub-Keplerian general advective accretion flows (GAAF; Rajesh & Mukhopadhyay 2010a) around black holes allowing evolution of the gas and radiation content into flows self-consistently.",
"We have, in one hand, considered radiation to be optically thick blackbody in the geometrically thin/slim vertically integrated disks.",
"On the other hand, we have also considered optically thin geometrically thicker flows in the assumption of bremsstrahlung radiation.",
"Hence, as matter advances towards a black hole, the ratio of the gas pressure to total pressure and the corresponding polytropic index vary significantly with the radial coordinate.",
"We have found that in several occasions, the accretion solutions as obtained with a constant $\\beta $ by the earlier models do not exist when $\\beta $ is allowed to vary self-consistently.",
"It has a very important implication when most of the existing models of accretion disks assume $\\beta $ and $\\gamma $ to be constant throughout the flow.",
"For example, an advection dominated class of solution, namely ADAF (Narayan & Yi 1994, 1995; Quataert & Narayan 1999), has been modeled based on gas dominated flows with a constant $\\gamma $ ($=1.5$ ) throughout, when $\\gamma $ is chosen to be ratio of specific heats.",
"Other models (e.g.",
"Abramowicz et al.",
"1995; Chakrabarti 1996; Rajesh & Mukhopadhyay 2010a), describing gas dominated optically thin flows, as well have assumed $\\gamma $ , which is polytropic index therein, to be constant.",
"In the present work, the impact of varying $\\gamma $ has been particularly demonstrated in the framework of later model.",
"Note, however, that polytropic index turns out to be a ratio of specific heats only when the flow is of pure gas with $\\gamma =5/3$ .",
"The results argue for correction to the earlier models based on constant $\\beta $ , when particularly $\\beta $ varies more than $100\\%$ during the infall of an optically thin gas dominated matter.",
"Hence, the present results imply that the entire branch of optically thin hot solutions of accretion disk needs to be re-investigated including appropriate variation of $\\gamma $ and $\\beta $ .",
"However, for simplicity we have considered the optically thin radiation arising due to bremsstrahlung mechanism only, neglecting synchrotron, inverse-Comptonization which are likely to operate in the inner hot regime of the flow.",
"However, inclusion of such effects would only favor the present results.",
"In future, we have plan to include such physics in studying very hot flows.",
"This will have immediate influence to the computed spectra (e.g.",
"Yuan et al.",
"2003; Mandal & Chakrabarti 2005) and subsequent inferences about sources."
],
[
"Acknowledgments",
"This work is partly supported by a project, Grant No.",
"SR/S2HEP12/2007, funded by DST, India.",
"One of the authors (PD) thanks the KVPY, DST, India, for providing a fellowship."
]
] | 1204.1274 |
[
[
"Price Jump Prediction in Limit Order Book"
],
[
"Abstract A limit order book provides information on available limit order prices and their volumes.",
"Based on these quantities, we give an empirical result on the relationship between the bid-ask liquidity balance and trade sign and we show that liquidity balance on best bid/best ask is quite informative for predicting the future market order's direction.",
"Moreover, we define price jump as a sell (buy) market order arrival which is executed at a price which is smaller (larger) than the best bid (best ask) price at the moment just after the precedent market order arrival.",
"Features are then extracted related to limit order volumes, limit order price gaps, market order information and limit order event information.",
"Logistic regression is applied to predict the price jump from the limit order book's feature.",
"LASSO logistic regression is introduced to help us make variable selection from which we are capable to highlight the importance of different features in predicting the future price jump.",
"In order to get rid of the intraday data seasonality, the analysis is based on two separated datasets: morning dataset and afternoon dataset.",
"Based on an analysis on forty largest French stocks of CAC40, we find that trade sign and market order size as well as the liquidity on the best bid (best ask) are consistently informative for predicting the incoming price jump."
],
[
"Introduction",
"The determination of jumps in financial time series already has a long history as a challenging, theoretically interesting and practically important problem.",
"Be it from the point of view of the statistician trying to separate, in spot prices, those moves corresponding to \"jumps\" from those who are compatible with the hypothesis of a process with continuous paths, or from the point of view of the practitioner: market maker, algorithmic trader, arbitrageur, who is in dire need of knowing the direction and the amplitude of the next price change, there is a vast, still unsatisfied interest for this question.",
"Several attempts have been made at theorizing the observability of the difference between processes with continuous or discontinuous paths, and the major breakthrough in that direction is probably due to Barndorff-Nielsen and Shephard [3], who introduced the concept of bi-power variation, and showed that - in a nutshell - the occurrence of jumps can be seen in the limiting behavior as the time step goes to zero of the bi-power variation: for a process with continuous paths, this quantity should converge to (a multiple of) the instantaneous variance, and the existence of a possibly different limit will be caused by the occurrence of jumps.",
"Since then, many authors, in particular Aït-Sahalia and Jacod (2009) [2] have contributed to shed a better light on this phenomenon, and one can safely say that rigorous statistical tests for identifying continuous-time, real-valued processes with discontinuous paths are now available to the academic community as well as the applied researcher.",
"However, it is a fact that the physics of modern, electronic, order-driven markets is not easily recast in the setting of real-valued, continuous-time processes, and it is also a fact that the time series of price, no matter how high the sampling frequency, is not anymore the most complete and accurate type of information one can get from the huge set of financial data at our hands.",
"In fact, a relatively recent trend of studies has emerged over the past 10 years, where the limit order book became the center of interest, and the price changes are but a by-product of the more complicated set of changes on limit orders, market orders, cancellation of orders, ... see e.g.",
"Chakraborti et al.",
"(2009) [6], Abergel et al.",
"(2011) [1] for the latest developments in the econophysics of order-driven markets.",
"This new standpoint is quite enlightening, in that the physics of price formation becomes much more apparent, but it calls for a drastic change in the basic modeling tools: prices now live on a discrete grid with a step size given by the tick, the changes in price occur at discrete times.",
"Furthermore, a host of important events that affect the order book rather than the price itself, events which are therefore essential in understanding the driving forces of the price changes, now become observable, and their role in the price dynamics must be taken into account when one is interested in understanding the latter.",
"Our point of view is slightly different: rather than concentrate on the one-dimensional price time series, we want to model the dynamics in event time of the whole order book, and focus on some specific events that can be interpreted in terms of jumps.",
"To do so, we shall depart from the classical definitions - if any such thing exists - of a jump in a financial time series, and restrict ourselves to the more natural, more realistic and also more prone to experimental validation, concept of a inter-trade price jump and trade-through.",
"By definition, an inter-trade price jump is defined as an event where a market order is executed at a price which is smaller (larger) than the best limit price on the Bid (Ask) just after the precedent market order arrival.",
"An inter-trade price jump permits a limit order submitted at the best bid (best ask) just after a market order arrival to be surely executed by the next market order arrival.",
"A trade-through corresponds to the arrival of a new market order, the size of which is larger than the quantity available at the best limit on the Bid (for a sell order) or Ask (for a buy order) side of the order book.",
"By nature, such an order will imply an automatic and instantaneous price change, the value of which will be exactly the difference in monetary units between the best limit price before and after transaction on the relevant side of the order book.",
"Trade-through can be interpreted as the instantaneous price change triggered by a market order, meanwhile, inter-trade price jump is post-trade market impact.",
"Most of researches on limit order book are based on stocks and often relates to characterizing features such as liquidity, volatility and bid-ask spread instead of making prediction, see Hasbrouck (1991) [10], Hausman et al.",
"(1992) [12], Keim and Madhavan (1996) [14], Lo et al.",
"(2002) [17], Lillo et al.",
"(2003) [15], Hasbrouck (2006) [13], Parlour and Seppi (2008) [18] and Jondeau et al.",
"(2008) and Linnainmaa and Rosu (2009) [16].",
"Trade-through has also been the object of several recent studies in the econometrics and finance literature, see e.g.",
"Foucault and Menkveld (2008) [8](for cross-sectional relationship study) and Pomponio and Abergel (2011) [19].",
"In this work, we investigate whether the order book shape is informative for the inter-trade price jump prediction and whether trade-through contributes to this prediction.",
"Recently, many researchers propose machine learning methods to make prediction on limit order book.",
"Blazejewski and Coggins (2005) [4] present a non-parametric model for trade sign (market order initiator) inference and they show that limit order book shape and historical trades size are informative for the trade sign prediction.",
"Fletcher et al.",
"(2010) [7] applied multi kernel learning with support vector machine in predicting the EURUSD price evolution from the limit order book information.",
"Here, logistic regression is introduced to predict the occurrence of inter-trade price jump.",
"Variable selection by lasso logistic regression provides us an insight into the dynamics of limit order book and allows us to select the most informative features for predicting relevant events.",
"We will show that some features of the limit order book have strong predictive and explanatory power, allowing one to make a sound prediction of the occurrence of inter-trade price jump knowing the state of the limit order book.",
"Trade-through is also confirmed to be quite informative for inter-trade price jump prediction.",
"This result in itself is interesting in that it allows one to use the full set of available information in order to do some prediction: whereas the history of the price itself is known not to be a good predictor of the next price moves - the so-called efficiency of the market is relatively hard to beat when one only uses the price information - we shall show that the limit order volumes contain more information, and the market order size contributes also to an accurate prediction of inter-trade price jump.",
"This paper is organized as follows.",
"Section describes the main notations in limit order book.",
"Section gives an empirical result on the relationship between Bid-Ask liquidity balance and trade sign.",
"Section introduces logistic regression for inter-trade price jump prediction and lasso logistic regression for variable selection.",
"The conclusion is in Section ."
],
[
"Description and data notation",
"The Euronext market adopts NSC (Nouveau Système de Cotation) for electronic trading.",
"During continuous trading from $9h00$ to $17h30$ , NSC matches market orders against the best limit order on the opposite side.",
"Various order types are accepted in NSC such as limit orders (an order to be traded at a fixed price with certain amount), market orders (order execution without price constraint), stop orders (issuing limit orders or market orders when a triggered price is reached) and iceberg orders (only a part of the size is visible in the book).",
"Limit order is posted to electronic trading system and they are placed into the book according to their prices, see Figure REF .",
"Market order is an order to be executed at the best available price in limit order book.",
"The lowest price of limit sell orders is called Best Ask; the highest price of limit buy orders is called Best Bid.",
"The gap between the Best Bid and the Best Ask is called the Spread.",
"When a market buy order with price higher/equal than the best ask price, a trade occurs and the limit order book is updated accordingly.",
"Limit orders can also be cancelled if there have not been executed, so the limit order book can be modified due to limit order cancellation, limit order arrival or market order arrival.",
"In case of iceberg orders, the disclosed part has the same priority as a regular of limit order while the hidden part has lower priority.",
"The hidden part will become visible as soon as the disclosed part is executed.",
"The case that the hidden part is consumed by a market order without being visible before is quite rarely.",
"In this study, we neglect stop orders and iceberg orders which are relatively rare compared to limit order and market order events.",
"In a limit order book, as shown in figure $~\\ref {LOB}$ , only a certain number of best buy/sell limit orders are available for public.",
"We denote the number of available bid/ask limit prices by $L$ .",
"Figure: Limit Order Book description.",
"Limit order price is discretized by tick price.In this study, for simplicity, we focus on limit order arrival events, limit order cancellation events and market order arrival events, see Figure REF .",
"The number of visible limit order levels is chosen to be five $L = 5$ .",
"Our dataset is provided by NATIXIS via Thomson Reuter's `Reuters Tick Capture Engine' and comprises of trades and limit order activities of the 40 member stocks of index $CAC40$ between April 1st 2011 and April 30th 2011.",
"In order to get rid of open hour and close hour, we extract the data from $09h05$ to $17h25$ .",
"Every transaction and every limit order book modification are recorded in milliseconds.",
"The data contains information on the $L$ best quotes on both bid and ask sides.",
"The trade data and quotes data are matched.",
"Figure: Dynamics of limit order book.",
"The first event is a trade-through event where a market order consumes 60 stocks at the bid side, then a new ask limit order of size 20 arrives in the Spread.",
"Successively, a cancellation at the best bid price follows and the precedent second best bid price emerges the best bid price, the a regular market order triggers a transaction of size 60.Denote t as a time index indicating all limit order book events.",
"$P_t^{b, i}$ and $P_t^{a, i}$ for $i = 1, \\cdots , L$ define the $i^{th}$ best log bid/ask quote instantaneously after the $t^{th}$ event.",
"We denote $S_t = P_t^{a, 1} - P_t^{b, 1}$ the spread instantaneously after the $t^{th}$ event.",
"$G_t^{b, i} = P_t^{b, i} - P_t^{b, i+1}$ , $G_t^{a, i} = P_t^{a, i+1} - P_t^{a, i}$ for $i = 1, \\cdots , L-1$ define respectively the $i^{th}$ best bid(ask) limit price gap instantaneously after the $t^{th}$ event.",
"Besides, $V_t^{b, i}$ and $V_t^{a, i}$ for $i = 1, \\cdots , L$ denote the log limit order volume on the $i^{th}$ best bid/ask quote instantaneously after the $t^{th}$ event.",
"The volume of trade is denoted by $V^{mo}_t$ ($V^{mo}_t = 0$ when there is no trade) and the price of trade is denoted by $P^{mo}_t$ ($P^{mo}_t = 0$ when there is no trade, $P^{mo}_t = P_{t}^{b, 1}$ when a market order touches bid side and $P^{mo}_t = P_{t}^{a, 1}$ when a market order touches ask side).",
"Moreover, we introduce six dummy variables $BLO_t$ , $ALO_t$ , $BMO_t$ , $AMO_t$ , $BTT_t$ and $ATT_t$ to indicate the direction of each event : bid side or ask side, respectively for limit order event ($BLO_t$ and $ALO_t$ ), market order event ($BMO_t$ and $AMO_t$ ) and trade-through event ($BTT_t$ and $ATT_t$ ).",
"The definition of variables is detailed in Table REF .",
"In order to capture the high-frequency dynamics in quotes and depths, we define a $K$ -dimensional vector $\\mathbf {R}^1_t = [G_t^{b, L-1}, \\cdots , G_t^{b, 1}, S_t, G_t^{a, 1}, \\cdots , G_t^{a, L-1}, V_t^{b, L}, \\cdots , V_t^{b, 1}, V_t^{a, 1}, \\cdots , V_t^{a, L}]\\;.$ Modelling log prices and log volumes instead of absolute values is suggested by Potters and Bouchaud (2003) [20] studying the statistical properties of market impacts and trades and can be found in many other empirical studies.",
"Price and volume changes in log is interpreted as related changes in percentage.",
"Another vector of variables is denoted by $\\mathbf {R}^2_t = [BMO_t, AMO_t, BLO_t, ALO_t, BTT_t, ATT_t]\\;,$ indicating the nature of the $t^{th}$ event.",
"Table REF provides a descriptive statistics of the data used in this paper.",
"It comprises limit order events, market order events and inter-trade price jump events.",
"The analysis is done on two separated datasets: morning dataset (between $09h05$ and $13h15$ ) and afternoon dataset (between $13h15$ and $17h25$ ).",
"We observe that there are more market order events in the afternoon than in the morning.",
"Similarly, inter-trade price jump events are slightly more frequent in the afternoon than in the morning.",
"However, trade-through events are more frequent in the morning than in the afternoon.",
"Table: Variable definitionsTable: Summary of limit order events, market order events and inter-trade price jump events, CAC40 stocks, April, 2011."
],
[
"Empirical facts : Bid-Ask liquidity balance and trade sign",
"Before making an analysis on price jump prediction, we try to reveal whether the limit order volume information plays a role in determining the future market order's direction (trade sign).",
"In order to study the conditional probability given the knowledge about bid/ask limit order liquidity, we propose a Bid-Ask volume ratio corresponding to depth $i$ just before the $k^{th}$ trade, which is defined as $W_{t_k - 1}(i)$ ($i \\in \\lbrace 1, \\dots , L\\rbrace $ ), more precisely, $W_{t_k - 1}(i) & = \\log \\left(\\frac{\\sum _{j=1}^i \\exp (V^{b,j}_{t_k - 1})}{\\sum _{j=1}^i \\exp (V^{a, j}_{t_k - 1})}\\right) \\nonumber \\\\& = \\log \\left(\\sum _{j=1}^i \\exp (V^{b,j}_{t_k - 1})\\right) - \\log \\left(\\sum _{j=1}^i \\exp (V^{a, j}_{t_k - 1})\\right) \\;,$ where $t_k$ is time index of the $k^{th}$ market order event.",
"For all $x \\in \\mathbb {R}_+$ , the conditional probability of a future buy market order (positive trade sign) that the next trade is triggered by a buy market order given $V_{t_k}(i) \\ge x$ is defined as, $& \\mathbb {P}\\left(I^{ts}_{t_{k}}=1 | W_{t_k - 1}(i) \\ge x \\right) \\;.$ where we denote the TradeSign at time $t_k$ by $I^{ts}_{t_k}$ .",
"Similarly, for all $x \\in \\mathbb {R}_+$ , the conditional probability of a future sell market order (negative trade sign) that the next trade is triggered by a sell market order given $V_{t_k}(i) \\ge x$ is defined as, $& \\mathbb {P}\\left(I^{ts}_{t_k}=-1 | W_{t_k - 1}(i) \\le x \\right) \\;.$ Figure: The conditional probability of a buy market order vs bid-ask volume ratio, April, 2011.Figure: The conditional probability of a sell market order vs bid-ask volume ratio, April, 2011.The relationship between $\\mathbb {P}\\left(I^{ts}_{t_k}=1 | W_{t_k - 1}(i) \\ge x \\right)$ and $x$ for $i \\in \\lbrace 1, \\dots , L\\rbrace $ is shown in Figure REF and REF .",
"We observe that the conditional probability of the next trade sign is highly correlated with the Bid-Ask volume ratio corresponding to depth 1.",
"Nevertheless, the dependance between the conditional probability of the next trade sign and the Bid-Ask volume ratio corresponding to depth larger than 1 is much more noised.",
"Figure REF in Appendix shows the relationship between $\\mathbb {P}\\left(I^{ts}_{t_k}=1 | W_{t_k - 1}(1) \\ge x \\right)$ and $x$ for all stocks of CAC40.",
"It is worth remarking that the trade sign's conditional probability reaches $0.80$ in average when the liquidity on the best limit prices is quite unbalanced."
],
[
"Logistic regression analysis",
"The result shown in the previous section reveals that Bid-Ask liquidity balance provides important information on the incoming market order.",
"In this section, we introduce the standard logistic regression to predict the inter-trade price jump occurrence and use $LASSO$ select regularization to evidence the importance of each variable in this prediction.",
"We denote the number of market order events by $N$ and for each $i \\in \\lbrace 1, \\cdots , N\\rbrace $ , $\\mathbf {X}_i=[1, V^{mo}_{t_i}, \\mathbf {R}^1_{t_i}, \\mathbf {R}^1_{t_i - 1}, \\cdots , \\mathbf {R}^1_{t_{i}-m+1}, \\mathbf {R}^2_t, \\mathbf {R}^2_{t_i-1}, \\cdots , \\mathbf {R}^2_{t_{i}-n+1}]$ ($X \\in \\mathbb {R}^{(p+1) \\times 1}$ , $p = m(2L-1) + 6n$ ) the explanatory variables summarizing the available order book information when the $t^{th}$ event is a market order event, $y_i$ is a binary variable indicating whether the event is an bid/ask inter-trade price jump, $y_i$ is defined as follows, $\\text{Bid side inter-trade price jump indicator: } Y_i ={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } P_{t_{i+1}}^{mo} < P_{t_i}^{b, 1} \\\\0, & \\text{ otherwise}\\end{array}\\right.",
"}$ or $\\text{Ask side inter-trade price jump indicator: } Y_i ={\\left\\lbrace \\begin{array}{ll}1, & \\text{if } P_{t_{i+1}}^{mo} > P_{t_i}^{a, 1} \\\\0, & \\text{ otherwise}\\end{array}\\right.",
"}$ In the logistic model, the probability of the bid/ask inter-trade price jump occurrence is assumed to be given by: $\\log {\\frac{\\mathrm {P_{\\beta }}(Y=1|\\mathbf {X})}{1-\\mathrm {P_{\\beta }}(Y=1|\\mathbf {X})}}=\\beta ^{T}\\mathbf {X}\\;,$ where $\\beta =[\\beta _0, \\beta _1, \\cdots , \\beta _p]^T$ .",
"Observing that for $i = 1$ $& W_{t_k-1}(i)= V^{b,1}_{t_k-1} - V^{a, 1}_{t_k-1} \\;,$ we see that the linearity of the conditional probability $\\mathrm {P_{\\beta }}(Y=1|\\mathbf {X})$ on variables $V^{b,1}_{t_k}$ and $V^{a, 1}_{t_k}$ in Equation REF allows us to capture the contribution of $W_{t_k}(i)$ in the prediction.",
"The parameters $\\beta $ are unknown and should be estimated from the data.",
"We use the maximum likelihood to estimate the parameters.",
"It is well known that the log-likelihood function given by $\\mathcal {L}(\\beta )=\\sum _{i=1}^{N}{\\lbrace \\log (1+e^{\\beta ^T \\mathbf {X}_i})-y_i\\beta ^T \\mathbf {X}_i\\rbrace }\\;.$ The likelihood function is convex and therefore can be optimized using a standard optimization method."
],
[
"Variable selection by LASSO",
"Since the number of explanatory variables $p$ being quite large, it is of interest to perform a variable selection procedure to select the most important variables.",
"A classical variable selection procedure when the number of regressors is large is the LASSO procedure see Hastie et al (2003) [11].",
"Instead of using a $BIC$ penalization, the LASSO procedure adds to the likelihood the norm of the logistic coefficient, which is known to induce a sparse solution.",
"This penalization induces an automatic variable selection effect.",
"The LASSO estimate for logistic regression is defined by $\\hat{\\beta }^{lasso}(\\lambda ) &=& \\underset{\\beta }{\\mathrm {argmin}} \\sum _{i=1}^{N}\\left(\\left(-\\log (1+e^{\\beta ^T \\mathbf {X}_i})+ Y_i\\beta ^T \\mathbf {X}_i\\right) + \\lambda \\sum _{j=1}^{p}{|\\beta _j|}\\right)\\;.$ The constraint on $\\sum _{j=1}^{p}{|\\beta _j|}$ makes the solutions nonlinear in the $y_i$ and there is no closed form expression as in ridge regression.",
"Because of the nature of constraint, making $\\lambda $ sufficiently large will cause some of the coefficients to be exactly zero.",
"Germain et Roueff (2009) [9] gives the uniform consistency and a functional central limit theorem for the LASSO regularization path for the general linear model."
],
[
"Results",
"Choosing $L = 5$ , $m=5$ and $n=5$ , the dimension of limit order book's profile vector is $p=1 + m(2L-1) + 6n = 76$ .",
"The parameter $\\lambda $ in LASSO is estimated by cross-validation, then we calculate AUC value (area under ROC curve) to measure the prediction quality.",
"A ROC (receiver operating characteristic) curve is a graphical plot of the true positive rate vs. false positive rate.",
"The area under the ROC curve is a good measure to measuring the model prediction quality.",
"The AUC value is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative one.",
"We show the out-of-sample AUC value in Figure REF .",
"The stocks are sorted in alphabetic order.",
"We see that for each prediction task, the AUC value is around $0.80$ and it is consistently high over all datasets and all stocks of $CAC40$ .",
"Figure: AUC value, price jump prediction, CAC40, April, 2011.In order to discover the contribution of each variable to the prediction, we add an analysis on the five firstly selected variables for each prediction task of all stocks of $CAC40$ (with allday dataset).",
"Figure REF and REF show how many times a variable is selected as the first (second, third, forth, fifth) selected variable by $LASSO$ .",
"We denote the $i$ events lagged log volume on the $j^{th}$ bid (ask) limit price by $VBi_{j}$ ($VAj_{i}$ ).",
"Similarly, $i$ events lagged log market order volume is denoted by $VMO_i$ and $i$ events lagged binary variables are denoted by $BMO_i$ , $AMO_i$ , $BTT_i$ , $ATT_i$ etc.",
"For the sake of simplicity, for each selection order $i$ ($i \\in \\lbrace 1, \\dots , 5\\rbrace $ ), we show the frequency distribution of the five most frequently selected variables among 746 backtests in each figure.",
"We observe that $VB1_0$ , $BMO_0$ and $VMO_0$ are the most selected variables for predicting the future bidside inter-trade price jump and that $VA1_0$ , $AMO_0$ and $VMO_0$ are the most selected variables for predicting the future askside inter-trade price jump.",
"In contrast, trade-through is less informative and contributes few to the price jump prediction.",
"It implies that the market order is sensitive to the liquidity on the best limit price.",
"As soon as the liquidity on the best limit price becomes significantly low, the next market order may touch it immediately.",
"The information provided by $BMO_0$ ($AMO_0$ ) and $VMO_0$ recalls the phenomena of long memory of order flow, see Bouchaud et al.",
"(2008) [5].",
"When a trader tries to buy or sell a large quantity of assets, he may split it into small pieces and execute them by market order successively.",
"Consequently, precedent market order direction contributes to predict the next market order event.",
"Figure: Variable selection for BidJump prediction, CAC40, April, 2011.",
"From left to right, from top to bottom, each figure shows how many times a variable is selected as the k th k^{th} selected variable by LASSOLASSO, k={1,⋯,5}k=\\lbrace 1, \\dots , 5\\rbrace .Figure: Variable selection for AskJump prediction, CAC40, April, 2011.",
"From left to right, from top to bottom, each figure shows how many times a variable is selected as the k th k^{th} selected variable by LASSOLASSO, k={1,⋯,5}k=\\lbrace 1, \\dots , 5\\rbrace ."
],
[
"Conclusion",
"In this paper, we provide an empirical result on the relationship between bid-ask limit order liquidity balance and trade sign and an analysis on the prediction of the inter-trade price jump occurrence by logistic regression.",
"We show that limit order liquidity balance on best bid/best ask is informative to predict the next market order's direction.",
"We then use limit order volumes, limit order price gaps and market order size to construct limit order book's feature for the prediction of inter-trade price jump occurrence.",
"LASSO logistic regression is introduced to help us identify the most informative limit order book features for the prediction.",
"Numerical analysis is done on two separated datasets : morning dataset and afternoon dataset.",
"LASSO logistic regression gets very good prediction results in terms of AUC value.",
"The AUC value is consistently high on both datasets and all stocks whatever the liquidity is.",
"This good prediction quality implies that limit order book profile is quite informative for predicting the incoming market order event.",
"The variable selection by LASSO logistic regression shows that several variables are quite informative for inter-trade price jump prediction.",
"The trade sign and market order size and the liquidity on the best limit prices are the most informative variables.",
"Nevertheless, the aggressiveness of market order, measured by trade-through, has less important impact than we had expected.",
"These results confirm that the limit order book is quite sensitive to the liquidity on the best limit prices and there is a long memory of order flow like what is shown by other authors.",
"This paper is merely a first attempt to discover the information hidden in limit order book and further studies will be needed to understand better the full dynamics of limit order book."
]
] | 1204.1381 |
[
[
"The Kilometer-Sized Main Belt Asteroid Population as Revealed by Spitzer"
],
[
"Abstract Multi-epoch Spitzer Space Telescope 24 micron data is utilized from the MIPSGAL and Taurus Legacy surveys to detect asteroids based on their relative motion.",
"These infrared detections are matched to known asteroids and rotationally averaged diameters and albedos are derived using the Near Earth Asteroid Model (NEATM) in conjunction with Monte Carlo simulations for 1835 asteroids ranging in size from 0.2 to 143.6 km.",
"A small subsample of these objects was also detected by IRAS or MSX and the single wavelength albedo and diameter fits derived from this data are within 5% of the IRAS and/or MSX derived albedos and diameters demonstrating the robustness of our technique.",
"The mean geometric albedo of the small main belt asteroids in this sample is p_V = 0.138 with a sample standard deviation of 0.105.",
"The albedo distribution of this sample is far more diverse than the IRAS or MSX samples.",
"The cumulative size-frequency distribution of asteroids in the main belt at small diameters is directly derived.",
"Completeness limits of the optical and infrared surveys are discussed."
],
[
"INTRODUCTION",
"Planetesimals are increasingly recognized as the evolutionary lynch pins for models of planet formation within the solar system.",
"Their demographics, compositions, and dynamical attributes are imprints of our circumstellar ecosystem extant at the epoch of planet building that likely reflect the general conditions in exo-planetary disks.",
"From the study of asteroids as relics of the early period of planet building, insight can be gained into the accretion processes and the initial composition of the proto-planetary disk.",
"Previous asteroid surveys performed with IRAS [53] and MSX [54] enabled estimates of the albedo and diameter distributions of large main belt asteroids (MBAs).",
"These surveys were flux limited to an asteroid diameter threshold of $\\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}10$ km; however, they still produced albedo and diameter estimates for $\\simeq 2000$ asteroids.",
"Recently, the NEOWISE survey [37] has released a preliminary catalog of albedos and diameters for $\\simeq 10^{6}$ MBAs.",
"This ensemble provided critical observational constraints for collisional models used to follow the evolution of planetesimals over the lifetime of the solar system [1].",
"These models use mean albedos for asteroids and their optical absolute magnitudes to generate the current day size-frequency distribution of asteroids.",
"However, uncertainty exists regarding whether or not there is a tight and narrowly defined correlation between the albedos and diameters of asteroids.",
"For instance, the IRAS survey suggests that the range of asteroid albedos becomes more diverse with decreasing diameter.",
"Compositional studies of main belt asteroids are utilized to explore whether or not our proto-planetary disk was contaminated by supernova products such as $^{26}$ Al.",
"Early compositional studies [18] suggested evidence of a compositional gradient as a function of semi-major axis in the main belt - from highly thermally altered compositions in the inner belt to non-thermally altered compositions in the outer main belt.",
"This gradient was attributed to parent body melting due to heating by the decay of radioactive isotopes [19], [38], and many models invoking this mechanism produced significant numbers of small thermally unaltered bodies in the inner main belt with diameters less than 20 km.",
"However, this population has yet to be observed.",
"For instance, the IRAS survey is incomplete for asteroids $< 20$ km at any zone in the main belt.",
"Possibly, these small bodies were destroyed via mutual collisions [10], yet recent analysis of the Sloan Moving Object Catalog [4] indicates that many small ($\\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}10$ km) dark asteroids were missed in prior asteroid surveys.",
"In addition the SMOC data indicate that colors of small main belt asteroids display significant compositional diversity as a function of semi-major axis, rather than the zoning present in the large asteroid population.",
"The unique and unparalleled $\\mu $ Jy point-source flux density sensitivity of the Spitzer Space Telescope during the cryogenic mission has enabled detection of faint asteroids with diameters as small as $\\simeq 1$ km at high signal to noise in both targeted surveys and serendipitous fields along the ecliptic.",
"Here we utilize data from the MIPSGAL Galactic Plane survey and the Taurus Molecular Cloud survey to investigate the albedo behaviour of small asteroids, with a specific objective to determine whether or not the small ($\\simeq 1$ km) small main belt asteroids have the same mean albedo and spatial albedo distribution as the large ($\\ge 10$ km) main belt asteroids populations detected in earlier IRAS and MSX surveys.",
"We use derived diameters from our MIPSGAL and Taurus catalogs to establish the size-frequency distribution of small main belt asteroids, and to assess whether the size-frequency distribution is functionally dependent on the heliocentric distance and/or composition.",
"In section , we briefly describe the mid-infrared (IR) surveys that were data-mined from the Spitzer archive to produce our asteroid catalog.",
"Section discusses our approach to deriving asteroid albedos and diameters, while Section discusses our thermal modeling results, survey completeness limits, comparisons to prior IRAS albedo catalogs of MBAs, as well as an examination of main belt albedo gradients, dynamical family albedos within the main belt and the overall bulk size-frequency distribution of asteroids.",
"We conclude in Section ."
],
[
"MIPSGAL AND TAURUS SURVEYS ",
"The two Spitzer surveys studied in this paper were selected via three criteria: multi-epoch 24 data taken with epoch separations on the scale of hours at ecliptic latitudes $\\le 20$ .",
"More than 95% of all known main belt asteroids are found at inclinations $\\le 20$ , and studies have shown that the number counts of asteroids drop off by a factor of two from ecliptic latitudes of 0 to latitudes of 5 to 10 in the IR [55], [51].",
"In order to detect the smallest, and thus faintest, asteroids in the Spitzer data, multiple epochs were required such that images from two epochs could be subtracted to remove fixed objects and allow for multiple detections of a single asteroid in an image pair.",
"To properly derive diameters and albedos from thermal data, 24 fluxes are required as these fluxes are neither contaminated by reflected solar flux [51], [41], nor on the Wien side of the asteroid spectral energy distribution (SED) where thermal fitting errors are highest [50], [23].",
"Two Spitzer surveys fulfilled these requirements – the MIPSGAL and the Taurus surveys.",
"The MIPSGAL survey [5] was designed to survey 72 square degrees of the inner galactic plane at 24 and 70 with the Multiband Imaging Photometer for Spitzer [48].",
"At low ecliptic latitudes (ecliptic latitudes from -1 to +14.2), two epochs of MIPS Scan observations were taken with separations of 3 to 7 hours to allow for asteroid rejection from the final image stacks over a total ecliptic survey area of 29.4 square degrees.",
"The MIPSGAL data were obtained in Cycle-2 of the Spitzer cryogenic mission during the period 2005 September 27–29 UT.",
"The Taurus survey [47] was designed to survey approximately 30 square degrees in the Taurus Molecular Cloud at 24 and 70 with MIPS Scan observations.",
"The Taurus Molecular Cloud is centered at $\\sim $ 3 ecliptic latitude, and all data of this region were taken at separate epochs at 5 to 12 hour intervals to allow for asteroid rejection from the final image stacks.",
"This region was observed twice in two different years to obtain the required stacked survey depth; the total asteroid survey area is equal to 53.12 square degrees.",
"The Taurus data were obtained in Cycle-1 and Cycle-3 of the Spitzer cryogenic mission during the periods 2005 February 27–March 2 UT and 2007 February 23–28 UT.",
"The MIPS 24 band imager is a $128 \\times 128$ pixel Si:As impurity band conduction detector with an effective wavelength of 23.68 with a native pixel scale of $249 \\times 2.60$ .",
"All 24 data are diffraction limited.",
"All data obtained in the MIPSGAL and Taurus programs utilized the MIPS Scan Astronomical Observing Template with a Fast Scan Rate resulting in a total integration time per pixel of 15.7 secs in each AOR mosaic.",
"The image data files selected for our analysis from MIPSGAL consists of 42 Astronomical Observing Requests (AORs) (21 pairs) reprocessed with the MIPSGAL processing pipeline of [39], except that asteroids were not masked out of the AOR mosaics.",
"The image data files selected for our analysis from the Taurus surveys consists of 30 AORs (15 pairs) processed by the data processing pipeline at the Spitzer Science Center.",
"The pairs of images which shared a common image center were registered utilizing the world coordinate system (WCS) and differenced as illustrated in Figure REF .",
"Image subtraction allows for removal of fixed point sources and galactic background structure.",
"MOPEX [35] was employed in conjunction with single epoch uncertainty maps to produce point-spread function (PSF) fitting photometry of the positive and negative sources in each difference image consisting of object positions, fluxes and 1-sigma uncertainties in the point source fitted fluxes.Positive and negative source catalogs were constrained to only report objects detected with PSF chi-squared normalized by the degrees of freedom in the PSF fit greater than one, which results in only returning objects detected at a signal-to-noise ratio of five (5) or greater.",
"Due to small world coordinate system offsets between epochs, some fixed sources are detected in both the positive and negative source catalogs.",
"To fully remove these sources from the asteroid candidate catalogs, the positive and negative catalogs are cross matched.",
"Any object with a partner in the opposite catalog with a position within a radius of 1.5 pixels is rejected from the asteroids candidate catalogs.",
"Each candidate catalog is also searched for false sources present in the data due to increased sensitivity in small regions with increased areal coverage in an AOR which appear with predictable offsets given the scan rate.",
"These latent sources were also removed from the final candidate catalogs.",
"Initial asteroid identification was performed utilizing known asteroids in the field.",
"The JPL Horizons ISPYhttp://ssc.spitzer.caltech.edu/warmmission/propkit/sso/horizons.pdf - Appendix 3 tool was queried on 22 January 2011 to produce lists of all known asteroids present in the MIPSGAL and Taurus images and that time, the Horizons database contained the orbital elements for 543,357 known asteroids.",
"The ISPY tool requires input of observation time and image corners and produces a list of known asteroids which would be present in the field, along with predicted positions, the predicted apparent magnitude, and the instantaneous rates of change in Right Ascension (RA.)",
"and Declination (Decl.)",
"at the time of observation in arcseconds per hour.",
"The observation time given for all ISPY queries was the observation time of the first BCD image in the AOR mosaic.",
"ISPY queries were executed on an AOR basis, therefore for each subtracted image; two (2) ISPY queries were executed to predict the positions of the asteroids in each epoch.",
"The predicted position at the start of an AOR, the AOR duration and the orbital rates are then convolved to define a search box for known asteroid candidates in each epoch.",
"In 90% of cases, only one object from the candidate asteroid catalog is present in the search box.",
"We interpret this coincidence as a direct object match.",
"In the cases where multiple candidates are detected within a search box, the predicted position of the known object and the matched candidates are output to a file for visual inspection and recovery.",
"A list of non-detected asteroids in each field is also produced to estimate the completeness of the 24 MIPS dataset.",
"All matched known asteroids and their corresponding predicted and detected positions, fluxes and orbital parameters are reported in Table .",
"Columns in the flux table include asteroid name, Request Key of associated observation, date and time of observation, predicted RA.",
"and Decl., detected RA.",
"and Decl., a flux data flag, 24 flux and associated uncertainty, heliocentric distance ($r_{h}$ ) and Spitzer-to-asteroid distance, phase angle ($\\alpha $ ), and optical absolute magnitude ($H$ ).",
"The flux flag has a value of 1 for all objects except in cases where asteroid flux varies by $\\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}30$ % between two epochs, which is denoted with a flag of 2, or if an additional source such as a star is within 375, which is denoted with a flag of 3.",
"Five hundred eighty-eight (588) known asteroids were detected only once in the Spitzer data, 1035 known asteroids were detected twice and 208 known asteroids were detected 3 or more times.",
"Eight (8) bright blended asteroid sources are present in the MIPSGAL and Taurus datasets.",
"These asteroids are 103 Hera, 206 Hersilla, 2 33 Asterope, 318 Magdalena, 106 Dione, 1122 Neith, 283 Emma and 2007 McCuskey.",
"Due to the extreme brightness of these sources these sources are “soft-saturated,” and a single point source fitting result does not accurately measure the total flux from these objects.",
"Fluxes for these objects were recovered determining the relative flux ratio between the model 24 PSF central source and the first Airy ring and multiplying this ratio times asteroid fluxes in the first Airy ring.",
"Positions in Table are the nominal positions of the saturated PSF center and the uncertainty in the reported fluxes is assumed to be 15%.",
"A flux flag value of 4 in Table is used to denote the instances where the reported fluxes for these objects are reported from a saturated source.",
"Asteroid candidates with no association with known asteroids were not used for further diameter and albedo analysis.",
"Estimating diameters and albedos for these objects is highly uncertain without the availability of optical absolute magnitudes, and derivation of orbits for these objects is problematic with only two epochs of data as acceleration vectors cannot be derived from the positional data."
],
[
"ALBEDO DETERMINATION",
"We used the Near Earth Asteroid Thermal Model [22] to determine the rotationally averaged diameters and albedos of known asteroids in our MIPSGAL and Taurus samples.",
"The NEATM relies upon a basic radiometric method to determine both the diameter and albedo of an asteroid [50].",
"NEATM assumes balance between incident radiation and emitted radiation, where the emitted radiation has two components; a reflected and a thermal component.",
"The reflected component has approximately same spectral energy distribution (SED) as the incident radiation; i.e., the reflected component is dominant in the optical and peaks in V band commensurate with the spectral region in which the sun emits the greatest flux.",
"The reflected asteroid flux is proportional to the diameter of the body, $D (km)$ and the geometric albedo, $p_{V}$ .",
"To maintain energy balance the thermal flux is proportional to the amount of incident flux which is not reflected.",
"However, asteroids do not maintain one single body temperature, T(K), rather there is a temperature distribution across the surface which is then observed in the mid-IR.",
"The NEATM utilizes an assumed temperature distribution to model the total IR flux, which is related to $p_{V}$ .",
"The temperature distribution utilized by NEATM is: $T_{NEATM}(\\phi , \\theta )= \\left[ \\frac{(1-A)S_\\odot }{\\eta r_{h}^{2} \\epsilon \\sigma } \\right]^{\\frac{1}{4}} (cos \\phi )^{\\frac{1}{4}} (cos \\theta )^{\\frac{1}{4}}$ where the temperature, $T$ is in Kelvin, $A$ is the geometric Bond albedo, S$_{\\odot }$ is the solar constant (W m$^{-2}$ ), r$_{h}$ is the heliocentric distance (AU), $\\epsilon $ is the emissivity of the object (assumed to be 0.9), $\\sigma $ is the Stefan-Boltzmann constant, $\\eta $ is the beaming parameter, $\\phi $ is the latitude, and $\\theta $ is longitude of the coordinate grid superposed on the asteroid.",
"The derived temperatures are not phase angle dependent in the NEATM approach [22].",
"In the NEATM temperature distribution, $\\eta $ , the beaming parameter is utilized as a variable to characterize both shape and thermal inertia.",
"In an ideal case where an asteroid is a perfect sphere with zero thermal inertia, $\\eta $ equals unity.",
"Only one thermal photometric measurement is available from the 24 measurements; therefore, NEATM was run with a fixed beaming parameter of $\\eta =1.07$ .",
"This value of $\\eta $ was selected by averaging the value of $\\eta $ for 1584 main belt, Hilda and Trojan asteroids observed by IRAS and/or MSX [50].",
"In addition we adopt a value for the emissivity ($\\epsilon $ ) of 0.9, a value appropriate for rock [40], and a phase slope parameter ($G$ ) of 0.15 when computing the asteroid diameter and albedo.",
"To compute the geometric albedo and thus the temperature distribution on the illuminated face of the asteroid, one must anchor solutions to an optical data point.",
"We utilized optical absolute magnitudes ($H$ ) from the Minor Planet Center www.cfa.harvard.edu/iau/mpc.html(MPC) for the purposes of our solutions.",
"The validity of our thermal models to compute rotationally averaged parameters of asteroids is robust and as yielded model albedo and diameters that are consistent with radar and occultation measurements of many tens of asteroids .",
"All mean albedo and mean diameter solutions reported in Table are derived from Monte Carlo modeling for each asteroid per sighting.",
"A 500 data point distribution was created for each object observation such that the mean flux was equal to the flux measured by MOPEX and the standard deviation of the distribution was equivalent to the uncertainties in the flux measurement.",
"These flux points were then used in conjunction the known orbital parameters and the $H$ magnitude to produce albedo and diameter fitting results.",
"In this Monte Carlo modelling, the optical absolute magnitude ($H$ ) was also varied by up to 0.2 magnitudes; equal to the mean offset in asteroid absolute magnitudes as derived from the MPC and the Asteroid Orbital Elements Database [2].",
"Due to the wide width ($\\Delta \\lambda = 4.7$ ) of the MIPS 24 channel, a color correction is also required to accurately fit the albedo and diameter.",
"Our implementation of NEATM applies color corrections iteratively, such that a color correction is applied to the model asteroid flux with each refinement of the albedo [50].",
"Instead of using the subsolar temperature for the color correction, we calculate the mean of the temperature distribution for the application of the color correction, as described in [50].",
"The standard deviation of the albedos and diameters listed in Table are the 1-sigma statistical uncertainty ($\\pm $ ) in the results added in quadrature with a $\\pm 2\\%$ error in the absolute calibration of MIPS 24 data [17].",
"Results reported in Table are sorted by AOR Request Key and asteroid name/provisional designation allowing for direct matching of results with input data by line in Table .",
"Table is sorted in alphanumeric order and reports the mean albedo and diameter and associated 1-sigma uncertainties for 1831 asteroids, as well as the number of sightings used to arrive at these solutions."
],
[
"Albedo and Diameter Properties/Validity",
"Prior observations and thermal model fits of the nine brightest sources, 103 Hera, 206 Hersilla, 233 Asterope, 318 Madgalena, 106 Dione, 1122 Neith, 283 Emma, 2007 McCuskey and 106 Aethusa derived using the NEATM and IRAS or MSX photometry were compared to those obtained from the MIPS photometry.",
"Table summarizes the albedos and diameters computed from the MIPS 24 data, and their IRAS or MSX derived NEATM albedo, diameter and beaming parameter [50] and an occultation diameter if available from [14].",
"The diameter estimates from MIPSGAL photometry are within the uncertainties of those derived from fitting the SEDs produced by minimum of three wavelength specific fluxes from the IRAS and MSX surveys for all the objects.",
"The thermal model solutions for the three asteroids with occultation derived diameters also match within 1$\\%$ .",
"This overlap in the diameter estimates suggests that the MIPSGAL and Taurus 24 solutions are robust, which is not surprising as we are observing thermal emission from asteroids at the peak of, or on the Rayleigh-Jeans tail of the SED.",
"However, a slight variation in the mean diameters is present.",
"This small spread in the distribution can be attributed to the use of a single mean beaming parameter of $\\eta = 1.07$ for all asteroids in the MIPS data.",
"This value of $\\eta $ must be used as there is insufficient photometry to allow for independent fits of albedo, diameter, and beaming parameter simultaneously.",
"[59] in their analysis of IRAS Low Resolution Spectrometer (LRS) SEDs derive a mean $\\eta = 0.98$ with a one sigma uncertainty of 0.08, commensurate with the value we adopt in our work.",
"The agreement between IRAS and Spitzer results for these diameters is evidence that utilization of a single beaming parameter is appropriate for a bulk treatment of main belt asteroids.",
"Spitzer 70 data likely would provide an additional constraint on the derived characteristics of the sample asteroids.",
"Unfortunately half of the 70 array malfunctioned, and the default MIPS scan AOT used for these observations leaves large gaps in the mosaics, resulting in a striped 70 mosaic wherein useful data only exists for half of the areal coverage of a 24 mosaic.",
"This poor coverage coupled with the low sensitivity of 70 fast scan maps, which is insufficient to recover 90 km asteroids, led us to discard these data from our analysis.",
"Use of a single beaming parameter based on IRAS and MSX results assumes that small and large asteroid bodies have similar surface roughnesses and thermal inertia.",
"For main belt asteroids with diameters $> 20$ km, thermal inertia and diameter are inversely proportional [12] and this net effect would drive the beaming parameter to larger values.",
"A similar effect of increasing thermal inertia with decreased size is noted within the Near Earth Asteroid population as well [11].",
"If a beaming parameter of 1.31 (equal to the mean from IRAS and MSX plus a 1-sigma standard deviation) is assumed for the 24 thermal modeling, the albedos differ from those reported here by $\\le 10$ % and the albedos differ by $\\sim 5$ %.",
"If a beaming parameter of 0.77 (equal to the mean from IRAS and MSX minus a 1-sigma standard deviation) is used for the 24 thermal modelling, the albedos differ from those reported here by $\\sim 30$ % and the offsets are not systematic.",
"An additional source of albedo and diameter uncertainty is related to the reliability of the optical absolute magnitudes provided by the MPC.",
"A systematic color dependent offset was found between apparent V band magnitudes calculated using ASTORB [2] orbital elements and absolute magnitudes and the synthetic V band photometry derived from the Sloan Digital Sky Survey [29] of order 0.34 and 0.44 magnitude respectively for the blue and red populations of asteroids.",
"This magnitude discrepancy is lessened to a 0.2 magnitude offset when MPC absolute magnitudes are used to derive a projected apparent magnitude.",
"While a systematic offset in projected apparent magnitude could be the result of the SDSS using only two-body computations to calculate $r_{h}$ , geocentric distance ($\\Delta $ ) and phase angle, the relative offset of 0.1 magnitudes between red and blue objects is accounted for in our Monte Carlo modelling allowance of $\\pm $ 0.2 magnitude variation in the optical absolute magnitude $H$ ."
],
[
"Completeness",
"To place the albedos and diameters in the MIPSGAL and Taurus catalogs in context, the effects of optical and IR completeness must be considered.",
"To assess the completeness of optical asteroid surveys, we assume that they are complete to a V magnitude of 21.5, commensurate with the 95% completeness limit from the SDSS [29] and other surveys such as the Sub-Kilometer Asteroid Diameter Survey [16] and Spacewatch [32].",
"Assuming that an asteroid will be detected at opposition by one of a number of surveys, we utilize the relation $m_{V}=H + 5 log [r_{h}(r_{h}-1)],$ and calculate the completeness limits in terms of $H$ in each of the four main belt asteroids zones as defined in [62], adopting opposition and a heliocentric distance which corresponds to the outer semimajor axis range of each respective zone.",
"These values range from absolute magnitudes of 18.6 in the inner main belt to 16.57 in the outer main belt.",
"Unfortunately, many asteroid surveys are pencil beam surveys which do not cover the full sky, therefore we estimate the full sky completeness of asteroid surveys utilizing the H magnitude distributions from the Minor Planet Center.",
"We assume that all optical surveys are complete in each main belt region to the magnitude bin which contains the highest number of asteroid sources, and report those H magnitudes in Table .",
"Values in Table reflect both the completeness in terms of $H$ and diameters assuming a mean asteroid geometric albedo $p_{V}= 0.02$ commensurate with the darkest observed asteroid albedos from any survey.",
"Queries of the Horizons database via the ISPY tool predict a total number of 7598 asteroid appearances for 3429 individual asteroids in the MIPSGAL and Taurus surveys.",
"The catalog produced in this work contains 3486 sightings of 1831 individual asteroids, resulting in an overall object detection rate of $\\sim 53$ %.",
"There are three possible causes for this low recovery rate: rates of asteroid motion too low for the detection of movement between epochs; high rates of asteroid motion during a single AOR; and the mid-IR sensitivity completeness cut off.",
"Those asteroids whose rates of motion would make them appear as fixed targets in the two epoch MIPSGAL data are Centaurs or Kuiper Belt objects.",
"From the instantaneous rates of change in RA.",
"and Decl.",
"provided via the ISPY query, 32 objects are found to have rates of motion that would be insufficient for two epoch detection via the subtraction method for the shortest epoch separation of 3 hours.",
"Near Earth asteroids (NEAs) are objects which move at such high rates that they may not be matched in an AOR due to smearing of the flux along the direction of motion.",
"The rates of motion required for an asteroid source to move 1.2 (half of a native MIPS pixels) in an individual 5 sec BCD and a 15 sec stacked mosaic are $\\sim 1464$ arcseconds per hour and $\\sim 293$ arcseconds per hour, respectively.",
"The asteroid 2002 AL14 has the greatest instantaneous predicted rate of motion of 186 arcseconds per hour in this survey data and was recovered in all three epochs where sightings were predicted.",
"Therefore, the expected losses due to asteroid motion are biased towards non-detection of slowly moving objects in both the MIPSGAL and Taurus datasets and the losses are less than 1% in total.",
"To assess the completeness of the 24 data, synthetic sources were added to single epoch MIPSGAL and Taurus AORs which were subsequently subtracted following the data analysis techniques described in Section .",
"The 90% completeness limit was found to be 2 mJy for the MIPSGAL survey.",
"We adopt this as the 24 completeness limit of both the MIPSGAL and Taurus surveys, although the Taurus survey is complete to 1.5 mJy due to a lack of extended background emission when compared to the MIPSGAL regions.",
"At 24 , the fluxes of asteroids are most highly dependent upon diameter, not albedo.",
"This enables completeness estimates as a function of diameter to be derived within the same zones utilized to analyze the optical completeness as reported in Table .",
"These diameters were also calculated via the NEATM flux distribution with $\\eta = 1.07$ , assuming an object was observed at opposition (thus at a phase angle of zero degrees), and that the Spitzer-to-asteroid distance was 1 AU less than the heliocentric distance.",
"The asteroid completeness limits derived from the optical and the MIPSGAL and Taurus surveys are compatible; however, for all subsequent analysis, objects were removed which had fluxes less than the 24 completeness limit of 2 mJy or if they had an $H$ magnitude greater than the optical completeness limit in their region.",
"This constraint de-biases the Spitzer sample and resulted in the removal of 289 objects from the combined MIPSGAL and Taurus catalogs in the subsequent analysis; 280 for having $H$ values less than the optical completeness, 4 for having fluxes less than the 24 completeness limit, and 5 for having both $H$ values less than the optical completeness and fluxes less than the 24 completeness limit.",
"It is useful to compare the relative completeness of the MIPSGAL and Taurus surveys to the NEOWISE survey [34].",
"With a mid-IR completeness limit of the NEOWISE survey currently unavailable [37], we utilize the $H$ magnitude distributions as a function of semi-major axis to compare relative completeness between the NEOWISE, MIPSGAL, and Taurus surveys.",
"In the inner main belt, the $H$ magnitudes of the NEOWISE survey peak at $H \\simeq 15.5$ [34], whereas the mean $H$ magnitude of the MIPSGAL and Taurus catalogs in the inner main belt is 15.8 mag as illustrated in Figure REF once sources with H magnitudes greater than the optical completeness limit are removed.",
"This offset of $\\sim 0.3$ magnitudes between the NEOWISE and MIPSGAL and Taurus surveys is consistent in all semimajor axis zones.",
"This offset translates roughly into a diameter difference of 0.5 km at any given albedo indicating that the Spitzer data is detecting asteroids at least 0.5 km smaller than NEOWISE survey at any given region of the asteroid belt.",
"This offset is confirmed also by a simple estimate of the asteroid diameters which can be detected by the WISE mission [61].",
"Assuming a 5-$\\sigma $ limiting flux of 10 mJy at 22 , a beaming parameter, $\\eta = 1.07$ , geometric albedo $p_{V} = 0.14$ , and an asteroid observed at opposition (phase angle $\\alpha = 0$ ) at a heliocentric distance of 2.5 AU and a geocentric distance ($\\Delta $ ) of 1.5 AU, WISE can only detect asteroids with diameters $\\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}1.65$ km, whereas under these s ame orbital assumptions and a flux completeness limit of 2 mJy, our Spitzer data is sensitive to asteroids with diameters $D \\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle >}}{\\sim }\\,}0.79$ km."
],
[
"Albedo Catalog Comparision",
"The albedo distribution histogram from the MIPSGAL and Taurus surveys is presented in Figure REF .",
"The albedo distribution derived for small asteroids is more diverse than the albedo distribution for large asteroids derived from IRAS and MSX data.",
"The mean albedo for the complete Spitzer sample is 0.147 with a sample standard deviation of 0.104, whereas the mean albedo for the IRAS and MSX sample of 1584 objects in [50] is $p_{V} = 0.081$ with a sample standard deviation of 0.064.",
"To test if these small and large asteroid albedo distributions were selected from the same parent distribution, we performed a Komolgorov-Smirnov (K-S) test which rejected this hypothesis at the 99.99 percent level.",
"We also performed a Wilcoxon-Mann-Whitney test to determine if these two albedo distributions were selected from a population with the same mean and that hypothesis was also rejected at the 99.99 percent level.",
"Studies of space weathering [43], [8] indicate that young collisional fragments have different colors or higher albedos than old asteroids which have been subjected to solar wind exposure or micrometeorite impacts.",
"Lunar space weathering causes microscopic melting of the surfaces and the formation of agglutinates.",
"Impact melt causes submicroscopic metallic iron on the surfaces of lunar regolith particles (which in general are highly regular in shape, i.e., spherical) in cases where iron-bearing assemblages are extant on the surface [31], [30], [46].",
"The net effect is a reddened slope and decreased albedo with increasing exposure time.",
"This lunar-type space weathering is assumed to modify the surfaces of asteroids as well, as minimal lunar-type space weathering is needed to match ordinary chondrite spectra to the spectra of S-type asteroids [21], [20].",
"However, analysis of asteroid regolith from the Hayabusa 25143 Itokawa sample return mission [45], [58] seems to suggest that asteroid the optical properties of asteroid surfaces may be altered by a combination of radiation-induced amorphization in addition to in situ reduction of regolith iron by solar wind irradiation.",
"Whether or not the properties of the regolith dust from asteroid Itokawa are representative of all asteroids in general awaits further in situ sample return confirmation [3], [33].",
"Space weathering likely affects the albedo diversity we observe in our sample.",
"Such diversity can be explained if lunar-type space weathering effects, comprising micrometeor bombardment and solar wind irradiation, dominate.",
"The distribution peak for carbonaceous asteroids is at a similar albedo for both large and small asteroids as there is insufficient iron bearing minerals for this iron sputtering on regolith particles.",
"The high albedo tail of the small asteroid population also exhibits a greater diversity than the large asteroid population.",
"A complication to this interpretation is that lunar-type space weathering has only been well studied with asteroids and mineralogies characteristic of the S- taxonomic type in observations and lab studies; the effects of solar wind exposure on compositions similar to C-type asteroids have not been the subject of laboratory investigation.",
"Whether or not space weathering effects are germane is discussed further in Section REF , where characteristics of individual dynamical families within the main belt are compared."
],
[
"Albedo gradient across Main Belt",
"A population of small thermally unaltered asteroids should exist in the inner main belt if $^{26}$ Al melting models are correct [38], [19].",
"To critically examine this hypothesis, we have analyzed the albedo-orbital distribution of asteroids in the MIPSGAL and Taurus surveys, Figure REF , where the bulk albedo distribution of asteroids is color coded by albedo.",
"The bulk albedo distribution, Figure REF (left) can be contaminated by dynamical family members; for example a single family of many small S-type fragments can make the outer main belt appear silicate rich.",
"To determine the effects of dynamical families on the heliocentric distribution of albedo types, our MIPSGAL and Taurus catalogs were cross referenced with the Dynamical Family Catalog of [42] which utilized the proper elements for 293,368 asteroids to discriminate family memberships for 55 dynamical families.",
"Of these 55 dynamical families, 47 are represented in our data and only eight families have more than 20 members in our combined MIPSGAL and Taurus albedo catalogs.",
"Figure REF (right) shows the resultant bulk albedo-orbital distribution if these dynamical families are removed.",
"To compare the albedo distribution of small main belt asteroids in this dataset to those large asteroids detected by IRAS and MSX, we utilize the albedo definitions of S, X and C complex asteroids from [50] where C-types have $p_{V} \\le 0.08$ , X-types are described by a geometric albedo 0.08 $< p_{V} \\le $ 0.15, and S-types span the range of geometric albedos 0.15 $< p_{V} \\le $ 0.35.",
"The semimajor axis distribution of each classification from IRAS and MIPSGAL/Taurus is displayed in Figure REF .",
"Though the semimajor axis distributions of S- and X-type asteroids appear similar between the IRAS and Spitzer surveys, the C-type distributions show a marked enhancement within the inner main belt.",
"Removal of dynamical families, Figure REF , does not markedly change in the overall semimajor axis distribution of taxonomic type.",
"Twenty-two percent (22%) of all small dark ($p_{V} \\le 0.08$ ) , presumably carbonaceous, asteroids reside in the inner main belt (2 AU $\\le $ a $\\le $ 2.5 AU).",
"An enhancement of small C-type asteroids in the inner main belt is commensurate with the study of [4] who find that the distribution of small C- and X-type asteroids observed by the SDSS are fairly evenly distributed as a function of semimajor axis."
],
[
"Dynamical family albedos",
"Of the 47 dynamical families represented in the Spitzer MIPSGAL and Taurus albedo catalog, eight Main Belt families have more than 20 family members when combined with the IRAS and MSX albedo catalog.",
"When albedo and diameter are compared for each dynamical family, no trends of increasing albedo with decreasing diameter are seen within the Main Belt population (Figures REF - REF ) and the mean albedos of the families are consistent with the taxonomic type of their largest member, except in the case of the Nysa/Polana family (Table ).",
"The Nysa/Polana family (Figure REF ) shows albedo evidence for what may be two taxonomic types within the family – a very low albedo C-type asteroid grouping and a high albedo, S-type group.",
"This split between the compositions of the family has been detected in the optical, where spectroscopic results found Nysa to be an S-type asteroid and Polana to be a C-type asteroid [6].",
"Although spectroscopically it was unclear if this subdivision in compositional types extended to small diameters, we find evidence of both taxonomic types amongst the small family members.",
"The lack of albedo trend with decreasing diameter within the main belt potentially vitiates the origin of the albedo offset between the IRAS and Spitzer datasets arising from space weathering due to solar wind implantation on asteroid surfaces.",
"Although there is no direct way to measure asteroid age, a correlation between collisional timescale and asteroid diameter can be derived wherein the smaller asteroids are presumed to be on average younger than their larger neighbors [9].",
"To explain the color offsets between ordinary chondrites and S-type asteroids, lunar-type space weathering from the solar wind irradiation has been preferred mechanism invoked [52], [43], [36] to account for the reddening of S-type asteroid slopes and a decrease in albedo with increasing asteroid age.",
"The effects of space weathering are likely best understood by examining an asteroid population with a presumed common origin, such as the Koronis dynamical family, whose Spitzer derived albedos are presented in Figure REF , combined with the optical colors of asteroids within the Koronis dynamical family [57], which indicate a trend towards a redder optical slope with increasing diameter.",
"No trend towards an increased albedo is apparent in the 2 to 5 km diameter Koronis family population observed by Spitzer, although [57] argue that of a trend towards bluer colors exists in this size range.",
"This trend, where the color reddens as a function of age but the geometric albedo shows little to no modification, is commensurate with the interpretation Galileo flyby data of Koronis family member, 243 Ida [7], [24].",
"The Galileo imaging data of 243 Ida shows large variations in spectroscopic absorption band depths and 1 and 2 related to $^{+2}$ Fe; however, there is a lack of albedo variation commensurate with the varying surface color.",
"Individual $\\simeq 52$ nm diameter, irregularly shaped dust particle samples returned from the surface of the S-type NEA Itokawa have amorphous rims populated by small nanophase iron particles with an average size of $\\sim 2$ nm [45].",
"Laboratory measurements of small ($< 10$ nm) nanophase iron particles indicate that these particles only redden reflectance spectra, and their presence in asteroid regolith would not result in a decreased albedo [30].",
"To produce a reduction (darkening) of the albedo and a steeper slope to the reflectance spectra (reddening) requires vapor deposition of “larger” ($\\raisebox {-0.6ex}{\\,\\stackrel{\\raisebox {-.2ex}{\\textstyle <}}{\\sim }\\,}40$ nm) metallic iron nanoparticles on grain rims [44], [46].",
"The variation of spectroscopic band depths related to $^{+2}$ Fe on 243 Ida without a related color variation could also be a signature derived from small nanophase iron particles deposited on the surfaces of individual regolith dust particles.",
"In our Spitzer dataset, no trend is evident correlating an increasing albedo with decreasing diameter for the Koronis family and the other S-type families, including Flora, Eunomia and Eos.",
"Hence, invoking traditional lunar-type space weathering mechanisms alone may not be sufficient to explain the relatively large albedo diversity within the small main belt asteroid population.",
"Our results wherein albedo does not change with diameter, and therefore age, coupled with results from Galileo and the Hyabusa mission suggests that the dominant space weathering mechanism is one which produces small nanophase iron particles.",
"Lunar-type space weathering cannot be directly ruled out as the mechanism which causes space weathering within the asteroid belt.",
"However as small nanophase iron particles do not modify the albedo, space weathering which produces these particles is insufficient to explain the relatively large fraction of small MBAs in the high albedo ($p_{V} > 0.15$ ) tail of the main belt asteroid albedo distribution.",
"This observation suggests that the high albedo tail of the MBA albedo distribution is a function of composition, rather than space weathering."
],
[
"Size-Frequency Distributions",
"From the MIPSGAL and Taurus data we can directly derive a size-frequency distribution (SFD) slope for small asteroids.",
"Optical surveys such as the SDSS [27] and Spacewatch [28] have derived size-frequency distributions of main belt asteroids by assuming a mean albedo for all observed objects.",
"The slope of the cumulative size frequency distributions, $b$ , from the relation $N ( > D) \\propto D^{-b}$ , as derived by these two surveys ranges from $b= 1.3$ to 1.8 respectively over an optical magnitude range $V \\le 21$ .",
"The cumulative SFD for all main belt asteroids in the Spitzer MIPSGAL and Taurus surveys is presented in Figure REF .",
"The SFD between 7 and 25 km can be fit by a single power-law slope of b=2.34 $\\pm $ 0.05.",
"The measured SFD deviates from this fitted slope by 3$\\sigma $ starting at 8 km.",
"If one assumes that the break is a result of optical survey completeness, rather than a signature of the a transition between the regimes of asteroid strength dominated by material strength versus gravitational potential energy as predicted from laboratory studies [25], [26], then it can be said that current optical asteroid surveys are only complete to $\\sim $ 8 km.",
"Removal of dynamical families from the Spitzer dataset modifies the power-law slope slightly; however, these changes in $b$ are less than the derived uncertainties ($\\pm 0.03$ ).",
"A difference between power-law SFD slopes of asteroids was noted in the g$^{\\prime }$ and r$^{\\prime }$ filter surveys by [60].",
"Although it was unclear if this was an effect of color or albedo, the [60] result can be tested with the MIPSGAL and Taurus data by using albedo as a proxy for composition.",
"We have utilized the albedo ranges from [50] for S- and C-type taxonomic groups and present the SFDs in Figure REF .",
"The slope of the C-type SFD between 8 and 25 km is $ b = 2.49~\\pm ~0.07$ , far shallower than the SFD slope of $b = 2.20~\\pm ~0.18$ derived for the S-type asteroids between 8 and 25 km in the MIPSGAL and Taurus catalogs.",
"These Spitzer results are similar to the slopes derived by [60], indicating that difference between the SFD slopes derived in g$^{\\prime }$ and r$^{\\prime }$ filters were likely a function of composition/taxonomic type."
],
[
"CONCLUSIONS",
"From the study of small Main Belt asteroids with Spitzer, we find that some these objects are more diverse than the large main belt asteroids observed by IRAS and MSX.",
"The mean geometric albedo for small main belt asteroids is higher than that of large main belt asteroids and the overall range of albedo variation is greater for small asteroids by a factor of 2.",
"The distribution of low albedo asteroids in the solar system is also very different for small and large asteroids; only 9% of all large (D $>$ 10 km) asteroids with C-type albedos are found in the inner Main Belt, but 24% of all small ($D < 10$ km) asteroids with C-type albedos are found in the inner main belt.",
"Though the extreme diversity of main belt asteroid albedos could be attributed to space weathering effects, this interpretation is not supported by the albedo results within dynamical families.",
"Of the eight main belt dynamical families with more than 20 objects in the Spitzer and IRAS catalogs, none show the clear relationship of increasing albedo with decreasing diameter characteristic of lunar-type space weathering.",
"To determine if this diverse albedo range is caused by space weathering or compositional variations optical colors and/or spectra of these small main belt asteroids will be required to discriminate compositional taxonomies.",
"The bulk size-frequency distribution (SFD) of the Main Belt utilizing asteroid diameters was derived directly from the Spitzer survey data.",
"This bulk SFD shows evidence for a power-law break at 8.62 km.",
"This asteroid diameter is consistent with the break diameter found for the Hilda group asteroid population [50] and suggests that asteroid diameters of $\\simeq 8.5$ km lie at the transition boundary where smaller bodies are dominated by internal material strength, whereas larger bodies are bound by gravitational potential energy.",
"This SFD break derived from measures of the small asteroid population (diameters down to $\\sim 1$ km) occurs at larger diameters than those suggested from dynamical modeling of the evolution of these bodies [1].",
"Our Spitzer results therefore provide new observational constraint for collisional models that purport to follow the evolution of rocky planetesimals over the lifetime of the solar system.",
"ELR and CEW acknowledge support from National Science Foundation grant AST-0706980 to conduct this research.",
"This work is based, in part, on archival data obtained with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.",
"Support for this work was provided by an award issued by JPL/Caltech.",
"The authors also wish to thank the meticulous reading of our manuscript by an anonymous referee whose insight helped to improve the narrative.",
"lccccccccccccccr 0pt Orbital elements and 24 fluxes for asteroids detected in the MIPSGAL and Taurus Surveys Name or Request Date Time Predicted Predicted Detected Detected Flux 24 Flux Heliocentric Geocentric Phase Absolute Absolute Provisional Number RA Dec RA Dec Flag Flux Uncertainty Distance Distance Angle Magnitude Magnitude Designation (UT) (UT) (Deg) (Deg) (Deg) (Deg) (mJy) (mJy) (AU) (AU) (Deg) $\\pm $ 1998KQ42 15598848 2005-09-28 07:28:19.89 273.74063 -15.64690 273.75385 -15.65198 1 24.950 0.345 1.72 1.21 12.03 15.300 0.150 2000AF141 15598848 2005-09-28 07:28:19.89 273.74564 -17.95830 273.75189 -17.95803 1 10.390 0.337 2.17 1.73 16.66 15.100 0.150 1998QD70 15598848 2005-09-28 07:28:19.89 273.84332 -17.12060 273.84808 -17.12097 1 22.830 0.432 2.81 2.44 20.02 14.100 0.150 2001TT94 15598848 2005-09-28 07:28:19.89 273.79892 -15.39880 273.80466 -15.39959 1 11.340 0.315 2.87 2.50 22.72 15.000 0.150 2001TK102 15598848 2005-09-28 07:28:19.89 273.80942 -17.00150 273.81311 -17.00447 1 20.590 0.469 3.47 3.14 16.88 14.300 0.150 2006RG52 15598848 2005-09-28 07:28:19.89 273.74466 -18.40490 273.74939 -18.40561 1 5.754 0.338 2.67 2.29 22.16 16.900 0.150 In column 9, the flux flags are the following: 1= asteroid flux which matches within 30% of flux in subsequent epochs, 2= flux which varies $>$ 30% btwn epochs, 3= nearby bright source, 4=blended bright source.",
"The full catalog is will be available as a machine-readable table.",
"lccccr Sighting solutions for asteroids detected in the MIPSGAL and Taurus Surveys Name or Request Geometric Geometric Diameter Diameter Provisional Number Albedo Albedo Error Designation Error (km) (km) 1998KQ42 15598848 0.35 0.04 1.95 0.20 2000AF141 15598848 0.36 0.04 2.12 0.22 1998QD70 15598848 0.15 0.02 5.23 0.53 2001TT94 15598848 0.12 0.01 3.86 0.39 2001TK102 15598848 0.06 0.01 7.38 0.75 2006RG52 15598848 0.06 0.01 2.35 0.24 The full catalog is will be available as a machine-readable table.",
"lccccc Mean asteroid geometric albedos and diameters Name or Geometric Geometric Diameter Diameter Number Provisional Albedo Albedo Error Observations Designation Error (km) (km) 1321T-2 0.20 0.05 1.71 0.34 2 1413T-2 0.04 0.01 1.74 0.26 1 1978VE10 0.08 0.01 8.39 1.26 1 1978VE6 0.14 0.04 2.96 0.52 2 1978VZ5 0.18 0.03 4.79 0.70 2 1979MM2 0.39 0.06 1.69 0.25 1 1980FY2 0.212 0.03 5.46 0.82 1 1981EA29 0.05 0.01 8.29 1.26 2 The full catalog is will be available as a machine-readable table.",
"lccccl Comparison of Spitzer derived albedos and diameters to IRAS/MSX and Occultation Diameters Asteroid MIPS MIPS IRAS/MSX IRAS/MSX Occultation Name Geometric Diameter Geometric Diameter Diameter Albedo (km) Albedo (km) (km) 103 Hera 0.20 $\\pm $ 0.04 88.30 $\\pm $ 8.51 0.19 $\\pm $ 0.02 91.58 $\\pm $ 4.14 89.1 $\\pm $ 1.1 106 Dione 0.07 $\\pm $ 0.01 168.72 $\\pm $ 8.89 0.07 $\\pm $ 0.01 169.92 $\\pm $ 7.86 176.7 $\\pm $ 0.4 206 Hersilia 0.06 $\\pm $ 0.02 97.99 $\\pm $ 7.40 0.06 $\\pm $ 0.01 101.72 $\\pm $ 5.18 233 Asterope 0.10 $\\pm $ 0.01 97.54 $\\pm $ 10.32 0.08 $\\pm $ 0.01 109.56 $\\pm $ 5.04 283 Emma 0.03 $\\pm $ 0.01 145.44 $\\pm $ 7.72 0.03 $\\pm $ 0.01 145.70 $\\pm $ 5.89 148.00 $\\pm $ 16.26 318 Magdalena 0.03 $\\pm $ 0.01 105.32 $\\pm $ 11.11 0.03 $\\pm $ 0.01 106.08 $\\pm $ 0.25 1064 Aethusa 0.17 $\\pm $ 0.04 25.42 $\\pm $ 4.28 0.27 $\\pm $ 0.03 20.64 $\\pm $ 1.37 1122 Neith 0.34 $\\pm $ 0.02 13.81 $\\pm $ 0.73 0.34 $\\pm $ 0.07 13.84 $\\pm $ 1.46 2007 McCuskey 0.03 $\\pm $ 0.01 35.26 $\\pm $ 3.74 0.07 $\\pm $ 0.01 33.79 $\\pm $ 1.31 cccc Completeness limits in the optical and 24 Semimajor Optical Optical 24 Axis Completeness Completeness Completeness Range $H$ Diameter Diameter (AU) (mag) (km) (km) 2.06 - 2.5 17.25 3.33 0.79 2.5 - 2.82 16.75 4.20 1.05 2.82- 3.27 16.25 5.28 1.47 3.27 - 3.65 15.75 6.65 1.88 lccccr Geometric Albedos of Dynamical Families Derived from Spitzer Surveys Dynamical Heliocentric DeMeo Tholen Number of Mean Geometric Family Distance Taxonomic Taxonomic Members Albedo (AU) Type Type Flora 2.20 Sw S 47 0.207 $\\pm $ 0.092 Vesta 2.36 V V 42 0.272 $\\pm $ 0.156 Nysa/Polana 2.42 S/B 68 0.146 $\\pm $ 0.107 Eunomia 2.64 K S 26 0.175 $\\pm $ 0.101 Koronis 2.87 S S 30 0.174 $\\pm $ 0.051 Eos 3.01 K S 78 0.128 $\\pm $ 0.047 Themis 3.13 C C 71 0.063 $\\pm $ 0.035 Hygiea 3.14 C C 40 0.071 $\\pm $ 0.046"
]
] | 1204.1116 |
[
[
"Optical Phonon Anomaly in Bilayer Graphene with Ultrahigh Carrier\n Densities"
],
[
"Abstract Electron-phonon coupling (EPC) in bilayer graphene (BLG) at different doping levels is studied by first-principles calculations.",
"The phonons considered are long-wavelength high-energy symmetric (S) and antisymmetric (AS) optical modes.",
"Both are shown to have distinct EPC-induced phonon linewidths and frequency shifts as a function of the Fermi level $E_F$.",
"We find that the AS mode has a strong coupling with the lowest two conduction bands when the Fermi level $E_F$ is nearly 0.5 eV above the neutrality point, giving rise to a giant linewidth (more than 100 cm$^{-1}$) and a significant frequency softening ($\\sim$ 60 cm$^{-1}$).",
"Our \\emph{ab initio} calculations show that the origin of the dramatic change arises from the unusual band structure in BLG.",
"The results highlight the band structure effects on the EPC in BLG in the high carrier density regime."
],
[
"Optical Phonon Anomaly in Bilayer Graphene with Ultrahigh Carrier Densities Jia-An Yan [email protected] Department of Physics, Astronomy, and Geosciences, Towson University, 8000 York Road, Towson, MD 21252, USA K. Varga Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37035, USA M. Y. Chou [email protected] School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan Electron-phonon coupling (EPC) in bilayer graphene (BLG) at different doping levels is studied by first-principles calculations.",
"The phonons considered are long-wavelength high-energy symmetric (S) and antisymmetric (AS) optical modes.",
"Both are shown to have distinct EPC-induced phonon linewidths and frequency shifts as a function of the Fermi level $E_F$ .",
"We find that the AS mode has a strong coupling with the lowest two conduction bands when the Fermi level $E_F$ is nearly 0.5 eV above the neutrality point, giving rise to a giant linewidth (more than 100 cm$^{-1}$ ) and a significant frequency softening ($\\sim $ 60 cm$^{-1}$ ).",
"Our ab initio calculations show that the origin of the dramatic change arises from the unusual band structure in BLG.",
"The results highlight the band structure effects on the EPC in BLG in the high carrier density regime.",
"63.20.kd; 63.22.Rc; 73.22.Pr; 78.30.Na Electron-phonon coupling (EPC) is an important effect in monolayer graphene (MLG) [1].",
"Interesting phenomena such as the renormalization of the phonon energy [2], the Kohn anomalies [3], and the breakdown of the adiabatic (Born-Oppenheimer) approximation [5], [4] have been reported.",
"It is expected that even more intriguing effects should be found in AB-stacked bilayer graphene (BLG) where both the band structure and the doping level (i.e., the Fermi level $E_F$ ) can be tuned through the applied electrical gates [6], [7], [9], [8], allowing for the control of a delicate interplay between electrons, phonons, and photons [10], [13], [14], [11], [12], [15].",
"Previous investigations mainly focused on situations with charge carriers near the charge neutrality Dirac point $E_D$ (i.e., $|E_F-E_D|<$ 0.4 eV) [10], [11], [12], [15].",
"In this energy range, phenomena such as the phonon mode renormalization, Raman broadening, and the phonon frequency shift induced by EPC have been successfully predicted by a simplified tight-binding (TB) model and effective mass theory [13], [14].",
"Recent progress in fabricating electrolytic [16], [17], [18] and ionic-liquid gates [19] provides the possibilities of doping MLG and BLG with ultrahigh charge-carrier densities of $|n| > 10^{14}$ cm$^{-2}$ and of tuning the Fermi level close to the van Hove singularity (VHS) point at $M$ in the first Brillouin zone (BZ) [16], [20].",
"Such a high carrier density will embark interesting technological applications including supercapacitors [21], transparent electrodes [22], and high performance organic thin film transistors [23].",
"In this regard, understanding the carrier dynamics will be a crucial step for the potential electronic device applications.",
"Distinct many-body effects and superconducting instability have been observed in doped graphene when the Fermi energy approaches the VHS point [20].",
"Since superconductivity also occurs in graphene-related systems such as graphite-intercalation compounds (GICs), the study of EPC in BLG might shed new light on the underlying mechanism of superconductivity in GICs [24], [25], [26].",
"Raman and infrared (IR) spectra are powerful tools to probe the EPC in MLG and BLG by providing the information of the phonon linewidth and frequency shifts as a function of doping and field strengths [5], [4], [11], [12], [27].",
"Although the aforementioned models [13], [14] are known to work well for low-energy charge carriers, deviations between theoretical predictions and experimental observations become significant when $|E_F-E_D|$ $>$ 0.4 eV [27].",
"Such a discrepancy calls for a full consideration of band structure effects from first principles, as the electronic structure in this regime is expected to be considerably modified from its low energy part.",
"Figure: (Color online) Band structures of bilayer graphene calculated from (a) a simplified tight-binding model with only nearest-neighbor (NN) coupling and (b) density functional theory.",
"The xx axis is in units of 2π\\pi /a 0 a_0, where a 0 a_0 is the graphene lattice constant.",
"All possible transitions (I-VI) through the phonon modes are labeled in (b).",
"The neutrality point E D E_D has been shifted to energy zero.In this work, we show that the band structure of BLG at high doping levels plays a critical role in the EPC of the two long-wavelength high-energy symmetric (S) and antisymmetric (AS) optical modes in the high carrier density regime.",
"For both modes, the linewidth $\\gamma $ exhibits a dramatic dependence on the electron and hole doping level.",
"In particular, the AS ($E_u$ ) mode exhibits a giant EPC-induced phonon linewidth (more than 100 cm$^{-1}$ ) and significant softening (60 cm$^{-1}$ ) when the Fermi level $E_F$ is 0.5 eV above the neutrality point $E_D$ .",
"Despite an intensive study of the EPC in BLG [28], [13], [14], this feature has not been reported so far.",
"This giant enhancement originates from the strong coupling between the AS mode and two conduction bands.",
"The strong EPC in this high carrier-density regime might affect the performance of BLG-based devices.",
"The first-principles calculations reported in this work are performed using the Quantum ESPRESSO code [29], [30].",
"The electronic structure is obtained using the local density approximation (LDA) within the density-functional theory (DFT), and the core-valence interaction is modeled by norm-conserving pseudopotentials [31].",
"The wave functions of the valence electrons are expanded in plane waves with a kinetic energy cutoff of 70 Ry.",
"A vacuum region of 20 Å has been introduced to eliminate the artificial interaction between neighboring supercells along the $z$ direction.",
"The relaxed C-C bond length is 1.42 Å and the interlayer distance is 3.32 Å for BLG.",
"The phonon frequencies and associated eigenvectors were computed using the density-functional perturbation theory (DFPT) [29], details of which have been presented in our previous work [32], [33].",
"The self-energy $\\Pi _{\\mathbf {q}\\nu } (\\omega )$ of a phonon with wave vector $\\mathbf {q}$ , branch index $\\nu $ , and frequency $\\omega _{\\mathbf {q}\\nu }$ provides information on the renormalization and damping of that phonon due to the interaction with other elementary excitations.",
"Following the Migdal approximation, the self-energy induced by the EPC in BLG reads [13]: $\\Pi _{\\mathbf {q}\\nu }(\\omega )=2 \\sum _{mn}\\int \\frac{d\\mathbf {k}}{\\Omega _{\\mathrm {BZ}}}|g_{mn}^{\\nu }(\\mathbf {k},\\mathbf {q})|^2 \\frac{[f(\\epsilon _{n\\mathbf {k+q}})-f(\\epsilon _{m\\mathbf {k}})][\\epsilon _{n\\mathbf {k+q}}-\\epsilon _{m\\mathbf {k}}]}{(\\epsilon _{n\\mathbf {k+q}}-\\epsilon _{m\\mathbf {k}})^2-(\\hbar \\omega +i\\eta )^2},$ where $\\epsilon _{m\\mathbf {k}}$ is the energy of an electronic state $|m\\mathbf {k}\\rangle $ with crystal momentum $\\mathbf {k}$ and band index $m$ , $f(\\epsilon _{m\\mathbf {k}})$ the corresponding Fermi occupation, and $\\eta $ a positive infinitesimal.",
"For a given mode $\\omega =\\omega _0$ , the phonon linewidth is $\\gamma $ = $-2\\mathrm {Im}(\\Pi _{\\mathbf {q}\\nu } (\\omega _0))$ and the phonon frequency shift is $\\Delta \\omega $ = $\\frac{1}{\\hbar }[\\mathrm {Re}(\\Pi _{\\mathbf {q}\\nu } (\\omega _0)|_{E_F}-\\Pi _{\\mathbf {q}\\nu }(\\omega _0)|_{E_F=0}]$ .",
"The EPC matrix element in Eq.",
"(1) is given by $g_{mn}^{\\nu }(\\mathbf {k},\\mathbf {q})=\\sqrt{\\frac{\\hbar }{2M\\omega _{\\mathbf {q}}^{\\nu }}}\\langle m\\;\\mathbf {k+q}|\\frac{\\delta V_{scf}}{\\delta u_{\\mathbf {q}}^{\\nu }}|n\\;\\mathbf {k}\\rangle ,$ where $\\delta V_{scf}\\equiv V_{scf}(u_{\\mathbf {q}}^{\\nu })-V_{scf}(0)$ is the variation of the self-consistent potential field due to the perturbation of a phonon with wave vector $\\mathbf {q}$ and branch index $\\nu $ .",
"Variations of the potential field $\\delta V_{scf}$ are calculated through self-consistent calculations to find the potential field for both perturbed and unperturbed systems.",
"The perturbed phonon mode is handled with the frozen-phonon approach [33].",
"The DFT calculations have been carried out on a dense 201$\\times $ 201 $k$ -grid within a minizone (0.4$\\times $ 0.4) enclosing the BZ corner $K$ in the reciprocal space.",
"This is equivalent to 500$\\times $ 500 $k$ -grid sampling in the whole Brillouin zone.",
"Finally, the EPC matrix elements are computed using Eq. (2).",
"By changing the Fermi level $E_F$ in Eq.",
"(1), we can investigate the dependence of $\\gamma $ and $\\Delta \\omega $ on different doping levels, assuming the EPC matrix elements are unchanged.",
"This approximation is justified by the small dependence of the EPC matrix elements on doping for the $\\Gamma $ phonon modes in graphene [34].",
"For all the linewidths calculated below, we used a parameter $\\eta $ = 5 meV.",
"In Fig.",
"REF we show the band structure of BLG obtained by the TB model (with only nearest-neighbor (NN) interactions) and DFT calculations.",
"For BLG, the low-energy dispersions within a TB model can be well described by $E(k)$ = $\\pm t_2/2\\pm \\sqrt{t_2^2/4+t_1^2|f(k)|^2}$ , with $t_1$ = $-2.7$ eV and $t_2$ = 0.36 eV, the intralayer and interlayer hopping parameters, respectively [35].",
"The low-energy parabolic dispersions predicted by the TB model are in agreement with the DFT result.",
"Nevertheless, there is still a noticeable difference: In Fig.",
"REF (b), the band dispersions have an evident asymmetry between the conduction and valence bands, while the TB model shows nearly symmetric band structure relative to the neutrality point [Fig.",
"REF (a)].",
"This electron-hole asymmetry for the low-energy charge carriers has already been revealed by infrared spectroscopy [7] and becomes more significant for the high-energy carriers.",
"In particular, our first-principles calculations predict that there is a crossing between the two conduction bands (c1, c2) at about 0.1$\\times $$2\\pi /a_0$ from $K$ along the $K$ -$M$ direction in the BZ [36].",
"The energy is around 1.0 eV above the neutrality point, as shown in Fig.",
"REF (b).",
"This feature is not captured by the simple TB model, indicating the necessity of first-principles calculations.",
"As we will show later, the band structure plays a crucial role in understanding the physics of high-energy charge carriers.",
"Figure: (Color online) Calculated linewidths γ\\gamma for the (a) symmetric (S) and (b) antisymmetric (AS) mode as a function of E F E_F.",
"Note that the vertical scale has been changed in (b).",
"The neutrality point E D E_D is set to be energy zero.",
"Insets are schematic plots of the S and AS modes.The asymmetric band structure of BLG has important implications on the phonon linewidth and frequency shift because the S and AS modes couple with different bands in BLG [33].",
"As indicated in Fig.",
"REF (b), the allowed transitions [i.e., nonzero EPC matrix elements in Eq.",
"(2)] for the S mode are transitions I (v1-c1), II (v2-c2), V (v1-v2), and VI (c1-c2).",
"In contrast, the AS mode only couples with transitions III (v1-c2), IV (v2-c1), V (v1-v2), and VI (c1-c2).",
"When $E_F$ is tuned to be 0.5 eV above the neutrality point, the transition VI becomes active because the energy requirement is satisfied.",
"On the other hand, the crossing of the two conduction bands (c1, c2) in this energy range provides a larger electronic phase space in the Brillouin zone for the phonon mode to couple with, as will be shown later.",
"As a result, the linewidth of the AS mode is expected to be significantly enhanced.",
"In Figs.",
"REF (a) and REF (b), we show the calculated linewidths for the S and AS modes with respect to $E_F$ .",
"The S and AS modes ($\\hbar \\omega _0\\sim $ 0.2 eV) considered in this work have been schematically shown in the insets.",
"The corresponding frequency shifts $\\Delta \\omega $ are presented in Fig.",
"REF (a) and REF (b), respectively.",
"Figure: Calculated frequency shift Δω\\Delta \\omega for the (a) symmetric and (b) antisymmetric mode as a function of E F E_F.",
"The neutrality point E D E_D is set to be energy zero.In the low doping regime with $|E_F-E_D| < \\hbar \\omega _0/2 \\sim 0.1$ eV, the S mode can be in resonant coupling with the electron-hole pair from the the top valence band (v1) and the bottom conduction band (c1).",
"As a result, the linewidth of the S mode is constant, as depicted in Fig.",
"REF (a).",
"In contrast, the AS mode has a negligible linewidth in this range.",
"Our calculations of phonon linewidth and frequency shift in this energy region reproduce the features found in previous DFT calculations [28] and agree reasonably well with the experimental data [11].",
"This is also consistent with a previous study based on a continuum model [13].",
"In comparison, dramatic differences are found for the ultrahigh electron doping when $E_F -E_D > 0.5$ eV.",
"The calculated $\\gamma $ for the AS mode is significantly larger than that of the S mode.",
"For example, when $E_F-E_D$ = 0.70 eV, $\\gamma $ is 150 cm$^{-1}$ for the AS mode, around 20 times larger than that of the S mode (8.0 cm$^{-1}$ ).",
"Correspondingly, there is a significant softening ($\\sim 60$ cm$^{-1}$ ) for the frequency shift when $E_F = $ 0.6 eV, as shown in Fig.",
"REF (b).",
"Further increase of the Fermi level will result in an even more significant increase of the phonon linewidth (not shown).",
"In contrast, the phonon softening becomes smeared out as $E_F$ increases.",
"Although such a doping level requires large electron densities, it may be achieved either by directly applying electrical gate in experiment [17] or by chemical doping [20].",
"For $E_F-E_D$ = 0.5 eV, the desired electron density in BLG is estimated to be $n \\sim 6\\times $ 10$^{13}$ cm$^{-2}$ , still an order of magnitude smaller than the doping limit achieved in monolayer graphene ($\\sim $ 4$\\times $ 10$^{14}$ cm$^{-2}$ ) [16].",
"Therefore, we expect that this high electron doping regime will be realized in experiment [17] and that the giant phonon linewidth as well as the significant phonon mode softening may be verified by Raman or infrared measurements.",
"Figure: Calculated joint density of states (JDOS) N(E F ,ω 0 )N(E_F,\\omega _0) as a function of the Fermi level.",
"The neutrality point E D E_D is set to be energy zero.This giant linewidth enhancement mainly arises from the large phase space associated with the high doping range.",
"In Fig.",
"REF we plot the joint density of states (JDOS) $N(E_F,\\omega _0)=\\frac{4\\pi }{N_k}\\sum _{\\mathbf {k}jj^{\\prime }}[f(\\epsilon _{\\mathbf {k}j})-f(\\epsilon _{\\mathbf {k}j^{\\prime }})]\\delta [\\epsilon _{\\mathbf {k}j}-\\epsilon _{\\mathbf {k}j^{\\prime }}+\\hbar \\omega _0]$ as a function of $E_F$ .",
"The JDOS increases dramatically when the Fermi level is in the range of $E_F-E_D >$ 0.5 eV, which allows more electronic states to couple with the phonon modes.",
"The profile of $N(E_F)$ is similar to the phonon linewidth profile of the AS mode, indicating that the giant enhancement is mainly an electronic structure effect.",
"In summary, our first-principles calculations find that the phonon linewidths and frequency shifts for the long-wavelength high-energy optical modes in bilayer graphene exhibit a distinct dependence on the electron and hole doping due the intriguing interplay between the unique band structure and the phonon modes in this system.",
"In particular, we predict that the linewidth for the antisymmetric mode could be significantly enhanced when the Fermi level is tuned to be 0.5 eV above the neutrality point.",
"J.A.Y.",
"is grateful to Z. Jiang, F. Giustino, C. -H. Park, W. Duan, F. Liu and S. C. Zhang for fruitful discussions and thanks Mark A. Edwards for the support.",
"Part of this work was done at the Georgia Southern University in Statesboro, Georgia.",
"M.Y.C.",
"acknowledges support by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No.",
"DEFG02-97ER45632.",
"This research used computational resources at the National Energy Research Scientific Computing Center (supported by the Office of Science of the U.S. Department of Energy under Contract No.",
"DE-AC02-05CH11231) and at the National Institute for Computational Sciences under XSEDE startup allocation (Request No.",
"DMR110111)."
]
] | 1204.1584 |
[
[
"Fabrication and characterisation of ambipolar devices on an undoped\n AlGaAs/GaAs heterostructure"
],
[
"Abstract We have fabricated AlGaAs/GaAs heterostructure devices in which the conduction channel can be populated with either electrons or holes simply by changing the polarity of a gate bias.",
"The heterostructures are entirely undoped, and carriers are instead induced electrostatically.",
"We use these devices to perform a direct comparison of the scattering mechanisms of two-dimensional (2D) electrons ($\\mu_\\textrm{peak}=4\\times10^6\\textrm{cm}^2/\\textrm{Vs}$) and holes ($\\mu_\\textrm{peak}=0.8\\times10^6\\textrm{cm}^2/\\textrm{Vs}$) in the same conduction channel with nominally identical disorder potentials.",
"We find significant discrepancies between electron and hole scattering, with the hole mobility being considerably lower than expected from simple theory."
],
[
"Fabrication and characterization of ambipolar devices on an undoped AlGaAs/GaAs heterostructure J.C.H.",
"Chen [email protected] School of Physics, University of New South Wales, Sydney NSW 2052, Australia D.Q.",
"Wang School of Physics, University of New South Wales, Sydney NSW 2052, Australia O. Klochan School of Physics, University of New South Wales, Sydney NSW 2052, Australia A.P.",
"Micolich School of Physics, University of New South Wales, Sydney NSW 2052, Australia K. Das Gupta Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom [Now at ]Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India F. Sfigakis Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom D.A.",
"Ritchie Cavendish Laboratory, J J Thomson Avenue, Cambridge CB3 0HE, United Kingdom D. Reuter Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany A.D. Wieck Angewandte Festkörperphysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany A.R.",
"Hamilton [email protected] School of Physics, University of New South Wales, Sydney NSW 2052, Australia We have fabricated AlGaAs/GaAs heterostructure devices in which the conduction channel can be populated with either electrons or holes simply by changing the polarity of a gate bias.",
"The heterostructures are entirely undoped, and carriers are instead induced electrostatically.",
"We use these devices to perform a direct comparison of the scattering mechanisms of two-dimensional (2D) electrons ($\\mu _\\textrm {peak}=4\\times 10^6\\textrm {cm}^2/\\textrm {Vs}$ ) and holes ($\\mu _\\textrm {peak}=0.8\\times 10^6\\textrm {cm}^2/\\textrm {Vs}$ ) in the same conduction channel with nominally identical disorder potentials.",
"We find significant discrepancies between electron and hole scattering, with the hole mobility being considerably lower than expected from simple theory.",
"Modulation-doped $\\textrm {Al}_{x}\\textrm {Ga}_{1-x}\\textrm {As/GaAs}$ heterostructures have formed the starting point for innumerable studies of low dimensional electron and hole systems.",
"[1] In modulation doped heterostructures the spatial separation of the dopants and the channels significantly reduces scattering, so that very high electron or hole mobilities can be achieved at low temperatures.",
"[2] However, the type of charge carrier is determined by the dopants used during the heterostructure growth, which makes it very difficult to fabricate ambipolar devices that can operate with both electron and hole conduction.",
"The ability to switch seamlessly between electrons and holes in the same device would be of interest for studies of scattering, interaction effects and spin related phenomena, since the two types of charge carriers have very different effective masses, bandstructures and spin properties.",
"To create ambipolar devices we eschew conventional modulation doping techniques, using instead a gate electrode to populate the channel electrostatically.",
"The challenge with this approach is to make good electrical contact to the 2D electrons or holes in the channel without forming an unwanted contact to the gate electrode which must overlap the ohmic contact.",
"Hirayama et al.",
"[3], [4] used ion-implantation to overcome this problem and fabricate ambipolar devices on AlGaAs/GaAs heterostructures.",
"Thick high Al content AlGaAs diffusion barriers were used to suppress leakage between the top gate and the ohmic contacts.",
"However, a high temperature anneal ($\\sim 800^{\\circ }$ C) is required after the ion implantation to activate the dopants, which is higher than the wafer growth temperature and may have adverse effects on the heterostructure and the carrier mobility.",
"Here we describe ambipolar devices fabricated without the need for ion implantation and subsequent dopant activation anneals.",
"With this device design we are able to make high quality ambipolar 2D systems, and compare the transport lifetime of electrons and holes formed in the same channel, with the same scattering potential.",
"Figure: (a) Device schematic, showing top gate separated from the ohmics by the polyimide.",
"(b) Optical micrograph of a typical induced device used in this study.",
"(c) Two terminal conductance of electrons (V TG >0V_{\\textrm {TG}}>0) and holes (V TG <0V_{\\textrm {TG}}<0) through the device at 240 mK with an excitation voltage of 100μ100 \\mu V. (d) Carrier density as a function of applied top gate bias.",
"The densities are linear in top gate voltage and are highly reproducible between thermal cycles.We have fabricated devices from a number of different undoped heterostructures, extending the approach described in Refs.",
"Harrell and Sarkozy2D for making unipolar devices.",
"Ohmic contacts are fabricated by standard optical lithography techniques, so there is no need for ion implantation.",
"Here we present data from wafer B13520 grown by molecular beam epitaxy on a (100) GaAs substrate, although similar data was obtained on other wafers.",
"A $1\\mu $ m GaAs buffer layer was followed by 300 nm of undoped AlGaAs and capped by 17 nm of GaAs.",
"Standard UV photolithography was used to define the mesa, ohmics, polyimide and top gate patterns.",
"AuBe alloy was used for the hole contacts and NiAuGe for the electron contacts.",
"A 500 nm thick layer of polyimide was used as the insulator to isolate the top gate from the ohmic contacts, while allowing the top gates to overlap the ohmic contacts.",
"The top-gate was deposited on top of the polyimide by thermal evaporation.",
"A schematic of the device is shown in figure 1(a) and a micrograph of a typical ambipolar device is presented in figure 1(b).",
"The long rectangular gold pattern in the middle of the device is the Ti/Au top gate, which defines the conduction channel.",
"The device was characterised at 240 mK using standard lock-in techniques.",
"The two-terminal conductance is shown in Fig.",
"1(c) as a function of top-gate gate bias $V_{\\textrm {TG}}$ .",
"The channel is populated with holes for $V_{\\textrm {TG}}<0$ , and with electrons for $V_{\\textrm {TG}}>0$ .",
"The threshold voltages for electrons and holes depend on how well the three-dimensional ohmics contact to the 2D charge carriers, which can vary from device to device.",
"In contrast the 2D electron and hole densities were extremely reproducible between thermal cycles on the same device (see Fig.",
"1(d)), and consistent within $\\pm 5\\%$ between devices from the same wafer.",
"The range of densities accessible was limited by the requirement to keep the leakage current between gate and ohmic contacts below 1 nA at the high end, and by the threshold voltage of the ohmics at the low end.",
"Magnetotransport data were taken at different top gate voltages in a perpendicular magnetic field up to 2T.",
"Figure 2 shows typical transport data for both electrons and holes.",
"From the Shubnikov-de Haas oscillations the densities were calculated at each top-gate bias.",
"The densities are plotted in Fig.",
"$1(d)$ and increase linearly with top gate voltage.",
"The slope of $dn/dV_{\\textrm {TG}}$ was similar in all devices measured at $2.7 \\pm 0.1 \\times 10^{10}m^{-2}/V$ .",
"The slopes were used to calculate the thickness of the polyimide layer for each device, which was consistent with the experimental values measured with a Dektak surface profilometer ($500-600$ nm).",
"The top-gate biases at which the 2D electron and hole densities extrapolated to zero were 1.09 V and -0.52 V respectively, with the 1.61 V difference between these values being very close to the low temperature GaAs band gap of 1.52 eV.",
"Figure: Magnetoresistance measured at 240 mK, showing the Shubnikov de-Haas oscillations for similar electron and hole densities.The Shubnikov-de Haas oscillations shown in Fig.",
"2 are more pronounced for electrons than holes.",
"This is because the hole effective mass is $3-5$ times larger than the electron effective mass.",
"Since the Landau level separation is $\\hbar eB/m^{*}$ , the effects of disorder broadening and thermal smearing are more severe for holes than electrons at the same measurement temperature.",
"Figure: Symbols show mobility as a function of density for electrons (squares) and holes (circles).",
"Four sets of calculations of total mobility are shown as solid lines.",
"From the top down the first three are for electrons (blue), holes with m h * =0.2m e m_{h}^{*} = 0.2m_{e} (light green), and holes with m h * =0.9m e m_{h}^{*} = 0.9m_{e} (dark green), all calculated with the same interface roughness and background impurity levels.",
"The fourth solid line (red) is for holes with m h * =0.4m e m_{h}^{*} = 0.4m_{e} using a higher background impurity level in the GaAs.",
"Dashed and dash-dotted lines are the calculated mobilities accounting only for interface roughness (Δ\\Delta =2 Å Λ\\Lambda =4 nm) and background impurities ( N BI GaAs =2.42×10 13 cm -3 N^{GaAs}_{BI} = 2.42 \\times 10^{13} \\textrm {cm}^{-3}, N BI AlGaAs =7.25×10 13 cm -3 N^{AlGaAs}_{BI} = 7.25 \\times 10^{13} \\textrm {cm}^{-3}).",
"The total mobility includes both contributions.",
"The grey diagonal lines in the lower left corner indicate the region where 1<k F l<101 < k_{F}l < 10.The carrier density and mobility calculated from the magnetotransport data are plotted in Fig.",
"3.",
"For both types of charge carriers, the mobility initially increases with carrier density as shown by the squares (electrons) and circles (holes) in Fig.",
"3.",
"In the conventional picture of transport in high mobility 2D systems this is due to the increase in the Fermi velocity of the charge carriers as the density is increased.",
"At intermediate densities ($10^{11} cm^{-2}$ ) the mobility saturates, and then stays relatively constant to the highest density measured.",
"This occurs because at higher densities the carriers are pressed against the AlGaAs/GaAs interface, which enhances interface roughness scattering.",
"The peak mobility for electrons is $4\\times 10^6 \\textrm {cm}^{2}/\\textrm {Vs}$ compared to $7.9\\times 10^5 \\textrm {cm}^{2}/\\textrm {Vs}$ for holes at 235 mK.",
"To compare the scattering mechanisms between electrons and holes, we have modelled the $T=0$ mobility as a function of density with the approach detailed in Refs.",
"Warrick2D, Sarkozy2D, SarahMac, WMak.",
"The calculations took into account the wavefunction of the charge carriers, screening in the Hubbard approximation, background impurity scattering and interface roughness scattering.",
"Previous modelling of similar heterostructures has shown that the background doping is approximately three times higher in AlGaAs than GaAs, [9], [8] and this ratio was used to model the mobility data for electrons in Fig.",
"3.",
"The interface roughness is characterized by two parameters, the mean amplitude of the interface roughness $\\Delta $ and the roughness correlation length $\\Lambda $ .",
"The best fit to the measured electron mobility was obtained with $\\Delta $ =2 Å and $\\Lambda $ =4 nm, which is comparable to the values used in Ref.",
"Sarkozy2D.",
"The same values were used to calculate the hole mobility.",
"The grey lines delineate the region $k_{F}l < 10$ , where single particle scattering theory begins to break down.",
"The calculated mobilities are plotted as a function of density in Fig.",
"3, with blue lines for electrons and red/green for holes.",
"At high densities interface roughness (IR) is the main scattering mechanism, whereas background impurity (BI) scattering dominates at low densities.",
"The total calculated mobilities are shown with solid lines.",
"The best fit for the electron data was obtained with an impurity density of $N^{GaAs}_{BI} = 2.42 \\times 10^{13} \\textrm {cm}^{-3}$ and $N^{AlGaAs}_{BI} = 7.25 \\times 10^{13} \\textrm {cm}^{-3}$ .",
"After fitting the electron mobility data, we used the same impurity densities and interface roughness parameters to calculate the expected mobility of the holes, shown as the solid green line in Fig.",
"3.",
"Since the hole bands are non-parabolic, the hole mass depends both on the heterostructure details and the hole density.",
"We have therefore calculated the expected hole mobility for hole masses of $0.2m_0$ and $0.9m_0$ , which are at the extreme limits of values reported in the literature.",
"For all values of the hole mass, the calculated hole mobility (green solid lines in Fig 3) consistently exceeds the measured data.",
"This is an unexpected result, since at first sight the disorder potential should be exactly the same for electrons and holes in the same channel, measured in the same cooldown.",
"Our calculations show that this discrepancy cannot be due to uncertainties in the hole mass.",
"There are a number of possible origins of the discrepancy.",
"(i) We first look at the mobility at low densities, where the discrepancy between the measured and calculated hole mobility is largest.",
"For $p < 4 \\times 10^{10} \\textrm {cm}^{-2}$ the measured mobility drops off dramatically, with the slope $d\\mu _{h}/dp$ being much larger than the calculations (green lines in Fig.",
"3).",
"This sharp drop in mobility at low densities has also been observed in other induced hole systems, [7], [10] and has been taken as evidence that the system is becoming inhomogeneous with transport crossing over to percolation [11], [12].",
"Indeed we obtain a reasonable fit to the 2D percolation expression $\\sigma (p)=A(\\sigma -\\sigma _0)^{4/3}$ , with $\\sigma _0=1\\times 10^{10} cm^{-2}$ , for low hole densities (not shown).",
"(ii) At higher densities it is unlikely that percolation and inhomogeneities are relevant, since $k_F l > 100$ , yet the measured mobility remains lower than the expected mobility for all hole densities.",
"It is possible that bandstructure effects, arising from the asymmetric confining potential of the single heterojunction, may play a role.",
"The asymmetric confining potential leads to a Rashba splitting of the heavy hole band into two branches, which appear as if they have different masses [13], [14].",
"These bandstructure effects were not included in the modelling shown in Fig.",
"3.",
"However initial calculations using two independent heavy hole bands with $m^{+}_{hh} = 0.9m_{0}$ and $m^{-}_{hh} = 0.2m_{0}$ , and neglecting inter-band scattering, did not significantly improve the fit to the measured hole data.",
"This is to be expected since the resulting mobility will lie between the mobilities calculated in the absence of bandstructure effects for $m_h^*=0.2m_0$ and $0.9m_0$ (iii) The third possibility is that the number of background ionised impurities is larger in the GaAs channel when it is occupied with holes than when it is occupied with electrons.",
"This increase in the ionised impurity density could arise when switching from electrons to holes because the Fermi level in the triangular self-consistent quantum well moves from above the conduction band to below the valence band.",
"An approximate doubling of the impurity density in the channel moves the calculated mobility due to background impurities lower, resulting in good agreement with the measured mobility for $p > 5 \\times 10^{10} \\textrm {cm}^{-2}$ (red line in Fig.",
"3).",
"However it is difficult to reconcile this increase in backround ionised impurities with the fact that the background doping in the MBE chamber is p-type.",
"Overall, our data suggest that the scattering processes are very different, and considerably more complex, for holes than for electrons.",
"To summarise we have fabricated ambipolar devices on an undoped AlGaAs/GaAs heterostructure.",
"The devices were robust to thermal cycling, and showed high electron and hole mobilities.",
"The density dependence of the electron mobility was well described by standard Fermi golden rule scattering theory, but the hole data is much more complex, with additional scattering mechanisms involved.",
"Our results show that further work is needed to understand the scattering properties of 2D holes, even when the disorder potential is well characterised.",
"In the future such devices could be used for studies of the anomalous metallic behaviour in 2D systems, as well as transport in quantum wires and dots.",
"[15], [16], [17], [18] This work was supported by the Australian Research Council under the DP and LX schemes, and by the ARCNN.",
"ARH and APM acknowledge ARC Professorial and Future Fellowships respectively; JCHC acknowledges an Australian Postgraduate Award.",
"We thank U. Zülicke, R. Winkler and A. Croxall for helpful discussions."
]
] | 1204.0827 |
[
[
"Measurement of the ratio of prompt $\\chi_{c}$ to $J/\\psi$ production in\n $pp$ collisions at $\\sqrt{s}=7$ TeV"
],
[
"Abstract The prompt production of charmonium $\\chi_{c}$ and $J/\\psi$ states is studied in proton-proton collisions at a centre-of-mass energy of $\\sqrt{s}=7$ TeV at the Large Hadron Collider.",
"The $\\chi_{c}$ and $J/\\psi$ mesons are identified through their decays $\\chi_{c}\\rightarrow J/\\psi \\gamma$ and $J/\\psi\\rightarrow \\mu^+\\mu^-$ using 36 pb$^{-1}$ of data collected by the LHCb detector in 2010.",
"The ratio of the prompt production cross-sections for $\\chi_{c}$ and $J/\\psi$, $\\sigma (\\chi_{c}\\rightarrow J/\\psi \\gamma)/ \\sigma (J/\\psi)$, is determined as a function of the $J/\\psi$ transverse momentum in the range $2 < p_{\\mathrm T}^{J/\\psi} < 15$ GeV/$c$.",
"The results are in excellent agreement with next-to-leading order non-relativistic expectations and show a significant discrepancy compared with the colour singlet model prediction at leading order, especially in the low $p_{\\mathrm T}^{J/\\psi}$ region."
],
[
"Introduction",
"The study of charmonium production provides an important test of the underlying mechanisms described by Quantum Chromodynamics (QCD).",
"At the centre-of-mass energies of proton-proton collisions at the Large Hadron Collider, $c \\overline{c } $ pairs are expected to be produced predominantly via Leading Order (LO) gluon-gluon interactions, followed by the formation of bound charmonium states.",
"The former can be calculated using perturbative QCD and the latter is described by non-perturbative models.",
"Other, more recent, approaches make use of non-relativistic QCD factorization (NRQCD), which assumes the $c \\overline{c } $ pair to be a combination of colour-singlet and colour-octet states as it evolves towards the final bound system via the exchange of soft gluons [1].",
"The fraction of ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ produced through the radiative decay of $\\chi _{c}$ states is an important test of both the colour-singlet and colour-octet production mechanisms.",
"In addition, knowledge of this fraction is required for the measurement of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ polarisation, since the predicted polarisation is different for ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ mesons coming from the radiative decay of $\\chi _{c}$ state compared to those that are directly produced.",
"In this paper, we report the measurement of the ratio of the cross-sections for the production of $P$ -wave charmonia $\\chi _{cJ}(1P)$ , with $J=$ 0, 1, 2, to the production of ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ in promptly produced charmonium.",
"The ratio is measured as a function of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ transverse momentum in the range $2\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,15~{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c} $ and in the rapidity range $2.0\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5$ .",
"Throughout the paper we refer to the collection of $\\chi _{cJ}(1P)$ states as $\\chi _{c}$ .",
"The $\\chi _{c}$ and ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidates are reconstructed through their respective decays $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ and ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\rightarrow \\mu ^+ \\mu ^- $ using a data sample corresponding to an integrated luminosity of 36 $\\mbox{\\,pb}^{-1}$ collected during 2010.",
"Prompt (non-prompt) production refers to charmonium states produced at the interaction point (in the decay of $b$ -hadrons); direct production refers to prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ mesons that are not decay products of an intermediate resonant state, such as the $\\psi {(2S)}$ .",
"The measurements are complementary to the measurements of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ production cross-section [2] and the ratio of the prompt $\\chi _{c}$ production cross-sections for the $J=1$ and $J=2$ spin states [3], and extend the $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ coverage with respect to previous experiments [4], [5]."
],
[
"LHCb detector and selection requirements",
"The LHCb detector [6] is a single-arm forward spectrometer with a pseudo-rapidity range $2 \\,{<}\\,\\eta \\,{<}\\,5$ .",
"The detector consists of a silicon vertex detector, a dipole magnet, a tracking system, two ring-imaging Cherenkov (RICH) detectors, a calorimeter system and a muon system.",
"Of particular importance in this measurement are the calorimeter and muon systems.",
"The calorimeter system consists of a scintillating pad detector (SPD) and a pre-shower system, followed by electromagnetic (ECAL) and hadron calorimeters.",
"The SPD and pre-shower are designed to distinguish between signals from photons and electrons.",
"The ECAL is constructed from scintillating tiles interleaved with lead tiles.",
"Muons are identified using hits in muon chambers interleaved with iron filters.",
"The signal simulation sample used for this analysis was generated using the Pythia $6.4$ generator [7] configured with the parameters detailed in Ref. [8].",
"The EvtGen [9], Photos [10] and Geant4 [11] packages were used to decay unstable particles, generate QED radiative corrections and simulate interactions in the detector, respectively.",
"The sample consists of events in which at least one ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\rightarrow \\mu ^+ \\mu ^- $ decay takes place with no constraint on the production mechanism.",
"The trigger consists of a hardware stage followed by a software stage, which applies a full event reconstruction.",
"For this analysis, events are selected which have been triggered by a pair of oppositely charged muon candidates, where either one of the muons has a transverse momentum $p_{\\mathrm {T}}\\,{>}\\,\\mbox{${1.8}\\:{{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}}$}$ or one of the pair has $p_{\\mathrm {T}}\\,{>}\\,\\mbox{${0.56}\\:{{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}}$}$ and the other has $p_{\\mathrm {T}}\\,{>}\\,\\mbox{${0.48}\\:{{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}}$}$ .",
"The invariant mass of the candidates is required to be greater than ${2.9}\\:{{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c^2}}$.",
"The photons are not involved in the trigger decision for this analysis.",
"Photons are reconstructed using the electromagnetic calorimeter and identified using a likelihood-based estimator, $\\mathrm {CL}_{\\gamma }$ , constructed from variables that rely on calorimeter and tracking information.",
"For example, in order to reduce the electron background, candidate photon clusters are required not to be matched to the trajectory of a track extrapolated from the tracking system to the cluster position in the calorimeter.",
"For each photon candidate a value of $\\mathrm {CL}_{\\gamma }$ , with a range between 0 (background-like) and 1 (signal-like), is calculated based on simulated signal and background samples.",
"The photons are classified as one of two types: those that have converted to electrons in the material after the dipole magnet and those that have not.",
"Converted photons are identified as clusters in the ECAL with correlated activity in the SPD.",
"In order to account for the different energy resolutions of the two types of photons, the analysis is performed separately for converted and non-converted photons and the results are combined.",
"Photons that convert before the magnet require a different analysis strategy and are not considered here.",
"The photons used to reconstruct the $\\chi _{c}$ candidates are required to have a transverse momentum $p_{\\mathrm {T}}^{\\gamma }\\,{>}\\,\\mbox{${650}\\:{{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c}}$}$ , a momentum $p^{\\gamma }\\,{>}\\,\\mbox{${5}\\:{{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}}$}$ and $\\mathrm {CL}_{\\gamma }\\,{>}\\,0.5$ ; the efficiency of the $\\mathrm {CL}_{\\gamma }$ cut for photons from $\\chi _{c}$ decays is $72\\%$ .",
"All ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidates are reconstructed using the decay ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\rightarrow \\mu ^+ \\mu ^- $ .",
"The muon and ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ identification criteria are identical to those used in Ref.",
"[2]: each track must be identified as a muon with $p_{\\mathrm {T}}\\,{>}\\,700{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c} $ and have a track fit $\\chi ^{2}/\\mathrm {ndf}\\,{<}\\,4$ , where $\\mathrm {ndf}$ is the number of degrees of freedom.",
"The two muons must originate from a vertex with a probability of the vertex fit greater than $0.005$ .",
"In addition, the $\\mu ^+ \\mu ^- $ invariant mass is required to be in the range ${3062}\\,{-}\\,{\\mbox{${3120}\\:{{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2}}$}}$ .",
"The $\\chi _{c}$ candidates are formed from the selected ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidates and photons.",
"The non-prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ contribution arising from $b$ -hadron decays is taken from Ref. [2].",
"For the $\\chi _{c}$ candidates, the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ pseudo-decay time, $t_{z}$ , is used to reduce the contribution from non-prompt decays, by requiring $t_{z}\\,{=}\\,(z_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{-}\\,z_{PV})M_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{/}\\,p_{z}<\\mbox{${0.1}\\:{{\\rm \\,ps}}$}$ , where $M_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ is the reconstructed dimuon invariant mass, $z_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}-z_{\\mathrm {PV}}$ is the $z$ separation of the reconstructed production (primary) and decay vertices of the dimuon, and $p_{z}$ is the $z$ -component of the dimuon momentum.",
"The $z$ -axis is parallel to the beam line in the centre-of-mass frame.",
"Simulation studies show that, with this requirement applied, the remaining fraction of $\\chi _{c}$ from $b$ -hadron decays is about $0.1\\%$ .",
"This introduces an uncertainty much smaller than any of the other systematic or statistical uncertainties evaluated in this analysis and is not considered further.",
"The distributions of the $\\mu ^+ \\mu ^- $ mass of selected ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidates and the mass difference, $\\Delta M\\,{=}\\,M\\left(\\mu ^+ \\mu ^- \\,\\gamma \\right)\\,{-}\\,M\\left(\\mu ^+ \\mu ^- \\right)$ , of the selected $\\chi _{c}$ candidates for the converted and non-converted samples are shown in Fig.",
"REF .",
"The total number of prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidates observed in the data is $\\sim 2.6$ million.",
"The fit procedure to extract the three $\\chi _{c}$ signal yields using Gaussian functions and one common function for the combinatorial background is discussed in Ref. [3].",
"The total number of $\\chi _{c 0}$ , $\\chi _{c 1}$ and $\\chi _{c 2}$ candidates observed are 823, $38\\,630$ and $26\\,114$ respectively.",
"Since the $\\chi _{c 0} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ branching fraction is $\\sim 30$ (17) times smaller than that of the $\\chi _{c 1}$ ($\\chi _{c 2}$ ), the yield of $\\chi _{c 0}$ is small as expected [12].",
"Figure: (a) Invariant mass of the μ + μ - \\mu ^+ \\mu ^- pair forselected J/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} candidates.The solid red curve corresponds to the signal andthe background is shown as a dashed purple curve.",
"(b) and (c) show theΔM=Mμ + μ - γ-Mμ + μ - \\Delta M\\,{=}\\,M\\left(\\mu ^+ \\mu ^- \\,\\gamma \\right)\\,{-}\\,M\\left(\\mu ^+ \\mu ^- \\right)distributions of selected χ c \\chi _{c} candidateswith (b) converted and (c) non-convertedphotons.The upper solid blue curve corresponds to the overall fit functiondescribed in Ref.",
".The lower solid curves correspond to the fitted χ c0 \\chi _{c 0},χ c1 \\chi _{c 1} and χ c2 \\chi _{c 2} contributions from left to right, respectively(the χ c0 \\chi _{c 0} peak is barely visible).The background distribution isshown as a dashed purple curve."
],
[
"Determination of the cross-section ratio",
"The main contributions to the production of prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ arise from direct production and from the feed-down processes $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ and $\\psi {(2S)} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} X$ where $X$ refers to any final state.",
"The cross-section ratio for the production of prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ from $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ decays compared to all prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ can be expressed in terms of the three $\\chi _{cJ}$ $(J=0,1,2)$ signal yields, $N_{\\chi _{cJ}}$ , and the prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ yield, $N_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , as $\\dfrac{\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )}{\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})}& \\, \\approx \\,\\dfrac{\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )}{\\sigma ^{\\mathrm {dir}}({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})+\\sigma (\\psi {(2S)} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} X)+\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )} \\nonumber \\\\& \\nonumber \\\\& \\, \\,{=}\\, \\,\\dfrac{\\displaystyle {\\sum _{J=0}^{J=2}}\\;\\dfrac{N_{\\chi _{cJ}}}{\\epsilon ^{\\chi _{cJ}}_{\\gamma } \\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}} }\\cdot \\dfrac{\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}}{\\epsilon ^{\\chi _{cJ}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}}}{N_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}R_{2S}+\\displaystyle {\\sum _{J=0}^{J=2}}\\;\\dfrac{N_{\\chi _{cJ}}}{\\epsilon ^{\\chi _{cJ}}_{\\gamma } \\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}}}\\left[\\dfrac{\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}}{\\epsilon ^{\\chi _{cJ}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}}-R_{2S}\\right]} \\\\\\multicolumn{2}{l}{\\text{with}}\\\\R_{2S} & \\, \\,{=}\\, \\,\\dfrac{1+f_{2S}}{1+f_{2S}\\dfrac{\\epsilon _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}^{2S}}{\\epsilon _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}^{\\mathrm {dir}}}} \\\\\\multicolumn{2}{l}{\\text{and}}\\\\f_{2S} & \\, \\,{=}\\, \\, \\dfrac{\\sigma (\\psi {(2S)} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} X)}{\\sigma ^{\\mathrm {dir}}({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})}.$ The total prompt $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ cross-section is $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )=\\sum _{J=0}^{J=2} \\; \\sigma _{\\chi _{cJ}}\\cdot {\\cal B}(\\chi _{cJ}\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )$ where $\\sigma _{\\chi _{cJ}}$ is the production cross-section for each $\\chi _{cJ}$ state and $\\cal B$ ($\\chi _{cJ}\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ ) is the corresponding branching fraction.",
"The cross-section ratio $f_{2S}$ is used to link the prompt $\\psi {(2S)}$ contribution to the direct ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ contribution and $R_{2S}$ takes into account their efficiencies.",
"The combination of the trigger, reconstruction and selection efficiencies for direct ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ , for ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ from $\\psi {(2S)}$ decay, and for ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ from $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ decay are $\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , $\\epsilon ^{2S}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , and $\\epsilon ^{\\chi _{cJ}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ respectively.",
"The efficiency to reconstruct and select a photon from a $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ decay, once the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ is already selected, is $\\epsilon ^{\\chi _{cJ}}_{\\gamma }$ and the efficiency for the subsequent selection of the $\\chi _{cJ}$ is $\\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}}$ .",
"The efficiency terms in Eq.",
"(1) are determined using simulated events and are partly validated with control channels in the data.",
"The results for the efficiency ratios $\\epsilon ^{2S}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}/\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , $\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}/\\epsilon ^{\\chi _{cJ}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and the product $\\epsilon ^{\\chi _{cJ}}_{\\gamma }\\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}}$ are discussed in Sect. .",
"The prompt $N_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and $N_{\\chi _{cJ}}$ yields are determined in bins of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ in the range $2\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,15~{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c} $ using the methods described in Refs.",
"[2] and [3] respectively.",
"In Ref.",
"[2] a smaller data sample is used to determine the non-prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ fractions in bins of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and rapidity.",
"These results are applied to the present ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ sample without repeating the full analysis."
],
[
"Efficiencies",
"The efficiencies to reconstruct and select ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ candidates are taken from simulation.",
"The efficiency ratio $\\epsilon ^{2S}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}/\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ is consistent with unity for all $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bins; hence, $R_{2S}$ is set equal to 1 in Eq. ().",
"The ratio of efficiencies $\\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}/\\epsilon ^{\\chi _{cJ}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and the product of efficiencies $\\epsilon ^{\\chi _{cJ}}_{\\gamma }\\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}}$ for the $\\chi _{c 1}$ and $\\chi _{c 2}$ states are shown in Fig.",
"REF .",
"In general these efficiencies are the same for the two states, except at low $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ where the reconstruction and detection efficiencies for $\\chi _{c 2}$ are significantly larger than for $\\chi _{c 1}$ .",
"This difference arises from the effect of the requirement $p_{\\mathrm {T}}^{\\gamma }\\,{>}\\,\\mbox{${650}\\:{{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c}}$}$ which results in more photons surviving from $\\chi _{c 2}$ decays than from $\\chi _{c 1}$ decays.",
"Figure: (a) Ratio of the reconstruction and selection efficiency for directJ/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} compared to J/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} from χ c \\chi _{c} decays,ϵ J/ψ dir /ϵ J/ψ χ c \\epsilon ^{\\mathrm {dir}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}/\\epsilon ^{\\chi _{c}}_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}},and (b) the photon reconstruction and selection efficiencymultiplied by the χ c \\chi _{c} selection efficiency,ϵ γ χ cJ ϵ sel χ cJ \\epsilon ^{\\chi _{cJ}}_{\\gamma }\\epsilon ^{\\chi _{cJ}}_{\\mathrm {sel}},obtained from simulation.The efficiencies are presented separately for theχ c1 \\chi _{c 1} (red triangles) and χ c2 \\chi _{c 2} (inverted blue triangles) states,and as a function of p T J/ψ p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}.The photon detection efficiency obtained using simulation is validated using candidate $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ and $B^+\\rightarrow \\chi _{c} K^+$ (including charge conjugate) decays selected from the same data set as the prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ candidates.",
"The efficiency to reconstruct and select a photon from a $\\chi _{c}$ in $B^+\\rightarrow \\chi _{c} K^+$ decays, $\\epsilon _{\\gamma }$ , is evaluated using $\\epsilon _{\\gamma }\\,{=}\\,\\frac{N_{B^+\\rightarrow \\chi _{c} K^+}}{N_{B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+}}\\times \\frac{{\\cal B}(B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+)}{{\\cal B}(B^+\\rightarrow \\chi _{c} K^+)\\cdot {\\cal B}(\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )}\\times R_{\\epsilon }$ where $N_{B^+\\rightarrow \\chi _{c} K^+}$ and $N_{B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+}$ are the measured yields of $B^+\\rightarrow \\chi _{c} K^+$ and $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ and ${\\cal B}$ are the known branching fractions.",
"The factor $R_{\\epsilon }=1.04\\pm 0.02$ is obtained from simulation and takes into account any differences in the acceptance, trigger, selection and reconstruction efficiencies of the $K$ , ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ , $\\chi _{c}$ (except the photon detection efficiency) and $B^+$ in $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ and $B^+\\rightarrow \\chi _{c} K^+$ decays.",
"All branching fractions are taken from Ref. [12].",
"The $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ branching fraction is ${\\cal B}(B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+)= (1.013\\pm 0.034)\\times 10^{-3}$ .",
"The dominant process for $B^+\\rightarrow \\chi _{c} K^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma K^+$ decays is via the $\\chi _{c 1}$ state, with branching fractions ${\\cal B}(B^+\\rightarrow \\chi _{c 1} K^+)= (4.6\\pm 0.4)\\times 10^{-4}$ and ${\\cal B}(\\chi _{c 1} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )= (34.4\\pm 1.5)\\times 10^{-2}$ ; the contributions from the $\\chi _{c 0}$ and $\\chi _{c 2}$ modes are neglected.",
"The $B^+\\rightarrow \\chi _{c} K^+$ and $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ candidates are selected keeping as many of the selection criteria in common as possible with the main analysis.",
"The ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ selection criteria are the same as for the prompt analysis, apart from the pseudo-decay time requirement.",
"The bachelor kaon is required to have a well measured track ($\\chi ^2/\\mathrm {ndf}<5$ ), a minimum impact parameter $\\chi ^2 $ with respect to all primary vertices of greater than 9 and a momentum greater than $5\\, {\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c} $ .",
"The bachelor is identified as a kaon by the RICH detectors by requiring the difference in log-likelihoods between the kaon and pion hypotheses to be larger than 5.",
"The $B$ candidate is formed from the $\\chi _{c}$ or ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ candidate and the bachelor kaon.",
"The $B$ vertex is required to be well measured ($\\chi ^2/\\mathrm {ndf}<9$ ) and separated from the primary vertex (flight distance $\\chi ^2 >50$ ).",
"The $B$ momentum vector is required to point towards the primary vertex ($\\cos \\theta >0.9999$ , where $\\theta $ is the angle between the $B$ momentum and the direction between the primary and $B$ vertices) and have an impact parameter $\\chi ^2$ smaller than 9.",
"The combinatorial background under the $\\chi _{c}$ peak for the $B^+\\rightarrow \\chi _{c} K^+$ candidates is reduced by requiring the mass difference $\\Delta M_{\\chi _{c}}= M(\\mu ^+ \\mu ^- \\gamma )-M(\\mu ^+ \\mu ^-)< 600{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2} $ .",
"A small number of $B^+\\rightarrow \\chi _{c} K^+$ candidates which form a good $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ candidate are removed by requiring $|M(\\mu ^+ \\mu ^- \\gamma K)-M(\\mu ^+ \\mu ^- K)|>200{\\mathrm {\\,Me\\hspace{-1.00006pt}V\\!/}c^2} $ .",
"The $\\Delta M_{B^+}=M(\\mu ^+ \\mu ^- \\gamma K)-M(\\mu ^+ \\mu ^- \\gamma )$ mass distribution for the $B^+\\rightarrow \\chi _{c} K^+$ candidates is shown in Fig.",
"REF (a); $\\Delta M_{B^+}$ is computed to improve the resolution and hence the signal-to-background ratio.",
"The $B^+\\rightarrow \\chi _{c} K^+$ yield, $142\\pm 15$ candidates, is determined from a fit that uses a Gaussian function to describe the signal peak and a threshold function, $f(x)\\,{=}\\,x^{a}\\!\\left(1-e^{\\frac{m_{0}}{c}\\left(1-x\\right)}\\right)+b\\left(x-1\\right),$ where $x\\,{=}\\,\\Delta M_{B^+}\\,{/}\\,m_{0}$ and $m_0$ , $a$ , $b$ and $c$ are free parameters, to model the background.",
"The reconstructed $B^+$ mass distribution for the $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ candidates is shown in Fig.",
"REF (b).",
"The $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ yield, $8440\\pm 96$ candidates, is determined from a fit that uses a Crystal Ball function [13] to describe the signal peak and an exponential to model the background.",
"The photon efficiency from the observation of $B^+\\rightarrow \\chi _{c} K^+$ and $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ decays is measured to be $\\epsilon _{\\gamma }=(11.3\\pm 1.2\\pm 1.2)\\%$ where the first error is statistical and is dominated by the observed yield of $B^+\\rightarrow \\chi _{c} K^+$ candidates, and the second error is systematic and is given by the uncertainty on the branching fraction $\\cal B$ ($B^+\\rightarrow \\chi _{c 1} K^+$ ).",
"The photon efficiency measured in data can be compared to the photon efficiency, $(11.7\\pm 0.3)\\%$ , obtained using the same procedure on simulated events.",
"The measurements are in good agreement and the uncertainty on the difference between data and simulation is propagated as a $\\pm 14\\%$ relative systematic uncertainty on the photon efficiency in the measurement of $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ .",
"Figure: (a) ReconstructedΔM B + =M(μ + μ - γK)-M(μ + μ - γ)\\Delta M_{B^+}=M(\\mu ^+ \\mu ^- \\gamma K)-M(\\mu ^+ \\mu ^- \\gamma )mass distribution for B + →χ c K + B^+\\rightarrow \\chi _{c} K^+ candidates and(b) the reconstructed B + B^+ mass distribution forB + →J/ψK + B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+ candidates.The LHCb data are shown as solid black points,the full fit functions with a solid blue (upper) curve,the contribution from signal candidates with a dashed red(lower curve) andthe background with a dashed purple curve."
],
[
"Polarisation",
"The simulation used to calculate the efficiencies and, hence, extract the result of Eq.",
"(REF ) assumes that the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ are unpolarised.",
"The effect of polarised states is studied by reweighting the simulated events according to different polarisation scenarios; the results are shown in Table REF .",
"It is also noted that, since the $\\psi {(2S)}$ decays predominantly to ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\pi \\pi $ , with the $\\pi \\pi $ in an $S$ wave state [14], and the $\\psi {(2S)}$ polarisation should not differ significantly from the polarisation of directly produced ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ mesons, the effect of the polarisation can be considered independent of the $\\psi {(2S)} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} X$ contribution [15].",
"The ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ angular distributions are calculated in the helicity frame assuming azimuthal symmetry.",
"This choice of reference frame provides an estimate of the effect of polarisation on the results, pending the direct measurements of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ polarisations.",
"The ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ system is described by the angle $\\theta _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , which is the angle between the directions of the $\\mu ^+ $ in the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ rest frame and the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ in the laboratory frame.",
"The $\\theta _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ distribution depends on the parameter $\\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ which describes the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ polarisation; $\\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}=+1, -1, 0$ corresponds to pure transverse, pure longitudinal and no polarisation respectively.",
"The $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ system is described by three angles: $\\theta ^{\\prime }_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , $\\theta _{\\chi _{c}}$ and $\\phi $ , where $\\theta ^{\\prime }_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ is the angle between the directions of the $\\mu ^+ $ in the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ rest frame and the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ in the $\\chi _{c}$ rest frame, $\\theta _{\\chi _{c}}$ is the angle between the directions of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ in the $\\chi _{c}$ rest frame and the $\\chi _{c}$ in the laboratory frame, and $\\phi $ is the angle between the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ decay plane in the $\\chi _{c}$ rest frame and the plane formed by the $\\chi _{c}$ direction in the laboratory frame and the direction of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ in the $\\chi _{c}$ rest frame.",
"The general expressions for the angular distributions are independent of the choice of polarisation axis (here chosen as the direction of the $\\chi _{c}$ in the laboratory frame) and are detailed in Ref. [4].",
"The angular distributions of the $\\chi _{c}$ states depend on $m_{\\chi _{cJ}}$ which is the azimuthal angular momentum quantum number of the $\\chi _{cJ}$ state.",
"For each simulated event in the unpolarised sample, a weight is calculated from the distributions of $\\theta ^{\\prime }_{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , $\\theta _{\\chi _{c}}$ and $\\phi $ in the various polarisation hypotheses compared to the unpolarised distributions.",
"The weights shown in Table REF are then the average of these per-event weights in the simulated sample.",
"For a given ($|m_{\\chi _{c 1}}|$ , $|m_{\\chi _{c 2}}|$ , $\\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ ) polarisation combination, the central value of the determined cross-section ratio in each $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bin should be multiplied by the number in the table.",
"The maximum effect from the possible polarisation of the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ , $\\chi _{c 1}$ and $\\chi _{c 2}$ mesons is given separately from the systematic uncertainties in Table REF and Fig.",
"REF .",
"Table: Polarisation weights in p T J/ψ p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}} bins for different combinations of theJ/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}, χ c1 \\chi _{c 1} and χ c2 \\chi _{c 2} polarisations.λ J/ψ \\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}} is the J/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} polarisation parameter;λ J/ψ =+1,-1,0\\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}=+1, -1, 0 corresponds tofully transverse, fully longitudinal and no polarisation respectively.m χ cJ m_{\\chi _{cJ}} is theazimuthal angular momentum quantum numbercorresponding tototal angular momentum JJ; Unpol means the χ c \\chi _{c} is unpolarised."
],
[
"Systematic uncertainties",
"The systematic uncertainties detailed below are measured by repeatedly sampling from the distribution of the parameter under consideration.",
"For each sampled value, the cross-section ratio is calculated and the $68.3\\%$ probability interval is determined from the resulting distribution.",
"The statistical errors from the finite number of simulated events used for the calculation of the efficiencies are included as a systematic uncertainty in the final results.",
"The uncertainty is determined by sampling the efficiencies used in Eq.",
"REF according to their errors.",
"The relative systematic uncertainty due to the limited size of the simulation sample is found to be in the range ${(0.3}\\,{-}\\,{3.2)\\%}$ and is given for each $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bin in Table REF .",
"The efficiency extracted from the simulation sample for reconstructing and selecting a photon in $\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma $ decays has been validated using $B^+\\rightarrow \\chi _{c} K^+$ and $B^+\\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} K^+$ decays observed in the data, as described in Sect. .",
"The relative uncertainty between the photon efficiencies measured in the data and simulation, $\\pm 14\\%$ , arises from the finite size of the observed $B^+\\rightarrow \\chi _{c} K^+$ yield and the uncertainty on the known $B^+\\rightarrow \\chi _{c 1} K^+$ branching fraction, and is taken to be the systematic error assigned to the photon efficiency in the measurement of $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ .",
"The relative systematic uncertainty on the cross-section ratio used in Eq.",
"REF is determined by sampling the photon efficiency according to its systematic error.",
"It is found to be in the range ${(6.4}\\,{-}\\,{8.7)\\%}$ and is given for each $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bin in Table REF .",
"The ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ yield used in Eq.",
"REF is corrected for the fraction of non-prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ , taken from Ref. [2].",
"For those $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and rapidity bins used in this analysis and not covered by Ref.",
"[2] ($13\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,14$${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ and $3.5\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5$ ; $11\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,13$${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ and $4\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5$ ; and $14\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,15$${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ ), a linear extrapolation is performed, allowing for asymmetric errors.",
"The systematic uncertainty on the cross-section ratio is determined by sampling the non-prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ fraction according to a bifurcated Gaussian function.",
"The relative systematic uncertainty from the non-prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ fraction is found to be in the range ${(1.3}\\,{-}\\,{10.7)\\%}$ and is given for each $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bin in Table REF .",
"The method used to determine the systematic uncertainty due to the fit procedure in the extraction of the $\\chi _{c}$ yields is discussed in detail in Ref. [3].",
"The uncertainty includes contributions from uncertainties on the fixed parameters, the fit range and the shape of the overall fit function.",
"The overall relative systematic uncertainty from the fit is found to be in the range ${(0.4}\\,{-}\\,{3.2)\\%}$ and is given for each bin of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ in Table REF .",
"The systematic uncertainty related to the calibration of the simulation sample is evaluated by performing the full analysis using simulated events and comparing to the expected cross-section ratio from simulated signal events.",
"The results give an underestimate of $10.9\\%$ in the measurement of the $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ cross-section ratio.",
"This deviation is caused by non-Gaussian signal shapes in the simulation which arise from an untuned calorimeter calibration.",
"These are not seen in the data, which is well described by Gaussian signal shapes.",
"This deviation is included as a systematic error by sampling from the negative half of a Gaussian with zero mean and a width of $10.9\\%$ .",
"The relative uncertainty on the cross-section ratio is found to be in the range ${(6.3}\\,{-}\\,{8.2)\\%}$ and is given for each bin of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ in Table REF .",
"A second check of the procedure was performed using simulated events generated according to the distributions observed in the data, i.e.",
"three overlapping Gaussians and a background shape similar to that in Fig.",
"REF .",
"In this case no evidence for a deviation was observed.",
"Other systematic uncertainties due to the modelling of the detector in the simulation are negligible.",
"In summary, the overall systematic uncertainty is evaluated by simultaneously sampling the deviation of the cross-section ratio from the central value, using the distributions of the cross-section ratios described above.",
"The systematic uncertainty is then determined from the resulting distribution as described earlier in this section.",
"The separate systematic uncertainties are shown in bins of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ in Table REF and the combined uncertainties are shown in Table REF ."
],
[
"Results and conclusions",
"The cross-section ratio, $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ , measured in bins of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ is given in Table REF and shown in Fig.",
"REF .",
"The measurements are consistent with, but suggest a different trend to previous results from CDF using $p\\bar{p}$ collisions at $\\sqrt{s}=1.8$ TeV [5] as shown in Fig.",
"REF (a), and from HERA-$B$ in $p\\mathrm {A}$ collisions at $\\sqrt{s}=41.6$ GeV, with $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ below roughly 5${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ , which gave $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})= 0.188\\pm 0.013 ^{+0.024}_{-0.022}$ [4].",
"Table: Ratio σ(χ c →J/ψγ)/σ(J/ψ)\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}) in bins of p T J/ψ p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}} in the range2<p T J/ψ <15 Ge V/c2\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,15~{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c} and in therapidity range 2.0<y J/ψ <4.52.0\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5.The first error is statistical andthe second is systematic (apart from the polarisation).Also given is the maximum effect of the unknown polarisations on the resultsas described in Sect.",
".Figure: Ratio σ(χ c →J/ψγ)/σ(J/ψ)\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}) in bins ofp T J/ψ p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}} in the range 2<p T J/ψ <15 Ge V/c2\\,{<}\\,p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,15~{\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c} .The LHCb results,in the rapidity range 2.0<y J/ψ <4.52.0\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5 andassuming the production of unpolarised J/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} and χ c \\chi _{c} mesons,are shown with solid black circles and the internalerror bars correspond to the statistical error;the external error bars include the contributionfrom the systematic uncertainties (apart from the polarisation).The lines surrounding the data points show themaximum effect of the unknownJ/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} and χ c \\chi _{c} polarisations on the result.The upper and lower limits correspond to the spin states as described inthe text.The CDF data points, ats=1.8 Te V\\sqrt{s}\\,{=}\\,\\mbox{${1.8}\\:{\\mathrm {\\,Te\\hspace{-1.00006pt}V}}$}in pp ¯p\\bar{p} collisions andin the J/ψ{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} pseudo-rapidity range |η J/ψ |<1.0|\\eta ^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}|<1.0,are shown in (a) with open blue circles .The two hatched bands in (b) correspondto the ChiGen Monte Carlo generatorprediction and NLO NRQCD .Theory predictions, calculated in the LHCb rapidity range $2.0\\,{<}\\,y^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}\\,{<}\\,4.5$ , from the ChiGen Monte Carlo generator [16] and from the NLO NRQCD calculations [17] are shown as hatched bands in Fig.",
"REF (b).",
"The ChiGen Monte Carlo event generator is an implementation of the leading-order colour-singlet model described in Ref. [18].",
"However, since the colour-singlet model implemented in ChiGen does not reliably predict the prompt ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ cross-section, the $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ prediction uses the ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ cross-section measurement from Ref.",
"[2] as the denominator in the cross-section ratio.",
"Figure REF also shows the maximum effect of the unknown ${J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}$ and $\\chi _{c}$ polarisations on the result, shown as lines surrounding the data points.",
"In the first $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ bin, the upper limit corresponds to a spin state combination $(|m_{\\chi _{c 1}}|,|m_{\\chi _{c 2}}|,\\lambda _{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}})$ equal to $(1,2,-1)$ and the lower limit to $(0,1,1)$ .",
"For all subsequent bins, the upper and lower limits correspond to the spin state combinations $(0,2,-1)$ and $(1,0,1)$ respectively.",
"In summary, the ratio of the $\\sigma (\\chi _{c} \\rightarrow {J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}} \\gamma )\\,/\\,\\sigma ({J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}})$ prompt production cross-sections is measured using 36$\\mbox{\\,pb}^{-1}$ of data collected by LHCb during 2010 at a centre-of-mass energy $\\sqrt{s}\\,{=}\\,\\mbox{${7}\\:{\\mathrm {\\,Te\\hspace{-1.00006pt}V}}$}$ .",
"The results provide a significant statistical improvement compared to previous measurements [4], [5].",
"The results are in agreement with the NLO NRQCD model [17] over the full range of $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ .",
"However, there is a significant discrepancy compared to the leading-order colour-singlet model described by the ChiGen Monte Carlo generator [16].",
"At high $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , NLO corrections fall less slowly with $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ and become important, it is therefore not unexpected that the model lies below the data.",
"At low $p_{\\mathrm {T}}^{{J \\hspace{-1.66656pt}/\\hspace{-1.111pt}\\psi \\hspace{1.111pt}}}$ , the data appear to put a severe strain on the colour-singlet model."
],
[
"Acknowledgments",
"We would like to thank L. A. Harland-Lang, W. J. Stirling and K.-T. Chao for supplying the theory predictions for comparison to our data and for many helpful discussions.",
"We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC.",
"We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA).",
"We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.",
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[
[
"Characteristics of Two-Dimensional Quantum Turbulence in a Compressible\n Superfluid"
],
[
"Abstract Under suitable forcing a fluid exhibits turbulence, with characteristics strongly affected by the fluid's confining geometry.",
"Here we study two-dimensional quantum turbulence in a highly oblate Bose-Einstein condensate in an annular trap.",
"As a compressible quantum fluid, this system affords a rich phenomenology, allowing coupling between vortex and acoustic energy.",
"Small-scale stirring generates an experimentally observed disordered vortex distribution that evolves into large-scale flow in the form of a persistent current.",
"Numerical simulation of the experiment reveals additional characteristics of two-dimensional quantum turbulence: spontaneous clustering of same-circulation vortices, and an incompressible energy spectrum with $k^{-5/3}$ dependence for low wavenumbers $k$ and $k^{-3}$ dependence for high $k$."
],
[
"Characteristics of Two-Dimensional Quantum Turbulence in a Compressible Superfluid T. W. Neely College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA A. S. Bradley Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin 9016, New Zealand E. C. Samson College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA S. J. Rooney Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin 9016, New Zealand E. M. Wright College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA K. J. H. Law Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK R. Carretero-González Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA P. G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA M. J. Davis School of Mathematics and Physics, University of Queensland, Qld 4072, Australia B. P. [email protected] College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA Under suitable forcing a fluid exhibits turbulence, with characteristics strongly affected by the fluid's confining geometry.",
"Here we study two-dimensional quantum turbulence in a highly oblate Bose-Einstein condensate in an annular trap.",
"As a compressible quantum fluid, this system affords a rich phenomenology, allowing coupling between vortex and acoustic energy.",
"Small-scale stirring generates an experimentally observed disordered vortex distribution that evolves into large-scale flow in the form of a persistent current.",
"Numerical simulation of the experiment reveals additional characteristics of two-dimensional quantum turbulence: spontaneous clustering of same-circulation vortices, and an incompressible energy spectrum with $k^{-5/3}$ dependence for low wavenumbers $k$ and $k^{-3}$ dependence for high $k$ .",
"A critical distinction between hydrodynamic turbulence in a bulk fluid [1] and in one whose flows are restricted to two dimensions is that energy dissipation at small length scales is generally inhibited in the latter.",
"In two-dimensional (2D) flows subject to small-scale forcing, energy flux is blocked through the small length scales and, instead, energy is transferred towards larger scales, comprising the inverse energy cascade of 2D turbulence [2], [3].",
"Small-scale forcing may thus generate large-scale flows, as seen for instance in dispersal of atmospheric and oceanic particulates [4], flows of soap films [5], [6] and plasmas [7], and Jupiter's Great Red Spot [8], [9].",
"However, the nature of 2D turbulence in quantum fluids is less clear.",
"Progress in 2D quantum turbulence (2DQT) may offer innovative routes to understanding quantum fluid dynamics [10], [11] and aspects of the universality of 2D turbulence.",
"Here we describe an experimental and numerical study of forced and decaying 2DQT in a dilute-gas Bose-Einstein condensate (BEC).",
"Our primary result is the first clear evidence that three key characteristics of 2D turbulence may also simultaneously appear in systems exhibiting 2DQT: (i) emergence of large-scale flow from small-scale forcing, seen experimentally and numerically, (ii) numerical observation of the formation of coherent vortex structures accompanying approximate enstrophy conservation [12], and (iii) numerical observation of an incompressible kinetic energy spectrum with $k^{-5/3}$ dependence for low wavenumbers $k$ and $k^{-3}$ dependence for high $k$ .",
"Our observations are consistent with the notion that an inverse energy cascade can exist in this system.",
"Concepts of significance for 2D turbulence and quantum fluids share a common origin.",
"Analyzing point vortex motion in a bounded domain, Onsager proposed in 1949 that long-lived vortices may develop via mergers of smaller vortices in turbulent flows of a 2D fluid, enabling the remaining vortices to move more freely and thereby maximize system entropy [13], [14].",
"He also proposed that vortices in superfluids have quantized circulation, and implied that turbulent 2D vortex dynamics might be ideally studied in superfluids.",
"However, experimental demonstration of 2DQT has not been reported and has only recently been addressed numerically [15], [16], [17], [18], [19].",
"To experimentally reach the 2DQT regime, we utilize optical and magnetic confinement to create highly oblate BECs [20].",
"A harmonic potential with radial ($r$ ) and axial ($z$ ) trapping frequencies $(\\omega _r/2\\pi ,\\omega _z/2\\pi ) = (8,90)$ Hz confines BECs of up to $\\sim $ 2$\\times 10^6$ $^{87}$ Rb atoms having radial and axial radii of $52 \\,\\mu $ m and $5\\, \\mu $ m respectively.",
"For these conditions, vortex bending and tilting are suppressed [21], leading to 2D vortex dynamics.",
"An annular trap is created with a 23-$\\mu $ m radius, blue-detuned Gaussian laser beam directed axially through the trap center; the beam creates a barrier of height $U_0\\sim 1.5 \\mu _0$ , where $\\mu _0~\\sim 8 \\hbar \\omega _z$ is the BEC chemical potential in the purely harmonic trap.",
"Relative to the phase transition temperature $T_c \\sim 116$ nK, the initial temperature is $T \\sim 0.9 T_c$ .",
"Figure: (a) Timing sequence used to study 2DQT.",
"(b) and (c) Experimental in situ column-density images of the BEC immediately prior to the stir, as viewed (b) in the plane of 2D trapping and (c) along the zz axis.",
"Lighter grayscale shades indicate larger column densities, as in subsequent experimental and numerical density data.",
"(d) Illustration of stirring, the black arrow shows the trajectory of the harmonic trap center relative to the larger fluid-free region created by the laser barrier.",
"(e) In situ image of the BEC 10 s after stirring; vortices are not observable, necessitating an expansion stage to resolve them.At time $t=0$ of each experimental run, a magnetic bias field moves the center of the harmonic trap, but not the central barrier, in one complete 5.7-$\\mu $ m-diameter circle over 333 ms.",
"This motion induces small-scale forcing and nucleation of numerous vortices in a highly disordered distribution, which we identify with 2DQT much as the notion of a `vortex tangle' is identified with 3D quantum turbulence [10], [22], [23].",
"Afterwards, the BEC remains in the annular trap for a variable hold time $t_h$ up to 50 s while the 2DQT decays.",
"At $t_h=$ 1.17 s, the system temperature is reduced to $\\sim $ 0.6$T_c$ in order to decrease rates of thermal damping and vortex-antivortex recombination.",
"At the end of the hold period, the central barrier is ramped off over 250 ms, the BEC is released from the trap, and ballistic expansion of the BEC enlarges the vortex cores such that they are resolvable by absorption imaging.",
"Figure REF illustrates this sequence and shows images of the trapped BEC.",
"Two experimental time sequences of post-stir dynamics are shown in Figure REF (a) and (b), emphasizing the microscopic variability of vortex distributions.",
"Just after the stir ($t_h$ = 0 ms) a disordered vortex distribution appears.",
"Large-scale superflow is evident after $t_h \\approx 0.33$ s and with increasing $t_h$ , as indicated by the large fluid-free hole in the middle of the expanded BEC; this flow evolves into a persistent current by $t_h \\approx 8.17$ s. An optional 3-s hold between barrier ramp-down and BEC release gives the vortices pinned by the central barrier time to separate enough to determine the circulation winding number about the barrier; see Supplemental Material Fig.",
"S1 [27].",
"Our experiment demonstrates that under suitable conditions of forcing and dissipation, a highly disordered vortex distribution can evolve into a large scale flow in an annular trap.",
"However, measuring kinetic energy spectra and in situ vortex dynamics remain forefront experimental problems, motivating us to utilize numerical modeling and analysis for further characterizing 2DQT in a stirred, trapped BEC.",
"BECs admit a first-principles theoretical approach that is numerically tractable, enabling accurate modeling [24].",
"The physical system consists of a large non-condensate component close to thermal equilibrium and a BEC responding both to external forcing and to damping by the non-condensate component.",
"Numerically, we focus on the dynamics of just the BEC.",
"We simulate the experimental stirring procedure using damped Gross-Pitaevskii theory [25].",
"The parameters most readily measured are the total atom number $N$ and temperature $T$ .",
"We have developed an efficient Hartree-Fock scheme for determining the chemical potential $\\mu (N,T)$ and reservoir cutoff energy $\\epsilon _{\\rm cut}(N,T)$ in Ref.",
"[26], and adapt the same procedure to the present experiment, accounting for the shift in the trap minimum caused by the central barrier.",
"We thus find the parameters needed to model the experimental system [27].",
"Figure REF (c) shows simulations that correspond to experimental observations.",
"Here too vortices become pinned to the central barrier to form a persistent current; at $t_h$ =8.17 s, three vortices are pinned to the barrier, as indicated by the corresponding quantum phase profile (see Movie S1 [27]).",
"Ramping off the obstacle beam in the simulation (over 250 ms as in the experiment) gives the column densities shown in Figure REF (d), with density distributions more readily compared to (a) and (b).",
"Development of superflow at $t_h=8.17$ s in (c) leads to a large region of low density in the trap center after barrier ramp-down, as seen in (a), (b), (d).",
"Figure: (a) and (b) 200-μ\\mu m-square experimental column-density images acquired at the hold times t h t_h indicated above the images.",
"Each BEC undergoes ballistic expansion immediately after the central barrier ramp-down in order to resolve the vortex cores.",
"Each image is acquired from a separate experimental run.",
"(c) In situ numerical data (96-μ\\mu m-square images) for the hold times indicated.",
"See also Movie S1 .",
"For each state represented in (c), ramping off the laser barrier in 250ms gives the data shown in (d).Analysis of our numerical simulations further characterizes 2DQT through two distinct dynamical features of the system evolution, namely the development of a logarithmically bilinear kinetic energy spectrum, and the formation of tightly bound, long-lived clusters of vortices with the same sign of circulation.",
"To examine numerically the dependence of the kinetic energy on the wavenumber $k$ at any instant in time, we use the techniques of previous studies [16], [17], [18], [19], [28] for extracting $E^i(k)$ , the portion of a BEC's kinetic energy spectrum that corresponds to an incompressible superfluid component, derived by extracting the divergence-free density-weighted velocity field that embeds vorticity; the curl-free part of this field corresponds to sound waves and acoustic energy.",
"The spectra of Fig.",
"REF are obtained from various times of the simulation and calculated using spatial grids of $1811^2$ points separated by $\\xi /4=0.1\\,\\mu $ m, where the $\\xi $ is the healing length.",
"Each curve shows the spectrum of a 2D slice through $z=0$ , although the spectra are little changed by averaging slices through the BEC.",
"The ultraviolet (large $k$ ) $E^i(k)\\propto k^{-3}$ region of the spectrum is a conspicuous feature once vortices are present.",
"This power law is a universal property of a quantized vortex core in a compressible 2D quantum fluid, as analyzed in Ref.",
"[29], occurring for $k>k_s\\equiv \\xi ^{-1}$ .",
"The associated length scale $\\sim $$2\\pi \\xi $ thus serves to distinguish between scales where the system's physical characteristics are dominated by motion of point-like vortices ($k < k_s$ ), and those where characteristics derive from the structure of individual vortex cores ($k > k_s$ ).",
"The ultraviolet power law only plays a role in the energy spectrum through its amplitude, which is proportional to the total vortex number [29].",
"The only mechanisms that can appreciably change the incompressible energy for $k>k_s$ are creation and loss of free vortices.",
"As stirring injects kinetic energy into the system a Kolmogorov $k^{-5/3}$ power-law spectrum develops in the $k<k_s$ region, and is determined by the vortex configuration [29].",
"This spectrum spans a decade in $k$ -space and is established by the end of the stir ($\\sim $ 331 ms) which is also when total incompressible kinetic energy is maximal.",
"Logarithmically bilinear spectra are also obtained after ramping off the central barrier (not shown).",
"Post stir, Fig.",
"REF (inset) indicates a slow loss of energy with approximate preservation of the Kolmogorov power law.",
"Eventually the system spectrally condenses via generation of a metastable persistent current with three units of circulation.",
"Figure: Log-log plots of E i (k)E^i(k) (per atom) for times (128,160,181,208,331)(128, 160, 181, 208, 331) ms over which forcing occurs, plotted against kξk\\xi with healing length ξ=0.42μ\\xi =0.42\\, \\mu m. Vertical dashed lines indicate k T k_T, k F k_F, k s k_s, and k ξ k_{\\xi }, defined in the text.",
"Red and blue lines indicate E i (k)∝k -5/3 E^i(k) \\propto k^{-5/3} and k -3 k^{-3}, respectively.",
"A spectral peak at 181 ms appears at k F ≈2π/(11ξ)k_F \\approx 2\\pi /(11\\xi ).",
"Inset: log-log plot of E i (k)E^i(k) vs. kξk\\xi (labels omitted) over the same domain as the main plot.",
"From top to bottom, curves show E i (k)∝k -5/3 E^i(k) \\propto k^{-5/3} and k -3 k^{-3} (solid lines), and E i (k)E^i(k) at 331ms, after 14 s of free decay, and for a charge-3 persistent current.Three additional wavenumbers are indicated in Fig.",
"REF .",
"The cross-sectional radial thickness of the BEC approximately corresponds to the length scale $25\\, \\mu $ m $=2\\pi /k_T$ .",
"At high wavenumbers, $k_{\\xi }=2\\pi /\\xi $ corresponds to the scale of the healing length $\\xi =0.42\\, \\mu $ m, the approximate size of the smallest features (e.g.",
"vortex cores) supported by a BEC.",
"Finally, a wavenumber $k_F$ is associated with the peak in the incompressible spectrum at $\\sim 181$ ms, as we now describe.",
"In the classical theory of forced 2D turbulence [3], spectrally localized forcing is related to the injection rates of enstrophy ($\\eta $ ) and energy ($\\epsilon $ ) density as $k_F=\\sqrt{\\eta /\\epsilon }$ [1].",
"In our numerical results, the forcing that precedes the full development of the logarithmically bilinear spectrum is associated with the spectral peak at 181 ms in Fig.",
"REF .",
"At 208 ms the peak has dispersed and the spectrum is already approximately logarithmically bilinear.",
"We estimate the location of $k_F$ from the numerically computed changes in total incompressible kinetic energy, $\\Delta E^i \\approx 2.9\\times 10^{-3}\\,\\mu _0\\!\\cdot \\!",
"N$ , and enstrophy, $\\Delta \\Omega \\approx 0.963\\times 10^{-3}\\,\\mu _0\\!\\cdot \\!",
"N/\\xi ^{2}$ , occurring during this 27-ms time interval.",
"We find $k_F\\equiv \\sqrt{\\Delta \\Omega /\\Delta E^i}=0.57 \\xi ^{-1} \\simeq 2\\pi /(11\\xi )$ , shown in Fig.",
"REF , and coinciding with the spectral peak.",
"The physical mechanism for injection of energy and vorticity into our BECs involves coupling between pairs of opposite-circulation vortices and acoustic energy.",
"In the stirring process, density waves are first generated in the BEC, most prominently where the fluid density approaches zero.",
"These density waves decay to vortices (see Movie S1 [27]) in two characteristic ways.",
"First, a density wave can develop into a localized dark soliton and then decay to a vortex dipole.",
"A second injection mechanism involves the decay of a density wave near the central barrier into a single vortex within the fluid and a partner antivortex pinned by the barrier.",
"Empirically we find that the length scale $\\sim $ 2$\\pi /k_F$ coincides with the separation of phase singularities created from the decay of a localized sound pulse into a vortex dipole, visible in Movie S1 [27].",
"Examining the instances of vortex dipole creation from sound during the stir period, we find dipole lengths $d$ in the range 6.7$\\xi $ to 11$\\xi $ , suggesting an injection of incompressible energy near a wavenumber $k\\sim 2\\pi /d$ .",
"During the stir, there is one case of a vortex dipole annihilating irreversibly to sound at $t=0.23$ s, where the dipole length is $d \\sim 6.7\\xi $ .",
"Two transient events during the nominal constant enstrophy period discussed below correspond to dipole annihilation at $\\sim 6.7\\xi $ and $\\sim 8.9\\xi $ .",
"Furthermore, the superfluid density modulations preceding vortex nucleation at the 181-ms spectral peak have a length scale of approximately $11\\xi \\simeq 2\\pi /k_F$ ; see Fig.",
"S2 [27].",
"Taken together, these observations indicate that forcing involves efficient energy and enstrophy transfer from the compressible to the incompressible fluid components for wavenumbers $k_F\\lesssim k\\lesssim k_s$ .",
"The conservation of enstrophy in 2D turbulence corresponds to conservation of vortex number in 2DQT [17], [29].",
"Between $\\sim $ 300 ms and $\\sim $ 600 ms after the stir begins, the total vortex number is nominally constant in our simulation.",
"When vortex-antivortex annihilation occurs, the resulting sound pulses are quickly (within $\\sim $ 20 ms) refocused by the inhomogeneous density, regenerating the vortices.",
"We thus identify this 300-ms period of nominally constant vortex number with enstrophy conservation.",
"During this period, we numerically observe four instances of the formation of two-vortex clusters (same-sign vortices).",
"Fig.",
"REF shows three two-vortex clusters and a vortex dipole (opposite-sign vortices) present 30 ms after the end of stir.",
"This dipole exists for 20 ms, while the longest-lived of the two-vortex clusters exists for 630 ms.",
"The vortices of this cluster orbit each other 15 times, travel together halfway around the BEC, and eventually dissociate upon colliding with a vortex dipole; see Movie S1 [27] and Fig.",
"REF .",
"We see in Fig.",
"REF (a) and (b) that there are large regions free of vortices, and regions where many vortices are densely packed, which is indicative of clustering, but we are not able to measure the vortex circulation directions to directly confirm this.",
"Clustering was, however, experimentally observed in Ref.",
"[20] in the form of dipolar clusters reproducibly generated by a moving obstacle.",
"In the present case of circular forcing, numerically we observe long-lived clusters emerging intermittently within otherwise irregular flows, in a manner suggestive of Onsager's statistical hydrodynamics result [13].",
"Over the tens of seconds after stirring stops, free vortices decay either by being pinned to the obstacle beam, damping at the outer BEC boundary, or annihilating with free or pinned vortices of opposite sign.",
"This results in the formation of a persistent current, reflecting the net angular momentum imparted by the stirring.",
"The development of this superflow represents the growth of large-scale flow originating from small-scale forcing in an annular geometry, and serves as a check of our numerical procedures and interpretations.",
"The mean number of vortices (pinned and free) for $t_h = 23$ s is 3.5 in the experiment, and 5 in the simulation.",
"For $t_h = 43$ s, these values decline to 2.5 and 3 respectively.",
"Figure: Numerically obtained BEC column density shown for three hold times.",
"In each 96-μ\\mu m-square image, symbols indicate clusters of same-sign vortices, either co-rotating (red), or counter rotating (blue) with the stir.",
"Each shape represents the same cluster at different times.",
"Vortex dipoles and their propagation directions are indicated by green arrows.",
"At t h =648t_h=648 ms, one cluster remains after having traveled clockwise halfway around the BEC.",
"See Fig.",
"S3 .Previous investigations of 2DQT have centered on numerically obtaining kinetic energy spectra, but these have been inconclusive regarding the possibility of an inverse energy cascade in 2DQT, conservation of enstrophy, and correspondence between spectra and vortex dynamics.",
"In our study, simulation of the experimentally realized forcing shows the development of an inertial range.",
"Additionally, small-scale forcing within the trap enables the clustering of vortices; as clustering suppresses vortex-antivortex annihilation, it provides a mechanism to enable enstrophy conservation in a compressible superfluid.",
"Regarding the possibility of a compressible superfluid supporting an inverse energy cascade, energy flux calculations provide the most direct route to analyzing cascades, although such an approach for a trapped BEC with an inhomogeneous density distribution is an open problem [30].",
"However, the following observations are consistent with the existence of an inverse energy cascade in our system near the end of the stir: (i) vortex dipole recombination is suppressed and thus there is little dissipation over a forcing range $k_F$ to $k_s$ ; (ii) $E^i(k)\\propto k^{-3}$ for $k>k_s$ , a range that cannot support incompressible energy flux [29]; (iii) enstrophy is nominally conserved, and kinetic energy spectral developments occur primarily for $k<k_s$ ; and (iv) $E^i(k)\\propto k^{-5/3}$ for $k<k_s$ , a signature of conserved energy transfer across the associated scales.",
"Figure: NO_CAPTIONOur observations indicate that characteristics of forced and decaying 2DQT in compressible quantum fluids may be analogous to those of 2D turbulence in incompressible classical fluids.",
"In particular, growth of large-scale flow, aggregation of vorticity, nominal enstrophy conservation, and an energy spectrum with $k^{-5/3}$ and $k^{-3}$ spectral features occur with suitable forcing.",
"The vortex clusters are suggestive of Onsager's predictions, indicating a new link between Onsager's analysis of 2D point-vortex dynamics [13] and the theory of 2D turbulence initiated by Kraichnan, Leith, and Batchelor [2].",
"Our observations motivate further investigations of 2DQT, with future work focusing on energy fluxes, the roles of dissipation and boundary conditions, and direct experimental observations of turbulent vortex dynamics of a 2D quantum fluid.",
"We thank Colm Connaughton and Sergey Nazarenko for useful discussions, and the US National Science Foundation grants PHY-0855677 and DMS-0806762, the US Army Research Office, The Marsden Fund, The Royal Society of New Zealand, and The New Zealand Foundation for Research, Science, and Technology contracts UOOX0801 and NERF-UOOX0703 for funding.",
"References follow Supplemental Material.",
"Theoretical background.",
"The damped GPE theory used in our simulations can be derived rigorously for the dilute-gas BEC through a detailed treatment of reservoir interactions within the Wigner phase-space representation [31], by neglecting thermal noise.",
"An approximate and practical stochastic Gross-Piteavskii theory can be obtained [32] by: (i) neglecting the particle-conserving reservoir interaction processes (scattering terms) that are known to be small in the quasi-equilibrium regime; and (ii) neglecting the weak spatial and time dependence of the damping parameter.",
"This allows the damping parameter to be calculated a priori, once the reservoir parameters are known.",
"The resulting stochastic Gross-Pitaevskii equation (SGPE) has been used to study spontaneous vortex formation [33] during Bose condensation, and vortex dynamics [21, 26] at high temperature.",
"The SPGPE is derived by treating all atoms above an appropriately chosen energy cutoff $\\epsilon _{\\rm {cut}}$ as thermalized (incoherent region) with temperature $T$ and chemical potential $\\mu $ , leading to a grand-canonical description of the atoms below $\\epsilon _{\\rm {cut}}$ (coherent region).",
"A dimensionless rate $\\gamma \\equiv \\gamma (T,\\mu ,\\epsilon _{\\rm {cut}})$ describes Bose-enhanced collisions between thermal reservoir atoms and atoms in the BEC.",
"However, it is not known how to extract a well-defined condensate orbital from the SGPE trajectories in high-temperature systems containing vortices.",
"To extract physical information about the condensate dynamics we nelgelct the thermal noise to obtain the damped Gross-Piteavskii equation of motion.",
"We thus arrive at a tractable, microscopically determined equation for the condensate orbital, with a priori determined reservoir parameters.",
"After a trivial shift of energy by the chemical potential $\\mu $ we obtain the equation of motion for the wavefunction $\\psi (\\textbf {r},t)$ $i\\hbar \\frac{\\partial \\psi ({\\mathbf {r}},t)}{\\partial t}=(i\\gamma -1)(\\mu -L)\\psi ({\\mathbf {r}},t),$ where the operator $L$ $L\\psi ({\\mathbf {r}},t)\\equiv \\left(-\\frac{\\hbar ^2}{2m}\\nabla ^2+V({\\mathbf {r}})+g|\\psi ({\\mathbf {r}},t)|^2\\right)\\psi ({\\mathbf {r}},t),$ is the generator of GPE evolution for atoms of mass $m$ in an external potential $V({\\mathbf {r}})$ .",
"The interaction parameter is $g=4\\pi \\hbar ^2 a/m$ , for s-wave scattering length $a$ .",
"In most cases the damping parameter is small ($\\gamma \\ll 1$ ), and is typically much smaller than any other timescales characterizing the evolution.",
"In applying our approach to modeling the experiment of the main text, we find the self-consistent parameters $\\mu =34\\hbar \\bar{\\omega }$ , $\\epsilon _{\\rm cut}=83\\hbar \\bar{\\omega }$ , for geometric mean $\\bar{\\omega }=(\\omega _r^2\\omega _z)^{1/3}$ , to describe a system of $N=2.6\\times 10^6$ atoms held at temperature $T=0.9T_c$ in the combined trap, giving the damping parameter $\\gamma =7.96 \\times 10^{-4}$ .",
"The potential is characterized by half-width $\\sigma _0=16.3\\,\\mu $ m, and is well contained within the coherent region: $\\sigma _0\\ll R_{\\rm cut}=\\sqrt{2\\epsilon _{\\rm cut}/m\\omega _r^2}=73\\,\\mu $ m, which is the spatial cutoff imposed by $\\epsilon _{\\rm cut}$ .",
"It thus has no other significant effect on the incoherent region.",
"These parameters give an initial state containing $\\sim 6\\times 10^5$ coherent region atoms, with the remainder in the incoherent region.",
"There are some technical limitations of this approach.",
"First, the number of atoms in the simulations is approximately constant due to the fixed chemical potential of the thermal reservoir, while the BEC number in the experiment first grows as $T/T_c$ is reduced to $\\sim 0.6$ , then decays with a 1/$e$ lifetime of 24(3) s. A related issue is the slow spatial drift of the barrier beam of the experiment, which can decrease the number of vortices that can be stably pinned to the barrier, providing a loss mechanism for the persistent current at long times; this latter aspect of our work will be discussed in a separate publication.",
"However, our procedure is suitable for simulating the BEC conditions early in the stir and hold process, and we do not expect detailed modeling of the evaporative cooling stage to alter our main results.",
"Movie S1: Dynamics of a forced, damped BEC.",
"A movie of the damped Gross-Pitaevskii equation dynamics may be viewed online at [URL to be provided by Physical Review].",
"The movie shows the column density and phase profile in the $z=0$ plane generated for a simulation of damped Gross-Pitaevskii equation for the experimental parameters.",
"A number of events are visible, notably: $t=140$ ms: The first evidence of a boundary vortex appears at the inside boundary of the BEC.",
"This is most apparent in the phase profile.",
"$t=160$ ms: A prominent density modulation feature develops.",
"$t=166$ ms: This time is half-way through stir sequence.",
"$t=190-210$ ms: The first vortices injected into the BEC via sound decay are visible.",
"$t=220$ ms: Many vortices are now found in the bulk superfluid.",
"$t=230$ ms: Vortex dipole annihilation.",
"$t=260$ ms: Two long-lived vortex aggregates have formed ($\\sim $ 3 o'clock and 4 o'clock positions).",
"$t=390-410$ ms: A vortex approaches the inner BEC boundary ($\\sim $ 4 o'clock position) and is captured by the central barrier, increasing the circulation pinned to the barrier by one unit.",
"In the process, a sound pulse is released into the BEC.",
"$t=420-450$ ms: A vortex dipole forms as two vortices of opposite circulation approach each other ($\\sim $ 7 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=480-520$ ms: A vortex dipole forms ($\\sim $ 3 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=590-640$ ms: A vortex dipole forms ($\\sim $ 8 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=600$ ms: Thermalization of the sound field across the whole system has occurred by this time.",
"$t=790$ ms: Vortex dipole annihilation.",
"$t=1000$ ms: A two-vortex cluster, a dipole, and a free vortex collide.",
"Vortex exchange occurs; the collision results in the same vortex structures emerging.",
"$t=1.980$ s: A vortex collision results in a vortex being slightly tilted with respect to the $z$ axis ($\\sim $ 4 o'clock position), showing that this system is not strictly two-dimensional.",
"Note that this vortex returns to an orientation along the $z$ axis within about 50 ms.",
"Figure: Vortices observed in experimental data.",
"Here, BECs are held in the annular trap for 20 s prior to the central barrier ramp down, trap release, expansion, and imaging.",
"In (a) the expansion and imaging procedure takes place immediately after the barrier ramp.",
"The dark region in the center of the BEC corresponds to multiple units of circulation, and is not due to the presence of the barrier itself; i.e., with no circulation about the barrier, a hole would not appear in the BEC after ballistic expansion.",
"In (b) and (c), BECs are held for an additional 3 s in the trap before the expansion imaging procedure.",
"This additional hold time allows the vortices that comprise the region of vorticity shown in (a) to dissociate and become experimentally observable.",
"(d)-(f) show similar images for t h =30t_h = 30 s.Figure: Numerically obtained column density 181 ms after stirring begins.",
"The bar indicates the scale of forcing calculated from the energy and enstrophy injected during the stir (see main text).40-μ\\mu m-square field of view.Figure: Vortex clusters seen for a main sequence hold time of t h =0t_h= 0 ms, followed by the 250-ms barrier ramp-down, and an additional hold time of 0 ms, 230 ms, and 1.23 s in the harmonic trap (96-μ\\mu m-square images).",
"In the left-most image, five two-vortex clusters are seen.",
"Only two or three of these clusters existed at t h =0t_h=0 ms prior to the beam ramp (red circle, blue triangle and square of Fig.",
"4 of the main text), while the remainder formed from vortices that were pinned to the beam and then released into the quantum fluid.",
"By 230 ms after the laser barrier has been ramped completely off, one of the two-vortex clusters has acquired a third vortex (red circle of middle image), and these three vortices orbit each other.",
"This structure eventually grows to form a loose five-vortex cluster shown 1.23 s after the barrier ramp.",
"This cluster persists for about 200 ms.",
"The remaining clusters have dissociated by this time, but three new two-vortex clusters have formed (rectangles).$t=140$ ms: The first evidence of a boundary vortex appears at the inside boundary of the BEC.",
"This is most apparent in the phase profile.",
"$t=160$ ms: A prominent density modulation feature develops.",
"$t=166$ ms: This time is half-way through stir sequence.",
"$t=190-210$ ms: The first vortices injected into the BEC via sound decay are visible.",
"$t=220$ ms: Many vortices are now found in the bulk superfluid.",
"$t=230$ ms: Vortex dipole annihilation.",
"$t=260$ ms: Two long-lived vortex aggregates have formed ($\\sim $ 3 o'clock and 4 o'clock positions).",
"$t=390-410$ ms: A vortex approaches the inner BEC boundary ($\\sim $ 4 o'clock position) and is captured by the central barrier, increasing the circulation pinned to the barrier by one unit.",
"In the process, a sound pulse is released into the BEC.",
"$t=420-450$ ms: A vortex dipole forms as two vortices of opposite circulation approach each other ($\\sim $ 7 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=480-520$ ms: A vortex dipole forms ($\\sim $ 3 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=590-640$ ms: A vortex dipole forms ($\\sim $ 8 o'clock position), then undergoes self-annihilation followed by revival at the outer BEC boundary.",
"$t=600$ ms: Thermalization of the sound field across the whole system has occurred by this time.",
"$t=790$ ms: Vortex dipole annihilation.",
"$t=1000$ ms: A two-vortex cluster, a dipole, and a free vortex collide.",
"Vortex exchange occurs; the collision results in the same vortex structures emerging.",
"$t=1.980$ s: A vortex collision results in a vortex being slightly tilted with respect to the $z$ axis ($\\sim $ 4 o'clock position), showing that this system is not strictly two-dimensional.",
"Note that this vortex returns to an orientation along the $z$ axis within about 50 ms.",
"Experimental winding number determination.",
"Multi-quantum vortices are energetically unstable, thus loose clustering is favorable to perfect co-location of multiple vortices.",
"We make use of this energetic instability in order to determine the size of the persistent current formed and its subsequent time evolution in the experiment, as shown in Figure S1.",
"By enabling extra hold time between beam ramp down and expansion, we can count the total number of free and pinned vortices for any hold time.",
"For long-enough hold times where the number of free vortices has dropped to much less than one per image (on average), the observed vortices can be attributed to current pinned at the barrier, particularly if they appear clustered about the position of the barrier as shown in Fig. S1.",
"We note that observations of such regular structures of vortices, as shown, do not always occur after beam ramp down, and the vortex distribution is often more irregular.",
"As stated in the main text, the mean number of vortices experimentally observed at $t_h = 23$ s is 3.5.",
"If instead we remove the central barrier at the beginning of the hold period, and let the system evolve in a purely harmonic trap for 23 s, the mean numbers of vortices observed becomes 1.2.",
"The fluid circulation is thus maintained at significantly higher levels in the annular trap, justifying a description of this state as a persistent current.",
"Density modulations during forcing.",
"Figure S2 shows the stir-induced density modulations observed in the numerical simulations.",
"These density modulations decay to vortices.",
"As density modulations at a wavenumber $k$ correspond to compressible energy at that wavenumber, we expect an influx of energy into the incompressible regime at a wavenumber that corresponds to these modulations, as discussed in the main text.",
"Vortex clusters and dynamics.",
"In Fig.",
"3 of the main text we show images from our simulation that contain vortex clusters that occur after the completion of the stir in the annular trap.",
"Alternatively, we may examine the distribution of vortex clusters in the gas after the ramp down of the stirring potential to determine whether such clusters might occur in the experimental system subsequent to beam ramp down.",
"Clusters do indeed appear in such numerical data, as Fig.",
"S3 shows, and are not limited to just two vortices per cluster."
]
] | 1204.1102 |
[
[
"The UBV(RI)c colors of the Sun"
],
[
"Abstract Photometric data in the UBV(RI)c system have been acquired for 80 solar analog stars for which we have previously derived highly precise atmospheric parameters Teff, log g, and [Fe/H] using high resolution, high signal-to-noise ratio spectra.",
"UBV and (RI)c data for 46 and 76 of these stars, respectively, are published for the first time.",
"Combining our data with those from the literature, colors in the UBV(RI)c system, with ~0.01 mag precision, are now available for 112 solar analogs.",
"Multiple linear regression is used to derive the solar colors from these photometric data and the spectroscopically derived Teff, log g, and [Fe/H] values.",
"To minimize the impact of systematic errors in the model-dependent atmospheric parameters, we use only the data for the ten stars that most closely resemble our Sun, i.e., the solar twins, and derive the following solar colors: (B-V)=0.653+/-0.005, (U-B)=0.166+/-0.022, (V-R)=0.352+/-0.007, and (V-I)=0.702+/-0.010.",
"These colors are consistent, within the 1 sigma errors, with those derived using the entire sample of 112 solar analogs.",
"We also derive the solar colors using the relation between spectral line-depth ratios and observed stellar colors, i.e., with a completely model-independent approach, and without restricting the analysis to solar twins.",
"We find: (B-V)=0.653+/-0.003, (U-B)=0.158+/-0.009, (V-R)=0.356+/-0.003, and (V-I)=0.701+/-0.003, in excellent agreement with the model-dependent analysis."
],
[
"INTRODUCTION",
"Our Sun is the primary reference in stellar astrophysics.",
"Its fundamental parameters are known with a precision and accuracy far greater than those of any other astronomical object known.",
"Observationally, however, comparing the Sun with the distant stars is not an easy task.",
"Unless dedicated to solar observation, or carefully adapted for that purpose, telescopes and their instruments are designed to collect as much light as possible from faint targets.",
"Any attempt to observe the Sun with the same instrumental setup used to observe the distant stars will suffer from saturation.",
"Fortunately, the Sun as a star can be studied indirectly, in particular using stars that have spectral features very similar to those observed in the solar spectrum, i.e., solar analog stars [6].",
"A wealth of useful information on the physical properties of stars can be inferred from their photometry.",
"Narrow band systems such as Strömgren's $uvby$ -$\\beta $ [33] and systems designed for very large, all-sky surveys such as the $ugriz$ system [10] are in many ways superior, or at least complementary, to the Johnson-Cousins UBV(RI)$_\\mathrm {C}$ system [18], [9].",
"Nevertheless, for historical reasons, one could argue that the latter is still one of the most important ones [2].",
"Much of our knowledge on stars is based on this type of observational data, and it is no surprise that whenever a new photometric system is introduced, transformation equations to the UBV(RI)$_\\mathrm {C}$ system must be determined.",
"Theoretical models can be used to translate photometric data into physical parameters, and vice versa.",
"These relationships, however, must be able to reproduce very well the solar values, given the high precision and accuracy with which the solar properties are known.",
"The problem is that the solar colors cannot be measured directly, i.e., in an identical fashion as those of the distant stars, as explained before.",
"Since they need to be derived indirectly, they are typically very uncertain and not very useful for the calibration of stellar models.",
"Thus the need for refinement in the derivation of the solar colors whenever possible.",
"The solar colors in the UBV(RI)$_\\mathrm {C}$ system, in particular $(B-V)_\\odot $ , have been a subject of debate for many decades.",
"Values found in the literature, as derived by many different authors using a variety of techniques, range from about 0.62 to 0.69.",
"Using the effective temperature ($T_\\mathrm {eff}$ ) versus $(B-V)$ relation by [5], and adopting $\\mathrm {[Fe/H]}=0$ , one finds that this range of $(B-V)$ color corresponds to a $T_\\mathrm {eff}$ range of 216 K. Such large uncertainty in a fundamental zero point calibration represents a severe limitation for reliably constraining stellar models.",
"A few direct measurements of the $(B-V)$ solar color have been made [31], [34], but the range of $(B-V)_\\odot $ values reported is essentially the same as that corresponding to the indirect measurements, suggesting that instrumental effects are very difficult to control [35].",
"Indirectly, the solar colors can be measured using samples of stars with known physical properties and interpolating the correlation between these parameters and observed colors to the solar values , , [16], .",
"In some cases, other types of observations, for example spectroscopic or spectrophotometric, of the Sun and the distant stars, are used, in addition to the stellar photometry, to interpolate to the solar values [8], [32], .",
"The large range of $(B-V)_\\odot $ values found in the literature (0.62–0.69), and the fact that the average error in the $(B-V)$ values typically measured with present-day instrumentation for the distant stars is only about 0.01 mag, suggest that systematic errors are still the dominant source of uncertainty for indirect determinations of $(B-V)_\\odot $ .",
"For older reviews and a complete list of references on $(B-V)_\\odot $ , we refer the reader to [7] and [12].",
"In a more recent revival of the $(B-V)_\\odot $ debate, [28], [29] and [4] have both used the so-called infrared flux method [3] to derive the effective temperatures of large samples of nearby stars with accurate $\\log g$ and $\\mathrm {[Fe/H]}$ values, which were then used to calibrate $\\mathrm {[Fe/H]}$ -dependent $T_\\mathrm {eff}$ -color relations.",
"Using the latter, interpolation to the solar $T_\\mathrm {eff}=5777$ K and $\\mathrm {[Fe/H]}=0$ allowed them to infer $(B-V)_\\odot $ , among other solar colors.",
"Interestingly, even though both groups used the same technique to derive the star's $T_\\mathrm {eff}$ values, their inferred solar colors differ by about 0.03 mag.",
"While [29] suggest a “blue” $(B-V)_\\odot =0.619$ , [4] find a more “red” $(B-V)_\\odot =0.651$ .",
"Although in principle nearly consistent within the 1 $\\sigma $ uncertainties, which are about 0.02 mag for each, this discrepancy has been traced back to a difference in the zero point of the absolute flux calibration in the IRFM.",
"[5] have fine-tuned this absolute calibration and validated their IRFM $T_\\mathrm {eff}$ scale using interferometrically measured stellar angular diameters and HST spectrophotometry.",
"Their implementation of the IRFM gives us the most reliable $T_\\mathrm {eff}$ scale available today, from which they infer $(B-V)_\\odot =0.641\\pm 0.024$ .",
"The relatively large size of the error bar compared to the typical error in $(B-V)$ measurements ($\\simeq 0.01$ mag) is due to the fact that $T_\\mathrm {eff}$ -color relations of a sample of stars covering a wide range of stellar parameters was used, thus propagating small, but non-negligible, systematic errors into the analysis.",
"In recent years, we have undertaken the task of studying solar twin and analog stars, i.e., stars with atmospheric parameters $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ identical and very similar to those of our Sun, respectively.",
"We have carried out spectroscopic surveys in both the southern and northern hemispheres, searching for these stars and performing unprecedentedly high precision spectroscopic analysis [22], , [23], .",
"Surprisingly for us, before the present work, photometric data in the UBV(RI)$_\\mathrm {C}$ system for the solar twins and analogs that we identified were scarce in the literature.",
"For example, only about half of the stars of interest were found in the UBV section of the General Catalogue of Photometric Data and the Hipparcos catalog $(B-V)$ compilation [27].",
"Motivated by this lack of fundamental, very important astronomical data, we have carried out campaigns to measure colors of solar analog stars in the UBV(RI)$_\\mathrm {C}$ system at three different locations, which allowed us to cover the entire sky.",
"In this paper, we present the photometric data acquired and use them along with our spectroscopically determined stellar atmospheric parameters, as well as the high quality spectra themselves, to derive the solar UBV(RI)$_\\mathrm {C}$ colors.",
"We expect these solar colors to be both very precise and accurate because the sample selection guarantees that the impact of systematic errors is small.",
"For the first time, a statistically significant sample of solar twins and analogs with highly precise differential stellar parameters derived from high quality spectra, and homogeneously measured photometry, are available to derive the UBV(RI)$_\\mathrm {C}$ colors of the Sun.",
"lcccccc 0pc SAAO Photometry HIP $V$ $(B-V)$ $(U-B)$ $(V-R)$ $(V-I)$ $N_\\mathrm {obs}$ 348 $8.602\\pm 0.015$ $0.669\\pm 0.015$ $0.124\\pm 0.015$ $0.346\\pm 0.015$ $0.691\\pm 0.015$ 1 996 $8.215\\pm 0.015$ $0.664\\pm 0.015$ $0.163\\pm 0.015$ $0.352\\pm 0.015$ $0.694\\pm 0.015$ 1 1499 $6.474\\pm 0.012$ $0.687\\pm 0.008$ $0.257\\pm 0.008$ $0.368\\pm 0.005$ $0.715\\pm 0.005$ 2 4909 $8.505\\pm 0.025$ $0.636\\pm 0.006$ $0.133\\pm 0.015$ $0.363\\pm 0.013$ $0.689\\pm 0.016$ 2 5134 $8.979\\pm 0.009$ $0.640\\pm 0.004$ $0.081\\pm 0.015$ $0.345\\pm 0.008$ $0.706\\pm 0.013$ 2 6407 $8.625\\pm 0.004$ $0.656\\pm 0.004$ $0.144\\pm 0.015$ $0.360\\pm 0.011$ $0.704\\pm 0.015$ 2 8507 $8.898\\pm 0.004$ $0.651\\pm 0.006$ $0.141\\pm 0.023$ $0.359\\pm 0.011$ $0.730\\pm 0.007$ 2 8841 $9.246\\pm 0.006$ $0.669\\pm 0.004$ $0.157\\pm 0.014$ $0.378\\pm 0.004$ $0.729\\pm 0.015$ 2 9349 $7.991\\pm 0.004$ $0.650\\pm 0.004$ $0.147\\pm 0.008$ $0.343\\pm 0.004$ $0.691\\pm 0.012$ 2 11915 $8.615\\pm 0.008$ $0.653\\pm 0.004$ $0.134\\pm 0.004$ $0.354\\pm 0.004$ $0.699\\pm 0.004$ 2 28336 $8.995\\pm 0.015$ $0.642\\pm 0.015$ $0.130\\pm 0.015$ $0.360\\pm 0.015$ $0.710\\pm 0.015$ 1 30037 $9.162\\pm 0.015$ $0.682\\pm 0.015$ $0.213\\pm 0.015$ $0.361\\pm 0.015$ $0.706\\pm 0.015$ 1 30502 $8.667\\pm 0.015$ $0.664\\pm 0.015$ $0.152\\pm 0.015$ $0.368\\pm 0.015$ $0.707\\pm 0.015$ 1 36512 $7.733\\pm 0.004$ $0.655\\pm 0.005$ $0.120\\pm 0.004$ $0.355\\pm 0.005$ $0.696\\pm 0.015$ 2 38072 $9.222\\pm 0.004$ $0.648\\pm 0.004$ $0.151\\pm 0.011$ $0.363\\pm 0.006$ $0.701\\pm 0.004$ 2 39748 $8.591\\pm 0.006$ $0.615\\pm 0.011$ $0.050\\pm 0.004$ $0.340\\pm 0.008$ $0.681\\pm 0.004$ 2 41317 $7.809\\pm 0.006$ $0.664\\pm 0.008$ $0.159\\pm 0.004$ $0.367\\pm 0.004$ $0.714\\pm 0.011$ 2 43190 $8.508\\pm 0.015$ $0.670\\pm 0.015$ $0.232\\pm 0.015$ $0.370\\pm 0.015$ $0.696\\pm 0.015$ 1 44935 $8.688\\pm 0.015$ $0.654\\pm 0.015$ $0.182\\pm 0.015$ $0.345\\pm 0.015$ $0.684\\pm 0.015$ 1 44997 $8.325\\pm 0.015$ $0.666\\pm 0.015$ $0.191\\pm 0.015$ $0.344\\pm 0.015$ $0.685\\pm 0.015$ 1 46126 $8.514\\pm 0.006$ $0.653\\pm 0.010$ $0.167\\pm 0.006$ $0.354\\pm 0.006$ $0.704\\pm 0.023$ 2 49756 $7.525\\pm 0.015$ $0.644\\pm 0.015$ $0.181\\pm 0.015$ $0.349\\pm 0.015$ $0.672\\pm 0.015$ 1 51258 $7.874\\pm 0.004$ $0.730\\pm 0.004$ $0.344\\pm 0.008$ $0.386\\pm 0.006$ $0.735\\pm 0.004$ 2 54102 $8.653\\pm 0.004$ $0.649\\pm 0.004$ $0.142\\pm 0.015$ $0.346\\pm 0.004$ $0.698\\pm 0.004$ 2 55409 $8.001\\pm 0.010$ $0.657\\pm 0.011$ $0.174\\pm 0.017$ $0.368\\pm 0.007$ $0.720\\pm 0.008$ 2 57291 $7.466\\pm 0.008$ $0.740\\pm 0.004$ $0.354\\pm 0.006$ $0.375\\pm 0.016$ $0.732\\pm 0.013$ 2 59357 $8.655\\pm 0.008$ $0.627\\pm 0.008$ $0.076\\pm 0.004$ $0.344\\pm 0.004$ $0.684\\pm 0.007$ 2 60081 $8.023\\pm 0.007$ $0.696\\pm 0.006$ $0.290\\pm 0.008$ $0.373\\pm 0.007$ $0.702\\pm 0.007$ 2 60370 $6.703\\pm 0.004$ $0.651\\pm 0.004$ $0.148\\pm 0.013$ $0.349\\pm 0.008$ $0.674\\pm 0.008$ 2 60653 $8.731\\pm 0.015$ $0.638\\pm 0.015$ $0.109\\pm 0.015$ $0.358\\pm 0.015$ $0.715\\pm 0.015$ 1 64150 $6.761\\pm 0.007$ $0.688\\pm 0.016$ $0.200\\pm 0.004$ $0.349\\pm 0.017$ $0.694\\pm 0.004$ 2 64497 $8.920\\pm 0.004$ $0.653\\pm 0.004$ $0.176\\pm 0.004$ $0.357\\pm 0.004$ $0.686\\pm 0.013$ 2 64713 $9.250\\pm 0.004$ $0.648\\pm 0.004$ $0.138\\pm 0.004$ $0.355\\pm 0.010$ $0.690\\pm 0.010$ 2 64794 $8.421\\pm 0.006$ $0.640\\pm 0.006$ $0.150\\pm 0.016$ $0.343\\pm 0.010$ $0.696\\pm 0.016$ 2 64993 $8.878\\pm 0.004$ $0.650\\pm 0.007$ $0.155\\pm 0.013$ $0.352\\pm 0.006$ $0.697\\pm 0.007$ 2 66885 $9.309\\pm 0.005$ $0.630\\pm 0.012$ $0.067\\pm 0.018$ $0.366\\pm 0.004$ $0.729\\pm 0.004$ 2 69063 $8.882\\pm 0.006$ $0.632\\pm 0.004$ $0.068\\pm 0.004$ $0.352\\pm 0.004$ $0.706\\pm 0.010$ 2 73815 $8.181\\pm 0.011$ $0.668\\pm 0.008$ $0.171\\pm 0.011$ $0.360\\pm 0.006$ $0.698\\pm 0.004$ 2 74389 $7.768\\pm 0.010$ $0.640\\pm 0.014$ $0.149\\pm 0.007$ $0.349\\pm 0.004$ $0.689\\pm 0.004$ 2 75923 $9.171\\pm 0.014$ $0.664\\pm 0.006$ $0.138\\pm 0.004$ $0.367\\pm 0.018$ $0.718\\pm 0.007$ 2 77883 $8.727\\pm 0.006$ $0.681\\pm 0.004$ $0.214\\pm 0.011$ $0.368\\pm 0.004$ $0.719\\pm 0.004$ 2 79304 $8.670\\pm 0.006$ $0.629\\pm 0.007$ $0.166\\pm 0.006$ $0.353\\pm 0.011$ $0.680\\pm 0.004$ 2 79578 $6.533\\pm 0.033$ $0.678\\pm 0.028$ $0.145\\pm 0.012$ $0.352\\pm 0.008$ $0.699\\pm 0.004$ 2 79672 $5.503\\pm 0.015$ $0.644\\pm 0.015$ $0.157\\pm 0.015$ $0.354\\pm 0.015$ $0.704\\pm 0.015$ 1 82853 $8.993\\pm 0.026$ $0.660\\pm 0.020$ $0.181\\pm 0.004$ $0.396\\pm 0.004$ $0.728\\pm 0.006$ 2 83707 $8.606\\pm 0.015$ $0.655\\pm 0.004$ $0.181\\pm 0.004$ $0.348\\pm 0.011$ $0.699\\pm 0.007$ 2 85042 $6.287\\pm 0.004$ $0.669\\pm 0.004$ $0.207\\pm 0.017$ $0.364\\pm 0.020$ $0.707\\pm 0.051$ 2 85272 $9.121\\pm 0.010$ $0.640\\pm 0.011$ $0.095\\pm 0.012$ $0.368\\pm 0.011$ $0.718\\pm 0.008$ 2 85285 $8.378\\pm 0.019$ $0.632\\pm 0.008$ $0.076\\pm 0.018$ $0.363\\pm 0.019$ $0.715\\pm 0.005$ 2 86796 $5.124\\pm 0.006$ $0.681\\pm 0.028$ $0.296\\pm 0.017$ $0.386\\pm 0.005$ $0.706\\pm 0.005$ 2 89162 $8.903\\pm 0.007$ $0.658\\pm 0.005$ $0.176\\pm 0.004$ $0.363\\pm 0.010$ $0.698\\pm 0.011$ 2 89650 $8.943\\pm 0.010$ $0.644\\pm 0.006$ $0.126\\pm 0.004$ $0.354\\pm 0.010$ $0.679\\pm 0.013$ 2 91332 $7.971\\pm 0.008$ $0.696\\pm 0.009$ $0.263\\pm 0.004$ $0.365\\pm 0.015$ $0.705\\pm 0.004$ 2 102152 $9.188\\pm 0.013$ $0.671\\pm 0.018$ $0.196\\pm 0.023$ $0.382\\pm 0.004$ $0.727\\pm 0.004$ 2 104504 $8.542\\pm 0.015$ $0.640\\pm 0.015$ $0.081\\pm 0.015$ $0.366\\pm 0.015$ $0.696\\pm 0.015$ 1 108996 $8.856\\pm 0.015$ $0.640\\pm 0.015$ $0.165\\pm 0.015$ $0.350\\pm 0.015$ $0.677\\pm 0.015$ 1 118159 $9.017\\pm 0.004$ $0.633\\pm 0.018$ $0.090\\pm 0.015$ $0.355\\pm 0.014$ $0.679\\pm 0.005$ 2 lcccccc 0pc SPM Photometry HIP $V$ $(B-V)$ $(U-B)$ $(V-R)$ $(V-I)$ $N_\\mathrm {obs}$ 348 $8.595\\pm 0.020$ $0.636\\pm 0.022$ $0.095\\pm 0.021$ $0.376\\pm 0.036$ $0.706\\pm 0.022$ 1 996 $8.189\\pm 0.009$ $0.630\\pm 0.012$ $0.114\\pm 0.021$ $0.372\\pm 0.011$ $0.689\\pm 0.011$ 1 2131 $8.923\\pm 0.008$ $0.642\\pm 0.011$ $0.106\\pm 0.023$ $0.376\\pm 0.009$ $0.720\\pm 0.009$ 1 2894 $8.651\\pm 0.018$ $0.659\\pm 0.028$ $0.200\\pm 0.039$ $0.371\\pm 0.034$ $0.703\\pm 0.019$ 1 4909 $8.515\\pm 0.009$ $0.633\\pm 0.011$ $0.106\\pm 0.014$ $0.372\\pm 0.012$ $0.688\\pm 0.013$ 1 5134 $8.969\\pm 0.007$ $0.624\\pm 0.009$ $0.066\\pm 0.017$ $0.365\\pm 0.010$ $0.702\\pm 0.008$ 1 6407 $8.613\\pm 0.021$ $0.649\\pm 0.022$ $0.124\\pm 0.017$ $0.370\\pm 0.035$ $0.704\\pm 0.021$ 1 7245 $8.361\\pm 0.009$ $0.667\\pm 0.013$ $0.186\\pm 0.022$ $0.376\\pm 0.011$ $0.691\\pm 0.011$ 1 8507 $0.126\\pm 0.014$ 1 9349 $8.220\\pm 0.054$ 1 18261 $7.980\\pm 0.027$ $0.616\\pm 0.029$ $0.083\\pm 0.017$ $0.348\\pm 0.047$ $0.670\\pm 0.028$ 1 25670 $8.275\\pm 0.021$ $0.663\\pm 0.023$ $0.167\\pm 0.012$ $0.362\\pm 0.036$ $0.698\\pm 0.024$ 1 28336 $8.998\\pm 0.005$ $0.647\\pm 0.013$ $0.091\\pm 0.013$ $0.366\\pm 0.017$ $0.687\\pm 0.026$ 1 36512 $7.700\\pm 0.011$ $0.668\\pm 0.019$ $0.134\\pm 0.019$ $0.406\\pm 0.029$ $0.685\\pm 0.099$ 1 38072 $9.214\\pm 0.016$ $0.660\\pm 0.025$ $0.125\\pm 0.027$ $0.362\\pm 0.026$ $0.693\\pm 0.021$ 2 41317 $7.798\\pm 0.015$ $0.673\\pm 0.023$ $0.127\\pm 0.029$ $0.386\\pm 0.034$ $0.730\\pm 0.024$ 1 44324 $7.943\\pm 0.010$ $0.620\\pm 0.027$ $0.083\\pm 0.035$ $0.342\\pm 0.011$ $0.674\\pm 0.016$ 4 44935 $8.743\\pm 0.004$ $0.643\\pm 0.006$ $0.203\\pm 0.006$ $0.364\\pm 0.006$ $0.691\\pm 0.006$ 1 44997 $8.370\\pm 0.015$ $0.650\\pm 0.016$ $0.204\\pm 0.007$ $0.383\\pm 0.027$ $0.713\\pm 0.016$ 1 46066 $8.928\\pm 0.007$ $0.664\\pm 0.017$ $0.188\\pm 0.016$ $0.381\\pm 0.008$ $0.717\\pm 0.008$ 2 49572 $9.288\\pm 0.006$ $0.640\\pm 0.007$ $0.138\\pm 0.006$ $0.357\\pm 0.008$ $0.702\\pm 0.007$ 1 49756 $7.540\\pm 0.004$ $0.647\\pm 0.006$ $0.186\\pm 0.006$ $0.361\\pm 0.006$ $0.691\\pm 0.006$ 1 55459 $7.646\\pm 0.004$ $0.646\\pm 0.006$ $0.153\\pm 0.006$ $0.359\\pm 0.006$ $0.692\\pm 0.006$ 1 56948 $8.669\\pm 0.004$ $0.646\\pm 0.006$ $0.180\\pm 0.006$ $0.360\\pm 0.006$ $0.680\\pm 0.006$ 1 59357 $8.752\\pm 0.004$ $0.662\\pm 0.010$ $0.090\\pm 0.011$ $0.388\\pm 0.006$ $0.746\\pm 0.006$ 1 60314 $8.780\\pm 0.008$ $0.665\\pm 0.009$ $0.155\\pm 0.007$ $0.358\\pm 0.011$ $0.676\\pm 0.009$ 1 62175 $8.011\\pm 0.005$ $0.656\\pm 0.006$ $0.194\\pm 0.007$ $0.366\\pm 0.006$ $0.682\\pm 0.006$ 1 64150 $6.883\\pm 0.007$ $0.717\\pm 0.008$ $0.217\\pm 0.006$ $0.402\\pm 0.009$ $0.724\\pm 0.008$ 1 64497 $9.035\\pm 0.004$ $0.701\\pm 0.006$ $0.194\\pm 0.006$ $0.397\\pm 0.006$ 1 64713 $9.297\\pm 0.008$ $0.669\\pm 0.020$ $0.187\\pm 0.019$ $0.367\\pm 0.012$ $0.727\\pm 0.009$ 1 64794 $8.461\\pm 0.020$ $0.667\\pm 0.020$ $0.143\\pm 0.006$ $0.377\\pm 0.030$ $0.710\\pm 0.020$ 1 64993 $8.921\\pm 0.004$ $0.666\\pm 0.009$ $0.172\\pm 0.009$ $0.365\\pm 0.006$ $0.712\\pm 0.006$ 1 66885 $9.274\\pm 0.010$ $0.628\\pm 0.023$ $0.096\\pm 0.024$ $0.353\\pm 0.016$ $0.741\\pm 0.011$ 1 73815 $8.173\\pm 0.005$ $0.668\\pm 0.006$ $0.161\\pm 0.008$ $0.363\\pm 0.006$ $0.683\\pm 0.009$ 2 74341 $8.857\\pm 0.005$ $0.673\\pm 0.014$ $0.165\\pm 0.018$ $0.354\\pm 0.016$ $0.684\\pm 0.009$ 3 74389 $7.760\\pm 0.021$ $0.623\\pm 0.024$ $0.202\\pm 0.024$ $0.352\\pm 0.022$ $0.667\\pm 0.023$ 1 75923 $9.149\\pm 0.005$ $0.651\\pm 0.006$ $0.134\\pm 0.006$ $0.363\\pm 0.007$ $0.689\\pm 0.006$ 1 77883 $8.770\\pm 0.004$ $0.700\\pm 0.006$ $0.227\\pm 0.006$ $0.395\\pm 0.006$ $0.751\\pm 0.006$ 1 78028 $8.651\\pm 0.012$ $0.638\\pm 0.019$ $0.118\\pm 0.024$ $0.355\\pm 0.016$ $0.683\\pm 0.016$ 5 78680 $8.243\\pm 0.013$ $0.626\\pm 0.018$ $0.079\\pm 0.021$ $0.358\\pm 0.016$ $0.698\\pm 0.016$ 3 79186 $8.341\\pm 0.014$ $0.675\\pm 0.028$ $0.140\\pm 0.033$ $0.377\\pm 0.022$ $0.724\\pm 0.018$ 3 79304 $8.718\\pm 0.004$ $0.661\\pm 0.007$ $0.148\\pm 0.008$ $0.365\\pm 0.006$ $0.705\\pm 0.011$ 2 79672 $5.522\\pm 0.019$ $0.680\\pm 0.019$ $0.182\\pm 0.006$ $0.401\\pm 0.019$ $0.773\\pm 0.019$ 2 81512 $9.245\\pm 0.015$ $0.652\\pm 0.017$ $0.140\\pm 0.019$ $0.374\\pm 0.019$ $0.712\\pm 0.016$ 3 85285 $8.356\\pm 0.010$ $0.642\\pm 0.020$ $0.049\\pm 0.021$ $0.362\\pm 0.013$ $0.689\\pm 0.017$ 3 88194 $7.084\\pm 0.011$ $0.656\\pm 0.012$ $0.132\\pm 0.016$ $0.390\\pm 0.016$ $0.723\\pm 0.018$ 3 88427 $9.329\\pm 0.006$ $0.638\\pm 0.013$ $0.089\\pm 0.022$ $0.357\\pm 0.018$ $0.704\\pm 0.007$ 1 89443 $8.843\\pm 0.004$ $0.660\\pm 0.007$ $0.147\\pm 0.013$ $0.380\\pm 0.006$ $0.715\\pm 0.006$ 1 100963 $7.081\\pm 0.009$ $0.651\\pm 0.014$ $0.128\\pm 0.024$ $0.359\\pm 0.010$ $0.708\\pm 0.010$ 1 102152 $9.220\\pm 0.010$ $0.667\\pm 0.017$ $0.158\\pm 0.022$ $0.383\\pm 0.032$ $0.715\\pm 0.016$ 1 104504 $8.594\\pm 0.020$ $0.636\\pm 0.022$ $0.057\\pm 0.014$ $0.361\\pm 0.039$ $0.703\\pm 0.021$ 1 108708 $8.945\\pm 0.017$ $0.659\\pm 0.020$ $0.162\\pm 0.012$ $0.379\\pm 0.036$ $0.707\\pm 0.021$ 1 108996 $8.889\\pm 0.009$ $0.659\\pm 0.010$ $0.103\\pm 0.029$ $0.357\\pm 0.010$ $0.688\\pm 0.014$ 2 109931 $8.956\\pm 0.019$ $0.674\\pm 0.020$ $0.204\\pm 0.020$ $0.388\\pm 0.034$ $0.710\\pm 0.020$ 1 118159 $9.004\\pm 0.004$ $0.627\\pm 0.007$ $0.090\\pm 0.014$ $0.358\\pm 0.007$ $0.681\\pm 0.006$ 1 lccccc 0pc OPD Photometry HIP $V$ $(B-V)$ $(V-R)$ $(V-I)$ $N_\\mathrm {obs}$ 348 $8.604\\pm 0.026$ $0.348\\pm 0.017$ $0.688\\pm 0.035$ 1 996 $8.216\\pm 0.028$ $0.347\\pm 0.018$ $0.666\\pm 0.037$ 1 4909 $8.498\\pm 0.024$ $0.647\\pm 0.015$ $0.356\\pm 0.016$ $0.674\\pm 0.032$ 1 5134 $8.970\\pm 0.025$ $0.623\\pm 0.016$ $0.352\\pm 0.016$ $0.702\\pm 0.034$ 1 6407 $8.617\\pm 0.025$ $0.633\\pm 0.016$ $0.361\\pm 0.016$ $0.705\\pm 0.034$ 1 7245 $8.366\\pm 0.032$ $0.350\\pm 0.021$ $0.685\\pm 0.043$ 1 8507 $8.924\\pm 0.024$ $0.654\\pm 0.016$ $0.370\\pm 0.016$ $0.721\\pm 0.033$ 1 39748 $8.675\\pm 0.058$ $0.614\\pm 0.067$ 2 49756 $7.577\\pm 0.020$ $0.347\\pm 0.010$ $0.678\\pm 0.012$ 2 55409 $8.149\\pm 0.102$ $0.366\\pm 0.020$ $0.713\\pm 0.025$ 3 59357 $8.676\\pm 0.020$ $0.353\\pm 0.009$ $0.682\\pm 0.012$ 3 60653 $8.742\\pm 0.020$ $0.361\\pm 0.009$ $0.694\\pm 0.012$ 3 64497 $8.995\\pm 0.020$ $0.357\\pm 0.009$ $0.688\\pm 0.012$ 3 64713 $9.429\\pm 0.057$ $0.674\\pm 0.065$ 2 64794 $8.867\\pm 0.064$ 3 73815 $8.137\\pm 0.108$ $0.364\\pm 0.021$ $0.708\\pm 0.026$ 1 74341 $8.917\\pm 0.022$ $0.355\\pm 0.011$ $0.699\\pm 0.013$ 3 74389 $7.803\\pm 0.020$ $0.349\\pm 0.009$ $0.678\\pm 0.012$ 3 75923 $9.182\\pm 0.012$ $0.658\\pm 0.011$ $0.368\\pm 0.009$ $0.725\\pm 0.014$ 3 77883 $8.734\\pm 0.011$ $0.691\\pm 0.011$ $0.377\\pm 0.009$ $0.738\\pm 0.013$ 3 79304 $8.703\\pm 0.013$ $0.656\\pm 0.014$ $0.351\\pm 0.011$ $0.692\\pm 0.016$ 3 82853 $8.018\\pm 0.151$ $0.376\\pm 0.031$ $0.779\\pm 0.038$ 1 83707 $8.528\\pm 0.108$ $0.358\\pm 0.021$ $0.699\\pm 0.026$ 1 85272 $9.120\\pm 0.030$ $0.600\\pm 0.022$ $0.347\\pm 0.019$ $0.689\\pm 0.039$ 1 85285 $8.400\\pm 0.025$ $0.606\\pm 0.017$ $0.347\\pm 0.017$ $0.682\\pm 0.034$ 2 88194 $7.171\\pm 0.022$ $0.356\\pm 0.011$ $0.706\\pm 0.014$ 2 89162 $8.882\\pm 0.031$ $0.347\\pm 0.020$ $0.675\\pm 0.041$ 2 89650 $8.946\\pm 0.011$ $0.641\\pm 0.010$ $0.357\\pm 0.009$ $0.697\\pm 0.013$ 5 100963 $7.140\\pm 0.022$ $0.346\\pm 0.011$ $0.694\\pm 0.014$ 2 104504 $8.532\\pm 0.010$ $0.617\\pm 0.008$ $0.363\\pm 0.008$ $0.703\\pm 0.012$ 3 108708 $8.937\\pm 0.010$ $0.659\\pm 0.008$ $0.368\\pm 0.008$ $0.712\\pm 0.012$ 3 108996 $8.881\\pm 0.010$ $0.643\\pm 0.008$ $0.360\\pm 0.008$ $0.703\\pm 0.012$ 3 118159 $9.005\\pm 0.025$ $0.623\\pm 0.016$ $0.344\\pm 0.016$ $0.667\\pm 0.033$ 1"
],
[
"SAMPLE AND PHOTOMETRIC DATA",
"The stars used in this work are listed in Table 4 of [1], who studied the evolution of lithium abundances in Sun-like stars using high resolution, high signal-to-noise ratio spectra acquired by [30] and [21].",
"These spectra were taken using the R. G. Tull coudé spectrograph on the 2.7 m Telescope at McDonald Observatory and the MIKE spectrograph on the 6.5 m Clay/Magellan Telescope at Las Campanas Observatory.",
"The spectral resolution $R=\\lambda /\\Delta \\lambda $ of the spectroscopic data is about 60,000 while the signal-to-noise ratios range from about 150 to 600, with a median value closer to 400.",
"The stellar parameters $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ used in this work are those listed in [1] and they were determined by forcing excitation/ionization equilibrium of iron lines in the stellar spectra.",
"Given the high quality of the data and the careful sample selection, the average errors in the stellar parameters are only $\\Delta T_\\mathrm {eff}=41$ K, $\\Delta \\log g=0.06$ , and $\\Delta \\mathrm {[Fe/H]}=0.03$ , although they are significantly smaller for the stars that are most similar to the Sun.",
"Systematic errors are not included in these error estimates, but we expect them to be very small because of the strictly differential approach we used to derive the atmospheric parameters.",
"All of the objects analyzed in the present study are main-sequence stars, as confirmed by their $\\log g$ values.",
"We refer the reader to [21], [30], and [1] for details on the spectroscopic data reduction, the determination of stellar parameters, and the assessment of errors.",
"UBV(RI)$_\\mathrm {C}$ magnitudes and colors for as many as possible of the stars in [1] were measured at three sites: SAAO (South African Astronomical Observatory), SPM (San Pedro Martir, in México), and OPD (Observatório do Pico dos Dias, in Brazil); 57 stars were observed at SAAO, 55 at SPM, and 33 at OPD.",
"A number of stars were observed at more than one location; the total number of unique stars observed is 80.",
"Below we describe briefly our photometric observations.",
"The SAAO UBV(RI)$_\\mathrm {C}$ observations were made using the 0.5 m telescope and a photomultiplier tube (PMT) based modular photometer at Sutherland [19].",
"The PMT is a Hamamatsu R943-02 Gallium Arsenide tube and it is thermoelectrically cooled to low temperatures to reduce dark counts to minimum levels.",
"Observations were carried out throughout 2010 and 2011 in blocks of several weeks spread over the two years.",
"Observations of both the target objects and E-region standard stars [25] were made each night through the UBV(RI)$_\\mathrm {C}$ filters, mostly alternating between a standard and target objects.",
"The observations were later corrected to the UBV(RI)$_\\mathrm {C}$ system using nightly observations of the E-region standards and current transformation equations that are maintained and regularly updated (about twice a year) at the Observatory.",
"Observations were done only during photometric nights, and any standard star observations that deviate from the standard magnitude by more than $\\pm 0.05$ is not used in the reduction or determination of zero points for transformation to the UBV(RI)$_\\mathrm {C}$ system.",
"Based on the observations of standard stars made in multiple observing nights and/or runs, we estimate an accuracy of about 0.01 mag for the SAAO measurements of visual magnitudes and colors.",
"The E-region stars used in our reductions have visual magnitudes from about $V=5$ to $V=10$ , which is similar to the magnitude range of our observed program stars.",
"The SPM observations were carried out during two runs; eight nights in May 2011 (from the 21st to the 28th) and five nights in October 2011 (from the 20th to the 24th).",
"The San Pedro Martir 0.84 m telescope was used, along with the Mexman filter wheel.",
"During the May run a SITe CCD was used (1024x1024 pixels, gain=4.8 e$^-$ /ADU, readout noise=13e$^-$ ) while in October an e2v-4290 CCD was used (4.5Kx2K pixels, gain=1.7 e$^-$ /ADU, readout noise=3.8e$^-$ ).",
"Sky flat fields were taken at the beginning and end of each night, and bias frames were taken between each observed field.",
"Landolt standards were observed both at the meridian and at large air masses.",
"All the images were bias subtracted and flat field corrected.",
"Cosmic rays were removed using the L.A. Cosmic task [36].",
"Instrumental magnitudes were calculated using the IRAF photcal package and the observations of the standard stars.",
"The OPD photometric data were acquired using the Zeiss 0.6 m telescope at Pico dos Dias Observatory, operated by the Laboratório Nacional de Astrofísica, in Brazil, during the years 2009 and 2010.",
"The instrument used for the observation was the FOTRAP , which consists of a wheel with 6 filters (Johnson-Cousins UBV(RI)$_\\mathrm {C}$ and clear) running at 20 Hz and acquiring data almost simultaneously in all filters.",
"Light from the telescope passes through the filter wheel and then by a set of diaphragms that is used for limiting interference of light from the sky and/or nearby objects.",
"Then the light reaches the Hamamatsu photomultiplier operating at $-25$ degrees Celsius.",
"Throughout the night, various [11] standard stars are observed.",
"Usually, one in every three stars observed was a standard, making sure no star with an airmass greater than 1.5 was observed, following the suggestion by [15], whose photometric reduction method is used in the reduction software of this instrument.",
"The reduction is made using the software “mags.exe,” which was specially written for the instrument FOTRAP, as described in [17].",
"Figure: Difference in color measured at the SPM and SAAO observatories as a function of apparent visual magnitude, as observed from SAAO.Figure: Difference in color measured at the OPD and SAAO observatories as a function of apparent visual magnitude, as observed from SAAO.cccc 0pc Photometry Offsets $\\Delta (\\mathrm {color})$ mean $\\sigma $ $n$ 1 4cSPM–SAAO $\\Delta (B-V)$ 0.011 0.022 28 $\\Delta (U-B)$ 0.001 0.021 29 $\\Delta (V-R)$ 0.021 0.016 28 $\\Delta (V-I)$ 0.011 0.024 27 OPD–SAAO $\\Delta (B-V)$ -0.004 0.016 17 $\\Delta (V-R)$ 0.001 0.007 25 $\\Delta (V-I)$ 0.002 0.015 25 TW–LIT $\\Delta (B-V)$ 0.002 0.011 34 $\\Delta (U-B)$ 0.013 0.034 14 $\\Delta (V-R)$ 0.007 0.006 4 $\\Delta (V-I)$ 0.006 0.013 4 1$n$ is the number of stars in common between the two samples.",
"Figure: Difference in color measured by us and those found in the literature a function of apparent visual magnitude, as found in the literature.The photometric data collected at the three sites described above are given in Tables to .",
"Figure REF shows the comparison of colors measured at the SPM and SAAO observatories for the stars in common.",
"Similarly, Figure REF shows the comparison of OPD and SAAO data.",
"In Table we list the mean offsets and the star-to-star standard deviation of the difference between colors measured at different sites, determined using data for stars in common between the different samples.",
"In most cases the mean differences are compatible with zero within the 1 $\\sigma $ uncertainties, suggesting that any offsets that could be a product of employing different sets of photometric standard stars and/or data reduction procedures are smaller than the observational errors.",
"The only exception is the SPM $(V-R)$ data set, which shows a non-zero mean offset of $0.021\\pm 0.016$ relative to the SAAO data.",
"To prevent this offset from introducing unwanted noise in our solar colors analysis, we corrected the SPM $(V-R)$ colors so that their mean difference with the SAAO data is exactly zero.",
"The values listed in Table , however, are the original ones.",
"For the stars that were observed from more than one location, we adopted a weighted mean of the colors given from each site.",
"The error associated to these average colors corresponds to the sample variance.",
"However, we adopted a minimum photometric error of 0.004 mag to prevent unreasonably small errors arising from numerical artifacts, i.e., from coincidental agreement between the (statistically few) mean values reported from different sites.",
"We also searched for UBV(RI)$_\\mathrm {C}$ photometry in the GCPD [26] and $(B-V)$ colors in the Hipparcos catalog [27] for the stars in [1].",
"We used the latter only if not available in the GCDP.",
"These Hipparcos $(B-V)$ colors correspond to those compiled by the mission team from previously published standard UBV system data sets (i.e., those with flag G in column 39 of the Hipparcos catalog), and not to the colors inferred from transformation equations using Tycho photometry (flag T instead).",
"Sixty-six (66) stars were found with either UBV and/or RI$_\\mathrm {(C)}$ colors previously reported in the literature.",
"Thirty four (34) of these stars were observed by us.",
"However, only four of them have RI$_\\mathrm {(C)}$ data in the literature.",
"Thus, a proper comparison of our measured colors with previously published values can only be done for $(B-V)$ and $(U-B)$ .",
"This comparison is shown in Figure REF .",
"The average difference in $(B-V)$ color between our measurements and those found in the literature is $\\Delta (B-V)=0.002\\pm 0.011$ , i.e., consistent with zero within the 1 $\\sigma $ uncertainty.",
"Moreover, the star-to-star scatter in this comparison (0.011 mag) suggests that the mean error in the measurements of $(B-V)$ is about $0.008$ mag$=0.011/\\sqrt{2}$ , which is identical to the average $(B-V)$ error given in our Table .",
"Thus, our $(B-V)$ error estimates appear to be very reliable.",
"For $(U-B)$ , we derive $\\Delta (U-B)=0.013\\pm 0.034$ , also consistent with zero within the uncertainties.",
"We also computed offsets for $(V-R)$ and $(V-I)$ , but they are based on data for only four stars in common, and are therefore not so reliable.",
"In any case, this comparison suggests that our color measurements are consistent with the UBV(RI)$_\\mathrm {C}$ color scales found in the literature, and therefore with the historically adopted photometric zero points.",
"The color offsets between our data (TW) and previously published values (LIT) are given in the lower section of Table .",
"Figure: Effective temperature versus color relations for our sample of solar twins and analogs.",
"Open circles represent stars with near solar metallicity (-0.05<[ Fe /H]<+0.05-0.05<\\mathrm {[Fe/H]}<+0.05).",
"Upside-down and regular triangles correspond to stars with [ Fe /H]<-0.05\\mathrm {[Fe/H]}<-0.05 and [ Fe /H]>+0.05\\mathrm {[Fe/H]}>+0.05, respectively.",
"Our sample of solar twin stars is shown with filled circles.",
"Average error bars are shown at bottom left of each panel.The stellar parameters and photometry adopted in this work are given in Table .",
"Here we have combined our photometric data with those found in the literature, giving equal weight to each when available for the same star.",
"Objects for which photometric data are published for the first time are assigned mean values and errors from our measurements only.",
"Stars not observed photometrically by us, but found in the literature, are assigned those previously published values.",
"The average errors, in mag, of the measured colors given in Table are $\\Delta (B-V)=0.008$ , $\\Delta (U-B)=0.012$ , $\\Delta (V-R)=0.010$ , and $\\Delta (V-I)=0.010$ ."
],
[
"Color-$T_\\mathrm {eff}$ Relations",
"As is well known, colors are good indicators of $T_\\mathrm {eff}$ , although in many cases they can also be sensitive to other stellar parameters.",
"Figure REF , for example, shows the relation between our UBV(RI)$_\\mathrm {C}$ colors and $T_\\mathrm {eff}$ , which clearly reveals a dependence on a second parameter, namely $\\mathrm {[Fe/H]}$ , although this is much more pronounced for $(B-V)$ than $(V-I)$ .",
"The sensitivity of the UBV(RI)$_\\mathrm {C}$ colors to $\\log g$ is very weak, as will be shown quantitatively later in this section.",
"As noted in Section , the $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ values we used were derived from our high quality spectra, using the standard excitation/ionization equilibrium balance condition for Fe i and Fe ii lines .",
"Although this technique is heavily model-dependent, the fact that our sample stars are all very similar to our Sun allows us to minimize the impact of systematic errors, because they are nearly the same for all of these objects, and because we employ a strictly differential analysis using the solar spectrum as reference.",
"We used the data from Table to perform a multiple linear regression of the following form: $\\mathrm {color}=a_0+a_1(T_\\mathrm {eff}-5777\\,\\mathrm {K})+a_1(\\log g-4.44)+a_2\\mathrm {[Fe/H]}\\ ,$ from which the solar colors are inferred: $\\mathrm {color}_\\odot =a_0$ .",
"The error in the solar color is derived by adding in quadrature the 1 $\\sigma $ scatter of the regression, which takes into account the errors in the observed stellar colors, and the error due to the stellar parameter uncertainties.",
"To calculate the error due to $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ uncertainties, we computed 5000 solar colors using $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ values modified randomly from their mean values, assuming a Gaussian distribution for each of the three stellar atmospheric parameters.",
"The individual errors in these parameters for each star, as listed in Table , were adopted as the standard deviations of these distributions.",
"The 1 $\\sigma $ scatter from the 5000 tests described above was finally adopted as the uncertainty due to errors in the stellar parameters.",
"Two sets of solar colors were derived, a first one using the entire sample of 112 solar analogs, and a second set inferred using only the data for the 10 stars that most closely resemble the Sun (hereafter referred to as the solar twins).",
"The solar twin sample was defined as those stars having their stellar parameters $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ within 1.4 $\\sigma $ from the solar values, where $\\sigma $ is the average error in the stellar parameters of the sample.",
"Modifying slightly the definition of solar twin star has little impact on our results.",
"The multiplicative factor of 1.4 was chosen arbitrarily so that the sample of solar twins could have exactly ten elements.",
"ccc 0pc Solar Colors Inferred from $T_\\mathrm {eff}$ and $\\mathrm {[Fe/H]}$ Measurements color solar twins solar analogs $(B-V)$ $0.653\\pm 0.005$ $0.658\\pm 0.014$ $(U-B)$ $0.166\\pm 0.022$ $0.163\\pm 0.026$ $(V-R)$ $0.352\\pm 0.007$ $0.361\\pm 0.011$ $(V-I)$ $0.702\\pm 0.010$ $0.707\\pm 0.012$ The solar colors derived using the method described above are listed in Table REF .",
"In particular, we find $(B-V)_\\odot =0.653\\pm 0.005$ using only the solar twin data.",
"This value is consistent within the $1\\,\\sigma $ errors with that derived using the entire sample of solar analogs, $(B-V)_\\odot =0.658\\pm 0.014$ .",
"We note, however, that the mean $(B-V)_\\odot $ value increases by 0.005 mag when using the full sample, which suggests that the effective temperatures of non solar twin stars may be slightly overestimated, making the Sun appear redder than it actually is.",
"We find that the mean $(B-V)_\\odot $ value obtained using only solar twins would be in perfect agreement with that derived using the full sample if the $T_\\mathrm {eff}$ values of non solar twin stars were cooler by about 20 K. This implies that systematic errors in the model-dependent determination of stellar parameters from iron line analysis (excitation/ionization equilibrium), although small, are non-negligible for solar analogs, but not so important for the solar twins.",
"This is of course true only when dealing with very high quality spectroscopic data such as those used by [21], [30], and [1], where effective temperatures with a precision comparable to 20 K are possible to achieve.",
"The $(U-B)_\\odot $ color has the largest error of all UBV(RI)$_\\mathrm {C}$ solar colors derived; it is greater than 0.02 mag.",
"This is not at all surprising because $U$ -filter observations and their standardization are known to be very challenging.",
"For $(V-R)$ and $(V-I)$ we derive solar colors with errors below or about 0.01 mag.",
"As with $(B-V)$ , the solar $(V-R)$ and $(V-I)$ colors inferred using solar twins are slightly bluer than those obtained using the full sample of solar analogs.",
"Decreasing the $T_\\mathrm {eff}$ values of non solar twin stars by 20 K gives agreement within 0.001 mag for the mean $(V-I)_\\odot $ values, but the $(V-R)_\\odot $ colors still differ by 0.007 mag.",
"A $T_\\mathrm {eff}$ decrease of about 70 K is necessary to make the $(V-R)_\\odot $ colors agree perfectly.",
"This is highly unlikely given the high precision of our $T_\\mathrm {eff}$ determinations, and therefore suggests that there are small systematic errors affecting our $(V-R)$ colors.",
"Even though $\\log g$ is included in the regression formula (Eq.",
"REF ) for completeness, we find that the precision of our results is not compromised if we choose to neglect it.",
"For example, a regression using only $T_\\mathrm {eff}$ and $\\mathrm {[Fe/H]}$ gives us the same solar $(B-V)$ colors of solar twins or analogs within 0.001 mag.",
"Moreover, the errors are identical to the case when $\\log g$ is also included.",
"This is likely the result of having selected only main-sequence stars for our sample.",
"The impact of $\\mathrm {[Fe/H]}$ on these calculations, however, must not be ignored.",
"A regression on $T_\\mathrm {eff}$ only, or even $T_\\mathrm {eff}$ and $\\log g$ , results in a solar $(B-V)$ color with an error that is about twice as large as that obtained using Eq.",
"REF .",
"As mentioned earlier, the metallicity dependence of UBV(RI)$_\\mathrm {C}$ colors is clearly seen in Figure REF .",
"We also tested regression formulae including quadratic terms, i.e., $T_\\mathrm {eff}^2$ , $\\mathrm {[Fe/H]}^2$ , and $\\log g^2$ , as well as mixed terms such as $T_\\mathrm {eff}\\times \\mathrm {[Fe/H]}$ , but found no noticeable improvements; the 1 $\\sigma $ scatter of the regression (i.e., data minus fit value residuals) did not change by more than 0.001 mag, and the same was true for the mean values obtained for the solar colors."
],
[
"Spectral Line-Depth Ratios",
"The strength of a spectral line depends on many parameters.",
"In addition to the physical conditions of the gas in which the line is formed, which makes the line strength sensitive to the model atmosphere adopted, the properties of the atom, ion, or molecule responsible for the absorption, and those of the transition that produces the line are all directly related to the line strength.",
"Of particular interest for our work is the excitation potential (EP) of the feature.",
"Spectral lines with very different EP values show significantly different sensitivity to $T_\\mathrm {eff}$ .",
"Thus, ratios of depths of spectral line pairs with very different EP values are known to be excellent $T_\\mathrm {eff}$ indicators [14], [13], and therefore they are expected to correlate well with observed colors.",
"cccccccc 0pc $(B-V)_\\odot $ Color Inferred from LDR Measurements $\\lambda _1$ (Å) species $\\lambda _2$ (Å) species $N_\\star $ $\\sigma _\\mathrm {fit}$ $(B-V)_\\odot $ $\\sigma _\\mathrm {ss}$ 5490.15 TiI 5517.53 SiI 90 0.014 0.649 0.005 5690.43 SiI 5727.05 VI 90 0.015 0.649 0.006 5701.11 SiI 5727.05 VI 90 0.012 0.651 0.006 5727.05 VI 5753.65 SiI 90 0.015 0.653 0.002 6007.31 NiI 6046.00 SI 90 0.013 0.654 0.005 6039.73 VI 6046.00 SI 90 0.014 0.645 0.006 6039.73 VI 6052.68 SI 90 0.013 0.654 0.004 6046.00 SI 6062.89 FeI 90 0.012 0.651 0.005 6046.00 SI 6085.27 FeI 42 0.011 0.652 0.007 6046.00 SI 6091.18 TiI 90 0.015 0.645 0.006 6052.68 SI 6062.89 FeI 90 0.012 0.656 0.003 6052.68 SI 6081.44 VI 34 0.014 0.654 0.005 6052.68 SI 6091.18 TiI 90 0.014 0.652 0.004 6090.21 VI 6091.92 SiI 90 0.013 0.655 0.004 6090.21 VI 6106.60 SiI 90 0.011 0.657 0.004 6090.21 VI 6125.03 SiI 90 0.013 0.654 0.003 6090.21 VI 6131.86 SiI 83 0.012 0.656 0.006 6091.92 SiI 6128.99 NiI 90 0.014 0.657 0.006 6106.60 SiI 6119.53 VI 90 0.011 0.656 0.004 6106.60 SiI 6126.22 TiI 90 0.013 0.653 0.003 6106.60 SiI 6135.36 VI 89 0.014 0.653 0.005 6108.12 NiI 6155.14 SiI 90 0.015 0.656 0.003 6119.53 VI 6131.86 SiI 83 0.014 0.651 0.006 6125.03 SiI 6128.99 NiI 90 0.014 0.656 0.002 6128.99 NiI 6131.86 SiI 83 0.013 0.658 0.007 6175.42 NiI 6224.51 VI 84 0.015 0.655 0.003 6176.81 NiI 6224.51 VI 84 0.015 0.655 0.003 6186.74 NiI 6224.51 VI 32 0.014 0.656 0.004 6199.19 VI 6215.15 FeI 74 0.015 0.656 0.004 6204.64 NiI 6224.51 VI 84 0.015 0.655 0.002 6204.64 NiI 6243.11 VI 90 0.013 0.649 0.003 6215.15 FeI 6224.51 VI 84 0.013 0.655 0.003 6215.15 FeI 6251.82 VI 90 0.015 0.655 0.004 6223.99 NiI 6224.51 VI 84 0.014 0.656 0.003 6223.99 NiI 6243.11 VI 89 0.014 0.648 0.006 6224.51 VI 6230.09 NiI 84 0.014 0.655 0.004 6230.09 NiI 6243.11 VI 90 0.013 0.651 0.005 6240.66 FeI 6243.81 SiI 90 0.014 0.654 0.010 6240.66 FeI 6244.48 SiI 90 0.014 0.654 0.008 6243.11 VI 6243.81 SiI 90 0.015 0.649 0.002 6327.60 NiI 6414.99 SiI 35 0.013 0.655 0.005 6414.99 SiI 6498.95 FeI 35 0.013 0.651 0.007 6710.31 FeI 6748.84 SI 89 0.012 0.651 0.006 6710.31 FeI 6757.17 SI 90 0.012 0.650 0.007 6757.17 SI 6806.85 FeI 34 0.011 0.656 0.005 [12] was the first to use line-depth ratios (LDRs) to infer solar colors.",
"As pointed out by him, one of the great advantages of using LDRs is that they are nearly insensitive to the stellar metallicity, at least for nearby thin-disk stars, because, to first approximation, the line strengths scale with $\\mathrm {[Fe/H]}$ regardless of the element producing the line.",
"If, in addition, the line pairs have similar wavelengths and the spectroscopic data used are very homogeneous, particularly concerning the continuum normalization, line-depth ratios are also independent of spectral resolution.",
"Using LDRs to infer the solar colors has also the great advantage of being a completely model-independent approach.",
"Figure: Two examples of observed (B-V)(B-V) color as a function of line-depth ratio measured in our spectra.",
"The wavelengths of the lines used and the 1σ1\\,\\sigma of the linear fit shown with a solid line are given in the upper part of each panel.We used our high resolution, high signal-to-noise ratio spectra to measure as many as possible LDRs for all line pairs listed in the study by [20], and inspected the LDR versus color relations obtained using our photometric data.",
"We fitted a straight line to each of these relations, and computed the standard deviation ($1\\,\\sigma $ ) of the fit minus data residuals.",
"Two examples of these fits are shown in Figure REF .",
"Then we measured the line-depth ratios in our solar spectra, which are in fact reflected Sun-light observations of bright asteroids, and used the LDR versus color fits to infer a solar color for each line pair.",
"The weighted mean and average values obtained from all line pairs used were adopted as the final solar colors.",
"Not all line-pairs listed in the [20] study were used in the end.",
"Line-pairs for which the linear fits had a $1\\,\\sigma $ value with a significant contribution from observational errors in the spectra, were discarded.",
"For example, for $(B-V)$ we excluded the fits with $1\\,\\sigma >0.015$ mag, because the typical $(B-V)$ error is 0.01 mag, and adopting only the pairs with $1\\,\\sigma <0.015$ mag implies that the only pairs that are used are those in which the spectroscopic errors (i.e., the errors in LDR), when propagated to the photometric data in this relation, are similar to the photometric ones, or smaller.",
"Although somewhat arbitrary, this automated procedure eliminates line-pairs which may be affected by blends, continuum normalization issues intrinsic to our data, and/or instrumental imperfections.",
"As an example, in Table REF we list all the line pairs used to derive the solar $(B-V)$ color from LDR versus color relations.",
"For each pair, we provide the number of stars, $N_\\star $ , used to construct the empirical relation and the standard deviation of the fit minus data residuals ($\\sigma _\\mathrm {fit}$ ).",
"Also, for each pair we provide the mean and standard deviation of the $(B-V)$ color that corresponds to the nine reflected Sun-light asteroid observations used for solar reference ($\\sigma _\\mathrm {ss}$ ).",
"This is because each solar spectrum gives us a slightly different value for the line-depth ratio of each pair.",
"Note that the standard deviation from the mean color of our nine solar spectra is very small; in many cases it is below 0.005 mag.",
"The weighted mean and sample variance of the $(B-V)$ solar colors inferred from the 45 line pairs used is finally adopted as the solar color.",
"We used as weights ($w$ ) the inverse of the standard deviations of the LDR versus color fits and the $1\\,\\sigma $ scatter in the colors obtained for the nine solar spectra, added in quadrature, i.e., $1/w=\\sigma _\\mathrm {fit}^2+\\sigma _\\mathrm {ss}^2$ (see Table REF ).",
"Some of the scatter seen in Figure REF could in principle be attributed to $\\mathrm {[Fe/H]}$ and/or $\\log g$ effects.",
"To test this hypothesis, we repeated the procedures described above, but using, instead of a simple linear fit of LDR versus color, a linear regression similar to Eq.",
"REF , replacing the ($T_\\mathrm {eff}-5777$ K) term with the LDR values.",
"The exact same mean value and error was obtained for $(B-V)_\\odot $ , suggesting that the impact of $\\mathrm {[Fe/H]}$ and $\\log g$ on the LDR versus color relations is below the 0.001 mag level.",
"This implies that the scatter seen in Figure REF is dominated by observational errors in both LDR and $(B-V)$ .",
"We computed solar colors for each line pair and for each solar spectrum available.",
"Our spectra come from two different sources and were taken on several different observing runs.",
"The results given in Table REF were obtained using all available data.",
"We made sure that analyzing the data separately per run or per observing site does not improve these results in a significant manner.",
"In fact, due to the lower number of stars available to derive the solar colors, this approach typically gives us larger errors, in some cases about twice as large for $(B-V)$ , for example.",
"Thus, we conclude that small differences in the spectral resolution, sky conditions, and/or instrumental setup have a negligible impact in our derivation of the solar colors.",
"This observation also suggests that the continuum normalization of all our available data is robust and consistent across different data sets as well as observing runs and sites.",
"ccc 0pc Solar Colors Inferred from LDR Measurements color value $N_\\mathrm {pairs}$ $(B-V)$ $0.653\\pm 0.003$ 45 $(U-B)$ $0.158\\pm 0.009$ 42 $(V-R)$ $0.356\\pm 0.003$ 47 $(V-I)$ $0.701\\pm 0.003$ 53 We performed a similar exercise to the one described above to derive the other UBV(RI)$_\\mathrm {C}$ solar colors, which are listed in Table REF .",
"Using the LDR technique, we find $(B-V)_\\odot =0.653\\pm 0.003$ , $(U-B)_\\odot =0.158\\pm 0.009$ , $(V-R)_\\odot =0.356\\pm 0.003$ , and $(V-I)=0.701\\pm 0.003$ , in excellent agreement with the solar colors obtained with the method described in Section REF .",
"The significantly smaller error bars obtained with the LDR method are probably due to the fact that no systematic uncertainties similar to those of the stellar atmospheric parameters affect the LDR measurements, in addition to the fact that the spectroscopic data are very homogeneous and of extremely high quality."
],
[
"CONCLUSIONS",
"The problem of the lack of important photometric data for solar analog stars in the UBV(RI)$_\\mathrm {C}$ system has been addressed and solved with our UBV(RI)$_\\mathrm {C}$ observations of 80 stars very similar to our Sun, for which previously obtained high resolution, high signal-to-noise ratio spectra are available.",
"The combined use of high-quality photometric and spectroscopic data of Sun-like stars allows us to study the Sun as a star without the need to modify or design instruments specifically for the direct observation of the Sun.",
"We have derived the solar colors in the UBV(RI)$_\\mathrm {C}$ system using two different methods.",
"The first one uses the atmospheric parameters $T_\\mathrm {eff}$ , $\\log g$ , and $\\mathrm {[Fe/H]}$ derived using a model-dependent analysis, whereas the second method employs only measurements of spectral line-depth ratios (LDRs) and the observed photometry, thus being completely model-independent.",
"We find excellent agreement for the solar colors derived using these two techniques.",
"In particular, we derive $(B-V)_\\odot =0.653\\pm 0.005$ , but the LDR method gives a smaller error of 0.003 mag.",
"An uncertainty of 0.005 mag in $(B-V)_\\odot $ translates into an error of $\\pm 16$ K in $T_\\mathrm {eff}$ whereas a 0.003 mag uncertainty corresponds to only 9 K. Thus, our highly precise solar colors can be used to constrain stellar models and calibrate effective temperature scales or color-$T_\\mathrm {eff}$ relations at the 10 K level.",
"With respect to the recent debate in the literature concerning the solar $(B-V)$ color, our results favor the “red” value closer to 0.65 mag over the “blue” solar color of about 0.62 mag.",
"Given the high quality of our photometric and spectroscopic data, as well as our careful sample selection and derivation of solar colors from the wealth of available data, we argue that our solar UBV(RI)$_\\mathrm {C}$ colors are the most precise and reliable ones published to date.",
"Along with the solar uvby-$\\beta $ colors derived by [24], precise and accurate solar colors in the historically most important photometric systems are now available.",
"I.R.",
"'s work was performed under contract with the California Institute of Technology (Caltech) funded by NASA through the Sagan Fellowship Program.",
"This paper uses observations made at the South African Astronomical Observatory (SAAO).",
"J.M.",
"acknowledges support from FAPESP (2010/17510-3), CNPq, and USP.",
"lccrccccccl 0pc Adopted Stellar Parameters and Photometry HIP $T_\\mathrm {eff}$ (K) $\\log g$ $\\mathrm {[Fe/H]}$ $V$ $(B-V)$ $(U-B)$ $(V-R)$ $(V-I)$ $N_\\mathrm {obs}$ 1 Source2 348 $5777\\pm 40$ $4.41\\pm 0.07$ $-0.130\\pm 0.024$ $8.600\\pm 0.004$ $0.644\\pm 0.008$ $0.151\\pm 0.026$ $0.348\\pm 0.004$ $0.695\\pm 0.007$ 3 SPM+SAAO+OPD+LIT 996 $5860\\pm 41$ $4.38\\pm 0.07$ $0.000\\pm 0.022$ $8.197\\pm 0.012$ $0.643\\pm 0.019$ $0.146\\pm 0.023$ $0.351\\pm 0.004$ $0.689\\pm 0.006$ 3 SPM+SAAO+OPD 1499 $5756\\pm 44$ $4.37\\pm 0.05$ $0.189\\pm 0.015$ $6.474\\pm 0.012$ $0.680\\pm 0.004$ $0.265\\pm 0.011$ $0.368\\pm 0.004$ $0.714\\pm 0.004$ 2 SAAO+LIT 2131 $5720\\pm 41$ $4.38\\pm 0.07$ $-0.210\\pm 0.026$ $8.923\\pm 0.008$ $0.643\\pm 0.004$ $0.106\\pm 0.023$ $0.355\\pm 0.009$ $0.720\\pm 0.009$ 1 SPM+LIT 2894 $5820\\pm 44$ $4.54\\pm 0.07$ $-0.030\\pm 0.025$ $8.651\\pm 0.018$ $0.659\\pm 0.028$ $0.200\\pm 0.039$ $0.350\\pm 0.034$ $0.703\\pm 0.019$ 1 SPM 4909 $5836\\pm 54$ $4.44\\pm 0.07$ $0.020\\pm 0.024$ $8.512\\pm 0.006$ $0.637\\pm 0.004$ $0.119\\pm 0.013$ $0.357\\pm 0.005$ $0.687\\pm 0.004$ 4 SPM+SAAO+OPD 5134 $5779\\pm 38$ $4.49\\pm 0.07$ $-0.190\\pm 0.023$ $8.973\\pm 0.005$ $0.637\\pm 0.007$ $0.074\\pm 0.007$ $0.346\\pm 0.004$ $0.703\\pm 0.004$ 4 SPM+SAAO+OPD 6407 $5787\\pm 25$ $4.47\\pm 0.03$ $-0.090\\pm 0.011$ $8.624\\pm 0.002$ $0.652\\pm 0.005$ $0.135\\pm 0.010$ $0.360\\pm 0.004$ $0.704\\pm 0.004$ 4 SPM+SAAO+OPD+LIT 7245 $5843\\pm 47$ $4.53\\pm 0.07$ $0.100\\pm 0.023$ $8.361\\pm 0.001$ $0.675\\pm 0.006$ $0.148\\pm 0.017$ $0.354\\pm 0.004$ $0.691\\pm 0.004$ 2 SPM+OPD+LIT 8507 $5720\\pm 55$ $4.44\\pm 0.08$ $-0.080\\pm 0.026$ $8.899\\pm 0.004$ $0.651\\pm 0.004$ $0.130\\pm 0.007$ $0.363\\pm 0.005$ $0.730\\pm 0.004$ 4 SPM+SAAO+OPD 8841 $5676\\pm 45$ $4.50\\pm 0.06$ $-0.120\\pm 0.021$ $9.246\\pm 0.006$ $0.674\\pm 0.004$ $0.157\\pm 0.014$ $0.378\\pm 0.004$ $0.729\\pm 0.015$ 2 SAAO+LIT 9349 $5825\\pm 28$ $4.49\\pm 0.06$ $0.010\\pm 0.017$ $7.992\\pm 0.017$ $0.650\\pm 0.004$ $0.147\\pm 0.008$ $0.343\\pm 0.004$ $0.691\\pm 0.004$ 3 SPM+SAAO 11072 $5897\\pm 84$ $4.01\\pm 0.06$ $-0.037\\pm 0.057$ $5.190\\pm 0.007$ $0.597\\pm 0.004$ $0.120\\pm 0.004$ $0.355\\pm 0.020$ $0.692\\pm 0.020$ 0 LIT 11728 $5738\\pm 30$ $4.37\\pm 0.05$ $0.045\\pm 0.019$ $0.666\\pm 0.015$ 0 LIT 11915 $5793\\pm 43$ $4.45\\pm 0.06$ $-0.050\\pm 0.021$ $8.615\\pm 0.008$ $0.649\\pm 0.004$ $0.134\\pm 0.004$ $0.354\\pm 0.004$ $0.699\\pm 0.004$ 2 SAAO+LIT 12186 $5812\\pm 34$ $4.09\\pm 0.05$ $0.094\\pm 0.040$ $5.785\\pm 0.006$ $0.654\\pm 0.007$ $0.180\\pm 0.028$ $0.360\\pm 0.010$ $0.700\\pm 0.010$ 0 LIT 14614 $5803\\pm 28$ $4.47\\pm 0.03$ $-0.104\\pm 0.016$ $7.840\\pm 0.010$ $0.620\\pm 0.010$ $0.130\\pm 0.010$ 0 LIT 14632 $6026\\pm 42$ $4.28\\pm 0.05$ $0.136\\pm 0.019$ $4.047\\pm 0.008$ $0.595\\pm 0.007$ $0.118\\pm 0.010$ 0 LIT 15457 $5771\\pm 65$ $4.56\\pm 0.02$ $0.078\\pm 0.041$ $4.836\\pm 0.010$ $0.679\\pm 0.007$ $0.188\\pm 0.008$ $0.384\\pm 0.005$ $0.726\\pm 0.008$ 0 LIT 18261 $5891\\pm 34$ $4.44\\pm 0.05$ $0.002\\pm 0.016$ $7.980\\pm 0.027$ $0.628\\pm 0.006$ $0.083\\pm 0.017$ $0.327\\pm 0.047$ $0.670\\pm 0.028$ 1 SPM+LIT 22263 $5826\\pm 48$ $4.54\\pm 0.01$ $0.005\\pm 0.029$ $5.497\\pm 0.012$ $0.632\\pm 0.012$ $0.136\\pm 0.007$ $0.359\\pm 0.005$ $0.691\\pm 0.005$ 0 LIT 22528 $5683\\pm 52$ $4.33\\pm 0.10$ $-0.350\\pm 0.035$ $9.540\\pm 0.010$ $0.630\\pm 0.010$ $0.090\\pm 0.010$ 0 LIT 23835 $5723\\pm 33$ $4.16\\pm 0.05$ $-0.184\\pm 0.017$ $4.920\\pm 0.034$ $0.645\\pm 0.005$ $0.142\\pm 0.006$ 0 LIT 25670 $5755\\pm 37$ $4.38\\pm 0.05$ $0.071\\pm 0.017$ $8.275\\pm 0.021$ $0.659\\pm 0.004$ $0.167\\pm 0.012$ $0.341\\pm 0.036$ $0.698\\pm 0.024$ 1 SPM+LIT 28336 $5713\\pm 61$ $4.53\\pm 0.08$ $-0.170\\pm 0.027$ $8.998\\pm 0.001$ $0.654\\pm 0.007$ $0.108\\pm 0.019$ $0.354\\pm 0.007$ $0.704\\pm 0.010$ 2 SPM+SAAO+LIT 29525 $5715\\pm 61$ $4.41\\pm 0.04$ $-0.005\\pm 0.036$ $6.442\\pm 0.014$ $0.660\\pm 0.015$ $0.160\\pm 0.004$ 0 LIT 30037 $5690\\pm 30$ $4.42\\pm 0.06$ $0.050\\pm 0.030$ $9.162\\pm 0.015$ $0.682\\pm 0.015$ $0.213\\pm 0.015$ $0.361\\pm 0.015$ $0.706\\pm 0.015$ 1 SAAO 30502 $5745\\pm 25$ $4.47\\pm 0.05$ $-0.010\\pm 0.020$ $8.667\\pm 0.015$ $0.664\\pm 0.015$ $0.152\\pm 0.015$ $0.368\\pm 0.015$ $0.707\\pm 0.015$ 1 SAAO 36512 $5740\\pm 15$ $4.50\\pm 0.03$ $-0.092\\pm 0.020$ $7.729\\pm 0.011$ $0.656\\pm 0.004$ $0.121\\pm 0.004$ $0.356\\pm 0.005$ $0.696\\pm 0.004$ 3 SPM+SAAO 38072 $5839\\pm 68$ $4.53\\pm 0.11$ $0.060\\pm 0.037$ $9.222\\pm 0.002$ $0.648\\pm 0.004$ $0.147\\pm 0.009$ $0.362\\pm 0.005$ $0.701\\pm 0.004$ 4 SPM+SAAO 38228 $5693\\pm 58$ $4.52\\pm 0.07$ $0.007\\pm 0.025$ $6.900\\pm 0.010$ $0.682\\pm 0.004$ 0 LIT 39748 $5835\\pm 30$ $4.48\\pm 0.06$ $-0.200\\pm 0.030$ $8.592\\pm 0.009$ $0.615\\pm 0.004$ $0.050\\pm 0.004$ $0.340\\pm 0.023$ $0.681\\pm 0.004$ 4 SAAO+OPD 41317 $5724\\pm 15$ $4.46\\pm 0.03$ $-0.044\\pm 0.020$ $7.807\\pm 0.004$ $0.668\\pm 0.004$ $0.158\\pm 0.004$ $0.365\\pm 0.005$ $0.712\\pm 0.008$ 3 SPM+SAAO+LIT 42438 $5864\\pm 47$ $4.46\\pm 0.09$ $-0.052\\pm 0.026$ $5.631\\pm 0.009$ $0.619\\pm 0.004$ $0.070\\pm 0.004$ 0 LIT 43190 $5775\\pm 30$ $4.37\\pm 0.06$ $0.120\\pm 0.030$ $8.508\\pm 0.015$ $0.670\\pm 0.015$ $0.232\\pm 0.015$ $0.370\\pm 0.015$ $0.696\\pm 0.015$ 1 SAAO 44324 $5934\\pm 49$ $4.51\\pm 0.06$ $-0.020\\pm 0.019$ $7.943\\pm 0.010$ $0.620\\pm 0.027$ $0.083\\pm 0.035$ $0.321\\pm 0.011$ $0.674\\pm 0.016$ 4 SPM 44713 $5784\\pm 35$ $4.36\\pm 0.03$ $0.096\\pm 0.024$ $7.306\\pm 0.006$ $0.668\\pm 0.005$ $0.201\\pm 0.006$ $0.371\\pm 0.005$ $0.713\\pm 0.005$ 0 LIT 44935 $5800\\pm 25$ $4.41\\pm 0.05$ $0.070\\pm 0.020$ $8.739\\pm 0.014$ $0.645\\pm 0.004$ $0.200\\pm 0.007$ $0.344\\pm 0.004$ $0.690\\pm 0.004$ 2 SPM+SAAO 44997 $5782\\pm 29$ $4.52\\pm 0.04$ $0.033\\pm 0.020$ $8.347\\pm 0.023$ $0.659\\pm 0.008$ $0.202\\pm 0.005$ $0.348\\pm 0.008$ $0.698\\pm 0.014$ 2 SPM+SAAO 46066 $5709\\pm 65$ $4.49\\pm 0.12$ $-0.070\\pm 0.039$ $8.928\\pm 0.007$ $0.664\\pm 0.017$ $0.188\\pm 0.016$ $0.360\\pm 0.008$ $0.717\\pm 0.008$ 2 SPM 46126 $5890\\pm 30$ $4.48\\pm 0.06$ $0.140\\pm 0.030$ $8.514\\pm 0.006$ $0.651\\pm 0.004$ $0.149\\pm 0.030$ $0.354\\pm 0.006$ $0.704\\pm 0.023$ 2 SAAO+LIT 49572 $5831\\pm 52$ $4.33\\pm 0.06$ $0.010\\pm 0.021$ $9.288\\pm 0.006$ $0.640\\pm 0.007$ $0.138\\pm 0.006$ $0.336\\pm 0.008$ $0.702\\pm 0.007$ 1 SPM 49756 $5804\\pm 52$ $4.45\\pm 0.07$ $0.041\\pm 0.023$ $7.540\\pm 0.008$ $0.647\\pm 0.004$ $0.185\\pm 0.004$ $0.343\\pm 0.004$ $0.687\\pm 0.007$ 4 SPM+SAAO+OPD+LIT 51258 $5720\\pm 25$ $4.23\\pm 0.05$ $0.360\\pm 0.030$ $7.874\\pm 0.004$ $0.725\\pm 0.004$ $0.344\\pm 0.008$ $0.386\\pm 0.006$ $0.735\\pm 0.004$ 2 SAAO+LIT 52137 $5842\\pm 69$ $4.56\\pm 0.08$ $0.070\\pm 0.026$ $8.640\\pm 0.010$ $0.640\\pm 0.010$ $0.190\\pm 0.010$ 0 LIT 53721 $5916\\pm 53$ $4.48\\pm 0.01$ $0.027\\pm 0.038$ $5.049\\pm 0.015$ $0.606\\pm 0.010$ $0.124\\pm 0.007$ 0 LIT 54102 $5870\\pm 30$ $4.51\\pm 0.06$ $0.040\\pm 0.030$ $8.653\\pm 0.004$ $0.649\\pm 0.004$ $0.142\\pm 0.015$ $0.346\\pm 0.004$ $0.698\\pm 0.004$ 2 SAAO 55409 $5760\\pm 25$ $4.52\\pm 0.05$ $-0.010\\pm 0.020$ $8.002\\pm 0.014$ $0.659\\pm 0.004$ $0.193\\pm 0.011$ $0.368\\pm 0.004$ $0.719\\pm 0.004$ 5 SAAO+OPD+LIT 55459 $5838\\pm 21$ $4.42\\pm 0.03$ $0.038\\pm 0.012$ $7.646\\pm 0.004$ $0.644\\pm 0.004$ $0.147\\pm 0.010$ $0.338\\pm 0.006$ $0.692\\pm 0.006$ 1 SPM+LIT 56948 $5795\\pm 23$ $4.43\\pm 0.03$ $0.023\\pm 0.014$ $8.669\\pm 0.004$ $0.646\\pm 0.006$ $0.180\\pm 0.006$ $0.339\\pm 0.006$ $0.680\\pm 0.006$ 1 SPM 56997 $5559\\pm 65$ $4.53\\pm 0.08$ $-0.030\\pm 0.027$ $5.321\\pm 0.015$ $0.723\\pm 0.013$ $0.261\\pm 0.018$ 0 LIT 57291 $5690\\pm 22$ $4.30\\pm 0.04$ $0.304\\pm 0.030$ $7.466\\pm 0.008$ $0.740\\pm 0.004$ $0.354\\pm 0.006$ $0.375\\pm 0.016$ $0.732\\pm 0.013$ 2 SAAO 59357 $5810\\pm 30$ $4.45\\pm 0.06$ $-0.240\\pm 0.030$ $8.731\\pm 0.039$ $0.618\\pm 0.004$ $0.078\\pm 0.004$ $0.351\\pm 0.010$ $0.715\\pm 0.031$ 6 SPM+SAAO+OPD+LIT 59610 $5899\\pm 62$ $4.34\\pm 0.04$ $-0.034\\pm 0.041$ $7.360\\pm 0.010$ $0.640\\pm 0.010$ 0 LIT 60081 $5811\\pm 21$ $4.38\\pm 0.04$ $0.315\\pm 0.030$ $8.023\\pm 0.007$ $0.696\\pm 0.006$ $0.290\\pm 0.008$ $0.373\\pm 0.007$ $0.702\\pm 0.007$ 2 SAAO 60314 $5874\\pm 72$ $4.52\\pm 0.10$ $0.110\\pm 0.033$ $8.780\\pm 0.008$ $0.649\\pm 0.017$ $0.155\\pm 0.007$ $0.337\\pm 0.011$ $0.676\\pm 0.009$ 1 SPM+LIT 60370 $5897\\pm 25$ $4.46\\pm 0.05$ $0.171\\pm 0.030$ $6.703\\pm 0.004$ $0.641\\pm 0.005$ $0.148\\pm 0.013$ $0.349\\pm 0.008$ $0.674\\pm 0.008$ 2 SAAO+LIT 60653 $5725\\pm 30$ $4.38\\pm 0.06$ $-0.290\\pm 0.030$ $8.735\\pm 0.005$ $0.638\\pm 0.014$ $0.109\\pm 0.015$ $0.360\\pm 0.004$ $0.702\\pm 0.010$ 4 SAAO+OPD 62175 $5849\\pm 51$ $4.43\\pm 0.06$ $0.140\\pm 0.021$ $8.011\\pm 0.005$ $0.661\\pm 0.004$ $0.194\\pm 0.007$ $0.345\\pm 0.006$ $0.682\\pm 0.006$ 1 SPM+LIT 64150 $5755\\pm 41$ $4.39\\pm 0.05$ $0.056\\pm 0.016$ $6.822\\pm 0.061$ $0.676\\pm 0.020$ $0.204\\pm 0.004$ $0.374\\pm 0.013$ $0.700\\pm 0.012$ 3 SPM+SAAO+LIT 64497 $5860\\pm 110$ $4.56\\pm 0.11$ $0.120\\pm 0.037$ $8.978\\pm 0.057$ $0.668\\pm 0.022$ $0.182\\pm 0.008$ $0.362\\pm 0.009$ $0.687\\pm 0.004$ 6 SPM+SAAO+OPD 64713 $5815\\pm 25$ $4.52\\pm 0.05$ $-0.010\\pm 0.020$ $9.260\\pm 0.022$ $0.649\\pm 0.004$ $0.140\\pm 0.010$ $0.351\\pm 0.023$ $0.710\\pm 0.019$ 5 SPM+SAAO+OPD 64794 $5743\\pm 61$ $4.33\\pm 0.08$ $-0.100\\pm 0.027$ $8.428\\pm 0.041$ $0.637\\pm 0.006$ $0.141\\pm 0.008$ $0.344\\pm 0.028$ $0.701\\pm 0.013$ 6 SPM+SAAO+OPD+LIT 64993 $5875\\pm 30$ $4.56\\pm 0.06$ $0.090\\pm 0.030$ $8.900\\pm 0.022$ $0.656\\pm 0.008$ $0.166\\pm 0.008$ $0.348\\pm 0.013$ $0.706\\pm 0.009$ 4 SPM+SAAO+OPD 66618 $5951\\pm 25$ $4.35\\pm 0.05$ $0.135\\pm 0.030$ $6.962\\pm 0.004$ $0.622\\pm 0.004$ $0.175\\pm 0.005$ 0 LIT 66885 $5685\\pm 30$ $4.48\\pm 0.06$ $-0.380\\pm 0.030$ $9.302\\pm 0.014$ $0.635\\pm 0.004$ $0.077\\pm 0.014$ $0.364\\pm 0.014$ $0.730\\pm 0.005$ 4 SPM+SAAO+OPD+LIT 69063 $5670\\pm 30$ $4.31\\pm 0.06$ $-0.450\\pm 0.030$ $8.882\\pm 0.006$ $0.623\\pm 0.004$ $0.068\\pm 0.004$ $0.352\\pm 0.004$ $0.706\\pm 0.010$ 2 SAAO+LIT 71683 $5840\\pm 22$ $4.33\\pm 0.04$ $0.228\\pm 0.030$ $0.002\\pm 0.008$ $0.653\\pm 0.023$ $0.230\\pm 0.004$ $0.362\\pm 0.010$ $0.693\\pm 0.010$ 0 LIT 72659 $5517\\pm 67$ $4.56\\pm 0.09$ $-0.117\\pm 0.033$ $4.718\\pm 0.008$ $0.748\\pm 0.019$ $0.231\\pm 0.019$ 0 LIT 73815 $5803\\pm 33$ $4.34\\pm 0.05$ $0.020\\pm 0.016$ $8.174\\pm 0.003$ $0.663\\pm 0.006$ $0.164\\pm 0.005$ $0.352\\pm 0.009$ $0.696\\pm 0.006$ 5 SPM+SAAO+OPD+LIT 74341 $5853\\pm 57$ $4.51\\pm 0.08$ $0.090\\pm 0.026$ $8.860\\pm 0.013$ $0.673\\pm 0.013$ $0.165\\pm 0.018$ $0.348\\pm 0.010$ $0.689\\pm 0.007$ 6 SPM+OPD 74389 $5859\\pm 24$ $4.48\\pm 0.04$ $0.105\\pm 0.030$ $7.773\\pm 0.014$ $0.636\\pm 0.014$ $0.153\\pm 0.014$ $0.349\\pm 0.004$ $0.687\\pm 0.005$ 6 SPM+SAAO+OPD 75923 $5775\\pm 25$ $4.56\\pm 0.05$ $-0.020\\pm 0.020$ $9.156\\pm 0.012$ $0.658\\pm 0.006$ $0.137\\pm 0.004$ $0.353\\pm 0.013$ $0.704\\pm 0.015$ 6 SPM+SAAO+OPD 77052 $5697\\pm 33$ $4.54\\pm 0.02$ $0.035\\pm 0.023$ $5.868\\pm 0.011$ $0.686\\pm 0.011$ $0.234\\pm 0.010$ $0.380\\pm 0.010$ $0.740\\pm 0.010$ 0 LIT 77466 $5700\\pm 56$ $4.40\\pm 0.09$ $-0.280\\pm 0.028$ $9.204\\pm 0.009$ $0.647\\pm 0.005$ $0.120\\pm 0.014$ 0 LIT 77740 $5900\\pm 19$ $4.45\\pm 0.04$ $0.125\\pm 0.030$ $0.628\\pm 0.012$ 0 LIT 77883 $5695\\pm 25$ $4.39\\pm 0.05$ $0.040\\pm 0.020$ $8.755\\pm 0.020$ $0.687\\pm 0.008$ $0.224\\pm 0.005$ $0.371\\pm 0.004$ $0.729\\pm 0.014$ 6 SPM+SAAO+OPD 78028 $5879\\pm 98$ $4.57\\pm 0.12$ $-0.030\\pm 0.041$ $8.651\\pm 0.012$ $0.638\\pm 0.019$ $0.118\\pm 0.024$ $0.334\\pm 0.016$ $0.683\\pm 0.016$ 5 SPM 78680 $5923\\pm 67$ $4.57\\pm 0.08$ $-0.000\\pm 0.027$ $8.243\\pm 0.013$ $0.626\\pm 0.018$ $0.079\\pm 0.021$ $0.337\\pm 0.016$ $0.698\\pm 0.016$ 3 SPM 79186 $5709\\pm 48$ $4.27\\pm 0.08$ $-0.120\\pm 0.024$ $8.341\\pm 0.014$ $0.676\\pm 0.004$ $0.140\\pm 0.033$ $0.356\\pm 0.022$ $0.724\\pm 0.018$ 3 SPM+LIT 79304 $5945\\pm 30$ $4.53\\pm 0.06$ $0.110\\pm 0.030$ $8.703\\pm 0.021$ $0.646\\pm 0.015$ $0.160\\pm 0.009$ $0.347\\pm 0.004$ $0.683\\pm 0.008$ 7 SPM+SAAO+OPD 79578 $5860\\pm 33$ $4.53\\pm 0.07$ $0.072\\pm 0.030$ $6.533\\pm 0.033$ $0.647\\pm 0.007$ $0.145\\pm 0.012$ $0.352\\pm 0.008$ $0.699\\pm 0.004$ 2 SAAO+LIT 79672 $5822\\pm 9$ $4.45\\pm 0.02$ $0.051\\pm 0.020$ $5.510\\pm 0.009$ $0.650\\pm 0.004$ $0.177\\pm 0.004$ $0.357\\pm 0.005$ $0.691\\pm 0.011$ 3 SPM+SAAO+LIT 80337 $5881\\pm 33$ $4.53\\pm 0.02$ $0.033\\pm 0.022$ $5.391\\pm 0.012$ $0.628\\pm 0.011$ $0.108\\pm 0.048$ $0.353\\pm 0.005$ $0.681\\pm 0.006$ 0 LIT 81512 $5790\\pm 58$ $4.46\\pm 0.07$ $-0.020\\pm 0.025$ $9.245\\pm 0.015$ $0.652\\pm 0.017$ $0.140\\pm 0.019$ $0.353\\pm 0.019$ $0.712\\pm 0.016$ 3 SPM 82853 $5640\\pm 30$ $4.21\\pm 0.06$ $-0.180\\pm 0.030$ $8.965\\pm 0.163$ $0.660\\pm 0.007$ $0.181\\pm 0.004$ $0.396\\pm 0.004$ $0.729\\pm 0.008$ 3 SAAO+OPD 83601 $6071\\pm 43$ $4.38\\pm 0.08$ $0.048\\pm 0.028$ $6.013\\pm 0.008$ $0.575\\pm 0.005$ $0.041\\pm 0.012$ $0.325\\pm 0.010$ $0.635\\pm 0.010$ 0 LIT 83707 $5880\\pm 30$ $4.45\\pm 0.06$ $0.150\\pm 0.030$ $8.605\\pm 0.011$ $0.655\\pm 0.004$ $0.181\\pm 0.004$ $0.350\\pm 0.004$ $0.699\\pm 0.004$ 3 SAAO+OPD 85042 $5692\\pm 37$ $4.39\\pm 0.02$ $0.037\\pm 0.026$ $6.287\\pm 0.004$ $0.679\\pm 0.004$ $0.233\\pm 0.008$ $0.364\\pm 0.020$ $0.707\\pm 0.051$ 2 SAAO+LIT 85272 $5700\\pm 30$ $4.42\\pm 0.06$ $-0.340\\pm 0.030$ $9.121\\pm 0.000$ $0.632\\pm 0.016$ $0.095\\pm 0.012$ $0.363\\pm 0.009$ $0.717\\pm 0.006$ 3 SAAO+OPD 85285 $5730\\pm 30$ $4.43\\pm 0.06$ $-0.390\\pm 0.030$ $8.365\\pm 0.015$ $0.629\\pm 0.011$ $0.065\\pm 0.013$ $0.348\\pm 0.009$ $0.712\\pm 0.008$ 7 SPM+SAAO+OPD 86796 $5809\\pm 22$ $4.28\\pm 0.04$ $0.298\\pm 0.030$ $5.124\\pm 0.006$ $0.700\\pm 0.004$ $0.240\\pm 0.004$ $0.385\\pm 0.004$ $0.708\\pm 0.004$ 2 SAAO+LIT 88194 $5735\\pm 21$ $4.40\\pm 0.03$ $-0.071\\pm 0.010$ $7.101\\pm 0.035$ $0.639\\pm 0.008$ $0.126\\pm 0.004$ $0.360\\pm 0.006$ $0.712\\pm 0.008$ 5 SPM+OPD+LIT 88427 $5810\\pm 57$ $4.42\\pm 0.07$ $-0.160\\pm 0.025$ $9.329\\pm 0.006$ $0.638\\pm 0.013$ $0.089\\pm 0.022$ $0.336\\pm 0.018$ $0.704\\pm 0.007$ 1 SPM 89162 $5835\\pm 30$ $4.32\\pm 0.06$ $0.070\\pm 0.030$ $8.902\\pm 0.005$ $0.658\\pm 0.005$ $0.176\\pm 0.004$ $0.360\\pm 0.006$ $0.696\\pm 0.006$ 4 SAAO+OPD 89443 $5796\\pm 73$ $4.48\\pm 0.12$ $-0.020\\pm 0.038$ $8.843\\pm 0.004$ $0.660\\pm 0.007$ $0.147\\pm 0.013$ $0.359\\pm 0.006$ $0.715\\pm 0.006$ 1 SPM 89650 $5855\\pm 25$ $4.48\\pm 0.05$ $0.020\\pm 0.020$ $8.944\\pm 0.001$ $0.643\\pm 0.004$ $0.126\\pm 0.004$ $0.356\\pm 0.004$ $0.688\\pm 0.009$ 7 SAAO+OPD 91332 $5775\\pm 25$ $4.20\\pm 0.05$ $0.206\\pm 0.030$ $7.971\\pm 0.008$ $0.692\\pm 0.007$ $0.263\\pm 0.004$ $0.365\\pm 0.015$ $0.705\\pm 0.004$ 2 SAAO+LIT 96402 $5713\\pm 49$ $4.33\\pm 0.03$ $-0.029\\pm 0.030$ $7.560\\pm 0.010$ $0.678\\pm 0.007$ $0.154\\pm 0.010$ 0 LIT 96895 $5808\\pm 39$ $4.33\\pm 0.05$ $0.097\\pm 0.020$ $5.959\\pm 0.009$ $0.644\\pm 0.006$ $0.189\\pm 0.009$ 0 LIT 96901 $5737\\pm 28$ $4.34\\pm 0.04$ $0.055\\pm 0.016$ $6.228\\pm 0.019$ $0.663\\pm 0.005$ $0.191\\pm 0.016$ 0 LIT 100963 $5802\\pm 17$ $4.45\\pm 0.03$ $0.008\\pm 0.013$ $7.089\\pm 0.021$ $0.651\\pm 0.013$ $0.128\\pm 0.024$ $0.342\\pm 0.004$ $0.703\\pm 0.007$ 3 SPM+OPD 100970 $5823\\pm 40$ $4.23\\pm 0.03$ $0.083\\pm 0.025$ $6.895\\pm 0.015$ $0.645\\pm 0.005$ $0.180\\pm 0.020$ 0 LIT 109110 $5817\\pm 60$ $4.46\\pm 0.03$ $0.062\\pm 0.030$ $7.570\\pm 0.010$ $0.674\\pm 0.015$ 0 LIT 102152 $5737\\pm 47$ $4.35\\pm 0.06$ $-0.010\\pm 0.022$ $9.208\\pm 0.015$ $0.669\\pm 0.004$ $0.176\\pm 0.019$ $0.382\\pm 0.004$ $0.726\\pm 0.004$ 3 SPM+SAAO 104504 $5836\\pm 48$ $4.50\\pm 0.06$ $-0.160\\pm 0.022$ $8.544\\pm 0.021$ $0.622\\pm 0.004$ $0.068\\pm 0.012$ $0.363\\pm 0.004$ $0.701\\pm 0.004$ 5 SPM+SAAO+OPD+LIT 107350 $6015\\pm 50$ $4.48\\pm 0.07$ $-0.020\\pm 0.019$ $5.942\\pm 0.011$ $0.587\\pm 0.004$ $0.032\\pm 0.004$ 0 LIT 108708 $5875\\pm 51$ $4.51\\pm 0.07$ $0.150\\pm 0.024$ $8.939\\pm 0.003$ $0.659\\pm 0.004$ $0.162\\pm 0.012$ $0.368\\pm 0.004$ $0.711\\pm 0.004$ 4 SPM+OPD 108996 $5838\\pm 56$ $4.50\\pm 0.08$ $0.060\\pm 0.027$ $8.881\\pm 0.012$ $0.648\\pm 0.008$ $0.152\\pm 0.025$ $0.351\\pm 0.011$ $0.691\\pm 0.011$ 6 SPM+SAAO+OPD 109931 $5739\\pm 74$ $4.29\\pm 0.08$ $0.040\\pm 0.026$ $8.956\\pm 0.019$ $0.663\\pm 0.006$ $0.194\\pm 0.016$ $0.367\\pm 0.034$ $0.710\\pm 0.020$ 1 SPM+LIT 113357 $5803\\pm 47$ $4.38\\pm 0.05$ $0.221\\pm 0.017$ $5.467\\pm 0.020$ $0.665\\pm 0.012$ $0.233\\pm 0.028$ 0 LIT 118159 $5905\\pm 44$ $4.55\\pm 0.07$ $-0.010\\pm 0.022$ $9.010\\pm 0.006$ $0.627\\pm 0.004$ $0.090\\pm 0.004$ $0.341\\pm 0.007$ $0.680\\pm 0.004$ 4 SPM+SAAO+OPD 1Number of photometric observations made in this work.",
"2If the source includes LIT, which corresponds to previously published values, the LIT flag applies only to the UBV data.",
"The RI$_\\mathrm {(C)}$ data are, to the best of our knowledge, published here for the first time, except for stars with the following HIP numbers: 1499, 11072, 12186, 15457, 22263, 41317, 44713, 71683, 77052, 79672, 80337, 83601, and 86796."
]
] | 1204.0828 |
[
[
"Khovanov module and the detection of unlinks"
],
[
"Abstract We study a module structure on Khovanov homology, which we show is natural under the Ozsvath-Szabo spectral sequence to the Floer homology of the branched double cover.",
"As an application, we show that this module structure detects trivial links.",
"A key ingredient of our proof is that the H_1/Torsion module structure on Heegaard Floer homology detects S^1xS^2 connected summands."
],
[
"Introduction",
"The Jones polynomial [15] has had a tremendous impact since its discovery, leading to an array of invariants of knots and 3–manifolds.",
"The meaning of these invariants is rather elusive.",
"In fact it remains unknown whether there exists a non-trivial knot with the same Jones polynomial as the unknot.",
"Khovanov discovered a refinement of the Jones polynomial which assigns bigraded homology groups to a link [16].",
"The Jones polynomial is recovered by taking the Euler characteristic of Khovanov's homology and keeping track of the additional grading by the exponent of a formal variable $q$ .",
"One could hope that the geometry contained in this refinement is more transparent, and a step towards understanding the question above would be to determine whether Khovanov homology detects the unknot.",
"The pure combinatorial nature of Khovanov homology, however, makes direct inspection of its ability to detect the unknot quite difficult.",
"Surprisingly, the most fruitful approach to such questions has been through connections with the world of gauge theory and Floer homology.",
"Indeed, a recent result of Kronheimer and Mrowka uses an invariant of knots arising from instanton Floer homology to prove that Khovanov homology detects the unknot.",
"More precisely, [19] states that $\\mathrm {rk}\\ Kh^{{r}}(K)=1 \\Longleftrightarrow K \\mathrm {\\ is\\ the\\ unknot.",
"}$ Their result had numerous antecedents [1], [2], [3], [10], [11], [12], [13], [37], most aimed at generalizing or exploiting a spectral sequence discovered by Ozsváth and Szabó [31] which begins at Khovanov homology and converges to the Heegaard Floer homology of the branched double cover.",
"Kronheimer and Mrowka's theorem raises the natural question of whether Khovanov homology also detects an unlink of numerous components.",
"Interest in this question is heightened by the existence of infinite families of links which have the same Jones polynomial as unlinks with two or more components [4], [43].",
"One immediately observes that the rank of Khovanov homology does not detect unlinks.",
"This is demonstrated by the Hopf link, which has the same rank Khovanov homology as the 2–component unlink (see Subsection REF for a discussion).",
"Thus more information must be brought to the table if one hopes to use Khovanov homology to detect unlinks.",
"Using a small portion of the bigrading on Khovanov homology we were able to make initial progress on the question of unlink detection in [12].",
"There we showed [12] $ Kh(L) \\text{\\ detects the unknot }\\Longleftrightarrow \\newline Kh(L) \\text{\\ detects the two-component unlink}$ which, together with Kronheimer and Mrowka's theorem settles the question for unlinks of two components.",
"Unfortunately, the strategy in [12] could not be extended.",
"The purpose of this article is to show that Khovanov homology detects unlinks by exploiting a module structure inherent in the theory.",
"In the next section we define a module structure on the Khovanov chain complex, and prove that the induced module structure on the Khovanov homology groups with $\\mathbb {F}=\\mathbb {Z}/2\\mathbb {Z}$ coefficients is an invariant of the link.",
"More precisely, we have Proposition 1 Let $L\\subset S^3$ be a link of $n$ components.",
"The Khovanov homology $Kh(L;\\mathbb {F})$ is a module over the ring $\\mathbb {F}[X_0,...,X_{n-1}]/(X_0^2,..., X_{n-1}^2).$ The isomorphism type of this module is an invariant of $L$ .",
"The action of $X_i$ is defined analogously to reduced Khovanov homology [17], and is obtained from the chain map on $CKh(K)$ induced by the link cobordism which merges an unknotted circle to the $i$ -th component of $L$ .",
"Our main theorem shows that the Khovanov module detects unlinks: Theorem 2 Let $L$ be a link of $n$ components.",
"If there is an isomorphism of modules $Kh(L;\\mathbb {F})\\cong \\mathbb {F}[X_0,...,X_{n-1}]/(X_0^2,..., X_{n-1}^2),$ then $L$ is the unlink.",
"Theorem REF will be proved in the context of the spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover of $L$ .",
"This latter invariant also has a module structure.",
"Indeed $\\widehat{HF}(Y)$ is a module over the exterior algebra on $H^1(Y;\\mathbb {F})$ , the first singular cohomology of $Y$ with coefficients in $\\mathbb {F}$ .",
"A key ingredient in our proof, Theorem REF , is to refine Ozsváth–Szabó's spectral sequence to incorporate both module structures.",
"A consequence of this result is the following theorem, which indicates the flavor of our refinement: Theorem 3 There is a spectral sequence of modules, starting at the reduced Khovanov module of the mirror of a link and converging to the Floer homology of its branched double cover.",
"Armed with this structure, we show that if the Khovanov module is isomorphic to that of the unlink, then the Floer homology of the branched double cover is isomorphic to $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,..., X_{n-1}^2)$ as a module (see Proposition REF ).",
"The second main ingredient in our proof is the following theorem, which says that Floer homology detects $S^1\\times S^2$ summands in the prime decomposition of a 3–manifold.",
"Theorem 4 Suppose that $\\widehat{HF}(Y;\\mathbb {F})\\cong \\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,..., X_{n-1}^2)$ as a module.",
"Then $Y\\cong M\\#(\\#^{n-1}(S^1\\times S^2))$ , where $M$ is an integer homology sphere satisfying $\\widehat{HF}(M)\\cong \\mathbb {F}$ .",
"This theorem seems interesting in its own right, and complements an array of results on the faithfulness of the Floer invariants for particularly simple manifolds.",
"The proof of the main theorem follows quickly from Theorems REF and REF .",
"Indeed, if the Khovanov module of $L$ is isomorphic to that of the unlink, then the two results imply that the branched double cover of $L$ is homeomorphic to the connected sum of $S^1\\times S^2$ 's with an integer homology sphere whose Floer homology has rank one.",
"Using classical tools from equivariant topology, we then see that $L$ is a split link, each component of which has the Khovanov homology of the unknot.",
"Kronheimer and Mrowka's theorem then tells us that each component is unknotted.",
"Outline: This paper is organized as follows.",
"In Section 2 we will review Khovanov homology and its module structure.",
"There we prove that the module structure is an invariant.",
"In Section 3 we will give the necessary background on Heegaard Floer homology, especially the module structure and an $A_{\\infty }$ type relation.",
"In Section 4 we relate the Heegaard Floer module structure with Ozsváth and Szabó's link surgeries spectral sequence.",
"As an application, we connect the module structure on Khovanov homology of a link with the module structure on the Heegaard Floer homology of the branched double cover of the link.",
"Section 5 is devoted to a nontriviality theorem for the module structure on Heegaard Floer homology.",
"The proof is similar to the standard nontriviality theorem in Heegaard Floer theory.",
"The main theorem is proved in Section 6, using the results in the previous two sections as well as a homological algebra argument.",
"Acknowledgements.",
"This work was initiated when the authors participated the “Homology Theories of Knots and Links” program at MSRI, and was carried out further when the authors visited Simons Center for Geometry and Physics.",
"We are grateful to Ciprian Manolescu and Tomasz Mrowka for helpful conversations.",
"We also wish to thank Robert Lipshitz, Sucharit Sarkar and the referee for pointing out a mistake in the proof of Proposition REF .",
"Special thanks are due to Sucharit Sarkar for suggesting a way to fix the mistake.",
"The first author was partially supported by NSF grant numbers DMS-0906258 and DMS-1150872 and an Alfred P. Sloan Research Fellowship.",
"The second author was partially supported by an AIM Five-Year Fellowship, NSF grant numbers DMS-1021956, DMS-1103976, and an Alfred P. Sloan Research Fellowship."
],
[
"Preliminaries on Khovanov homology",
"Khovanov homology is a combinatorial invariant of links in the 3–sphere which refines the Jones polynomial.",
"In this section we briefly recall some background on this invariant, but will assume familiarity with [16].",
"Our primary purpose is to establish notation and define a module structure on the Khovanov chain complex (or homology) which is implicit in [16], [17], but which has not attracted much attention.",
"To a diagram $D$ of a link $L\\subset S^3$ , Khovanov associates a bigraded cochain complex, $(CKh_{i,j}(D),\\partial )$ [16].",
"The $i$ –grading is the cohomological grading, in the sense that is raised by one by the coboundary operator $\\partial \\colon \\,CKh_{i,j}(D)\\rightarrow CKh_{i+1,j}(D).$ The $j$ –grading is the so-called “quantum\" grading, and is preserved by the differential.",
"One can think of this object as a collection of complexes in the traditional sense, with the collection indexed by an additional grading.",
"The homology of these complexes does not depend on the particular diagram chosen for $L$ , and produces an invariant $Kh(L):=\\bigoplus _{i,j} Kh_{i,j}(L)$ called the Khovanov (co)homology of L. Taking the Euler characteristic in each quantum grading, and keeping track of this with a variable $q$ , we naturally obtain a Laurent polynomial $V_K(q)=\\sum _j {\\big (}\\sum _i(-1)^{i}\\mathrm {rank}\\ Kh_{i,j}(L){\\big )}\\cdot q^j.$ This polynomial agrees with the (properly normalized) Jones polynomial.",
"The complex $CKh(D)$ is obtained by applying a $(1+1)$ –dimensional TQFT to the hypercube of complete resolutions of the diagram, and the algebra assigned to a single unknotted circle by this structure is $\\mathcal {A}=\\mathbb {F}[X]/(X^2)$ .",
"The product on this algebra is denoted $m\\colon \\,\\mathcal {A}\\otimes \\mathcal {A}\\rightarrow \\mathcal {A}.$ There is also a coproduct $\\Delta \\colon \\,\\mathcal {A}\\rightarrow \\mathcal {A}\\otimes \\mathcal {A}$ which is defined by letting $\\Delta (\\mathbf {1})=\\mathbf {1}\\otimes X+X\\otimes \\mathbf {1},\\qquad \\Delta (X)=X\\otimes X.$ Our purposes will not require strict knowledge of gradings, and for convenience we relax Khovanov homology to a relatively $\\mathbb {Z}\\oplus \\mathbb {Z}$ –graded theory.",
"This means that we consider Khovanov homology up to overall shifts in either the homological or quantum grading.",
"In these terms, the quantum grading of $\\mathbf {1}\\in \\mathcal {A}$ is two greater than that of $X$ , and the homological grading is given by the number of crossings resolved with a 1–resolution, in a given complete resolution.",
"Section 3 of [17] describes a module structure on the Khovanov homology of a knot which we now recall.",
"Given a diagram $D$ for a knot $K$ , let $p\\in K$ be a marked point.",
"Now place an unknotted circle next to $p$ and consider the saddle cobordism that merges the circle with a segment of $D$ neighboring $p$ .",
"Cobordisms induce maps between Khovanov complexes and, as such, we have a map $\\mathcal {A}\\otimes CKh(D)\\overset{f_p}{\\longrightarrow }CKh(D).$ Figure: Merging an unknotted circle with DD.Explicitly, the map is the algebra multiplication $\\mathcal {A}\\otimes \\mathcal {A}\\rightarrow \\mathcal {A}$ between vectors associated to the additional unknot and the unknot in each complete resolution of the diagram which contains the marked point.",
"We denote the induced map on homology by $F_p$ .",
"It is independent of the pair $(D,p)$ , in a sense made precise by the following proposition.",
"Proposition 2.1 ([17]) Let $D$ and $D^{\\prime }$ be diagrams for a knot $K$ , and let $p\\in D$ , $p^{\\prime }\\in D^{\\prime }$ be marked points on each diagram.",
"Then there is a commutative diagram: ${\\begin{matrix}\\mathcal {A}\\otimes CKh(D) &\\xrightarrow{}& CKh(D)\\\\\\mathbox{mphantom}{\\scriptstyle 1\\otimes e}\\downarrow {\\scriptstyle 1\\otimes e}&& \\mathbox{mphantom}{\\scriptstyle e}\\downarrow {\\scriptstyle e}&&\\\\\\mathcal {A}\\otimes CKh(D^{\\prime }) &\\xrightarrow{}& CKh(D^{\\prime }),\\end{matrix}}$ where $e$ is a chain homotopy equivalence.",
"As observed in [17], the proof is quite simple provided one knows the proof of invariance for Khovanov homology.",
"According to [16], a sequence of Reidemeister moves from $D$ to $D^{\\prime }$ induces a chain homotopy equivalence $e$ between the associated complexes.",
"If these moves occur in the complement of a neighborhood of $p$ , then $e$ obviously commutes with the map $f_p$ , where $p$ is regarded as a point in both $D$ and $D^{\\prime }$ .",
"Now it suffices to observe that Reidemeister moves which cross under or over $p$ can be traded for a sequence of moves which do not (simply drag the segment of knot in question the opposite direction over the plane).",
"Thus a choice of marked point endows the Khovanov homology of a knot with the structure of an $\\mathcal {A}$ –module.",
"One immediate application of this module structure is that it allows for the definition of the reduced Khovanov homology.",
"Indeed, we can consider $\\mathbb {F}=X\\cdot \\mathcal {A}$ as an $\\mathcal {A}$ –module, and correspondingly define the reduced Khovanov complex This is not quite Khovanov's definition, but is isomorphic to it.",
"$CKh^{{r}}(D):= CKh(D)\\otimes _\\mathcal {A}\\mathbb {F}.$ The structure we use is a straightforward generalization of this construction to links.",
"Consider a diagram $D$ for a link $L\\subset S^3$ of $n$ components.",
"For each component $K_i\\subset L$ pick a marked point $p_i\\subset K_i$ and obtain a map: $ {\\begin{matrix} \\mathcal {A}\\otimes CKh(D) &\\xrightarrow{}&CKh(D).\\end{matrix}} $ Now for each $i$ , let $x_i(-):=f_{p_i}(X \\otimes -)$ denote the chain map induced from this module structure, and $ X_i\\colon \\,Kh(L) {\\longrightarrow } Kh(L), \\ \\ i=0,...,n-1,$ denote the map induced on homology.",
"From the definition we see that $x_i$ , and hence $X_i$ , commute and satisfy $x_i^2=0.$ Thus we can regard Khovanov homology as a module over $\\mathbb {F}[X_0,...,X_{n-1}]/(X_0^2,...,X_{n-1}^2)$ .",
"Showing that this module structure is a link invariant is not as straightforward as above.",
"This is due to the fact that we may not be able to connect two multi-pointed link diagrams by a sequence of Reidemeister moves which occur in the complement of all the basepoints.",
"One may expect, then, that the module structure is an invariant only of the pointed link isotopy class.",
"In fact the isomorphism type of the module structure on $Kh(L;\\mathbb {F})$ does not depend on the choice of marked points or link diagram, as the following proposition shows.",
"Proposition 2.2 Let $D$ and $D^{\\prime }$ be diagrams for a link $L$ of $n$ components and let $p_i\\in D_i$ , $p_i^{\\prime }\\in D_i^{\\prime }$ be marked points, one chosen on each component of the diagram.",
"Then there is a commutative diagram: ${\\begin{matrix}\\displaystyle \\frac{\\mathbb {F}[X_0,...,X_{n-1}]}{(X_0^2,...,X_{n-1}^2)}\\otimes Kh(D) &\\xrightarrow{}& Kh(D)\\\\\\mathbox{mphantom}{\\scriptstyle 1\\otimes e_*}\\downarrow {\\scriptstyle 1\\otimes e_*}&& \\mathbox{mphantom}{\\scriptstyle e_*}\\downarrow {\\scriptstyle e_*}&&\\\\\\displaystyle \\frac{\\mathbb {F}[X_0,...,X_{n-1}]}{(X_0^2,...,X_{n-1}^2)}\\otimes Kh(D^{\\prime }) &\\xrightarrow{}& Kh(D^{\\prime }),\\end{matrix}}$ where $e_*$ is an isomorphism induced by a chain homotopy equivalence.",
"The proof of Proposition REF uses an argument suggested by Sucharit Sarkar.",
"Any two diagrams $D,D^{\\prime }$ are related by a sequence of Reidemeister moves.",
"If a Reidemeister move occurs in the complement of neighborhoods of all marked points $p_i$ , then the commutative diagram obviously exists.",
"Thus we only need to consider the case of isotoping an arc over or under a marked point or, equivalently, of sliding a basepoint past a crossing.",
"This case is contained in the following lemma.",
"Lemma 2.3 Suppose that $D$ is a link diagram, and $\\chi _0$ is a crossing of $D$ .",
"Suppose that $p,q$ are two marked points on a strand passing $\\chi _0$ , such that $p,q$ are separated by $\\chi _0$ .",
"Let $x_p,x_q\\colon \\,CKh(D)\\rightarrow CKh(D)$ be the module multiplications induced by the marked points $p,q$ .",
"Then $x_p$ and $x_q$ are homotopy equivalent.",
"Consider the cube of resolutions of $D$ .",
"If $IJ$ is an oriented edge of the cube, let $\\partial _{IJ}\\colon \\,C(I)\\rightarrow C(J)$ be the map induced by the elementary cobordism corresponding to the edge.",
"Let $I_0,I_1$ be any two complete resolutions of $D$ that differ only at the crossing $\\chi _0$ , where $I_0$ is locally the 0–resolution and $I_1$ is locally the 1–resolution.",
"Suppose that the immediate successors of $I_0$ besides $I_1$ are $J_0^1,J_0^2,\\dots ,J_0^r$ .",
"Then the immediate successors of $I_1$ are $J_1^1,J_1^2,\\dots ,J_1^r$ , where $J_0^i$ and $J_1^i$ differ only at the crossing $\\chi _0$ .",
"We write $CKh(D)$ as the direct sum of two subgroups $CKh(D)=CKh(D_0)\\oplus CKh(D_1).$ Here $D_j$ is the $j$ –resolution of $D$ at $\\chi _0$ for $j=0,1$ .",
"Then $CKh(D_1)$ is a subcomplex of $CKh(D)$ , and $CKh(D_0)$ is a quotient complex of $CKh(D)$ .",
"We define $H\\colon \\,CKh(D)\\rightarrow CKh(D)$ as follows.",
"On $CKh(D_0)$ , $H=0$ ; on $CKh(D_1)$ , $H\\colon \\,CKh(D_1)\\rightarrow CKh(D_0)$ is defined by the map associated to the elementary cobordism from $D_1$ to $D_0$ .",
"We claim that $x_p-x_q=\\partial H+H\\partial .$ The claim obviously implies Lemma REF , and hence Proposition REF .",
"To prove the claim, we begin with another lemma.",
"Lemma 2.4 Let $I_0,I_1$ be as above.",
"Then for any $\\alpha \\in C(I_0)$ and $\\beta \\in C(I_1)$ , the following identities hold: $(x_p-x_q)(\\alpha )&=&H\\partial _{I_0I_1}(\\alpha ),\\\\(x_p-x_q)(\\beta )&=&\\partial _{I_0I_1}H(\\beta ).$ Proof.",
"It is easy to check the following identities in $\\mathcal {A}$ : $m\\Delta =0,\\quad \\Delta m(a\\otimes b)=Xa\\otimes b+a\\otimes Xb.$ Our conclusion then immediately follows.",
"By the definition of $H$ , we have $H(\\alpha )=0$ .",
"Moreover, $H\\partial (\\alpha )&=&H\\big (\\partial _{I_0I_1}\\alpha +\\sum _{i=1}^r\\partial _{I_0J_0^i}\\alpha \\big )\\\\&=&H\\partial _{I_0I_1}(\\alpha ),$ which is equal to $(x_p-x_q)(\\alpha )$ by Lemma REF .",
"Thus (REF ) holds for $\\alpha \\in C(I_0)$ .",
"Turning to $\\beta \\in C(I_1)$ , we have $\\partial H(\\beta )=\\partial _{I_0I_1}H(\\beta )+\\sum _{i=1}^r\\partial _{I_0J_0^i}H(\\beta ),$ and $H\\partial (\\beta )=H\\big (\\sum _{i=1}^r\\partial _{I_1J_1^i}(\\beta )\\big ).$ Let $\\overline{D}$ be the diagram (of a possibly different link) which is obtained from $D$ by changing the crossing $\\chi _0$ .",
"Then $H$ is the summand of the differential in $CKh(\\overline{D})$ which is induced by the edges parallel to $I_1I_0$ .",
"It is clear that $\\sum _{i=1}^r\\partial _{I_0J_0^i}H(\\beta )+H\\big (\\sum _{i=1}^r\\partial _{I_1J_1^i}(\\beta )\\big )$ is equal to $\\partial ^2\\beta $ in $CKh(\\overline{D})$ , which is zero.",
"So from (REF ) (REF ) and Lemma REF we have $(\\partial H+H\\partial )(\\beta )=\\partial _{I_0I_1}H(\\beta )=(x_p-x_q)(\\beta ).$ This finishes the proof of (REF ), hence Lemma REF follows.",
"Remark 2.5 Note that we make no claims about the naturality of the module structure.",
"The only result we need is that isotopic links have isomorphic Khovanov modules.",
"Indeed, it seems likely that the module structure on Khovanov homology is functorial, but only in the category of pointed links; that is, links with a choice of basepoint on each component.",
"When considering the relationship with Heegaard Floer homology, it will be more natural to work in the context of reduced Khovanov homology.",
"Note that reduced Khovanov homology can be interpreted as the homology of the kernel complex $H_*(\\mathrm {ker}\\lbrace CKh(L)\\overset{x_i}{\\rightarrow }CKh(L)\\rbrace )$ or, equivalently, as the kernel of the map on homology, $\\mathrm {ker}{X_i}$ [40].",
"In particular, the module structure from the proposition descends to an $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ –module structure on the reduced Khovanov homology.",
"Henceforth, we will let $p_0$ be the point chosen to define reduced Khovanov homology, so that $Kh^{{r}}(L)$ is a module over $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ .",
"Although the reduced Khovanov homology groups do not depend on the choice of component $L_0$ containing $p_0$ , their module structure will, in general, depend on this choice.",
"Thus we use $Kh^{{r}}(L,L_0)$ to emphasize the dependence of the module structure on the component $L_0$ and abuse the notation $Kh^{{r}}(L)$ when $L_0$ is understood.",
"As we shall see, this module structure on the reduced Khovanov homology is connected to a module structure on the Heegaard Floer homology of the branched double cover through a refinement of the Ozsváth–Szabó spectral sequence.",
"We also note that, since $X_i^2=0$ , the module structure equips the Khovanov homology (or reduced Khovanov homology) with the structure of a chain complex.",
"Thus it makes sense to talk about the homology of Khovanov homology with respect to $X_i$, which we frequently denote $H_*(Kh(L),X_i)$ .",
"It follows from [40] that $H_*(Kh(L),X_i)=0$ for any $i$ .",
"It is far from true, however, that $H_*(Kh^{{r}}(L),X_i)=0$ for each $i$ , in general."
],
[
"Example: The unlink versus the Hopf link",
"It is simple yet instructive to consider the distinction between the Khovanov module of the two component unlink and the Hopf link.",
"The former is represented by Khovanov chain complex $CKh(\\text{Unlink})=\\mathcal {A}\\langle X_0\\rangle \\otimes _\\mathbb {F}\\mathcal {A}\\langle X_1\\rangle $ with $\\partial \\equiv 0$ .",
"The homology is thus isomorphic to $\\mathbb {F}[X_0,X_1]/(X_0^2,X_1^2)$ as a module, and is supported in a single homological grading.",
"The reduced Khovanov homology is $Kh^{{r}}(\\mathrm {Unlink})\\cong \\mathbb {F}[X]/X^2$ .",
"Figure: Khovanov homology of the two-component unlink and the Hopf link, where a black dot stands for a copy of ℤ\\mathbb {Z}, the ii–coordinate is the homological grading and the jj–coordinate is the quantum grading.Figure: Module structure on the Khovanov homology of the two-component unlink and the Hopf link.",
"The X 0 X_0 and X 1 X_1 actions are different for the two-component unlink, but are equal for the Hopf link.The Hopf link, on the other hand, has Khovanov homology $Kh(\\mathrm {Hopf})\\cong \\mathbb {F}\\langle a\\rangle \\oplus \\mathbb {F}\\langle b\\rangle \\oplus \\mathbb {F}\\langle c\\rangle \\oplus \\mathbb {F}\\langle d\\rangle ,$ with the relative homological gradings of $c$ and $d$ equal, and two greater than those of $a$ and $b$ .",
"The relative quantum grading of $a$ is 2 lower than that of $b$ which is two lower than $c$ which, in turn, is two lower than $d$ .",
"The module structure is given by $ X_0(b)=X_1(b)=a\\ \\ \\ \\ \\ X_0(d)=X_1(d)=c.",
"$ More succinctly, the Khovanov module of the Hopf link is isomorphic to $\\frac{\\mathbb {F}[X]}{X^2}\\oplus \\frac{\\mathbb {F}[X]\\lbrace 2,4\\rbrace }{X^2},$ where we use notation $\\lbrace 2,4\\rbrace $ to denote a shift of 2 and 4 in the homological and quantum gradings respectively.",
"In this module each $X_i$ acts as $X$ .",
"It follows that the module structure on reduced Khovanov homology is trivial; that is $Kh^{{r}}(\\mathrm {Hopf})\\cong \\mathbb {F}\\langle a\\rangle \\oplus \\mathbb {F}\\langle c\\rangle ,$ where $X\\in \\mathbb {F}[X]/X^2$ acts as zero on both summands.",
"Thus, the homology of the reduced Khovanov homology with respect to $X$ is $H_*(Kh^{{r}}(\\mathrm {Hopf}),X)\\cong \\mathbb {F}\\langle a\\rangle \\oplus \\mathbb {F}\\langle c\\rangle ,$ while for the unlink it is trivial $ H_*(Kh^{{r}}(\\mathrm {Unlink}),X)=0.$"
],
[
"Preliminaries on Heegaard Floer homology",
"In this section, we recall the basic theory of Heegaard Floer homology, with emphasis on the action of $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ and twisted coefficients.",
"For a detailed account of the theory, we refer the reader to [26] (see also [35], [34], [33] for gentler introductions).",
"Suppose $Y$ is a closed oriented 3–manifold, together with a $\\mathrm {Spin}^c$ structure $\\mathfrak {s}\\in \\mathrm {Spin}^c(Y)$ .",
"Let $(\\Sigma ,\\mbox{${\\alpha }$},\\mbox{$\\beta $},z)$ be an admissible pointed Heegaard diagram for $(Y,\\mathfrak {s})$ , in the sense of [26].",
"To such a diagram we can associate the Ozsváth–Szabó infinity chain complex, ${CF}^{\\infty }(Y,\\mathfrak {s})$ .",
"This chain complex is freely generated over the ring $\\mathbb {F}[U,U^{-1}]$ by intersection points $\\mathbf {x}\\in {\\mathbb {T}}_{\\alpha }\\cap {\\mathbb {T}}_{\\beta }$ , where ${\\mathbb {T}}_{\\alpha }$ (resp.",
"${\\mathbb {T}}_{\\beta }$ ) is the Lagrangian torus in Sym$^g(\\Sigma )$ , the $g$ –fold symmetric product of $\\Sigma $ .",
"The boundary operator is defined by $\\partial ^\\infty \\mathbf {x}= \\sum _{\\mathbf {y}\\in \\mathbb {T}_{\\alpha }\\cap \\mathbb {T}_{\\beta }}\\sum _{\\lbrace \\phi \\in \\pi _2(\\mathbf {x},\\mathbf {y})|\\mu (\\phi )=1\\rbrace }\\#\\widehat{\\mathcal {M}}(\\phi )\\cdot U^{n_z(\\phi )}\\mathbf {y},$ where $\\pi _2(\\mathbf {x},\\mathbf {y})$ is the set of homotopy classes of Whitney disks connecting $\\mathbf {x}$ to $\\mathbf {y}$ , $\\mu $ is the Maslov index, $\\#\\widehat{\\mathcal {M}}(\\phi )$ denotes the reduction modulo two of the number of unparametrized pseudo-holomorphic maps in the homotopy class $\\phi $ , and $n_z(\\phi )$ denotes the algebraic intersection number of such a map with the holomorphic hypersurface in Sym$^g(\\Sigma )$ consisting of unordered $g$ –tuples of points in $\\Sigma $ at least one of which is the basepoint $z$ ."
],
[
"The $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ action",
"In this subsection we describe an action of $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ on the Floer homology of a 3–manifold $Y$ .",
"Let $\\zeta \\subset \\Sigma $ be a closed oriented (possibly disconnected) immersed curve which is in general position with the $\\alpha $ – and $\\beta $ –curves.",
"Namely, $\\zeta $ is transverse to these curves, and $\\zeta $ does not contain any intersection point of $\\alpha $ – and $\\beta $ –curves.",
"Note that any closed curve $\\zeta _0\\subset Y$ can be homotoped in $Y$ to be an immersed curve in $\\Sigma $ .",
"If $\\phi $ is a topological Whitney disk connecting $\\mathbf {x}$ to $\\mathbf {y}$ , let $\\partial _{\\alpha }\\phi =(\\partial \\phi )\\cap \\mathbb {T}_{\\alpha }$ .",
"We can also regard $\\partial _{\\alpha }\\phi $ as a multi-arc that lies on $\\Sigma $ and connects $\\mathbf {x}$ to $\\mathbf {y}$ .",
"Similarly, we define $\\partial _{\\beta }\\phi $ as a multi-arc connecting $\\mathbf {y}$ to $\\mathbf {x}$ .",
"Let $a^{\\zeta }\\colon \\,{CF}^{\\infty }(Y,\\mathfrak {s})\\rightarrow {CF}^{\\infty }(Y,\\mathfrak {s})$ be defined on generators as $a^{\\zeta }(\\mathbf {x})=\\sum _{\\mathbf {y}\\in \\mathbb {T}_{\\alpha }\\cap \\mathbb {T}_{\\beta }}\\sum _{\\lbrace \\phi \\in \\pi _2(\\mathbf {x},\\mathbf {y})|\\mu (\\phi )=1\\rbrace }\\big (\\zeta \\cdot (\\partial _{\\alpha }\\phi )\\big )\\:\\#\\widehat{\\mathcal {M}}(\\phi )\\cdot U^{n_z(\\phi )} \\mathbf {y},$ where $\\zeta \\cdot (\\partial _{\\alpha }\\phi )$ is the algebraic intersection number of $\\zeta $ and $\\partial _{\\alpha }\\phi $ .",
"We extend the map to the entire complex by requiring linearity and $U$ –equivariance.",
"As shown in Ozsváth–Szabó [26], Gromov compactness implies that $a^{\\zeta }$ is a chain map.",
"Moreover, this chain map clearly respects the sub-, quotient, and subquotient complexes $CF^-,CF^+,\\widehat{CF}$ , respectively.",
"Thus $a^\\zeta $ induces a map, denoted $A^\\zeta =(a^\\zeta )_*$ , on all versions of Heegaard Floer homology.",
"The following lemma shows that $A^\\zeta $ only depends on the homology class $[\\zeta ]\\in H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ .",
"Lemma 3.1 Suppose $\\zeta _1,\\zeta _2\\subset \\Sigma $ are two curves which are homologous in $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ , then $a^{\\zeta _1}$ is chain homotopic to $a^{\\zeta _2}$ .",
"Proof.",
"Since $\\zeta _1$ and $\\zeta _2$ are homologous in $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ , there exists a nonzero integer $m$ such that $m[\\zeta _1]=m[\\zeta _2]\\in H_1(Y;\\mathbb {Z})$ .",
"Using the fact that $H_1(Y)\\cong H_1(\\Sigma )/([\\alpha _1]\\dots ,[\\alpha _g],[\\beta _1],\\dots ,[\\beta _g]),$ we conclude that there is a 2–chain $B$ in $\\Sigma $ , such that $\\partial B$ consists of $m\\zeta _2$ , $m(-\\zeta _1)$ and copies of $\\alpha $ –curves and $\\beta $ –curves.",
"Perturbing $B$ slightly, we get a 2–chain $B^{\\prime }$ such that $\\partial B^{\\prime }=m\\zeta _2-m\\zeta _1+\\sum (a_i\\alpha ^{\\prime }_i+b_i\\beta ^{\\prime }_i),$ where $\\alpha _i^{\\prime },\\beta _i^{\\prime }$ are parallel copies of $\\alpha _i,\\beta _i$ .",
"Let $\\phi $ be a topological Whitney disk connecting $\\mathbf {x}$ to $\\mathbf {y}$ .",
"Since $\\alpha _i^{\\prime }$ is disjoint from all $\\alpha $ –curves, we have $\\alpha _i^{\\prime }\\cdot \\partial _{\\alpha }\\phi =0$ .",
"Similarly, $\\beta _i^{\\prime }\\cdot \\partial _{\\alpha }\\phi =-\\beta _i^{\\prime }\\cdot \\partial _{\\beta }\\phi =0.$ We have $ n_{\\mathbf {x}}(B^{\\prime })-n_{\\mathbf {y}}(B^{\\prime })=-\\partial B^{\\prime }\\cdot \\partial _{\\alpha }\\phi =m(\\zeta _1-\\zeta _2)\\cdot \\partial _{\\alpha }\\phi \\in m\\mathbb {Z}.$ Pick an intersection point $\\mathbf {x}_0$ representing the Spin$^c$ structure $\\mathfrak {s}$ .",
"After adding copies of $\\Sigma $ to $B^{\\prime }$ , we can assume that $n_{\\mathbf {x}_0}(B^{\\prime })$ is divisible by $m$ .",
"Since any two intersection points representing $\\mathfrak {s}$ are connected by a topological Whitney disk, (REF ) implies that $n_{\\mathbf {x}}(B^{\\prime })$ is divisible by $m$ for any $\\mathbf {x}$ representing $\\mathfrak {s}$ .",
"Now we define a map $H\\colon \\,CF^{\\infty }(Y,\\mathfrak {s})\\rightarrow CF^{\\infty }(Y,\\mathfrak {s})$ by letting $H(\\mathbf {x})=\\frac{n_{\\mathbf {x}}(B^{\\prime })}{m}\\cdot \\mathbf {x}$ on generators, and extending linearly.",
"It follows from (REF ) that $a^{\\zeta _1}-a^{\\zeta _2}=\\partial \\circ H-H\\circ \\partial .$ Namely, $a^{\\zeta _1},a^{\\zeta _2}$ are chain homotopic.",
"In light of the lemma, we will often denote the induced map on homology by $A^{[\\zeta ]}$ , where $[\\zeta ]\\in H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ .",
"Lemma 4.17 of [26] shows that $A^{[\\zeta ]}$ satisfies $A^{[\\zeta ]}\\circ A^{[\\zeta ]}=0$ , hence varying the class within $H_1(Y;\\mathbb {Z})/\\mathrm {Tor}$ endows the Floer homology groups with the structure of a $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ module.",
"From its definition, the action satisfies $A^{2[\\zeta ]}=A^{[\\zeta ]}+A^{[\\zeta ]}$ .",
"In light of the fact that we work with $\\mathbb {F}=\\mathbb {Z}/2\\mathbb {Z}$ coefficients, it will thus make more sense to regard the action as by $\\Lambda ^*(H_1(Y;\\mathbb {F}))$ , where it is understood that classes in $H_1(Y;\\mathbb {F})\\cong H_1(Y;\\mathbb {Z})\\otimes \\mathbb {F}$ arising from even torsion in $H_1(Y;\\mathbb {Z})$ act as zero.",
"An important example is $Y=\\#^n(S^1\\times S^2)$ .",
"The module structure of $\\widehat{HF}(Y)$ has been computed by Ozsváth and Szabó [27], [28]: As a module over $\\Lambda ^*(H_1(Y;\\mathbb {F}))$ , $\\widehat{HF}(Y)\\cong \\Lambda ^*(H^1(Y;\\mathbb {F})),$ where the action of $[\\zeta ]$ is given by the contraction operator $\\iota _{[\\zeta ]}$ defined using the natural hom pairing.",
"We remark that the ring $\\Lambda ^*(H_1(Y;\\mathbb {F}))$ is isomorphic to $\\mathbb {F}[X_0,\\dots ,X_{n-1}]/(X_0^2,\\dots ,X_{n-1}^2)$ , and the module $\\Lambda ^*(H^1(Y;\\mathbb {F}))$ is isomorphic to the free module $\\mathbb {F}[X_0,\\dots ,X_{n-1}]/(X_0^2,\\dots ,X_{n-1}^2)$ .",
"As with the module structure on Khovanov homology, we can consider the homology of the Heegaard Floer homology with respect to $A^{[\\zeta ]}$ : $H_*( HF^\\circ (Y,\\mathfrak {s}), A^{[\\zeta ]}).$ For $Y=\\#^n(S^1\\times S^2)$ , we have $H_*(\\widehat{HF}(Y), X_i)=0$ for all $i$ .",
"We conclude this subsection by analyzing the $H_1(Y;\\mathbb {F})$ action in the presence of essential spheres.",
"We begin with the case of separating spheres.",
"In this case the action splits according to a Künneth principle.",
"More precisely, let $(\\Sigma _i,\\mbox{${\\alpha }$}_i,\\mbox{$\\beta $}_i,z_i)$ be a Heegaard diagram for $Y_i$ , $i=1,2$ .",
"Let $\\Sigma $ be the connected sum of $\\Sigma _1$ and $\\Sigma _2$ , with the connected sum performed at $z_1$ and $z_2$ .",
"Let $\\zeta _i\\subset \\Sigma _i\\backslash \\lbrace z_i\\rbrace $ be a closed curve.",
"Suppose $\\mathfrak {s}_1\\in \\mathrm {Spin}^c(Y_1), \\mathfrak {s}_2\\in \\mathrm {Spin}^c(Y_2)$ .",
"Now $(\\Sigma ,\\mbox{${\\alpha }$}_1\\cup \\mbox{${\\alpha }$}_2,\\mbox{$\\beta $}_1\\cup \\mbox{${\\beta }$}_2,z_1=z_2)$ is a Heegaard diagram for $Y_1\\#Y_2$ .",
"Using the proof of the Künneth formula for $\\widehat{HF}$ of connected sums [27], one sees that the action of ${\\zeta _1\\cup \\zeta _2}$ on $\\widehat{CF}(Y_1\\#Y_2,\\mathfrak {s}_1\\#\\mathfrak {s}_2)\\cong \\widehat{CF}(Y_1,\\mathfrak {s}_1)\\otimes \\widehat{CF}(Y_2,\\mathfrak {s}_2)$ is given by $a^{\\zeta _1\\cup \\zeta _2}=a^{\\zeta _1}\\otimes \\mathrm {id}+\\mathrm {id}\\otimes a^{\\zeta _2}.$ Next, we turn to the case of non-separating spheres.",
"Here, we have a vanishing theorem for the homology of the Floer homology with respect to the action.",
"First we state a version of the Künneth theorem in this context.",
"Proposition 3.2 Let $\\mathfrak {s}$ be a Spin$^c$ structure on a closed oriented 3–manifold $Y$ .",
"Let $\\mathfrak {s}_0$ be the Spin$^c$ structure on $S^1\\times S^2$ with $c_1(\\mathfrak {s}_0)=0$ .",
"Suppose $\\zeta _1$ is a closed curve in $Y$ , and $\\zeta _2$ is a closed curve in $S^1\\times S^2$ .",
"Then Equation (REF ) holds on the chain complex $CF^{\\circ }(Y\\#(S^1\\times S^2),\\mathfrak {s}\\#\\mathfrak {s}_0)\\cong CF^{\\circ }(Y,\\mathfrak {s})\\otimes \\widehat{CF}(S^1\\times S^2,\\mathfrak {s}_0).$ Proof.",
"This follows from the proof of [27].",
"We now have the promised vanishing theorem.",
"Proposition 3.3 Let $\\sigma \\colon \\,\\mathbb {Z}\\rightarrow \\mathbb {F}$ be the natural quotient map which sends 1 to $\\mathbf {1}$ .",
"Let $[\\zeta ]\\in H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ be a class for which there exists a two-sphere $S\\subset Y$ satisfying $\\sigma ([\\zeta ]\\cdot [S])\\ne 0\\in \\mathbb {F}$ .",
"Then $H_*(\\widehat{HF}(Y;\\mathbb {F}),A^{[\\zeta ]})=0\\ ,\\ H_*({HF}^+(Y;\\mathbb {F}),A^{[\\zeta ]})=0.$ Proof.",
"Since the sphere $S$ is homologically nontrivial, it is non-separating.",
"Thus $Y\\cong Y_1\\#(S^1\\times S^2)$ , with $S$ isotopic to $*\\times S^2$ in $S^1\\times S^2$ .",
"We can express $[\\zeta ]$ as $[\\zeta ]=[\\zeta _1]\\oplus [\\zeta _2]\\in H_1(Y_1)\\oplus H_1(S^1\\times S^2)\\cong H_1(Y),$ where $\\sigma ([\\zeta _2]\\cdot [S])\\ne 0$ .",
"An explicit calculation with a genus one Heegaard diagram shows that $\\widehat{HF}(S^1\\times S^2)$ has two generators $x,y$ for which $A^{[\\zeta _2]}(x)=([\\zeta _2]\\cdot [S])y\\ ,\\ A^{[\\zeta _2]}(y)=0.$ Hence $H_*(\\widehat{HF}(S^1\\times S^2;\\mathbb {F}),A^{[\\zeta _2]})=0$ .",
"Our conclusion now follows from the previous proposition."
],
[
"Holomorphic polygons and the $\\Lambda ^*(H_1(Y;\\mathbb {F}))$ action ",
"In this subsection we consider operators on Heegaard Floer homology induced by counting pseudo-holomorphic Whitney polygons, and the interaction of these operators with the $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ action.",
"Our main result here is a compatibility relation, Theorem REF .",
"This relation will be useful in two ways.",
"First, it will allow us to understand the $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ action in the context of the cobordism maps constituting the $(3+1)$ –dimensional TQFT in Ozsváth–Szabó theory.",
"Second, it will be the key ingredient for showing that the spectral sequence from Khovanov homology to the Heegaard Floer homology of the branched double cover respects the module structure on both theories.",
"Let $(\\Sigma , \\mbox{${\\eta }$}^0,...,\\mbox{${\\eta }$}^n,z)$ be a pointed Heegaard $(n+1)$ –tuple diagram; that is, a surface of genus $g$ , together with $(n+1)$ distinct $g$ –tuples $\\mbox{${\\eta }$}^i=\\lbrace \\eta _1^i,...,\\eta _g^i\\rbrace $ of homologically linearly independent attaching curves as in [31].",
"A Heegaard $(n+1)$ –tuple diagram naturally gives rise to an oriented 4–manifold $W_{0,...,n}$ by the pants construction [26].",
"The boundary of this manifold is given by: $\\partial W_{0,...,n}= -Y_{0,1}\\sqcup -Y_{1,2} \\sqcup ... \\sqcup -Y_{n-1,n}\\sqcup Y_{0,n}, $ where $Y_{i,j}$ is the 3–manifold specified by the attaching curves $\\mbox{${\\eta }$}^i,\\mbox{${\\eta }$}^j$ .",
"Each $\\mbox{${\\eta }$}^i$ gives rise to a Lagrangian torus in Sym$^g(\\Sigma )$ which we denote $i$ .",
"We also require an admissibility hypotheses on two-chains with boundary that realize homological relations between curves in the different $g$ –tuples $\\mbox{${\\eta }$}^i$ , the so-called multi-periodic domains.",
"For this section it suffices to assume that all multi-periodic domains have positive and negative coefficients, a condition which can be achieved by winding their boundary on the Heegaard diagram.",
"To a Heegaard $(n+1)$ –tuple diagram one can associate a chain map: $ f_{0,...,n}: \\bigotimes _{i=1}^{n} \\widehat{CF}(Y_{i-1,i}) \\longrightarrow \\widehat{CF}(Y_{0,n}),$ defined by counting pseudo-holomorphic maps of Whitney $(n+1)$ –gons into Sym$^g$ with boundary conditions in the Lagrangian tori.",
"On generators, $f_{0,...,n}$ is given by $ f_{0,...,n}(\\mathbf {x}_1\\otimes ...\\otimes \\mathbf {x}_{n})=\\sum _{\\mathbf {y}}\\sum _{\\underset{\\mu (\\phi )=2-n, \\ n_z(\\phi )=0}{\\phi \\in \\pi _2(\\mathbf {x}_1,,...,\\mathbf {x}_{n},\\mathbf {y})}}{ \\#{ \\mathcal {M}}}(\\phi )\\cdot \\mathbf {y},$ and we extend linearly to the full complex.",
"Let us unpack the notation a bit.",
"The first summation is over all $\\mathbf {y}\\in \\mathbb {T}_{0}\\cap \\mathbb {T}_{n}$ .",
"In the second, we sum over all homotopy classes of Whitney $(n+1)$ –gons.",
"We restrict attention to only those homotopy classes with Maslov index $\\mu (\\phi )=2-n$ .",
"This condition ensures that as we vary within the $(n-2)$ –dimensional universal family of conformal $(n+1)$ –gons, we obtain a finite number of pseudo-holomorphic maps from these conformal polygons to Sym$^g(\\Sigma )$ .",
"We denote this number, reduced modulo 2, by $\\#\\mathcal {M}(\\phi )$ .",
"When $n=1$ the domain of our pseudo-holomorphic map has a 1–dimensional automorphism group isomorphic to $\\mathbb {R}$ , and we consider instead the unparametrized moduli space $\\widehat{\\mathcal {M}}={\\mathcal {M}}/\\mathbb {R}$ .",
"See [26] and [31] for more details.",
"The polygon operators satisfy an $A_\\infty $ -associativity relation (see [31]): $ \\sum _{0\\le i<j\\le n} f_{0,...i,j,...n} \\ (\\theta _1\\otimes ...\\otimes f_{i,...,j}(\\theta _{i+1}\\otimes ... \\otimes \\theta _{j})\\otimes ...\\otimes \\theta _{n})=0,$ where $\\theta _i\\in \\widehat{CF}(Y_{i-1,i})$ are chains in the Floer complexes associated to the pairs of Lagrangians assigned to the vertices of the $(n+1)$ –gon.",
"This relation breaks up into a collection of relations, one for each $n>0$ .",
"For $n=1$ , the relation simply states that $f_{0,1}: \\widehat{CF}(Y_{0,1})\\rightarrow \\widehat{CF}(Y_{0,1})$ is a differential.",
"Examining the definition of the $H_1(Y;\\mathbb {Z})/\\mathrm {Tor}$ action from the previous section, we see that it closely resembles the Floer differential.",
"Indeed, the action of $\\zeta $ is simply the Floer differential weighted by $\\zeta \\cdot \\partial _0 \\phi $ , the intersection of a curve representing a class in $H_1(Y;\\mathbb {Z})/\\mathrm {Tor}$ with the $\\mbox{${\\eta }$}^0$ –component of the boundary of the image of a pseudo-holomorphic Whitney disk.",
"Motivated by this similarity we define operators for any closed curve $\\zeta \\in \\Sigma $ $a^\\zeta _{0,...,n}\\colon \\,\\bigotimes _{i=1}^{n} \\widehat{CF}(Y_{{i-1},{i}}) \\longrightarrow \\widehat{CF}(Y_{{0},{n}}).$ On generators, $a^\\zeta _{0,...,n}$ is given by $ a^\\zeta _{0,...,n}(\\mathbf {x}_1\\otimes ...\\otimes \\mathbf {x}_{n})=\\sum _{\\mathbf {y}}\\sum _{\\underset{\\mu (\\phi )=2-n, n_z(\\phi )=0}{\\phi \\in \\pi _2(\\mathbf {x}_1,...,\\mathbf {x}_{n},\\mathbf {y})}}{ (\\zeta \\cdot (\\partial _0\\phi )) \\#{ \\mathcal {M}}}(\\phi )\\cdot \\mathbf {y},$ and we extend linearly to the full complex.",
"We have the following $A_\\infty $ -compatibility relation between the $a^\\zeta $ and $f$ maps.",
"Theorem 3.4 Let $\\zeta \\in \\Sigma $ be a curve on the Heegaard surface of a given $(n+1)$ –tuple diagram which is in general position with respect to all the attaching curves.",
"Let $f_{0,...n}$ and $a^\\zeta _{0,...n}$ be the polygon maps defined above.",
"Then we have the following relation: $ \\sum _{0\\le i<j\\le n} a^\\zeta _{0,...i,j,...n} \\ (\\theta _1\\otimes ...\\otimes f_{i,...,j}(\\theta _{i+1}\\otimes ... \\otimes \\theta _{j})\\otimes ...\\otimes \\theta _{n}) \\ \\ + $ $ \\sum _{0< j\\le n} f_{0,j,...n} \\ (a^\\zeta _{0,...j}(\\theta _1\\otimes ...\\otimes \\theta _{j})\\otimes \\theta _{j+1}\\otimes ... \\otimes \\theta _{n})=0,$ Where $\\theta _i\\in \\widehat{CF}(Y_{i-1,i})$ are any Heegaard Floer chains.",
"Proof.",
"We consider the $n=3$ version of the relation.",
"A guiding schematic is shown in Figure REF .",
"First we establish some notation.",
"Suppose $\\theta =\\sum _{l=1}^{N} \\mathbf {x}_l$ is a chain (recall we work mod 2, so that no coefficient is needed in front of $\\mathbf {x}_i$ ).",
"Let $\\pi _2(\\theta ,\\mathbf {p},\\mathbf {q},\\mathbf {s}):=\\coprod _{l=1,...,N} \\lbrace \\phi \\in \\pi _2(\\mathbf {x}_l,\\mathbf {p},\\mathbf {q},\\mathbf {s}) \\rbrace ,$ with similar notation to handle $\\pi _2(\\mathbf {p},\\theta ,\\mathbf {q},\\mathbf {s})$ , $\\pi _2(\\mathbf {s},\\mathbf {p},\\theta ,\\mathbf {q})$ , and $\\pi _2(\\mathbf {p},\\mathbf {q},\\mathbf {s},\\theta )$ Consider the moduli space $\\mathcal {M}(\\phi \\in \\pi _2(\\theta _1,\\theta _2,\\theta _3,\\theta _4), \\mu (\\phi )=0):=\\coprod _{\\underset{\\mu (\\phi )=0, n_z(\\phi )=0}{ \\phi \\in \\pi _2(\\theta _1,\\theta _2,\\theta _3,\\theta _4) } } \\mathcal {M}\\big (\\phi \\big ).$ For an appropriate family of almost complex structures on Sym$^g(\\Sigma )$ , this moduli space is a smooth 1–dimensional manifold with a Gromov compactification.",
"The boundary points of this compactification are in bijection with points in the following products of zero dimensional moduli spaces: $\\widehat{\\mathcal {M}}\\big (\\phi _1\\in \\pi _2(\\theta _1,\\rho ),\\mu (\\phi _1)=1\\big )\\times \\mathcal {M}\\big (\\psi _1\\in \\pi _2(\\rho ,\\theta _2,\\theta _3,\\theta _4),\\mu (\\psi _1)=-1\\big ),$ $\\widehat{\\mathcal {M}}\\big (\\phi _2\\in \\pi _2(\\theta _2,\\rho ),\\mu (\\phi _2)=1\\big )\\times \\mathcal {M}\\big (\\psi _2\\in \\pi _2(\\theta _1,\\rho ,\\theta _3,\\theta _4),\\mu (\\psi _2)=-1\\big ),$ $\\widehat{\\mathcal {M}}\\big (\\phi _3\\in \\pi _2(\\theta _3,\\rho ),\\mu (\\phi _3)=1\\big )\\times \\mathcal {M}\\big (\\psi _3\\in \\pi _2(\\theta _1,\\theta _2,\\rho ,\\theta _4),\\mu (\\psi _3)=-1\\big ),$ $\\mathcal {M}\\big (\\phi _4\\in \\pi _2(\\theta _1,\\theta _2,\\theta _3,\\rho ),\\mu (\\phi _4)=-1\\big ) \\times \\widehat{\\mathcal {M}}\\big (\\psi _4\\in \\pi _2(\\rho ,\\theta _4),\\mu (\\psi _4)=1\\big ),$ and ${\\mathcal {M}}\\big (\\phi _5\\in \\pi _2(\\theta _2,\\theta _3,\\rho ),\\mu (\\phi _5)=0\\big )\\times \\mathcal {M}\\big (\\psi _5\\in \\pi _2(\\theta _1,\\rho ,\\theta _4),\\mu (\\psi _5)=0\\big ),$ ${\\mathcal {M}}\\big (\\phi _6\\in \\pi _2(\\theta _1,\\theta _2,\\rho ),\\mu (\\phi _6)=0\\big )\\times \\mathcal {M}\\big (\\psi _6\\in \\pi _2(\\rho ,\\theta _3,\\theta _4),\\mu (\\psi _6)=0\\big ).$ In each case $\\rho $ is dummy variable ranging over all chains in the appropriate Floer complex.",
"Thus we are considering moduli spaces associated to all possible decompositions of the homotopy classes in $\\pi _2(\\theta _1,\\theta _2,\\theta _3,\\theta _4)$ with $\\mu (\\phi )=0$ and $n_z(\\phi )=0$ into homotopy classes of bigons concatenated with rectangles and triangles concatenated with triangles.",
"For each $\\phi \\in \\pi _2(\\theta _1,\\theta _2,\\theta _3,\\theta _4)$ with $\\mu (\\phi )=0$ we expand the equality $(\\zeta \\cdot \\partial _{0}\\phi )\\#(\\partial \\mathcal {M}(\\phi ))=0,$ and observe that since $\\phi _i*\\psi _i=\\phi $ for each $i=1,...,6$ , we have $\\zeta \\cdot \\partial _{0}\\phi =\\zeta \\cdot \\partial _{0}\\phi _i+\\zeta \\cdot \\partial _{0}\\psi _i.$ Figure: Schematic for n=3n=3 case of Theorem .",
"One dimensional moduli spaces of rectangles are considered, with boundary conditions in the numbered Lagrangians.",
"The compactified moduli spaces have six types of boundary points.",
"A dark circle on a boundary arc labeled “0\" indicates that we weight the count of points in a homotopy class by the intersection of the η 0 ⊂Σ\\mbox{${\\eta }$}^0\\subset \\Sigma portion of the boundary with the curve ζ⊂Σ\\zeta \\subset \\Sigma .",
"Conservation of intersection number ensures that the sum of the weights in any vertical column equals ζ·∂ 0 φ\\zeta \\cdot \\partial _0\\phi .",
"The 9 types of weighted boundary point counts are in bijection with the terms in the n=3n=3 A ∞ A_\\infty -relation for the ζ\\zeta -action.Summing over all $\\phi \\in \\pi _2(\\theta _1,\\theta _2,\\theta _3,\\theta _4)$ , the terms in the expansion correspond to the coefficient of $\\theta _4$ in $ a^\\zeta _{0,1,2,3}(f_{0,1}(\\theta _1)\\otimes \\theta _2\\otimes \\theta _3)+f_{0,1,2,3}(a^\\zeta _{0,1}(\\theta _1)\\otimes \\theta _2\\otimes \\theta _3)$ $+ a^\\zeta _{0,1,2,3}(\\theta _1\\otimes f_{1,2}(\\theta _2)\\otimes \\theta _3) $ $+a^\\zeta _{0,1,2,3}(\\theta _1\\otimes \\theta _2\\otimes f_{2,3}(\\theta _3))$ $+ a^\\zeta _{0,3}(f_{0,1,2,3}(\\theta _1\\otimes \\theta _2\\otimes \\theta _3)) + f_{0,3}(a^\\zeta _{0,1,2,3}(\\theta _1\\otimes \\theta _2\\otimes \\theta _3)) $ $ +a^\\zeta _{0,1,3}(\\theta _1\\otimes f_{1,2,3}(\\theta _2\\otimes \\theta _3)) $ $ +a^\\zeta _{0,2,3}(f_{0,1,2}(\\theta _1\\otimes \\theta _2)\\otimes \\theta _3)+ f_{0,2,3}(a^\\zeta _{0,1,2}(\\theta _1\\otimes \\theta _2)\\otimes \\theta _3).$ The terms in the six lines correspond to the contribution from the six types of boundary points for the 1–dimensional moduli space, listed above.",
"Note that the asymmetry between the $a\\circ f$ and $f\\circ a$ terms is a result of the fact that $\\zeta \\cdot \\partial _0 \\phi _2=\\zeta \\cdot \\partial _0\\phi _3=\\zeta \\cdot \\partial _0\\phi _5=0, $ which, in turn, is due to the fact that these homotopy classes have boundary conditions outside of 0.",
"This proves the proposition for $n=3$ .",
"The general case is a straightforward yet notationally cumbersome extension."
],
[
"Cobordisms",
"In this section we use the compatibility relation between the polygon operators $a^{\\zeta }$ and $f$ given by Theorem REF to understand the behavior of the Ozsváth–Szabó cobordism maps with respect to the $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ action.",
"Our main goal is Theorem REF , which shows that the maps on Floer homology induced by a cobordism $W$ from $Y_1$ to $Y_2$ commute with the action, provided that the curves in $Y_1$ and $Y_2$ inducing the action are homologous in $W$ .",
"The key is the $n=2$ compatibility relation.",
"Let $(\\Sigma ,\\mbox{${\\eta }$}^0,\\mbox{${\\eta }$}^1,\\mbox{${\\eta }$}^2,z)$ be a pointed Heegaard triple-diagram, and let $f_{0,1,2}\\colon \\,{\\widehat{CF}}(Y_{0,1}) \\otimes {\\widehat{CF}}(Y_{1,2})\\rightarrow {\\widehat{CF}}(Y_{0,2})$ denote the polygon map from the previous section, which counts pseudo-holomorphic Whitney triangles.",
"Let $\\Theta \\in {\\widehat{CF}}(Y_{1,2})$ be a cycle.",
"Then we define a map $f_{\\Theta }: \\widehat{CF}(Y_{0,1})\\longrightarrow \\widehat{CF}(Y_{0,2})$ by $ f_{\\Theta }(-):=f_{0,1,2}(-\\otimes \\Theta ).$ The $A_\\infty $ relation for $n=2$ states that $&&f_{0,1,2}(f_{0,1}(-)\\otimes \\Theta )+ f_{0,1,2}(-\\otimes f_{1,2}(\\Theta ))+ f_{0,2}(f_{0,1,2}(-\\otimes \\Theta ) \\\\&=& f_{\\Theta }\\circ \\partial _{0,1}+ 0 + \\partial _{0,2}\\circ f_{\\Theta }=0,$ where we have used the fact that $\\Theta $ is a cycle for the $f_{1,2}$ differential.",
"This shows that $f_\\Theta $ is a chain map.",
"Note that $Y_{1,2}$ could be any 3–manifold for the moment, and $\\Theta $ an arbitrary cycle.",
"In many concrete situations $Y_{1,2}\\cong \\#^n S^1\\times S^2$ and $\\Theta $ can be assumed to be a generator, $\\Theta ={\\mathbf {y}}\\in \\mathbb {T}_{1}\\cap \\mathbb {T}_{2}$ .",
"Given a closed curve $\\zeta \\subset \\Sigma $ that is in general position with respect to the attaching curves, we can regard $\\zeta $ as a curve in $Y_{0,1}$ and $Y_{0,2}$ .",
"As such, we have an action by $\\zeta $ on $\\widehat{CF}(Y_{0,1})$ and $\\widehat{CF}(Y_{0,2})$ .",
"The following lemma shows that the induced maps on homology commute with that of $f_{\\Theta }$ .",
"Lemma 3.5 Given a pointed Heegaard triple diagram, $(\\Sigma ,\\mbox{${\\eta }$}^0,\\mbox{${\\eta }$}^1,\\mbox{${\\eta }$}^2,z)$ , let $f_{\\Theta }=f_{0,1,2}(-\\otimes \\Theta )$ be the chain map from $\\widehat{CF}(Y_{0,1})$ to $\\widehat{CF}(Y_{0,2})$ associated to a cycle $\\Theta \\in \\widehat{CF}(Y_{1,2})$ .",
"Let $\\zeta \\subset \\Sigma $ be a curve and $a^{\\zeta _0},a^{\\zeta _1}$ denote the actions of $\\zeta $ on $\\widehat{CF}(Y_{0,1})$ and $\\widehat{CF}(Y_{0,2})$ , respectively.",
"Then $f_{\\Theta }\\circ a^{\\zeta _0}$ and $a^{\\zeta _1}\\circ f_{\\Theta }$ are chain homotopic.",
"Proof.",
"The $n=2$ version of Theorem REF states that $ a_{0,1,2}^\\zeta (f_{0,1}(-)\\otimes \\Theta )+ a_{0,1,2}^\\zeta (-\\otimes f_{1,2}(\\Theta ))+a_{0,2}^\\zeta (f_{0,1,2}(-\\otimes \\Theta )) +$ $+ f_{0,1,2}(a_{0,1}^\\zeta (-)\\otimes \\Theta )+ f_{0,2}(a_{0,1,2}^\\zeta (-)\\otimes \\Theta )=0.$ Let $H(-):=a_{0,1,2}^\\zeta (-\\otimes \\Theta )$ .",
"Using the fact that $f_{1,2}(\\Theta )=0$ , we get $H\\circ \\partial _{0,1} +a^{\\zeta _1}\\circ f_\\Theta + f_\\Theta \\circ a^{\\zeta _0}+ \\partial _{0,2}\\circ H=0.$ In other words, $a_{0,1,2}^\\zeta $ provides the requisite chain homotopy.",
"The following theorem concerns the naturality of the homological action under the homomorphisms induced by cobordisms.",
"Similar results can be found in [29].",
"Theorem 3.6 Suppose $Y_1,Y_2$ are two closed, oriented, connected 3–manifolds, and $W$ is a cobordism from $Y_1$ to $Y_2$ .",
"Let $\\widehat{F}_{W}\\colon \\,\\widehat{HF}(Y_1)\\longrightarrow \\widehat{HF}(Y_2)$ be the homomorphism induced by $W$ .",
"Suppose $\\zeta _1\\subset Y_1$ , $\\zeta _2\\subset Y_2$ are two closed curves which are homologous in $W$ .",
"Then $\\widehat{F}_{W}\\circ A^{[\\zeta _1]}= A^{[\\zeta _2]}\\circ \\widehat{F}_{W}.$ Proof.",
"Since $\\zeta _1$ and $\\zeta _2$ are homologous in $W$ , there exists an oriented proper surface $S\\subset W$ connecting $\\zeta _1$ to $\\zeta _2$ .",
"By adding tubes between the components of $S$ , we can assume that it is connected.",
"Let $S^{\\prime }$ denote the surface obtained by removing a collar neighborhood of the components of the boundary of $S$ lying in $Y_2$ .",
"Now let $W_1^{\\prime }$ be a neighborhood of $Y_1\\cup S^{\\prime }$ in $W$ .",
"Then $W_1^{\\prime }$ can be obtained from $Y_1\\times I$ by adding 1–handles.",
"Thus $\\partial W_1^{\\prime }=-Y_1\\sqcup Y_1^{\\prime }$ , where $Y_1^{\\prime }\\cong Y_1\\#^kS^1\\times S^2$ .",
"The boundary of $S^{\\prime }$ induces a curve $\\zeta _1^{\\prime }\\subset Y_1^{\\prime }$ .",
"Using Lemma REF and Proposition REF , we see that the conclusion of the theorem holds for the cobordism $W_1^{\\prime }$ and the curves $\\zeta _1,\\zeta _1^{\\prime }$ .",
"The cobordism $W$ can thus be decomposed as $W_1^{\\prime }\\cup _{Y_1^{\\prime }}W_2$ , where $W_2$ consists of 1–, 2–, and 3–handles which are disjoint from $S$ ; indeed, the portion of $S$ in $W_2$ is simply the trace of $\\zeta _1^{\\prime }$ in this cobordism.",
"The map on Floer homology induced by a cobordism is defined by associating chain maps to handle attachments in a handle decomposition [29].",
"As above, Lemma REF and Proposition REF show that the maps associated to the 1– and 3–handles in $W_2$ commute with the action.",
"Commutativity of the action with the maps associated to the 2–handles is ensured by Lemma REF (the 2–handle maps are defined as $f_\\Theta $ for an appropriate Heegaard triple diagram and choice of $\\Theta $ ).",
"Hence our conclusion for $W$ holds by the composition law for cobordism maps, [29].",
"Remark 3.7 The above theorem has obvious generalizations in two directions.",
"First, one could refine the theorem to account for $\\mathrm {Spin}^c$ structures: to a cobordism $W$ equipped with a $\\mathrm {Spin}^c$ structure $\\mathfrak {t}$ we have maps between the Floer homology groups $\\widehat{HF}(Y_1,\\mathfrak {t}|_{Y_1})$ and $\\widehat{HF}(Y_2,\\mathfrak {t}|_{Y_2})$ , and one can show that the commutativity theorem respects this structure.",
"Second, one can prove the same theorem for the other versions of Floer homology.",
"In this situation, however, one must take care.",
"We have been considering the sum of cobordism maps associated to all $\\mathrm {Spin}^c$ structures on $W$ .",
"This cannot, in general, be done with the minus and infinity versions of Floer homology, since these versions require admissibility hypotheses that cannot be simultaneously achieved for all $\\mathrm {Spin}^c$ structures with a single Heegaard diagram.",
"For that reason, the commutativity theorem in these versions must incorporate $\\mathrm {Spin}^c$ structures.",
"Alternatively, one can continue to sum over all $\\mathrm {Spin}^c$ structures if we consider the “completed\" versions of $CF^-$ and $CF^\\infty $ with base rings $\\mathbb {F}[[U]]$ and $\\mathbb {F}[[U,U^{-1}]$ , respectively.",
"Similar remarks hold for the $A_\\infty $ compatibility relation, Theorem REF ."
],
[
"Heegaard Floer homology with twisted\ncoefficients",
"Suppose $Y$ is a closed oriented 3–manifold, $\\mathfrak {s}\\in \\mathrm {Spin}^c(Y)$ .",
"Let $(\\Sigma ,\\mbox{${\\alpha }$},\\mbox{$\\beta $},z)$ be an admissible Heegaard diagram for $(Y,\\mathfrak {s})$ .",
"Let $\\mathcal {R}=\\mathbb {F}[T,T^{-1}]$ .",
"Given $[\\omega ]\\in H^2(Y;\\mathbb {Z})$ , let $\\eta \\subset \\Sigma $ be a closed curve which represents the Poincaré dual of $\\omega $ in $H_1(Y;\\mathbb {Z})$ .",
"Let $\\underline{CF}^{\\infty }(Y,\\mathfrak {s};\\mathcal {R}_{\\eta })$ be the $\\mathcal {R}[U,U^{-1}]$ –module freely generated by $\\mathbf {x}\\in \\mathbb {T}_{\\alpha }\\cap \\mathbb {T}_{\\beta }$ which represent $\\mathfrak {s}$ .",
"The differential $\\underline{\\partial }$ is defined by $\\underline{\\partial }(\\mathbf {x})=\\sum _{\\mathbf {y}}\\sum _{\\stackrel{\\scriptstyle \\phi \\in \\pi _2(\\mathbf {x},\\mathbf {y})}{\\mu (\\phi )=1}}\\#\\big (\\mathcal {M}(\\phi )/\\mathbb {R}\\big )T^{\\eta \\cdot \\partial _{\\alpha }\\phi }U^{n_z(\\phi )}\\mathbf {y}.$ The homology of this chain complex depends on $\\eta $ only through its homology class in $H_1(Y)$ .",
"Similarly, there are chain complexes $\\underline{CF}^{\\pm }(Y,\\mathfrak {s};\\mathcal {R}_{\\eta })$ and $\\underline{\\widehat{CF}}(Y,\\mathfrak {s};\\mathcal {R}_{\\eta })$ .",
"Their homologies are called the $\\omega $ –twisted Floer homologies, denoted $\\underline{HF}^{\\circ }(Y,\\mathfrak {s};\\mathcal {R}_{\\eta })\\quad \\text{or}\\quad \\underline{HF}^{\\circ }(Y,\\mathfrak {s};\\mathcal {R}_{[\\omega ]})$ when there is no confusion.",
"The field $\\mathbb {F}=\\mathcal {R}/(T-1)\\mathcal {R}$ is also an $\\mathcal {R}$ –module.",
"By definition $CF^{\\circ }(Y;\\mathbb {F})=\\underline{CF}^{\\circ }(Y;\\mathcal {R}_{[\\omega ]})\\otimes _{\\mathcal {R}}\\mathbb {F}$ is the usual untwisted Heegaard Floer chain complex over $\\mathbb {F}$ .",
"There are chain maps on the $\\omega $ –twisted chain complex induced by cobordisms, as in Ozsváth–Szabó [26], [27], Jabuka–Mark [14] and Ni [25].",
"More precisely, suppose $W\\colon \\,Y_1\\rightarrow Y_2$ is a cobordism, $[\\Omega ]\\in H^2(W;\\mathbb {Z})$ .",
"Let $[\\omega _1],[\\omega _2]$ be the restriction of $[\\Omega ]$ to $Y_1$ and $Y_2$ , respectively.",
"Then there is a map $\\underline{F}^{\\circ }_{W;[\\Omega ]}\\colon \\,\\underline{HF}^{\\circ }(Y_1;\\mathcal {R}_{[\\omega _1]})\\rightarrow \\underline{HF}^{\\circ }(Y_2;\\mathcal {R}_{[\\omega _2]}).$ We can also define the $\\Lambda ^*(H_1(Y)/\\mathrm {Tors})$ action on the $\\omega $ –twisted Floer homology by letting $a^{\\zeta }(\\mathbf {x})=\\sum _{\\mathbf {y}\\in \\mathbb {T}_{\\alpha }\\cap \\mathbb {T}_{\\beta }}\\sum _{\\lbrace \\phi \\in \\pi _2(\\mathbf {x},\\mathbf {y})|\\mu (\\phi )=1\\rbrace }\\big (\\zeta \\cdot (\\partial _{\\alpha }\\phi )\\big )\\:\\#\\widehat{\\mathcal {M}}(\\phi )T^{\\eta \\cdot \\partial _{\\alpha }\\phi }U^{n_z(\\phi )}\\mathbf {y}.$ As in Subsection REF , there are twisted versions of (REF ) and Theorem REF ."
],
[
"Module structures and the link surgeries spectral sequence",
"The connection between Khovanov homology and Heegaard Floer homology arises from a calculational tool called the link surgeries spectral sequence, [31].",
"Roughly, this device takes as input a framed link in a 3–manifold, with output a filtered complex.",
"The homology of the complex is isomorphic to the Heegaard Floer homology of the underlying 3–manifold, and the $E_1$ term of the associated spectral sequence splits as a direct sum of the Heegaard Floer groups of the manifolds obtained by surgery on the link with varying surgery slopes.",
"The differentials in the spectral sequence are induced by the holomorphic polygon maps of Subsection REF .",
"Applying this machinery to a particular surgery presentation of the 2-fold branched cover of a link produces the spectral sequence from Khovanov homology to the Heegaard Floer homology.",
"Our purpose in this section is to refine the link surgeries spectral sequence to incorporate the $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ –module structure on Floer homology, and subsequently relate this structure to the module structure on Khovanov homology.",
"In Subsection REF we prove that a curve in the complement of a framed link induces a filtered chain map acting on the complex giving rise to the link surgeries spectral sequence, Theorem REF .",
"In Subsection REF we use these filtered chain maps to endow the spectral sequence from Khovanov homology to the Floer homology of the branched double cover with a module structure.",
"This module structure allows us to prove a “collapse\" result for the spectral sequence (Proposition REF ).",
"This result states that if the Khovanov module of a link is isomorphic to that of the unlink then this module is also isomorphic to the Floer homology of the branched double cover.",
"Since our results involve filtered chain maps and the morphisms they induce on spectral sequences, we begin with a digression on spectral sequences and their morphisms.",
"The reader familiar with these concepts may wish to skip ahead to Subsection REF , but is warned that our perspective on spectral sequences, while equivalent to the standard treatment, is slightly non-standard."
],
[
"A review of spectral sequences and their morphisms",
"Suppose that a complex of vector spaces $(C,d)$ admits a decomposition $ C=\\underset{i\\ge 0}{\\bigoplus }\\ C^{(i)},$ which is respected by the differential, in the sense that $d=d^{(0)}+d^{(1)}+d^{(2)}+...$ , where $d^{(m)}: C^{(i)}\\rightarrow C^{(i+m)}$ for each $i$ .",
"From this structure, one can construct a spectral sequence; that is, a sequence of chain complexes $\\lbrace (E_r,\\delta _r)\\rbrace _{r=0}^{\\infty }$ satisfying $H_*(E_r,\\delta _r)\\cong E_{r+1}$ .",
"Under mild assumptions, one has $\\delta _r=0$ for all sufficiently large $r$ , and the resulting limit satisfies $E_\\infty \\cong H_*(C,d)$ .",
"Typically, one constructs such a spectral sequence by noting that the subcomplexes, $F^p= \\underset{i\\ge p}{\\bigoplus }\\ C^{(i)},$ form a filtration $C=F^0\\supset F^1\\supset ...,$ and then appealing to the well-known construction of the spectral sequence associated to a filtered complex (see, e.g.",
"[24]).",
"For the purpose of understanding morphisms of spectral sequence induced by filtered chain maps, we find it more transparent to construct the spectral sequence by a method called iterative cancellation or reduction, a procedure we briefly recall.",
"The method relies on the following well-known lemma: Lemma 4.1 Let $(C,d)$ be a chain complex of $R$ -modules, freely generated by chains $\\lbrace \\bf x_i\\rbrace _{i\\in I}$ , and suppose that $d({\\bf x_k}, {\\bf x_l})=1$ , where $d({\\bf a},{\\bf b})$ denotes the coefficient of $\\bf b$ in $d(\\bf a)$ .",
"Then we can define a complex $(C^{\\prime },d^{\\prime })$ , freely generated by $\\lbrace {\\bf x_i} | i\\ne k,l\\rbrace $ , which is chain homotopy equivalent to $(C,d)$ .",
"Proof.",
"Let $h:C\\rightarrow C$ be the module homomorphism defined by $h(\\mathbf {x}_l)=\\mathbf {x}_k$ and $h(\\mathbf {x}_i)=0$ if $i\\ne l$ .",
"Then the differential on $C^{\\prime }$ is given by $d^{\\prime }= \\pi \\circ (d-d h d)\\circ \\iota ,$ where $\\pi : C\\rightarrow C^{\\prime }$ and $\\iota : C^{\\prime }\\rightarrow C$ are the natural projection and inclusions.",
"Now the chain maps $f:C\\rightarrow C^{\\prime }$ and $g:C^{\\prime }\\rightarrow C$ defined by $f=\\pi \\circ ( \\mathbb {I} - d\\circ h) \\ \\ \\ \\ \\ g=(\\mathbb {I} - h\\circ d)\\circ \\iota , $ are mutually inverse chain homotopy equivalences.",
"Indeed, $f\\circ g = \\mathbb {I}_{C^{\\prime }}$ and $g\\circ f\\sim \\mathbb {I}_{C}$ via the homotopy $h$ .",
"Of course we can employ the lemma under the weaker assumption that $d({\\bf x_k}, {\\bf x_l})$ is a unit in $R$ , simply by rescaling the basis.",
"In the present situation $(C,d)$ is a complex of vector spaces, so this applies whenever $d({\\bf x_k}, {\\bf x_l})\\ne 0$ .",
"We will use the lemma in a filtered sense.",
"To make this precise, given a complex with increasing filtration as above, let $\\mathcal {F}(a)\\in \\mathbb {Z}^{\\ge 0}$ denote the filtration level of a chain, i.e.",
"$\\mathcal {F}(a)= \\text{max}\\lbrace i\\in \\mathbb {Z}^{\\ge 0} \\ | a\\in F^i\\rbrace ,$ and define $\\mathcal {F}(a,b)=\\mathcal {F}(a)-\\mathcal {F}(b)$ .",
"Note that the filtration of a linear combination of chains satisfies $\\mathcal {F}(a+b)\\ge \\text{min}\\lbrace \\mathcal {F}(a),\\mathcal {F}(b)\\rbrace .$ Lemma 4.2 With the notation from Lemma REF , suppose that $(C,d)$ is a filtered complex, that $d(\\mathbf {x}_k,\\mathbf {x}_l)=1$ and that $ \\mathcal {F}(d(\\mathbf {x}_k), \\mathbf {x}_l)\\ge 0.$ Then the reduced complex $(C^{\\prime },d^{\\prime })$ inherits a filtration $\\mathcal {F}^{\\prime }$ from $(C,d)$ by the formula $\\mathcal {F}^{\\prime }(a):=\\mathcal {F}(\\iota (a))$ and the filtration degree of $d^{\\prime }$ is no less than that of $d$ in the sense that $ \\mathcal {F}^{\\prime }(d^{\\prime }(a))\\ge \\mathcal {F}(d\\circ \\iota (a)), \\quad \\text{for all\\ } a\\in C^{\\prime }.$ Moreover, if $\\mathcal {F}(\\mathbf {x}_k,\\mathbf {x}_l)=0$ , then $(C,d)$ and $(C^{\\prime },d^{\\prime })$ are filtered chain homotopy equivalent.",
"Proof.",
"To prove that $\\mathcal {F}^{\\prime }$ defines a filtration on the reduced complex we must show $ \\mathcal {F}^{\\prime }(a,d^{\\prime }(a))\\le 0 \\quad \\text{for all\\ } a \\in C^{\\prime },$ which, by definition, is the same as showing $ \\mathcal {F}(\\iota (a))\\le \\mathcal {F}(\\iota (d^{\\prime }(a))=\\mathcal {F}(\\iota \\circ \\pi \\circ (d-dhd)\\circ \\iota (a)).$ To begin, observe that for any $x\\in C$ we have $\\mathcal {F}(\\iota \\circ \\pi (x))\\ge \\mathcal {F}(x)$ ; that is, dropping $\\mathbf {x}_k$ and $\\mathbf {x}_l$ from a chain can only increase the filtration level.",
"Thus the right-hand side of (REF ) satisfies $\\mathcal {F}(\\iota \\circ \\pi \\circ (d-dhd)\\circ \\iota (a))\\ge \\mathcal {F}((d-dhd)\\circ \\iota (a)).$ Let $r\\in R$ denote $d(\\iota (a),\\mathbf {x}_l)$ , the coefficient of $\\mathbf {x}_l$ in $d(\\iota (a))$ .",
"By the definition of $h$ , the right hand of (REF ) is equal to $\\mathcal {F}(d(\\iota (a))-r\\cdot d(\\mathbf {x}_k))$ , which satisfies the inequality: $\\mathcal {F}(d(\\iota (a))-r\\cdot d(\\mathbf {x}_k))\\ge \\text{min}\\lbrace \\mathcal {F}(d(\\iota (a)),\\mathcal {F}(rd(\\mathbf {x}_k))\\rbrace $ Now if $\\mathcal {F}(rd(\\mathbf {x}_k))\\ge \\mathcal {F}(d(\\iota (a))$ then the right hand side of (REF ) equals $\\mathcal {F}(d(\\iota (a))$ , which satisfies $\\mathcal {F}(d(\\iota (a))\\ge \\mathcal {F}(\\iota (a))$ because $(C,d)$ is filtered.",
"If $\\mathcal {F}(rd(\\mathbf {x}_k))<\\mathcal {F}(d(\\iota (a))$ , then $r\\ne 0$ and the right side of (REF ) equals $\\mathcal {F}(rd(\\mathbf {x}_k))$ which satisfies $\\mathcal {F}(rd(\\mathbf {x}_k))\\ge \\mathcal {F}(\\mathbf {x}_l)$ by assumption.",
"In this case, though, we have $\\mathcal {F}(\\mathbf {x}_l)\\ge \\mathcal {F}(\\iota (a))$ since $d(\\iota (a),\\mathbf {x}_l)\\ne 0$ and $(C,d)$ is filtered.",
"Thus in both cases, we have verified (REF ).",
"The statement about filtration degree is straightforward.",
"To see that $C$ and $C^{\\prime }$ are filtered homotopy equivalent when the filtration levels of the cancelled chains agree, it suffices to show that the chain maps $f$ and $g$ and chain homotopy $h$ from the previous lemma are filtered, in the sense that $ \\mathcal {F}^{\\prime } \\circ f\\ge \\mathcal {F}, \\quad \\mathcal {F}\\circ g \\ge \\mathcal {F}^{\\prime }, \\quad \\text{and} \\quad \\mathcal {F}\\circ h\\ge \\mathcal {F}.$ These are immediate from the definition of $\\mathcal {F}^{\\prime }$ and the fact that $h$ preserves $\\mathcal {F}$ (if the value of $h$ is nonzero) when $\\mathcal {F}(\\mathbf {x}_k,\\mathbf {x}_l)=0$ .",
"We can use the lemmas to easily produce a spectral sequence.",
"To begin, define $(E_0, d_0)= (C,d)$ .",
"Now use Lemma REF to cancel all of the non-zero terms in the differential of order zero i.e.",
"the $d^{(0)}$ terms.",
"Lemma REF implies that the result is a filtered chain homotopy equivalent complex, $(E_1,d_1)$ for which the lowest order terms in the differential are of order one; that is, the differential can be written as $d_1=d_1^{(1)}+ d_1^{(2)} +\\cdots $ with respect to the natural direct sum decomposition of $E_1$ induced by (REF ).",
"Now cancel the $d_1^{(1)}$ components of $d_1$ .",
"The result is a chain homotopy equivalent complex $(E_2,d_2)$ satisfying $d_2=d_2^{(2)}+d_2^{(3)}+\\cdots $ .",
"Assuming the complex is finitely generated in each homological degree, we can iterate this process until all differentials of all orders have been cancelled.",
"Denote the lowest order term of $d_r$ by $\\delta _r:= d_r^{(r)}.$ The fact that $d_r\\circ d_r=0$ implies $\\delta _r\\circ \\delta _r=0$ , and canceling $d_r^{(r)}$ is equivalent to taking homology with respect to $\\delta _r$ .",
"We represent the process schematically: $\\begin{array}{cc}E_0, d_0=d^{(0)}+d^{(1)}+d^{(2)}+d^{(3)}+\\cdots & \\\\\\downarrow & \\text{Homology with respect to }\\delta _0=d^{(0)} \\\\E_1, d_1= \\ \\ 0 \\ + d_1^{(1)} +d_1^{(2)}+d_1^{(3)}+\\cdots & \\\\\\downarrow & \\text{Homology with respect to }\\delta _1=d_1^{(1)}\\\\E_2, d_2= \\ \\ 0 \\ + \\ \\ 0 \\ +d_2^{(2)}+d_2^{(3)}+\\cdots & \\\\\\downarrow & \\text{Homology with respect to }\\delta _2=d_2^{(2)}\\\\\\vdots & \\\\E_\\infty , d_\\infty \\equiv \\ \\ 0 \\ \\ \\hspace{93.95122pt}& \\\\\\end{array}$ The resulting structure is a spectral sequence with $r$ -th page given by $(E_r,\\delta _r)$ .",
"By construction, $E_\\infty \\cong H_*(C,d)$ , since $(E_\\infty ,d_\\infty )$ is chain homotopy equivalent to $(C,d)$ .",
"The concerned reader can take comfort in the knowledge that the spectral sequence described here is isomorphic to that produced by the standard construction, [24].",
"The proof of equivalence is straightforward but rather notationally cumbersome, and since our results make no use of it we leave it for the interested reader.",
"Let $C$ and $\\overline{C}$ be complexes with filtrations induced by decompositions: $ C=\\underset{i\\ge 0}{\\bigoplus }\\ C^{(i)} \\ \\ \\ \\ \\ \\overline{C}= \\underset{i\\ge 0}{\\bigoplus }\\ {\\overline{C}}^{(i)},$ and let $a:C\\rightarrow \\overline{C}$ be a filtered chain map, i.e.",
"$a(F^i)\\subset \\overline{F}^i$ .",
"Such a map is well-known to induce a morphism between the associated spectral sequences; that is a sequence of chain maps: $ \\alpha _r: (E_r,\\delta _r) \\longrightarrow (\\overline{E}_r,\\overline{\\delta }_r), \\ \\ \\text{satisfying} \\ \\ (\\alpha _r)_*=\\alpha _{r+1} $ A standard treatment of this construction can be found in [24].",
"The perspective of reduction offers a concrete construction of this morphism as follows.",
"To begin, note that a filtered chain map decomposes into homogeneous summands in a manner similar to the differential $ a=a^{(0)}+a^{(1)}+a^{(2)}+\\cdots ,\\ \\ \\text{where} \\ \\ a^{(m)}: C^{(i)}\\rightarrow \\overline{C}^{(i+m)}.",
"$ Recalling that $E_0=C$ and $\\overline{E}_0=\\overline{C}$ , let $a_0:E_0\\rightarrow \\overline{E}_0$ be defined as $a_0:=a$ .",
"Let $g_r:(E_r,d_r)\\rightarrow (E_{r+1},d_{r+1}) \\ \\ \\text{and} \\ \\ \\overline{g}_r:(\\overline{E}_r,\\overline{d}_r)\\rightarrow (\\overline{E}_{r+1},\\overline{d}_{r+1})$ denote the chain homotopy equivalences that cancel the $r$ -th order terms in the differential, and inductively define $a_{r+1}: (E_{r+1},d_{r+1})\\rightarrow (\\overline{E}_{r+1},\\overline{d}_{r+1})$ by $a_{r+1}=\\overline{g}_r\\circ a_r\\circ (g_r)^{-1},$ where $(g_r)^{-1}$ is the chain homotopy inverse of $g_r$ .",
"Now $a_r$ decomposes into summands according to the filtration, and it is easy to see that each $a_r$ respects the filtration, i.e.",
"$ a_r= a_r^{(0)}+ a_r^{(1)}+ a_r^{(2)}+\\cdots .$ Indeed, this holds by assumption for $a_0=a$ , and the homotopy equivalences $(g_r)^{-1}$ and $\\overline{g}_r$ provided by the cancellation lemma are all non-decreasing in filtration degree (they involve terms of the form $h\\circ d_r$ and $\\overline{d}_r\\circ \\overline{h}$ , respectively, where $h,\\overline{h}$ have degree $-r$ and $d_r,\\overline{d}_r$ have degree $r$ .)",
"Let the 0–th order term of $a_r$ be denoted by: $ \\alpha _r:= a_r^{(0)}.$ The 0–th order terms of the equality $a_r\\circ d_r=\\overline{d}_r\\circ a_r$ must themselves be equal, which implies $\\alpha _r \\circ \\delta _r= \\overline{\\delta }_r\\circ \\alpha _r$ .",
"Thus $\\alpha _r$ is a chain map between $(E_r,\\delta _r)$ and $(\\overline{E}_r,\\overline{\\delta }_r)$ , and by the definition of the map induced on homology, we have $(\\alpha _r)_*=\\alpha _{r+1}$ .",
"This provides the desired morphism of spectral sequences.",
"It should be pointed out that one can verify that the morphism constructed above agrees with the more traditional construction.",
"Since we will not make use of this fact we again omit the proof.",
"We should also point out that in many situations, one or all of the intermediate chain maps $a_r$ may have vanishing 0–th order terms; that is $a_r^{(0)}=0$ .",
"The same rational which shows $a_r^{(0)}=\\alpha _r$ is a chain map shows that the lowest order non-vanishing term is a chain map.",
"This allows one to construct a morphism of spectral sequences where the filtration shift of the maps between successive pages is monotonically increasing with the page index."
],
[
"An action on the link surgeries spectral sequence",
"Let $L=K_1\\cup \\cdots \\cup K_l \\subset Y$ be a framed link.",
"In this subsection we prove that a curve $\\gamma \\subset Y\\setminus L$ gives rise to a filtered endomorphism of the filtered complex that produces the link surgeries spectral sequence.",
"To state the theorem, we establish a bit of notation.",
"A multi-framing is an $l-$ tuple $I=\\lbrace m_1,...,m_l\\rbrace \\in \\hat{\\mathbb {Z}}^l$ where $\\hat{\\mathbb {Z}}=\\mathbb {Z}\\cup \\infty $ .",
"A multi-framing specifies a 3-manifold, which we denote $Y(I)$ , by performing surgery on $L$ with slope on the $i$ -th component given by $m_i$ (defined with respect to the base framing).",
"Thus $Y(\\infty ,...,\\infty )=Y$ corresponds to not doing surgery at all.",
"Theorem 4.3 Let $L\\subset Y$ be a framed link.",
"There is a filtered complex $(C(L),d)$ whose homology is isomorphic to $\\widehat{HF}(Y)$ .",
"The $E_1$ page of the associated spectral sequence satisfies $E_1\\cong \\bigoplus _{I\\in \\lbrace 0,1\\rbrace ^{|L|}} \\widehat{HF}(Y(I)).$ A curve $\\zeta \\subset Y\\setminus L$ induces a filtered chain map $ a^\\zeta \\colon \\,C(L)\\longrightarrow C(L),$ whose induced map is isomorphic to the action $ (a^{\\zeta })_*=A^{[\\zeta ]}\\colon \\,\\widehat{HF}(Y)\\longrightarrow \\widehat{HF}(Y).$ The induced map on the $E_1$ page of the spectral sequence is given by the sum ${\\begin{matrix}\\underset{I}{\\bigoplus }\\ \\widehat{HF}(Y(I)) &\\xrightarrow{}& \\underset{I}{\\bigoplus }\\ \\widehat{HF}(Y(I)),\\end{matrix}}$ where $A_I^{[\\zeta ]}$ is the map on $\\widehat{HF}(Y(I))$ induced by $\\zeta $ , viewed as a curve in $Y(I)$ .",
"Proof.",
"The existence of the filtered complex computing $\\widehat{HF}(Y)$ is [31].",
"The refinement here is the existence of a filtered chain map associated to a curve in the framed link diagram.",
"This map will be defined as a sum of the holomorphic polygon operators $a^\\zeta _{0,1,...,n}$ from Subsection REF .",
"In order to make this precise, we must recall the proof of Ozsváth and Szabó's theorem.",
"There are essentially two main steps.",
"The first is to construct a filtered chain complex $(X,D)$ from a framed link $L\\subset Y$ , and the second is to show that a natural (filtered) subcomplex $(C(L),d)$ has homology isomorphic to $\\widehat{HF}(Y)$ .",
"In both steps, we will take care to introduce our refinement at the appropriate time.",
"The first step in the proof is to associate a filtered complex to a Heegaard multi-diagram adapted to a framed link $L=K_1\\cup ... \\cup K_l \\subset Y$ .",
"More precisely, we have a Heegaard multi-diagram $(\\Sigma , \\mbox{${\\eta }$}^0,\\mbox{${\\eta }$}^1,...,\\mbox{${\\eta }$}^{k},w)$ with the property that the 3–manifolds $Y_{0,i}$ ($i=1,...,k$ ) are in one-to-one correspondence with the 3–manifolds $Y(I)$ associated to all possible multi-framings $I=\\lbrace m_1,...,m_l\\rbrace $ , with $m_i\\in \\lbrace 0,1,\\infty \\rbrace $ .",
"The 3–manifolds $Y_{i,j}$ for $i,j>0$ are all diffeomorphic to the connected sum of a number of $S^1\\times S^2$ 's.",
"As usual, we require the multi-diagram to be admissible.",
"From such a Heegaard diagram one constructs a filtered complex $(X,D)$ .",
"As a group, the complex splits as a direct sum over the set of multiframings: $X= \\bigoplus _{I\\in \\lbrace 0,1,\\infty \\rbrace ^l} \\widehat{CF}(Y(I)).$ The differential on X is given as a sum of holomorphic polygon maps associated to certain sub multi-diagrams.",
"Call a multi-framing $J$ an immediate successor of a multi-framing $I$ if the two framings differ on exactly one component $K_i\\subset L$ and if the restriction of $J$ to $K_i$ is one greater than the restriction of $I$ , taken with respect to the length two ordering $0<1<\\infty $ .",
"Write $I<J$ if $J$ is an immediate successor of $I$ .",
"Given any sequence of immediate successors $I^0<I^1<\\dots <I^m$ , there is a map $D_{I^0<\\cdots <I^m}\\colon \\,\\widehat{CF}(Y(I^0))\\longrightarrow \\widehat{CF}(Y(I^m))$ defined by the pseudo-holomorphic polygon map of Subsection REF $ D_{I^0<\\dots <I^m}(-):=f_{0,i_0,\\dots ,i_m}(-\\otimes \\Theta _1\\otimes \\dots \\otimes \\Theta _{m}).$ The indexing $(0,i_0,\\dots ,i_m)$ which specifies the sub multi-diagram $(\\Sigma ,\\mbox{${\\eta }$}^0,\\mbox{${\\eta }$}^{i_0},\\dots ,\\mbox{${\\eta }$}^{i_m},w)$ is determined by the requirement that $Y_{0,i_j}= Y(I^j)$ for each $j=0,1,\\dots ,m$ , under the bijection between the 3-manifolds $Y_{0,i}$ and those associated to the multi-framings.",
"Here and throughout, $\\Theta _j$ is a cycle generating the highest graded component of $\\widehat{HF}_*(Y_{i_{j-1},i_{j}})$ .",
"For a sequence of successors of length zero, $D_I\\colon \\,\\widehat{CF}(Y(I))\\rightarrow \\widehat{CF}(Y(I))$ is simply the Floer differential.",
"Ozsváth and Szabó define an endomorphism $D\\colon \\,X\\rightarrow X$ by $D= \\sum _{\\lbrace I^0<I^1<\\cdots <I^m\\rbrace } D_{I^0<I^1<\\cdots <I^m},$ where the sum is over all sequences of immediate successors (of all lengths).",
"They show that $D\\circ D=0$ [31].",
"The key tool in their proof is the $A_\\infty $ relation for pseudo-holomorphic polygon maps, together with a vanishing theorem for these maps in the present context, [31].",
"The resulting complex $(X,D)$ has a natural filtration by the totally ordered set $\\lbrace 0,1,\\infty \\rbrace ^l$ which arises from its defining decomposition along multiframings.",
"We can also collapse this to a $\\mathbb {Z}$ -filtration, and it is this latter filtration which will be of primary interest.",
"To do this, let the norm of a multi-framing $I=(m_1,...,m_l)$ be given by $|I|= \\sum _{i=1}^{l} m^{\\prime }_i,$ where $m^{\\prime }_i=m_i$ if $m_i\\in \\lbrace 0,1\\rbrace $ and $m^{\\prime }_i=2$ if $m_i=\\infty $ .",
"Then $X= \\bigoplus _{i\\ge 0 } \\ C^{(i)},$ where $C^{(i)}=\\bigoplus _{\\lbrace I\\in \\lbrace 0,1,\\infty \\rbrace ^l | \\ |I|=i\\rbrace } \\widehat{CF}(Y(I)).$ The differential $D$ clearly respects the decomposition, giving rise to a decreasing filtration of $X$ by sub complexes $F^p=\\oplus _{i\\ge p} C^{(i)}$ , as in the previous subsection.",
"The 0–th page of the spectral sequence is given by $E_0= \\bigoplus _{I\\subset \\lbrace 0,1,\\infty \\rbrace ^l} \\widehat{CF}(Y(I)),$ with $\\delta _0$ differential given simply as the Floer differential; that is, $\\delta _0|_{\\widehat{CF}(Y(I))}=D_I$ .",
"The $E_1$ page of the spectral sequence splits as $E_1\\cong \\bigoplus _{I} \\widehat{HF}(Y(I)),$ with $\\delta _1|_{\\widehat{HF}(Y(I))}$ given by $\\sum (D_{I<J})_*$ , the sum over all immediate successors of $I$ , of the maps induced on homology by $D_{I<J}$ .",
"These are the maps $\\widehat{F}_W\\colon \\,\\widehat{HF}(Y(I))\\rightarrow \\widehat{HF}(Y(J))$ associated to the 2–handle cobordism between $Y(I)$ and $Y(J)$ (here we are using the fact that $J$ is an immediate successor of $I$ ).",
"We are now in a position to understand the action of a curve $\\zeta \\subset Y\\setminus L$ .",
"Without loss of generality, we may assume that $\\zeta $ lies on the Heegaard surface, and is in general position with all attaching curves.",
"Define maps $a^\\zeta _{I^0<\\cdots <I^m}\\colon \\,\\widehat{CF}(Y(I^0))\\rightarrow \\widehat{CF}(Y(I^m))$ by the polygon operators from Subsection REF , $a^\\zeta _{I^0<\\cdots <I^m}(-):=a^\\zeta _{0,i_0,...,i_m}(-\\otimes \\Theta _1\\otimes \\cdots \\otimes \\Theta _{m}).$ The sum of all these maps (over all sequences of immediate successors) is an endomorphism $a^\\zeta \\colon \\,X\\rightarrow X$ , which the following lemma shows is a chain map.",
"This map forms the basis of our refinement.",
"Lemma 4.4 The map $a^\\zeta : X\\rightarrow X$ given by $a^\\zeta = \\sum _{\\lbrace I^0<I^1<\\dots <I^m\\rbrace } a^\\zeta _{I^0<I^1<\\dots <I^m}$ is a chain map.",
"Moreover, $a^\\zeta $ respects the collapsed filtration $F^i$ and the filtration of $(X,D)$ by the totally ordered set $\\lbrace 0,1,\\infty \\rbrace ^l$ .",
"Proof.",
"We want to prove that $D\\circ a^{\\zeta }+a^{\\zeta }\\circ D=0$ .",
"The proof is analogous to [31].",
"Given $I,J$ such that $|I|<|J|$ , we expand the component map $D\\circ a^{\\zeta }+a^{\\zeta }\\circ D\\colon \\,\\widehat{CF}(Y(I))\\rightarrow \\widehat{CF}(Y(J))$ to get $ \\sum _{I=I^0<\\cdots <I^m=J}\\sum _{0\\le r\\le m}(D_{I^{r}<\\cdots <I^m}\\circ a^{\\zeta }_{I^0<\\cdots <I^r}+a^{\\zeta }_{I^{r}<\\cdots <I^m}\\circ D_{I^0<\\cdots <I^r}),$ where the first sum is over all sequences of immediate successors connecting $I$ to $J$ .",
"Such sequences must be of length $m=|J|-|I|$ .",
"Pick one of these sequences $I^0<\\cdots < I^m$ , and consider the term on the left in the second summation, applied to a chain $\\mathbf {x}$ $\\sum _{0\\le r\\le m}D_{I^{r}<\\cdots <I^m}\\circ a^{\\zeta }_{I^0<\\cdots <I^r}(\\mathbf {x}):=\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad $ $\\quad \\quad \\quad \\sum _{0\\le r\\le m} f_{0,i_r,\\dots ,i_m}(a^\\zeta _{0,i_0,\\dots ,i_r}(\\mathbf {x}\\otimes \\Theta _1\\otimes \\dots \\otimes \\Theta _{r})\\otimes \\Theta _{r+1}\\otimes \\dots \\otimes \\Theta _{m}),$ where the indices $i_n$ refer to the sets of attaching curves for which $Y_{0,i_n}\\simeq Y(I^n)$ , as above.",
"Theorem REF indicates that this is equal to $&&\\sum _{0\\le r<s\\le m} a^\\zeta _{0,i_0,\\dots ,i_r,i_s,\\dots ,i_m} (\\mathbf {x}\\otimes \\dots \\otimes f_{i_r,\\dots ,i_s}(\\Theta _{r+1}\\otimes ... \\otimes \\Theta _{s})\\otimes ...\\otimes \\Theta _{m})\\\\&&+\\sum _{0\\le s\\le m} a^\\zeta _{0,i_s,\\dots ,i_m}( f_{0,i_0,\\dots ,i_s}(\\mathbf {x}\\otimes \\dots \\otimes \\Theta _{s})\\otimes \\Theta _{s+1}\\otimes \\dots \\otimes \\Theta _{m}).$ But the second term in the above sum is equal to, and hence cancels over $\\mathbb {F}$ , the $a^\\zeta \\circ D$ term from the inner sum in (REF ).",
"Thus (REF ) becomes $\\sum _{I^0<\\dots <I^m}\\sum _{0\\le r<s\\le m} a^\\zeta _{0,i_0,\\dots ,i_r,i_s,\\dots ,i_m} (\\mathbf {x}\\otimes \\dots \\otimes f_{i_r,\\dots ,i_s}(\\Theta _{r+1}\\otimes ... \\otimes \\Theta _{s})\\otimes ...\\otimes \\Theta _{m}),$ This expression can be rewritten as $\\sum _{\\begin{array}{c}\\scriptstyle I^{\\prime }=:I^r\\\\\\scriptstyle J^{\\prime }=:I^s\\end{array}}\\!\\!\\!\\!\\!",
"\\sum _{\\begin{array}{c}\\scriptstyle I^0<\\cdots <I^r\\\\\\scriptstyle I^s<\\cdots <I^m\\end{array}}\\!\\!",
"\\!\\!\\!\\!\\!\\!\\!",
"\\!\\!",
"a^{\\zeta }_{0,i_0,..,i_r,i_s,..,i_m}\\big (\\mathbf {x}\\otimes ..\\otimes \\!\\!\\!\\!\\!",
"\\!\\!\\!\\!\\!",
"\\sum _{\\begin{array}{c}\\scriptstyle I^r<\\cdots <I^s\\\\\\scriptstyle \\end{array}}\\!\\!\\!\\!\\!",
"\\!\\!\\!\\!\\!",
"f_{i_r,..,i_s}(\\Theta _{r+1}\\otimes ..\\otimes \\Theta _{s})\\otimes ..\\otimes \\Theta _{m}\\big )\\\\$ where $I^{\\prime }$ and $J^{\\prime }$ in the first sum range over all multi-framings between $I$ and $J$ (which we subsequently relabel $I^{\\prime }=I^r$ and $J^{\\prime }=I^s$ ).",
"The second sum ranges over all sequences of successors connecting $I$ to $I^{\\prime }$ and $J^{\\prime }$ to $J$ .",
"The third sum ranges over all sequences of successors connecting $I^{\\prime }$ to $J^{\\prime }$ .",
"However, [31] states that $\\sum _{I^r<\\dots < I^s} f_{i_r,..,i_s}(\\Theta _{r+1}\\otimes ..\\otimes \\Theta _{s})\\equiv 0,$ for any $I^r$ and $I^s$ .",
"This shows that $a^\\zeta $ is a chain map.",
"It clearly respects the filtration $F^i$ (since it splits into homogeneous summands by the length of the sequence of successors) and the subcomplexes of the filtration of $(X,D)$ by the totally ordered set $\\lbrace 0,1,\\infty \\rbrace ^l$ .",
"The second step in the proof of Ozsváth–Szabó's theorem is to show that $(X,D)$ has a natural quotient complex $(C(L),d)$ whose homology is isomorphic to $\\widehat{HF}(Y)$ .",
"Given a subset of multiframings $S\\subset \\lbrace 0,1,\\infty \\rbrace ^l$ , let $X(S)$ denote the subgroup of $X$ given by $X(S)=\\bigoplus _{I\\in S}\\ \\widehat{CF}(Y(I)).$ Let $(C(L),d)=X(\\lbrace 0,1\\rbrace ^l)$ be the group generated by all multiframings which do not contain $\\infty $ .",
"This is clearly a quotient complex, with associated subcomplex generated by those multiframings which contain $\\infty $ .",
"The complex $X(\\lbrace 0,1\\rbrace ^l)$ inherits the filtration from $X$ , and the curve map $a^\\zeta $ induces a filtered chain map from $X(\\lbrace 0,1\\rbrace ^l)$ to itself.",
"Ozsváth and Szabó show that the homology of $(C(L),d)$ is isomorphic to $\\widehat{HF}(Y)$ , and our task is to show that their isomorphism fits into the following diagram ${\\begin{matrix}\\widehat{HF}(Y) &\\xrightarrow{}& H_*(C(L),d)\\\\\\mathbox{mphantom}{\\scriptstyle A^{[\\zeta ]}}\\downarrow {\\scriptstyle A^{[\\zeta ]}}&& \\mathbox{mphantom}{\\scriptstyle (a^{\\zeta })_*}\\downarrow {\\scriptstyle (a^{\\zeta })_*}&&\\\\\\widehat{HF}(Y)&\\xrightarrow{}&H_*(C(L),d).\\end{matrix}}$ The key ingredient is a refined version of the strong form of the surgery exact triangle, [31].",
"Let $K$ be a framed knot in a 3–manifold $Y$ , and let $f=D_{0<1}\\colon \\,\\widehat{CF}(Y(0))\\longrightarrow \\widehat{CF}(Y(1))$ be the map induced by the 2–handle cobordism.",
"Ozsváth–Szabó [31] show that the mapping cone complex $M(f)$ is quasi-isomorphic to $\\widehat{CF}(Y)$ .",
"We must account for the $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ action.",
"To do this let $\\zeta \\subset Y\\setminus K$ be a curve, as usual.",
"Consider the complex $X(\\lbrace 0,1,\\infty \\rbrace )=\\widehat{CF}(Y(0))\\oplus \\widehat{CF}(Y(1))\\oplus \\widehat{CF}(Y(\\infty )),$ endowed with the differential $D=\\left( \\begin{array}{ccc}D_0 & 0 & 0 \\\\D_{0<1} & D_1 & 0 \\\\D_{0<1<\\infty } & D_{1<\\infty } & D_\\infty \\end{array} \\right).",
"$ There is a natural short exact sequence $ 0\\rightarrow X(\\lbrace \\infty \\rbrace )\\rightarrow X(\\lbrace 0,1,\\infty \\rbrace )\\rightarrow X(\\lbrace 0,1\\rbrace )\\rightarrow 0 ,$ where the sub and quotient complexes are identified with $\\widehat{CF}(Y)$ and $M(f)$ , respectively.",
"Ozsváth and Szabó show that $X(\\lbrace 0,1,\\infty \\rbrace )$ is acyclic [31], proving that $M(f)$ is quasi-isomorphic to $\\widehat{CF}(Y)$ .",
"We have the map $ a^{\\zeta }: X(\\lbrace 0,1,\\infty \\rbrace )\\longrightarrow X(\\lbrace 0,1,\\infty \\rbrace )$ given by $a^\\zeta =\\left( \\begin{array}{ccc}a^\\zeta _0 & 0 & 0 \\\\a^\\zeta _{0<1} & a^\\zeta _1 & 0 \\\\a^\\zeta _{0<1<\\infty } & a^\\zeta _{1<\\infty } & a^\\zeta _\\infty \\end{array} \\right),$ which Lemma REF shows is a chain map.",
"Moreover, $a^\\zeta $ respects the short exact sequence (REF ), thus inducing a map on the sub and quotient complex.",
"The map on the subcomplex is simply $a^\\zeta _\\infty \\colon \\,\\widehat{CF}(Y_\\infty )\\rightarrow \\widehat{CF}(Y_\\infty )$ and the map on the quotient complex is $ a^\\zeta _{M(f)} = \\begin{pmatrix} a^\\zeta _0 & 0 \\\\ a^\\zeta _{0<1} & a^\\zeta _1 \\end{pmatrix}\\colon \\,M(f)\\rightarrow M(f).$ Considering the corresponding long exact sequence in homology, we obtain a commutative diagram: ${\\begin{matrix}... &\\xrightarrow{}& H_*(M(f)) &\\xrightarrow{}& H_*(X(\\lbrace 0,1,\\infty \\rbrace )&\\xrightarrow{}& \\widehat{HF}_*(Y_\\infty ) &\\xrightarrow{}&...\\\\&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta _{M(f)})_*}\\downarrow {\\scriptstyle (a^\\zeta _{M(f)})_*}&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta )_*}\\downarrow {\\scriptstyle (a^\\zeta )_*}&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta _\\infty )_*}\\downarrow {\\scriptstyle (a^\\zeta _\\infty )_*}&& && \\\\... &\\xrightarrow{}& H_*(M(f)) &\\xrightarrow{}& H_*(X(\\lbrace 0,1,\\infty \\rbrace )&\\xrightarrow{}& \\widehat{HF}_*(Y_\\infty ) &\\xrightarrow{}&...,\\end{matrix}}$ By Ozsváth and Szabó's theorem, $H_*(X(\\lbrace 0,1,\\infty \\rbrace )=0$ , showing that ${\\begin{matrix}\\widehat{HF}_{*+1}(Y)&\\xrightarrow{}& H_*(M(f)) \\\\\\mathbox{mphantom}{\\scriptstyle (a^{\\zeta })_{*+1}}\\downarrow {\\scriptstyle (a^{\\zeta })_{*+1}}&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta _{M(f)})_*}\\downarrow {\\scriptstyle (a^\\zeta _{M(f)})_*}&& \\\\\\widehat{HF}_{*+1}(Y)&\\xrightarrow{}& H_*(M(f)).\\end{matrix}}$ This can be interpreted as saying that the Floer homology of $Y$ is isomorphic to the mapping cone of the 2–handle map, as a module over $\\Lambda ^*(H_1(Y;\\mathbb {Z})/\\mathrm {Tors})$ .",
"Given this refinement of the surgery exact triangle, the proof of Theorem REF proceeds quickly by the same inductive argument used in the proof of [31].",
"Specifically, we return to the complex $X$ coming from a framed link diagram of $l$ components.",
"If $l=1$ , the preceding discussion proves the theorem, showing that the filtered complex $X(\\lbrace 0,1\\rbrace )$ computes the homology of $X(\\infty )=\\widehat{CF}(Y)$ , and that a curve $\\zeta \\subset Y\\setminus K$ induces a filtered chain map whose induced map agrees with that of $a^{\\zeta }\\colon \\,\\widehat{CF}(Y)\\rightarrow \\widehat{CF}(Y)$ .",
"Assume that this remains true for the complex $X(\\lbrace 0,1\\rbrace ^{l-1})$ associated to an $(l-1)$ –component link and the induced map $a^\\zeta $ .",
"That is, we have a commutative diagram: ${\\begin{matrix}\\widehat{HF}_{*+1}(Y)&\\xrightarrow{}& H_*(X(\\lbrace 0,1\\rbrace ^{l-1})) \\\\\\mathbox{mphantom}{\\scriptstyle (a^{\\zeta })_{*+1}}\\downarrow {\\scriptstyle (a^{\\zeta })_{*+1}}&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta )_*}\\downarrow {\\scriptstyle (a^\\zeta )_*}&& \\\\\\widehat{HF}_{*+1}(Y)&\\xrightarrow{}& H_*(X(\\lbrace 0,1\\rbrace ^{l-1})).\\end{matrix}}$ Turn now to an $l$ –component link.",
"In this case, we consider the complex $X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace 0,1,\\infty \\rbrace )$ (that this is a complex follows from the fact that it is a quotient of $X$ by the subcomplex consisting of multiframings with at least one of the first $l-1$ parameters equals to $\\infty $ ) .",
"There is the short exact sequence, compatible with the maps induced by $a^\\zeta $ : ${15pt}{\\begin{matrix}0 &\\xrightarrow{}& X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace \\infty \\rbrace )&\\xrightarrow{}&X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace 0,1,\\infty \\rbrace )&\\xrightarrow{}& X(\\lbrace 0,1\\rbrace ^l) &\\xrightarrow{}&0\\\\&& \\mathbox{mphantom}{\\scriptstyle a^\\zeta }\\downarrow {\\scriptstyle a^\\zeta }&& \\mathbox{mphantom}{\\scriptstyle a^\\zeta }\\downarrow {\\scriptstyle a^\\zeta }&& \\mathbox{mphantom}{\\scriptstyle a^\\zeta }\\downarrow {\\scriptstyle a^\\zeta }&& && \\\\0 &\\xrightarrow{}& X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace \\infty \\rbrace )&\\xrightarrow{}&X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace 0,1,\\infty \\rbrace )&\\xrightarrow{}& X(\\lbrace 0,1\\rbrace ^l) &\\xrightarrow{}&0.\\end{matrix}}$ The middle term has a natural filtration coming from the total ordering on $\\lbrace 0,1\\rbrace ^{l-1}$ .",
"The associated graded groups of this filtration are each isomorphic to $H_*(X(I\\times \\lbrace 0,1,\\infty \\rbrace ))$ for some multiframing $I\\subset \\lbrace 0,1\\rbrace ^{l-1}$ .",
"The strong form of the surgery exact triangle, however, implies that these groups are all zero.",
"It follows that $H_*(X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace 0,1,\\infty \\rbrace ))=0$ .",
"Thus we have the diagram ${\\begin{matrix}H_{*+1}(X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace \\infty \\rbrace ))&\\xrightarrow{}& H_*(X(\\lbrace 0,1\\rbrace ^{l})) \\\\\\mathbox{mphantom}{\\scriptstyle (a^{\\zeta })_{*+1}}\\downarrow {\\scriptstyle (a^{\\zeta })_{*+1}}&& \\mathbox{mphantom}{\\scriptstyle (a^\\zeta )_*}\\downarrow {\\scriptstyle (a^\\zeta )_*}&& \\\\H_{*+1}(X(\\lbrace 0,1\\rbrace ^{l-1}\\times \\lbrace \\infty \\rbrace ))&\\xrightarrow{}& H_*(X(\\lbrace 0,1\\rbrace ^{l})).\\end{matrix}}$ Our inductive hypothesis equates the left hand side with $A^{[\\zeta ]}\\colon \\,\\widehat{HF}(Y)\\rightarrow \\widehat{HF}(Y)$ .",
"This completes the proof that the map induced on homology by $a^\\zeta $ is isomorphic to $A^{[\\zeta ]}$ .",
"To see that the induced map $\\alpha _1^{\\zeta }$ on the $E_1$ page is given by ${\\begin{matrix}\\underset{I}{\\bigoplus }\\ \\widehat{HF}(Y(I)) &\\xrightarrow{}& \\underset{I}{\\bigoplus }\\ \\widehat{HF}(Y(I)),\\end{matrix}}$ it suffices to recall that $\\alpha _1^{\\zeta }$ is the induced map on the homology of $(E_0,\\delta _0)$ by $\\alpha _0^{\\zeta }$ , where $\\alpha ^\\zeta _0$ is the map on the $E_0$ page induced by $a^{\\zeta }$ .",
"But $\\alpha ^\\zeta _0$ , in turn, is simply the lowest order term of $a^{\\zeta }$ and is given by ${\\begin{matrix}\\underset{I}{\\bigoplus }\\ \\widehat{CF}(Y(I)) &\\xrightarrow{}& \\underset{I}{\\bigoplus }\\ \\widehat{CF}(Y(I)),\\end{matrix}}$ where $a_I^\\zeta :\\widehat{CF}(Y(I))\\rightarrow \\widehat{CF}(Y(I))$ is the operator obtained by viewing $\\zeta $ as a curve in $Y(I))$ .",
"By definition, we have $A_I^{[\\zeta ]}=(a_I^\\zeta )_*$ ."
],
[
"Connecting the Khovanov module to the Floer module",
"As discussed in Section , a marked point $p_0$ on one component of a link diagram gives rise to the reduced Khovanov chain complex, $CKh^{{r}}$ .",
"Choosing a marked point on each remaining component gives the reduced Khovanov homology an $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ –module structure.",
"More precisely, additional marked points give rise to chain maps $x_i\\colon \\,CKh^{{r}}(L)\\rightarrow CKh^{{r}}(L), \\ \\ i=1,...,n-1$ which satisfy $x_i\\circ x_j=x_j\\circ x_i$ and $x_i\\circ x_i=0$ .",
"Consider a properly embedded arc $t_i\\subset (S^3,L)$ connecting $p_0$ to the $i$ –th additional marked point.",
"The preimage of this arc in the branched double cover is a closed curve $\\zeta _i= \\pi ^{-1}(t_i)\\subset \\Sigma (L)$ .",
"As in Subsection REF , we can assume that $\\zeta _i$ lies on the Heegaard surface of a Heegaard diagram of $\\Sigma (L)$ , and hence we obtain a chain map on the associated Floer complex: $ a^{\\zeta _i}\\colon \\,\\widehat{CF}(\\Sigma (L))\\longrightarrow \\widehat{CF}(\\Sigma (L)).$ This chain map is related to the chain map $x_i$ on the Khovanov complex by the following theorem.",
"Theorem 4.5 Let $D$ be a diagram for a link $L=K_0\\cup ...\\cup K_{n-1}\\subset S^3$ , together with a base point $p_0\\subset K_0$ .",
"There is a filtered chain complex, $(C(D),d)$ , whose homology is isomorphic to $\\widehat{HF}(\\Sigma (L))$ .",
"The associated spectral sequence satisfies $(E_1, \\delta _1)\\cong (CKh^{{r}}(\\overline{D}),\\partial )$ where $\\overline{D}$ is the mirror of $D$ and the reduced Khovanov complex is defined with $p_0$ .",
"Let $t_i$ be a proper arc connecting $p_0\\subset K_0$ to $p_i\\subset K_i$ , and $\\zeta _i= \\pi ^{-1}(t_i)\\subset \\Sigma (L)$ its lift to the branched double cover.",
"Then there is a filtered chain map $ a^{\\zeta _i}\\colon \\,(C(D),d)\\longrightarrow (C(D),d),$ whose induced map on homology satisfies $(a^{\\zeta _i})_*=A^{[\\zeta _i]}\\colon \\,\\widehat{HF}(\\Sigma (L))\\rightarrow \\widehat{HF}(\\Sigma (L)).$ The induced map $\\alpha _1^{\\zeta _i}$ on the $E_1$ page of the spectral sequence satisfies ${\\begin{matrix}(E_1,\\delta _1) &\\xrightarrow{}& (E_1,\\delta _1) \\\\\\mathbox{mphantom}{\\scriptstyle \\cong }\\downarrow {\\scriptstyle \\cong }&& \\mathbox{mphantom}{\\scriptstyle \\cong }\\downarrow {\\scriptstyle \\cong }&& \\\\(CKh^{{r}}(\\overline{D}),\\partial )&\\xrightarrow{}& (CKh^{{r}}(\\overline{D}),\\partial ).\\end{matrix}}$ Proof.",
"The first part of the theorem is the content of [31], which is a rather immediate consequence of the link surgeries spectral sequence, applied to a particular surgery presentation of $\\Sigma (L)$ coming from the link diagram.",
"The key point is that the branched double covers of links which differ by the unoriented skein relation differ by a triad of surgeries along a framed knot in that manifold.",
"Moreover, the branched double cover of the natural saddle cobordism passing between the zero and one resolution is the 2–handle cobordism between the branched double covers which appears in the surgery exact triangle.",
"The branched double cover of a complete resolution is diffeomorphic to $\\#^{k}S^1\\times S^2$ , where $k+1$ is the number of components of the resolution, and the Heegaard Floer homology of $\\#^{k}S^1\\times S^2$ is isomorphic to the reduced Khovanov homology of the complete resolution.",
"Moreover, the maps between the Floer homology of the connected sums of $S^1\\times S^2$ 's induced by the 2–handle cobordisms agree with the Frobenius algebra defining the reduced Khovanov differential.",
"In the present situation, we wish to keep track of an action.",
"In the case of Khovanov homology it is the action $x_i$ induced by a point $p_i\\in K_i$ , whereas in the Floer setting it is the action of the curve $\\zeta _i=\\pi ^{-1}(t_i)$ arising as the lift of an arc $t_i$ connecting $p_0$ to $p_i$ .",
"Theorem REF shows that $\\zeta _i$ induces a filtered chain map $a^{\\zeta _i}\\colon \\,X(\\lbrace 0,1\\rbrace ^l)\\rightarrow X(\\lbrace 0,1\\rbrace ^l),$ whose induced map on homology agrees with $A^{[\\zeta _i]}\\colon \\,\\widehat{HF}(\\Sigma (L))\\rightarrow \\widehat{HF}(\\Sigma (L))$ .",
"Thus it suffices to see that the induced map on the $E_1$ page of the spectral sequence agrees with the Khovanov action.",
"This follows from the fact that the induced map $\\alpha _1^{\\zeta _i}: (E_1,\\delta _1)\\rightarrow (E_1,\\delta _1)$ is simply the sum of the actions of $\\zeta _i$ on the Floer homology groups at each individual vertex in the cube; that is, ${\\begin{matrix}E_1 &\\xrightarrow{}& E_1 \\\\\\mathbox{mphantom}{\\scriptstyle \\cong }\\downarrow {\\scriptstyle \\cong }&& \\mathbox{mphantom}{\\scriptstyle \\cong }\\downarrow {\\scriptstyle \\cong }&& \\\\\\underset{I\\subset \\lbrace 0,1\\rbrace ^l}{\\bigoplus }\\widehat{HF}(Y(I))&\\xrightarrow{}& \\underset{I\\subset \\lbrace 0,1\\rbrace ^l}{\\bigoplus }\\widehat{HF}(Y(I)),\\end{matrix}}$ where $A_I^{[\\zeta _i]}: \\widehat{HF}(Y(I))\\rightarrow \\widehat{HF}(Y(I))$ is the map induced by $\\zeta _i$ , viewed as curve in $Y(I)$ .",
"Now each $Y(I)$ is diffeomorphic to $\\#^k S^1\\times S^2$ for some $k$ , and $\\widehat{HF}(\\#^k S^1\\times S^2) \\cong \\mathbb {F}[x_1,...,x_{k}]/(x_1^2,...,x_k^2)$ as a module over $\\Lambda ^*(H_1(\\#^{k}S^1\\times S^2))$ .",
"Here a basis for $H_1(\\#^{k} S^1\\times S^2)$ is identified with the variables $x_1,...,x_{k}$ , and such a basis is given by the lift of any $k$ proper arcs connecting the unknot containing $p_0$ to each of the $k$ other unknots.",
"Under this correspondence, the action of $\\zeta _i$ on the Floer homology of the branched double cover of a complete resolution agrees with the action of $x_i$ on the reduced Khovanov homology of the complete resolution.",
"This completes the proof.",
"The preceding theorem allows us to endow the entire Ozsváth–Szabó spectral sequence with a module structure.",
"To describe this, we say that a ring $R$ acts on a spectral sequence $\\lbrace (E_i,\\delta _i)\\rbrace _{i=0}^\\infty $ if for each $i$ we have $E_i$ is an $R$ –module.",
"The differential is $R$ –linear: $x\\cdot \\delta _i(\\beta )=\\delta _i(x\\cdot \\beta )$ for all $x\\in R, \\beta \\in E_i$ .",
"Equivalently, $R$ acts on $E_i$ by chain maps.",
"The $R$ –module structure on $E_{i+1}$ is induced through homology by the module structure on $E_i$ .",
"If the above conditions hold only for $i\\ge r$ , we say that the action begins at the $r$ –th page.",
"The following is now an easy corollary of Theorem REF .",
"Corollary 4.6 Let $L=K_1\\cup ...\\cup K_n\\subset S^3$ be a link.",
"Then the spectral sequence from the reduced Khovanov homology of $\\overline{L}$ to the Floer homology of $\\Sigma (L)$ is acted on by $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ .",
"The resulting module structure on Khovanov homology is isomorphic to the module structure induced by the basepoint maps.",
"Remark 4.7 Reduced Khovanov homology appears as the $E_2$ page of a spectral sequence.",
"As our proof indicates, the action begins at $E_1$ .",
"The action on $E_0$ holds only up to chain homotopy.",
"Proof.",
"For each component $K_i\\subset L$ , choose a basepoint $p_i\\subset K_i$ on a diagram for $L$ .",
"Pick a system of arcs $t_i$ connecting $p_0$ to $p_i$ , and consider the closed curves $\\zeta _i=\\pi ^{-1}(t_i), i=1,...,n-1,$ arising from their lifts to $\\Sigma (L)$ .",
"According to Theorem REF we obtain a collection of filtered chain maps, $ a^{\\zeta _i}: (C,d)\\rightarrow (C,d), \\ \\ i=1,...,n-1,$ acting on the filtered chain complex which gives rise to the Ozsváth–Szabó spectral sequence.",
"Consider the free group $F_{n-1}$ on $n-1$ generators, $X_1,...,X_{n-1}$ .",
"There is an obvious action of the group algebra $\\mathbb {F}[F_{n-1}]$ on the spectral sequence: simply define the action of $X_i$ on $E_r$ to be $\\alpha _r^{\\zeta _i}$ , the map induced by $a^{\\zeta _i}$ on $E_r$ and extend this to $\\mathbb {F}[F_{n-1}]$ in the natural way.",
"Thus an element such as $X_1X_2 + X_6$ acts on $E_r$ by the chain map $\\alpha _r^{\\zeta _1}\\circ \\alpha _r^{\\zeta _2}+ \\alpha _r^{\\zeta _6}$ .",
"The fact that for each curve $\\zeta _i$ and each $r$ , $\\alpha _r^{\\zeta _i}: (E_r,\\delta _r)\\rightarrow (E_r,\\delta _r),$ is a chain map satisfying, $\\alpha _{r+1}^{\\zeta _i}= (\\alpha _{r}^{\\zeta _i})_*$ , implies that this is indeed an action by $\\mathbb {F}[F_{n-1}]$ .",
"To see that the action descends to $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ , it suffices to check that $X_iX_j+X_jX_i$ and $X_i^2$ act as zero on each page of the spectral sequence, for all $i,j$ .",
"These relations clearly hold on the $E_1$ page, however, since Theorem REF implies that the map induced by $a^{\\zeta _i}$ on $E_1$ agrees with $x_i$ , the map on the reduced Khovanov complex induced by $p_i$ .",
"But if an endomorphism of a spectral sequence is zero on some page, it is zero for all subsequent pages, since $\\alpha _{r+1}=(\\alpha _{r})_*$ .",
"It follows that $X_iX_j+X_jX_i$ acts as zero on $E_r$ for all $r\\ge 1$ .",
"To show that the module structure on $E_2$ agrees with the module structure on Khovanov homology induced by the basepoint maps it suffices to note, again, that $\\alpha _1^{\\zeta _i}=x_i$ and $\\alpha _2^{\\zeta _i}=(\\alpha _1^{\\zeta _i})_*=(x_i)_*$ .",
"It is natural to ask about convergence of the $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ –action.",
"In particular, since the homology of the spectral sequence converges to the Floer homology of $\\Sigma (L)$ , one could hope that the action converges to the $\\Lambda ^*(H_1(Y;\\mathbb {F}))$ -action (where the homology classes $[\\zeta _i]$ serve as a spanning set for $H_1(Y;\\mathbb {F})$ .)",
"While this is true it is not necessarily as useful as one may think, since convergence is phrased in terms of the associated graded module (see [24]) and this module may have many extensions.",
"Put differently, the module structure on the associated graded is sensitive only to the lowest order terms of the filtered chain maps $a^{\\zeta _i}$ , and the higher order terms may contribute in a non-trivial way to the module structure on $\\widehat{HF}(\\Sigma (L))$ .",
"Despite this, the corollary can still be used to prove the following “collapse result\" for the spectral sequence.",
"This theorem will be one of our main tools for showing that the Khovanov module detects unlinks.",
"Proposition 4.8 Suppose $Kh^{{r}}(\\overline{L})\\cong \\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ as a module over $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ .",
"Then $\\widehat{HF}(\\Sigma (L))\\cong \\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ as a module over $\\Lambda ^*(H_1(\\Sigma (L);\\mathbb {F}))$ .",
"Proof.",
"The assumption on the reduced Khovanov homology implies, in particular, that the entire Khovanov homology is supported in homological grading zero.",
"This is because $1\\in Kh^{{r}}(L)$ generates the homology as a module over $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ , and the module action preserves the homological grading (it is induced by chain maps of degree zero).",
"In the proof of Theorem REF , however, the filtration on the complex computing $\\widehat{HF}(\\Sigma (L))$ arises from a graded decomposition coming from the norm of a multi-framing $ X(\\lbrace 0,1\\rbrace ^l)= \\bigoplus _{i\\ge 0} \\ C^{(i)}, \\ \\ \\ \\ \\ \\ C^{(i)}= \\bigoplus _{ \\lbrace I \\in \\lbrace 0,1\\rbrace ^l | \\ |I|=i\\rbrace } \\ \\widehat{CF}(Y(I))$ and the norm $|I|$ corresponds to the homological grading on the Khovanov complex.",
"Thus the $E_2$ page of the spectral sequence, which is identified with $Kh^{{r}}(L)$ , is supported in a single filtration.",
"Since the higher differentials strictly lower the filtration, it follows that the spectral sequence has collapsed and that $Kh^{{r}}(\\overline{L})\\cong E_2=E_\\infty \\cong \\widehat{HF}(\\Sigma (L))$ .",
"Moreover, since the module structure on $E_\\infty $ is induced by $E_2$ through homology, it follows that this is an isomorphism of $\\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,...,X_{n-1}^2)$ modules.",
"We claim that the latter module structure agrees with the $\\Lambda ^*(H_1(\\Sigma (L);\\mathbb {F}))$ –module structure on Floer homology.",
"To see this, note first that the curves $\\zeta _i$ span $H_1(\\Sigma (L);\\mathbb {F})$ .",
"Now on one hand we have the filtered chain maps $a^{\\zeta _i}$ whose induced maps on homology agree with $A^{[\\zeta _i]}:\\widehat{HF}(\\Sigma (L))\\rightarrow \\widehat{HF}(\\Sigma (L))$ .",
"On the other we have $X_i$ , the maps induced by $a^{\\zeta _i}$ on the $E_\\infty $ page of the spectral sequence.",
"To see that $A^{[\\zeta ]}=X_i$ , it suffices to recall the construction from Section REF of the morphism of spectral sequences induced by $a^{\\zeta _i}$ .",
"By construction, each filtered map $a^{\\zeta _i}$ is chain homotopic to a map $a_\\infty ^{\\zeta _i}: E_\\infty \\rightarrow E_\\infty , \\ \\ \\ a_\\infty ^{\\zeta _i}= a_\\infty ^{(0)}+ a_\\infty ^{(1)}+a_\\infty ^{(2)}+...,$ and the morphism induced on the $E_\\infty $ is, by definition, the lowest order term in this map: $\\alpha ^{\\zeta _i}_\\infty :=a_\\infty ^{(0)}$ .",
"But the discussion above indicates that $E_\\infty $ is supported in a single filtration summand, hence the higher order terms in the decomposition of $a_\\infty ^{\\zeta _i}$ vanish.",
"It follows that the maps $A^{[\\zeta ]}=(a^{\\zeta _i})_*=a_\\infty ^{\\zeta _i}$ and $X_i= a_\\infty ^{(0)}$ are equal.",
"The proposition follows."
],
[
"A nontriviality theorem for homology actions",
"In this section we prove a non-triviality result for Floer homology, Theorem REF .",
"Roughly speaking, the theorem says that the homology of the Floer homology with respect to the action of any curve is non-trivial, provided a manifold does not contain a homologically essential 2–sphere.",
"This detection theorem for Floer homology will transfer through the spectral sequence of the previous section to our detection theorem for Khovanov homology.",
"Following Kronheimer and Mrowka [18], let $HF^{\\circ }(Y|R)=\\bigoplus _{\\lbrace \\mathfrak {s}\\in \\mathrm {Spin}^c(Y) | \\langle c_1(\\mathfrak {s}),[R]\\rangle =x(R) \\rbrace }HF^{\\circ }(Y,\\mathfrak {s}),$ where $R$ is a Thurston norm minimizing surface in $Y$ and $x(R)$ is its Thurston norm.",
"Theorem 5.1 Suppose $Y$ is a closed, oriented and irreducible 3–manifold with $b_1(Y)>0$ .",
"Let $R\\subset Y$ be a Thurston norm minimizing connected surface.",
"Then there exists a cohomology class $[\\omega ]\\in H^2 ( Y;\\mathbb {Z} )$ with $\\langle [\\omega ],[R]\\rangle >0$ , such that for any $[\\zeta ]\\subset H_1(Y;\\mathbb {Z})$ the homology group with respect to $A^{[\\zeta ]}$ $H(\\underline{\\widehat{HF}}(Y|R;\\mathcal {R}_{[\\omega ]}),A^{[\\zeta ]})$ has positive rank as an $\\mathcal {R}$ –module.",
"Proof.",
"We adapt the argument of Ozsváth and Szabó [32].",
"By Gabai [8], there exists a taut foliation $F$ of $Y$ , such that $R$ is a compact leaf of $F$ .",
"The work of Eliashberg and Thurston [6] shows that $F$ can be approximated by a weakly symplectically semi-fillable contact structure $\\xi $ , where $Y\\times [-1,1]$ is the weak semi-filling.",
"By Giroux [9], $Y$ has an open book decomposition which supports $\\xi $ .",
"By plumbing positive Hopf bands to the page of such an open book, we may assume that the binding is connected and that the genus of the page is greater than one.",
"Let $K\\subset Y$ denote the binding, and let $Y_0$ be the fibered 3–manifold obtained from $Y$ by 0–surgery on $K$ .",
"There is a 2–handle cobordism $W_0\\colon \\,Y\\rightarrow Y_0$ .",
"Similarly, there is a 2–handle cobordism $-W_0\\colon \\,Y_0\\rightarrow Y$ , obtained by reversing the orientation of $W_0$ and viewing it “backwards\".",
"Eliashberg [5] shows that the weak semi-filling $Y\\times [-1,1]$ can be embedded in a closed symplectic 4–manifold $X$ (see also Etnyre [7], for an alternative construction).",
"This is done by constructing symplectic caps for the boundary components.",
"Eliashberg's caps are produced by first equipping the 2-handle cobordisms $W_0$ and $-W_0$ with appropriate symplectic structures, and then extending these structures over Lefschetz fibrations $V_0$ and $V_0^{\\prime }$ whose boundaries are the fibered 3–manifolds $-Y_0$ and $Y_0$ , respectively (note $V_0$ and $V_0^{\\prime }$ are not, in general, orientation-reversing diffeomorphic).",
"Moreover, [5] says that we can choose $V_0$ and $V_0^{\\prime }$ so that $H_1(V_0)=H_1(V_0^{\\prime })=0.$ The result of the construction is a closed symplectic 4–manifold, $(X,\\omega )$ , which decomposes as $X=V_0^{\\prime }\\underset{-Y_0}{\\cup }-W_0\\underset{-Y=Y\\times \\lbrace -1\\rbrace }{\\cup }Y\\times [-1,1]\\underset{-Y}{\\cup }W_0\\underset{-Y_0}{\\cup }V_0.$ We view the cobordism from right to left, so that the orientation shown on a 3–manifold is that which it inherits as the oriented boundary of the 4–manifold to the right of the union in the decomposition.",
"By perturbing $\\omega $ slightly and multiplying by an integer, we may assume $[\\omega ]\\in H^2(X;\\mathbb {Z})$ .",
"We can also arrange that $b_2^+(V_0^{\\prime })>1$ and $b_2^+(V_0)>1$ , and hence can decompose $V_0$ by an admissible cut along a 3–manifold, $N$ .",
"Thus $X=X_1\\cup _N X_2$ with $b_2^+(X_i)>0$ , and $X_2=V_0^{\\prime }\\underset{-Y_0}{\\cup }-W_0\\underset{-Y}{\\cup }Y\\times [-1,1]\\underset{-Y}{\\cup }W_0\\underset{-Y_0}{\\cup }(V_0\\setminus X_1).",
"$ Denote the canonical Spin$^c$ structure associated to $\\omega $ by $\\mathfrak {k}(\\omega )$ , and the restriction of $\\mathfrak {k}(\\omega )$ to $Y_0$ by $\\mathfrak {t}$ .",
"Let $c(\\xi ;[\\omega ])\\in \\underline{\\widehat{HF}}(-Y|R;\\mathcal {R}_{[\\omega ]})/\\mathcal {R}^\\times $ be the twisted Ozsváth–Szabó contact invariant defined in [30], [32], and let $c^+(\\xi ;[\\omega ])\\in \\underline{HF}^+(-Y|R;\\mathcal {R}_{[\\omega ]})/\\mathcal {R}^\\times $ be its image under the natural map $\\iota _*:\\underline{\\widehat{HF}}\\rightarrow \\underline{HF}^+$ .",
"Let $\\pi \\colon \\,Y_0\\rightarrow S^1$ be the fibration on the 0–surgery induced by the open book decomposition of $Y$ , and let $c(\\pi )$ be a generator of $\\underline{\\widehat{HF}}(-Y_0,\\mathfrak {t};\\mathcal {R}_{[\\omega ]})$ whose image in $\\underline{HF}^+(-Y_0,\\mathfrak {t};\\mathcal {R}_{[\\omega ]})\\cong \\mathcal {R}$ is a generator $c^+(\\pi ;[\\omega ])$ .",
"By [30] $\\underline{\\widehat{F}}_{W_0;[\\omega ]}(c(\\pi ))\\doteq c(\\xi ;[\\omega ]),$ where “$\\doteq $ \" denotes equality, up to multiplication by a unit.",
"The following commutative diagram summarizes the relationship between the contact invariants: ${\\begin{matrix}c(\\pi ) &\\xrightarrow{}& c(\\xi ;[\\omega ])\\\\{\\scriptstyle \\iota _*}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\iota _*}&& {\\scriptstyle \\iota _*}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\iota _*}&&\\\\c^+(\\pi ;[\\omega ])&\\xrightarrow{}& c^+(\\xi ;[\\omega ]).\\end{matrix}}$ Let $W=V_0^{\\prime }\\cup -W_0$ , and let $B$ be an open 4-ball in $V_0^{\\prime }$ .",
"As $W$ is a symplectic filling of $(Y,\\xi )$ , the argument in [32] shows that $\\underline{F}^+_{W\\setminus B;[\\omega ]}(c^+(\\xi ;[\\omega ]))$ is a non-torsion element in $\\underline{HF}^+(S^3;\\mathcal {R}_{[\\omega ]})$ .",
"Note that here, as above, we regard $W$ from right to left; namely, as a cobordism from $-Y$ to $S^3$ .",
"It follows that $\\underline{\\widehat{F}}_{W\\setminus B;[\\omega ]}(c(\\xi ;[\\omega ]))\\quad \\text{is non-torsion.",
"}$ One can construct a Heegaard diagram for $-Y_0$ such that there are only two intersection points representing $\\mathfrak {t}$ , and such that there are no holomorphic disks connecting them which avoid the hypersurface specified by the basepoint.",
"Indeed, such a Heegaard diagram is constructed in the course of the proof of [30], and its desired properties are verified in the proof of [30] (here we use the fact that the page of the open book has genus greater than one, though [44] indicates that the same technique can be adapted for genus one open books).",
"Since $H_1(Y;\\mathbb {Z})$ is naturally a subgroup of $H_1(Y_0;\\mathbb {Z})$ , any $[\\zeta ]\\in H_1(Y)$ can be viewed as an element in $H_1(Y_0;\\mathbb {Z})$ , and the preceding discussion implies that $A^{[\\zeta ]}=0$ on $\\underline{\\widehat{HF}}(-Y_0,\\mathfrak {t};\\mathcal {R}_{[\\omega ]})$ for every $[\\zeta ]$ .",
"Using (REF ) and Theorem REF , $A^{[\\zeta ]}(c(\\xi ;[\\omega ]))&\\doteq &A^{[\\zeta ]}\\circ \\underline{\\widehat{F}}_{W_0;[\\omega ]}( c(\\pi ))\\\\&=&\\underline{\\widehat{F}}_{W_0;[\\omega ]}\\circ A^{[\\zeta ]}( c(\\pi ))\\\\&=&\\underline{\\widehat{F}}_{W_0;[\\omega ]}(0)=0.$ Hence $c(\\xi ;[\\omega ])\\in \\mathrm {ker}(A^{[\\zeta ]})$ .",
"Now if $kc(\\xi ;[\\omega ])\\in \\mathrm {im}(A^{[\\zeta ]})$ for some nonzero $k\\in \\mathcal {R}$ , then there is an element $a\\in \\underline{\\widehat{HF}}(Y;\\mathcal {R}_{[\\omega ]})$ such that $A^{[\\zeta ]}(a)=kc(\\xi ;[\\omega ])$ .",
"Using (REF ) and Theorem REF , $\\underline{\\widehat{F}}_{W\\setminus B;[\\omega ]}(kc(\\xi ;[\\omega ]))&=&\\underline{\\widehat{F}}_{W\\setminus B;[\\omega ]}\\circ A^{[\\zeta ]}(a)\\\\&=&A^{0}\\circ \\underline{\\widehat{F}}_{W\\setminus B;[\\omega ]}(a)\\\\&=&0,$ a contradiction to (REF ).",
"Hence $kc(\\xi ;[\\omega ])\\notin \\mathrm {im}(A^{[\\zeta ]})$ .",
"It follows that $c(\\xi ;[\\omega ])$ represents a non-torsion element in $H(\\underline{\\widehat{HF}}(Y|R;\\mathcal {R}_{[\\omega ]}),A^{[\\zeta ]})$ , so our conclusion holds.",
"Corollary 5.2 Suppose $Y$ is a closed, oriented 3–manifold which does not contain $S^1\\times S^2$ connected summands.",
"Then there exists a cohomology class $[\\omega ]\\in H^2(Y;\\mathbb {Z})$ such that for any $[\\zeta ]\\subset H_1(Y;\\mathbb {Z})$ the homology group with respect to $A^{[\\zeta ]}$ $H(\\underline{\\widehat{HF}}(Y;\\mathcal {R}_{[\\omega ]}),A^{[\\zeta ]})$ has positive rank as an $\\mathcal {R}$ –module.",
"Proof.",
"In the case that $b_1(Y)>0$ , this follows from Theorem REF and the twisted version of (REF ).",
"When $b_1(Y)=0$ , the theorem holds with $[\\omega ]=0$ .",
"Indeed, when $[\\omega ]=0$ the corresponding Floer homology group $\\widehat{HF}(Y;\\mathcal {R}_{[\\omega ]})$ is simply the homology of $\\widehat{CF}(Y)\\otimes _\\mathbb {F}\\mathbb {F}[T,T^{-1}]$ ; that is, we take the untwisted complex and tensor over $\\mathbb {F}$ with $\\mathbb {F}[T,T^{-1}]$ .",
"Now [27] indicates that the Euler characteristic of $\\widehat{HF}(Y)$ , and hence its rank over $\\mathbb {F}$ , is non-trivial.",
"The universal coefficient theorem then implies $H_*(\\widehat{CF}(Y)\\otimes _\\mathbb {F}\\mathbb {F}[T,T^{-1}])$ has positive rank as an $\\mathbb {F}[T,T^{-1}]$ –module.",
"Finally, since every class $[\\zeta ]\\in H_1(Y;\\mathbb {Z})$ is torsion, Lemma REF shows that the $A^{[\\zeta ]}=0$ , as an operator on $\\widehat{HF}(Y;\\mathcal {R})$ ."
],
[
"Links with the Khovanov module of an unlink",
"We now bring together the results from previous sections to prove our main theorems.",
"The first task is to prove Theorem REF , which states that Heegaard Floer homology, as a module over $\\Lambda (H^1(Y;\\mathbb {F}))$ , detects $S^1\\times S^2$ summands in the prime decomposition of a closed oriented 3–manifold.",
"By way of the module structure on the spectral sequence from Khovanov homology to Heegaard Floer homology (specifically Proposition REF ), this detection theorem will quickly lead to the Khovanov module's detection of unlinks, Theorem REF .",
"The detection theorem for the Heegaard Floer module makes use of of Corollary REF from the previous section.",
"The main challenge is to take this corollary, which is a non-vanishing result for homology actions on Heegaard Floer homology with twisted coefficients, and use it to obtain a characterization result for the Floer homology module with untwisted, i.e.",
"$\\mathbb {F}$ , coefficients.",
"Not surprising, to pass from twisted coefficients to untwisted coefficients we will need the universal coefficients theorem.",
"Let us recall its statement from Spanier [41].",
"Theorem 6.1 Let $C$ be a free chain complex over a principal ideal domain, $R$ , and suppose that $M$ is an $R$ –module.",
"Then there is a functorial short exact sequence $0\\rightarrow H_q(C)\\otimes _R M\\rightarrow H_q(C;M)\\rightarrow \\mathrm {Tor}_R(H_{q-1}(C),M)\\rightarrow 0.$ This exact sequence is split, but the splitting may not be functorial.",
"We wish to apply the universal coefficients theorem to understand the Heegaard Floer homology with $\\mathbb {F}$ coefficients through an understanding of the Floer homology with twisted coefficients, where the twisted coefficient ring is $\\mathcal {R}=\\mathbb {F}[T,T^{-1}]$ .",
"Viewing $\\mathbb {F}$ as the trivial $\\mathcal {R}$ module (where $T$ acts as 1), the following lemma analyzes the tensor product in the universal coefficient splitting.",
"Lemma 6.2 Suppose $M$ is a finitely generated module over $\\mathcal {R}=\\mathbb {F}[T,T^{-1}]$ , $M^{\\mathrm {tors}}$ is the submodule of $M$ consisting of all torsion elements, and $M^{\\mathrm {free}}=M/M^{\\mathrm {tors}}$ .",
"Then there is a short exact sequence $0\\rightarrow M^{\\mathrm {tors}}\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow M\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow M^{\\mathrm {free}}\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow 0.$ Moreover, $M^{\\mathrm {tors}}\\otimes _{\\mathcal {R}}\\mathbb {F}\\cong \\mathrm {Tor}_{\\mathcal {R}}(M,\\mathbb {F}).$ Proof.",
"The short exact sequence $0\\rightarrow M^{\\mathrm {tors}}\\rightarrow M\\rightarrow M^{\\mathrm {free}}\\rightarrow 0$ gives rise to a long exact sequence $\\cdots \\rightarrow \\mathrm {Tor}_1^{\\mathcal {R}}(M^{\\mathrm {free}},\\mathbb {F})\\rightarrow M^{\\mathrm {tors}}\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow M\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow M^{\\mathrm {free}}\\otimes _{\\mathcal {R}}\\mathbb {F}\\rightarrow 0.$ Since $M^{\\mathrm {free}}$ is free, $\\mathrm {Tor}_{\\mathcal {R}}(M^{\\mathrm {free}},\\mathbb {F})=0$ , hence we have the desired short exact sequence.",
"Since $\\mathcal {R}$ is a principal ideal domain $M^{\\mathrm {tors}}\\cong \\bigoplus _i \\mathcal {R}/(p_i^{k_i}),$ where $p_i$ 's are prime elements in $\\mathcal {R}$ , $k_i\\in \\mathbb {Z}_{\\ge 1}$ .",
"Note that $\\mathbb {F}\\cong \\mathcal {R}/(T-1)$ .",
"If $p_i\\ne (T-1)$ up to a unit, then $\\mathcal {R}/(p_i^{k_i})\\otimes _{\\mathcal {R}}\\mathbb {F}=0,\\quad \\mathrm {Tor}_{\\mathcal {R}}(\\mathcal {R}/(p_i^{k_i}),\\mathbb {F})=0.$ If $p_i=(T-1)$ up to a unit, then $\\mathcal {R}/(p_i^{k_i})\\otimes _{\\mathcal {R}}\\mathbb {F}\\cong \\mathbb {F},\\quad \\mathrm {Tor}_{\\mathcal {R}}(\\mathcal {R}/(p_i^{k_i}),\\mathbb {F})\\cong p_i^{k_i-1}\\mathcal {R}/(p_i^{k_i})\\cong \\mathbb {F}.$ Hence our result follows.",
"Theorem REF states that if $\\widehat{HF}(Y;\\mathbb {F})\\cong \\mathbb {F}[X_1,...,X_{n-1}]/(X_1^2,..., X_{n-1}^2)$ as a module, then $Y\\cong M\\#(\\#^{n-1}(S^1\\times S^2))$ , where $M$ is an integer homology sphere satisfying $\\widehat{HF}(M)\\cong \\mathbb {F}$ .",
"We turn to the proof of this theorem.",
"Proof of Theorem REF .",
"We first reduce to the case that $Y$ is irreducible.",
"Suppose that $Y$ is a nontrivial connected sum.",
"Then we can apply (REF ) to restrict our attention to a connected summand.",
"If $Y=S^1\\times S^2$ , then our conclusion holds.",
"So we may assume that $Y$ is irreducible.",
"Let $\\Lambda _{n-1}=\\mathbb {F} [X_1,\\dots ,X_{n-1}]/(X_1^2,\\dots ,X_{n-1}^2).$ Let $\\zeta _1,\\dots ,\\zeta _{n-1}$ be elements in $H_1(Y;\\mathbb {Z})/\\mathrm {Tors}$ such that $A^{\\zeta _i}(\\mathbf {1})=X_i$ .",
"If $S=\\lbrace i_1,\\dots ,i_k\\rbrace \\subset \\lbrace 1,\\dots ,n\\rbrace $ , let $A^S=A^{\\zeta _{i_1}}\\circ \\cdots \\circ A^{\\zeta _{i_k}}$ and let $X_S=X_{i_1}X_{i_2}\\cdots X_{i_k}\\in \\widehat{HF}(Y;\\mathbb {F} )\\cong \\Lambda _{n-1}.$ It follows that $A^S(\\mathbf {1})=X_S$ .",
"Since $A^{[\\zeta _i]}$ decreases the Maslov grading by 1, we can give a relative Maslov grading to $\\widehat{HF}(Y;\\mathbb {F} )$ such that the grading of $X_S$ is $n-1-|S|$ .",
"By Corollary REF , there exists $[\\omega ]\\in H^2(Y;\\mathbb {Z})$ such that $H(\\underline{\\widehat{HF}}(Y;\\mathcal {R}_{[\\omega ]}),A^{[\\zeta ]})$ is non-torsion for any $[\\zeta ]\\in H_1(Y)$ .",
"Let $C=\\underline{\\widehat{CF}}(Y;\\mathcal {R}_{[\\omega ]})$ .",
"Let $H^{\\mathrm {tors}}(C)$ be the submodule of $H(C)$ which consists of all torsion elements in $H(C)$ , and let $H^{\\mathrm {free}}(C)=H(C)/H^{\\mathrm {tors}}(C)$ .",
"By Theorem REF , there is a short exact sequence ${\\begin{matrix}0\\rightarrow H_*(C)\\otimes _{\\mathcal {R}}\\mathbb {F} &\\xrightarrow{}&H_*(C\\otimes _{\\mathcal {R}}\\mathbb {F} )&\\xrightarrow{}&\\mathrm {Tor}_{\\mathcal {R}}(H_{*-1}(C),\\mathbb {F} )\\rightarrow 0.\\end{matrix}}$ Moreover, by the functoriality of (REF ), the three groups are $\\Lambda _{n-1}$ –modules such that the exact sequence respects the module structure.",
"Recall that $H_*(C\\otimes _{\\mathcal {R}}\\mathbb {F})\\cong \\widehat{HF}(Y;\\mathbb {F})\\cong \\Lambda _{n-1}.$ Claim 1.",
"The map $\\mu _{n-1}$ is zero.",
"If the map $\\mu _{n-1}$ is nonzero, then it is an isomorphism since $H_{n-1}(C\\otimes _{\\mathcal {R}}\\mathbb {F} )$ is one-dimensional.",
"Since $\\mathbf {1}\\in H_{n-1}(C\\otimes _{\\mathcal {R}}\\mathbb {F} )$ generates the module $H_{*}(C\\otimes _{\\mathcal {R}}\\mathbb {F} )$ , the map $\\mu _*$ is surjective.",
"Hence $\\mu _*$ is an isomorphism and $\\mathrm {Tor}_{\\mathcal {R}}(H(C),\\mathbb {F} )=0$ .",
"Lemma REF implies that $H^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} =0$ and the module $H(C)\\otimes _{\\mathcal {R}}\\mathbb {F} $ is isomorphic to the module $H^{\\mathrm {free}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} $ .",
"Now we have $H(H^{\\mathrm {free}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} ,A^{\\zeta _1})\\cong H(H(C)\\otimes _{\\mathcal {R}}\\mathbb {F} ,A^{\\zeta _1})\\cong H(H(C\\otimes _{\\mathcal {R}}\\mathbb {F} ),A^{\\zeta _1})\\cong 0,$ a contradiction to the fact that $H(H(C),A^{\\zeta _1})$ has positive rank.",
"Claim 2.",
"The map $\\tau _{1}$ is zero.",
"If $\\tau _1$ is nonzero, then there exists an $(n-2)$ –element subset $S\\subset \\lbrace 1,\\dots ,n-1\\rbrace $ such that $\\tau _1(X_S)\\ne 0$ .",
"We claim that the $2^{n-2}$ elements $A^{S^{\\prime }}\\circ \\tau (\\mathbf {1}),\\quad S^{\\prime }\\subset S$ are linearly independent over $\\mathbb {F} $ .",
"In fact, suppose $S_1,\\dots ,S_m$ are subsets of $S$ with $|S_i|=k$ , we want to show that $\\sum _iA^{S_i}\\circ \\tau (\\mathbf {1})\\ne 0.$ Apply $A^{S\\backslash S_1}$ to the left hand side of (REF ).",
"Since $(S\\backslash S_1)\\cap S_i\\ne \\emptyset $ for all $i\\ne 1$ , $A^{S\\backslash S_1}A^{S_i}=0$ when $i\\ne 1$ .",
"So we get $A^{S\\backslash S_1}(\\sum _iA^{S_i}\\circ \\tau (\\mathbf {1}))&=&A^{S\\backslash S_1}A^{S_1}\\circ \\tau (\\mathbf {1})\\\\&=&A^S\\circ \\tau (\\mathbf {1})\\\\&=&\\tau \\circ A^S(\\mathbf {1})\\\\&=&\\tau (X_S)\\\\&\\ne &0.$ So (REF ) holds.",
"Now we have proved that the rank of $\\mathrm {Tor}_{\\mathcal {R}}(H(C),\\mathbb {F} )$ is at least $2^{n-2}$ , which is half of the rank of $H_*(C\\otimes _{\\mathcal {R}}\\mathbb {F} )$ .",
"By (REF ), we have $ H_*(C\\otimes _{\\mathcal {R}}\\mathbb {F} )\\cong (H(C)\\otimes _{\\mathcal {R}}\\mathbb {F})\\oplus \\mathrm {Tor}_{\\mathcal {R}}(H(C),\\mathbb {F}).$ Using Lemma REF , we see that $H^{\\mathrm {free}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} =0$ , which contradicts the fact that $H(C)$ has positive rank.",
"This finishes the proof of Claim 2.",
"By Claim 2 we have $\\mathrm {Tor}_{\\mathcal {R}}(H_0(C),\\mathbb {F} )=0$ .",
"So $H_0^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} =0$ by Lemma REF .",
"By Claim 1 $H_{n-2}^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} \\cong \\mathrm {Tor}_{\\mathcal {R}}(H_{n-2}(C),\\mathbb {F} )\\ne 0.$ Let $u\\in H_{n-2}^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} $ be a nonzero element, then $\\mu (u)\\ne 0$ and there exists an $(n-2)$ –element subset $S\\subset \\lbrace 1,\\dots ,n-1\\rbrace $ such that $A^S\\circ \\mu (u)$ is the generator of $H_0(C\\otimes _{\\mathcal {R}}\\mathbb {F})$ .",
"Thus $\\mu \\circ A^S(u)\\ne 0$ .",
"Since $u\\in H_{n-2}^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} $ , $A^S(u)\\in H_0^{\\mathrm {tors}}(C)\\otimes _{\\mathcal {R}}\\mathbb {F} \\cong 0$ by (REF ).",
"Thus $\\mu \\circ A^S(u)=0$ , a contradiction.",
"$\\Box $ With the detection theorem in hand, we can easily prove that the Khovanov module detects unlinks: Proof of Theorem REF .",
"It follows from the module structure of $Kh(L)$ that $Kh^{{r}}(L,L_0)\\cong \\Lambda _{n-1}=\\mathbb {F}[X_1,\\dots ,X_{n-1}]/(X_1^2,\\dots ,X_{n-1}^2)$ .",
"By Proposition REF , $\\widehat{HF}(\\Sigma (L))\\cong \\Lambda _{n-1}$ as a module.",
"By Theorem REF , $\\Sigma (L)\\cong M\\#(\\#^{n-1}(S^1\\times S^2))$ , where $M$ is an integral homology sphere with $\\widehat{HF}(M)\\cong \\mathbb {F}$ .",
"If a link $J$ is non-split, then $\\Sigma (J)$ does not contain an $S^1\\times S^2$ connected summand; on the other hand, if $J=J_1\\sqcup J_2$ , then $\\Sigma (J)=\\Sigma (J_1)\\#\\Sigma (J_2)\\#(S^1\\times S^2)$ [12].",
"Applying this fact to the link $L$ at hand, it follows that $L=L_0\\sqcup L_1\\sqcup \\cdots \\sqcup L_{n-1}$ .",
"Since $Kh(J_1\\sqcup J_2)\\cong Kh(J_1)\\otimes Kh(J_2)$ , each $L_i$ has $\\mathrm {rank}\\:Kh(L_i)=2$ .",
"It follows from [19] that each $L_i$ is an unknot, so $L$ is an unlink.",
"$\\Box $"
]
] | 1204.0960 |
[
[
"Manifolds over Cayley-Dickson algebras and their immersions"
],
[
"Abstract Holomorphic manifolds over Cayley-Dickson algebras are defined and their embeddings and immersions are studied."
],
[
"Introduction.",
"Real and complex manifolds are widely used in different branches of mathematics [10], [13], [20].",
"On the other hand, Cayley-Dickson algebras ${\\cal A}_r$ , particularly, the quaternion skew field ${\\bf H}={\\cal A}_2$ and the octonion algebra ${\\bf O}={\\cal A}_3$ , have found many-sided applications not only in mathematics, but also in theoretical physics [2], [5], [6], [7], [9], [14], [21].",
"Functions of Cayley-Dickson variables were studied earlier [15], [16], [17], [19].",
"Their super-differentiability was defined in terms of representing them words and phrases as a differentiation which is real-linear, additive and satisfying Leibniz' rule on an algebra of phrases over ${\\cal A}_r$ .",
"A super-differentiable function on a domain $U$ in ${\\cal A}_r^n$ or $l_2({\\cal A}_r)$ of ${\\cal A}_r$ -variables is also called holomorphic.",
"This article is devoted to investigations of ${\\cal A}_r$ -holomorphic manifolds.",
"Their embeddings and immersions are studied.",
"Results and notations of previous papers [15], [16], [17], [19] are used below.",
"Main results of this paper are obtained for the first time."
],
[
"Manifolds over Cayley-Dickson algebras",
"1.",
"Definitions and Notes.",
"An ${\\bf R}$ linear space $X$ which is also left and right ${\\cal A}_r$ module will be called an ${\\cal A}_r$ vector space.",
"We present $X$ as the direct sum $(DS)$ $\\quad X=X_0i_0\\oplus ... \\oplus X_m i_m\\oplus ...$ , where $X_0$ ,...,$X_m,...$ are pairwise isomorphic real linear spaces, where $i_0,...,i_{2^r-1}$ are generators of the Cayley-Dickson algebra ${\\cal A}_r$ such that $i_0=1$ , $i_k^2=-1$ and $i_ki_j=-i_ji_k$ for each $k\\ge 1$ and $j\\ge 1$ so that $k\\ne j$ , $~2\\le r$ .",
"Let $X$ and $Y$ be two $\\bf R$ linear normed spaces which are also left and right ${\\cal A}_r$ modules, where $1\\le r$ , such that $(1)$ $0\\le \\Vert ax\\Vert _X \\le |a| \\Vert x \\Vert _X $ and $ \\Vert x a \\Vert _X\\le |a| \\Vert x \\Vert _X $ and $(2)$ $\\Vert ax_j \\Vert _X =|a| \\Vert x_j \\Vert _X$ and $(3)$ $\\Vert x+y \\Vert _X \\le \\Vert x \\Vert _X + \\Vert y \\Vert _X $ for all $x, y\\in X$ and $a\\in {\\cal A}_r$ and $x_j\\in X_j$ .",
"Such spaces $X$ and $Y$ will be called ${\\cal A}_r$ normed spaces.",
"Suppose that $X$ and $Y$ are two normed spaces over the Cayley-Dickson algebra ${\\cal A}_v$ .",
"A continuous $\\bf R$ linear mapping $\\theta : X\\rightarrow Y$ is called an $\\bf R$ linear homomorphism.",
"If in addition $\\theta (bx)=b\\theta (x)$ and $\\theta (xb)=\\theta (x)b$ for each $b \\in {\\cal A}_v$ and $x\\in X$ , then $\\theta $ is called a homomorphism of ${\\cal A}_v$ (two sided) modules $X$ and $Y$ .",
"If a homomorphism is injective, then it is called an embedding ($\\bf R$ linear or for ${\\cal A}_v$ modules correspondingly).",
"If a homomorphism $h$ is bijective and from $X$ onto $Y$ so that its inverse mapping $h^{-1}$ is also continuous, then it is called an isomorphism ($\\bf R$ linear or of ${\\cal A}_v$ modules respectively).",
"2.",
"Definitions.",
"We say that a real vector space $Z$ is supplied with a scalar product if a bi-$\\bf R$ -linear bi-additive mapping $<,>: Z^2\\rightarrow {\\bf R}$ is given satisfying the conditions: $(1)$ $<x,x> ~ \\ge 0$ , $ ~ <x,x>=0$ if and only if $x=0$ ; $(2)$ $<x,y>=<y,x>$ ; $(3)$ $<ax+by,z>=a<x,z>+b<y,z>$ for each real numbers $a, b\\in {\\bf R}$ and vectors $x, y,z\\in Z$ .",
"Then an ${\\cal A}_r$ vector space $X$ is supplied with an ${\\cal A}_r$ valued scalar product, if a bi-${\\bf R}$ -linear bi-${\\cal A}_r$ -additive mapping $<*,*>: X^2\\rightarrow {\\cal A}_r$ is given such that $(4)$ $\\quad <f,g> = \\sum _{j,k} <f_j,g_k>i_j^*i_k$ , where $f=f_0i_0+...+f_mi_m+...$ , $ ~f, g\\in X$ , $ ~ f_j, g_j \\in X_j$ , each $X_j$ is a real linear space with a real valued scalar product, $(X_j, <*,*>)$ is real linear isomorphic with $(X_k,<*,*>)$ and $<f_j,g_k>\\in {\\bf R}$ for each $j, k$ .",
"The scalar product induces the norm: $(5)$ $\\Vert f \\Vert := \\sqrt{<f,f>}$ .",
"An ${\\cal A}_r$ normed space or an ${\\cal A}_r$ vector space with ${\\cal A}_r$ scalar product complete relative to its norm will be called an ${\\cal A}_r$ Banach space or an ${\\cal A}_r$ Hilbert space respectively.",
"A Hilbert space $X$ over ${\\cal A}_r$ is denoted by $l_2(\\lambda ,{\\cal A}_r)$ , where $\\lambda $ is a set of the cardinality $card (\\lambda )\\ge \\aleph _0$ which is the topological weight of $X_0$ , i.e.",
"$X_0=l_2(\\lambda ,{\\bf R})$ .",
"A mapping $f: U\\rightarrow l_2(\\lambda ,{\\cal A}_r)$ can be written in the form $f(z) = \\sum _{j\\in \\lambda } f^j(z)e_j,$ where $\\lbrace e_j: j \\in \\lambda \\rbrace $ is an orthonormal basis in the Hilbert space $l_2(\\lambda ,{\\cal A}_r)$ , $~U$ is a domain in $l_2(\\psi ,{\\cal A}_r)$ , $~f^j(z)\\in {\\cal A}_r$ for each $z\\in U$ and every $j\\in \\lambda $ .",
"If $f$ is Frechét differentiable over $\\bf R$ and each function $f^j(z)$ is holomorphic by each Cayley-Dickson variable $\\mbox{}_kz$ on $U$ , then $f$ is called holomorphic on $U$ , where $z = \\sum _{k\\in \\psi }\\mbox{}_kzq_k,$ while $\\lbrace q_k: ~ k\\in \\psi \\rbrace $ denotes the standard orthonormal basis in $l_2(\\psi ,{\\cal A}_r)$ , $\\mbox{}_kz\\in {\\cal A}_r$ .",
"3.",
"Definition.",
"Let $M$ be a set such that $M=\\bigcup _jU_j$ , $M$ is a Hausdorff topological space, each $U_j$ is open in $M$ , $\\phi _j: U_j\\rightarrow \\phi _j(U_j)\\subset X$ are homeomorphisms, $\\phi _j(U_j)$ is open in ${\\cal A}_r$ for each $j$ , if $U_i\\cap U_j\\ne \\emptyset $ , the transition mapping $\\phi _i\\circ \\phi _j^{-1}$ is ${\\cal A}_r$ -holomorphic on its domain, where $X$ is either ${\\cal A}_r^m$ with $m\\in {\\bf N}$ or a Hilbert space $l_2(\\lambda ,{\\cal A}_r)$ over the Cayley-Dickson algebra ${\\cal A}_r$ .",
"Then $M$ is called the ${\\cal A}_r$ -holomorphic manifold.",
"4.",
"Proposition.",
"Let $M$ be an ${\\cal A}_r$ holomorphic manifold.",
"Then there exists a tangent bundle $TM$ which has the structure of an ${\\cal A}_r$ holomorphic manifold such that each fibre $T_xM$ is the vector space over the Cayley-Dickson algebra ${\\cal A}_r$ .",
"Proof.",
"The Cayley-Dickson algebra ${\\cal A}_r$ has the real shadow, which is the Euclidean space $\\bf R^{2^r}$ , since ${\\cal A}_r$ is the algebra over $\\bf R$ .",
"Therefore, a manifold $M$ has also a real manifold structure.",
"Each ${\\cal A}_r$ holomorphic mapping is infinite differentiable in accordance with Theorems 2.15 and 3.10 in [15], [16].",
"Then the tangent bundle $TM$ exists, which is $C^{\\infty }$ -manifold such that each fibre $T_xM$ is a tangent space, where $x\\in M$ , $~T$ is the tangent functor.",
"If $At (M)= \\lbrace (U_j,\\phi _j): j \\rbrace $ , then $At (TM)= \\lbrace (TU_j, T\\phi _j): j \\rbrace $ , $TU_j=U_j\\times X$ , where $X$ is the ${\\cal A}_r$ vector space on which $M$ is modeled, $T(\\phi _j\\circ \\phi _k^{-1})=(\\phi _j\\circ \\phi _k^{-1}, D(\\phi _j\\circ \\phi _k^{-1}))$ for each $U_j\\cap U_k\\ne \\emptyset $ .",
"Each transition mapping $\\phi _j\\circ \\phi _k^{-1}$ is ${\\cal A}_r$ holomorphic on its domain, then its (strong) differential coincides with the super-differential $D(\\phi _j\\circ \\phi _k^{-1})= D_z(\\phi _j\\circ \\phi _k^{-1})$ , since ${\\tilde{\\partial }} (\\phi _j\\circ \\phi _k^{-1})=0$ .",
"Therefore, the super-differential $D(\\phi _j\\circ \\phi _k^{-1})$ is $\\bf R$ -linear and ${\\cal A}_r$ -additive, hence it is the automorphism of the ${\\cal A}_r$ vector space $X$ .",
"But $D_z(\\phi _j\\circ \\phi _k^{-1})$ is ${\\cal A}_r$ holomorphic as well, consequently, $TM$ is the ${\\cal A}_r$ holomorphic manifold.",
"5.",
"Definitions.",
"A $C^1$ -mapping $f: M\\rightarrow N$ is called an immersion, if $rang (df|_x: T_xM\\rightarrow T_{f(x)}N) = m_M$ for each $x\\in M$ , where $m_M := dim_{\\bf R}M$ .",
"An immersion $f: M\\rightarrow N$ is called an embedding, if $f$ is bijective.",
"6.",
"Theorem.",
"Let $M$ be a compact ${\\cal A}_r$ holomorphic manifold, $dim_{{\\cal A}_r}M = m<\\infty $ , where $2\\le r$ .",
"Then there exists an ${\\cal A}_r$ holomorphic embedding $\\tau : M\\hookrightarrow {\\cal A}_r^{2m+1}$ and an ${\\cal A}_r$ holomorphic immersion $\\theta : M\\rightarrow {\\cal A}_r^{2m}$ correspondingly.",
"Each continuous mapping $f: M\\rightarrow {\\cal A}_r^{2m+1}$ or $f: M\\rightarrow {\\cal A}_r^{2m}$ can be approximated by $\\tau $ or $\\theta $ relative to the norm $\\Vert * \\Vert _{C^0}$ .",
"If $M$ is a paracompact ${\\cal A}_r$ holomorphic manifold with countable atlas on $l_2(\\lambda ,{\\cal A}_r)$ , where $card (\\lambda )\\ge \\aleph _0$ , then there exists a holomorphic embedding $\\tau : M\\hookrightarrow l_2(\\lambda ,{\\cal A}_r)$ .",
"Proof.",
"Let at first $M$ be compact.",
"Since $M$ is compact, then it is finite dimensional over ${\\cal A}_r$ , $dim_{{\\cal A}_r}M = m\\in \\bf N$ , such that $dim_{{\\cal A}_r}M = 2^rm$ is its real dimension.",
"Take an atlas $At^{\\prime } (M)$ refining the initial atlas $At (M)$ of $M$ such that $({U^{\\prime }}_j, \\phi _j)$ are charts of $M$ , where each ${U^{\\prime }}_j$ is ${\\cal A}_r$ holomorphic diffeomorphic to an interior of the unit ball $Int(B({\\cal A}_r^m,0,1))$ , where $B({\\cal A}_r^m,y,\\rho ) := \\lbrace z\\in {\\cal A}_r^m: |z-y|\\le \\rho \\rbrace $ .",
"In view of compactness of the manifold $M$ a covering $\\lbrace {U^{\\prime }}_j: j \\rbrace $ has a finite subcovering, hence $At^{\\prime } (M)$ can be chosen finite.",
"Denote for convenience the latter atlas as $At (M)$ .",
"Let $(U_j, \\phi _j)$ be the chart of the atlas $At (M)$ , where $U_j$ is open in $M$ , hence $M\\setminus U_j$ is closed in $M$ .",
"Consider the space ${\\cal A}_r^m\\times {\\bf R}$ as the $\\bf R$ -linear space $\\bf R^{2^rm+1}$ .",
"The unit sphere $S^{2^rm}:=S ({\\bf R}^{2^rm+1},0,1) := \\lbrace z\\in {\\bf R}^{2^rm+1}:$ $|z|=1 \\rbrace $ in ${\\cal A}_r^m\\times \\bf R$ can be supplied with two charts $(V_1, \\phi _1)$ and $(V_2, \\phi _2)$ such that $V_1:=S^{2^rm}\\setminus \\lbrace 0,...,0, 1 \\rbrace $ and $V_2:=S^{2^rm}\\setminus \\lbrace 0,...,0, - 1 \\rbrace $ , where $\\phi _1$ and $\\phi _2$ are stereographic projections from poles $\\lbrace 0,...,0, 1 \\rbrace $ and $ \\lbrace 0,...,0, -1 \\rbrace $ of $V_1$ and $V_2$ respectively onto ${\\cal A}_r^m$ .",
"The conjugation is given by the formula: $z^* = - (2^r-2)^{-1}\\sum _{p=0}^{2^r-1}(i_pz)i_p $ in ${\\cal A}_r^m$ , which provides $z^*$ in the $z$ -representation.",
"Therefore $\\phi _1\\circ \\phi _2^{-1}$ written in the $z$ -representation is the ${\\cal A}_r$ holomorphic diffeomorphism in ${\\cal A}_r^m\\setminus \\lbrace 0 \\rbrace $ , i.e.",
"the super-differential $D_z (\\phi _1\\circ \\phi _2^{-1})$ exists, where $i_0,...,i_{2^r-1}$ are the standard generators of ${\\cal A}_r$ .",
"Thus the unit sphere $S^{2^rm}$ can be supplied with the structure of the ${\\cal A}_r$ holomorphic manifold.",
"Therefore, there exists an ${\\cal A}_r$ holomorphic mapping $\\psi _j$ .",
"That is locally $z$ -analytic of $M$ into the unit sphere $S^{2^rm}$ such that $\\psi _j: U_j\\rightarrow \\psi _j(U_j)$ is the ${\\cal A}_r$ holomorphic diffeomorphism onto the subset $\\psi _j(U_j)$ in $S^{2^rm}$ .",
"Using of such mappings $\\psi _j$ is sufficient, where $\\psi _j$ can be considered as components of a holomorphic diffeomorphism: $\\psi : M\\rightarrow (S^{2^rm})^n$ with $n$ equal to the number of charts.",
"There is an embedding of ${\\cal A}_r^m\\times \\bf R$ into ${\\cal A}_r^{m+1}$ .",
"Then the mapping $\\psi (z):=(\\psi _1(z),...,\\psi _n(z))$ is the embedding into $(S^{2^rm})^n$ and hence into ${\\bf K}^{n{m+1}}$ , since the rank is $rank [d_z\\psi (z)]=2^rm$ at each point $z\\in M$ .",
"Indeed, the rank is $rank[d_z\\psi _j(z)]=2^rm$ for each $z\\in U_j$ and the dimension is bounded from above $dim_{{\\cal A}_r}\\psi (U_j)\\le dim_{{\\cal A}_r} M= m$ .",
"Moreover, $\\psi (z)\\ne \\psi (y)$ for each $z\\ne y\\in U_j$ , since $\\psi _j(z)\\ne \\psi _j(y)$ .",
"If $z\\in U_j$ and $y\\in M\\setminus U_j$ , then there exists a number $l\\ne j$ so that $y\\in U_l\\setminus U_j$ , $\\psi _j(z)\\ne \\psi _j(y)=x_j$ .",
"Let $M\\hookrightarrow {\\cal A}_r^N$ be the ${\\cal A}_r$ holomorphic embedding as above.",
"There is also the ${\\cal A}_r$ holomorphic embedding of $M$ into $(S^{2^rm})^n$ as it is shown above, where $(S^{2^rm})^n$ is the ${\\cal A}_r$ holomorphic manifold as the product of ${\\cal A}_r$ holomorphic manifolds.",
"Let $P{\\bf R}^n$ denote the real projective space formed from the Euclidean space ${\\bf R}^{n+1}$ , denote by $\\phi : {\\bf R}^{n+1}\\rightarrow P{\\bf R}^n$ the corresponding projective mapping.",
"Geometrically $P{\\bf R}^n$ is considered as $S^n/\\tau $ , where $S^n:= \\lbrace y\\in {\\bf R}^{n+1}: ~ \\Vert y \\Vert =1 \\rbrace $ is the unit sphere in ${\\bf R}^{n+1}$ , while $\\tau $ is the equivalence relation making identical two spherically symmetric points, i.e.",
"points belonging to the same straight line containing zero and intersecting the unit sphere.",
"Consider ${\\cal A}_r^n$ as the algebra of all $n\\times n$ diagonal matrices $A = diag (a_1,...,a_n)$ with entries $a_1,...,a_n\\in {\\cal A}_r$ .",
"Then ${\\cal A}_r^n$ is isomorphic with the tensor product of algebras ${\\cal A}_r^n = {\\cal A}_r \\otimes _{\\bf R} {\\bf R}^n$ over the real field, where ${\\bf R}^n$ is considered as the algebra of all diagonal $n\\times n$ matrices $C=diag (b_1,..,b_n)$ with entries $b_1,...,b_n\\in {\\bf R}$ .",
"Put $P{\\cal A}_r^n=\\phi ({\\cal A}_r\\otimes _{\\bf R} {\\bf R}^{n+1})$ to be the Cayley-Dickson projective space, where $\\phi $ is extended from ${\\bf R}^{n+1}$ onto ${\\cal A}_r \\otimes _{\\bf R} {\\bf R}^{n+1}$ so that $\\phi (ax)=a\\phi (x)$ and $\\phi (xa)=\\phi (x)a$ for each $a\\in {\\cal A}_r$ with $|a|=1$ and every $x\\in {\\bf R}^{n+1}$ , also $\\phi (x_0i_0+...+x_{2^r-1}i_{2^r-1})=\\phi (x_0)i_0\\alpha _0+...+\\phi (x_{2^r-1})i_{2^r-1}\\alpha _{2^r-1}$ for each non-zero vector $x=x_0i_0+...+x_{2^r-1}i_{2^r-1}\\in {\\cal A}_r^{n+1}$ , where $\\alpha _j := \\Vert x_j \\Vert / \\Vert x \\Vert $ , $x_j\\in {\\bf R}^{n+1}$ for each $j$ , $~\\Vert x \\Vert ^2 = \\Vert x_0 \\Vert ^2 +... + \\Vert x_{2^r-1} \\Vert ^2$ .",
"If $z\\in P{\\cal A}_r^n$ , then by our definition $\\phi ^{-1}(z)$ is the ${\\cal A}_r$ straight line in ${\\cal A}_r^{n+1}$ .",
"To each element $x\\in {\\cal A}_r^{n+1}$ we pose an ${\\cal A}_r$ straight line $< {\\cal A}_r,x \\rbrace := \\phi ^{-1}(\\phi (x))$ .",
"That is the bundle of all ${\\cal A}_r$ straight lines $< {\\cal A}_r,x \\rbrace $ in ${\\cal A}_r^N$ is considered, where $x\\in {\\cal A}_r^N$ , $x\\ne 0$ , so that $<{\\cal A}_r,x \\rbrace $ is the ${\\cal A}_r$ vector space of dimension 1 over ${\\cal A}_r$ , which has the real shadow isomorphic with ${\\bf R}^{2^r}$ .",
"Fix the standard orthonormal base $ \\lbrace e_1,...,e_N \\rbrace $ in ${\\cal A}_r^N$ and projections on ${\\cal A}_r$ -vector subspaces relative to this base $P^L(x):=\\sum _{e_j\\in L}x_je_j$ for the ${\\cal A}_r$ vector span $L=span_{{\\cal A}_r} \\lbrace e_i:$ $i\\in \\Lambda _L \\rbrace $ , $ ~ \\Lambda _L\\subset \\lbrace 1,...,N \\rbrace $ , where $x=\\sum _{j=1}^Nx_je_j,$ $x_j\\in {\\cal A}_r$ for each $j$ , $ ~ e_j=(0,...,0,1,0,...,0)$ with 1 at $j$ -th place.",
"In this base consider the ${\\cal A}_r$ -Hermitian scalar product $<x,y> := \\sum _{j=1}^Nx_j^*y_j.$ Let $l\\in {\\cal A}_rP^{N-1}$ , take an ${\\cal A}_r$ -hyperplane denoted by $({\\cal A}_r^{N-1})_l$ and given by the condition: $<x,y>=0$ for each $x\\in ({\\cal A}_r^{N-1})_l$ and $y\\in l$ .",
"Take a vector $0\\ne [l]\\in {\\cal A}_r^N$ as a representative characterizes the equivalence class $l=<{\\cal A}_r, [l] \\rbrace $ of unit norm $\\Vert [l] \\Vert =1$ .",
"Then the orthonormal base $\\lbrace q_1,...,q_{N-1} \\rbrace $ in $({\\cal A}_r^{N-1})_l$ and the vector $[l]=:q_N$ compose the orthonormal base $\\lbrace q_1,...,q_N \\rbrace $ in ${\\cal A}_r^N$ .",
"This provides the ${\\cal A}_r$ holomorphic projection $\\pi _l: {\\cal A}_r^N\\rightarrow ({\\cal A}_r^{N-1})_l$ relative to the orthonormal base $ \\lbrace q_1,...,q_N \\rbrace $ .",
"Indeed, the operator $\\pi _l$ is ${\\cal A}_r$ left $\\pi _l(bx_0)=b\\pi _l(x_0)$ and also right $\\pi _l(x_0b)=\\pi _l(x_0)b$ linear for each $x_0\\in X_0$ and $b\\in {\\cal A}_r$ , but certainly non-linear relative to ${\\cal A}_r$ .",
"Therefore the mapping $\\pi _l$ is ${\\cal A}_r$ holomorphic.",
"To construct an immersion it is sufficient, that each projection $\\pi _l: T_xM\\rightarrow ({\\cal A}_r^{N-1})_l$ has $ker [d(\\pi _l(x))]= \\lbrace 0 \\rbrace $ for each $x\\in M$ .",
"The set of all points $x\\in M$ for which $ker [d(\\pi _l(x))] \\ne \\lbrace 0 \\rbrace $ is called the set of forbidden directions of the first kind.",
"Forbidden are those and only those directions $l\\in {\\cal A}_rP^{N-1}$ for which there exists a point $x\\in M$ such that $l^{\\prime }\\subset T_xM$ , where $l^{\\prime }=[l]+z$ , $z\\in {\\cal A}_r^N$ .",
"The set of all forbidden directions of the first kind forms the ${\\cal A}_r$ holomorphic manifold $Q$ of ${\\cal A}_r$ dimension $(2m-1)$ with points $(x,l)$ , where $x\\in M$ , $ ~ l\\in {\\cal A}_rP^{N-1}$ , $ ~ [l]\\in T_xM$ .",
"Take the mapping $g: Q\\rightarrow {\\cal A}_rP^{N-1}$ given by $g(x,l):=l$ .",
"Then this mapping $g$ is ${\\cal A}_r$ holomorphic.",
"Each paracompact manifold $A$ modeled on ${\\cal A}_r^p$ can be supplied with the Riemann manifold structure also.",
"Therefore, on a manifold $A$ there exists a Riemann volume element.",
"In view of the Morse theorem $\\mu (g(Q))=0$ , if $N-1>2m-1$ , that is, $2m<N$ , where $\\mu $ is the Riemann volume element in ${\\cal A}_rP^{N-1}$ .",
"In particular, $g(Q)$ is not contained in ${\\cal A}_rP^{N-1}$ and there exists $l_0\\notin g(Q)$ , consequently, there exists $\\pi _{l_0}: M\\rightarrow ({\\cal A}_r^{N-1})_{l_0}$ .",
"This procedure can be prolonged, when $2m<N-k$ , where $k$ is the number of the step of projection.",
"Hence $M$ can be immersed into ${\\cal A}_r^{2m}$ .",
"Consider now the forbidden directions of the second type: $l\\in {\\cal A}_rP^{N-1}$ , for which there exist $x\\ne y\\in M$ simultaneously belonging to $l$ after suitable parallel translation $[l]\\mapsto [l]+z$ , $z\\in {\\cal A}_r^N$ .",
"The set of the forbidden directions of the second type forms the manifold $\\Phi :=M^2\\setminus \\Delta $ , where $\\Delta := \\lbrace (x,x):$ $x\\in M \\rbrace $ .",
"Consider $\\psi : \\Phi \\rightarrow {\\cal A}_rP^{N-1}$ , where $\\psi (x,y)$ is the straight ${\\cal A}_r$ -line with the direction vector $[x,y]$ in the orthonormal base.",
"Then $\\mu (\\psi (\\Phi ))=0$ in ${\\cal A}_rP^{N-1}$ , if $2m+1<N$ .",
"Then the closure $cl (\\psi (\\Phi ))$ coincides with $\\psi (\\Phi )\\cup g(Q)$ in ${\\cal A}_rP^{N-1}$ .",
"Hence there exists $l_0\\notin cl (\\psi (\\Phi ))$ .",
"Then consider $\\pi _{l_0}: M\\rightarrow ({\\cal A}_r)_{l_0}^{N-1}$ .",
"This procedure can be prolonged, when $2m+1<N-k$ , where $k$ is the number of the step of projection.",
"Hence $M$ can be embedded into ${\\cal A}_r^{2m+1}$ .",
"The approximation property follows from compactness of $M$ and the non-commutative analog of the Stone-Weierstrass theorem (see also Theorem 2.7 in [15], [16]).",
"Let now $M$ be a paracompact ${\\cal A}_r$ holomorphic manifold with countable atlas on $l_2(\\lambda ,{\\bf K})$ .",
"Spaces $l_2(\\lambda ,{\\cal A}_r)\\oplus {\\cal A}_r^m$ and $l_2(\\lambda ,{\\cal A}_r)\\oplus l_2(\\lambda ,{\\cal A}_r)$ are isomorphic as ${\\cal A}_r$ Hilbert spaces with $l_2(\\lambda ,{\\cal A}_r)$ , since $card (\\lambda )\\ge \\aleph _0$ .",
"Take an additional variable $z\\in {\\cal A}_r$ , when $z=j\\in \\bf N$ .",
"Then it gives a number of a chart.",
"Each $TU_j$ is ${\\cal A}_r$ holomorphically diffeomorphic with $U_j\\times l_2(\\lambda ,{\\cal A}_r)$ .",
"Consider ${\\cal A}_r$ holomorphic functions $\\psi $ on domains in $l_2(\\lambda ,{\\cal A}_r)\\oplus l_2(\\lambda ,{\\cal A}_r)\\oplus {\\cal A}_r$ .",
"Then there exists an ${\\cal A}_r$ holomorphic mapping $\\psi _j: M\\rightarrow l_2(\\lambda ,{\\cal A}_r)$ such that $\\psi _j: U_j\\rightarrow \\psi _j(U_j)\\subset l_2(\\lambda ,{\\cal A}_r)$ is an ${\\cal A}_r$ holomorphic diffeomorphism.",
"Then the mapping $(\\psi _1,\\psi _2,...)$ provides the ${\\cal A}_r$ holomorphic embedding of $M$ into $l_2(\\lambda ,{\\cal A}_r)$ ."
]
] | 1204.1545 |
[
[
"Two-fluid evolving Lorentzian wormholes"
],
[
"Abstract We investigate the evolution of a family of wormholes sustained by two matter components: one with homogeneous and isotropic properties $\\rho(t)$ and another inhomogeneous and anisotropic $\\rho_{in}(t,r)$.",
"The rate of expansion of these evolving wormholes is only determined by the isotropic and homogeneous matter component $\\rho(t)$.",
"Particularly, we consider a family of exact two-fluid evolving wormholes expanding with constant velocity and satisfying the dominant and the strong energy conditions in the whole spacetime.",
"In general, for the case of vanishing isotropic fluid $\\rho(t)$ and cosmological constant $\\Lambda$ the space expands with constant velocity, and for $\\rho(t)=0$ and $\\Lambda \\neq 0$ the rate of expansion is determined by the cosmological constant.",
"The considered here two-fluid evolving wormholes are a generalization of single fluid models discussed in previous works of the present authors [Phys.\\ Rev.\\ D {\\bf 78}, 104006 (2008); Phys.\\ Rev.\\ D {\\bf 79}, 024005 (2009)]."
],
[
"Introduction",
"Wormhole spacetimes have become one of the most popular and intensively studied topics in general relativity.",
"Throughout the last decades there has been an accumulating volume of works on the analytic wormhole geometries.",
"The various approaches include both static [1] and evolving relativistic versions [2].",
"They principally consider static wormhole spacetimes sustained by a single fluid component which requires the violation of the null energy condition (NEC), and the interest has been focused on traversable wormholes, which have no horizons, allowing two-way passage through them.",
"These hypothetical tunnels in spacetime allow effective superluminal travels, although the speed of light is not locally surpassed [3].",
"It is interesting to note that for constructing wormhole geometries in general is adopted the reverse approach for solving the Einstein field equations.",
"This means that one first fixes the form of the spacetime metric (such as the redshift and shape functions) and then, by computing the field equations, one finds the energy-momentum tensor components needed to support such a spacetime geometry.",
"The obtained in such a way stress components automatically satisfy local conservation equations, by virtue of the Bianchi identities.",
"This reverse method helps us to find that a static traversable wormhole violates the NEC [4], [5], thus in general relativity an exotic type of matter is required for sustaining a static traversable wormhole.",
"It is interesting to note that there are explicit static wormholes solutions respecting the energy conditions in the whole spacetime in Einstein-Gauss-Bonnet gravity [6].",
"Notice also that in higher dimensions, the presence of terms with higher powers in the curvature provided by certain class of Lovelock theories, allows to remove the possibility of violating energy conditions even in vacuum, since the whole spacetime is devoid of any kind of stress-energy tensor [7].",
"However, it is well known that in Einstein gravity there are nonstatic Lorentzian wormholes which do not require WEC violating matter to sustain them.",
"Such wormholes may exist for arbitrarily small or large intervals of time [8].",
"On the purely gravitational side, most of the efforts are directed to study Lorentzian wormholes sustained by a single exotic fluid in classical general relativity.",
"However, one can consider also gravitational configurations filled with two or more fluids [9].",
"For example, in cosmology such two-fluid models are widely considered today in order to explain the observed accelerated expansion of the Universe [10].",
"In this paper we shall study evolving wormholes sustained by two fluids: one with homogeneous and isotropic properties $\\rho (t)$ and another inhomogeneous and anisotropic $\\rho _{in}(t,r)$ .",
"The theoretical construction of these wormholes will be performed by imposing conditions on the stress-energy tensor threading the evolving wormhole geometry.",
"Specifically, we shall consider the radial and tangential pressures of the inhomogeneous and anisotropic matter to obey barotropic equations of state with constant state parameters.",
"On the other hand, the homogeneous and isotropic fluid is taken to be that of a perfect fluid described by the energy-momentum tensor $T_{\\alpha \\beta }=(\\rho +p) u_{\\alpha }u_{\\beta }-p g_{\\alpha \\beta },$ where $u_{\\alpha }$ is the four-velocity of the fluid, $\\rho $ and $p$ are the energy density and the pressure of the cosmic fluid respectively We shall suppose that the dynamics of the gravitational fields is governed by Einstein field equations $R_{\\alpha \\beta }-\\frac{R}{2} g_{\\alpha \\beta }=-\\kappa T_{\\alpha \\beta }-\\Lambda g_{\\alpha \\beta },$ where $\\kappa =8 \\pi G$ and $\\Lambda $ is the cosmological constant, and the evolving wormhole metric will be given by $ds^2=-e^{2\\Phi (t,r)}dt^2+ a(t)^2 \\left(\\frac{dr^2}{1-\\frac{b(r)}{r}}+r^2 d \\Omega ^2 \\right),$ where $\\Phi (t,r)$ is the redshift function, $a(t)$ is the scale factor of the wormhole universe, $b(r)$ is the shape function and $d\\Omega ^2=d\\theta ^2+sin^2 \\theta d \\varphi ^2$ .",
"Note that the essential characteristics of a wormhole geometry are encoded in the spacelike section of the metric (REF ).",
"It is clear that this metric becomes a static wormhole if $a(t)\\rightarrow const$ and $\\Phi (t,r)=\\Phi (r)$ and, as $b(r) \\rightarrow 0$ and $\\Phi (t,r) \\rightarrow 0$ it becomes a flat Friedmann-Robertson-Walker metric.",
"The organization of the paper is as follows: In Sec.",
"II we present the dynamical field equations for wormhole models with a matter source composed of an ideal isotropic cosmic fluid and an anisotropic and inhomogeneous one.",
"In Sec.",
"III some aspects of the geometry of the general solution are discussed.",
"In Sec.",
"IV expanding wormholes are discussed, and in Sec.",
"V we conclude with some remarks."
],
[
"Field equations",
"Let us now consider the dynamical field equations describing evolving wormhole models (REF ).",
"We shall be interested in studying wormhole scenarios filled with two fluids $\\rho =\\rho (t)$ and $\\rho _{_{in}}=\\rho _{_{in}}(t,r)$ , where the first cosmic fluid always remains homogeneous and isotropic and the other component is in general an inhomogeneous and anisotropic fluid.",
"Since we have a spherically symmetric space-time, the cosmic fluid $\\rho _{_{in}}(t,r)$ in general may have anisotropic pressures, which we shall define as $p_{r}(t,r)$ and $p_{l}(t,r)$ for the radial and lateral components respectively.",
"If $p_{r}(t,r)=p_{l}(t,r)$ we have an isotropic inhomogeneous pressure.",
"Thus, for spherically symmetric spacetimes written in comoving coordinates (REF ) and filled with these two kinds of cosmic fluids, the Einstein field equations may be written in the following form: $3 e^{-2\\phi (t,r)} H^2+\\frac{b^{\\prime }}{a^2 r^2}=\\kappa \\rho _{_{in}}(t,r)+\\kappa \\rho (t)+\\Lambda , \\\\ - e^{-2\\phi (t,r)}\\, \\left( 2\\frac{\\ddot{a}}{a}+ H^2\\right)- \\frac{b}{a^2 r^3} + 2 e^{-2\\phi (t,r)} H \\frac{\\partial \\phi }{\\partial t}+ \\\\\\nonumber \\frac{2}{r^2 a^2} (r-b ) \\frac{\\partial \\phi }{\\partial r}=\\kappa p_r(t,r)+\\kappa p(t)-\\Lambda , \\\\ -e^{-\\phi (t,r)} \\left(2 \\frac{\\ddot{a}}{a}+ H^2\\right) + \\frac{b-r b^{\\prime }}{2 a^2 r^3}+ \\\\ \\nonumber 2e^{-\\phi (t,r)} H \\frac{\\partial \\phi }{\\partial t} +\\frac{1}{2a^2r^2} (2r-b-rb^{\\prime }) \\frac{\\partial \\phi }{\\partial r} + \\\\ \\nonumber \\frac{1}{a^2r} (r-b)\\left( \\left(\\frac{\\partial \\phi }{\\partial r} \\right)^2+\\frac{\\partial ^2 \\phi }{\\partial r^2}\\right)=\\kappa p_{_l}(t,r)+\\kappa p(t) -\\Lambda , \\\\2 e^{-\\phi (t,r)} \\sqrt{\\frac{r-b(r)}{r}} \\, \\frac{\\partial \\phi }{\\partial r} \\, \\dot{a}=0, $ where it was assumed that the 4-velocity of both fluids is the timelike vector $u^{\\alpha }=(e^{-\\phi },0,0,0)$ , $\\kappa =8 \\pi G$ , $H=\\dot{a}/a$ , and an overdot and a prime denote differentiation $d/dt$ and $d/dr$ respectively.",
"In attempting to find solutions to the field equations we first note that Eq.",
"() gives some constraints on relevant metric functions which separate the wormhole solutions into two branches: one static branch given by the condition $\\dot{a}=0$ and another non-static branch for $\\partial \\phi /\\partial r=0$ .",
"In what follows we shall restrict our discussion to nonstatic branch.",
"The condition $\\partial \\phi /\\partial r=0$ implies that the redshift function can only be a function of $t$ , i.e.",
"$\\phi (t,r)=f(t)$ so, without any loss of generality, by rescaling the time coordinate we can set $\\phi (t,r)=0$ .",
"Thus we shall look for the solutions to Einstein equations described by the metric form $ds^2=-dt^2+ a(t)^2 \\left( \\frac{dr^2}{1-\\frac{b(r)}{r}}+r^2 d\\Omega ^2 \\right),$ so we have to put $\\Phi =0$ into the Eqs.",
"(REF )-().",
"As we shall see below this will imply that the anisotropic and inhomogeneous matter component $\\rho _{_{in}}(t,r)$ cannot be isotropic.",
"Notice that the field equations (REF )-() generalize the Einstein equations considered in Refs.",
"[11] and [12].",
"Thus, in order to quickly solve the field equations we shall use the conservation equations $T^\\mu _{\\,\\,\\,\\,\\nu ;\\mu }=0$ .",
"By supposing that each fluid satisfies the standard conservation equation separately we obtain $\\frac{\\partial \\rho }{\\partial t}+3H (\\rho +p)=0, \\\\\\frac{\\partial \\rho _{_{in}}}{\\partial t}+H (3 \\rho _{_{in}}+p_r+2p_{_l})=0, \\\\\\frac{2(p_{_l}-p_r)}{r}=\\frac{\\partial p_r}{\\partial r},$ where Eq.",
"(REF ) states the conservation of the isotropic and homogeneous component and Eqs.",
"() and () are valid for the anisotropic and inhomogeneous cosmic fluid and may be interpreted as the conservation equation and the relativistic Euler equation (or the hydrostatic equation for equilibrium for the anisotropic matter supporting the gravitational configuration) respectively.",
"Notice that from equations () and () we see that for an isotropic but still inhomogeneous matter component $\\rho _{_{in}}$ , i.e.",
"$p_{_l}=p_r=p_{_{in}}$ , we have to require $\\partial p_r/\\partial r=0$ , so the pressure will depend only on time $t$ , obtaining the standard cosmological conservation equation $\\dot{\\rho }_{_{in}}(t)+3 H [\\rho _{_{in}}(t)+p_{_{in}}(t)]=0$ .",
"Thus in this case we have a noninteracting superposition of two homogeneous and isotropic fluids $\\rho $ and $\\rho _{_{in}}$ .",
"This leads us to conclude that if we want to study evolving wormholes filled with a mixture of homogeneous, isotropic fluid $\\rho (t)$ and an inhomogeneous fluid $\\rho _{_{in}}(t,r)$ , we must consider only anisotropic matter component $\\rho _{_{in}}(t,r)$ with $ p_r \\ne p_{_l}$ .",
"For solving the field equations of the considered gravitational configuration we shall consider the following anzats: we shall require that the radial and the lateral pressures have barotropic equations of state.",
"Thus we shall write for them $ p_r(t,r)=\\omega _r \\, \\rho _{_{in}}(t,r), \\nonumber \\\\p_{_l}(t,r)=\\omega _{_l} \\, \\rho _{_{in}}(t,r),$ where $\\omega _r$ and $\\omega _{_l}$ are constant state parameters (note that in this case we have that $p_{_{\\theta }}=p_{_{\\phi }}=p_{_l}$ due to the spherical symmetry).",
"In the following, for solving the field equations, we shall require that $p_r(t,r) \\ne p_{_l}(t,r)$ .",
"By taking into account Eq.",
"(REF ), from Eq.",
"() we get $\\rho _{_{in}}(t,r)=F(t) \\, r^{2(\\omega _{_l}-\\omega _r)/\\omega _r},$ where $F(t)$ is an integration function, and introducing this expression into Eq.",
"() we have for the energy density of the anisotropic matter $\\rho _{_{in}}(t,r)=\\frac{C \\, r^{2(\\omega _{_l}-\\omega _r)/\\omega _r}}{a^{3+\\omega _r+2\\omega _{_l}}},$ where $C$ is an integration constant.",
"Now, by subtracting Eqs.",
"() and (), and using the full energy density (REF ), we obtain the differential equation $\\frac{\\kappa (\\omega _{_l}-\\omega _r)\\, C \\,r^{2(\\omega _{_l}-\\omega _r)/\\omega _r}}{a^{(3+\\omega _r+2\\omega _{_l})}}=\\frac{3b-rb^{\\prime }}{2a^2r^3}.$ It is straightforward to see that in order to have a solution for the shape function $b=b(r)$ we must impose the constraint $\\omega _r+2\\omega _{_l}+1=0$ on the state parameters $\\omega _r$ and $\\omega _{_l}$ , thus obtaining for the shape function $b(r)=C_3 r^3-\\kappa \\, C \\, \\omega _r \\, r^{-1/\\omega _r},$ where $C_3$ is a new integration constant.",
"Notice that the constraint (REF ) implies that the radial and tangential pressures are given by $p_r=\\omega _r \\rho _{_{in}}, \\,\\,\\, p_{_l}= -\\frac{1}{2} \\,(1+\\omega _r) \\rho _{_{in}},$ so the energy density and pressures satisfy the following relation: $\\rho _{_{in}}+p_r+2 p_{_l}=0.$ This equation implies that the inhomogeneous and anisotropic component satisfies the strong energy condition.",
"In this case for $\\omega _{_r} \\le -1$ , $-1 \\le \\omega _{_r} \\le 1$ and $\\omega _{_r} \\ge 1$ we have that $\\omega _{_l} \\ge 0$ , $-1 \\le \\omega _{_l} \\le 0$ and $\\omega _{_l} \\le -1$ respectively.",
"Now, from Eqs.",
"(REF ), (REF ), (REF ) and taking into account the constraint (REF ) we obtain the following master equation for the scale factor: $3 H^2=-\\frac{3C_3}{a^2}+ \\kappa \\rho + \\Lambda .$ Note that by taking into account the metric () we conclude that $C_3$ may be absorbed by rescaling the $r$ -coordinate as follows: $C_3=1$ for $C_3>0$ and $C_3=-1$ for $C_3<0$ , so without any loss of generality we can identify it with the spatial curvature parameter $k$ by putting $C_3=k$ , with $k=-1,0,1$ .",
"Thus the master equation (REF )) may be written in the form $ 3 H^2+\\frac{3k}{a^2} = \\kappa \\rho (t)+\\Lambda .$ Summarizing, we have shown that for the gravitational configuration $ds^2=dt^2-a(t)^2 \\times \\nonumber \\\\ \\left( \\frac{dr^2}{1-kr^2+\\kappa \\, C \\, \\omega _r\\, r^{-1-1/\\omega _r}}+ r^2 (d\\theta ^2+sin^2 \\theta d \\varphi ^2)\\right), \\nonumber \\\\$ filled with the inhomogeneous cosmic fluid $\\rho _{_{in}}(t,r)=\\frac{C \\, r^{-3-1/\\omega _r}}{ a^{2}(t)},$ (with anisotropic pressures $p_r=\\omega _r \\rho _{_{in}}$ and $p_l=-\\frac{1}{2}(1+\\omega _r) \\rho _{_{in}}$ ) and another noninteracting arbitrary homogeneous and isotropic $\\rho (t)$ , the evolution of the scale factor $a(t)$ is governed by the standard Friedmann equation (REF ) and the conservation equation (REF ).",
"In conclusion, the rate of expansion of these evolving wormholes is only determined by the matter component $\\rho (t)$ which may be in principle an ideal barotropic fluid, a scalar field or any other isotropic and homogeneous cosmic fluid considered in literature.",
"Notice that if $\\rho (t)=\\Lambda =0$ the space expands with constant velocity [11], and for $\\rho (t)=0$ and $\\Lambda \\ne 0$ the rate of expansion is determined by the cosmological constant [12].",
"It is worth noticing that for $\\omega _r=-1/3$ we have a limiting case since the 3-space becomes isotropic and homogeneous and the anisotropic matter behaves as an ideal string gas (i.e.",
"$p_r=p_{_l}=-\\rho _{_{in}}(t)/3$ ).",
"From Eq.",
"(REF ) we conclude that in this case the matter component behaves as $\\rho _{_{in}}=C/a^2(t)$ , implying that we have a FRW cosmology filled with a mixture of a curvature fluid with a cosmic fluid $\\rho (t)$ ."
],
[
"Some remarks on the geometry of the space-time",
"It is clear that the gravitational configuration () is sustained via a matter source made of the inhomogeneous and anisotropic cosmic fluid (REF ).",
"Let us point out some properties of the discussed geometry.",
"In general the metric (REF ) is not conformally flat since the Weyl tensor does not vanish for this metric, except for $C=0$ or $C\\ne 0$ and $\\omega _r=-1/3$ .",
"On the other hand, this acceleration-free space-time ($g_{tt}=1$ ) is characterized by zero anisotropic stress $\\sigma _{\\alpha \\beta }(t,r)$ and zero heat-flux vector $q_{\\alpha }$ .",
"Note that the metric (REF ) is conformal to the following static metric: $ds^2=d\\tau ^2- \\left( \\frac{dr^2}{1-k r^2+\\kappa \\, C \\, \\omega _r \\,r^{-1-1/\\omega _r}}+ r^2 d \\Omega ^2 \\right), \\hspace{8.5359pt}$ where $d \\Omega ^2=d\\theta ^2+sin^2 \\theta d \\varphi ^2$ and $\\tau =\\int dt/a(t)$ is the conformal time.",
"In general the component $g_{rr}^{-1}$ of the metric (REF ) may be valid for all $r>0$ or vanish for some value of the radial coordinate $r_0>0$ ; however this does not mean that this space-time contains an event horizon at $r_0$ since $g_{\\tau \\tau }=1$ .",
"So in principle, for some sets of the model parameters, this space-time may contain a naked singularity at $r=0$ which may be observable from the outside.",
"In general this geometry admits three-dimensional slices $t=t_0=const$ with a variable curvature.",
"In this case the 3-curvature may be written as ${}^3R=-a_0^{-2}\\left(6k+2\\kappa C r^{-3-\\frac{1}{\\omega _r}}\\right),$ where $a_0=a(t_0)$ .",
"If $\\omega _r<-1/3$ or $\\omega _r>0$ these slices are asymptotically flat for $k=0$ , asymptotically de-Sitter for $k=1$ or asymptotically anti de-Sitter for $k=-1$ .",
"Note that for these ranges of $\\omega _r$ there may arise a naked singularity at $r=0$ , and on the other hand the energy density of the inhomogeneous matter component vanishes as $r \\rightarrow \\infty $ since from Eq.",
"(REF ) we have that $\\rho _{_{in}}(t_0,r) \\rightarrow 0$ .",
"It is remarkable that the 3-dimensional slices $t=t_0$ of metric (REF ) include as a particular case the 3-dimensional slices $t=t_0$ of the Kottler metric $ds^2=\\left(1-\\frac{2M}{r}-\\frac{\\Lambda }{3} r^2 \\right) dt^2-\\frac{dr^2}{1-\\frac{2M}{r}-\\frac{\\Lambda }{3} r^2}+ r^2 d \\Omega ^2,\\nonumber \\\\$ which includes the de-Sitter ($\\Lambda >0$ ) and anti de-Sitter ($\\Lambda <0$ ) space-times.",
"The 3-curvature of its slices $t=t_0$ is given by ${}^3R=-2\\Lambda $ , so they are also spaces of constant curvature.",
"By comparing metrics (REF ) and (REF ) we conclude that Kottler slices $t=t_0$ are obtained from slices of metric (REF ) by putting $a_0^{-2}k=\\Lambda /3$ , $\\kappa C \\omega _r=-2M$ and $\\omega _r\\rightarrow \\pm \\infty $ .",
"From Eq.",
"(REF ) we see that ${}^3R a_0^{2}=-6k-2\\kappa C r^{-3-\\frac{1}{\\omega _r}} \\equiv \\nonumber \\\\-6k-2\\frac{\\kappa C \\omega _r}{\\omega _r} \\, r^{-3-\\frac{1}{\\omega _r}}\\longrightarrow -6k,$ for $\\omega _r \\rightarrow \\pm \\infty $ , thus effectively we have in this case a space of constant curvature.",
"It can be shown that for this limit the inhomogeneous energy density (REF ) vanishes since $\\rho _{_{in}}(t,r)=\\frac{C \\, r^{-3-1/\\omega _r}}{ a^{2}(t)} \\equiv \\frac{C \\omega _r \\, r^{-3-1/\\omega _r}}{ \\omega _r a^{2}(t)}\\longrightarrow 0$ for $\\omega _r \\rightarrow \\pm \\infty $ , while the anisotropic pressures take the forms $p_r(t,r) \\rightarrow \\frac{\\omega _rC}{a^2(t) r^3}$ and $p_{_l}(t,r) \\rightarrow \\frac{\\omega _rC}{2a^2(t) r^3}$ .",
"Unfortunately this model with a vanishing inhomogeneous and anisotropic energy density $\\rho _{_{in}}(t,r)$ and non-vanishing pressures $p_r(t,r)$ and $p_{_l}(t,r)$ is non-physical, so we rule it out from consideration.",
"If $-1/3<\\omega _r<0$ the curvature increases with radius and as $r\\rightarrow \\infty $ we have the asymptotic forms ${}^3R \\sim r^\\alpha $ , $\\rho _{_{in}} \\sim r^\\alpha $ and $f^2(r) \\sim r^\\beta $ with $\\alpha >0$ and $\\beta >2$ respectively.",
"These gravitational configurations do not contain any singularity at $r=0$ for slices $t=t_0$ ."
],
[
"Expanding Wormhole Universes",
"Now we shall study gravitational configurations where the considered solution represents an expanding wormhole geometry [4], [5].",
"Before treating Lorentzian wormhole geometries in more detail, let us note again that the metric ansatz (REF ) provides an explicit class of dynamic wormholes that generalize the static, spherically symmetric ones first considered by Morris and Thorne [4].",
"Several other aspects of static and evolving wormhole spacetimes are analyzed in Refs.",
"[1] and [2] respectively.",
"In order to have a wormhole geometry the functions $\\Phi (r)$ and $b(r)$ must satisfy some constraints defined by the authors of Ref. [4].",
"However, these authors originally were interested only in wormhole geometries featuring two asymptotically flat regions connected by a bridge.",
"In our case we are interested in a more general asymptotic behavior.",
"Due to the presence of the constant $k=-1,0,1$ ; besides the asymptotically flat wormholes ($k=0$ ) we may have anti–de Sitter ($k=-1$ ) asymptotic wormholes which also may be of particular interest [13].",
"Thus the main constraints may be defined as follows: Constraint 1: A no–horizon condition, i.e.",
"$e^{\\Phi (r)}$ is finite throughout the space–time in order to ensure the absence of horizons and singularities.",
"Constraint 2: The shape function $b(r)$ must obey at the throat $r =r_0$ the following condition: $b(r_0) = r_0$ , being $r_0$ the minimum value of the $r$ –coordinate.",
"In other words $g^{-1}_{rr}(r_0)=0$ .",
"Constraint 3: Finiteness of the proper radial distance $l(r)=\\pm \\int ^r_{r_0} \\frac{dr}{\\sqrt{1-b(r)/r}}$ for $r \\ge r_0$ throughout the space–time.",
"The $\\pm $ signs refer to the two regions which are connected by the wormhole.",
"Let us now consider the possibility of having a wormhole geometry.",
"From the metric (REF ) we conclude that $e^{\\Phi (r)}=1$ , thus the first constraint is automatically fulfilled.",
"It must be remarked that, for a general dynamical wormhole, the definition of the location of a wormhole throat is not straightforward.",
"In this case, the position of the wormhole throat depends on the time-slicing, and for a time-dependent wormhole it may not be possible to locate the entire throat within one time slice, as the dynamic throat is an extended object in spacetime [14].",
"In our case, the evolving metric (REF ) is conformal to the static spacetime (REF ), which represents a static wormhole for $\\omega _r < -1$ or $\\omega _r>0$ , and then we always may determine the location of the wormhole throat on the hypersurface with $t=t_0=const$ .",
"In general, for time-dependent spherically symmetric wormhole spacetimes, alternative definitions of a wormhole throat are required, equally valid for static as well as for dynamical wormholes.",
"Several different definitions have been given in Refs.",
"[6], [14], [15], [16].",
"For example Hochberg and Visser [14] and Hayward [16] have introduced two independent quasilocal definitions of a throat for dynamical wormholes.",
"These authors do not consider global properties of the wormholes, i.e.",
"they make no assumptions about symmetries, asymptotic flatness, topology, etc., and the wormhole throat is a two-dimensional surface of nonvanishing minimal area on a null hypersurface.",
"On the other hand, in the Ref.",
"[15] the authors have defined a wormhole throat quasilocally in terms of a surface of nonvanishing minimal area on a spacelike hypersurface.",
"Some properties of these three definitions are compared in [15].",
"From the second constraint, $g^{-1}_{rr}(r_0)=0$ , we can find the minimum value $r_0$ of the radial coordinate where the wormhole throat must be located.",
"From this throat condition, and by taking into account the metric (REF ), we obtain for the integration constant $C=\\frac{(kr^2_0-1)}{\\kappa \\omega _r} \\, r_0^{(1+\\omega _r)/\\omega _r},$ yielding for the shape function $b(r)$ and the metric component $g_{rr}$ $b(r)=r_0\\left(\\frac{r}{r_0}\\right)^{-1/\\omega _r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\nonumber \\\\ +kr^3_0 \\left(\\frac{r}{r_0}\\right)^3\\left(1-\\left(\\frac{r}{r_0}\\right)^{-(1+3\\omega _r)/\\omega _r}\\right),\\nonumber \\\\ a^2(t) g_{rr}^{-1}= 1-\\left(\\frac{r}{r_0}\\right)^{-(1+\\omega _r)/\\omega _r}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\nonumber \\\\ -kr^2_0 \\left(\\frac{r}{r_0}\\right)^2\\left(1-\\left(\\frac{r}{r_0}\\right)^{-(1+3\\omega _r)/\\omega _r}\\right),$ respectively.",
"It is easy to verify that the wormhole throat is located at $r_0$ since $b(r_0)=r_0$ .",
"In this case the energy density of the matter threading the wormhole takes the following form: $\\kappa \\rho _{_{in}}(t,r)=\\frac{kr^2_0-1}{r^2_0 \\omega _r a(t)^2}\\left(\\frac{r}{r_0}\\right)^{-(1+3\\omega _r)/\\omega _r}.$ Clearly, in order to have an evolving wormhole we must require $\\omega _r < -1$ or $\\omega _r >0$ (in both of these cases, in the $g_{rr}$ metric component, $(1+\\omega _r)/\\omega _r>0$ and $(1+3\\omega _r)/\\omega _r>0$ ), implying that the inhomogeneous and anisotropic cosmic fluid (REF ) can support the existence of evolving wormholes.",
"It can be shown that the form of the wormhole is preserved during all evolution.",
"Notice that for an anisotropic matter with $\\omega _r<-1$ we have that $\\rho _{_{in}}(t,r)>0$ , while for an anisotropic matter with $\\omega _r>0$ we have that $\\rho _{_{in}}(t,r)<0$ .",
"For $\\rho (t)=\\Lambda =0$ the shape of the wormhole expands with constant velocity [11].",
"In the presence of a cosmological constant with $\\rho _{_{in}}(t,r)=0$ the wormhole configurations have an accelerated expansion (contraction) [12].",
"Other properties of such evolving wormholes are discussed by authors of the Ref.",
"[11], [12] and references cited therein.",
"As explicit examples of evolving wormholes, let us first consider the case where the isotropic and homogeneous component is given by a perfect fluid with the barotropic state equation $p(t)=\\omega \\rho (t)$ and $k=\\Lambda =0$ .",
"Thus the scale factor is given by $a(t)=a_0 \\, t^{2/3(1+\\omega )}$ and the energy density by $\\rho (t)={\\frac{4}{3\\kappa \\left( 1+\\omega \\right) ^{2}\\,{t}^{2}}},$ while the metric takes the form $ds^2=dt^2-a_0^2 \\, t^{4/3(1+\\omega )} \\times \\nonumber \\\\ \\left(\\frac{dr^2}{1-\\left(\\frac{r}{r_0}\\right)^{-(1+\\omega _r)/\\omega _r}}+ r^2 (d\\theta ^2+sin^2 \\theta d \\varphi ^2)\\right).",
"\\nonumber \\\\$ In this case, the energy density of the anisotropic matter and its pressure components are given by $\\kappa \\rho _{_{in}}(t,r)=-\\frac{1}{\\omega _r r^2_0 a_0^2 \\,t^{4/3(1+\\omega )}}\\left(\\frac{r}{r_0}\\right)^{-(1+3\\omega _r)/\\omega _r}$ and (REF ) respectively.",
"Note that if $\\omega _r<-1$ or $\\omega _r>0$ we have asymptotically flat FRW regions for $r\\longrightarrow \\infty $ and $\\omega >-1$ .",
"Let us suppose that the isotropic fluid satisfies the dominant energy condition, i.e.",
"$ |p|\\le \\rho $ , $\\rho \\ge 0$ .",
"Thus if $-1 \\le \\omega < -1/3$ the expansion of the evolving wormhole is accelerated, i.e.",
"both universes and the throat of the wormhole are simultaneously expanding with acceleration, while for $-1/3<\\omega \\le 1$ the expansion is decelerated.",
"On the other hand, the total matter content is given by $\\rho _{_{total}}= \\rho (t)+\\rho _{_{in}}(t,r)$ and for any $\\omega _r<0$ we have that $\\rho _{_{total}} \\ge 0$ .",
"For $\\omega _r>0$ we can have in general time intervals where the total energy is positive or negative.",
"For $\\omega >-1/3$ the wormhole model starts with a positive total energy density (since for a fixed $r=const$ the isotropic component dominates over the another one), then decreases till zero at certain $t_{eq}$ , and becomes negative for $t> t_{eq}$ .",
"For $\\omega < -1/3$ the total energy density starts negative, then increases till zero at certain $t_{eq}$ , and becomes positive for $t> t_{eq}$ .",
"The case $\\omega =-1/3$ is more interesting, since it allows us to consider evolving wormhole models satisfying the dominant energy condition (DEC) in the whole spacetime.",
"In this case the total energy density and the corresponding total pressure components are given by $\\rho _{_T}=\\rho +\\rho _{in}=\\left( 3-\\frac{(r/r_0)^{-\\frac{1+3\\omega _r}{\\omega _r}}}{\\omega _r r_0^2 a_0^2} \\right) t^{-2}, \\nonumber \\\\p_{_{r,T}}=-\\frac{1}{3} \\rho +\\omega _r \\rho _{in}, \\nonumber \\\\p_{_{l,T}}=-\\frac{1}{3} \\rho -\\frac{1}{2}(1+\\omega _r) \\rho _{in}.$ From this expressions we conclude that the total matter content satisfies the strong energy condition since $\\rho _{_T}+p_{_{r,T}}+2p_{_{l,T}} \\equiv 0$ .",
"In order to fulfill the DEC we need to satisfy the following conditions: $\\rho _{_T} \\ge 0$ , $\\rho _{_T} + p_{_{r,T}} \\ge 0$ , $\\rho _{_T} - p_{_{r,T}} \\ge 0$ , $\\rho _{_T} + p_{_{l,T}} \\ge 0 $ and $\\rho _{_T} - p_{_{l,T}} \\ge 0$ .",
"These conditions imply that the following constraints must be satisfied: $\\frac{1}{3 \\omega _r } \\le a_0^2 r_0^2, \\\\\\frac{1+\\omega _r}{2 \\omega _r } \\le a_0^2 r_0^2, \\\\\\frac{1-\\omega _r}{4 \\omega _r } \\le a_0^2 r_0^2, \\\\\\frac{3+\\omega _r}{8 \\omega _r } \\le a_0^2 r_0^2.",
"$ Hence, for any given $\\omega _r > 0$ , the DEC is fulfilled by choosing the parameters $a_0^2$ and $r_0^2$ satisfying the conditions (REF )-().",
"Note that in this case the isotropic fluid gives a scale factor given by $a(t)=t$ .",
"Thus, the class of analytic two-fluid evolving wormholes with $\\omega =-1/3$ satisfies the dominant and strong energy conditions in the whole spacetime and expands with constant velocity.",
"It is interesting to note that the violation or not of the NEC, at and near the throat, of a dynamical wormhole, may be properly connected to the generalizations of the flare-out condition for an arbitrary wormhole discussed by above cited authors of the Refs.",
"[14], [15], [16].",
"In the definitions of [14], [16] dynamical wormhole throats are trapping horizons, i.e.",
"hypersurfaces foliated by “marginally trapped surfaces\", and the violation of the NEC is a generic property of such wormhole throats.",
"While in the Ref.",
"[15] a dynamical spherically symmetric wormhole throat is defined as a “trapped sphere\", and the NEC can still be satisfied for some wormhole configurations.",
"As a second example, we shall consider an evolving wormhole with $k=0$ and filled with a minimally coupled scalar field with the exponential potential $V(\\phi )=V_0 e^{-\\lambda \\phi }$ , where $\\lambda $ and $V_0$ are constant parameters.",
"In this case the energy density and the pressure of the homogeneous and isotropic component are given by $\\rho _{\\phi }=\\dot{\\phi }^2/2+V(\\phi )$ and $p_{\\phi }=\\dot{\\phi }^2/2-V(\\phi )$ respectively.",
"Thus Eqs.",
"(REF ) and (REF ) imply that $\\ddot{\\phi }(t) + 3H \\dot{\\phi } - \\lambda V_0 e^{-\\lambda \\phi }=0, \\\\3 H^2=\\kappa \\left(\\frac{\\dot{\\phi }^2}{2}+V(\\phi ) \\right).$ A particular exact solution, describing the power-law expansion $a(t)=a_0 t^p$ of the evolving wormhole, is given by $\\phi (t)=\\frac{2}{\\lambda } \\ln \\, t,$ where $\\lambda ^2=\\frac{2 \\kappa }{p}, V_0=\\frac{p(3p-1)}{\\kappa }.$ The first relation implies that $p>0$ , and from the second one we have that $V_0<0$ if $0<p<1/3$ and $V_0>0$ if $p>1/3$ .",
"Finally, as a last example we shall consider evolving wormholes with $k=0$ and filled with a tachyon field giving the power-law expansion (REF ), where $p$ is a constant parameter.",
"In this case the energy density and the pressure of the homogeneous and isotropic component are given by $\\rho _{_T}=\\frac{V(\\phi )}{\\sqrt{1-\\dot{\\phi }^2}}$ and $p_{_T}=-V(\\phi ) \\sqrt{1-\\dot{\\phi }^2}$ respectively.",
"Thus Eqs.",
"(REF ) and (REF ) imply that $\\frac{\\ddot{\\phi }(t)}{1-\\dot{\\phi }^2} + 3H \\dot{\\phi } +\\frac{1}{V(\\phi )} \\frac{dV(\\phi )}{d\\phi }=0, \\\\3 H^2=\\frac{\\kappa V(\\phi )}{\\sqrt{1-\\dot{\\phi }^2}}.$ It can be shown that in order to have the power-law expansion (REF ) the tachyon potential takes the form $V(\\phi )=\\alpha \\phi ^{-2}$ , where $\\phi (t)=\\phi _0 t$ , $\\phi _0=\\sqrt{2/3p}$ and $\\kappa \\alpha = 2p \\sqrt{1-2/(3p)}$ .",
"Notice that for discussed expanding wormholes, filled with a scalar and tachyon fields, the wormhole geometry is given by $ds^2=dt^2-a_0^2 \\, t^{2p} \\times \\nonumber \\\\ \\left( \\frac{dr^2}{1-\\left(\\frac{r}{r_0}\\right)^{-(1+\\omega _r)/\\omega _r}}+ r^2 (d\\theta ^2+sin^2 \\theta d \\varphi ^2)\\right).",
"\\nonumber \\\\$ In this case, the energy density of the anisotropic component and its pressure are given by $\\kappa \\rho _{_{in}}(t,r)=-\\frac{1}{\\omega _r r^2_0 a_0^2 \\, t^{ 2p}}\\left(\\frac{r}{r_0}\\right)^{-(1+3\\omega _r)/\\omega _r}$ and (REF ) respectively."
],
[
"Conclusions",
"We have developed models for evolving wormholes sustained by two cosmic fluids: one with homogeneous and isotropic properties and another inhomogeneous and anisotropic.",
"It is remarkable that the energy density of the matter threading and sustaining such a wormhole is the inhomogeneous and anisotropic component, while the rate of expansion of the evolving wormhole is determined by the isotropic and homogeneous component.",
"This matter component may be in principle an ideal barotropic fluid, a scalar field or any other cosmic fluid satisfying the homogeneity and isotropy requirements.",
"For the case where the cosmological constant and the isotropic and homogenous component are absent the inhomogeneous space expands with constant velocity, and when only the isotropic and homogenous component is absent the rate of expansion is determined by the cosmological constant.",
"In general, we have wormhole universes for $\\omega _{r}<-1$ or $\\omega _{r}>0$ .",
"The present results generalize our previous works [11] and [12].",
"The case when the studied inhomogeneous geometry represents wormhole configurations expanding with constant velocity, i.e.",
"$\\omega _{r}<-1$ or $\\omega _{r}>0$ and $\\rho (t)=\\Lambda =0$ , was discussed in Ref.",
"[11], while the scenarios where the expansion rate is determined by the cosmological constant, i.e.",
"$\\rho (t)=0$ , were discussed in Ref. [12].",
"If now $\\rho (t) \\ne 0$ and $\\Lambda =0$ , the expansion rate of the wormhole is determined by this isotropic and homogeneous fluid.",
"It is interesting to note that the results of Ref.",
"[12] were generalized to the case of evolving wormholes sustained by a single inhomogeneous and anisotropic fluid $\\varrho (t,r)$ , by imposing the generalized equation of state $\\varrho +\\alpha P_r+2 \\beta P_t=0$ , where $\\alpha $ and $\\beta $ are constant parameters, and $P_r$ and $P_t$ are the radial and transverse pressures, respectively [19].",
"All particular wormhole solutions discussed in this Ref.",
"are related to solutions reported in Ref. [12].",
"Note that the wormhole geometries (REF )-(REF ) far from the throat look like a flat FRW Universe.",
"At first glance, if the wormhole throat is located outside of the cosmological horizon of any observer, then he is not in causal contact with the throat.",
"Thus, for late times, an observer in this wormhole Universe located too far from the wormhole throat will see the Universe isotropic and homogeneous and it will be in principle unable for him to make a decision about whether he lives in a space of constant curvature or in a space of a wormhole spacetime.",
"An interesting feature of the discussed here solutions is that one can consider evolving wormholes with positive total energy density, i.e.",
"$\\rho _{_{total}}=\\rho (t)+\\rho _{_{in}}(t,r)>0$ .",
"We always are free to consider positive homogeneous and isotropic energy density $\\rho (t)$ .",
"In order to have positive $\\rho _{_{in}}(t,r)$ we must choose $\\omega _{_r} \\le -1$ .",
"In this case the inhomogeneous and anisotropic matter component sustaining the evolving wormhole may be considered a generalization of the hypothetical phantom energy used in cosmology in order to explain accelerated expansion of the Universe.",
"Effectively, this cosmic phantom source is characterized by a positive homogeneous energy density, i.e.",
"$\\rho _{_{DE}}(t)>0$ , and by an isotropic pressure satisfying $p_{_{DE}}(t)<-\\rho _{_{DE}}(t)$ .",
"Clearly for $p_{_{DE}}= \\gamma \\rho _{_{DE}}$ we have that $\\gamma <-1$ .",
"In our case the phantom energy is realized by the inhomogeneous matter component $\\rho _{_{in}}(t,r)>0$ with linear but highly anisotropic equation of state $p_{_{r}}<-\\rho _{_{in}}<0$ , $p_{_l}>0$ .",
"Spherically symmetric distribution of phantom energy, depending only on the radial coordinate $r$ , with such linear equation of state for the radial pressure were considered in Refs.",
"[17], where the authors constructed static wormholes sustained by a positive energy density $\\rho (r)>0$ .",
"Lastly, let us note that, to our knowledge, the results described in this article, for evolving lorentzian wormholes, with energy-momentum tensor associated with a mixture of one isotropic and homogeneous fluid with an inhomogeneous and anisotropic component, are firstly reported here.",
"There are in the literature many works reporting on evolving wormhole solutions to the Einstein field equations, however, a large number of these papers discuss dynamic wormhole solutions with a single fluid source [2], [8], [18], [19].",
"It is worth to mention here that a general class of higher evolving dimensional wormholes sustained by a single fluid was studied in Ref. [20].",
"Most specifically, it was considered a quasi-static spherically symmetric evolving wormholes, with static four non-compact dimensions and an arbitrary number of extra time-dependent compact dimensions.",
"The results of the study show that the WEC cannot be satisfied at the throat.",
"This is mainly due to that the matter content is not distributed in the whole space and is matched to the vacuum.",
"The presence of this matter-vacuum boundary places restrictions on the time dependence of extra compact dimensions, consequently implying the violation of the WEC.",
"On the contrary, as shown in the previous Sec., we can have evolving wormhole configurations satisfying the WEC.",
"On the other hand, with respect to wormholes involving a mixture of two fluids, a dynamical wormhole, filled with a perfect fluid and a ghost scalar field, is provided in Ref. [15].",
"The considered in this Ref.",
"dynamical wormhole metric (4.53), $ds^2=dt^2-a^2(t) \\left( dx^2-(x^2+\\tilde{b}^2) d \\Omega ^2 \\right),$ may be rewritten, by using the transformation $x^2=r^2-\\tilde{b}^2$ , as $ds^2=dt^2-a^2(t) \\left( \\frac{dr^2}{1-\\tilde{b}^2/r^2}-r^2 d\\Omega ^2 \\right),$ where $\\tilde{b}$ is a constant parameter and $a(t)=t/t_0$ .",
"Thus the solution (4.58)-(4.60) of the Ref.",
"[15] is a particular case of the discussed in this paper wormhole geometries ()-(), with $\\omega =-1/3$ , $\\omega _r=1$ , $r_0=\\tilde{b}$ and $a_0=1/t_0$ .",
"In this case the anisotropic and inhomogeneous component with $\\omega _r=1$ may be identified as a massless ghost scalar field (note that the energy density () becomes negative).",
"The conditions (REF )-() imply the inequalities $a_0^2r_0^2 \\ge 1/3$ , $a_0^2r_0^2 \\ge 1$ , $a_0^2r_0^2 \\ge 0$ and $a_0^2r_0^2 \\ge 1/2$ respectively.",
"Hence, for this particular solution the DEC is satisfied in the whole space for $a_0 r_0 \\ge 1$ .",
"Notice that the wormhole throat definitions of Hochberg-Visser [14] or Hayward [16] do not apply to this evolving wormhole since the whole spacetime is foliated by trapped surfaces and there is no trapping horizon [15].",
"As far as we know, there is only one more paper where two-fluid evolving lorentzian wormholes are considered.",
"In Ref.",
"[21] FRW models with a traversable wormhole are considered.",
"In this case the matter content is divided into two parts: the cosmic part $\\rho $ that depends on cosmological time only, and the wormhole part $\\rho _w$ that depends on the radial coordinate only.",
"This wormholes can be finally connected with the particular wormholes solutions (REF )-(REF )."
],
[
"Acknowledgements",
"This work was supported by CONICYT through Grants FONDECYT N$^0$ 1080530 and 1110230 (MC, SdC), and by Dirección de Investigación de la Universidad del Bío-Bío (MC).",
"SdC also was supported by PUCV grant N$^0$ 123.710/2011."
]
] | 1204.0753 |
[
[
"The volume of K\\\"ahler-Einstein Fano varieties and convex bodies"
],
[
"Abstract We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points.",
"The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes.",
"The case of spherical varieties/multiplicity free symplectic manifolds is also discussed.",
"The proof uses Moser-Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in C^n of independent interest.",
"The paper supersedes our previous preprint concerning the case of toric Fano manifolds."
],
[
"Complex geometry",
"Let $X$ be an $n-$ dimensional complex manifold $X$ which is Fano, i.e.",
"its first Chern class $c_{1}(X)$ is ample (positive) and in particular $X$ is a projective algebraic variety.",
"For some time it was expected that the top-intersection number $c_{1}(X)^{n},$ also called the (anti-canonical) degree of $X,$ is maximal for the $n-$ dimensional complex projective space, i.e.",
"$c_{1}(X)^{n}\\le (n+1)^{n},$ There are now counterexamples to this bound.",
"For example, as shown by Debarre (see page 139 in [16]), even in the case when $X$ is toric (i.e.",
"$X$ admits an effective holomorphic action of the complex torus $({C}^{*})^{n}$ with an open dense orbit) there is no universal polynomial upper bound on the $n-$ th root of the degree of $X.$ A more recent conjecture says that the bound above holds for any Fano manifold whose Picard number is one [37].",
"Given the special role of KÀhler-Einstein metrics in complex geometry - in particular in connection to Chern number inequalities [59] - it is also natural to ask if the bound above holds for any Fano manifold admitting a KÀhler-Einstein metric $\\omega $ ?",
"Then $c_{1}(X)^{n}/n!$ is the volume of $X$ in the metric $\\omega .$ One step in this direction was taken by Gauntlett-Martelli-Sparks-Yau [27], who showed, using Bishop's volume inequality, that if $X$ is a Fano KÀhler-Einstein manifold then the inequality REF holds when the right hand side is multiplied by $(n+1)/I(X),$ where $I(X)$ is the Fano index of $X,$ i.e.",
"the largest positive integer $I$ such that $c_{1}(X)/I$ is an integral class in the Picard group of $X.$ As is well-known $I(X)\\le n+1$ with equality precisely for $X={P}^{n}$ and hence the latter result leaves the question of the maximization property of ${P}^{n}$ open.",
"The main result in this paper shows that the inequality REF indeed holds for KÀhler-Einstein Fano manifolds in the presence of a certain amount of symmetry: Theorem 1.1 Let $X$ be a Fano manifold which admits a KÀhler-Einstein metric and a holomorphic ${C}^{*}-$ action with a finite number of fixed points.",
"Then the first Chern class $c_{1}(X)$ satisfies the following upper bound $c_{1}(X)^{n}\\le (n+1)^{n}$ In other words, the complex projective space ${P}^{n}$ has maximal degree among all Fano manifolds $X$ as above.",
"The starting point of the proof of the theorem is the fact that, under the assumptions in the theorem, there is a holomorphic $S^{1}-$ action on $X,$ preserving the KÀhler-Einstein metric and with an attractive fixed point $p.$ The key point of the proof, which builds on our previous work, is then to study $S^{1}-$ invariant Moser-Trudinger type inequalities in a sufficently large $S^{1}-$ invariant Stein domain $\\Omega $ in $X$ containing the fixed point $p$ (compare section REF below).",
"The simplest class of varieties in which the assumption in the previous theorem are satisfied is the class of (generalized) flag varieties, i.e.",
"rational $G-$ homogeneous spaces.",
"As is well-known these are all Fano manifolds and by homogeneity they also carry KÀhler-Einstein metrics, invariant under the maximal compact subgroup $K$ of $G.$ In this case the bound in the previous theorem was first obtained by Snow [54], using representation theory and quite elaborate calculcations.",
"More generally, the previous theorem applies to any Fano manifold $X$ on which a reductive connected complex algebraic group $G$ (i.e.",
"$G$ is the complexification of compact Lie group $K)$ acts algebraically with finitely many orbits (see Remark REF ).",
"A particularly rich class of such $G-$ varieties is given by spherical varieties (i.e.",
"a Borel subgroup $B$ of $G$ has an open dense orbit in $X)$ [43], [11], [10].",
"In case a spherical variety is Fano it may or may not admit a KÀhler-Einstein metrics and the inequality in the previous theorem can hence be viewed as a new obstruction for the existence of a KÀhler-Einstein metric on spherical Fano varities.",
"According to a formula of Brion [11] the top-intersection number $c_{1}(L)$ of a polarized spherical variety $(X,L)$ can be expressed as an explicit integral over a certain polytope $P$ naturally associated to $X.$ We will recall the symplecto-geometric description of Brion's formula in section .",
"Let us also point out that an interesting classical subclass of spherical varieties is offered by Schubert varieties and it is well-known that any smooth Schubert variety in a Grassmannian is Fano [58].",
"Another rich subclass is given by $G-$ equivariant compactifications of symmetric spaces and in particular the so called wonderful compactifications, which are often Fano [53].",
"In fact, smooth wonderful compactifications are always weakly Fano, i.e.$-K_{X}$ is nef and big [53] and, in fact, Theorem REF is still valid when $X$ is merely weakly Fano if one uses the notion of (singular) KÀhler-Einstein metrics introduced in [7] (see Remark REF ).",
"Of course, in Theorem REF it is enough to assume that $X$ can be deformed to a complex manifold satisfying the assumptions in the theorem.",
"Moreover, in the absense of a KÀhler-Einstein metric we show that the inequality in the Theorem REF still holds when the right hand side is multiplied by $1/R(X)^{n},$ where $R(X)$ is the greatest lower bound on the Ricci curvature of $X$ (see Theorem REF )."
],
[
"Toric Fano varieties",
"In the special spherical case when $X$ is a toric manifold, i.e.",
"the groups $G$ and $B$ both coincide with the complex torus ${C}^{*n},$ the inequality in Theorem REF was conjectured to hold by Nill-Paffenholz [47].",
"We expect that the previous theorem can be extended to singular spherical Fano varieties admitting (singular) KÀhler-Einstein metrics, but we will only show this for toric varieties.",
"First recall that, by definition, $X$ is a Fano variety if $K_{X}$ is an ample ${Q}-$ line bundle and $\\omega $ is a singular KÀhler-Einstein metric on $X$ if its is a bona fide KÀhler-Einstein metric on the regular locus of $X$ such that $\\omega $ extends to a global current in $c_{1}(X)\\in H^{2}(X,{Q})$ with continuous local potentials (see [7]).",
"Theorem 1.2 Let $X$ be an $n-$ dimensional toric Fano variety which admits a (singular) KÀhler-Einstein metric.",
"Then its first Chern class $c_{1}(X)$ satisfies the following upper bound which is attained when $X$ is the complex projective space ${P}^{n}:$ $c_{1}(X)^{n}\\le (n+1)^{n}$ The universal bound in the previous theorem should be contrasted with the well-known fact that the volume of a general Fano variety $X$ of a fixed dimension $n\\ge 2$ can be arbitrarily large unless conditions on the singularities of $X$ are imposed (see [24] and references therein and example 4 in [17] for a simple toric example).",
"According to the fundamental Yau-Tian-Donaldson conjecture in KÀhler geometry the existence of a KÀhler-Einstien metric on a Fano manifold $X$ is equivalent to $X$ being K-stable (see the recent survey [50]).",
"This notion of stability is of an algebro-geometric nature.",
"The case of toric Fano manifolds was settled by Wang-Zhou [57], who more precisely showed that a toric Fano manifold $X$ admits a KÀhler-Einstein metric precisely when 0 is the barycenter of the canonical lattice polytope $P_{X}$ associated to $X$ (see below).",
"In the paper [6] we extend the result of Wang-Zhou to the setting of general (possibly singular) Fano varieties: Theorem 1.3 [6] Let $X$ be an $n-$ dimensional toric Fano variety.",
"Then the following is equivalent: $X$ admits a (singular) KÀhler-Einstein metric The barycenter is the unique interior lattice point of the polytope $P_{X}$ associated to $X.$ More generally, the result is shown to hold in the setting of toric log Fano varieties $(X,\\Delta )$ familiar from the Minimal Model Program (MMP), i.e.",
"$X$ is a toric variety and $\\Delta $ is a torus invariant ${Q}-$ divisor on $X$ with coefficents $<1$ such that the anti-canonical divisor $-(K_{X}+\\Delta )$ of $(X,\\Delta )$ defines an ample ${Q}-$ line bundle on $X.$ In this general setting Theorem REF holds for the log first Chern class $c_{1}(-(K_{X}+\\Delta ))$ of $(X,\\Delta )$ if the coefficients of are positive $\\Delta $ and Theorem REF holds for any toric log Fano variety $(X,\\Delta ).$ The barycenter condition for the existence of a KÀhler-Einstein metric on a toric variety is the link to Ehrhart's volume conjecture in convex geometry, to which we next turn."
],
[
"Convex geometry",
"There is a well-known dictionary relating toric polarized varieties $(X,L)$ and rational polytopes $P$ [17], [21], [15].",
"In particular, the top intersection number $c_{1}(L)^{n}$ coincides with $n!$ times the volume of the corresponding polytope $P.$ As pointed out in [47] one of the motivations for the bound REF on $c_{1}(X)^{n}$ in the toric setting is another more general conjecture of Ehrhart in the realm of convex geometry, which can be seen as a variant of Minkowski’s first theorem for non-symmetric convex bodies: Conjecture (Ehrhart).",
"Let $P$ be an n-dimensional convex body which contains precisely one interior lattice point.",
"If the point coincides with the barycenter of $P$ then $\\mbox{Vol}(P)\\le \\frac{(n+1)^{n}}{n!",
"}$ The case when $n=2$ was settled by Ehrhart [22], as well as the special case of simplices in arbitrary dimensions [23].",
"As explained in the survey [28] the best upper bound in Ehrhart's conjecture, to this date, is $\\mbox{Vol}(P)\\le (n+1)^{n}/n^{n}.$ As we will next explain Theorem REF (or rather its more general Log version) confirms Ehrhart's conjecture for a large class of rational polytopes.",
"First recall that any rational polytope in ${R}^{n},$ containing zero in its interior, may be written uniquely as $P=\\lbrace p\\in {R}^{n}:\\,\\,\\left\\langle l_{F},p\\right\\rangle \\ge -a_{F}\\rbrace ,$ where the index $F$ ranges over the facets of $P,$ $a_{F}$ is a positive rational number and the vector $l_{F}$ is a primitive lattice vector, i.e.",
"it has integer coefficients with no common factors (geometrically, $l_{F}$ is the inward normal vector of the facet $F$ normalized with respect to the integer structure).",
"As is well-known [17] toric Fano varieties with $-K_{X}$ an ample ${Q}-$ line bundle correspond to rational polytopes $P$ as above with $a_{F}=1.$ More generally, toric log Fano varieties $(X,\\Delta )$ with $\\Delta $ an effective ${Q}-$ divisor on $X$ correspond to polytopes $P$ with $a_{F}\\le 1$ [15], [6].",
"Combining the Log version of Theorem REF above with the existence result for KÀhler-Einstein metrics on (possibly singular) toric varieties hence gives the following Corollary 1.4 The bound in the Ehrhart conjecture holds for all rational polytopes satisfing $a_{F}\\le 1$ in the representation REF and such that 0 is the barycenter of $P.$ In any polytope such that $a_{F}\\le 1$ the origin is indeed the unique lattice point (see the foot note on page 105 in [17]).",
"The class of such rational polytopes $P$ is vast and appears naturally both in algebraic geometry and in combinatorics.",
"For example, the dual (polar) $P=Q^{*}$ of any lattice polytope $Q$ containing 0 in its interior is in this class (see [26] for the combinatorics of such polytopes).",
"In particular, this shows that the bound in the Ehrhart conjecture holds for any reflexive convex lattice polytope (i.e.",
"a lattice polytope $P$ containing 0 such that its dual $Q$ is also a lattice polytope).",
"Such polytopes correspond to Gorenstein toric Fano varieties and were introduced and studied by Batyrev [3] in connection to mirror symmetry of pairs of Calabi-Yau manifolds.",
"In dimension $n\\le 8$ the Ehrhart conjecture up to $n\\le 8$ for reflexive Delzant polytopes (i.e.",
"those corresponding to smooth toric Fano varieties) has previously been confirmed by computer assistance (as announced in [47]), using the classification of such polytopes for $n\\le 8$ [60]).",
"As explained in section the arguments in the proof of Theorem REF and its Corollary above can be carried out directly in terms of convex analysis in ${R}^{n}$ without any reference to the corresponding toric variety.",
"In fact, it is enough to assume that $P$ is a convex body and one then obtains the following Theorem 1.5 Let $P$ be a convex body contained in the positive octant with barycenter $b_{P}=(1,1,...,1).$ Then the volume of $P$ is maximal when $P$ is a regular simplex, i.e.",
"for $(n+1)$ times the unit-simplex: $\\mbox{Vol}(P)\\le \\frac{(n+1)^{n}}{n!",
"}$ Applying the previous theorem to a suitable affine transformation of a given rational polytope gives us back REF .",
"After we had proved the Theorem REF by the arguments outlined above, Bo'az Klartag showed us a short and elegant direct proof of this statement using tools from convex geometry.",
"His argument, that we will reproduce below as Remark REF , is based on Grunbaum's inequality [29].",
"Nevertheless, we have decided to keep here our original argument as well since it exemplifies the relation of this kind of inequalities in convex geometry to KÀhler geometry.",
"Also, our argument may also be useful when dealing with other singular varities than toric ones, such as spherical varities.",
"Interestingly, the proof of Grunbaum's inequality is based on a clever application of the Brunn-Minkowski inequality for convex bodies and the latter inequality, or more precisely its functional form due to Prekopa [52] also plays a key role in our proof (since it is used in the proof of the Moser-Trudinger type inequalities).",
"In fact, using the positivity of the direct image bundles in [8] we will obtain a complex geometric generalization of Prekopa's result in the presence of a suitable action of a compact Lie group on a Stein manifold (Theorem REF ).",
"This also leads to a generalization of the Prekopa theorem in ${R}^{n}$ to non-compact real symmetric spaces of independent interest (Corollary REF ).",
"Since such spaces typically have negatively sectional curve this latter result appears to be rather intruiging, when contrasted with the results in [14], which demand non-negative Ricci curvature."
],
[
"Critical mean field type equations\nfor $S^{1}-$ invariant domains in {{formula:17d1c4bf-0e1a-4cf5-89b0-f8c826e828c1}}",
"As explained in section REF the starting point of the proof of Theorem REF is that an $S^{1}-$ invariant KÀhler-Einstein metric on $X$ induces a solution $\\phi $ to a mean field type equation on a Stein domain $\\Omega $ of $X$ admitting an $S^{1}-$ action with an attractive fixed point.",
"The proof is thus reduced to establishing a criticality result for solutions of such equations.",
"For concreteness, here we will only state the result in the case when the domain $\\Omega $ is contained in ${C}^{n}$ with its standard action by $S^{1}$ (or more generally any linear action of $S^{1}$ with positive weights $m_{i},$ i.e.",
"defined by $(e^{i\\theta },(z_{1},...,z_{n}))\\mapsto (e^{im_{1}\\theta }z_{1},...,e^{im_{n}\\theta }z_{n})).$ Denote by $dV$ the Euclidean volume element and write $dd^{c}=i\\partial \\bar{\\partial }/2\\pi ,$ so that $(dd^{c}\\phi )^{n}$ is the Monge-AmpÚre measure of $\\phi ,$ whose density is equal to $(\\frac{i}{2\\pi })^{n}$ times the determinant of the complex Hessian $(\\frac{\\partial ^{2}\\phi }{\\partial z_{i}\\partial \\bar{z}_{j}}).$ Theorem 1.6 Let $\\Omega $ be a connected smoothly bounded $S^{1}-$ invariant pseudoconvex domain in ${C}^{n},$ containing $0.$ For any given positive real number $\\gamma $ a necessary condition for the existence of an $S^{1}-$ symmetric plurisubharmonic solution $\\phi \\in \\mathcal {C}^{\\infty }(\\bar{\\Omega })$ to the equation $(dd^{c}\\phi )^{n}=\\frac{e^{-\\gamma \\phi }dV}{\\int _{\\Omega }e^{-\\gamma \\phi }dV}\\,\\,\\mbox{in\\,\\,$\\Omega $}\\,\\,\\,\\,\\phi =0\\,\\,\\mbox{on\\,$\\partial \\Omega $}$ is that $\\gamma \\le (n+1).$ More generally, the proof of the previous theorem shows that it is enough to assume that $\\phi $ is a continuous solution in the sense of pluripotential theory.",
"As shown in [4] $\\gamma <(n+1)$ is a sufficient condition for existence of such weak solutions for any pseudoconvex domain (by [30] any such continuous solution is in fact smooth in the interior of $\\Omega ).$ Hence $\\gamma =n+1$ appears to be a critical parameter for the equations REF on $\\Omega $ in the presence of an $S^{1}-$ action as above.",
"Moreover, the condition in the previous theorem that the fixed point 0 be contained in $\\Omega $ is crucial.",
"For example, if $\\Omega $ is an annulus in ${C}$ then it is well-known that there exist $S^{1}-$ invariant solutions for $\\gamma $ arbitrarily large (see section 5 in [13]).",
"To prove the theorem we first show that any solution $\\phi $ as in the previous theorem is an extremal for a Moser-Trudinger type inequality on $\\Omega ,$ which becomes stronger as the parameter $\\gamma $ increases.",
"We then go on to show that when $\\gamma >(n+1),$ a suitable regularization of the pluricomplex Green function $g$ of $\\Omega $ with a logarithmic pole at 0 violates the corresponding Moser-Trudinger type inequality.",
"This approach should be compared with a result of Ding-Tian [20] saying that for any Fano manifold $X$ admitting a KÀhler-Einstein metric there is a corresponding Moser-Trudinger type inequality for positively curved metrics on the anti-canonical line bundle $-K_{X}$ which has the metric on $-K_{X}$ induced by the KÀhler-Einsten metric as an extremal.",
"However, it is well-known that there are global obstructions to the existence of metrics on $-K_{X}$ with prescribed logarithmic poles and this is the reason that we need to replace $X$ with a Stein subdomain $\\Omega .$ The prize we have to pay is that an appropriate symmetry assumption is then needed to deduce the corresponding Moser-Trudinger type inequalities on $\\Omega $ (compare the discussion on symmetry breaking in section REF ).",
"We conjecture that the inequality $\\gamma \\le n+1$ in the previous theorem is in fact a strict inequality and that if $\\phi _{j}$ is a sequence of $S^{1}-$ symmetric solutions $\\phi _{j}$ associated to a sequence of parameters $\\gamma _{j}$ converging to the critical value $n+1,$ then $\\phi _{j}$ converges weakly to the pluricomplex Green function $g_{\\Omega }$ of $\\Omega $ with a logarithmic pole at $0.$ This is easy to verify in the case when $\\Omega $ is the unit-ball and $\\phi _{j}$ is the radial solution (see [4]).",
"The motivation for this conjecture comes from the concentration-compactness principles extensively studied in the one-dimensional situation (see [13] and references therein).",
"As explained in [13] the equations above then appear as mean field equations for statistical mechanical models with $\\gamma $ playing the role of the inverse temperature and the critical value corresponding to a phase transition."
],
[
"Organization",
"After having set up the complex geometric and group-theoretic frame work in the beginning of Section we establish Prekopa type convexity inequalites and give the proof of Theorem REF , by reducing it to the proof of Theorem REF .",
"Then in Section the singular setting on a toric variety is considered and the proof of Theorem REF is explained, using real convex analysis.",
"Finally, in Section Theorem REF is applied to spherical varieties and rephrased in terms of Lie algebras and symplectic geometry."
],
[
"Acknowledgments",
"We are grateful to Benjamin Nill for helpful comments on the toric setting, Michel Brion for his help with spherical varieties, Gabor Székelyhidi for encouraging us to consider the relation to the invariant $R(X)$ and Bo'az Klartag for allowing us to include here his beautiful reduction to Grunbaum's inequality."
],
[
"KÀhler-Einstein metrics and Monge-AmpÚre equations",
"Let $L\\rightarrow X$ be a holomorphic line bundle over an $n-$ dimensional compact complex manifold.",
"We will denote by $H^{0}(X,L)$ the space of all global holomorphic sections with values in $L.$ A Hermitian metric $\\left\\Vert \\cdot \\right\\Vert $ on $L$ may be represented by a collection of local functions $\\phi (:=\\lbrace \\phi _{U}\\rbrace )$ defined as follows: given a local trivializing section $s$ of $L$ on an open subset $U\\subset X$ we define the local weights $\\phi _{U}:=-\\log \\left\\Vert s\\right\\Vert ^{2}$ of the metric.",
"Of course, $\\phi _{U}$ depends on $s,$ but the (normalized) curvature form of the metric $dd^{c}\\phi _{U}:=\\frac{i}{2\\pi }\\partial \\bar{\\partial }\\phi $ is a globally well-defined two form on $X$ representing the first Chern class $c_{1}(L).$ The normalizations have been chosen so that $c_{1}(L)$ is an integral class.",
"Note that the metric on $L$ has semi-positive curvature form precisely when the local weights $\\phi _{U}$ are plurisubharmonic (psh, for short).",
"We recall that according to the Kodaira embedding theorem the line bundle $L$ is ample, i.e.",
"the Kodaira map $X\\rightarrow {P}(H^{0}(X,L))^{*}$ is an embedding for $k$ sufficiently large, precisely when $L$ admits a metric with positive curvature.",
"For any such $k$ the open manifold $S:=\\lbrace s=0\\rbrace ,$ for $s$ a given non-trivial element in $H^{0}(X,kL),$ is a Stein manifold, i.e.",
"$S$ admits a smooth and strictly plurisubharmonic exhaustion-function $\\phi _{S}.$ Indeed, $\\phi _{S}$ can be taken as $-\\log \\left\\Vert s\\right\\Vert ^{2}$ for any positively curved metric on $L.$ A KÀhler metric $\\omega $ on $X$ is said to be KÀhler-Einstein if it has constant Ricci curvature, i.e.",
"$\\mbox{Ric $\\omega =\\Lambda \\omega $}$ for some constant $\\Lambda .$ We will be interested in the case when $\\Lambda $ is positive and after a scaling we may as well assume that $\\Lambda =1.$ Then $X$ is necessarily a Fano manifold, i.e.",
"the dual $-K_{X}$ of the canonical line bundle $K_{X}:=\\Lambda ^{n}(T^{*}X)$ is ample.",
"Equivalently, $\\omega $ is a KÀhler-Einstein metric on the Fano manifold $X$ iff $\\omega $ is the curvature of a positively curved metric $\\left\\Vert \\cdot \\right\\Vert $ on $-K_{X}$ such that, if $s$ is a local trivialization of $-K_{X}$ over $U,$ then there is a positive constant $C$ such that the following Monge-AmpÚre equation holds on $U:$ $(dd^{c}\\phi )^{n}=Ce^{-\\phi }dV,\\,\\,\\,\\, dV:=i^{n^{2}}\\theta \\wedge \\bar{\\theta }$ where $\\phi $ is the corresponding weight of the metric and $\\theta $ is the holomorphic $n-$ form which is dual to $s.$ More generally, given a positive integer $k$ the local section $s$ can be replaced by a local non-vanishing holomorphic section $s_{k}$ of $-kK_{X}\\rightarrow U$ and one then sets $\\phi :=-\\frac{1}{k}\\log \\left\\Vert s_{k}\\right\\Vert ^{2},$ replacing the dual $\\theta $ above with its $k$ th root.",
"We remark that this latter flexibility allows one to define KÀhler-Einstien metrics on a singular (normal) variety $X$ (see [7]).",
"Indeed, in general $K_{X}$ is then merely defined as a Weil divisor, but assuming that $X$ is a Fano variety, i.e.",
"$-kK_{X}$ is an ample line bundle for some positive integer $k$ (in other words $K_{X}$ is an ample ${Q}-$ Cartier divisor) the previous definition makes sense on the regular locus of $X$ and one then adds the global condition that the corresponding metric be continuous on all of $X.$ Anyway, in this paper we will mainly stick to the case when $X$ is smooth."
],
[
"Group actions",
"Let $G$ be a complex Lie group acting by holomorphisms on a compact complex manifold $X.$ In other words, $X$ is a compact complex $G-$ manifold.",
"If $G$ acts linearly on a vector space $V,$ we write $V^{G}$ for the subspace of $G-$ invariant vectors and $V^{(G)}$ for the subspace of $G-$ eigenvectors, i.e.",
"$v\\in V^{(G)}$ iff $gv=\\chi (g)v,$ where $\\chi $ is a character, i.e.",
"a homomorphism from $G$ to ${C}^{*}.$ In particular, if $L$ is a $G-$ equivariant line bundle over $X$ then $G$ acts linearly on the vector space $H^{0}(X,L)$ by setting $(g\\cdot s)(p):=g(s(g^{-1}p))$ for any $s\\in H^{0}(X,L).$ In the proof of Theorem REF we will have great use for the following Lemma 2.1 Let $X$ be a smooth projective variety admitting a holomorphic action by the circle $S^{1}$ with isolated fixed points.",
"If the first Betti number of $X$ vanishes and $L$ is a given $S^{1}-$ equivariant ample line bundle over $X,$ then there exists a fixed point $p\\in X$ and an $S^{1}-$ eigenvector $s\\in H^{0}(X,kL)$ for some $k>0$ such that $s(p)\\ne 0$ and $H^{0}(S)^{T}={C},$ where $S$ is the Stein manifold $S:=X-\\lbrace s=0\\rbrace $ containing $p.$ Since $L$ is ample it admits a metric with positive curvature form $\\omega ,$ defining a symplectic form on $X.$ Moreover, averaging over the compact group $S^{1}$ we may assume that $\\omega $ is $S^{1}-$ invariant, i.e.",
"the action is symplectic.",
"Since the first Betti number of $X$ vanishes the action admits a Hamiltonian function $f,$ i.e.",
"$df=\\omega (V,\\cdot )$ where $V$ is the vector field on $X$ generating the $S^{1}-$ action.",
"It then follows from general principles that the action lifts to $L.$ Anyway, in our setting $L$ will be equal to $-K_{X}$ which admits a canonical lift of the $S^{1}-$ action on $X.$ We let $p$ be a point where the minimum of $f$ is attained.",
"Then $V$ vanishes at $f$ and hence $p$ is a fixed point.",
"Next, we pick a positive number $k$ such that $kL$ is globally generated and decompose $H^{0}(X,kL)=\\oplus V_{m}$ in the one-dimensional eigenspaces for the $S^{1}-$ action.",
"By the assumption of global generation there is a section $s\\in H^{0}(X,kL)$ such that $s(p)\\ne 0$ and by the previous decomposition we can thus take $s\\in V_{m}$ for some $m.$ Let now $S:=X-\\lbrace s=0\\rbrace ,$ which, as explained above, is a Stein manifold.",
"To prove REF we note that the action of $S^{1}$ on the tangent space at the fixed point $p$ has positive weights in the following sense (i.e.",
"it is an attractive fixed point) : by a general result for compact Lie group (see Satz 4.4 in [32]) we may linearize the action in an invariant neighborhood of $U$ of $p$ so that $e^{i\\theta }\\cdot (z_{1},...,z_{n})=(e^{im_{1}\\theta }z_{1},...,e^{im_{n}\\theta }z_{n}).$ The positivity referred to above then amounts to having $m_{i}>0$ for all $i.$ Indeed, after performing a linear change of coordinates we may assume that $f(z)=\\sum m_{i}|z_{i}|^{2}+o(|z|^{2})$ and since $f$ has a minimum at $p$ (corresponding to $z=0)$ and the fixed point $p$ is isolated it must be that $m_{i}>0.$ Taylor expanding a given holomorphic function $g$ on $U$ wrt the variables $z_{i}$ then reveals that $g$ is $S^{1}-$ invariant only if it is constant on $U.$ But since $S$ is connected this concludes the proof of the lemma.",
"Example 2.2 Set $X={P}^{1}$ and $L=-K_{X}$ with its usual $S^{1}-$ action obtained by identifying $X$ with the two-sphere and considering rotations around a fixed axes.",
"Then $X$ has two fixed points (the north and the south pole) and fixing the standard affine chart $U_{0}\\unknown.",
"{C}$ containing the south pole, where the action is given by $(e^{i\\theta },z)\\mapsto e^{i\\theta }z,$ we can write any section $s\\in H^{0}(X,-K_{X})$ over $U_{0}$ as $s=f(z)\\frac{\\partial }{\\partial z}$ so that $(e^{i\\theta }\\cdot s)(z)=e^{i\\theta }f(e^{-i\\theta }z)dz.$ Hence, we can take the section $s$ in the previous lemma as the one determined by $f(z)=1$ (which has weight $m=1)$ so that $S=U_{0}.$ Even though if it will not - strictly speaking - be needed for the proof of Theorem REF , we make a brief digression to explain how, using the Bialynicki- Birula decomposition (see Theorem 4.4 in [9]), one may, essentially, take the Stein domain $S$ to be equivariantly isomorphic to ${C}^{n}$ with a linear ${C}^{*}-$ action.",
"First recall that the Bialynicki- Birula decomposition says that any non-singular $n-$ dimensional projective variety $X$ with a ${C}^{*}-$ action having only isolated fixed points may be written as the disjoint union $X=\\coprod _{p}X_{p}$ where $p$ ranges over the fixed point $p$ in $X$ and where $X_{p}$ is ${C}^{*}-$ invariant set equivariantly isomorphic to the affine space $T^{+}X_{|p},$ i.e.",
"the the direct sum of the positive weight spaces in $TX_{|p}$ with the induced linear ${C}^{*}-$ action.",
"Concretely, $X_{p}$ is the attracting set for $p$ under the ${C}^{*}-$ action , i.e.",
"$x\\in X_{p}$ iff $\\lim _{\\lambda \\rightarrow 0}\\lambda \\cdot x=p.$ In particular, if the all the weights at $p$ are all positive (as for $p$ in the previous lemma) then $X_{p}$ is a Zariski open subvariety of $X$ which is equivariantly isomorphic to ${C}^{n}$ with a linear ${C}^{*}-$ action.",
"We note that $X_{p}\\subset S,$ where $S=\\lbrace s\\ne 0\\rbrace $ is the Stein manifold appearing in the previous lemma.",
"Moreover, given a positively curved $S^{1}-$ invariant metric on $L$ the function $\\phi :=-\\log \\left\\Vert s\\right\\Vert ^{2}$ is a psh exhaustion function of $X_{p}.$ To see that $\\phi $ is indeed proper we note that since $\\phi $ is strictly convex along the ${C}^{*}-$ orbits and (since $s(p)\\ne 0)$ is bounded from below close to $p$ it follows that $\\phi \\rightarrow \\infty $ as $\\left|\\lambda \\right|\\rightarrow \\infty $ along a given ${C}^{*}-$ orbit in $X_{p},$ which implies properness by a basic compactness argument.",
"As for the $S^{1}-$ invariance it follows from the fact that $s$ is an eigenvector (as explained in the beginning of section REF below)."
],
[
"A Prekopa type convexity result on Stein manifolds under group actions",
"One of the key ingredients in the proof of Theorem REF is a convexity result of Prekopa type.",
"Let us first recall the Prekopa inequality in its original form [52]: If $\\phi _{t}(x)$ is a convex function on $I\\times {R}^{n}$ , where $I$ is an open interval in ${R}$ with coordinate $t,$ then the function $t\\mapsto -\\log \\int _{{R}^{n}}e^{-\\phi _{t}}dx$ is convex.",
"Here we will obtain a complex geometric generalization of this result.",
"We let $S$ be an $n-$ dimensional Stein manifold with trivial canonical line bundle $K_{S},$ i.e.",
"$S$ admits a non-vanishing holomorphic $n-$ form $\\theta $ (also called a holomorphic volume form).",
"Theorem 2.3 Let $K$ be a compact group acting on a bounded Stein domain $\\Omega $ with a holomorphic volume form $\\theta $ and such that all $K-$ invariant holomorphic functions are constant, i.e.",
"$H^{0}(\\Omega )^{K}={C}.$ If $\\phi _{t}(z)$ is a $K-$ invariant bounded psh function on $D\\times \\Omega ,$ where $D$ is the unit-disc in ${C}$ then the function $t\\mapsto -\\log i^{n^{2}}\\int _{\\Omega }e^{-\\phi _{t}}\\theta \\wedge \\bar{\\theta },$ is subharmonic in $t.$ More generally, if $\\Omega $ is unbounded and $e^{-\\phi _{t}}$ is integrable on $\\Omega $ for $t$ fixed, the same conclusion holds if the space $H^{0}(\\Omega )$ is replaced by $H^{0}(\\Omega )\\cap L^{2}(e^{-\\phi _{t}}\\theta \\wedge \\bar{\\theta }).$ Consider the (infinite dimensional) Hermitian holomorphic vector bundle $E\\rightarrow D$ whose fiber $E_{t}$ is the Hilbert space of all holomorphic functions on $\\Omega $ of finite $L^{2}-$ norm $\\left\\Vert f\\right\\Vert _{\\phi _{t}}^{2}:=i^{n^{2}}\\int _{\\Omega }|f|^{2}e^{-\\phi _{t}}\\theta \\wedge \\bar{\\theta },$ As shown in [8] this bundle has positive curvature in the following sense: for any given holomorphic section $\\Lambda $ of the dual bundle $E^{*}$ the function $t\\mapsto -\\log (\\left\\Vert \\Lambda \\right\\Vert _{\\phi _{t}}^{2}):=-\\log (\\sup _{f\\in E_{t}}\\left|\\left\\langle \\Lambda ,f\\right\\rangle \\right|^{2}/\\left\\Vert f\\right\\Vert _{\\phi _{t}}^{2})$ is subharmonic.",
"Strictly speaking the proof in [8] concerned the case when $\\Omega $ is a pseudoconvex domain in ${C}^{n},$ but the proof can be repeated word for word in the Stein case, using that Hörmander's $L^{2}-$ estimates for $\\bar{\\partial }$ are still valid.",
"We now let $\\sigma $ be the invariant probability measure on $K$ (i.e.",
"the Haar measure) and set $\\Lambda (f):=\\int _{K}(k^{*}f)\\sigma $ Since the rhs above is a $K-$ invariant holomorphic function on $\\Omega $ it is, by assumption, constant (for $t$ fixed) and hence $\\Lambda $ indeed defines a holomorphic section of $E^{*}.$ Thus it will be enough to show that, under the condition REF , $\\left\\Vert \\Lambda \\right\\Vert _{\\phi _{t}}^{2}=1/\\int _{\\Omega }e^{-\\phi _{t}}\\theta \\wedge \\bar{\\theta },$ i.e.",
"the sup above is attained for constant functions.",
"But this follows immediately from estimating $\\int _{\\Omega }e^{-\\phi _{t}}\\theta \\wedge \\bar{\\theta }\\left\\Vert \\Lambda \\right\\Vert _{\\phi _{t}}^{2}\\le \\int _{K}d\\sigma \\int _{\\Omega }e^{-\\phi _{t}}|k^{*}f|^{2}\\theta \\wedge \\bar{\\theta }=\\int _{\\Omega }e^{-\\phi _{t}}|f|^{2}\\theta \\wedge \\bar{\\theta },$ using that $\\phi _{t}$ is $K-$ invariant in the last equality.",
"For example, if $\\Omega =S={C}^{n},$ $\\theta =dz$ and $\\phi _{t}$ grows as $(n+1)\\log (|z|^{2})+O(1)$ at infinity one can take the compact group $K$ in the theorem to be trivial, i.e.",
"no symmetry assumption is needed.",
"The point is that, viewing ${C}^{n}$ as a Zariski open set in $X:={P}^{n}$ the space $H^{0}({C}^{n})\\cap L^{2}(e^{-\\phi _{t}})$ can be identified with $H^{n,0}(X,-K_{X})\\unknown.",
"H^{0}({P}^{n},{C})$ which is one-dimensional.",
"However, for a general psh function $\\phi _{t}$ in ${C}^{n}$ one can construct counter-examples to the subharmonicity in formula REF , by adapting an example of Kiselman [34] to the present setting.",
"Remark 2.4 Replacing $\\phi $ by $m\\phi $ and letting $m\\rightarrow \\infty $ in the previous theorem shows that the function $t\\mapsto \\inf _{\\Omega }\\phi _{t}$ is also subharmonic in $t.$ This property is a well-known instance of the Kiselman minimum principle [34] extended to the setting of Lie groups by Loeb [42].",
"It should however be pointed out that the setting in [42] is more general as it applies to certain non-compact groups $K.$"
],
[
"An application to symmetric spaces",
"Before continuing we make a brief digression, showing how the previous theorem implies a Prekopa type convexity inequality for real symmetric spaces.",
"More precisely, we consider a non-compact symmetric space with compact dual $K,$ i.e.",
"the symmetric space may be written as $G/K$ where $G$ is the complexification of the compact group $K$ (so that $G$ is a connected reductive complex Lie group and for simplicity we will assume that $G$ is semi-simple [25]).",
"As is well-known $G$ is a Stein manifold.",
"More precisely, by results of Chevalley $G$ is an affine algebraic variety (see the appendix in [46]) with a $G$ -bi-invariant pseudo-Riemannian metric (induced by the Killing form on the Lie algebra of $G$ ).",
"The corresponding $K-$ principal fiber bundle $\\pi :\\, G\\rightarrow G/K$ induces a $G-$ bi-invariant symmetric Riemannian metric on $G/K.$ There is a $G-$ bi-invariant volume form $\\mu _{G}$ on $G$ and it can be written as $\\mu _{G}=\\theta \\wedge \\bar{\\theta },$ where $\\theta $ is a $G-$ bi-invariant holomorphic top form on $G.$ The push-forward of $\\mu _{G}$ under the projection $\\pi $ clearly coincides with the $G-$ bi-invariant volume form $\\mu _{G/K}$ on the symmetric space $G/K.$ Corollary 2.5 Let $\\phi _{t}$ be a geodesically convex function on the Riemannian product ${R}\\times G/K,$ where ${R}$ is the real line with its Euclidean Riemannian metric.",
"Then the function $t\\mapsto -\\log \\int _{G/K}e^{-\\phi _{t}}\\mu _{G/K}$ is convex on ${R}.$ It is well-known that the convex functions on the symmetric space $G/K$ may be written as $\\pi _{*}\\psi $ where $\\psi $ is a $K-$ invariant psh function on $G$ (see for example Lemma 2, page 34 in [21]).",
"In particular, we may identify $\\phi _{t}$ with a $S^{1}\\times K-$ invariant psh function $\\psi _{t}$ on ${C}^{*}\\times G.$ Hence, the corollary will follow from the previous theorem once we have checked that any $K-$ invariant holomorphic function on $G$ is constant.",
"But this follows immediately from the basic fact that a holomorphic function on an $n-$ dimensional connected complex manifold is uniquely determined by its restriction to a totally real submanifold (here $K)$ of real dimension $n.$ Example 2.6 If $G=SL(2,{C})$ and $K=SU(2),$ then the symmetric space $G/K$ may be identified with the three-dimensional real hyperbolic space.",
"It is interesting to compare the previous corollary with the results in [14], where a so called Prekopa-Leindler inequality is obtained valid for any (possibly non-symmetric) Riemannian manifold $(M,g)$ with non-negative Ricci curvature.",
"This inequality is in fact stronger than the Prekopa inequality on $(M,g),$ i.e.",
"the analog of Corollary REF for $(M,g).$ For example, in the case when $(M,g)$ is Euclidean space (which may be obtain as above by taking $G$ to be abelian) the Prekopa-Leindler inequality implies the Brunn-Minkowski inequality for general sets, while the Prekopa inequality a priori only implies the Brunn-Minkowski inequality for convex sets.",
"Conversely, as shown in [14] the validity of the Prekopa-Leindler inequality on $(M,g)$ actually implies that the Ricci curvature of $(M,g)$ is non-negative and hence it cannot hold on a general symmetric space as above, since it is well-known that an irreducible symmetric space $G/K$ has negative sectional curvatures when $G$ is non-abelian."
],
[
"Proofs of Theorems ",
"We will start by reducing the proof of Theorem REF to proving Theorem REF .",
"In the following we will use the notation $T=S^{1}$ for the one-dimensional real torus inbedded in ${C}^{*}$ and hence acting on $X.$ Since $X$ admits a KÀhler-Einstein metric it also admits a $T-$ invariant KÀhler-Einstein metric $\\omega $ [2], which is the curvature form of a $T-$ invariant metric $\\left\\Vert \\cdot \\right\\Vert $ on $-K_{X}.$ To simplify the notation we assume that $-K_{X}$ is globally generated (otherwise just replace $-K_{X}$ with a large tensor power) and take $p$ to be the $T-$ fix point and $s$ the holomorphic section of $-K_{X},$ furnished by Lemma REF .",
"As explained above $\\phi :=-\\log \\left\\Vert s\\right\\Vert ^{2}$ is then a strictly psh smooth exhaustion function of the Stein manifold $S:=\\lbrace s\\ne 0\\rbrace $ containing the fixed point $p.$ Moreover, $\\phi $ is $T-$ invariant.",
"Indeed, by construction $g\\cdot s=\\chi (g)s$ where $\\chi $ is a character on the compact group $T.$ Hence $\\chi $ takes values in the unit-circle $S^{1}\\subset {C},$ showing that $\\phi $ is $T-$ invariant, as desired.",
"As explained above the KÀhler-Einstein equation for $\\omega $ (and the correspond metric on $-K_{X})$ then translates to the Monge-AmpÚre equation on $S$ $(dd^{c}\\phi )^{n}=Ce^{-\\phi }dV$ for a positive constant $C,$ where $dV$ is the volume form induced by the trivialization $s.$ We may rewrite $C=V_{X}/\\int _{S}e^{-\\phi }dV$ , where $V_{X}=\\int _{S}(dd^{c}\\phi )^{n}$ which coincides with the top-intersection number $c_{1}(X)^{n}.$ Let $\\Omega _{R}$ be the set where $\\phi <R$ and note that the sets $\\Omega _{R}$ exhaust ${C}^{n},$ since $\\phi $ is proper.",
"Assume now, to get a contradiction, that the bound in the theorem to be proved is not valid for $X.$ Then we can fix $R$ sufficiently large so that $V_{\\Omega _{R}}:=\\int _{\\Omega _{R}}(dd^{c}\\phi )^{n}>(n+1)^{n}.$ Writing $\\Omega :=\\Omega _{R}$ and replacing $\\phi $ by $\\phi -R$ we then obtain a $T-$ invariant smooth plurisubharmonic function $\\phi $ solving the following equation: $(dd^{c}\\phi )^{n}=V_{\\Omega }\\frac{e^{-\\phi }dV}{\\int _{\\Omega }e^{-\\phi }dV},\\,\\,\\,\\,\\phi =0\\,\\,\\mbox{on\\,$\\partial \\Omega $}$ on the $T-$ invariant domain $\\Omega $ which is a hyperconvex domain, i.e.",
"it admits a negative continuous psh exhaustion function (namely $\\phi ).$ More precisely, since $\\phi $ is smooth and strictly psh the domain $\\Omega $ is a Stein domain.",
"Finally, rescaling, i.e.",
"replacing $\\phi $ with $V_{\\Omega }^{1/n}\\phi $ we have thus obtained a solution to the equation in Theorem REF with a parameter $\\gamma :=V_{\\Omega }^{1/n}>(n+1)$ and all that remains is thus establishing Theorem REF in the slightly more general case when ${C}^{n}$ is replaced with a Stein domain with a holomorphic $S^{1}-$ action admitting an attractive fixed point."
],
[
"Proof of Theorem ",
"We will denote by $\\mathcal {H}_{0}(\\Omega )$ the space of all psh functions on $\\Omega $ which are continuous up to the boundary, where they are assumed to vanish.",
"Its $T-$ invariant subspace will be denoted by $\\mathcal {H}_{0}(\\Omega )^{T}.$ Let $\\phi _{0}$ be a solution to equation REF with a fixed parameter $\\gamma .$ The following Moser-Trudinger type inequality can then be established on $\\Omega $ (which has $\\phi _{0}$ as an extremal): there is a positive constant $C$ such that $\\frac{1}{\\gamma }\\log \\int _{\\Omega }e^{-\\gamma \\phi }dV\\le \\frac{1}{(n+1)}\\int _{\\Omega }(-\\phi )(dd^{c}\\phi )^{n}+C$ for any $\\phi \\in \\mathcal {H}_{0}(\\Omega )^{T}.$ We also write the previous inequality as $\\mathcal {G}_{\\gamma }(\\phi )\\le C$ for the corresponding Moser-Trudinger type functional $\\mathcal {G}_{\\gamma }.$ Given the convexity property in Theorem REF the proof may be obtained by repeating the proof of Theorem 1.4 in [4], but for completeness we have recalled the proof in section REF below.",
"The desired contradiction will now be obtained by exhibiting a function violating the previous Moser-Trudinger type inequality.",
"To this end we first recall the definition of the pluricomplex Green function $g_{p}$ of a pseudoconvex domain $\\Omega $ with a pole at a given point $p:$ $g_{p}:=\\sup \\left\\lbrace \\phi :\\,\\,\\,\\phi \\in (PSH)(\\Omega )\\cap \\mathcal {C}^{0})(\\overline{\\Omega }-\\lbrace p\\rbrace ):\\,\\,\\phi \\le 0,\\,\\,\\,\\phi \\le \\log |z|^{2}+O(1)\\right\\rbrace $ where $z$ denotes fixed local holomorphic coordinates centered at $p.$ We will often write $g=g_{p}$ to simpify the notation.",
"As is well-known [36] $g$ is continuous up to the boundary on $\\Omega $ apart from a singularity at $p$ and satisfies $(dd^{c}g)^{n}=\\delta _{p}\\,\\mbox{\\,\\ on\\,}\\Omega -\\lbrace p\\rbrace ,\\,\\,\\,\\, g=\\log |z|^{2}+O(1)$ In particular $\\int (dd^{c}g)^{n}=1$ and $\\int _{\\Omega }e^{-ng}dV=\\infty .$ We note that if $T$ acts, as above, on $\\Omega $ and $p$ is taken as a fixed point for the action, then $g$ is $T-$ invariant.",
"Indeed, since $p$ is invariant under the action of $T$ so is the the convex class of functions where the sup in REF is taken and hence the sup $g$ must be $T-$ invariant.",
"The contradiction is now obtained by showing that there is a family of functions $g_{t}$ in $\\mathcal {H}_{0}(\\Omega )^{T}$ decreasing to $g$ such that, for $t$ sufficently large $g_{t}$ violates the Moser-Trudinger type inequality REF if $\\gamma >(n+1).$ To this end we set $g_{t}:=\\log (e^{-2t}+e^{g})-C_{t},\\,\\,\\,\\,\\phi _{t}=(e^{-2t}+Ce^{\\log |z|^{2}})$ where $C$ is a constant such that $-g_{t}\\le -\\phi _{t}$ and $C_{t}$ is the constant ensuring that $g_{t}$ vanishes on the boundary, i.e.",
"$C_{t}=\\log (e^{-2t}+1)=o(1/t).$ In fact, this part of the argument does not require any symmetry assumptions.",
"We fix local holomorphic coordinates $z$ centered at $p$ such that, locally around $p,$ $\\phi (z)=\\log |z|^{2}+O(1).$ Trivially we have, by replacing $\\Omega $ with a coordinate ball $B,$ that $-\\frac{1}{\\gamma }\\log \\int _{\\Omega }e^{-\\gamma g_{t}}dV\\le -\\frac{1}{\\gamma }\\log \\int _{B}e^{-\\gamma \\phi _{t}}dz\\wedge d\\bar{z}+C^{\\prime }$ Denoting by $F_{t}$ the scaling map defined by $F_{t}(z)=e^{t}z$ we have $\\phi _{t}=F_{t}^{*}\\phi _{0}-2t+o(1/t).$ Hence, we may write $-\\frac{1}{\\gamma }\\log \\int _{B}e^{-\\gamma \\phi _{t}}dz\\wedge d\\bar{z}=-2t+o(1/t)+-\\frac{1}{\\gamma }\\log \\int _{B}e^{-\\gamma F_{t}^{*}\\phi _{0}}dz\\wedge d\\bar{z}$ Rewriting $dz\\wedge \\bar{dz}$ as $e^{-2tn}F_{t}^{*}dz\\wedge \\bar{dz}$ gives $-\\frac{1}{\\gamma }\\log \\int _{B}e^{-\\gamma F_{t}\\phi _{0}}dz\\wedge d\\bar{z}=2tn\\frac{1}{\\gamma }-I_{t},\\,\\,\\, I_{t}:=\\frac{1}{\\gamma }\\log \\int _{e^{t}B}e^{-\\gamma \\phi _{0}}dz\\wedge d\\bar{z}$ Next, we consider the energy term: since $-g_{t}\\le -\\log (e^{-2t}+0)-C_{t}=2t+o(1/t)$ we get $\\int _{\\Omega }(-g_{t})(dd^{c}g_{t})^{n}\\le (2t+o(1/t))\\int _{\\Omega }(dd^{c}g_{t})^{n}.$ But since $g_{t}$ is a sequence of bounded psh functions tending to zero at the boundary of $\\Omega $ and decreasing to $g$ it follows from well-known convergence properties that $\\int _{\\Omega }(dd^{c}g_{t})^{n}\\rightarrow \\int _{\\Omega }(dd^{c}g)^{n}=1$ (in fact, in our case $g$ may be taken to be smooth close to the boundary and then the convergence follows immediately from Stokes theorem).",
"All in all this means that $-\\mathcal {G}(g_{t})\\le \\frac{1}{(n+1)}2t+(2t)(-1+\\frac{n}{\\gamma })-\\log I_{t}+o(1/t),$ Note that when $\\gamma =(n+1)$ we have $(-1+\\frac{n}{\\gamma })=\\frac{1}{(n+1)}$ and hence when $\\gamma >(n+1)$ first term above, which is linear in $t,$ tends to $-\\infty $ as $t\\rightarrow \\infty .$ Moreover, since $I_{t}\\ge -C$ it follows that $-\\mathcal {G}(g_{t})\\rightarrow -\\infty $ when $t\\rightarrow \\infty $ and hence $\\mathcal {G}$ is not bounded from above, which contradicts the Moser-Trudinger inequality.",
"This completes the proof of Theorem REF and thus of Theorem REF , as well.",
"Remark 2.7 There is an alternative way of obtaining a contradiction to the Moser-Trudinger inequaly with parameter $\\gamma >(n+1)$ (see our previous preprint [5]).",
"Indeed, as shown in [4] the inequality induces another inequality of Brezis-Merle-Demailly type on the $(n+1)-$ dimensional product $\\Omega ^{\\prime }=\\Omega \\times D$ of $\\Omega $ with the unit-disc which in particular implies that $\\int _{\\Omega ^{\\prime }}e^{-(n+1)\\phi }dV<\\infty ,$ for any $S^{1}-$ invariant plurisubharmonic function on $\\Omega ^{\\prime },$ say with isolated singularities compactly contained in $\\Omega ^{\\prime },$ vanishing on the boundary and with unit Monge-AmpÚre mass on $\\Omega ^{\\prime }.$ But this is immediately seen to be contradicted by the pluricomplex Green function of $\\Omega ^{\\prime }$ with a pole at the origin."
],
[
"Symmetry breaking",
"The assumption in Theorem REF that 0 be contained in $\\Omega $ is crucial.",
"For example, when $\\Omega $ is an annulus, $r<|z|<1,$ in ${C},$ it is well-known [13] that there exists a (uniquely determined) $S^{1}-$ invariant solution $\\phi _{\\gamma }$ for any value of $\\gamma .$ Moreover, by the method of moving planes any solution of the equations is necesseraly $S^{1}-$ invariant [13] and thus coincides with $\\phi _{\\gamma }.$ In the range $\\gamma <2$ the solution $\\phi _{\\gamma }$ is an extremal for the corresponding Moser-Trudinger inequalities which are known to hold for general funtions $\\phi $ in $\\mathcal {H}_{0}(\\Omega ).$ We note however that at the critical value $\\gamma =2$ an interesting instance of symmetry breaking appears.",
"Indeed, since Theorem REF applies for any $\\gamma >0,$ we deduce, as before, that $\\phi _{\\gamma }$ is always an extremal for the corresponding Moser-Trudinger type inequality for $S^{1}-$ invariant functions in $\\mathcal {H}_{0}(\\Omega ).$ But when $\\gamma >2$ the corresponding Moser-Trudinger inequality does not hold on all of $\\mathcal {H}_{0}(\\Omega ).$ Indeed, as before it is violated by a suitable regularization of a Green function with a pole in any given point in $\\Omega .$ This also shows that Theorem REF cannot hold general if the invariance assumption is removed.",
"For completeness we will next recall the proof in [4] of the inequality used above.",
"The arguments carry over verbatim to the setting of Stein domains, even if as explained above the ${C}^{n}$ - setting is adequate for our purposes."
],
[
"Proof of the Moser-Trudinger type inequality\n",
"Let $\\mathcal {G}(\\phi ):=\\frac{1}{\\gamma }\\log \\int _{\\Omega }e^{-\\gamma \\phi }dV+\\frac{1}{(n+1)}\\int _{\\Omega }\\phi (dd^{c}\\phi )^{n},$ whose Euler-Lagrange equation (i.e.",
"the critical point equation $d\\mathcal {G}_{|\\phi }=0)$ is precisely the complex Monge-AmpÚre equation REF .",
"Given $\\phi _{0}$ and $\\phi _{1}$ in $\\mathcal {H}_{0}(\\Omega )$ there is a unique geodesic $\\phi _{t}$ connecting them in $\\mathcal {H}_{0}(\\Omega ).$ It may be defined as the following envelope: setting $\\Phi (z.t):=\\phi _{t}(z),$ where now $t$ has been extended to a complex strip $\\mathcal {T}$ by imposing invariance in the imaginary $t-$ direction, we set $M:=\\Omega \\times \\mathcal {T}$ the boundary data $\\Phi _{\\partial M}$ is determined by $\\phi _{0}$ and $\\phi _{1}$ and we set $\\Phi (z,t):=\\sup \\left\\lbrace \\Psi (z,t):\\,\\,\\,\\Psi \\in \\mathcal {C}^{0}(\\bar{M})\\cap PSH(M),\\,\\,\\,\\Psi _{\\partial M}\\le \\Phi _{\\partial M}\\right\\rbrace $ Since $M$ is hyperconvex it follows that $\\Phi \\in \\mathcal {C}^{0}(\\bar{M})\\cap PSH(M)$ is the unique solution of the following Dirichlet problem: $(dd^{c}\\Phi )^{n+1}=0,\\,\\,\\,\\mbox{in\\,\\ $M$}$ and on $\\partial M$ the function $\\Phi $ coincides with the boundary data determined by $\\phi _{i}$ (see [4] and references therein).",
"We also note that if $\\phi _{t}$ is $T-$ invariant for $t=0,1$ then it is in fact $T-$ invariant for any $t,$ as follows from its definition REF as an envelope (just as in the similar case of the Green function discussed above).",
"By Theorem REF the functional $t\\mapsto \\frac{1}{\\gamma }\\log \\int _{\\Omega }e^{-\\gamma \\phi _{t}}dV$ is concave along any geodesic (indeed, the assumption REF follows immediately from Taylor expanding a holomorphic function on $\\Omega $ around the origin and using the positivity of the weights of the action).",
"Next, we note that $\\mathcal {E}(\\mathbf {\\phi }):=\\int _{\\Omega }\\phi (dd^{c}\\phi )^{n}$ is affine along a geodesic $\\phi _{t}.$ Indeed, letting $t$ be complex a direct calculation gives $dd^{c}\\mathcal {E}(\\phi _{t})=\\int _{\\Omega }(dd^{c}\\phi )^{n+1}$ which, by definition, vanishes if $\\phi _{t}$ is a geodesic.",
"All in all this means that $\\mathcal {G}(\\phi _{t})$ is concave along a geodesic.",
"Letting now $\\phi $ be an arbitrary element in $\\mathcal {H}_{0}(\\Omega )^{T}$ we take $\\phi _{t}$ to be the geodesic connecting the solution $\\phi _{0}$ of equation REF (obtained from the invariant KÀhler-Einstein metric on $X)$ and $\\phi _{1}=\\phi .$ Heuristically, $\\mathcal {G}(\\phi _{t})$ has a critical point at $t=0,$ i.e.",
"its right derivative vanishes for $t=0$ and hence by concavity $\\mathcal {G}(\\phi _{1})\\le \\mathcal {G}(\\phi _{0}).$ However, since $\\phi _{t}$ is not, a priori, smooth one has to be a bit careful when differentiating $\\mathcal {G}(\\phi _{t}).$ However, by the affine concavity of $\\mathcal {E}$ it is not hard too see that $\\frac{1}{(n+1)}\\frac{d}{dt}_{t=0^{+}}\\mathcal {E}(\\phi _{t})\\le \\int _{\\mathcal {B}}\\dot{\\phi }_{0}(dd^{c}\\phi _{0})^{n}/n!$ (see Lemma 3.4 in [4]) and hence, by concavity, $\\mathcal {G}(\\phi _{1})\\le 0+\\mathcal {G}(\\phi _{0}),$ which concludes the proof of the M-T inequality with $C=\\mathcal {G}(\\phi _{0}).$ Remark 2.8 Theorem is still valid when $X$ is merely weakly Fano, i.e $-K_{X}$ is nef and big, if one uses the notion of (singular) KÀhler-Einstein metrics introduced in [7].",
"Indeed, by [7] such a metric is smooth on a Zariski open subset of $X$ and hence, by the KÀhler-Einstein equation, strictly positively curved there.",
"Moreover, by the Kawamata-Shokurov basepoint free theorem ([38], Theorem 3.3), $-mK_{X}$ is base point free for $m$ sufficently large and hence Lemma REF still applies and together with the Bialynicki- Birula decomposition one gets an exhaustion by $T-$ invariant Stein domains of a dense open embedding of ${C}^{n}$ in $X.$ The rest of the proof then proceeds exactly as before."
],
[
"Bounds on the volume in the absense of a KÀhler-Einstein metric.",
"Recall that for a general Fano manifold $X$ the “greatest lower bound on the Ricci curvature” is the invariant $R(X)\\in ]0,1]$ defined as the sup of all positive numbers $r$ such that $\\mbox{Ric $\\omega \\ge r\\omega $}$ (see [56]).",
"A simple modification of the proof of Theorem REF then gives the following more general statement: Theorem 2.9 Let $X$ be a Fano manifold admitting a holomorphic ${C}^{*}-$ action with a finite number of fix points.",
"Then the first Chern class $c_{1}(X)$ satisfies the following upper bound $c_{1}(X)^{n}\\le \\left(\\frac{n+1}{R(X)}\\right)^{n}$ The point is that, as shown in [56], the invariant $R(X)$ coincides with the sup of all $r\\in [0,1[$ such that, for any given KÀhler form $\\eta $ there exists $\\omega _{r}$ such that $\\mbox{Ric $\\omega _{r}=r\\omega _{r}+(1-r)\\eta $}$ As is well-known $\\omega _{r}$ is uniquely determined.",
"In particular, if $X$ admits a holomorphic action by $S^{1}$ then, by averaging over $S^{1},$ we may take $\\eta $ to be $S^{1}-$ invariant.",
"By the uniqueness of solutions to the previous equation it then follows that $\\omega _{r}$ is also $S^{1}-$ invariant.",
"For any fixed $r$ we can now repeat the proof of Theorem REF and obtain an $S^{1}-$ invariant psh solution $\\phi $ to $(dd^{c}\\phi )^{n}=V_{\\Omega }\\frac{e^{-r\\phi }e^{-(1-r)\\phi _{0}}dV}{\\int _{\\Omega }e^{-r\\phi }e^{-(1-r)\\phi _{0}}dV},\\,\\,\\,\\,\\phi =0\\,\\,\\mbox{on\\,$\\partial \\Omega $},$ where $\\eta =dd^{c}\\phi _{0}$ on $\\Omega .$ The Moser-Trudinger inequality still applies when $dV$ is replaced with $e^{-(1-r)\\phi _{0}}dV.$ Indeed, we just apply Theorem REF to the curve $(1-r)\\phi _{t}+r\\phi _{0},$ where $\\phi _{t}$ is a geodesic as before.",
"In fact, since $\\phi _{0}$ is bounded this gives the same Moser-Trudinger inequality as before, but with the integrand $e^{-\\phi }$ replaced with $e^{-r\\phi }.$ Hence, rescaling we obtain a contradiction if $r^{n}V(X)>(n+1)^{n},$ just as before.",
"In particular, taking the sup over all $r<R(X)$ then concludes the proof of the previous theorem."
],
[
"The volume of (singular)\nToric varieties and convex bodies",
"We will next briefly explain how to carry out the proof of Theorem REF possibly singular toric Fano varieties directly using convex analysis in ${R}^{n}.$ This approach has the advantage of bypassing some technical difficulties related to the singularities of the toric variety in question.",
"The motivation comes from the well-known correspondence between $T-$ invariant positively curved metrics on toric line bundles $L\\rightarrow X_{P}$ and convex functions in ${R}^{n}$ whose (sub-gradient) image is contained in the corresponding polytope $P$ (see [21], [6] and references therein) More precisely, a psh function $\\Phi (z)$ on the complex torus ${C}^{*n},$ embedded in $X,$ is the weight of a, possibly singular, positively curved metric on $L\\rightarrow X_{P}$ iff $\\Phi $ is the pull-back under the Log map $\\mbox{Log }{C}^{*n}\\rightarrow {R}^{n}:\\,\\,\\,\\, z\\mapsto x:=(\\log (|z_{1}|^{2},...,\\log (|z_{n}|^{2}),$ of a convex function $\\phi (x)$ on ${R}^{n}$ such that the (sub-) gradient image $d\\phi ({R}^{n})$ is contained in $P.$ This correspondence has been mostly studied in the case when $X_{P}$ is smooth and the metric on $L$ is smooth and positively curved.",
"Then the corresponding convex function has the property that the differential (gradient) $d\\phi $ defines a diffeomorphism from ${R}^{n}$ to the interior of $P$ and moreover its Legendre transform satisfies Guillemin's boundary conditions (see [21] and references therein).",
"However, we stress that these refined regularity properties will play no role here.",
"Our normalizations are such that $\\mbox{(Log)}_{*}MA(\\Phi )=MA_{{R}}(\\phi ),$ where $MA_{{R}}(\\phi )$ denotes $n!$ times the usual real Monge-AmpÚre measure of the convex function $\\phi ,$ i.e.",
"$MA_{{R}}\\phi )(E):=n!\\int _{d\\phi (E)}dp,$ for any Borel measure $E.$ We also point out that in the case when $X_{P}$ is smooth one may assume that $P$ is contained in the positive octant with a vertex at 0 and the psh function $\\Phi $ can then by taken to be defined on all of ${C}^{n},$ as in the proof of Theorem REF .",
"The technical difficulity in the general case when $X_{P}$ may be singular - when seen from the complex point of view - is that, even if we may still arrange that $P$ is in the positive octant with a vertex at $0,$ the function $\\phi $ will a priori only be continuous on ${C}^{*n},$ i.e.",
"away from the coordinate axes.",
"Anyway, from the real point of view the latter theorem can be rephrased entirely in terms of convex analysis on convex domains in ${R}^{n}$ of the form $\\Omega :=\\lbrace \\psi <0\\rbrace $ where $\\psi $ is a convex function on ${R}^{n}$ such that its (sub-) gradient image of a convex body $P$ in the positive octant.",
"We propose to call such a domain monotone.",
"Note that, since $\\partial \\psi /\\partial x_{j}\\ge 0$ any monotone convex domain $\\Omega $ is invariant under the action of the additive semi-group $]-\\infty ,0]^{n}$ by translations.",
"In particular, $\\Omega $ is unbounded in contrast to the standard setting of bounded convex domains in convex analysis.",
"Still, it is not hard to generalize the usual properties valid for convex functions on bounded domains, as long as one works with bounded convex functions on $\\Omega .$ More precisely, a convenient function space to work with is the space $\\mathcal {H}(\\Omega ,\\psi )$ of all bounded convex functions $\\phi $ on $\\Omega $ such that $\\phi \\ge \\delta \\psi $ for some positive number $\\delta $ (depending on $\\phi ).$ For example, it is not hard to see that for any such $\\phi $ we have $\\phi =0$ on $\\partial \\Omega $ and the total real Monge-AmpÚre mass of $\\phi $ on $\\Omega $ is finite.",
"Now all the usual notions concerning psh functions on bounded pseudoconvex domains can be transported to the setting of monotone convex domains.",
"For example, we can define the Green function $g$ of $\\Omega $ (playing the role of the usual pluricomplex Green function for a domain in ${C}^{n}$ with a pole at the origin) by $g(x):=\\sup \\left\\lbrace \\phi (x):\\,\\,\\,\\phi \\in \\mathcal {H}(\\Omega ,\\psi ):\\,\\,\\phi (x)\\le \\log (\\sum _{i=1}^{n}e^{x_{i}})+O(1)\\right\\rbrace $ Then standard arguments show that $g_{\\Omega }$ is convex and continuous on $\\bar{\\Omega }$ and $g_{\\Omega }=0$ on $\\partial \\Omega $ $g(x)=\\log (\\sum _{i=1}^{n}e^{x_{i}})+O(1)$ $MA_{{R}}(g)=0$ and if $g_{t}:=\\log (e^{-2t}+e^{g_{\\Omega }})$ then $g_{t}\\in \\mathcal {H}(\\Omega ,\\psi )$ with $\\lim _{t\\rightarrow \\infty }\\int _{\\Omega }MA_{{R}}(g_{t})=1$ and $\\lim _{t\\rightarrow \\infty }\\int _{\\Omega }e^{-ng_{t}}d\\nu _{n}=\\infty ,$ where $d\\nu _{n}(x)=e^{\\sum _{i=1}^{n}x_{i}}dx,$ i.e.",
"$d\\nu _{n}$ is a multiple of the push-forward under the Log map of the Lebesgue measure on ${C}^{n}.$ Moreover, if $\\Phi $ is a $T^{n}-$ invariant psh function in ${C}^{n}$ satisfying the KÀhler-Einstien equation REF then its push-forward $\\psi $ satisfies the following real Monge-AmpÚre equation $MA_{{R}}(\\psi )=e^{-\\psi (x)}d\\nu _{n}(x),$ The following theorem, proved in [6], shows that the previous equation admits a solution iff $(1,...1)$ is the barycenter of $P:$ Theorem 3.1 [6] Let $P$ be a convex body and fix an element $p_{0}$ in $P.$ Then there is a smooth convex function $\\phi $ such that $d\\phi $ defines a diffeomorphism from ${R}^{n}$ to the interior of $P$ and such that $\\phi $ solves the equation $MA_{{R}}(\\phi )=e^{-\\phi (x)+\\left\\langle p_{0},x\\right\\rangle }dx$ on ${R}^{n}$ iff $p_{0}$ is the barycenter of $P.$ One can now go on to obtain Moser-Trudinger type inequalities (and Brezis-Merle-Demailly type inequalities) essentially as before, using Prekopa's convexity theorem in ${R}^{n}.$ The only technical difficulty is to make sure that the corresponding energy type functional $\\mathcal {E}_{{R}}(\\phi ):=\\frac{1}{(n+1)}\\int _{\\Omega }\\phi MA_{{R}}(\\phi )$ still has the appropriate properties (for its first and second derivatives along geodesics).",
"To this end one can first establish that if $M(\\phi _{1},...,\\phi _{n})$ denotes the real mixed Monge-AmpÚre measure obtained by polarizing the usual real Monge-AmpÚre measure, then the pairing $(\\phi _{0},\\phi _{1},...,\\phi _{n})\\mapsto \\int _{\\Omega }\\phi _{0}M(\\phi _{1},...,\\phi _{n}),$ is finite and symmetric on $\\mathcal {H}(\\Omega ,\\psi )^{n+1}.$ The new technical difficulty compared to the classical situation (see for example [35] and references therein) comes from the unboundedness of $\\Omega .$ But using that $\\Omega $ is complete “at infinity” in $\\Omega $ (wrt the Euclidean metric) one can use standard cut-off function argument to carry out the required integrations by parts.",
"Concretely, given a smooth compactly supported function $f$ on ${R}$ one can use the sequence $\\chi _{j}(x):=f(|x|/j)^{2}$ as cut-off functions on $\\Omega $ (this is sometimes referred to as “the Gaffney trick”).",
"The advantage of the ${R}^{n}-$ approach is that it applies equally well to the case when the corresponding toric variety $X_{P}$ is singular.",
"In fact, one may as well replace $P$ with any convex body, even though there is then no corresponding toric variety.",
"Mimicking the proof of Theorem REF , replacing $\\phi $ with a solution of the equation in Theorem REF one then obtains a proof of Theorem REF essentially as above.",
"Next, we explain how to deduce Cor REF from Theorem REF Given a real polytope $P$ with non-empty interior we can write it as $P=\\lbrace p\\in {R}^{n}:\\,\\,\\alpha _{F}(p)\\ge 0\\rbrace ,$ where $F$ is an index running over the facets $\\lbrace \\alpha _{F}=0\\rbrace $ of $P.$ We fix a vertex $v$ and affine functions $\\alpha _{F_{1}},....,\\alpha _{F_{N}}$ cutting out $n$ faces of $P$ meeting $v$ and spanning a cone of maximal dimension.",
"Then $P^{\\prime }=\\alpha (P),\\,\\,\\,\\,\\,\\,\\alpha (p):=(\\alpha _{F_{1}}(p)/\\alpha _{F_{1}}(b_{P}),...,\\alpha _{F_{n}}(p)/\\alpha _{F_{n}}(b_{P}))$ is an $n-$ dimensional polytope in the positive octant $[0,\\infty [^{n}$ such that $b_{P^{\\prime }}=(1,1,...,1).$ Moreover, if $P$ is a rational polytope as in the statement of Cor REF , then $\\mbox{Vol}(P^{\\prime })=\\frac{d}{a_{F_{1}}\\cdots a_{F_{n}}}\\mbox{Vol}(P)$ where $d$ is the determinant of the linear map $p\\mapsto (l_{F_{1}}(p),...,l_{F_{n}}(p)).$ But the map is represented by an invertible matrix with integer coefficients and hence $d$ is a positive integer and in particular $d\\ge 1.$ Since, by assumption, $a_{F_{i}}\\le 1$ it follows that $\\mbox{Vol}(P^{\\prime })\\ge \\mbox{Vol}(P)$ and hence it will be enough to prove Theorem REF .",
"Remark 3.2 As kindly pointed out to us by Bo'az Klartag Theorem REF can also be deduced from Grunbaum's inequality.",
"We thank him for allowing us to reproduce his elegant argument here.",
"First recall that the Grunbaum inequality says that if $P$ is a convex body, and $P_{-}$ denotes the intersection of $P$ with an affine half-space defined by one side of a hyperplane $H$ passing through the barycenter of $P,$ then $\\mbox{Vol}(P_{-})\\ge \\left(\\frac{n}{(n+1)}\\right)^{n}\\mbox{Vol}(P).$ In particular, if $P$ is a convex body as in the statement of Theorem REF we can take $P_{-}$ to be $n$ times the unit-simplex $\\Delta $ in the positive octant with one vertex at $0.$ Since, $\\mbox{Vol}(P_{-})\\le \\mbox{Vol}(n\\Delta )=n^{n}/n!$ this gives the desired inequality."
],
[
"The homogeneous case",
"Let us start by specializing Theorem REF to the case when $X$ is a rational homogenuous space.",
"Even though this is the simples case the corresponding degree bound, when rephrased in terms of representation theory, is highly non-trival and was first obtained by Snow [54].",
"Let us first recall some basic representation theory (see [54] and references therein).",
"Let $K$ be a compact complex semi-simple Lie group and denote by $G$ its complexification.",
"Under the adjoint action of a fixed maximal torus $T$ the Lie algebra $L(G)$ decomposes as $L(G)=L(T_{c})\\oplus E_{+}\\oplus \\overline{E}_{+},\\,\\,\\, E_{+}=\\bigoplus _{\\alpha \\in R^{+}}E_{\\alpha },$ where $R_{+}$ is a consistent choice of positive roots $\\alpha \\in L(T)^{*}$ and $E_{\\alpha }$ denote the corresponding weight spaces, i.e.",
"$E_{\\alpha }$ is generated by a vector $Z_{\\alpha }$ such that $[t,Z_{\\alpha }]=i\\left\\langle t,\\alpha \\right\\rangle Z_{\\alpha }$ for any $t\\in L(T).$ A consistent choice of positive roots $R_{+}$ corresponds to a choice of Borel group $B$ with Lie algebra $L(B)=L(T)\\oplus \\overline{E}_{+}$ The corresponding (complete) flag variety is defined as the $G-$ homogenuous compact complex manifold $G/B(=K/T).$ As is well-known any rational $G-$ homogeneous compact complex manifold $X$ may be written as $X=G/\\mathcal {P},$ for some $G$ as above and a parabolic subgroup $\\mathcal {P}$ of $G$ (i.e.",
"a subgroup containing a Borel group which we may assume is $B).$ We recall that any $G-$ homogeneous line bundle $L\\rightarrow G/\\mathcal {P}$ is determined by a weight $\\lambda ,$ i.e.",
"an element in the weight lattice of $L(T_{c})^{*}.$ Indeed, $L_{\\lambda }=G\\times _{(P,\\rho _{\\lambda })}{C},$ where $\\rho _{\\lambda }$ is a homomorphism $P\\rightarrow {C}^{*},$ which is uniquely detetermined by its restriction to the complex torus in $P$ and hence determined by an element $\\lambda $ in the weight lattice of $L(T_{c})^{*}.$ By construction, the tangent bundle $TX$ is generated at the identity coset by root vectors $Z_{\\alpha }\\in E_{\\alpha }$ for $\\alpha $ in a subset $R_{X}^{+}$ of the positive roots.",
"In particular, the weight $\\lambda _{X}$ of the anti-canonical line bundle $-K_{X}$ may be written as $\\lambda _{X}=\\sum _{\\alpha \\in R_{X}^{+}}\\alpha ,$ defining an ample line bundle, so that $X$ is Fano.",
"Moreover, by the general Weyl character formula, $c_{1}(L_{\\lambda })^{n}=n!\\prod _{\\alpha \\in R_{X}^{+}}\\frac{\\left\\langle \\alpha ,\\lambda \\right\\rangle }{\\left\\langle \\alpha ,\\rho \\right\\rangle },$ with $\\rho $ denoting, as usual, the half-sum of all the positive roots $\\alpha $ and where $\\left\\langle \\cdot ,\\cdot \\right\\rangle $ denotes the Killing form.",
"Hence, Theorem REF applied to $G/\\mathcal {P}$ translates to a Lie algebra statement first shown by Snow [54].",
"Indeed, as shown in [54] (Prop 1) formula REF can be simplified using the Dynkin diagram of $G.$ Using this latter formula and the Weyl character formula above together with classification theory for semi-simple Lie groups, Snow then shows, using quite elaborate calculations, how to deduce the bound in Theorem REF for the Fano manifold $X=G/\\mathcal {P}.$ It should also be pointed out that the results in [54] also give that ${P}^{n}$ is the unique maximizer of the degree among all partial flag manifolds."
],
[
"The case of spherical varities and multiplicity free actions",
"In this section we will reformulate Theorem REF in the case when $X$ is spherical variety, using Brion's formula for the volume [11] of a line bundle on $X.$ We will use the symplecto-geometric formulation (see the end of [11] and [10] where further references can be found).",
"Let us start by recalling the definition of the Duistermaat-Heckman measure in symplectic geometry.",
"Let $(X,\\omega )$ be a symplectic manifold and $K$ a compact connected Lie group acting by symplectomorphisms on $X$ and fix a compact maximal torus $T$ in $K.$ Assume for simplicity that the first Betti number of $X$ vanishes (which will be the case here since $X$ will be a Fano manifold).",
"Then there is a moment map $\\mu _{T}:\\,\\, X\\rightarrow L(T)^{*},$ where $L(T)$ denotes the real vector space given by the Lie algebra of $T.$ The image $P:=\\mu _{T}(X)$ is a convex rational polytope and the density $v(p)$ of the Duistermaat-Heckman measure $(\\mu _{T})_{*}\\omega ^{n}/n!$ on $\\mu _{T}(X)$ is continuous and piecewise polynomial.",
"The convex polytope obtained by intersecting $\\mu _{T}(X)$ with a fixed positive Weyl chamber $\\Lambda _{+}$ in $L(T)^{*}$ is called the moment polytope.",
"The action of $K$ is said to be multiplicity-free if the group of symplectomorphisms of $(X,\\omega )$ which commute with $K$ is abelian ( equivalently, all $K-$ invariant functions on $X$ Poisson commute).",
"As is well-known [10] any $G-$ spherical non-singular complex algebraic variety $X$ with a $G-$ equivariant ample line bundle $L\\rightarrow X$ equipped with a fixed positive curvature form $\\omega $ in $c_{1}(L)$ corresponds to a symplectic manifold $(X,\\omega )$ with a symplectic action by the real form $K$ of $G$ which is multiplicity-free.",
"Moreover, as shown by Brion [11], in the spherical (i.e.",
"multiplicity free) case the density $v$ of the Duistermaat-Heckman measure is explicitly given by $v(p)=\\prod _{\\alpha }\\frac{\\left\\langle \\alpha ,p\\right\\rangle }{\\left\\langle \\alpha ,\\rho \\right\\rangle },$ where $\\rho $ denotes the half-sum of all the positive roots $\\alpha $ and the products runs over all positive roots $\\alpha $ such that $\\left\\langle \\alpha ,p\\right\\rangle >0.$ Moreover, the Lesbesgue measure $dp$ has been normalized to give unit-volume to the fundamental domain of the lattice in $P$ (see section REF ).",
"By the definition of $v$ as the density of the Duistermaat-Heckman measure $\\int _{X}\\omega ^{n}/n!=\\int _{P}v(p)dp$ Hence, applying Theorem REF to a spherical non-singular Fano variety gives the following Corollary 4.1 Let $(X,\\omega )$ be symplectic manifold with a symplectic action by a compact Lie group $K$ which is multiplicity-free and denote by $P$ the image of the moment map associated to a fixed maximal torus $T$ in $K.$ If $X$ admits an integrable $\\omega -$ compatible complex structure $J$ preserved by $K$ and such that the cohomology class $[\\omega ]$ contains a KÀhler-Einstein metric on $(X,J)$ then $\\int _{P}v(p)dp\\le (n+1)^{n}/n!$ Equality hold when $(X,\\omega )$ is complex projective space equipped and $\\omega $ is the standard suitably normalized $SU(n)-$ invariant symplectic form (i.e.",
"$\\omega $ is $(n+1)$ times the Fubini-Study form).",
"We recall that the condition that a spherical variety $X$ be Fano can be expressed rather explicitly in algebro-geometric terms [12].",
"Remark 4.2 It is well-known that any spherical variety $X$ has a holomorphic ${C}^{*}-$ action such that the fixed point set.",
"$X^{{C}^{*}}$ is finite, so that Theorem REF indeed can be applied to $X.$ Let us briefly recall the reason that $X^{{C}^{*}}$ is finite (as kindly explained to us by Michel Brion).",
"The starting point is the fundamental fact that any $G-$ spherical variety $X$ can be covered by a finite number of $G-$ orbits [43].",
"Next one shows that if $G$ is a reductive group acting transitively on a set $Y$ (here an orbit of $G)$ then the fixed point set $Y^{T_{c}}$ is finite for any maximal complex torus $T_{c}$ in $G.$ Indeed, the Weyl group $N_{G}(T_{c})/T_{c}$ is finite if $G$ is reductive and its acts transitively on $Y^{T_{c}},$ as follows from the definition of the normalizer $N_{G}(T_{c})$ (see Proposition 7.2 in [18]).",
"Finally, it is a general fact that for a generic (regular) one-parameter subgroup ${C}^{*}$ in $T_{c}$ one has that $Y^{T}=Y^{{C}^{*}}$ (as can be proved by reducing the problem to the linear action of $T$ on a vector space using [55]) In particular, the previous corollary applies to horospherical Fano varieties.",
"These are homogenous toric bundles over a rational homogenous variety [51], [49].",
"Let us for simplicity consider the case of homogenous fibrations $X$ over a (complete) flag variety: $X\\rightarrow G/B$ where any fiber is biholomorphic to a given toric variety $F.$ Then the corresponding polytope $P$ is contained in the interiour of a positive Weyl chamber, coinciding with the moment polytope of $F$ under the induced torus action.",
"Moreover, $X$ is Fano with a KÀhler-Einstein metric iff $F$ is Fano and the sum of the positive roots $\\sum \\alpha $ coincides with the barycenter of the reflexive polytope $P$ wrt the Duistermaat-Heckman measure $vdp$ (this was first shown in [51], but see also the illuminating discussion in section 4.1 in [21]).",
"Hence, we arrive at the following Corollary 4.3 Let $G$ be a semi-simple complex Lie group and fix a maximal torus $T$ in $G$ and a set $R^{+}$ of $n$ positive roots $\\alpha _{i}$ for the Lie algebra of $G.$ Let $P$ be a reflexive lattice polytope in the positive Weyl chamber of $L(T)^{*}$ which is Delzant and such that $\\sum _{\\alpha \\in R^{+}}\\alpha $ is the barycenter of $P$ wrt the Duistermaat-Heckman measure $vdp.$ Then $\\int _{P}vdp\\le (n+1)^{n}/n!$ We expect that the condition that $P$ be Delzant, i.e.",
"the corresponding toric variety is smooth, can be removed.",
"Finally, let us mention the connection to Okounkov bodies.",
"As shown by Okounkov [48] one can associate another convex polytope $\\Delta $ to a polarized spherical variety $X,$ such that $\\Delta $ fibers over the moment polytope $P$ (the fibers being the Gelfand-Cetlin string polytopes).",
"The definition is made so that $c_{1}(L)^{n}/n!=\\mbox{Vol}(\\Delta )$ More generally, to any polarized projective variety $(X,L)$ there is convex body $\\Delta $ associated (further depending on an auxiliary choice of flag in $X)$ such that the previous formula for $c_{1}(L)^{n}/n!$ holds [40], [33].",
"In the light of Theorem REF and the toric case it would be interesting to know if the condition that $X$ be Fano (and $L=-K_{X})$ with a KÀhler-Einstein metric can be naturally expressed in terms of properties of $\\Delta ?$ See also [1] for the case of reductive spherical varieties."
]
] | 1204.1308 |
[
[
"Flows at the Edge of an Active Region: Observation and Interpretation"
],
[
"Abstract Upflows observed at the edges of active regions have been proposed as the source of the slow solar wind.",
"In the particular case of Active Region (AR) 10942, where such an upflow has been already observed, we want to evaluate the part of this upflow that actually remains confined in the magnetic loops that connect AR10942 to AR10943.",
"Both active regions were visible simultaneously on the solar disk and were observed by STEREO/SECCHI EUVI.",
"Using Hinode/EIS spectra, we determine the Doppler shifts and densities in AR10943 and AR10942, in order to evaluate the mass flows.",
"We also perform magnetic field extrapolations to assess the connectivity between AR10942 and AR10943.",
"AR10943 displays a persistent downflow in Fe XII.",
"Magnetic extrapolations including both ARs show that this downflow can be connected to the upflow in AR10942.",
"We estimate that the mass flow received by AR10943 areas connected to AR10942 represents about 18% of the mass flow from AR10942.",
"We conclude that the upflows observed on the edge of active regions represent either large-scale loops with mass flowing along them (accounting for about one-fifth of the total mass flow in this example) or open magnetic field structures where the slow solar wind originates."
],
[
"Introduction",
"The Sun interacts with the whole heliosphere, and in particular with the planets of the solar system, through the solar wind.",
"As the plasma $\\beta $ (ratio of the plasma pressure to the magnetic pressure) is low in the low corona, the dynamics of the plasma is dominated by the magnetic field (frozen-in condition) implying that the plasma material is flowing along magnetic field lines.",
"In particular, the fast solar wind follows open magnetic field lines from solar coronal holes to the interplanetary space but the sources of the slow wind remain an open issue.",
"Fast and slow winds can be distinguished according to their speeds (around 600 and $300\\,\\textrm {km}\\cdot \\textrm {s}^{-1}$ respectively) and their composition, but this is not the only difference between both these types.",
"The fast wind is expelled from coronal holes, especially at the poles , and possibly from the intersections of chromospheric network boundaries in coronal holes .",
"The slow solar wind is not as well understood as the fast one, for various reasons probably related to its time variability.",
"Its transient nature and its relation with large-scale coronal structures, have been revealed from both in-situ (e.g. )",
"and remote-sensing observations (e.g.",
"the blobs of ).",
"As for the coronal sources, the edges of coronal holes (Coronal Hole Boundaries) have been proposed as the location of reconnection between coronal hole (CH) and non-CH magnetic fields, because of the differential rotation between these two kinds of regions , .",
"Recent spectrocopic and imaging observations with SUMER/SOHO and XRT/Hinode seem to support this mechanism (, ).",
"However, other locations and mechanisms have been proposed, such as streamer boundaries and the edges of Active Regions , , .",
"As we shall see below, this latter possibility has been very recently put forward in the context of an Active Region close to an \"open field\" region, an issue we focus on in this Paper.",
"Active regions (ARs) in the Sun's atmosphere are composed of closed multi-temperature loops in the solar corona.",
"Recently a specific flows distribution has been shown for some ARs.",
"It corresponds to redshifts inside loops (, ) and blueshifts at the edge of active regions (, , , , ).",
"In the last two papers, according to magnetic field extrapolations, blueshifts are observed along fanning out, far-reaching or even open field lines.",
"Then the observed flows seem to occur at the boundaries between active regions and coronal holes.",
"Thus the outflows can supply mass to the solar wind, as suggested by scintillation measurements at 2.5 solar radii, these outflows coming from an actually open region at the edge of the active region .",
"Active Region 10942 has already been extensively studied: Fast upflows have been observed at the North-East edge of this AR from Hinode/EIS Doppler shifts in Fe12 195.12 Å , , and from apparent flows in Hinode/XRT and TRACE time series.",
"These upflows have been proposed as a source of the slow solar wind.",
"Linear force-free , and potential field source surface extrapolations show open field lines in agreement with this hypothesis.",
"have also noticed that AR 10942 was connected by large scale loops to a magnetic dipole, approximately 400 arcsec away, which actually is AR 10943.",
"This is a clear evidence for magnetic connection, but matter exchange has never been quantified.",
"Large scale loops connecting two active regions have been observed since Skylab and more recently with SDO .",
"The connection between far-distance active regions has also been observed in the case of transequatorial loops whether they are elongated stable structures , or ephemeral loops related to flare and filament eruption .",
"Of special interest here are the spectroscopic observations of a transequatorial loop which indicate that the loop plasma was multithermal and covered roughly 2 orders of magnitude in temperature .",
"Moreover line-of-sight steady flows of the order of 30 to 40 km.s$^{-1}$ were detected and interpreted as a necessary condition for maintaining the loop structure.",
"The above discussions on flows in active regions should not lead us to forget the issue of the \"rest\" wavelengths used to define absolutely the flows, even if these flows are relatively important in AR.",
"Because the region taken as a reference is often the quiet Sun (QS), the issue of average temperature-dependent flows in the QS is critical.",
"mention average line shifts at 1 MK < T < 1.8 MK bluer than those observed at 1 MK (about -1.8 $\\pm $ 0.6 km.s$^{-1}$ ) translating into a maximum Doppler shift of -4.4 $\\pm $ 2.2 km.s$^{-1}$ around 1.8 MK.",
"If we assume that the actual uncertainties are of the order of 2 km.s$^{-1}$ , one immediately sees that the sign itself of the velocities (flows) may be changed.",
"This clearly shows the need of a very careful determination of the wavelength reference.",
"In this paper we set out to further explore the link between Active Regions 10942 and 10943.",
"In Sec.",
"we use EUV image and spectroscopic observations to analyze the flows in AR 10943.",
"A special attention has been paid to the determination of the velocities taking account of the global flow velocities dependance on temperature in the solar atmosphere , .",
"To understand the magnetic connection between both regions, we compute magnetic field extrapolations in Sec. .",
"Finally, the results are discussed in Sec.",
"and we conclude in Sec. .",
"On 2007 February 20, less than four months after its launch, the heliocentric separation angle between the STEREO B probe and the Earth was still negligible ($0.1\\,\\deg $ ).",
"This means that STEREO B/SECCHI EUVI images , have the same viewing angle than SoHO and Hinode and that we can use these images in combination with SoHO/MDI and Hinode/EIS.",
"STEREO/SECCHI EUVI was in its normal mode, with full-disk observations at a cadence of 10 min in the EUV channel $\\lambda $ 195Å.",
"We selected the 8:05 UT observation corrected by EUVI_prep from the SolarSoft library, shown in Fig.REF .",
"In this image, both active regions can be seen simultaneously.",
"On the eastern side, the EUVI image in 195 Å displays the AR 10942 complex made of a set of loops connecting the two extreme polarities (see Fig.",
"REF ) of the AR, mainly in the southern side.",
"Some straight (mostly fan-like) structures are also clearly seen on the north-eastern side (X = -400, Y = -50 to 0, see Fig.",
"REF ) which are candidate for open magnetic fields.",
"At South, below AR 10942, internal loops, rather compact structures (X = -300, Y = -300) seem to be the feet of (apparently) very sheared loops.",
"On the western side of the image, another smaller and compact AR (10943) does not seem to be connected with its neighbouring regions, except for a set of diffuse loops at the East of AR 10943 whose feet seem to be located in between the two ARs.",
"One also notes that the lower side of these diffuse loops is very sharp.",
"The overall picture is that the two ARs seem to be very disconnected on one hand, and that the eastern feet of the diffuse loops mentioned above could coincide with the QSL labelled “e” in Fig.4 of , on the other hand.",
"The EUVI image in the hot 284Å line confirms this picture, contrary to the He2 image where the chromosphere between the two AR does not seem to be very perturbed.",
"Finally, the 171 image is more puzzling because it does not display the above (too) hot loops but also shows some dark area on the western side of AR 10942, which could be a coronal hole or a filament channel."
],
[
"Spectroscopic observations",
"Spectroscopic information can be obtained from the Hinode/EUV imaging spectrometer (EIS; ).",
"We select two raster scans on 2007 February 20 that covers part of AR 10943 (at 05:47 UT, study ID 45) and part of AR 10942 (at 11:16 UT, study ID 57) respectively in order to have full spectra (around selected spectral lines) as a function of both solar dimensions in this region.",
"The slit positions during these raster scans are shown in the STEREO image (Fig.",
"REF ) and in Table REF .",
"Both raster scans are partial on the active regions.",
"Nevertheless, as we show with magnetic field extrapolation in Sec , the feet of the interconnecting loops between the two active regions are located at the respective edges of the active regions in the FOV of EIS.",
"So the raster scans are sufficient for our study.",
"The delay of the scans and between the scans are negligible in comparison with the time of the continuous flows which are visible on a few days.",
"We apply the correction procedures eis_prep and eis_slit_tilt from the SolarSoft library.",
"An additional correction must be applied for the orbital temperature variation of EIS; for this we have chosen to develop a specific method which is described in Appendix : we use the orbital variation in the He2 256.32 Å line to correct the orbital lineshifts in other lines.",
"Indeed, this line is chromospheric and optically thick, the insensitivity of its centroid with respect to activity allows us to better isolate orbital variations.",
"We focus on the Fe12 195.12 Å line which is emitted around $\\log T = 6.1$ .",
"We produce intensity (Fig.",
"REF a for AR 10943 and Fig.",
"REF a for AR 10942) and Doppler velocity (Fig.",
"REF b for AR 10943 and Fig.",
"REF b for AR 10942) maps deduced from the parameters of a single Gaussian fit of this line using the correction of orbital variation.",
"We choose to ignore the self-blending of Fe12 195.12 Å with Fe12 195.18 Å because we do not focus on the width (which is the most influenced parameter) and our study concerns low density structures where the contribution of Fe12 195.18 Å is negligible .",
"The level 2 data show that the velocities in the active region are of the same order in the other EIS windows available except for Fe13 196.54 Å where that pattern is reversed.",
"In the Fe12 186.88 Å and Ca17 192.82 Å lines, the core of the AR is blue but the other structures are the same.",
"We cannot conclude about the Fe13 196.54 Å and Fe12 186.88 Å lines velocities because they are far different from the Fe13 202.04 Å and Fe12 195.12 Å respectively.",
"The automatic analysis could not be sufficient.",
"The Ca17 192.82 Å line shows that the flow is upward in the core of the AR at very high temperature ($\\log T_{max}=6.7$ ) but there are still downflows in the vicinity of the active region.",
"However the flow rate through a surface perpendicular to the line-of-sight is higher there than in the rest of the whirl.",
"These downflows are persistent for a few days, therefore an impulsive event such as a jet is excluded as a viable mechanism producing these localised redshifts.",
"One also notes a redshifted area in the top third of the raster FOV.",
"This area is not studied here for two reasons: it is not magnetically connected to AR 10942 (see Sec.",
"REF ) and the redshift is rather weak (a few km.s$^{-1}$ ).",
"We get a density map (Fig.",
"REF c for AR 10943 and Fig.",
"REF c for AR 10942) by also fitting the Fe12 196.64 Å line, and computing the Fe12 196.64 Å/ Fe12 195.12 Å intensity ratio, which is sensitive to density.",
"The density is deduced from the theoretical intensity ratio produced by the CHIANTI atomic database , , as shown in Fig.",
"REF .",
"In order to derive the flow rate through a surface perpendicular to the line-of-sight (Fig.",
"REF d for AR 10943 and Fig.",
"REF d for AR 10942), we multiply the density by the Doppler velocity.",
"The core of AR 10943 is characterized by hot loops bright in intensity (with a maximum of 9500 counts/pix, Fig.",
"REF a), downward Doppler velocities and high densities between $4\\times 10^9$ cm$^{-3}$ and $1\\times 10^{10}$ cm$^{-3}$ .",
"These densities are consistent with values in active region loops by ($1.3\\times 10^9$ and $9.5\\times 10^{10}$ cm$^{-3}$ for Fe12) and with ($3\\times 10^8$ cm$^{-3}$ $\\le n_e \\le $ $1\\times 10^{11}$ cm$^{-3}$ ).",
"The whirl of faint plasma around the core of AR 10943 is mostly blueshifted, with electron density between $6.3\\times 10^8$ and $1\\times 10^9$ cm$^{-3}$ ; moreover there is a clear straight redshifted (up to 16 km.s$^{-1}$ ) structure cutting the whirl in the South-East edge where the density is notably low (around $5\\times 10^8$ cm$^{-3}$ ) but higher than Quiet Sun densities for Fe12 ($n_e=2.5-3.2\\times 10^8$ cm$^{-3}$ , )."
],
[
"Magnetic field observations",
"We investigate the possible magnetic connectivity between the two active regions AR 10942 and AR 10943.",
"We use a SOHO/MDI level 1.8 96 minutes line-of-sight magnetogram (see Fig.REF ) to study the distribution of the photospheric magnetic field.",
"The SOHO/MDI magnetograms have been recorded on 2007 February 20 at $T_{ref}=08:05$ UT (between the raster scans analysed in Sec.",
"and simultaneous with the STEREO SECCHI/EUVI image of fig.",
"REF ).",
"We select an area encompassing the active regions (heliocentric coordinate $x$ from -720 to 275 arcsec and $y$ from -321 to 177 arcsec).",
"The total unsigned magnetic flux for this area is $4.05\\times 10^{22}$ Mx and the flux unbalance is only about 1.7% (a very low value for extrapolation).",
"The total unsigned flux for AR 10942 is $9.90\\times 10^{21}$ Mx with a negative flux of $4.44\\times 10^{21}$ Mx and a positive flux of $5.46\\times 10^{21}$ Mx.",
"For AR 10943, the total unsigned flux is $5.53\\times 10^{21}$ Mx with a negative flux of $4.12\\times 10^{21}$ Mx and a positive flux of $1.41\\times 10^{21}$ Mx.",
"The net magnetic flux for AR 10943 is about 50% of the total flux.",
"AR 10943 is in excess of negative flux, while AR 10942 is in excess of positive flux.",
"If a magnetic connection exists between the two active regions, thus this connection is between the positive polarity of AR 10942 and the negative polarity of AR 10943."
]
] | 1204.1377 |
[
[
"Measurement of the charge asymmetry in top quark pair production in pp\n collision data at sqrt(s) = 7 TeV using the ATLAS detector"
],
[
"Abstract A measurement of the charge asymmetry in the production of top quark pairs in the semileptonic decay channel has been performed.",
"A dataset corresponding to an integrated luminosity of 1.04 inverse femtobarn, obtained at a centre-of-mass energy of 7 TeV with the ATLAS experiment at the LHC, was used.",
"After performing a selection of events with one isolated lepton, at least four jets and missing transverse energy, a kinematic fit was performed to reconstruct the top-antitop event topology.",
"The charge asymmetry was determined using the differential distribution of the difference of absolute reconstructed rapidities of the top and antitop quark.",
"An unfolding procedure was applied to correct for detector acceptance and resolution effects and to obtain the corresponding distribution at parton level.",
"The total charge asymmetry after unfolding was measured to be A_C = -0.018 +/- 0.028 (stat.)",
"+/- 0.023 (syst.)",
"in agreement with the Standard Model prediction.",
"In addition, a simultaneous unfolding in the difference of absolute reconstructed rapidities of the top and antitop quark and the invariant mass of the top-antitop pair was performed."
],
[
"Introduction",
"The field of Elementary Particle Physics is concerned with understanding the most fundamental building blocks of Nature and their interactions.",
"The Standard Model of Particle Physics[1], [2], [3], [4], [5], [6] is one of the most successful and thorough theories in physics, its predictions being in astonishing agreement with the observed phenomena to highest precision.",
"Regardless of its success in explaining the interactions of all fundamental particles, the Standard Model is not without shortcomings.",
"The fine-tuning of radiative corrections to the Higgs boson mass[7], [8], the strong evidence for the existence of Dark Matter and Dark Energy, and a missing mechanism to describe gravity within the framework of a quantum field theory are only a few of the remaining ambiguities.",
"According to current knowledge, there exist six quarks in Nature: the up, down, strange, charm, bottom and the top quark along with the six known leptons (electron, muon and tau together with their corresponding neutrinos).",
"Quarks and leptons are grouped into three subsets or generations of a quark and lepton doublet each.",
"The top quark is the heaviest known elementary particle and has been the focus of studies for several decades, from indirect searches using electroweak precision data to its discovery by the CDF[9] and DØ[10] experiments at the Tevatron in 1995, the observation of the single top quark[11], [12] in 2009, to newest precision measurements of its properties at the Tevatron and the LHC.",
"Its unique characteristics, namely the large mass of about $173\\,\\text{GeV}$[13] and its short lifetime, provide the opportunity to perform precise measurements of electroweak interactions.",
"Due to the affinity of its mass to the electroweak scale and the potential link to the vacuum expectation value of the Higgs field, the top quark also allows indirect constraints of the Higgs mass in combination with precision measurements of the $W$ boson mass.",
"Furthermore, since the top quark is the only quark which has a decay width larger than the hadronisation scale, it does not form hadronic bound states.",
"As a result, top quark properties, such as its spin, are accessible without being obscured by the process of hadronisation.",
"At hadron colliders, such as the Tevatron or the Large Hadron Collider, top quark pairs are mainly produced via the strong interaction, either via gluon-gluon fusion or via quark-antiquark annihilation: $q + \\bar{q} & \\rightarrow & t + \\bar{t}\\text{,} \\nonumber \\\\g + g & \\rightarrow & t + \\bar{t}\\text{.}",
"\\nonumber $ In the Born approximation, these production mechanisms are entirely symmetric under the exchange of the final state top and antitop quark.",
"Consequently, there is no angular discrimination between the top and antitop quark and the resulting predicted differential distributions are identical for both particles.",
"In the Standard Model, an asymmetry in the production of top quark pairs arises due to radiative corrections from virtual and real gluon emission if higher order corrections are taken into account.",
"These higher order corrections introduce interferences between amplitudes which are odd under the exchange of the final state quark and antiquark.",
"Interference terms of final state and initial state gluon bremsstrahlung, and of higher order amplitudes with Born level amplitudes contribute to an overall imbalance of the differential distributions of the final state top quark and antitop quark.",
"An additional small contribution originates from the interference of different amplitudes in quark-gluon scattering: $g + q & \\rightarrow & t + \\bar{t} + q^{\\prime }\\text{.}",
"\\nonumber $ The measurement of the charge asymmetry in top quark pair production provides the opportunity to verify perturbative Quantum Chromodynamics and consequently, the Standard Model.",
"Moreover, a similar effect is predicted and observed in Quantum Electrodynamics[14], [15], [16], where radiative corrections lead to an asymmetry in the electroweak production of fermion-antifermion pairs.",
"Since this effect has been studied to very high precision, the verification of its counterpart in Quantum Chromodynamics would be yet another confirmation of the Standard Model and its predictions.",
"Furthermore, potential new physics, in particular theories involving the breaking of electroweak symmetry, could lead to deviations from the Standard Model expectation due to large anomalous couplings to the top quark predicted in numerous theoretical models.",
"As a matter of fact, current independent measurements performed by CDF[17], [18] and DØ[19], [20] suggest a possible discrepancy between the predicted and observed charge asymmetry in proton-antiproton collisions.",
"This effect is observed in particular for high invariant $\\mbox{$t\\bar{t}$}$ masses and high $\\mbox{$t\\bar{t}$}$ rapidity differences, which is supported by several models predicting physics beyond the Standard Model.",
"Since the predicted charge asymmetry is small at hadron colliders due to the probabilistic nature of the initial state parton kinematics, precise knowledge of the detection mechanisms, sophisticated analysis methods and detailed understanding of potential systematic effects are crucial to accomplish such a measurement.",
"This is in particular true for the Large Hadron Collider, where a high centre-of-mass energy and a symmetric hadronic initial state ($pp$ ) make this measurement even more difficult.",
"The increased fraction of top quark pairs produced via (charge symmetric) gluon-gluon fusion lead to a dilution of the measured asymmetry.",
"In addition, there is no preferred initial state quark direction in proton-proton collisions and hence no resulting forward-backward asymmetry which could be measured directly as it is the case at the Tevatron.",
"Consequently, a new analysis concept and new observables have to be considered to perform this measurement under the conditions of the LHC.",
"The charge asymmetry has been measured in top quark pair production at the Tevatron by both the CDF[17] and DØ[19] collaborations and preliminary results have also been shown by the CMS[21] experiment at the LHC.",
"This thesis describes a measurement of the charge asymmetry in top quark pair production which has been performed with the ATLAS experiment for the first time in 2011[22].",
"This document is organised as follows: Chapter gives an introduction to the theoretical aspects of the Standard Model and theories beyond, paying special attention to the top quark and the charge asymmetry in top quark pair production.",
"Chapter covers technical aspects of the LHC and the ATLAS detector.",
"Definition and description of the objects taken into account for the described analysis and the trigger strategy is given in Chapter , followed by a summary of the event selection performed to increase the fraction of relevant signal events with respect to various background processes in Chapter .",
"A summary of the data and Monte Carlo samples used, and a description of data driven methods to determine the background contributions from $W$ +jets and QCD multijets, are given in Chapter .",
"A detailed explanation of the reconstruction method used to obtain parton level information of the top and antitop quark based on measured quantities follows in Chapter .",
"An unfolding approach performed to account for detector acceptance and resolution is described in Chapter , followed by Chapter , covering relevant systematic uncertainties affecting the analysis.",
"Finally, the results of the analysis are presented in Chapter and a summary of this thesis is given in Chapter .",
"In Nature, all observed matter consists of building blocks which are considered to be elementary, the leptons and quarks, shown schematically in Figure REF.",
"They are classified into three generations or families with increasing order of quark masses.",
"Figure: Summary of elementary particles.",
"Aside from the quarks (upper left box) and leptons (lower left box), the gauge bosons (right vertical box) and the hypothetical Higgs boson are shown.Leptons and quarks obey Fermi-Dirac statistics and hence carry a non-integer spin.",
"Leptons are described by their quantum numbers, electric charge ($Q$ ), the third component of the weak isospin ($T_3$ ), and the weak hypercharge ($Y_{\\rm {W}} = 2(Q - T_3)$ ), summarised in Table REF.",
"Table: Leptons and their properties and quantum numbers, ordered by generation.",
"Where no uncertainty on the mass is given, it is negligible at the given precision.Similarly, quarks are assigned the quantum numbers charge ($Q$ ), the third component of the weak isospin ($T_3$ ), and hypercharge ($Y = 2(Q - T_3)$ ), describing qualities of related hadronic bound states.",
"A summary of these properties can be found in Table REF.",
"Table: Quarks and their properties and quantum numbers, ordered by generation.Furthermore, all quarks carry a colour charge, denoted by a respective quantum number which can take the values red, green and blue.",
"The first generation is constituted by the up ($u$ ) and down ($d$ ) quark doublet (the building blocks of protons and neutrons) alongside the electron ($e$ ) and the electron-neutrino ($\\nu _e$ ).",
"The second generation contains the charm ($c$ ), strange ($s$ ) quarks, the muon ($\\mu $ ) and muon-neutrino ($\\nu _{\\mu }$ ).",
"Finally, the top ($t$ ) and bottom ($b$ ) quark compose the third generation, together with the tau ($\\tau $ ) and the tau-neutrino ($\\nu _{\\tau }$ ) in the lepton sector.",
"The top quark assumes a quite distinct role among the other quarks due to its large mass of $(173.2 \\pm 0.9)\\,\\text{GeV}$[13].",
"Within the Standard Model, particle interactions are described by a quantum field theory consistent with both quantum mechanics and special relativity, combining the electroweak theory and Quantum Chromodynamics into a structure denoted by the gauge symmetry group $\\text{SU(3)}_C \\times \\text{SU(2)}_L \\times \\text{U(1)}_Y$.",
"This structure describes colour charge ($C$ ), weak isospin ($L$ ) and hypercharge ($Y$ ) gauge groups.",
"The underlying gauge theory is non-Abelian due to the non-commutative nature of the SU(3) and SU(2) field strength tensors.",
"Given the matter and gauge fields, and requiring local gauge invariance and renormalisability, the Standard Model Lagrangian can be constructed as $\\mathcal {L}_{\\text{SM}} & = & \\mathcal {L}_{\\text{SU(3)}} + \\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}} \\\\& = & \\mathcal {L}_{\\text{SU(3)}}^{\\text{Gauge}} + \\mathcal {L}_{\\text{SU(3)}}^{\\text{Matter\\phantom{g}\\!\\!}}",
"+ \\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Gauge}} + \\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Matter\\phantom{g}}} + \\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Higgs}} + \\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Yukawa\\phantom{g}}}.$ The first term, describing strong gauge interactions, is given by $\\mathcal {L}_{\\text{SU(3)}}^{\\text{Gauge}} = \\frac{1}{2 g_{S}^{2}} \\operatorname{Tr}{G^{\\mu \\nu } G_{\\mu \\nu }},$ where $G^{\\mu \\nu }$ is the gluon field strength tensor and $g_{S}$ is the strong gauge coupling constant.",
"Strong interactions are described by perturbative Quantum Chromodynamics (QCD).",
"The matter term of the SU(3) Lagrangian contains the gauge covariant derivatives of the quarks: $\\mathcal {L}_{\\text{SU(3)}}^{\\text{Matter\\phantom{g}\\!\\!}}",
"= i \\bar{q}_{i \\alpha } \\mbox{$\\lnot \\!\\!D$}_{\\beta }^{\\alpha } q_{i}^{\\beta },$ where $\\alpha , \\beta \\in [1,2,3]$ are the quark colour indices.",
"A summation over the quark flavour index $i$ is implied and it is $\\mbox{$\\lnot \\!\\!D$}_{\\mu \\beta }^{\\alpha } = \\partial _{\\mu } \\delta _{\\beta }^{\\alpha } + i g_{S} G_{\\mu \\beta }^{\\alpha }.$ The first term of the electroweak Lagrangian describes the corresponding gauge interactions of the electroweak theory: $\\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Gauge}} = \\frac{1}{2 g^{2}} \\operatorname{Tr}{W^{\\mu \\nu } W_{\\mu \\nu }} - \\frac{1}{4 g^{\\prime \\,2}} B^{\\mu \\nu } B_{\\mu \\nu },$ with the weak isospin and hypercharge gauge field strength tensors $W^{\\mu \\nu }$ and $B^{\\mu \\nu }$ , respectively.",
"The SU(2) and U(1) gauge couplings are represented by the constants $g$ and $g^{\\prime }$ .",
"Through mixing of the $B$ and $W_3$ fields, the photon and $Z$ boson are generated: $\\begin{pmatrix}\\gamma \\\\Z\\end{pmatrix}=\\begin{pmatrix}\\phantom{-}\\cos \\theta _W & \\sin \\theta _W \\\\- \\sin \\theta _W & \\cos \\theta _W\\end{pmatrix}\\begin{pmatrix}B \\\\W_3\\end{pmatrix},$ where $\\theta _W$ denotes the weak mixing angle.",
"Similarly, the $W^{\\pm }$ bosons are generated through mixing of the $W_1$ and $W_2$ fields.",
"The matter term of the electroweak Lagrangian contains the kinetic energy terms from the fermions and their gauge field interactions: $\\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Matter\\phantom{g}}} = i \\bar{q}_{L}^{i} \\mbox{$\\lnot \\!\\!D$}q_{L}^{i} + i \\bar{u}_{R}^{i} \\mbox{$\\lnot \\!\\!D$}u_{R}^{i} + i \\bar{d}_{R}^{i} \\mbox{$\\lnot \\!\\!D$}d_{R}^{i} + i \\bar{l}_{L}^{i} \\mbox{$\\lnot \\!\\!D$}l_{L}^{i} + i \\bar{e}_{R}^{i} \\mbox{$\\lnot \\!\\!D$}e_{R}^{i}.$ A summation over the index $i$ is implied.",
"The indices $L$ and $R$ refer to the left and right-handed chiral projections $\\psi _L = (1-\\gamma _5) \\frac{\\psi }{2} \\quad \\text{and} \\quad \\psi _R = (1+\\gamma _5) \\frac{\\psi }{2}.$ The left-handed quark and lepton fields are represented by the SU(2) doublets $q_{L}^{i} = {u^i \\atopwithdelims ()d^i}_L \\quad \\text{and} \\quad l_{L}^{i} = {\\nu ^i \\atopwithdelims ()e^i}_L$ while the right-handed fields are represented by the singlets $u_{R}^{i}$ , $d_{R}^{i}$ and $e_{R}^{i}$ .",
"The gauge covariant terms, describing the electroweak gauge interactions of the fermions, are given by $D_{\\mu } q_{L}^{i} & = & (\\partial _{\\mu } + \\frac{i g}{2} \\tau W_{\\mu } + i \\frac{g^{\\prime }}{6} B_{\\mu }) q_{L}^{i}, \\\\D_{\\mu } l_{L}^{i} & = & (\\partial _{\\mu } + \\frac{i g}{2} \\tau W_{\\mu } - i \\frac{g^{\\prime }}{2} B_{\\mu }) l_{L}^{i}, \\\\D_{\\mu } u_{R}^{i} & = & (\\partial _{\\mu } + i \\frac{2}{3} g^{\\prime } B_{\\mu }) u_{R}^{i}, \\\\D_{\\mu } d_{R}^{i} & = & (\\partial _{\\mu } - i \\frac{g^{\\prime }}{3} B_{\\mu }) d_{R}^{i}, \\\\D_{\\mu } e_{R}^{i} & = & (\\partial _{\\mu } - i g^{\\prime } B_{\\mu }) e_{R}^{i}.",
"$ Note that there are no mass terms for the fermions in either the SU(3) or the SU(2) $\\times $ U(1) gauge theory since such terms are forbidden by the gauge invariance of the Standard Model.",
"A Dirac mass term for a fermion field is not invariant under a chiral transformation and hence would violate the requirement of gauge invariance and renormalisability.",
"However, since weak interactions are observed to be short ranged, the gauge bosons must obtain non-vanishing masses through a different mechanism.",
"Both gauge boson and fermion masses are generated by spontaneous symmetry breaking of the Higgs term in the Standard Model Lagrangian, $\\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Higgs}} = (D^{\\mu } \\phi )^{\\dag } D_{\\mu } \\phi + \\mu ^2 \\phi ^{\\dag } \\phi - \\lambda (\\phi ^{\\dag } \\phi )^2 ,$ containing the kinetic energy of the Higgs field, which is represented by the complex scalar field $\\phi = {\\phi ^{+} \\atopwithdelims ()\\phi ^{0}}.$ The first term describes the Higgs field interactions with the gauge fields and the latter two denote the Higgs potential, shown in Figure REF.",
"Figure: The Higgs potential.",
"Electroweak symmetry breaking induces a non-zero vacuum expectation value in the minima of the Higgs potential, leading to effective masses for the gauge bosons and fermions.The final term in the Standard Model Lagrangian describes the Yukawa interactions of the fermions with the Higgs field: $\\mathcal {L}_{\\text{SU(2)} \\times \\text{U(1)}}^{\\text{Yukawa\\phantom{g}}} = - \\Gamma _{u}^{ij} \\bar{q}_{L}^{i} \\epsilon \\phi ^{\\ast } u_R^j - \\Gamma _{d}^{ij} \\bar{q}_{L}^{i} \\epsilon \\phi d_R^j - \\Gamma _{e}^{ij} \\bar{l}_{L}^{i} \\epsilon \\phi e_R^j + \\text{h.c.},$ where $\\epsilon = i \\sigma _2$ denotes the two dimensional total antisymmetric tensor which ensures electrical neutrality of the individual Yukawa terms and $\\sigma _2$ represents the Pauli matrix $\\sigma _2 = \\left(\\begin{array}{cc}0 & -i \\\\i & \\phantom{-}0\\end{array} \\right).$ Furthermore, the Yukawa couplings $\\Gamma _u$ , $\\Gamma _d$ and $\\Gamma _e$ denote the respective complex $3 \\times 3$ matrices in generation space, describing the interactions between the Higgs doublet and the different fermion flavours.",
"As they are not required to be diagonal, a mixing amongst the three generations is allowed.",
"Image taken from wikipedia (public domain).",
"The electroweak symmetry is spontaneously broken by acquisition of a non-zero vacuum expectation value of the neutral Higgs field component $\\langle \\phi ^0 \\rangle = \\frac{\\mu }{\\sqrt{2 \\lambda }} \\equiv \\frac{v}{\\sqrt{2}}$ which, consequently, generates masses $M_Z$ and $M_W$ for the electroweak gauge bosons through unitarity gauge, and $M_f$ for the fermions from gauge invariant Yukawa couplings $\\Gamma _f$ of the Higgs boson to fermions: $M_Z & = & \\frac{1}{2} v g, \\\\M_W & = & \\frac{1}{2} v \\sqrt{g^2 + g^{\\prime \\,2}}, \\\\M_f & = & \\Gamma _f \\frac{v}{\\sqrt{2}}.$ A summary of the gauge bosons and their properties can be found in Table REF.",
"Table: Gauge bosons and their properties and quantum numbers."
],
[
"The Top Quark",
"In the following, the production and decay of top quark pairs and singly produced top quarks (single tops) within the Standard Model will be discussed in detail.",
"Furthermore, an overview of important properties of the top quark and their measurement will be given.",
"In particular, the charge asymmetry in the production of top quarks pairs within the Standard Model and in theories beyond will be covered."
],
[
"Top Quark Production at Hadron Colliders",
"At hadron colliders, $t\\bar{t}$ pairs are mainly produced through strong interactions described by perturbative QCD.",
"Interactions between the quark and gluon constituents of the colliding hadrons (either protons or antiprotons) participate in a hard scattering process and produce a top quark and an antitop quark in the final state.",
"At Born level approximation, top quark pairs can be produced via gluon-gluon fusion ($gg$ ) or via the annihilation of quark-antiquark pairs ($q\\bar{q}$ ).",
"The relevant leading order Feynman diagrams for the contributing processes are shown in Figure REF.",
"Figure: Lowest order diagrams contributing to top quark pair production at hadron colliders.",
"Top quarks are produced via strong interaction, either in quark-antiquark annihilation (top) or gluon-gluon fusion (bottom).Due to the fact that hadrons are composite particles, consisting of partons with unknown fractions $x$ of the initial hadron momenta, the initial state of the parton interaction is not precisely known.",
"However, hadron interactions in $pp$ and $p\\bar{p}$ collisions can be described by separating the partonic reactions into a short distance and a long distance contribution.",
"The long distance part can be factorised into longitudinal parton momentum distribution functions (PDFs) $f_i (x_i , \\mu _{F}^{2})$ , where $\\mu _{F}^{2}$ denotes an (arbitrary) factorisation scale describing the separation of the long and short distance contributions.",
"An additional renormalisation scale $\\mu _{R}^{2}$ is introduced to account for higher order corrections, where ultraviolet divergent terms may emerge and a renormalisation approach can be used to absorb such divergences into corresponding counter terms.",
"Both scales $\\mu _{F}^{2}$ and $\\mu _{R}^{2}$ are commonly chosen to correspond to the momentum transfer $\\mu _{F}^{2} = \\mu _{R}^{2} = Q^2$ .",
"Furthermore, for the calculation and simulation of top quark processes, $Q^2$ is typically chosen such that $\\mu _{F} = \\mu _{R} = Q = m_t$ corresponds to the top pole mass $m_t$ and the associated scale variation dependency is studied.",
"The PDFs represent the probability distribution of observing a parton of type $i$ at a given scale $\\mu _{F}^{2}$ with a longitudinal parton momentum fraction $x_i$ .",
"Since these probabilities cannot be universally derived from QCD, they have to be provided from experimental studies of the proton structure, mostly from deep inelastic lepton-proton scattering experiments at the H1[26], [27], [28], [29] and ZEUS[30], [31], [32], [33] experiments at the HERA electron-proton collider.",
"As an example, the $e^{+}p$ and $e^{-}p$ production cross-sections measured in deep-inelastic scattering experiments at HERA can be found in Figure REF[34] in comparison to the CTEQ10 PDF next-to-leading order (NLO) prediction[35].",
"Figure: Comparison of CTEQ10 NLO predictions for reduced cross-sections in e + pe^{+}p (left) and e - pe^{-}p (right) neutral-current deep inelastic scattering experiments from combined HERA-1 data, with correlated systematic shifts included.The short distance term arises from the hard scattering process of the respective partons, denoted by the partonic cross-section for partons $i$ and $j$ , $\\sigma _{ij}$ .",
"This contribution is characterised by high momentum transfer.",
"Hence, it is not dependent on the incoming hadron type or the respective wave functions and can be described by perturbative QCD, as indicated by the leading order diagrams in Figure REF.",
"At a given centre-of-mass energy $\\sqrt{s}$ and for a top mass parameter $m_t$ , the total top quark pair production cross-section can be calculated from the short distance and long distance terms as $\\sigma _{t \\bar{t}} \\left( \\sqrt{s}, m_t \\right) = \\sum _{i,j} \\iint \\text{d}x_i \\text{d}x_j f_i (x_i , Q^2) f_j (x_j , Q^2) \\times \\sigma _{ij \\rightarrow t \\bar{t}} \\left( \\rho , m_t^2 , x_i , x_j , \\alpha _s (Q^2), Q^2 \\right),$ where the summation is performed over all permutations of $i,j = \\lbrace q, \\bar{q}, g \\rbrace $ .",
"The PDFs of the initial state protons are denoted by $f_i (x_i , Q^2)$ and $f_j (x_j , Q^2)$ , respectively and the parameter $\\rho $ is given by $\\rho = \\frac{4 m_t^2}{\\sqrt{x_i x_j s}} = \\frac{4 m_t^2}{\\sqrt{\\hat{s}}},$ where $x_i x_j s \\equiv \\hat{s}$ denotes the effective centre-of-mass energy in the partonic reaction.",
"The probability of a parton $i$ to be carrying a momentum fraction of $x_i$ decreases significantly with rising $x_i$ , as can be seen in Figure REF, where two PDFs from the CTEQ10 PDF set[35] are shown as an example.",
"Figure: CTEQ10 parton distribution functions at different momentum transfers for gluons and different quark/antiquark flavours.",
"Shown are the PDF sets for μ=5GeV\\mu = 5\\,\\text{GeV} (left) and μ=m t \\mu = m_t (right).The PDFs have been evaluated at scales $\\mu = 5\\,\\text{GeV}$ and $\\mu = m_t$ , respectively, where $\\mu \\equiv Q$ .",
"The minimal energy carried by the two incoming partons to produce a top quark pair at the threshold (i.e.",
"at rest), is given by $\\sqrt{x_i x_j s} \\ge 2m_t$ and hence, assuming both partons carrying the same momentum fractions as an approximation, $x_i \\approx x_j \\equiv x$ : $x \\approx \\frac{2m_t}{\\sqrt{s}}.$ This corresponds to a typical value of $x \\approx 0.05$ at the LHC for a centre-of-mass energy of $\\sqrt{s} = 7\\,\\text{TeV}$ .",
"As shown in Figure REF, the gluon PDFs dominate significantly over any other parton in the corresponding range of $x$ .",
"Consequently, the production of top quark pairs at the LHC is dominated by gluon-gluon fusion.",
"At the Tevatron (where the typical value of $x$ is of the order of 0.2) the production of top quarks is dominated by quark-antiquark annihilation processes, in particular involving up and down valence quarks.",
"Since the centre-of-mass energy at the LHC is significantly higher, top quark pairs are typically produced above the threshold, but still within the gluon-gluon fusion dominated range of the PDFs.",
"The total $t\\bar{t}$ cross-section at the LHC is predicted in an approximate next-to-next-to-leading order (NNLO) calculation to be $165_{-16}^{+11}\\,\\text{pb}$[36], [37], [38] for a centre-of-mass energy of $\\sqrt{s} = 7\\,\\text{TeV}$ and $m_t = 172.5\\,$ GeV.",
"Preliminary measurements have been performed at both ATLAS and CMS, yielding $\\sigma _{t \\bar{t}}\\,(\\sqrt{s} = 7\\,\\text{TeV}) & = & 179.0_{-9.7}^{+9.8}\\text{\\,(stat.+syst.)}",
"\\pm 6.6\\text{\\,(lumi.",
")}\\,\\text{pb} \\text{\\cite {ATLAS-CONF-2011-121}}, \\\\\\sigma _{t \\bar{t}}\\,(\\sqrt{s} = 7\\,\\text{TeV}) & = & 165.8 \\pm 2.2\\text{\\,(stat.)}",
"\\pm 10.6\\text{\\,(syst.)}",
"\\pm 7.8\\text{\\,(lumi.",
")}\\,\\text{pb} \\text{\\cite {CMS-PAS-TOP-11-024}},$ respectively.",
"Both measurements are in agreement with the Standard Model prediction.",
"This cross-section is several orders of magnitude lower than, for example, the SM $Z$ and $W$ boson production cross-sections or the inclusive QCD multijet production cross-section at comparable values of $Q^2$ .",
"This can be seen in Figure REF, where the total production cross-sections for several SM processes are shown as a function of centre-of-mass energy of the colliding (anti)protons.",
"Figure: QCD predictions for hard-scattering cross-sections at the Tevatron and the LHC.",
"The top quark pair production cross-section is denoted by σ t \\sigma _t.",
"The discontinuities in the different curves denote the change from pp ¯p\\bar{p} collisions at the Tevatron to pppp collisions at the LHC.",
"For the LHC, different centre-of-mass energies are highlighted by three vertical dashed lines, where the leftmost line corresponds to a centre-of-mass energy of s=7TeV\\sqrt{s} = 7\\,\\text{TeV}.Consequently, a sophisticated real-time selection to identify the relevant final state particles and obtain a good signal to background separation with respect to other SM processes and, more importantly, the dominant QCD multijet background, is crucial for all top quark related measurements at the LHC.",
"Furthermore, an extensive theoretical understanding and modelling of these backgrounds is necessary to facilitate the measurement of top quark properties to highest precision and in order to achieve a sensitivity to potential deviations from the Standard Model expectations."
],
[
"Production of Single Top Quarks",
"In addition to the production of top quark pairs via the strong interaction, single top quarks can be produced in electroweak charged current interactions.",
"Three mechanisms for this production exist, as shown in Figure REF.",
"Single top quarks can emerge in the fusion of $W$ bosons and gluons ($t$ -channel process) similar to the production of heavy flavour quarks in deep-inelastic scattering via charged current interactions.",
"In addition, they can be produced by annihilation of quark-antiquark pairs ($s$ -channel process) and exchange of an off-shell $W^{\\ast }$ , or by $Wt$ production in quark-gluon interactions.",
"Figure: Electroweak single top production diagrams via WW-gluon fusion (a), exchange of an off-shell W * W^{\\ast } (b) and WtWt production (c).The corresponding production cross-sections have been approximated at next-to-next-to-leading-order to be $\\sigma ^{\\text{t-ch}\\phantom{Wt}}_{t}\\!\\!\\!\\!\\!\\!\\,(\\sqrt{s} = 7\\,\\text{TeV}) & = & 64.57_{-2.01}^{+2.71}\\,\\text{pb} \\text{\\cite {st_tchan_pred}}, \\\\\\sigma ^{\\text{s-ch}\\phantom{Wt}}_{t}\\!\\!\\!\\!\\!\\!\\,(\\sqrt{s} = 7\\,\\text{TeV}) & = & \\phantom{0}4.63_{-0.17}^{+0.19}\\,\\text{pb} \\text{\\cite {st_schan_pred}}, \\\\\\sigma ^{Wt\\phantom{\\text{t-ch}}}_{t}\\!\\!\\!\\!\\!\\!\\,(\\sqrt{s} = 7\\,\\text{TeV}) & = & 15.74_{-1.08}^{+1.06}\\,\\text{pb} \\text{\\cite {st_tchan_pred}},$ for $m_t = 172.5\\,$ GeV in the $t$ -channel, in the $s$ -channel, and for $Wt$ production, respectively.",
"Since in all production channels a top charged current is involved, the respective cross-sections behave as $\\sigma _{t} \\propto |V_{tb}|^2 g^2,$ where $|V_{tb}|$ denotes the relative probability that the top quark decays into a bottom quark via the exchange of a $W$ boson.",
"This is described by the Cabbibo-Kobayashi-Maskawa (CKM) matrix $V_{\\text{CKM}}$[23] which summarises the relative transition probabilities in charged weak interactions.",
"The magnitudes of the CKM matrix elements, here denoted by the matrix $M_{\\text{CKM}}$ , are $M_{\\text{CKM}} & = &\\begin{pmatrix}|V_{ud}| & |V_{us}| & |V_{ub}| \\\\|V_{cd}| & |V_{cs}| & |V_{cb}| \\\\|V_{td}| & |V_{ts}| & |V_{tb}|\\end{pmatrix} \\\\& = &\\begin{pmatrix}0.97428 \\pm 0.00015 & 0.2253 \\pm 0.0007 & 0.00347^{+0.00016}_{-0.00012} \\\\0.2252 \\pm 0.0007 & 0.97345^{+0.00015}_{-0.00016} & 0.0410^{+0.0011}_{-0.0007} \\\\0.00862^{+0.00026}_{-0.00020} & 0.0403^{+0.0011}_{-0.0007} & 0.999152^{+0.000030}_{-0.000045}\\end{pmatrix}.$ As a consequence of this dependency, measurements of the single top quark production cross-section provide an implicit sensitivity to the CKM matrix element $V_{tb}$ .",
"At the LHC, single tops are predominantly produced via the $t$ -channel interaction, followed by $Wt$ production, due to the large initial state gluon contribution at the LHC centre-of-mass energy, while the $s$ -channel process is suppressed.",
"First direct evidence for single top quarks was found in 2006 by the DØ collaboration at the Tevatron[44], followed by its observation[45], [46] in 2009 by the DØ and CDF experiments.",
"First preliminary measurements of the inclusive single top production cross-section have been conducted at the LHC, yielding $\\sigma ^{t\\text{-ch}}_{t}\\,(\\sqrt{s} = 7\\,\\text{TeV}) = 90_{-22}^{+32}\\,\\text{pb} \\text{\\cite {ATLAS-CONF-2011-101}}$ and $\\sigma ^{t\\text{-ch}}_{t}\\,(\\sqrt{s} = 7\\,\\text{TeV}) = 83.6 \\pm 29.8\\text{\\,(stat.+syst.)}",
"\\pm 3.3\\text{\\,(lumi.)}",
"\\,\\text{pb} \\text{\\cite {PhysRevLett.107.091802}}$ in the $t$ -channel as measured by ATLAS and CMS, respectively.",
"The $Wt$ production cross-section has been measured at CMS to be $\\sigma ^{Wt}_{t}\\,(\\sqrt{s} = 7\\,\\text{TeV}) = 22_{-7}^{+9}\\text{\\,(stat.+syst.)}",
"\\,\\text{pb} \\text{\\cite {CMS-PAS-TOP-11-022}}$ and a limit has been set by a corresponding measurement at ATLAS, corresponding to $\\sigma ^{Wt}_{t}\\,(\\sqrt{s} = 7\\,\\text{TeV}) < 39\\,\\text{pb} \\text{\\cite {ATLAS-CONF-2011-104}}$ at 95 % C.L.",
"Furthermore, searches for single top quarks produced in $s$ -channel interactions have been conducted at ATLAS, limiting the corresponding production cross-section to $\\sigma ^{s\\text{-ch}}_{t}\\,(\\sqrt{s} = 7\\,\\text{TeV}) < 26.5 \\,\\text{pb} \\text{\\cite {ATLAS-CONF-2011-118}}$ at 95 % C.L.",
"All measurements are in agreement with Standard Model predictions."
],
[
"Top Quark Decay",
"Due to the magnitude of the CKM matrix element $V_{tb}$ being close to unity, the top quark decays in almost $100\\,\\%$ of the cases via electroweak charged current interaction into a $b$ quark and $W$ boson, which then in turn decays either leptonically into a charged lepton and the corresponding (anti)neutrino or hadronically into a quark-antiquark pair.",
"At leading order, the Standard Model prediction for the total decay width of the top quark, $\\Gamma _t^0$ , is given by $\\Gamma _t^0 = \\frac{G_F m_t^3}{8 \\pi \\sqrt{2}} |V_{tb}|^2,$ where $G_F$ denotes the Fermi coupling constant $G_F = \\frac{\\sqrt{2}}{8} \\frac{g^2}{M_W^2}.$ Taking into account higher order corrections at next-to-leading order, the total top quark decay width becomes $\\Gamma _t = \\Gamma _t^0 \\left( 1 - \\frac{M_W^2}{m_t^2} \\right)^2 \\left( 1 + 2 \\frac{M_W^2}{m_t^2} \\right)^2 \\left[ 1 - \\frac{2 \\alpha _s}{3 \\pi } \\left( \\frac{2 \\pi ^2}{3} - \\frac{5}{2} \\right) \\right],$ where terms of order $m_b^2 / m_t^2$ and $(\\alpha _s / \\pi ) M_W^2 / m_t^2$ have been neglected.",
"At a top mass of $170\\,\\text{GeV}$ and $\\alpha _s$ evaluated at the $Z$ scale, this yields an approximate predicted decay width of $\\Gamma _t \\approx 1.3\\,\\text{GeV},$ and a corresponding mean lifetime of $\\tau _t \\approx 0.5 \\cdot 10^{-24}\\,\\text{s},$ which is significantly lower than the time scale corresponding to the strong hadronisation scale $\\Lambda _{\\text{QCD}} \\approx 250\\,\\text{MeV}$ .",
"Hence, the top quark decays before being able to form hadronic bound states such as the $t\\bar{t}$-quarkonium.",
"Consequently, the top quark spin/polarisation properties are preserved in its decay and are transferred to the decay products.",
"Since the top quark decays almost exclusively into a $b$ quark and $W$ boson, the resulting final state decay channels are well defined and can be separated into three cases, characterised by the final state particles: Full hadronic final state (alljets): Both $W$ bosons from the $t\\bar{t}$ pair further decay into quarks, leading to a total amount of six quarks including the $b$ quarks from the initial top and antitop decays.",
"Semileptonic final state (lepton+jets): One $W$ boson from the $t\\bar{t}$ pair decays into quarks, while the second one decays leptonically, leading to a total of four quarks including the $b$ quarks from the initial top and antitop decays, and one charged lepton and its corresponding (anti-)neutrino.",
"Dileptonic final state (dilepton): Both $W$ bosons from the $t\\bar{t}$ decay into a charged lepton and the corresponding (anti-)neutrino, respectively.",
"In addition, two remaining $b$ quarks from the top and antitop decays are produced.",
"If the charged lepton is a $tau$ in the semileptonic or dileptonic channel, either a muon or electron and the corresponding (anti-)neutrino, or further quarks from the hadronic decay of the $tau$ lepton are produced.",
"The respective $W$ branching ratios (BR) at leading order (LO) can be found in Table REF.",
"Table: Theoretical (LO) and measured WW branching ratios.Taking these branching fractions into account, possible $t\\bar{t}$ final states and their approximate relative probabilities are shown schematically in Figure REF.",
"Figure: Top quark pair decay channels (left) and branching fractions (right)."
],
[
"Top Quark Properties",
"Several properties of the top quark have been studied in collider experiments such as the Tevatron and the LHC.",
"Amongst them, the top quark mass has been determined with a relative uncertainty of only $0.5\\,\\%$[13] by combining the most recent measurements from DØ and CDF.",
"This combination constitutes the most precise (in relative terms) mass measurement of any quark so far.",
"Since the top quark does not form hadronic bound states, the top quark mass $m_t$ is defined as the pole mass in this context and is measured to be $m_t = 173.18 \\pm 0.56\\,\\text{(stat.)}",
"\\pm 0.75\\,\\text{(syst.",
")}.$ In addition, measurements to exclude an exotic top quark carrying a charge of $4e/3$Here, $e$ denotes the electron charge., which is predicted to be $2e/3$ in the Standard Model, have been performed.",
"This model has been excluded with approximately $95\\,\\%$ C.L.",
"[52] and $90\\,\\%$ C.L.",
"[53], respectively, in independent measurements at CDF and DØ.",
"A corresponding measurement at ATLAS excludes a top quark charge of $4e/3$ at more than five standard deviations[54].",
"Images taken from http://www-d0.fnal.gov/Run2Physics/top/.",
"Since the top quark does not form bound hadronic states due to its small lifetime, it provides the unique opportunity to measure quark properties which are usually concealed by hadronisation, such as spin correlations between quark-antiquark pairs produced at hadron colliders.",
"Since all other quarks depolarise due to QCD interactions before fragmentation, it is not possible to gain knowledge about the spin from the final state, while the final state particles in the top quark decay preserve significant amount of information about the spins of the initial top and antitop quarks to potentially allow their measurement.",
"First studies at the Tevatron using the angular distributions of the final state particles indicate a correlation strength $C$ of $C = 0.57 \\pm 0.31\\,\\text{(stat.+syst.)}",
"\\text{\\cite {Abazov:2011ka}}$ in the beam basis, compatible with the next-to-leading order Standard Model prediction of $C_{\\text{SM}} = 0.78_{-0.04}^{+0.03}$[56] and excluding a non-correlation hypothesis at $97.7\\,\\%$ C.L.",
"Similar studies at ATLAS indicate a correlation strength $A_{\\text{helicity}}$ in the helicity basis of $A_{\\text{helicity}} = 0.34_{-0.11}^{+0.15}\\,\\text{(stat.+syst.)}",
"\\text{\\cite {ATLAS-CONF-2011-117}},$ which is compatible with the corresponding next-to-leading order Standard Model prediction.",
"Furthermore, polarisations of $W$ bosons from the top quark decay are predicted to arise due to the different possible helicity states of the produced on-shell $W$ bosons.",
"Consequently, the $W$ helicity has been measured and the obtained left-handed, longitudinal and right-handed polarisations have been found to be consistent with the Standard Model predictions[58], [59].",
"The charge asymmetry in top quark pair production is key topic of this thesis and will be discussed in detail in the following section."
],
[
"Charge Asymmetry in Top Quark Pair Production",
"As discussed in Section REF, top quark pairs at hadron colliders are produced via gluon-gluon fusion or quark-antiquark annihilation in Born approximation: $q + \\bar{q} & \\rightarrow & t + \\bar{t}\\text{,} \\nonumber \\\\g + g & \\rightarrow & t + \\bar{t}\\text{.}",
"\\nonumber $ These leading order processes obviously do not discriminate between the final state top and antitop, as can be seen from the respective Born partonic differential cross-sections for the two production mechanisms, $\\frac{d \\sigma _{q\\bar{q} \\rightarrow t\\bar{t}}}{d \\cos {\\hat{\\theta }}} & = & \\alpha _s^2 \\frac{\\pi \\beta }{3 \\hat{s} N_C} \\left( 1 + c^2 + 4m_t^2 \\right), \\\\\\frac{d \\sigma _{gg \\rightarrow t\\bar{t}}}{d \\cos {\\hat{\\theta }}} & = & \\alpha _s^2 \\frac{\\pi \\beta }{2 \\hat{s}} \\left( \\frac{1}{N_C (1-c^2)} - \\frac{3}{16} \\right) \\times \\left(1 + c^2 + 8m_t^2 - \\frac{32m_t^4}{1-c^2} \\right),$ where $\\hat{\\theta }$ denotes the polar angle of the top quark with respect to the incoming parton in the centre-of-mass rest frame, $N_C = 3$ , $\\beta = \\sqrt{1-4m_t^2}$ and $c = \\beta \\cos {\\hat{\\theta }}$ .",
"Consequently, the same holds for the full $pp \\rightarrow t \\bar{t}$ differential cross-sections.",
"If, however, the processes are considered at higher order in the Standard Model, where radiative corrections from real or virtual gluon emission are introduced, a significant asymmetry can be generated in the differential $t\\bar{t}$ cross-section, leading to a charge asymmetry in the production of top quark pairs[60].",
"This effect is caused by the interference of amplitudes which are odd under the exchange of the final state top quark and antitop quark.",
"The dominant contribution to an overall charge asymmetry stems from interference between the leading order amplitude for quark-antiquark annihilation and the corresponding one-loop corrections (box diagrams), creating a positive contribution to the total charge asymmetry.",
"The two diagrams are shown in Figure REF.",
"Figure: Box (left) and Born (right) diagrams contributing to the production of top quark pairs through quark-antiquark annihilation.In addition, interferences between initial state and final state gluon bremsstrahlung have to be taken into account, contributing negatively but typically of lower magnitude than the box-Born interference to the overall asymmetry.",
"The respective diagrams are shown in Figure REF.",
"Figure: Final state (left) and initial state (right) bremsstrahlung diagrams contributing to the production of top quark pairs through quark-antiquark annihilation.The asymmetry arising from the combination of these contributions can be described by comparing the colour factor terms arising in the differential cross-sections from the two cut diagrams[61] shown in Figure REF after averaging over initial states and summing over final states.",
"Figure: Cut diagrams contributing to the asymmetry in the production of top quark pairs through quark-antiquark annihilation, arising from virtual and real gluon emission.",
"The respective contributions to the cross-section are odd under the exchange of the final state top and antitop quarks.The respective colour factors $C_A$ and $C_B$ for the two diagrams (a) and (b), respectively, can be expressed as[60] $C_A = \\frac{1}{N_C^2} \\operatorname{Tr}{\\frac{\\lambda ^a}{2} \\frac{\\lambda ^b}{2} \\frac{\\lambda ^c}{2}} \\operatorname{Tr}{\\frac{\\lambda ^a}{2} \\frac{\\lambda ^c}{2} \\frac{\\lambda ^b}{2}} = \\frac{1}{16 N_C^2} \\left( f_{abc}^2 + d_{abc}^2 \\right)\\phantom{-}\\phantom{,}$ and $C_B = \\frac{1}{N_C^2} \\operatorname{Tr}{\\frac{\\lambda ^a}{2} \\frac{\\lambda ^b}{2} \\frac{\\lambda ^c}{2}} \\operatorname{Tr}{\\frac{\\lambda ^b}{2} \\frac{\\lambda ^c}{2} \\frac{\\lambda ^a}{2}} = \\frac{1}{16 N_C^2} \\left( -f_{abc}^2 + d_{abc}^2 \\right),$ where $f_{abc}^2 = 24$ and $d_{abc}^2 = 40/3$ .",
"Consequently, the respective contributions $d\\sigma _A$ and $d\\sigma _B$ to the cross-section are odd under the exchange of the final state top and antitop quarks: $d\\sigma _A \\left( t, \\bar{t} \\right) = - d\\sigma _B \\left( \\bar{t}, t \\right).$ A small contribution is introduced at the order of $\\alpha _s^3$ through interferences of different terms in quark-gluon scattering, $g + q & \\rightarrow & t + \\bar{t} + q^{\\prime }\\text{,} \\nonumber $ which is shown in Figure REF.",
"Figure: Quark-gluon scattering diagrams contributing to the hadronic production of top quark pairs introduced at order of α s 3 \\alpha _s^3.The individual charge asymmetric contributions from quark-antiquark annihilation and quark-gluon scattering can be found in Figure REF[60] as a function of the partonic centre-of-mass energy, quantified by the integrated forward-backward contributions $\\sigma _{A}^i = \\int _0^1 \\frac{d \\sigma _{A}^i}{d \\cos {\\hat{\\theta }}} d \\cos {\\hat{\\theta }} - \\int _{-1}^0 \\frac{d \\sigma _{A}^i}{d \\cos {\\hat{\\theta }}} d \\cos {\\hat{\\theta }},$ where $i$ denotes the two contributions from quark-antiquark annihilation ($q \\bar{q}$ ) and quark-gluon scattering ($qg$ ).",
"Figure: Integrated charge asymmetric parts of the top quark pair production cross-section from quark-antiquark annihilation (qq ¯q \\bar{q}) and quark-gluon scattering (qgqg) initiated processes as a function of the partonic centre-of-mass energy.The resulting overall QCD charge asymmetry $A_{t\\bar{t}}$ can be expressed in terms of the asymmetric contributions $\\sigma _{\\text{A}}$ and symmetric contributions $\\sigma _{\\text{S}}$ to the total production cross-section as ratio $A_{t\\bar{t}}^{\\text{QCD}} = \\frac{\\sigma _{\\text{A}}}{\\sigma _{\\text{S}}} = \\frac{ \\alpha _s^3 \\sigma _{\\text{A}}^{\\text{(1)}} + \\alpha _s^4 \\sigma _{\\text{A}}^{\\text{(2)}} + ... }{ \\alpha _s^2 \\sigma _{\\text{S}}^{\\text{(0)}} + \\alpha _s^3 \\sigma _{\\text{S}}^{\\text{(1)}} + ... } = \\frac{ \\alpha _s^3 \\sigma _{\\text{A,} q\\bar{q}}^{\\text{(1)}} + \\alpha _s^3 \\sigma _{\\text{A,} qg}^{\\text{(1)}} + \\alpha _s^4 \\sigma _{\\text{A,} q\\bar{q}}^{\\text{(2)}} + ... }{ \\alpha _s^2 \\sigma _{\\text{S}}^{\\text{(0)}} + \\alpha _s^3 \\sigma _{\\text{S}}^{\\text{(1)}} + ... },$ where $\\sigma _{\\text{S}}^{\\text{($i$)}} = \\sigma _{\\text{S,} gg}^{\\text{($i$)}} + \\sigma _{\\text{S,} q\\bar{q}}^{\\text{($i$)}} + \\sigma _{\\text{S,} qg}^{\\text{($i$)}}$ contains the respective symmetric contributions at a given order $i$ .",
"This asymmetry can also be parametrised as $A_{t\\bar{t}}^{\\text{QCD}} = \\alpha _s A_{t\\bar{t}}^{(0)} + \\alpha _s^2 A_{t\\bar{t}}^{(1)} + ...$ Higher order corrections arising from QCD which are not taken into account in Equation REF, such as $A_{t\\bar{t}}^{(1)}$ , can be evaluated in next-to-next-to-leading log approximation using soft gluon resummation techniques[62], [63], [64] in order to improve the theoretical predictions on the respective differential cross-section contributions and the associated uncertainties.",
"In addition to the asymmetry arising from the Standard Model QCD terms, further contributions originate from mixed QCD-electroweak interference terms to the quark-antiquark annihilation process[60], [65].",
"The $t\\bar{t}$ colour-singlet configuration of the box diagram in Figure REF can interfere with the production of top quark pairs through a photon or a $Z$ boson (and similarly for the interference between initial state and final state radiation), as shown in Figure REF.",
"Figure: Mixed QCD-electroweak cut diagrams contributing to the asymmetry in the production of top quark pairs through quark-antiquark annihilation, arising from interferences between the singlet-state box diagram and initial and final state radiation diagrams with electroweak tt ¯t\\bar{t} production (left) and from the interference of the gluon-γ\\gamma and gluon-ZZ box diagrams with the Born diagram (right).Furthermore, interferences of the gluon-$\\gamma $ and gluon-$Z$ box diagrams with the leading order QCD amplitude as indicated in Figure REF contribute to an additional asymmetry, which together with Figure REF leads to a total increase of asymmetry by a factor of about 1.09[60] with respect to the QCD contributions.",
"The overall charge asymmetry, including both the QCD and electroweak contributions, can be expressed as $A_{t\\bar{t}} = \\frac{ \\alpha ^2 \\tilde{\\sigma }_{\\text{A}}^{\\text{(0)}} + \\alpha _s^3 \\sigma _{\\text{A}}^{\\text{(1)}} + \\alpha _s^2 \\alpha \\tilde{\\sigma }_{\\text{A}}^{\\text{(1)}} + \\alpha _s^4 \\sigma _{\\text{A}}^{\\text{(2)}} + ... }{ \\alpha ^2 \\tilde{\\sigma }_{\\text{S}}^{\\text{(0)}} + \\alpha _s^2 \\sigma _{\\text{S}}^{\\text{(0)}} + \\alpha _s^3 \\sigma _{\\text{S}}^{\\text{(1)}} + \\alpha _s^2 \\alpha \\tilde{\\sigma }_{\\text{S}}^{\\text{(1)}} + ... },$ where $\\tilde{\\sigma }_{\\text{A}}^{\\text{($i$)}}$ and $\\tilde{\\sigma }_{\\text{S}}^{\\text{($i$)}}$ denote the asymmetric and symmetric contributions from QCD-electroweak mixing, respectively.",
"Note that a similar effect emerges in Quantum Electrodynamics, where interferences at the order of $\\alpha ^3$ create an asymmetry in the angular distribution of final state particles produced in $e^{+}e^{-}$ collisions due to virtual radiative corrections and soft and hard photon emission[14], [15], [16]."
],
[
"Charge Asymmetry Beyond the Standard Model",
"Numerous theoretical models predicting the manifestation of physics beyond the Standard Model (BSM) exist, several of which can have implications on the charge asymmetry observed in the production of top quark pairs at hadron colliders.",
"The most popular models will be explained in the following.",
"In chiral colour models[66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], SM colour charge is extended to contain a right-handed and left-handed contribution $\\text{SU(3)}_R \\times \\text{SU(3)}_L$ to reflect the chirality.",
"This extension implies a breaking of the symmetry of the diagonal $\\text{SU(3)}_C$ group and the generation of a heavy colour-octet gauge boson, the axigluon.",
"The associated coupling to quarks has a pure axial-vector structure and is of the same magnitude as the QCD coupling.",
"Alternative models with non-chiral structure imply the existence of massive gauge bosons with pure vector-like couplings to quarks (colorons)[77], [78], [79], [80] or Kaluza-Klein[81], [82] excited states arising from models including extra dimensions[83], [84], [85], [86], [87], [88], [89], [90], [91].",
"Further generalisations include the assumption of different coupling strengths for the different $\\text{SU(3)}$ contributions[92], [93], [94], [95], [96], [97], [98], [99], [100], leading to both vector and axial-vector couplings in the interactions of the respective colour-octet resonance $G_{\\mu }^{a}$ and quarks.",
"The vector and axial-vector coupling strengths, $g_{V}^{q_i}$ and $g_{A}^{q_i}$ , respectively, lead to the following generalised term $\\mathcal {L}_{G^{\\prime }}$[101] in the modified SM Lagrangian: $\\mathcal {L}_{G^{\\prime }} = g_s t^a \\bar{q}_i \\left( g_{V}^{q_i} + g_{A}^{q_i} \\gamma _{5} \\right) \\gamma ^{\\mu } G_{\\mu }^{a} q_i.$ The corresponding leading-order cross-section for top quark pair production in the annihilation of quark-antiquark pairs is given by[102], [103] $\\frac{d \\sigma _{q\\bar{q} \\rightarrow t\\bar{t}}}{d \\cos {\\hat{\\theta }}} & = & \\alpha _s^2 \\frac{\\pi \\beta }{3 \\hat{s} N_C} \\Biggl ( C_{+} + \\frac{2 \\hat{s} \\left( \\hat{s} - m_G^2 \\right)}{\\left( \\hat{s} - m_G^2 \\right)^2 + m_G^2 \\Gamma _G^2} \\left[ g_V^q g_V^t \\left( C_{+} \\right) + 2 g_A^q g_A^t c \\right] \\\\& + & \\frac{\\hat{s}^2}{\\left( \\hat{s} - m_G^2 \\right)^2 + m_G^2 \\Gamma _G^2} \\left[ \\left( (g_V^q)^2 + (g_A^q)^2\\right) \\times \\left( (g_V^t)^2 C_{+} + (g_A^t)^2 C_{-} \\right) + 8 g_V^q g_A^q g_V^t g_A^t c\\right] \\Biggr ), $ where $C_{\\pm } = 1 + c^2 \\pm 4m_t^2$ .",
"The vector and axial-vector couplings of the resonances to the light quarks and top quarks are given by the constants $g_V^q$ and $g_A^q$ , and by $g_V^t$ and $g_A^t$ , respectively.",
"Compared to the SM cross-section, an additional asymmetric contribution is introduced by terms which are odd in c. Hence, a large positive asymmetry can be generated in models where $g_A^q g_A^t < 0$ or where the term $8 g_V^q g_A^q g_V^t g_A^t c$ is dominant.",
"A negative asymmetry on the other hand can be created in flavour universal models where $g_A^q = g_A^t$ .",
"Models involving a colour-octet resonance typically require the resonance to be off-shell, either heavy[98], or below the $t\\bar{t}$ production threshold[100], since in the intermediate mass range a distinct excess would emerge in the $t\\bar{t}$ mass spectrum, which is not observed.",
"Alternatively, a very broad resonance[100], [104] would be concealed due to limited statistics in the tail of the $M_{t \\bar{t}}$ distribution."
],
[
"Extra Weak Gauge Bosons",
"Different theoretical models, such as some Grand Unified Theories (GUTs), topcolor or left-right[105] models predict the existence of extra weak gauge bosons, such as the $W^{\\prime }$ or $Z^{\\prime }$ .",
"Furthermore, these states can appear as Kaluza-Klein excitations in extra dimensional models[106], [107].",
"A sizable contribution to the charge asymmetry can only be introduced in $W^{\\prime }$ or $Z^{\\prime }$ $t$ -channel interactions[108], [109], [110], [105], [111], [112], [113], [114], [115], [116], [117], [118], [119], [120], [121], [122], [123], [124], [125], [126].",
"The $s$ -channel $t\\bar{t}$ production through a $Z^{\\prime }$ is suppressed due to the fact that the corresponding amplitudes do not create interferences with the SM amplitude[101].",
"A potentially significant asymmetry can be created by the introduction of flavour violating couplings into the SM Lagrangian by a term $\\mathcal {L}_{W^{\\prime }/Z^{\\prime }}$[101], such as $\\mathcal {L}_{W^{\\prime }/Z^{\\prime }} = \\bar{t} \\left( g_V^{Z^{\\prime }} + g_A^{Z^{\\prime }} \\gamma _5 \\right) \\gamma ^{\\mu } Z^{\\prime }_{\\mu } u + \\bar{t} \\left( g_V^{W^{\\prime }} + g_A^{W^{\\prime }} \\gamma _5 \\right) \\gamma ^{\\mu } W^{\\prime }_{\\mu } d.$ A real $Z^{\\prime }$ contribution is constrained by the absence of like-sign top quark pair production[122], except for very light $Z^{\\prime }$ resonances.",
"Furthermore, for a $Z^{\\prime }$ leading to a sizable positive charge asymmetry, a large corresponding $Z^{\\prime }$ coupling has to be assumed and a corresponding excess in the tail of the $t\\bar{t}$ mass distribution would be generated.",
"Such excess is not observed, however.",
"Furthermore, $W^{\\prime }$ and $Z^{\\prime }$ left-handed couplings are disfavoured by precision measurements in $B$ hadron systems[127]."
],
[
"Coloured Scalars",
"In addition to gauge bosons, neutral or charged coloured scalars[128] can be introduced in particular in the presence of larger gauge groups such as SU(5) or SO(10)[129].",
"These occur primarily in GUT models close to the unification scale.",
"However, some of the coloured scalar states can manifest at lower energy scales[130], [131], for example due to gauge coupling unification.",
"Similar to extra weak gauge bosons, $t\\bar{t}$ production via coloured scalars in the $s$ -channel is not affected by any charge asymmetric contributions due to the absence of interferences with the SM amplitudes[101].",
"Consequently, only $t$ -channel flavour-violating couplings can introduce a significant asymmetry in the production cross-section, such as the exchange of scalar colour singlets[132], [108], [133], triplets[131], [132], [134], [108], [135], [136], [127], [137], [138], sextets[132], [127], [139], [138] and octets[131], [132].",
"Furthermore, a light scalar contribution is disfavoured due to the implied existence of a highly constrained new top quark decay channel ($t \\rightarrow S^{\\prime } u$ )[23], [140].",
"A generalised contribution from a coloured scalar SU(2) doublet $S^{\\prime }$ to the SM Lagrangian can be described by an additional term $\\mathcal {L}_{S^{\\prime }}$[101], [132], given by $\\mathcal {L}_{S^{\\prime }} = t^a \\bar{t} \\left( g_S + g_P \\gamma _5 \\right) \\phi ^a u,$ where $g_S$ and $g_P$ denote the scalar and pseudoscalar coupling constants, respectively.",
"The resulting asymmetry $y_{S^{\\prime }}$ created in the exchange is hence given by $y_{S^{\\prime }} = \\sqrt{g_S^2 + g_P^2}.$ This contribution is typically negative for a heavy coloured scalar.",
"However, due to the destructive interference of the scalar contribution with the SM, a positive overall asymmetry can be generated."
],
[
"Top Quark Charge asymmetry at Hadron Colliders",
"In order to quantify a potential charge asymmetry created in the Standard Model or BSM models, the natural choice of observable would be the production angle $\\theta _t$ of the top/antitop quarks with respect to the incoming partons from the hard scattering process, as depicted in Figure REF.",
"Figure: Top quark pair production kinematics in quark-antiquark annihilation.",
"The initial state partons and their momenta p → q \\vec{p}_q and p → q ¯ \\vec{p}_{\\bar{q}} and the produced top quarks and their momenta p → t \\vec{p}_t and p → t ¯ \\vec{p}_{\\bar{t}}, respectively, are shown.",
"In addition, the production angle of the top quark θ t \\theta _t is shown.The corresponding differential charge asymmetry $A(\\cos {\\theta _t})$ at the partonic level is given by $A(\\cos {\\theta _t}) = \\frac{N_t(\\cos {\\theta _t}) - N_{\\bar{t}}(\\cos {\\theta _t})}{N_t(\\cos {\\theta _t}) + N_{\\bar{t}}(\\cos {\\theta _t})},$ in the $q\\bar{q}$ rest frame, where $N_t(\\cos {\\theta _t}) = \\frac{d \\sigma }{d \\Omega }(\\cos {\\theta _t})$ and $N_{\\bar{t}}(\\cos {\\theta _t}) = N_t(- \\cos {\\theta _t})$ due to the symmetry of charge conjugation.",
"Consequently, the integrated charge asymmetry $A$ can be quantified such that $A = \\frac{N_t(\\cos {\\theta _t} \\ge 0) - N_{\\bar{t}}(\\cos {\\theta _t} \\ge 0)}{N_t(\\cos {\\theta _t} \\ge 0) + N_{\\bar{t}}(\\cos {\\theta _t} \\ge 0)}.$ However, in the strong production of top quark pairs at hadron colliders, the production angle as such is not accessible experimentally due to the fact that the initial state of the partonic reaction is of probabilistic nature.",
"Since the available information is limited to the hadronic initial state ($pp$ or $p\\bar{p}$ ) and the PDFs of the protons and/or antiprotons, respectively, different methods to measure the charge asymmetry, making use only of the final state information of the hadronic collision, must be employed.",
"At non-symmetric hadron colliders such as the Tevatron, where protons are brought to collision with antiprotons, a charge asymmetry in the production of top quark pairs as introduced in Equation REF, observed in the $t\\bar{t}$ rest frame, corresponds directly to an equal-sized geometric forward-backward asymmetry, $A_{\\text{FB}}$ , since $N_t(y) = N_{\\bar{t}}(- y)$ .",
"Since this quantity is experimentally accessible in a direct way due to the fact that the initial directions of the proton and antiproton are known, the measurement of the underlying charge asymmetry in the laboratory frame is possible.",
"At $pp$ colliders such as the LHC, no forward-backward asymmetry is visible in the laboratory frame due to the intrinsic charge conjugation symmetry of the initial state collisions.",
"However, since top quarks are preferentially emitted in the direction of the incoming parton, and quarks in the proton on average carry a larger momentum fraction than antiquarks, an excess of top quarks in the forward and backward regions is expected in the laboratory frame.",
"Consequently, different widths of the corresponding rapidity distributions of top quarks and antitop quarks, and hence, the respective decay products, are predicted.",
"The underlying charge asymmetry in the $t\\bar{t}$ rest frame can be extracted either from the final state particles directly or by performing a kinematic reconstruction of the $t\\bar{t}$ decay signature.",
"Since an asymmetry can solely be created from the quark-antiquark annihilation contribution to the top quark pair production cross-section, the total charge asymmetry in both $pp$ and $p\\bar{p}$ collisions can be significantly diluted due to the (symmetric) gluon-gluon fusion contribution.",
"The magnitude of the overall asymmetry depends strongly on the centre-of-mass energy since the fraction of soft gluons in the proton/antiproton PDFs and hence the probability of gluon interactions in the partonic reaction increases with rising hadron momentum.",
"In addition, the top quark pair production cross-section in $pp$ collisions shows a higher contribution from gluon-gluon fusion since interactions of sea quarks are dominant.",
"This differs from $p\\bar{p}$ collisions, where potential interactions of valence (anti)quarks from the colliding (anti)protons lead to an increased contribution from quark-antiquark annihilation in the overall cross-section.",
"In order to quantify the charge asymmetry at the LHC, a proper observable, taking into account the potential differences in the rapidity distributions of the top and antitop quark in the laboratory frame has to be chosen.",
"Different frame-invariant variables based on rapidity or pseudorapidity differences of the final state top and antitop quarks are typically used to measure the asymmetry.",
"Potential observables to quantify the inclusive charge asymmetry in the $t\\bar{t}$ rest frame $A_{t\\bar{t}} = \\frac{N^{+} - N^{-}}{N^{+} + N^{-}}$ based on rapidities and pseudorapidities of the final state top and antitop quarks include parametrisations such as $\\begin{array}{lcl}N^{+} = N(\\eta _t - \\eta _{\\bar{t}} \\ge 0) & , & N^{-} = N(\\eta _t - \\eta _{\\bar{t}} \\le 0); \\\\N^{+} = N(y_t - y_{\\bar{t}} \\ge 0) & , & N^{-} = N(y_t - y_{\\bar{t}} \\le 0)\\text{\\cite {PhysRevLett.101.202001,Aaltonen:2011kc,Abazov:2011rq}}; \\\\N^{+} = N(|\\eta _t| - |\\eta _{\\bar{t}}| \\ge 0) & , & N^{-} = N(|\\eta _t| - |\\eta _{\\bar{t}}| \\le 0)\\text{\\cite {CMS-PAS-TOP-11-014}}; \\\\N^{+} = N(|y_t| - |y_{\\bar{t}}| \\ge 0) & , & N^{-} = N(|y_t| - |y_{\\bar{t}}| \\le 0)\\text{\\cite {CONFNote}}.\\end{array}$ Since $\\Delta y = y_t - y_{\\bar{t}} = 2 y_t^{t\\bar{t}}$ and correspondingly for the pseudorapidities of the final state top and antitop quarks, it follows that $A_{t\\bar{t}} = \\frac{N(\\Delta y \\ge 0) - N(\\Delta y \\le 0)}{N(\\Delta y \\ge 0) + N(\\Delta y \\le 0)}.$ At the Tevatron, the predicted charge asymmetry within the Standard Model for the given observable $A_{t\\bar{t}}$ , evaluated for a centre-of-mass energy of $1.96\\,\\text{TeV}$ , is $A_{t\\bar{t}} = 0.087 \\pm 0.010 \\text{\\cite {Kuhn:2011ri}}.$ In the following analysis, the charge asymmetry will be quantified using a parametrisation based on the difference of absolute rapidities of the top and antitop quarks, defined as $A_C = \\frac{N(|y_t| - |y_{\\bar{t}}| \\ge 0) - N(|y_t| - |y_{\\bar{t}}| \\le 0)}{N(|y_t| - |y_{\\bar{t}}| \\ge 0) + N(|y_t| - |y_{\\bar{t}}| \\le 0)}.$ At the LHC, the predicted charge asymmetry within the Standard Model for the given observable $A_C$ , evaluated for a centre-of-mass energy of $7\\,\\text{TeV}$ , is $A_C = 0.0115 \\pm 0.0006 \\text{\\cite {Kuhn:2011ri}}.$ A summary of the predicted charge asymmetries for Tevatron and LHC measurements for various BSM models can be found in Figure REF.",
"Potential regions in the phase space of inclusive charge asymmetry from new physics, indicated by the variable $A_C^{\\text{new}}$ at the LHC plotted against the associated forward-backward asymmetry $A_{\\text{FB}}^{\\text{new}}$ at the Tevatron, are highlightedNote that $A_{\\text{FB}}^{\\text{new}}$ and $A_C^{\\text{new}}$ denote only the respective contributions to the overall charge asymmetry and forward-backward asymmetry originating from the corresponding BSM model.",
"The predicted Standard Model contribution in the respective variables is subtracted (and hence corresponds to $A_{\\text{FB}}^{\\text{new}} = 0$ and $A_C^{\\text{new}} = 0$ , respectively)..",
"Different generalised predictions and parametrisations[127], [116] of $Z^{\\prime }$ and $W^{\\prime }$ models, scalar triplet ($\\omega ^4$ ) and sextet ($\\Omega ^4$ ) models, a generalised colour-octet resonance model ($\\mathcal {G}_{\\mu }$ ) and a colour-singlet Higgs-like isodoublet $\\phi $ are shown.",
"The same model predictions are shown for a high $t\\bar{t}$ invariant mass region.",
"Figure: Predicted charge asymmetries at the Tevatron and LHC for various BSM models, .",
"The inclusive charge asymmetry originating from new physics, A C new A_C^{\\text{new}}, at the LHC vs. the forward-backward asymmetry A FB new A_{\\text{FB}}^{\\text{new}} at the Tevatron (left) is shown.",
"Furthermore, the identical predictions in a high invariant mass region where M tt ¯ >450GeVM_{t \\bar{t}} > 450\\,\\text{GeV} (right) for the different models in the created phase space is shown.",
"The Standard Model prediction corresponds to A FB new =0A_{\\text{FB}}^{\\text{new}} = 0 and A C new =0A_C^{\\text{new}} = 0, respectively.Depending on the observed asymmetries at the Tevatron and the LHC, the exclusion of different theories can be possible.",
"Recent measurements of the charge asymmetry by the CDF and DØ collaborations at the Tevatron indicate large partonic asymmetries $A_{t \\bar{t}}$ in the $t\\bar{t}$ rest frame of $A_{t \\bar{t}} = 0.201 \\pm 0.065\\,\\text{(stat.)}",
"\\pm 0.018\\,\\text{(syst.)}",
"\\text{\\cite {CDFCONF-10584}}$ in the combination of the dileptonic and semileptonic decay channel, indicating a $2.9 \\sigma $ excess, and $A_{t \\bar{t}} = 0.196 \\pm 0.065\\,\\text{(stat.+syst.)}",
"\\text{\\cite {Abazov:2011rq}},$ respectively, indicating a $1.9 \\sigma $ excess above the Standard Model prediction.",
"Furthermore, larger deviations for high $t \\bar{t}$ invariant masses[141] and for high rapidity differences[143] have been observed.",
"In particular, an asymmetry of $A_{t \\bar{t}} = 0.475 \\pm 0.114\\,\\text{(stat.+syst.)}",
"\\text{\\cite {Aaltonen:2011kc}}$ for $t \\bar{t}$ invariant masses above 450 GeV has been observed, indicating a $3.4 \\sigma $ deviation from the Standard Model prediction, as shown in Figure REF.",
"Figure: Summary of results obtained in a tt ¯t \\bar{t} invariant mass dependent measurement of the charge asymmetry at CDF.",
"A deviation of 3.4σ3.4 \\sigma is observed in the high M tt ¯ M_{t \\bar{t}} region.Similar measurements have been performed by the ATLAS and CMS collaborations at the LHC.",
"In previous ATLAS measurements, a charge asymmetry in the variable $A_C$ of $A_C = -0.024 \\pm 0.016\\,\\text{(stat.)}",
"\\pm 0.023\\,\\text{(syst.)}",
"\\text{\\cite {CONFNote}}$ has been measured, while a measurement at CMS using the corresponding observable based on pseudorapidities instead of rapidities indicates a charge asymmetry of $A_C = -0.016 \\pm 0.030\\,\\text{(stat.)}",
"_{-0.021}^{+0.026}\\,\\text{(syst.)}",
"\\text{\\cite {CMS-PAS-TOP-11-014}},$ in compatibility with the Standard Model prediction."
],
[
"Experimental Setup",
"This chapter describes the technical details of the Large Hadron Collider and the ATLAS experiment, focusing on the detector subsystems and their properties after a short general overview.",
"In addition, the ATLAS trigger system is explained in more detail concerning technical and functional parameters."
],
[
"LHC and ATLAS Technical Overview",
"The Large Hadron Collider (LHC)[144] is situated at CERN, the European Centre for Nuclear Research near Geneva, Switzerland, and is the technologically most advanced particle accelerator so far.",
"It is designed as a proton-proton accelerator with the potential to accelerate and collide heavy ions as well.",
"Its construction started in 1999 and physics operation commenced in November 2009 with the first proton-proton collisions.",
"The collider has been constructed in the former accelerator ring of the Large Electron Positron Collider (LEP), which is about 27 kilometres in circumference and 100 to 125 metres below ground level, partly making use of the already existing infrastructure of the LEP ring as well as two existing caverns for experiments.",
"A schematic view of the accelerator ring is shown in Figure REF.",
"Figure: Layout of the LHC accelerator complex, including four of the LHC experiments.Image taken from wikipedia (public domain).",
"The accelerator complex incorporates six experiments, two of them being multi-purpose experiments, ATLAS[145] and CMS[146], while the other four are designed for more specific fields of research.",
"Having two independently designed multi-purpose detectors is vital for cross-confirmation of any potential discoveries made and naturally allows to combine the results of both experiments.",
"Among the four other detectors, the LHCb[147] experiment focuses on $b$ physics, while ALICE[148] has been designed for heavy ion physics.",
"TOTEM[149] and LHCf[150] are both designed to conduct studies of forward physics, including soft and hard diffractive processes and low-$x$ QCD.",
"Protons are extracted from hydrogen atoms in a duoplasmatron[151] and then accelerated by a linear accelerator (up to 50 MeV) before being injected into the first circular accelerator, the Proton Synchrotron Booster ($50\\,\\text{MeV} \\rightarrow 1.4\\,\\text{GeV}$ ).",
"Afterwards, they are injected into the Proton Synchrotron (PS, $1.4\\,\\text{GeV} \\rightarrow 26\\,\\text{GeV}$ ) and subsequently into the Super Proton Synchrotron (SPS, $26\\,\\text{GeV} \\rightarrow 450\\,\\text{GeV}$ ), before being transferred into the main LHC ring.",
"The two LHC proton beams deliver an energy of 3.5 TeV each (7 TeV at design specifications) and bunches of about $10^{11}$ protons can be brought to collision at a bunch crossing rate (BCR) of 40 MHz within one of the various detectors.",
"Collision interactions are typically characterised by the instantaneous luminosity $\\mathcal {L}$ , which relates the cross-section $\\sigma $ of a given process to the corresponding event rate $\\dot{N}$ , given the experimental acceptance $A$ and measurement efficiency $\\varepsilon $ : $\\mathcal {L} = \\frac{\\dot{N}}{\\sigma A \\varepsilon }.$ At the design luminosity of $\\mathcal {L} = 10^{34}\\,\\rm {cm}^{-2}\\rm {s}^{-1}$ and the given collision parameters, this leads to a total of about 23 proton - proton collisions per bunch crossing on average, which implies an overall interaction rate in the GHz regime.",
"The ATLAS detector is designed to investigate a wide range of physics processes, including the measurement of well-known Standard Model processes, the search for the Higgs boson, extra dimensions, and particles that could constitute dark matter.",
"It is 44 m in length, 25 m in height and 25 m in width, with a total weight of about 7000 tonnes.",
"As a comparison, CMS, the second LHC multi-purpose experiment, weighs about 12500 tonnes while being 21 m long, 15 m wide, and 15 m high.",
"ATLAS features an onion-like structure, which will be discussed in detail in Section .",
"A summary of the ATLAS and the LHC specifications compared to other accelerator/detector combinations is shown in Table REF.",
"Table: Comparison of LHC/ATLAS to otheraccelerators/detectors.",
"The particle types brought to collision, the centre-of-mass energy s\\sqrt{s}, the bunch crossing rate (BCR), the amount of readout channels (N CH N_{\\rm {CH}}), the average amount of data per event and the year of startup of the accelerator are shown.Values refer to Tevatron Run II.",
"As can be seen from the comparison table, the LHC exceeds the bunch crossing rate of the largest particle accelerator up to 2009 (Tevatron in Run II, Fermi National Accelerator Laboratory) by a factor of seven.",
"Furthermore, the amount of detector readout channels used at ATLAS increased by two orders of magnitude with respect to CDF/DØ, together with the average amount of data per event increasing by about one order of magnitude.",
"This results in a much higher total raw data throughput rate to be processed by the readout and data acquisition system.",
"As a consequence, a sophisticated trigger system designed for the reduction of data rates is needed to facilitate analyses.",
"This can only be achieved by selecting a subset of potentially relevant events out of the large amount of collisions that take place at LHC/ATLAS, rejecting a large fraction of raw data."
],
[
"ATLAS Coordinate System",
"The ATLAS coordinate system is a right-handed coordinate frame with the $x$ -axis pointing towards the centre of the LHC ring and the $z$ -axis being directed along the beam pipe, while the $y$ -axis points upwards (slightly tilted with respect to the vertical direction ($0.704\\,^{\\circ } $ ) due to the general tilt of the LHC tunnel).",
"In this context, the pseudorapidity can be introduced as $\\eta = - \\ln \\tan \\frac{\\theta }{2}$ with $\\theta $ being the polar angle with respect to the positive $y$ -axis.",
"For massive objects such as jets, the rapidity is used, given by $y = \\frac{1}{2} \\ln { \\left( \\frac{E + p_z}{E - p_z} \\right)}.$ In addition, the transverse momentum $p_{\\rm {T}}$ of a particle in the detector is defined as the momentum perpendicular to the $z$ -axis: $p_{\\rm {T}} = \\sqrt{p_x^2 + p_y^2}.$ Furthermore, the azimuthal angle $\\phi $ is defined around the beam axis."
],
[
"The ATLAS Detector Subsystems",
"In order to allow for reliable detection of particles and measurement of their properties, the ATLAS detector requirements include a good hermiticity with respect to detector acceptance, a high spatial and timing resolution, in particular to minimise occupancy of individual detector components, to measure $p_{\\rm {T}}$ with high resolution and to allow for distinction between different particles, a low material budget to minimise particle interaction and energy loss with non-active detector materials.",
"ATLAS has a cylindrical shape with layers of detector components arranged in axial succession.",
"Each of these layers is designed to detect different types of particles which are mostly originating from the primary Figure: The ATLAS detector subsystems.interaction point of the proton beams at the centre of ATLAS.",
"As they travel throughout the detector, they can be measured by its successive layers.",
"The different detector subsystems are shown in Figure REF and constitute, from the innermost to the outermost layer, the inner detector, the solenoid magnet, the electromagnetic calorimeter, the hadronic calorimeter, the toroidal magnet, and the muon spectrometer.",
"The detectors are complementary: charged particles are detected in the innermost layers by their hits in the tracking chambers, where the particle trajectory is bent by the magnetic field of the superconducting solenoid magnet.",
"Using this tracking information, the momentum of the particles can be determined.",
"Around the magnet, the electromagnetic and hadronic calorimeters are designed to measure the energy of particles.",
"These are brought to a stop in the calorimeter by interaction with the detector material[23] (ionisation), thus depositing all of their energy, which is measured in the calorimeter cells.",
"Finally, the muon chambers allow for additional momentum measurements of muons, which penetrate all other layers of the detector only depositing very little energy in the detector material.",
"The measurements in the muon chambers are performed using the shape of the tracks, which are bent by the magnetic field of the toroidal magnets.",
"Image taken from http://www.atlas.ch/."
],
[
"Inner Detector",
"In the following, the inner detector[145], [152] is described in more detail.",
"It is situated near the interaction point to allow for high precision measurement of charged particle trajectories.",
"It covers a pseudorapidity range of $|\\eta | < 2.5$ and consists of three subsystems, the silicon pixel detector, the semiconductor tracker (SCT) and the transition radiation tracker (TRT).",
"The innermost layers of the inner detector (three in the cylindrical barrel region, three endcap disks on each side of the forward region) constitute the pixel detector, which is designed to measure particle vertices and extract track momenta from the reconstructed particle hits in the detector layers.",
"Due to its close proximity to the primary interaction point, a very high spatial resolution of the pixel detector is required, which is achieved by very small pixel sizes of 50 $\\umu $ m $\\times $ 400 $\\umu $ m and 50 $\\umu $ m $\\times $ 600 $\\umu $ m (about $8 \\cdot 10^7$ readout channels in total), with the pixel detector covering a total area of $2.3\\,\\rm {m}^2$ .",
"Around the pixel detector, the semiconductor tracker (or silicon strip tracker) measures the momentum of charged particles.",
"It consists of four barrel layers and nine endcap wheels on each side, covering a total area of $61.1\\,\\rm {m}^2$ and making use of about $6.3 \\cdot 10^6$ readout channels.",
"Figure: Cross-sectional view of a quarter-section of the ATLAS inner detector showing each of the major detector elements alongside with its active dimensions and envelopes.The outermost part of the inner detector is constituted by the TRT, which consists of straw tubes with a diameter of 4 mm and a maximum length of 80 cm, filled with an ionisable gas.",
"The barrel tubes are divided in two at the centre and read out at each end to reduce occupancy.",
"The ionisation charges created by charged particles travelling through the gas filled tubes are used to enhance track pattern recognition and to improve momentum resolution of the objects identified in the pixel detector and semiconductor tracker with an additional average of 36 hits per track.",
"Furthermore, its function is to distinguish electrons and pions making use of the different amount of transition radiation emitted by these particles when crossing the boundary surface of two media with different dielectric constants (in this case a special radiator foam with a large amount of air bubbles to achieve a maximum material transition surface).",
"The transition radiation tracker has a total of 351000 readout channels.",
"The resulting tracking performance of the inner detector subsystems for single particles and particles in jets can be found in Table REF.",
"Table: Track parameter resolutions at infinite momentum σ X (∞)\\sigma _X(\\infty ) and the transverse momentum p X p_X for which the intrinsic and multiple-scattering contribution equalsthe intrinsic resolution.",
"The momentum and angles correspond to muons, while the impact parameters correspond topions.",
"The values are shown for two η\\eta regions, one in the barrel inner detector (where the amount of material isclose to its minimum) and one in the endcap (where the amount of material is close to its maximum)."
],
[
"Calorimeters",
"Around the solenoid magnet (which is described in more detail in Chapter REF), an electromagnetic (EM) liquid argon sampling calorimeter is used to detect and identify electromagnetically interacting particles and to measure their energy[145], [153], [154], [155].",
"It also allows, in combination with the hadronic calorimeters, for reconstruction of hadronic jets and the measurement of the missing energy of an event.",
"Figure: Overall layout of the ATLAS calorimeters.The EM calorimeter covers a pseudorapidity range of $|\\eta | < 3.2$ and comprises several layers of accordion-shaped kapton-copper electrodes and lead absorber plates shrouded in stainless steel, with the gaps in between filled with liquid argon at a temperature of 87 K. Whenever an electromagnetically interacting particle passes through one of the lead absorber plates, it creates a particle shower that ionises the liquid argon.",
"Exposed to an electric field, the drifting ionisation charges induce a signal in the electrodes due to capacitive coupling.",
"The resulting signal is sampled and digitised at 40 MHz, corresponding to the bunch crossing rate.",
"The EM calorimeter has a total of 170000 readout channels, and provides an energy resolution of $\\dfrac{\\sigma _{\\rm {EM}}^{\\rm {sp}}}{E} = \\dfrac{(10.1 \\pm 0.4)\\,\\%}{\\sqrt{E\\,[\\rm {GeV}]}}~\\text{(stochastic)}~\\oplus ~0.2 \\pm 0.1\\,\\%~\\text{(constant)}\\text{ \\cite {AtlasExperiment}},$ as measured in particle test beams.",
"The total module thickness in the barrel region corresponds to at least 22 radiation lengths ($X_0$ ), increasing from $22\\,X_0$ to $30\\,X_0$ between $|\\eta |=0$ and $|\\eta |=0.8$ and from $24\\,X_0$ to $33\\,X_0$ between $|\\eta |=0.8$ and $|\\eta |=1.3$ .",
"In the endcaps, the total thickness is greater than $24\\,X_0$ except for $|\\eta |<1.475$ , increasing from $24\\,X_0$ to $38\\,X_0$ in the outer wheel ($1.475 < |\\eta | < 2.5$ ) and from $26\\,X_0$ to $36\\,X_0$ in the inner wheel ($2.5 < |\\eta | < 3.2$ ).",
"In analogy, the hadronic calorimeter (hCAL)[145], [153], [154] is designed to measure the energy of hadronic particles that can penetrate the EM calorimeter.",
"This sampling calorimeter consists of iron absorbers for showering which are interleaved with plastic scintillator tiles (thus being referred to as tile calorimeter) in the barrel part of the detector ($|\\eta |<1.7$ ).",
"The scintillator tiles emit a shower of photons whenever charged particles pass through them due to excitation of the atoms in the scintillating material and subsequent emission of visible or UV photons.",
"These light pulses are carried by optical fibres to photomultiplier tubes and converted to electric signals.",
"The total number of tile calorimeter readout channels is of the order of 10000.",
"Since the scintillating tiles are very sensitive to radiation damage, liquid argon is used as sampling medium together with copper absorbers in the forward endcap regions ($1.5<|\\eta |<3.2$ ) in close proximity to the proton beams.",
"This provides improved radiation hardness in the region of increased particle flux.",
"The total number of channels for both endcaps is 5632.",
"For the same reason, a high density copper/tungsten absorber liquid argon forward calorimeter (FCAL) covers the pseudorapidity region $3.1<|\\eta |<4.9$ , with an additional 3524 channels for both forward regions together.",
"In order to estimate the performance of the hadronic calorimeter, test beam studies were conducted, showing an energy resolution of $\\dfrac{\\sigma _{\\pi }^{\\rm {HAD}}}{E_{\\pi }} = \\dfrac{(56.4 \\pm 0.4)\\,\\%}{\\sqrt{E\\,[\\rm {GeV}]}}~\\text{(stochastic)}~\\oplus ~5.5 \\pm 0.1\\,\\%~\\text{(constant)}\\text{ \\cite {AtlasExperiment}}$ for pions, and a radial depth of approximately 7.4 interactions lengths ($\\lambda $ ) for the tile calorimeter.",
"The hadronic endcaps show an energy resolution of $\\dfrac{\\sigma _{e}^{\\rm {HEC}}}{E_{e}} = \\dfrac{(21.4 \\pm 0.1)\\,\\%}{\\sqrt{E\\,[\\rm {GeV}]}}~\\text{(stochastic)}\\text{ \\cite {AtlasExperiment}}$ for electrons (the constant term being compatible with zero), and $\\dfrac{\\sigma _{\\pi }^{\\rm {HEC}}}{E_{\\pi }} = \\dfrac{(70.6 \\pm 1.5)\\,\\%}{\\sqrt{E\\,[\\rm {GeV}]}}~\\text{(stochastic)}~\\oplus ~5.8 \\pm 0.2\\,\\%~\\text{(constant)}\\text{ \\cite {AtlasExperiment}}$ for pions.",
"The jet energy resolution for the overall calorimeter system is described by the parametrisation $\\dfrac{\\sigma _{\\rm {jet}}}{E_{\\rm {jet}}^{\\phantom{2}}} = \\sqrt{ \\dfrac{a^2}{E_{\\rm {jet}}^{\\phantom{2}}} +\\dfrac{b^2}{E_{\\rm {jet}}^2} + c^2 }.$ For central jets in the region $0.2<|\\eta |<0.4$ , it is $a \\approx 60\\,\\%\\,\\sqrt{\\rm {GeV}}$ (stochastic), $c \\approx 3\\,\\%$ (constant) and the noise term $b$ increases from 0.5 GeV to 1.5 GeV from barrel to endcap ranges[145]."
],
[
"Muon Chambers",
"The muon chambers[145], [156], [154] are designed to detect muons which are able to pass all other detector systems depositing only a small amount of energy in the material due to the fact that muons in the GeV regime are approximately minimum ionising particles.",
"Being deflected by the magnetic field in the detector, it is possible to determine the muon momentum and sign of electric charge by measuring its trajectory as it passes through the tracking chambers.",
"The muon spectrometer is also designed to trigger on these particles, utilising dedicated trigger chambers.",
"The driving performance goal is a standalone transverse momentum resolution of approximately $10\\,\\%$ for 1 TeV tracks, which translates into a sagitta along the z axis of about $\\chi = 500\\,\\umu $ m, to be measured with a resolution of $\\sigma _{\\chi }\\le 50\\,\\umu $ m. The sagitta $\\chi $ is given by $\\chi = R - R \\cos {\\frac{\\theta }{2}},$ where $R$ is the radius of the track curvature and $\\theta $ is the angle enclosed by the outermost of three equidistant points along the track, as can be seen in Figure REF.",
"To achieve high spatial tracking resolution, three layers (stations) of drift chambers (precision chambers) are employed both in the barrel and in the endcap region.",
"In the barrel these chambers are arranged in concentric cylinders, with the radii of the detector layers being at about 5 m, $7.5$ m and 10 m, covering a pseudorapidity range of $|\\eta | < 1.0$ .",
"Two of these layers are placed near the inner and outer field boundary, while the third is situated within the field volume.",
"The muon momentum is determined from the track sagitta.",
"In this region, exclusively monitored drift tube chambers (MDTs) are used.",
"Due to the magnet cryostats in the endcap region, however, placing one station within the field is not possible.",
"Hence it is necessary to rely on a point-angle measurement to determine the track momentum in this detector region (a point in the inner station and an angle in the combined middle-outer stations).",
"The endcap layers are arranged in four concentric discs at 7 m, 10 m, 14 m, and 21-23 m from the detector origin, covering a pseudorapidity range of $1.0<|\\eta |<2.7$ .",
"Here, in addition to MDT chambers, cathode strip chambers (CSCs) are used in the innermost layer of the inner station due to radiation hardness requirements.",
"Figure: Illustration of the track sagitta χ\\chi and its geometricrelation to the bending radius RR for a track of a particle with a given momentum p →\\vec{p} travelling througha homogeneous magnetic field, shown for the case of three equidistant track hits along the curvature.The only exception to the continuous $\\eta $ coverage of the muon chambers is made at $|\\eta | < 0.05$ in the $R$ -$\\phi $ plane to allow for cable and service outlets for the inner detector, the central solenoid and the calorimeters (central gap).",
"Further regions of reduced acceptance can be found at the feet of the detector.",
"Figure: The ATLAS muon system.",
"The trigger chambers (RPC, TGC) and the precisionchambers (MDT, CSC) are shown.Regardless of the high spatial resolutions, the timing resolution of the precision chambers as shown in Table REF is too low due to the drift time to ensure differentiation between muons from subsequent bunch crossings for the trigger (where the scale of the required timing resolution is set by the bunch crossing interval of 25 ns).",
"Thus it is necessary to employ additional drift chambers with high timing resolution (at cost of spatial resolution), the trigger chambers.",
"These provide a fast momentum estimate and are primarily used for the trigger (since their spatial resolution is too low with respect to the precision chambers), which will be described in Chapter .",
"In the barrel region, two out of three resistive plate chambers (RPCs) are placed directly in front of and behind the central MDT, while the third is situated directly below or above (according to the mechanical constraints in the respective region) the outermost precision chamber.",
"The RPCs are furthermore used to determine the second coordinate for the MDT chambers (in tube wire direction).",
"In the endcaps, three layers of thin gap chambers (TGCs) are located near the central endcap MDT layer for triggering.",
"A comparison of the different chamber technologies, necessary to allow for both fast triggering and high precision measurements, is shown in Table REF.",
"Table: Comparison of muon chamber technologies.",
"Both the trigger chambers (RPC and TGC) and precision chambers (MDT and CSC) and their respective resolution and timing performance are shown."
],
[
"Magnet System",
"Outside the inner detector follows a solenoid magnet[145], [157], which is used to bend the trajectories of charged particles on their way through the inner detector, making it possible to measure the particle momentum with Figure: Schematic view of the ATLAS solenoidal (inner cylinder) andtoroidal magnets (outer coils).high resolution.",
"Its axial field strength is about 2 T (peak $2.6$ T) using low-temperature superconducting cables cooled down to 1.8 K with liquid helium during operation with a nominal current flow of 8000 A.",
"In addition, a large toroidal magnet[145], [158] consisting of eight superconducting coils in the barrel region and eight more at each of the forward regions extends throughout the muon chambers, providing a magnetic field strength of 4 T (peak $4.7$ T).",
"The whole toroid system contains over 70 km of superconducting cable, allowing for a design current of 20000 A with a stored energy of above 1 GJ.",
"To minimise multiple scattering of the muons, the toroid design incorporates an air core.",
"Similar to the solenoid magnet, its purpose is to bend the trajectories of muons in order to measure their transverse momentum (in combination with the tracking information from the inner detector).",
"The design luminosity of $10^{34}\\,\\rm {cm}^{-2}\\rm {s}^{-1}$ , in combination with the bunch crossing rate of 40 MHz and the amount of protons contained in each single bunch, leads to a proton-proton collision rate in the GHz regime.",
"This corresponds to an extremely high theoretical raw data rate of about 1.5 PBs$^{-1}$ .",
"Being able to store only a fraction of this amount of data on storage media ($\\sim (300-500)$ MBs$^{-1}$ ) and only a small fraction of these collisions being useful for analysis, the ATLAS trigger system[145], [159], [154] has been designed to reduce the initial data rate by several orders of magnitude.",
"Figure REF shows the total rates of several physics processes in comparison to the total interaction rate.",
"As an example, the frequency of the Standard Model $t\\bar{t}$ production at $\\sqrt{s}$ = 7 TeV is of the order of 1 mHz, constituting only a small fraction of the total amount of the raw data rate, making the selection of this and other physics processes a crucial task.",
"In order to achieve such a reduction and to select only relevant physics events / processes, ATLAS uses a three-level trigger system for real-time event selection (with the last two levels being referred to as high-level trigger), while each trigger level refines the decisions of its predecessor.",
"An overview of the different trigger levels and the global structure is shown in Figure REF and will be discussed in more detail in the following sections.",
"Figure: Block diagram of the trigger/DAQ system.",
"On the left side the typical collision and the data equivalent at the different stages of triggering are shown, while in the middle section the different components of the trigger system are shown schematically.",
"The right side of the graphic gives a short summary of the operations and the technologies used at the respective level."
],
[
"Level 1 Trigger (LVL1)",
"The LVL1 trigger is completely hardware-based, where highly specialised components are deployed, including field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs) and reduced instruction set computing (RISC) chips.",
"Since most of this hardware is integrated directly into the particular detector components in order to reduce material occurrence from cabling and additional readout electronics, the LVL1 trigger system is required to be highly parallelised.",
"Being based on the muon and calorimeter system only, the LVL1 trigger[160] performs an initial selection on the basis of the hits in the muon trigger chambers and calorimeters.",
"In the muon chambers, low-$p_{\\rm {T}}$ and high-$p_{\\rm {T}}$ muons are identified by measuring the tracks in the trigger chambers (RPCs and TGCs) using coincidence windows for discrimination.",
"Low-$p_{\\rm {T}}$ muons have a smaller bending radius, thus allowing for detection by two layers in close proximity to each other (Moreover, track-hit matching can be difficult if the in-plane hit distance in the coincidence layers is too large due to the curvature of the track).",
"In the barrel region, low-$p_{\\rm {T}}$ muons are identified by the consecutive layers RPC1 and RPC2 (The number after the chamber type identifies the layer).",
"In contrast, high-$p_{\\rm {T}}$ muons produce an almost straight track and therefore the coincidence layers used should be as separated as possible to allow for the measurement of the track radius.",
"Consequently, high-$p_{\\rm {T}}$ muons are measured by the combination of the outermost RPC1 and RPC3 hits in the barrel.",
"In the endcaps, low-$p_{\\rm {T}}$ muons are identified by the TGC2 - TGC3 coincidence window, while TGC1 and TGC3 are used to identify high-$p_{\\rm {T}}$ tracks.",
"This is shown in Figure REF, where a quadrant layout of the muon trigger chambers is shown.",
"Figure: Layout of the muon trigger chambers.",
"A quarter cross-section in the bending plane with typical low p T p_{\\rm {T}} and high p T p_{\\rm {T}} muon tracks and the corresponding coincidence windows in the different layers for barrel (RPC1-3) and endcap (TGC1-3) are shown.Classification of transverse momentum is achieved by using large lookup tables of track hits to find an estimate for the track momentum.",
"Different exclusive low and high $p_{\\rm {T}}$ thresholds are defined (e.g.",
"10, 11, 20 and 40 GeV) and can be modified if necessary.",
"The calorimeter trigger selection is based on low resolution information from all ATLAS calorimeters and is designed to identify high $E_{\\rm {T}}$ electrons and photons, hadron jets and the total transverse energy alongside with large missing transverse energy $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$, where this is defined by the sum of all vectored energy depositions $\\vec{E}_T$ in the transversal plane: $\\lnot \\!\\!\\vec{E}_{\\rm {T}} = - \\sum \\vec{E}_{\\rm {T}}.$ This definition arises from the fact that the initial transverse momentum of the incoming protons is approximately zero and due to conservation of energy in the transverse plane, the total vector sum of final state transverse energies has to be zero as well.",
"Furthermore, isolation requirements based on calorimeter information are available for the calorimeter trigger on this trigger level.",
"Since the LVL1 trigger operates synchronously with data taking, the latency for a decision is about $2.5\\,\\umu \\rm {s}$ , which is achieved by making use of pipeline memories despite the bunch crossing and data taking frequency of 40 MHz.",
"The trigger decision is derived while the information from the sensors is kept in the buffer memory for about 100 bunch crossings.",
"At the end of the latency time, the readout data is rejected or accepted (after leaving the pipeline memory).",
"Only in the latter case the geometric information of the triggered object is forwarded to the next trigger level as region of interest (ROI), which includes spatial position ($\\eta $ and $\\phi $ ) and $E_{\\rm {T}}$ estimates of the embedded objects (e, $\\mu $ , $\\tau $ , $\\gamma $ and jet candidates) as well as global energy information ($E_{\\rm {T}}$ and $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$) for further analysis.",
"This way, the data rate is reduced to about (75-100) kHz at LVL1."
],
[
"Level 2 Trigger (LVL2)",
"After a LVL1 accept, events are read out from the pipeline memories and stored in readout buffers (ROBs) until being processed by the LVL2 trigger[161].",
"This trigger level uses the ROIs from the previous level to further reduce the data rate to about 1 kHz.",
"This is achieved by analysing each ROI in the detector system from which it originated, accessing the data from other detector subsystems in the ROI in addition, including the inner detector tracking information, the full-granularity calorimeter hits and the precision chamber measurements from the muon spectrometer.",
"With an increased processing latency of 10 ms and the data rate already reduced at LVL1, the LVL2 trigger allows for more complex algorithms being applied to the trigger objects and the information from the respective ROIs.",
"As a part of the high-level trigger, the LVL2 stage is completely software-based and runs on dedicated computing farms.",
"The trigger decision is performed by an event-driven sequential selection procedure using the detector data.",
"Despite the LVL2 trigger providing more time to conduct the trigger decision than the LVL1 stage, the selection algorithms still have to be kept simple and efficient.",
"Hence, the sequence of the different algorithms / requirements is determined by their complexity, simple ones (with respect to CPU time and memory) are executed first, while each algorithm uses the result of its predecessor (seeded reconstruction)."
],
[
"Event Filter (EF)",
"In the final execution level of the trigger, which is software based and runs on CPU farms, the global event is collected from the ROBs.",
"The EF accesses the complete event information and all detector subsystems using full granularity.",
"An event reconstruction at trigger level is possible by using similar algorithms as it is the case for offline reconstruction, accessing calibration and alignment information from databases.",
"In addition to the reconstruction of the event as a whole, the EF performs extended tasks that are not possible at earlier trigger levels, such as vertex reconstruction, final track fitting and algorithms requiring larger ROIs than available at LVL2 (e.g.",
"the calculation of global $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$).",
"The latency of the EF is of the order of seconds, after which events passing this final trigger level are stored permanently for further analysis or, in some cases, may be redirected to special storage elements as well (if needed for calibration or alignment exclusively)."
],
[
"Trigger Implementation",
"The information used to conduct the LVL1 decision is given in terms of multiplicities of trigger items resembling candidates for physical objects (like electrons, muons, jets,...) detected in the calorimeters or muon trigger chambers which have sufficiently high $p_{\\rm {T}}$.",
"These are sent to the central trigger processor (CTP) together with threshold information on global energy sums.",
"The delivered multiplicities are discriminated against corresponding requirements or conditions, leading to logical values for each condition within so-called trigger menus.",
"Examples for LVL1 trigger items are Table: NO_CAPTIONThe final LVL1 event decision is derived from the values of the defined trigger items by applying a logical OR.",
"If the event is accepted, the ROIs for all trigger items and the contained trigger elements above the defined thresholds are delivered to the high-level trigger, where they are used as seeds for further processing.",
"The step-by-step execution of trigger algorithms as it is performed on LVL2 and EF level is called a trigger chain, consisting of different intermediate trigger signatures (cf.",
"trigger items on LVL1), where the successive trigger algorithms keep refining these signatures in the course of a trigger chain.",
"Examples for such trigger signatures, as they have been used in this analysis, are Table: NO_CAPTIONThe individual decisions can also be logically combined to more complex trigger items.",
"As an additional requirement for the high-level trigger, parallel execution of different trigger chains without interference is required to be possible in independent slices (e.g.",
"electron slice and muon slice) to ensure transparency and scalability of the trigger system.",
"The information of individual high-level trigger signatures is organised in trigger streams, grouping similar triggers based on the respective purpose, priority, and procedure of processing of the respective events.",
"An inclusive model is chosen to allow for trigger items to be contained in multiple trigger streams in parallel.",
"Raw data or physics streams contain the data events selected for full reconstruction and later analysis and correspond to the respective types of trigger items, such as the physics_Egamma and physics_Muon streams which contain events triggered by electron/photon or muon triggers, respectively.",
"Further trigger streams exist for calibration and monitoring purposes, such as the Express stream, which contains a small subset of relevant events for very fast reconstruction in order to allow for real-time monitoring of the data taking and trigger system."
],
[
"Underlying Event and Pile-Up",
"At hadron colliders such as the LHC, a multitude of partons is involved in the hadronic collisions due to the substructure of the colliding particles, while typically only one parton from each of the incoming hadrons is involved in the hard scattering process and has sufficient energy to create high energy/momentum final state particles.",
"Nevertheless, the remaining partons still contribute to the final event signature through low momentum transfer interactions.",
"These additional reactions which occur in parallel to the hard scattering event are denoted underlying event.",
"Furthermore, additional contributions to the final event signature arise from the interactions of other protons within the colliding bunches.",
"Despite the fact that the probability of multiple hard scatterings occurring within one bunch crossing is relatively low, the likelihood of soft interactions between the constituent partons originating from additional proton-proton collisions in the same bunch crossing increases significantly with instantaneous luminosity $\\mathcal {L}$ .",
"In addition to this in-time pile-up contribution, final state particles from different bunch crossings can lead to additional signatures if the identification of the correct bunch crossing is unsuccessful (out-of-time pile-up).",
"Both the underlying event and the pile-up contributions play a significant role in the final state of the observed collisions and hence have to be included in the simulation of signal and background processes to achieve a proper modelling of the data taken.",
"This modelling is in particular difficult due to the fact that both the underlying event and the pile-up cannot fully be described in perturbative QCD, since the particles created in such processes typically have very low momentum/energy.",
"Furthermore, due to time-dependent changes in the LHC environment parameters, the simulation has to be adapted continuously to match the respective setup for a given subset of data taken."
],
[
"Trigger Strategy and Object Definition",
"This chapter covers the determination and different approaches of applying trigger and reconstruction efficiencies in analyses.",
"In addition, the object definitions used in this analysis are described."
],
[
"Data Quality",
"In order to ensure that only data taken under well defined and stable conditions is taken into account for physics analyses, dedicated online[162] and offline[163] monitoring systems ensure data integrity and quality.",
"The online data quality monitoring accesses real-time detector status information and makes use of events from the Express trigger stream to provide several low-level quantities and distributions.",
"This allows for a quick response to problems with the LHC beam conditions or the detector that may arise during operation.",
"The data quality offline monitoring uses a first reconstruction performed in order to identify and record problems in the detector hardware and the data acquisition and processing.",
"All relevant information from the individual detector systems and reconstructed event quantities are combined into a small set of key numbers and distributions to allow for both automatic and manual monitoring.",
"Information from the online and offline data quality monitoring as well as feedback from the individual shift crews is combined into a database containing LHC beam conditions, detector status and data flow information which can be used to create lists of runs usable for analyses (GoodRunsLists), containing a set of data taking run and luminosity block (LB) information."
],
[
"Trigger and Reconstruction Efficiencies",
"Trigger and detector acceptance and response are represented by the respective trigger and reconstruction efficiencies, which are typically both estimated in Monte Carlo simulations and measured with data driven methods.",
"However, both the reconstruction and trigger simulation used for Monte Carlo samples typically do not fully reflect the actual conditions due to limitations of the simulation modelling.",
"Hence, discrepancies have to be corrected for in the Monte Carlo.",
"This ensures that the actual detector performance is reflected in the corrected Monte Carlo distributions of relevant physical quantities.",
"Since for Monte Carlo samples the detector simulation/reconstruction does not retain the information which links objects from the Monte Carlo generator level to reconstructed objects or trigger objects, it can be necessary to perform a matching procedure to identify corresponding objects and their affiliation at the different trigger levels and at offline reconstruction level.",
"This is achieved by a geometric matching in $\\Delta R$ , defined for example for the matching of objects at reconstruction level to the corresponding generated (true) object as $\\Delta R & = & \\sqrt{ (\\Delta \\eta )^2 + (\\Delta \\phi )^2 } \\nonumber \\\\& = & \\sqrt{ (\\eta _{\\text{true}} - \\eta _{\\text{reco}})^2 + (\\phi _{\\text{true}} - \\phi _{\\text{reco}})^2 }$ and for the different trigger levels, accordingly.",
"A positive match of the given objects is denoted by a $\\Delta R$ below a predefined threshold.",
"This threshold is typically chosen according to the spatial resolution of the measurement for the respective objects.",
"Both trigger and reconstruction efficiencies are commonly taken into account in physics analyses using scale factors, relating the respective efficiencies measured with data driven methods to the expected performance from Monte Carlo simulations.",
"Alternatively, in particular trigger efficiencies can be applied by implementing a reweighting approach, taking into account detector and trigger acceptance by direct application of data driven efficiencies in the form of event weights.",
"In this context, the reconstruction efficiency $\\varepsilon _{\\text{reco}}$ is typically defined as the fraction of objects which have been identified by a specific reconstruction algorithm, $\\varepsilon _{\\text{reco}} = \\frac{N_{\\text{reco}}}{N_{\\text{candidate}}},$ where $N_{\\text{candidate}}$ denotes the amount of reconstruction candidates (e.g.",
"tracks or energy depositions) considered for reconstruction.",
"In analogy, the trigger efficiency $\\varepsilon _{\\text{trig}}$ refers to the fraction of reconstructed objects which have been selected by a given trigger item or chain, $\\varepsilon _{\\text{trig}} = \\frac{N_{\\text{trig}}}{N_{\\text{reco}}}.$ These efficiencies are typically parametrised or binned in kinematic quantities of the given objects, such as $p_{\\rm {T}}$ and $\\eta $ ."
],
[
"Measurement of Trigger Efficiencies",
"In order to obtain trigger efficiencies from data, a simple Monte Carlo based counting method is not applicable as events rejected by the trigger are typically no longer available offline.",
"Furthermore, it is not desirable to rely on the trigger simulation.",
"To measure trigger efficiencies, there exist several data driven methods, including: orthogonal triggers: the trigger efficiency $\\varepsilon _{\\rm {A}}$ for a given trigger item A is determined with respect to a different, ideally orthogonal (i.e.",
"completely uncorrelated) trigger B, whose efficiency $\\varepsilon _{\\rm {B}}$ is known, minimum bias datasets: the trigger efficiency for an arbitrary trigger item is determined on a minimum bias dataset, where only a minimal trigger selection has been applied and thus a trigger efficiency measurement can be performed with minimal bias, the Tag & Probe method: the trigger efficiency for a given trigger is measured by selecting a specific process from a given data sample (if possible) of which the kinematics are known, allowing for a decoupling of event selection and trigger efficiency determination.",
"In the following, the Tag & Probe method is explained in more detail since it is commonly used to determine both trigger and reconstruction efficiencies for physics analyses.",
"To measure the trigger efficiency from data, it is possible to select a specific process, e.g.",
"${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ events for the determination of muon trigger efficiencies by application of the Tag & Probe data method, which is shown schematically in Figure REF.",
"Figure: The Tag & Probe method.",
"An event selection is performed using the tag object, while the actual trigger efficiency is determined only with the probe object to ensure independence of the two processes.The selection is achieved by making use of the fact that if one isolated muon is present in a ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ event, there has to be a second muon in the observed event that is isolated as well, on which the actual efficiency measurement can be performed.",
"The basic concept of this method focuses on decoupling event selection and the actual determination of the trigger efficiency.",
"Events are selected by first identifying a muon that has triggered the event, the tag muon.",
"Moreover, several requirements are applied to the tag muon, e.g.",
"$p_{\\rm {T}}$ and isolation criteria.",
"To ensure the ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ sample to be as pure as possible, these requirements should be very tight, leading to an enrichment of ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ events.",
"If the tag muon meets all requirements, a second muon (if present) in the event is selected, the probe muon, which must comply to requirements that typically correspond to the selection performed in the analysis for which the obtained efficiencies are used.",
"In addition, the dimuon invariant mass $M_{\\mu \\mu }$ has to be sufficiently close to the $Z$ pole mass.",
"If that is the case, there is a high probability that the probe muon does not originate from a background process, but constitutes a muon from the ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ signal.",
"The actual trigger efficiency $\\varepsilon _{\\rm {TP}}$ for the Tag & Probe method can be determined by the fraction of probe muons that have been triggered: $\\varepsilon _{\\rm {TP}} = \\frac{\\rm {N}_{\\mu }^{\\rm {probe\\,\\&\\,trigger}}}{\\rm {N}_{\\mu }^{\\rm {probe}}}.$ Since the invariant mass constraint directly relates the tag and the probe muon, it is obvious that the geometric correlation can create a bias in the estimated trigger efficiency as well.",
"This can, for example, be the case if one of the two muons produced in the $Z$ decay emerges into an acceptance gap of the detector, while the other muon is properly reconstructed.",
"In the context of the Tag & Probe method, this muon could pass the tag muon criteria, but since there is no second muon in the event, no efficiency can be determined, the event not being taken into account at all.",
"Hence, the Tag & Probe method does not cover events where only one of the muons was found, neglecting the corresponding contribution to the overall efficiency.",
"This effect, however, is typically very small."
],
[
"Scale Factors and Event Rejection Approach",
"If the data driven efficiency $\\varepsilon _{\\text{data}}$ for a specific trigger item or trigger chain has been measured and the corresponding Monte Carlo trigger simulation efficiency $\\varepsilon _{\\text{MC}}$ is known, the respective scale factor $f$ is defined by $f = \\frac{\\varepsilon _{\\text{data}}}{\\varepsilon _{\\text{MC}}},$ which quantifies the discrepancies between measured and predicted trigger or reconstruction efficiency.",
"Typically, both the underlying efficiencies and scale factors are parametrised according to actual dependencies on geometric and/or analysis related quantities such as transverse momentum or pseudorapidity.",
"The scale factors are applied by rejecting events on MC which do not pass the trigger criteria of the trigger simulation and applying the scale factors as an event weight based on all relevant objects on MC events passing the trigger simulation.",
"For events requiring exactly one reconstructed object, the event weight equals the scale factor corresponding to the respective object properties.",
"However, if more than one object is required in the final state, the probability $p$ of an event to be triggered is determined by the logical OR of the trigger probabilities of the all $N$ objects taken into account, $p = 1 - \\prod _{i=1}^N \\left( 1 - \\varepsilon _i \\right),$ where $\\varepsilon _i$ corresponds to the trigger efficiency/probability of the $i$ -th object.",
"Correlations can be taken into account by an appropriate parametrisation of the underlying efficiencies and scale factors.",
"The corresponding scale factor for events requiring more than one final state object is given by $f = \\frac{1 - \\prod \\left( 1 - \\varepsilon _i^{\\text{data}} \\right)}{1 - \\prod \\left( 1 - \\varepsilon _i^{\\text{MC}} \\right)}.$ Consequently, this scale factor depends on the individual object properties, even if the respective efficiencies $\\varepsilon _i^{\\text{data}}$ and $\\varepsilon _i^{\\text{MC}}$ do not, since Equation REF does not factorise in these quantities.",
"Due to limited statistics, taking into account all possible combinations of parameters for objects $i,j,$ necessary for an accurate representation of scale factors, is not feasible.",
"In particular, the treatment of correlations in the statistical uncertainties of the scale factors becomes increasingly difficult both analytically and computationally.",
"Furthermore, depending on the magnitude of the underlying efficiencies, a significant amount of MC events is rejected when requiring the trigger simulation to have selected events, leading to a reduction of available MC statistics and practical loss of CPU time spent on the generation of the rejected events.",
"This can be of great importance, in particular if analyses are limited by MC statistics.",
"However, since efficiencies of some of the relevant objects, such as electrons, usually are relatively high ($>95\\,\\%$ ) in the used signal regions, the corresponding loss of MC statistics can be negligible for those objects, depending on the amount of available generated events."
],
[
"Reweighting Approach",
"As an alternative to the determination and application of scale factors, a trigger reweighting approach, making exclusive use of data driven measurements of trigger efficiencies, can be applied to MC events.",
"If the data driven efficiency $\\varepsilon _{\\text{data}}$ for a specific trigger item or trigger chain is known, the corresponding event weights can be determined regardless of the amount of required final state objects.",
"The event weight $w$ is calculated directly from the respective parametrisation of the object efficiencies: $w = 1 - \\prod _{i=1}^N \\left( 1 - \\varepsilon _i^{\\text{data}} \\right),$ which factorises in the trigger (in)efficiencies.",
"Note that only objects following identical selection criteria as used in the efficiency determination should be taken into account in the event weight calculation to ensure that the respective parametrisation is valid.",
"Furthermore, the choice of a proper parametrisation of the efficiencies is crucial due to the fact that neglecting potential dependencies of the efficiencies can lead to significant mis-modelling of the trigger response on MC.",
"This effect is typically smaller for the scale factor approach, where the weights are typically close to one even if the overall efficiencies significantly differ from one (and hence, the impact of the parametrisation is smaller).",
"However, the trigger reweighting approach offers several additional advantages over a scale factor approach, most prominently due to the fact that no trigger simulation is involved in either the determination or application of the trigger efficiencies.",
"Since the simulated trigger decision is not taken into account, all MC events are preserved regardless of the magnitude of the underlying trigger efficiencies, retaining the full MC statistics.",
"In addition, efficiencies obtained from data can be applied to MC independent of potential changes or errors in the implementation of the trigger simulation, which would make repeated extraction of scale factors or alternative solutions necessary.",
"Since the trigger requirements for a given data period typically do not change, the obtained efficiencies have to be determined only once for a set of reconstruction algorithm parameters and remain unchanged as well.",
"Possible correlations in the statistical uncertainties derived for the object efficiencies can be modelled properly in a relatively simple way due to the fact that all objects obtain efficiencies from the same parametrisation.",
"Hence, a propagation of the statistical uncertainties to the final event weights, event yields or even distributions of object and event quantities is possible.",
"However, since the uncertainties on trigger efficiencies are usually relatively small compared to other sources of uncertainties in physics analyses, the impact is expected to be small when neglecting the respective correlations.",
"Typically, uncertainties on both scale factors and efficiencies are regarded as fully correlated in a conservative approximation.",
"In the following, a summary of objects used in this analysis and their definition will be given.",
"The algorithms used to reconstruct the individual objects and the corresponding performance will be discussed."
],
[
"Jets",
"Jets represent collimated collections of long-lived hadrons, originating mostly from partons after the fragmentation and hadronisation process[164].",
"Jets were reconstructed by associating objects from the EM calorimeter and hCAL, and tracks reconstructed in the detector into physical jet objects representing the underlying fragmentation and parton shower processes.",
"Jet reconstruction algorithms are preferably collinear and infra-red safe, i.e.",
"collinear splitting and soft gluon emissions should not change the algorithm response of the final reconstructed jet.",
"In this context, the jet reconstruction was performed using the anti-$k_t$ ($R=0.4$ ) jet clustering algorithm[165], [166].",
"Tracks and energy depositions (topological clusters[145]) in the calorimeter identified at the electromagnetic energy scale[145] (EM scale), calibrated to yield a correct energy response for electrons and photons, were associated into combined jets of particles.",
"This was achieved by iterative combination of pairs of objects into proto-jets according to the respective geometric distance of the individual constituents and the proto-jets until a convergent state was reached.",
"Jet quality criteria were applied to identify jets which do not correspond to physical in-time energy deposits in the calorimeter.",
"Possible sources for such bad jets are for example hardware problems, calorimeter showers induced by cosmic rays and beam remnants.",
"The jet reconstruction efficiencies were determined from data with a Tag & Probe method, using jets from charged tracks in the ID, where the efficiency was defined as the fraction of probe track jets matching a corresponding calorimeter jet.",
"The jet energy resolution $\\sigma (E_{\\rm {T}}) / E_{\\rm {T}}$ for jets in the region $|\\eta | < 2.8$ ranged from about 0.2 (for 20 GeV jets) to 0.05 (for 900 GeV jetsUpdated plots at https://twiki.cern.ch/twiki/bin/view/AtlasPublic/JetEtmissApproved2011JetResolution)[167].",
"Differences between measured efficiencies and efficiencies obtained in Monte Carlo simulations were taken into account by randomly removing jets from events according to a Gaussian probability distribution corresponding to the jet reconstruction efficiency uncertainty.",
"In addition, a calibration of the jet energy scale[145] (JES) was performed to obtain the energy of the final state particle-level jet from the measured energy of the reconstructed jet.",
"The differences in response from the EM calorimeter and hCAL and additional effects such as energy mis-measurements of particles not coming to a stop in the calorimeter were taken into account.",
"The following requirements were applied to reconstructed jets (calibrated at the electromagnetic + JES (EM+JES) scale unless stated otherwise) after electron/muon overlap removal: Jet transverse momentum: $p_{\\rm {T}}(\\text{jet}) > 25\\,\\text{GeV}$ , Jet pseudorapidity at the EM scale, including JES $\\eta $ correction: $|\\eta _{\\text{EMscale}}(\\text{jet})| < 2.5$ .",
"Due to the dominant decay of top quarks into a $W$ boson and a $b$ quark, events containing top quarks are characterised by hard $b$ jets, which have the distinction of possessing a long lifetime, a large corresponding $B$ hadron mass and a large branching ratio into leptons.",
"Hence, the identification of $b$ jets and the discrimination against light quark jets can significantly increase the signal to background ratio for top quark decays.",
"The identification (or tagging) of $b$ jets is mostly driven by exploiting the increased lifetime with respect to light jets and the associated significant flight path length $l$ , leading to secondary vertices and measurable transversal and longitudinal impact parameters $d_0$ and $z_0$ of the final state particles.",
"These denote the spatial distance of closest approach with respect to the associated primary vertex in the transversal respectively longitudinal plane.",
"For this analysis, the JetFitterCombNN tagging algorithm[168] was used, which determines a $b$ tag probability/weight $w$ for a given jet according to a Neural Network combination of the weights from the IP3D and JetFitter tagging algorithms[168].",
"The IP3D algorithm used the significance of $d_0$ and $z_0$ of each track contained in the respective jet to determine a likelihood corresponding to the $b$ jet probability.",
"The JetFitter algorithm implemented a Kalman Filter[169] to identify a common line of primary vertex and weak $b$ and $c$ hadron decay vertices within the jets.",
"Using the approximated hadron flight path, a $b$ jet likelihood was obtained based on the masses, momenta, flight length and track multiplicities of the reconstructed vertices.",
"A cut on the $b$ tag weight of $w > 0.35$ was applied to classify $b$ jets, corresponding to an approximate $b$ tagging efficiency of $70\\,\\%$ in simulated $t\\bar{t}$ decays and a rejection rate of about 5 for $c$ jets and about 100 for light flavour jets.",
"The performance of the $b$ tagging algorithm was determined on specific data samples and compared to the MC prediction.",
"A set of scale factors, parametrised in jet $p_{\\rm {T}}$ was determined and applied to MC events for both the $b$ tagging efficiency and the light jet mis-tag rate to take into account differences between data and MC.",
"The respective event weight was determined as logical OR from the scale factors of all jets taken into account."
],
[
"Muons",
"Muons were identified and selected both at online (trigger) and offline (reconstruction) level in accordance with the respective object quality criteria.",
"Muon reconstruction was performed with the MuId[170] algorithm, taking into account the tracks of the ID and the MS and creating a combined reconstructed track.",
"Particle hits identified in the MS chambers were associated to the corresponding track segments in the individual layers of the ID.",
"A combined fit was performed to obtain an optimised reconstructed trajectory from the joint information of both systems and to optimise geometrical and transverse momentum resolution.",
"The MS momentum resolution in $\\sigma (p_{\\rm {T}}) / p_{\\rm {T}}$ ranges from 0.04 (for 20 GeV muons) to 0.06 (for 200 GeV muons) in the central detector region and from 0.05 (for 20 GeV muons) to 0.20 (for 200 GeV muons) in the forward detector region.",
"The ID momentum resolution in $\\sigma (p_{\\rm {T}}^{-1})$ ranges from 0.001 (for 20 GeV muons) to 0.0005 (for 200 GeV muons) in the central detector region and from 0.004 (for 20 GeV muons) to 0.005 (for 200 GeV muons) in the forward detector region[171].",
"In addition, differences in the muon momentum resolution between data and Monte Carlo were corrected for by randomly changing the muon $p_{\\rm {T}}$ on Monte Carlo according to a Gaussian resolution function to reflect the resolution observed in data.",
"Muon trigger and reconstruction efficiencies were measured with data driven methods on ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ events using the Tag & Probe method and were compared to Monte Carlo simulations.",
"Both the respective efficiencies and scale factors were parametrised in muon transverse momentum, pseudorapidity and azimuthal angle to reflect differences in the detector acceptance and the $p_{\\rm {T}}$ dependency for both the trigger and reconstruction efficiencies.",
"Muons were selected at trigger level by requiring the EF_mu18 trigger chain (corresponding to a muon with a $p_{\\rm {T}} > 18\\,$ GeV at trigger level), which was seeded at LVL1 by the L1_MU10 trigger item and at LVL2 by the L2_mu18 trigger item, to have fired.",
"This requirement was applied to data only, while the trigger efficiency was taken into account for Monte Carlo events by determining an event weight based directly on the trigger efficiency associated with the reconstructed and selected muons.",
"Hence, no explicit (simulated) trigger decision was required.",
"Furthermore, no geometric matching of trigger objects and reconstructed muons has been applied due to a software error resulting in incorrect trigger object representation within the used data and Monte Carlo samples.",
"The resulting mis-modelling of the trigger response has been estimated and was taken into account as an additional systematic uncertainty (c.f.",
"Chapter ).",
"Reconstructed muons from the MuId algorithm were required to be combined muons, i.e.",
"to have a combined ID and MS track passing the respective track criteria.",
"In addition, cuts specific to select semileptonic top quark decays were applied in order to reject muons from heavy and light flavour decays such as $b$ and $c$ hadrons or in-flight decays of kaons and pions.",
"In particular, isolation requirements were applied based on the momentum and energy deposition within a $\\Delta R$ cone of size 0.3 around the muon track, $E_{\\text{T,cone30}}(\\mu )$ and $p_{\\text{T,cone30}}(\\mu )$ , respectively.",
"The following requirements have been used: Muon transverse momentum: $p_{\\rm {T}}(\\mu ) > 20\\,\\text{GeV}$ , Muon pseudorapidity: $|\\eta (\\mu )| < 2.5$ , Jet overlap removal by requiring $\\Delta R (\\mu ,\\text{closest reconstructed jet}) > 0.4$ , where jets reconstructed by the anti-$k_t$ algorithm with $R=0.4$ were taken into account, calibrated to the EM+JES scale and with $p_{\\rm {T}}(\\text{jet}) > 20\\,\\text{GeV}$ , Isolation: $E_{\\text{T,cone30}}(\\mu ) < 4\\,\\text{GeV}$ and $p_{\\text{T,cone30}}(\\mu ) < 4\\,\\text{GeV}$ ."
],
[
"Electrons",
"Similarly to muons, electrons were identified and selected according to the respective electron quality criteria.",
"Electrons were reconstructed by matching energy cluster hits (or seeds) above a threshold of about $3\\,\\text{GeV}$ in the electromagnetic calorimeter to corresponding extrapolated ID tracks, vetoing tracks from photon conversion pairs.",
"In addition, a matching of the track momentum and cluster energy was performed by application of a cut on $E/p$ .",
"Information from the TRT chambers was used to enhance the separation of electron candidates from pions in the reconstruction process.",
"The electron energy resolution was about $(1.2 \\pm 0.1\\,\\text{(stat.)}",
"\\pm 0.3\\,\\text{(syst.",
")})\\,\\%$ in the central detector region, about 1.8 % in the endcaps, and of the order of 3 % in the forward detector region[172].",
"The electron energy scale and resolution were obtained with data driven methods in kinematic regions similar to top quark pair production events, using events from ${Z/\\gamma ^* \\rightarrow e e}$ decays.",
"On data, the energy scale was corrected as a function of electron transverse energy $E_{\\rm {T}}(e)$ and cluster pseudorapidity $|\\eta _{\\text{cluster}}|$ .",
"On MC, the electron energy was corrected by applying a randomised Gaussian resolution function to the electron energy, following a similar parametrisation.",
"Statistical and systematic uncertainties on both scale and energy corrections were taken into account and were assigned to the MC.",
"Electron trigger and reconstruction efficiencies were measured with data driven methods on ${Z/\\gamma ^* \\rightarrow e e}$ and ${W^{\\pm } \\rightarrow e \\nu _e}$ events using the Tag & Probe method and compared to Monte Carlo simulations.",
"The respective electron scale factors were parametrised in electron pseudorapidity $|\\eta _{\\text{cluster}}|$ (using cluster quantities) to reflect differences in the detector acceptance for both the trigger and reconstruction efficiencies.",
"Electrons were selected at trigger level by requiring the EF_e20_medium trigger chain (corresponding to an electron with a $E_{\\rm {T}} > 20\\,$ GeV at trigger level, and medium denoting a specific set of object criteria at trigger level, respectively), which was seeded at LVL1 by the L1_EM14 trigger item and at LVL2 by the L2_e20_medium trigger item, to have fired.",
"This requirement was applied to both data and MC events.",
"In addition, a geometric matching of reconstructed electrons to the corresponding trigger objects in a given event was performed to ensure that the reconstructed electron had fired the trigger.",
"Data/MC scale factors were applied as event weights to take into account differences in the efficiencies between data and trigger simulation.",
"In this analysis, reconstructed electrons were required to pass the tight[173] electron quality requirements.",
"In addition, the following requirements were applied to electron candidates: Electron transverse energy: $E_{\\rm {T}}(e) > 25\\,\\text{GeV}$ , Electron cluster pseudorapidity: $|\\eta _{\\text{cluster}}(e)| < 2.47$ , Exclusion of the non-covered transition region $1.37 < |\\eta _{\\text{cluster}}(e)| < 1.52$ , Jet overlap removal by requiring $\\Delta R (e,\\text{closest reconstructed jet}) > 0.2$ , where all jets reconstructed by the anti-$k_t$ algorithm with $R=0.4$ were taken into account, Isolation: $E_{\\text{T,cone20}}(e) < 3.5\\,\\text{GeV}$ after leakage and pile-up correction[173], where $E_{\\text{T,cone20}}(e)$ denotes the energy deposition within a $\\Delta R$ cone of size 0.2 around the electron cluster.",
"The electron transverse energy $E_{\\rm {T}}(e)$ was determined from the energy deposited in the cluster in the calorimeter $E_{\\text{cluster}}(e)$ and the associated track pseudorapidity $\\eta _{\\text{track}}(e)$ , $E_{\\rm {T}}(e) = \\frac{E_{\\text{cluster}}(e)}{\\cosh {\\left( \\eta _{\\text{track}}(e) \\right)}}.$"
],
[
"Missing Transverse Energy",
"The missing transverse energy $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ is an object-based quantity derived from the topological clusters in the calorimeter calibrated at the EM scale, corrected for the energy scale of the corresponding object associated to the cluster.",
"Objects taken into account were electrons and jets, where jets were separated into high- and low-$p_{\\rm {T}}$ (soft) jets.",
"The electron contribution used electrons passing the tight electron quality criteria with $p_{\\rm {T}} > 10\\,\\text{GeV}$ , while jets with $p_{\\rm {T}} > 20\\,\\text{GeV}$ were corrected to the EM+JES scale and soft jets with $7\\,\\text{GeV} < p_{\\rm {T}} < 20\\,\\text{GeV}$ were included at the EM scale.",
"In addition, muons were included in the definition using the transverse momentum of the corresponding tracks due to the fact that muons typically only deposit small amounts of energy in the calorimeter.",
"Muons reconstructed with the MuId reconstruction algorithm with $|\\eta | < 2.5$ were taken into account, distinguishing between isolated (requiring $\\Delta R (\\mu ,\\text{closest jet}) > 0.3$, where all jets reconstructed by the anti-$k_t$ algorithm with $R=0.4$ were taken into account) and non-isolated (where the energy deposited in the calorimeter was taken into account in the jet term).",
"Furthermore, muons in detector regions with low acceptance ($|\\eta | < 0.1$ and $1.0 < |\\eta | < 1.3$ ) were taken into account making use of the corresponding calorimeter response.",
"Clusters not associated to any object were included in an additional (CellOut) term, calibrated at the EM scale, as well as the energy deposition of isolated muons in the calorimeter.",
"The missing transverse energy terms in the $x$ and $y$ directions were calibrated and combined into an overall missing energy in the respective dimension: $\\lnot \\!\\!E_i = E_i^{\\text{electrons}} + E_i^{\\text{jets}} + E_i^{\\text{softjets}} + E_i^{\\text{muons}} + E_i^{\\text{CellOut}},$ where $i = \\lbrace x,y \\rbrace $ .",
"Consequently, the respective scalar transverse missing energy $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ is given by $\\lnot \\!\\!E_{\\rm {T}} = \\sqrt{\\lnot \\!\\!E_x^2 + \\lnot \\!\\!E_y^2}.$ The resolution of the $\\lnot \\hspace{-2.84544pt} E_x$ and $\\lnot \\hspace{-2.84544pt} E_y$ components ranged from about 2 GeV (for a total transverse energy of 20 GeV) to 14 GeV (for a total transverse energy of 700 GeV)[174]."
],
[
"Event Selection",
"The event selection used for the top charge asymmetry measurement was aimed at maximising the signal contribution from the $t\\bar{t}$ decay in the semileptonic decay channel.",
"At the same time, the background contribution from different sources was minimised.",
"The two distinct observable final states, muon+jets and electron+jets, were treated independently.",
"As discussed in Chapter , top quark pairs are dominantly produced at the LHC by gluon-gluon fusion.",
"Both the top and antitop quark then decay into a $W$ boson and a $b$ quark in almost 100 % of all cases.",
"The $W$ boson decays into two jets or a charged lepton and a neutrino, which can only be measured indirectly as missing transverse energy.",
"Hence, the event selection for the semileptonic decay channel was focused on topologies with at least four reconstructed jets, exactly one isolated lepton (muon or electron) and missing transverse energy.",
"A preselection of the delivered raw data based on a common GoodRunsList (c.f.",
"Chapter ), ensuring stable beam conditions and data quality, has been applied prior to the event selection.",
"This selection included global data quality flags, e.g.",
"requiring stable beams at a centre-of-mass energy of 7 TeV and the LVL1 central trigger and luminosity measurement to be functional, indicating stable running and data taking conditions of the LHC and ATLAS, respectively.",
"Furthermore, the data quality of the individual detector subsystems has been verified.",
"These criteria correspond to a data quality selection efficiency of 84.1 % in the electron+jets channel and 84.3 % in the muon+jets channel.",
"In addition to the baseline selection, which will be described in the following, several corrections have been applied to Monte Carlo samples on an event-by-event basis to account for potential mis-matches in the detector simulation with respect to data, as described in Chapter .",
"Muon trigger efficiencies have been taken into account directly by performing a trigger reweighting using trigger efficiencies obtained with a Tag & Probe method, while discrepancies between the simulation and data for electron and muon reconstruction efficiencies and the electron trigger efficiencies as well as the $b$ tagging efficiencies have been taken into account by applying the corresponding scale factors to the Monte Carlo events.",
"Furthermore, each Monte Carlo event has been assigned a weight according to the average amount of $pp$ interactions per bunch crossing in the respective sample to account for differences in the modelling of pile-up.",
"The following event selection has been used to enhance the signal to background ratio in the recorded samples.",
"The described selections have been applied to both data and Monte Carlo samples unless stated otherwise: The electron or muon trigger was required to have fired.",
"The trigger item used was EF_e20_medium in the electron+jets channel for both Monte Carlo and data.",
"In the muon+jets channel, the EF_mu18 trigger item was required for data, while a trigger reweighting using EF_mu18 trigger efficiencies obtained with data driven methods was applied to Monte Carlo events, made necessary by a mis-modelling of the trigger simulation in the used Monte Carlo samples.",
"A primary vertex with at least five tracks associated to it was required to improve rejection of non-collision background from the underlying event, pile-up and cosmic radiation.",
"Exactly one isolated lepton (one electron and no muon or vice versa) passing the respective object quality criteria with $E_{\\rm {T}}>25\\,\\rm {GeV}$ (electron+jets) or $p_{\\rm {T}}>20\\,\\rm {GeV}$ (muon+jets), respectively, was required.",
"The selected lepton was required to match the object of the fired trigger (electron+jets only, since due to a software problem, the muon trigger matching requirement was dropped from the selection).",
"Any event where a reconstructed electron and muon share a common track was rejected.",
"Any event containing a bad jet (c.f.",
"Chapter REF) with $p_{\\rm {T}}>20\\,\\rm {GeV}$ at the EM+JES scale was rejected.",
"A missing transverse energy $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}} > 35\\,\\rm {GeV}$ (electron+jets) or $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}} > 20\\,\\rm {GeV}$ (muon+jets) was required.",
"A cut of $m_{\\rm {T}}(W)\\footnote {In this context, \\mbox{$m_{\\rm {T}}(W)$}~denotes the W boson transverse mass, defined as\\begin{equation}\\mbox{$m_{\\rm {T}}(W)$}~= \\sqrt{2 p_{\\rm {T}}^l p_{\\rm {T}}^{\\nu } \\left( 1 - \\cos {\\left(\\phi ^l - \\phi ^{\\nu }\\right)} \\right)},\\end{equation}where p_{\\rm {T}}^{l/\\nu } and \\phi ^{l/\\nu } describe the lepton and neutrino transverse momentum and azimuthal angle, respectively.",
"The neutrino information is represented by the measured missing transverse energy, \\mbox{$\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$}.}",
"> 25\\,\\rm {GeV}$ (electron+jets) or a triangular cut of $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}} + m_{\\rm {T}}(W) > 60\\,\\rm {GeV}$ (muon+jets) was applied in order to suppress the QCD multijet background contribution, since these events typically have low $m_{\\rm {T}}(W)$ and low $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ .",
"At least four jets with $p_{\\rm {T}}>25\\,\\rm {GeV}$ passing the jet quality criteria were required.",
"Any event where a jet was found in an area of LAr calorimeter defects was rejected and electrons which were affected were removed from the respective events, correcting the measured $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ accordingly.",
"For Monte Carlo, a randomised subset of events was dropped according to the relative fraction of the data sample affected by these defects in order to correct for the created mis-match between data and simulation.",
"At least one jet which has been $b$ tagged using the JetFitterCombinedNN algorithm with a weight $w > 0.35$ (corresponding to an overall $b$ tagging efficiency of about 70 % in simulated $t\\bar{t}$ events) was required to further improve the signal to background ratio."
],
[
"Samples and Process Modelling",
"This chapter describes the data sample used in this analysis and the Monte Carlo samples generated to simulate the signal contribution and most of the backgrounds.",
"Furthermore, the data driven estimations of the dominant background contributions, $W$ +jets and QCD multijet production, are described."
],
[
"Data Sample",
"A set of ATLAS data taken in the course of the year 2011, corresponding to an integrated luminosity of $\\int \\mathcal {L} \\, dt = (1.04 \\pm 0.04)\\,\\text{fb}^{-1}$ has been analysed, after the preselection of the delivered raw data using the corresponding GoodRunsList.",
"The data has been recorded between March, 22rd, 2011 and June, 28th, 2011.",
"A peak instantaneous luminosity of about $\\mathcal {L} = 1.3 \\cdot 10^{33}\\,\\rm {cm}^{-2}\\rm {s}^{-1}$ was reached and a bunch spacing of 50 ns was used."
],
[
"Signal and Background Monte Carlo Samples",
"Several Monte Carlo samples have been generated to facilitate this analysis, including nominal samples for the signal contribution and various background processes as well as several additional samples used in the evaluation of systematic uncertainties.",
"All samples correspond to the mc10b production commonly performed for all ATLAS analyses using a generalised set of parameters to match the data taking conditions during the time period that was considered.",
"In particular, the contribution from in-time and out-of-time pile-up was added to all generated Monte Carlo events in the parton showering simulation process after the generation of the initial hard scattering.",
"A fixed configuration of average proton-proton interactions per bunch crossing was used, while the actual data taking conditions with respect to pile-up changed over the course of time.",
"In order to correct for the mis-match between individual data taking periods and the simulated pile-up contribution in the MC, an event-based reweighting was performed, taking into account the expected and observed distribution of the average number of interactions per bunch crossing.",
"This pile-up reweighting was performed for all MC samples.",
"The $t\\bar{t}$ signal process has been simulated using the mc@nlo generator [175] (v3.41) which incorporates the CTEQ6.6 [176] parton distribution function set and makes use of a next-to-leading order calculation approach for QCD processes.",
"Both the parton showering and fragmentation processes, and the underlying event have been modelled using the herwig v6.510 [177] and jimmy [178] generators utilising the CTEQ6.6 and AUET1[179] tunes to match the ATLAS data, respectively.",
"The inclusive $t\\bar{t}$ cross section has been estimated to approximately next-to-next-to-leading order using the Hathor tool [180] to be $165_{-16}^{+11}\\,\\text{pb}$[36], [37], [38] for $m_t = 172.5\\,$ GeV and the MC has been scaled accordingly.",
"For this analysis, only semileptonic decays of the top quark pairs were considered.",
"The respective cross-section, taking into account the proper branching fractions, was 89.3 pb, including a $k$ -factor of 1.117 to rescale the next-to-leading order perturbative QCD cross section in mc@nlo to the approximate next-to-next-to-leading order cross-section.",
"The signal sample contained 15000000 simulated events, corresponding to an integrated luminosity of about $150\\,\\text{fb}^{-1}$ .",
"The electroweak single top production was simulated using the mc@nlo and jimmy generators and the respective cross-sections have been calculated at approximately next-to-next-to-leading order to be $64.57_{-2.01}^{+2.71}$ pb in the t-channel, $4.63_{-0.17}^{+0.19}$ pb in the s-channel, and $15.74_{-1.08}^{+1.06}$ pb for $Wt$ production, as introduced in Chapter REF.",
"The background contribution from the production of heavy gauge bosons with additional jets was modelled using the leading order alpgen generator [181], interfaced to herwig and jimmy for the purposes of parton shower and hadronisation simulation.",
"The CTEQ6.1[182] parton distribution functions and the AUET1 tune were employed for proper ATLAS data matching for both the matrix element evaluations and the parton showering.",
"The production of additional partons was taken into account by generating different subsamples with different final state parton multiplicities, where additional partons can be either light ($u$ ,$d$ ,$s$ ) partons (simulated in $W$ +jets and $Z/\\gamma $ +jets light flavour samples) or heavy quarks (simulated in $W$ +$c$ +jets, $W$ +$c\\bar{c}$ +jets, $W$ +$b\\bar{b}$ +jets, and $Z$ +$b\\bar{b}$ +jets samples, respectively).",
"Since the inclusive $W$ +jets and $Z/\\gamma $ +jets samples included contributions from both light partons and heavy quarks in the matrix element and parton shower simulation, the created overlap in phase space between the inclusive samples and the heavy quark contribution was taken into account by removing double counted events from the respective samples.",
"The production cross-sections of the used alpgen samples were normalised to the corresponding approximate next-to-next-to-leading order cross-sections using $k$ -factors of 1.20 ($W$ +jets) and 1.25 ($Z$ +jets), respectively.",
"Furthermore, the relative fractions of the individual $W$ +jets heavy quark contributions to the overall $W$ +jets sample have been determined in data driven methods[183], [184], and were accounted for by applying corresponding scale factors to the individual samples.",
"They have been found to be $1.63 \\pm 0.76$ for $W$ +$c\\bar{c}$ +jets and $W$ +$b\\bar{b}$ +jets, and $1.11 \\pm 0.35$ for the $W$ +$c$ +jets contribution, respectively.",
"The $W$ +jets light quark contributions were scaled accordingly to conserve the overall predicted cross-section.",
"Contributions from diboson ($WW$ , $WZ$ and $ZZ$ ) production and decays was simulated using herwig at leading order, and the corresponding production cross-sections were normalised to the next-to-next-to leading order predictions, using $k$ -factors of 1.48 ($WW$ ), 1.60 ($WZ$ ) and 1.30 ($ZZ$ ), respectively.",
"Each sample was inclusive and has been filtered to include only events containing at least one lepton (electron or muon) with $p_{\\rm {T}} > 10\\,\\text{GeV}$ and $|\\eta | < 2.8$ at parton level.",
"The $k$ -factors have been determined such that the unfiltered herwig cross sections agree with the next-to-next-to-leading order calculations.",
"For the evaluation of the systematic uncertainties of the various generators and the simulation of hadronisation and fragmentation, alternative samples have been used for the signal contribution and the $Z$ +jets background contribution.",
"The $t\\bar{t}$ production and decay has been simulated with the powheg generator [185], and the corresponding parton showering and fragmentation processes have been modelled using both herwig and jimmy (as used for the nominal mc@nlo sample), and pythia in order to evaluate systematic effects from the parton showering.",
"Furthermore, additional mc@nlo signal samples using different top mass hypotheses of 170 GeV and 180 GeV have been used in order to quantify systematics arising from the uncertainties on the top mass prediction.",
"Finally, several samples using different strengths of initial state radiation (ISR) and final state radiation (FSR) have been generated at leading order using the acermc generator [186], corresponding to different contributions of ISR and FSR based on observations in data[187].",
"An alternative inclusive $Z$ +jets background modelling was performed using the sherpa [188] generator and the CTEQ6.6 parton distribution functions.",
"Details on the evaluation of systematic uncertainties can be found in Chapter .",
"All background samples and the samples used for the evaluation of systematic uncertainties corresponded to an integrated luminosity of about $10-30\\,\\text{fb}^{-1}$ before analysis specific selection."
],
[
"Data Driven Estimation of the QCD Multijet Contribution",
"The identification of top quark pairs decaying semileptonically relies on the identification of one lepton in the final state, carrying a large transverse momentum.",
"Hence, mis-identified leptons (fake leptons), which can originate from various sources, pose a non-negligible background to the identification of $t\\bar{t}$ signal events.",
"Potential sources for mis-identified leptons include semileptonic decays of $b$ quarks into $b$ jets containing leptons with mis-identified isolation properties, long lived weakly decaying particles such as $\\pi ^{\\pm }$ or $K$ mesons, $\\pi ^{0}$ mesons which are mis-identified as electrons, direct photon conversion and reconstruction of electrons produced in the process.",
"These processes are most dominant in regions where the contribution from real leptons are small, most prominently for the background contribution from QCD multijets.",
"Despite the fact that object and event selection are designed to ensure the suppression of these backgrounds by requiring stringent criteria, the QCD multijet production cross-section is orders of magnitudes higher than the top quark pair production cross-section.",
"Since the simulation of these backgrounds in Monte Carlo simulations is highly difficult and several of the described contributions are detector dependent, data driven methods are necessary to obtain reliable estimates for the fake lepton background contribution from QCD multijet events.",
"In order to estimate the contribution from QCD multijet fake muons and electrons, a data driven method, the Matrix Method, was applied to data using different control regions for electrons and muons dominated by QCD multijet processes.",
"The Matrix Method allows to statistically separate two contributions of a data sample based on the impact of a defined selection on the respective subsamples.",
"The Matrix Method defines two subsets, $N_{\\rm {loose}}$ and $N_{\\rm {tight}}$, of the data sample before and after application of a particular requirement applied in the event selection, typically with a large discrimination power between signal (in this case real leptons) and background (in this case fake leptons) contributions.",
"The number of events in the loose sample, $N_{\\rm {loose}}$, is given by the sum of the signal and the background contributions $N^{\\rm {sig}}$ and $N^{\\rm {fake}}$ in the given sample: $N_{\\rm {loose}} = N^{\\rm {sig}} + N^{\\rm {fake}}.$ After the application of the defined requirement, which is passed by signal events with a probability (or efficiency) of $\\varepsilon ^{\\rm {sig}}$ and by background events with a probability of $\\varepsilon ^{\\rm {fake}}$, the number of events $N_{\\rm {tight}}$ in the tight sample follows by taking into account the respective probabilities for the imposed requirement for the signal and background contributions: $N_{\\rm {tight}} = \\varepsilon ^{\\rm {sig}} N^{\\rm {sig}} + \\varepsilon ^{\\rm {fake}} N^{\\rm {fake}}.$ This situation is illustrated in Figure REF.",
"Figure: An illustration ofthe Matrix Method and the effect of the applied selection on the underlying sample subsets.The linear system of two equations with two unknown variables can be rewritten as a matrix equation: $\\left( \\begin{array}{c}N_{\\rm {loose}} \\\\N_{\\rm {tight}}\\end{array} \\right)=\\left( \\begin{array}{cc}1 & 1 \\\\\\varepsilon ^{\\rm {sig}} & \\varepsilon ^{\\rm {fake}}\\end{array} \\right)\\left( \\begin{array}{c}N^{\\rm {sig}} \\\\N^{\\rm {fake}}\\end{array} \\right)$ Solving the matrix equations yields the signal and background contributions in the data sample prior to the isolation cut: $N^{\\rm {fake}} & = & \\frac{N_{\\rm {tight}} - \\varepsilon ^{\\rm {sig}} N_{\\rm {loose}}}{\\varepsilon ^{\\rm {fake}} -\\varepsilon ^{\\rm {sig}}}, \\\\N^{\\rm {sig}} & = & \\frac{\\varepsilon ^{\\rm {fake}} N_{\\rm {loose}} - N_{\\rm {tight}}}{\\varepsilon ^{\\rm {fake}} -\\varepsilon ^{\\rm {sig}}}.$ In order to determine the background contribution in the signal region (which corresponds to the tight selection by construction), the fraction of $N^{\\rm {fake}}$ in the tight sample can be calculated using Equation REF as $N_{\\rm {tight}}^{\\rm {fake}} & = & \\varepsilon ^{\\rm {fake}} N^{\\rm {fake}} \\\\& = & \\frac{\\varepsilon ^{\\rm {fake}}}{\\varepsilon ^{\\rm {sig}} - \\varepsilon ^{\\rm {fake}}} \\left(\\varepsilon ^{\\rm {sig}} N_{\\rm {loose}} - N_{\\rm {tight}} \\right).$ If the selection probabilities $\\varepsilon ^{\\rm {sig}}$ and $\\varepsilon ^{\\rm {fake}}$ for signal and background, respectively, are sufficiently different, the overall contribution of the QCD multijet background can be used to determine event based weights for the used data sample in order to obtain the distributions of the QCD multijet background contribution in arbitrary variables.",
"This is done by assigning a weight to each data event based on the chosen requirement and the corresponding signal and fake probabilities of the objects taken into account for a given event.",
"If it passes the loose selection only, the event weight is given by setting $N_{\\rm {loose}} = 1$ , $N_{\\rm {tight}} = 0$ in Equation REF, yielding $w_{\\text{loose}} = \\frac{\\varepsilon ^{\\rm {sig}} \\varepsilon ^{\\rm {fake}}}{\\varepsilon ^{\\rm {sig}} - \\varepsilon ^{\\rm {fake}}}.$ Similarly, if both the loose and tight requirements are fulfilled, the event weight is given by setting $N_{\\rm {loose}} = 1$ , $N_{\\rm {tight}} = 1$ in Equation REF: $w_{\\text{tight}} = \\frac{\\left( \\varepsilon ^{\\rm {sig}} -1 \\right) \\varepsilon ^{\\rm {fake}}}{\\varepsilon ^{\\rm {sig}} - \\varepsilon ^{\\rm {fake}}}.$ This approach allows for a purely data driven estimation of both the normalisation and the shape of the QCD multijet background in semileptonic decays of top quark pairs.",
"The individual parameters that have been used in the estimation for both the muon and electron channels in this analysis will be covered in the following sections.",
"In addition, detailed studies of the performance and stability of the methods used will be demonstrated for the muon+jets channel alongside with an approach to obtain well-defined statistical and systematic uncertainties on the estimate."
],
[
"Muon+jets Channel",
"In order to estimate the contribution from QCD multijet fake muons, the matrix method was applied to data in the QCD multijet-enriched low-$m_{\\rm {T}}(W)$ control region.",
"Furthermore, an inverted triangular cut was imposed to achieve orthogonality to the signal region in the determination of the fake probabilities $\\varepsilon ^{\\rm {fake}}$ used in the Matrix Method: $m_{\\rm {T}}(W) < 20\\,\\rm {GeV} \\text{ and } \\lnot \\!\\!E_{\\rm {T}} + m_{\\rm {T}}(W) < 60\\,\\rm {GeV}.$ The impact of the described requirements on the QCD multijet estimate and the simulated $t\\bar{t}$ signal is shown in Figure REF.",
"Figure: Impact of the requirement of m T (W)<20 GeV m_{\\rm {T}}(W) < 20\\,\\rm {GeV} and the triangular cut, ¬E T \\lnot \\hspace{-2.84544pt} E_{\\rm {T}} +m T (W)<60 GeV +\\,m_{\\rm {T}}(W) < 60\\,\\rm {GeV}, on both the QCD multijet estimate and the tt ¯t\\bar{t} signal contribution, taken from Monte Carlo simulations.",
"The imposed cuts are illustrated by the black lines.The signal probabilities $\\varepsilon ^{\\rm {sig}}$ were determined in the signal region using a ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ Tag & Probe method in order to select prompt muons from $Z$ decays.",
"The loose selection was identical to the full selection applied in the signal region (for details on the individual requirements, refer to Chapter and Chapter ), except for the muon isolation.",
"The tight selection in addition requires isolation criteria based on both momentum and energy depositions around the muon tracks, $p_{\\rm {T,cone30}} < 4.0\\,\\rm {GeV} \\text{ and } E_{\\rm {T,cone30}} < 4.0\\,\\rm {GeV},$ corresponding to the full analysis event selection.",
"The fake probabilities have been obtained separately both with and without explicitly requiring at least one $b$ tagged jet ($\\ge 0$ $b$ tags and $\\ge 1$ $b$ tags, respectively).",
"In addition, the fake probabilities have been determined requiring at least two $b$ tagged jets, for completeness.",
"For the latter cases, the JetFitCombNN $b$ tagging algorithm with a working point of $w = 0.35$ (corresponding to an overall $b$ tagging efficiency of 70 %) has been used in accordance with the signal region event selection.",
"Furthermore, the signal muon contribution from $W$ +jets and $Z$ +jets in the control region was obtained from Monte Carlo and subtracted to obtain a purer QCD multijet estimation.",
"This contamination was of the order of 1.7 % ($\\ge 0$ $b$ tags) and 1.8 % ($\\ge 1$ $b$ tags).",
"In order to verify the stability of the method, the signal and fake (both without and with the requirement of at least one $b$ tagged jet) probabilities are shown as a function of the relative run number (full dataset with GoodRunsList applied) in Figure REF.",
"Figure: Integrated signal and fake (without and with the requirement of at least one bb tagged jet) probabilities, shown as a function of relative run number (full dataset with GoodRunsList applied).",
"The observed probabilities were stable over the whole data taking period under consideration.As can be seen, neither signal nor fake probabilities showed any significant trend with respect to run number and therefore, to instantaneous luminosity and different pile-up conditions.",
"In order to take into account dependencies on object kinematics, the signal and fake probabilities have been determined as a function of muon pseudorapidity $\\eta $ to reflect the dependency on the muon detector acceptance.",
"Furthermore, they have been parametrised as a function of the leading jet transverse momentum $p_{\\rm {T}}(j_1)$ in order to take into account the effects of hard jets and hence increased hadronic activity on the muon isolation.",
"The respective projections for muon $\\eta $ and $p_{\\rm {T}}(j_1)$ can be found in Figure REF.",
"Figure: Signal and fake probabilitiesas a function of muon η\\eta and leading jet p T (j 1 )p_{\\rm {T}}(j_1).The event yields and fractions in the signal region, obtained on a dataset corresponding to $1.04\\,\\rm {fb}^{-1}$ for different amounts of reconstructed jets required in the event selection (jet bins) can be found in Table REF.",
"Table: Event yields and fractions in the signal region for a dataset corresponding to 1.04 fb -1 1.04\\,\\rm {fb}^{-1}.",
"Uncertainties correspond to the overall normalisation uncertainty due to statistical uncertainties from the obtained signal and fake probabilities, a second control region, the systematic shift of the m T (W)m_{\\rm {T}}(W) cut and the MC uncertainty on the WW/ZZ+jets normalisation used in the subtraction of real leptons in the control region.",
"The tagged probabilities correspond to the JetFitCombNN bb tagging algorithm with a working point of w=0.35w = 0.35.The following sources of systematic uncertainty have been taken into account in the estimate and were combined to a single uncertainty on the event weight, which can hence be propagated into a bin-by-bin normalisation and shape uncertainty on an arbitrary variable: Statistical uncertainty on the signal and fake probabilities: Takes into account the uncertainties on both the signal and fake probabilities.",
"The resulting uncertainty on the obtained event weights was evaluated using Gaussian error propagation.",
"Note that the assumption of a symmetric probability distribution function was valid for the signal probabilities as well despite the closeness to unity, owing to the high statistics available from ${Z/\\gamma ^* \\rightarrow \\mu \\mu }$ and the resulting small impact on the overall statistical uncertainty.",
"Systematic uncertainty from second control region: A second control region (high $d_0$ significance) was used for the fake probabilities to determine a bin-by-bin systematic discrepancy which was quoted as additional uncertainty on the event weight.",
"Systematic uncertainty due to choice of control region cut: The low transverse $W$ mass control region cut was varied by $5\\,$ GeV up and down to estimate the impact on the obtained probabilities and event weights.",
"Uncertainties on $W$ /$Z$ +jets Monte Carlo normalisation: For the 1 jet inclusive bin, the $W$ /$Z$ +jets Monte Carlo normalisation uncertainties (25 %) were used to quantify the effect on the subtraction of real leptons in the control region.",
"The resulting overall normalisation uncertainties are shown in Table REF for the signal region ($\\ge 4$ jets selection)."
],
[
"Electron+jets Channel",
"The QCD multijet contribution in the electron+jets channel has been estimated in analogy to the muon+jets channel, following slightly different criteria corresponding to the loose and tight selections[183].",
"The signal probabilities used in the Matrix Method were determined in the signal region with a ${Z/\\gamma ^* \\rightarrow e e}$ Tag & Probe method in order to select prompt electrons from the $Z$ decay.",
"The fake probabilities have been determined in the QCD multijet-enriched low-$\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ control region: $5\\,\\rm {GeV}~<~\\lnot \\!\\!E_{\\rm {T}}~<~20\\,\\rm {GeV}.",
"$ Similar to the evaluation in the muon+jets channel, the loose selection was identical to the full selection applied in the signal region, except for the electron isolation criteria, which was modified in the loose selection: $E_{\\rm {T,cone20}} & < & 6.0\\,\\rm {GeV}, $ while the tight selection requires $E_{\\rm {T,cone20}} & < & 3.5\\,\\rm {GeV}.",
"$ In addition, slightly less stringent track quality criteria and the corresponding $\\lnot \\hspace{-2.84544pt} E_{\\rm {T}}$ definition were applied to the reconstructed electron candidates in the loose sample with respect to the tight sample.",
"Furthermore, a subtraction of the real lepton contribution in the control region has been performed using the corresponding Monte Carlo predictions, in analogy to the muon+jets channel treatment."
],
[
"Data Driven Estimation of the $W$ +jets Contribution",
"Due to the fact that the parton density of $u$ quarks in the protons brought to collision in the LHC is on average higher than the parton density of $d$ quarks (which can be observed already at lower momentum transfers, as depicted in Figure REF), a higher rate of $W^{+}$ than of $W^{-}$ is expected at the LHC.",
"Since the production rates and the asymmetry in the production of $W^{+}$ and $W^{-}$ events has been determined to a higher theoretical precision[189], [190] than the overall $W$ +jets rate at the LHC, the observed $W^{+}/W^{-}$ asymmetry in data can be used to obtain an estimate for the rate of the $W$ +jets background contribution in the signal region[183].",
"Assuming that all processes other than the $W$ +jets production are symmetric in the final state lepton charge, the total number of $W$ +jets events, $N_W$ , can be extracted from the amount of observed data events passing the selection criteria described in Chapter (except for the requirement of at least one $b$ tagged jet) with a positively (negatively) charged lepton, given by $D^{+}$ and $D^{-}$ , respectively: $N_W & = & N_{W^{+}} + N_{W^{-}} \\\\& = & \\left( \\frac{r_{\\text{MC}} + 1}{r_{\\text{MC}} - 1} \\right) \\left( D^{+} - D^{-} \\right).$ The fraction $r_{\\text{MC}} = \\frac{\\sigma _{pp \\rightarrow W^{+}}}{\\sigma _{pp \\rightarrow W^{-}}}$ has been evaluated on Monte Carlo, based on the same event selection, and has been determined to be $1.56 \\pm 0.06$ in the electron+jets channel and $1.65 \\pm 0.08$ in the muon+jets channel, respectively.",
"The dominant contributions to the overall uncertainty were due to PDF and jet energy scale uncertainties, and by the uncertainties on the heavy quark contribution fractions (i.e.",
"the relative contributions from $W b\\bar{b}$ +jets, $W c\\bar{c}$ +jets and $W c$ +jets).",
"Furthermore, the obtained overall $W$ +jets rate $N_W$ has been extrapolated to the full event selection by determining the relative fraction of events passing the requirement of at least one $b$ tagged jet after requiring exactly two reconstructed jets, $f_{2,\\ge 1 b \\text{ tag}}$ , on data, and by determining the ratio $k_{2 \\rightarrow \\ge 4}$ of the same fraction for the sample where at least four reconstructed jets were required with respect to $f_{2,\\ge 1 b \\text{ tag}}$ , using the $W$ +jets Monte Carlo prediction[184]: $N_{W,\\ge 1 b \\text{ tag}} = N_W \\cdot f_{2,\\ge 1 b \\text{ tag}} \\cdot k_{2 \\rightarrow \\ge 4}.$ The fraction $f_{2,\\ge 1 b \\text{ tag}}$ has been measured to be $0.063 \\pm 0.005$ in the electron+jets channel and $0.068 \\pm 0.005$ in the muon+jets channel, including statistical and systematic uncertainties.",
"The extrapolation factor $k_{2 \\rightarrow \\ge 4}$ has been determined to be $2.52 \\pm 0.36$ in the electron+jets channel and $2.35 \\pm 0.34$ in the muon+jets channel, respectively.",
"The uncertainties include contributions from the statistical limitation of the used Monte Carlo samples and systematic uncertainties."
],
[
"Kinematic Event Reconstruction",
"As described in Chapter , the signature of a $t\\bar{t}$ event in the semileptonic decay channel at leading order is the observation of four reconstructed jets, one isolated lepton and missing transverse energy.",
"A reconstruction of the full $t\\bar{t}$ final state was performed following a likelihood approach.",
"A probability for the observation of a set of measured quantities under the assumption of a specific model and a set of model parameters was assigned.",
"In this particular case, the model described the $t\\bar{t}$ decay and the input quantities were the measured energies of the four jets, the measured energy of the lepton, and the missing transverse energy.",
"The fit parameters of the likelihood were the parton energies, the lepton transverse momentum and the three neutrino momentum components.",
"The likelihood was used to assign the measured jets to the decay products of the $t\\bar{t}$ system.",
"For this study, all permutations with four out of the five leading jets (if exist) were taken into account for the event reconstruction to increase the probability of identifying the proper combination in the presence of additional jets (e.g.",
"from ISR or pile-up).",
"Moreover, the (non-Gaussian) partonic energy resolution (the resolution of the particle jets with respect to the partons) of the final state objects were taken into account through the use of object specific transfer functions in order to obtain the final likelihood: $L & = & \\mathcal {B}(\\widetilde{E}_{\\rm {p,}1}, \\widetilde{E}_{\\rm {p,}2} | m_W, \\Gamma _W) \\cdot \\mathcal {B}(\\widetilde{E}_{\\rm l}, \\widetilde{E}_{\\nu } | m_W, \\Gamma _W) \\cdot \\nonumber \\\\& & \\mathcal {B}(\\widetilde{E}_{\\rm {p,}1}, \\widetilde{E}_{\\rm {p,}2}, \\widetilde{E}_{\\rm {p,}3} | m_t, \\Gamma _t) \\cdot \\mathcal {B}(\\widetilde{E}_{\\rm l}, \\widetilde{E}_{\\nu }, \\widetilde{E}_{\\rm {p,}4} | m_t, \\Gamma _t) \\cdot \\nonumber \\\\& & \\mathcal {W}( \\hat{E}_{x}^{miss}| \\widetilde{p}_{x, \\nu }) \\cdot \\mathcal {W}(\\hat{E}_{y}^{miss} | \\widetilde{p}_{y, \\nu }) \\cdot \\mathcal {W}(\\hat{E}_{\\rm {lep}} | \\widetilde{E}_{\\rm {lep}}) \\cdot \\nonumber \\\\& & \\prod _{\\rm {i=1}}^4 \\mathcal {W}(\\hat{E}_{\\rm {jet,}i} | \\widetilde{E}_{\\rm {p,}i}) \\cdot P(\\textrm {b tag} ~| ~\\textrm {quark}),$ where: the $\\mathcal {B}$ s represent the Breit-Wigner parametrisation of the parton (from which the associated jets originated) energies $\\widetilde{E}_{\\rm {p,}i}$ and lepton energies $\\widetilde{E}_{\\rm {lep}}$ with respect to the fitted ones, the $\\mathcal {W}$ s are the transfer functions associating the measured jets/leptons to the partonic objects, where the mapping functions of the objects are parametrised with a double Gaussian, the $m_W$ and $\\Gamma _W$ denote the $W$ boson mass and its decay width.",
"The parameters are fixed to $m_W = 80.4\\,\\rm {GeV}$ and $\\Gamma _W = 2.1\\,\\rm {GeV}$ , respectively, the $m_t$ and $\\Gamma _t$ denote the top quark pole mass and its decay width.",
"The parameters were fixed to $m_t = 172.5\\,\\rm {GeV}$ and $\\Gamma _t = 1.5\\,\\rm {GeV}$ , respectively, the $\\widetilde{X}$ are the partonic object quantities and $\\hat{X}$ their corresponding measured values, $P(\\textrm {$ b$ tag} ~| ~\\textrm {quark})$ is a $b$ tag probability or rejection efficiency, depending on the quark flavor.",
"The probability $P(\\textrm {$ b$ tag} ~| ~\\textrm {quark})$ was used to take into account the tagging efficiency and rejection rate of the used $b$ tagging algorithms at a specific working point.",
"The most probable event topology hypothesis was chosen by iterating over all possible permutations of reconstructed jets, the lepton and the missing energy and by maximising the logarithmic likelihood, $\\log {L}$ .",
"The permutation with the highest event probability was used for all further studies.",
"The reconstruction efficiency, obtained in simulations, is shown for both the muon+jets and electron+jets channel in Figure REF .",
"Figure: Reconstruction efficiencies for the muon+jets (left) andelectron+jets (right) channel.",
"The indicated efficiencies denote the probability of reconstructing the correct (or true) combination of objects (only matched events takeninto account).",
"The bars marked pure statistical indicate the efficiency which is expected by choosing a combination at random.As can be seen, the overall efficiencies for the reconstruction of the correct event topology (All Correct) in both channels were 62 % (74 %) with a fixed mass parameter, without (with) $b$ tagging information taken into account.",
"In order to associate the reconstructed objects with the corresponding truth quarks and leptons, a simple $\\Delta R$ matching was applied, using cone sizes of $0.3$ for jets and $0.1$ for leptons.",
"An event was considered matched if all truth partons originating from the hard scattering process could successfully be identified with reconstructed jets and the truth lepton was matched to a reconstructed one.",
"For the shown performance evaluation, only events where the four reconstructed jets and the lepton were successfully matched to corresponding truth level objects (contributing positively to the reconstruction efficiency) were taken into account.",
"The matching efficiency on simulated $t\\bar{t}$ events was about 30 % in both channels.",
"Examples for the transfer functions used in the likelihood can be found in Figure REF and Figure REF, where the fit functions in different energy regions for $b$ jets in the pseudorapidity range $|\\eta |<0.8$ and for electrons in the pseudorapidity range $0.8<|\\eta |<1.37$ are shown.",
"Figure: The transfer functions mapping the measured bb jets to the corresponding partonic objects, in the range |η|<0.8|\\eta |<0.8, for different energies.Figure: The transfer functions mapping the measured electrons to the corresponding partonic objects, in the range 0.8<|η|<1.370.8<|\\eta |<1.37, for different energies.A double Gaussian function was used in the fit of the transfer functions: $W(E_{\\mathrm {true}}, E_{\\mathrm {reco}}) = \\frac{1}{2 \\pi (p_2 + p_3 p_5)} (e^{- \\frac{(\\Delta E - p_1)^2}{2 p_2^2} } + p_3 e^{- \\frac{(\\Delta E - p_4)^2}{2p_5^2} } ) ,$ where the parameters $p_1$ , $p_2$ , $p_3$ , $p_4$ and $p_5$ are functions of the true energy of the respective particle and $\\Delta E = E_{\\mathrm {true}} - E_{\\mathrm {reco}}$ ."
],
[
"Unfolding",
"Any measured observable is influenced by imperfections of the used measurement apparatus and procedure itself, such as limited resolution of the detector response, the detector acceptance and possible object and event selections which are applied to the data.",
"Due to these distortions, any measurement of such observable does not fully represent the original (or true) quantity.",
"Mathematically, the actual measurement can be considered to be a convolution of the true quantity with a function representing the overall detector and selection acceptance and the detector response.",
"Let the true quantity be represented by a vector $\\smash{\\vec{t}}$ (with entries $t_j$ and $j = 1,2,...,n_t$ ) describing the bin contents of a histogram, and the measured or reconstructed distribution by a corresponding vector $\\smash{\\vec{k}}$ (with entries $k_i$ and $i = 1,2,...,n_k$ ), respectively.",
"The underlying detector resolution effects can be described by a transition or response matrix $R^{\\text{res}}$ , which contains the individual transition probabilities and hence the migrations between the observed elements of the distribution and the corresponding true values.",
"Furthermore, the detector acceptance and applied selection can be quantified by an additional weight factor for each element of $R^{\\text{res}}$ , taking into account the probability of events from a particular entry of $\\smash{\\vec{t}}$ being observed at all in the measurement process.",
"This additional correction, which can be described by a second matrix $R^{\\text{acc}}$ , together with the response matrix describing the resolution effects, yields the overall response matrix $R$ : $R = R^{\\text{acc}} R^{\\text{res}},$ denoting the transition probabilities between the observed distribution and the true distribution: $\\vec{k} = R \\vec{t},$ where $R_{ij} & = & P(\\text{observed in bin }i~|~\\text{expected in bin }j) \\\\& = & P(k_i|t_j).$ The concept of unfolding is illustrated in Figure REF, where an example distribution for an arbitrary variable $x$ is shown at the different stages in the measurement process.",
"Figure: Schematic overview of the measurement and unfolding process.",
"The true distribution of an arbitrary variable xx (left) is affected by acceptance effects (centre) and resolution effects (right) in the measurement process.",
"The unfolding procedure attempts to reverse these effects to obtain the most probable true distribution corresponding to the given measured distribution.In order to find an estimator for the true distribution given the measured distribution, an unfolding method[191] can be applied to correct for the respective acceptance and resolution effects.",
"In this process, the response matrix is derived from an arbitrary reference sample, typically using Monte Carlo simulations.",
"This procedure is called the training step of the unfolding.",
"The obtained response matrix has to be inverted in order to allow the unfolding of any measured distribution to its corresponding true distribution.",
"Since the response matrix represents the full resolution and acceptance information of the underlying measurement, the unfolding procedure can be performed model-independently, assuming that the detector simulation used in the training sample is sufficiently accurate.",
"However, in most situations where unfolding is applied, an exact and unique inverse response matrix $R^{-1}$ does not necessarily have to exist, such that $R R^{-1} = I,$ where $I$ is the unity matrix.",
"Hence, approximations are needed to perform the above matrix inversion to acceptable accuracy.",
"Limited statistics in the reference sample and the resulting statistical fluctuations can lead to additional and inadvertent bin migration effects in the response matrix, which do not represent the underlying resolution and acceptance effects.",
"Consequently, these contributions have to be suppressed in the matrix inversion process, achieved by applying a regularisation procedure in order to limit the propagation of statistical fluctuations into the unfolded distribution or quantity.",
"This regularisation typically involves a cut-off or weight parameter representing the sensitivity of the unfolding approach with respect to short-ranged bin-by-bin changes.",
"Hence, the regularisation can be regarded as a constraint on the smoothness of the response matrix and hence the unfolded distribution.",
"The obtained approximate inverse matrix $R^{-1}$ is applied to the distribution measured from data, $\\smash{\\vec{m}}$ , and the respective unfolded distribution, $\\smash{\\vec{u}}$ , is obtained as estimator for the true distribution based on the given measurement: $\\vec{u} = R^{-1} \\vec{m}.$ Several procedures have been developed to perform the inversion of the response matrix and the necessary regularisation.",
"These unfolding methods will be briefly explained and evaluated with respect to their value and applicability for the measurement of the charge asymmetry using the observable $A_C$ in the following.",
"Bin-by-bin correction - Neglecting bin migrations, a simple reweighting can be performed by defining a correction factor for each bin of the distribution measured on data based on the true and measured reference distributions[192]: $c_i = \\frac{t_i}{k_i}.$ Consequently, the unfolding is performed by application of the respective correction factors to the corresponding bins of the distribution $\\smash{\\vec{m}}$ obtained from the data measurement to obtain the unfolded distribution $\\smash{\\vec{u}}$ : $u_i = c_i \\cdot m_i.$ Despite the simplicity of this method, it is prone to biases due to the reference Monte Carlo used in the training step, in particular the shape of the respective distributions.",
"Since a priori no bin migrations are taken into account, it relies strongly on the correct description of the underlying physics and hence does not provide a model-independent approach.",
"Furthermore, ambiguities in the determination of the statistical uncertainties may arise for cases where $k_i > t_i$ , where the obtained relative uncertainty can be underestimated, being smaller than the expected uncertainty for an ideal detector (i.e.",
"where $m_i$ would itself be an estimator of $t_i$ ).",
"Singular value decomposition - This method employs a singular value decomposition (SVD) approach[193] in order to perform the inversion of the response matrix in the unfolding procedure.",
"Singular value decomposition of a given real $m \\times n$ matrix $R$ involves a factorisation such that $R = U S V^T,$ where $U$ and $V$ denote $m \\times m$ and $n \\times n$ orthogonal matrices, respectively, such that $U U^T = U^T U = I_m$ and $V V^T = V^T V = I_n$ (with $I_m$ and $I_n$ being the corresponding $m \\times m$ and $n \\times n$ unity matrices).",
"Furthermore, $S$ denotes an $m \\times n$ diagonal matrix with non-negative diagonal elements, i.e.",
"$S_{ij} = 0$ for $i \\ne j$ and $S_{ii} = s_i \\ge 0$ .",
"The matrix entries $s_i$ are called singular values of the matrix $R$ .",
"This form can be used in order to decompose the given measured distribution $\\smash{\\vec{k}}$ and the unknown true distribution $\\smash{\\vec{t}}$ into a series of orthogonal and normalised functions of the respective $m$ and $n$ classes by performing an appropriate rotation of the respective vectors.",
"Consequently, the given initial system of linear equations as shown in Equation REF is reduced to a diagonal system of equations, such that $U^T \\vec{k} = S V^T \\vec{t} \\quad \\quad \\Leftrightarrow \\quad \\quad \\vec{d} = S \\vec{z},$ where $\\smash{\\vec{d}}$ and $\\smash{\\vec{z}}$ denote the rotated measured and true vectors, respectively.",
"This procedure is in particular effective if the respective singular values $s_i$ are small and the statistical uncertainties on the entries of the measured distribution are large, since in such a case any exact inversion solution is dominated by statistical fluctuations and thus physically meaningless.",
"By transformation of the given system of linear equations into the form of a weighted least squares minimisation, taking into account measurement uncertainties, a regularisation of the unfolding procedure can be achieved by the addition of a corresponding regularisation or stabilisation term to the expression to be minimised[194], [195], [196].",
"This introduces prior knowledge of the given problem and involves the requirement of minimal curvature of the obtained unfolded solution (i.e.",
"the smoothness of the resulting distribution), eliminating statistical bin-to-bin fluctuations similar to the suppression of high-frequency harmonics in Fourier analysis.",
"A regularisation parameter $\\tau $ defines the relative weight of the additional regularisation term with respect to the terms originating from the given system of linear equation in the minimisation: $\\left( R \\vec{t} - \\vec{k} \\right)^T \\left( R \\vec{t} - \\vec{k} \\right) + \\tau \\left( C \\vec{t} \\right)^T C \\vec{t},$ where $C$ denotes a matrix representing the prior condition on the solution.",
"Bayesian iterative unfolding - A procedure based on an iterative approach to perform the inversion of the response matrix following Bayes' theorem is applied[197].",
"This approach allows incorporating new knowledge to update a prior probability of observation of a given event[198], [199], [200] in an iterative procedure.",
"In order to obtain the inverted response matrix, the posterior probability of obtaining the true distribution $\\vec{t}$ given the measured distribution $\\vec{k}$ is calculated accordingly, assuming prior knowledge $P_{0}(t_j)$ for the individual components of $\\vec{t}$ based on the true distribution obtained in the Monte Carlo training step of the unfolding: $P(t_j|k_i) = \\frac{P(k_i|t_j)P_{0}(t_j)}{\\sum _{l=1}^{n_t}P(k_i|t_l)P_{0}(t_l)}.$ Note that in this context, the probability $P(k_i|t_j)$ is identical to the transition probability $R_{ij}$ contained in the response matrix.",
"The obtained posterior probability distribution function is used as a prior in the next iteration, consecutively updating the existing knowledge about the respective probabilities with increasing number of iterations: $P_{1}(t_j) & \\propto & \\sum _{i} P(t_j|k_i) \\cdot k_i \\propto \\sum _{i} P(k_i|t_j)\\cdot P_{0}(t_j) \\cdot k_i \\\\P_{2}(t_j) & \\propto & \\sum _{i} P(t_j|k_i) \\cdot k_i \\propto \\sum _{i} P(k_i|t_j)\\cdot P_{1}(t_j) \\cdot k_i \\\\P_{3}(t_j) & \\propto & \\sum _{i} P(t_j|k_i) \\cdot k_i \\propto \\sum _{i} P(k_i|t_j)\\cdot P_{2}(t_j) \\cdot k_i \\\\& \\vdots & $ Regularisation of the Bayes iterative unfolding procedure can be achieved naturally by restricting the number of iterations $N_{\\text{It}}$ such that the underlying true distribution is recovered within the statistical uncertainties, and bin-to-bin fluctuations which are of purely statistical nature are suppressed.",
"For a large number of iterations, a convergent state is reached, yielding the true, but strongly fluctuating inverse of the response matrix (thus minimising any remaining systematic bias of the unfolded distribution at the cost of larger statistical uncertainties).",
"The number of iterations necessary to reach convergence depends on different conditions, including the choice of binning, the strength of bin migrations in the response matrix (i.e.",
"the magnitude of its off-diagonal elements), and the choice of prior.",
"The optimal choice of $N_{\\text{It}}$ is case dependent and must be determined following a well-defined procedure, balancing remaining bias and statistical uncertainty of the obtained result.",
"Due to the fact that the bin-by-bin unfolding cannot account for bin migrations and does not allow for a regularised unfolding procedure, and hence is expected to be heavily model-dependent, this procedure was not eligible for the usage in this analysis.",
"It only allows for a comparison of the measured asymmetry with the Standard Model prediction (unless the unfolding is performed for different model hypotheses), which can be achieved using the measured asymmetry directly as well.",
"The SVD unfolding approach, despite providing a well defined methodology, cannot perform an unfolding in more than one parameter in its current technical implementations, which would limit the analysis to an inclusive unfolding in a single parameter.",
"Given the requirements of the measurement, a Bayesian iterative unfolding was performed in this analysis in order to recover the inclusive $|y_t| - |y_{\\bar{t}}|$ distribution and the resulting charge asymmetry observable $A_C$ at the parton level.",
"In particular, this approach allowed for a simultaneous unfolding in multiple observables due to the fact that Bayesian unfolding is independent of the ordering of the classes/bins.",
"Since many of the BSM models summarised in Chapter predict different magnitudes of the charge asymmetry for low and high $M_{t\\bar{t}}$ regions, a simultaneous unfolding of $|y_t| - |y_{\\bar{t}}|$ and the invariant $t\\bar{t}$ mass, $M_{t\\bar{t}}$ , has been performed, taking into account bin migrations in both dimensions.",
"Semileptonic $t\\bar{t}$ events generated with the mc@nlo generator have been used as reference sample to obtain the response matrix based on the detector simulation.",
"Furthermore, a simple extraction of the covariance matrix of the unfolded distributions[201] was possible.",
"The SVD unfolding was performed for the inclusive measurement as a cross-check to verify the stability and consistency of the Bayes iterative approach (c.f.",
"Appendix ).",
"The techniques utilised were available in the RooUnfold package[201], which provided simple interfaces and efficient implementations for all three mentioned unfolding methods."
],
[
"Systematic Uncertainties",
"In addition to the statistical uncertainty originating from limited data statistics, there were a multitude of systematic effects that can have an impact on the performed measurement.",
"These effects were studied individually and a corresponding systematic uncertainty on the measurement result was assigned for each contribution.",
"In order to evaluate the individual effects, the analysis was performed for each systematic under consideration with a modified response matrix and/or background contribution depending on the modelled effect.",
"The changes typically corresponded to an uncertainty of one or more parameters (e.g.",
"a shift of the muon trigger efficiencies according to the respective uncertainties) or an alternate model (e.g.",
"a different MC generator used for the simulation of $t\\bar{t}$ events).",
"The uncertainty was extracted in each case based on the shift in the measurement central value when changing the parameters accordingly, and was symmetrised.",
"In order to suppress the statistical component of the obtained uncertainty inherent in the re-evaluation of the central value by changing different parameters of the performed measurement, the requirement of at least one $b$ tagged jet in the event selection was replaced by a reweighting method.",
"The $|y_t| - |y_{\\bar{t}}|$ distribution for the Monte Carlo based background contributions ($W$ +jets, $Z$ +jets, single top and diboson background) was obtained by direct application of the $b$ tag weights to the events passing the nominal event selection without the requirement of at least one $b$ tagged jet.",
"This approach is very similar to the trigger reweighting approach described in Chapter REF.",
"The same procedure was applied for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ .",
"Furthermore, the normalisation of the resulting distributions was modified to match the distribution obtained using the nominal event selection to avoid the introduction of a potential bias in the background subtraction.",
"Control plots showing the agreement of the two approaches can be found in Appendix .",
"Since the weighted distributions were compatible within statistical uncertainties with the nominal distributions, no additional systematic uncertainty was assigned.",
"The bin-by-bin statistical uncertainty in the background distribution is reduced by up to 25 %.",
"The following systematics were considered and evaluated for this analysis.",
"All contributions were assigned to the Monte Carlo prediction (signal and/or background, were applicable).",
"QCD multijet uncertainty - The QCD multijet background was estimated with data driven methods in both the electron and muon channel.",
"Since both the shape and the overall normalisation of the estimation can only be verified to a limited extend, a very conservative systematic uncertainty of 100 % was quoted despite the availability of more advanced estimates (c.f.",
"Chapter REF), following the recommendations of the performance groups.",
"Results following a less conservative approach are discussed in Appendix .",
"The QCD multijet background normalisation was shifted up and down by 100% to quantify this uncertainty.",
"Since the QCD multijet background contribution is intrinsically charge symmetric, but enters the background subtraction, the normalisation and not shape was the dominant source of systematic uncertainty.",
"Jet energy scale - The jet energy scale (JES) uncertainty was derived using information from test beam data, LHC collision data and simulation and was taken into account by scaling up and down the energy of all considered jets by $1\\,\\sigma $ of the associated transverse momentum uncertainty based on the jet $p_{\\rm {T}}$ and $\\eta $[202].",
"The full event selection and kinematic reconstruction has been re-run with the scaled jets.",
"In addition, the missing transverse energy has been re-evaluated, taking into account the scaled contributions of the jets in $p_{x}$ and $p_{y}$ .",
"For jets within the acceptance range, the JES uncertainty varied from about 2.5 % for high $p_{\\rm {T}}$ jets in the central detector region to about 14 % for low $p_{\\rm {T}}$ jets in the forward region.",
"Pile-up (JES) - Depending on the instantaneous luminosity and the amount of vertices, an event weight was assigned to match the pile-up contribution in the simulation to data.",
"An additional systematic uncertainty of 5% (7%) for low $p_{\\rm {T}}$ jets or 2% (3%) for high $p_{\\rm {T}}$ jets in the $|\\eta |<2.1$ ($2.1<|\\eta |<4.5$ ) region was added to the JES uncertainty in quadrature[203].",
"$b$ jet energy scale - In order to account for the difference of the energy scale for $b$ jets with respect to light quark jets, all $b$ jets (i.e.",
"jets which have a matched truth $b$ quark in simulations) were scaled by an additional fraction ranging from 2.5 % in the low $p_{\\rm {T}}$ jet region to 0.76 % in the high $p_{\\rm {T}}$ jet region and added to the JES uncertainty in quadrature[202].",
"Jet reconstruction efficiency - The jet reconstruction efficiency (JRE) was evaluated by randomly dropping jets from events with a probability of about 2%[167].",
"The resulting difference with respect to the nominal case was symmetrised and quoted as JRE systematic uncertainty.",
"Jet energy resolution - A smearing of the jet transverse momentum corresponding to a resolution of about 10 % was applied to Monte Carlo events as systematic to reflect the difference between the jet energy resolution (JER) observed on data and Monte Carlo[167].",
"The resulting discrepancies were symmetrised and quoted as JER systematic uncertainty.",
"Muon efficiencies - In order to account for the trigger and reconstruction efficiencies for muons[204], [205], global and object based scale factors and efficiencies were taken into account and a systematic uncertainty was assigned on an event-by-event basis (for global scale factors) or on an object basis.",
"These were combined into an overall muon efficiency uncertainty of the order of 1 %.",
"In addition, a one-sided uncertainty of 1.5 % (events with 0 or 1 $b$ tagged jet) or 2.2 % (events with more than 1 $b$ tagged jet) was assigned to the muon trigger efficiency to account for a mis-modelled and as a consequence discarded trigger object matching algorithm in the Monte Carlo samples.",
"Muon scales and resolution - Since the Monte Carlo muon momentum scales and resolution differed from the ones observed in data, the muon momentum was smeared and a scaling of up to 1.5 % was applied on object level to account for this discrepancy[171].",
"A systematic uncertainty at the sub-percent level was assigned by scaling up and down both the momentum scaling and smearing by 1 $\\sigma $ according to the respective uncertainty.",
"In addition, the missing energy was re-evaluated with the modified four-vector information.",
"The full event selection and kinematic reconstruction was performed for the different scales, resulting in a symmetrised systematic uncertainty based on the comparison of the different results of the measurement.",
"Electron efficiencies - In order to account for the trigger and reconstruction efficiencies for electrons, global and object based scale factors were taken into account[172].",
"A systematic uncertainty was assigned on an event basis (for global scale factors) or on an object basis, which were combined into an overall electron efficiency uncertainty of the order of 1 %.",
"Electron scales and resolution - In order to take into account discrepancies between the electron energy resolution on Monte Carlo and data, a Gaussian smearing procedure was applied to the electron energy for Monte Carlo events to reflect the resolution in data[172].",
"In addition, the electron energy in data was corrected to account for a scaling mis-match between data and Monte Carlo.",
"Both systematic uncertainties were of the order of 1 % to 2 % and were assigned to the Monte Carlo prediction for consistency.",
"$b$ tag scale factors - Due to discrepancies in the $b$ tagging efficiencies and fake rates between data and simulation, all Monte Carlo jets were assigned a specific weight to account for this effect[206], [168].",
"The obtained $b$ tag weights for each jet were combined into an event weight by multiplication (corresponding to a logical AND of all jets taken into account).",
"The provided scale factors contained uncertainties which result in small shape variations.",
"In order to determine the deviation in the shapes from the nominal case due to the $b$ tag scaling and to quantify the corresponding systematic uncertainty on the measurement, the resulting samples were shifted up and down by the provided uncertainties.",
"These were of the order of 8 %, depending on the jet $p_{\\rm {T}}$ and $\\eta $ .",
"PDF uncertainty - For the mc@nlo signal Monte Carlo, CTEQ6.6 PDFs were utilised to model the incoming partons to the hard scattering process, as described in Chapter .",
"The impact of the choice of PDFs was evaluated by varying the eigenvalues of the CTEQ parametrisation[207] or by comparison with the respective MRST2001 parametrisation[208], using the Lhapdf tool[209].",
"Event weights were determined and the variations in the resulting pseudosamples were taken into account as PDF systematic uncertainty.",
"LAr defects - Parts of the LAr calorimeter readout electronics were inoperative during a significant time period of data taking due to a technical problem.",
"Having occurred after the production of the used Monte Carlo samples, it was necessary to correct for the resulting mismatch between data and simulated events at the analysis level.",
"Monte Carlo events were dropped with a probability corresponding to the relative amount of data affected (84.0 %) if an electron or a jet entered the region of degraded acceptance (taking into account the isolation requirements).",
"A systematic uncertainty was assigned based on different transverse momentum requirements for the jets taken into account for this procedure after symmetrisation.",
"ISR and FSR - In order to take into account initial and final state radiation, which can introduce additional jets in the observed events, different Monte Carlo samples with varying ISR and FSR contributions[187] (ISR and FSR contributions scaled up and down independently and in combination) were evaluated by replacing the nominal signal sample in the measurement.",
"The systematic uncertainty was quoted as the maximum relative deviation from the nominal leading order sample observed in these variations and applied to the mc@nlo prediction.",
"The parameters were varied in a range comparable to those used in the Perugia Soft/Hard tune variations[210].",
"$t \\bar{t}$ modelling - The impact of using different MC generators for the signal process modelling was studied.",
"In addition to the mc@nlo generator, the powheg generator was used for comparison and a symmetrised systematic uncertainty was assigned based on the variations in the measurement results for the alternate modelling.",
"Parton shower / fragmentation - In addition to the matrix element level MC generator, the effect of different showering models was taken into account by comparing the results for the powheg generator with showering performed by pythia and by herwig, and a symmetrised systematic uncertainty was assigned based on the variations in the measurement results for the alternate shower modelling.",
"Top mass - Since the top mass parameter was considered fixed, the uncertainty on the measurement of the mass was taken into account separately.",
"Two different Monte Carlo samples generated with different mass parameters (scaled up and down to 180 GeV and 170 GeV, respectively) were used and the observed deviations were linearly interpolated according to the actual uncertainty of 0.5 % of the top mass measurement[13] and a symmetrised systematic uncertainty was quoted.",
"$W$ +jets background uncertainty - The W+jets background normalisation was estimated with a data driven method and a systematic uncertainty based on the limited statistics available and several systematic contributions to the method have been evaluated (see Chapter for details).",
"An overall $W$ +jets normalisation uncertainty of 22.4 % and 22.7 % was obtained in the muon+jets and electron+jets channel, respectively.",
"In addition, the $W$ +jets background shape uncertainty has been evaluated by modifying several generator parameters such as the renormalisation scale or the functional form of the factorisation scale compared to the nominal alpgen parameters, based on the leading jet $p_{\\rm {T}}$.",
"A symmetrised systematic uncertainty on the charge asymmetry measurement has been assigned based on the deviations for two different sets of parameters with respect to the nominal results.",
"$Z$ +jets background uncertainty - In order to quantify the uncertainty on the $Z$ +jets contribution normalisation, a Berends-Giele scaling uncertainty[211] was calculated, corresponding to an overall normalisation uncertainty of 34 %.",
"In addition, the $Z$ +jets background was determined independently from both the alpgen and sherpa generator to quantify the shape uncertainty, which was quoted based on the symmetrised discrepancy of the results obtained with sherpa with respect to the nominal case, in which alpgen samples were used.",
"Charge mis-identification - Since the detector momentum resolution is finite, and the lepton charge was identified by taking into account the bending radius of the particle track, a certain probability for mis-identifying the lepton charge exists, especially for high transverse momentum leptons due to their almost straight trajectories.",
"This probability was evaluated on Monte Carlo and on data to be of the order of 0.2 % to 0.5 % in the central detector region, and up to 2.5 % in the forward region, respectively, in the electron+jets channel[183].",
"In the muon+jets channel, the probability was found to be below 0.003 % in all cases.",
"A corresponding symmetrised systematic uncertainty on the measurement of the charge asymmetry was determined.",
"$b$ tag charge - A dependency of the $b$ tag efficiencies on the $b$ quark charge can lead to a bias in the measurement due to the requirement of at least one $b$ tagged jet.",
"Hence, a simple Monte Carlo study on parton level was performed by simulating a difference in the $b$ tag efficiency of 5 % between $b$ and $\\bar{b}$ quarks.",
"The resulting impact on the charge asymmetry on parton level was studied and the difference with respect to the nominal case (assuming identical tagging efficiencies for $b$ and $\\bar{b}$ quarks) was quoted as systematic uncertainty.",
"MC generator statistics - Since the signal Monte Carlo sample entered directly into the unfolding response matrix and statistical fluctuations in the bins of this matrix can have an impact on the unfolding process, an additional ensemble test was performed by fluctuating the obtained nominal response matrix on a bin-by-bin basis following a Gaussian probability distribution (since the statistics from the mc@nlo sample were very high a Gaussian model could safely be assumed).",
"Uncertainties of the order of 0.3 % to 3 % have been obtained, depending on the statistics in each bin of the response matrix.",
"Unfolding convergence - Based on the convergence evaluation which was used to determine the optimal amount of regularisation in the unfolding process (for details refer to Chapter ), a remaining absolute uncertainty of 1 , corresponding directly to the choice of convergence criterion, was assigned, representing the potential remaining change with respect to further increase in regularisation.",
"Unfolding bias - Closure tests have been performed using ensembles of pseudodata to quantify any remaining bias from the unfolding at the chosen regularisation strengths by obtaining pull distributions for the measured asymmetry, normalised with respect to the respective unfolding statistical uncertainty.",
"The corresponding distributions can be found in Figure REF in Appendix .",
"The remaining bias was extracted from the residuals in the pull distributions in the closure tests and was taken into account as an additional systematic uncertainty.",
"The residual bias after unfolding was extracted from the respective pull distributions corresponding to the regularisation used in the individual cases and taken into account as an additional relative uncertainty on the unfolded results, which was of the order of 1 % to 11 %.",
"Other backgrounds - For the small backgrounds from single top and diboson production, only normalisation uncertainties were considered.",
"For the single top contribution, the uncertainties of the approximate next-to-next-to-leading order prediction (for details, refer to Chapter REF) were taken into account (corresponding to an uncertainty of about 11 %), while for the diboson production, an overall uncertainty of 5 % was assumed.",
"Luminosity - The relative uncertainty on the measurement of the integrated luminosity of the used data sample was estimated to be 3.7 %[212] and was taken into account for the measurement.",
"Pile-up - In order to take into account the difference in pile-up conditions between Monte Carlo and data, an event based reweighting was performed to reflect the distribution of average observed number of interactions per bunch crossing on data.",
"The impact of the pile-up conditions on the measurement before unfolding is shown in detail in Chapter .",
"Since no significant pile-up dependency was observed, no additional systematic uncertainty was assigned."
],
[
"Event Yields & Control Plots",
"The final expected and observed number of events in both the muon+jets and the electron+jets channel after performing the event selection can be found in Table REF, both without and with the requirement of at least one $b$ tagged jet.",
"Table: Observed number of data events in comparison to the expected number of Monte Carlo signal events and different background contributions for the event selection, both with and without the requirement of at least one bb tagged jet.",
"The QCD multijet and WW+jets contributions were estimated using data driven methods (c.f.",
"Chapter ).",
"Uncertainties are statistical and include the respective systematic uncertainties on the normalisation for QCD multijet and WW+jets and the cross-section uncertainties on all other contributions.",
"For the QCD multijet background, a conservative 50 % (100 %) overall normalisation uncertainty for the selection without (with) requiring at least one bb tagged jet was assumed.Control plots for the full event selection as described in Chapter , showing a comparison of the observation and expectation for several object and event quantities, can be found in Figure REF and Figure REF, for both the muon+jets and the electron+jets channel, respectively.",
"The uncertainties on the expectation include statistical and the leading systematic uncertainties, that is QCD multijet background and $W$ +jets normalisation, luminosity, jet energy scale, $b$ tag scale factors and $t\\bar{t}$ cross-section uncertainty.",
"Figure: Control plots for the muon+jets channel.",
"From the top left to the bottom right, the transverse momentum p T p_{\\rm {T}}, the pseudorapidity η\\eta and the azimuthal angle φ\\phi of the selected muon are shown.",
"Additional plots show the transverse momentum of the leading jet (p T (j 1 )p_{\\rm {T}}(j_1)), the WW transverse mass m T (W)m_{\\rm {T}}(W) and the transverse missing energy ¬E T \\lnot \\hspace{-2.84544pt} E_{\\rm {T}}.",
"Uncertainties are statistical and for WW+jets also include systematic uncertainties on normalisation.",
"For the QCD multijet background, a conservative 100 % systematic uncertainty was assumed.",
"In addition, the uncertainties on luminosity, jet energy scale, bb tag scale factors and tt ¯t\\bar{t} cross-section are shown.Figure: Control plots for the electron+jets channel.",
"From the top left to the bottom right, the transverse momentum p T p_{\\rm {T}}, the pseudorapidity η\\eta and the azimuthal angle φ\\phi of the selected electron are shown.",
"Additional plots show the transverse momentum of the leading jet (p T (j 1 )p_{\\rm {T}}(j_1)), the WW transverse mass m T (W)m_{\\rm {T}}(W) and the transverse missing energy ¬E T \\lnot \\hspace{-2.84544pt} E_{\\rm {T}}.",
"Uncertainties are statistical and for WW+jets also include systematic uncertainties on normalisation.",
"For the QCD multijet background, a conservative 100 % systematic uncertainty was assumed.",
"In addition, the uncertainties on luminosity, jet energy scale, bb tag scale factors and tt ¯t\\bar{t} cross-section are shown.Figure: Control plots for the tt ¯t\\bar{t} event reconstruction, on the left for the muon+jets, on the right for the electron+jets channel.",
"The top row shows the logarithmic likelihood logL\\log {L} of the kinematic fit, followed by the invariant mass M tt ¯ M_{t \\bar{t}} and the transverse momentum p T p_{\\rm {T}} of the tt ¯t\\bar{t} system.",
"Uncertainties are statistical and for WW+jets also include systematic uncertainties on normalisation.",
"For the QCD multijet background, a conservative 100 % systematic uncertainty was assumed.",
"In addition, the uncertainties on luminosity, jet energy scale, bb tag scale factors and tt ¯t\\bar{t} cross-section are shown.Due to the tighter definition of selection criteria in the electron+jets channel, the number of events in the electron+jets channel was significantly lower than it was the case for the muon+jets channel.",
"This was necessary in order to reduce the contribution from the increased number of electron fakes originating from the QCD multijet background with respect to the muon channel, where the expected fake rate was significantly lower.",
"The overall agreement between Monte Carlo prediction and data was very good in both channels.",
"Additional control plots for the same quantities without the explicit requirement of at least one $b$ tagged jet can be found in Figure REF and Figure REF of Appendix for completeness.",
"In addition to the control plots showing basic object and event kinematics, several more complex quantities based on the kinematic event reconstruction, as described in Chapter , can be found in Figure REF.",
"The respective distributions for the logarithmic likelihood of the kinematic fit, and the invariant mass and transverse momentum of the reconstructed $t\\bar{t}$ system in both the muon+jets and the electron+jets channel are shown.",
"The agreement between data and prediction is very good, indicating a proper modelling of the $t\\bar{t}$ signal and background kinematics in the various Monte Carlo samples and data driven background estimates."
],
[
"Measurement of the Charge Asymmetry",
"Events passing the described event selection were taken into account to determine the differential and integrated charge asymmetry based on the distribution of $|y_t| - |y_{\\bar{t}}|$ .",
"A subtraction of the predicted background contributions was performed on the distributions measured on data to obtain an estimate for the $t\\bar{t}$ signal contribution only.",
"The corresponding distributions are shown in Figure REF for both the inclusive measurement of the $|y_t| - |y_{\\bar{t}}|$ distribution and the corresponding measurement for the two cases of $M_{t \\bar{t}} < 450$ GeV and $M_{t \\bar{t}} > 450$ GeV.",
"For the $M_{t \\bar{t}}$ dependent measurement, an additional requirement on the event reconstruction logarithmic likelihood of $\\log {L} > -52$ (see Figure REF) has been applied in order to improve the resolution in the $t\\bar{t}$ invariant mass.",
"The respective relative resolution in $M_{t \\bar{t}}$ for both channels before and after the application of the additional requirement on $\\log {L}$ can be found in Figure REF and Figure REF in Appendix .",
"The relative resolution improves from 28.8 % to 18.4 % in the muon+jets channel and from 28.8 % to 17.7 % in the electron+jets channel.",
"The resulting charge asymmetries for the observable $A_C^{\\text{reco}}$ obtained after background subtraction can be found in Table REF for both the muon+jets channel and the electron+jets channel.",
"The observed results are shown alongside the predicted $t\\bar{t}$ charge asymmetries obtained with mc@nlo for comparison.",
"The uncertainties on the prediction correspond to the limited Monte Carlo statistics in the used sample after the applied selection.",
"Figure: Distributions for |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}|, on the left for the muon+jets, on the right for the electron+jets channel.",
"The top row shows the distribution for the inclusive measurement, while the lower rows show the corresponding distributions requiring M tt ¯ <450M_{t\\bar{t}} < 450 GeV and M tt ¯ >450M_{t\\bar{t}} > 450 GeV, respectively.",
"Uncertainties are statistical and for WW+jets also include systematic uncertainties on normalisation.",
"For the QCD multijet background, a conservative 100 % systematic uncertainty was assumed.",
"In addition, the uncertainties on luminosity, jet energy scale, bb tag scale factors and tt ¯t\\bar{t} cross-section are shown.Table: Measured values of the charge asymmetry observable A C reco A_C^{\\text{reco}} for the muon+jets and electron+jets channel after subtraction of the various background contributions.",
"The results for the inclusive measurement and the respective measurements for M tt ¯ <450M_{t \\bar{t}} < 450 GeV and M tt ¯ >450M_{t \\bar{t}} > 450 GeV are shown together with the mc@nlo predictions.As can be seen in the table, the obtained results were compatible with the mc@nlo Standard Model prediction in all cases within the statistical and systematic uncertainties.",
"However, in the electron+jets channel, a tendency towards more negative asymmetries was observed, in particular for the measurement of $A_C^{\\text{reco}}$ for $M_{t \\bar{t}} < 450$ GeV.",
"A breakdown of the individual systematic uncertainties taken into account can be found in Table REF and Table REF of Appendix for completeness.",
"In addition, the integrated charge asymmetry $A_C^{\\text{reco}}$ after background subtraction as a function of the invariant $t\\bar{t}$ mass, $M_{t \\bar{t}}$ is shown in Figure REF together with the Standard Model prediction obtained from simulated $t\\bar{t}$ events.",
"Figure: Integrated charge asymmetry A C reco A_C^{\\text{reco}} after background subtraction as a function of M tt ¯ M_{t \\bar{t}}.",
"The observed asymmetries (red, dashed) and the asymmetries predicted by mc@nlo (blue) are shown.",
"Uncertainties are statistical only.The observed asymmetry values show no statistically significant deviation from the Standard Model prediction.",
"Since, in addition, no systematic uncertainties were included, the observed discrepancies are negligible."
],
[
"Unfolding",
"In order to perform the Bayes iterative unfolding and to correct the measured differential and integrated charge asymmetry for detector resolution and acceptance effects, the corresponding response matrix has been obtained using the full set of $t\\bar{t}$ signal Monte Carlo events generated with mc@nlo (15000000 simulated events).",
"A unified binning for the true and reconstructed distributions (and consequently for the response matrices) in the variable $|y_t| - |y_{\\bar{t}}|$ has been used, employing six bins with variable width to ensure sufficient statistics in the tails of the respective distributions.",
"Bin edges at $\\lbrace -3.0, -1.2, -0.6, 0.0, 0.6, 1.2, 3.0 \\rbrace $ for the inclusive unfolding and $\\lbrace -3.00, -0.96, -0.48, 0.0, 0.48, 0.96, 3.00 \\rbrace $ for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ were chosen.",
"Furthermore, for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , an additional requirement on the event reconstruction logarithmic likelihood of $\\log {L} > -52$ (see Figure REF) has been applied in order to improve the resolution of the $t\\bar{t}$ invariant mass in the same way as for the measurement of the charge asymmetry before unfolding.",
"The response matrix representation used in the inclusive measurement of the distribution of $|y_t| - |y_{\\bar{t}}|$ and of $A_C^{\\text{unf}}$ is shown in Figure REF for both the muon+jets and electron+jets channel.",
"Figure: Unfolding response matrices for the inclusive unfolding of the charge asymmetry distribution |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}|.",
"The bin migration probability corresponds to the box sizes displayed in the response matrix, and is shown independently for the muon+jets channel (left) and the electron+jets channel (right).Similarly, the corresponding matrices for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ can be found in Figure REF.",
"The transition probability information of the corresponding two-dimensional distributions (with $2 \\times 6$ bins) was encoded in the matrix by linearisation of the respective histograms, concatenating the bins from the two $M_{t \\bar{t}}$ regions into a single one dimensional histogram (with $1 \\times 12$ bins).",
"Hence, the associated response matrix contained $2 \\times 2$ quadrants, corresponding to the transition probabilities for the true and reconstructed $|y_t| - |y_{\\bar{t}}|$ distributions in the low and high $M_{t \\bar{t}}$ regions of phase space, respectively.",
"Figure: Unfolding response matrices for the simultaneous unfolding of the distribution in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}} showing the discrete bin correspondence of the truth distribution with respect to the distribution obtained after event selection and reconstruction.",
"The bin migration probability corresponds to the box sizes displayed in the response matrix, and is shown independently for the muon+jets channel (left) and the electron+jets channel (right).",
"The four quadrants represent the respective transition probabilities for the true and reconstructed |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| distributions in the low and high M tt ¯ M_{t \\bar{t}} regions of phase space.A closure test has been performed to verify that the Bayesian iterative unfolding approach can be used to recover an arbitrary asymmetry present in the true distribution.",
"The full mc@nlo $t\\bar{t}$ signal sample was used as a basis to verify the correct unfolding response for different artificially injected truth level asymmetries.",
"This was achieved by performing an event-by-event reweighting of the $t\\bar{t}$ signal sample to asymmetries of -10 %, -5 %, 0 %, 5 % and 10 % in the variable $A_C^{\\text{true}}$ by systematically increasing the weights of events with $|y_t| - |y_{\\bar{t}}| < 0$ and decreasing the weights of events with $|y_t| - |y_{\\bar{t}}| > 0$ by the same amount.",
"Pseudoexperiments, using statistically independent sets of pseudodata corresponding to the statistics expected in the used data sample (after background subtraction), were conducted based on a Poissonian fluctuation of the respective distribution in $|y_t| - |y_{\\bar{t}}|$ , taking into account the additional event weights determined in the asymmetry reweighting.",
"Ensemble tests were performed to confirm the linearity of the unfolding response in the true value of the chosen charge asymmetry observable $A_C^{\\text{true}}$ .",
"Furthermore, the dependency of the obtained results on the regularisation of the unfolding procedure was studied.",
"The average unfolded value of $A_C^{\\text{unf}}$ obtained from the sets of pseudodata as a function of the injected true value of $A_C^{\\text{true}}$ can be found for both the inclusive unfolding and the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ in Figure REF for different regularisation strengths, using $N_{\\text{It}} = 5$ , 10, 20, 40 and 80 iterations in the Bayesian unfolding.",
"Figure: The obtained overall inclusive asymmetry after unfolding as a function of injected true asymmetry A C true A_C^{\\text{true}} for different regularisation parameters N It N_{\\text{It}} for both the muon+jets channel (left) and electron+jets channel (right).",
"The top row shows the respective distribution for the inclusive measurement, while the lower rows show the corresponding distributions for M tt ¯ <450M_{t\\bar{t}} < 450 GeV and M tt ¯ >450M_{t\\bar{t}} > 450 GeV, respectively.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.A straight line fit using a parametrisation of the form $A_C^{\\text{unf}} = a \\cdot A_C^{\\text{true}} + b$ has been performed, where $a$ and $b$ denote the slope and offset parameters, respectively.",
"The procedure has been repeated for each of the individual choices of $N_{\\text{It}}$ to obtain the calibration curves in order to verify the linearity and determine the respective slopes and offsets of the fit.",
"These provided a measure for the average remaining bias in the unfolded differential and integrated asymmetry after unfolding for different regularisation strengths.",
"As can be seen in the fits, a slope close to one could be achieved in all cases, indicating a proper average correspondence of the unfolded asymmetry to the injected true value, independent of the strength of the injected asymmetry.",
"In addition, the expected statistical uncertainty on $A_C^{\\text{unf}}$ after the unfolding procedure is shown in Figure REF in Appendix as a function of $N_{\\text{It}}$ .",
"As expected, the uncertainty increases with the number of iterations used in the regularisation due to the increasing sensitivity to bin-to-bin statistical fluctuations in the inversion process of the response matrix.",
"The expected uncertainties are independent of the injected value of $A_C^{\\text{true}}$ .",
"In order to achieve a convergent state in the iterative unfolding process, the dependency of the unfolded result as a function of the number of iterations has been studied with respect to a defined convergence criterion.",
"For individual ensembles of pseudodata, the unfolding procedure was considered to be converged if the absolute change in the unfolded asymmetry in terms of $A_C^{\\text{unf}}$ , $\\Delta A_C^{\\text{unf}}$ , for a given amount of iterations $N_{\\text{It}} = i$ with respect to the previous amount of iterations $N_{\\text{It}} = i-1$ was lower than 1 , i.e.",
"if $\\Delta A_C^{\\text{unf}} = |A_C^{\\text{unf}}(N_{\\text{It}} = i) - A_C^{\\text{unf}}(N_{\\text{It}} = i-1)| < 0.001.$ Figure REF in Appendix shows the percentage of ensembles which have reached the defined convergence criterion as a function of regularisation strength, parametrised by the amount of iterations $N_{\\text{It}}$ .",
"At $N_{\\text{It}} = 40$ (inclusive unfolding), and $N_{\\text{It}} = 80$ (simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ ) iterations, all ensembles of pseudodata were ensured to have reached convergence for all injected value of $A_C^{\\text{true}}$ .",
"Hence, these regularisation strengths were chosen for the inclusive unfolding and the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ , respectively.",
"An additional systematic uncertainty of 1 , corresponding directly to the choice of convergence criterion was assigned to the unfolded value of $A_C^{\\text{unf}}$ (c.f.",
"Chapter ).",
"In addition, a further cross-check was performed to verify that the unfolding procedure reached a convergent state by determining the average standard deviation of the variable $\\Delta A_C^{\\text{unf}}$ for the used set of ensembles as a function of $N_{\\text{It}}$ , which shows a monotonous falling behaviour in all cases as expected.",
"The corresponding additional control plots can be found in Figure REF in Appendix for completeness.",
"Despite the large amount of iterations necessary to reach convergence, in particular for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ and the associated expected statistical uncertainties, this approach allowed performance of the unfolding following a well-defined and model-independent procedure.",
"Furthermore, the remaining expected bias in the unfolding procedure was minimised, as can be seen in the calibration curves in Figure REF for the used choices of $N_{It} = 40$ and $N_{It} = 80$ , respectively.",
"The corresponding slope and offset parameters extracted from the straight line fit in the calibration can be found in Table REF for completeness.",
"Table: Slopes and offsets from the linear fit in the unfolding calibration.",
"The parameters were obtained for a linear fit of the average unfolded value of A C unf A_C^{\\text{unf}} as a function of the true A C true A_C^{\\text{true}} value, obtained from sets of pseudoexperiments for 40 (inclusive unfolding) and 80 (simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}) iterations, respectively.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.",
"The shown fit parameter uncertainties are statistical only.Additional closure tests have been performed using ensembles of pseudodata to quantify any remaining bias from the unfolding at the chosen regularisation strengths and a corresponding systematic uncertainty was assigned to the unfolded result accordingly (c.f.",
"Chapter ), while no correction of the unfolded results for the obtained calibration was performed.",
"The unfolded integrated asymmetries in $A_C^{\\text{unf}}$ are shown in Table REF for both the inclusive unfolding and for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , using 40 and 80 iterations in the unfolding process, respectively.",
"Table: Unfolded values of the charge asymmetry observable A C unf A_C^{\\text{unf}} for the muon+jets and electron+jets channel.",
"The results for the inclusive measurement and the respective results for the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}} for M tt ¯ <450M_{t \\bar{t}} < 450 GeV and M tt ¯ >450M_{t \\bar{t}} > 450 GeV are shown.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.",
"Furthermore, the respective mc@nlo predictions are shown.Figure: Unfolded distribution of |y t |-|y t ¯ ||y_t|-|y_{\\bar{t}}|, normalised to unity for both the muon+jets channel (left) and electron+jets channel (right).",
"The top row shows the distributions for the inclusive measurement, while the lower rows show the corresponding distributions for M tt ¯ <450M_{t\\bar{t}} < 450 GeV and M tt ¯ >450M_{t\\bar{t}} > 450 GeV, respectively.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.",
"The uncertainties include both statistical and systematic shape components.In addition, Figure REF shows the obtained distributions after unfolding for both the inclusive unfolding and the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ .",
"The unfolded distributions have been normalised to unity and the shape uncertainties obtained from all systematic effects described in Chapter and the bin-by-bin statistical uncertainties have been included.",
"Furthermore, the covariance matrices corresponding to the unfolded $|y_t|-|y_{\\bar{t}}|$ distributions are shown in Figure REF and REF for the inclusive unfolding and the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , respectively.",
"Figure: Covariance matrices corresponding to the unfolded |y t |-|y t ¯ ||y_t|-|y_{\\bar{t}}| distribution for the muon+jets channel (left) and the electron+jets channel (right) for the inclusive unfolding.",
"The numbers inside the boxes represent the values and the sign of the correlation among the different bins and have been scaled by a factor of 1000 for readability.Figure: Covariance matrices corresponding to the unfolded |y t |-|y t ¯ ||y_t|-|y_{\\bar{t}}| distribution for the muon+jets channel (left) and the electron+jets channel (right) for the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}.",
"The numbers inside the boxes represent the values and the sign of the correlation among the different bins and have been scaled by a factor of 1000 for readability.A summarised list of all systematics and their contribution to the overall systematic uncertainties can be found in Table REF for the inclusive unfolding and in Table REF for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ .",
"Table: List of all systematic uncertainties taken into account for the unfolding procedure in the measurement of the top charge asymmetry.",
"The numbers in brackets denote the uncertainties before using the larger uncertainty of both channels as conservative estimate.Table: List of all systematic uncertainties taken into account for the unfolding procedure in the measurement of the top charge asymmetry.",
"The numbers in brackets denote the uncertainties before using the larger uncertainty of both channels as conservative estimate.All systematic uncertainties based on a replacement of the unfolding matrix which were of the same order of magnitude or lower than the respective MC statistics uncertainty could not be resolved to full extent due to the inherent fluctuations from limited statistics in the response matrix.",
"In those cases, the larger of the systematic uncertainties in both channels was used for the final systematic uncertainty on the unfolded charge asymmetry.",
"The resulting total systematic uncertainties were 0.023 in the muon+jets channel and 0.028 in the electron+jets channel for the inclusive unfolding.",
"For the simultaneous unfolding in $|y_t|-|y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ the resulting total systematic uncertainties were 0.049 ($M_{t\\bar{t}} < 450\\,\\text{GeV}$ ) and 0.034 ($M_{t\\bar{t}} > 450\\,\\text{GeV}$ ) in the muon+jets channel, and 0.091 ($M_{t\\bar{t}} < 450\\,\\text{GeV}$ ) and 0.035 ($M_{t\\bar{t}} > 450\\,\\text{GeV}$ ) in the electron+jets channel, respectively.",
"The overall uncertainty in the individual channels was dominated by the statistical uncertainty.",
"The systematic uncertainties were dominated by the contributions from ISR/FSR, top mass and jet energy resolution in the muon+jets channel, and by the uncertainties originating from QCD multijet background, $t\\bar{t}$ modelling and parton shower / fragmentation in the electron+jets channel.",
"As described in Chapter , the QCD multijet background contribution has been estimated very conservatively, assuming a 100 % normalisation uncertainty.",
"Most of the other large contributions can be traced to the available Monte Carlo statistics in the used samples.",
"Since only 3000000 ($t\\bar{t}$ modelling, parton shower / fragmentation), or 1000000 (top mass, ISR/FSR) Monte Carlo events were available for the respective samples used in the evaluation of the systematic uncertainties (as opposed to 15000000 for the nominal signal sample), the statistical component in the evaluation of the response matrix uncertainty and the unfolding procedure was larger by factors of two to four with respect to the nominal case.",
"In addition to the evaluation of the systematic uncertainties in the unfolding process, the effect of pile-up on the measured quantity before the unfolding was studied in order to ensure the stability of the method for different pile-up conditions.",
"Figure REF in Appendix shows the measured integrated asymmetry before background subtraction, $A_C^{\\text{data}}$ , as a function of the number of primary vertices and of the bunch timing, i.e.",
"at which relative position in the respective bunch the $pp$ collision corresponding to the respective event occurred.",
"As there was no statistically significant dependence of the measured asymmetry on the bunch timing or on the number of primary vertices in either channel, no additional systematic uncertainty due to pile-up was assigned.",
"Both the differential and the integrated asymmetries after unfolding were in agreement with the mc@nlo Standard Model prediction within the estimated uncertainties.",
"The provided differential distributions alongside with the respective covariance matrices can directly be put into context with analogue measurements at other experiments and theoretical predictions."
],
[
"Combination",
"A best linear unbiased estimator (Blue) method[213], [214] has been used to combine the results from the muon+jets and electron+jets channel after unfolding, taking into account systematic uncertainties and the associated correlationsNote that since the exact correlations are unknown for most of the systematic uncertainties contributing to the overall result, correlation coefficients between the muon+jets and electron+jets channel of either zero or one have been assumed.. All systematic uncertainties have been considered to be fully correlated between the muon+jets and the electron+jets channel except for the contributions from the Monte Carlo statistics of the response matrix, LAr defects (since the treatment of electrons does not affect the muon+jets channel), QCD multijet and $W$ +jets normalisation (since both have been determined with data driven methods based on orthogonal datasets), charge mis-identification, and the systematic uncertainty from unfolding convergence and remaining bias.",
"No correlation was assumed for the statistical uncertainties.",
"For the inclusive unfolding, a combined value of $A_C^{\\text{unf}} = -0.018 \\pm 0.028\\,\\text{(stat.)}",
"\\pm 0.023\\,\\text{(syst.",
")}$ was obtained, where the relative weight of the muon+jets channel result was 63.9 %.",
"The combination of the results for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ yielded $A_C^{\\text{unf}} (M_{t\\bar{t}} < 450\\,\\text{GeV}) & = & -0.053 \\pm 0.070\\,\\text{(stat.)}",
"\\pm 0.054\\,\\text{(syst.",
")},\\\\A_C^{\\text{unf}} (M_{t\\bar{t}} > 450\\,\\text{GeV}) & = & -0.008 \\pm 0.035\\,\\text{(stat.)}",
"\\pm 0.032\\,\\text{(syst.",
")},$ where the relative weights of the muon+jets channel were 73.7 % and 59.5 %, respectively.",
"Note that the combined systematic uncertainty was slightly lower than the systematic uncertainties in both the muon+jets and the electron+jets channel.",
"This effect is inherent to the Blue uncertainty propagation and is due to the assumed correlations of the individual contributions, which can be regarded as additional prior information in the propagation of the uncertainties.",
"The combined results were compatible with the mc@nlo Standard Model expectation (c.f.",
"Table REF) within the estimated uncertainties, not indicating any significant deviation."
],
[
"Summary & Conclusion",
"A measurement of the charge asymmetry in the production of top quark pairs at the ATLAS experiment was performed, using a dataset corresponding to an integrated luminosity of 1.04 fb$^{-1}$ taken over the course of 2011 at a centre-of-mass energy of $\\sqrt{s}$ = 7 TeV.",
"An object and event selection was employed in the lepton+jets decay channel in order to identify events with a signature corresponding to a semileptonic decay of a $t\\bar{t}$ pair, given by one isolated lepton (muon or electron) with large transverse momentum, at least four reconstructed jets and large missing transverse momentum.",
"The selection was furthermore chosen such that various background contributions, including the production of single top quarks, heavy gauge bosons in association with jets, the contribution from diboson production and fake leptons predominantly produced in QCD multijet events, were reduced.",
"A set of background contributions was estimated using Monte Carlo simulations, while the $W$ +jets and QCD multijet normalisation was determined using data driven methods.",
"A kinematic fit was performed in order to reconstruct $t\\bar{t}$ events based on the measured objects after application of the selection, yielding the most probable object kinematics at the parton level under the assumption of a semileptonic top quark decay event topology.",
"The reconstructed objects were used to obtain the differential distribution of the difference of absolute rapidities of the reconstructed top and antitop, $|y_t| - |y_{\\bar{t}}|$ .",
"A subtraction of the various background contributions was performed in order to obtain the integrated charge asymmetry of the $t\\bar{t}$ signal contribution at reconstruction level, $A_C^{\\text{reco}}$ .",
"Furthermore, $A_C^{\\text{reco}}$ was determined as a function of the invariant $t\\bar{t}$ mass, $M_{t \\bar{t}}$ .",
"An unfolding procedure was applied to the reconstructed $|y_t| - |y_{\\bar{t}}|$ distribution in order to correct for resolution and acceptance effects and to obtain the corresponding distribution at parton level.",
"A Bayesian iterative unfolding procedure was used and cross-checks and calibrations were performed to verify the linearity, convergence and stability of the approach.",
"The unfolding was performed both for the inclusive $|y_t| - |y_{\\bar{t}}|$ distribution and in addition simultaneously in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ .",
"A combination of obtained results in the muon+jets and electron+jets channel was performed using the Blue method, yielding a combined integrated charge asymmetry after unfolding of $A_C^{\\text{unf}} = -0.018 \\pm 0.028\\,\\text{(stat.)}",
"\\pm 0.023\\,\\text{(syst.",
")},$ while the mc@nlo prediction was $0.0056 \\pm 0.0003\\,\\text{(stat.",
")}$ .",
"For the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , combined values of $A_C^{\\text{unf}} (M_{t\\bar{t}} < 450\\,\\text{GeV}) & = & -0.053 \\pm 0.070\\,\\text{(stat.)}",
"\\pm 0.054\\,\\text{(syst.",
")},\\\\A_C^{\\text{unf}} (M_{t\\bar{t}} > 450\\,\\text{GeV}) & = & -0.008 \\pm 0.035\\,\\text{(stat.)}",
"\\pm 0.032\\,\\text{(syst.",
")}$ were obtained.",
"The mc@nlo predictions were $0.0024 \\pm 0.0004\\,\\text{(stat.",
")}$ and $0.0086 \\pm 0.0004\\,\\text{(stat.",
")}$ , respectively.",
"A summary of the results obtained in the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ can be found in Figure REF.",
"Figure: Unfolded asymmetries in two regions of M tt ¯ M_{t \\bar{t}} compared to the prediction from mc@nlo.",
"The error bands on the mc@nlo prediction include uncertainties from parton distribution functions and renormalisation and factorisation scales.The obtained results were in agreement with the Standard Model prediction.",
"However, both for the measured asymmetry after reconstruction and the corresponding unfolded values a tendency to more negative integrated asymmetries was observed in the electron channel.",
"This effect is most dominant for the simultaneous unfolding in the region where $M_{t\\bar{t}} < 450\\,\\text{GeV}$ .",
"Nevertheless, the discrepancy is covered by the estimated overall uncertainties including systematics.",
"The estimated uncertainties were dominated by the statistical uncertainty, in particular for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ .",
"The largest contributions to the systematic uncertainties originated from ISR/FSR, top mass and jet energy resolution in the muon+jets channel, and from the uncertainties from QCD multijet background, $t\\bar{t}$ modelling and parton shower/fragmentation in the electron+jets channel.",
"Most of these contributions, except for the QCD multijet background uncertainty, can be attributed to large parts to the statistical component inherent in the evaluation of the systematic uncertainties, which involves a replacement of the unfolding response matrix with a corresponding matrix obtained from a different sample.",
"Since only a fraction of events of about 10 % with respect to the nominal $t\\bar{t}$ sample was available in those cases, statistical fluctuations in the response matrix dominated over the actual systematic shift to be evaluated.",
"This also explains the discrepancies observed in the comparison of the individual contributions between the muon+jets and electron+jets channel.",
"Furthermore, the QCD multijet normalisation uncertainty was conservatively estimated to be 100 % following the recommendations of the performance groups, which led to a large contribution from this uncertainty, in particular in the electron+jets channel.",
"A summary of the obtained results alongside with recent results obtained by DØ, CDF and CMS in comparison to several theoretical models beyond the Standard Model is shown in Figure REF.",
"Potential regions in the phase space of inclusive charge asymmetry from new physics, indicated by the variable $A_C^{\\text{new}}$ at the LHC plotted against the associated forward-backward asymmetry $A_{\\text{FB}}^{\\text{new}}$ at the Tevatron, are highlighted (c.f.",
"Chapter REF).",
"Figure: Measurement results and predicted charge asymmetries at the Tevatron and LHC for various BSM models, .",
"The inclusive charge asymmetry from new physics A C new A_C^{\\text{new}} at the LHC vs. corresponding the forward-backward asymmetry A FB new A_{\\text{FB}}^{\\text{new}} at the Tevatron (left) and the identical predictions in a high invariant mass region where M tt ¯ >450GeVM_{t \\bar{t}} > 450\\,\\text{GeV} (right) for the different models in the created phase space is shown.",
"The labelled lines indicate the central values of the results measured at different experiments, while the dashed lines indicate the 1σ1\\,\\sigma uncertainty regions (gray area denotes the ATLAS measurement uncertainty).",
"The Standard Model prediction corresponds to A FB new =0A_{\\text{FB}}^{\\text{new}} = 0 and A C new =0A_C^{\\text{new}} = 0, respectively.The measurement performed in this analysis together with the results from other experiments already puts pressure on several of the proposed models.",
"This is most prominent for $Z^{\\prime }$ models, which are disfavoured at $2\\,\\sigma $ to $3\\,\\sigma $ by this measurement and a similar measurement by CMS, while being favoured in the high invariant mass region by CDF.",
"Furthermore, disagreements between favoured regions from the measurements at the Tevatron and at the LHC are observed, in particular for the high invariant $t\\bar{t}$ mass region.",
"These measurements of the charge asymmetry at hadron colliders provide the first step towards a better understanding of Quantum Chromodynamics and possible extensions of the Standard Model.",
"Despite the limitation of the measurement by the statistical uncertainty, in particular for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , an increased dataset from the ATLAS experiment will quickly reduce this uncertainty, significantly increasing the sensitivity.",
"This will allow to make much more precise statements especially in regions where an increased asymmetry is expected, making it possible to fully exclude several of the available BSM models with sufficient significance.",
"Furthermore, non-excluded models could be further constrained in their respective parameters.",
"Further improvements can be achieved for the systematic uncertainties, where methods will evolve and in particular existing estimates can be replaced by more precise studies and methods.",
"As an example, the QCD multijet systematic uncertainty has been evaluated in Appendix using more advanced methods, as described in Chapter REF.",
"However, one of the most prominent sources of uncertainty is the limited statistics of the used Monte Carlo samples, in particular for the evaluation of the systematic uncertainties.",
"These could be reduced significantly if larger samples would be available.",
"Optimisations in the unfolding procedure could lead to further reductions of the statistical uncertainties.",
"Further studies of the unfolding procedure and the associated statistical and systematic uncertainties are discussed in Appendix .",
"Finally, an additional cross-check has been performed for the inclusive charge asymmetry, comparing the obtained results to an SVD unfolding approach, using otherwise identical parameters.",
"Comparable values have been obtained."
],
[
"SVD Unfolding",
"As an additional cross-check, the inclusive unfolding has been performed with the SVD unfolding procedure (for details, refer to Chapter ), using otherwise identical analysis parameters.",
"The same binning and the same response matrix as determined for the Bayesian iterative unfolding (c.f.",
"Chapter ) has been used.",
"A closure test has been performed to verify that the SVD unfolding approach can be used to recover an arbitrary asymmetry present in the true distribution.",
"Similar to the Bayesian iterative unfolding, pseudoexperiments using statistically independent sets of pseudodata corresponding to the statistics expected in the used data sample were performed.",
"Ensemble tests were conducted to confirm the linearity of the unfolding response in the true value of the chosen charge asymmetry observable, $A_C^{\\text{true}}$ , and the dependency of the obtained results on the regularisation of the unfolding procedure was studied.",
"The average unfolded value of $A_C^{\\text{unf}}$ obtained from the respective sets of pseudodata as a function of the injected true value of $A_C^{\\text{true}}$ can be found in Figure REF for different regularisation strengths corresponding to choices of the regularisation parameter of $\\tau = 2$ , 3, 4, 5 and 6.",
"Since $\\tau $ corresponds directly to a fraction of the number of bins, it can not exceed the number of bins chosen for the unfolding by construction[193].",
"Figure: The obtained inclusive asymmetry A C unf A_C^{\\text{unf}} after SVD unfolding as a function of injected true asymmetry A C true A_C^{\\text{true}} for different regularisation parameters τ\\tau for both the muon+jets channel (left) and electron+jets channel (right).A straight line fit has been performed as described in Chapter .",
"As can be seen in the respective fits, a slope close to one can be achieved in all cases for $\\tau = 6$ , indicating a proper correspondence of the unfolded asymmetry to the respective true value, independent of the strength of the injected asymmetry.",
"Since the SVD unfolding does not rely on an iterative procedure, no convergence criterion was imposed to choose a proper regularisation.",
"Instead, the parameter $\\tau $ was chosen such that the remaining unfolding bias was minimised, i.e.",
"such that the slope parameter $a$ of the linear fit in the calibration was as close to unity as possible.",
"As can be seen in Figure REF, the optimal choice was given by $\\tau = 6$ for both the muon+jets and electron+jets channel.",
"Additional closure tests have been performed as described in Chapter , using ensembles of pseudodata to quantify any remaining bias from the unfolding at the chosen regularisation strengths.",
"A corresponding systematic uncertainty was assigned to the unfolded result accordingly from the residuals of the respective pull distributions (c.f.",
"Chapter ).",
"The unfolded integrated asymmetries $A_C^{\\text{unf}}$ for the inclusive unfolding, using $\\tau = 6$ in the SVD unfolding process, were determined to be $A_C^{\\text{unf}} = -0.005 \\pm 0.034\\,\\text{(stat.)}",
"\\pm 0.024\\,\\text{(syst.",
")}$ in the muon channel and $A_C^{\\text{unf}} = -0.056 \\pm 0.043\\,\\text{(stat.)}",
"\\pm 0.029\\,\\text{(syst.",
")}$ in the electron channel, respectively.",
"In addition, the obtained distributions after unfolding can be found in Figure REF.",
"Figure: Unfolded distribution of |y t |-|y t ¯ ||y_t|-|y_{\\bar{t}}| using an SVD unfolding procedure, normalised to unity for both the muon+jets channel (left) and electron+jets channel (right).",
"The uncertainties include both statistical and systematic shape components.A summarised list of all systematics and their contribution to the overall systematic uncertainty can be found in Table REF.",
"Table: List of all systematic uncertainties taken into account for the unfolding procedure in the measurement of the top charge asymmetry.",
"The numbers in brackets denote the uncertainties before using the largest uncertainty of both channels as conservative estimate.All systematic uncertainties based on a replacement of the unfolding matrix which were of the same order of magnitude or lower than the respective MC statistics uncertainty could not be resolved to full extent due to the inherent fluctuations from limited statistics in the response matrix.",
"In those cases, the largest of the systematic uncertainties in both channels was used for the final systematic uncertainty on the unfolded charge asymmetry.",
"The resulting combined systematic uncertainties were 0.024 in the muon+jets channel and 0.029 in the electron+jets channel for the inclusive unfolding.",
"The Blue method was used to combine the measurements after performing the SVD unfolding in the muon+jets and electron+jets channel.",
"The same assumptions about correlations as described in Chapter were used.",
"A combined value of $A_C^{\\text{unf}} = -0.024 \\pm 0.027\\,\\text{(stat.)}",
"\\pm 0.024\\,\\text{(syst.",
")}$ was obtained, where the relative weight of the muon+jets channel result was 64.3 %.",
"The obtained results were compatible with the results from the Bayesian unfolding shown in Chapter ."
],
[
"Additional Studies of Unfolding and Systematics",
"As described in Chapter , the unfolding procedure employed used a strong regularisation in order to achieve a convergent state and to reduce the remaining bias after unfolding.",
"However, this implies large statistical uncertainties on the obtained results.",
"Furthermore, as described in Chapter , the QCD multijet background contribution systematic uncertainty was conservatively assumed to be 100 % and no explicit shape uncertainty was included despite the availability of more advanced estimates.",
"As an additional cross-check, the unfolding procedure has been repeated, discarding the explicit requirement of convergence of the Bayesian iterative procedure.",
"Instead, smaller values of $N_{\\text{It}}$ were used, implying lower statistical uncertainties, and the expected remaining bias of the unfolded value $A_C^{\\text{unf}}$ was extracted from pseudoexperiments and corrected for.",
"This, however, implies larger assumptions about the underlying physics since the remaining bias is determined and corrected for based on the observable $A_C$ only.",
"Consequently, no fully model-independent calibration can be performed and a bias not covered by the uncertainties can remain.",
"Furthermore, the described approach does not allow for the extraction of the full unfolded distribution and the associated bin-by-bin uncertainties since the calibration is performed with respect to the integrated asymmetry observable $A_C$ only.",
"A closure test has been performed to calibrate the unfolded asymmetry $A_C^{\\text{unf}}$ with respect to the true asymmetry $A_C^{\\text{true}}$ .",
"Similar to the procedure described in Chapter , pseudoexperiments using statistically independent sets of pseudodata corresponding to the statistics expected in the used data sample (after background subtraction) were created based on a Poissonian fluctuation of the respective reconstructed distributions of $|y_t| - |y_{\\bar{t}}|$ .",
"Ensemble tests were performed to confirm the linearity of the unfolding response in the true value of the chosen charge asymmetry observable, $A_C^{\\text{true}}$ , and the dependence of the obtained results on the regularisation of the unfolding procedure was studied.",
"The average value of $A_C^{\\text{unf}}$ obtained from the respective sets of pseudodata as a function of the injected true asymmetry, $A_C^{\\text{true}}$ , can be found for both the inclusive unfolding and the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ in Figure REF for different regularisation strengths, using $N_{\\text{It}} = 2$ , 4, 6, 7, 8 and 10 iterations in the Bayesian unfolding.",
"Figure: The obtained overall inclusive asymmetry after unfolding, A C unf A_C^{\\text{unf}}, as a function of injected true asymmetry A C true A_C^{\\text{true}} for different regularisation parameters N It N_{\\text{It}} for both the muon+jets channel (left) and electron+jets channel (right).",
"The top row shows the respective distribution for the inclusive measurement, while the lower rows show the corresponding distributions for M tt ¯ <450M_{t\\bar{t}} < 450 GeV and M tt ¯ >450M_{t\\bar{t}} > 450 GeV, respectively.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.A straight line fit has been performed as described in Chapter .",
"As can be seen in the respective fits, the slopes differ from one, in particular for $M_{t\\bar{t}} < 450$ .",
"In addition, the respective expected statistical uncertainty on $A_C^{\\text{unf}}$ is shown in Figure REF as a function of $N_{\\text{It}}$ , taking into account the parameters for the slope and offset $a$ and $b$ , respectively, from the straight line fits of the calibration curves.",
"Figure: Expected statistical uncertainties on A C unf A_C^{\\text{unf}} as a function of the regularisation parameter N It N_{\\text{It}} for different injected true asymmetries A C true A_C^{\\text{true}} for both the muon+jets channel (left) and electron+jets channel (right).",
"The top row shows the respective distribution for the inclusive measurement, while the lower rows show the corresponding distributions for M tt ¯ <450M_{t\\bar{t}} < 450 GeV and M tt ¯ >450M_{t\\bar{t}} > 450 GeV, respectively.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.",
"The arrows indicate the chosen values for N It N_{\\text{It}}.For such small changes in $N_{\\text{It}}$ , the expected statistical uncertainty after correction increases only slightly.",
"The number of iterations for the unfolding of the used data set was chosen such that the statistical uncertainty was minimised, while requiring that any injected asymmetry in the tested range was recoverable within the expected statistical uncertainties.",
"This approach, however, led to a remaining bias of the unfolded asymmetry $A_C^{\\text{unf}}$ with respect to the true asymmetry $A_C^{\\text{true}}$ .",
"Consequently, this bias was corrected for by taking into account the slope and offset parameters $a$ and $b$ of the straight line fit, respectively, to obtain a corrected value of the unfolded asymmetry, $A_C^{\\text{unf,corr}}$ , given by $A_C^{\\text{unf,corr}} = \\frac{A_C^{\\text{unf}} - b}{a}.$ Consequently, the corresponding statistical and systematic uncertainties after correction, $\\sigma _{A_C}^{\\text{unf,corr}}$ , can be determined by $\\sigma _{A_C}^{\\text{unf,corr}} = \\frac{\\sigma _{A_C}^{\\text{unf}}}{a}.$ Following the described requirements, regularisation parameters of $N_{\\text{It}} = 6$ for the muon+jets and $N_{\\text{It}} = 4$ in the electron+jets channel were chosen for the inclusive unfolding, while for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ , $N_{\\text{It}} = 7$ and $N_{\\text{It}} = 6$ were chosen for the muon+jets and electron+jets channel, respectively.",
"The corresponding slope and offset parameters extracted from the straight line fit in the calibration can be found in Table REF for completeness.",
"Table: Slopes and offsets from the linear fit in the unfolding calibration.",
"The parameters were obtained for a linear fit of the average unfolded value of A C unf A_C^{\\text{unf}} as a function of the true value A C true A_C^{\\text{true}}, obtained from sets of pseudoexperiments.",
"For the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}}, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.The unfolded integrated asymmetries in $A_C^{\\text{unf}}$ are shown in Table REF for both the inclusive unfolding and for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ , using the chosen regularisation strength in the unfolding process.",
"Table: Unfolded values of the charge asymmetry observable A C unf A_C^{\\text{unf}} for the muon+jets and electron+jets channel.",
"The results for the inclusive measurement and the respective results for the simultaneous unfolding in |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| and M tt ¯ M_{t \\bar{t}} for M tt ¯ <450M_{t \\bar{t}} < 450 GeV and M tt ¯ >450M_{t \\bar{t}} > 450 GeV, taking into account the correction for unfolding bias, are shown.",
"For the simultaneous unfolding, a cut on the event reconstruction likelihood logL\\log {L} was applied to improve the M tt ¯ M_{t\\bar{t}} resolution of the selected events.",
"Furthermore, the respective mc@nlo predictions are shown.A summarised list of all systematics and their contribution to the overall systematic uncertainties can be found in Table REF for the inclusive unfolding and in Table REF for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ .",
"Table: List of all systematic uncertainties taken into account for the unfolding procedure in the measurement of the top charge asymmetry.",
"The numbers in brackets denote the uncertainties before using the larger uncertainty of both channels as conservative estimate.Table: List of all systematic uncertainties taken into account for the unfolding procedure in the measurement of the top charge asymmetry.",
"The numbers in brackets denote the uncertainties before using the larger uncertainty of both channels as conservative estimate.All systematic uncertainties based on a replacement of the unfolding matrix which were of the same order of magnitude or lower than the respective MC statistics uncertainty could not be resolved to full extent due to the inherent fluctuations from limited statistics in the response matrix.",
"Hence, for those cases, the largest of the systematic uncertainties in both channels was used for the final systematic uncertainty on the unfolded charge asymmetry.",
"The resulting combined systematic uncertainties were 0.022 in the muon+jets channel and 0.029 in the electron+jets channel for the inclusive unfolding.",
"For the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ the resulting total systematics were 0.034 ($M_{t\\bar{t}} < 450\\,\\text{GeV}$ ) and 0.021 ($M_{t\\bar{t}} > 450\\,\\text{GeV}$ ) in the muon+jets channel, and 0.046 ($M_{t\\bar{t}} < 450\\,\\text{GeV}$ ) and 0.034 ($M_{t\\bar{t}} > 450\\,\\text{GeV}$ ) in the electron+jets channel, respectively.",
"The Blue method was used to combine the measurements after performing the unfolding for the described alternative approach in the muon+jets and electron+jets channel, taking into account the respective systematic uncertainties and the associated correlations.",
"The same assumptions about correlations as described in Chapter were used and a combined value of $A_C^{\\text{unf}} = -0.035 \\pm 0.021\\,\\text{(stat.)}",
"\\pm 0.021\\,\\text{(syst.",
")}$ for the inclusive unfolding was obtained, where the relative weight of the muon+jets channel result was 65.7 %.",
"The combination of the results for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t \\bar{t}}$ yielded $A_C^{\\text{unf}} (M_{t\\bar{t}} < 450\\,\\text{GeV}) & = & -0.058 \\pm 0.043\\,\\text{(stat.)}",
"\\pm 0.035\\,\\text{(syst.",
")},\\\\A_C^{\\text{unf}} (M_{t\\bar{t}} > 450\\,\\text{GeV}) & = & -0.025 \\pm 0.028\\,\\text{(stat.)}",
"\\pm 0.022\\,\\text{(syst.",
")},$ where the relative weight of the muon+jets channel was 65.6 % and 69.4 %, respectively.",
"As expected, the resulting statistical uncertainties are significantly lower than the ones obtained in the procedure described in Chapter .",
"This reduction is due to the smaller number of iterations used.",
"The obtained central values from the two different approaches were compatible within the respective statistical uncertainties.",
"The systematic uncertainties in the individual channels were of the same order for both approaches for the inclusive unfolding.",
"Larger deviations were observed for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ , where the statistical component in the evaluation of the systematic uncertainties was larger.",
"These statistical fluctuations are expected to be reduced by using a smaller amount of iterations in the unfolding procedure.",
"This effect can be seen in particular for the (dominant) contributions involving the replacement of the response matrix with samples with a smaller number of events.",
"For most of these cases, the obtained systematic uncertainties are significantly lower than the ones obtained in the procedure described in Chapter .",
"Consequently, the unfolding procedure used in the main part of this analysis, despite being more conservative, is expected to be significantly more stable, in particular due to the reduced model dependency and the requirement of a convergent unfolding process.",
"In addition, the conservative systematic uncertainties assumed for the QCD multijet normalisation in the muon+jets channel have been replaced by a combined normalisation and shape systematic uncertainty as described in Chapter REF.",
"All other parameters of the analysis have not been changed with respect to the nominal procedure described in Chapter .",
"This approach yielded uncertainties on the measurement which were significantly lower than for the assumption of a 100 % normalisation uncertainty, as expected.",
"Uncertainties of 0.0007 for the inclusive unfolding, and 0.002 ($M_{t\\bar{t}} < 450\\,\\text{GeV}$ ) and 0.001 ($M_{t\\bar{t}} > 450\\,\\text{GeV}$ ) for the simultaneous unfolding in $|y_t| - |y_{\\bar{t}}|$ and $M_{t\\bar{t}}$ were obtained (compared to 0.0011, 0.018 and 0.004, respectively, as shown in Chapter )."
],
[
"$\\mathbf {b}$ Tag Weighting Control Plots",
"It was verified that the direct application of $b$ tag efficiencies to distributions obtained without the requirement of at least one $b$ tagged jet yields effective distributions comparable to the ones obtained with the full event selection.",
"A comparison was performed for the distributions of $|y_t| - |y_{\\bar{t}}|$ both for the background contributions (c.f.",
"Figure REF) only and for the data distribution after background subtraction (c.f.",
"Figure REF).",
"The distributions are in excellent agreement within the shown statistical uncertainties.",
"Figure: Comparison of the distribution for |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| for the Monte Carlo background contribution for the full event selection (blue) and without bb tag requirement, but with bb tag efficiencies applied directly in a reweighting approach (red, dashed).",
"The bin-by-bin statistical uncertainty is reduced by up to 25 %.Figure: Comparison of the distribution for |y t |-|y t ¯ ||y_t| - |y_{\\bar{t}}| for data after background subtraction for the full event selection (blue) and without bb tag requirement, but with bb tag efficiencies applied directly in a reweighting approach to the subtracted background contribution (red, dashed).tocchapterBibliography Acknowledgements tocchapterAcknowledgements Foremost, I would like to express my sincere gratitude to Prof. Dr. Arnulf Quadt and Dr. Carsten Hensel, who have been my advisers during my whole thesis and the accompanying studies at the 2nd Institute of Physics.",
"I also like to thank Carsten and Kevin for the constant support and for continuously providing constructive advice, and most importantly, for always having an open ear.",
"Sincere thanks are extended to everyone involved in the extensive proofreading process of my thesis draft, Adam, Erik, Kerim, and especially Kevin, who spent a lot of his time (and most likely several red pencils) correcting this document at various draft stages.",
"The many suggestions I received from all of them were a great help.",
"Thanks also go to Frederic, Kerim and Umberto for the extremely productive work we did together on the summer conference note, the paper and the two internal ATLAS notes on the top quark charge asymmetry.",
"In addition, Frederic kindly provided me with his Fortran implementation of the Blue method, for which I would like to thank him in particular.",
"A special thank goes to all of my colleagues and friends at the 2nd Institute of Physics, especially Adam, Elisabeth, Matthias and Philipp for providing diversion from work whenever I was in need.",
"I cannot finish without extending a very personal thank you to Martina for constantly supporting and motivating me, in particular during the last year, with her always finding the right words to cheer me up.",
"Furthermore, it was her who helped keeping away real life problems from me during my time of working on this analysis and writing my thesis.",
"Finally, I would like to thank my parents for providing me with the opportunity to be where I am.",
"Without them, none of this would have been possible.",
"Their support and encouragement during school and my studies have been priceless.",
"Science.",
"It works, bitches.",
"– Randall Munroe, xkcd #54 Five card stud, nothing wild.",
"And the sky is the limit.",
"– Capt.",
"Jean-Luc Picard, U.S.S.",
"Enterprise"
],
[
"Acknowledgements",
"tocchapterAcknowledgements Foremost, I would like to express my sincere gratitude to Prof. Dr. Arnulf Quadt and Dr. Carsten Hensel, who have been my advisers during my whole thesis and the accompanying studies at the 2nd Institute of Physics.",
"I also like to thank Carsten and Kevin for the constant support and for continuously providing constructive advice, and most importantly, for always having an open ear.",
"Sincere thanks are extended to everyone involved in the extensive proofreading process of my thesis draft, Adam, Erik, Kerim, and especially Kevin, who spent a lot of his time (and most likely several red pencils) correcting this document at various draft stages.",
"The many suggestions I received from all of them were a great help.",
"Thanks also go to Frederic, Kerim and Umberto for the extremely productive work we did together on the summer conference note, the paper and the two internal ATLAS notes on the top quark charge asymmetry.",
"In addition, Frederic kindly provided me with his Fortran implementation of the Blue method, for which I would like to thank him in particular.",
"A special thank goes to all of my colleagues and friends at the 2nd Institute of Physics, especially Adam, Elisabeth, Matthias and Philipp for providing diversion from work whenever I was in need.",
"I cannot finish without extending a very personal thank you to Martina for constantly supporting and motivating me, in particular during the last year, with her always finding the right words to cheer me up.",
"Furthermore, it was her who helped keeping away real life problems from me during my time of working on this analysis and writing my thesis.",
"Finally, I would like to thank my parents for providing me with the opportunity to be where I am.",
"Without them, none of this would have been possible.",
"Their support and encouragement during school and my studies have been priceless.",
"Science.",
"It works, bitches.",
"– Randall Munroe, xkcd #54 Five card stud, nothing wild.",
"And the sky is the limit.",
"– Capt.",
"Jean-Luc Picard, U.S.S.",
"Enterprise"
]
] | 1204.0952 |
[
[
"Effects of transition metal substitutions on the incommensurability and\n spin fluctuations in BaFe2As2 by elastic and inelastic neutron scattering"
],
[
"Abstract The spin fluctuation spectra from nonsuperconducting Cu-substituted, and superconducting Co-substituted, BaFe2As2 are compared quantitatively by inelastic neutron scattering measurements and are found to be indis- tinguishable.",
"Whereas diffraction studies show the appearance of incommensurate spin-density wave order in Co and Ni substituted samples, the magnetic phase diagram for Cu substitution does not display incommensu- rate order, demonstrating that simple electron counting based on rigid-band concepts is invalid.",
"These results, supported by theoretical calculations, suggest that substitutional impurity effects in the Fe plane play a signifi- cant role in controlling magnetism and the appearance of superconductivity, with Cu distinguished by enhanced impurity scattering and split-band behavior."
],
[
"Effects of transition metal substitutions on the incommensurability and spin fluctuations in BaFe$_2$ As$_2$ by elastic and inelastic neutron scattering M. G. Kim,$^{1,2}$ J. Lamsal,$^{1,2}$ T. W. Heitmann,$^{3}$ G. S. Tucker,$^{1,2}$ D. K. Pratt,$^{1,2}$ S. N. Khan,$^{1,4}$ Y.",
"B. Lee,$^{1,2}$ A. Alam,$^{1}$ A. Thaler,$^{1,2}$ N. Ni,$^{1,2}$ S. Ran,$^{1,2}$ S. L. Bud'ko,$^{1,2}$ K. J. Marty,$^5$ M. D. Lumsden,$^5$ P. C. Canfield,$^{1,2}$ B. N. Harmon,$^{1,2}$ D. D. Johnson,$^{1,6}$ A. Kreyssig,$^{1,2}$ R. J. McQueeney,$^{1,2}$ A. I. Goldman$^{1,2,}$ [email protected] $^1$ Ames Laboratory, U.S. DOE, Iowa State University, Ames, IA 50011, USA $^2$ Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA $^3$ The Missouri Research Reactor, University of Missouri, Columbia, MO 65211, USA $^4$ Department of Physics, University of Illinois, Urbana, IL 61801, USA $^5$ Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA $^6$ Department of Materials Science & Engineering, Iowa State University, Ames, IA 50011, USA 74.70.Xa, 75.25.-j, 75.30.Fv, 75.30.Kz The spin fluctuation spectra from nonsuperconducting Cu-substituted, and superconducting Co-substituted, BaFe$_2$ As$_2$ are compared quantitatively by inelastic neutron scattering measurements and are found to be indistinguishable.",
"Whereas diffraction studies show the appearance of incommensurate spin-density wave order in Co and Ni substituted samples, the magnetic phase diagram for Cu substitution does not display incommensurate order, demonstrating that simple electron counting based on rigid-band concepts is invalid.",
"These results, supported by theoretical calculations, suggest that substitutional impurity effects in the Fe plane play a significant role in controlling magnetism and the appearance of superconductivity, with Cu distinguished by enhanced impurity scattering and split-band behavior.",
"The role of chemical substitution and its effects on structure, magnetism and superconductivity have become central issues in studies of the iron-pnictide superconductors.",
"[1], [2], [3], [4] This is particularly true for transition-metal ($M$ ) substitution on Fe sites, resulting, nominally, in electron doping of the FeAs layers.",
"When low concentrations of Co,[5], [6] Ni,[7], [8] Rh,[9], [10] Pt[11] and Pd[9], [10] replace Fe, the structural transition temperature ($T_S$ ) and the antiferromagnetic (AFM) transition temperature ($T_N$ ) are both suppressed to lower values and split with $T_S$ $>$ $T_N$ .",
"[5], [6], [12], [7], [9], [13], [14] When the structural and magnetic transitions are suppressed to sufficiently low temperatures, superconductivity emerges below $T_c$ and coexists with antiferromagnetism over some range of concentration.",
"Moreover, for Co, Rh and Ni substitutions in BaFe$_2$ As$_2$ , neutron diffraction measurements manifest a distinct suppression of the magnetic order parameter in the superconducting regime $(T < T_c$ ), which clearly indicates competition between AFM order and superconductivity.",
"[15], [16], [13], [14], [17] Cu substitution in BaFe$_2$ As$_2$ , in contrast, suppresses the magnetic and structural transitions, but does not support superconductivity[8], [2] except, perhaps, below 2 K over a very narrow range in composition.",
"[18] This dichotomy between Co and Ni substitutions and that of Cu is also realized in quaternary fluoroarsenides.",
"[19] However, for Co/Cu co-substitutions in BaFe$_2$ As$_2$ , at a fixed non-superconducting Co concentration, the addition of Cu first promotes and then suppresses $T_c$ .",
"[18] It has been suggested that previously neglected impurity effects play an important role in this behavior.",
"[8], [20] The effects of impurity scattering are also neglected in a simple rigid-band picture for $M$ substitutions, which, at least for Co substitution in BaFe$_2$ As$_2$ , seems to adequately account for the evolution of angle-resolved photoemission spectroscopy (ARPES)[21], Hall effect, and thermoelectric power (TEP) measurements with concentration.",
"[22] The rigid-band model has also been used successfully to model the suppression of the AFM transition temperature and ordered moment in Ba(Fe$_{1-x}$ Co$_x$ )$_2$ As$_2$ for “underdoped\" samples.",
"[17] Nevertheless, this approach now faces strong challenges from recent theoretical and experimental studies.",
"[23], [24], [25] Further comparative studies of Co, Ni and Cu substitutions are needed and may provide clues regarding both the nature of unconventional superconductivity in the iron pnictides and clarify the effects of $M$ substitutions.",
"Because a strong link between superconductivity and spin fluctuations in iron pnictides is generally acknowledged,[1], [3], [4] it is important to establish first whether there are any differences between the spin fluctuation spectra between superconducting (e.g., Co) and nonsuperconducting (e.g., Cu) substituted samples.",
"Here we report on single crystal inelastic scattering measurements of the spin fluctuation spectra of Co and Cu substituted samples, with similar suppressions of the magnetic and structural ordering temperatures relative to the parent BaFe$_2$ As$_2$ compound.",
"We show that there are no quantitative differences in the normal state spin fluctuation spectra.",
"Therefore, we must search elsewhere for evidence of differences between Co/Ni and Cu substitutions in relation to their superconducting properties.",
"To this end, we performed single crystal neutron diffraction measurements of the magnetic ordering in Ba(Fe$_{1-x}M_x$ )$_2$ As$_2$ with $M$ either Ni or Cu.",
"Observations of incommensurate spin-density-wave order, in particular, are a very sensitive probe of the nature of Fermi-surface nesting in the iron pnictides and, therefore, may be used to study impurity effects as a function of the $M$ doping.",
"We find that, like the Co-substituted compound,[26] Ni substitution also manifests incommensurate (IC) AFM order over a narrow range of $x$ at approximately half of the critical concentration for IC order in Co.",
"However, the AFM ordering for Cu substitution remains commensurate (C) up to $x \\approx 0.044$ , where AFM order is absent, demonstrating that a rigid-band view is not appropriate.",
"We propose that the absence of incommensurability and superconductivity for Ba(Fe$_{1-x}$ Cu$_x$ )$_2$ As$_2$ arises from enhanced impurity scattering associated with Cu, consistent with the behavior of $T_c$ with Cu substitution in Ba(Fe$_{1-x-y}$ Co$_x$ Cu$_y$ )$_2$ As$_2$ .",
"Single crystals of Ba(Fe$_{1-x}M_x$ )$_2$ As$_2$ with $M$ = Co, Ni and Cu were grown out of a FeAs self-flux using the high temperature solution growth technique described in Ref. canfield2009,nicu2010.",
"Using wavelength-dispersive spectroscopy, the combined statistical and systematic error on the $M$ composition is not greater than 5%.",
"Inelastic neutron scattering experiments were performed on the HB3 spectrometer at the High-Flux Isotope Reactor at Oak Ridge National Laboratory at a fixed final energy of 14.7 meV.",
"The data here are described in terms of the orthorhombic indexing, $\\textbf {Q}$ = ($\\frac{2\\pi \\emph {H}}{a}$ $\\frac{2\\pi \\emph {K}}{b}$ $\\frac{2\\pi \\emph {L}}{c}$ ), where $a \\ge b \\approx 5.6$ Å and $c \\approx 13$ Å.",
"Samples were aligned in the orthorhombic ($H 0 L$ ) plane and mounted in a closed-cycle refrigerator for low-temperature studies.",
"Diffraction measurements were done on the TRIAX triple-axis spectrometer at the University of Missouri Research Reactor employing an incident neutron energy of $14.7~m$ eV.",
"Samples were studied in the vicinity of $\\textbf {Q}_\\mathrm {AFM}$ = (1 0 3) in the ($\\zeta $ $K$ 3$\\zeta $ ) plane, allowing a search for incommensurability along the b axis ([0 $K$ 0], transverse direction) as found for Ba(Fe$_{1-x}$ Co$_x$ )$_2$ As$_2$ .",
"[26] The inelastic neutron scattering spectra were measured for single-crystals of underdoped Ba(Fe$_{0.953}$ Co$_{0.047}$ )$_2$ As$_2$ and Ba(Fe$_{0.972}$ Cu$_{0.028}$ )$_2$ As$_2$ .",
"These two samples were chosen because they have similar tetragonal-orthorhombic transition temperatures [$T_S$ (Co) = 63 K, $T_S$ (Cu) = 73 K] and AFM transition temperatures [$T_N$ (Co) = 47 K, $T_N$ (Cu) = 64 K], which are comparably suppressed relative to the parent BaFe$_2$ As$_2$ compound [$T_N~\\approx ~T_S = 140$ K].",
"Bulk transport measurements show a superconducting transition for the Co substituted sample at 17 K, whereas no superconducting transition is observed in the Cu substituted sample down to 2 K. The Co(Cu) sample consisted of 9(2) co-aligned crystals weighing a total of 1.88(1.52) grams and a total mosaic width of 1.5$^\\circ $ (0.6$^\\circ $ ) full-width-at-half-maximum (FWHM).",
"Figure: (Color online) Inelastic neutron scattering from the Co and Cu substituted samples.",
"(a) Normalized dynamic magnetic susceptibility at 25 K determined from constant-Q E-scans at the (1 0 1) magnetic Bragg point.",
"(b)-(i) Q-scans at several fixed values of the energy loss.",
"The feature at (1.3 0 1) results from spurious scattering not related to spin excitations.Figure REF compares the inelastic magnetic scattering from the Co and Cu substituted samples measured at $T$ = 25 K. The data are plotted in terms of the dynamic magnetic susceptibility, $\\chi ^{\\prime \\prime }(\\textbf {Q},\\omega ) = [I(\\textbf {Q},\\omega )-C(\\textbf {Q},\\omega )](1-e^{-\\hbar \\omega /kT})$ , where $I(Q,\\omega )$ is the raw neutron intensity and $C(\\textbf {Q},\\omega )$ is the nonmagnetic background determined from averaged inelastic scattering at positions well away from the magnetic signal [e.g.",
"Q = (0.79 0 1.72) and (0.72 0 1.88)].",
"The data for these samples were normalized to each other using measurements of several transverse phonon peaks, and this was found to be consistent with the ratio of the masses of the two samples.",
"The constant-$\\textbf {Q}$ energy scan measured at Q$_\\mathrm {AFM}$ = (1 0 1) [Fig.",
"REF (a)] as well as the constant-$E$ $\\textbf {Q}$ -scans along the [1 0 0] and [0 0 1] directions [Figs.",
"REF (b)-(i)] show that the normal state (above $T_c$ ) dynamic susceptibility for Co and Cu substituted samples are indistinguishable.",
"Below $T_c$ , the spectrum of the Co substituted sample manifests a magnetic resonance feature above 4 meV (not shown here) in the superconducting state as observed previously by many groups.",
"[1], [27] However, the dynamic susceptibility for the nonsuperconducting Cu substituted sample is temperature independent down to 5 K. The close similarity of the normal state susceptibility for single substitutions of Co and Cu show that we must look beyond the spin fluctuation spectra to understand the absence of superconductivity in Ba(Fe$_{1-x}$ Cu$_x$ )$_2$ As$_2$ , motivating a closer look at the effects of $M$ substitutions upon magnetism in BaFe$_2$ As$_2$ .",
"Figure: (Color online) Scattering near the (1 0 3) magnetic Bragg point for Ba(Fe 1-x M x _{1-x}M_x) 2 _2As 2 _2 where MM is (a) Ni and (b) Cu.",
"(c) Temperature dependence of the scattering near the (1 0 3) magnetic Bragg point for Ba(Fe 0.963 _{0.963}Ni 0.037 _{0.037}) 2 _2As 2 _2.",
"Intensities are normalized by mass of the samples to facilitate comparisons.",
"Lines are fits to the data, as described in the text.We have shown previously that IC-AFM order is found in Ba(Fe$_{1-x}$ Co$_x$ )$_2$ As$_2$ for $x~\\ge ~0.056$ , providing a measure of the effect of Co substitution on the Fermi surface.",
"Co substitution detunes the electron- and hole-like Fermi surfaces[21] and eventually results in a mismatch that favors IC-AFM order.",
"This suggests that Fermi surface nesting is a crucial factor in stabilizing both C and IC phases in the magnetic phase diagram of the $A$ Fe$_2$ As$_2$ ($A$ = Ba, Sr, Ca) compounds.",
"[26] Figures REF (a) and (b) show the low-$T$ scattering for transverse (0 $K$ 0) scans through the (1 0 3) magnetic Bragg point for several Ni and Cu compositions.",
"For Ba(Fe$_{1-x}$ Ni$_x$ )$_2$ As$_2$ , a transition from a C-AFM order for $x < 0.035$ (with resolution limited magnetic Bragg peaks) to IC-AFM order for $x \\ge 0.035$ is clearly demonstrated by the symmetric pair of peaks at (1 $\\pm \\epsilon $ 3).",
"For $x > 0.037$ , no long-range AFM order was observed.",
"The lines in Fig.",
"REF (a) are fits to the data using a single Gaussian for $x = 0.029$ , a convoluted Gaussian + Lorentzian line shape for $x = 0.031$ , three Gaussians for $x = 0.035$ (to account for the presence of the dominant IC and residual C components), and two Gaussians for $x = 0.037$ .",
"The detailed description of the IC structure based on these fits is very similar for Co and Ni substitution.",
"The incommensurability, $\\epsilon $ , derived from fits to these data was $0.033\\pm 0.003$ reciprocal lattice units (r.l.u.",
"), close to the value found for $\\epsilon $ for Co samples.",
"[26] These data show that, as previously observed for Co substitution, Ni substitution results in an abrupt change from C to IC AFM order at $x_c = 0.035 \\pm 0.002$ .",
"The ratio ($\\approx 0.6$ ) of this critical concentration to $x_c = 0.056$ for Co[26], is consistent with Ni “donating” roughly twice the number of electrons as Co. As discussed previously for Co substitutions, the abrupt transition between C and IC magnetic structures is similar to what has been observed for dilute substitutions of Mn or Ru in the canonical spin-density-wave (SDW) system, Cr.",
"[26] Detailed theoretical studies of the nesting and free energy of the competing C and IC-SDW states in BaFe$_2$ As$_2$ may shed further light on this behavior.",
"There is a significant broadening of the IC magnetic diffraction peaks as compared to the C magnetic peaks indicating a much reduced magnetic correlation length ($\\xi \\sim $ 60 Å), again consistent with the broadening found for the Co substituted samples.",
"[26] The peak widths obtained from these fits are given in Fig.",
"REF (a) and show that the C component remains resolution limited, whereas the IC peaks are more than 5 times broader.",
"Recent measurements on Ni-substituted samples by Luo et al.",
"[28] are consistent with our results.",
"The temperature dependence of the transverse (0 $K$ 0) scans through the magnetic scattering for superconducting Ba(Fe$_{0.963}$ Ni$_{0.037}$ )$_2$ As$_2$ is illustrated in Fig.",
"REF (c).",
"The integrated intensity of the magnetic scattering increases below $T_N$ , reaches a maximum at the superconducting transition temperature ($T_c$ ), and decreases monotonically below $T_c$ as observed previously for Co substituted samples,[15], [16], [17], [26] demonstrating, again, that magnetic order competes with superconductivity.",
"The positions and widths of the IC magnetic peaks appear to be temperature independent within the resolution of our measurement.",
"Figure: (Color online) Trends in the FWHM and maximum ordered moment for MM substitution.",
"(a) Evolution of the FWHM of the magnetic peaks vs. concentration.",
"The solid(open) circles represent the FWHM of the C-AFM(IC-AFM) peaks.",
"(b) Measured ordered moment derived from the integrated intensity of the magnetic Bragg peaks as a function of the extra electron count, assuming that Co donates 1, Ni 2, and Cu 3, extra-electrons to the dd-band.",
"The data for Ba(Fe 1-x _{1-x}Co x _x) 2 _2As 2 _2 are taken from references Pratt2011,fernandesunconventional2010.In striking contrast to the data for Co samples[26] and here for Ni, Figure REF (b) shows no evidence of a C-to-IC transition versus $x$ for Ba(Fe$_{1-x}$ Cu$_x$ )$_2$ As$_2$ .",
"Instead, the C magnetic Bragg peak is well described by a single Lorentzian lineshape that broadens strongly for $x \\ge 0.039$ [see Figs.",
"REF (b) and REF (a)], and no AFM long-range order is found for $x \\ge 0.044$ .",
"To further emphasize the differences between Co, Ni and Cu substitutions, Fig.",
"REF (b) displays the maximum ordered magnetic moment (at $T_c$ for Co and Ni substitution and at our base temperature, 5 K for Cu substitution) as a function of extra electron count under the oft-used assumption that Co, Ni, and Cu donate 1, 2, and 3, respectively, to the $d$ -bands.",
"The maximum ordered moment was estimated from the integrated intensity of the magnetic Bragg peaks using the C magnetic structure factor normalized by the mass of the samples, as described previously.",
"[17] Under the stated assumption, Co and Ni act similarly to suppress the moment over a range of $x$ that mimics a rigid-band picture.",
"This is clearly not the case for Cu substitution (although rescaling the electron count by an additional factor of two would move the results on top of Co and Ni).",
"Nevertheless, the IC magnetic order found for Ni and Co substitutions in this regime is not found for Cu substitution.",
"Figure: (Color online) For Ba(Fe 1-x M x )_{1-x}M_x) 2 _2As 2 _2, the KKR-CPA (a) site-projected DOS versus E-E F E - E_F at 6% Co, 3% Ni (fixed e - /Fee^{-}/Fe), and 2% Cu (the Fe DOS changes negligibly with MM); and Bloch spectral functions, A(𝐤;E F )A({\\bf k};E_{F}), along specific 𝐤{\\bf k}-directions versus at.",
"% MM for (b) electrons, and (c) holes.",
"Insets: 𝐤{\\bf k}-direction of the cut across electron (centered at XX) and hole (centered at ZZ) states.Peak locations of electron/hole states are compared to the \"rigid-band\" expectations (vertical dashed lines) from parent compound at fixed e - /Fee^-/Fe and three at.",
"% Cu values.To further elucidate the differences between Co/Ni and Cu substitution in BaFe$_2$ As$_2$ we employed the Korringa-Kohn-Rostoker method using the Coherent-Potential Approximation (KKR-CPA) to address the effects of substitution on the density of states (DOS), and solute disorder (impurity) scattering on the Fermi surfaces [i.e., the Bloch spectral functions $A({\\bf k};E_{F})$ at the Fermi energy $E_F$ ].",
"[29], [30], [31] First, Figure REF (a) shows that the $d$ -band partial DOS of Co and Ni are common-band-like (e.g.",
"overlap with the Fe $d$ -bands), whereas Cu exhibits split-band behavior with its $d$ -states located $\\sim $ 4 eV below $E_F$ .",
"Only $s$ -$p$ states participate at $E_F$ and, therefore, Cu behaves almost as a +1 $s$ -$p$ valence with very different scattering behavior from Co and Ni.",
"We note that these results are consistent with ordered DFT calculations at large $x$ .",
"[23] For nesting-driven ordering,[32], [33], [34], [35] the convolution between the electron- and hole-like Fermi surfaces dictate the location of peaks in the susceptibility.",
"[32] Figures REF (b) and (c) illustrate the behavior of the Fermi-surfaces for electrons and holes at a fixed solute concentrations for Co (6%) and Ni (3%) [red and green lines] compared to the \"rigid-band\" expectation from the parent compound at a fixed $e^{-}/Fe$ (0.06).",
"These solute concentrations are close to the respective $x_c$ for the observed C to IC magnetic ordering, and a rigid-band treatment [vertical dashed lines in Figs.",
"REF (b) and (c)] provides an estimate for $\\varepsilon $ of $\\sim 0.021$ , similar to that observed in our measurements.",
"As solute content increases, the electron(hole) surfaces expand(contract) and the spectral broadening due to chemical disorder is evident.",
"Due to common $d$ -band behavior for Co and Ni, spectral peaks for the electrons clearly mimic rigid-band expectations at fixed $e^{-}/Fe$ , but the holes less so.",
"In contrast, with a split Cu $d$ -band, rigid-band concepts are rendered invalid.",
"States well below $E_F$ (due to hybridization and band-filling) and at $E_F$ contribute to the total susceptibility.",
"[36], [37], [34], [38] As a stronger scatterer than Fe, Co, or Ni, $\\sim $ 1% Cu (rather than 2% Cu assuming a +3 valence) acts like 6% Co or 3% Ni in terms of broadening of the spectral features.",
"Most importantly, the hole states are especially sensitive to the Cu content, with a rapid loss of intensity and increased disorder broadening evident, as shown for up to 4% Cu for comparison with our experiments.",
"The convolution of the electron- and hole-like Fermi surfaces is dramatically diminished and, therefore, so is the impetus for incommensurability.",
"[34] We propose that the absence of IC-AFM order in Ba(Fe$_{1-x}$ Cu$_x$ )$_2$ As$_2$ arises from enhanced impurity scattering effects associated with the stronger potential for Cu.",
"The small incommensurability measured for Co and Ni substituted BaFe$_2$ As$_2$ requires relatively sharp and well-defined features in the Fermi surface topology.",
"Disorder due to impurity scattering introduces spectral broadening in both energy and momentum to the extent that the magnetic structure remains C rather than IC for Cu.",
"This is in substantial agreement with recent work by Berlijn et al.,[39] for Zn substitutions in BaFe$_2$ As$_2$ .",
"Finally, we note that such impurity effects are expected to impact superconductivity in the iron pnictides as well.",
"Essential elements of the under-doped regions of the phase diagram for electron-doped BaFe$_2$ As$_2$ are captured by considering both inter- and intra-band impurity scattering.",
"[40], [20] Although impurity scattering introduced by $M$ substitution causes pair breaking and suppresses $T_c$ , it can be even more damaging for spin-density wave ordering so that $T_N$ is suppressed more rapidly, allowing superconductivity to emerge at finite substitution levels.",
"Interestingly, the phenomenological model by Fernandes et al.",
"[20] indicates that the behavior of $T_c$ for s$^{\\pm }$ pairing is a non-monotonic function of impurity concentration, depending on the strength of the impurity potential and the ratio of the intra-band ($\\Gamma _0$ ) to inter-band ($\\Gamma _\\pi $ ) impurity scattering, which may vary strongly between Co and Cu.",
"Indeed, they find a range in $\\frac{\\Gamma _0}{\\Gamma _\\pi }$ where $T_c$ first increases and then decreases with impurity concentration, very similar to that observed for Co/Cu co-substitutions in BaFe$_2$ As$_2$ .",
"This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (OBES), Division of Materials Sciences and Engineering.",
"Work at the High Flux Isotope Reactor, Oak Ridge National Laboratory, was sponsored by the Scientific User Facilities Division, DOE/OBES.",
"SNK and DDJ acknowledge partial support from ORNL's Center for Defect Physics, Energy Frontier Research Center."
]
] | 1204.1538 |
[
[
"A hyperbolic metric and stability conditions on K3 surfaces with \\rho=1"
],
[
"Abstract In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\\rho (X) =1$.",
"And we show that all walls are geodesic in the normalized space with respect to the hyperbolic metric.",
"Furthermore we demonstrate how the hyperbolic metric is helpful for us by discussing mainly three topics.",
"We first make a study of so called Bridgeland's conjecture.",
"In the second topic we prove a famous Orlov's theorem without the global Torelli theorem.",
"In the third topic we give an explicit example of stable complexes in large volume limits by using the hyperbolic metric.",
"Though Bridgeland's conjecture may be well-known for algebraic geometers, we would like to start from the review of it."
],
[
"Introduction",
"In this article we introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank $\\rho (X) =1$ .",
"And we show that all walls are geodesic in the normalized space with respect to the hyperbolic metric.",
"Furthermore we demonstrate how the hyperbolic metric is helpful for us by discussing mainly three topics.",
"We first make a study of so called Bridgeland's conjecture.",
"In the second topic we prove a famous Orlov's theorem without the global Torelli theorem.",
"In the third topic we give an explicit example of stable complexes in large volume limits by using the hyperbolic metric.",
"Though Bridgeland's conjecture may be well-known for algebraic geometers, we would like to start from the review of it."
],
[
"Bridgeland's conjecture",
"In [4] Bridgeland introduced the notion of stability conditions on arbitrary triangulated categories ${\\mathcal {D}}$ .",
"By virtue of this we could define the notion of “$\\sigma $ -stability” for objects $E \\in {\\mathcal {D}}$ with respect to a stability condition $\\sigma $ on ${\\mathcal {D}}$ .",
"Bridgeland also showed that each connected component of the space $\\mathop {\\mathrm {Stab}}\\nolimits ({\\mathcal {D}})$ consisting of stability conditions on ${\\mathcal {D}}$ is a complex manifold unless $\\mathop {\\mathrm {Stab}}\\nolimits ({\\mathcal {D}})$ is empty.",
"Hence the non-emptiness of $\\mathop {\\mathrm {Stab}}\\nolimits ({\\mathcal {D}})$ is one of the biggest problem.",
"Many researchers study this problem in various situations.",
"For instance suppose ${\\mathcal {D}}$ is the bounded derived category $D(M)$ of coherent sheaves on a projective manifold $M$ .",
"In the case of $\\mathop {\\mathrm {dim}}\\nolimits M=1$ , the non-emptiness of $\\mathop {\\mathrm {Stab}}\\nolimits (D(M))$ was proven in the original article [4].",
"Furthermore the space $\\mathop {\\mathrm {Stab}}\\nolimits (D(M))$ was studied in detail by [17] (the genus is 0), [4] (the genus is 1) and [15] (the genus is greater than 1).",
"In the case of $\\mathop {\\mathrm {dim}}\\nolimits M=2$ , the non-emptiness was proven by [5] (K3 or abelian surfaces) and [1] (other surfaces).",
"In the case of $\\mathop {\\mathrm {dim}}\\nolimits M=3$ it is discussed by [2].",
"These are just a handful of many studies.",
"As we stated before, the space $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ of stability conditions on the derived category $D(X)$ of a projective K3 surface $X$ is not empty by [5].",
"This fact is proven by finding a distinguished connected component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"For $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ Bridgeland conjectured the following: Conjecture 1.1 (Bridgeland) The space $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ is connected, that is, $\\mathop {\\mathrm {Stab}}\\nolimits (X)= \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Furthermore the distinguished component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected.",
"As was proven by [5] and [10], if the conjecture holds then we can determine the group structure of $\\mathop {\\mathrm {Aut}}\\nolimits (D(X))$ as follows: We have the covering map $\\pi \\colon \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X)$ by [5] (See also Theorem REF ).",
"Here ${\\mathcal {P}}^+_0(X)$ is a subset of $H^*(X, {\\mathbb {C}})$ (See also Section REF ).",
"By virtue of [5] and [10], if Conjecture REF holds we have the exact sequence of groups: $1 \\rightarrow \\pi _1 ({\\mathcal {P}}^+_0(X)) \\rightarrow \\mathop {\\mathrm {Aut}}\\nolimits (D(X)) \\stackrel{\\kappa }{\\rightarrow } O^{+}_{{\\mathrm {Hodge}}} (H^*(X, {\\mathbb {Z}})) \\rightarrow 1, $ where $O^{+}_{{\\mathrm {Hodge}}} (H^*(X, {\\mathbb {Z}}))$ is the Hodge isometry group of $H^*(X, {\\mathbb {Z}})$ preserving the orientation of $H^*(X,{\\mathbb {Z}})$ .",
"Hence Conjecture REF predicts that the kernel $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa )$ of the representation $\\kappa $ is given by the fundamental group $\\pi _1 ({\\mathcal {P}}^+_0(X))$ and that $\\mathop {\\mathrm {Aut}}\\nolimits (D(X))$ is given by an extension of $\\pi _1({\\mathcal {P}}^+_0(X))$ and $O^+_{{\\mathrm {Hodge}}}(H^*(X, {\\mathbb {Z}}))$ ."
],
[
"First theorem",
"Recall the right $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}) $ -action on $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ where $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ is the universal cover of ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ .",
"We define $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ by the quotient of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ by the right $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ action.",
"We call it a normalized stability manifold.",
"For a projective K3 surface with $\\rho (X) =1$ , we first introduce a hyperbolic metric on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"We also show that the hyperbolic metric is independent of the choice of Fourier-Mukai partners of $X :$ Theorem 1.2 (=Theorem REF ) Assume that $\\rho (X) =1$ .",
"(1) $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is a hyperbolic 2 dimensional manifold.",
"(2) Let $Y$ be a Fourier-Mukai partner of $X$ and $\\Phi \\colon D(Y) \\rightarrow D(X)$ an equivalence which preserves the distinguished component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Then the induced morphism $\\Phi _*^{\\mathrm {n}} \\colon \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(Y) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is an isometry with respect to the hyperbolic metric.",
"Clearly if $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ is connected it is unnecessary to assume that $\\Phi $ preserves the distinguished component.",
"We remark that there is another study by Woolf which focuses on the metric on $\\mathop {\\mathrm {Stab}}\\nolimits ({\\mathcal {D}})$ (not normalized!).",
"In [20], he showed that $\\mathop {\\mathrm {Stab}}\\nolimits ({\\mathcal {D}})$ is complete with respect to the original metric introduced by Bridgeland.",
"Our study is the first work which focuses on a different structure from Bridgeland's original framework."
],
[
"Second theorem",
"Next, by using the hyperbolic structure, we observe the simply connectedness of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ : Theorem 1.3 (=Theorem REF ) Let $X$ be a projective K3 surface with $\\rho (X)=1$ .",
"The following three conditions are equivalent.",
"(1) $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected.",
"(2) $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is isomorphic to the upper half plane ${\\mathbb {H}}$ .",
"(3) Let $W(X)$ be the subgroup of $\\mathop {\\mathrm {Aut}}\\nolimits (D(X))$ generated by two times compositions of the spherical twist $T_A$ by spherical locally free sheaves $A$ .",
"$W(X)$ is isomorphic to the free group generated by $T_A^2 $ : $W(X)= \\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{A} ({\\mathbb {Z}}\\cdot T_A^2) ,$ where $A$ runs through all spherical locally free sheaves and $\\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }$ is the free product.",
"We give two remarks on Theorem REF .",
"Firstly we could not prove the simply connectedness.",
"However by using the hyperbolic structure on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ , we can deduce the global geometry not only of $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ but also of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ as follows.",
"Since $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is a $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -bundle on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ , and we see $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected if and only if it is a $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -bundle over the upper half plane ${\\mathbb {H}}$ .",
"Secondly, if Conjecture REF holds then we see the kernel $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa )$ is generated by $W(X)$ and the double shift $[2]$ .",
"Since the double shift $[2]$ commutes with any equivalence, the freeness of $W(X)$ implies $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa ) / {\\mathbb {Z}}[2]$ is free.",
"However in higher Picard rank cases, it is thought that the generators of $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa )/ {\\mathbb {Z}} [2]$ have relations (See also Remark REF ).",
"Hence the freeness of $W(X)$ is a special phenomena."
],
[
"Third theorem",
"In the third theorem, we study chamber structures on $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ in terms of the hyperbolic structure on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Before we state the third theorem, let us recall chamber structures.",
"For a set ${\\mathcal {S}} \\subset D(X)$ of objects which has bounded mass and an arbitrary compact subset $B \\subset \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ , we can define a finite collection of real codimension 1 submanifolds $\\lbrace W_{\\gamma } \\rbrace _{\\gamma \\in \\Gamma }$ satisfying the following property: Let $C \\subset B \\setminus \\bigcup _{\\gamma \\in \\Gamma }W_{\\gamma }$ be an arbitrary connected component.",
"If $E \\in {\\mathcal {S}}$ is $\\sigma $ -semistable for some $\\sigma \\in C$ then $E$ is $\\tau $ -semistable for all $\\tau \\in C$ .",
"Each $W_{\\gamma }$ is said to be a wall and each connected component $C$ is said to be a chamber.",
"In this paper we call all data of chambers and walls a chamber structure.",
"We have to remark that chamber structures on $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ descend to the normalized stability manifold $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Namely $C/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ and $\\lbrace W_{\\gamma }/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}) \\rbrace $ also define a chamber structure on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Our third theorem is the following: Theorem 1.4 (=Theorem REF ) All walls of chamber structures of $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ are geodesic."
],
[
"Revisit of Orlov's theorem",
"Generally speaking Fourier-Mukai transformations on $X$ may change chamber structures (This does not mean Fourier-Mukai transformations just permute chambers).",
"By Theorems REF and REF , we see that the image of walls by Fourier-Mukai transformations is also geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Applying this observation we show the following: Proposition 1.5 (=Proposition REF ) Let $X$ be a projective K3 surface with $\\rho (X)=1$ and $Y$ a Fourier-Mukai partner of $X$ with an equivalence $\\Phi \\colon D(Y) \\rightarrow D(X)$ .",
"If the induced morphism $\\Phi _* \\colon \\mathop {\\mathrm {Stab}}\\nolimits (Y) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits (X)$ preserves the distinguished component, then $Y$ is isomorphic to the fine moduli space of Gieseker stable torsion free sheaves.",
"We have to mention that a more stronger statement was already proven by Orlov in [18]; Any Fourier-Mukai partner of projective K3 surfaces is isomorphic to the fine moduli space of Gieseker stable sheaves.",
"Our proof never needs the global Torelli theorem which was essential for Orlov's proof.",
"Hence our proof gives a new feature of stability condition; The theory of stability conditions substitutes for the global Torelli theorem.",
"Since the strategy of Proposition REF is technical, we will explain it in §REF ."
],
[
"Stable complexes in the large volume limit",
"We also discuss the stability of complexes in large volume limits by using Lemma REF which is crucial for Theorem REF .",
"More precisely in Corollary REF we prove that the complexes $T_A({\\mathcal {O}}_x)$ are stable in the large volume limit where $T_A({\\mathcal {O}}_x)$ is a spherical twist of ${\\mathcal {O}}_x$ by a spherical locally free sheaf.",
"Originally it was expected that the $\\sigma $ -stability in the large volume limit is equivalent to Gieseker twisted stability (See also [5]).",
"However the possibility of stable complexes in the large volume limit is referred in [3].",
"We give an answer to this problem."
],
[
"Contents",
"In Section we prepare some basic terminologies.",
"In Section we prove the first main theorem.",
"In Section we prove the second main theorem.",
"The third theorem will be proven in Section .",
"The analysis of $\\partial U(X)$ , which is necessary for Theorem REF , will be also done in Section .",
"In Section we revisit Orlov's theorem.",
"In Section we discuss the stability of $T_A^{-1}({\\mathcal {O}}_x)$ in the large volume limit."
],
[
"Preliminaries",
"In this section we prepare basic notations and lemmas.",
"Let $(X,L)$ be a pari of a projective K3 surface with ${\\mathrm {NS}}(X) = {\\mathbb {Z}} L$ .",
"Almost all notions are defined for general projective K3 surfaces.",
"To simplify the explanations we focus on K3 surfaces with $\\rho (X) =1$ ."
],
[
"Terminologies",
"The abelian category of coherent sheaves on $X$ is denoted by ${\\mathrm {Coh}}(X)$ .",
"Note that the numerical Grothendieck group ${\\mathcal {N}}(X)$ is isomorphic to $H^0(X, {\\mathbb {Z}}) \\mathop {\\oplus }\\nolimits {\\mathrm {NS}}(X) \\mathop {\\oplus }\\nolimits H^4(X, {\\mathbb {Z}}).$ We put $v(E) = ch (E) \\sqrt{td_X}$ for $E \\in D(X)$ .",
"Then we see $v(E) = r_E \\mathop {\\oplus }\\nolimits c_E \\mathop {\\oplus }\\nolimits s_E \\in {\\mathcal {N}}(X).$ One can easily check that $r_E= \\mathop {\\mathrm {rank}}\\nolimits E$ , $c_E $ is the first Chern class $c_1(E)$ and $s_E = \\chi (X, E)- \\mathop {\\mathrm {rank}}\\nolimits E$ .",
"Hence for a vector $v = r\\mathop {\\oplus }\\nolimits c\\mathop {\\oplus }\\nolimits s \\in {\\mathcal {N}}(X)$ , the component $r$ is called the rank of $v$ .",
"The Mukai pairing ${\\langle }, {\\rangle }$ on $H^*(X, {\\mathbb {Z}}) $ is given by ${\\langle } r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s , r^{\\prime }\\mathop {\\oplus }\\nolimits c^{\\prime }\\mathop {\\oplus }\\nolimits s^{\\prime } {\\rangle } = c c^{\\prime } -r s^{\\prime } - r^{\\prime }s.$ By Riemann-Roch theorem we see $\\chi (E,F) = \\sum _{i} (-1)^i\\mathop {\\mathrm {dim}}\\nolimits \\mathop {\\mathrm {Hom}}\\nolimits _{D(X)}^i(E,F) = - {\\langle } v(E), v(F) {\\rangle }.$ An object $A \\in D(X)$ is said to be spherical if $A$ staisfies $\\mathop {\\mathrm {Hom}}\\nolimits _{D(X)}^i(A,A) = {\\left\\lbrace \\begin{array}{ll} {\\mathbb {C}} & (i=0,2) \\\\ 0 & (\\text{otherwise}).",
"\\end{array}\\right.",
"}$ We note that $v(A)^2=-2$ if $A$ is spherical.",
"By the effort of [19], for a spherical object $A$ we could define the autoequivalence $T_A$ called a spherical twist (See also [7]).",
"By the definition of $T_A$ we have the following distinguished triangle for $E \\in D(X)$ : ${\\begin{matrix}\\mathop {\\mathrm {Hom}}\\nolimits _{D(X)}^*(A, E) \\otimes A &\\xrightarrow{}& E &\\xrightarrow{}& T_A(E) ,\\end{matrix}}$ where ${\\mathrm {ev}}$ is the evaluation map.",
"We call the above triangle a spherical triangle.",
"We note that the vector of $T_A(E)$ can be calculated as follows $v(T_A(E)) = v(E) + {\\langle } v(E), v(A) {\\rangle } v(A).$ Let $\\Delta (X)$ be the set of $(-2)$ -vectors: $\\Delta (X) =\\lbrace \\delta \\in {\\mathcal {N}}(X) | \\delta ^2 =-2 \\rbrace $ and let $\\Delta ^+(X) $ be the set $\\lbrace \\delta \\in \\Delta (X) |\\delta = r \\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s, r>0 \\rbrace $ .",
"Following [5], we put ${\\mathcal {P}}(X) = \\lbrace v \\in {\\mathcal {N}}(X)\\otimes {\\mathbb {C}} | {\\mathfrak {Re}}(v) \\mbox{ and }{\\mathfrak {Im}}(v) \\mbox{ span a positive 2 plane} \\rbrace $ Since ${\\mathcal {P}}(X)$ has two connected components, we define ${\\mathcal {P}}^+(X)$ by the connected component containing $\\exp (\\sqrt{-1}\\omega )$ where $\\omega $ is an ample class.",
"Then ${\\mathcal {P}}^+(X)$ has the right ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ action as the change of basis of the planes.",
"This action is free.",
"Hence there exists the quotient ${\\mathcal {P}}^+(X) \\rightarrow {\\mathcal {P}}^+(X) /{\\mathrm {GL}}^+(2, {\\mathbb {R}})$ which gives a principle ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ -bundle with a global section.",
"Under the assumption $\\rho (X)=1$ , ${\\mathcal {P}}^+(X)/{\\mathrm {GL}}^+(2, {\\mathbb {R}})$ is isomorphic to the set ${\\mathfrak {H}}(X)$ where ${\\mathfrak {H}}(X) = \\lbrace (\\beta ,\\omega )=(xL, yL) | x+\\sqrt{-1} \\in {\\mathbb {H}}\\rbrace .$ Clearly ${\\mathfrak {H}}(X)$ is canonically isomorphic to ${\\mathbb {H}}$ .",
"Then the global section ${\\mathfrak {H}}(X) \\rightarrow {\\mathcal {P}}^+(X)$ is given by ${\\mathfrak {H}}(X) \\ni (x,y) \\mapsto \\exp (\\beta + \\sqrt{-1}\\omega ) \\in {\\mathcal {P}}^+(X).$ In particular ${\\mathcal {P}}^+(X)$ is isomorphic to ${\\mathbb {H}} \\times GL^+(2, {\\mathbb {R}})$ .",
"We put ${\\mathcal {P}}^+_0(X)$ by ${\\mathcal {P}}^+_0(X) = {\\mathcal {P}}^+(X) \\setminus \\bigcup _{\\delta \\in \\Delta (X) }{\\langle } \\delta {\\rangle }^{\\perp }$ where ${\\langle } \\delta {\\rangle }^{\\perp }$ is the orthogonal complement of $\\delta $ with respect to the Mukai pairing on $H^*(X, {\\mathbb {Z}})$We remark that the definition of ${\\mathcal {P}}^+_0(X)$ is independent of the assumption $\\rho (X)=1$ ..",
"Define ${\\mathfrak {H}}_0(X) = \\lbrace v \\in {\\mathfrak {H}} (X) | {\\langle } \\exp (v), \\delta {\\rangle } \\ne 0\\ (\\forall \\delta \\in \\Delta (X)) \\rbrace .$ Then we see ${\\mathcal {P}}^+_0(X)$ is isomorphic to ${\\mathfrak {H}}_0(X) \\times {\\mathrm {GL}}^+(2, {\\mathbb {R}})$ ."
],
[
"Stability conditions on K3 surfaces",
"Let $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ be the set of numerical locally finite stability conditions on $D(X)$ .",
"We put $\\sigma = ({\\mathcal {A}}, Z) \\in \\mathop {\\mathrm {Stab}}\\nolimits (X)$ where ${\\mathcal {A}}$ is the heart of a bounded t-structure on ${\\mathcal {D}}$ and $Z$ is a central charge.",
"Since the Mukai paring is non-degenerate on ${\\mathcal {N}}(X)$ we have the natural map: $\\pi \\colon \\mathop {\\mathrm {Stab}}\\nolimits (X) \\rightarrow {\\mathcal {N}}(X)\\otimes {\\mathbb {C}},\\ \\pi (\\sigma )= Z^{\\vee }$ where $Z(E) = {\\langle } Z^{\\vee }, v(E){\\rangle }$ .",
"In $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ , there is a connected component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ which contains the set $U(X):$ $U(X) =\\lbrace \\sigma = ({\\mathcal {A}}, Z) \\in \\mathop {\\mathrm {Stab}}\\nolimits (X) |Z^{\\vee } \\in {\\mathcal {P}}(X)\\setminus \\bigcup _{\\delta \\in \\Delta (X) }{\\langle } \\delta {\\rangle }^{\\perp }, \\\\ {\\mathcal {O}}_x \\mbox{ is $\\sigma $-stable in the same phase for all }x \\in X \\rbrace .$ Let $\\bar{U}(X)$ be the closure of $U(X)$ in $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ .",
"Then we see that $\\bar{U}(X)$ be the set of stability conditions $\\sigma $ such that ${\\mathcal {O}}_x$ ($\\forall x\\in X$ ) is $\\sigma $ -semistable in the same phase with $Z^{\\vee } \\in {\\mathcal {P}}(X) \\setminus \\bigcup _{\\delta \\in \\Delta (X) }{\\langle } \\delta {\\rangle }^{\\perp }$ .",
"Define $\\partial U(X)$ by $\\bar{U}(X) \\setminus U(X)$ and call it the boundary of $U(X)$.",
"We define the set $V(X)$ by $V(X) = \\lbrace \\sigma = ({\\mathcal {A}}, Z) \\in U(X) | Z({\\mathcal {O}}_x)=-1, \\ {\\mathcal {O}}_x\\mbox{ is $\\sigma $-stable with phase $1$} \\rbrace .$ One can see $U(X) = V(X) \\cdot \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}) \\cong V(X) \\times \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ by [5].",
"Furthermore the set $V(X)$ is parametrized by $(\\beta , \\omega ) \\in {\\mathfrak {H}}(X)$ in the following way: For the pair $(\\beta , \\omega )$ , put ${\\mathcal {A}}_{(\\beta , \\omega )}$ and $Z_{(\\beta , \\omega )}$ as follows $:$ ${\\mathcal {A}}_{(\\beta ,\\omega )} &:=& \\big \\lbrace E^{\\bullet } \\in D(X) \\big |H^i(E^{\\bullet }) {\\left\\lbrace \\begin{array}{ll}\\in {\\mathcal {T}}_{(\\beta ,\\omega )} & (i=0) \\\\\\in {\\mathcal {F}}_{(\\beta ,\\omega )} & (i=-1) \\\\= 0 & (\\text{otherwise})\\end{array}\\right.}",
"\\big \\rbrace \\\\ Z_{(\\beta , \\omega )} (E) &:=& {\\langle } \\exp (\\beta + \\sqrt{-1}\\omega ), v(E) {\\rangle },$ where ${\\mathcal {T}}_{(\\beta ,\\omega )} &:=& \\lbrace E \\in {\\mathrm {Coh}}(X) | E\\mbox{ is a torsion sheaf or }\\mu _{\\omega }^-(E/{\\mathrm {torsion}} ) > \\beta \\omega \\rbrace \\mbox{ and }\\\\{\\mathcal {F}}_{(\\beta ,\\omega )} &:=& \\lbrace E \\in {\\mathrm {Coh}}(X) | E\\mbox{ is torsion free and }\\mu _{\\omega }^+ (E) \\le \\beta \\omega \\rbrace .$ Here $\\mu _{\\omega }^+ (E)$ (respectively $\\mu _{\\omega }^-(E)$ ) is the maximal slope (respectively minimal slope) of semistable factors of a torsion free sheaf $E$ with respect to the slope stability.",
"Since the pair $({\\mathcal {T}}_{(\\beta , \\omega )}, {\\mathcal {F}}_{(\\beta , \\omega )})$ gives a torsion pair on ${\\mathrm {Coh}}(X)$ , ${\\mathcal {A}}_{(\\beta , \\omega )}$ is the heart of a bounded $t$ -structure on $D(X)$ .",
"We denote the pair $({\\mathcal {A}}_{(\\beta , \\omega )}, Z_{(\\beta , \\omega )})$ by $\\sigma _{(\\beta , \\omega )}$ .",
"Proposition 2.1 ([5]) Assume that $(\\beta , \\omega )$ satisfies the condition ${\\langle } \\exp (\\beta +\\sqrt{-1}\\omega ), \\delta {\\rangle } \\notin {\\mathbb {R}}_{\\le 0}, (\\forall \\delta \\in \\Delta ^+(X)) $ Then the pair $\\sigma _{(\\beta , \\omega )}$ gives a numerical locally finite stability condition on $D(X)$ .",
"Furthermore we have $V(X) = \\lbrace \\sigma _{(\\beta , \\omega )} \\in \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) | (\\beta , \\omega )\\mbox{ satisfies the condition (\\ref {BG-condition})} \\rbrace .$ Remark 2.2 We put $v(E) = r_E \\mathop {\\oplus }\\nolimits c_1(E) \\mathop {\\oplus }\\nolimits s_E$ for $E \\in D(X)$ .",
"As the author remarked in [11], for objects $E \\in D(X)$ with $\\mathop {\\mathrm {rank}}\\nolimits E \\ne 0$ , we can rewrite $Z_{(\\beta , \\omega )} (E)$ as follows, $Z_{(\\beta , \\omega )}(E) = \\frac{v(E)^2}{2 r_E} + \\frac{r_E}{2}\\Big ( \\omega + \\sqrt{-1} \\big (\\frac{c_1(E)}{r_E}- \\beta \\big ) \\Big )^2 .$ This equation (REF ) plays an important role in Lemma REF which is crucial for Theorem REF .",
"Definition 2.3 For a projective K3 surface with $\\rho (X)=1$ we define the subgroup $W(X)$ of $\\mathop {\\mathrm {Aut}}\\nolimits (D(X))$ generated by $W(X) = {\\langle } T_A^2 | A=\\mbox{spherical locally free sheaf}{\\rangle }.$ Then by using $U(X)$ and $W(X)$ we can describe $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ in a explicit way: Proposition 2.4 ([5]) Let $X$ be a projective K3 with $\\rho (X)=1$ .",
"The distinguished connected component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is given by $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) = \\bigcup _{\\Phi \\in W(X)} \\Phi _* (\\bar{U}(X)).$ Theorem 2.5 ([5]) The natural map $\\pi : \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {N}}(X) \\otimes {\\mathbb {C}}$ has the image ${\\mathcal {P}}^+_0(X)$ .",
"Furthermore $\\pi $ is a Galois covering.",
"The covering transformation group is the subgroup generated by equivalences in $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa ) $ which preserve $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Corollary 2.6 For a pair $(X,L)$ , the induced map $\\pi ^{{\\mathrm {n}}}\\colon \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) \\rightarrow {\\mathfrak {H}}^+_0(X)$ is also a Galois covering map.",
"We have the following ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ -equivariant diagram: ${\\begin{matrix}\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)/ {\\mathbb {Z}}[2] &\\xrightarrow{}& {\\mathcal {P}}^+_0(X) \\\\\\downarrow && \\downarrow &&\\\\\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) &\\xrightarrow{}& {\\mathfrak {H}}^+_0(X) .\\end{matrix}}$ We note that both vertical maps are ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ -bundles and that $\\pi ^{\\prime }$ is also a Galois covering.",
"By Theorem REF the covering transformation group of $\\pi ^{\\prime }$ is a subgroup of $\\mathop {\\mathrm {Aut}}\\nolimits (D(X))/ {\\mathbb {Z}}[2]$ .",
"Hence the right ${\\mathrm {GL}}^+(2, {\\mathbb {R}})$ -action on $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)/ {\\mathbb {Z}}[2]$ commutes with the covering transformations.",
"Hence $\\pi ^{{\\mathrm {n}}}$ is also a Galois covering."
],
[
"On the fundamental group of ${\\mathcal {P}}^+_0(X)$",
"We are interested in the fundamental group $\\pi _1 ({\\mathcal {P}}^+_0(X))$ .",
"Generally speaking, it is highly difficult to describe the above condition (REF ) explicitly.",
"Because of this difficulty, it becomes difficult to determine the relation between generators of $\\pi _1({\\mathcal {P}}^+_0(X))$ .",
"Hence it seems impossible to determine the group structure of $\\pi _1({\\mathcal {P}}^+_0(X))$ .",
"However, under the assumption $\\rho (X)=1$ it becomes easier.",
"Definition 2.7 Let $\\delta = r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s \\in \\Delta (X)$ .",
"An associated point $p \\in {\\mathfrak {H}}(X)$ with $\\delta \\in \\Delta (X)$ is the point $p \\in {\\mathfrak {H}}(X)$ such that ${\\langle }\\exp (p),\\delta {\\rangle } = 0$ .",
"We also denote the point by $p(\\delta )$ and call it a spherical point.",
"If $\\delta $ is the Mukai vector of a spherical object $A$ we denote simply $p(v(A))$ by $p(A)$ .",
"Remark 2.8 Let $\\delta \\in \\Delta (X)$ and we put $\\delta = r\\mathop {\\oplus }\\nolimits c\\mathop {\\oplus }\\nolimits s$ .",
"Since $c^2 \\ge 0$ we see $r \\ne 0$ .",
"Thus we have the disjoint sum $\\Delta (X) = \\Delta ^+(X) \\sqcup (-\\Delta ^+ (X))$ .",
"Now we have the explicit description of $p(\\delta ) $ as follows: $p (\\delta )= (\\frac{c}{r},\\frac{1}{\\sqrt{d}|r|} L) \\in {\\mathfrak {H}}(X),$ where we put $L^2 =2d$ .",
"Moreover one sees $p(\\delta )= p(-\\delta )$ .",
"The key lemma of this subsection is that the set $\\lbrace p(\\delta ) \\in {\\mathfrak {H}}(X) | \\delta \\in \\Delta (X) \\rbrace $ is discreet in ${\\mathfrak {H}}(X)$ .",
"To show this claim we introduce some notations.",
"Definition 2.9 Let $\\delta = r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s \\in \\Delta ^+(X)$ .",
"1.",
"We define the set $\\Delta ^{(i)}(X)$ by $\\Delta ^{(i)}(X) = \\lbrace r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s \\in \\Delta ^+(X) | r\\mbox{ is the $i$-th smallest in }\\Delta ^+(X) \\rbrace .$ We also define the rank associated to $\\Delta ^{(i)}(X)$ by $r$ for some $\\delta = r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s \\in \\Delta ^{(i)}(X)$ .",
"2.",
"We define the subset ${\\mathcal {V}}(X)$ of ${\\mathfrak {H}}(X)$ as follows.",
"${\\mathcal {V}}(X) = \\lbrace (\\beta , \\omega ) \\in {\\mathfrak {H}}(X) | (\\beta , \\omega )\\mbox{ satisfies the condition }(\\ref {BG-condition}) \\rbrace .$ As we remarked in Proposition REF this set is isomorphic to $V(X)$ consisting of stability conditions by the natural morphism $\\pi $ .",
"3.",
"Let $r_i$ be the rank associated to $\\Delta ^{(i)}(X)$ .",
"We define the subset ${\\mathcal {V}}^{(i)}(X)$ of ${\\mathcal {V}}(X)$ by ${\\mathcal {V}}^{(i)}(X) = \\lbrace (\\beta , \\omega ) \\in {\\mathcal {V}}(X) | \\omega ^2 > \\frac{2}{r_i^2} \\rbrace .$ Remark 2.10 Let $X$ be a projective (not necessary Picard rank one) K3 surface.",
"For any $\\delta =r\\mathop {\\oplus }\\nolimits c\\mathop {\\oplus }\\nolimits s \\in \\Delta (X)$ with $r\\ge 0$ , there exists a spherical sheaf $A$ on $X$ such that $v(A)= \\delta $ by [13].",
"In particular if $r>0$ then we can take $A$ as a locally free sheaf.",
"In addition if we assume ${\\mathrm {NS}}(X) = {\\mathbb {Z}}L$ then we see $A$ is Gieseker-stable by [16].",
"Since we see $\\gcd (r, n)=1$ where $n$ satisfies $nL =c$ , $A$ is $\\mu $ -stable by [9].",
"Remark 2.11 For instance $\\Delta ^{(1)}(X)$ is the set of Mukai vectors of line bundles on $X$ .",
"Thus $\\mathop {\\mathrm {rank}}\\nolimits \\Delta ^{(1)}(X)=1$ for any $(X,L)$ .",
"However for $i>1$ , the rank of $\\Delta ^{(i)}(X)$ depends on the degree $L^2$ .",
"Since $\\mathop {\\mathrm {rank}}\\nolimits \\Delta ^{(1)}(X)=1$ , we see $(\\beta ,\\omega )$ is in ${\\mathcal {V}}^{(1)}(X)$ if and only if $\\omega ^2 >2$ .",
"We have the following infinite filtration of ${\\mathcal {V}}^{(i)}(X)$ ($i =1,2 ,3 \\cdots $ ) ${\\mathcal {V}}^{(1)}(X) \\subset {\\mathcal {V}}^{(2)}(X) \\subset \\cdots \\subset {\\mathcal {V}}^{(n)}(X) \\subset \\cdots \\subset {\\mathcal {V}}(X).$ Lemma 2.12 Notations being as above, (1) the set ${\\mathfrak {S}} = \\lbrace p(\\delta ) \\in {\\mathfrak {H}}(X) | \\delta \\in \\Delta (X) \\rbrace $ is a discreet set in ${\\mathfrak {H}}(X)$ .",
"(2) Furthermore the set ${\\mathcal {V}}(X)$ is open in ${\\mathfrak {H}}(X)$ .",
"Suppose that ${\\mathrm {NS}}(X) = {\\mathbb {Z}} L $ with $L^2=2d$ .",
"Let $p(\\delta ) $ be the spherical point of $\\delta \\in \\Delta ^+(X)$ .",
"We put $\\delta = r\\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s$ where $c = n L$ for some $n \\in {\\mathbb {Z}}$ .",
"Recall that $p(\\delta ) $ is given by $p(\\delta ) = (\\frac{nL}{r}, \\frac{1}{\\sqrt{d}r}L).$ We also note that $\\gcd (r,n)=1$ since $\\delta ^2 =-2$ and ${\\mathrm {NS}}(X) = {\\mathbb {Z}} L$ .",
"Let $B_{\\epsilon }$ be the open ball whose center is $p(\\delta )$ and the radius is $\\epsilon $ (with respect to the usual metric).",
"Since $r_{i+1} \\ge r_i +1$ (where $r_i$ is the rank of $\\Delta ^{(i)}(X)$ ) if $\\epsilon $ is smaller than $\\frac{1}{\\sqrt{d}} (\\frac{1}{r}- \\frac{1}{r +1})$ we see $B_{\\epsilon } \\cap {\\mathfrak {S}} = \\lbrace p(\\delta ) \\rbrace $ .",
"We prove the second assertion.",
"We define $S(\\delta )$ for $\\delta \\in \\Delta ^+(X)$ as follows: $S(\\delta ) = \\lbrace (\\beta , \\omega ) \\in {\\mathfrak {H}}(X) | \\beta = \\frac{c}{r}, 0< \\omega ^2 \\le \\frac{2}{r^2} \\rbrace .$ Then one can check that ${\\mathcal {V}}(X) = {\\mathfrak {H}}(X) \\setminus \\bigcup _{\\delta \\in \\Delta ^+(X)} S(\\delta ).$ Hence we see ${\\mathcal {V}}^{(i)}(X) = \\lbrace (\\beta , \\omega ) \\in {\\mathfrak {H}}(X) | \\omega ^2 > \\frac{2}{r_i^2 } \\rbrace \\setminus \\bigcup _{\\delta \\in \\Delta ^{(\\le i-1)} }S(\\delta ),$ where $\\Delta ^{(\\le i )} = \\bigcup _{j=1}^i \\Delta ^{(j)}(X)$ .",
"Since the set $\\lbrace \\frac{c}{r} | \\delta = r \\mathop {\\oplus }\\nolimits c \\mathop {\\oplus }\\nolimits s \\in \\Delta ^{(\\le i)} \\rbrace $ is discreet in ${\\mathbb {R}} L$ , the set ${\\mathcal {V}}^{(i)}(X)$ is open in ${\\mathfrak {H}}(X)$ .",
"Since we have ${\\mathcal {V}}(X) = \\bigcup _{i \\in {\\mathbb {N}}} {\\mathcal {V}}^{(i)}(X),$ the set ${\\mathcal {V}}(X)$ is open in ${\\mathfrak {H}} (X)$ .",
"Definition 2.13 We set elements of the fundamental groups $\\pi _1({\\mathfrak {H}}_0(X))$ and of $\\pi _1(GL^+(2, {\\mathbb {R}}))$ as follows.",
"We define $\\ell _{\\delta } $ by the loop which turns round only the spherical point $p(\\delta ) \\in {\\mathfrak {H}}(X)$ counterclockwise; Figure: For p(δ)p(\\delta )we define the loop ℓ δ \\ell _{\\delta } as the above direction.",
"We also assume that there are no spherical points p(δ ' )p(\\delta ^{\\prime }) in the inside of ℓ δ \\ell _{\\delta } except for p(δ)p(\\delta ) itself.",
"We define $g \\in \\pi _1(GL^+(2, {\\mathbb {R}}))$ by $g\\colon [0,1]\\ni t \\mapsto \\begin{pmatrix} \\cos (2 \\pi t) & -\\sin (2\\pi t) \\\\ \\sin (2 \\pi t) & \\cos (2 \\pi t) \\end{pmatrix} \\in GL^+(2, {\\mathbb {R}}).$ We note that $g$ is a generator of $\\pi _1(GL^+(2, {\\mathbb {R}}))$ since $\\pi _1(GL^+(2, {\\mathbb {R}})) \\cong \\pi _1(SO(2)) \\cong {\\mathbb {Z}}$ .",
"Proposition 2.14 The fundamental group $\\pi _1({\\mathcal {P}}^+_0(X))$ is isomorphic to $\\Big (\\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{\\delta \\in \\Delta ^+(X) } {\\mathbb {Z}} \\cdot \\ell _{\\delta } \\Big )\\times {\\mathbb {Z}} \\cdot g$ where $\\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{\\delta \\in \\Delta ^{+} } {\\mathbb {Z}} \\cdot \\ell _{\\delta }$ is a free product of infinite cyclic groups ${\\mathbb {Z}}$ generated by $\\ell _{\\delta }$ .",
"Since ${\\mathcal {P}}^+_0(X)$ is isomorphic to ${\\mathfrak {D}}^+_0 (X) \\times GL^+(2, {\\mathbb {R}})$ we see $\\pi _1({\\mathcal {P}}^+_0(X) ) \\cong \\pi _1 ({\\mathfrak {D}}^+_0(X)) \\times {\\mathbb {Z}}\\cdot g$ .",
"As we remarked before we have $\\Delta (X) = \\Delta ^+(X) \\sqcup (-\\Delta ^+(X))$ .",
"Hence we see ${\\mathfrak {D}}^+_0(X) = {\\mathfrak {D}}^+(X) \\setminus \\bigcup _{\\delta \\in \\Delta (X)} {\\langle } \\delta {\\rangle }^{\\perp } = {\\mathfrak {D}}^+(X) \\setminus \\bigcup _{\\delta \\in \\Delta ^+(X)} {\\langle } \\delta {\\rangle }^{\\perp }$ Since ${\\mathfrak {D}}^+_0(X)$ is isomorphic to ${\\mathfrak {H}}_0(X)$ it is enough to show that $\\pi _1({\\mathfrak {H}}_0(X)) = \\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{\\delta \\in \\Delta ^+} {\\mathbb {Z}} \\cdot \\ell _{\\delta }$ We choose a base point $p$ of ${\\mathfrak {H}}_0(X)$ so that $p = \\sqrt{-1}\\omega $ with $\\omega ^2 \\gg 2$ .",
"Let $\\ell $ be the loop whose base point is $p$ .",
"Then there is a compact contractible subset $C$ whose interior $C^{\\mathit {in}}$ contains $\\ell $ .",
"Then the following set is finite: $\\lbrace p(\\delta ) \\in C^{\\mathit {in}} | \\delta \\in \\Delta ^+(X) \\rbrace .$ Since the fundamental group of the complement of $n$ -points in $C$ is the free group of rank $n$ , we see the homotopy equivalence class of $\\ell $ is uniquely given by $\\ell _{\\delta _1 } ^{k^1} \\ell _{\\delta _2}^{k_2} \\cdots \\ell _{\\delta _m}^{k_m}$ where each $k_i \\in {\\mathbb {Z}}$ .",
"In fact if another loop $m$ is homotopy equivalent to $\\ell $ by $H\\colon [0,1] \\times [0,1] \\rightarrow {\\mathfrak {H}}_0(X)$ , then there is a contractible compact set $C^{\\prime }$ such that $(C^{\\prime })^{\\mathit {in}}$ contains the image of $H$ .",
"Since there are at most finite spherical point in $(C^{\\prime })^{\\mathit {in}}$ , we see the above representation is unique.",
"Thus we have finished the proof.",
"To simplify the notations we denote $\\ell _{v(A)}$ by $\\ell _{A}$ .",
"By Remark REF , we see $\\pi _1({\\mathfrak {H}}_0(X)) = {\\langle } \\ell _A | A\\mbox{ is spherical and locally free} {\\rangle } = \\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{A} {\\mathbb {Z}} \\ell _A.$"
],
[
"Hyperbolic structure on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$",
"Let $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ be the connected components of $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ introduced in $§$ 2.",
"In this section we discuss a hyperbolic structure on the normalized stability manifold $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"To simplify explanations of this section we always use the following notations.",
"Let $(X_i, L_i)$ ($i=1,2$ ) be projective K3 surfaces with ${\\mathrm {NS}}(X_i)= {\\mathbb {Z}}L_i$ and let $\\Phi \\colon D(X_2) \\rightarrow D(X_1)$ be an equivalence between them.",
"The induced isometry ${\\mathcal {N}}(X_2) \\rightarrow {\\mathcal {N}}(X_1)$ by $\\Phi $ is denoted by $\\Phi ^{{\\mathcal {N}}}$ .",
"For a closed point $p_i \\in X_i$ we set $v(\\Phi ({\\mathcal {O}}_{p_2})) = r_1 \\mathop {\\oplus }\\nolimits n_1L_1 \\mathop {\\oplus }\\nolimits s_1 \\mbox{ and } v(\\Phi ^{-1}({\\mathcal {O}}_{p_1})) = r_2 \\mathop {\\oplus }\\nolimits n_2L_2 \\mathop {\\oplus }\\nolimits s_2.$ Since $X_1$ and $X_2$ are Fourier-Mukai partners each other, we see $L_1^2 = L_2^2 = 2d$ for some $d \\in {\\mathbb {N}}$ .",
"Lemma 3.1 Notations being as above, (1) $r_1=0$ if and only if $r_2=0$ .",
"In particular if $r_2=0$ then $\\Phi ^{{\\mathcal {N}}}({\\mathcal {O}}_{p_2}) = \\pm v({\\mathcal {O}}_{p_1}) =\\pm (0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1)$ .",
"(2) If $\\Phi ^{{\\mathcal {N}}}({\\mathcal {O}}_{p_2}) = 0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1$ then $\\Phi ^{{\\mathcal {N}}}$ is numerically equivalent to $(M \\otimes ) ^{{\\mathcal {N}}}$ where $M$ is in ${\\mathrm {Pic}}(X_1)$ under the canonical identification ${\\mathcal {N}}(X_2) \\cong {\\mathcal {N}}(X_1)$ .",
"By the symmetry it is enough to show that $r_2=0$ under the assumption $r_1=0$ .",
"If $r_1=0$ , since $v(\\Phi ({\\mathcal {O}}_{p_2}))$ is isotropic, we see $n_1^2 L_1^2 = 0$ .",
"Thus $n_1=0$ .",
"Moreover since $v(\\Phi ({\\mathcal {O}}_{p_2}))$ is primitive, $s_1$ should be $\\pm 1$ .",
"Hence $\\Phi ^{{\\mathcal {N}}}(0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1) = \\pm (0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1)$ .",
"This gives the proof of the first assertion.",
"Second assertion essentially follows from the argument in the proof for [7].",
"Hence we recall his arguments.",
"Since $\\rho (X_i)=1$ , there is the canonical isomorphism $f \\colon {\\mathcal {N}}(X_2) \\rightarrow {\\mathcal {N}}(X_1)$ where $f(0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1) =0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1, f(0\\mathop {\\oplus }\\nolimits L_{2}\\mathop {\\oplus }\\nolimits 0)=0\\mathop {\\oplus }\\nolimits L_{1}\\mathop {\\oplus }\\nolimits 0$ and $f(0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1)=0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1$ .",
"We show that $\\Phi ^{{\\mathcal {N}}}=(\\otimes M)^{{\\mathcal {N}} }$ ($\\exists M \\in {\\mathrm {Pic}}(X_1)$ ) under the canonical identification $f\\colon {\\mathcal {N}}(X_2)\\rightarrow {\\mathcal {N}}(X_1)$ .",
"One can check easily $v(\\Phi ^{{\\mathcal {N}}} (1\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 0)) = 1 \\mathop {\\oplus }\\nolimits M \\mathop {\\oplus }\\nolimits \\frac{M^2}{2} \\ (\\exists M \\in {\\mathrm {Pic}}(X_1)),$ by using the facts ${\\langle } 1\\mathop {\\oplus }\\nolimits 0 \\mathop {\\oplus }\\nolimits 0, v({\\mathcal {O}}_{p_2}) {\\rangle } =-1$ and ${\\langle } 1\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 0{\\rangle }^2 =0$ .",
"Now consider the functor $\\Psi = (\\otimes M^{-1}\\circ \\Phi )\\colon D(X_2) \\rightarrow D(X_1) \\rightarrow D(X_1).$ Then we see $\\Psi ^{{\\mathcal {N}}}(0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1) = 0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1$ and $\\Psi ^{{\\mathcal {N}}}(1\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 0) = 1\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 0$ .",
"Thus $\\Psi ^{{\\mathcal {N}}}$ induces the isomorphism $\\Psi ^{{\\mathcal {N}}} \\colon {\\mathrm {NS}}(X_2) \\rightarrow {\\mathrm {NS}}(X_1).$ Since ${\\mathrm {NS}}(X_i) = {\\mathbb {Z}} L_i$ we see $\\Psi ^{{\\mathcal {N}}} (L_2) = \\pm L_1$ .",
"Since any equivalence preserves the orientations by [10] we see $\\Psi ^{{\\mathcal {N}}} (L_2) = L_1$ .",
"This gives the proof of the second assertion.",
"Lemma 3.2 For $(\\beta _i, \\omega _i) \\in {\\mathfrak {H}}(X_i)$ $(i=1,2)$ , we put $\\beta _i + \\sqrt{-1} \\omega _i = (x_i + \\sqrt{-1}y_i) L_i$ .",
"(1) For any $\\beta _2 + \\sqrt{-1}\\omega _2\\in {\\mathfrak {H}}(X_2)$ , there exist $\\beta _1 + \\sqrt{-1}\\omega _1 \\in {\\mathfrak {H}}(X_1)$ and $\\lambda \\in {\\mathbb {C}}^*$ such that $\\Phi ^{{\\mathcal {N}}} (\\exp (\\beta _2 + \\sqrt{-1} \\omega _2)) = \\lambda \\exp (\\beta _1 + \\sqrt{-1}\\omega _1)$ .",
"(2) If $r_1\\ne 0$ then $r_1 r_2 >0$ .",
"Furthermore we have $x_1 + \\sqrt{-1}y_1 = \\frac{1}{d \\sqrt{r_1 r_2}} \\cdot \\frac{-1}{(x_2+ \\sqrt{-1}y_2)- \\frac{n_2}{r_2} } +\\frac{n_1}{r_1}.$ In particular this gives a linear fractional transformation on ${\\mathbb {H}}$ .",
"We put $\\mho _2 = \\exp (\\beta _2 + \\sqrt{-1}\\omega _2)$ and $\\Phi ^{{\\mathcal {N}}} (\\mho _2) = u\\mathop {\\oplus }\\nolimits v \\mathop {\\oplus }\\nolimits w$ .",
"Since we have $\\mho _2^2=0$ and $\\mho _2 \\bar{\\mho }_2 >0$ , we see the following: (a) $v ^2 =2u w$ and (b) $v \\bar{v} - u \\bar{w} - \\bar{u} w >0$ .",
"If $u=0$ then $v^2 $ should be 0.",
"Since we have $v^2 \\ge 0$ by the assumption, we see $\\Phi ^{{\\mathcal {N}}}(\\mho _2) = 0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits w$ .",
"This contradicts the second inequality.",
"Thus $u$ should not be 0 and we see $\\Phi ^{{\\mathcal {N}}}(\\mho _2) &=& u (1\\mathop {\\oplus }\\nolimits \\frac{v}{u} \\mathop {\\oplus }\\nolimits \\frac{w}{u}) \\\\&=& u \\Big (1\\mathop {\\oplus }\\nolimits \\frac{v}{u} \\mathop {\\oplus }\\nolimits \\frac{1}{2}\\Big (\\frac{v}{u}\\Big )^2\\Big )_.$ Since $\\frac{v}{u}$ is in ${\\mathrm {NS}}(X)\\otimes {\\mathbb {C}}$ we can put $\\frac{v}{u} = (x + \\sqrt{-1}y) L_1$ for some $(x, y) \\in {\\mathbb {R}}^2$ .",
"By the inequality of (b), we see $y \\ne 0$ .",
"Since $\\Phi $ preserves the orientation by [10], we see $y >0$ .",
"Thus we have proved the first assertion.",
"We prove the second assertion.",
"By the first assertion we put $\\Phi ^{{\\mathcal {N}}} (\\exp (\\beta _2 + \\sqrt{-1}\\omega _2)) = \\lambda \\exp (\\beta _1 + \\sqrt{-1}\\omega _1).$ Then we see $\\lambda &=& - {\\langle } \\Phi ^{{\\mathcal {N}}} (\\exp (\\beta _2 + \\sqrt{-1} \\omega _2)), v({\\mathcal {O}}_{p_1}) {\\rangle } \\\\&=& -{\\langle } \\exp (\\beta _2+ \\sqrt{-1}\\omega _2), v(\\Phi ^{-1}({\\mathcal {O}}_{p_1}) ){\\rangle } \\\\&=& - Z_{(\\beta _2, \\omega _2)} (\\Phi ^{-1}({\\mathcal {O}}_{p_1})),$ and $-1 &=& {\\langle } \\exp (\\beta _2 + \\sqrt{-1}\\omega _2), v ({\\mathcal {O}}_{p_2}) {\\rangle }\\\\&=& {\\langle } \\Phi ^{{\\mathcal {N}}}(\\exp (\\beta _2 + \\sqrt{-1}\\omega _2)), v(\\Phi ({\\mathcal {O}}_{p_2})) {\\rangle } \\\\&=& \\lambda \\cdot Z_{(\\beta _1, \\omega _1)} (\\Phi ({\\mathcal {O}}_{p_2})).$ Thus we have $1 = Z_{(\\beta _2, \\omega _2)}(\\Phi ^{-1}({\\mathcal {O}}_{p_1})) \\cdot Z_{(\\beta _1, \\omega _1)} (\\Phi ({\\mathcal {O}}_{p_2}))$ By Lemma REF we see $r_1 \\ne 0$ and $r_2 \\ne 0$ .",
"Now recall Remark REF .",
"Since $v(\\Phi ({\\mathcal {O}}_{p_2}))^2= v(\\Phi ^{-1}({\\mathcal {O}}_{p_1}))^2 =0$ , we have $Z_{(\\beta _2, \\omega _2)} (\\Phi ^{-1}({\\mathcal {O}}_{p_1})) = \\frac{r_2}{2} \\Big ( y_2+ \\sqrt{-1} \\big ( \\frac{n_2}{r_2} - x_2 \\big ) \\Big )^2 L_2^2$ and $Z_{(\\beta _1, \\omega _1)} (\\Phi ({\\mathcal {O}}_{p_2})) = \\frac{r_1}{2} \\Big ( y_1+ \\sqrt{-1} \\big ( \\frac{n_1}{r_1} - x_1 \\big ) \\Big )^2 L_1^2.$ Since $L_1^2 = L_2 ^2 = 2d$ we see $(x_1- \\frac{n_1}{r_1}) + \\sqrt{-1}y_1 = \\frac{\\pm 1}{d \\sqrt{r_1r_2}} \\cdot \\frac{1}{(x_2- \\frac{n_2}{r_2})+\\sqrt{-1}y_2}.",
"$ Since the left hand side is in the upper half plane ${\\mathbb {H}}$ , $\\sqrt{r_1r_2}$ should be a real number.",
"Thus we see $r_1 r_2 >0$ .",
"Furthermore, since the imaginary part of the left hand side is positive we have $(x_1- \\frac{n_1}{r_1}) + \\sqrt{-1}y_1 = \\frac{-1}{d \\sqrt{r_1r_2}} \\cdot \\frac{1}{(x_2- \\frac{n_2}{r_2})+\\sqrt{-1}y_2}.$ Thus we have finished the proof.",
"Recall that $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) = \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) / \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ .",
"Theorem 3.3 Assume that $\\rho (X) =1$ .",
"(1) $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is a hyperbolic 2 dimensional manifold.",
"(2) Let $Y$ be a Fourier-Mukai partner of $X$ and $\\Phi \\colon D(Y) \\rightarrow D(X)$ an equivalence.",
"Suppose that $\\Phi $ preserves the distinguished component.",
"Then the induced morphism $\\Phi _*^{{\\mathrm {n}}} \\colon \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(Y) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is an isometry with respect to the hyperbolic metric.",
"By Corollary REF , we have the normalized covering map $\\pi ^{{\\mathrm {n}}}\\colon \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) \\rightarrow {\\mathfrak {H}}_0(X).$ Since ${\\mathfrak {H}}_0(X)$ is isomorphic to the open subset of ${\\mathbb {H}}$ by Lemma REF , we can define the hyperbolic metric on ${\\mathfrak {H}}_0(X)$ which is given by $ds^2 = \\frac{dx^2 + dy^2}{y^2},$ where $x+\\sqrt{-1}y \\in {\\mathbb {H}}$ .",
"Since $\\pi ^{{\\mathrm {n}}}$ is a covering map, we can also define the hyperbolic metric on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Thus $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is hyperbolic.",
"Now we prove the second assertion.",
"If $v(\\Phi ({\\mathcal {O}}_y))$ is not $\\pm (0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1)$ by Lemma REF , we see that the induced morphism between ${\\mathfrak {H}}_0(Y) \\rightarrow {\\mathfrak {H}}_0(X)$ is given by the linearly fractional transformation.",
"Since $\\pi ^{{\\mathrm {n}}}$ is an isometry, $\\Phi ^{{\\mathrm {n}}}_*$ is also an isometry.",
"Suppose that $v(\\Phi ({\\mathcal {O}}_y)) = \\pm (0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1)$ .",
"If necessary by taking a shift $[1]$ which gives the trivial action on ${\\mathfrak {H}}(X)$ we can assume that $v(\\Phi ({\\mathcal {O}}_y)) = 0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1$ .",
"Then, by Lemma REF , the induced action on ${\\mathbb {H}}$ is given by a parallel transformation $z \\mapsto z +n$ for some $n \\in {\\mathbb {Z}}$ .",
"Thus we have finished the proof."
],
[
"Simply connectedness of $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$",
"In this section we always assume $\\rho (X) =1$ .",
"Then, as was shown in the previous section, $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is a hyperbolic manifold.",
"By using the hyperbolic structure, we shall discuss the simply connectedness of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Namely we show the following: Theorem 4.1 The following conditions are equivalent.",
"(1) $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected.",
"(2) $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is isomorphic to the upper half plane ${\\mathbb {H}}$ .",
"(3) $W(X)$ is isomorphic to the free group generated by $T_A^2$ : $W(X)= \\operatornamewithlimits{\\mathop {\\mbox{\\huge $\\ast $}}\\nolimits }_{A} ({\\mathbb {Z}}\\cdot T_A^2) ,$ where $A$ runs through all spherical locally free sheaves.",
"We first show that $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected if and only if $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is simply connected.",
"Since the right action of $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ on $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is free, the natural map $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ gives the $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+ (2, {\\mathbb {R}})$ -bundle on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ .",
"Thus there is an exact sequence of fundamental groups: ${\\begin{matrix}\\pi _1(\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})) &\\xrightarrow{}& \\pi _1 (\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)) &\\xrightarrow{}& \\pi _1(\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)) &\\xrightarrow{}& 1.\\end{matrix}}$ Since $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ is simply connected we see that $\\pi _1 (\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) ) =\\lbrace 1\\rbrace $ if and only if $\\pi _1 (\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X))=\\lbrace 1\\rbrace $ .",
"Since $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X)$ is a hyperbolic and complex manifold, $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is isomorphic to ${\\mathbb {H}}$ if and only if $\\pi _1 (\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X))=\\lbrace 1\\rbrace $ by Riemann's mapping theorem.",
"Thus we have proved that the first condition is equivalent to the second one.",
"We secondly show the first condition is equivalent to the third one.",
"Let ${\\mathrm {Cov}}(\\pi )$ be the covering transformation group of $\\pi \\colon \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X)$ .",
"We put $\\tilde{W}(X)$ by the group generated by $W(X)$ and the double shift $[2]$ .",
"Note that $\\tilde{W}(X)$ is isomorphic to $W(X) \\times {\\mathbb {Z}} \\cdot [2]$ .",
"We claim that $\\tilde{W}(X)$ is isomorphic to ${\\mathrm {Cov}}(\\pi )$ .",
"Recall that all spherical sheaf $A$ on $X$ with $\\rho (X)=1$ is $\\mu $ -stable by Remark REF .",
"Hence any $\\Phi \\in \\tilde{W}(X)$ gives a trivial action on $H^*(X, {\\mathbb {Z}})$ and preserves the connected component $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Thus $\\Phi $ gives the covering transformation by [5].",
"Thus we have the group homomorphism $\\tilde{W}(X) \\rightarrow {\\mathrm {Cov}}(X)$ .",
"In particular by Proposition REF , we see this morphism is a surjection.",
"Furthermore as is shown in [5], this is injective.",
"Thus we have proved our claim.",
"Since the covering $\\pi \\colon \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X)$ is a Galois covering, we have the exact sequence of groups: ${\\begin{matrix}1&\\xrightarrow{}& \\pi _1(\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)) &\\xrightarrow{}& \\pi _1 ({\\mathcal {P}}^+_0 (X)) &\\xrightarrow{}& {\\mathrm {Cov}}(\\pi ) &\\xrightarrow{}&1.\\end{matrix}}$ As will be shown in Proposition REF we see $\\varphi (\\ell _{A}) = T_A^2$ and $\\varphi (g)= [2]$ .",
"If $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected then $\\varphi $ is the isomorphism.",
"Hence $W(X)$ is a free group generated by $T_A^2$ .",
"Conversely if $W(X)$ is a free group generated by $T_A^2$ , then $\\varphi $ is an isomorphism.",
"Hence $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected.",
"Remark 4.2 Since the quotient map $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is a $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -bundle, we see that $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is simply connected if and only if $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is a $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -bundle over ${\\mathbb {H}}$ .",
"Thus we can deduce the global geometry of the stability manifold $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Remark 4.3 We give some remarks for $W(X)$ .",
"Recall that any equivalence $\\Phi \\in \\mathop {\\mathrm {Aut}}\\nolimits (D(X))$ induces the Hodge isometry $\\Phi ^{H}$ of $H^*(X, {\\mathbb {Z}})$ in a canonical way.",
"If Bridgeland's conjecture holds, the group $W(X) \\times {\\mathbb {Z}}[2]$ is the kernel $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa )$ of the natural map $\\kappa \\colon \\mathop {\\mathrm {Aut}}\\nolimits (D(X)) \\rightarrow O^+_{\\text{Hodge}} (H^*(X, {\\mathbb {Z}}))\\colon \\Phi \\rightarrow \\Phi ^{H}.$ Moreover $\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa )$ is given by $\\pi _1 ({\\mathcal {P}}^+_0(X))$ .",
"The freeness of $W(X)$ means any two orthogonal complements ${\\langle } \\delta _1 {\\rangle }^{\\perp }$ and ${\\langle } \\delta _2 {\\rangle }^{\\perp }$ (where $\\delta _1$ and $\\delta _2 \\in \\Delta (X)$ ) do not meet each other in ${\\mathcal {P}}^+_0(X)$ .",
"In more general situations (namely the case of $\\rho (X) \\ge 2$ ) there should be some orthogonal complements such that ${\\langle } \\delta _1 {\\rangle }^{\\perp }$ and $ {\\langle }\\delta _2{\\rangle }^{\\perp }$ meet each other.",
"Hence we expect that the quotient group ${\\mathop {\\mathrm {Ker}}\\nolimits (\\kappa ) /{\\mathbb {Z}} \\cdot [2]}$ is not a free group."
],
[
"Wall and the hyperbolic structure",
"Let $X$ be a projective K3 surface with Picard rank one.",
"We have two goals of this section.",
"The first aim is to show Proposition REF which is necessary for Theorem REF .",
"The second aim is to show that any wall is geodesic.",
"Now we start this section from the following key lemma.",
"Lemma 5.1 Any $\\sigma \\in \\partial U(X)$ is in a general position (See also [5]).",
"Namely the point $\\sigma $ lies on only one irreducible component of $\\partial U(X)$ .",
"Before we start the proof, we remark that Maciocia proved a similar assertion in a slightly different situation in [14].",
"Suppose that there is an element $\\sigma =({\\mathcal {A}}, Z) \\in \\partial U(X)$ which is not general.",
"Let $W_1$ and $W_2$ be two irreducible components of $\\partial U(X)$ such that $\\sigma \\in W_1 \\cap W_2$ .",
"By [5] we may assume $\\forall \\tau _1 \\in W_1\\setminus \\lbrace \\sigma \\rbrace $ and $\\forall \\tau _2 \\in W_2 \\setminus \\lbrace \\sigma \\rbrace $ are in general positions in a sufficiently small neighborhood of $\\sigma $ .",
"Hence by [5] there are two $(-2)$ -vectors $\\delta _i \\in \\Delta ^+(X)$ ($i=1,2$ ) such that for any $\\tau _i =({\\mathcal {A}}_i, Z_i) \\in W_i\\setminus \\lbrace \\sigma \\rbrace $ the imaginary part ${\\mathfrak {Im}} Z_i({\\mathcal {O}}_x)\\overline{Z_i(\\delta _i)}$ is 0 where $i \\in \\lbrace 1, 2\\rbrace $ and $x\\in X$ .",
"Since these are closed conditions, the central charge $Z$ of $\\sigma $ also satisfies the following condition: ${\\mathfrak {Im}}Z({\\mathcal {O}}_x) \\overline{Z(\\delta _1)} = {\\mathfrak {Im}} Z({\\mathcal {O}}_x) \\overline{Z(\\delta _2)} =0.",
"$ By the assumption ${\\mathrm {NS}}(X) = {\\mathbb {Z}} L$ , there exists $ g \\in {GL}^+(2, {\\mathbb {R}})$ such that $Z^{\\prime }(E) := g^{-1} \\circ Z (E) = {\\langle } \\exp (\\beta + \\sqrt{-1}\\omega ), v(E) {\\rangle }$ where $(\\beta , \\omega ) \\in {\\mathfrak {H}}(X)$ .",
"Now we put $\\delta _i = r_i \\mathop {\\oplus }\\nolimits n_i L \\mathop {\\oplus }\\nolimits s_i$ .",
"Note that $r_i \\ne 0$ since $n_i^2 L_i^2 \\ge 0$ .",
"Since $Z^{\\prime }({\\mathcal {O}}_x) =-1$ we see ${\\mathfrak {Im}}Z^{\\prime }(\\delta _i)$ is zero by the condition (REF ).",
"Thus we see $\\frac{n_1L}{r_1} = \\frac{n_2 L }{r_2} = \\beta .$ Since $\\delta _i^2 =-2$ we see $\\gcd (r_i, n_i)=1$ .",
"Hence we have $\\delta _1 = \\delta _2$ .",
"This contradicts $W_1 \\ne W_2$ .",
"By Lemma REF and [5] we see $\\partial U(X)$ is a disjoint union of real codimension 1 submanifolds: $\\partial U(X) = \\coprod _{A:\\text{spherical locally free}} (W_A^+ \\sqcup W_A^-),$ where $W_A^+$ (respectively $W_A^-$ ) is the set of stability conditions whose type is $(A^+)$ (respectively $(A^-)$ ).",
"In the following we give an explicit description of each component $W_{A}^{\\pm }$ .",
"Lemma 5.2 Let $X$ be a projective K3 surface with ${\\mathrm {NS}}(X) = {\\mathbb {Z}}L$ and let $A$ be a spherical locally free sheaf.",
"We put $v(A) = r_A\\mathop {\\oplus }\\nolimits n_A L \\mathop {\\oplus }\\nolimits s_A$ and define the set $S(v(A))$ by $S(v(A)) = \\lbrace (\\beta ,\\omega )\\in {\\mathfrak {H}}(X) | \\beta =\\frac{n_A L}{r_A}, 0< \\omega ^2 < \\frac{2}{r_A^2} \\rbrace .$ Then $W_A^{\\pm }$ is isomorphic to $S(v(A)) \\times \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ .",
"In particular $W_A^{\\pm }/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ is a hyperbolic segment spanned by two points in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ which is isomorphic to $S(v(A))$ .",
"We have to consider two cases: $\\sigma \\in W_A^+$ or $\\sigma \\in W_A^-$ .",
"Since the proof is similar, we give the proof only for the case $\\sigma \\in W_A^+$ .",
"Since $\\sigma \\in W_A^+$ , the Jordan-Hölder filtration of ${\\mathcal {O}}_x$ is given by the spherical triangle (REF ) ${\\begin{matrix}A^{\\oplus r_A}&\\xrightarrow{}& {\\mathcal {O}}_x &\\xrightarrow{}& T_A ({\\mathcal {O}}_x).\\end{matrix}}$ By taking $T_A^{-1}$ to the triangle (REF ) we have ${\\begin{matrix}A^{\\oplus r_A}[1] &\\xrightarrow{}& T_A^{-1}({\\mathcal {O}}_x) &\\xrightarrow{}& {\\mathcal {O}}_x.\\end{matrix}}$ Thus ${\\mathcal {O}}_x$ is $T_{A*}^{-1} \\sigma $ -stable.",
"Hence $T_{A*}^{-1}\\sigma $ is in $U(X)$ .",
"Now we put $T_{A*}^{-1} \\sigma = \\tau = ({\\mathcal {A}}, Z)$ .",
"Since $Z(A[1])/ Z({\\mathcal {O}}_x) \\in {\\mathbb {R}}_{>0}$ , we see that $\\tau $ is in the set $W^{\\prime } = \\lbrace \\sigma _{(\\beta , \\omega )} \\in V(X) | \\beta = \\frac{n_A L}{r_A}, \\frac{2}{r_A^2} < \\omega ^2 \\rbrace \\cdot \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}).$ Thus we see $W_A^+ \\subset T_{A*}W^{\\prime }$ .",
"To show the inverse inclusion, let $\\tau ^{\\prime } = ({\\mathcal {A}}^{\\prime }, Z^{\\prime }) $ be in $W^{\\prime }$ .",
"As we remarked in Remark REF , $A$ is $\\mu $ -stable locally free sheaf.",
"Then $A[1]$ has no nontrivial subobject in ${\\mathcal {A}}^{\\prime }$ by [8].",
"Hence $A[1]$ is $\\tau ^{\\prime }$ -stable, in particular, with phase 1.",
"Since $T_A^{-1} ({\\mathcal {O}}_x)$ is given by the extension (REF ) of ${\\mathcal {O}}_x$ and $A^{\\mathop {\\oplus }\\nolimits r_A}[1]$ , the object $T_A^{-1}({\\mathcal {O}}_x)$ is strictly $\\tau ^{\\prime }$ -semistable.",
"Thus by taking $T_A$ to the triangle (REF ), we obtain the Jordan-Hölder filtration (REF ).",
"Hence we see $W_A^+ = T_{A*}W^{\\prime }$ .",
"Since the induced morphism between ${\\mathfrak {H}}(X)$ by $T_A$ is given by Lemma REF , we see $W_A^+ = T_{A*} W^{\\prime } \\cong S(v(A)) \\times \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}).$ For a spherical locally free sheaf $A$ we define the point $q = p(T_A({\\mathcal {O}}_x)) \\in \\bar{{\\mathfrak {H}}}(X)$ by $(\\beta , \\omega )= (\\frac{c_1(A)}{r_A}, 0)$ .",
"By the simple calculation we see that ${\\langle } \\exp (q) , v(T_A({\\mathcal {O}}_x)) {\\rangle }=0.$ Thus in the sense of Definition REF , $p(T_A({\\mathcal {O}}_x))$ could be regarded as the associated point of the isotropic vector $v(T_A({\\mathcal {O}}_x))$ .",
"In view of this we define the following notion: Definition 5.3 An associated point $p \\in \\bar{{\\mathfrak {H}}}(X)$ with a primitive isotropic vector $v \\in {\\mathcal {N}}(X)$ is the point which satisfies ${\\langle } \\exp (p), v {\\rangle }=0.$ Clearly if $v = r\\mathop {\\oplus }\\nolimits nL \\mathop {\\oplus }\\nolimits s$ then $p$ is given by $\\frac{n}{r}$ .",
"In particuclar if $v = 0\\mathop {\\oplus }\\nolimits 0\\mathop {\\oplus }\\nolimits 1$ the associated point is $\\infty \\in \\bar{{\\mathfrak {H}}}(X)$ .",
"We denote the point by $p(v)$ .",
"As an application of Lemma REF we give the proof of a remained proposition: Proposition 5.4 Let $\\varphi \\colon \\pi _1({\\mathcal {P}}^+_0(X)) \\rightarrow {\\mathrm {Cov}}(\\pi )$ be the morphism in the proof of Theorem REF .",
"Then $\\varphi (\\ell _{A}) = T_A^2$ and $\\varphi (g)=[2]$ .",
"We set a base point of $\\pi _1 ({\\mathfrak {H}}_0(X))$ as $\\sqrt{-1}\\omega _0$ with $\\omega _0^2 \\gg 2$ .",
"We also define a base point of $\\pi _1({\\mathcal {P}}^+_0(X))$ by $\\exp (\\sqrt{-1}\\omega _0)$ .",
"Let $\\sigma _0 = \\sigma _{(0, \\omega _0)} \\in V(X)$ be a base point of the covering map $\\pi \\colon \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X)$ .",
"Let $\\ell _A\\colon [0,1] \\rightarrow {\\mathfrak {H}}_0(X)$ be the loop defined in Definition REF which turns round the point $p(v(A))$ and let $\\tilde{\\ell }_A$ be the lift of $\\ell _A$ to $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"The second assertion is almost obvious.",
"In Definitions REF we choosed $g$ as $g\\colon [0,1] \\rightarrow GL^+(2, {\\mathbb {R}})\\colon t\\mapsto \\begin{pmatrix} \\cos (2\\pi t) & -\\sin (2\\pi t) \\\\ \\sin (2\\pi t)& \\cos (2 \\pi t) \\end{pmatrix}.$ Then the induced action of $g$ on $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is given by the double shift $[2]$ .",
"Hence it is enough to show that $\\tilde{\\ell }_A (1) = T_{A*}^2 \\sigma _0$ .",
"Since there are no spherical point $p(\\delta )$ inside the loop $\\ell _A$ except for $p(v(A))$ itself, the intersection $\\ell _A ([0,1]) \\cap \\pi (\\partial U(X))$ consists of only one point.",
"We may assume the point is given by $\\ell _A(1/2)$ .",
"Since we have $\\tilde{\\ell }_A([0, 1/2)) \\subset U(X)$ we see that $\\tilde{\\ell }_A (1/2) = \\tau $ is in $\\partial U(X)$ and that $\\tau $ is of type $(A^+)$ or $(A^-)$ by Lemma REF and [5].",
"We finally claim that $\\tau $ is of type $(A^+)$ .",
"To prove the claim we put $\\tilde{\\ell }_A \\Big (\\frac{1}{2}-\\epsilon \\Big ) =\\sigma _{\\epsilon }= ({\\mathcal {A}}_{\\epsilon }, Z_{\\epsilon }) \\in \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) ,$ for $0< \\epsilon \\ll 1$ .",
"In fact suppose to the contrary that $\\tau $ is of type $(A^-)$ .",
"By Proposition [5] we may assume both $A$ and $T_A^{-1}({\\mathcal {O}}_x)$ are $\\sigma _{\\epsilon }$ -stable for any $\\epsilon $ .",
"Since we see ${\\mathfrak {Im}}Z_{\\epsilon }({\\mathcal {O}}_x)/ Z_{\\epsilon }(A[2]) >0$ , the distinguished triangle ${\\begin{matrix}T_A^{-1} ({\\mathcal {O}}_x) &\\xrightarrow{}& {\\mathcal {O}}_x &\\xrightarrow{}& A^{\\mathop {\\oplus }\\nolimits r_A}[2]\\end{matrix}}$ gives the Harder-Narasimhan filtration of ${\\mathcal {O}}_x$ in $\\sigma _{\\epsilon }$ .",
"This contradicts the fact that ${\\mathcal {O}}_x$ is $\\sigma _{\\epsilon }$ -stable.",
"Hence $\\ell _A (1/2)$ is of type $(A^+)$ and $\\tilde{\\ell }_A(1/2 + \\epsilon ) $ is in $T_{A*}^2 U(X)$ .",
"For $t> 1/2$ , since $\\ell _A$ does not meet $\\pi (\\partial U(X))$ , we see $\\tilde{\\ell }_A(1) = T_{A*}^2 \\sigma _0$ .",
"Finally we observe so called walls in terms of the hyperbolic structure.",
"As we showed in Lemma REF each boundary components of $\\partial V(X)$ is geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"More generally we show that any wall is geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Let ${\\mathcal {S}}$ be the set objects which have bounded mass in $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ , and $B$ a compact subset of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Then by [5] we have a finite set $\\lbrace W_{\\gamma } \\rbrace _{\\gamma \\in \\Gamma }$ of real codimension 1 submanifolds satisfying the property in the proposition.",
"For the set $\\lbrace W_{\\gamma } \\rbrace _{\\gamma \\in \\Gamma }$ we put ${\\mathfrak {W}}({\\mathcal {S}},B) = \\Big ( \\bigcup _{\\gamma \\in \\Gamma } W_{\\gamma } \\Big )/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}).$ Note that ${\\mathfrak {W}}({\\mathcal {S}},B)$ is a subset of $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Theorem 5.5 The set ${\\mathfrak {W}}({\\mathcal {S}}, B)$ is geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Following [5] let ${\\mathcal {T}}$ be the set of objects ${\\mathcal {T}} =\\lbrace A \\in D(X) | \\exists E \\in {\\mathcal {S}}, \\exists \\sigma \\in B\\text{ such that }m_{\\sigma }(A) \\le m_{\\sigma }(E) \\rbrace .$ We put the set of Mukai vectors in ${\\mathcal {T}}$ by $I =\\lbrace v(A) | A \\in {\\mathcal {T}} \\rbrace $ and let $\\gamma $ be the pair $\\gamma = (v_i, v_j) \\in I \\times I$ which are not proportional.",
"As was shown in [5], each wall component $W_{\\gamma }$ is given by $W_{\\gamma } = \\lbrace \\sigma =({\\mathcal {A}}, Z) \\in \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) | Z(v_i)/Z(v_j) \\in {\\mathbb {R}}_{>0} \\rbrace .$ We put $W_{\\gamma }/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ by ${\\mathfrak {W}}_{\\gamma }$ .",
"It is enough to prove that ${\\mathfrak {W}}_{\\gamma }$ is geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Since $I$ is finite set (Recall that ${\\mathcal {T}}$ has bounded mass) we can take a sufficiently large $m \\in {\\mathbb {Z}} $ so that the rank of all vectors in $T_{mL}^H(I)$ are not 0.",
"For the set $T_{mL}^H(I)$ we define ${\\mathfrak {W}}_{\\gamma }^T$ by ${\\mathfrak {W}}_{\\gamma }^T= \\lbrace [\\sigma ] =[({\\mathcal {A}}, Z)] \\in \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) | Z(T_{mL}^H(v_i))/ Z(T^H_{mL}(v_j)) \\in {\\mathbb {R}}_{>0} \\rbrace .$ We may assume the central charge of $[\\sigma ] \\in {\\mathfrak {W}}_{\\gamma }^T$ is given by $Z(E)= {\\langle } \\exp (\\beta + \\sqrt{-1}\\omega ), v(E) {\\rangle }$ where $(\\beta , \\omega ) \\in {\\mathfrak {H}}(X)$ .",
"We note that $\\sigma \\in {\\mathfrak {W}}_{\\gamma }^{T}$ satisfies the following equation ${\\mathfrak {Im}} Z(T_{mL}^H(v_i)) \\overline{Z(T_{mL}^H(v_j))}=0.",
"$ Then one can easily check that the equation (REF ) defines hyperbolic line in ${\\mathfrak {H}}(X)$ .",
"Since the hyperbolic structure is induced from ${\\mathfrak {H}}(X)$ the set ${\\mathfrak {W}}_{\\gamma }^T$ is geodesic also in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Since we have $T_{mL}^{{\\mathrm {n}}} {\\mathfrak {W}}_{\\gamma }^T = {\\mathfrak {W}}_{\\gamma }$ the set ${\\mathfrak {W}}_{\\gamma }$ is also geodesic in $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ by Theorem REF ."
],
[
"Revisit of Orlov's theorem via hyperbolic structure",
"In this section we demonstrate applications of the hyperbolic structure on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ .",
"Mainly we prove Orlov's theorem without the global Torelli theorem but with assuming the connectedness of $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ in Proposition REF .",
"Hence our application suggests that Bridgeland's theory substitutes for the global Torelli theorem."
],
[
"Strategy for Proposition ",
"Since the proof of Proposition REF is technical, we explain the strategy and the roles of some lemmas which we prepare in §REF .",
"Proposition REF will be proved in §REF .",
"If we have an equivalence $\\Phi \\colon D(Y) \\rightarrow D(X)$ preserving the distinguished component then there exists $\\Psi \\in W(X)$ such that $(\\Psi \\circ \\Phi )_* U(Y) \\cap V(X) \\ne \\emptyset $ by Proposition REF .",
"We want to take the large volume limit in the domain $(\\Psi \\circ \\Phi )_* U(Y) \\cap V(X)$ .",
"Because of the complicatedness of the set $V(X)$ , we consider the subset $V(X)_{>2}=\\lbrace \\sigma _{(\\beta , \\omega )} \\in V(X) | \\omega ^2 >2 \\rbrace $ and focus on the domain $D_{>2}= (\\Psi \\circ \\Phi )_* U(Y) \\cap V(X)_{>2}$ .",
"To take the large volume limit, we have to know the shape of the domain $D_{>2}$ .",
"To know the shape of $D_{>2}$ we have to see where the boundary $(\\Psi \\circ \\Phi )_* \\partial U(Y)$ appears in $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"As we showed in Lemma REF , any connected component of $\\partial U(Y)$ is the product of $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ and a hyperbolic segment spanned by two associated points.",
"Since any equivalence $D(Y) \\rightarrow D(X)$ induces an isometry between the normalized spaces $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(Y) \\rightarrow \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ by Theorem REF , we see that the image $(\\Psi \\circ \\Phi )_* \\partial U(Y)$ is also the products of $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^{+}(2, {\\mathbb {R}})$ and hyperbolic segments spanned by two associated points (See also Lemma REF below).",
"This is the reason why the hyperbolic metric on $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)$ is important for us.",
"Here we have to recall that $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ is conjecturally $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -bundle over the upper half plane ${\\mathbb {H}}$ .",
"Since we don't have the explicit isomorphism $\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) \\rightarrow {\\mathbb {H}}$ yet, it is impossible to observe the place $(\\Psi \\circ \\Phi )_* \\partial U(Y)$ in $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"Instead of this observation, we study the numerical information of $(\\Psi \\circ \\Phi )_* \\partial U(Y)$ , namely the image of $(\\Psi \\circ \\Phi )_* \\partial U(Y)$ by the quotient map $\\pi _{{\\mathfrak {H}}} \\colon \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X) \\rightarrow {\\mathfrak {H}}_0(X)$ .",
"Set ${\\mathfrak {W}}= \\pi _{{\\mathfrak {H}}} \\big ((\\Psi \\circ \\Phi )_* \\partial U(Y) \\big )$ .",
"As we showed in Lemma REF , ${\\mathfrak {W}}$ is the disjoint sum of hyperbolic segments.",
"As we show in Lemma REF later, there are two types (I) and (II) of components of ${\\mathfrak {W}}$ .",
"The type (I) is a hyperbolic segment which does not intersect the domain $\\pi _{{\\mathfrak {H}}}(V(X)_{>2})$ and the type (II) is a hyperbolic segment which does intersect $\\pi _{{\\mathfrak {H}}}(V(X)_{>2})$ .",
"Recall that our basic strategy is to take the limit in the domain $V(X)_{>2}$ .",
"If the family of type (II) components is unbounded in $\\pi _{{\\mathfrak {H}}}(V(X)_{>2})$ , it may be impossible to take the large volume limit.",
"Hence we have to show the boundedness of type (II) components (Proposition REF and Corollary REF )."
],
[
"Technical lemmas",
"We prepare some technical lemmas.",
"Throughout this section we use the following notations.",
"For a K3 surface $(X, L)$ we put $L^2 =2d$ .",
"Suppose that $E \\in D(X)$ satisfies $v(E)^2=0$ and $A \\in D(X)$ is spherical.",
"We put their Mukai vectors respectively $v(E) = r_E \\mathop {\\oplus }\\nolimits n_E L \\mathop {\\oplus }\\nolimits s_E\\text{ and }v(A)= r_A \\mathop {\\oplus }\\nolimits n_A L \\mathop {\\oplus }\\nolimits s_A.$ We denote $(\\beta , \\omega ) \\in \\bar{{\\mathfrak {H}}}(X)$ by $(xL, yL)$ .",
"The main object is the following set ${\\mathfrak {W}}(A,E) = \\lbrace (\\beta , \\omega )\\in \\bar{{\\mathfrak {H}}}(X) | {\\mathfrak {Im}}Z_{(\\beta , \\omega )}(E) \\overline{Z_{(\\beta , \\omega )}(A)} =0 \\rbrace .$ One can easily check that the condition ${\\mathfrak {Im}}Z_{(\\beta , \\omega )}(E) \\overline{Z_{(\\beta , \\omega )}(A)} =0 $ is equivalent to $N_{A,E}(x,y)= \\lambda _E (\\frac{-1}{r_A} + dr_A y^2 - \\frac{d \\lambda _A^2}{r_A})- \\lambda _A (d r_E y^2 - \\frac{\\lambda _E^2}{r_E} )=0,$ where $\\lambda _E = n_E - r_E x$ and $\\lambda _A = n_A - r_A x$ .",
"We also have $N_{A, E}(x,y)= d (r_A n_E -r_E n_A)y^2 + d\\lambda _E \\lambda _A (\\frac{n_E}{r_E}- \\frac{n_A}{r_A}) -\\frac{\\lambda _E}{r_A}.",
"$ Lemma 6.1 Suppose that $0< r_E $ and $\\frac{n_E}{r_E}\\ne \\frac{n_A}{r_A} $ .",
"Then ${\\mathfrak {W}}(A,E)$ is the half circle passing through the following 4 points: $(x, y)=(\\alpha _E, 0), (\\frac{n_E}{r_E}, 0), (\\frac{n_A}{r_A}, \\frac{1}{\\sqrt{d}|r_A|})\\text{ and }(\\alpha _A, \\frac{1}{\\sqrt{d}|r_A|}),$ where $\\alpha _E = \\frac{n_A}{r_A}- \\frac{1}{dr_A^2(\\frac{n_E}{r_E}-\\frac{n_A}{r_A})}$ and $\\alpha _A = \\frac{n_E}{r_E}- \\frac{1}{dr_A^2(\\frac{n_E}{r_E}-\\frac{n_A}{r_A})}$ .",
"In particular the set ${\\mathfrak {W}}(A, E)$ is a hyperbolic line passing through above 4 points.",
"We can prove Lemma REF by the simple calculation of (REF ).",
"In particular the first two points are associated points with respectively $T_A(E)$ and $E$ .",
"Hence we put them respectively $p(T_A(E))= (\\alpha _E, 0)$ , $p(E)= (\\frac{n_E}{r_E}, 0)$ , $ p(A) = (\\frac{n_A}{r_A}, \\frac{1}{\\sqrt{d}|r_A|})$ and $q=(\\alpha _A, \\frac{1}{\\sqrt{d}|r_A|})$ .",
"We remark that if $\\frac{n_E}{r_E} = \\frac{n_A}{r_A}$ then ${\\mathfrak {W}}(A,E)$ is a hyperbolic line defined by $x= \\frac{n_E}{r_E}$ .",
"Lemma 6.2 Suppose that $0< r_E $ and $0< \\frac{n_E}{r_E}- \\frac{n_A}{r_A} $ .",
"Then there two types of the configuration of the above four points on ${\\mathfrak {W}}(A,E)$ : (I) If $\\frac{1}{d|r_A|} \\le \\frac{n_E}{r_E}- \\frac{n_A}{r_A}$ then we have $\\alpha _E < \\frac{n_A}{r_A}\\le \\alpha _A < \\frac{n_E}{r_E}$ .",
"See also Figure REF below.",
"(II) If $0< \\frac{n_E}{r_E}- \\frac{n_A}{r_A} < \\frac{1}{d|r_A|} $ then we have $\\alpha _E < \\alpha _A < \\frac{n_A}{r_A} < \\frac{n_E}{r_E}$ .",
"See also Figure REF below.",
"Figure: figure for for type (II) in Lemma Similarly to Lemma REF we could prove the assertion by simple calculations.",
"Let $\\Phi \\colon D(Y) \\rightarrow D(X)$ be an equivalence preserving the distinguished component.",
"Suppose $E= \\Phi ({\\mathcal {O}}_y)$ .",
"By Lemma REF , $\\pi _{{\\mathfrak {H}}} (\\Phi _* \\partial U(Y))$ is the direct sum of hyperbolic segments $\\overline{p(A) p(T_A(E))}$ spanned by two points $p(A)$ and $p(T_A(E))$ .",
"Clearly the segment $\\overline{p(A) p(T_A(E))}$ is a subset of ${\\mathfrak {W}}(A,E)$ .",
"Following Lemma REF we have the disjoint sum : $\\pi _{{\\mathfrak {H}}} (\\Phi _* \\partial U(Y)) = \\coprod _{{\\mathrm {type(I)}}}\\overline{p(A^{\\prime }) p(T_{A^{\\prime }}(E)) } \\sqcup \\coprod _{{\\mathrm {type(II)}}}\\overline{p(A) p(T_{A}(E))} .",
"$ Since the type (II) segments become obstructions when we take the large volume limit in $V(X)_{>2}$ .",
"Hence we have to show the boundedness of type (II) segments.",
"To show this, we give an upper bound of the diameter of the type (II) half circle ${\\mathfrak {W}}(A,E)$ in the following proposition.",
"Clearly from Lemma REF the diameter is given by $\\frac{n_E}{r_E}- \\alpha _E$ .",
"Proposition 6.3 Suppose that $r_E>0$ and $0< \\frac{n_E}{r_E}- \\frac{n_A}{r_A} < \\frac{1}{\\sqrt{d}|r_A|}$ .",
"Then we have $0 < \\frac{n_E}{r_E} - \\alpha _E \\le \\frac{1}{r_E} + \\frac{r_E}{d}.$ By the assumption one easily sees $r_A \\cdot ( r_A n_E - r_E n_A) >0$ .",
"Hence we see $\\frac{n_E}{r_E} - \\alpha _E &=& \\Big ( \\frac{n_E}{r_E} - \\frac{n_A}{r_A} \\Big ) + \\frac{1}{dr_A^2 \\Big ( \\frac{n_E}{r_E} - \\frac{n_A}{r_A} \\Big )} \\\\&=& \\Big | \\frac{1}{r_A} \\Big | \\cdot \\Big ( \\frac{ |r_An_E - r_E n_A |}{r_E} + \\frac{r_E}{d | r_A n_E - r_E n_A|} \\Big ) \\\\&\\le & \\frac{|r_A n_E - r_E n_A|}{r_E} + \\frac{r_E}{d |r_A n_E- r_E n_A|}.",
"$ By the assumption we have $\\frac{|r_A n_E - r_E n_A|}{r_E} < \\frac{r_E}{d |r_A n_E - r_E n_A|}.$ Since the continuous function $f(t)= \\frac{1}{t}+\\frac{t}{d}$ on ${\\mathbb {R}}_ {>0}$ is an increasing function for $\\frac{1}{t} < \\frac{t}{d}$ .",
"Since we have $\\frac{r_E}{| r_A n_E - r_E n_A|} \\le r_E$ the following inequality holds: $(\\ref {aa}) \\le \\frac{1}{r_E} + \\frac{r_E}{d}.$ Thus we have proved the inequality.",
"The following corollary is a simple paraphrase of Proposition REF .",
"However it is crucial for the proof of our main result, Proposition REF .",
"Corollary 6.4 Let $\\Phi \\colon D(Y) \\rightarrow D(X)$ be an equivalence which preserves the distinguished component.",
"Set $v(\\Phi ({\\mathcal {O}}_y)) = r \\mathop {\\oplus }\\nolimits n L_X \\mathop {\\oplus }\\nolimits s$ and $L_X ^2 =2d$ and assume $r >0$ .",
"Then the image $\\pi _{{\\mathfrak {H}}} (\\Phi _* \\partial U(Y))$ is in the following shaded closed region $R(Y, \\Phi )$ where $\\pi _{{\\mathfrak {H}}}\\colon \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}} (X) \\rightarrow {\\mathfrak {H}}_0(X)$ $:$ $R(Y, \\Phi ) =\\lbrace (xL_X, yL_X) \\in {\\mathfrak {H}}(X)| \\Big (x-\\frac{n}{r} + \\frac{1}{2}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big ) \\Big )^2 + y^2 \\le \\frac{1}{4}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big )^2 , \\\\\\Big (x-\\frac{n}{r} - \\frac{1}{2}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big ) \\Big )^2 + y^2 \\le \\frac{1}{4}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big )^2\\mbox{ or }y^2 \\le \\frac{1}{d} \\rbrace .$ Figure: Figure for the region R(Y,Φ)R(Y, \\Phi ).As we explained in (REF ), we see that $\\pi _{{\\mathfrak {H}}} (\\Phi _* \\partial U(Y)) = \\coprod _{{\\mathrm {type(I)}}}\\overline{p(A^{\\prime }) p(T_{A^{\\prime }}(\\Phi ({\\mathcal {O}}_y)))} \\sqcup \\coprod _{{\\mathrm {type(II)}}}\\overline{p(A) p(T_{A}(\\Phi ({\\mathcal {O}}_y)))}$ where $A$ and $A^{\\prime }$ are spherical object of $D(X)$ .",
"Clearly type (I) hyperbolic segments $\\overline{p(A^{\\prime }) p(T_{A^{\\prime }}(\\Phi ({\\mathcal {O}}_y)))}$ are in the following region: $\\lbrace (xL_X, y L_X ) | y^2 \\le \\frac{1}{d} \\rbrace .$ By Proposition REF , the type (II) hyperbolic segments are in the region $\\lbrace (xL_X, yL_X) \\in {\\mathfrak {H}}(X)| \\Big (x-\\frac{n}{r} + \\frac{1}{2}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big ) \\Big )^2 + y^2 \\le \\frac{1}{4}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big )^2 \\mbox{ or } \\\\\\Big (x-\\frac{n}{r} - \\frac{1}{2}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big ) \\Big )^2 + y^2 \\le \\frac{1}{4}\\Big ( \\frac{1}{d}+ \\frac{r}{d} \\Big )^2 \\rbrace .$ This gives the proof."
],
[
"Revisit of Orlov's theorem",
"We prove the main result of this section.",
"Proposition 6.5 Let $(X,L_X)$ be a projective K3 surface with $\\rho (X)=1$ and $(Y, L_Y)$ a Fourier-Mukai partner of $(X,L_X)$ .",
"If an equivalence $\\Phi \\colon D(Y) \\rightarrow D(X)$ preserves the distinguished component, then $Y$ is isomorphic to the fine moduli space of Gieseker stable torsion free sheaves.",
"We first put $L_X^2 = L_Y^2 =2d$ and $v_0 = v(\\Phi ({\\mathcal {O}}_y)) = r \\mathop {\\oplus }\\nolimits n L_X \\mathop {\\oplus }\\nolimits s$ .",
"If necessary by taking $T_{{\\mathcal {O}}_X}$ and $[1]$ , we may assume $r>0$ .",
"We denote the composition of two morphisms $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X) \\rightarrow {\\mathcal {P}}^+_0(X) \\rightarrow {\\mathfrak {H}}_0(X)$ by $\\pi _{{\\mathfrak {H}}}$ .",
"By the assumption we have $\\Phi _*U(Y) \\subset \\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ .",
"We can take a stability condition $\\tau \\in U(Y)$ so that $\\pi _{{\\mathfrak {H}}}(\\Phi _* \\tau )= (\\beta _0, \\omega _0)= (aL_X, bL_X)$ with (i) $ a < \\frac{n}{r} -\\big (\\frac{1}{r} + \\frac{r}{d} \\big ) $ and (ii) $2 < \\omega _0^2$ .",
"By the second condition (ii) and Lemma REF we see $\\pi _{{\\mathfrak {H}}} \\circ \\Phi _*(\\tau )$ does not lie on $\\pi _{{\\mathfrak {H}}}( \\partial U(X))$ .",
"Hence $\\Phi _* (\\tau )$ is in a chamber of $\\mathop {\\mathrm {Stab}^{\\dagger }}\\nolimits (X)$ by Proposition REF .",
"Hence we see $\\exists \\Psi \\in W(X) \\times {\\mathbb {Z}} [2]\\text{ such that }(\\Psi \\circ \\Phi ) _* (\\tau ) \\in U(X).$ Now we put $\\Phi ^{\\prime } = \\Psi \\circ \\Phi $ and take $\\sigma _0 \\in V(X)$ as $\\sigma _{(\\beta _0, \\omega _0)}$ .",
"Since $\\Phi ^{\\prime } _*(\\tau )$ and $\\sigma _0$ belong to the same $\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}})$ -orbit, $\\sigma _0$ is in $V(X) \\cap \\Phi ^{\\prime }_* (U(Y))$ .",
"We define a family ${\\mathcal {F}}$ of stability conditions as follows: ${\\mathcal {F}} = \\lbrace \\sigma _{(\\beta _0 , t \\omega _0)} \\in V(X) | 1 <t \\in {\\mathbb {R}} \\rbrace .$ Then we see $\\pi _{{\\mathfrak {H}}}({\\mathcal {F}}) \\cap R(Y,\\Phi ^{\\prime } )=\\emptyset $ by Corollary REF .",
"Hence ${\\mathcal {F}}$ does not meet $\\Phi ^{\\prime }_* (\\partial U(Y))$ .",
"Since $\\sigma _0 \\in \\Phi ^{\\prime }_* (U(Y))$ we see ${\\mathcal {F}} \\subset \\Phi ^{\\prime }_* (U(Y))$ and the object $\\Phi ^{\\prime }({\\mathcal {O}}_y)$ is $\\sigma $ -stable for all $\\sigma \\in {\\mathcal {F}}$ .",
"By Bridgeland's large volume limit argument [5] we see that $\\Phi ^{\\prime } ({\\mathcal {O}}_y) $ is a Gieseker semistable torsion free sheafSince we are assuming $\\rho (X)=1$ , the Gieseker stability is equivalent to the twisted stability..",
"Moreover by [16] (or the argument of [11]) $\\Phi ^{\\prime }({\\mathcal {O}}_y)$ is Gieseker stable.",
"Since $v_0 = v(\\Phi ^{\\prime }({\\mathcal {O}}_y))$ is isotropic and there is $ u \\in {\\mathcal {N}}(X)$ such that ${\\langle } v_0,u {\\rangle }=1$ , there exists the fine moduli space ${\\mathcal {M}}$ of Gieseker stable sheaves (See also [7]).",
"Hence $Y$ is isomorphic to ${\\mathcal {M}}$ .",
"Remark 6.6 Clearly the key ingredient of Proposition REF is Corollary REF .",
"The role of Corollary REF is to detect the place of the numerical image of walls $\\pi _{{\\mathfrak {H}}} (\\Phi (\\partial U(Y)))$ .",
"Without Theorems REF and REF , it was difficult to detect the place of $\\pi _{{\\mathfrak {H}}} (\\Phi (\\partial U(Y)))$ .",
"By virtue of these theorems, the problem is reduced to the problem with two associated points $p(A)$ and $p(T_A(\\Phi ({\\mathcal {O}}_y)))$ .",
"Remark 6.7 We explain the relation between author's work and Huybrechts's question in [8].",
"In [8], it was proven that all non-trivial Fourier-Mukai partners of projective K3 surfaces are given by the fine moduli spaces of $\\mu $ -stable locally free sheaves (See also [8]).",
"We note that this proposition holds for all projective K3 surfaces.",
"If the Picard rank is one, the proof of the proposition is based on the lattice argument.",
"In the proof of [8] Huybrechts asks whether there is a geometric proof.",
"In the previous work [12], the author gave an answer of Huybrechts's question, that is a geometric proof.",
"However our proof is not completely independent of lattice theories, because it is based on Orlov's theorem which strongly depends on the global Torelli theorem.",
"As a consequence of Proposition REF and [12], we could give the another proof of [8] which is completely independent of the global Torelli theorem with assuming the connectedness of $\\mathop {\\mathrm {Stab}}\\nolimits (X)$ ."
],
[
"Stable complexes in large volume limits",
"Let $A$ be a spherical sheaf in $D(X)$ .",
"At the end of this paper we discuss the stability of the complex $T_A^{-1}({\\mathcal {O}}_x)$ in the large volume limitWe remark that $T_A^{-1}(O_x)$ is a 2-terms complex such that $H^0(T_A^{-1}({\\mathcal {O}}_x)) = {\\mathcal {O}}_x$ and $H^{-1}(T_A^{-1}({\\mathcal {O}}_x))= A^{\\mathop {\\oplus }\\nolimits r_A}$ .. More precisely we shall show that $T_A^{-1}({\\mathcal {O}}_x)$ is $\\sigma _{(\\beta , \\omega )}$ -stable if $\\beta \\omega < \\mu _{\\omega }(A)$ and $\\omega ^2 >2$ .",
"The possibility of stable complexes in the large volume limit is predicted in [3].",
"For the vector $v(A) = r_A \\mathop {\\oplus }\\nolimits n_A L \\mathop {\\oplus }\\nolimits s_A$ we define the subset ${\\mathfrak {D}}_A \\subset {\\mathfrak {H}}(X)$ as follows: ${\\mathfrak {D}}_A= \\lbrace (xL , yL) \\in {\\mathfrak {H}}(X) | (x-\\frac{n_A}{r_A})^2 + (y-\\frac{1}{2\\sqrt{d}r_A})^2 < \\frac{1}{4dr_A^2} \\rbrace $ Lemma 7.1 Notations being as above.",
"In the domain ${\\mathfrak {D}}_A$ , there are no spherical point $p(\\delta )$ with $(-2)$ -vectors $\\delta $ .",
"Moreover ${\\mathfrak {D}}_A$ does not intersect $\\pi _{{\\mathfrak {H}}} \\circ T_{A*} (\\partial U(X))$ .",
"By the spherical twist $T_A$ , we have the diagram: ${\\begin{matrix}\\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X) &\\xrightarrow{}& \\mathop {\\mathrm {Stab}}\\nolimits ^{{\\mathrm {n}}}(X)\\\\\\downarrow && \\downarrow && \\\\{\\mathfrak {H}}_0(X) &\\xrightarrow{}& {\\mathfrak {H}}_0(X).\\end{matrix}}$ By Lemma REF , $T_A^{{\\mathfrak {H}}}$ is given by the liner fractional transformation $T_A^{{\\mathfrak {H}}}(x+\\sqrt{-1}y) = \\frac{1}{dr_A}\\cdot \\frac{-1}{x+ \\sqrt{-1}y- \\frac{n_A}{r_A}} + \\frac{n_A}{r_A}.$ We remark that $T_A^{{\\mathfrak {H}}}$ is conjugate to the transformation $z \\mapsto -1/dr_A z$ .",
"Now we recall there are no spherical point $p(\\delta )$ in the domain ${\\mathfrak {H}}(X)_{>2}= \\lbrace (\\beta , \\omega ) \\in {\\mathfrak {H}}(X) | \\omega ^2 >2 \\rbrace .",
"$ One can easily check that $T_A^{{\\mathfrak {H}}}({\\mathfrak {H}}(X)_{>2})= {\\mathfrak {D}}_A$ .",
"Moreover it is clear that $\\pi _{{\\mathfrak {H}}} (\\partial U(X)) \\cap {\\mathfrak {H}}(X)_{>2} $ .",
"This gives the proof.",
"Define the subset $D_A^+ \\subset V(X) $ by $D_A^+= \\lbrace \\sigma _{(xL, yL)} \\in V(X) | x < \\frac{n_A}{r_A}, (xL , yL) \\in {\\mathfrak {D}}_A \\rbrace .$ In the following proposition, we discuss the stability of sheaves $T_A({\\mathcal {O}}_x)$ in the “small” volume limit $D_A^+$ .",
"Proposition 7.2 For any $\\sigma \\in D_A^+$ , $T_A({\\mathcal {O}}_x)$ is $\\sigma $ -stable.",
"In particular $D_A^+ \\subset T_{A*} U(X) \\cap V(X)$ .",
"To simplify the notation we set $A(x)= T_A({\\mathcal {O}}_x)[-1]$ .",
"It is enough to show that $A(x)$ is $\\sigma $ -stable for all $\\sigma \\in D_A^+$ .",
"One can see $A(x)$ is the kernel of the evaluation map $\\mathop {\\mathrm {Hom}}\\nolimits (A, {\\mathcal {O}}_x)\\otimes A \\rightarrow {\\mathcal {O}}_x$ and is Gieseker stable.",
"We note that there exists $\\sigma \\in D_A^+$ such that $A(x)$ is $\\sigma $ -stable by [12].",
"In particular we see $D_A^+ \\cap T_{A*}(U(X)) \\ne \\emptyset $ .",
"Hence it is enough to show $D_A^+ \\cap T_{A*}(\\partial U(X)) = \\emptyset $ .",
"This is obvious by Lemma REF .",
"We set $(D_A^+)^{\\vee } = \\lbrace \\sigma _{(xL , yL)} \\in V(X) | (yL)^2 > 2, x> \\frac{n_A}{r_A} \\rbrace .$ Corollary 7.3 For any $\\sigma \\in (D_A^+)^{\\vee }$ , $T_A^{-1}({\\mathcal {O}}_x)$ is $\\sigma $ -stable.",
"In particular $(D_A^+)^{\\vee } \\subset T_{A*}^{-1} (U(X))$ .",
"Since $D_A^+ \\subset T_{A*} (U(X)) \\cap U(X)$ by Proposition REF , we see $T_{A*}^{-1} (D_A^+) \\subset U(X) \\cap T_{A*}^{-1}(U(X)).$ Since the $\\sigma $ -stability is equivalent to the $\\sigma \\cdot \\tilde{g}$ -stability for any $\\tilde{g} \\in \\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^{+}(2, {\\mathbb {R}})$ , it is enough to show that $T_{A*}^{-1} (D_A^+)/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2, {\\mathbb {R}}) = (D_A^+)^{\\vee }/\\mathop {\\widetilde{\\mathrm {GL}}}\\nolimits ^+(2,{\\mathbb {R}})$ .",
"This is obvious from Lemma REF .",
"Remark 7.4 In the article [3], the possibility of the stable complexes in large volume limits is referred.",
"Hence Corollary REF gives the proof of this prediction."
],
[
"Acknowledgement",
"The author is partially supported by Grant-in-Aid for Scientific Research (S), No 22224001."
]
] | 1204.1128 |
[
[
"Reclaiming the energy of a schedule: models and algorithms"
],
[
"Abstract We consider a task graph to be executed on a set of processors.",
"We assume that the mapping is given, say by an ordered list of tasks to execute on each processor, and we aim at optimizing the energy consumption while enforcing a prescribed bound on the execution time.",
"While it is not possible to change the allocation of a task, it is possible to change its speed.",
"Rather than using a local approach such as backfilling, we consider the problem as a whole and study the impact of several speed variation models on its complexity.",
"For continuous speeds, we give a closed-form formula for trees and series-parallel graphs, and we cast the problem into a geometric programming problem for general directed acyclic graphs.",
"We show that the classical dynamic voltage and frequency scaling (DVFS) model with discrete modes leads to a NP-complete problem, even if the modes are regularly distributed (an important particular case in practice, which we analyze as the incremental model).",
"On the contrary, the VDD-hopping model leads to a polynomial solution.",
"Finally, we provide an approximation algorithm for the incremental model, which we extend for the general DVFS model."
],
[
"Introduction",
"The energy consumption of computational platforms has recently become a critical problem, both for economic and environmental reasons [25].",
"As an example, the Earth Simulator requires about 12 MW (Mega Watts) of peak power, and PetaFlop systems may require 100 MW of power, nearly the output of a small power plant (300 MW).",
"At $100 per MW.Hour, peak operation of a PetaFlop machine may thus cost $10,000 per hour [12].",
"Current estimates state that cooling costs $1 to $3 per watt of heat dissipated [31].",
"This is just one of the many economical reasons why energy-aware scheduling has proved to be an important issue in the past decade, even without considering battery-powered systems such as laptops and embedded systems.",
"As an example, the Green500 list (www.green500.org) provides rankings of the most energy-efficient supercomputers in the world, therefore raising even more awareness about power consumption.",
"To help reduce energy dissipation, processors can run at different speeds.",
"Their power consumption is the sum of a static part (the cost for a processor to be turned on) and a dynamic part, which is a strictly convex function of the processor speed, so that the execution of a given amount of work costs more power if a processor runs in a higher mode [15].",
"More precisely, a processor running at speed $s$ dissipates $s^3$ watts [17], [28], [7], [2], [10] per time-unit, hence consumes $s^3 \\times d$ joules when operated during $d$ units of time.",
"Faster speeds allow for a faster execution, but they also lead to a much higher (supra-linear) power consumption.",
"Energy-aware scheduling aims at minimizing the energy consumed during the execution of the target application.",
"Obviously, it makes sense only if it is coupled with some performance bound to achieve, otherwise, the optimal solution always is to run each processor at the slowest possible speed.",
"In this paper, we investigate energy-aware scheduling strategies for executing a task graph on a set of processors.",
"The main originality is that we assume that the mapping of the task graph is given, say by an ordered list of tasks to execute on each processor.",
"There are many situations in which this problem is important, such as optimizing for legacy applications, or accounting for affinities between tasks and resources, or even when tasks are pre-allocated [29], for example for security reasons.",
"In such situations, assume that a list-schedule has been computed for the task graph, and that its execution time should not exceed a deadline $D$ .",
"We do not have the freedom to change the assignment of a given task, but we can change its speed to reduce energy consumption, provided that the deadline $D$ is not exceeded after the speed change.",
"Rather than using a local approach such as backfilling [32], [27], which only reclaims gaps in the schedule, we consider the problem as a whole, and we assess the impact of several speed variation models on its complexity.",
"More precisely, we investigate the following models: Continuous model.",
"Processors can have arbitrary speeds, and can vary them continuously: this model is unrealistic (any possible value of the speed, say $\\sqrt{e^{^\\pi }}$ , cannot be obtained) but it is theoretically appealing [3].",
"A maximum speed, $s_{\\mathit {max}}$ , cannot be exceeded.",
"Discrete model.",
"Processors have a discrete number of predefined speeds (or frequencies), which correspond to different voltages that the processor can be subjected to [26].",
"Switching frequencies is not allowed during the execution of a given task, but two different tasks scheduled on a same processor can be executed at different frequencies.",
"Vdd-Hopping model.",
"This model is similar to the Discrete one, except that switching modes during the execution of a given task is allowed: any rational speed can be simulated, by simply switching, at the appropriate time during the execution of a task, between two consecutive modes [24].",
"Incremental model.",
"In this variant of the Discrete model, we introduce a value $\\delta $ that corresponds the minimum permissible speed increment, induced by the minimum voltage increment that can be achieved when controlling the processor CPU.",
"This new model aims at capturing a realistic version of the Discrete model, where the different modes are spread regularly instead of arbitrarily chosen.",
"Our main contributions are the following.",
"For the Continuous model, we give a closed-form formula for trees and series-parallel graphs, and we cast the problem into a geometric programming problem [6] for general DAGs.",
"For the Vdd-Hopping model, we show that the optimal solution for general DAGs can be computed in polynomial time, using a (rational) linear program.",
"Finally, for the Discrete and Incremental models, we show that the problem is NP-complete.",
"Furthermore, we provide approximation algorithms which rely on the polynomial algorithm for the Vdd-Hopping model, and we compare their solution with the optimal Continuous solution.",
"The paper is organized as follows.",
"We start with a survey of related literature in Section .",
"We then provide the formal description of the framework and of the energy models in Section , together with a simple example to illustrate the different models.",
"The next two sections constitute the heart of the paper: in Section , we provide analytical formulas for continuous speeds, and the formulation into the convex optimization problem.",
"In Section , we assess the complexity of the problem with all the discrete models: Discrete, Vdd-Hopping and Incremental, and we discuss approximation algorithms.",
"Finally we conclude in Section ."
],
[
"Related work",
"Reducing the energy consumption of computational platforms is an important research topic, and many techniques at the process, circuit design, and micro-architectural levels have been proposed [23], [21], [14].",
"The dynamic voltage and frequency scaling (DVFS) technique has been extensively studied, since it may lead to efficient energy/performance trade-offs [18], [12], [3], [9], [20], [34], [32].",
"Current microprocessors (for instance, from AMD [1] and Intel [16]) allow the speed to be set dynamically.",
"Indeed, by lowering supply voltage, hence processor clock frequency, it is possible to achieve important reductions in power consumption, without necessarily increasing the execution time.",
"We first discuss different optimization problems that arise in this context.",
"Then we review energy models."
],
[
"DVFS and optimization problems",
"When dealing with energy consumption, the most usual optimization function consists in minimizing the energy consumption, while ensuring a deadline on the execution time (i.e., a real-time constraint), as discussed in the following papers.",
"In [26], Okuma et al.",
"demonstrate that voltage scaling is far more effective than the shutdown approach, which simply stops the power supply when the system is inactive.",
"Their target processor employs just a few discretely variable voltages.",
"De Langen and Juurlink [22] discuss leakage-aware scheduling heuristics which investigate both DVS and processor shutdown, since static power consumption due to leakage current is expected to increase significantly.",
"Chen et al.",
"[8] consider parallel sparse applications, and they show that when scheduling applications modeled by a directed acyclic graph with a well-identified critical path, it is possible to lower the voltage during non-critical execution of tasks, with no impact on the execution time.",
"Similarly, Wang et al.",
"[32] study the slack time for non-critical jobs, they extend their execution time and thus reduce the energy consumption without increasing the total execution time.",
"Kim et al.",
"[20] provide power-aware scheduling algorithms for bag-of-tasks applications with deadline constraints, based on dynamic voltage scaling.",
"Their goal is to minimize power consumption as well as to meet the deadlines specified by application users.",
"For real-time embedded systems, slack reclamation techniques are used.",
"Lee and Sakurai [23] show how to exploit slack time arising from workload variation, thanks to a software feedback control of supply voltage.",
"Prathipati [27] discusses techniques to take advantage of run-time variations in the execution time of tasks; it determines the minimum voltage under which each task can be executed, while guaranteeing the deadlines of each task.",
"Then, experiments are conducted on the Intel StrongArm SA-1100 processor, which has eleven different frequencies, and the Intel PXA250 XScale embedded processor with four frequencies.",
"In [33], the goal of Xu et al.",
"is to schedule a set of independent tasks, given a worst case execution cycle (WCEC) for each task, and a global deadline, while accounting for time and energy penalties when the processor frequency is changing.",
"The frequency of the processor can be lowered when some slack is obtained dynamically, typically when a task runs faster than its WCEC.",
"Yang and Lin [34] discuss algorithms with preemption, using DVS techniques; substantial energy can be saved using these algorithms, which succeed to claim the static and dynamic slack time, with little overhead.",
"Since an increasing number of systems are powered by batteries, maximizing battery life also is an important optimization problem.",
"Battery-efficient systems can be obtained with similar techniques of dynamic voltage and frequency scaling, as described by Lahiri et al.",
"in [21].",
"Another optimization criterion is the energy-delay product, since it accounts for a trade-off between performance and energy consumption, as for instance discussed by Gonzalez and Horowitz in [13].",
"We do not discuss further these latter optimization problems, since our goal is to minimize the energy consumption, with a fixed deadline.",
"In this paper, the application is a task graph (directed acyclic graph), and we assume that the mapping, i.e., an ordered list of tasks to execute on each processor, is given.",
"Hence, our problem is closely related to slack reclamation techniques, but instead on focusing on non-critical tasks as for instance in [32], we consider the problem as a whole.",
"Our contribution is to perform an exhaustive complexity study for different energy models.",
"In the next paragraph, we discuss related work on each energy model."
],
[
"Energy models",
"Several energy models are considered in the literature, and they can all be categorized in one of the four models investigated in this paper, i.e., Continuous, Discrete, Vdd-Hopping or Incremental.",
"The Continuous model is used mainly for theoretical studies.",
"For instance, Yao et al.",
"[35], followed by Bansal et al.",
"[3], aim at scheduling a collection of tasks (with release time, deadline and amount of work), and the solution is the time at which each task is scheduled, but also, the speed at which the task is executed.",
"In these papers, the speed can take any value, hence following the Continuous model.",
"We believe that the most widely used model is the Discrete one.",
"Indeed, processors have currently only a few discrete number of possible frequencies [1], [16], [26], [27].",
"Therefore, most of the papers discussed above follow this model.",
"Some studies exploit the continuous model to determine the smallest frequency required to run a task, and then choose the closest upper discrete value, as for instance [27] and [36].",
"Recently, a new local dynamic voltage scaling architecture has been developed, based on the Vdd-Hopping model [24], [4], [5].",
"It was shown in [23] that significant power can be saved by using two distinct voltages, and architectures using this principle have been developed (see for instance [19]).",
"Compared to traditional power converters, a new design with no needs for large passives or costly technological options has been validated in a STMicroelectronics CMOS 65nm low-power technology [24].",
"To the best of our knowledge, this paper introduces the Incremental model for the first time.",
"The main rationale is that future technologies may well have an increased number of possible frequencies, and these will follow a regular pattern.",
"For instance, note that the SA-1100 processor, considered in [27], has eleven frequencies which are equidistant, i.e., they follow the Incremental model.",
"Lee and Sakurai [23] exploit discrete levels of clock frequency as $f$ , $f/2$ , $f/3$ , ..., where $f$ is the master (i.e., the higher) system clock frequency.",
"This model is closer to the Discrete model, although it exhibits a regular pattern similarly to the Incremental model.",
"Our work is the first attempt to compare these different models: on the one hand, we assess the impact of the model on the problem complexity (polynomial vs NP-hard), and on the other hand, we provide approximation algorithms building upon these results.",
"The closest work to ours is the paper by Zhang et al.",
"[36], in which the authors also consider the mapping of directed acyclic graphs, and compare the Discrete and the Continuous models.",
"We go beyond their work in this paper, with an exhaustive complexity study, closed-form formulas for the continuous model, and the comparison with the Vdd-Hopping and Incremental models."
],
[
"Framework",
"First we detail the optimization problem in Section REF .",
"Then we describe the four energy models in Section REF .",
"Finally, we illustrate the models and motivate the problem with an example in Section REF ."
],
[
"Optimization problem",
"Consider an application task graph $\\mathcal {G}=(V,\\mathcal {E})$ , with $n=|V|$ tasks denoted as $V= \\lbrace T_1, T_2, \\dots , T_n\\rbrace $ , and where the set $\\mathcal {E}$ denotes the precedence edges between tasks.",
"Task $T_i$ has a cost $w_i$ for $1 \\le i \\le n$ .",
"We assume that the tasks in $\\mathcal {G}$ have been allocated onto a parallel platform made up of identical processors.",
"We define the execution graph generated by this allocation as the graph $G=(V,E)$ , with the following augmented set of edges: $\\mathcal {E} \\subseteq E$ : if an edge exists in the precedence graph, it also exists in the execution graph; if $T_1$ and $T_2$ are executed successively, in this order, on the same processor, then $(T_1,T_2)\\in ~\\!E$ .",
"The goal is to the minimize the energy consumed during the execution while enforcing a deadline $D$ on the execution time.",
"We formalize the optimization problem in the simpler case where each task is executed at constant speed.",
"This strategy is optimal for the Continuous model (by a convexity argument) and for the Discrete and Incremental models (by definition).",
"For the Vdd-Hopping model, we reformulate the problem in Section REF .",
"Let $d_i$ be the duration of the execution of task $T_i$ , $t_i$ its completion time, and $s_i$ the speed at which it is executed.",
"We obtain the following formulation of the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem, given an execution graph $G= (V,E)$ and a deadline $D$ ; the $s_i$ values are variables, whose values are constrained by the energy model (see Section REF ).",
"$\\begin{array}{lrl}\\text{~~~Minimize}& & \\sum _{i=1}^{n} s_i^3 \\times d_i\\\\\\text{~~~subject to} &\\text{(i)} & w_i = s_i \\times d_i \\text{ for each task } T_i \\in V\\\\&\\text{(ii)} & t_i + d_j \\le t_j \\text{ for each edge } (T_i,T_j) \\in E\\\\&\\text{(iii)} & t_i \\le D \\text{ for each task } T_i \\in V \\end{array}$ Constraint (i) states that the whole task can be executed in time $d_i$ using speed $s_i$ .",
"Constraint (ii) accounts for all dependencies, and constraint (iii) ensures that the execution time does not exceed the deadline $D$ .",
"The energy consumed throughout the execution is the objective function.",
"It is the sum, for each task, of the energy consumed by this task, as we detail in the next section.",
"Note that $d_i=w_i/s_i$ , and therefore the objective function can also be expressed as $\\sum _{i=1}^{n} s_i^2 \\times w_i$ .",
"Energy models In all models, when a processor operates at speed $s$ during $d$ time-units, the corresponding consumed energy is $s^3 \\times d$ , which is the dynamic part of the energy consumption, following the classical models of the literature [17], [28], [7], [2], [10].",
"Note that we do not take static energy into account, because all processors are up and alive during the whole execution.",
"We now detail the possible speed values in each energy model, which should be added as a constraint in Equation (REF ).",
"In the Continuous model, processors can have arbitrary speeds, from 0 to a maximum value $s_{\\mathit {max}}$ , and a processor can change its speed at any time during execution.",
"In the Discrete model, processors have a set of possible speed values, or modes, denoted as $s_1,...,s_m$ .",
"There is no assumption on the range and distribution of these modes.",
"The speed of a processor cannot change during the computation of a task, but it can change from task to task.",
"In the Vdd-Hopping model, a processor can run at different speeds $s_1,...,s_m$ , as in the previous model, but it can also change its speed during a computation.",
"The energy consumed during the execution of one task is the sum, on each time interval with constant speed $s$ , of the energy consumed during this interval at speed $s$ .",
"In the Incremental model, we introduce a value $\\delta $ that corresponds to the minimum permissible speed (i.e., voltage) increment.",
"That means that possible speed values are obtained as $s=s_{\\mathit {min}}+ i\\times \\delta $ , where $i$ is an integer such that $0\\le i \\le \\frac{s_{\\mathit {max}}-s_{\\mathit {min}}}{\\delta }$ .",
"Admissible speeds lie in the interval $[s_{\\mathit {min}},s_{\\mathit {max}}]$ .",
"This new model aims at capturing a realistic version of the Discrete model, where the different modes are spread regularly between $s_1=s_{\\mathit {min}}$ and $s_m=s_{\\mathit {max}}$ , instead of being arbitrarily chosen.",
"It is intended as the modern counterpart of a potentiometer knob!",
"Example Consider an application with four tasks of costs $w_1=3$ , $w_2=2$ , $w_3=1$ and $w_4=2$ , and one precedence constraint $T_1\\rightarrow T_3$ .",
"We assume that $T_1$ and $T_2$ are allocated, in this order, onto processor $P_1$ , while $T_3$ and $T_4$ are allocated, in this order, on processor $P_2$ .",
"The resulting execution graph $G$ is given in Figure REF , with two precedence constraints added to the initial task graph.",
"The deadline on the execution time is $D=1.5$ .",
"We set the maximum speed to $s_{\\mathit {max}}=6$ for the Continuous model.",
"For the Discrete and Vdd-Hopping models, we use the set of speeds $s^{(d)}_1=2$ , $s^{(d)}_2=5$ and $s^{(d)}_3=6$ .",
"Finally, for the Incremental model, we set $\\delta =2$ , $s_{\\mathit {min}}=2$ and $s_{\\mathit {max}}=6$ , so that possible speeds are $s^{(i)}_1=2$ , $s^{(i)}_2=4$ and $s^{(i)}_3=6$ .",
"We aim at finding the optimal execution speed $s_i$ for each task $T_i$ ($1\\le i \\le 4$ ), i.e., the values of $s_i$ which minimize the energy consumption.",
"With the Continuous model, the optimal speeds are non rational values, and we obtain $s_1 = \\frac{2}{3}(3+35^{1/3})\\simeq 4.18;\\quad s_2 = s_1\\times \\frac{2}{35^{1/3}}\\simeq 2.56;\\quad s_3 = s_4 = s_1\\times \\frac{3}{35^{1/3}}\\; \\simeq 3.83.$ Figure: Execution graph for the example.Note that all speeds are lower than the maximum $s_{\\mathit {max}}$ .",
"These values are obtained thanks to the formulas derived in Section .",
"The energy consumption is then $E^{(c)}_{opt} = \\sum _{i=1}^4 w_i \\times s_i^2= 3.s_1^2+2.s_2^2+ 3.s_3^2 \\simeq 109.6$ .",
"The execution time is $\\frac{w_1}{s_1} + \\max \\left(\\frac{w_2}{s_2},\\frac{w_3+w_4}{s_3}\\right)$ , and with this solution, it is equal to the deadline $D$ (actually, both processors reach the deadline, otherwise we could slow down the execution of one task).",
"For the Discrete model, if we execute all tasks at speed $s^{(d)}_2=5$ , we obtain an energy $E=8\\times 5^2=200$ .",
"A better solution is obtained with $s_1=s^{(d)}_3=6$ , $s_2=s_3=s^{(d)}_1=2$ and $s_4=s^{(d)}_2=5$ , which turns out to be optimal: $E^{(d)}_{opt}=3\\times 36+(2+1)\\times 4+2\\times 25=170$ .",
"Note that $E^{(d)}_{opt}>E^{(c)}_{opt}$ , i.e., the optimal energy consumption with the Discrete model is much higher than the one achieved with the Continuous model.",
"Indeed, in this case, even though the first processor executes during $3/6 + 2/2 = D$ time units, the second processor remains idle since $3/6 + 1/2 + 2/5=1.4<D$ .",
"The problem turns out to be NP-hard (see Section REF ), and the solution has been found by performing an exhaustive search.",
"With the Vdd-Hopping model, we set $s_1=s^{(d)}_2=5$ ; for the other tasks, we run part of the time at speed $s^{(d)}_2=5$ , and part of the time at speed $s^{(d)}_1=2$ in order to use the idle time and lower the energy consumption.",
"$T_2$ is executed at speed $s^{(d)}_1$ during time $\\frac{5}{6}$ and at speed $s^{(d)}_2$ during time $\\frac{2}{30}$ (i.e., the first processor executes during time $3/5 + 5/6 + 2/30 = 1.5 = D$ , and all the work for $T_2$ is done: $2\\times 5/6 + 5\\times 2/30 = 2 = w_2$ ).",
"$T_3$ is executed at speed $s^{(d)}_2$ (during time $1/5$ ), and finally $T_4$ is executed at speed $s^{(d)}_1$ during time $0.5$ and at speed $s^{(d)}_2$ during time $1/5$ (i.e., the second processor executes during time $3/5 + 1/5 + 0.5 + 1/5 = 1.5 = D$ , and all the work for $T_4$ is done: $2\\times 0.5 + 5\\times 1/5 = 2 = w_4$ ).",
"This set of speeds turns out to be optimal (i.e., it is the optimal solution of the linear program introduced in Section REF ), with an energy consumption $E^{(v)}_{opt}=(3/5 + 2/30 + 1/5 + 1/5) \\times 5^3 + (5/6 + 0.5)\\times 2^3 = 144$ .",
"As expected, $E^{(c)}_{opt}\\le E^{(v)}_{opt} \\le E^{(d)}_{opt}$ , i.e., the Vdd-Hopping solution stands between the optimal Continuous solution, and the more constrained Discrete solution.",
"For the Incremental model, the reasoning is similar to the Discrete case, and the optimal solution is obtained by an exhaustive search: all tasks should be executed at speed $s^{(i)}_2=4$ , with an energy consumption $E^{(i)}_{opt}=8 \\times 4^2 = 128 > E^{(c)}_{opt}$ .",
"It turns out to be better than Discrete and Vdd-Hopping, since it has different discrete values of energy which are more appropriate for this example.",
"The Continuous model With the Continuous model, processor speeds can take any value between 0 and $s_{\\mathit {max}}$ .",
"First we prove that, with this model, the processors do not change their speed during the execution of a task (Section REF ).",
"Then, we derive in Section REF the optimal speed values for special execution graph structures, expressed as closed form algebraic formulas, and we show that these values may be irrational (as already illustrated in the example in Section REF ).",
"Finally, we formulate the problem for general DAGs as a convex optimization program in Section REF .",
"Preliminary lemma Lemma 1 (constant speed per task) With the Continuous model, each task is executed at constant speed, i.e., a processor does not change its speed during the execution of a task.",
"Suppose that in the optimal solution, there is a task whose speed changes during the execution.",
"Consider the first time-step at which the change occurs: the computation begins at speed $s$ from time $t$ to time $t^{\\prime }$ , and then continues at speed $s^{\\prime }$ until time $t^{\\prime \\prime }$ .",
"The total energy consumption for this task in the time interval $[t;t^{\\prime \\prime }]$ is $E=(t^{\\prime }-t)\\times s^3+(t^{\\prime \\prime }-t^{\\prime })\\times (s^{\\prime })^3$ .",
"Moreover, the amount of work done for this task is $W=(t^{\\prime }-t)\\times s+(t^{\\prime \\prime }-t^{\\prime })\\times s^{\\prime }$ .",
"If we run the task during the whole interval $[t;t^{\\prime \\prime }]$ at constant speed $W/(t^{\\prime \\prime }-t)$ , the same amount of work is done within the same time.",
"However, the energy consumption during this interval of time is now $E^{\\prime }=(t^{\\prime \\prime }-t)\\times (W/(t^{\\prime \\prime }-t))^3$ .",
"By convexity of the function $x \\mapsto x^3$ , we obtain $E^{\\prime }<E$ since $t<t^{\\prime }<t^{\\prime \\prime }$ .",
"This contradicts the hypothesis of optimality of the first solution, which concludes the proof.",
"Special execution graphs Independent tasks Consider the problem of minimizing the energy of $n$ independent tasks (i.e., each task is mapped onto a distinct processor, and there are no precedence constraints in the execution graph), while enforcing a deadline $D$ .",
"Proposition 1 (independent tasks) When $G$ is composed of independent tasks $\\lbrace T_1, \\dots , T_n\\rbrace $ , the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained when each task $T_i$ ($1\\le i \\le n$ ) is computed at speed $s_i=\\frac{w_i}{D}$ .",
"If there is a task $T_i$ such that $s_i > s_{\\mathit {max}}$ , then the problem has no solution.",
"For task $T_i$ , the speed $s_i$ corresponds to the slowest speed at which the processor can execute the task, so that the deadline is not exceeded.",
"If $s_i > s_{\\mathit {max}}$ , the corresponding processor will never be able to complete its execution before the deadline, therefore there is no solution.",
"To conclude the proof, we note that any other solution would have higher values of $s_i$ because of the deadline constraint, and hence a higher energy consumption.",
"Therefore, this solution is optimal.",
"Linear chain of tasks This case corresponds for instance to $n$ independent tasks $\\lbrace T_1,\\dots , T_n\\rbrace $ executed onto a single processor.",
"The execution graph is then a linear chain (order of execution of the tasks), with $T_i \\rightarrow T_{i+1}$ , for $1\\le i < n$ .",
"Proposition 2 (linear chain) $\\;\\;$ When $G$ is a linear chain of tasks, the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained when each task is executed at speed $s=\\frac{W}{D}$ , with $W=\\sum _{i=1}^n w_i$ .",
"If $s>s_{\\mathit {max}}$ , then there is no solution.",
"Suppose that in the optimal solution, tasks $T_i$ and $T_j$ are such that $s_i < s_j$ .",
"The total energy consumption is $E_{opt}$ .",
"We define $s$ such that the execution of both tasks running at speed $s$ takes the same amount of time than in the optimal solution, i.e., $(w_i+w_j)/s = w_i/s_i + w_j/s_j$ : $s = \\frac{(w_i+w_j)}{w_is_j + w_js_i} \\times s_is_j$ .",
"Note that $s_i < s < s_j$ (it is the barycenter of two points with positive mass).",
"We consider a solution such that the speed of task $T_k$ , for $1\\le k \\le n$ , with $k\\ne i$ and $k \\ne j$ , is the same as in the optimal solution, and the speed of tasks $T_i$ and $T_j$ is $s$ .",
"By definition of $s$ , the execution time has not been modified.",
"The energy consumption of this solution is $E$ , where $E_{opt}-E = w_is_i^{2} + w_js_j^{2} -(w_i+w_j)s^{2}$ , i.e., the difference of energy with the optimal solution is only impacted by tasks $T_i$ and $T_j$ , for which the speed has been modified.",
"By convexity of the function $x \\mapsto x^{2}$ , we obtain $E_{opt}>E$ , which contradicts its optimality.",
"Therefore, in the optimal solution, all tasks have the same execution speed.",
"Moreover, the energy consumption is minimized when the speed is as low as possible, while the deadline is not exceeded.",
"Therefore, the execution speed of all tasks is $s=W/D$ .",
"Corollary 1 A linear chain with $n$ tasks is equivalent to a single task of cost $W=\\sum _{i=1}^n w_i$ .",
"Indeed, in the optimal solution, the $n$ tasks are executed at the same speed, and they can be replaced by a single task of cost $W$ , which is executed at the same speed and consumes the same amount of energy.",
"Fork and join graphs Let $V\\!=\\!\\lbrace T_1,\\dots ,T_n\\rbrace $ .",
"We consider either a fork graph $G = (V\\cup \\lbrace T_{0}\\rbrace , E)$ , with $E=\\lbrace (T_{0},T_i),T_i \\in V\\rbrace $ , or a join graph $G =(V\\cup \\lbrace T_{0}\\rbrace , E)$ , with $E=\\lbrace (T_i,T_{0}),T_i\\in V\\rbrace $ .",
"$T_{0}$ is either the source of the fork or the sink of the join.",
"Theorem 1 (fork and join graphs) When $G$ is a fork (resp.",
"join) execution graph with $n+1$ tasks $T_0,T_1,\\dots ,T_n$ , the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is the following: ${ }\\quad \\bullet $ the execution speed of the source (resp.",
"sink) $T_0$ is $s_{0} = \\dfrac{\\left(\\sum _{i=1}^n w_i^{3}\\right)^{\\frac{1}{3}} +w_{0}}{D}$ ; ${ }\\quad \\bullet $ for the other tasks $T_i$ , $1\\le i \\le n$ , we have $s_i = s_{0} \\times \\dfrac{w_i}{\\left(\\sum _{i=1}^nw_i^{3}\\right)^{\\frac{1}{3}}}$ if $s_{0}\\le s_{\\mathit {max}}$ .",
"Otherwise, $T_{0}$ should be executed at speed $s_0=s_{\\mathit {max}}$ , and the other speeds are $s_i = \\frac{w_i}{D^{\\prime }}$ , with $D^{\\prime }=D-\\frac{w_{0}}{s_{\\mathit {max}}}$ , if they do not exceed $s_{\\mathit {max}}$ (Proposition REF for independent tasks).",
"Otherwise there is no solution.",
"If no speed exceeds $s_{\\mathit {max}}$ , the corresponding energy consumption is ${\\bf minE}(G,D)=\\frac{\\left((\\sum _{i=1}^n w_i^3)^{\\frac{1}{3}} +w_0\\right)^3}{D^2}\\; .$ Let $t_0=\\frac{w_{0}}{s_0}$ .",
"Then, the source or the sink requires a time $t_0$ for execution.",
"For $1\\le i \\le n$ , task $T_i$ must be executed within a time $D-t_0$ so that the deadline is respected.",
"Given $t_0$ , we can compute the speed $s_i$ for task $T_i$ using Theorem REF , since the tasks are independent: $s_i = \\frac{w_i}{D-t_0} = w_i \\cdot \\frac{s_0}{s_0 D - w_0}$ .",
"The objective is therefore to minimize $\\sum _{i=0}^n w_i s_i^{2}$ , which is a function of $s_0$ : $\\sum _{i=0}^n w_i s_i^{2} = w_0s_0^2 +\\sum _{i=1}^n w_i^3 \\cdot \\frac{s_0^2}{(s_0D - w_0)^2}= s_0^2 \\left( w_0 + \\frac{\\sum _{i=1}^n w_i^3}{(s_0 D - w_0)^2}\\right)= f(s_0).$ Let $W_3 = \\sum _{i=1}^n w_i^3$ .",
"In order to find the value of $s_0$ which minimizes this function, we study the function $f(x)$ , for $x>0$ .",
"$f^{\\prime }(x) = 2x \\left(w_{0} + \\frac{W_3}{(x D-w_{0})^2}\\right) -2D \\cdot x^{2} \\cdot \\frac{W_3}{(x D-w_{0})^{3}} $ , and therefore $f^{\\prime }(x)=0$ for $x=(W_3^{\\frac{1}{3}} + w_0)/D$ .",
"We conclude that the optimal speed for task $T_0$ is $s_0 = \\frac{\\left(\\sum _{i=1}^n w_i^{3} \\right)^{\\frac{1}{3}} +w_{0}}{D}$ , if $s_0 \\le s_{\\mathit {max}}$ .",
"Otherwise, $T_0$ should be executed at the maximum speed $s_0 =s_{\\mathit {max}}$ , since it is the bottleneck task.",
"In any case, for $1\\le i \\le n$ , the optimal speed for task $T_i$ is $s_i = w_i \\frac{s_0}{s_0 D - w_0}$ .",
"Finally, we compute the exact expression of minE$(G,D) =f(s_0)$ , when $s_0\\le s_{\\mathit {max}}$ : $f(s_0) = s_0^2 \\left( w_0 + \\frac{W_3}{(s_0 D - w_0)^2}\\right)=\\left(\\frac{W_3^{\\frac{1}{3}} +w_0}{D}\\right)^2 \\left(\\frac{W_3}{W_3^{2/3}}+w_0 \\right)=\\frac{\\left(W_3^{\\frac{1}{3}} +w_0\\right)^3}{D^2}, $ which concludes the proof.",
"Corollary 2 (equivalent tasks for speed) Consider a fork or join graph with tasks $T_i$ , $0\\le i \\le n$ , and a deadline $D$ , and assume that the speeds in the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ do not exceed $s_{\\mathit {max}}$ .",
"Then, these speeds are the same as in the optimal solution for $n+1$ independent tasks $T^{\\prime }_0, T^{\\prime }_1, \\dots , T^{\\prime }_n$ , where $w^{\\prime }_0 = \\left(\\sum _{i=1}^n w_{i}^{3} \\right)^{\\frac{1}{3}} + w_{0}$ , and, for $1\\le i \\le n$ , $w^{\\prime }_i=w^{\\prime }_0 \\cdot \\frac{w_i}{\\left(\\sum _{i=1}^n w_{i}^{3}\\right)^{\\frac{1}{3}}}\\;$ .",
"Corollary 3 (equivalent task for energy) Consider a fork or join graph $G$ and a deadline $D$ , and assume that the speeds in the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ do not exceed $s_{\\mathit {max}}$ .",
"We say that the graph $G$ is equivalent to the graph $G^{(eq)}$ , consisting of a single task $T^{(eq)}_0$ of weight $w^{(eq)}_0 = \\left(\\sum _{i=1}^n w_{i}^{3} \\right)^{\\frac{1}{3}} + w_{0}$ , because the minimum energy consumption of both graphs are identical: minE$(G,D)$ =minE$(G^{(eq)},D)$ .",
"Trees We extend the results on a fork graph for a tree $G=(V,E)$ with $|V|=n+1$ tasks.",
"Let $T_0$ be the root of the tree; it has $k$ children tasks, which are each themselves the root of a tree.",
"A tree can therefore be seen as a fork graph, where the tasks of the fork are trees.",
"The previous results for fork graphs naturally lead to an algorithm that peels off branches of the tree, starting with the leaves, and replaces each fork subgraph in the tree, composed of a root $T_0$ and $k$ children, by one task (as in Corollary REF ) which becomes the unique child of $T_0$ 's parent in the tree.",
"We say that this task is equivalent to the fork graph, since the optimal energy consumption will be the same.",
"The computation of the equivalent cost of this task is done thanks to a call to the eq procedure, while the tree procedure computes the solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ (see Algorithm REF ).",
"Note that the algorithm computes the minimum energy for a tree, but it does not return the speeds at which each task must be executed.",
"However, the algorithm returns the speed of the root task, and it is then straightforward to compute the speed of each children of the root task, and so on.",
"Theorem 2 (tree graphs) When $G$ is a tree rooted in $T_0$ ($T_0 \\in V$ , where $V$ is the set of tasks), the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ can be computed in polynomial time $O(|V|^2)$ .",
"Let $G$ be a tree graph rooted in $T_0$ .",
"The optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained with a call to tree $(G,T_0,D)$ , and we prove its optimality recursively on the depth of the tree.",
"Similarly to the case of the fork graphs, we reduce the tree to an equivalent task which, if executed alone within a deadline $D$ , consumes exactly the same amount of energy.",
"The procedure eq is the procedure which reduces a tree to its equivalent task (see Algorithm REF ).",
"If the tree has depth 0, then it is a single task, eq $(G,T_0)$ returns the equivalent cost $w_0$ , and the optimal execution speed is $\\frac{w_0}{D}$ (see Proposition REF ).",
"There is a solution if and only if this speed is not greater than $s_{\\mathit {max}}$ , and then the corresponding energy consumption is $\\frac{w_0^3}{D^2}$ , as returned by the algorithm.",
"Assume now that for any tree of depth $i<p$ , eq computes its equivalent cost, and tree returns its optimal energy consumption.",
"We consider a tree $G$ of depth $p$ rooted in $T_0$ : $G = T_0 \\cup \\lbrace G_i\\rbrace $ , where each subgraph $G_i$ is a tree, rooted in $T_i$ , of maximum depth $p-1$ .",
"As in the case of forks, we know that each subtree $G_i$ has a deadline $D-x$ , where $x=\\frac{w_0}{s_0}$ , and $s_0$ is the speed at which task $T_0$ is executed.",
"By induction hypothesis, we suppose that each graph $G_i$ is equivalent to a single task, $T^{\\prime }_i$ , of cost $w^{\\prime }_i$ (as computed by the procedure eq).",
"We can then use the results obtained on forks to compute $w^{(eq)}_0$ (see proof of Theorem REF ): $w^{(eq)}_0 = \\displaystyle \\left( \\sum _{i} (w^{\\prime }_{i})^{3} \\right)^{\\frac{1}{3}} + w_{0}$ .",
"Finally the tree is equivalent to one task of cost $w^{(eq)}_0$ , and if $\\frac{w^{(eq)}_0}{D}\\le s_{\\mathit {max}}$ , the energy consumption is $\\frac{\\left(w^{(eq)}_0\\right)^3}{D^2}$ , and no speed exceeds $s_{\\mathit {max}}$ .",
"Note that the speed of a task is always greater than the speed of its successors.",
"Therefore, if $\\frac{w^{(eq)}_0}{D}> s_{\\mathit {max}}$ , we execute the root of the tree at speed $s_{\\mathit {max}}$ and then process each subtree $G_i$ independently.",
"Of course, there is no solution if $\\frac{w_0}{s_{\\mathit {max}}}>D$ , and otherwise we perform the recursive calls to tree to process each subtree independently.",
"Their deadline is then $D-\\frac{w_0}{s_{\\mathit {max}}}$ .",
"To study the time complexity of this algorithm, first note that when calling tree $(G,T_0,D)$ , there might be at most $|V|$ recursive calls to tree, once at each node of the tree.",
"Without accounting for the recursive calls, the tree procedure performs one call to the eq procedure, which computes the cost of the equivalent task.",
"This eq procedure takes a time $O(|V|)$ , since we have to consider the $|V|$ tasks, and we add the costs one by one.",
"Therefore, the overall complexity is in $O(|V|^2)$ .",
"Solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ for trees.",
"procedure tree (tree $G$ , root $T_0$ , deadline $D$ ) Let $w$ =eq (tree $G$ , root $T_0$ ) $\\frac{w}{D}\\le s_{\\mathit {max}}$ $\\frac{w^3}{D^2}$ ; $\\frac{w_0}{s_{\\mathit {max}}} > D$ Error:No Solution; /* $T_0$ is executed at speed $s_{\\mathit {max}}$ */ $w_0 \\times s_{\\mathit {max}}^2 + \\displaystyle \\!\\!\\!\\sum _{G_i\\mbox{ subtreerooted in }T_i\\in \\mbox{children}(T_0)} \\!\\!\\!",
"{\\bf tree} \\left(G_i,T_i,D-\\frac{w_0}{s_{\\mathit {max}}}\\right)$ procedure eq (tree $G$ , root $T_0$ ) children($T_0$ )=$\\emptyset $ $w_0$ ; $\\left(\\displaystyle \\sum _{G_i\\mbox{ subtree rooted in }T_i\\in \\mbox{children}(T_0)}\\left(\\mbox{\\bf eq} (G_i,T_i)\\right)^3 \\right)^{\\frac{1}{3}} +w_0$ ; Series-parallel graphs We can further generalize our results to series-parallel graphs (SPGs), which are built from a sequence of compositions (parallel or series) of smaller-size SPGs.",
"The smallest SPG consists of two nodes connected by an edge (such a graph is called an elementary SPG).",
"The first node is the source, while the second one is the sink of the SPG.",
"When composing two SGPs in series, we merge the sink of the first SPG with the source of the second one.",
"For a parallel composition, the two sources are merged, as well as the two sinks, as illustrated in Figure REF .",
"Figure: Composition of series-parallel graphs (SPGs).We can extend the results for tree graphs to SPGs, by replacing step by step the SPGs by an equivalent task (procedure cost in Algorithm REF ): we can compute the equivalent cost for a series or parallel composition.",
"However, since it is no longer true that the speed of a task is always larger than the speed of its successor (as was the case in a tree), we have not been able to find a recursive property on the tasks that should be set to $s_{\\mathit {max}}$ , when one of the speeds obtained with the previous method exceeds $s_{\\mathit {max}}$ .",
"The problem of computing a closed form for a SPG with a finite value of $s_{\\mathit {max}}$ remains open.",
"Still, we have the following result when $s_{\\mathit {max}}= +\\infty $ : Theorem 3 (series-parallel graphs) When $G$ is a SPG, it is possible to compute recursively a closed form expression of the optimal solution of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ , assuming $s_{\\mathit {max}}=+\\infty $ , in polynomial time $O(|V|)$ , where $V$ is the set of tasks.",
"Let $G$ be a series-parallel graph.",
"The optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained with a call to SPG $(G,D)$ , and we prove its optimality recursively.",
"Similarly to trees, the main idea is to peel the graph off, and to transform it until there remains only a single equivalent task which, if executed alone within a deadline $D$ , would consume exactly the same amount of energy.",
"The procedure cost is the procedure which reduces a tree to its equivalent task (see Algorithm REF ).",
"The proof is done by induction on the number of compositions required to build the graph $G$ , $p$ .",
"If $p=0$ , $G$ is an elementary SPG consisting in two tasks, the source $T_0$ and the sink $T_1$ .",
"It is therefore a linear chain, and therefore equivalent to a single task whose cost is the sum of both costs, $w_0+w_1$ (see Corollary REF for linear chains).",
"The procedure cost returns therefore the correct equivalent cost, and SPG returns the minimum energy consumption.",
"Let us assume that the procedures return the correct equivalent cost and minimum energy consumption for any SPG consisting of $i<p$ compositions.",
"We consider a SPG $G$ , with $p$ compositions.",
"By definition, $G$ is a composition of two smaller-size SPGs, $G_1$ and $G_2$ , and both of these SPGs have strictly fewer than $p$ compositions.",
"We consider $G^{\\prime }_1$ and $G^{\\prime }_2$ , which are identical to $G_1$ and $G_2$ , except that the cost of their source and sink tasks are set to 0 (these costs are handled separately), and we can reduce both of these SPGs to an equivalent task, of respective costs $w^{\\prime }_1$ and $w^{\\prime }_2$ , by induction hypothesis.",
"There are two cases: If $G$ is a series composition, then after the reduction of $G^{\\prime }_1$ and $G^{\\prime }_2$ , we have a linear chain in which we consider the source $T_0$ of $G_1$ , the sink $T_1$ of $G_1$ (which is also the source of $G_2$ ), and the sink $T_2$ of $G_2$ .",
"The equivalent cost is therefore $w_0 + w^{\\prime }_1 + w_1 + w^{\\prime }_2 + w_2$ , thanks to Corollary REF for linear chains.",
"If $G$ is a parallel composition, the resulting graph is a fork-join graph, and we can use Corollaries REF and REF to compute the cost of the equivalent task, accounting for the source $T_0$ and the sink $T_1$ : $w_0 + \\left((w^{\\prime }_1)^3+(w^{\\prime }_2)^3\\right)^\\frac{1}{3} + w_1$ .",
"Once the cost of the equivalent task of the SPG has been computed with the call to cost $(G)$ , the optimal energy consumption is $\\frac{\\left({\\bf cost}(G)\\right)^3}{D^2}$ .",
"Contrarily to the case of tree graphs, since we never need to call the SPG procedure again because there is no constraint on $s_{\\mathit {max}}$ , the time complexity of the algorithm is the complexity of the cost procedure.",
"There is exactly one call to cost for each composition, and the number of compositions in the SPG is in $O(|V|)$ .",
"All operations in cost can be done in $O(1)$ , hence a complexity in $O(|V|)$ .",
"Solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ for series-parallel graphs.",
"procedure SPG (series-parallel graph $G$ , deadline $D$ ) $\\frac{\\left(\\mbox{\\bf cost}(G)\\right)^3}{D^2}$ ; procedure cost (series-parallel graph $G$ ) Let $T_0$ be the source of $G$ and $T_1$ its sink $G$ is composed of only two tasks, $T_0$ and $T_1$ $w_0+w_1$ ; /* $G$ is a composition of two SPGs $G_1$ and $G_2$ .",
"*/ For $i=1,2$ , let $G^{\\prime }_i=G_i$ where the cost of source and sink tasks is set to 0 $w^{\\prime }_1=\\mbox{\\bf cost}(G^{\\prime }_1); \\; w^{\\prime }_2=\\mbox{\\bf cost}(G^{\\prime }_2)$ $G$ is a series composition Let $T_0$ be the source of $G_1$ , $T_1$ be its sink, and $T_2$ be the sink of $G_2$ $w_0 + w^{\\prime }_1 + w_1 + w^{\\prime }_2 + w_2$ ; /* It is a parallel composition.",
"*/ Let $T_0$ be the source of $G$ , and $T_1$ be its sink $w_0 + \\left((w^{\\prime }_1)^3+(w^{\\prime }_2)^3\\right)^\\frac{1}{3} + w_1$ ; General DAGs For arbitrary execution graphs, we can rewrite the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem as follows: $\\begin{array}{lrl}\\text{~~~Minimize}& & \\sum _{i=1}^{n} u_i^{-2} \\times w_i\\\\\\text{~~~subject to}&\\text{(i)} & t_i + w_j \\times u_j \\le t_j \\text{ for each edge }(T_i,T_j) \\in E\\\\&\\text{(ii)} & t_i \\le D \\text{ for each task } T_i \\in V\\\\&\\text{(iii)} & u_i \\ge \\frac{1}{s_{\\mathit {max}}} \\text{ for each task } T_i \\in V\\\\\\end{array}$ Here, $u_i = 1/s_i$ is the inverse of the speed to execute task $T_i$ .",
"We now have a convex optimization problem to solve, with linear constraints in the non-negative variables $u_i$ and $t_i$ .",
"In fact, the objective function is a posynomial, so we have a geometric programming problem (see [6]) for which efficient numerical schemes exist.",
"However, as illustrated on simple fork graphs, the optimal speeds are not expected to be rational numbers but instead arbitrarily complex expressions (we have the cubic root of the sum of cubes for forks, and nested expressions of this form for trees).",
"From a computational complexity point of view, we do not know how to encode such numbers in polynomial size of the input (the rational task weights and the execution deadline).",
"Still, we can always solve the problem numerically and get fixed-size numbers which are good approximations of the optimal values.",
"In the following, we show that the total power consumption of any optimal schedule is constant throughout execution.",
"While this important property does not help to design an optimal solution, it shows that a schedule with large variations in its power consumption is likely to waste a lot of energy.",
"We need a few notations before stating the result.",
"Consider a schedule for a graph $G=(V,E)$ with $n$ tasks.",
"Task $T_i$ is executed at constant speed $s_i$ (see Lemma REF ) and during interval $[b_i,c_i]$ : $T_i$ begins its execution at time $b_i$ and completes it at time $c_i$ .",
"The total power consumption $P(t)$ of the schedule at time $t$ is defined as the sum of the power consumed by all tasks executing at time $t$ : $P(t) = \\sum _{1 \\le i \\le n, \\; t \\in [b_i,c_i]} s_i^3\\; .",
"$ Theorem 4 Consider an instance of Continuous, and an optimal schedule for this instance, such that no speed is equal to $s_{\\mathit {max}}$ .",
"Then the total power consumption of the schedule throughout execution is constant.",
"We prove this theorem by induction on the number of tasks of the graph.",
"First we prove a preliminary result: Lemma 2 Consider a graph $G=(V,E)$ with $n \\ge 2$ tasks, and any optimal schedule of deadline $D$ .",
"Let $t_1$ be the earliest completion time of a task in the schedule.",
"Similarly, let $t_2$ be the latest starting time of a task in the schedule.",
"Then, either $G$ is composed of independent tasks, or $0<t_1\\le t_2<D$ .",
"Task $T_i$ is executed at speed $s_i$ and during interval $[b_i,c_i]$ .",
"We have $t_1 = \\min _{1\\le i \\le n} c_i$ and $t_2 = \\max _{1\\le i \\le n} b_i$ .",
"Clearly, $0 \\le t_1,t_2 \\le D$ by definition of the schedule.",
"Suppose that $t_2 < t_1$ .",
"Let $T_1$ be a task that ends at time $t_1$ , and $T_2$ one that starts at time $t_2$ .",
"Then: $\\nexists T\\in V,~(T_1,T)Â~\\in E$ (otherwise, $T$ would start after $t_2$ ), therefore, $t_1=D$ ; $\\nexists T\\in V,~(T,T_2)Â~\\in E$ (otherwise, $T$ would finish before $t_1$ ); therefore $t_2=0$ .",
"This also means that all tasks start at time 0 and end at time $D$ .",
"Therefore, $G$ is only composed of independent tasks.",
"Back to the proof of the theorem, we consider first the case of a graph with only one task.",
"In an optimal schedule, the task is executed in time $D$ , and at constant speed (Lemma REF ), hence with constant power consumption.",
"Suppose now that the property is true for all DAGs with at most $n-1$ tasks.",
"Let $G$ be a DAG with $n$ tasks.",
"If $G$ is exactly composed of $n$ independent tasks, then we know that the power consumption of $G$ is constant (because all task speeds are constant).",
"Otherwise, let $t_1$ be the earliest completion time, and $t_2$ the latest starting time of a task in the optimal schedule.",
"Thanks to Lemma REF , we have $0<t_1\\le t_2<D$ .",
"Suppose first that $t_1=t_2=t_0$ .",
"There are three kinds of tasks: those beginning at time 0 and ending at time $t_0$ (set $S_1$ ), those beginning at time $t_0$ and ending at time $D$ (set $S_2$ ), and finally those beginning at time 0 and ending at time $D$ (set $S_3$ ).",
"Tasks in $S_3$ execute during the whole schedule duration, at constant speed, hence their contribution to the total power consumption $P(t)$ is the same at each time-step $t$ .",
"Therefore, we can suppress them from the schedule without loss of generality.",
"Next we determine the value of $t_0$ .",
"Let $A_1=\\sum _{T_i\\in S_1} w_i^3$ , and $A_2=\\sum _{T_i\\in S_2} w_i^3$ .",
"The energy consumption between 0 and $t_0$ is $\\frac{A_1}{t_0^2}$ , and between $t_0$ and $D$ , it is $\\frac{A_2}{(D-t_0)^2}$ .",
"The optimal energy consumption is obtained with $t_0=\\frac{A_1^{\\frac{1}{3}}}{A_1^{\\frac{1}{3}}+A_2^{\\frac{1}{3}}}$ .",
"Then, the total power consumption of the optimal schedule is the same in both intervals, hence at each time-step: we derive that $P(t) = \\left(\\frac{A_1^{\\frac{1}{3}}+A_2^{\\frac{1}{3}}}{D}\\right)^3$ , which is constant.",
"Suppose now that $t_1<t_2$ .",
"For each task $T_i$ , let $w^{\\prime }_i$ be the number of operations executed before $t_1$ , and $w^{\\prime \\prime }_i$ the number of operations executed after $t_1$ (with $w^{\\prime }_i+w^{\\prime \\prime }_i=w_i$ ).",
"Let $G^{\\prime }$ be the DAG $G$ with execution costs $w^{\\prime }_i$ , and $G^{\\prime \\prime }$ be the DAG $G$ with execution costs $w^{\\prime \\prime }_i$ .",
"The tasks with a cost equal to 0 are removed from the DAGs.",
"Then, both $G^{\\prime }$ and $G^{\\prime \\prime }$ have strictly fewer than $n$ tasks.",
"We can therefore apply the induction hypothesis.",
"We derive that the power consumption in both DAGs is constant.",
"Since we did not change the speeds of the tasks, the total power consumption $P(t)$ in $G$ is the same as in $G^{\\prime }$ if $t < t_1$ , hence a constant.",
"Similarly, the total power consumption $P(t)$ in $G$ is the same as in $G^{\\prime \\prime }$ if $t > t_1$ , hence a constant.",
"Considering the same partitioning with $t_2$ instead of $t_1$ , we show that the total power consumption $P(t)$ is a constant before $t_2$ , and also a constant after $t_2$ .",
"But $t_1 < t_2$ , and the intervals $[0,t_2]$ and $[t_1,D]$ overlap.",
"Altogether, the total power consumption is the same constant throughout $[0,D]$ , which concludes the proof.",
"Discrete models In this section, we present complexity results on the three energy models with a finite number of possible speeds.",
"The only polynomial instance is for the Vdd-Hopping model, for which we write a linear program in Section REF .",
"Then, we give NP-completeness results in Section REF , and approximation results in Section REF , for the Discrete and Incremental models.",
"The Vdd-Hopping model Theorem 5 With the Vdd-Hopping model, $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ can be solved in polynomial time.",
"Let $G$ be the execution graph of an application with $n$ tasks, and $D$ a deadline.",
"Let $s_1,...,s_m$ be the set of possible processor speeds.",
"We use the following rational variables: for $1\\le i \\le n$ and $1\\le j \\le m$ , $b_i$ is the starting time of the execution of task $T_i$ , and $\\alpha _{(i,j)}$ is the time spent at speed $s_j$ for executing task $T_i$ .",
"There are $n + n\\times m = n(m+1)$ such variables.",
"Note that the total execution time of task $T_i$ is $\\sum _{j=1}^m \\alpha _{(i,j)}$ .",
"The constraints are: $\\forall 1\\le i\\le n, \\; b_i\\ge 0$ : starting times of all tasks are non-negative numbers; $\\forall 1\\le i \\le n, \\; b_i+\\sum _{j=1}^m\\alpha _{(i,j)}\\le D$ : the deadline is not exceeded by any task; $\\forall 1 \\le i,i^{\\prime } \\le n$ such that $T_i\\rightarrow T_{i^{\\prime }}$ , $\\; t_i+\\sum _{j=1}^m \\alpha _{(i,j)}\\le t_{i^{\\prime }}$ : a task cannot start before its predecessor has completed its execution; $\\forall 1\\le i \\le n, \\; \\sum _{j=1}^m\\alpha _{(i,j)} \\times s_j\\ge w_i$ : task $T_i$ is completely executed.",
"The objective function is then $\\min \\left(\\sum _{i=1}^n\\sum _{j=1}^m \\alpha _{(i,j)}s_j^3\\right)$ .",
"The size of this linear program is clearly polynomial in the size of the instance, all $n(m+1)$ variables are rational, and therefore it can be solved in polynomial time [30].",
"NP-completeness results Theorem 6 With the Incremental model (and hence the Discrete model), $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is NP-complete.",
"We consider the associated decision problem: given an execution graph, a deadline, and a bound on the energy consumption, can we find an execution speed for each task such that the deadline and the bound on energy are respected?",
"The problem is clearly in NP: given the execution speed of each task, computing the execution time and the energy consumption can be done in polynomial time.",
"To establish the completeness, we use a reduction from 2-Partition [11].",
"We consider an instance $\\mathcal {I} _1$ of 2-Partition: given $n$ strictly positive integers $a_1, \\ldots , a_n$ , does there exist a subset $I$ of $\\lbrace 1, \\ldots , n\\rbrace $ such that $\\sum _{i\\in I}a_i=\\sum _{i \\notin I}a_i$ ?",
"Let $T=\\frac{1}{2}\\sum _{i=1}^n a_i$ .",
"We build the following instance $\\mathcal {I} _2$ of our problem: the execution graph is a linear chain with $n$ tasks, where: task $T_i$ has size $w_i=a_i$ ; the processor can run at $m=2$ different speeds; $s_1=1$ and $s_2=2$ , (i.e., $s_{\\mathit {min}}=1, s_{\\mathit {max}}=2, \\delta =1$ ); $L=3T/2$ ; $E=5T$ .",
"Clearly, the size of $\\mathcal {I} _2$ is polynomial in the size of $\\mathcal {I} _1$ .",
"Suppose first that instance $\\mathcal {I} _1$ has a solution $I$ .",
"For all $i\\in I$ , $T_i$ is executed at speed 1, otherwise it is executed at speed 2.",
"The execution time is then $\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2=\\frac{3}{2}T = D$ , and the energy consumption is $E=\\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 5T = E$ .",
"Both bounds are respected, and therefore the execution speeds are a solution to $\\mathcal {I} _2$ .",
"Suppose now that $\\mathcal {I} _2$ has a solution.",
"Since we consider the Discrete and Incremental models, each task run either at speed 1, or at speed 2.",
"Let $I=\\lbrace i\\; |\\; T_i\\mbox{ is executed at speed }1\\rbrace $ .",
"Note that we have $\\sum _{i\\notin I}a_i = 2T - \\sum _{i\\in I}a_i$ .",
"The execution time is $D^{\\prime }=\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2 = T + (\\sum _{i\\in I}a_i)/2$ .",
"Since the deadline is not exceeded, $D^{\\prime } \\le D = 3T/2$ , and therefore $\\sum _{i\\in I}a_i \\le T$ .",
"For the energy consumption of the solution of $\\mathcal {I} _2$ , we have $E^{\\prime } = \\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 2T + 3\\sum _{i\\notin I}a_i$ .",
"Since $E^{\\prime }\\le E=5T$ , we obtain $3\\sum _{i\\notin I}a_i\\le 3T$ , and hence $\\sum _{i\\notin I}a_i\\le T$ .",
"Since $\\sum _{i\\in I}a_i + \\sum _{i\\notin I}a_i = 2T$ , we conclude that $\\sum _{i\\in I}a_i = \\sum _{i\\notin I}a_i =T$ , and therefore $\\mathcal {I} _1$ has a solution.",
"This concludes the proof.",
"Approximation results Here we explain, for the Incremental and Discrete models, how the solution to the NP-hard problem can be approximated.",
"Note that, given an execution graph and a deadline, the optimal energy consumption with the Continuous model is always lower than that with the other models, which are more constrained.",
"Theorem 7 With the Incremental model, for any integer $K>0$ , the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2$ , in a time polynomial in the size of the instance and in $K$ .",
"Consider an instance $\\mathcal {I} _{inc}$ of the problem with the Incremental model.",
"The execution graph $G$ has $n$ tasks, $D$ is the deadline, $\\delta $ is the minimum permissible speed increment, and $s_{\\mathit {min}}, s_{\\mathit {max}}$ are the speed bounds.",
"Moreover, let $K>0$ be an integer, and let $E_{inc}$ be the optimal value of the energy consumption for this instance $\\mathcal {I} _{inc}$ .",
"We construct the following instance $\\mathcal {I} _{vdd}$ with the Vdd-Hopping model: the execution graph and the deadline are the same as in instance $\\mathcal {I} _{inc}$ , and the speeds can take the values $\\left\\lbrace s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^i\\right\\rbrace _{0\\le i\\le N}\\;,$ where $N$ is such that $s_{max}$ is not exceeded: $N=\\left\\lfloor (\\ln (s_{\\mathit {max}})-\\ln (s_{\\mathit {min}}))/\\ln \\left(1+\\frac{1}{K}\\right)\\right\\rfloor $ .",
"As $N$ is asymptotically of order $O(K\\ln (s_{\\mathit {max}}))$ , the number of possible speeds in $\\mathcal {I} _{vdd}$ , and hence the size of $\\mathcal {I} _{vdd}$ , is polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Next, we solve $\\mathcal {I} _{vdd}$ in polynomial time thanks to Theorem REF .",
"For each task $T_i$ , let $s^{(vdd)}_i$ be the average speed of $T_i$ in this solution: if the execution time of the task in the solution is $d_i$ , then $s^{(vdd)}_i = w_i/d_i$ ; $E_{vdd}$ is the optimal energy consumption obtained with these speeds.",
"Let $s^{(algo)}_i=\\min _u\\lbrace s_{\\mathit {min}}+u\\times \\delta \\; | \\; u\\times \\delta \\ge s^{(vdd)}_i\\rbrace $ be the smallest speed in $\\mathcal {I} _{inc}$ which is larger than $s^{(vdd)}_i$ .",
"There exists such a speed since, because of the values chosen for $\\mathcal {I} _{vdd}$ , $s^{(vdd)}_i\\le s_{\\mathit {max}}$ .",
"The values $s^{(algo)}_i$ can be computed in time polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Let $E_{algo}$ be the energy consumption obtained with these values.",
"In order to prove that this algorithm is an approximation of the optimal solution, we need to prove that $E_{algo} \\le (1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2\\times E_{inc}$ .",
"For each task $T_i$ , $s^{(algo)}_i-\\delta \\le s^{(vdd)}_i\\le s^{(algo)}_i$ .",
"Since $s_{\\mathit {min}}\\le s^{(vdd)}_i$ , we derive that $s^{(algo)}_i \\le s^{(vdd)}_i \\times (1+\\frac{\\delta }{s_{\\mathit {min}}})$ .",
"Summing over all tasks, we get $\\text{~~~~~~~~~~~~~~} E_{algo}=\\sum _iw_i\\left(s^{(algo)}_i\\right)^2\\le \\sum _iw_i\\left(s^{(vdd)}_i\\times (1+\\frac{\\delta }{s_{\\mathit {min}}})\\right)^2\\le E_{vdd} \\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Next, we bound $E_{vdd}$ thanks to the optimal solution with the Continuous model, $E_{con}$ .",
"Let $\\mathcal {I} _{con}$ be the instance where the execution graph $G$ , the deadline $D$ , the speeds $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ are the same as in instance $\\mathcal {I} _{inc}$ , but now admissible speeds take any value between $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ .",
"Let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ , and let $0\\le u\\le N$ be the value such that: $ \\text{~~~~~~~~~~~~~~} s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^u \\le s^{(con)}_i\\le s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^{u+1}=s^*_i~$ .",
"In order to bound the energy consumption for $I_{vdd}$ , we assume that $T_i$ runs at speed $s^*_i$ , instead of $s^{(vdd)}_i$ .",
"The solution with these speeds is a solution to $I_{vdd}$ , and its energy consumption is $E^* \\ge E_{vdd}$ .",
"From the previous inequalities, we deduce that $s^*_i \\le s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)$ , and by summing over all tasks, $E_{vdd} \\le E^* = \\sum _i w_i \\left(s^*_i\\right)^2\\le \\sum _i w_i \\left(s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)\\right)^2\\le E_{con} \\times \\left(1+\\frac{1}{K}\\right)^2\\le E_{inc} \\times \\left(1+\\frac{1}{K}\\right)^2\\; .$ Proposition 3 ${ }$ For any integer $\\delta >0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Continuous model can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2$ in the Incremental model with speed increment $\\delta $ .",
"For any integer $K>0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Discrete model can be approximated within a factor $(1+\\frac{\\alpha }{s_1})^2(1+\\frac{1}{K})^2$ , with $\\alpha =\\max _{1 \\le i < m}\\lbrace s_{i+1}-s_i\\rbrace $ , in a time polynomial in the size of the instance and in $K$ .",
"For the first part, let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ in instance $\\mathcal {I} _{con}$ ; $E_{con}$ is the optimal energy consumption.",
"For any task $T_i$ , let $s_i$ be the speed of $\\mathcal {I} _{inc}$ such that $s_i-\\delta <s^{con}_i\\le s_i$ .",
"Then, $s^{(con)}_i\\le s_i\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)$ .",
"Let $E$ be the energy with speeds $s_i$ .",
"$E_{con}\\le E\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Let $E_{inc}$ be the optimal energy of $\\mathcal {I} _{inc}$ .",
"Then, $E_{con}\\le E_{inc}\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"For the second part, we use the same algorithm as in Theorem REF .",
"The same proof leads to the approximation ratio with $\\alpha $ instead of $\\delta $ .",
"Conclusion In this paper, we have assessed the tractability of a classical scheduling problem, with task preallocation, under various energy models.",
"We have given several results related to Continuous speeds.",
"However, while these are of conceptual importance, they cannot be achieved with physical devices, and we have analyzed several models enforcing a bounded number of achievable speeds, a.k.a.",
"modes.",
"In the classical Discrete model that arises from DVFS techniques, admissible speeds can be irregularly distributed, which motivates the Vdd-Hopping approach that mixes two consecutive modes optimally.",
"While computing optimal speeds is NP-hard with discrete modes, it has polynomial complexity when mixing speeds.",
"Intuitively, the Vdd-Hopping approach allows for smoothing out the discrete nature of the modes.",
"An alternate (and simpler in practice) solution to Vdd-Hopping is the Incremental model, where one sticks with unique speeds during task execution as in the Discrete model, but where consecutive modes are regularly spaced.",
"Such a model can be made arbitrarily efficient, according to our approximation results.",
"Altogether, this paper has laid the theoretical foundations for a comparative study of energy models.",
"In the recent years, we have observed an increased concern for green computing, and a rapidly growing number of approaches.",
"It will be very interesting to see which energy-saving technological solutions will be implemented in forthcoming future processor chips!"
],
[
"The ",
"With the Continuous model, processor speeds can take any value between 0 and $s_{\\mathit {max}}$ .",
"First we prove that, with this model, the processors do not change their speed during the execution of a task (Section REF ).",
"Then, we derive in Section REF the optimal speed values for special execution graph structures, expressed as closed form algebraic formulas, and we show that these values may be irrational (as already illustrated in the example in Section REF ).",
"Finally, we formulate the problem for general DAGs as a convex optimization program in Section REF ."
],
[
"Preliminary lemma",
"Lemma 1 (constant speed per task) With the Continuous model, each task is executed at constant speed, i.e., a processor does not change its speed during the execution of a task.",
"Suppose that in the optimal solution, there is a task whose speed changes during the execution.",
"Consider the first time-step at which the change occurs: the computation begins at speed $s$ from time $t$ to time $t^{\\prime }$ , and then continues at speed $s^{\\prime }$ until time $t^{\\prime \\prime }$ .",
"The total energy consumption for this task in the time interval $[t;t^{\\prime \\prime }]$ is $E=(t^{\\prime }-t)\\times s^3+(t^{\\prime \\prime }-t^{\\prime })\\times (s^{\\prime })^3$ .",
"Moreover, the amount of work done for this task is $W=(t^{\\prime }-t)\\times s+(t^{\\prime \\prime }-t^{\\prime })\\times s^{\\prime }$ .",
"If we run the task during the whole interval $[t;t^{\\prime \\prime }]$ at constant speed $W/(t^{\\prime \\prime }-t)$ , the same amount of work is done within the same time.",
"However, the energy consumption during this interval of time is now $E^{\\prime }=(t^{\\prime \\prime }-t)\\times (W/(t^{\\prime \\prime }-t))^3$ .",
"By convexity of the function $x \\mapsto x^3$ , we obtain $E^{\\prime }<E$ since $t<t^{\\prime }<t^{\\prime \\prime }$ .",
"This contradicts the hypothesis of optimality of the first solution, which concludes the proof."
],
[
"Independent tasks",
"Consider the problem of minimizing the energy of $n$ independent tasks (i.e., each task is mapped onto a distinct processor, and there are no precedence constraints in the execution graph), while enforcing a deadline $D$ .",
"Proposition 1 (independent tasks) When $G$ is composed of independent tasks $\\lbrace T_1, \\dots , T_n\\rbrace $ , the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained when each task $T_i$ ($1\\le i \\le n$ ) is computed at speed $s_i=\\frac{w_i}{D}$ .",
"If there is a task $T_i$ such that $s_i > s_{\\mathit {max}}$ , then the problem has no solution.",
"For task $T_i$ , the speed $s_i$ corresponds to the slowest speed at which the processor can execute the task, so that the deadline is not exceeded.",
"If $s_i > s_{\\mathit {max}}$ , the corresponding processor will never be able to complete its execution before the deadline, therefore there is no solution.",
"To conclude the proof, we note that any other solution would have higher values of $s_i$ because of the deadline constraint, and hence a higher energy consumption.",
"Therefore, this solution is optimal."
],
[
"Linear chain of tasks",
"This case corresponds for instance to $n$ independent tasks $\\lbrace T_1,\\dots , T_n\\rbrace $ executed onto a single processor.",
"The execution graph is then a linear chain (order of execution of the tasks), with $T_i \\rightarrow T_{i+1}$ , for $1\\le i < n$ .",
"Proposition 2 (linear chain) $\\;\\;$ When $G$ is a linear chain of tasks, the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained when each task is executed at speed $s=\\frac{W}{D}$ , with $W=\\sum _{i=1}^n w_i$ .",
"If $s>s_{\\mathit {max}}$ , then there is no solution.",
"Suppose that in the optimal solution, tasks $T_i$ and $T_j$ are such that $s_i < s_j$ .",
"The total energy consumption is $E_{opt}$ .",
"We define $s$ such that the execution of both tasks running at speed $s$ takes the same amount of time than in the optimal solution, i.e., $(w_i+w_j)/s = w_i/s_i + w_j/s_j$ : $s = \\frac{(w_i+w_j)}{w_is_j + w_js_i} \\times s_is_j$ .",
"Note that $s_i < s < s_j$ (it is the barycenter of two points with positive mass).",
"We consider a solution such that the speed of task $T_k$ , for $1\\le k \\le n$ , with $k\\ne i$ and $k \\ne j$ , is the same as in the optimal solution, and the speed of tasks $T_i$ and $T_j$ is $s$ .",
"By definition of $s$ , the execution time has not been modified.",
"The energy consumption of this solution is $E$ , where $E_{opt}-E = w_is_i^{2} + w_js_j^{2} -(w_i+w_j)s^{2}$ , i.e., the difference of energy with the optimal solution is only impacted by tasks $T_i$ and $T_j$ , for which the speed has been modified.",
"By convexity of the function $x \\mapsto x^{2}$ , we obtain $E_{opt}>E$ , which contradicts its optimality.",
"Therefore, in the optimal solution, all tasks have the same execution speed.",
"Moreover, the energy consumption is minimized when the speed is as low as possible, while the deadline is not exceeded.",
"Therefore, the execution speed of all tasks is $s=W/D$ .",
"Corollary 1 A linear chain with $n$ tasks is equivalent to a single task of cost $W=\\sum _{i=1}^n w_i$ .",
"Indeed, in the optimal solution, the $n$ tasks are executed at the same speed, and they can be replaced by a single task of cost $W$ , which is executed at the same speed and consumes the same amount of energy."
],
[
"Fork and join graphs",
"Let $V\\!=\\!\\lbrace T_1,\\dots ,T_n\\rbrace $ .",
"We consider either a fork graph $G = (V\\cup \\lbrace T_{0}\\rbrace , E)$ , with $E=\\lbrace (T_{0},T_i),T_i \\in V\\rbrace $ , or a join graph $G =(V\\cup \\lbrace T_{0}\\rbrace , E)$ , with $E=\\lbrace (T_i,T_{0}),T_i\\in V\\rbrace $ .",
"$T_{0}$ is either the source of the fork or the sink of the join.",
"Theorem 1 (fork and join graphs) When $G$ is a fork (resp.",
"join) execution graph with $n+1$ tasks $T_0,T_1,\\dots ,T_n$ , the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is the following: ${ }\\quad \\bullet $ the execution speed of the source (resp.",
"sink) $T_0$ is $s_{0} = \\dfrac{\\left(\\sum _{i=1}^n w_i^{3}\\right)^{\\frac{1}{3}} +w_{0}}{D}$ ; ${ }\\quad \\bullet $ for the other tasks $T_i$ , $1\\le i \\le n$ , we have $s_i = s_{0} \\times \\dfrac{w_i}{\\left(\\sum _{i=1}^nw_i^{3}\\right)^{\\frac{1}{3}}}$ if $s_{0}\\le s_{\\mathit {max}}$ .",
"Otherwise, $T_{0}$ should be executed at speed $s_0=s_{\\mathit {max}}$ , and the other speeds are $s_i = \\frac{w_i}{D^{\\prime }}$ , with $D^{\\prime }=D-\\frac{w_{0}}{s_{\\mathit {max}}}$ , if they do not exceed $s_{\\mathit {max}}$ (Proposition REF for independent tasks).",
"Otherwise there is no solution.",
"If no speed exceeds $s_{\\mathit {max}}$ , the corresponding energy consumption is ${\\bf minE}(G,D)=\\frac{\\left((\\sum _{i=1}^n w_i^3)^{\\frac{1}{3}} +w_0\\right)^3}{D^2}\\; .$ Let $t_0=\\frac{w_{0}}{s_0}$ .",
"Then, the source or the sink requires a time $t_0$ for execution.",
"For $1\\le i \\le n$ , task $T_i$ must be executed within a time $D-t_0$ so that the deadline is respected.",
"Given $t_0$ , we can compute the speed $s_i$ for task $T_i$ using Theorem REF , since the tasks are independent: $s_i = \\frac{w_i}{D-t_0} = w_i \\cdot \\frac{s_0}{s_0 D - w_0}$ .",
"The objective is therefore to minimize $\\sum _{i=0}^n w_i s_i^{2}$ , which is a function of $s_0$ : $\\sum _{i=0}^n w_i s_i^{2} = w_0s_0^2 +\\sum _{i=1}^n w_i^3 \\cdot \\frac{s_0^2}{(s_0D - w_0)^2}= s_0^2 \\left( w_0 + \\frac{\\sum _{i=1}^n w_i^3}{(s_0 D - w_0)^2}\\right)= f(s_0).$ Let $W_3 = \\sum _{i=1}^n w_i^3$ .",
"In order to find the value of $s_0$ which minimizes this function, we study the function $f(x)$ , for $x>0$ .",
"$f^{\\prime }(x) = 2x \\left(w_{0} + \\frac{W_3}{(x D-w_{0})^2}\\right) -2D \\cdot x^{2} \\cdot \\frac{W_3}{(x D-w_{0})^{3}} $ , and therefore $f^{\\prime }(x)=0$ for $x=(W_3^{\\frac{1}{3}} + w_0)/D$ .",
"We conclude that the optimal speed for task $T_0$ is $s_0 = \\frac{\\left(\\sum _{i=1}^n w_i^{3} \\right)^{\\frac{1}{3}} +w_{0}}{D}$ , if $s_0 \\le s_{\\mathit {max}}$ .",
"Otherwise, $T_0$ should be executed at the maximum speed $s_0 =s_{\\mathit {max}}$ , since it is the bottleneck task.",
"In any case, for $1\\le i \\le n$ , the optimal speed for task $T_i$ is $s_i = w_i \\frac{s_0}{s_0 D - w_0}$ .",
"Finally, we compute the exact expression of minE$(G,D) =f(s_0)$ , when $s_0\\le s_{\\mathit {max}}$ : $f(s_0) = s_0^2 \\left( w_0 + \\frac{W_3}{(s_0 D - w_0)^2}\\right)=\\left(\\frac{W_3^{\\frac{1}{3}} +w_0}{D}\\right)^2 \\left(\\frac{W_3}{W_3^{2/3}}+w_0 \\right)=\\frac{\\left(W_3^{\\frac{1}{3}} +w_0\\right)^3}{D^2}, $ which concludes the proof.",
"Corollary 2 (equivalent tasks for speed) Consider a fork or join graph with tasks $T_i$ , $0\\le i \\le n$ , and a deadline $D$ , and assume that the speeds in the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ do not exceed $s_{\\mathit {max}}$ .",
"Then, these speeds are the same as in the optimal solution for $n+1$ independent tasks $T^{\\prime }_0, T^{\\prime }_1, \\dots , T^{\\prime }_n$ , where $w^{\\prime }_0 = \\left(\\sum _{i=1}^n w_{i}^{3} \\right)^{\\frac{1}{3}} + w_{0}$ , and, for $1\\le i \\le n$ , $w^{\\prime }_i=w^{\\prime }_0 \\cdot \\frac{w_i}{\\left(\\sum _{i=1}^n w_{i}^{3}\\right)^{\\frac{1}{3}}}\\;$ .",
"Corollary 3 (equivalent task for energy) Consider a fork or join graph $G$ and a deadline $D$ , and assume that the speeds in the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ do not exceed $s_{\\mathit {max}}$ .",
"We say that the graph $G$ is equivalent to the graph $G^{(eq)}$ , consisting of a single task $T^{(eq)}_0$ of weight $w^{(eq)}_0 = \\left(\\sum _{i=1}^n w_{i}^{3} \\right)^{\\frac{1}{3}} + w_{0}$ , because the minimum energy consumption of both graphs are identical: minE$(G,D)$ =minE$(G^{(eq)},D)$ ."
],
[
"Trees",
"We extend the results on a fork graph for a tree $G=(V,E)$ with $|V|=n+1$ tasks.",
"Let $T_0$ be the root of the tree; it has $k$ children tasks, which are each themselves the root of a tree.",
"A tree can therefore be seen as a fork graph, where the tasks of the fork are trees.",
"The previous results for fork graphs naturally lead to an algorithm that peels off branches of the tree, starting with the leaves, and replaces each fork subgraph in the tree, composed of a root $T_0$ and $k$ children, by one task (as in Corollary REF ) which becomes the unique child of $T_0$ 's parent in the tree.",
"We say that this task is equivalent to the fork graph, since the optimal energy consumption will be the same.",
"The computation of the equivalent cost of this task is done thanks to a call to the eq procedure, while the tree procedure computes the solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ (see Algorithm REF ).",
"Note that the algorithm computes the minimum energy for a tree, but it does not return the speeds at which each task must be executed.",
"However, the algorithm returns the speed of the root task, and it is then straightforward to compute the speed of each children of the root task, and so on.",
"Theorem 2 (tree graphs) When $G$ is a tree rooted in $T_0$ ($T_0 \\in V$ , where $V$ is the set of tasks), the optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ can be computed in polynomial time $O(|V|^2)$ .",
"Let $G$ be a tree graph rooted in $T_0$ .",
"The optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained with a call to tree $(G,T_0,D)$ , and we prove its optimality recursively on the depth of the tree.",
"Similarly to the case of the fork graphs, we reduce the tree to an equivalent task which, if executed alone within a deadline $D$ , consumes exactly the same amount of energy.",
"The procedure eq is the procedure which reduces a tree to its equivalent task (see Algorithm REF ).",
"If the tree has depth 0, then it is a single task, eq $(G,T_0)$ returns the equivalent cost $w_0$ , and the optimal execution speed is $\\frac{w_0}{D}$ (see Proposition REF ).",
"There is a solution if and only if this speed is not greater than $s_{\\mathit {max}}$ , and then the corresponding energy consumption is $\\frac{w_0^3}{D^2}$ , as returned by the algorithm.",
"Assume now that for any tree of depth $i<p$ , eq computes its equivalent cost, and tree returns its optimal energy consumption.",
"We consider a tree $G$ of depth $p$ rooted in $T_0$ : $G = T_0 \\cup \\lbrace G_i\\rbrace $ , where each subgraph $G_i$ is a tree, rooted in $T_i$ , of maximum depth $p-1$ .",
"As in the case of forks, we know that each subtree $G_i$ has a deadline $D-x$ , where $x=\\frac{w_0}{s_0}$ , and $s_0$ is the speed at which task $T_0$ is executed.",
"By induction hypothesis, we suppose that each graph $G_i$ is equivalent to a single task, $T^{\\prime }_i$ , of cost $w^{\\prime }_i$ (as computed by the procedure eq).",
"We can then use the results obtained on forks to compute $w^{(eq)}_0$ (see proof of Theorem REF ): $w^{(eq)}_0 = \\displaystyle \\left( \\sum _{i} (w^{\\prime }_{i})^{3} \\right)^{\\frac{1}{3}} + w_{0}$ .",
"Finally the tree is equivalent to one task of cost $w^{(eq)}_0$ , and if $\\frac{w^{(eq)}_0}{D}\\le s_{\\mathit {max}}$ , the energy consumption is $\\frac{\\left(w^{(eq)}_0\\right)^3}{D^2}$ , and no speed exceeds $s_{\\mathit {max}}$ .",
"Note that the speed of a task is always greater than the speed of its successors.",
"Therefore, if $\\frac{w^{(eq)}_0}{D}> s_{\\mathit {max}}$ , we execute the root of the tree at speed $s_{\\mathit {max}}$ and then process each subtree $G_i$ independently.",
"Of course, there is no solution if $\\frac{w_0}{s_{\\mathit {max}}}>D$ , and otherwise we perform the recursive calls to tree to process each subtree independently.",
"Their deadline is then $D-\\frac{w_0}{s_{\\mathit {max}}}$ .",
"To study the time complexity of this algorithm, first note that when calling tree $(G,T_0,D)$ , there might be at most $|V|$ recursive calls to tree, once at each node of the tree.",
"Without accounting for the recursive calls, the tree procedure performs one call to the eq procedure, which computes the cost of the equivalent task.",
"This eq procedure takes a time $O(|V|)$ , since we have to consider the $|V|$ tasks, and we add the costs one by one.",
"Therefore, the overall complexity is in $O(|V|^2)$ .",
"Solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ for trees.",
"procedure tree (tree $G$ , root $T_0$ , deadline $D$ ) Let $w$ =eq (tree $G$ , root $T_0$ ) $\\frac{w}{D}\\le s_{\\mathit {max}}$ $\\frac{w^3}{D^2}$ ; $\\frac{w_0}{s_{\\mathit {max}}} > D$ Error:No Solution; /* $T_0$ is executed at speed $s_{\\mathit {max}}$ */ $w_0 \\times s_{\\mathit {max}}^2 + \\displaystyle \\!\\!\\!\\sum _{G_i\\mbox{ subtreerooted in }T_i\\in \\mbox{children}(T_0)} \\!\\!\\!",
"{\\bf tree} \\left(G_i,T_i,D-\\frac{w_0}{s_{\\mathit {max}}}\\right)$ procedure eq (tree $G$ , root $T_0$ ) children($T_0$ )=$\\emptyset $ $w_0$ ; $\\left(\\displaystyle \\sum _{G_i\\mbox{ subtree rooted in }T_i\\in \\mbox{children}(T_0)}\\left(\\mbox{\\bf eq} (G_i,T_i)\\right)^3 \\right)^{\\frac{1}{3}} +w_0$ ;"
],
[
"Series-parallel graphs",
"We can further generalize our results to series-parallel graphs (SPGs), which are built from a sequence of compositions (parallel or series) of smaller-size SPGs.",
"The smallest SPG consists of two nodes connected by an edge (such a graph is called an elementary SPG).",
"The first node is the source, while the second one is the sink of the SPG.",
"When composing two SGPs in series, we merge the sink of the first SPG with the source of the second one.",
"For a parallel composition, the two sources are merged, as well as the two sinks, as illustrated in Figure REF .",
"Figure: Composition of series-parallel graphs (SPGs).We can extend the results for tree graphs to SPGs, by replacing step by step the SPGs by an equivalent task (procedure cost in Algorithm REF ): we can compute the equivalent cost for a series or parallel composition.",
"However, since it is no longer true that the speed of a task is always larger than the speed of its successor (as was the case in a tree), we have not been able to find a recursive property on the tasks that should be set to $s_{\\mathit {max}}$ , when one of the speeds obtained with the previous method exceeds $s_{\\mathit {max}}$ .",
"The problem of computing a closed form for a SPG with a finite value of $s_{\\mathit {max}}$ remains open.",
"Still, we have the following result when $s_{\\mathit {max}}= +\\infty $ : Theorem 3 (series-parallel graphs) When $G$ is a SPG, it is possible to compute recursively a closed form expression of the optimal solution of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ , assuming $s_{\\mathit {max}}=+\\infty $ , in polynomial time $O(|V|)$ , where $V$ is the set of tasks.",
"Let $G$ be a series-parallel graph.",
"The optimal solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is obtained with a call to SPG $(G,D)$ , and we prove its optimality recursively.",
"Similarly to trees, the main idea is to peel the graph off, and to transform it until there remains only a single equivalent task which, if executed alone within a deadline $D$ , would consume exactly the same amount of energy.",
"The procedure cost is the procedure which reduces a tree to its equivalent task (see Algorithm REF ).",
"The proof is done by induction on the number of compositions required to build the graph $G$ , $p$ .",
"If $p=0$ , $G$ is an elementary SPG consisting in two tasks, the source $T_0$ and the sink $T_1$ .",
"It is therefore a linear chain, and therefore equivalent to a single task whose cost is the sum of both costs, $w_0+w_1$ (see Corollary REF for linear chains).",
"The procedure cost returns therefore the correct equivalent cost, and SPG returns the minimum energy consumption.",
"Let us assume that the procedures return the correct equivalent cost and minimum energy consumption for any SPG consisting of $i<p$ compositions.",
"We consider a SPG $G$ , with $p$ compositions.",
"By definition, $G$ is a composition of two smaller-size SPGs, $G_1$ and $G_2$ , and both of these SPGs have strictly fewer than $p$ compositions.",
"We consider $G^{\\prime }_1$ and $G^{\\prime }_2$ , which are identical to $G_1$ and $G_2$ , except that the cost of their source and sink tasks are set to 0 (these costs are handled separately), and we can reduce both of these SPGs to an equivalent task, of respective costs $w^{\\prime }_1$ and $w^{\\prime }_2$ , by induction hypothesis.",
"There are two cases: If $G$ is a series composition, then after the reduction of $G^{\\prime }_1$ and $G^{\\prime }_2$ , we have a linear chain in which we consider the source $T_0$ of $G_1$ , the sink $T_1$ of $G_1$ (which is also the source of $G_2$ ), and the sink $T_2$ of $G_2$ .",
"The equivalent cost is therefore $w_0 + w^{\\prime }_1 + w_1 + w^{\\prime }_2 + w_2$ , thanks to Corollary REF for linear chains.",
"If $G$ is a parallel composition, the resulting graph is a fork-join graph, and we can use Corollaries REF and REF to compute the cost of the equivalent task, accounting for the source $T_0$ and the sink $T_1$ : $w_0 + \\left((w^{\\prime }_1)^3+(w^{\\prime }_2)^3\\right)^\\frac{1}{3} + w_1$ .",
"Once the cost of the equivalent task of the SPG has been computed with the call to cost $(G)$ , the optimal energy consumption is $\\frac{\\left({\\bf cost}(G)\\right)^3}{D^2}$ .",
"Contrarily to the case of tree graphs, since we never need to call the SPG procedure again because there is no constraint on $s_{\\mathit {max}}$ , the time complexity of the algorithm is the complexity of the cost procedure.",
"There is exactly one call to cost for each composition, and the number of compositions in the SPG is in $O(|V|)$ .",
"All operations in cost can be done in $O(1)$ , hence a complexity in $O(|V|)$ .",
"Solution to $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ for series-parallel graphs.",
"procedure SPG (series-parallel graph $G$ , deadline $D$ ) $\\frac{\\left(\\mbox{\\bf cost}(G)\\right)^3}{D^2}$ ; procedure cost (series-parallel graph $G$ ) Let $T_0$ be the source of $G$ and $T_1$ its sink $G$ is composed of only two tasks, $T_0$ and $T_1$ $w_0+w_1$ ; /* $G$ is a composition of two SPGs $G_1$ and $G_2$ .",
"*/ For $i=1,2$ , let $G^{\\prime }_i=G_i$ where the cost of source and sink tasks is set to 0 $w^{\\prime }_1=\\mbox{\\bf cost}(G^{\\prime }_1); \\; w^{\\prime }_2=\\mbox{\\bf cost}(G^{\\prime }_2)$ $G$ is a series composition Let $T_0$ be the source of $G_1$ , $T_1$ be its sink, and $T_2$ be the sink of $G_2$ $w_0 + w^{\\prime }_1 + w_1 + w^{\\prime }_2 + w_2$ ; /* It is a parallel composition.",
"*/ Let $T_0$ be the source of $G$ , and $T_1$ be its sink $w_0 + \\left((w^{\\prime }_1)^3+(w^{\\prime }_2)^3\\right)^\\frac{1}{3} + w_1$ ; General DAGs For arbitrary execution graphs, we can rewrite the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem as follows: $\\begin{array}{lrl}\\text{~~~Minimize}& & \\sum _{i=1}^{n} u_i^{-2} \\times w_i\\\\\\text{~~~subject to}&\\text{(i)} & t_i + w_j \\times u_j \\le t_j \\text{ for each edge }(T_i,T_j) \\in E\\\\&\\text{(ii)} & t_i \\le D \\text{ for each task } T_i \\in V\\\\&\\text{(iii)} & u_i \\ge \\frac{1}{s_{\\mathit {max}}} \\text{ for each task } T_i \\in V\\\\\\end{array}$ Here, $u_i = 1/s_i$ is the inverse of the speed to execute task $T_i$ .",
"We now have a convex optimization problem to solve, with linear constraints in the non-negative variables $u_i$ and $t_i$ .",
"In fact, the objective function is a posynomial, so we have a geometric programming problem (see [6]) for which efficient numerical schemes exist.",
"However, as illustrated on simple fork graphs, the optimal speeds are not expected to be rational numbers but instead arbitrarily complex expressions (we have the cubic root of the sum of cubes for forks, and nested expressions of this form for trees).",
"From a computational complexity point of view, we do not know how to encode such numbers in polynomial size of the input (the rational task weights and the execution deadline).",
"Still, we can always solve the problem numerically and get fixed-size numbers which are good approximations of the optimal values.",
"In the following, we show that the total power consumption of any optimal schedule is constant throughout execution.",
"While this important property does not help to design an optimal solution, it shows that a schedule with large variations in its power consumption is likely to waste a lot of energy.",
"We need a few notations before stating the result.",
"Consider a schedule for a graph $G=(V,E)$ with $n$ tasks.",
"Task $T_i$ is executed at constant speed $s_i$ (see Lemma REF ) and during interval $[b_i,c_i]$ : $T_i$ begins its execution at time $b_i$ and completes it at time $c_i$ .",
"The total power consumption $P(t)$ of the schedule at time $t$ is defined as the sum of the power consumed by all tasks executing at time $t$ : $P(t) = \\sum _{1 \\le i \\le n, \\; t \\in [b_i,c_i]} s_i^3\\; .",
"$ Theorem 4 Consider an instance of Continuous, and an optimal schedule for this instance, such that no speed is equal to $s_{\\mathit {max}}$ .",
"Then the total power consumption of the schedule throughout execution is constant.",
"We prove this theorem by induction on the number of tasks of the graph.",
"First we prove a preliminary result: Lemma 2 Consider a graph $G=(V,E)$ with $n \\ge 2$ tasks, and any optimal schedule of deadline $D$ .",
"Let $t_1$ be the earliest completion time of a task in the schedule.",
"Similarly, let $t_2$ be the latest starting time of a task in the schedule.",
"Then, either $G$ is composed of independent tasks, or $0<t_1\\le t_2<D$ .",
"Task $T_i$ is executed at speed $s_i$ and during interval $[b_i,c_i]$ .",
"We have $t_1 = \\min _{1\\le i \\le n} c_i$ and $t_2 = \\max _{1\\le i \\le n} b_i$ .",
"Clearly, $0 \\le t_1,t_2 \\le D$ by definition of the schedule.",
"Suppose that $t_2 < t_1$ .",
"Let $T_1$ be a task that ends at time $t_1$ , and $T_2$ one that starts at time $t_2$ .",
"Then: $\\nexists T\\in V,~(T_1,T)Â~\\in E$ (otherwise, $T$ would start after $t_2$ ), therefore, $t_1=D$ ; $\\nexists T\\in V,~(T,T_2)Â~\\in E$ (otherwise, $T$ would finish before $t_1$ ); therefore $t_2=0$ .",
"This also means that all tasks start at time 0 and end at time $D$ .",
"Therefore, $G$ is only composed of independent tasks.",
"Back to the proof of the theorem, we consider first the case of a graph with only one task.",
"In an optimal schedule, the task is executed in time $D$ , and at constant speed (Lemma REF ), hence with constant power consumption.",
"Suppose now that the property is true for all DAGs with at most $n-1$ tasks.",
"Let $G$ be a DAG with $n$ tasks.",
"If $G$ is exactly composed of $n$ independent tasks, then we know that the power consumption of $G$ is constant (because all task speeds are constant).",
"Otherwise, let $t_1$ be the earliest completion time, and $t_2$ the latest starting time of a task in the optimal schedule.",
"Thanks to Lemma REF , we have $0<t_1\\le t_2<D$ .",
"Suppose first that $t_1=t_2=t_0$ .",
"There are three kinds of tasks: those beginning at time 0 and ending at time $t_0$ (set $S_1$ ), those beginning at time $t_0$ and ending at time $D$ (set $S_2$ ), and finally those beginning at time 0 and ending at time $D$ (set $S_3$ ).",
"Tasks in $S_3$ execute during the whole schedule duration, at constant speed, hence their contribution to the total power consumption $P(t)$ is the same at each time-step $t$ .",
"Therefore, we can suppress them from the schedule without loss of generality.",
"Next we determine the value of $t_0$ .",
"Let $A_1=\\sum _{T_i\\in S_1} w_i^3$ , and $A_2=\\sum _{T_i\\in S_2} w_i^3$ .",
"The energy consumption between 0 and $t_0$ is $\\frac{A_1}{t_0^2}$ , and between $t_0$ and $D$ , it is $\\frac{A_2}{(D-t_0)^2}$ .",
"The optimal energy consumption is obtained with $t_0=\\frac{A_1^{\\frac{1}{3}}}{A_1^{\\frac{1}{3}}+A_2^{\\frac{1}{3}}}$ .",
"Then, the total power consumption of the optimal schedule is the same in both intervals, hence at each time-step: we derive that $P(t) = \\left(\\frac{A_1^{\\frac{1}{3}}+A_2^{\\frac{1}{3}}}{D}\\right)^3$ , which is constant.",
"Suppose now that $t_1<t_2$ .",
"For each task $T_i$ , let $w^{\\prime }_i$ be the number of operations executed before $t_1$ , and $w^{\\prime \\prime }_i$ the number of operations executed after $t_1$ (with $w^{\\prime }_i+w^{\\prime \\prime }_i=w_i$ ).",
"Let $G^{\\prime }$ be the DAG $G$ with execution costs $w^{\\prime }_i$ , and $G^{\\prime \\prime }$ be the DAG $G$ with execution costs $w^{\\prime \\prime }_i$ .",
"The tasks with a cost equal to 0 are removed from the DAGs.",
"Then, both $G^{\\prime }$ and $G^{\\prime \\prime }$ have strictly fewer than $n$ tasks.",
"We can therefore apply the induction hypothesis.",
"We derive that the power consumption in both DAGs is constant.",
"Since we did not change the speeds of the tasks, the total power consumption $P(t)$ in $G$ is the same as in $G^{\\prime }$ if $t < t_1$ , hence a constant.",
"Similarly, the total power consumption $P(t)$ in $G$ is the same as in $G^{\\prime \\prime }$ if $t > t_1$ , hence a constant.",
"Considering the same partitioning with $t_2$ instead of $t_1$ , we show that the total power consumption $P(t)$ is a constant before $t_2$ , and also a constant after $t_2$ .",
"But $t_1 < t_2$ , and the intervals $[0,t_2]$ and $[t_1,D]$ overlap.",
"Altogether, the total power consumption is the same constant throughout $[0,D]$ , which concludes the proof.",
"Discrete models In this section, we present complexity results on the three energy models with a finite number of possible speeds.",
"The only polynomial instance is for the Vdd-Hopping model, for which we write a linear program in Section REF .",
"Then, we give NP-completeness results in Section REF , and approximation results in Section REF , for the Discrete and Incremental models.",
"The Vdd-Hopping model Theorem 5 With the Vdd-Hopping model, $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ can be solved in polynomial time.",
"Let $G$ be the execution graph of an application with $n$ tasks, and $D$ a deadline.",
"Let $s_1,...,s_m$ be the set of possible processor speeds.",
"We use the following rational variables: for $1\\le i \\le n$ and $1\\le j \\le m$ , $b_i$ is the starting time of the execution of task $T_i$ , and $\\alpha _{(i,j)}$ is the time spent at speed $s_j$ for executing task $T_i$ .",
"There are $n + n\\times m = n(m+1)$ such variables.",
"Note that the total execution time of task $T_i$ is $\\sum _{j=1}^m \\alpha _{(i,j)}$ .",
"The constraints are: $\\forall 1\\le i\\le n, \\; b_i\\ge 0$ : starting times of all tasks are non-negative numbers; $\\forall 1\\le i \\le n, \\; b_i+\\sum _{j=1}^m\\alpha _{(i,j)}\\le D$ : the deadline is not exceeded by any task; $\\forall 1 \\le i,i^{\\prime } \\le n$ such that $T_i\\rightarrow T_{i^{\\prime }}$ , $\\; t_i+\\sum _{j=1}^m \\alpha _{(i,j)}\\le t_{i^{\\prime }}$ : a task cannot start before its predecessor has completed its execution; $\\forall 1\\le i \\le n, \\; \\sum _{j=1}^m\\alpha _{(i,j)} \\times s_j\\ge w_i$ : task $T_i$ is completely executed.",
"The objective function is then $\\min \\left(\\sum _{i=1}^n\\sum _{j=1}^m \\alpha _{(i,j)}s_j^3\\right)$ .",
"The size of this linear program is clearly polynomial in the size of the instance, all $n(m+1)$ variables are rational, and therefore it can be solved in polynomial time [30].",
"NP-completeness results Theorem 6 With the Incremental model (and hence the Discrete model), $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is NP-complete.",
"We consider the associated decision problem: given an execution graph, a deadline, and a bound on the energy consumption, can we find an execution speed for each task such that the deadline and the bound on energy are respected?",
"The problem is clearly in NP: given the execution speed of each task, computing the execution time and the energy consumption can be done in polynomial time.",
"To establish the completeness, we use a reduction from 2-Partition [11].",
"We consider an instance $\\mathcal {I} _1$ of 2-Partition: given $n$ strictly positive integers $a_1, \\ldots , a_n$ , does there exist a subset $I$ of $\\lbrace 1, \\ldots , n\\rbrace $ such that $\\sum _{i\\in I}a_i=\\sum _{i \\notin I}a_i$ ?",
"Let $T=\\frac{1}{2}\\sum _{i=1}^n a_i$ .",
"We build the following instance $\\mathcal {I} _2$ of our problem: the execution graph is a linear chain with $n$ tasks, where: task $T_i$ has size $w_i=a_i$ ; the processor can run at $m=2$ different speeds; $s_1=1$ and $s_2=2$ , (i.e., $s_{\\mathit {min}}=1, s_{\\mathit {max}}=2, \\delta =1$ ); $L=3T/2$ ; $E=5T$ .",
"Clearly, the size of $\\mathcal {I} _2$ is polynomial in the size of $\\mathcal {I} _1$ .",
"Suppose first that instance $\\mathcal {I} _1$ has a solution $I$ .",
"For all $i\\in I$ , $T_i$ is executed at speed 1, otherwise it is executed at speed 2.",
"The execution time is then $\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2=\\frac{3}{2}T = D$ , and the energy consumption is $E=\\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 5T = E$ .",
"Both bounds are respected, and therefore the execution speeds are a solution to $\\mathcal {I} _2$ .",
"Suppose now that $\\mathcal {I} _2$ has a solution.",
"Since we consider the Discrete and Incremental models, each task run either at speed 1, or at speed 2.",
"Let $I=\\lbrace i\\; |\\; T_i\\mbox{ is executed at speed }1\\rbrace $ .",
"Note that we have $\\sum _{i\\notin I}a_i = 2T - \\sum _{i\\in I}a_i$ .",
"The execution time is $D^{\\prime }=\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2 = T + (\\sum _{i\\in I}a_i)/2$ .",
"Since the deadline is not exceeded, $D^{\\prime } \\le D = 3T/2$ , and therefore $\\sum _{i\\in I}a_i \\le T$ .",
"For the energy consumption of the solution of $\\mathcal {I} _2$ , we have $E^{\\prime } = \\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 2T + 3\\sum _{i\\notin I}a_i$ .",
"Since $E^{\\prime }\\le E=5T$ , we obtain $3\\sum _{i\\notin I}a_i\\le 3T$ , and hence $\\sum _{i\\notin I}a_i\\le T$ .",
"Since $\\sum _{i\\in I}a_i + \\sum _{i\\notin I}a_i = 2T$ , we conclude that $\\sum _{i\\in I}a_i = \\sum _{i\\notin I}a_i =T$ , and therefore $\\mathcal {I} _1$ has a solution.",
"This concludes the proof.",
"Approximation results Here we explain, for the Incremental and Discrete models, how the solution to the NP-hard problem can be approximated.",
"Note that, given an execution graph and a deadline, the optimal energy consumption with the Continuous model is always lower than that with the other models, which are more constrained.",
"Theorem 7 With the Incremental model, for any integer $K>0$ , the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2$ , in a time polynomial in the size of the instance and in $K$ .",
"Consider an instance $\\mathcal {I} _{inc}$ of the problem with the Incremental model.",
"The execution graph $G$ has $n$ tasks, $D$ is the deadline, $\\delta $ is the minimum permissible speed increment, and $s_{\\mathit {min}}, s_{\\mathit {max}}$ are the speed bounds.",
"Moreover, let $K>0$ be an integer, and let $E_{inc}$ be the optimal value of the energy consumption for this instance $\\mathcal {I} _{inc}$ .",
"We construct the following instance $\\mathcal {I} _{vdd}$ with the Vdd-Hopping model: the execution graph and the deadline are the same as in instance $\\mathcal {I} _{inc}$ , and the speeds can take the values $\\left\\lbrace s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^i\\right\\rbrace _{0\\le i\\le N}\\;,$ where $N$ is such that $s_{max}$ is not exceeded: $N=\\left\\lfloor (\\ln (s_{\\mathit {max}})-\\ln (s_{\\mathit {min}}))/\\ln \\left(1+\\frac{1}{K}\\right)\\right\\rfloor $ .",
"As $N$ is asymptotically of order $O(K\\ln (s_{\\mathit {max}}))$ , the number of possible speeds in $\\mathcal {I} _{vdd}$ , and hence the size of $\\mathcal {I} _{vdd}$ , is polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Next, we solve $\\mathcal {I} _{vdd}$ in polynomial time thanks to Theorem REF .",
"For each task $T_i$ , let $s^{(vdd)}_i$ be the average speed of $T_i$ in this solution: if the execution time of the task in the solution is $d_i$ , then $s^{(vdd)}_i = w_i/d_i$ ; $E_{vdd}$ is the optimal energy consumption obtained with these speeds.",
"Let $s^{(algo)}_i=\\min _u\\lbrace s_{\\mathit {min}}+u\\times \\delta \\; | \\; u\\times \\delta \\ge s^{(vdd)}_i\\rbrace $ be the smallest speed in $\\mathcal {I} _{inc}$ which is larger than $s^{(vdd)}_i$ .",
"There exists such a speed since, because of the values chosen for $\\mathcal {I} _{vdd}$ , $s^{(vdd)}_i\\le s_{\\mathit {max}}$ .",
"The values $s^{(algo)}_i$ can be computed in time polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Let $E_{algo}$ be the energy consumption obtained with these values.",
"In order to prove that this algorithm is an approximation of the optimal solution, we need to prove that $E_{algo} \\le (1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2\\times E_{inc}$ .",
"For each task $T_i$ , $s^{(algo)}_i-\\delta \\le s^{(vdd)}_i\\le s^{(algo)}_i$ .",
"Since $s_{\\mathit {min}}\\le s^{(vdd)}_i$ , we derive that $s^{(algo)}_i \\le s^{(vdd)}_i \\times (1+\\frac{\\delta }{s_{\\mathit {min}}})$ .",
"Summing over all tasks, we get $\\text{~~~~~~~~~~~~~~} E_{algo}=\\sum _iw_i\\left(s^{(algo)}_i\\right)^2\\le \\sum _iw_i\\left(s^{(vdd)}_i\\times (1+\\frac{\\delta }{s_{\\mathit {min}}})\\right)^2\\le E_{vdd} \\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Next, we bound $E_{vdd}$ thanks to the optimal solution with the Continuous model, $E_{con}$ .",
"Let $\\mathcal {I} _{con}$ be the instance where the execution graph $G$ , the deadline $D$ , the speeds $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ are the same as in instance $\\mathcal {I} _{inc}$ , but now admissible speeds take any value between $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ .",
"Let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ , and let $0\\le u\\le N$ be the value such that: $ \\text{~~~~~~~~~~~~~~} s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^u \\le s^{(con)}_i\\le s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^{u+1}=s^*_i~$ .",
"In order to bound the energy consumption for $I_{vdd}$ , we assume that $T_i$ runs at speed $s^*_i$ , instead of $s^{(vdd)}_i$ .",
"The solution with these speeds is a solution to $I_{vdd}$ , and its energy consumption is $E^* \\ge E_{vdd}$ .",
"From the previous inequalities, we deduce that $s^*_i \\le s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)$ , and by summing over all tasks, $E_{vdd} \\le E^* = \\sum _i w_i \\left(s^*_i\\right)^2\\le \\sum _i w_i \\left(s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)\\right)^2\\le E_{con} \\times \\left(1+\\frac{1}{K}\\right)^2\\le E_{inc} \\times \\left(1+\\frac{1}{K}\\right)^2\\; .$ Proposition 3 ${ }$ For any integer $\\delta >0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Continuous model can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2$ in the Incremental model with speed increment $\\delta $ .",
"For any integer $K>0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Discrete model can be approximated within a factor $(1+\\frac{\\alpha }{s_1})^2(1+\\frac{1}{K})^2$ , with $\\alpha =\\max _{1 \\le i < m}\\lbrace s_{i+1}-s_i\\rbrace $ , in a time polynomial in the size of the instance and in $K$ .",
"For the first part, let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ in instance $\\mathcal {I} _{con}$ ; $E_{con}$ is the optimal energy consumption.",
"For any task $T_i$ , let $s_i$ be the speed of $\\mathcal {I} _{inc}$ such that $s_i-\\delta <s^{con}_i\\le s_i$ .",
"Then, $s^{(con)}_i\\le s_i\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)$ .",
"Let $E$ be the energy with speeds $s_i$ .",
"$E_{con}\\le E\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Let $E_{inc}$ be the optimal energy of $\\mathcal {I} _{inc}$ .",
"Then, $E_{con}\\le E_{inc}\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"For the second part, we use the same algorithm as in Theorem REF .",
"The same proof leads to the approximation ratio with $\\alpha $ instead of $\\delta $ .",
"Conclusion In this paper, we have assessed the tractability of a classical scheduling problem, with task preallocation, under various energy models.",
"We have given several results related to Continuous speeds.",
"However, while these are of conceptual importance, they cannot be achieved with physical devices, and we have analyzed several models enforcing a bounded number of achievable speeds, a.k.a.",
"modes.",
"In the classical Discrete model that arises from DVFS techniques, admissible speeds can be irregularly distributed, which motivates the Vdd-Hopping approach that mixes two consecutive modes optimally.",
"While computing optimal speeds is NP-hard with discrete modes, it has polynomial complexity when mixing speeds.",
"Intuitively, the Vdd-Hopping approach allows for smoothing out the discrete nature of the modes.",
"An alternate (and simpler in practice) solution to Vdd-Hopping is the Incremental model, where one sticks with unique speeds during task execution as in the Discrete model, but where consecutive modes are regularly spaced.",
"Such a model can be made arbitrarily efficient, according to our approximation results.",
"Altogether, this paper has laid the theoretical foundations for a comparative study of energy models.",
"In the recent years, we have observed an increased concern for green computing, and a rapidly growing number of approaches.",
"It will be very interesting to see which energy-saving technological solutions will be implemented in forthcoming future processor chips!"
],
[
"Discrete models",
"In this section, we present complexity results on the three energy models with a finite number of possible speeds.",
"The only polynomial instance is for the Vdd-Hopping model, for which we write a linear program in Section REF .",
"Then, we give NP-completeness results in Section REF , and approximation results in Section REF , for the Discrete and Incremental models."
],
[
"The ",
"Theorem 5 With the Vdd-Hopping model, $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ can be solved in polynomial time.",
"Let $G$ be the execution graph of an application with $n$ tasks, and $D$ a deadline.",
"Let $s_1,...,s_m$ be the set of possible processor speeds.",
"We use the following rational variables: for $1\\le i \\le n$ and $1\\le j \\le m$ , $b_i$ is the starting time of the execution of task $T_i$ , and $\\alpha _{(i,j)}$ is the time spent at speed $s_j$ for executing task $T_i$ .",
"There are $n + n\\times m = n(m+1)$ such variables.",
"Note that the total execution time of task $T_i$ is $\\sum _{j=1}^m \\alpha _{(i,j)}$ .",
"The constraints are: $\\forall 1\\le i\\le n, \\; b_i\\ge 0$ : starting times of all tasks are non-negative numbers; $\\forall 1\\le i \\le n, \\; b_i+\\sum _{j=1}^m\\alpha _{(i,j)}\\le D$ : the deadline is not exceeded by any task; $\\forall 1 \\le i,i^{\\prime } \\le n$ such that $T_i\\rightarrow T_{i^{\\prime }}$ , $\\; t_i+\\sum _{j=1}^m \\alpha _{(i,j)}\\le t_{i^{\\prime }}$ : a task cannot start before its predecessor has completed its execution; $\\forall 1\\le i \\le n, \\; \\sum _{j=1}^m\\alpha _{(i,j)} \\times s_j\\ge w_i$ : task $T_i$ is completely executed.",
"The objective function is then $\\min \\left(\\sum _{i=1}^n\\sum _{j=1}^m \\alpha _{(i,j)}s_j^3\\right)$ .",
"The size of this linear program is clearly polynomial in the size of the instance, all $n(m+1)$ variables are rational, and therefore it can be solved in polynomial time [30].",
"NP-completeness results Theorem 6 With the Incremental model (and hence the Discrete model), $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ is NP-complete.",
"We consider the associated decision problem: given an execution graph, a deadline, and a bound on the energy consumption, can we find an execution speed for each task such that the deadline and the bound on energy are respected?",
"The problem is clearly in NP: given the execution speed of each task, computing the execution time and the energy consumption can be done in polynomial time.",
"To establish the completeness, we use a reduction from 2-Partition [11].",
"We consider an instance $\\mathcal {I} _1$ of 2-Partition: given $n$ strictly positive integers $a_1, \\ldots , a_n$ , does there exist a subset $I$ of $\\lbrace 1, \\ldots , n\\rbrace $ such that $\\sum _{i\\in I}a_i=\\sum _{i \\notin I}a_i$ ?",
"Let $T=\\frac{1}{2}\\sum _{i=1}^n a_i$ .",
"We build the following instance $\\mathcal {I} _2$ of our problem: the execution graph is a linear chain with $n$ tasks, where: task $T_i$ has size $w_i=a_i$ ; the processor can run at $m=2$ different speeds; $s_1=1$ and $s_2=2$ , (i.e., $s_{\\mathit {min}}=1, s_{\\mathit {max}}=2, \\delta =1$ ); $L=3T/2$ ; $E=5T$ .",
"Clearly, the size of $\\mathcal {I} _2$ is polynomial in the size of $\\mathcal {I} _1$ .",
"Suppose first that instance $\\mathcal {I} _1$ has a solution $I$ .",
"For all $i\\in I$ , $T_i$ is executed at speed 1, otherwise it is executed at speed 2.",
"The execution time is then $\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2=\\frac{3}{2}T = D$ , and the energy consumption is $E=\\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 5T = E$ .",
"Both bounds are respected, and therefore the execution speeds are a solution to $\\mathcal {I} _2$ .",
"Suppose now that $\\mathcal {I} _2$ has a solution.",
"Since we consider the Discrete and Incremental models, each task run either at speed 1, or at speed 2.",
"Let $I=\\lbrace i\\; |\\; T_i\\mbox{ is executed at speed }1\\rbrace $ .",
"Note that we have $\\sum _{i\\notin I}a_i = 2T - \\sum _{i\\in I}a_i$ .",
"The execution time is $D^{\\prime }=\\sum _{i\\in I}a_i+\\sum _{i\\notin I}a_i/2 = T + (\\sum _{i\\in I}a_i)/2$ .",
"Since the deadline is not exceeded, $D^{\\prime } \\le D = 3T/2$ , and therefore $\\sum _{i\\in I}a_i \\le T$ .",
"For the energy consumption of the solution of $\\mathcal {I} _2$ , we have $E^{\\prime } = \\sum _{i\\in I} a_i+\\sum _{i\\notin I}a_i\\times 2^2 = 2T + 3\\sum _{i\\notin I}a_i$ .",
"Since $E^{\\prime }\\le E=5T$ , we obtain $3\\sum _{i\\notin I}a_i\\le 3T$ , and hence $\\sum _{i\\notin I}a_i\\le T$ .",
"Since $\\sum _{i\\in I}a_i + \\sum _{i\\notin I}a_i = 2T$ , we conclude that $\\sum _{i\\in I}a_i = \\sum _{i\\notin I}a_i =T$ , and therefore $\\mathcal {I} _1$ has a solution.",
"This concludes the proof.",
"Approximation results Here we explain, for the Incremental and Discrete models, how the solution to the NP-hard problem can be approximated.",
"Note that, given an execution graph and a deadline, the optimal energy consumption with the Continuous model is always lower than that with the other models, which are more constrained.",
"Theorem 7 With the Incremental model, for any integer $K>0$ , the $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ problem can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2$ , in a time polynomial in the size of the instance and in $K$ .",
"Consider an instance $\\mathcal {I} _{inc}$ of the problem with the Incremental model.",
"The execution graph $G$ has $n$ tasks, $D$ is the deadline, $\\delta $ is the minimum permissible speed increment, and $s_{\\mathit {min}}, s_{\\mathit {max}}$ are the speed bounds.",
"Moreover, let $K>0$ be an integer, and let $E_{inc}$ be the optimal value of the energy consumption for this instance $\\mathcal {I} _{inc}$ .",
"We construct the following instance $\\mathcal {I} _{vdd}$ with the Vdd-Hopping model: the execution graph and the deadline are the same as in instance $\\mathcal {I} _{inc}$ , and the speeds can take the values $\\left\\lbrace s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^i\\right\\rbrace _{0\\le i\\le N}\\;,$ where $N$ is such that $s_{max}$ is not exceeded: $N=\\left\\lfloor (\\ln (s_{\\mathit {max}})-\\ln (s_{\\mathit {min}}))/\\ln \\left(1+\\frac{1}{K}\\right)\\right\\rfloor $ .",
"As $N$ is asymptotically of order $O(K\\ln (s_{\\mathit {max}}))$ , the number of possible speeds in $\\mathcal {I} _{vdd}$ , and hence the size of $\\mathcal {I} _{vdd}$ , is polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Next, we solve $\\mathcal {I} _{vdd}$ in polynomial time thanks to Theorem REF .",
"For each task $T_i$ , let $s^{(vdd)}_i$ be the average speed of $T_i$ in this solution: if the execution time of the task in the solution is $d_i$ , then $s^{(vdd)}_i = w_i/d_i$ ; $E_{vdd}$ is the optimal energy consumption obtained with these speeds.",
"Let $s^{(algo)}_i=\\min _u\\lbrace s_{\\mathit {min}}+u\\times \\delta \\; | \\; u\\times \\delta \\ge s^{(vdd)}_i\\rbrace $ be the smallest speed in $\\mathcal {I} _{inc}$ which is larger than $s^{(vdd)}_i$ .",
"There exists such a speed since, because of the values chosen for $\\mathcal {I} _{vdd}$ , $s^{(vdd)}_i\\le s_{\\mathit {max}}$ .",
"The values $s^{(algo)}_i$ can be computed in time polynomial in the size of $\\mathcal {I} _{inc}$ and $K$ .",
"Let $E_{algo}$ be the energy consumption obtained with these values.",
"In order to prove that this algorithm is an approximation of the optimal solution, we need to prove that $E_{algo} \\le (1+\\frac{\\delta }{s_{\\mathit {min}}})^2(1+\\frac{1}{K})^2\\times E_{inc}$ .",
"For each task $T_i$ , $s^{(algo)}_i-\\delta \\le s^{(vdd)}_i\\le s^{(algo)}_i$ .",
"Since $s_{\\mathit {min}}\\le s^{(vdd)}_i$ , we derive that $s^{(algo)}_i \\le s^{(vdd)}_i \\times (1+\\frac{\\delta }{s_{\\mathit {min}}})$ .",
"Summing over all tasks, we get $\\text{~~~~~~~~~~~~~~} E_{algo}=\\sum _iw_i\\left(s^{(algo)}_i\\right)^2\\le \\sum _iw_i\\left(s^{(vdd)}_i\\times (1+\\frac{\\delta }{s_{\\mathit {min}}})\\right)^2\\le E_{vdd} \\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Next, we bound $E_{vdd}$ thanks to the optimal solution with the Continuous model, $E_{con}$ .",
"Let $\\mathcal {I} _{con}$ be the instance where the execution graph $G$ , the deadline $D$ , the speeds $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ are the same as in instance $\\mathcal {I} _{inc}$ , but now admissible speeds take any value between $s_{\\mathit {min}}$ and $s_{\\mathit {max}}$ .",
"Let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ , and let $0\\le u\\le N$ be the value such that: $ \\text{~~~~~~~~~~~~~~} s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^u \\le s^{(con)}_i\\le s_{\\mathit {min}}\\times \\left(1+\\frac{1}{K}\\right)^{u+1}=s^*_i~$ .",
"In order to bound the energy consumption for $I_{vdd}$ , we assume that $T_i$ runs at speed $s^*_i$ , instead of $s^{(vdd)}_i$ .",
"The solution with these speeds is a solution to $I_{vdd}$ , and its energy consumption is $E^* \\ge E_{vdd}$ .",
"From the previous inequalities, we deduce that $s^*_i \\le s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)$ , and by summing over all tasks, $E_{vdd} \\le E^* = \\sum _i w_i \\left(s^*_i\\right)^2\\le \\sum _i w_i \\left(s^{(con)}_i \\times \\left(1+\\frac{1}{K}\\right)\\right)^2\\le E_{con} \\times \\left(1+\\frac{1}{K}\\right)^2\\le E_{inc} \\times \\left(1+\\frac{1}{K}\\right)^2\\; .$ Proposition 3 ${ }$ For any integer $\\delta >0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Continuous model can be approximated within a factor $(1+\\frac{\\delta }{s_{\\mathit {min}}})^2$ in the Incremental model with speed increment $\\delta $ .",
"For any integer $K>0$ , any instance of $\\textsc {Min\\-En\\-er\\-gy}(G,D)$ with the Discrete model can be approximated within a factor $(1+\\frac{\\alpha }{s_1})^2(1+\\frac{1}{K})^2$ , with $\\alpha =\\max _{1 \\le i < m}\\lbrace s_{i+1}-s_i\\rbrace $ , in a time polynomial in the size of the instance and in $K$ .",
"For the first part, let $s^{(con)}_i$ be the optimal continuous speed for task $T_i$ in instance $\\mathcal {I} _{con}$ ; $E_{con}$ is the optimal energy consumption.",
"For any task $T_i$ , let $s_i$ be the speed of $\\mathcal {I} _{inc}$ such that $s_i-\\delta <s^{con}_i\\le s_i$ .",
"Then, $s^{(con)}_i\\le s_i\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)$ .",
"Let $E$ be the energy with speeds $s_i$ .",
"$E_{con}\\le E\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"Let $E_{inc}$ be the optimal energy of $\\mathcal {I} _{inc}$ .",
"Then, $E_{con}\\le E_{inc}\\times \\left(1+\\frac{\\delta }{s_{\\mathit {min}}}\\right)^2$ .",
"For the second part, we use the same algorithm as in Theorem REF .",
"The same proof leads to the approximation ratio with $\\alpha $ instead of $\\delta $ .",
"Conclusion In this paper, we have assessed the tractability of a classical scheduling problem, with task preallocation, under various energy models.",
"We have given several results related to Continuous speeds.",
"However, while these are of conceptual importance, they cannot be achieved with physical devices, and we have analyzed several models enforcing a bounded number of achievable speeds, a.k.a.",
"modes.",
"In the classical Discrete model that arises from DVFS techniques, admissible speeds can be irregularly distributed, which motivates the Vdd-Hopping approach that mixes two consecutive modes optimally.",
"While computing optimal speeds is NP-hard with discrete modes, it has polynomial complexity when mixing speeds.",
"Intuitively, the Vdd-Hopping approach allows for smoothing out the discrete nature of the modes.",
"An alternate (and simpler in practice) solution to Vdd-Hopping is the Incremental model, where one sticks with unique speeds during task execution as in the Discrete model, but where consecutive modes are regularly spaced.",
"Such a model can be made arbitrarily efficient, according to our approximation results.",
"Altogether, this paper has laid the theoretical foundations for a comparative study of energy models.",
"In the recent years, we have observed an increased concern for green computing, and a rapidly growing number of approaches.",
"It will be very interesting to see which energy-saving technological solutions will be implemented in forthcoming future processor chips!"
],
[
"Conclusion",
"In this paper, we have assessed the tractability of a classical scheduling problem, with task preallocation, under various energy models.",
"We have given several results related to Continuous speeds.",
"However, while these are of conceptual importance, they cannot be achieved with physical devices, and we have analyzed several models enforcing a bounded number of achievable speeds, a.k.a.",
"modes.",
"In the classical Discrete model that arises from DVFS techniques, admissible speeds can be irregularly distributed, which motivates the Vdd-Hopping approach that mixes two consecutive modes optimally.",
"While computing optimal speeds is NP-hard with discrete modes, it has polynomial complexity when mixing speeds.",
"Intuitively, the Vdd-Hopping approach allows for smoothing out the discrete nature of the modes.",
"An alternate (and simpler in practice) solution to Vdd-Hopping is the Incremental model, where one sticks with unique speeds during task execution as in the Discrete model, but where consecutive modes are regularly spaced.",
"Such a model can be made arbitrarily efficient, according to our approximation results.",
"Altogether, this paper has laid the theoretical foundations for a comparative study of energy models.",
"In the recent years, we have observed an increased concern for green computing, and a rapidly growing number of approaches.",
"It will be very interesting to see which energy-saving technological solutions will be implemented in forthcoming future processor chips!"
]
] | 1204.0939 |
[
[
"Generalized Hausdorff measure for generic compact sets"
],
[
"Abstract Let $X$ be a Polish space.",
"We prove that the generic compact set $K\\subseteq X$ (in the sense of Baire category) is either finite or there is a continuous gauge function $h$ such that $0<\\mathcal{H}^{h}(K)<\\infty$, where $\\mathcal{H}^h$ denotes the $h$-Hausdorff measure.",
"This answers a question of C. Cabrelli, U.",
"B. Darji, and U. M. Molter.",
"Moreover, for every weak contraction $f\\colon K\\to X$ we have $\\mathcal{H}^{h} (K\\cap f(K))=0$.",
"This is a measure theoretic analogue of a result of M. Elekes."
],
[
"Introduction",
"Hausdorff dimension is one of the most important concepts to measure the size of a metric space, but there are some cases when a finer notion of dimension is needed.",
"An important example is the trail of the $n$ -dimensional $(n\\ge 2)$ Brownian motion defined on $[0,1]$ .",
"It has Hausdorff dimension 2 almost surely, but its $\\mathcal {H}^2$ measure is 0 with probability 1.",
"It is well-known that there is a gauge function $h$ such that the $h$ -Hausdorff measure of the trail is positive and finite almost surely, where $h(x)=x^2 \\log \\log (1/x)$ if $n\\ge 3$ and $h(x)=x^2 \\log (1/x) \\log \\log \\log (1/x)$ if $n=2$ .",
"Thus the exact dimension is logarithmically smaller than 2.",
"R. O. Davies [3] constructed a Cantor set $K\\subseteq \\mathbb {R}$ that is either null or non-$\\sigma $ -finite for every translation invariant Borel measure on $\\mathbb {R}$ .",
"This implies that there is no gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ , where $\\mathcal {H}^h$ denotes the $h$ -Hausdorff measure.",
"C. Cabrelli, U. B.",
"Darji, and U. M. Molter [2] dealt with the problem that for `how many' compact sets $K\\subseteq \\mathbb {R}$ exist a translation invariant Borel measure $\\mu $ or a gauge function $h$ such that $0<\\mu (K)<\\infty $ or $0<\\mathcal {H}^{h}(K)<\\infty $ , respectively.",
"They proved that the generic compact set $K\\subseteq \\mathbb {R}$ (see Definition REF ) admits a translation invariant Borel measure $\\mu $ such that $0<\\mu (K)<\\infty $ .",
"They defined a compact set $K\\subseteq \\mathbb {R}$ to be $\\mathcal {H}$ -visible if there is a gauge function $h$ such that $0<\\mathcal {H}^h(K)<\\infty $ .",
"They showed that the set of $\\mathcal {H}$ -visible compact sets is dense in the space of all non-empty compact subsets of $\\mathbb {R}$ endowed with the Hausdorff metric.",
"They posed the problem whether the generic compact set $K\\subseteq \\mathbb {R}$ is $\\mathcal {H}$ -visible.",
"We answer this question affirmatively by the following more general result.",
"Theorem 1.1 Let $X$ be a Polish space.",
"The generic compact set $K\\subseteq X$ is either finite or there is a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ .",
"We remark here that for every fixed gauge function $h$ the generic compact set $K\\subseteq X$ has zero $\\mathcal {H}^h$ measure.",
"If $X$ is a perfect Polish space then the set of finite compact subsets of $X$ form a meager set in the metric space of all non-empty compact subsets of $X$ endowed with the Hausdorff metric.",
"Therefore Theorem REF implies the following result.",
"Corollary 1.2 Let $X$ be a perfect Polish space.",
"For the generic compact set $K\\subseteq X$ there is a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ .",
"M. Elekes [4] studied metric spaces $X$ which are not complete but possess the Banach Fixed Point Theorem, that is, every contraction $f\\colon X\\rightarrow X$ has a fixed point.",
"He proved the following theorem which is interesting in its own right.",
"Theorem 1.3 (M. Elekes) For the generic compact set $K\\subseteq \\mathbb {R}$ for any contraction $f\\colon K\\rightarrow \\mathbb {R}$ the set $f(K)$ does not contain a non-empty relatively open subset of $K$ .",
"The first author of the present paper [1] constructed metric spaces $X$ such that every weak contraction $f\\colon X\\rightarrow X$ is constant, where he used measure theoretic methods.",
"Based on [1], we prove the (somewhat stronger) measure theoretic analogue of Theorem REF .",
"Theorem REF (Main Theorem) Let $X$ be a Polish space.",
"The generic compact set $K\\subseteq X$ is either finite or there is a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ , and for every weak contraction $f\\colon K\\rightarrow X$ we have $\\mathcal {H}^{h} \\left(K\\cap f(K)\\right)=0$ .",
"In Section we recall some notions from metric spaces which we use in this paper.",
"In Section we introduce the notion of balanced compact sets.",
"It is shown in [1] that for every balanced compact set there is a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ and that $\\mathcal {H}^{h} \\left(K\\cap f(K)\\right)=0$ for every weak contraction $f\\colon K\\rightarrow X$ .",
"In Section we prove that in a perfect Polish space the generic compact set is a balanced compact set, and we conclude the proof of Theorem REF and Theorem REF ."
],
[
"Preliminaries",
"Let $(X,d)$ be a metric space, and let $A,B\\subseteq X$ be arbitrary sets.",
"We denote by $\\operatorname{cl}A$ and $\\operatorname{diam}A$ the closure and the diameter of $A$ , respectively.",
"We use the convention $\\operatorname{diam}\\emptyset = 0$ .",
"The distance of the sets $A$ and $B$ is defined by $\\operatorname{dist}(A,B)=\\inf \\lbrace d(x,y): x\\in A, \\, y\\in B\\rbrace $ .",
"Let $B(x,r)=\\lbrace y\\in X: d(x,y)\\le r\\rbrace $ and $U(x,r)=\\lbrace y\\in X: d(x,y)< r\\rbrace $ for all $x\\in X$ and $r>0$ .",
"More generally, consider $B(A,r)=\\lbrace x\\in X: \\operatorname{dist}(A,\\lbrace x\\rbrace )\\le r\\rbrace $ .",
"The function $h\\colon [0,\\infty )\\rightarrow [0,\\infty )$ is defined to be a gauge function if it is non-decreasing, right-continuous, and $h(x)=0$ iff $x=0$ .",
"For $A\\subseteq X$ and $\\delta >0$ consider $\\mathcal {H}^{h}_{\\delta }(A)&=\\inf \\left\\lbrace \\sum _{i=1}^\\infty h\\left(\\operatorname{diam}A_{i}\\right): A \\subseteq \\bigcup _{i=1}^{\\infty } A_{i},\\, \\forall i \\,\\operatorname{diam}A_i \\le \\delta \\right\\rbrace , \\\\\\mathcal {H}^{h}(A)&=\\lim _{\\delta \\rightarrow 0+}\\mathcal {H}^{h}_{\\delta }(A).$ We call $\\mathcal {H}^{h}$ the $h$ -Hausdorff measure.",
"For more information on these concepts see [6].",
"Let $X$ be a complete metric space.",
"A set is somewhere dense if it is dense in a non-empty open set, otherwise it is called nowhere dense.",
"We say that $M \\subseteq X$ is meager if it is a countable union of nowhere dense sets, and a set is co-meager if its complement is meager.",
"Baire's Category Theorem implies that a set is co-meager if and only if it contains a dense $G_{\\delta }$ set.",
"We say that the generic element $x \\in X$ has property $\\mathcal {P}$ if $\\lbrace x \\in X : x \\textrm { has property } \\mathcal {P} \\rbrace $ is co-meager.",
"A metric space $X$ is perfect if it has no isolated points.",
"A metric space $X$ is Polish if it is complete and separable.",
"Given two metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ , a function $f\\colon X\\rightarrow Y$ is called a weak contraction if $d_{Y}(f(x_{1}),f(x_{2}))<d_{X}(x_{1},x_{2})$ for every $x_{1},x_{2}\\in X$ , $x_1\\ne x_2$ .",
"Let $\\mathbb {N}^{<\\omega }$ stand for the set of finite sequences of natural numbers.",
"Let us denote the set of positive odd numbers by $2\\mathbb {N}+1$ ."
],
[
"The definition of balanced compact sets",
"Following [1] we define balanced compact sets.",
"Definition 3.1 If $a_n$ $(n\\in \\mathbb {N}^+)$ are positive integers then let us consider, for every $n\\in \\mathbb {N}^+$ , $\\mathcal {I}_{n}=\\prod _{k=1}^{n}\\lbrace 1,2,\\dots ,a_k\\rbrace \\quad \\textrm {and} \\quad \\mathcal {I}=\\bigcup _{n=1}^{\\infty } \\mathcal {I}_{n}.$ We say that a map $\\Phi \\colon 2\\mathbb {N} +1 \\rightarrow \\mathcal {I}$ is an index function according to the sequence $\\langle a_n \\rangle $ if it is surjective and $\\Phi (n) \\in \\bigcup _{k=1}^{n} \\mathcal {I}_{k}$ for every odd $n$ .",
"Definition 3.2 Let $X$ be a Polish space.",
"A compact set $K\\subseteq X$ is balanced if it is of the form $ K=\\bigcap _{n=1}^{\\infty }\\left(\\bigcup _{i_1=1}^{a_1}\\dots \\bigcup _{i_n=1}^{a_n}C_{i_1 \\dots i_n} \\right),$ where the $a_{n}$ are positive integers and $C_{i_1\\dots i_n}\\subseteq X$ are non-empty closed sets with the following properties.",
"There are positive reals $b_n$ and there is an index function $\\Phi \\colon 2 \\mathbb {N}+1 \\rightarrow \\mathcal {I}$ according to the sequence $\\langle a_n\\rangle $ such that for all $n\\in \\mathbb {N}^+$ and $(i_1,\\dots ,i_{n}),(j_1,\\dots ,j_{n})\\in \\mathcal {I}_{n}$ $a_1\\ge 2$ and $a_{n+1}\\ge n a_{1}\\cdots a_{n}$ , $C_{i_{1}\\dots i_{n+1}}\\subseteq C_{i_1 \\dots i_{n}}$ , $\\operatorname{diam}C_{i_{1} \\dots i_n}\\le b_n$ , $\\operatorname{dist}(C_{i_1 \\dots i_n},C_{j_1\\dots j_n})>2b_n$ if $(i_1,\\dots ,i_n)\\ne (j_1, \\dots ,j_n)$ .",
"If $n$ is odd, $C_{i_1 \\dots i_{n}}\\subseteq C_{\\Phi (n)}$ and $C_{j_1 \\dots j_{n}}\\nsubseteq C_{\\Phi (n)}$ , then for all $s,t\\in \\lbrace 1,\\dots ,a_{n+1}\\rbrace $ , $s\\ne t$ , we have $\\operatorname{dist}\\left(C_{i_1\\dots i_{n}s},C_{i_1 \\dots i_{n}t}\\right)> \\operatorname{diam}\\left(\\bigcup _{j_{n+1}=1}^{a_{n+1}} C_{j_1 \\dots j_{n}j_{n+1}}\\right).$ Remark 3.3 Property (REF ) and the notion of an index function $\\Phi $ are not needed for the proof of Theorem REF , only for Theorem REF .",
"Note that we cannot require property (REF ) for every positive integer.",
"The proof of Lemma REF only works if we restrict this property to odd numbers.",
"Remark 3.4 In a countable Polish space $X$ there is no balanced compact set $K\\subseteq X$ , since every balanced compact set has cardinality $2^{\\aleph _0}$ ."
],
[
"The Main Theorem",
"Definition 4.1 If $X$ is a Polish space then let $(\\mathcal {K}(X),d_{H})$ be the set of non-empty compact subsets of $X$ endowed with the Hausdorff metric; that is, for each $K_1,K_2\\in \\mathcal {K}(X)$ , $d_{H}(K_1,K_2)=\\min \\left\\lbrace r: K_1\\subseteq B(K_2,r) \\textrm { and } K_2\\subseteq B(K_1,r)\\right\\rbrace .$ It is well-known that $(\\mathcal {K}(X),d_{H})$ is a Polish space, see e.g.",
"[5], hence we can use Baire category arguments.",
"Let $B_{H}(K,r)\\subseteq \\mathcal {K}(X)$ denote the closed ball around $K$ with radius $r$ .",
"The main goal of this paper is to prove the following theorem.",
"Theorem 4.2 (Main Theorem) Let $X$ be a Polish space.",
"The generic compact set $K\\subseteq X$ is either finite or there is a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ , and for every weak contraction $f\\colon K\\rightarrow X$ we have $\\mathcal {H}^{h} \\left(K\\cap f(K)\\right)=0$ .",
"Remark 4.3 If $X$ is a Polish space and $h$ is a fixed gauge function then it is easy to see that for the generic compact set $K\\subseteq X$ we have $\\mathcal {H}^{h}(K)=0$ .",
"If $X$ is uncountable then infinite compact sets form a second category subset in $\\mathcal {K}(X)$ , therefore the gauge function $h$ must depend on $K$ in the Main Theorem.",
"The first author of the paper proved the following theorem [1].",
"Theorem 4.4 Let $X$ be a Polish space, and let $K\\subseteq X$ be a balanced compact set.",
"Then there exists a continuous gauge function $h$ such that $0<\\mathcal {H}^{h}(K)<\\infty $ , and for every weak contraction $f\\colon K\\rightarrow X$ we have $\\mathcal {H}^{h}\\left(K\\cap f(K)\\right)=0$ .",
"If $h$ is a gauge function then finite sets have zero $\\mathcal {H}^h$ measure, so Theorem REF also holds for compact sets $K\\subseteq X$ that can be written as a union of a balanced compact set and a finite set.",
"Therefore the following theorem implies our Main Theorem.",
"Theorem 4.5 If $X$ is a Polish space then the generic compact set $K\\subseteq X$ is either finite or it can be written as the union of a balanced compact set and a finite set.",
"To prove Theorem REF first we give definitions and prove two key lemmas.",
"Definition 4.6 Let us fix an onto map $\\Psi \\colon 2\\mathbb {N}+1\\rightarrow \\mathbb {N}^{<\\omega }$ such that $\\Psi (n)$ has at most $n$ coordinates for every odd $n$ .",
"For $n\\in \\mathbb {N}^{+}$ and sequence $(a_1, a_2,\\ldots , a_{2n-1})$ , we define the function $\\Phi =\\Phi _{a_1 a_2 \\ldots a_{2n-1}} :\\lbrace 2k-1: 1\\le k\\le n\\rbrace \\rightarrow \\bigcup _{m=1}^{2n-1} \\mathcal {I}_{m}$ by setting $\\Phi (2k-1)={\\left\\lbrace \\begin{array}{ll} \\Psi (2k-1) & \\textrm { if } \\Psi (2k-1)\\in \\bigcup _{m=1}^{2k-1} \\mathcal {I}_{m} \\\\1\\in \\mathcal {I}_{1} & \\textrm { otherwise}.\\end{array}\\right.",
"}$ Remark 4.7 If $\\langle a_n \\rangle _{n\\in \\mathbb {N}^+}$ is a sequence of positive integers then the above definition implies that the functions $\\Phi _{a_1 \\dots a_{2n-1}}$ have a common extension $\\Phi \\colon 2\\mathbb {N} +1 \\rightarrow \\mathcal {I}$ , and $\\Phi $ is an index function according to the sequence $\\langle a_n \\rangle $ .",
"Let $X$ be a Polish space.",
"Definition 4.8 Let $n\\in \\mathbb {N}^+$ .",
"We call the pair of $(a_1, \\ldots , a_{2n})$ and $\\left\\lbrace \\big ((i_1,\\ldots , i_k), \\,U_{i_1\\ldots i_k}\\big ) \\,:\\, (i_1, \\dots , i_k ) \\in \\mathcal {I}_k, \\ 1\\le k \\le 2n \\right\\rbrace $ a balanced scheme of size $n$ if the numbers $a_k$ are positive integers, the sets $U_{i_1\\ldots i_k}$ are non-empty open subsets of $X$ , and there exist positive reals $b_k$ for which $a_1\\ge 2$ and $a_{k}\\ge (k-1)a_{1}\\cdots a_{k-1}$ for all $2\\le k \\le 2n$ , $\\operatorname{cl}U_{i_{1}\\dots i_{k}} \\subseteq U_{i_1 \\dots i_{k-1}}$ for all $(i_1,\\dots , i_k) \\in \\mathcal {I}_k$ and $2\\le k\\le 2n$ , $\\operatorname{diam}U_{i_{1} \\dots i_k}\\le b_k$ for all $(i_1, \\dots ,i_k) \\in \\mathcal {I}_k$ and $1\\le k\\le 2n$ , $\\operatorname{dist}(U_{i_1 \\dots i_k}, \\,U_{j_1\\dots j_k})>2b_k$ if $(i_1,\\dots , i_k)\\ne (j_1, \\dots ,j_k) \\in \\mathcal {I}_k$ and $1\\le k\\le 2n$ .",
"Let $\\Phi =\\Phi _{a_1 \\ldots a_{2n-1}}$ .",
"If $k<2n$ is odd, $U_{i_1 \\dots i_{k}}\\subseteq U_{\\Phi (k)}$ and $U_{j_1 \\dots j_{k}} \\nsubseteq U_{\\Phi (k)}$ , then for all $s,t\\in \\lbrace 1,\\dots ,a_{k+1}\\rbrace $ , $s\\ne t$ , we have $\\operatorname{dist}\\left(U_{i_1\\dots i_{k}s},U_{i_1 \\dots i_{k}t}\\right)> \\operatorname{diam}\\left(\\bigcup _{j_{k+1}=1}^{a_{k+1}} U_{j_1 \\dots j_{k}j_{k+1}}\\right).$ Let $(\\emptyset , \\emptyset )$ be the balanced scheme of size 0.",
"Definition 4.9 If $n\\in \\mathbb {N}^+$ and $\\pi $ is a balanced scheme of size $n$ as in Definition REF , then we define a non-empty open subset of $\\mathcal {K}(X)$ , $ \\mathcal {U}(\\pi ) \\!",
"= \\!",
"\\left\\lbrace K\\in \\mathcal {K}(X) : K\\subseteq \\!",
"\\bigcup _{i_1=1}^{a_1} \\!",
"\\cdots \\!",
"\\bigcup _{i_{2n}=1}^{a_{2n}}U_{i_1 \\ldots i_{2n}}, \\ \\forall (i_1, \\ldots ,i_{2n})\\in \\mathcal {I}_{2n} \\ K\\cap U_{i_1 \\ldots i_{2n}}\\ne \\emptyset \\right\\rbrace .$ For $\\pi =(\\emptyset , \\emptyset )$ we define $\\mathcal {U}(\\pi )=\\mathcal {K}(X)$ .",
"Assume $n\\in \\mathbb {N}$ , and let $\\pi $ and $\\pi ^{\\prime }$ be balanced schemes of size $n$ and $n+1$ , respectively.",
"We say that $\\pi ^{\\prime }$ is consistent with $\\pi $ if $a_k(\\pi ^{\\prime })=a_k(\\pi )$ and $U_{i_1\\dots i_k}(\\pi ^{\\prime })=U_{i_1\\dots i_k}(\\pi )$ for all $k\\in \\lbrace 1,\\dots , 2n\\rbrace $ and $(i_1,\\dots ,i_k)\\in \\mathcal {I}_{k}$ .",
"Remark 4.10 Let $\\pi $ and $\\pi ^{\\prime }$ be balanced schemes of size $n$ and $n+1$ , respectively.",
"If $\\pi ^{\\prime }$ is consistent with $\\pi $ then $\\mathcal {U}(\\pi ^{\\prime })\\subseteq \\mathcal {U}(\\pi )$ , and we may assume $b_k(\\pi ^{\\prime })=b_k(\\pi )$ for every $k\\in \\lbrace 1,\\dots , 2n\\rbrace $ .",
"Lemma 4.11 Assume $n\\in \\mathbb {N}$ .",
"Let $X$ be a non-empty perfect Polish space, let $\\pi $ be a balanced scheme of size $n$ , and let $\\mathcal {V}\\subseteq \\mathcal {U}(\\pi )$ be a non-empty open subset of $\\mathcal {K}(X)$ .",
"There exists a balanced scheme $\\pi ^{\\prime }$ of size $n+1$ such that $\\pi ^{\\prime }$ is consistent with $\\pi $ and $\\mathcal {U}(\\pi ^{\\prime })\\subseteq \\mathcal {V}$ .",
"Let $a_k(\\pi ^{\\prime })=a_k(\\pi )=a_k$ , $b_k(\\pi ^{\\prime })=b_k(\\pi )=b_k$ , $U_{i_1\\dots i_k}(\\pi ^{\\prime })= U_{i_1\\dots i_k}(\\pi )=U_{i_1\\dots i_k}$ for every $k\\le 2n$ and $(i_1,\\dots ,i_k)\\in \\mathcal {I}_{k}$ .",
"Then $\\pi ^{\\prime }$ will satisfy properties (REF )-(REF ) for all $k\\le 2n$ , since the map $\\Phi _{a_1 \\dots a_{2n+1}}$ extends $\\Phi _{a_1 \\dots a_{2n-1}}$ by Definition REF .",
"Therefore it is enough to construct $a_{k}(\\pi ^{\\prime })=a_{k}$ , $b_{k}(\\pi ^{\\prime })=b_{k}$ , and $U_{i_1\\dots i_{k}}(\\pi ^{\\prime })=U_{i_1\\dots i_{k}}$ for $k\\in \\lbrace 2n+1,2n+2\\rbrace $ and $(i_1,\\dots ,i_{k})\\in \\mathcal {I}_{k}$ .",
"As finite compact sets form a dense subset in $\\mathcal {K}(X)$ and $X$ is perfect, it is easy to see that there is a finite set $K_0\\in \\mathcal {V}$ with the following property.",
"There is an integer $N\\ge 2$ such that $N\\ge 2n(a_1\\cdots a_{2n})$ and $\\# (K_0\\cap U_{i_1\\dots i_{2n}})=N$ for every $(i_1,\\dots ,i_{2n})\\in \\mathcal {I}_{2n}$ .",
"Set $a_{2n+1}=N$ , then (REF ) holds for $k=2n+1$ .",
"For $(i_1,\\dots ,i_{2n})\\in \\mathcal {I}_{2n}$ let $ K_0\\cap U_{i_1\\dots i_{2n}}=\\left\\lbrace x_{i_1\\dots i_{2n+1}}: 1\\le i_{2n+1}\\le a_{2n+1}\\right\\rbrace .$ For $(i_1,\\dots ,i_{2n+1})\\in \\mathcal {I}_{2n+1}$ consider the non-empty open sets $ U_{i_1\\dots i_{2n+1}}=U(x_{i_1\\dots i_{2n+1}},b_{2n+1}/2),$ where $b_{2n+1}>0$ is sufficiently small.",
"Then the sets $U_{i_1\\dots i_{2n+1}}$ satisfy properties (REF )–(REF ), and $B_{H}(K_0,b_{2n+1})\\subseteq \\mathcal {V}$ .",
"(Notice that we did not require property (REF ) to hold for even numbers, and indeed, we could not satisfy it here for an arbitrary $\\mathcal {V}$ .)",
"Let $a_{2n+2}=(2n+1)(a_1\\cdots a_{2n+1})$ , so (REF ) holds for $k=2n+2$ .",
"First consider those $(i_1, \\ldots , i_{2n+1})$ for which $U_{i_1\\dots i_{2n+1}}\\subseteq U_{\\Phi (2n+1)}$ , where $\\Phi =\\Phi _{a_1 \\ldots a_{2n+1}}$ .",
"Then by the perfectness of $X$ we can fix distinct points $x_{i_1\\dots i_{2n+2}}\\in U_{i_1\\dots i_{2n+1}}$ ($i_{2n+2}\\in \\lbrace 1,\\dots ,a_{2n+2}\\rbrace $ ).",
"Let $\\delta $ be the minimum distance between the points $x_{i_1 \\ldots i_{2n+2}}$ we have defined so far.",
"Now consider those $(i_1, \\ldots , i_{2n+1})$ for which $U_{i_1\\dots i_{2n+1}}\\nsubseteq U_{\\Phi (2n+1)}$ .",
"For each of them, fix distinct points $x_{i_1\\dots i_{2n+2}}\\in U_{i_1\\dots i_{2n+1}}$ ($i_{2n+2}\\in \\lbrace 1,\\dots ,a_{2n+2}\\rbrace $ ) such that $ \\operatorname{diam}\\left(\\bigcup _{i_{2n+2}=1}^{a_{2n+2}} \\lbrace x_{i_1\\dots i_{2n+2}}\\rbrace \\right)\\le \\frac{\\delta }{2}.$ For $(i_1,\\dots ,i_{2n+2})\\in \\mathcal {I}_{2n+2}$ consider the non-empty open sets $ U_{i_1\\dots i_{2n+2}}=U(x_{i_1\\dots i_{2n+2}},b_{2n+2}/2),$ where $b_{2n+2}>0$ is sufficiently small.",
"Then the sets $U_{i_1\\dots i_{2n+2}}$ satisfy properties (REF )–(REF ).",
"Therefore $\\pi ^{\\prime }$ is a balanced scheme of size $n+1$ , and $\\pi ^{\\prime }$ is consistent with $\\pi $ .",
"Finally, we need to prove that $\\mathcal {U}(\\pi ^{\\prime })\\subseteq \\mathcal {V}$ .",
"We show that for every $K\\in \\mathcal {U}(\\pi ^{\\prime })$ , $ d_{H}(K,K_0)\\le b_{2n+1}.$ Let $K\\in \\mathcal {U}(\\pi ^{\\prime })$ .",
"By the definition of $\\mathcal {U}(\\pi ^{\\prime })$ we have $K \\subseteq \\bigcup _{i_1=1}^{a_1} \\cdots \\bigcup _{i_{2n+1}=1}^{a_{2n+1}}U_{i_1 \\ldots i_{2n+1}}$ and $K\\cap U_{i_1\\dots i_{2n+1}}\\ne \\emptyset $ for all $(i_1,\\dots ,i_{2n+1})\\in \\mathcal {I}_{2n+1}$ .",
"The set $K_0$ has the above properties by its definition, too.",
"As $\\operatorname{diam}U_{i_1\\dots i_{2n+1}}\\le b_{2n+1}$ for all $(i_1,\\dots ,i_{2n+1})\\in \\mathcal {I}_{2n+1}$ , (REF ) follows.",
"Equation (REF ) implies $\\mathcal {U}(\\pi ^{\\prime })\\subseteq B_{H}(K_0,b_{2n+1})$ , therefore $B_{H}(K_0,b_{2n+1})\\subseteq \\mathcal {V}$ yields $\\mathcal {U}(\\pi ^{\\prime })\\subseteq \\mathcal {V}$ .",
"Lemma 4.12 Assume $n\\in \\mathbb {N}$ .",
"Let $X$ be a non-empty perfect Polish space, and let $\\pi $ be a balanced scheme of size $n$ .",
"Then there are balanced schemes $\\pi _j$ $(j\\in \\mathbb {N})$ of size $n+1$ such that each $\\pi _j$ is consistent with $\\pi $ , the sets $\\mathcal {U}(\\pi _j)$ $(j\\in \\mathbb {N})$ are pairwise disjoint, and $\\bigcup _{j=0}^{\\infty } \\mathcal {U}(\\pi _j)$ is dense in $\\mathcal {U}(\\pi )$ .",
"Let $\\mathcal {U}_{i}\\subseteq \\mathcal {U}(\\pi )$ $(i\\in \\mathbb {N})$ be non-empty disjoint open sets such that $\\bigcup _{i=0}^{\\infty } \\mathcal {U}_i$ is dense in $\\mathcal {U}(\\pi )$ .",
"For all $i\\in \\mathbb {N}$ let $\\mathcal {B}_i$ be a countable basis of $\\mathcal {U}_i$ , and let $\\mathcal {B}=\\bigcup _{i=0}^{\\infty } \\mathcal {B}_{i}$ .",
"We may assume $\\emptyset \\notin \\mathcal {B}$ and let us consider an enumeration $\\mathcal {B}=\\lbrace \\mathcal {V}_n: n\\in \\mathbb {N}\\rbrace $ .",
"Let $j\\in \\mathbb {N}$ and assume that $\\pi _k$ and $n(k)\\in \\mathbb {N}$ ($k<j$ ) are already defined such that $\\mathcal {U}(\\pi _k)\\subseteq \\mathcal {V}_{n(k)}$ for $k<j$ .",
"Consider $n(j)=\\min \\left\\lbrace n\\in \\mathbb {N}: \\mathcal {V}_n\\cap \\left(\\cup _{k<j} \\mathcal {U}(\\pi _{k})\\right)=\\emptyset \\right\\rbrace .$ The definition of $\\mathcal {B}$ and the induction hypothesis easily imply that $\\bigcup _{k<j} \\mathcal {U}(\\pi _{k})$ can intersect at most $j$ open sets $\\mathcal {U}_i$ , so $n(j)<\\infty $ exists.",
"Lemma REF implies that there is a balanced scheme $\\pi _j$ of size $n+1$ such that $\\pi _j$ is consistent with $\\pi $ and $\\mathcal {U}(\\pi _j)\\subseteq \\mathcal {V}_{n(j)}$ .",
"The construction yields that $\\bigcup _{j=0}^{\\infty } \\mathcal {U}(\\pi _j)$ intersects each $\\mathcal {V}_i$ , thus it is dense in each $\\mathcal {U}_i$ , therefore it is dense in $\\mathcal {U}(\\pi )$ , and the union is clearly a disjoint union.",
"Now we are ready to prove Theorem REF that implies our Main Theorem.",
"First assume that $X$ is perfect, we prove that the generic compact set $K\\subseteq X$ is balanced.",
"We may assume that $X\\ne \\emptyset $ .",
"Let $\\mathcal {G}_0=\\mathcal {K}(X)$ .",
"Lemma REF implies that there are balanced schemes $\\pi _j$ $(j\\in \\mathbb {N}$ ) of size 1 such that the disjoint union $\\mathcal {G}_{1}=\\bigcup _{j_1=0}^{\\infty } \\mathcal {U}(\\pi _{j_1})$ is a dense open set in $\\mathcal {K}(X)$ .",
"Assume by induction that the balanced schemes $\\pi _{j_1\\dots j_n}$ of size $n$ and the dense open set $\\mathcal {G}_{n}$ are already defined.",
"Lemma REF implies that for every $j_1,\\dots ,j_{n}\\in \\mathbb {N}$ there exist balanced schemes $\\pi _{j_1\\dots j_{n+1}}$ $(j_{n+1}\\in \\mathbb {N})$ of size $n+1$ such that $\\pi _{j_1\\dots j_{n+1}}$ is consistent with $\\pi _{j_1\\dots j_{n}}$ and the disjoint union $\\bigcup _{j_{n+1}=0}^{\\infty } \\mathcal {U}(\\pi _{j_1\\dots j_{n+1}})$ is dense in $\\mathcal {U}(\\pi _{j_1\\dots j_{n}})$ .",
"Then the disjoint union $\\mathcal {G}_{n+1}=\\bigcup _{j_1=0}^{\\infty } \\cdots \\!",
"\\bigcup _{j_{n+1}=0}^{\\infty } \\mathcal {U}(\\pi _{j_1 \\dots j_{n+1}})$ is dense in $\\mathcal {G}_{n}$ , and the induction hypothesis yields that $\\mathcal {G}_{n+1}$ is a dense open set in $\\mathcal {K}(X)$ .",
"Consider $\\mathcal {G}=\\bigcap _{n=0}^{\\infty } \\mathcal {G}_{n}.$ As a countable intersection of dense open sets $\\mathcal {G}$ is co-meager in $\\mathcal {K}(X)$ .",
"Let $K\\in \\mathcal {G}$ be arbitrary fixed, it is enough to prove that $K$ is balanced.",
"Since the $n$ th level open sets $\\mathcal {U}(\\pi _{j_1\\dots j_n})$ are pairwise disjoint, there is a (unique) sequence $\\langle j_{n} \\rangle _{n\\in \\mathbb {N}^+}$ such that $K\\in \\mathcal {U}(\\pi _{j_1\\dots j_n})$ for all $n\\in \\mathbb {N}^+$ .",
"As the balanced scheme $\\pi _{j_1\\dots j_{n+1}}$ is consistent with $\\pi _{j_1\\dots j_{n}}$ for every $n\\in \\mathbb {N}^+$ , there are positive integers $a_n$ and non-empty open sets $U_{i_1\\dots i_n}$ witnessing this fact.",
"By Remark REF , the functions $\\Phi _{a_1 a_2 \\ldots a_{2n-1}}$ have a common extension $\\Phi \\colon 2\\mathbb {N}+1 \\rightarrow \\mathcal {I}$ , and $\\Phi $ is an index function according to the sequence $\\langle a_n \\rangle $ .",
"For $n\\in \\mathbb {N}^+$ and $(i_1,\\dots ,i_n)\\in \\mathcal {I}_{n}$ let us define $C_{i_1\\dots i_n}=\\operatorname{cl}U_{i_1\\dots i_n}.$ Since $K\\in \\mathcal {U}(\\pi _{j_1\\dots j_n})$ for every $n$ , Definition REF implies that $K=\\bigcap _{n=1}^{\\infty }\\left(\\bigcup _{i_1=1}^{a_1}\\cdots \\bigcup _{i_n=1}^{a_n}C_{i_1 \\dots i_n} \\right).$ From Definition REF it follows that the positive integers $a_n$ and the non-empty closed sets $C_{i_1 \\dots i_n}$ satisfy properties $(\\ref {01})$ –$(\\ref {05})$ of Definition REF .",
"Therefore $K$ is balanced.",
"Now let $X$ be an arbitrary non-empty Polish space.",
"Then there is a perfect set $X^{*}\\subseteq X$ such that $U=X\\setminus X^{*}$ is countable open, see [5].",
"Let $S$ be the set of isolated points of $X$ .",
"Then $S$ is open, and $S\\subseteq U$ .",
"We claim that $S$ is dense in $U$ , thus $U\\subseteq \\operatorname{cl}S$ .",
"Indeed, assume to the contrary that there is a non-empty open set $V\\subseteq U$ such that $V\\cap S=\\emptyset $ .",
"By shrinking $V$ , we may suppose that $\\operatorname{cl}V \\subseteq U$ .",
"Then $\\operatorname{cl}V\\subseteq U$ is a non-empty perfect set, so it has cardinality $2^{\\aleph _0}$ by [5], which is a contradiction.",
"For a set $A\\subseteq X$ let us denote by $\\mathcal {K}(A)$ the metric space of non-empty compact subsets of $A$ , similarly as in Definition REF .",
"Since $S$ is open, compact non-empty subsets of $S$ form a dense open subset of $\\mathcal {K}(\\operatorname{cl}S)$ .",
"As $S$ is the set of isolated points, every compact subset of $S$ is finite.",
"The first part of the proof implies that there is a dense $G_{\\delta }$ set $\\mathcal {F}^{*}\\subseteq \\mathcal {K}(X^{*})$ such that every $K^{*}\\in \\mathcal {F}^{*}$ is balanced.",
"Let $\\mathcal {F}\\subseteq \\mathcal {K}(X)$ be the the set of those non-empty compact subsets $K\\subseteq X$ for which $K\\cap \\operatorname{cl}S\\subseteq S$ and $K\\cap X^* \\in \\mathcal {F}^* \\cup \\lbrace \\emptyset \\rbrace $ .",
"Clearly, every $K\\in \\mathcal {F}$ is a union of $\\emptyset $ or a balanced compact set in $X^*$ and finitely many points in $S$ .",
"We claim that $\\mathcal {F}$ is a dense $G_\\delta $ subset of $\\mathcal {K}(X)$ .",
"Let us define the continuous map $R\\colon \\mathcal {K}(X) \\rightarrow \\mathcal {K}(X^{*}) \\cup \\lbrace \\emptyset \\rbrace , \\quad R(K)=K\\cap X^{*},$ where the distance of $\\emptyset $ to points of $\\mathcal {K}(X^{*})$ is defined to be 1.",
"We show that the map $R$ is open.",
"Let $K\\in \\mathcal {K}(X)$ and $C^{*}\\in \\mathcal {K}(X^{*}) \\cup \\lbrace \\emptyset \\rbrace $ be arbitrary, and set $K^{*}=K\\cap X^{*}$ .",
"It is enough to construct $C\\in \\mathcal {K}(X)$ such that $C\\cap X^{*}=C^{*}$ and $d_{H}(K,C)\\le d_{H}(K^{*},C^{*})$ .",
"If $K\\subseteq X^{*}$ or $K^{*}=C^{*}$ , then $C=C^{*}$ or $C=K$ works, respectively.",
"Thus we may assume that $K\\setminus X^{*}\\ne \\emptyset $ and $d_{H}(K^{*},C^{*})>0$ .",
"The compactness of $K$ implies that there are finitely many open sets $V_i$ such that $K\\setminus X^{*}\\subseteq \\bigcup _{i=1}^{m} V_i$ , $V_i\\cap (K\\setminus X^{*})\\ne \\emptyset $ , and $\\operatorname{diam}V_i \\le d_{H}(K^{*},C^{*})$ for all $i\\in \\lbrace 1,\\dots ,m\\rbrace $ .",
"Let us choose $x_i\\in V_i\\setminus X^{*}$ for all $i\\in \\lbrace 1,\\dots , m\\rbrace $ arbitrarily, and consider $C=C^{*}\\cup \\bigcup _{i=1}^{m} \\lbrace x_i\\rbrace $ .",
"It is easy to see that $C\\in \\mathcal {K}(X)$ fulfills the required properties.",
"Since $R$ is open, $R^{-1}(\\mathcal {F}^*\\cup \\lbrace \\emptyset \\rbrace )$ is dense $G_\\delta $ in $\\mathcal {K}(X)$ .",
"We clearly have $\\mathcal {F}=R^{-1}(\\mathcal {F}^*\\cup \\lbrace \\emptyset \\rbrace ) \\cap \\mathcal {K}((X\\setminus \\operatorname{cl}S)\\cup S).$ As $(X\\setminus \\operatorname{cl}S)\\cup S$ is dense open in $X$ , $\\mathcal {K}((X\\setminus \\operatorname{cl}S)\\cup S)$ is dense open in $\\mathcal {K}(X)$ .",
"Thus $\\mathcal {F}$ is dense $G_\\delta $ in $\\mathcal {K}(X)$ , which concludes the proof.",
"Acknowledgement.",
"The authors are indebted to M. Elekes and to an anonymous referee for their valuable comments."
]
] | 1204.1100 |
[
[
"Search for the decay Bs0 -> mu mu with the ATLAS detector"
],
[
"Abstract A blind analysis searching for the decay Bs0 -> mumu has been performed using proton-proton collisions at a centre-of-mass energy of 7 TeV recorded with the ATLAS detector at the LHC.",
"With an integrated luminosity of 2.4 fb^(-1) no excess of events over the background expectation is found and an upper limit is set on the branching fraction BR(Bs0 -> mu mu) < 2.2 (1.9) x10^(-8) at 95% (90%) confidence level."
],
[
"The ATLAS Collaboration G. Aad$^{\\rm 48}$ , B. Abbott$^{\\rm 111}$ , J. Abdallah$^{\\rm 11}$ , S. Abdel Khalek$^{\\rm 115}$ , A.A. Abdelalim$^{\\rm 49}$ , O. Abdinov$^{\\rm 10}$ , B. Abi$^{\\rm 112}$ , M. Abolins$^{\\rm 88}$ , O.S.",
"AbouZeid$^{\\rm 158}$ , H. Abramowicz$^{\\rm 153}$ , H. Abreu$^{\\rm 136}$ , E. Acerbi$^{\\rm 89a,89b}$ , B.S.",
"Acharya$^{\\rm 164a,164b}$ , L. Adamczyk$^{\\rm 37}$ , D.L.",
"Adams$^{\\rm 24}$ , T.N.",
"Addy$^{\\rm 56}$ , J. Adelman$^{\\rm 176}$ , S. Adomeit$^{\\rm 98}$ , P. Adragna$^{\\rm 75}$ , T. Adye$^{\\rm 129}$ , S. Aefsky$^{\\rm 22}$ , J.A.",
"Aguilar-Saavedra$^{\\rm 124b}$$^{,a}$ , M. Aharrouche$^{\\rm 81}$ , S.P.",
"Ahlen$^{\\rm 21}$ , F. Ahles$^{\\rm 48}$ , A. Ahmad$^{\\rm 148}$ , M. Ahsan$^{\\rm 40}$ , G. Aielli$^{\\rm 133a,133b}$ , T. Akdogan$^{\\rm 18a}$ , T.P.A.",
"Åkesson$^{\\rm 79}$ , G. Akimoto$^{\\rm 155}$ , A.V.",
"Akimov $^{\\rm 94}$ , A. Akiyama$^{\\rm 66}$ , M.S.",
"Alam$^{\\rm 1}$ , M.A.",
"Alam$^{\\rm 76}$ , J. Albert$^{\\rm 169}$ , S. Albrand$^{\\rm 55}$ , M. Aleksa$^{\\rm 29}$ , I.N.",
"Aleksandrov$^{\\rm 64}$ , F. Alessandria$^{\\rm 89a}$ , C. Alexa$^{\\rm 25a}$ , G. Alexander$^{\\rm 153}$ , G. Alexandre$^{\\rm 49}$ , T. Alexopoulos$^{\\rm 9}$ , M. Alhroob$^{\\rm 164a,164c}$ , M. Aliev$^{\\rm 15}$ , G. Alimonti$^{\\rm 89a}$ , J. Alison$^{\\rm 120}$ , B.M.M.",
"Allbrooke$^{\\rm 17}$ , P.P.",
"Allport$^{\\rm 73}$ , S.E.",
"Allwood-Spiers$^{\\rm 53}$ , J. Almond$^{\\rm 82}$ , A. Aloisio$^{\\rm 102a,102b}$ , R. Alon$^{\\rm 172}$ , A. Alonso$^{\\rm 79}$ , B. Alvarez Gonzalez$^{\\rm 88}$ , M.G.",
"Alviggi$^{\\rm 102a,102b}$ , K. Amako$^{\\rm 65}$ , C. Amelung$^{\\rm 22}$ , V.V.",
"Ammosov$^{\\rm 128}$ , A. Amorim$^{\\rm 124a}$$^{,b}$ , N. Amram$^{\\rm 153}$ , C. Anastopoulos$^{\\rm 29}$ , L.S.",
"Ancu$^{\\rm 16}$ , N. Andari$^{\\rm 115}$ , T. Andeen$^{\\rm 34}$ , C.F.",
"Anders$^{\\rm 20}$ , G. Anders$^{\\rm 58a}$ , K.J.",
"Anderson$^{\\rm 30}$ , A. Andreazza$^{\\rm 89a,89b}$ , V. Andrei$^{\\rm 58a}$ , X.S.",
"Anduaga$^{\\rm 70}$ , A. Angerami$^{\\rm 34}$ , F. Anghinolfi$^{\\rm 29}$ , A. Anisenkov$^{\\rm 107}$ , N. Anjos$^{\\rm 124a}$ , A. Annovi$^{\\rm 47}$ , A. Antonaki$^{\\rm 8}$ , M. Antonelli$^{\\rm 47}$ , A. Antonov$^{\\rm 96}$ , J. Antos$^{\\rm 144b}$ , F. Anulli$^{\\rm 132a}$ , S. Aoun$^{\\rm 83}$ , L. Aperio Bella$^{\\rm 4}$ , R. Apolle$^{\\rm 118}$$^{,c}$ , G. Arabidze$^{\\rm 88}$ , I. Aracena$^{\\rm 143}$ , Y. Arai$^{\\rm 65}$ , A.T.H.",
"Arce$^{\\rm 44}$ , S. Arfaoui$^{\\rm 148}$ , J-F. Arguin$^{\\rm 14}$ , E. Arik$^{\\rm 18a}$$^{,*}$ , M. Arik$^{\\rm 18a}$ , A.J.",
"Armbruster$^{\\rm 87}$ , O. Arnaez$^{\\rm 81}$ , V. Arnal$^{\\rm 80}$ , C. Arnault$^{\\rm 115}$ , A. Artamonov$^{\\rm 95}$ , G. Artoni$^{\\rm 132a,132b}$ , D. Arutinov$^{\\rm 20}$ , S. Asai$^{\\rm 155}$ , R. Asfandiyarov$^{\\rm 173}$ , S. Ask$^{\\rm 27}$ , B. Åsman$^{\\rm 146a,146b}$ , L. Asquith$^{\\rm 5}$ , K. Assamagan$^{\\rm 24}$ , A. Astbury$^{\\rm 169}$ , B. Aubert$^{\\rm 4}$ , E. Auge$^{\\rm 115}$ , K. Augsten$^{\\rm 127}$ , M. Aurousseau$^{\\rm 145a}$ , G. Avolio$^{\\rm 163}$ , R. Avramidou$^{\\rm 9}$ , D. Axen$^{\\rm 168}$ , G. Azuelos$^{\\rm 93}$$^{,d}$ , Y. Azuma$^{\\rm 155}$ , M.A.",
"Baak$^{\\rm 29}$ , G. Baccaglioni$^{\\rm 89a}$ , C. Bacci$^{\\rm 134a,134b}$ , A.M. Bach$^{\\rm 14}$ , H. Bachacou$^{\\rm 136}$ , K. Bachas$^{\\rm 29}$ , M. Backes$^{\\rm 49}$ , M. Backhaus$^{\\rm 20}$ , E. Badescu$^{\\rm 25a}$ , P. Bagnaia$^{\\rm 132a,132b}$ , S. Bahinipati$^{\\rm 2}$ , Y. Bai$^{\\rm 32a}$ , D.C. Bailey$^{\\rm 158}$ , T. Bain$^{\\rm 158}$ , J.T.",
"Baines$^{\\rm 129}$ , O.K.",
"Baker$^{\\rm 176}$ , M.D.",
"Baker$^{\\rm 24}$ , S. Baker$^{\\rm 77}$ , E. Banas$^{\\rm 38}$ , P. Banerjee$^{\\rm 93}$ , Sw. Banerjee$^{\\rm 173}$ , D. Banfi$^{\\rm 29}$ , A. Bangert$^{\\rm 150}$ , V. Bansal$^{\\rm 169}$ , H.S.",
"Bansil$^{\\rm 17}$ , L. Barak$^{\\rm 172}$ , S.P.",
"Baranov$^{\\rm 94}$ , A. Barbaro Galtieri$^{\\rm 14}$ , T. Barber$^{\\rm 48}$ , E.L. Barberio$^{\\rm 86}$ , D. Barberis$^{\\rm 50a,50b}$ , M. Barbero$^{\\rm 20}$ , D.Y.",
"Bardin$^{\\rm 64}$ , T. Barillari$^{\\rm 99}$ , M. Barisonzi$^{\\rm 175}$ , T. Barklow$^{\\rm 143}$ , N. Barlow$^{\\rm 27}$ , B.M.",
"Barnett$^{\\rm 129}$ , R.M.",
"Barnett$^{\\rm 14}$ , A. Baroncelli$^{\\rm 134a}$ , G. Barone$^{\\rm 49}$ , A.J.",
"Barr$^{\\rm 118}$ , F. Barreiro$^{\\rm 80}$ , J. Barreiro Guimarães da Costa$^{\\rm 57}$ , P. Barrillon$^{\\rm 115}$ , R. Bartoldus$^{\\rm 143}$ , A.E.",
"Barton$^{\\rm 71}$ , V. Bartsch$^{\\rm 149}$ , R.L.",
"Bates$^{\\rm 53}$ , L. Batkova$^{\\rm 144a}$ , J.R. Batley$^{\\rm 27}$ , A. Battaglia$^{\\rm 16}$ , M. Battistin$^{\\rm 29}$ , F. Bauer$^{\\rm 136}$ , H.S.",
"Bawa$^{\\rm 143}$$^{,e}$ , S. Beale$^{\\rm 98}$ , T. Beau$^{\\rm 78}$ , P.H.",
"Beauchemin$^{\\rm 161}$ , R. Beccherle$^{\\rm 50a}$ , P. Bechtle$^{\\rm 20}$ , H.P.",
"Beck$^{\\rm 16}$ , S. Becker$^{\\rm 98}$ , M. Beckingham$^{\\rm 138}$ , K.H.",
"Becks$^{\\rm 175}$ , A.J.",
"Beddall$^{\\rm 18c}$ , A. Beddall$^{\\rm 18c}$ , S. Bedikian$^{\\rm 176}$ , V.A.",
"Bednyakov$^{\\rm 64}$ , C.P.",
"Bee$^{\\rm 83}$ , M. Begel$^{\\rm 24}$ , S. Behar Harpaz$^{\\rm 152}$ , P.K.",
"Behera$^{\\rm 62}$ , M. Beimforde$^{\\rm 99}$ , C. Belanger-Champagne$^{\\rm 85}$ , P.J.",
"Bell$^{\\rm 49}$ , W.H.",
"Bell$^{\\rm 49}$ , G. Bella$^{\\rm 153}$ , L. Bellagamba$^{\\rm 19a}$ , F. Bellina$^{\\rm 29}$ , M. Bellomo$^{\\rm 29}$ , A. Belloni$^{\\rm 57}$ , O. Beloborodova$^{\\rm 107}$$^{,f}$ , K. Belotskiy$^{\\rm 96}$ , O. Beltramello$^{\\rm 29}$ , O. Benary$^{\\rm 153}$ , D. Benchekroun$^{\\rm 135a}$ , K. Bendtz$^{\\rm 146a,146b}$ , N. Benekos$^{\\rm 165}$ , Y. Benhammou$^{\\rm 153}$ , E. Benhar Noccioli$^{\\rm 49}$ , J.A.",
"Benitez Garcia$^{\\rm 159b}$ , D.P.",
"Benjamin$^{\\rm 44}$ , M. Benoit$^{\\rm 115}$ , J.R. Bensinger$^{\\rm 22}$ , K. Benslama$^{\\rm 130}$ , S. Bentvelsen$^{\\rm 105}$ , D. Berge$^{\\rm 29}$ , E. Bergeaas Kuutmann$^{\\rm 41}$ , N. Berger$^{\\rm 4}$ , F. Berghaus$^{\\rm 169}$ , E. Berglund$^{\\rm 105}$ , J. Beringer$^{\\rm 14}$ , P. Bernat$^{\\rm 77}$ , R. Bernhard$^{\\rm 48}$ , C. Bernius$^{\\rm 24}$ , T. Berry$^{\\rm 76}$ , C. Bertella$^{\\rm 83}$ , A. Bertin$^{\\rm 19a,19b}$ , F. Bertolucci$^{\\rm 122a,122b}$ , M.I.",
"Besana$^{\\rm 89a,89b}$ , N. Besson$^{\\rm 136}$ , S. Bethke$^{\\rm 99}$ , W. Bhimji$^{\\rm 45}$ , R.M.",
"Bianchi$^{\\rm 29}$ , M. Bianco$^{\\rm 72a,72b}$ , O. Biebel$^{\\rm 98}$ , S.P.",
"Bieniek$^{\\rm 77}$ , K. Bierwagen$^{\\rm 54}$ , J. Biesiada$^{\\rm 14}$ , M. Biglietti$^{\\rm 134a}$ , H. Bilokon$^{\\rm 47}$ , M. Bindi$^{\\rm 19a,19b}$ , S. Binet$^{\\rm 115}$ , A. Bingul$^{\\rm 18c}$ , C. Bini$^{\\rm 132a,132b}$ , C. Biscarat$^{\\rm 178}$ , U. Bitenc$^{\\rm 48}$ , K.M.",
"Black$^{\\rm 21}$ , R.E.",
"Blair$^{\\rm 5}$ , J.-B.",
"Blanchard$^{\\rm 136}$ , G. Blanchot$^{\\rm 29}$ , T. Blazek$^{\\rm 144a}$ , C. Blocker$^{\\rm 22}$ , J. Blocki$^{\\rm 38}$ , A. Blondel$^{\\rm 49}$ , W. Blum$^{\\rm 81}$ , U. Blumenschein$^{\\rm 54}$ , G.J.",
"Bobbink$^{\\rm 105}$ , V.B.",
"Bobrovnikov$^{\\rm 107}$ , S.S. Bocchetta$^{\\rm 79}$ , A. Bocci$^{\\rm 44}$ , C.R.",
"Boddy$^{\\rm 118}$ , M. Boehler$^{\\rm 41}$ , J. Boek$^{\\rm 175}$ , N. Boelaert$^{\\rm 35}$ , J.A.",
"Bogaerts$^{\\rm 29}$ , A. Bogdanchikov$^{\\rm 107}$ , A. Bogouch$^{\\rm 90}$$^{,*}$ , C. Bohm$^{\\rm 146a}$ , J. Bohm$^{\\rm 125}$ , V. Boisvert$^{\\rm 76}$ , T. Bold$^{\\rm 37}$ , V. Boldea$^{\\rm 25a}$ , N.M. Bolnet$^{\\rm 136}$ , M. Bomben$^{\\rm 78}$ , M. Bona$^{\\rm 75}$ , M. Bondioli$^{\\rm 163}$ , M. Boonekamp$^{\\rm 136}$ , C.N.",
"Booth$^{\\rm 139}$ , S. Bordoni$^{\\rm 78}$ , C. Borer$^{\\rm 16}$ , A. Borisov$^{\\rm 128}$ , G. Borissov$^{\\rm 71}$ , I. Borjanovic$^{\\rm 12a}$ , M. Borri$^{\\rm 82}$ , S. Borroni$^{\\rm 87}$ , V. Bortolotto$^{\\rm 134a,134b}$ , K. Bos$^{\\rm 105}$ , D. Boscherini$^{\\rm 19a}$ , M. Bosman$^{\\rm 11}$ , H. Boterenbrood$^{\\rm 105}$ , D. Botterill$^{\\rm 129}$ , J. Bouchami$^{\\rm 93}$ , J. Boudreau$^{\\rm 123}$ , E.V.",
"Bouhova-Thacker$^{\\rm 71}$ , D. Boumediene$^{\\rm 33}$ , C. Bourdarios$^{\\rm 115}$ , N. Bousson$^{\\rm 83}$ , A. Boveia$^{\\rm 30}$ , J. Boyd$^{\\rm 29}$ , I.R.",
"Boyko$^{\\rm 64}$ , N.I.",
"Bozhko$^{\\rm 128}$ , I. Bozovic-Jelisavcic$^{\\rm 12b}$ , J. Bracinik$^{\\rm 17}$ , P. Branchini$^{\\rm 134a}$ , A. Brandt$^{\\rm 7}$ , G. Brandt$^{\\rm 118}$ , O. Brandt$^{\\rm 54}$ , U. Bratzler$^{\\rm 156}$ , B. Brau$^{\\rm 84}$ , J.E.",
"Brau$^{\\rm 114}$ , H.M. Braun$^{\\rm 175}$ , B. Brelier$^{\\rm 158}$ , J. Bremer$^{\\rm 29}$ , K. Brendlinger$^{\\rm 120}$ , R. Brenner$^{\\rm 166}$ , S. Bressler$^{\\rm 172}$ , D. Britton$^{\\rm 53}$ , F.M.",
"Brochu$^{\\rm 27}$ , I. Brock$^{\\rm 20}$ , R. Brock$^{\\rm 88}$ , E. Brodet$^{\\rm 153}$ , F. Broggi$^{\\rm 89a}$ , C. Bromberg$^{\\rm 88}$ , J. Bronner$^{\\rm 99}$ , G. Brooijmans$^{\\rm 34}$ , W.K.",
"Brooks$^{\\rm 31b}$ , G. Brown$^{\\rm 82}$ , H. Brown$^{\\rm 7}$ , P.A.",
"Bruckman de Renstrom$^{\\rm 38}$ , D. Bruncko$^{\\rm 144b}$ , R. Bruneliere$^{\\rm 48}$ , S. Brunet$^{\\rm 60}$ , A. Bruni$^{\\rm 19a}$ , G. Bruni$^{\\rm 19a}$ , M. Bruschi$^{\\rm 19a}$ , T. Buanes$^{\\rm 13}$ , Q. Buat$^{\\rm 55}$ , F. Bucci$^{\\rm 49}$ , J. Buchanan$^{\\rm 118}$ , P. Buchholz$^{\\rm 141}$ , R.M.",
"Buckingham$^{\\rm 118}$ , A.G. Buckley$^{\\rm 45}$ , S.I.",
"Buda$^{\\rm 25a}$ , I.A.",
"Budagov$^{\\rm 64}$ , B. Budick$^{\\rm 108}$ , V. Büscher$^{\\rm 81}$ , L. Bugge$^{\\rm 117}$ , O. Bulekov$^{\\rm 96}$ , A.C. Bundock$^{\\rm 73}$ , M. Bunse$^{\\rm 42}$ , T. Buran$^{\\rm 117}$ , H. Burckhart$^{\\rm 29}$ , S. Burdin$^{\\rm 73}$ , T. Burgess$^{\\rm 13}$ , S. Burke$^{\\rm 129}$ , E. Busato$^{\\rm 33}$ , P. Bussey$^{\\rm 53}$ , C.P.",
"Buszello$^{\\rm 166}$ , B. Butler$^{\\rm 143}$ , J.M.",
"Butler$^{\\rm 21}$ , C.M.",
"Buttar$^{\\rm 53}$ , J.M.",
"Butterworth$^{\\rm 77}$ , W. Buttinger$^{\\rm 27}$ , S. Cabrera Urbán$^{\\rm 167}$ , D. Caforio$^{\\rm 19a,19b}$ , O. Cakir$^{\\rm 3a}$ , P. Calafiura$^{\\rm 14}$ , G. Calderini$^{\\rm 78}$ , P. Calfayan$^{\\rm 98}$ , R. Calkins$^{\\rm 106}$ , L.P. Caloba$^{\\rm 23a}$ , R. Caloi$^{\\rm 132a,132b}$ , D. Calvet$^{\\rm 33}$ , S. Calvet$^{\\rm 33}$ , R. Camacho Toro$^{\\rm 33}$ , P. Camarri$^{\\rm 133a,133b}$ , D. Cameron$^{\\rm 117}$ , L.M.",
"Caminada$^{\\rm 14}$ , S. Campana$^{\\rm 29}$ , M. Campanelli$^{\\rm 77}$ , V. Canale$^{\\rm 102a,102b}$ , F. Canelli$^{\\rm 30}$$^{,g}$ , A. Canepa$^{\\rm 159a}$ , J. Cantero$^{\\rm 80}$ , L. Capasso$^{\\rm 102a,102b}$ , M.D.M.",
"Capeans Garrido$^{\\rm 29}$ , I. Caprini$^{\\rm 25a}$ , M. Caprini$^{\\rm 25a}$ , D. Capriotti$^{\\rm 99}$ , M. Capua$^{\\rm 36a,36b}$ , R. Caputo$^{\\rm 81}$ , R. Cardarelli$^{\\rm 133a}$ , T. Carli$^{\\rm 29}$ , G. Carlino$^{\\rm 102a}$ , L. Carminati$^{\\rm 89a,89b}$ , B. Caron$^{\\rm 85}$ , S. Caron$^{\\rm 104}$ , E. Carquin$^{\\rm 31b}$ , G.D. Carrillo Montoya$^{\\rm 173}$ , A.A. Carter$^{\\rm 75}$ , J.R. Carter$^{\\rm 27}$ , J. Carvalho$^{\\rm 124a}$$^{,h}$ , D. Casadei$^{\\rm 108}$ , M.P.",
"Casado$^{\\rm 11}$ , M. Cascella$^{\\rm 122a,122b}$ , C. Caso$^{\\rm 50a,50b}$$^{,*}$ , A.M. Castaneda Hernandez$^{\\rm 173}$$^{,i}$ , E. Castaneda-Miranda$^{\\rm 173}$ , V. Castillo Gimenez$^{\\rm 167}$ , N.F.",
"Castro$^{\\rm 124a}$ , G. Cataldi$^{\\rm 72a}$ , P. Catastini$^{\\rm 57}$ , A. Catinaccio$^{\\rm 29}$ , J.R. Catmore$^{\\rm 29}$ , A. Cattai$^{\\rm 29}$ , G. Cattani$^{\\rm 133a,133b}$ , S. Caughron$^{\\rm 88}$ , P. Cavalleri$^{\\rm 78}$ , D. Cavalli$^{\\rm 89a}$ , M. Cavalli-Sforza$^{\\rm 11}$ , V. Cavasinni$^{\\rm 122a,122b}$ , F. Ceradini$^{\\rm 134a,134b}$ , A.S. Cerqueira$^{\\rm 23b}$ , A. Cerri$^{\\rm 29}$ , L. Cerrito$^{\\rm 75}$ , F. Cerutti$^{\\rm 47}$ , S.A. Cetin$^{\\rm 18b}$ , A. Chafaq$^{\\rm 135a}$ , D. Chakraborty$^{\\rm 106}$ , I. Chalupkova$^{\\rm 126}$ , K. Chan$^{\\rm 2}$ , B. Chapleau$^{\\rm 85}$ , J.D.",
"Chapman$^{\\rm 27}$ , J.W.",
"Chapman$^{\\rm 87}$ , E. Chareyre$^{\\rm 78}$ , D.G.",
"Charlton$^{\\rm 17}$ , V. Chavda$^{\\rm 82}$ , C.A.",
"Chavez Barajas$^{\\rm 29}$ , S. Cheatham$^{\\rm 85}$ , S. Chekanov$^{\\rm 5}$ , S.V.",
"Chekulaev$^{\\rm 159a}$ , G.A.",
"Chelkov$^{\\rm 64}$ , M.A.",
"Chelstowska$^{\\rm 104}$ , C. Chen$^{\\rm 63}$ , H. Chen$^{\\rm 24}$ , S. Chen$^{\\rm 32c}$ , X. Chen$^{\\rm 173}$ , A. Cheplakov$^{\\rm 64}$ , R. Cherkaoui El Moursli$^{\\rm 135e}$ , V. Chernyatin$^{\\rm 24}$ , E. Cheu$^{\\rm 6}$ , S.L.",
"Cheung$^{\\rm 158}$ , L. Chevalier$^{\\rm 136}$ , G. Chiefari$^{\\rm 102a,102b}$ , L. Chikovani$^{\\rm 51a}$ , J.T.",
"Childers$^{\\rm 29}$ , A. Chilingarov$^{\\rm 71}$ , G. Chiodini$^{\\rm 72a}$ , A.S. Chisholm$^{\\rm 17}$ , R.T. Chislett$^{\\rm 77}$ , M.V.",
"Chizhov$^{\\rm 64}$ , G. Choudalakis$^{\\rm 30}$ , S. Chouridou$^{\\rm 137}$ , I.A.",
"Christidi$^{\\rm 77}$ , A. Christov$^{\\rm 48}$ , D. Chromek-Burckhart$^{\\rm 29}$ , M.L.",
"Chu$^{\\rm 151}$ , J. Chudoba$^{\\rm 125}$ , G. Ciapetti$^{\\rm 132a,132b}$ , A.K.",
"Ciftci$^{\\rm 3a}$ , R. Ciftci$^{\\rm 3a}$ , D. Cinca$^{\\rm 33}$ , V. Cindro$^{\\rm 74}$ , C. Ciocca$^{\\rm 19a}$ , A. Ciocio$^{\\rm 14}$ , M. Cirilli$^{\\rm 87}$ , M. Citterio$^{\\rm 89a}$ , M. Ciubancan$^{\\rm 25a}$ , A. Clark$^{\\rm 49}$ , P.J.",
"Clark$^{\\rm 45}$ , W. Cleland$^{\\rm 123}$ , J.C. Clemens$^{\\rm 83}$ , B. Clement$^{\\rm 55}$ , C. Clement$^{\\rm 146a,146b}$ , Y. Coadou$^{\\rm 83}$ , M. Cobal$^{\\rm 164a,164c}$ , A. Coccaro$^{\\rm 138}$ , J. Cochran$^{\\rm 63}$ , P. Coe$^{\\rm 118}$ , J.G.",
"Cogan$^{\\rm 143}$ , J. Coggeshall$^{\\rm 165}$ , E. Cogneras$^{\\rm 178}$ , J. Colas$^{\\rm 4}$ , A.P.",
"Colijn$^{\\rm 105}$ , N.J. Collins$^{\\rm 17}$ , C. Collins-Tooth$^{\\rm 53}$ , J. Collot$^{\\rm 55}$ , G. Colon$^{\\rm 84}$ , P. Conde Muiño$^{\\rm 124a}$ , E. Coniavitis$^{\\rm 118}$ , M.C.",
"Conidi$^{\\rm 11}$ , S.M.",
"Consonni$^{\\rm 89a,89b}$ , V. Consorti$^{\\rm 48}$ , S. Constantinescu$^{\\rm 25a}$ , C. Conta$^{\\rm 119a,119b}$ , G. Conti$^{\\rm 57}$ , F. Conventi$^{\\rm 102a}$$^{,j}$ , M. Cooke$^{\\rm 14}$ , B.D.",
"Cooper$^{\\rm 77}$ , A.M. Cooper-Sarkar$^{\\rm 118}$ , K. Copic$^{\\rm 14}$ , T. Cornelissen$^{\\rm 175}$ , M. Corradi$^{\\rm 19a}$ , F. Corriveau$^{\\rm 85}$$^{,k}$ , A. Cortes-Gonzalez$^{\\rm 165}$ , G. Cortiana$^{\\rm 99}$ , G. Costa$^{\\rm 89a}$ , M.J. Costa$^{\\rm 167}$ , D. Costanzo$^{\\rm 139}$ , T. Costin$^{\\rm 30}$ , D. Côté$^{\\rm 29}$ , L. Courneyea$^{\\rm 169}$ , G. Cowan$^{\\rm 76}$ , C. Cowden$^{\\rm 27}$ , B.E.",
"Cox$^{\\rm 82}$ , K. Cranmer$^{\\rm 108}$ , F. Crescioli$^{\\rm 122a,122b}$ , M. Cristinziani$^{\\rm 20}$ , G. Crosetti$^{\\rm 36a,36b}$ , R. Crupi$^{\\rm 72a,72b}$ , S. Crépé-Renaudin$^{\\rm 55}$ , C.-M. Cuciuc$^{\\rm 25a}$ , C. Cuenca Almenar$^{\\rm 176}$ , T. Cuhadar Donszelmann$^{\\rm 139}$ , M. Curatolo$^{\\rm 47}$ , C.J.",
"Curtis$^{\\rm 17}$ , C. Cuthbert$^{\\rm 150}$ , P. Cwetanski$^{\\rm 60}$ , H. Czirr$^{\\rm 141}$ , P. Czodrowski$^{\\rm 43}$ , Z. Czyczula$^{\\rm 176}$ , S. D'Auria$^{\\rm 53}$ , M. D'Onofrio$^{\\rm 73}$ , A.",
"D'Orazio$^{\\rm 132a,132b}$ , C. Da Via$^{\\rm 82}$ , W. Dabrowski$^{\\rm 37}$ , A. Dafinca$^{\\rm 118}$ , T. Dai$^{\\rm 87}$ , C. Dallapiccola$^{\\rm 84}$ , M. Dam$^{\\rm 35}$ , M. Dameri$^{\\rm 50a,50b}$ , D.S.",
"Damiani$^{\\rm 137}$ , H.O.",
"Danielsson$^{\\rm 29}$ , V. Dao$^{\\rm 49}$ , G. Darbo$^{\\rm 50a}$ , G.L.",
"Darlea$^{\\rm 25b}$ , W. Davey$^{\\rm 20}$ , T. Davidek$^{\\rm 126}$ , N. Davidson$^{\\rm 86}$ , R. Davidson$^{\\rm 71}$ , E. Davies$^{\\rm 118}$$^{,c}$ , M. Davies$^{\\rm 93}$ , A.R.",
"Davison$^{\\rm 77}$ , Y. Davygora$^{\\rm 58a}$ , E. Dawe$^{\\rm 142}$ , I. Dawson$^{\\rm 139}$ , R.K. Daya-Ishmukhametova$^{\\rm 22}$ , K. De$^{\\rm 7}$ , R. de Asmundis$^{\\rm 102a}$ , S. De Castro$^{\\rm 19a,19b}$ , S. De Cecco$^{\\rm 78}$ , J. de Graat$^{\\rm 98}$ , N. De Groot$^{\\rm 104}$ , P. de Jong$^{\\rm 105}$ , C. De La Taille$^{\\rm 115}$ , H. De la Torre$^{\\rm 80}$ , F. De Lorenzi$^{\\rm 63}$ , L. de Mora$^{\\rm 71}$ , L. De Nooij$^{\\rm 105}$ , D. De Pedis$^{\\rm 132a}$ , A.",
"De Salvo$^{\\rm 132a}$ , U.",
"De Sanctis$^{\\rm 164a,164c}$ , A.",
"De Santo$^{\\rm 149}$ , J.B. De Vivie De Regie$^{\\rm 115}$ , G. De Zorzi$^{\\rm 132a,132b}$ , W.J.",
"Dearnaley$^{\\rm 71}$ , R. Debbe$^{\\rm 24}$ , C. Debenedetti$^{\\rm 45}$ , B. Dechenaux$^{\\rm 55}$ , D.V.",
"Dedovich$^{\\rm 64}$ , J. Degenhardt$^{\\rm 120}$ , C. Del Papa$^{\\rm 164a,164c}$ , J. Del Peso$^{\\rm 80}$ , T. Del Prete$^{\\rm 122a,122b}$ , T. Delemontex$^{\\rm 55}$ , M. Deliyergiyev$^{\\rm 74}$ , A. Dell'Acqua$^{\\rm 29}$ , L. Dell'Asta$^{\\rm 21}$ , M. Della Pietra$^{\\rm 102a}$$^{,j}$ , D. della Volpe$^{\\rm 102a,102b}$ , M. Delmastro$^{\\rm 4}$ , P.A.",
"Delsart$^{\\rm 55}$ , C. Deluca$^{\\rm 148}$ , S. Demers$^{\\rm 176}$ , M. Demichev$^{\\rm 64}$ , B. Demirkoz$^{\\rm 11}$$^{,l}$ , J. Deng$^{\\rm 163}$ , S.P.",
"Denisov$^{\\rm 128}$ , D. Derendarz$^{\\rm 38}$ , J.E.",
"Derkaoui$^{\\rm 135d}$ , F. Derue$^{\\rm 78}$ , P. Dervan$^{\\rm 73}$ , K. Desch$^{\\rm 20}$ , E. Devetak$^{\\rm 148}$ , P.O.",
"Deviveiros$^{\\rm 105}$ , A. Dewhurst$^{\\rm 129}$ , B. DeWilde$^{\\rm 148}$ , S. Dhaliwal$^{\\rm 158}$ , R. Dhullipudi$^{\\rm 24}$$^{,m}$ , A.",
"Di Ciaccio$^{\\rm 133a,133b}$ , L. Di Ciaccio$^{\\rm 4}$ , A.",
"Di Girolamo$^{\\rm 29}$ , B.",
"Di Girolamo$^{\\rm 29}$ , S. Di Luise$^{\\rm 134a,134b}$ , A.",
"Di Mattia$^{\\rm 173}$ , B.",
"Di Micco$^{\\rm 29}$ , R. Di Nardo$^{\\rm 47}$ , A.",
"Di Simone$^{\\rm 133a,133b}$ , R. Di Sipio$^{\\rm 19a,19b}$ , M.A.",
"Diaz$^{\\rm 31a}$ , F. Diblen$^{\\rm 18c}$ , E.B.",
"Diehl$^{\\rm 87}$ , J. Dietrich$^{\\rm 41}$ , T.A.",
"Dietzsch$^{\\rm 58a}$ , S. Diglio$^{\\rm 86}$ , K. Dindar Yagci$^{\\rm 39}$ , J. Dingfelder$^{\\rm 20}$ , C. Dionisi$^{\\rm 132a,132b}$ , P. Dita$^{\\rm 25a}$ , S. Dita$^{\\rm 25a}$ , F. Dittus$^{\\rm 29}$ , F. Djama$^{\\rm 83}$ , T. Djobava$^{\\rm 51b}$ , M.A.B.",
"do Vale$^{\\rm 23c}$ , A.",
"Do Valle Wemans$^{\\rm 124a}$$^{,n}$ , T.K.O.",
"Doan$^{\\rm 4}$ , M. Dobbs$^{\\rm 85}$ , R. Dobinson $^{\\rm 29}$$^{,*}$ , D. Dobos$^{\\rm 29}$ , E. Dobson$^{\\rm 29}$$^{,o}$ , J. Dodd$^{\\rm 34}$ , C. Doglioni$^{\\rm 49}$ , T. Doherty$^{\\rm 53}$ , Y. Doi$^{\\rm 65}$$^{,*}$ , J. Dolejsi$^{\\rm 126}$ , I. Dolenc$^{\\rm 74}$ , Z. Dolezal$^{\\rm 126}$ , B.A.",
"Dolgoshein$^{\\rm 96}$$^{,*}$ , T. Dohmae$^{\\rm 155}$ , M. Donadelli$^{\\rm 23d}$ , M. Donega$^{\\rm 120}$ , J. Donini$^{\\rm 33}$ , J. Dopke$^{\\rm 29}$ , A. Doria$^{\\rm 102a}$ , A. Dos Anjos$^{\\rm 173}$ , A. Dotti$^{\\rm 122a,122b}$ , M.T.",
"Dova$^{\\rm 70}$ , A.D. Doxiadis$^{\\rm 105}$ , A.T. Doyle$^{\\rm 53}$ , M. Dris$^{\\rm 9}$ , J. Dubbert$^{\\rm 99}$ , S. Dube$^{\\rm 14}$ , E. Duchovni$^{\\rm 172}$ , G. Duckeck$^{\\rm 98}$ , A. Dudarev$^{\\rm 29}$ , F. Dudziak$^{\\rm 63}$ , M. Dührssen $^{\\rm 29}$ , I.P.",
"Duerdoth$^{\\rm 82}$ , L. Duflot$^{\\rm 115}$ , M-A.",
"Dufour$^{\\rm 85}$ , M. Dunford$^{\\rm 29}$ , H. Duran Yildiz$^{\\rm 3a}$ , R. Duxfield$^{\\rm 139}$ , M. Dwuznik$^{\\rm 37}$ , F. Dydak $^{\\rm 29}$ , M. Düren$^{\\rm 52}$ , J. Ebke$^{\\rm 98}$ , S. Eckweiler$^{\\rm 81}$ , K. Edmonds$^{\\rm 81}$ , C.A.",
"Edwards$^{\\rm 76}$ , N.C. Edwards$^{\\rm 53}$ , W. Ehrenfeld$^{\\rm 41}$ , T. Eifert$^{\\rm 143}$ , G. Eigen$^{\\rm 13}$ , K. Einsweiler$^{\\rm 14}$ , E. Eisenhandler$^{\\rm 75}$ , T. Ekelof$^{\\rm 166}$ , M. El Kacimi$^{\\rm 135c}$ , M. Ellert$^{\\rm 166}$ , S. Elles$^{\\rm 4}$ , F. Ellinghaus$^{\\rm 81}$ , K. Ellis$^{\\rm 75}$ , N. Ellis$^{\\rm 29}$ , J. Elmsheuser$^{\\rm 98}$ , M. Elsing$^{\\rm 29}$ , D. Emeliyanov$^{\\rm 129}$ , R. Engelmann$^{\\rm 148}$ , A. Engl$^{\\rm 98}$ , B. Epp$^{\\rm 61}$ , A. Eppig$^{\\rm 87}$ , J. Erdmann$^{\\rm 54}$ , A. Ereditato$^{\\rm 16}$ , D. Eriksson$^{\\rm 146a}$ , J. Ernst$^{\\rm 1}$ , M. Ernst$^{\\rm 24}$ , J. Ernwein$^{\\rm 136}$ , D. Errede$^{\\rm 165}$ , S. Errede$^{\\rm 165}$ , E. Ertel$^{\\rm 81}$ , M. Escalier$^{\\rm 115}$ , C. Escobar$^{\\rm 123}$ , X. Espinal Curull$^{\\rm 11}$ , B. Esposito$^{\\rm 47}$ , F. Etienne$^{\\rm 83}$ , A.I.",
"Etienvre$^{\\rm 136}$ , E. Etzion$^{\\rm 153}$ , D. Evangelakou$^{\\rm 54}$ , H. Evans$^{\\rm 60}$ , L. Fabbri$^{\\rm 19a,19b}$ , C. Fabre$^{\\rm 29}$ , R.M.",
"Fakhrutdinov$^{\\rm 128}$ , S. Falciano$^{\\rm 132a}$ , Y. Fang$^{\\rm 173}$ , M. Fanti$^{\\rm 89a,89b}$ , A. Farbin$^{\\rm 7}$ , A. Farilla$^{\\rm 134a}$ , J. Farley$^{\\rm 148}$ , T. Farooque$^{\\rm 158}$ , S. Farrell$^{\\rm 163}$ , S.M.",
"Farrington$^{\\rm 118}$ , P. Farthouat$^{\\rm 29}$ , P. Fassnacht$^{\\rm 29}$ , D. Fassouliotis$^{\\rm 8}$ , B. Fatholahzadeh$^{\\rm 158}$ , A. Favareto$^{\\rm 89a,89b}$ , L. Fayard$^{\\rm 115}$ , S. Fazio$^{\\rm 36a,36b}$ , R. Febbraro$^{\\rm 33}$ , P. Federic$^{\\rm 144a}$ , O.L.",
"Fedin$^{\\rm 121}$ , W. Fedorko$^{\\rm 88}$ , M. Fehling-Kaschek$^{\\rm 48}$ , L. Feligioni$^{\\rm 83}$ , D. Fellmann$^{\\rm 5}$ , C. Feng$^{\\rm 32d}$ , E.J.",
"Feng$^{\\rm 30}$ , A.B.",
"Fenyuk$^{\\rm 128}$ , J. Ferencei$^{\\rm 144b}$ , W. Fernando$^{\\rm 5}$ , S. Ferrag$^{\\rm 53}$ , J. Ferrando$^{\\rm 53}$ , V. Ferrara$^{\\rm 41}$ , A. Ferrari$^{\\rm 166}$ , P. Ferrari$^{\\rm 105}$ , R. Ferrari$^{\\rm 119a}$ , D.E.",
"Ferreira de Lima$^{\\rm 53}$ , A. Ferrer$^{\\rm 167}$ , D. Ferrere$^{\\rm 49}$ , C. Ferretti$^{\\rm 87}$ , A. Ferretto Parodi$^{\\rm 50a,50b}$ , M. Fiascaris$^{\\rm 30}$ , F. Fiedler$^{\\rm 81}$ , A. Filipčič$^{\\rm 74}$ , F. Filthaut$^{\\rm 104}$ , M. Fincke-Keeler$^{\\rm 169}$ , M.C.N.",
"Fiolhais$^{\\rm 124a}$$^{,h}$ , L. Fiorini$^{\\rm 167}$ , A. Firan$^{\\rm 39}$ , G. Fischer$^{\\rm 41}$ , M.J. Fisher$^{\\rm 109}$ , M. Flechl$^{\\rm 48}$ , I. Fleck$^{\\rm 141}$ , J. Fleckner$^{\\rm 81}$ , P. Fleischmann$^{\\rm 174}$ , S. Fleischmann$^{\\rm 175}$ , T. Flick$^{\\rm 175}$ , A. Floderus$^{\\rm 79}$ , L.R.",
"Flores Castillo$^{\\rm 173}$ , M.J. Flowerdew$^{\\rm 99}$ , T. Fonseca Martin$^{\\rm 16}$ , A. Formica$^{\\rm 136}$ , A. Forti$^{\\rm 82}$ , D. Fortin$^{\\rm 159a}$ , D. Fournier$^{\\rm 115}$ , H. Fox$^{\\rm 71}$ , P. Francavilla$^{\\rm 11}$ , S. Franchino$^{\\rm 119a,119b}$ , D. Francis$^{\\rm 29}$ , T. Frank$^{\\rm 172}$ , M. Franklin$^{\\rm 57}$ , S. Franz$^{\\rm 29}$ , M. Fraternali$^{\\rm 119a,119b}$ , S. Fratina$^{\\rm 120}$ , S.T.",
"French$^{\\rm 27}$ , C. Friedrich$^{\\rm 41}$ , F. Friedrich $^{\\rm 43}$ , R. Froeschl$^{\\rm 29}$ , D. Froidevaux$^{\\rm 29}$ , J.A.",
"Frost$^{\\rm 27}$ , C. Fukunaga$^{\\rm 156}$ , E. Fullana Torregrosa$^{\\rm 29}$ , B.G.",
"Fulsom$^{\\rm 143}$ , J. Fuster$^{\\rm 167}$ , C. Gabaldon$^{\\rm 29}$ , O. Gabizon$^{\\rm 172}$ , T. Gadfort$^{\\rm 24}$ , S. Gadomski$^{\\rm 49}$ , G. Gagliardi$^{\\rm 50a,50b}$ , P. Gagnon$^{\\rm 60}$ , C. Galea$^{\\rm 98}$ , E.J.",
"Gallas$^{\\rm 118}$ , V. Gallo$^{\\rm 16}$ , B.J.",
"Gallop$^{\\rm 129}$ , P. Gallus$^{\\rm 125}$ , K.K.",
"Gan$^{\\rm 109}$ , Y.S.",
"Gao$^{\\rm 143}$$^{,e}$ , A. Gaponenko$^{\\rm 14}$ , F. Garberson$^{\\rm 176}$ , M. Garcia-Sciveres$^{\\rm 14}$ , C. García$^{\\rm 167}$ , J.E.",
"García Navarro$^{\\rm 167}$ , R.W.",
"Gardner$^{\\rm 30}$ , N. Garelli$^{\\rm 29}$ , H. Garitaonandia$^{\\rm 105}$ , V. Garonne$^{\\rm 29}$ , J. Garvey$^{\\rm 17}$ , C. Gatti$^{\\rm 47}$ , G. Gaudio$^{\\rm 119a}$ , B. Gaur$^{\\rm 141}$ , L. Gauthier$^{\\rm 136}$ , P. Gauzzi$^{\\rm 132a,132b}$ , I.L.",
"Gavrilenko$^{\\rm 94}$ , C. Gay$^{\\rm 168}$ , G. Gaycken$^{\\rm 20}$ , E.N.",
"Gazis$^{\\rm 9}$ , P. Ge$^{\\rm 32d}$ , Z. Gecse$^{\\rm 168}$ , C.N.P.",
"Gee$^{\\rm 129}$ , D.A.A.",
"Geerts$^{\\rm 105}$ , Ch.",
"Geich-Gimbel$^{\\rm 20}$ , K. Gellerstedt$^{\\rm 146a,146b}$ , C. Gemme$^{\\rm 50a}$ , A. Gemmell$^{\\rm 53}$ , M.H.",
"Genest$^{\\rm 55}$ , S. Gentile$^{\\rm 132a,132b}$ , M. George$^{\\rm 54}$ , S. George$^{\\rm 76}$ , P. Gerlach$^{\\rm 175}$ , A. Gershon$^{\\rm 153}$ , C. Geweniger$^{\\rm 58a}$ , H. Ghazlane$^{\\rm 135b}$ , N. Ghodbane$^{\\rm 33}$ , B. Giacobbe$^{\\rm 19a}$ , S. Giagu$^{\\rm 132a,132b}$ , V. Giakoumopoulou$^{\\rm 8}$ , V. Giangiobbe$^{\\rm 11}$ , F. Gianotti$^{\\rm 29}$ , B. Gibbard$^{\\rm 24}$ , A. Gibson$^{\\rm 158}$ , S.M.",
"Gibson$^{\\rm 29}$ , D. Gillberg$^{\\rm 28}$ , A.R.",
"Gillman$^{\\rm 129}$ , D.M.",
"Gingrich$^{\\rm 2}$$^{,d}$ , J. Ginzburg$^{\\rm 153}$ , N. Giokaris$^{\\rm 8}$ , M.P.",
"Giordani$^{\\rm 164c}$ , R. Giordano$^{\\rm 102a,102b}$ , F.M.",
"Giorgi$^{\\rm 15}$ , P. Giovannini$^{\\rm 99}$ , P.F.",
"Giraud$^{\\rm 136}$ , D. Giugni$^{\\rm 89a}$ , M. Giunta$^{\\rm 93}$ , P. Giusti$^{\\rm 19a}$ , B.K.",
"Gjelsten$^{\\rm 117}$ , L.K.",
"Gladilin$^{\\rm 97}$ , C. Glasman$^{\\rm 80}$ , J. Glatzer$^{\\rm 48}$ , A. Glazov$^{\\rm 41}$ , K.W.",
"Glitza$^{\\rm 175}$ , G.L.",
"Glonti$^{\\rm 64}$ , J.R. Goddard$^{\\rm 75}$ , J. Godfrey$^{\\rm 142}$ , J. Godlewski$^{\\rm 29}$ , M. Goebel$^{\\rm 41}$ , T. Göpfert$^{\\rm 43}$ , C. Goeringer$^{\\rm 81}$ , C. Gössling$^{\\rm 42}$ , T. Göttfert$^{\\rm 99}$ , S. Goldfarb$^{\\rm 87}$ , T. Golling$^{\\rm 176}$ , A. Gomes$^{\\rm 124a}$$^{,b}$ , L.S.",
"Gomez Fajardo$^{\\rm 41}$ , R. Gonçalo$^{\\rm 76}$ , J. Goncalves Pinto Firmino Da Costa$^{\\rm 41}$ , L. Gonella$^{\\rm 20}$ , S. Gonzalez$^{\\rm 173}$ , S. González de la Hoz$^{\\rm 167}$ , G. Gonzalez Parra$^{\\rm 11}$ , M.L.",
"Gonzalez Silva$^{\\rm 26}$ , S. Gonzalez-Sevilla$^{\\rm 49}$ , J.J. Goodson$^{\\rm 148}$ , L. Goossens$^{\\rm 29}$ , P.A.",
"Gorbounov$^{\\rm 95}$ , H.A.",
"Gordon$^{\\rm 24}$ , I. Gorelov$^{\\rm 103}$ , G. Gorfine$^{\\rm 175}$ , B. Gorini$^{\\rm 29}$ , E. Gorini$^{\\rm 72a,72b}$ , A. Gorišek$^{\\rm 74}$ , E. Gornicki$^{\\rm 38}$ , B. Gosdzik$^{\\rm 41}$ , A.T. Goshaw$^{\\rm 5}$ , M. Gosselink$^{\\rm 105}$ , M.I.",
"Gostkin$^{\\rm 64}$ , I. Gough Eschrich$^{\\rm 163}$ , M. Gouighri$^{\\rm 135a}$ , D. Goujdami$^{\\rm 135c}$ , M.P.",
"Goulette$^{\\rm 49}$ , A.G. Goussiou$^{\\rm 138}$ , C. Goy$^{\\rm 4}$ , S. Gozpinar$^{\\rm 22}$ , I. Grabowska-Bold$^{\\rm 37}$ , P. Grafström$^{\\rm 29}$ , K-J.",
"Grahn$^{\\rm 41}$ , F. Grancagnolo$^{\\rm 72a}$ , S. Grancagnolo$^{\\rm 15}$ , V. Grassi$^{\\rm 148}$ , V. Gratchev$^{\\rm 121}$ , N. Grau$^{\\rm 34}$ , H.M. Gray$^{\\rm 29}$ , J.A.",
"Gray$^{\\rm 148}$ , E. Graziani$^{\\rm 134a}$ , O.G.",
"Grebenyuk$^{\\rm 121}$ , T. Greenshaw$^{\\rm 73}$ , Z.D.",
"Greenwood$^{\\rm 24}$$^{,m}$ , K. Gregersen$^{\\rm 35}$ , I.M.",
"Gregor$^{\\rm 41}$ , P. Grenier$^{\\rm 143}$ , J. Griffiths$^{\\rm 138}$ , N. Grigalashvili$^{\\rm 64}$ , A.A. Grillo$^{\\rm 137}$ , S. Grinstein$^{\\rm 11}$ , Y.V.",
"Grishkevich$^{\\rm 97}$ , J.-F. Grivaz$^{\\rm 115}$ , E. Gross$^{\\rm 172}$ , J. Grosse-Knetter$^{\\rm 54}$ , J. Groth-Jensen$^{\\rm 172}$ , K. Grybel$^{\\rm 141}$ , D. Guest$^{\\rm 176}$ , C. Guicheney$^{\\rm 33}$ , A. Guida$^{\\rm 72a,72b}$ , S. Guindon$^{\\rm 54}$ , H. Guler$^{\\rm 85}$$^{,p}$ , J. Gunther$^{\\rm 125}$ , B. Guo$^{\\rm 158}$ , J. Guo$^{\\rm 34}$ , V.N.",
"Gushchin$^{\\rm 128}$ , P. Gutierrez$^{\\rm 111}$ , N. Guttman$^{\\rm 153}$ , O. Gutzwiller$^{\\rm 173}$ , C. Guyot$^{\\rm 136}$ , C. Gwenlan$^{\\rm 118}$ , C.B.",
"Gwilliam$^{\\rm 73}$ , A. Haas$^{\\rm 143}$ , S. Haas$^{\\rm 29}$ , C. Haber$^{\\rm 14}$ , H.K.",
"Hadavand$^{\\rm 39}$ , D.R.",
"Hadley$^{\\rm 17}$ , P. Haefner$^{\\rm 99}$ , F. Hahn$^{\\rm 29}$ , S. Haider$^{\\rm 29}$ , Z. Hajduk$^{\\rm 38}$ , H. Hakobyan$^{\\rm 177}$ , D. Hall$^{\\rm 118}$ , J. Haller$^{\\rm 54}$ , K. Hamacher$^{\\rm 175}$ , P. Hamal$^{\\rm 113}$ , M. Hamer$^{\\rm 54}$ , A. Hamilton$^{\\rm 145b}$$^{,q}$ , S. Hamilton$^{\\rm 161}$ , L. Han$^{\\rm 32b}$ , K. Hanagaki$^{\\rm 116}$ , K. Hanawa$^{\\rm 160}$ , M. Hance$^{\\rm 14}$ , C. Handel$^{\\rm 81}$ , P. Hanke$^{\\rm 58a}$ , J.R. Hansen$^{\\rm 35}$ , J.B. Hansen$^{\\rm 35}$ , J.D.",
"Hansen$^{\\rm 35}$ , P.H.",
"Hansen$^{\\rm 35}$ , P. Hansson$^{\\rm 143}$ , K. Hara$^{\\rm 160}$ , G.A.",
"Hare$^{\\rm 137}$ , T. Harenberg$^{\\rm 175}$ , S. Harkusha$^{\\rm 90}$ , D. Harper$^{\\rm 87}$ , R.D.",
"Harrington$^{\\rm 45}$ , O.M.",
"Harris$^{\\rm 138}$ , K. Harrison$^{\\rm 17}$ , J. Hartert$^{\\rm 48}$ , F. Hartjes$^{\\rm 105}$ , T. Haruyama$^{\\rm 65}$ , A. Harvey$^{\\rm 56}$ , S. Hasegawa$^{\\rm 101}$ , Y. Hasegawa$^{\\rm 140}$ , S. Hassani$^{\\rm 136}$ , S. Haug$^{\\rm 16}$ , M. Hauschild$^{\\rm 29}$ , R. Hauser$^{\\rm 88}$ , M. Havranek$^{\\rm 20}$ , C.M.",
"Hawkes$^{\\rm 17}$ , R.J. Hawkings$^{\\rm 29}$ , A.D. Hawkins$^{\\rm 79}$ , D. Hawkins$^{\\rm 163}$ , T. Hayakawa$^{\\rm 66}$ , T. Hayashi$^{\\rm 160}$ , D. Hayden$^{\\rm 76}$ , H.S.",
"Hayward$^{\\rm 73}$ , S.J.",
"Haywood$^{\\rm 129}$ , M. He$^{\\rm 32d}$ , S.J.",
"Head$^{\\rm 17}$ , V. Hedberg$^{\\rm 79}$ , L. Heelan$^{\\rm 7}$ , S. Heim$^{\\rm 88}$ , B. Heinemann$^{\\rm 14}$ , S. Heisterkamp$^{\\rm 35}$ , L. Helary$^{\\rm 4}$ , C. Heller$^{\\rm 98}$ , M. Heller$^{\\rm 29}$ , S. Hellman$^{\\rm 146a,146b}$ , D. Hellmich$^{\\rm 20}$ , C. Helsens$^{\\rm 11}$ , R.C.W.",
"Henderson$^{\\rm 71}$ , M. Henke$^{\\rm 58a}$ , A. Henrichs$^{\\rm 54}$ , A.M. Henriques Correia$^{\\rm 29}$ , S. Henrot-Versille$^{\\rm 115}$ , F. Henry-Couannier$^{\\rm 83}$ , C. Hensel$^{\\rm 54}$ , T. Henß$^{\\rm 175}$ , C.M.",
"Hernandez$^{\\rm 7}$ , Y. Hernández Jiménez$^{\\rm 167}$ , R. Herrberg$^{\\rm 15}$ , G. Herten$^{\\rm 48}$ , R. Hertenberger$^{\\rm 98}$ , L. Hervas$^{\\rm 29}$ , G.G.",
"Hesketh$^{\\rm 77}$ , N.P.",
"Hessey$^{\\rm 105}$ , E. Higón-Rodriguez$^{\\rm 167}$ , J.C. Hill$^{\\rm 27}$ , K.H.",
"Hiller$^{\\rm 41}$ , S. Hillert$^{\\rm 20}$ , S.J.",
"Hillier$^{\\rm 17}$ , I. Hinchliffe$^{\\rm 14}$ , E. Hines$^{\\rm 120}$ , M. Hirose$^{\\rm 116}$ , F. Hirsch$^{\\rm 42}$ , D. Hirschbuehl$^{\\rm 175}$ , J. Hobbs$^{\\rm 148}$ , N. Hod$^{\\rm 153}$ , M.C.",
"Hodgkinson$^{\\rm 139}$ , P. Hodgson$^{\\rm 139}$ , A. Hoecker$^{\\rm 29}$ , M.R.",
"Hoeferkamp$^{\\rm 103}$ , J. Hoffman$^{\\rm 39}$ , D. Hoffmann$^{\\rm 83}$ , M. Hohlfeld$^{\\rm 81}$ , M. Holder$^{\\rm 141}$ , S.O.",
"Holmgren$^{\\rm 146a}$ , T. Holy$^{\\rm 127}$ , J.L.",
"Holzbauer$^{\\rm 88}$ , T.M.",
"Hong$^{\\rm 120}$ , L. Hooft van Huysduynen$^{\\rm 108}$ , C. Horn$^{\\rm 143}$ , S. Horner$^{\\rm 48}$ , J-Y.",
"Hostachy$^{\\rm 55}$ , S. Hou$^{\\rm 151}$ , A. Hoummada$^{\\rm 135a}$ , J. Howarth$^{\\rm 82}$ , I. Hristova $^{\\rm 15}$ , J. Hrivnac$^{\\rm 115}$ , I. Hruska$^{\\rm 125}$ , T. Hryn'ova$^{\\rm 4}$ , P.J.",
"Hsu$^{\\rm 81}$ , S.-C. Hsu$^{\\rm 14}$ , Z. Hubacek$^{\\rm 127}$ , F. Hubaut$^{\\rm 83}$ , F. Huegging$^{\\rm 20}$ , A. Huettmann$^{\\rm 41}$ , T.B.",
"Huffman$^{\\rm 118}$ , E.W.",
"Hughes$^{\\rm 34}$ , G. Hughes$^{\\rm 71}$ , M. Huhtinen$^{\\rm 29}$ , M. Hurwitz$^{\\rm 14}$ , U. Husemann$^{\\rm 41}$ , N. Huseynov$^{\\rm 64}$$^{,r}$ , J. Huston$^{\\rm 88}$ , J. Huth$^{\\rm 57}$ , G. Iacobucci$^{\\rm 49}$ , G. Iakovidis$^{\\rm 9}$ , M. Ibbotson$^{\\rm 82}$ , I. Ibragimov$^{\\rm 141}$ , L. Iconomidou-Fayard$^{\\rm 115}$ , J. Idarraga$^{\\rm 115}$ , P. Iengo$^{\\rm 102a}$ , O. Igonkina$^{\\rm 105}$ , Y. Ikegami$^{\\rm 65}$ , M. Ikeno$^{\\rm 65}$ , D. Iliadis$^{\\rm 154}$ , N. Ilic$^{\\rm 158}$ , T. Ince$^{\\rm 20}$ , J. Inigo-Golfin$^{\\rm 29}$ , P. Ioannou$^{\\rm 8}$ , M. Iodice$^{\\rm 134a}$ , K. Iordanidou$^{\\rm 8}$ , V. Ippolito$^{\\rm 132a,132b}$ , A. Irles Quiles$^{\\rm 167}$ , C. Isaksson$^{\\rm 166}$ , A. Ishikawa$^{\\rm 66}$ , M. Ishino$^{\\rm 67}$ , R. Ishmukhametov$^{\\rm 39}$ , C. Issever$^{\\rm 118}$ , S. Istin$^{\\rm 18a}$ , A.V.",
"Ivashin$^{\\rm 128}$ , W. Iwanski$^{\\rm 38}$ , H. Iwasaki$^{\\rm 65}$ , J.M.",
"Izen$^{\\rm 40}$ , V. Izzo$^{\\rm 102a}$ , B. Jackson$^{\\rm 120}$ , J.N.",
"Jackson$^{\\rm 73}$ , P. Jackson$^{\\rm 143}$ , M.R.",
"Jaekel$^{\\rm 29}$ , V. Jain$^{\\rm 60}$ , K. Jakobs$^{\\rm 48}$ , S. Jakobsen$^{\\rm 35}$ , J. Jakubek$^{\\rm 127}$ , D.K.",
"Jana$^{\\rm 111}$ , E. Jansen$^{\\rm 77}$ , H. Jansen$^{\\rm 29}$ , A. Jantsch$^{\\rm 99}$ , M. Janus$^{\\rm 48}$ , G. Jarlskog$^{\\rm 79}$ , L. Jeanty$^{\\rm 57}$ , I. Jen-La Plante$^{\\rm 30}$ , P. Jenni$^{\\rm 29}$ , A. Jeremie$^{\\rm 4}$ , P. Jež$^{\\rm 35}$ , S. Jézéquel$^{\\rm 4}$ , M.K.",
"Jha$^{\\rm 19a}$ , H. Ji$^{\\rm 173}$ , W. Ji$^{\\rm 81}$ , J. Jia$^{\\rm 148}$ , Y. Jiang$^{\\rm 32b}$ , M. Jimenez Belenguer$^{\\rm 41}$ , S. Jin$^{\\rm 32a}$ , O. Jinnouchi$^{\\rm 157}$ , M.D.",
"Joergensen$^{\\rm 35}$ , D. Joffe$^{\\rm 39}$ , L.G.",
"Johansen$^{\\rm 13}$ , M. Johansen$^{\\rm 146a,146b}$ , K.E.",
"Johansson$^{\\rm 146a}$ , P. Johansson$^{\\rm 139}$ , S. Johnert$^{\\rm 41}$ , K.A.",
"Johns$^{\\rm 6}$ , K. Jon-And$^{\\rm 146a,146b}$ , G. Jones$^{\\rm 118}$ , R.W.L.",
"Jones$^{\\rm 71}$ , T.J. Jones$^{\\rm 73}$ , C. Joram$^{\\rm 29}$ , P.M. Jorge$^{\\rm 124a}$ , K.D.",
"Joshi$^{\\rm 82}$ , J. Jovicevic$^{\\rm 147}$ , T. Jovin$^{\\rm 12b}$ , X. Ju$^{\\rm 173}$ , C.A.",
"Jung$^{\\rm 42}$ , R.M.",
"Jungst$^{\\rm 29}$ , V. Juranek$^{\\rm 125}$ , P. Jussel$^{\\rm 61}$ , A. Juste Rozas$^{\\rm 11}$ , S. Kabana$^{\\rm 16}$ , M. Kaci$^{\\rm 167}$ , A. Kaczmarska$^{\\rm 38}$ , P. Kadlecik$^{\\rm 35}$ , M. Kado$^{\\rm 115}$ , H. Kagan$^{\\rm 109}$ , M. Kagan$^{\\rm 57}$ , E. Kajomovitz$^{\\rm 152}$ , S. Kalinin$^{\\rm 175}$ , L.V.",
"Kalinovskaya$^{\\rm 64}$ , S. Kama$^{\\rm 39}$ , N. Kanaya$^{\\rm 155}$ , M. Kaneda$^{\\rm 29}$ , S. Kaneti$^{\\rm 27}$ , T. Kanno$^{\\rm 157}$ , V.A.",
"Kantserov$^{\\rm 96}$ , J. Kanzaki$^{\\rm 65}$ , B. Kaplan$^{\\rm 176}$ , A. Kapliy$^{\\rm 30}$ , J. Kaplon$^{\\rm 29}$ , D. Kar$^{\\rm 53}$ , M. Karagounis$^{\\rm 20}$ , K. Karakostas$^{\\rm 9}$ , M. Karnevskiy$^{\\rm 41}$ , V. Kartvelishvili$^{\\rm 71}$ , A.N.",
"Karyukhin$^{\\rm 128}$ , L. Kashif$^{\\rm 173}$ , G. Kasieczka$^{\\rm 58b}$ , R.D.",
"Kass$^{\\rm 109}$ , A. Kastanas$^{\\rm 13}$ , M. Kataoka$^{\\rm 4}$ , Y. Kataoka$^{\\rm 155}$ , E. Katsoufis$^{\\rm 9}$ , J. Katzy$^{\\rm 41}$ , V. Kaushik$^{\\rm 6}$ , K. Kawagoe$^{\\rm 69}$ , T. Kawamoto$^{\\rm 155}$ , G. Kawamura$^{\\rm 81}$ , M.S.",
"Kayl$^{\\rm 105}$ , V.A.",
"Kazanin$^{\\rm 107}$ , M.Y.",
"Kazarinov$^{\\rm 64}$ , R. Keeler$^{\\rm 169}$ , R. Kehoe$^{\\rm 39}$ , M. Keil$^{\\rm 54}$ , G.D. Kekelidze$^{\\rm 64}$ , J.S.",
"Keller$^{\\rm 138}$ , J. Kennedy$^{\\rm 98}$ , M. Kenyon$^{\\rm 53}$ , O. Kepka$^{\\rm 125}$ , N. Kerschen$^{\\rm 29}$ , B.P.",
"Kerševan$^{\\rm 74}$ , S. Kersten$^{\\rm 175}$ , K. Kessoku$^{\\rm 155}$ , J. Keung$^{\\rm 158}$ , F. Khalil-zada$^{\\rm 10}$ , H. Khandanyan$^{\\rm 165}$ , A. Khanov$^{\\rm 112}$ , D. Kharchenko$^{\\rm 64}$ , A. Khodinov$^{\\rm 96}$ , A. Khomich$^{\\rm 58a}$ , T.J. Khoo$^{\\rm 27}$ , G. Khoriauli$^{\\rm 20}$ , A. Khoroshilov$^{\\rm 175}$ , V. Khovanskiy$^{\\rm 95}$ , E. Khramov$^{\\rm 64}$ , J. Khubua$^{\\rm 51b}$ , H. Kim$^{\\rm 146a,146b}$ , M.S.",
"Kim$^{\\rm 2}$ , S.H.",
"Kim$^{\\rm 160}$ , N. Kimura$^{\\rm 171}$ , O. Kind$^{\\rm 15}$ , B.T.",
"King$^{\\rm 73}$ , M. King$^{\\rm 66}$ , R.S.B.",
"King$^{\\rm 118}$ , J. Kirk$^{\\rm 129}$ , A.E.",
"Kiryunin$^{\\rm 99}$ , T. Kishimoto$^{\\rm 66}$ , D. Kisielewska$^{\\rm 37}$ , T. Kittelmann$^{\\rm 123}$ , A.M. Kiver$^{\\rm 128}$ , E. Kladiva$^{\\rm 144b}$ , M. Klein$^{\\rm 73}$ , U. Klein$^{\\rm 73}$ , K. Kleinknecht$^{\\rm 81}$ , M. Klemetti$^{\\rm 85}$ , A. Klier$^{\\rm 172}$ , P. Klimek$^{\\rm 146a,146b}$ , A. Klimentov$^{\\rm 24}$ , R. Klingenberg$^{\\rm 42}$ , J.A.",
"Klinger$^{\\rm 82}$ , E.B.",
"Klinkby$^{\\rm 35}$ , T. Klioutchnikova$^{\\rm 29}$ , P.F.",
"Klok$^{\\rm 104}$ , S. Klous$^{\\rm 105}$ , E.-E. Kluge$^{\\rm 58a}$ , T. Kluge$^{\\rm 73}$ , P. Kluit$^{\\rm 105}$ , S. Kluth$^{\\rm 99}$ , N.S.",
"Knecht$^{\\rm 158}$ , E. Kneringer$^{\\rm 61}$ , E.B.F.G.",
"Knoops$^{\\rm 83}$ , A. Knue$^{\\rm 54}$ , B.R.",
"Ko$^{\\rm 44}$ , T. Kobayashi$^{\\rm 155}$ , M. Kobel$^{\\rm 43}$ , M. Kocian$^{\\rm 143}$ , P. Kodys$^{\\rm 126}$ , K. Köneke$^{\\rm 29}$ , A.C. König$^{\\rm 104}$ , S. Koenig$^{\\rm 81}$ , L. Köpke$^{\\rm 81}$ , F. Koetsveld$^{\\rm 104}$ , P. Koevesarki$^{\\rm 20}$ , T. Koffas$^{\\rm 28}$ , E. Koffeman$^{\\rm 105}$ , L.A. Kogan$^{\\rm 118}$ , S. Kohlmann$^{\\rm 175}$ , F. Kohn$^{\\rm 54}$ , Z. Kohout$^{\\rm 127}$ , T. Kohriki$^{\\rm 65}$ , T. Koi$^{\\rm 143}$ , G.M.",
"Kolachev$^{\\rm 107}$ , H. Kolanoski$^{\\rm 15}$ , V. Kolesnikov$^{\\rm 64}$ , I. Koletsou$^{\\rm 89a}$ , J. Koll$^{\\rm 88}$ , M. Kollefrath$^{\\rm 48}$ , A.A. Komar$^{\\rm 94}$ , Y. Komori$^{\\rm 155}$ , T. Kondo$^{\\rm 65}$ , T. Kono$^{\\rm 41}$$^{,s}$ , A.I.",
"Kononov$^{\\rm 48}$ , R. Konoplich$^{\\rm 108}$$^{,t}$ , N. Konstantinidis$^{\\rm 77}$ , A. Kootz$^{\\rm 175}$ , S. Koperny$^{\\rm 37}$ , K. Korcyl$^{\\rm 38}$ , K. Kordas$^{\\rm 154}$ , A. Korn$^{\\rm 118}$ , A. Korol$^{\\rm 107}$ , I. Korolkov$^{\\rm 11}$ , E.V.",
"Korolkova$^{\\rm 139}$ , V.A.",
"Korotkov$^{\\rm 128}$ , O. Kortner$^{\\rm 99}$ , S. Kortner$^{\\rm 99}$ , V.V.",
"Kostyukhin$^{\\rm 20}$ , S. Kotov$^{\\rm 99}$ , V.M.",
"Kotov$^{\\rm 64}$ , A. Kotwal$^{\\rm 44}$ , C. Kourkoumelis$^{\\rm 8}$ , V. Kouskoura$^{\\rm 154}$ , A. Koutsman$^{\\rm 159a}$ , R. Kowalewski$^{\\rm 169}$ , T.Z.",
"Kowalski$^{\\rm 37}$ , W. Kozanecki$^{\\rm 136}$ , A.S. Kozhin$^{\\rm 128}$ , V. Kral$^{\\rm 127}$ , V.A.",
"Kramarenko$^{\\rm 97}$ , G. Kramberger$^{\\rm 74}$ , M.W.",
"Krasny$^{\\rm 78}$ , A. Krasznahorkay$^{\\rm 108}$ , J. Kraus$^{\\rm 88}$ , J.K. Kraus$^{\\rm 20}$ , F. Krejci$^{\\rm 127}$ , J. Kretzschmar$^{\\rm 73}$ , N. Krieger$^{\\rm 54}$ , P. Krieger$^{\\rm 158}$ , K. Kroeninger$^{\\rm 54}$ , H. Kroha$^{\\rm 99}$ , J. Kroll$^{\\rm 120}$ , J. Kroseberg$^{\\rm 20}$ , J. Krstic$^{\\rm 12a}$ , U. Kruchonak$^{\\rm 64}$ , H. Krüger$^{\\rm 20}$ , T. Kruker$^{\\rm 16}$ , N. Krumnack$^{\\rm 63}$ , Z.V.",
"Krumshteyn$^{\\rm 64}$ , A. Kruth$^{\\rm 20}$ , T. Kubota$^{\\rm 86}$ , S. Kuday$^{\\rm 3a}$ , S. Kuehn$^{\\rm 48}$ , A. Kugel$^{\\rm 58c}$ , T. Kuhl$^{\\rm 41}$ , D. Kuhn$^{\\rm 61}$ , V. Kukhtin$^{\\rm 64}$ , Y. Kulchitsky$^{\\rm 90}$ , S. Kuleshov$^{\\rm 31b}$ , C. Kummer$^{\\rm 98}$ , M. Kuna$^{\\rm 78}$ , J. Kunkle$^{\\rm 120}$ , A. Kupco$^{\\rm 125}$ , H. Kurashige$^{\\rm 66}$ , M. Kurata$^{\\rm 160}$ , Y.A.",
"Kurochkin$^{\\rm 90}$ , V. Kus$^{\\rm 125}$ , E.S.",
"Kuwertz$^{\\rm 147}$ , M. Kuze$^{\\rm 157}$ , J. Kvita$^{\\rm 142}$ , R. Kwee$^{\\rm 15}$ , A.",
"La Rosa$^{\\rm 49}$ , L. La Rotonda$^{\\rm 36a,36b}$ , L. Labarga$^{\\rm 80}$ , J. Labbe$^{\\rm 4}$ , S. Lablak$^{\\rm 135a}$ , C. Lacasta$^{\\rm 167}$ , F. Lacava$^{\\rm 132a,132b}$ , H. Lacker$^{\\rm 15}$ , D. Lacour$^{\\rm 78}$ , V.R.",
"Lacuesta$^{\\rm 167}$ , E. Ladygin$^{\\rm 64}$ , R. Lafaye$^{\\rm 4}$ , B. Laforge$^{\\rm 78}$ , T. Lagouri$^{\\rm 80}$ , S. Lai$^{\\rm 48}$ , E. Laisne$^{\\rm 55}$ , M. Lamanna$^{\\rm 29}$ , L. Lambourne$^{\\rm 77}$ , C.L.",
"Lampen$^{\\rm 6}$ , W. Lampl$^{\\rm 6}$ , E. Lancon$^{\\rm 136}$ , U. Landgraf$^{\\rm 48}$ , M.P.J.",
"Landon$^{\\rm 75}$ , J.L.",
"Lane$^{\\rm 82}$ , C. Lange$^{\\rm 41}$ , A.J.",
"Lankford$^{\\rm 163}$ , F. Lanni$^{\\rm 24}$ , K. Lantzsch$^{\\rm 175}$ , S. Laplace$^{\\rm 78}$ , C. Lapoire$^{\\rm 20}$ , J.F.",
"Laporte$^{\\rm 136}$ , T. Lari$^{\\rm 89a}$ , A. Larner$^{\\rm 118}$ , M. Lassnig$^{\\rm 29}$ , P. Laurelli$^{\\rm 47}$ , V. Lavorini$^{\\rm 36a,36b}$ , W. Lavrijsen$^{\\rm 14}$ , P. Laycock$^{\\rm 73}$ , O.",
"Le Dortz$^{\\rm 78}$ , E. Le Guirriec$^{\\rm 83}$ , C. Le Maner$^{\\rm 158}$ , E. Le Menedeu$^{\\rm 11}$ , T. LeCompte$^{\\rm 5}$ , F. Ledroit-Guillon$^{\\rm 55}$ , H. Lee$^{\\rm 105}$ , J.S.H.",
"Lee$^{\\rm 116}$ , S.C. Lee$^{\\rm 151}$ , L. Lee$^{\\rm 176}$ , M. Lefebvre$^{\\rm 169}$ , M. Legendre$^{\\rm 136}$ , B.C.",
"LeGeyt$^{\\rm 120}$ , F. Legger$^{\\rm 98}$ , C. Leggett$^{\\rm 14}$ , M. Lehmacher$^{\\rm 20}$ , G. Lehmann Miotto$^{\\rm 29}$ , X. Lei$^{\\rm 6}$ , M.A.L.",
"Leite$^{\\rm 23d}$ , R. Leitner$^{\\rm 126}$ , D. Lellouch$^{\\rm 172}$ , B. Lemmer$^{\\rm 54}$ , V. Lendermann$^{\\rm 58a}$ , K.J.C.",
"Leney$^{\\rm 145b}$ , T. Lenz$^{\\rm 105}$ , G. Lenzen$^{\\rm 175}$ , B. Lenzi$^{\\rm 29}$ , K. Leonhardt$^{\\rm 43}$ , S. Leontsinis$^{\\rm 9}$ , F. Lepold$^{\\rm 58a}$ , C. Leroy$^{\\rm 93}$ , J-R. Lessard$^{\\rm 169}$ , C.G.",
"Lester$^{\\rm 27}$ , C.M.",
"Lester$^{\\rm 120}$ , J. Levêque$^{\\rm 4}$ , D. Levin$^{\\rm 87}$ , L.J.",
"Levinson$^{\\rm 172}$ , A. Lewis$^{\\rm 118}$ , G.H.",
"Lewis$^{\\rm 108}$ , A.M. Leyko$^{\\rm 20}$ , M. Leyton$^{\\rm 15}$ , B. Li$^{\\rm 83}$ , H. Li$^{\\rm 173}$$^{,u}$ , S. Li$^{\\rm 32b}$$^{,v}$ , X. Li$^{\\rm 87}$ , Z. Liang$^{\\rm 118}$$^{,w}$ , H. Liao$^{\\rm 33}$ , B. Liberti$^{\\rm 133a}$ , P. Lichard$^{\\rm 29}$ , M. Lichtnecker$^{\\rm 98}$ , K. Lie$^{\\rm 165}$ , W. Liebig$^{\\rm 13}$ , C. Limbach$^{\\rm 20}$ , A. Limosani$^{\\rm 86}$ , M. Limper$^{\\rm 62}$ , S.C. Lin$^{\\rm 151}$$^{,x}$ , F. Linde$^{\\rm 105}$ , J.T.",
"Linnemann$^{\\rm 88}$ , E. Lipeles$^{\\rm 120}$ , A. Lipniacka$^{\\rm 13}$ , T.M.",
"Liss$^{\\rm 165}$ , D. Lissauer$^{\\rm 24}$ , A. Lister$^{\\rm 49}$ , A.M. Litke$^{\\rm 137}$ , C. Liu$^{\\rm 28}$ , D. Liu$^{\\rm 151}$ , H. Liu$^{\\rm 87}$ , J.B. Liu$^{\\rm 87}$ , M. Liu$^{\\rm 32b}$ , Y. Liu$^{\\rm 32b}$ , M. Livan$^{\\rm 119a,119b}$ , S.S.A.",
"Livermore$^{\\rm 118}$ , A. Lleres$^{\\rm 55}$ , J. Llorente Merino$^{\\rm 80}$ , S.L.",
"Lloyd$^{\\rm 75}$ , E. Lobodzinska$^{\\rm 41}$ , P. Loch$^{\\rm 6}$ , W.S.",
"Lockman$^{\\rm 137}$ , T. Loddenkoetter$^{\\rm 20}$ , F.K.",
"Loebinger$^{\\rm 82}$ , A. Loginov$^{\\rm 176}$ , C.W.",
"Loh$^{\\rm 168}$ , T. Lohse$^{\\rm 15}$ , K. Lohwasser$^{\\rm 48}$ , M. Lokajicek$^{\\rm 125}$ , V.P.",
"Lombardo$^{\\rm 4}$ , R.E.",
"Long$^{\\rm 71}$ , L. Lopes$^{\\rm 124a}$ , D. Lopez Mateos$^{\\rm 57}$ , J. Lorenz$^{\\rm 98}$ , N. Lorenzo Martinez$^{\\rm 115}$ , M. Losada$^{\\rm 162}$ , P. Loscutoff$^{\\rm 14}$ , F. Lo Sterzo$^{\\rm 132a,132b}$ , M.J. Losty$^{\\rm 159a}$ , X. Lou$^{\\rm 40}$ , A. Lounis$^{\\rm 115}$ , K.F.",
"Loureiro$^{\\rm 162}$ , J. Love$^{\\rm 21}$ , P.A.",
"Love$^{\\rm 71}$ , A.J.",
"Lowe$^{\\rm 143}$$^{,e}$ , F. Lu$^{\\rm 32a}$ , H.J.",
"Lubatti$^{\\rm 138}$ , C. Luci$^{\\rm 132a,132b}$ , A. Lucotte$^{\\rm 55}$ , A. Ludwig$^{\\rm 43}$ , D. Ludwig$^{\\rm 41}$ , I. Ludwig$^{\\rm 48}$ , J. Ludwig$^{\\rm 48}$ , F. Luehring$^{\\rm 60}$ , G. Luijckx$^{\\rm 105}$ , W. Lukas$^{\\rm 61}$ , D. Lumb$^{\\rm 48}$ , L. Luminari$^{\\rm 132a}$ , E. Lund$^{\\rm 117}$ , B. Lund-Jensen$^{\\rm 147}$ , B. Lundberg$^{\\rm 79}$ , J. Lundberg$^{\\rm 146a,146b}$ , J. Lundquist$^{\\rm 35}$ , M. Lungwitz$^{\\rm 81}$ , D. Lynn$^{\\rm 24}$ , J. Lys$^{\\rm 14}$ , E. Lytken$^{\\rm 79}$ , H. Ma$^{\\rm 24}$ , L.L.",
"Ma$^{\\rm 173}$ , J.A.",
"Macana Goia$^{\\rm 93}$ , G. Maccarrone$^{\\rm 47}$ , A. Macchiolo$^{\\rm 99}$ , B. Maček$^{\\rm 74}$ , J. Machado Miguens$^{\\rm 124a}$ , R. Mackeprang$^{\\rm 35}$ , R.J. Madaras$^{\\rm 14}$ , W.F.",
"Mader$^{\\rm 43}$ , R. Maenner$^{\\rm 58c}$ , T. Maeno$^{\\rm 24}$ , P. Mättig$^{\\rm 175}$ , S. Mättig$^{\\rm 41}$ , L. Magnoni$^{\\rm 29}$ , E. Magradze$^{\\rm 54}$ , K. Mahboubi$^{\\rm 48}$ , S. Mahmoud$^{\\rm 73}$ , G. Mahout$^{\\rm 17}$ , C. Maiani$^{\\rm 132a,132b}$ , C. Maidantchik$^{\\rm 23a}$ , A. Maio$^{\\rm 124a}$$^{,b}$ , S. Majewski$^{\\rm 24}$ , Y. Makida$^{\\rm 65}$ , N. Makovec$^{\\rm 115}$ , P. Mal$^{\\rm 136}$ , B. Malaescu$^{\\rm 29}$ , Pa. Malecki$^{\\rm 38}$ , P. Malecki$^{\\rm 38}$ , V.P.",
"Maleev$^{\\rm 121}$ , F. Malek$^{\\rm 55}$ , U. Mallik$^{\\rm 62}$ , D. Malon$^{\\rm 5}$ , C. Malone$^{\\rm 143}$ , S. Maltezos$^{\\rm 9}$ , V. Malyshev$^{\\rm 107}$ , S. Malyukov$^{\\rm 29}$ , R. Mameghani$^{\\rm 98}$ , J. Mamuzic$^{\\rm 12b}$ , A. Manabe$^{\\rm 65}$ , L. Mandelli$^{\\rm 89a}$ , I. Mandić$^{\\rm 74}$ , R. Mandrysch$^{\\rm 15}$ , J. Maneira$^{\\rm 124a}$ , P.S.",
"Mangeard$^{\\rm 88}$ , L. Manhaes de Andrade Filho$^{\\rm 23a}$ , A. Mann$^{\\rm 54}$ , P.M. Manning$^{\\rm 137}$ , A. Manousakis-Katsikakis$^{\\rm 8}$ , B. Mansoulie$^{\\rm 136}$ , A. Mapelli$^{\\rm 29}$ , L. Mapelli$^{\\rm 29}$ , L. March $^{\\rm 80}$ , J.F.",
"Marchand$^{\\rm 28}$ , F. Marchese$^{\\rm 133a,133b}$ , G. Marchiori$^{\\rm 78}$ , M. Marcisovsky$^{\\rm 125}$ , C.P.",
"Marino$^{\\rm 169}$ , F. Marroquim$^{\\rm 23a}$ , Z. Marshall$^{\\rm 29}$ , F.K.",
"Martens$^{\\rm 158}$ , S. Marti-Garcia$^{\\rm 167}$ , B. Martin$^{\\rm 29}$ , B. Martin$^{\\rm 88}$ , J.P. Martin$^{\\rm 93}$ , T.A.",
"Martin$^{\\rm 17}$ , V.J.",
"Martin$^{\\rm 45}$ , B. Martin dit Latour$^{\\rm 49}$ , S. Martin-Haugh$^{\\rm 149}$ , M. Martinez$^{\\rm 11}$ , V. Martinez Outschoorn$^{\\rm 57}$ , A.C. Martyniuk$^{\\rm 169}$ , M. Marx$^{\\rm 82}$ , F. Marzano$^{\\rm 132a}$ , A. Marzin$^{\\rm 111}$ , L. Masetti$^{\\rm 81}$ , T. Mashimo$^{\\rm 155}$ , R. Mashinistov$^{\\rm 94}$ , J. Masik$^{\\rm 82}$ , A.L.",
"Maslennikov$^{\\rm 107}$ , I. Massa$^{\\rm 19a,19b}$ , G. Massaro$^{\\rm 105}$ , N. Massol$^{\\rm 4}$ , P. Mastrandrea$^{\\rm 132a,132b}$ , A. Mastroberardino$^{\\rm 36a,36b}$ , T. Masubuchi$^{\\rm 155}$ , P. Matricon$^{\\rm 115}$ , H. Matsunaga$^{\\rm 155}$ , T. Matsushita$^{\\rm 66}$ , C. Mattravers$^{\\rm 118}$$^{,c}$ , J. Maurer$^{\\rm 83}$ , S.J.",
"Maxfield$^{\\rm 73}$ , A. Mayne$^{\\rm 139}$ , R. Mazini$^{\\rm 151}$ , M. Mazur$^{\\rm 20}$ , L. Mazzaferro$^{\\rm 133a,133b}$ , M. Mazzanti$^{\\rm 89a}$ , S.P.",
"Mc Kee$^{\\rm 87}$ , A. McCarn$^{\\rm 165}$ , R.L.",
"McCarthy$^{\\rm 148}$ , T.G.",
"McCarthy$^{\\rm 28}$ , N.A.",
"McCubbin$^{\\rm 129}$ , K.W.",
"McFarlane$^{\\rm 56}$ , J.A.",
"Mcfayden$^{\\rm 139}$ , H. McGlone$^{\\rm 53}$ , G. Mchedlidze$^{\\rm 51b}$ , T. Mclaughlan$^{\\rm 17}$ , S.J.",
"McMahon$^{\\rm 129}$ , R.A. McPherson$^{\\rm 169}$$^{,k}$ , A. Meade$^{\\rm 84}$ , J. Mechnich$^{\\rm 105}$ , M. Mechtel$^{\\rm 175}$ , M. Medinnis$^{\\rm 41}$ , R. Meera-Lebbai$^{\\rm 111}$ , T. Meguro$^{\\rm 116}$ , R. Mehdiyev$^{\\rm 93}$ , S. Mehlhase$^{\\rm 35}$ , A. Mehta$^{\\rm 73}$ , K. Meier$^{\\rm 58a}$ , B. Meirose$^{\\rm 79}$ , C. Melachrinos$^{\\rm 30}$ , B.R.",
"Mellado Garcia$^{\\rm 173}$ , F. Meloni$^{\\rm 89a,89b}$ , L. Mendoza Navas$^{\\rm 162}$ , Z. Meng$^{\\rm 151}$$^{,u}$ , A. Mengarelli$^{\\rm 19a,19b}$ , S. Menke$^{\\rm 99}$ , E. Meoni$^{\\rm 11}$ , K.M.",
"Mercurio$^{\\rm 57}$ , P. Mermod$^{\\rm 49}$ , L. Merola$^{\\rm 102a,102b}$ , C. Meroni$^{\\rm 89a}$ , F.S.",
"Merritt$^{\\rm 30}$ , H. Merritt$^{\\rm 109}$ , A. Messina$^{\\rm 29}$$^{,y}$ , J. Metcalfe$^{\\rm 103}$ , A.S. Mete$^{\\rm 63}$ , C. Meyer$^{\\rm 81}$ , C. Meyer$^{\\rm 30}$ , J-P. Meyer$^{\\rm 136}$ , J. Meyer$^{\\rm 174}$ , J. Meyer$^{\\rm 54}$ , T.C.",
"Meyer$^{\\rm 29}$ , W.T.",
"Meyer$^{\\rm 63}$ , J. Miao$^{\\rm 32d}$ , S. Michal$^{\\rm 29}$ , L. Micu$^{\\rm 25a}$ , R.P.",
"Middleton$^{\\rm 129}$ , S. Migas$^{\\rm 73}$ , L. Mijović$^{\\rm 41}$ , G. Mikenberg$^{\\rm 172}$ , M. Mikestikova$^{\\rm 125}$ , M. Mikuž$^{\\rm 74}$ , D.W. Miller$^{\\rm 30}$ , R.J. Miller$^{\\rm 88}$ , W.J.",
"Mills$^{\\rm 168}$ , C. Mills$^{\\rm 57}$ , A. Milov$^{\\rm 172}$ , D.A.",
"Milstead$^{\\rm 146a,146b}$ , D. Milstein$^{\\rm 172}$ , A.A. Minaenko$^{\\rm 128}$ , M. Miñano Moya$^{\\rm 167}$ , I.A.",
"Minashvili$^{\\rm 64}$ , A.I.",
"Mincer$^{\\rm 108}$ , B. Mindur$^{\\rm 37}$ , M. Mineev$^{\\rm 64}$ , Y. Ming$^{\\rm 173}$ , L.M.",
"Mir$^{\\rm 11}$ , G. Mirabelli$^{\\rm 132a}$ , A. Misiejuk$^{\\rm 76}$ , J. Mitrevski$^{\\rm 137}$ , V.A.",
"Mitsou$^{\\rm 167}$ , S. Mitsui$^{\\rm 65}$ , P.S.",
"Miyagawa$^{\\rm 139}$ , K. Miyazaki$^{\\rm 66}$ , J.U.",
"Mjörnmark$^{\\rm 79}$ , T. Moa$^{\\rm 146a,146b}$ , S. Moed$^{\\rm 57}$ , V. Moeller$^{\\rm 27}$ , K. Mönig$^{\\rm 41}$ , N. Möser$^{\\rm 20}$ , S. Mohapatra$^{\\rm 148}$ , W. Mohr$^{\\rm 48}$ , R. Moles-Valls$^{\\rm 167}$ , J. Molina-Perez$^{\\rm 29}$ , J. Monk$^{\\rm 77}$ , E. Monnier$^{\\rm 83}$ , S. Montesano$^{\\rm 89a,89b}$ , F. Monticelli$^{\\rm 70}$ , S. Monzani$^{\\rm 19a,19b}$ , R.W.",
"Moore$^{\\rm 2}$ , G.F. Moorhead$^{\\rm 86}$ , C. Mora Herrera$^{\\rm 49}$ , A. Moraes$^{\\rm 53}$ , N. Morange$^{\\rm 136}$ , J. Morel$^{\\rm 54}$ , G. Morello$^{\\rm 36a,36b}$ , D. Moreno$^{\\rm 81}$ , M. Moreno Llácer$^{\\rm 167}$ , P. Morettini$^{\\rm 50a}$ , M. Morgenstern$^{\\rm 43}$ , M. Morii$^{\\rm 57}$ , J. Morin$^{\\rm 75}$ , A.K.",
"Morley$^{\\rm 29}$ , G. Mornacchi$^{\\rm 29}$ , J.D.",
"Morris$^{\\rm 75}$ , L. Morvaj$^{\\rm 101}$ , H.G.",
"Moser$^{\\rm 99}$ , M. Mosidze$^{\\rm 51b}$ , J. Moss$^{\\rm 109}$ , R. Mount$^{\\rm 143}$ , E. Mountricha$^{\\rm 9}$$^{,z}$ , S.V.",
"Mouraviev$^{\\rm 94}$ , E.J.W.",
"Moyse$^{\\rm 84}$ , F. Mueller$^{\\rm 58a}$ , J. Mueller$^{\\rm 123}$ , K. Mueller$^{\\rm 20}$ , T.A.",
"Müller$^{\\rm 98}$ , T. Mueller$^{\\rm 81}$ , D. Muenstermann$^{\\rm 29}$ , Y. Munwes$^{\\rm 153}$ , W.J.",
"Murray$^{\\rm 129}$ , I. Mussche$^{\\rm 105}$ , E. Musto$^{\\rm 102a,102b}$ , A.G. Myagkov$^{\\rm 128}$ , M. Myska$^{\\rm 125}$ , J. Nadal$^{\\rm 11}$ , K. Nagai$^{\\rm 160}$ , K. Nagano$^{\\rm 65}$ , A. Nagarkar$^{\\rm 109}$ , Y. Nagasaka$^{\\rm 59}$ , M. Nagel$^{\\rm 99}$ , A.M. Nairz$^{\\rm 29}$ , Y. Nakahama$^{\\rm 29}$ , K. Nakamura$^{\\rm 155}$ , T. Nakamura$^{\\rm 155}$ , I. Nakano$^{\\rm 110}$ , G. Nanava$^{\\rm 20}$ , A. Napier$^{\\rm 161}$ , R. Narayan$^{\\rm 58b}$ , M. Nash$^{\\rm 77}$$^{,c}$ , T. Nattermann$^{\\rm 20}$ , T. Naumann$^{\\rm 41}$ , G. Navarro$^{\\rm 162}$ , H.A.",
"Neal$^{\\rm 87}$ , P.Yu.",
"Nechaeva$^{\\rm 94}$ , T.J. Neep$^{\\rm 82}$ , A. Negri$^{\\rm 119a,119b}$ , G. Negri$^{\\rm 29}$ , S. Nektarijevic$^{\\rm 49}$ , A. Nelson$^{\\rm 163}$ , T.K.",
"Nelson$^{\\rm 143}$ , S. Nemecek$^{\\rm 125}$ , P. Nemethy$^{\\rm 108}$ , A.A. Nepomuceno$^{\\rm 23a}$ , M. Nessi$^{\\rm 29}$$^{,aa}$ , M.S.",
"Neubauer$^{\\rm 165}$ , A. Neusiedl$^{\\rm 81}$ , R.M.",
"Neves$^{\\rm 108}$ , P. Nevski$^{\\rm 24}$ , P.R.",
"Newman$^{\\rm 17}$ , V. Nguyen Thi Hong$^{\\rm 136}$ , R.B.",
"Nickerson$^{\\rm 118}$ , R. Nicolaidou$^{\\rm 136}$ , L. Nicolas$^{\\rm 139}$ , B. Nicquevert$^{\\rm 29}$ , F. Niedercorn$^{\\rm 115}$ , J. Nielsen$^{\\rm 137}$ , N. Nikiforou$^{\\rm 34}$ , A. Nikiforov$^{\\rm 15}$ , V. Nikolaenko$^{\\rm 128}$ , I. Nikolic-Audit$^{\\rm 78}$ , K. Nikolics$^{\\rm 49}$ , K. Nikolopoulos$^{\\rm 24}$ , H. Nilsen$^{\\rm 48}$ , P. Nilsson$^{\\rm 7}$ , Y. Ninomiya $^{\\rm 155}$ , A. Nisati$^{\\rm 132a}$ , T. Nishiyama$^{\\rm 66}$ , R. Nisius$^{\\rm 99}$ , L. Nodulman$^{\\rm 5}$ , M. Nomachi$^{\\rm 116}$ , I. Nomidis$^{\\rm 154}$ , M. Nordberg$^{\\rm 29}$ , P.R.",
"Norton$^{\\rm 129}$ , J. Novakova$^{\\rm 126}$ , M. Nozaki$^{\\rm 65}$ , L. Nozka$^{\\rm 113}$ , I.M.",
"Nugent$^{\\rm 159a}$ , A.-E. Nuncio-Quiroz$^{\\rm 20}$ , G. Nunes Hanninger$^{\\rm 86}$ , T. Nunnemann$^{\\rm 98}$ , E. Nurse$^{\\rm 77}$ , B.J.",
"O'Brien$^{\\rm 45}$ , S.W.",
"O'Neale$^{\\rm 17}$$^{,*}$ , D.C. O'Neil$^{\\rm 142}$ , V. O'Shea$^{\\rm 53}$ , L.B.",
"Oakes$^{\\rm 98}$ , F.G. Oakham$^{\\rm 28}$$^{,d}$ , H. Oberlack$^{\\rm 99}$ , J. Ocariz$^{\\rm 78}$ , A. Ochi$^{\\rm 66}$ , S. Oda$^{\\rm 155}$ , S. Odaka$^{\\rm 65}$ , J. Odier$^{\\rm 83}$ , H. Ogren$^{\\rm 60}$ , A. Oh$^{\\rm 82}$ , S.H.",
"Oh$^{\\rm 44}$ , C.C.",
"Ohm$^{\\rm 146a,146b}$ , T. Ohshima$^{\\rm 101}$ , S. Okada$^{\\rm 66}$ , H. Okawa$^{\\rm 163}$ , Y. Okumura$^{\\rm 101}$ , T. Okuyama$^{\\rm 155}$ , A. Olariu$^{\\rm 25a}$ , A.G. Olchevski$^{\\rm 64}$ , S.A. Olivares Pino$^{\\rm 31a}$ , M. Oliveira$^{\\rm 124a}$$^{,h}$ , D. Oliveira Damazio$^{\\rm 24}$ , E. Oliver Garcia$^{\\rm 167}$ , D. Olivito$^{\\rm 120}$ , A. Olszewski$^{\\rm 38}$ , J. Olszowska$^{\\rm 38}$ , A. Onofre$^{\\rm 124a}$$^{,ab}$ , P.U.E.",
"Onyisi$^{\\rm 30}$ , C.J.",
"Oram$^{\\rm 159a}$ , M.J. Oreglia$^{\\rm 30}$ , Y. Oren$^{\\rm 153}$ , D. Orestano$^{\\rm 134a,134b}$ , N. Orlando$^{\\rm 72a,72b}$ , I. Orlov$^{\\rm 107}$ , C. Oropeza Barrera$^{\\rm 53}$ , R.S.",
"Orr$^{\\rm 158}$ , B. Osculati$^{\\rm 50a,50b}$ , R. Ospanov$^{\\rm 120}$ , C. Osuna$^{\\rm 11}$ , G. Otero y Garzon$^{\\rm 26}$ , J.P. Ottersbach$^{\\rm 105}$ , M. Ouchrif$^{\\rm 135d}$ , E.A.",
"Ouellette$^{\\rm 169}$ , F. Ould-Saada$^{\\rm 117}$ , A. Ouraou$^{\\rm 136}$ , Q. Ouyang$^{\\rm 32a}$ , A. Ovcharova$^{\\rm 14}$ , M. Owen$^{\\rm 82}$ , S. Owen$^{\\rm 139}$ , V.E.",
"Ozcan$^{\\rm 18a}$ , N. Ozturk$^{\\rm 7}$ , A. Pacheco Pages$^{\\rm 11}$ , C. Padilla Aranda$^{\\rm 11}$ , S. Pagan Griso$^{\\rm 14}$ , E. Paganis$^{\\rm 139}$ , F. Paige$^{\\rm 24}$ , P. Pais$^{\\rm 84}$ , K. Pajchel$^{\\rm 117}$ , G. Palacino$^{\\rm 159b}$ , C.P.",
"Paleari$^{\\rm 6}$ , S. Palestini$^{\\rm 29}$ , D. Pallin$^{\\rm 33}$ , A. Palma$^{\\rm 124a}$ , J.D.",
"Palmer$^{\\rm 17}$ , Y.B.",
"Pan$^{\\rm 173}$ , E. Panagiotopoulou$^{\\rm 9}$ , N. Panikashvili$^{\\rm 87}$ , S. Panitkin$^{\\rm 24}$ , D. Pantea$^{\\rm 25a}$ , A. Papadelis$^{\\rm 146a}$ , Th.D.",
"Papadopoulou$^{\\rm 9}$ , A. Paramonov$^{\\rm 5}$ , D. Paredes Hernandez$^{\\rm 33}$ , W. Park$^{\\rm 24}$$^{,ac}$ , M.A.",
"Parker$^{\\rm 27}$ , F. Parodi$^{\\rm 50a,50b}$ , J.A.",
"Parsons$^{\\rm 34}$ , U. Parzefall$^{\\rm 48}$ , S. Pashapour$^{\\rm 54}$ , E. Pasqualucci$^{\\rm 132a}$ , S. Passaggio$^{\\rm 50a}$ , A. Passeri$^{\\rm 134a}$ , F. Pastore$^{\\rm 134a,134b}$ , Fr.",
"Pastore$^{\\rm 76}$ , G. Pásztor $^{\\rm 49}$$^{,ad}$ , S. Pataraia$^{\\rm 175}$ , N. Patel$^{\\rm 150}$ , J.R. Pater$^{\\rm 82}$ , S. Patricelli$^{\\rm 102a,102b}$ , T. Pauly$^{\\rm 29}$ , M. Pecsy$^{\\rm 144a}$ , M.I.",
"Pedraza Morales$^{\\rm 173}$ , S.V.",
"Peleganchuk$^{\\rm 107}$ , D. Pelikan$^{\\rm 166}$ , H. Peng$^{\\rm 32b}$ , B. Penning$^{\\rm 30}$ , A. Penson$^{\\rm 34}$ , J. Penwell$^{\\rm 60}$ , M. Perantoni$^{\\rm 23a}$ , K. Perez$^{\\rm 34}$$^{,ae}$ , T. Perez Cavalcanti$^{\\rm 41}$ , E. Perez Codina$^{\\rm 159a}$ , M.T.",
"Pérez García-Estañ$^{\\rm 167}$ , V. Perez Reale$^{\\rm 34}$ , L. Perini$^{\\rm 89a,89b}$ , H. Pernegger$^{\\rm 29}$ , R. Perrino$^{\\rm 72a}$ , P. Perrodo$^{\\rm 4}$ , S. Persembe$^{\\rm 3a}$ , V.D.",
"Peshekhonov$^{\\rm 64}$ , K. Peters$^{\\rm 29}$ , B.A.",
"Petersen$^{\\rm 29}$ , J. Petersen$^{\\rm 29}$ , T.C.",
"Petersen$^{\\rm 35}$ , E. Petit$^{\\rm 4}$ , A. Petridis$^{\\rm 154}$ , C. Petridou$^{\\rm 154}$ , E. Petrolo$^{\\rm 132a}$ , F. Petrucci$^{\\rm 134a,134b}$ , D. Petschull$^{\\rm 41}$ , M. Petteni$^{\\rm 142}$ , R. Pezoa$^{\\rm 31b}$ , A. Phan$^{\\rm 86}$ , P.W.",
"Phillips$^{\\rm 129}$ , G. Piacquadio$^{\\rm 29}$ , A. Picazio$^{\\rm 49}$ , E. Piccaro$^{\\rm 75}$ , M. Piccinini$^{\\rm 19a,19b}$ , S.M.",
"Piec$^{\\rm 41}$ , R. Piegaia$^{\\rm 26}$ , D.T.",
"Pignotti$^{\\rm 109}$ , J.E.",
"Pilcher$^{\\rm 30}$ , A.D. Pilkington$^{\\rm 82}$ , J. Pina$^{\\rm 124a}$$^{,b}$ , M. Pinamonti$^{\\rm 164a,164c}$ , A. Pinder$^{\\rm 118}$ , J.L.",
"Pinfold$^{\\rm 2}$ , B. Pinto$^{\\rm 124a}$ , C. Pizio$^{\\rm 89a,89b}$ , M. Plamondon$^{\\rm 169}$ , M.-A.",
"Pleier$^{\\rm 24}$ , E. Plotnikova$^{\\rm 64}$ , A. Poblaguev$^{\\rm 24}$ , S. Poddar$^{\\rm 58a}$ , F. Podlyski$^{\\rm 33}$ , L. Poggioli$^{\\rm 115}$ , T. Poghosyan$^{\\rm 20}$ , M. Pohl$^{\\rm 49}$ , F. Polci$^{\\rm 55}$ , G. Polesello$^{\\rm 119a}$ , A. Policicchio$^{\\rm 36a,36b}$ , A. Polini$^{\\rm 19a}$ , J. Poll$^{\\rm 75}$ , V. Polychronakos$^{\\rm 24}$ , D.M.",
"Pomarede$^{\\rm 136}$ , D. Pomeroy$^{\\rm 22}$ , K. Pommès$^{\\rm 29}$ , L. Pontecorvo$^{\\rm 132a}$ , B.G.",
"Pope$^{\\rm 88}$ , G.A.",
"Popeneciu$^{\\rm 25a}$ , D.S.",
"Popovic$^{\\rm 12a}$ , A. Poppleton$^{\\rm 29}$ , X. Portell Bueso$^{\\rm 29}$ , G.E.",
"Pospelov$^{\\rm 99}$ , S. Pospisil$^{\\rm 127}$ , I.N.",
"Potrap$^{\\rm 99}$ , C.J.",
"Potter$^{\\rm 149}$ , C.T.",
"Potter$^{\\rm 114}$ , G. Poulard$^{\\rm 29}$ , J. Poveda$^{\\rm 173}$ , V. Pozdnyakov$^{\\rm 64}$ , R. Prabhu$^{\\rm 77}$ , P. Pralavorio$^{\\rm 83}$ , A. Pranko$^{\\rm 14}$ , S. Prasad$^{\\rm 29}$ , R. Pravahan$^{\\rm 24}$ , S. Prell$^{\\rm 63}$ , K. Pretzl$^{\\rm 16}$ , D. Price$^{\\rm 60}$ , J. Price$^{\\rm 73}$ , L.E.",
"Price$^{\\rm 5}$ , D. Prieur$^{\\rm 123}$ , M. Primavera$^{\\rm 72a}$ , K. Prokofiev$^{\\rm 108}$ , F. Prokoshin$^{\\rm 31b}$ , S. Protopopescu$^{\\rm 24}$ , J. Proudfoot$^{\\rm 5}$ , X. Prudent$^{\\rm 43}$ , M. Przybycien$^{\\rm 37}$ , H. Przysiezniak$^{\\rm 4}$ , S. Psoroulas$^{\\rm 20}$ , E. Ptacek$^{\\rm 114}$ , E. Pueschel$^{\\rm 84}$ , J. Purdham$^{\\rm 87}$ , M. Purohit$^{\\rm 24}$$^{,ac}$ , P. Puzo$^{\\rm 115}$ , Y. Pylypchenko$^{\\rm 62}$ , J. Qian$^{\\rm 87}$ , Z. Qin$^{\\rm 41}$ , A. Quadt$^{\\rm 54}$ , D.R.",
"Quarrie$^{\\rm 14}$ , W.B.",
"Quayle$^{\\rm 173}$ , F. Quinonez$^{\\rm 31a}$ , M. Raas$^{\\rm 104}$ , V. Radescu$^{\\rm 41}$ , P. Radloff$^{\\rm 114}$ , T. Rador$^{\\rm 18a}$ , F. Ragusa$^{\\rm 89a,89b}$ , G. Rahal$^{\\rm 178}$ , A.M. Rahimi$^{\\rm 109}$ , D. Rahm$^{\\rm 24}$ , S. Rajagopalan$^{\\rm 24}$ , M. Rammensee$^{\\rm 48}$ , M. Rammes$^{\\rm 141}$ , A.S. Randle-Conde$^{\\rm 39}$ , K. Randrianarivony$^{\\rm 28}$ , F. Rauscher$^{\\rm 98}$ , T.C.",
"Rave$^{\\rm 48}$ , M. Raymond$^{\\rm 29}$ , A.L.",
"Read$^{\\rm 117}$ , D.M.",
"Rebuzzi$^{\\rm 119a,119b}$ , A. Redelbach$^{\\rm 174}$ , G. Redlinger$^{\\rm 24}$ , R. Reece$^{\\rm 120}$ , K. Reeves$^{\\rm 40}$ , E. Reinherz-Aronis$^{\\rm 153}$ , A. Reinsch$^{\\rm 114}$ , I. Reisinger$^{\\rm 42}$ , C. Rembser$^{\\rm 29}$ , Z.L.",
"Ren$^{\\rm 151}$ , A. Renaud$^{\\rm 115}$ , M. Rescigno$^{\\rm 132a}$ , S. Resconi$^{\\rm 89a}$ , B. Resende$^{\\rm 136}$ , P. Reznicek$^{\\rm 98}$ , R. Rezvani$^{\\rm 158}$ , R. Richter$^{\\rm 99}$ , E. Richter-Was$^{\\rm 4}$$^{,af}$ , M. Ridel$^{\\rm 78}$ , M. Rijpstra$^{\\rm 105}$ , M. Rijssenbeek$^{\\rm 148}$ , A. Rimoldi$^{\\rm 119a,119b}$ , L. Rinaldi$^{\\rm 19a}$ , R.R.",
"Rios$^{\\rm 39}$ , I. Riu$^{\\rm 11}$ , G. Rivoltella$^{\\rm 89a,89b}$ , F. Rizatdinova$^{\\rm 112}$ , E. Rizvi$^{\\rm 75}$ , S.H.",
"Robertson$^{\\rm 85}$$^{,k}$ , A. Robichaud-Veronneau$^{\\rm 118}$ , D. Robinson$^{\\rm 27}$ , J.E.M.",
"Robinson$^{\\rm 77}$ , A. Robson$^{\\rm 53}$ , J.G.",
"Rocha de Lima$^{\\rm 106}$ , C. Roda$^{\\rm 122a,122b}$ , D. Roda Dos Santos$^{\\rm 29}$ , A. Roe$^{\\rm 54}$ , S. Roe$^{\\rm 29}$ , O. Røhne$^{\\rm 117}$ , S. Rolli$^{\\rm 161}$ , A. Romaniouk$^{\\rm 96}$ , M. Romano$^{\\rm 19a,19b}$ , G. Romeo$^{\\rm 26}$ , E. Romero Adam$^{\\rm 167}$ , L. Roos$^{\\rm 78}$ , E. Ros$^{\\rm 167}$ , S. Rosati$^{\\rm 132a}$ , K. Rosbach$^{\\rm 49}$ , A. Rose$^{\\rm 149}$ , M. Rose$^{\\rm 76}$ , G.A.",
"Rosenbaum$^{\\rm 158}$ , E.I.",
"Rosenberg$^{\\rm 63}$ , P.L.",
"Rosendahl$^{\\rm 13}$ , O. Rosenthal$^{\\rm 141}$ , L. Rosselet$^{\\rm 49}$ , V. Rossetti$^{\\rm 11}$ , E. Rossi$^{\\rm 132a,132b}$ , L.P. Rossi$^{\\rm 50a}$ , M. Rotaru$^{\\rm 25a}$ , I. Roth$^{\\rm 172}$ , J. Rothberg$^{\\rm 138}$ , D. Rousseau$^{\\rm 115}$ , C.R.",
"Royon$^{\\rm 136}$ , A. Rozanov$^{\\rm 83}$ , Y. Rozen$^{\\rm 152}$ , X. Ruan$^{\\rm 32a}$$^{,ag}$ , F. Rubbo$^{\\rm 11}$ , I. Rubinskiy$^{\\rm 41}$ , B. Ruckert$^{\\rm 98}$ , N. Ruckstuhl$^{\\rm 105}$ , V.I.",
"Rud$^{\\rm 97}$ , C. Rudolph$^{\\rm 43}$ , G. Rudolph$^{\\rm 61}$ , F. Rühr$^{\\rm 6}$ , F. Ruggieri$^{\\rm 134a,134b}$ , A. Ruiz-Martinez$^{\\rm 63}$ , L. Rumyantsev$^{\\rm 64}$ , K. Runge$^{\\rm 48}$ , Z. Rurikova$^{\\rm 48}$ , N.A.",
"Rusakovich$^{\\rm 64}$ , J.P. Rutherfoord$^{\\rm 6}$ , C. Ruwiedel$^{\\rm 14}$ , P. Ruzicka$^{\\rm 125}$ , Y.F.",
"Ryabov$^{\\rm 121}$ , P. Ryan$^{\\rm 88}$ , M. Rybar$^{\\rm 126}$ , G. Rybkin$^{\\rm 115}$ , N.C. Ryder$^{\\rm 118}$ , A.F.",
"Saavedra$^{\\rm 150}$ , I. Sadeh$^{\\rm 153}$ , H.F-W. Sadrozinski$^{\\rm 137}$ , R. Sadykov$^{\\rm 64}$ , F. Safai Tehrani$^{\\rm 132a}$ , H. Sakamoto$^{\\rm 155}$ , G. Salamanna$^{\\rm 75}$ , A. Salamon$^{\\rm 133a}$ , M. Saleem$^{\\rm 111}$ , D. Salek$^{\\rm 29}$ , D. Salihagic$^{\\rm 99}$ , A. Salnikov$^{\\rm 143}$ , J. Salt$^{\\rm 167}$ , B.M.",
"Salvachua Ferrando$^{\\rm 5}$ , D. Salvatore$^{\\rm 36a,36b}$ , F. Salvatore$^{\\rm 149}$ , A. Salvucci$^{\\rm 104}$ , A. Salzburger$^{\\rm 29}$ , D. Sampsonidis$^{\\rm 154}$ , B.H.",
"Samset$^{\\rm 117}$ , A. Sanchez$^{\\rm 102a,102b}$ , V. Sanchez Martinez$^{\\rm 167}$ , H. Sandaker$^{\\rm 13}$ , H.G.",
"Sander$^{\\rm 81}$ , M.P.",
"Sanders$^{\\rm 98}$ , M. Sandhoff$^{\\rm 175}$ , T. Sandoval$^{\\rm 27}$ , C. Sandoval $^{\\rm 162}$ , R. Sandstroem$^{\\rm 99}$ , D.P.C.",
"Sankey$^{\\rm 129}$ , A. Sansoni$^{\\rm 47}$ , C. Santamarina Rios$^{\\rm 85}$ , C. Santoni$^{\\rm 33}$ , R. Santonico$^{\\rm 133a,133b}$ , H. Santos$^{\\rm 124a}$ , J.G.",
"Saraiva$^{\\rm 124a}$ , T. Sarangi$^{\\rm 173}$ , E. Sarkisyan-Grinbaum$^{\\rm 7}$ , F. Sarri$^{\\rm 122a,122b}$ , G. Sartisohn$^{\\rm 175}$ , O. Sasaki$^{\\rm 65}$ , N. Sasao$^{\\rm 67}$ , I. Satsounkevitch$^{\\rm 90}$ , G. Sauvage$^{\\rm 4}$ , E. Sauvan$^{\\rm 4}$ , J.B. Sauvan$^{\\rm 115}$ , P. Savard$^{\\rm 158}$$^{,d}$ , V. Savinov$^{\\rm 123}$ , D.O.",
"Savu$^{\\rm 29}$ , L. Sawyer$^{\\rm 24}$$^{,m}$ , D.H. Saxon$^{\\rm 53}$ , J. Saxon$^{\\rm 120}$ , C. Sbarra$^{\\rm 19a}$ , A. Sbrizzi$^{\\rm 19a,19b}$ , O. Scallon$^{\\rm 93}$ , D.A.",
"Scannicchio$^{\\rm 163}$ , M. Scarcella$^{\\rm 150}$ , J. Schaarschmidt$^{\\rm 115}$ , P. Schacht$^{\\rm 99}$ , D. Schaefer$^{\\rm 120}$ , U. Schäfer$^{\\rm 81}$ , S. Schaepe$^{\\rm 20}$ , S. Schaetzel$^{\\rm 58b}$ , A.C. Schaffer$^{\\rm 115}$ , D. Schaile$^{\\rm 98}$ , R.D.",
"Schamberger$^{\\rm 148}$ , A.G. Schamov$^{\\rm 107}$ , V. Scharf$^{\\rm 58a}$ , V.A.",
"Schegelsky$^{\\rm 121}$ , D. Scheirich$^{\\rm 87}$ , M. Schernau$^{\\rm 163}$ , M.I.",
"Scherzer$^{\\rm 34}$ , C. Schiavi$^{\\rm 50a,50b}$ , J. Schieck$^{\\rm 98}$ , M. Schioppa$^{\\rm 36a,36b}$ , S. Schlenker$^{\\rm 29}$ , E. Schmidt$^{\\rm 48}$ , K. Schmieden$^{\\rm 20}$ , C. Schmitt$^{\\rm 81}$ , S. Schmitt$^{\\rm 58b}$ , M. Schmitz$^{\\rm 20}$ , A. Schöning$^{\\rm 58b}$ , M. Schott$^{\\rm 29}$ , D. Schouten$^{\\rm 159a}$ , J. Schovancova$^{\\rm 125}$ , M. Schram$^{\\rm 85}$ , C. Schroeder$^{\\rm 81}$ , N. Schroer$^{\\rm 58c}$ , M.J. Schultens$^{\\rm 20}$ , J. Schultes$^{\\rm 175}$ , H.-C. Schultz-Coulon$^{\\rm 58a}$ , H. Schulz$^{\\rm 15}$ , J.W.",
"Schumacher$^{\\rm 20}$ , M. Schumacher$^{\\rm 48}$ , B.A.",
"Schumm$^{\\rm 137}$ , Ph.",
"Schune$^{\\rm 136}$ , C. Schwanenberger$^{\\rm 82}$ , A. Schwartzman$^{\\rm 143}$ , Ph.",
"Schwemling$^{\\rm 78}$ , R. Schwienhorst$^{\\rm 88}$ , R. Schwierz$^{\\rm 43}$ , J. Schwindling$^{\\rm 136}$ , T. Schwindt$^{\\rm 20}$ , M. Schwoerer$^{\\rm 4}$ , G. Sciolla$^{\\rm 22}$ , W.G.",
"Scott$^{\\rm 129}$ , J. Searcy$^{\\rm 114}$ , G. Sedov$^{\\rm 41}$ , E. Sedykh$^{\\rm 121}$ , S.C. Seidel$^{\\rm 103}$ , A. Seiden$^{\\rm 137}$ , F. Seifert$^{\\rm 43}$ , J.M.",
"Seixas$^{\\rm 23a}$ , G. Sekhniaidze$^{\\rm 102a}$ , S.J.",
"Sekula$^{\\rm 39}$ , K.E.",
"Selbach$^{\\rm 45}$ , D.M.",
"Seliverstov$^{\\rm 121}$ , B. Sellden$^{\\rm 146a}$ , G. Sellers$^{\\rm 73}$ , M. Seman$^{\\rm 144b}$ , N. Semprini-Cesari$^{\\rm 19a,19b}$ , C. Serfon$^{\\rm 98}$ , L. Serin$^{\\rm 115}$ , L. Serkin$^{\\rm 54}$ , R. Seuster$^{\\rm 99}$ , H. Severini$^{\\rm 111}$ , A. Sfyrla$^{\\rm 29}$ , E. Shabalina$^{\\rm 54}$ , M. Shamim$^{\\rm 114}$ , L.Y.",
"Shan$^{\\rm 32a}$ , J.T.",
"Shank$^{\\rm 21}$ , Q.T.",
"Shao$^{\\rm 86}$ , M. Shapiro$^{\\rm 14}$ , P.B.",
"Shatalov$^{\\rm 95}$ , K. Shaw$^{\\rm 164a,164c}$ , D. Sherman$^{\\rm 176}$ , P. Sherwood$^{\\rm 77}$ , A. Shibata$^{\\rm 108}$ , H. Shichi$^{\\rm 101}$ , S. Shimizu$^{\\rm 29}$ , M. Shimojima$^{\\rm 100}$ , T. Shin$^{\\rm 56}$ , M. Shiyakova$^{\\rm 64}$ , A. Shmeleva$^{\\rm 94}$ , M.J. Shochet$^{\\rm 30}$ , D. Short$^{\\rm 118}$ , S. Shrestha$^{\\rm 63}$ , E. Shulga$^{\\rm 96}$ , M.A.",
"Shupe$^{\\rm 6}$ , P. Sicho$^{\\rm 125}$ , A. Sidoti$^{\\rm 132a}$ , F. Siegert$^{\\rm 48}$ , Dj.",
"Sijacki$^{\\rm 12a}$ , O. Silbert$^{\\rm 172}$ , J. Silva$^{\\rm 124a}$ , Y. Silver$^{\\rm 153}$ , D. Silverstein$^{\\rm 143}$ , S.B.",
"Silverstein$^{\\rm 146a}$ , V. Simak$^{\\rm 127}$ , O. Simard$^{\\rm 136}$ , Lj.",
"Simic$^{\\rm 12a}$ , S. Simion$^{\\rm 115}$ , B. Simmons$^{\\rm 77}$ , R. Simoniello$^{\\rm 89a,89b}$ , M. Simonyan$^{\\rm 35}$ , P. Sinervo$^{\\rm 158}$ , N.B.",
"Sinev$^{\\rm 114}$ , V. Sipica$^{\\rm 141}$ , G. Siragusa$^{\\rm 174}$ , A. Sircar$^{\\rm 24}$ , A.N.",
"Sisakyan$^{\\rm 64}$ , S.Yu.",
"Sivoklokov$^{\\rm 97}$ , J. Sjölin$^{\\rm 146a,146b}$ , T.B.",
"Sjursen$^{\\rm 13}$ , L.A. Skinnari$^{\\rm 14}$ , H.P.",
"Skottowe$^{\\rm 57}$ , K. Skovpen$^{\\rm 107}$ , P. Skubic$^{\\rm 111}$ , M. Slater$^{\\rm 17}$ , T. Slavicek$^{\\rm 127}$ , K. Sliwa$^{\\rm 161}$ , V. Smakhtin$^{\\rm 172}$ , B.H.",
"Smart$^{\\rm 45}$ , S.Yu.",
"Smirnov$^{\\rm 96}$ , Y. Smirnov$^{\\rm 96}$ , L.N.",
"Smirnova$^{\\rm 97}$ , O. Smirnova$^{\\rm 79}$ , B.C.",
"Smith$^{\\rm 57}$ , D. Smith$^{\\rm 143}$ , K.M.",
"Smith$^{\\rm 53}$ , M. Smizanska$^{\\rm 71}$ , K. Smolek$^{\\rm 127}$ , A.A. Snesarev$^{\\rm 94}$ , S.W.",
"Snow$^{\\rm 82}$ , J. Snow$^{\\rm 111}$ , S. Snyder$^{\\rm 24}$ , R. Sobie$^{\\rm 169}$$^{,k}$ , J. Sodomka$^{\\rm 127}$ , A. Soffer$^{\\rm 153}$ , C.A.",
"Solans$^{\\rm 167}$ , M. Solar$^{\\rm 127}$ , J. Solc$^{\\rm 127}$ , E.Yu.",
"Soldatov$^{\\rm 96}$ , U. Soldevila$^{\\rm 167}$ , E. Solfaroli Camillocci$^{\\rm 132a,132b}$ , A.A. Solodkov$^{\\rm 128}$ , O.V.",
"Solovyanov$^{\\rm 128}$ , N. Soni$^{\\rm 2}$ , V. Sopko$^{\\rm 127}$ , B. Sopko$^{\\rm 127}$ , M. Sosebee$^{\\rm 7}$ , R. Soualah$^{\\rm 164a,164c}$ , A. Soukharev$^{\\rm 107}$ , S. Spagnolo$^{\\rm 72a,72b}$ , F. Spanò$^{\\rm 76}$ , R. Spighi$^{\\rm 19a}$ , G. Spigo$^{\\rm 29}$ , F. Spila$^{\\rm 132a,132b}$ , R. Spiwoks$^{\\rm 29}$ , M. Spousta$^{\\rm 126}$ , T. Spreitzer$^{\\rm 158}$ , B. Spurlock$^{\\rm 7}$ , R.D. St.",
"Denis$^{\\rm 53}$ , J. Stahlman$^{\\rm 120}$ , R. Stamen$^{\\rm 58a}$ , E. Stanecka$^{\\rm 38}$ , R.W.",
"Stanek$^{\\rm 5}$ , C. Stanescu$^{\\rm 134a}$ , M. Stanescu-Bellu$^{\\rm 41}$ , S. Stapnes$^{\\rm 117}$ , E.A.",
"Starchenko$^{\\rm 128}$ , J. Stark$^{\\rm 55}$ , P. Staroba$^{\\rm 125}$ , P. Starovoitov$^{\\rm 41}$ , A. Staude$^{\\rm 98}$ , P. Stavina$^{\\rm 144a}$ , G. Steele$^{\\rm 53}$ , P. Steinbach$^{\\rm 43}$ , P. Steinberg$^{\\rm 24}$ , I. Stekl$^{\\rm 127}$ , B. Stelzer$^{\\rm 142}$ , H.J.",
"Stelzer$^{\\rm 88}$ , O. Stelzer-Chilton$^{\\rm 159a}$ , H. Stenzel$^{\\rm 52}$ , S. Stern$^{\\rm 99}$ , G.A.",
"Stewart$^{\\rm 29}$ , J.A.",
"Stillings$^{\\rm 20}$ , M.C.",
"Stockton$^{\\rm 85}$ , K. Stoerig$^{\\rm 48}$ , G. Stoicea$^{\\rm 25a}$ , S. Stonjek$^{\\rm 99}$ , P. Strachota$^{\\rm 126}$ , A.R.",
"Stradling$^{\\rm 7}$ , A. Straessner$^{\\rm 43}$ , J. Strandberg$^{\\rm 147}$ , S. Strandberg$^{\\rm 146a,146b}$ , A. Strandlie$^{\\rm 117}$ , M. Strang$^{\\rm 109}$ , E. Strauss$^{\\rm 143}$ , M. Strauss$^{\\rm 111}$ , P. Strizenec$^{\\rm 144b}$ , R. Ströhmer$^{\\rm 174}$ , D.M.",
"Strom$^{\\rm 114}$ , J.A.",
"Strong$^{\\rm 76}$$^{,*}$ , R. Stroynowski$^{\\rm 39}$ , J. Strube$^{\\rm 129}$ , B. Stugu$^{\\rm 13}$ , I. Stumer$^{\\rm 24}$$^{,*}$ , J. Stupak$^{\\rm 148}$ , P. Sturm$^{\\rm 175}$ , N.A.",
"Styles$^{\\rm 41}$ , D.A.",
"Soh$^{\\rm 151}$$^{,w}$ , D. Su$^{\\rm 143}$ , HS.",
"Subramania$^{\\rm 2}$ , A. Succurro$^{\\rm 11}$ , Y. Sugaya$^{\\rm 116}$ , C. Suhr$^{\\rm 106}$ , K. Suita$^{\\rm 66}$ , M. Suk$^{\\rm 126}$ , V.V.",
"Sulin$^{\\rm 94}$ , S. Sultansoy$^{\\rm 3d}$ , T. Sumida$^{\\rm 67}$ , X. Sun$^{\\rm 55}$ , J.E.",
"Sundermann$^{\\rm 48}$ , K. Suruliz$^{\\rm 139}$ , G. Susinno$^{\\rm 36a,36b}$ , M.R.",
"Sutton$^{\\rm 149}$ , Y. Suzuki$^{\\rm 65}$ , Y. Suzuki$^{\\rm 66}$ , M. Svatos$^{\\rm 125}$ , S. Swedish$^{\\rm 168}$ , I. Sykora$^{\\rm 144a}$ , T. Sykora$^{\\rm 126}$ , J. Sánchez$^{\\rm 167}$ , D. Ta$^{\\rm 105}$ , K. Tackmann$^{\\rm 41}$ , A. Taffard$^{\\rm 163}$ , R. Tafirout$^{\\rm 159a}$ , N. Taiblum$^{\\rm 153}$ , Y. Takahashi$^{\\rm 101}$ , H. Takai$^{\\rm 24}$ , R. Takashima$^{\\rm 68}$ , H. Takeda$^{\\rm 66}$ , T. Takeshita$^{\\rm 140}$ , Y. Takubo$^{\\rm 65}$ , M. Talby$^{\\rm 83}$ , A. Talyshev$^{\\rm 107}$$^{,f}$ , M.C.",
"Tamsett$^{\\rm 24}$ , J. Tanaka$^{\\rm 155}$ , R. Tanaka$^{\\rm 115}$ , S. Tanaka$^{\\rm 131}$ , S. Tanaka$^{\\rm 65}$ , A.J.",
"Tanasijczuk$^{\\rm 142}$ , K. Tani$^{\\rm 66}$ , N. Tannoury$^{\\rm 83}$ , S. Tapprogge$^{\\rm 81}$ , D. Tardif$^{\\rm 158}$ , S. Tarem$^{\\rm 152}$ , F. Tarrade$^{\\rm 28}$ , G.F. Tartarelli$^{\\rm 89a}$ , P. Tas$^{\\rm 126}$ , M. Tasevsky$^{\\rm 125}$ , E. Tassi$^{\\rm 36a,36b}$ , M. Tatarkhanov$^{\\rm 14}$ , Y. Tayalati$^{\\rm 135d}$ , C. Taylor$^{\\rm 77}$ , F.E.",
"Taylor$^{\\rm 92}$ , G.N.",
"Taylor$^{\\rm 86}$ , W. Taylor$^{\\rm 159b}$ , M. Teinturier$^{\\rm 115}$ , M. Teixeira Dias Castanheira$^{\\rm 75}$ , P. Teixeira-Dias$^{\\rm 76}$ , K.K.",
"Temming$^{\\rm 48}$ , H. Ten Kate$^{\\rm 29}$ , P.K.",
"Teng$^{\\rm 151}$ , S. Terada$^{\\rm 65}$ , K. Terashi$^{\\rm 155}$ , J. Terron$^{\\rm 80}$ , M. Testa$^{\\rm 47}$ , R.J. Teuscher$^{\\rm 158}$$^{,k}$ , J. Therhaag$^{\\rm 20}$ , T. Theveneaux-Pelzer$^{\\rm 78}$ , M. Thioye$^{\\rm 176}$ , S. Thoma$^{\\rm 48}$ , J.P. Thomas$^{\\rm 17}$ , E.N.",
"Thompson$^{\\rm 34}$ , P.D.",
"Thompson$^{\\rm 17}$ , P.D.",
"Thompson$^{\\rm 158}$ , A.S. Thompson$^{\\rm 53}$ , L.A. Thomsen$^{\\rm 35}$ , E. Thomson$^{\\rm 120}$ , M. Thomson$^{\\rm 27}$ , R.P.",
"Thun$^{\\rm 87}$ , F. Tian$^{\\rm 34}$ , M.J. Tibbetts$^{\\rm 14}$ , T. Tic$^{\\rm 125}$ , V.O.",
"Tikhomirov$^{\\rm 94}$ , Y.A.",
"Tikhonov$^{\\rm 107}$$^{,f}$ , S. Timoshenko$^{\\rm 96}$ , P. Tipton$^{\\rm 176}$ , F.J. Tique Aires Viegas$^{\\rm 29}$ , S. Tisserant$^{\\rm 83}$ , T. Todorov$^{\\rm 4}$ , S. Todorova-Nova$^{\\rm 161}$ , B. Toggerson$^{\\rm 163}$ , J. Tojo$^{\\rm 69}$ , S. Tokár$^{\\rm 144a}$ , K. Tokunaga$^{\\rm 66}$ , K. Tokushuku$^{\\rm 65}$ , K. Tollefson$^{\\rm 88}$ , M. Tomoto$^{\\rm 101}$ , L. Tompkins$^{\\rm 30}$ , K. Toms$^{\\rm 103}$ , A. Tonoyan$^{\\rm 13}$ , C. Topfel$^{\\rm 16}$ , N.D. Topilin$^{\\rm 64}$ , I. Torchiani$^{\\rm 29}$ , E. Torrence$^{\\rm 114}$ , H. Torres$^{\\rm 78}$ , E. Torró Pastor$^{\\rm 167}$ , J. Toth$^{\\rm 83}$$^{,ad}$ , F. Touchard$^{\\rm 83}$ , D.R.",
"Tovey$^{\\rm 139}$ , T. Trefzger$^{\\rm 174}$ , L. Tremblet$^{\\rm 29}$ , A. Tricoli$^{\\rm 29}$ , I.M.",
"Trigger$^{\\rm 159a}$ , S. Trincaz-Duvoid$^{\\rm 78}$ , M.F.",
"Tripiana$^{\\rm 70}$ , W. Trischuk$^{\\rm 158}$ , B. Trocmé$^{\\rm 55}$ , C. Troncon$^{\\rm 89a}$ , M. Trottier-McDonald$^{\\rm 142}$ , M. Trzebinski$^{\\rm 38}$ , A. Trzupek$^{\\rm 38}$ , C. Tsarouchas$^{\\rm 29}$ , J.C-L. Tseng$^{\\rm 118}$ , M. Tsiakiris$^{\\rm 105}$ , P.V.",
"Tsiareshka$^{\\rm 90}$ , D. Tsionou$^{\\rm 4}$$^{,ah}$ , G. Tsipolitis$^{\\rm 9}$ , V. Tsiskaridze$^{\\rm 48}$ , E.G.",
"Tskhadadze$^{\\rm 51a}$ , I.I.",
"Tsukerman$^{\\rm 95}$ , V. Tsulaia$^{\\rm 14}$ , J.-W. Tsung$^{\\rm 20}$ , S. Tsuno$^{\\rm 65}$ , D. Tsybychev$^{\\rm 148}$ , A. Tua$^{\\rm 139}$ , A. Tudorache$^{\\rm 25a}$ , V. Tudorache$^{\\rm 25a}$ , J.M.",
"Tuggle$^{\\rm 30}$ , M. Turala$^{\\rm 38}$ , D. Turecek$^{\\rm 127}$ , I. Turk Cakir$^{\\rm 3e}$ , E. Turlay$^{\\rm 105}$ , R. Turra$^{\\rm 89a,89b}$ , P.M. Tuts$^{\\rm 34}$ , A. Tykhonov$^{\\rm 74}$ , M. Tylmad$^{\\rm 146a,146b}$ , M. Tyndel$^{\\rm 129}$ , G. Tzanakos$^{\\rm 8}$ , K. Uchida$^{\\rm 20}$ , I. Ueda$^{\\rm 155}$ , R. Ueno$^{\\rm 28}$ , M. Ugland$^{\\rm 13}$ , M. Uhlenbrock$^{\\rm 20}$ , M. Uhrmacher$^{\\rm 54}$ , F. Ukegawa$^{\\rm 160}$ , G. Unal$^{\\rm 29}$ , A. Undrus$^{\\rm 24}$ , G. Unel$^{\\rm 163}$ , Y. Unno$^{\\rm 65}$ , D. Urbaniec$^{\\rm 34}$ , G. Usai$^{\\rm 7}$ , M. Uslenghi$^{\\rm 119a,119b}$ , L. Vacavant$^{\\rm 83}$ , V. Vacek$^{\\rm 127}$ , B. Vachon$^{\\rm 85}$ , S. Vahsen$^{\\rm 14}$ , J. Valenta$^{\\rm 125}$ , P. Valente$^{\\rm 132a}$ , S. Valentinetti$^{\\rm 19a,19b}$ , S. Valkar$^{\\rm 126}$ , E. Valladolid Gallego$^{\\rm 167}$ , S. Vallecorsa$^{\\rm 152}$ , J.A.",
"Valls Ferrer$^{\\rm 167}$ , H. van der Graaf$^{\\rm 105}$ , E. van der Kraaij$^{\\rm 105}$ , R. Van Der Leeuw$^{\\rm 105}$ , E. van der Poel$^{\\rm 105}$ , D. van der Ster$^{\\rm 29}$ , N. van Eldik$^{\\rm 29}$ , P. van Gemmeren$^{\\rm 5}$ , I. van Vulpen$^{\\rm 105}$ , M. Vanadia$^{\\rm 99}$ , W. Vandelli$^{\\rm 29}$ , A. Vaniachine$^{\\rm 5}$ , P. Vankov$^{\\rm 41}$ , F. Vannucci$^{\\rm 78}$ , R. Vari$^{\\rm 132a}$ , T. Varol$^{\\rm 84}$ , D. Varouchas$^{\\rm 14}$ , A. Vartapetian$^{\\rm 7}$ , K.E.",
"Varvell$^{\\rm 150}$ , V.I.",
"Vassilakopoulos$^{\\rm 56}$ , F. Vazeille$^{\\rm 33}$ , T. Vazquez Schroeder$^{\\rm 54}$ , G. Vegni$^{\\rm 89a,89b}$ , J.J. Veillet$^{\\rm 115}$ , F. Veloso$^{\\rm 124a}$ , R. Veness$^{\\rm 29}$ , S. Veneziano$^{\\rm 132a}$ , A. Ventura$^{\\rm 72a,72b}$ , D. Ventura$^{\\rm 84}$ , M. Venturi$^{\\rm 48}$ , N. Venturi$^{\\rm 158}$ , V. Vercesi$^{\\rm 119a}$ , M. Verducci$^{\\rm 138}$ , W. Verkerke$^{\\rm 105}$ , J.C. Vermeulen$^{\\rm 105}$ , A. Vest$^{\\rm 43}$ , M.C.",
"Vetterli$^{\\rm 142}$$^{,d}$ , I. Vichou$^{\\rm 165}$ , T. Vickey$^{\\rm 145b}$$^{,ai}$ , O.E.",
"Vickey Boeriu$^{\\rm 145b}$ , G.H.A.",
"Viehhauser$^{\\rm 118}$ , S. Viel$^{\\rm 168}$ , M. Villa$^{\\rm 19a,19b}$ , M. Villaplana Perez$^{\\rm 167}$ , E. Vilucchi$^{\\rm 47}$ , M.G.",
"Vincter$^{\\rm 28}$ , E. Vinek$^{\\rm 29}$ , V.B.",
"Vinogradov$^{\\rm 64}$ , M. Virchaux$^{\\rm 136}$$^{,*}$ , J. Virzi$^{\\rm 14}$ , O. Vitells$^{\\rm 172}$ , M. Viti$^{\\rm 41}$ , I. Vivarelli$^{\\rm 48}$ , F. Vives Vaque$^{\\rm 2}$ , S. Vlachos$^{\\rm 9}$ , D. Vladoiu$^{\\rm 98}$ , M. Vlasak$^{\\rm 127}$ , A. Vogel$^{\\rm 20}$ , P. Vokac$^{\\rm 127}$ , G. Volpi$^{\\rm 47}$ , M. Volpi$^{\\rm 86}$ , G. Volpini$^{\\rm 89a}$ , H. von der Schmitt$^{\\rm 99}$ , J. von Loeben$^{\\rm 99}$ , H. von Radziewski$^{\\rm 48}$ , E. von Toerne$^{\\rm 20}$ , V. Vorobel$^{\\rm 126}$ , V. Vorwerk$^{\\rm 11}$ , M. Vos$^{\\rm 167}$ , R. Voss$^{\\rm 29}$ , T.T.",
"Voss$^{\\rm 175}$ , J.H.",
"Vossebeld$^{\\rm 73}$ , N. Vranjes$^{\\rm 136}$ , M. Vranjes Milosavljevic$^{\\rm 105}$ , V. Vrba$^{\\rm 125}$ , M. Vreeswijk$^{\\rm 105}$ , T. Vu Anh$^{\\rm 48}$ , R. Vuillermet$^{\\rm 29}$ , I. Vukotic$^{\\rm 115}$ , W. Wagner$^{\\rm 175}$ , P. Wagner$^{\\rm 120}$ , H. Wahlen$^{\\rm 175}$ , S. Wahrmund$^{\\rm 43}$ , J. Wakabayashi$^{\\rm 101}$ , S. Walch$^{\\rm 87}$ , J. Walder$^{\\rm 71}$ , R. Walker$^{\\rm 98}$ , W. Walkowiak$^{\\rm 141}$ , R. Wall$^{\\rm 176}$ , P. Waller$^{\\rm 73}$ , C. Wang$^{\\rm 44}$ , H. Wang$^{\\rm 173}$ , H. Wang$^{\\rm 32b}$$^{,aj}$ , J. Wang$^{\\rm 151}$ , J. Wang$^{\\rm 55}$ , R. Wang$^{\\rm 103}$ , S.M.",
"Wang$^{\\rm 151}$ , T. Wang$^{\\rm 20}$ , A. Warburton$^{\\rm 85}$ , C.P.",
"Ward$^{\\rm 27}$ , M. Warsinsky$^{\\rm 48}$ , A. Washbrook$^{\\rm 45}$ , C. Wasicki$^{\\rm 41}$ , P.M. Watkins$^{\\rm 17}$ , A.T. Watson$^{\\rm 17}$ , I.J.",
"Watson$^{\\rm 150}$ , M.F.",
"Watson$^{\\rm 17}$ , G. Watts$^{\\rm 138}$ , S. Watts$^{\\rm 82}$ , A.T. Waugh$^{\\rm 150}$ , B.M.",
"Waugh$^{\\rm 77}$ , M. Weber$^{\\rm 129}$ , M.S.",
"Weber$^{\\rm 16}$ , P. Weber$^{\\rm 54}$ , A.R.",
"Weidberg$^{\\rm 118}$ , P. Weigell$^{\\rm 99}$ , J. Weingarten$^{\\rm 54}$ , C. Weiser$^{\\rm 48}$ , H. Wellenstein$^{\\rm 22}$ , P.S.",
"Wells$^{\\rm 29}$ , T. Wenaus$^{\\rm 24}$ , D. Wendland$^{\\rm 15}$ , Z. Weng$^{\\rm 151}$$^{,w}$ , T. Wengler$^{\\rm 29}$ , S. Wenig$^{\\rm 29}$ , N. Wermes$^{\\rm 20}$ , M. Werner$^{\\rm 48}$ , P. Werner$^{\\rm 29}$ , M. Werth$^{\\rm 163}$ , M. Wessels$^{\\rm 58a}$ , J. Wetter$^{\\rm 161}$ , C. Weydert$^{\\rm 55}$ , K. Whalen$^{\\rm 28}$ , S.J.",
"Wheeler-Ellis$^{\\rm 163}$ , A. White$^{\\rm 7}$ , M.J. White$^{\\rm 86}$ , S. White$^{\\rm 122a,122b}$ , S.R.",
"Whitehead$^{\\rm 118}$ , D. Whiteson$^{\\rm 163}$ , D. Whittington$^{\\rm 60}$ , F. Wicek$^{\\rm 115}$ , D. Wicke$^{\\rm 175}$ , F.J. Wickens$^{\\rm 129}$ , W. Wiedenmann$^{\\rm 173}$ , M. Wielers$^{\\rm 129}$ , P. Wienemann$^{\\rm 20}$ , C. Wiglesworth$^{\\rm 75}$ , L.A.M.",
"Wiik-Fuchs$^{\\rm 48}$ , P.A.",
"Wijeratne$^{\\rm 77}$ , A. Wildauer$^{\\rm 167}$ , M.A.",
"Wildt$^{\\rm 41}$$^{,s}$ , I. Wilhelm$^{\\rm 126}$ , H.G.",
"Wilkens$^{\\rm 29}$ , J.Z.",
"Will$^{\\rm 98}$ , E. Williams$^{\\rm 34}$ , H.H.",
"Williams$^{\\rm 120}$ , W. Willis$^{\\rm 34}$ , S. Willocq$^{\\rm 84}$ , J.A.",
"Wilson$^{\\rm 17}$ , M.G.",
"Wilson$^{\\rm 143}$ , A. Wilson$^{\\rm 87}$ , I. Wingerter-Seez$^{\\rm 4}$ , S. Winkelmann$^{\\rm 48}$ , F. Winklmeier$^{\\rm 29}$ , M. Wittgen$^{\\rm 143}$ , M.W.",
"Wolter$^{\\rm 38}$ , H. Wolters$^{\\rm 124a}$$^{,h}$ , W.C. Wong$^{\\rm 40}$ , G. Wooden$^{\\rm 87}$ , B.K.",
"Wosiek$^{\\rm 38}$ , J. Wotschack$^{\\rm 29}$ , M.J. Woudstra$^{\\rm 82}$ , K.W.",
"Wozniak$^{\\rm 38}$ , K. Wraight$^{\\rm 53}$ , C. Wright$^{\\rm 53}$ , M. Wright$^{\\rm 53}$ , B. Wrona$^{\\rm 73}$ , S.L.",
"Wu$^{\\rm 173}$ , X. Wu$^{\\rm 49}$ , Y. Wu$^{\\rm 32b}$$^{,ak}$ , E. Wulf$^{\\rm 34}$ , B.M.",
"Wynne$^{\\rm 45}$ , S. Xella$^{\\rm 35}$ , M. Xiao$^{\\rm 136}$ , S. Xie$^{\\rm 48}$ , C. Xu$^{\\rm 32b}$$^{,z}$ , D. Xu$^{\\rm 139}$ , B. Yabsley$^{\\rm 150}$ , S. Yacoob$^{\\rm 145b}$ , M. Yamada$^{\\rm 65}$ , H. Yamaguchi$^{\\rm 155}$ , A. Yamamoto$^{\\rm 65}$ , K. Yamamoto$^{\\rm 63}$ , S. Yamamoto$^{\\rm 155}$ , T. Yamamura$^{\\rm 155}$ , T. Yamanaka$^{\\rm 155}$ , J. Yamaoka$^{\\rm 44}$ , T. Yamazaki$^{\\rm 155}$ , Y. Yamazaki$^{\\rm 66}$ , Z. Yan$^{\\rm 21}$ , H. Yang$^{\\rm 87}$ , U.K. Yang$^{\\rm 82}$ , Y. Yang$^{\\rm 60}$ , Z. Yang$^{\\rm 146a,146b}$ , S. Yanush$^{\\rm 91}$ , L. Yao$^{\\rm 32a}$ , Y. Yao$^{\\rm 14}$ , Y. Yasu$^{\\rm 65}$ , G.V.",
"Ybeles Smit$^{\\rm 130}$ , J. Ye$^{\\rm 39}$ , S. Ye$^{\\rm 24}$ , M. Yilmaz$^{\\rm 3c}$ , R. Yoosoofmiya$^{\\rm 123}$ , K. Yorita$^{\\rm 171}$ , R. Yoshida$^{\\rm 5}$ , C. Young$^{\\rm 143}$ , C.J.",
"Young$^{\\rm 118}$ , S. Youssef$^{\\rm 21}$ , D. Yu$^{\\rm 24}$ , J. Yu$^{\\rm 7}$ , J. Yu$^{\\rm 112}$ , L. Yuan$^{\\rm 66}$ , A. Yurkewicz$^{\\rm 106}$ , B. Zabinski$^{\\rm 38}$ , R. Zaidan$^{\\rm 62}$ , A.M. Zaitsev$^{\\rm 128}$ , Z. Zajacova$^{\\rm 29}$ , L. Zanello$^{\\rm 132a,132b}$ , A. Zaytsev$^{\\rm 107}$ , C. Zeitnitz$^{\\rm 175}$ , M. Zeman$^{\\rm 125}$ , A. Zemla$^{\\rm 38}$ , C. Zendler$^{\\rm 20}$ , O. Zenin$^{\\rm 128}$ , T. Ženiš$^{\\rm 144a}$ , Z. Zinonos$^{\\rm 122a,122b}$ , S. Zenz$^{\\rm 14}$ , D. Zerwas$^{\\rm 115}$ , G. Zevi della Porta$^{\\rm 57}$ , Z. Zhan$^{\\rm 32d}$ , D. Zhang$^{\\rm 32b}$$^{,aj}$ , H. Zhang$^{\\rm 88}$ , J. Zhang$^{\\rm 5}$ , X. Zhang$^{\\rm 32d}$ , Z. Zhang$^{\\rm 115}$ , L. Zhao$^{\\rm 108}$ , T. Zhao$^{\\rm 138}$ , Z. Zhao$^{\\rm 32b}$ , A. Zhemchugov$^{\\rm 64}$ , J. Zhong$^{\\rm 118}$ , B. Zhou$^{\\rm 87}$ , N. Zhou$^{\\rm 163}$ , Y. Zhou$^{\\rm 151}$ , C.G.",
"Zhu$^{\\rm 32d}$ , H. Zhu$^{\\rm 41}$ , J. Zhu$^{\\rm 87}$ , Y. Zhu$^{\\rm 32b}$ , X. Zhuang$^{\\rm 98}$ , V. Zhuravlov$^{\\rm 99}$ , D. Zieminska$^{\\rm 60}$ , R. Zimmermann$^{\\rm 20}$ , S. Zimmermann$^{\\rm 20}$ , S. Zimmermann$^{\\rm 48}$ , M. Ziolkowski$^{\\rm 141}$ , R. Zitoun$^{\\rm 4}$ , L. Živković$^{\\rm 34}$ , V.V.",
"Zmouchko$^{\\rm 128}$$^{,*}$ , G. Zobernig$^{\\rm 173}$ , A. Zoccoli$^{\\rm 19a,19b}$ , M. zur Nedden$^{\\rm 15}$ , V. Zutshi$^{\\rm 106}$ , L. Zwalinski$^{\\rm 29}$ .",
"$^{1}$ University at Albany, Albany NY, United States of America $^{2}$ Department of Physics, University of Alberta, Edmonton AB, Canada $^{3}$ $^{(a)}$ Department of Physics, Ankara University, Ankara; $^{(b)}$ Department of Physics, Dumlupinar University, Kutahya; $^{(c)}$ Department of Physics, Gazi University, Ankara; $^{(d)}$ Division of Physics, TOBB University of Economics and Technology, Ankara; $^{(e)}$ Turkish Atomic Energy Authority, Ankara, Turkey $^{4}$ LAPP, CNRS/IN2P3 and Université de Savoie, Annecy-le-Vieux, France $^{5}$ High Energy Physics Division, Argonne National Laboratory, Argonne IL, United States of America $^{6}$ Department of Physics, University of Arizona, Tucson AZ, United States of America $^{7}$ Department of Physics, The University of Texas at Arlington, Arlington TX, United States of America $^{8}$ Physics Department, University of Athens, Athens, Greece $^{9}$ Physics Department, National Technical University of Athens, Zografou, Greece $^{10}$ Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{11}$ Institut de Física d'Altes Energies and Departament de Física de la Universitat Autònoma de Barcelona and ICREA, Barcelona, Spain $^{12}$ $^{(a)}$ Institute of Physics, University of Belgrade, Belgrade; $^{(b)}$ Vinca Institute of Nuclear Sciences, University of Belgrade, Belgrade, Serbia $^{13}$ Department for Physics and Technology, University of Bergen, Bergen, Norway $^{14}$ Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA, United States of America $^{15}$ Department of Physics, Humboldt University, Berlin, Germany $^{16}$ Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland $^{17}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom $^{18}$ $^{(a)}$ Department of Physics, Bogazici University, Istanbul; $^{(b)}$ Division of Physics, Dogus University, Istanbul; $^{(c)}$ Department of Physics Engineering, Gaziantep University, Gaziantep; $^{(d)}$ Department of Physics, Istanbul Technical University, Istanbul, Turkey $^{19}$ $^{(a)}$ INFN Sezione di Bologna; $^{(b)}$ Dipartimento di Fisica, Università di Bologna, Bologna, Italy $^{20}$ Physikalisches Institut, University of Bonn, Bonn, Germany $^{21}$ Department of Physics, Boston University, Boston MA, United States of America $^{22}$ Department of Physics, Brandeis University, Waltham MA, United States of America $^{23}$ $^{(a)}$ Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; $^{(b)}$ Federal University of Juiz de Fora (UFJF), Juiz de Fora; $^{(c)}$ Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei; $^{(d)}$ Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil $^{24}$ Physics Department, Brookhaven National Laboratory, Upton NY, United States of America $^{25}$ $^{(a)}$ National Institute of Physics and Nuclear Engineering, Bucharest; $^{(b)}$ University Politehnica Bucharest, Bucharest; $^{(c)}$ West University in Timisoara, Timisoara, Romania $^{26}$ Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina $^{27}$ Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom $^{28}$ Department of Physics, Carleton University, Ottawa ON, Canada $^{29}$ CERN, Geneva, Switzerland $^{30}$ Enrico Fermi Institute, University of Chicago, Chicago IL, United States of America $^{31}$ $^{(a)}$ Departamento de Fisica, Pontificia Universidad Católica de Chile, Santiago; $^{(b)}$ Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile $^{32}$ $^{(a)}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; $^{(b)}$ Department of Modern Physics, University of Science and Technology of China, Anhui; $^{(c)}$ Department of Physics, Nanjing University, Jiangsu; $^{(d)}$ School of Physics, Shandong University, Shandong, China $^{33}$ Laboratoire de Physique Corpusculaire, Clermont Université and Université Blaise Pascal and CNRS/IN2P3, Aubiere Cedex, France $^{34}$ Nevis Laboratory, Columbia University, Irvington NY, United States of America $^{35}$ Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark $^{36}$ $^{(a)}$ INFN Gruppo Collegato di Cosenza; $^{(b)}$ Dipartimento di Fisica, Università della Calabria, Arcavata di Rende, Italy $^{37}$ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland $^{38}$ The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland $^{39}$ Physics Department, Southern Methodist University, Dallas TX, United States of America $^{40}$ Physics Department, University of Texas at Dallas, Richardson TX, United States of America $^{41}$ DESY, Hamburg and Zeuthen, Germany $^{42}$ Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany $^{43}$ Institut für Kern- und Teilchenphysik, Technical University Dresden, Dresden, Germany $^{44}$ Department of Physics, Duke University, Durham NC, United States of America $^{45}$ SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom $^{46}$ Fachhochschule Wiener Neustadt, Johannes Gutenbergstrasse 3 2700 Wiener Neustadt, Austria $^{47}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy $^{48}$ Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg i.Br., Germany $^{49}$ Section de Physique, Université de Genève, Geneva, Switzerland $^{50}$ $^{(a)}$ INFN Sezione di Genova; $^{(b)}$ Dipartimento di Fisica, Università di Genova, Genova, Italy $^{51}$ $^{(a)}$ E.Andronikashvili Institute of Physics, Tbilisi State University, Tbilisi; $^{(b)}$ High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia $^{52}$ II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany $^{53}$ SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom $^{54}$ II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany $^{55}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier and CNRS/IN2P3 and Institut National Polytechnique de Grenoble, Grenoble, France $^{56}$ Department of Physics, Hampton University, Hampton VA, United States of America $^{57}$ Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA, United States of America $^{58}$ $^{(a)}$ Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(b)}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(c)}$ ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany $^{59}$ Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan $^{60}$ Department of Physics, Indiana University, Bloomington IN, United States of America $^{61}$ Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria $^{62}$ University of Iowa, Iowa City IA, United States of America $^{63}$ Department of Physics and Astronomy, Iowa State University, Ames IA, United States of America $^{64}$ Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia $^{65}$ KEK, High Energy Accelerator Research Organization, Tsukuba, Japan $^{66}$ Graduate School of Science, Kobe University, Kobe, Japan $^{67}$ Faculty of Science, Kyoto University, Kyoto, Japan $^{68}$ Kyoto University of Education, Kyoto, Japan $^{69}$ Department of Physics, Kyushu University, Fukuoka, Japan $^{70}$ Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina $^{71}$ Physics Department, Lancaster University, Lancaster, United Kingdom $^{72}$ $^{(a)}$ INFN Sezione di Lecce; $^{(b)}$ Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy $^{73}$ Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom $^{74}$ Department of Physics, Jožef Stefan Institute and University of Ljubljana, Ljubljana, Slovenia $^{75}$ School of Physics and Astronomy, Queen Mary University of London, London, United Kingdom $^{76}$ Department of Physics, Royal Holloway University of London, Surrey, United Kingdom $^{77}$ Department of Physics and Astronomy, University College London, London, United Kingdom $^{78}$ Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France $^{79}$ Fysiska institutionen, Lunds universitet, Lund, Sweden $^{80}$ Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid, Madrid, Spain $^{81}$ Institut für Physik, Universität Mainz, Mainz, Germany $^{82}$ School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom $^{83}$ CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{84}$ Department of Physics, University of Massachusetts, Amherst MA, United States of America $^{85}$ Department of Physics, McGill University, Montreal QC, Canada $^{86}$ School of Physics, University of Melbourne, Victoria, Australia $^{87}$ Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{88}$ Department of Physics and Astronomy, Michigan State University, East Lansing MI, United States of America $^{89}$ $^{(a)}$ INFN Sezione di Milano; $^{(b)}$ Dipartimento di Fisica, Università di Milano, Milano, Italy $^{90}$ B.I.",
"Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Republic of Belarus $^{91}$ National Scientific and Educational Centre for Particle and High Energy Physics, Minsk, Republic of Belarus $^{92}$ Department of Physics, Massachusetts Institute of Technology, Cambridge MA, United States of America $^{93}$ Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{94}$ P.N.",
"Lebedev Institute of Physics, Academy of Sciences, Moscow, Russia $^{95}$ Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia $^{96}$ Moscow Engineering and Physics Institute (MEPhI), Moscow, Russia $^{97}$ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia $^{98}$ Fakultät für Physik, Ludwig-Maximilians-Universität München, München, Germany $^{99}$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München, Germany $^{100}$ Nagasaki Institute of Applied Science, Nagasaki, Japan $^{101}$ Graduate School of Science, Nagoya University, Nagoya, Japan $^{102}$ $^{(a)}$ INFN Sezione di Napoli; $^{(b)}$ Dipartimento di Scienze Fisiche, Università di Napoli, Napoli, Italy $^{103}$ Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, United States of America $^{104}$ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands $^{105}$ Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, Netherlands $^{106}$ Department of Physics, Northern Illinois University, DeKalb IL, United States of America $^{107}$ Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, Russia $^{108}$ Department of Physics, New York University, New York NY, United States of America $^{109}$ Ohio State University, Columbus OH, United States of America $^{110}$ Faculty of Science, Okayama University, Okayama, Japan $^{111}$ Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK, United States of America $^{112}$ Department of Physics, Oklahoma State University, Stillwater OK, United States of America $^{113}$ Palacký University, RCPTM, Olomouc, Czech Republic $^{114}$ Center for High Energy Physics, University of Oregon, Eugene OR, United States of America $^{115}$ LAL, Université Paris-Sud and CNRS/IN2P3, Orsay, France $^{116}$ Graduate School of Science, Osaka University, Osaka, Japan $^{117}$ Department of Physics, University of Oslo, Oslo, Norway $^{118}$ Department of Physics, Oxford University, Oxford, United Kingdom $^{119}$ $^{(a)}$ INFN Sezione di Pavia; $^{(b)}$ Dipartimento di Fisica, Università di Pavia, Pavia, Italy $^{120}$ Department of Physics, University of Pennsylvania, Philadelphia PA, United States of America $^{121}$ Petersburg Nuclear Physics Institute, Gatchina, Russia $^{122}$ $^{(a)}$ INFN Sezione di Pisa; $^{(b)}$ Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy $^{123}$ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA, United States of America $^{124}$ $^{(a)}$ Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, Lisboa, Portugal; $^{(b)}$ Departamento de Fisica Teorica y del Cosmos and CAFPE, Universidad de Granada, Granada, Spain $^{125}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic $^{126}$ Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic $^{127}$ Czech Technical University in Prague, Praha, Czech Republic $^{128}$ State Research Center Institute for High Energy Physics, Protvino, Russia $^{129}$ Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{130}$ Physics Department, University of Regina, Regina SK, Canada $^{131}$ Ritsumeikan University, Kusatsu, Shiga, Japan $^{132}$ $^{(a)}$ INFN Sezione di Roma I; $^{(b)}$ Dipartimento di Fisica, Università La Sapienza, Roma, Italy $^{133}$ $^{(a)}$ INFN Sezione di Roma Tor Vergata; $^{(b)}$ Dipartimento di Fisica, Università di Roma Tor Vergata, Roma, Italy $^{134}$ $^{(a)}$ INFN Sezione di Roma Tre; $^{(b)}$ Dipartimento di Fisica, Università Roma Tre, Roma, Italy $^{135}$ $^{(a)}$ Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies - Université Hassan II, Casablanca; $^{(b)}$ Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat; $^{(c)}$ Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; $^{(d)}$ Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; $^{(e)}$ Faculté des sciences, Université Mohammed V-Agdal, Rabat, Morocco $^{136}$ DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat a l'Energie Atomique), Gif-sur-Yvette, France $^{137}$ Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, United States of America $^{138}$ Department of Physics, University of Washington, Seattle WA, United States of America $^{139}$ Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{140}$ Department of Physics, Shinshu University, Nagano, Japan $^{141}$ Fachbereich Physik, Universität Siegen, Siegen, Germany $^{142}$ Department of Physics, Simon Fraser University, Burnaby BC, Canada $^{143}$ SLAC National Accelerator Laboratory, Stanford CA, United States of America $^{144}$ $^{(a)}$ Faculty of Mathematics, Physics & Informatics, Comenius University, Bratislava; $^{(b)}$ Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic $^{145}$ $^{(a)}$ Department of Physics, University of Johannesburg, Johannesburg; $^{(b)}$ School of Physics, University of the Witwatersrand, Johannesburg, South Africa $^{146}$ $^{(a)}$ Department of Physics, Stockholm University; $^{(b)}$ The Oskar Klein Centre, Stockholm, Sweden $^{147}$ Physics Department, Royal Institute of Technology, Stockholm, Sweden $^{148}$ Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America $^{149}$ Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom $^{150}$ School of Physics, University of Sydney, Sydney, Australia $^{151}$ Institute of Physics, Academia Sinica, Taipei, Taiwan $^{152}$ Department of Physics, Technion: Israel Institute of Technology, Haifa, Israel $^{153}$ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel $^{154}$ Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece $^{155}$ International Center for Elementary Particle Physics and Department of Physics, The University of Tokyo, Tokyo, Japan $^{156}$ Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo, Japan $^{157}$ Department of Physics, Tokyo Institute of Technology, Tokyo, Japan $^{158}$ Department of Physics, University of Toronto, Toronto ON, Canada $^{159}$ $^{(a)}$ TRIUMF, Vancouver BC; $^{(b)}$ Department of Physics and Astronomy, York University, Toronto ON, Canada $^{160}$ Institute of Pure and Applied Sciences, University of Tsukuba,1-1-1 Tennodai,Tsukuba, Ibaraki 305-8571, Japan $^{161}$ Science and Technology Center, Tufts University, Medford MA, United States of America $^{162}$ Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia $^{163}$ Department of Physics and Astronomy, University of California Irvine, Irvine CA, United States of America $^{164}$ $^{(a)}$ INFN Gruppo Collegato di Udine; $^{(b)}$ ICTP, Trieste; $^{(c)}$ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Udine, Italy $^{165}$ Department of Physics, University of Illinois, Urbana IL, United States of America $^{166}$ Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden $^{167}$ Instituto de Física Corpuscular (IFIC) and Departamento de Física Atómica, Molecular y Nuclear and Departamento de Ingeniería Electrónica and Instituto de Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC, Valencia, Spain $^{168}$ Department of Physics, University of British Columbia, Vancouver BC, Canada $^{169}$ Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada $^{170}$ Department of Physics, University of Warwick, Coventry, United Kingdom $^{171}$ Waseda University, Tokyo, Japan $^{172}$ Department of Particle Physics, The Weizmann Institute of Science, Rehovot, Israel $^{173}$ Department of Physics, University of Wisconsin, Madison WI, United States of America $^{174}$ Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany $^{175}$ Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany $^{176}$ Department of Physics, Yale University, New Haven CT, United States of America $^{177}$ Yerevan Physics Institute, Yerevan, Armenia $^{178}$ Domaine scientifique de la Doua, Centre de Calcul CNRS/IN2P3, Villeurbanne Cedex, France $^{a}$ Also at Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, Lisboa, Portugal $^{b}$ Also at Faculdade de Ciencias and CFNUL, Universidade de Lisboa, Lisboa, Portugal $^{c}$ Also at Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{d}$ Also at TRIUMF, Vancouver BC, Canada e Also at Department of Physics, California State University, Fresno CA, United States of America $^{f}$ Also at Novosibirsk State University, Novosibirsk, Russia $^{g}$ Also at Fermilab, Batavia IL, United States of America $^{h}$ Also at Department of Physics, University of Coimbra, Coimbra, Portugal $^{i}$ Also at Department of Physics, UASLP, San Luis Potosi, Mexico $^{j}$ Also at Università di Napoli Parthenope, Napoli, Italy $^{k}$ Also at Institute of Particle Physics (IPP), Canada $^{l}$ Also at Department of Physics, Middle East Technical University, Ankara, Turkey $^{m}$ Also at Louisiana Tech University, Ruston LA, United States of America $^{n}$ Also at Dep Fisica and CEFITEC of Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal $^{o}$ Also at Department of Physics and Astronomy, University College London, London, United Kingdom $^{p}$ Also at Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{q}$ Also at Department of Physics, University of Cape Town, Cape Town, South Africa $^{r}$ Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{s}$ Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany $^{t}$ Also at Manhattan College, New York NY, United States of America $^{u}$ Also at School of Physics, Shandong University, Shandong, China $^{v}$ Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{w}$ Also at School of Physics and Engineering, Sun Yat-sen University, Guanzhou, China $^{x}$ Also at Academia Sinica Grid Computing, Institute of Physics, Academia Sinica, Taipei, Taiwan $^{y}$ Also at Dipartimento di Fisica, Università La Sapienza, Roma, Italy $^{z}$ Also at DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat a l'Energie Atomique), Gif-sur-Yvette, France $^{aa}$ Also at Section de Physique, Université de Genève, Geneva, Switzerland $^{ab}$ Also at Departamento de Fisica, Universidade de Minho, Braga, Portugal $^{ac}$ Also at Department of Physics and Astronomy, University of South Carolina, Columbia SC, United States of America $^{ad}$ Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary $^{ae}$ Also at California Institute of Technology, Pasadena CA, United States of America $^{af}$ Also at Institute of Physics, Jagiellonian University, Krakow, Poland $^{ag}$ Also at LAL, Université Paris-Sud and CNRS/IN2P3, Orsay, France $^{ah}$ Also at Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{ai}$ Also at Department of Physics, Oxford University, Oxford, United Kingdom $^{aj}$ Also at Institute of Physics, Academia Sinica, Taipei, Taiwan $^{ak}$ Also at Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{*}$ Deceased"
]
] | 1204.0735 |
[
[
"Present-Day Descendants of z=3 Ly-{\\alpha} Emitting Galaxies in the\n Millennium-II Halo Merger Trees"
],
[
"Abstract Using the Millennium-II Simulation dark matter sub-halo merger histories, we created mock catalogs of Lyman Alpha Emitting (LAE) galaxies at z=3.1 to study the properties of their descendants.",
"Several models were created by selecting the sub-halos to match the number density and typical dark matter mass determined from observations of these galaxies.",
"We used mass-based and age-based selection criteria to study their effects on descendant populations at z~2, 1 and 0.",
"For the models that best represent LAEs at z=3.1, the z=0 descendants have a median dark matter halo mass of 10^12.7 M_Sun, with a wide scatter in masses (50% between 10^11.8 and 10^13.7 M_Sun).",
"Our study differentiated between central and satellite sub-halos and found that ~55% of z=0 descendants are central sub-halos with M_Median~10^12 M_Sun.",
"This confirms that central z=0 descendants of z=3.1 LAEs have halo masses typical of L* type galaxies.",
"The satellite sub-halos reside in group/cluster environments with dark matter masses around 10^14 M_Sun.",
"The median descendant mass is robust to various methods of age determination, but it could vary by a factor of 5 due to current observational uncertainties in the clustering of LAEs used to determine their typical z=3.1 dark matter mass."
],
[
"Introduction",
"Narrow band surveys have been used to discover Ly$\\alpha $ -emitting (LAE) galaxies at high redshift (e.g., [25], [9], [44], [40]) and to study their properties.",
"The galaxies' strong emission lines reveal a set of young, potentially dust-free galaxies theorized by [38].",
"LAEs allow us to study galaxy formation beginning with one of the smallest building blocks found so far.",
"At $z\\simeq 3.1$ , typical LAEs are low mass galaxies with $\\textup {M}_{Stellar}\\simeq 10^9\\rm \\, M_\\odot $ and little dust extinction, $A_V\\le 0.2$ [16], [33], [1].",
"These objects have been observed at $z\\ge 3$ [45], [18], [21], [33] and as far out as $z\\sim 7$ [26].",
"Previous papers have studied the evolution of LAEs and other, generally more massive, high-z galaxy populations (e.g.",
"[20], [39], [41], [46]) by using clustering properties as a technique to make evolutionary claims between redshifts.",
"The connection between $z=3.1$ LAEs and present-day galaxies was determined by [17] (hereafter Ga07) by measuring the clustering properties of the $z=3.1$ LAEs from the sample of [21].",
"Ga07 used the formalism devised by [31] to compute the median dark matter mass of their host halos to be $\\log _{10}\\textup {M}_{\\textup {{\\tiny DM}}}=10.9^{+0.5}_{-0.9}\\,\\textup {M}_\\odot $ .",
"These authors claimed evolution into present-day $L^*$ type galaxies based upon the analytical conditional mass function (e.g.",
"[24], [13]), a result echoed for LAEs at $z=2.1$ by [23] and LAEs at $z\\simeq 6.6$ by [36].",
"A weakness of the analytical conditional mass function is its inability to distinguish individual substructures within a halo or to predict the masses of these substructures.",
"N-body simulations fill this gap by producing halo merger trees that allow us to determine properties for the descendant halos and their substructures.",
"Spectral energy distribution fitting for $z=3.1$ LAEs reveals a large range in stellar ages (Ga07; [27], [33], [35]).",
"Ga07 determined the young stellar component to have an age of $20^{+30}_{-10}\\,\\textup {Myr}$ (Ga07) using a two-stellar population model.",
"This could have two simple interpretations: 1) LAEs occur as a galaxy's first burst of star formation, which lasts $\\sim 40 \\rm \\, Myr$ , or 2) LAEs are a recurring phase, where each burst of star formation lasts $\\sim 40\\,\\textup {Myr}$ (Ga07).",
"In both cases, stellar evolution produces dust, which ultimately quenches Ly$\\alpha $ emission.",
"The old stellar population in the two stellar population model is not well constrained, with an age of up to $2\\rm \\,Gyr$ .",
"A single stellar population model, which observations cannot dismiss, has an age of $50-100\\,\\textup {Myr}$ ([27]; see also [1]).",
"[2] determined stellar ages of $\\sim 1\\,\\textup {Gyr}$ for $z\\simeq 3.1$ LAEs.",
"[33] found $z\\simeq 3.1$ LAE stellar ages of $0.85^{+0.13}_{-0.42}\\,\\textup {Gyr}$ .",
"These clustering and spectral energy distribution results allow us to create mock LAE catalogs within the Millennium-II Simulation[1] ([5], hereafter MS-II) to study the dark matter mass evolution to the present day within a large cosmological simulation.",
"A similar investigation by [6] used star-forming galaxies at $z\\sim 2$ as a starting point and studied their dark matter halo evolution until the present-day within the Millennium simulation.",
"Section REF describes the MS-II.",
"Section presents specifics of the mock catalogs, while sections and report on the clustering analysis and descendants of the catalogs respectively.",
"All distances reported are comoving, $H_0=100h\\,\\textup {km/s/Mpc}$ and $h=0.73$ is assumed throughout.",
"[1]Data created from the MS-II can be accessed from the Max-Planck Institute and Durham University servers at http://www.mpa-garching.mpg.de/galform/millennium-II using a Structured Query Language (SQL) query.",
"For a merger tree explanation see http://www.g-vo.org/Millennium-II/Help?page=mergertrees.",
"Our study uses the results of the MS-II run by the Virgo Consortium.",
"The MS-II contains $2160^3$ particles in a cube of 100 $h^{-1}$ Mpc on a side.",
"The particle mass is $6.9\\times 10^6h^{-1}\\textup {M}_\\odot $ with a minimum halo mass of $1.38\\times 10^8h^{-1}\\textup {M}_\\odot $ .",
"MS-II gives us the ability to resolve 125 times less massive sub-halos than those observed within the Millennium Simulation [42], [30].",
"LAEs at $z=3.1$ appear to be hosted within low mass halos, making the improved mass resolution necessary.",
"Although the Millennium Simulation provides 125 times larger volume, MS-II offers robust statistics for spatial clustering of halos in the mass range of interest.",
"Both simulations use a $\\Lambda \\textup {CDM}$ cosmology with values: $&\\Omega _{\\textup {tot}} =1.0, \\Omega _m=0.25, \\Omega _b=0.045, \\Omega _{\\Lambda } =0.75\\\\&h=0.73, \\sigma _8 =0.9, n_s=1.$ All values except $n_s$ and $\\sigma _8$ are within $1\\sigma $ of the values reported in the 7-year WMAP results [29].",
"MS-II also offers improved temporal resolution, with dark matter halos selected in 67 timesteps using a friends-of-friends algorithm [10] with a linking length of $b=0.2$ [5].",
"Each FOF group was analyzed for sub-halos using the SUBFIND algorithm [43], which identifies gravitationally bound sub-halos within the FOF group.",
"We defined a central sub-halo to be the most massive substructure within an FOF group, and other sub-halos within the group were classified as satellites.",
"During our age selection (described in §REF ) we found that some of the youngest dark matter halos exhibited rapid mass growth in the range of a factor of 10-1000 during one timestep (approx.",
"200 Myr).",
"We believe this mass growth is unrealistic and is caused by misidentification of ownership of dark matter particles by the SUBFIND algorithm between neighboring sub-halos.",
"The errant merger trees appear to normalize by merging with the mass theft victim after a couple of timesteps.",
"This allows us to trust the results from our descendants that have had a few timesteps to normalize.",
"Our selection method, that removes fast growing merger trees, computes the dark matter mass ratios between the $z=3.1$ dark matter halo and its most massive predecessor and between the most massive $z=3.1$ predecessor and its most massive predecessor.",
"If either of these ratios are greater than 10, then we remove this merger tree from our catalogs.",
"With this filtered set of dark matter halos we produce our models."
],
[
"Sub-Halo Abundance Matching",
"We use Sub-Halo Abundance Matching (SHAM) to determine stellar masses for the descendants of our models.",
"The main principle of the SHAM algorithm is to assign luminosities or stellar masses from a luminosity or stellar mass function to dark matter masses from N-body dark matter simulations monotonically [8].",
"Dark matter sub-halos are assigned stellar masses by matching the number densities of halos such that $n_h(>\\textup {M}_{\\tiny DM})=n_g(>\\textup {M}_{\\tiny Stellar}(\\textup {M}_{\\tiny DM})).$ The dark matter masses used in eqn.",
"REF are sub-halo masses.",
"If the sub-halo is a satellite substructure then we modify the dark matter mass to be the infall mass.",
"Infall mass is defined as the larger of the mass of a central sub-halo before it falls into a larger halo to become a satellite or the satellite mass in the following timestep [8].",
"We use the infall mass because it gives a better representation of the current stellar component.",
"Unlike the dark matter particles that are disrupted during infall, the stellar component is deeper in the gravitational potential well and therefore is less likely to be disrupted during infall.",
"Our algorithm solves eqn.",
"REF by using a Newton's method approach where the function and its derivative are $0 & =& f(\\textup {M}_{\\tiny Stellar}) \\nonumber \\\\& = & n_h(>\\textup {M}_{\\tiny DM})- \\int ^\\infty _{\\frac{\\textup {\\footnotesize M}_{\\tiny Stellar}}{\\textup {\\footnotesize M}_*}}\\phi _*\\left(\\frac{\\textup {M}}{\\textup {M}_*}\\right)^\\alpha e^{-\\frac{\\textup {\\footnotesize M}}{\\textup {\\footnotesize M}_*}}d\\left(\\frac{\\textup {M}}{\\textup {M}_*}\\right) \\nonumber \\\\& = & n_h(>\\textup {M}_{\\tiny DM})-\\phi _* \\Gamma \\left[1+\\alpha ,\\frac{\\textup {\\footnotesize M}_{\\tiny Stellar}}{\\textup {\\footnotesize M}_*}\\right]$ $f^{\\prime }(\\textup {M}_{\\tiny Stellar}) = \\left(\\frac{\\phi _*}{\\textup {M}_*}\\right)\\left(\\frac{\\textup {M}_{\\tiny Stellar}}{\\textup {M}_*}\\right)^\\alpha e^{-\\frac{\\textup {\\footnotesize M}_{\\tiny Stellar}}{\\textup {\\footnotesize M}_*}}$ In the above we have used a Schecter function with parameters $\\phi _*$ , $M_*$ and $\\alpha $ and $\\Gamma $ is the incomplete gamma function.",
"This method allows us to assign stellar masses by solving eqn.",
"REF for $\\textup {M}_{\\tiny Stellar}$ .",
"The large number of sub-halos in the simulation forces us to use a sub-selection technique to determine the stellar mass to dark matter sub-halo mass function.",
"Afterwards we interpolate this function to determine the stellar masses of the $z=0$ descendants from their sub-halo dark matter masses."
],
[
"LAE Mock Catalogs",
"We created ten LAE mock catalogs from the MS-II using sub-halos at $z=3.06$ (snapnum[2] 31) based on mass and age selection.",
"The catalog names and properties are listed in Tables and .",
"In the following sections we will discuss the selection criteria and motivation for the different catalogs.",
"[2]Snapnum is the snapshot (timestep) number from the MS-II."
],
[
"Mass-based Catalogs",
"All mass-based catalogs were chosen to reproduce the $z=3.1$ LAE number density ($1.5\\times 10^{-3}\\textup {Mpc}^{-3}$ ) determined by [21], thus generating models with 3856 sub-halos in the MS-II volume.",
"The mass limit criterion was selected to have a minimum mass of $\\log _{10}\\textup {M}_{\\textup {\\tiny Min}}=10.6\\,\\textup {M}_\\odot $ to match that inferred by Ga07 from the observed clustering of $z= 3.1$ LAEs.",
"This approach is similar to the one used by [6]; their mass limit was chosen to reproduce the clustering of $z\\sim 2$ star-forming galaxies.",
"We randomly selected 5.24% of these sub-halos to match the above number density.",
"The Mass limit catalog was designed to reproduce the observed correlation length of $z= 3.1$ LAEs.",
"The Median catalog was selected using the median mass reported by Ga07, $\\log _{10}\\textup {M}_{\\textup {\\tiny Med}}=10.9\\,\\textup {M}_\\odot $ , as the catalog's median mass value.",
"We expanded evenly around the median mass to obtain the observed LAE number density, choosing halos with a range of $7.56\\times 10^{10}\\,\\rm M_\\odot $ –$8.36\\times 10^{10}\\,\\rm M_\\odot $ .",
"As expected, this catalog and the Mass Limit catalog have similar median masses; it should also reproduce the observed correlation length.",
"The $-\\sigma $ and $+\\sigma $ catalogs were selected and named based on the $\\pm 1\\sigma $ uncertainty reported by Ga07 in the observed LAE median mass.",
"The median masses of the catalogs are $\\log _{10}\\textup {M}_\\textup {\\tiny Med}=10.0\\,\\textup {M}_\\odot $ and $11.4\\,\\textup {M}_\\odot $ , respectively.",
"We expand evenly around their respective median masses to obtain the observed LAE number density.",
"The mass ranges for the $-\\sigma $ and $+\\sigma $ catalogs are $9.93\\times 10^{9}\\,\\rm M_\\odot $ –$1.01\\times 10^{10}\\,\\rm M_\\odot $ and $2.15\\times 10^{11}\\,\\rm M_\\odot $ –$3.04\\times 10^{11}\\,\\rm M_\\odot $ , respectively.",
"These catalogs were created to study the uncertainty in the descendant properties propagated from the observed clustering uncertainties and are not expected to reproduce the best-fit observed correlation length at $z=3.1$ ."
],
[
"Age-based Catalogs",
"Three age definitions were chosen to study the dependence of descendants' properties on age selection.",
"The age definitions use merger trees rooted in central sub-halos within FOF halos with $\\textup {M}_\\textup {{\\tiny FOF}}\\ge 3.98\\times 10^{10}\\,\\textup {M}_\\odot $ , the mass limit criteria.",
"Figure REF shows an example merger tree showing the formation, assembly and merger ages assigned.",
"Ages are defined as the difference of lookback time chosen and the lookback time at $z=3.1$ .",
"The three age definitions are as follows: The Formation age [15], [14] quantifies the timescale for mass growth of the most massive dark matter structure.",
"It is assigned by finding the most recent timestep where the sub-halo mass of the most massive progenitor is less than half of the maximum mass in the entire merger tree.",
"If this occurs between two timesteps, we linearly interpolate between these two to estimate the time when half the mass was accreted.",
"The Assembly age [32] measures when half the maximum mass of a galaxy is present in collapsed sub-halos even if they have not yet merged.",
"This is assigned by finding the most recent timestep where the sum of progenitor sub-halo masses is less than half of the maximum mass in the entire merger tree.",
"If this occurs between two timesteps, we linearly interpolate between these two to estimate the time when half the mass has assembled.",
"Because of these definitions, the Assembly ages are always equal to or greater than the Formation ages.",
"Figure REF a) shows the relation between Formation and Assembly ages.",
"Merger age searches for the most recent major merger.",
"Our definition was designed to to use a main sub-halo to follow the major merger which is similar to the methods used by [19].",
"We define a major merger to occur when two central sub-halos have a mass ratio of 3:1 or less in a timestep and the most massive halo within that timestep is involved.",
"The major merger begins when one of the descendants in the next timestep is a satellite of the other descendant or both centrals merge into a single sub-halo.",
"In Figure REF we have two central sub-halos, around $1.2\\,\\textup {Gyr}$ , where the lower mass central sub-halo descends into a satellite belonging to the descendant of the more massive central sub-halo.",
"We track the descendants until the two sub-halos merge, possibly triggering star formation.",
"We average the timestep of this sub-halo merger and the previous timestep to assign the merger time.",
"The infall timescale from the beginning of the 3:1 merger of the two centrals to the final merger of their descendants is comparable to the dynamical friction timescale used by [19] based on work from [4].",
"Figure REF b) & c) show little to no correlation between Merger age and Assembly or Formation ages.",
"The age distributions are presented in Figure REF a).",
"The LAE mock catalogs are then created from the ages calculated for all halos with the mass limit criteria by selecting the 5.24% youngest and 5.24% closest to median aged sub-halos.",
"See Figure REF b), c) & d) for age distributions for all halos and the age distributions of the selected catalogs.",
"Table lists the age properties of the different catalogs with the bold values being age statistics used to select that particular catalog.",
"All the median age-selected catalogs have median ages of $\\sim 1 \\,\\rm Gyr$ .",
"For star formation triggered by accretion of sub-halos in minor mergers, Formation and Assembly ages should roughly track the age of stars in a galaxy i.e.",
"the population dominating its stellar mass [12].",
"Since SED fits allow $\\sim 1\\,\\textup {Gyr}$ old populations to comprise the majority of stellar mass in LAEs (Ga07; [2], [33]), the median Formation and Assembly models are feasible models.",
"However, because the merger age should track a young stellar population born in a starburst triggered by the major merger, we cannot reconcile a $\\sim 1\\,\\textup {Gyr}$ median merger age with SED results; this rules out the Median Merger model at high confidence.",
"The young Formation and Assembly catalogs are viable models which have median ages of 240 Myr, consistent with ages found for single-population models.",
"The young Merger catalog has a median age of 91 Myr and is consistent with the observed age of the young stellar component."
],
[
"Clustering Analysis of Mock Catalogs",
"We used the naive estimator $\\xi _N$ (e.g., [28]) to calculate the two point auto-correlation function (2PCF) for the models.",
"$\\hat{\\xi }_{N}(r)=\\frac{DD(r)}{RR(r)}-1$ $DD(r)&=&\\frac{2\\times \\textup {\\footnotesize Number of data-data pairs within a radial bin}}{n_D(n_D-1)}\\nonumber \\\\RR(r)&=&\\frac{\\textup {\\footnotesize Volume contained within a radial bin}}{\\textup {\\footnotesize Volume of simulation}} \\nonumber $ Data-data pairs are unique pairs of sub-halos, from a catalog, that are separated and binned by radius $r$ .",
"The simple geometry of the simulation, in conjunction with the periodic boundary conditions, allow us to use a geometrical formula for RR, which eliminates uncertainty in the estimator caused by binning random-random pairs.",
"We applied a correction to our data abscissas to match the effective center of the radial bin using $<\\xi >_{bin}=\\frac{\\int _{r_L}^{r_L+\\Delta r} \\left(\\frac{r}{r_0}\\right)^{-\\gamma }r^2dr}{\\int _{r_L}^{r_L+\\Delta r}r^2 dr} \\\\ \\nonumber \\\\r_{bin}=r_0<\\xi >_{bin}^{-1/\\gamma }.$ The correction was applied by our fitting algorithm to shift the data point to the radius which corresponds to the average value of the bin.",
"We fit our naive estimator with the power law given by $\\xi (r)=\\frac{(r/r_0)^{-\\gamma }-\\omega _{\\Omega }}{1+\\omega _\\Omega } \\\\\\omega _{\\Omega }=\\int ^{R_{max}}_0RR(r)\\left( \\frac{r}{r_0}\\right)^{-\\gamma } dr,$ where $\\omega _{\\Omega }$ is the integral constraint found from the power-law term, $(\\frac{r}{r_0})^{-\\gamma }$ for the parameters during fitting.",
"We minimized $\\chi ^2$ to determine the best fit for the parameters $r_0$ and $\\gamma $ using $\\sigma _{N}^2 (r) & = & \\left(\\frac{1+\\xi (r)}{1+\\omega _\\Omega }\\right)^2\\left(\\frac{1-RR(r)}{(n_D(n_D-1)/2)RR(r)}\\right)$ for the variance of the naive estimator [28].",
"In the following discussion we will fix $\\gamma =1.8$ to compare our results to the observed correlation length for $z=3.1$ LAEs, $r_0=3.6^{+0.8}_{-1.0}\\,\\textup {Mpc}$ (Ga07).",
"Table lists the values found for $r_0$ fixing $\\gamma =1.8$ and also fitting both $r_0$ and $\\gamma $ as free parameters, while Figures REF and REF show the best fit 2PCF for the catalogs."
],
[
"Clustering of Mass-Based Catalogs",
"Figure REF and Table show the mass-based catalogs' 2PCFs, correlation lengths, $\\gamma $ and $\\chi ^2$ values.",
"We confirm that correlation lengths increase with median mass in our mass-based catalogs for both fixed $\\gamma =1.8$ and when $\\gamma $ is allowed to be a free parameter.",
"The Median and Mass limit catalogs have similar median masses and were expected to have the same correlation lengths when $\\gamma =1.8$ ; the correlation lengths are consistent with one another.",
"These two models also have correlation lengths that are consistent with the observed LAE correlation length of $r_0=3.6^{+0.8}_{-1.0}\\,\\textup {Mpc}$ , making them good representations of LAEs at $z=3.1$ .",
"The $+\\sigma $ and$-\\sigma $ models have correlation lengths close to the $\\pm 1\\sigma $ uncertainty from observed values, as expected.",
"Fixing $\\gamma =1.8$ allows us to compare our results to previous works, but the best fit models are not consistent with this value.",
"We find an average value of $\\gamma =1.33$ , which is consistent with the results from [5] at $z=2.07$ using our fitting range."
],
[
"Clustering of Age-Based Catalogs",
"The similarities of the mass properties of the age-based catalogs (see Table ) imply that any difference in the clustering properties between age-based catalogs are due to the age definition and selection.",
"The Formation Median and Assembly Median models have similar correlation lengths of $r_0\\simeq 3.6\\,\\textup {Mpc}$ and are consistent with the observed LAE correlation length.",
"The Assembly Young and Formation Young have higher correlation lengths of $r_0\\simeq 4.4\\,\\textup {Mpc}$ and are barely consistent at the 68% level with the $z\\simeq 3.1$ LAE correlation length.",
"We cannot rule out the Young Merger model at 95% confidence.",
"The Median Merger model has a correlation length large enough to be ruled out at 93% confidence.",
"This large correlation length and the high merger ages make the Median Merger model an unrealistic $z\\simeq 3.1$ LAE model.",
"No other models are ruled out based on clustering.",
"The observed trend for the Formation and Assembly ages that young aged models have a larger clustering length compared to the median aged models does not disagree with the findings of [14] that the youngest $20\\%$ of halos at $z=0$ cluster less strongly than the oldest $20\\%$ .",
"Once we use a similar selection we also find that the oldest halos have higher clustering compared to the youngest."
],
[
"Descendants of Mock LAEs",
"We used the MS-II merger trees to find the $z=2.07$ (snapnum 36), $z=0.99$ (snapnum 45), and $z=0$ (snapnum 67) descendants of the mock LAE catalogs.",
"We report the mass distributions of the descendants based on the mass of the host FOF group, $\\rm M_{\\textup {\\tiny FOF}}$ .",
"We prefer M$_{\\textup {\\tiny FOF}}$ because it determines the evolution of dark matter halos and is used in the Press-Schechter formalism, while the individual sub-halo mass traces the evolution of individual galaxies.",
"We classify the most massive sub-halo within each FOF group as a central and other sub-halos as satellites.",
"All other smaller sub-halos within the FOF group are satellites.",
"The following sub-sections will discuss the descendants of our mock catalogs as reported in Tables -."
],
[
"Mass Limit and Median Mass Model Descendants",
"Figure REF shows an example histograms demonstrating the evolution of the Mass Limit and Median models from $z=3.1$ to $z=0$ using FOF halo masses.",
"Both models have the same median masses, $\\log _{10} \\textup {M}_{med}\\simeq 10.9\\,\\textup {M}_\\odot $ , at $z=3.1$ , as also seen in Table .",
"All other models evolve in a similar manner.",
"Figure REF (bottom panel) summarizes this same evolution for all of the mass selected models.",
"The models' descendants also have similar satellite mass distributions (dashed histograms in figure REF ) as they evolve with time.",
"The satellite median masses grow from $\\textup {M}_{\\textup {{\\tiny FOF; Med}}}=10^{12}\\textup {M}_\\odot $ at $z=2$ to $10^{13.7}\\textup {M}_\\odot $ at $z=0$ .",
"The central descendants show evolution towards higher masses, as expected for bottom-up halo growth [10], though the 10th percentile in mass stays roughly constant at its $z\\simeq 3.1$ value.",
"We find that the catalogs' central population, which comprise 55-60% of the descendants, has mass growth from $\\textup {M}_{\\textup {{\\tiny FOF; Med}}}=10^{10.9}\\,\\textup {M}_\\odot $ at $z= 3.1$ to Milky Way-sized dark matter halos, $\\simeq 10^{11.8}\\textup {M}_\\odot $ , at $z=0$ .",
"The median FOF mass for all descendants grows from $\\textup {M}_{\\textup {{\\tiny FOF; Med}}}=10^{10.9}\\,\\textup {M}_\\odot $ at $z= 3.1$ to $\\textup {M}_{\\textup {{\\tiny FOF; Med}}}=10^{12.6}\\,\\textup {M}_\\odot $ at $z=0$ .",
"For comparison, Figure REF top panel shows these descendants in terms of the individual sub-halo mass instead of the mass of the FOF group that the sub-halo resides in.",
"The evolution of mass in this sense is similar to the FOF mass evolution; sub-halos tend to grow in mass towards $z=0$ , but mass loss is seen in the satellite populations.",
"Central sub-halo masses are similar to the FOF masses; therefore, we expect a similar median mass at $z=0$ .",
"We find the central sub-halo median mass to be $\\textup {M}=10^{11.8}\\textup {M}_\\odot $ at $z=0$ .",
"Comparing the top and bottom panels of figure REF we see that descendants that are satellites reside in massive halos, but the satellite sub-halos are less massive, with a median mass of $10^{10.8}\\,\\rm M_\\odot $ at $z=0$ ."
],
[
"Age-Based Model Descendants",
"The similarities in the six age-based models' mass distributions at $z=3.1$ suggests that we should expect similar descendant distributions.",
"Figure REF top and bottom panel confirm the similarities among these catalogs' descendants.",
"The mass evolution represented in figure REF bottom panel is the mass of the FOF halo where the sub-halo resides while the top panel uses sub-structure mass.",
"The evolution of the mass distributions are all similar and therefore independent of age definition or selection.",
"The descendant central median masses are $\\textup {M}_{\\textup {{\\tiny FOF}}}=10^{11.2}\\,\\textup {M}_\\odot $ at $z=2.1$ , $10^{11.5}\\,\\textup {M}_\\odot $ at $z=1$ , and $10^{12}\\,\\textup {M}_\\odot $ at $z=0$ with a small scatter of less than a factor two.",
"We find that the fraction of central sub-halos for all age-based models is independent of age definition or selection (Tables -).",
"The satellite descendants at $z=0$ comprise $42-44\\%$ of the population and have a median FOF mass of $10^{13.7}\\textup {M}_\\odot $ .",
"For all the models the full distribution of descendants have a median FOF mass within a factor of two of $10^{12.7}\\textup {M}_\\odot $ at $z=0$ .",
"These FOF masses are in agreement with those found for the Median and Mass limit catalogs described in section REF .",
"The sub-halo masses of the age-based models are also similar to the values obtained for the Median and Mass limit models."
],
[
"$+\\sigma $ and {{formula:1c773970-a6ed-4f66-b46f-9284ac80bb1e}} Model Descendants",
"The $+\\sigma $ and $-\\sigma $ models were created to quantify the uncertainties within the descendant distributions caused by the uncertainties in the observed clustering analysis.",
"Figure REF shows the $+\\sigma $ and $-\\sigma $ with the other mass-selected models for comparison using the FOF mass.",
"The central descendant sub-halos have median masses that are a factor of four higher for the $+\\sigma $ model and a factor of ten lower for the $-\\sigma $ model compared with the Median mass catalog's descendants at $z=0$ .",
"The satellite distributions at $z=0$ have median masses that are a factor of three larger for the $+\\sigma $ and a factor of five smaller for the $-\\sigma $ than the Median model.",
"When we consider the full descendant distribution the median mass of the $+\\sigma $ model is a factor of three larger and the $-\\sigma $ model is six times lower than the Median model.",
"The fraction of central descendant sub-halos is similar between the two models."
],
[
"Determining Stellar Properties of Descendants",
"Using the SHAM algorithm discussed in section REF , we determine the stellar masses for all sub-halos at $z=0$ .",
"We use the Schecter function parameters found by the Sloan Digital Sky Survey [37].",
"The parameters are $\\phi ^* & = & 2.2\\pm 0.5_{stat}\\pm 1_{sys}\\times 10^{-3}\\,\\textup {Mpc}^{-3} \\\\\\textup {M}^* & = & 1.005\\pm 0.004_{stat}\\pm 0.200_{sys}\\times 10^{11}\\,\\textup {M}_\\odot \\\\\\alpha & = & -1.222\\pm 0.002_{stat}\\pm 0.1_{sys}.$ Figure REF shows the relationship between infall and stellar masses determined from the SHAM algorithm.",
"After application of the SHAM algorithm to the entire $z=0$ MSII sub-halo catalog covering dark matter masses ranging from $10^{8.28}\\,\\textup {M}_\\odot $ to $10^{14.94}\\,\\textup {M}_\\odot $ , we obtained stellar masses ranging from $0.14\\,\\textup {M}_\\odot $ to $10^{11.80}\\,\\textup {M}_\\odot $ .",
"The very low stellar masses are attributable to the excess of dark matter sub-structures at low mass compared to the number of galaxies from the stellar mass function, which remains an unsolved problem in galaxy formation.",
"The $-1\\sigma $ model is the only model with a significant number of halos in this range.",
"Figure REF shows the median stellar masses for all the models.",
"The error bars denote the 10th and 90th percentiles.",
"All the models except the $+1\\sigma $ and $-1\\sigma $ have similar stellar mass distributions.",
"The $+1\\sigma $ and $-1\\sigma $ models have corresponding shifts in their median stellar masses due to their selection."
],
[
"Discussion and Conclusions",
"We find that the Median, Mass Limit, Young Formation, Median Formation, Young Assembly and Median Assembly models have correlation lengths within the 68% confidence interval for LAEs at $z=3.1$ , all though the Young Formation and Young Assembly models have correlation lengths near the $+1\\sigma $ limit.",
"As designed, the $+\\sigma $ and $-\\sigma $ catalogs have $z=3.1$ correlation lengths near the observed $\\pm 1\\sigma $ correlation length uncertainty.",
"The Young Merger and Median Merger models have large correlation lengths and are not consistent with the $z=3.1$ clustering measurement.",
"We eliminated the Merger Median model due to the age of $\\sim 1$ Gyr at $z=3.1$ , in contrast with starburst ages of 20-100 Myr determined from SED fitting.",
"However, SED modeling has not yet constrained the age of LAEs sufficiently to rule out the other age-based catalogs.",
"We studied the connection between LAEs at $z=3.1$ and galaxy populations at $z=2.1$ for which previous studies have determined typical dark matter halo masses.",
"[23] found LAEs at $z=2.1$ to have a median dark matter halo mass of $\\log (\\textup {M/M}_\\odot )=11.5^{+0.4}_{-0.5}$ .",
"Their study used bias evolution from the conditional mass function to show that $z=2.1$ LAEs are possible descendants of $z=3.1$ LAEs observed by Ga07.",
"All of our models, except the $+\\sigma $ and $-\\sigma $ models, have median dark matter halo masses of the full descendant distributions around $10^{11.3}\\textup {M}_\\odot $ at $z=2.1$ .",
"Hence, $z=2.1$ LAEs could be direct descendants of $z=3.1$ LAEs, rather than a new set of halos undergoing their first phase of star formation.",
"However, SED modeling by [22] and [34] have found typical starburst ages for $z=2.1$ LAEs of $12^{+149}_{-3}\\,\\textup {Myr}$ and $80^{+10}_{-20}\\,\\textup {Myr}$ .",
"Given these starburst ages, the $z=2.1$ LAEs would have to be experiencing a subsequent burst of star formation to be descendants of $z=3.1$ LAEs.",
"Another study by ([3], see their Fig.",
"2) determined the clustering of BX galaxies at $z\\simeq 2$ .",
"The clustering result for the least luminous ($K_s>21.5$ ) subset implies a median dark matter mass of $\\log _{10}\\textup {M}_{\\textup {\\tiny Med}}/\\textup {M}_\\odot =11.0^{+0.6}_{-0.9}$ .",
"This result is consistent with our $z=2.1$ descendant median dark matter mass, except for the $+\\sigma $ and $-\\sigma $ models.",
"We find that $\\sim 75\\%$ of the descendants of $z=3.1$ LAEs are centrals at $z=2.1$ ; it is unclear whether $\\simeq 2$ LAEs and BX galaxies represent a similar mix of centrals and satellites.",
"We can make a similar comparison from $z=2.1$ to $z=0.9$ .",
"The Deep Extragalactic Evolutionary Probe 2 (DEEP2, [11]) measured the clustering bias of color selected galaxies at $z\\simeq 0.9$ .",
"For our chosen cosmology, their results [7] imply median dark matter halo masses of $\\textup {M}_{\\textup {\\tiny Med}}=10^{12.9\\pm 0.1}\\,\\textup {M}_\\odot $ and $\\textup {M}_{\\textup {\\tiny Med}}=10^{12.0\\pm 0.1}\\,\\textup {M}_\\odot $ for red and blue galaxies, respectively.",
"The Mass limit, Median, and age-based models have $z=1$ descendant median dark matter halo masses of $\\textup {M}\\simeq 10^{11.9}\\,\\textup {M}_\\odot $ , consistent with the blue galaxies from the DEEP2 survey.",
"The DEEP2 red galaxy subset are not consistent with being descendants of our LAE catalogs due to their larger dark matter mass.",
"LAEs at $z=3.1$ are therefore possible progenitors of blue (late-type) galaxies residing in dark matter halos with mass $10^{12}\\,\\textup {M}_\\odot $ at $z\\simeq 1$ .",
"We find that $\\sim 65\\%$ of the descendants of $z=3.1$ LAEs are centrals at $z=1$ ; it is unclear whether the blue DEEP2 galaxies represent a similar mix of centrals and satellites.",
"The models that best represent LAEs at $z=3.1$ produce $z=0$ descendants with a median mass around $10^{12.7}\\,\\textup {M}_\\odot $ .",
"We find that $\\sim 55\\%$ of the descendants at $z=0$ are central sub-halos, with little variation on the fraction of the central descendants based on model selection.",
"When we study only centrals, the age-based models have median dark matter halo masses at $z=0$ of $10^{12}\\,\\textup {M}_\\odot $ while the Median Mass and Mass limit catalogs have slightly lower values of $10^{11.8}\\,\\textup { M}_\\odot $ .",
"We see no significant dependence on age definition or selection of the descendants' mass distributions.",
"These results show that the Mass Limit, Median Mass, and age-based catalogs' descendants have a majority of central sub-halos which reside in $L^*$ type dark matter halos at $z=0$ .",
"However, the $+\\sigma $ and $-\\sigma $ models have descendant masses that are a factor of three larger and seven times smaller than the median mass of the Median model.",
"The factor of five spread in median mass of the $+\\sigma $ and $-\\sigma $ catalogs' present-day descendants shows that a more precise measurement of the $z\\simeq 3$ LAE bias is needed to place a stronger constraint on the descendant properties.",
"J. W. would like to thank Peter Kurczynski, Viviana Acquaviva, Caryl Gronwall, Lucia Guaita and Michael J. Berry for helpful comments of numerous drafts.",
"This study was supported by grants from the National Science Foundation AST-0807570, 1055919 and Department of Energy DE-FG02-08ER41561.",
"The Millennium Simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory.",
"E.G.",
"thanks the U. C. Davis Physics Department for hospitality during the completion of this research.",
"lccccccccc 10 0pc Mass Properties and correlation lengths for $z= 3.1$ LAE models Namea 25th % Massb Median Massb 75th % Massb $r_0$ c $\\gamma $ c $\\chi ^2$ /D.O.F.",
"$r_0$ d $\\gamma $ d $\\chi ^2$ /D.O.F.",
"$\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) Mpc Mpc +1$\\sigma $ 11.36 11.40 11.44 4.67$^{+0.28}_{-0.25}$ 1.80 66.53/18 4.37$^{+0.16}_{-0.15}$ 1.34$^{+0.10}_{-0.09}$ 9.29/17 Mass Limit 10.71 10.87 11.15 3.88$^{+0.30}_{-0.28}$ 1.80 67.63/18 3.33$^{+0.17}_{-0.17}$ 1.24$^{+0.10}_{-0.09}$ 6.80/17 Median Mass 10.89 10.90 10.91 3.52$^{+0.21}_{-0.21}$ 1.80 37.42/18 3.15$^{+0.16}_{-0.18}$ 1.40$^{+0.12}_{-0.12}$ 8.95/17 -1$\\sigma $ 10.00 10.00 10.00 2.38$^{+0.25}_{-0.25}$ 1.80 44.43/18 1.60$^{+0.32}_{-0.35}$ 1.14$^{+0.19}_{-0.19}$ 10.40/17 10lFormation Age Young Formation 10.74 10.95 11.27 4.25$^{+0.28}_{-0.28}$ 1.80 67.12/18 3.83$^{+0.17}_{-0.18}$ 1.29$^{+0.11}_{-0.10}$ 9.65/17 Median Formation 10.72 10.89 11.17 3.58$^{+0.25}_{-0.25}$ 1.80 54.55/18 3.08$^{+0.17}_{-0.18}$ 1.29$^{+0.11}_{-0.10}$ 7.59/17 10lAssembly Age Young Assembly 10.75 10.96 11.28 4.37$^{+0.32}_{-0.30}$ 1.80 84.68/18 3.90$^{+0.20}_{-0.21}$ 1.25$^{+0.12}_{-0.12}$ 13.05/17 Median Assembly 10.73 10.90 11.18 3.56$^{+0.18}_{-0.21}$ 1.80 35.41/18 3.19$^{+0.17}_{-0.18}$ 1.41$^{+0.11}_{-0.11}$ 8.41/17 10lMerger Age Young Merger 10.72 10.91 11.19 4.65$^{+0.25}_{-0.28}$ 1.80 62.75/18 4.31$^{+0.14}_{-0.14}$ 1.33$^{+0.09}_{-0.09}$ 7.42/17 Merger Median 10.71 10.87 11.15 5.04$^{+0.39}_{-0.37}$ 1.80 121.25/18 4.66$^{+0.16}_{-0.16}$ 1.17$^{+0.09}_{-0.09}$ 8.61/17 aAll models have a number density of $1.5\\times 10^{-3}\\,\\textup {Mpc}^{-3}$ .",
"$+1\\sigma $ : Centered around $\\log _{10}\\textup {M}=11.4\\textup {M}_\\odot $ ; Mass Limit: Halos with M$\\ge 3.98\\times 10^{10}\\,\\textup {M}_\\odot $ ; Median Mass: Centered around log$_{10}$ M=10.9 M$_{\\odot }$ ; $-1\\sigma $ : Centered around $\\log _{10}\\textup {M}=10.0\\,\\textup {M}_{\\odot }$ ; Young Formation: Youngest sub-halos based on Formation age; Median Formation: Centered around the median 0.744 Gyr.",
"; Young Assembly: Youngest sub-halos based on Assembly age; Median Assembly: Centered around the median 0.943 Gyr.",
"; Young Merger: Youngest sub-halos based on Merger age.",
"; Median Merger: Centered around the median 1.090 Gyr.",
"bMass reported is the mass of the FOF halo where the sub-halo resides, $\\textup {M}_{\\textup {FOF}}$ .",
"cParameters for fits using fixed $\\gamma =1.8$ .",
"dParameters for fits allowing both $r_0$ and $\\gamma $ as free parameters.",
"lccccccccc 10 Age Properties of $z=3.1$ LAE models Namea Form.",
"25th Form.",
"50th Form.",
"75th Assem.",
"25th Assem.",
"50th Assem.",
"75th Merg.",
"25th Merg.",
"50th Merg.",
"75th Gyr Gyr Gyr Gyr Gyr Gyr Gyr Gyr Gyr +1$\\sigma $ 0.523 0.696 0.857 0.644 0.864 1.033 0.583 0.978 1.454 Mass Limit 0.569 0.748 0.921 0.724 0.946 1.110 0.583 1.090 1.527 Median Mass 0.580 0.766 0.942 0.754 0.964 1.128 0.583 1.090 1.527 -1$\\sigma $ 0.675 0.877 1.066 0.915 1.080 1.218 0.725 1.288 1.802 10lFormation Age Young Formation 0.156 0.258 0.301 0.164 0.289 0.354 0.267 0.856 1.454 Median Formation 0.738 0.747 0.755 0.831 0.905 1.018 0.583 0.978 1.454 10lAssembly Age Young Assembly 0.158 0.270 0.324 0.164 0.289 0.345 0.583 1.090 1.593 Median Assembly 0.684 0.785 0.847 0.936 0.946 0.956 0.583 0.978 1.454 10lMerger Age Young Merger 0.419 0.553 0.686 0.693 0.897 1.069 0.091 0.091 0.091 Merger Median 0.640 0.832 0.981 0.749 0.970 1.120 0.978 1.090 1.090 aModel descriptions found in Table .",
"Table reports the 25th, 50th and 75th percentiles for each model using the Formation (Form.",
"), Assembly (Assem.)",
"and Merger (Merg.)",
"age definitions (See section REF ).",
"Bold entries mark the values in the age definition used to select the age-based models.",
"lccccc 6 0pc $z=2.1$ Descendant Mass Distributions Namea Distribution Typeb Fractionc 25th % Massd Median Massd 75th % Massd $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) +1$\\sigma $ Full 100% 11.54 11.66 11.94 Central 76% 11.51 11.60 11.72 Satellite 24% 12.05 12.33 12.80 Mass Limit Full 100% 10.94 11.25 11.77 Central 76% 10.88 11.10 11.42 Satellite 24% 11.64 12.06 12.57 Median Mass Full 100% 11.01 11.13 11.50 Central 73% 10.99 11.06 11.18 Satellite 27% 11.58 11.92 12.41 -1$\\sigma $ Full 100% 10.09 10.20 10.65 Central 72% 10.06 10.13 10.24 Satellite 28% 10.76 11.24 11.95 Young Formation Full 100% 11.01 11.36 11.88 Central 72% 10.93 11.18 11.59 Satellite 28% 11.50 11.91 12.51 Median Formation Full 100% 10.94 11.25 11.74 Central 77% 10.88 11.11 11.43 Satellite 23% 11.65 12.06 12.52 Young Assembly Full 100% 11.01 11.36 11.91 Central 72% 10.92 11.19 11.58 Satellite 28% 11.52 11.94 12.55 Median Assembly Full 100% 10.97 11.26 11.71 Central 77% 10.91 11.12 11.44 Satellite 23% 11.57 11.97 12.47 Young Merger Full 100% 10.96 11.30 11.79 Central 75% 10.90 11.12 11.48 Satellite 25% 11.64 12.04 12.56 Merger Median Full 100% 10.94 11.24 11.76 Central 76% 10.89 11.10 11.40 Satellite 24% 11.65 12.06 12.58 aModel descriptions found in Table .",
"bCentrals are the most massive sub-halos within their FOF halo.",
"Satellites are all other sub-halos.",
"Full describes the properties of all descendant sub-halos.",
"The median mass values for the central and satellite distributions are shown in Figures REF and REF .",
"cFraction of objects within full, central or satellite subset.",
"dMass reported is the mass of the FOF halo where the sub-halo resides, $\\textup {M}_{\\textup {FOF}}$ .",
"lccccc 6 0pc $z=1$ Descendant Mass Distributions Namea Distribution Typeb Fractionc 25th % Massd Median Massd 75th % Massd $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) +1$\\sigma $ Full 100% 11.83 12.21 12.90 Central 63% 11.74 11.92 12.20 Satellite 37% 12.70 13.10 13.54 Mass Limit Full 100% 11.30 11.86 12.68 Central 62% 11.12 11.43 11.88 Satellite 38% 12.23 12.81 13.42 Median Mass Full 100% 11.27 11.68 12.53 Central 61% 11.18 11.34 11.63 Satellite 39% 12.21 12.74 13.33 -1$\\sigma $ Full 100% 10.29 10.70 11.93 Central 61% 10.21 10.34 10.61 Satellite 39% 11.43 12.18 13.04 Young Formation Full 100% 11.40 12.00 12.79 Central 60% 11.17 11.54 12.05 Satellite 40% 12.27 12.80 13.44 Median Formation Full 100% 11.30 11.84 12.63 Central 63% 11.13 11.45 11.89 Satellite 37% 12.21 12.78 13.38 Young Assembly Full 100% 11.40 12.00 12.79 Central 60% 11.18 11.55 12.06 Satellite 40% 12.26 12.79 13.44 Median Assembly Full 100% 11.32 11.83 12.61 Central 63% 11.15 11.46 11.87 Satellite 37% 12.19 12.71 13.38 Young Merger Full 100% 11.34 11.91 12.67 Central 62% 11.16 11.49 11.97 Satellite 38% 12.19 12.75 13.39 Merger Median Full 100% 11.27 11.83 12.61 Central 63% 11.12 11.42 11.87 Satellite 37% 12.23 12.78 13.45 aModel descriptions found in Table .",
"bAs defined in Table 3.",
"The median mass values for the central and satellite distributions are shown in Figures REF and REF .",
"cFraction of objects within full, central or satellite subset.",
"dMass reported is the mass of the FOF halo where the sub-halo resides, $\\textup {M}_{\\textup {FOF}}$ .",
"lccccc 6 0pc $z=0$ Descendant Mass Distributions Namea Distribution Typeb Fractionc 25th % Massd Median Massd 75th % Massd $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) $\\log _{10}$ (M$_\\odot $ ) +1$\\sigma $ Full 100% 12.26 12.95 13.83 Central 60% 12.05 12.38 12.93 Satellite 40% 13.41 13.88 14.36 Mass Limit Full 100% 11.75 12.65 13.69 Central 58% 11.42 11.93 12.61 Satellite 42% 13.04 13.70 14.32 Median Mass Full 100% 11.68 12.51 13.57 Central 55% 11.44 11.77 12.32 Satellite 45% 12.97 13.60 14.13 -1$\\sigma $ Full 100% 10.63 11.71 13.26 Central 57% 10.42 10.73 11.50 Satellite 43% 12.29 13.24 13.96 Young Formation Full 100% 11.91 12.84 13.74 Central 57% 11.53 12.07 12.86 Satellite 43% 13.08 13.70 14.23 Median Formation Full 100% 11.79 12.64 13.66 Central 56% 11.45 11.94 12.55 Satellite 44% 12.99 13.63 14.22 Young Assembly Full 100% 11.92 12.85 13.76 Central 57% 11.56 12.08 12.86 Satellite 43% 13.09 13.72 14.32 Median Assembly Full 100% 11.79 12.60 13.58 Central 58% 11.47 11.92 12.52 Satellite 42% 13.00 13.63 14.20 Young Merger Full 100% 11.83 12.71 13.70 Central 58% 11.47 11.99 12.67 Satellite 42% 13.03 13.71 14.20 Merger Median Full 100% 11.75 12.56 13.73 Central 58% 11.42 11.88 12.51 Satellite 42% 12.96 13.77 14.37 aModel descriptions found in Table .",
"bAs defined in Table 3.",
"The median mass values for the central and satellite distributions are shown in Figures REF and REF .",
"cFraction of objects within full, central or satellite subset.",
"dMass reported is the mass of the FOF halo where the sub-halo resides, $\\textup {M}_{\\textup {FOF}}$ ."
]
] | 1204.1368 |
[
[
"Herschel and JCMT observations of the early-type dwarf galaxy NGC 205"
],
[
"Abstract We present Herschel dust continuum, James Clerk Maxwell Telescope CO(3-2) observations and a search for [CII] 158 micron and [OI] 63 micron spectral line emission for the brightest early-type dwarf satellite of Andromeda, NGC 205.",
"While direct gas measurements (Mgas ~ 1.5e+6 Msun, HI + CO(1-0)) have proven to be inconsistent with theoretical predictions of the current gas reservoir in NGC 205 (> 1e+7 Msun), we revise the missing interstellar medium mass problem based on new gas mass estimates (CO(3-2), [CII], [OI]) and indirect measurements of the interstellar medium content through dust continuum emission.",
"Based on Herschel observations, covering a wide wavelength range from 70 to 500 micron, we are able to probe the entire dust content in NGC 205 (Mdust ~ 1.1-1.8e+4 Msun at Tdust ~ 18-22 K) and rule out the presence of a massive cold dust component (Mdust ~ 5e+5 Msun, Tdust ~ 12 K), which was suggested based on millimeter observations from the inner 18.4 arcsec.",
"Assuming a reasonable gas-to-dust ratio of ~ 400, the dust mass in NGC 205 translates into a gas mass Mgas ~ 4-7e+6 Msun.",
"The non-detection of [OI] and the low L_[CII]-to-L_CO(1-0) line intensity ratio (~ 1850) imply that the molecular gas phase is well traced by CO molecules in NGC 205.",
"We estimate an atomic gas mass of 1.5e+4 Msun associated with the [CII] emitting PDR regions in NGC 205.",
"From the partial CO(3-2) map of the northern region in NGC 205, we derive a molecular gas mass of M_H2 ~ 1.3e+5 Msun.",
"[abridged]"
],
[
"Introduction",
"[1]Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.",
"Eighty-five percent of all galaxies are located outside galaxy clusters, among which half reside in groups [68].",
"Therefore, studying group environments is of great importance to learn more about the habitats for an important fraction of galaxies.",
"The Local Group is of particular interest for studies of low surface brightness galaxies, since galaxies at the low luminosity end often remain undetected in more distant group structures.",
"Moreover, the Local Group allows probing the interstellar medium (ISM) of its residents at high resolution and, at the same time, offers a wealth of ancillary data being one of the best studied areas on the sky.",
"Studying the properties of the ISM and metal-enrichment in metal-poor dwarf galaxies offers a promising way to learn more about the conditions in the early Universe and the evolution of dwarf galaxies throughout the history of the universe.",
"Among the low surface brightness galaxies in the Local Group, the dwarf satellite NGC 205 ($\\sim $ 824 kpc, [94]) of the Andromeda galaxy is of particular importance due to its relatively low metal abundance (Z $\\sim $ 0.13 Z$_{\\odot }$ , [112]), interesting star formation history and indications of a tidal encounter with its massive companion M31.",
"Although photometrically classified as a dwarf elliptical galaxy [32], the formation processes for NGC 205 seem more closely related to the dwarf spheroidal galaxy population (transformed from late-type galaxies through internal and environmental processes) rather than to merger remnants, thought to be the main driver for the formation of genuine ellipticals [73].",
"The star formation history in NGC 205 has been studied extensively [1], [60], [8], [26].",
"An old stellar population (10 Gyr, [9]) dominates the overall stellar content of the dwarf galaxy and a plume of bright blue star clusters in the central region of NGC 205 was already identified $\\sim $ 60 years ago [1], [60].",
"Combining observations of this young stellar population with adequate model assumptions has provided several independent theoretical predictions of the current gas content in NGC 205.",
"Such predictions need to account for both the left-over gas reservoir after an epoch of star formation activity and the build-up of gas returned to the ISM by the evolved stellar population since the last starburst episode.",
"The left-over gas reservoir is probed through observations of the young stellar population providing an estimate of the total gas mass consumed during the last epoch of star formation.",
"Usually, it is assumed that a star formation efficiency of 10$\\%$ is reasonable (e.g.",
"[88], [18]).",
"From observations of both the nucleus and a region about 1$$ north of the nucleus with the International Ultraviolet Explorer (IUE) within an aperture of 10$$ $\\times $ 20$$ , [133] report a mass of young ($\\sim $ 10 Myr old) stars $M_{\\star }$ $\\sim $ 7 $\\times $ 10$^5$ M$_{\\odot }$ in NGC 205, which increases to $M_{\\star }$ $\\sim $ 1.4 $\\times $ 10$^6$ M$_{\\odot }$ when extrapolating to the whole galaxy based on the stellar light contribution from OB stars [133] and also taking stellar masses $<$ 1 M$_{\\odot }$ into account [87].",
"According to [8] NGC 205 was the host of a starburst starting $\\sim $ 500 Myr ago involving a total stellar burst mass of $M_{\\star }$ $\\sim $ 5.3 $\\times $ 10$^7$ M$_{\\odot }$ , which is considerably higher than the estimate in [133].",
"More recently, [98] estimate $M_{\\star }$ $\\sim $ 1.9 $\\times $ 10$^5$ M$_{\\odot }$ of stars to be produced between $\\sim $ 62 Myr and $\\sim $ 335 Myr ago from observations of the nuclear 29$$ $\\times $ 26$$ region of NGC 205 with the Advanced Camera for Survey (ACS) on board the Hubble Space Telescope (HST), when assuming a $\\Lambda $ CDM cosmological model.",
"Although the estimated burst mass in [98] also includes lower mass stars ($M_{\\star }$ $<$ 1 M$_{\\odot }$ ), their value should be considered a lower limit of the total burst mass, since only a limited period (from 62 to 335 Myr ago) in the star formation history of NGC 205 is analyzed and the observed area only corresponds to part of the region where the most recent star formation epoch took place.",
"The total amount of gas returned to the ISM by planetary nebulae is predicted to be $\\sim $ 1.8 $\\times $ 10$^6$ M$_{\\odot }$ [132], following the prescriptions in [38] and assuming a time lapse of $\\sim $ 500 Myr since the trigger of the last star formation activity.",
"With the lower limit for the stellar burst estimates relying on 10 Myr old stars [133], the corresponding mass returned to the ISM since the formation of those young stellar objects can be predicted in a similar way ($\\sim $ 3.6 $\\times $ 10$^4$ M$_{\\odot }$ ).",
"We only account for the mass loss from planetary nebulae since the mass lost from more massive stars is considered negligible due to their lower mass loss rate and shorter lifetime.",
"Considering that the estimated burst mass critically depends on the model assumptions and is often biased by the sensitivity and coverage of the observations, we calculate a total burst mass during the last episode of star formation in the range 1.4 $\\times $ 10$^6$ M$_{\\odot }$ $\\le $ $M_{\\star }$ $\\le $ 5.3 $\\times $ 10$^7$ M$_{\\odot }$ , where the lower and upper limits correspond to stellar burst mass predictions from [133] and [8], respectively.",
"A burst mass of 1.4 $\\times $ 10$^6$ M$_{\\odot }$ would predict that the initial gas reservoir before the star formation (SF) episode was $M_{g}$ $\\sim $ 1.4 $\\times $ 10$^7$ M$_{\\odot }$ for a star formation efficiency (SFE) of $\\sim $ 10$\\%$ .",
"Subtracting the 10$\\%$ of gas consumed into stars results in the left-over gas reservoir $M_{\\text{g}}$ $\\sim $ 1.3 $\\times $ 10$^7$ M$_{\\odot }$ after the star formation epoch.",
"When combining the left-over gas reservoir (i.e.",
"the majority of the ISM mass) and mass loss by planetary nebulae (depending on the assumed time lapse), we estimate a current gas content for NGC 205 ranging between 1.3 $\\times $ 10$^7$ M$_{\\odot }$ $\\le $ $M_{\\text{g}}$ $\\le $ 4.8 $\\times $ 10$^8$ M$_{\\odot }$ ."
],
[
"Observations of the ISM content",
"Up to now, the total gas mass in NGC 205 was estimated from Hi, CO(1-0) and dust continuum observations.",
"A total Hi mass of 4.0 $\\times $ 10$^5$ M$_{\\odot }$ , scaled to a distance $D$ = 824 kpc for NGC 205, is reported in [137] based on VLA observations covering the whole galaxy.",
"[132] detect CO(1-0) and CO(2-1) emission above the 3$\\sigma $ level from 3 and 4 positions, respectively, across the plane of NGC 205.",
"Although a partial beam overlap occurs for the CO(1-0) observations (see red circles in Figure REF ), the covered area in the CO(1-0) observations is 7 times larger than for the second CO(2-1) transition and, thus, provides a better estimate of the molecular gas content in NGC 205.",
"While [132] assumed a close to solar metallicity (implying a CO-to-$H_{\\text{2}}$ conversion factor $X_{\\text{CO}}$ $=$ 2.3 $\\times $ 10$^{20}$ cm$^{-2}$ (K km s$^{-1}$ )$^{-1}$ , [124]), we apply a conversion factor of 6.6 $\\times $ 10$^{20}$ cm$^{-2}$ (K km s$^{-1}$ )$^{-1}$ , determined from the expression reported in [14] relating the $X_{\\text{CO}}$ factor to the oxygen abundance in a galaxy.",
"This value for the $X_{\\text{CO}}$ factor is based on a metal abundance of $Z$ $\\sim $ 0.3 Z$_{\\odot }$ in the inner regions of NGC 205.",
"Whereas the earlier reported metallicity value (Z $\\sim $ 0.13 Z$_{\\odot }$ ) was obtained from averaging the oxygen abundances for 13 planetary nebulae in NGC 205 [112] and thus refers to the whole galaxy, [117] determined a mean metallicity [Z/H] $\\sim $ - 0.5 $\\pm $ 0.2 for the central regions in NGC 205 from Lick indices.",
"Considering that the last episode of star formation mainly occurred in the inner most regions, this gradient in metallicity and/or age is not surprising.",
"A similar central increase in colour and metallicity has been noticed in several early-type dwarf galaxies in the Fornax cluster [72] and a population of dEs with central blue cores has been observed in the Virgo cluster [79].",
"Additional CO(1-0) line emission was detected from a small area (beamsize $\\sim $ 21$$ ) observed in the south of NGC 205 [136] (see green circle in Figure REF ).",
"Combining those CO(1-0) observations, we obtain an estimate of $M_{\\text{H}_{2}}$ $=$ 6.9 $\\times $ 10$^5$ M$_{\\odot }$ .",
"This value for the molecular gas mass is however only a lower limit of the total H$_{2}$ mass, since the southern part of the galaxy is poorly covered by current CO(1-0) observations.",
"Combining both Hi and CO(1-0) observations, which were scaled by a factor of 1.4 to include helium, we derive a total gas mass $M_{g}$ $\\sim $ 1.5 $\\times $ 10$^{6}$ M$_{\\odot }$ for NGC 205, which is about one order of magnitude lower than the more modest theoretical predictions for the gas content.",
"This deficiency in the gas content of NGC 205 is often referred to as the problem of the “missing ISM mass\" [132].",
"Although gas observations directly probe the ISM component of interest, the dependence of the $X_{\\text{CO}}$ conversion factor on the metallicity [134], [13] and density of the gas [118], and the optical thickness of the CO(1-0) line introduce an uncertainty on the estimate of the total gas mass.",
"In particular for metal-poor galaxies the molecular gas phase could be poorly traced by CO [134], [82], [75], [76].",
"An alternative and promising method to measure the ISM mass in galaxies is to use observations of the continuum emission from dust ([58], [55], [56], [14], [66], [37], Eales et al.",
"in prep.).",
"Dust emission from NGC 205 was first detected with IRAS [111].",
"NGC 205 was also the first early-type dwarf galaxy detected at millimeter wavelengths [39].",
"Based on these IRAS data and 1.1 mm observations for the central 18$$ (21 $\\pm $ 5 mJy) in NGC 205, [39] estimated a dust mass of $M_{\\text{d}}$ $\\sim $ 3 $\\times $ 10$^{3}$ M$_{\\odot }$ at a temperature of $\\sim $ 19K.",
"Using ISO observations, [57] obtained a total dust mass estimate of 4.9 $\\times $ 10$^3$ M$_{\\odot }$ at a temperature of $\\sim $ 20 K. Recently, [87] probed the dust emission from NGC 205 with Spitzer and found a dust mass in the range $M_{\\text{d}}$ = 3.2-6.1 $\\times $ 10$^{4}$ M$_{\\odot }$ at a temperature of $\\sim $ 18 K. Taking the 1.1 mm observation of the core region into account, [87] found a dust component in the central regions at a temperature of $T_{d}$ $\\sim $ 11.6, sixteen times more massive, suggesting that a substantial amount of cold dust might be overlooked if one only takes IRAS, ISO and Spitzer observations into consideration.",
"Unfortunately only the central region was observed at 1.1 mm, and it could not be investigated whether such a putative cold dust component is present over the entire galaxy.",
"Under the assumption that the colder dust is not only distributed in the central region, but is abundantly present in the entire galaxy, they estimated a total gas mass of 5 $\\times $ 10$^{7}$ M$_{\\odot }$ for a gas-to-dust ratio of 100.",
"To probe this cold dust component, we need observations at wavelengths longwards of 160 $\\mu $ m (e.g.",
"[52], [42]).",
"Longer wavelength data also allow us to constrain the Rayleigh-Jeans side of the dust SED, from which a more robust temperature estimate for our SED model can be obtained.",
"Probing this cold dust component is now possible with the Herschel Space Telescope [108], covering a wavelength range from 70 up to 500 $\\mu $ m. Recent Herschel observations of nearby galaxies have demonstrated the presence of significant amounts of cold dust (e.g.",
"[5], [119], [6], [12]; Fritz et al., subm.).",
"In this paper, we present Herschel observations for NGC 205 taken in the frame of the VNGS and HELGA projects, with the aim of making an inventory of all the dust in NGC 205.",
"Furthermore, we report new gas mass measurements from James Clerk Maxwell Telescope (JCMT) CO(3-2) observations and Herschel [Cii] 158 $\\mu $ m and [Oi] 63 $\\mu $ m line spectroscopic mapping.",
"From those new dust and gas mass estimates, we are able to revise the “missing ISM\" problem in NGC 205.",
"In Section , the data and observing strategy from Herschel and JCMT observations are discussed.",
"The corresponding data reduction procedures are outlined and a brief overview of the ancillary dataset is given.",
"Section discusses the spatial distribution of gas and dust in NGC 205 ($§$ REF ), the global flux measurements ($§$ REF ) and the basic principles and results of the SED fitting procedure ($§$ REF ).",
"The JCMT CO(3-2) and PACS spectroscopy data are analyzed in Section REF and new estimates for the gas mass are determined.",
"Section reanalyses the missing ISM mass problem in NGC 205 ($§$ REF ), discusses the implications of our results for the SF conditions and properties of the ISM in the galaxy ($§$ REF ) and makes a comparison with the other early-type dwarf companions of Andromeda ($§$ REF ).",
"Finally, Section summarizes our conclusions."
],
[
"PACS Photometry",
"We use data for NGC 205 taken as part of two Herschel Guaranteed Time Projects: the Very Nearby Galaxy Survey (VNGS, PI: C. Wilson) and the Herschel Exploitation of Local Galaxy Andromeda (HELGA, PI: J. Fritz).",
"From the VNGS, we obtained PACS photometry [110] at 70 and 160 $\\mu $ m (ObsID 1342188692, 1342188693 ) and SPIRE photometry [53] at 250, 350 and 500 $\\mu $ m. PACS data were observed on the 29th of December 2009 and cover an area of 1$^{\\circ }$ by 1$^{\\circ }$ centered on NGC 205.",
"This area was observed in nominal and orthogonal scan direction with four repetitions at a medium scan speed (20$$ /s).",
"The main scientific objective of HELGA (Fritz et al. subm.)",
"focuses on dust in the extreme outskirts of the Andromeda Galaxy.",
"Thanks to its large survey area, NGC 205 was covered in the field of two overlapping scan observations.",
"HELGA observations were performed in the parallel fast scan mapping mode (60$$ /s), obtaining PACS 100, 160 $\\mu $ m and SPIRE 250, 350 and 500 $\\mu $ m photometry (ObsID 1342211294, 1342211309, 1342211319, 1342213207).",
"To reduce the PACS data, we used version 13 of the Scanamorphos (Roussel et al.",
"in prep., http://www2.iap.fr/users/roussel/herschel) map making technique.",
"Before applying Scanamorphos to the level 1 data, the raw data were pre-processed in Herschel Interactive Processing Environment (HIPE, [105]) version HIPE 6.0.1196.",
"Due to the different observing set-ups, the depth of the PACS observations is inhomogeneous among the different wavebands.",
"For PACS observations at 70 and 100 $\\mu $ m, we only have data available from one survey (either VNGS or HELGA, respectively).",
"This dataset was finally reduced to obtain maps with a pixel size of 2$$ .",
"The FWHM of the PACS beams are $\\sim $ 6$$ and $\\sim $ 7$$$\\times $ 13$$ at 70 and 100 $\\mu $ m, respectively.",
"Due to the lower level of redundancy and the fast scan speed, the PACS 100 $\\mu $ m waveband was observed in the least favorable conditions, resulting in the largest uncertainty values and interference patterns affecting the observations (see Figure REF , second panel on the top row).",
"In the red filter (PACS 160 $\\mu $ m), NGC 205 was covered by both VNGS and HELGA in medium and fast scan speed, resulting in a FWHM for the PACS beam of $\\sim $ 12$$ and $\\sim $ 12$$$\\times $ 16$$ , respectively.",
"Our final photometry map combines data from both Herschel projects at this overlapping wavelength with the aim of increasing the signal to noise ratio.",
"The unmatched scan speeds prevent reducing both observations simultaneously, since the drift correction is calculated over a certain stability length which depends on the scan speed of the observation.",
"Therefore, both datasets were reduced individually in Scanamorphos, using the same astrometry for the final maps.",
"Before combining both maps we convolved them to the same resolution of the PSF, to avoid issues with the different beam sizes in both observations.",
"Finally, the separately reduced and convolved VNGS and HELGA maps at 160 $\\mu $ m were combined into one single map in IRAF, with the imcombine task.",
"The images were produced with the latest version of the PACS calibration files (version 26) and divided by the appropriate colour correction factors (1.016, 1.034 and 1.075 at 70, 100 and 160 $\\mu $ m, see [101] for a power-law spectrum with $\\beta $ $=$ 2).",
"The background was also subtracted from the final PACS images.",
"An estimate for the background was obtained from averaging the background flux within 100 random apertures (diameter $=$ 4$\\times $ FWHM).",
"Random apertures were selected within an annulus centered on NGC 205 with inner radius of 5$$ and outer radius of 20$$ , avoiding regions with bright emission from background sources and M31.",
"Once the random background apertures were selected, those same positions for aperture photometry were applied at all wavelengths."
],
[
"SPIRE Photometry",
"SPIRE data were observed on the 27th of December 2009 (ObsID 1342188661), obtaining two repetitions of nominal and orthogonal scans at medium scan speed (30$$ /s).",
"For all SPIRE bands, datasets from both VNGS and HELGA projects were available and combined into one frame.",
"In the same way as for Scanamorphos, the continuous temperature variations are different for VNGS and HELGA observations.",
"Therefore, the corresponding data are reduced separately before combining them into one map.",
"The SPIRE data were largely reduced according to the standard pipeline (POF5_pipeline.py, dated 8 Jun 2010), provided by the SPIRE Instrument Control Centre (ICC).",
"Divergent from the standard pipeline were the use of the sigmaKappaDeglitcher (instead of the ICC-default waveletDeglitcher) and the BriGAdE method (Smith et al.",
"in prep.)",
"to remove the temperature drift and bring all bolometers to the same level (instead of the default temperatureDriftCorrection and the residual, median baseline subtraction).",
"Reduced SPIRE maps have pixel sizes of 6$$ , 8$$ and 12$$ at 250, 350 and 500 $\\mu $ m, respectively.",
"The FWHM of the SPIRE beams are 18.2$$ , 24.5$$ and 36.0$$ at 250, 350 and 500 $\\mu $ m, respectively.",
"SPIRE images are converted from MJy/beam to Jy/pixel units assuming beam areas of 423, 751 and 1587 arcsec$^2$ at 250, 350 and 500 $\\mu $ m, respectively.",
"Since the calibration in the standard pipeline is optimized for point sources, we apply correction factors (0.9828, 0.9834 and 0.9710 at 250, 350 and 500 $\\mu $ m) to convert the K4 colour correction factors from point source to extended source calibration [122].",
"Additionally, multiplicative colour correction factors (0.9924, 0.9991 and 1.0249 at 250, 350 and 500 $\\mu $ m) were applied and a correction factor of 1.0067 was used to update the fluxes in the 350 $\\mu $ m image to the latest v7 calibration product [122].",
"SPIRE images were also background subtracted, in a similar way as for the PACS images."
],
[
"PACS spectroscopy",
"PACS spectroscopy maps, [Cii] 157.74 $\\mu $ m and [Oi] 63 $\\mu $ m, were observed on the 14th of February 2011 (ObsID 1342214374, 1342214375, 1342214376).",
"The [Cii] observations cover the northern and the central area of NGC 205 (see Figure REF ), while two smaller [Oi] maps are centered on the CO peak in the north and on the centre of NGC 205 (see Figure REF ).",
"The PACS [110] spectroscopic observations of NGC205 were done in the chop/nod mode, and cover an area of $95 \\times 95$ arcsec ($3 \\times 3$ pointings) and $47 \\times 47$ arcsec (1 pointing) for the [Cii] (157.74 $\\mu $ m) and both [Oi] (63 $\\mu $ m) maps, respectively.",
"We used the largest chop throw of 6 arcmin to ensure we were not chopping onto extended source emission.",
"The maps were processed from Level 0 to Level 2 using the standard pipeline in HIPE (version 7.0.0), with version (FM, 32) of the calibration files.",
"Once the data are processed to Level2, we used the PACSman programPACSman is available for download at http://www.myravian.fr/Homepage/Softwares.html.",
"[74] to perform line fits to the unbinned spectral data in each spatial pixel using a least-squares fitting routine, and to create integrated flux density maps of the results.",
"For [Cii], the flux map mosaic was created by projecting the individual rasters onto an oversampled grid.",
"More details can be found in [74].",
"Figure: Left panel: KK band image of NGC 205.",
"Right panel: PACS 160 μ\\mu m image overlaid with the AORs for the [Oi] (black, solid lineboxes) and [Cii] (raster with white, solid lines) PACS line spectroscopy observations.",
"The [Oi] line was observed in the north of NGC 205, where the CO(1-0) emission peaks in the north, and in the central region of NGC 205.",
"The [Cii] observations cover both areas."
],
[
"Noise calculations",
"To determine the uncertainty on the PACS and SPIRE photometry data, we need to take into account three independent noise measurements.",
"Aside from the most important uncertainty factor due to the calibration, the random background noise and the map making uncertainty contribute to the flux uncertainty as well.",
"For the calibration uncertainty, values of 3, 3, and 5 $\\%$ were considered for the PACS 70, 100 and 160 $\\mu $ m data, respectively (PACS Observer's Manual 2011).",
"All SPIRE wavebands were assumed to have a calibration uncertainty of 7 $\\%$ ([125], SPIRE Observer's Manual 2010).",
"An estimate for the background noise is derived by taking 100 random apertures in the field around the galaxy (i.e.",
"the same apertures used to determine the background), calculating the mean pixel value within each aperture as well the average mean over all apertures and, finally, computing the standard deviation of the mean pixel values in those individual apertures.",
"Map making uncertainties are derived from the error map, which is produced during the data reduction procedure.",
"Specific uncertainties in map making are determined from this error map.",
"The total uncertainty on the flux value in a pixel is calculated as the square root of the sum of the three squared error contributions.",
"Calibrations and background uncertainties are summarized in Table REF .",
"The noise level in PACS spectroscopy observations is determined from the uncertainty in the integrated intensity map, calculated from the the formula: $\\Delta I = \\left[ \\Delta v \\sigma \\sqrt{N_{line}} \\right] \\sqrt{1+\\frac{N_{line}}{N_{base}}}$ where $\\Delta v$ corresponds to the channel width in km s$^{-1}$ , $\\sigma $ is the uncertainty in K and $N_{line}$ and $N_{base}$ represent the total number of channels covering the spectral line and the channels used for the baseline fitting, respectively.",
"A 1$\\sigma $ uncertainty value is estimated from taking the mean over 15 random apertures in this uncertainty map (see Table REF ).",
"Table: The 1σ\\sigma noise levels for the PACS spectroscopy observations."
],
[
"JCMT observations and ancillary data",
"The JCMT observations of the $^{12}$ CO (3-2) transition (rest frequency 345.79 GHz) were obtained with the HARP-B instrument [20] as part of project M10AC07 (PI: Tara Parkin) over eight nights in May, June and September of 2010, with a telescope beam size of 14.5 arcsec.",
"We obtained a single map in jiggle-chop mode with a footprint of 2 arcmin $\\times $ 2 arcmin on the sky, and a total integration time of 1350 sec for each of the 16 jiggle positions.",
"The observations were carried out using beam-switching with a chop throw of 150 arcsec from the centre of NGC 205.",
"We used the Auto-Correlation Spectrometer Imaging System (ACSIS) as our backend receiver, and it was set to a bandwidth of 1 GHz with 2048 channels, resulting in a resolution of 0.43 km s$^{-1}$ .",
"The data were then reduced using the StarlinkThe Starlink package is available for download at http://starlink.jach.hawaii.edu.",
"software package [24], maintained by the Joint Astronomy Centre.",
"For a full description of our data reduction and map making methods see [130] and [107].",
"We obtained raw MIPS data for NGC 205 from the Spitzer archive, which were reprocessed according to the procedure outlined in [7].",
"An Hi map for NGC 205 obtained from VLA observations [137] was kindly provided to us by Lisa Young."
],
[
"Distribution of gas and dust in the ISM of NGC 205",
"Figure REF displays the Herschel maps for NGC 205 in the PACS 70, 100, 160 $\\mu $ m and SPIRE 250, 350 and 500 $\\mu $ m wavebands.",
"In all bands, we are able to distinguish three dominant emission regions (north, central and south), which were first identified using MIPS data by [87].",
"A substantial amount of dust also resides in between those three distinctive emission regions.",
"Towards the southeast of NGC 205 there is also an indication for a tentative detection of a colder dust component in the SPIRE maps (see the red, dashed ellipse in Figure REF ).",
"The detection is below the 3$\\sigma $ level, but it coincides with an optical tidal tail reported in [116].",
"However, we cannot rule out the possibility that the faint blob corresponds to foreground Galactic cirrus emission similar to the emission identified in the surroundings of M81 [121], [28] or to emission originating from one or multiple background sources.",
"Figure: An overview of the Herschel maps.",
"From left to right: PACS 70, 100, 160 μ\\mu m (first row) and SPIRE 250, 350, 500 μ\\mu m (second row).The displayed maps cover an area of 8.0 ×\\times 6.5, with north up and east to the left.",
"Besides the PACS 70 μ\\mu m (VNGS) and PACS 100 μ\\mu m (HELGA) maps, all images in the other bands were produced using both VNGS and HELGA observations.The FWHM of the PSF is indicated as a white circle in the lower left corner of each image.",
"The red dashed line indicates the area where there is an indication for a tentative detection of a colder dust component.When comparing the Hi, H$_{2}$ and dust distribution in NGC 205 (see Figure REF ), we find a remarkable correspondence between the peaks in Hi and H$_{2}$ (as derived from the CO(3-2) observations, see Section REF ) column density and dust emission.",
"This seems to imply that the dust component in NGC 205 is well mixed with the atomic and molecular gas at the observed spatial scales of $\\sim $ 100 pc for SPIRE.",
"While a significant part of the atomic gas resides in the area south of the galaxy's center (see Figure REF ), current CO observations only cover most of the northern part of the galaxy ([67], [137], [132], JCMT CO(3-2) map from this work) and a minor part in the south of NGC 205 [136].",
"This lack of data makes it difficult to draw conclusions about the correlation of the molecular gas component with the dust or Hi gas in the southern area of NGC 205.",
"At least for the northern part of NGC 205, it seems that also the molecular gas component correlates well with the Hi gas and dust.",
"Figure: Hi column density contours (white, solid line, ), H 2 _{2} column density contours (black, solid contours, derived from the JCMT CO(3-2) map), CO(1-0) pointings (red circles: , green circle: ) and JCMT 1.1 mm 18 beam (yellow circle, ) overlaid on the SPIRE 250 μ\\mu m image.",
"The Hi contours range from 2 ×\\times 10 19 ^{19} to 3.5 ×\\times 10 20 ^{20} cm -2 ^{-2} in intervals of 6.6 ×\\times 10 19 ^{19} cm -2 ^{-2}, while the CO(3-2) contours represent a H 2 _{2} column density range 2.6 ×\\times 10 20 ^{20} ≤\\le N H 2 N_{\\text{H}_{2}} ≤\\le 1.9 ×\\times 10 21 ^{21} cm -2 ^{-2} increased in steps of 3.3 ×\\times 10 20 ^{20} cm -2 ^{-2}."
],
[
"Global fluxes",
"Global fluxes in all wavebands were determined from summing over all pixels in the background subtracted image with $>$ 3$\\sigma $ detection in the SPIRE 250 $\\mu $ m image.",
"Those global fluxes and the corresponding noise measurements (calibration uncertainty and random background noise) in each waveband are summarized in Table REF .",
"Flux measurements have been updated to the last PACS and SPIRE calibration products, converted to the extended source calibration and color corrected for each filter.",
"When comparing our fluxes at PACS 70 $\\mu $ m (2.2 $\\pm $ 0.2 Jy) and PACS 160 $\\mu $ m (4.0 $\\pm $ 0.3 Jy) to the MIPS fluxes at those overlapping wavelengths reported in [87] (MIPS 70 $\\mu $ m: 1.4 $\\pm $ 0.3 Jy; MIPS 160 $\\mu $ m: 8.8 $\\pm $ 4.7 Jy), a large discrepancy between PACS and MIPS fluxes is found.",
"In view of this large difference, we determined MIPS fluxes (MIPS 70 $\\mu $ m: 1.3 $\\pm $ 0.3 Jy; MIPS 160 $\\mu $ m: 4.6 $\\pm $ 0.9 Jy) from our reprocessed archival MIPS data by summing over the same pixels (uncertainty values only refer to calibration uncertainties).",
"To determine the corresponding PACS fluxes, the Herschel maps at 70 and 160 $\\mu $ m wavelengths were convolved with the appropriate kernels to match the elongated wings of the MIPS PSFs.",
"The customized kernels were created following the procedure in [6] (see also [51]).",
"Summing the flux values for the same pixels, we find the corresponding flux measurements for PACS 70 $\\mu $ m (2.3 $\\pm $ 0.2 Jy) and PACS 160 $\\mu $ m (3.8 $\\pm $ 0.3 Jy).",
"Upon comparison of the fluxes, we find a relatively good agreement between the PACS measurement and the flux determined from the archival MIPS image at 160 $\\mu $ m. The PACS 70 $\\mu $ m flux is sufficiently higher, which might either be a calibration issue or a dissimilarity in the background determination.",
"We argue that the deviation from the MIPS 160 $\\mu $ m measurement reported in [87] is either due to a flux calibration issue or an overestimated aperture correction.",
"Indeed, the flux calibration for MIPS [50], [123] was only finalized after the analysis in [87].",
"A similar comparison at 100 $\\mu $ m is possible between PACS and IRAS fluxes.",
"[111] report a total flux density of 3.78 $\\pm $ 0.57 Jy for NGC 205, which agrees well with the flux density 3.6 $\\pm $ 0.5 Jy determined from Herschel observations.",
"Table: Global fluxes (F ν F_{\\nu }) and contributions from the random background noise (σ back \\sigma _{back}) and the calibration uncertainty (σ cal \\sigma _{cal}) are provided for every waveband.",
"All tabulated fluxes have been multiplied by extended source calibration and filter colour correction factors."
],
[
"SED fitting method",
"For the SED fitting procedure, we apply the DustEm code [22], which predicts the emission of dust grains given the strength of the interstellar radiation field (ISRF) and a certain composition of grain types, with a specific size distribution, optical and thermal dust properties.",
"DustEm derives the local dust emissivity from computing explicitly the temperature distribution for every grain type of particular size and composition.",
"For the analysis in this paper, we adopt two different dust compositions ([35] and [22]), each containing polycyclic aromatic hydrocarbons (PAHs) and amorphous silicate grains complemented with either graphite or amorphous carbon dust particles, respectively.",
"The dust composition from [35] corresponds to the typical dust mixture found in our own Galaxy.",
"The amorphous carbonaceous grains are in the form of hydrogenated amorphous carbon, better known as a-C:H or HAC [22].",
"The spectral shape of the ISRF is assumed to be the same as determined in the solar neighborhood [90] (MMP83).",
"Although the shape and hardness of the ISRF in low-metallicity dwarf galaxies might differ from the Galactic ISRF [84], altering the spectral shape of the ISRF will in particular influence the radiation of transiently heated PAHs and very small grains but has been shown to only affect the total dust mass estimate by a factor of $<$ 10$\\%$ [40].",
"Moreover, based on the number and spectral type of young massive stars in NGC 205, [136] found an UV field corresponding well to the ISRF in the solar neighborhood.",
"Under these assumptions, we have only 2 free parameters: the dust mass $M_{\\text{d}}$ and the intensity of the ISRF $X_{\\text{ISRF}}$ relative to the Galactic ISRF.",
"During the SED fitting procedure, we explore a parameter grid in $X_{\\text{ISRF}}$ and dust mass by increasing them stepwise by a factor of 1.05.",
"In order to facilitate the least-square fitting procedure, we construct a pre-calculated library of dust models in DustEm, each with a different scaling of the ISRF and dust mass.",
"The model SED at those wavelengths is convolved with the response function of the filter-band passes, to include the appropriate colour correction (pixel values were not colour corrected in this case).",
"Finally, the model with the best fitting parameters is determined from a least-square fitting routine.",
"To estimate the uncertainties on the best fitting parameters, we perform a bootstrapping procedure.",
"Hereto, the same fitting routine is applied on a dataset of 100 flux densities, randomly determined from a Gaussian distribution for which the maximum value and width are chosen to correspond to the observed fluxes and uncertainty values.",
"The 1$\\sigma $ uncertainties on our best fitting model parameters correspond to the 16 and 84 percentiles of this Gaussian distribution.",
"Since the primary goal of this analysis is quantifying the total dust content in NGC 205, we restrict our SED fitting procedure to the wavelength range from 24 to 500 $\\mu $ m. The spectral shape throughout this wavelength range is determined from the flux measurements in the six available Herschel wavebands (PACS 70, 100, 160 $\\mu $ m and SPIRE 250, 350, 500 $\\mu $ m) and the MIPS 24 $\\mu $ m flux.",
"Prior to SED fitting, all images (Herschel+ancillary MIPS data) were convolved with the appropriate kernels, according to the procedure in [6], to match the resolution of the 500 $\\mu $ m images.",
"All images were also rebinned to the pixel scale (12$$ /pixel) of the 500 $\\mu $ m image."
],
[
"Pixel-by-pixel fitting",
"Dust masses and ISRF scaling factors are computed from a SED fitting procedure to every pixel with fluxes in at least three different bands above the 3$\\sigma $ level to constrain the spectral shape of the energy distribution.",
"In this way, we obtain 155 pixels with a sufficient signal to noise level and avoid contribution from noisy pixels (in particular at 100 $\\mu $ m) to the SED fitting procedure, which might bias our estimate of the total dust mass.",
"The uncertainty on the flux values in every pixel are calculated following the procedure outlined in Section REF .",
"Figure: The X ISRF X_{\\text{ISRF}} (left panel) and dust mass (right panel) obtained from a multi-component DustEm SED fitting procedure for every pixel with at least 3 detections above the 3σ\\sigma level across the 24 μ\\mu m to 500 μ\\mu m wavelength range.Figure REF shows the maps with the best fitting ISRF scaling factors (left) and dust column densities (right), determined from SED fitting with the silicate+amorphous carbon dust composition [22].",
"From those maps, we clearly notice that the dust in the center of NGC 205 has a higher temperature, while most of the dust mass resides in the northern and southern regions of NGC 205, corresponding to peaks in the Hi+H$_{2}$ and Hi column density, respectively.",
"For the three distinctive dust emission regions in NGC 205 the northern and southern areas are brighter because of the large amounts of dust residing in these areas, while the dust grains in the central region emit more prominently because they are exposed to a stronger ISRF.",
"Also outside these three emission regions, a substantial amount of dust is found to be present in the galaxy.",
"The total dust mass in NGC 205 is determined by adding the dust masses in all pixels, resulting in $M_{\\text{d}}$ $\\sim $ 1.8 $\\times $ 10$^{4}$ M$_{\\odot }$ .",
"An estimate for the average strength of the ISRF is calculated as the median of the values for $X_{\\text{ISRF}}$ in all pixels.",
"This median value of $X_{\\text{ISRF}}$ $\\sim $ 1.13 is translated into a large grain dust temperature $T_{d}$ $\\sim $ 17.8 K by averaging over all mean temperatures for grains with sizes between 4 $\\times $ 10$^{-3}$ and 2 $\\mu $ m for this specific strength of the ISRF.",
"When repeating the SED fitting procedure with a silicate+graphite dust composition [35], we find values for the dust mass ($M_{\\text{d}}$ $\\sim $ 1.1 $\\times $ 10$^{4}$ M$_{\\odot }$ ) and the median value of the scaling factor $X_{\\text{ISRF}}$ $\\sim $ 3.12 (or $T_{d}$ $\\sim $ 21.2 K) consistent with the results obtained for a silicate+amorphous carbon dust composition within the uncertainties of the SED fitting procedure.",
"The fitting results for both dust compositions are summarized in Table REF .",
"To check whether our results are hampered by the resolution in the SPIRE 500 $\\mu $ m waveband ($\\sim $ 36$$ ), we perform the same pixel-by-pixel SED fitting procedure at a resolution of the SPIRE 350 $\\mu $ m waveband ($\\sim $ 24.5 $$ ).",
"Whereas SED fitting on higher resolution data have shown to probe a more massive dust reservoir in the Large and Small Magellanic Clouds [47], we find best fitting values for the average scaling factor $X_{\\text{ISRF}}$ $\\sim $ 0.61$^{+0.31}_{-0.31}$ and the total dust mass $M_{d}$ $\\sim $ 0.9$^{+0.5}_{-0.5}$ $\\times $ 10$^{4}$ M$_{\\odot }$ .",
"Since the dust mass is somewhat lower than the results obtained from the SED fit with the 500 $\\mu $ m measurement, we conclude that a gain in resolution does not better trace the dust content in NGC 205.",
"On the contrary, including the 500 $\\mu $ m data in the SED fit results in a more massive dust component at a colder temperature in the outer regions of NGC 205.",
"A SED fitting procedure for the global fluxes was performed as well, giving similar results as for the pixel-by-pixel dust masses and $X_{\\text{ISRF}}$ factors (see Table REF ).",
"Figure REF displays the best fitting DustEm model for the silicate+amorphous carbon dust composition overlaid with the global Herschel fluxes and other data from the literature."
],
[
"Submm/mm excess",
"The dust masses obtained from our Herschel observations ($M_{\\text{d}}$ $\\sim $ 1.1-1.8 $\\times $ 10$^{4}$ M$_{\\odot }$ ) are comparable to the dust masses derived from the MIPS observations ($M_{\\text{d}}$ $\\sim $ 3 $\\times $ 10$^{4}$ M$_{\\odot }$ ) within the uncertainties of the observations and fitting procedure, but more than one order of magnitude lower than the predicted dust mass ($M_{\\text{d}}$ $\\sim $ 5 $\\times $ 10$^{5}$ M$_{\\odot }$ ) at a temperature of $\\sim $ 12 K based on mm+Spitzer observations [87].",
"Since the strength of the ISRF and therefore the heating of the dust grains is variable throughout the plane of the galaxy (see Figure REF , left panel), we calculate an upper limit for the dust mass at a dust temperature of $T_{d}$ $\\sim $ 12 K by scaling the SED until fitting the upper limit of the 500 $\\mu $ m flux measurement ($F_{\\nu }$ $\\sim $ 612 mJy).",
"We derive an upper mass limit $M_{d}$ $<$ 4.9 $\\times $ 10$^{4}$ M$_{\\odot }$ for the cold dust reservoir ($T_{d}$ $\\sim $ 12 K).",
"Therefore, our Herschel observations do not seem to support the presence of a massive cold dust component, implying that the millimeter flux is unlikely to originate from a cold dust reservoir.",
"With an upper limit of 0.06 mJy at 21 cm [80] and the absence of emission lines characteristic for LINER or Seyfert galaxies [89], the JCMT 1.1 mm flux measured by [39] is unlikely to be caused by synchrotron emission from either supernova remnants or an AGN-like nucleus.",
"Also the contribution from CO(2-1) line emission seems unable to account for the high mm measurement.",
"Indeed, based on the JCMT CO(2-1) line intensity (0.43 K km/s) for the inner region [132], we derive a flux density of 0.724 Jy at 1.3 mm.",
"Relying on the narrow CO(2-1) line width (13 km/s or 10 MHz) and the bandwidth of the IRAM UKT14 receiver (74 GHz), a contribution from CO(2-1) line emission to the 1.1 mm continuum observations is found to be negligible.",
"This implies that other explanations (calibration issues, background source, bad weather conditions) need to be invoked to explain the high 1.1 mm flux in the center of NGC 205.",
"Also Herschel observations at 500 $\\mu $ m do not show any indication for excess emission at submm or mm wavelengths, reminiscent of the submm excess observed in many star-forming dwarf galaxies and blue compact dwarfs [44], [36], [40], [41], [42], [54], [104].",
"In some cases, the observed excess even extends up to millimeter and centimetre wavelengths, such as observed in the Large (LMC) and Small Magellanic Clouds (SMC) [65], [17], [95], [109].",
"Several reasons have been invoked to account for this excess emission in the submm/mm wavebands.",
"Either large amounts of very cold dust (e.g.",
"[44], [45], [40], [104]), dust grains with optical properties diverging from the typical Galactic dust characteristics (e.g.",
"[77], [96]) or spinning dust grains (e.g.",
"[17], [109]) are thought to be responsible for the excess emission.",
"The fact that our SED model can account for the observed 500 $\\mu $ m emission in NGC 205 is interesting, because it is in contrast with the submillimeter excess observed in several other low metallicity dwarf galaxies.",
"This might be an indication for different ISM properties and star-forming conditions in this early-type dwarf galaxy, compared to the typical star-forming dwarfs revealing an excess submm emission.",
"However, an excess in wavebands longwards of 500 $\\mu $ m (see also [42]) cannot be ruled out based on the currently available dataset for NGC 205.",
"Table: Overview of the parameters (scaling factor for the ISRF X ISRF X_{\\text{ISRF}}, dust mass M d M_{\\text{d}}) for the best fitting DustEm model, either with a silicate+amorphous carbon or silicate+graphite dust composition, determined from a pixel-by-pixel or global SED fit.",
"The temperature estimate is obtained by taking the average over all mean temperatures for dust particles with sizes between 4 ×\\times 10 -3 ^{-3} and 2 μ\\mu m.Figure: The SED for the best fitting DustEm model (X ISRF X_{\\text{ISRF}} = 2.74 or T d T_{d} ∼\\sim 20.7 K, M d M_{\\text{d}} ∼\\sim 1.1 ×\\times 10 4 ^4 M ⊙ _{\\odot }) with an silicate+amorphous carbon dust composition (black, solid line), overlaid with the measured MIPS 24, PACS 70, 100, 160 μ\\mu m and SPIRE 250, 350 and 500 μ\\mu m flux densities.",
"Also the other flux densities from the literature are indicated.The dotted-dashed, dotted, triple dotted-dashed and dashed lines correspond to the PaH dust mixtures, small and large amorphous carbon and silicate dust grains, respectively.Since the SED fitting procedure with the silicate+graphite dust composition gave very similar results and uncertainties, the plot with this SED model is not explicitly shown here.",
"Also a stellar component for NGC 205, parametrized as a , Single Stellar Population with an age of 13 Gyr and a metallicity of Z = 0.002 and modeled as a Sersic profile of index n = 1 and effective radius 130 '' ^{\\prime \\prime } , was included on the SED fit, to allow a comparison of our SED model to observations at NIR/MIR wavelengths (blue dotted line)."
],
[
"JCMT CO(3-2) observations",
"In view of the limited coverage of previous CO(1-0) and CO(2-1) observations with only few pointings across the galaxy, we derive a molecular gas mass estimate for NGC 205 from a CO(3-2) map covering a larger part of the galaxy, however still only probing the H$_{2}$ gas in the northern region of the galaxy (see Figure REF ).",
"For all pixels with detections $>$ 3$\\sigma $ , the integrated CO(3-2) line intensity is converted to a H$_{2}$ column density according to the formula: $N_{\\text{H}_{2}} = \\frac{X_{\\text{CO}} I_{\\text{CO(3-2)}}}{\\eta _{\\text{mb}}\\left( \\frac{I_{\\text{CO(3-2)}}}{I_{\\text{CO(1-0)}}} \\right)}$ where $I_{\\text{CO(3-2)}}$ is the total integrated line intensity expressed in units of K km s$^{-1}$ .",
"The scaling factor to convert an antenna temperature T$_{A}^{\\star }$ into a main beam temperature $T_{mb}$ at the JCMT is $\\eta _{mb}$ = 0.6.",
"We assume a value of $\\sim $ 0.3 for the CO(3-2)-to-CO(1-0) line intensity ratio corresponding to the typical ratios found in the diffuse ISM of other nearby galaxies [135].",
"Since we derive a line ratio of $\\sim $ 0.34 for the central pointing reported in [132], we argue that this ratio serves as a good approximation for the entire CO(3-2) emitting region.",
"The same $X_{\\text{CO}}$ conversion factor (6.6 $\\times $ 10$^{20}$ cm$^{-2}$ (K km s$^{-1}$ )$^{-1}$ ) as introduced in Section REF is applied here.",
"The total molecular gas mass is derived from the column density following the equation: $M_{\\text{H}_{2}} = A N_{\\text{H}_{2}} m_{\\text{H}_{2}}$ where $A$ represents the surface of the CO(3-2) emitting region and $m_{\\text{H}_{2}}$ is the mass of a molecular hydrogen atom.",
"Inserting the correct values in equation REF results in an estimate for the total molecular gas mass $M_{\\text{H}_{2}}$ $\\sim $ 1.3 $\\times $ 10$^5$ M$_{\\odot }$ .",
"Considering that this value is a factor of $\\sim $ 5 lower than the $M_{\\text{H}_{2}}$ $\\sim $ 6.9 $\\times $ 10$^5$ M$_{\\odot }$ inferred from CO(1-0) detections, we argue that most of the H$_{2}$ in NGC 205 resides in regions of colder temperature ($T$ $\\le $ $T_{\\text{crit,CO(3-2)}}$ $\\sim $ 33 K) and/or lower density ($n$ $\\le $ $n_{\\text{crit,CO(3-2)}}$ $\\sim $ 2 $\\times $ 10$^4$ cm$^{-3}$ )."
],
[
"PACS spectroscopy observations",
"Relying on the low metal abundance of the ISM in the inner regions of NGC 205, there might be a significant fraction of molecular gas in NGC 205 which remains undetected by current CO observations, since CO is often a poor diagnostic of the H$_{2}$ content in low abundance environments exposed to hard radiation fields.",
"Based on the high values for the $L_{\\text{[CII]}}$ -to-$L_{\\text{CO}}$ ratio found in several low metallicity dwarf galaxies (e.g.",
"[83], [23]), [Cii] is often claimed to be a better tracer for the molecular gas in such environments.",
"Also the [Oi] fine-structure line, which is considered an important coolant of the neutral gas together with [Cii], can be used as an alternative probe for the molecular gas in a low metallicity ISM (e.g.",
"[126]).",
"Although both [Cii] and [Oi] are considered good tracers of molecular gas in a metal-poor ISM, the interpretation of their line fluxes is hampered by a lack of knowledge about the exact origin of the line emission from within a galaxy.",
"[Cii] emission is thought to arise either from the ionized (Hii regions) or the neutral (PDRs) medium, while [Oi] emission originates mainly from the neutral ISM.",
"In those neutral PDRs, the [Cii] line provides cooling for gas clouds with a moderate density ($n_{\\text{H}_{2}}$ $<$ 10$^4$ cm$^{-3}$ ), while the [Oi] line cools the higher density regions.",
"Table: [Cii] line measurements within the elliptical apertures overlaid on the [Cii] map in Figure Figure: Left: PACS [Cii] map of NGC 205.",
"The elliptical apertures for photometry are indicated as white circles.",
"The CO(1-0) pointing reported in covering the center of NGC 205 as well as the [Cii] intensity peak is color-coded in green.",
"Right: Zoom on the central bright region in the [Cii] map, overlaid with contours of MIPS 24 μ\\mu m surface brightnesses (blue, dashed-dotted curves) and Hi and CO(3-2) column densities (yellow, dashed and green, solid lines, respectively).",
"The Hi contours range from 2 ×\\times 10 19 ^{19} cm -2 ^{-2} to 2.6 ×\\times 10 20 ^{20} cm -2 ^{-2} in intervals of 4 ×\\times 10 19 ^{19} cm -2 ^{-2}, while the CO(3-2) contours represent a H 2 _{2} column density range 1.83 ×\\times 10 20 ^{20} ≤\\le N H 2 N_{\\text{H}_{2}} ≤\\le 1.28 ×\\times 10 21 ^{21} cm -2 ^{-2} increased in steps of 2.75 ×\\times 10 20 ^{20} cm -2 ^{-2}.",
"The contours representing the MIPS 24 μ\\mu m surface brightnesses range from 0.68 to 2 MJy/sr, stepwise increased by 0.33 MJy/sr.Figure: PACS spectral line plots.",
"First three panels show the [Cii] line emission summed over the individual apertures shown in Figure and described in Table , respectively.",
"The last two panels show the non-detection of the [Oi] line, with the emission added up within the observed regions covering the center and the CO peak in NGC 205, respectively.From our PACS spectroscopy observations, we detected [Cii] line emission in the center of NGC 205 (see Figure REF , left panel), while the [Oi] line was not detected in either of the two covered regions in NGC 205 (see Figure REF ).",
"The faint [Cii] emission in NGC 205 originates mainly from the nuclear region, with the peak intensity residing from a dust cloud west of the nucleus.",
"From the gas and dust contours in Figure REF (right panel) it becomes evident that the brightest [Cii] emission region is located at the boundary of the most massive Hi and molecular clouds in the center of NGC 205.",
"The spatial correlation with the CO reservoir in the center of NGC 205 and the surrounding star formation regions (see contours of hot dust emission from MIPS 24 $\\mu $ m data, Figure REF , right panel, blue, dashed-dotted curves) suggests that the [Cii] emission in NGC 205 originates from photodissociation regions in the outer layers of molecular cloud structures.",
"Although the [Cii] emission in galaxies might have a significant contribution from ionized media, the unavailability of ionized gas tracers (e.g.",
"[Nii], [Oiii]) impede a direct quantification of the [Cii] emission from ionized regions.",
"The non-detection of H$\\alpha $ line emission [137] seems to point towards a low fraction of ionized gas in the ISM of NGC 205.",
"To determine the intensity of the [Cii] line emission in NGC 205, we performed aperture photometry within ellipses matching the shape of prominent [Cii] emission regions in NGC 205 (see Figure REF , left panel).",
"The spectral line emission within each emission region is shown in Figure REF (first three panels).",
"The fluxes within those three apertures are summarized in Table REF .",
"From equation 1 in [81], we can derive the column density of atomic hydrogen, when adopting a C$^{+}$ abundance per hydrogen atom of $X_{\\text{C}^{+}}$ $=$ 1.4 $\\times $ 10$^{-4}$ [115] and $n_{\\text{crit}}$ $\\sim $ 2.7 $\\times $ 10$^{3}$ cm$^{-3}$ .",
"When assuming $X_{\\text{ISRF}}$ $\\sim $ 3 (or thus $G_{0}$ $\\sim $ 3) for the strength of the ISRF field (see Section REF ) and an average density $n_{\\text{H}}$ $\\sim $ 10$^{4}$ cm$^{-3}$ , we estimate a surface density temperature of T $\\sim $ 40 K from the PDR models in [69].",
"Inserting those values in equation (1.)",
"from [81], we obtain average column densities for the three apertures within the range 2.2 $\\times $ 10$^{19}$ cm$^{-2}$ $\\le $ $N_{\\text{H}}$ $\\le $ 3.7 $\\times $ 10$^{19}$ cm$^{-2}$ .",
"Summing over all three apertures, the total atomic gas mass for the [Cii] emitting regions in NGC 205 is derived to be $M_{\\text{g}}$ $\\sim $ 1.54 $\\times $ 10$^{4}$ M$_{\\odot }$ .",
"Since this value is negligible compared to the total gas mass $M_{g}$ $\\sim $ 6.9 $\\times $ 10$^{6}$ M$_{\\odot }$ derived from CO(1-0) observations (see Section REF ), we conclude that the molecular gas reservoir is probably well-traced by the CO lines in the low metallicity ISM of NGC 205.",
"This argument is furthermore supported by the relatively low $L_{\\text{[CII]}}$ -to-$L_{\\text{CO(1-0)}}$ line intensity ratio $\\sim $ 1850 within the central CO(1-0) pointing from [132] (see green aperture overlaid on Figure REF ).",
"This line intensity ratio is substantially lower than the values observed in star-forming dwarf galaxies ranging from 4000 to 80000 [83], [23].",
"Based on PDR models for values $X_{\\text{ISRF}}$ $\\sim $ 3 and $n_{\\text{H}}$ $\\sim $ 10$^{4}$ cm$^{-3}$ in the center of NGC 205, we would expect [Cii] line intensities of a few 10$^{-6}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ and $L_{\\text{[OI]}}$ -to-$L_{\\text{[CII]}}$ line intensity ratios close to 0.5 (see Figure 3 and 4 from [69]).",
"With a peak [Cii] line intensity of 3 $\\times $ 10$^{-6}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ and an upper limit of $L_{\\text{[OI]}}$ -to-$L_{\\text{[CII]}}$ $<$ 1.75, the observed [Cii] line emission in the center of NGC 205 is in agreement with PDR models.",
"The lack of any bright [Cii] emission from other regions in NGC 205 might be an indication for the strength of the radiation field being insufficient to photo-dissociate CO.",
"This argument is supported by the pixel-by-pixel analysis in Section REF , which indicates the stronger radiation field in the brightest [Cii] region compared to the rest of the galaxy (see Figure REF , left panel).",
"Alternatively, a large reservoirs of photodissociated CO molecules might be present in the outer layers of gas clouds, where a deficiency of ionizing photons might impede the formation of C$^{+}$ atoms.",
"A lack of ionizing photons is also supported by the non-detection of H$\\alpha $ in this galaxy [137].",
"According to this latter scenario, we would expect the majority of dissociated carbon to be locked in [Ci] rather than C$^{+}$ ."
],
[
"Missing ISM mass problem: revised",
"From our Herschel observations of the dust continuum and far-infrared fine structure lines [Cii] and [Oi] and JCMT CO(3-2) data, we are able to revisit the missing ISM mass problem in NGC 205.",
"We estimate an atomic gas mass of 1.54 $\\times $ 10$^{4}$ M$_{\\odot }$ associated with the [Cii] emitting PDRs in NGC 205.",
"From the CO(3-2) emitting regions in the northern part of NGC 205, we could derive a molecular gas mass of $M_{\\text{H}_{2}}$ $\\sim $ 1.3 $\\times $ 10$^5$ M$_{\\odot }$ .",
"In comparison with the $M_{\\text{H}_{2}}$ $\\sim $ 6.9 $\\times $ 10$^5$ M$_{\\odot }$ obtained from pointed CO(1-0) observations [132] covering the main CO(1-0) emission regions in the north of the galaxy, our CO(3-2) measurements indicate a low fraction of dense molecular gas in the diffuse ISM of NGC 205, where star formation is currently only occurring spontaneously in localized dense clouds.",
"Including Hi observations (4.0 $\\times $ 10$^5$ M$_{\\odot }$ ) and scaling the sum of the molecular and atomic hydrogen mass by a factor $\\sim $ 1.4 to include heavier elements, we obtain a total gas mass of $M_{g}$ $\\sim $ 0.7-1.5 $\\times $ 10$^6$ M$_{\\odot }$ from CO(3-2) or CO(1-0) observations, respectively.",
"Alternatively, we probed the ISM content through the galaxy's dust continuum emission.",
"This approach avoids introducing uncertainty factors arising from the $X_{\\text{CO}}$ conversion factor, but additional errors on the gas mass estimates result from the SED fitting procedure to calculate the dust mass and, even more importantly, the assumption on the value for the gas-to-dust ratio.",
"Since it has been shown that the gas-to-dust ratio in metal-poor galaxies deviates from the Galactic ratio ($\\sim $ 160, [138]), a gas-to-dust fraction of $\\sim $ 400 [46] is considered more realistic for the central ISM in NGC 205 in view of the recent star formation episode occurring in those inner regions.",
"This value is obtained from an extrapolation of the dust evolution model in [46], when assuming little or no dust destruction and a first-order trend of the gas-to-dust ratio with metallicity (Z $\\sim $ 0.3 Z$_{\\odot }$ ).",
"Relying on this gas-to-dust ratio of $\\sim $ 400, the total dust mass detected from our Herschel observations corresponds to a gas mass $M_{g}$ = $\\sim $ 4-7 $\\times $ 10$^6$ M$_{\\odot }$ .",
"Upon comparison with the theoretical gas content in the range [1.3 $\\times $ 10$^7$ M$_{\\odot }$ , 4.8 $\\times $ 10$^8$ M$_{\\odot }$ ] (see Section REF ), gas mass estimates from either dust continuum (4-7 $\\times $ 10$^6$ M$_{\\odot }$ ) or Hi+CO(1-0)+[Cii] (1.5 $\\times $ 10$^6$ M$_{\\odot }$ ) observations both confirm the missing ISM mass problem in NGC 205.",
"The lower ISM mass inferred from direct gas observations (1.5 $\\times $ 10$^6$ M$_{\\odot }$ ) in comparison with the indirect gas estimates from dust continuum observations (4-7 $\\times $ 10$^6$ M$_{\\odot }$ ) most likely results from the poor coverage of molecular gas tracers in NGC 205 and/or the uncertainty on the assumed gas-to-dust fraction (i.e.",
"a gas-to-dust ratio closer to the Galactic value would bring the ISM masses inferred from Hi+CO(1-0)+[Cii] and dust continuum observations in better agreement).",
"With the JCMT CO(3-2) and previous CO(1-0) and CO(2-1) observations mainly covering the northern part of the galaxy, the lack of knowledge about the molecular gas content for a large southern region in NGC 205, where the Hi emission dominates, prevents us from determining the entire molecular gas content.",
"If we assume a similar molecular-to-atomic gas mass ratio for the southern part of NGC 205 as observed in the northern part, the gas component in NGC 205 could be twice as massive than measured by current CO observations.",
"The observed peak in Hi (4 $\\times $ 10$^{20}$ cm$^{-2}$ or 3.2 M$_{\\odot }$ pc$^{-2}$ ) and dust (0.1 M$_{\\odot }$ pc$^{-2}$ ) column density would imply a Hi-to-dust ratio of $\\sim $ 32 in the south of NGC 205.",
"However, a more realistic gas-to-dust ratio would require a large molecular gas reservoir residing in those southern areas.",
"This argumentation is also confirmed by the H$_{2}$ column densities of 1.8 and 2.4 $\\times $ 10$^{20}$ cm$^{-2}$ measured from two JCMT CO(2-1) pointings near the southern peak in Hi, when assuming a CO(2-1)-to-CO(1-0) line ratio of $\\sim $ 0.9 $\\pm $ 0.2 as found by [136] in a spatially resolved molecular cloud.",
"In addition to this unexplored molecular gas reservoir in the southern part of the galaxy, a substantial fraction of the gas component might remain undetected by current observations, either because a fraction of the gas is locked in hot X-ray or ionized gas haloes (similar to the heated gas returned by evolved stars in giant massive ellipticals) and/or the present molecular gas is traced neither by CO nor [Cii] molecules.",
"Alternatively, we could invoke an internal/environmental mechanism solely removing the gas from NGC 205 and leaving the dust unharmed to explain the higher gas masses inferred from dust continuum observations.",
"In view of the good correlation found for the gas and dust component, there is however no reason to assume a higher inertia for the dust and thus to imply that gas particles are more easily transported out of the gravitationally bound regions in NGC 205.",
"In the next paragraphs, we discuss possible explanations for the inconsistency between the observed ISM content (either from Hi+CO(1-0)+[Cii] or dust continuum observations) in NGC 205 and theoretical predictions, such as non-standard conditions for the SFE or initial mass function (IMF) or environmental processes able to remove part of the ISM during the last episode of star formation in NGC 205.",
"Non-standard conditions such as a top-heavy IMF (more massive stars are produced, requiring less input gas mass) or a higher star formation efficiency ($>$ 10 $\\%$ of the gas is converted into stars) could be invoked to explain the missing ISM in NGC 205.",
"Harmonizing the observed ISM content with the theoretical predictions would require an IMF deviating significantly from the general assumptions and/or an increase in the SFE from 10$\\%$ to at least 65$\\%$ .",
"Although such a top-heavy IMF or increased SFE was thought to be present in some ULIRGS or starbursts (e.g.",
"M82, [71]), the non-standard conditions in those galaxies are owing to the presence of high density gas, introducing a different mode of star formation [25].",
"With a SFR $\\sim $ 7.0 $\\times $ 10$^{-4}$ M$_{\\odot }$ yr$^{-1}$ [98] during the last starburst episode, NGC 205 does not mimic the typical star formation activity ($>$ 10 M$_{\\odot }$ yr$^{-1}$ ) in those starburst galaxies.",
"Also under less extreme circumstances, local variations in the SFE [11] and perturbations at the upper mass end of the IMF [97] have been claimed.",
"However, the latter IMF variations have been questioned on its turn invoking poor extinction corrections and a variable SFR in those galaxies [16], [131].",
"Relying on recent results from [88] and [18], finding no evidence for extreme star formation efficiencies in star-forming dwarf galaxies and therefore rather supporting a moderate SFE of 10-20$\\%$ or less, we argue that non-standard conditions (top-heavy IMF and/or increased SFE) are not likely to occur in NGC 205. and we should invoke efficient mechanisms of gas removal to explain the low observed gas content in NGC 205."
],
[
"Supernova feedback",
"Feedback from supernovae [29], [2], [33] or potentially an AGN are capable of removing a significant amount of gas from NGC 205.",
"Relying on the absence of any LINER or Seyfert diagnostics [59], we argue that the effect of AGN feedback in NGC 205 is currently negligible.",
"Based on the 3$\\sigma $ upper limit of 3.8 $\\times $ 10$^4$ M$_{\\odot }$ for the mass of the SMBH in NGC 205 from studies of the stellar kinematics based on deep HST images [128], we believe the role of an AGN was also limited throughout the recent history of NGC 205.",
"On the contrary, supernova feedback and/or stellar winds could be responsible for expelling gas from the central regions, during the last episode of star formation.",
"Examining the effectiveness of SN II feedback during the last episode of star formation, [132] confirm that supernova winds or blasts were capable of removing a significant amount of gas from the central regions of NGC 205.",
"Indeed, applying the dwarf galaxy model for $M_{galaxy}$ $\\sim $ 10$^9$ M$_{\\odot }$ from [33], a minimum energy of $\\sim $ 10$^{55}$ f$_{gas}$ ergs would be required to expel most of the gas from the inner regions.",
"Following the burst mass $\\sim $ 7 $\\times $ 10$^5$ M$_{\\odot }$ reported in [133], the gas fraction is found to be f$_{gas}$ $\\sim $ 0.007, when assuming a 10$\\%$ star formation efficiency.",
"This would require an energy of at least 7 $\\times $ 10$^{52}$ ergs to remove gas.",
"This level is easily achieved with the energy $\\sim $ 6 $\\times $ 10$^{54}$ ergs released by the SNe II associated with the last burst [132].",
"Keeping in mind the clumpy gas distribution and the offset of the local Hi column density peaks from the locations of young stars [137] in addition to those theoretical arguments, we argue that both supernova feedback and/or stellar winds have likely disturbed the ISM of NGC 205.",
"Energy feedback from supernovae has furthermore been proven to have an important share in the formation of dwarf spheroidal systems [73].",
"Despite theoretical arguments supporting a history of violent supernova explosions, a previous attempt to detect supernova remnants in NGC 205 failed ($F_{\\nu }$ $<$ 0.06 mJy, [80]).",
"Relying on the low radio continuum detection rate for low surface brightness dwarfs [61], [80], it might not be surprising that the detection of supernova remnants at 20 cm was unsuccessful.",
"The majority of the 20 cm emission is thought to arise from synchrotron emission originating from electrons accelerated in the expanding shells of Type II and Type Ib supernova remnants.",
"Due to the increased cosmic ray diffusion timescales [70], [102] (i.e.",
"electrons are escaping more easily from the ISM) for galaxies with a low star formation activity (SFR $\\le $ 0.2 M$_{\\odot }$ yr$^{-1}$ ), those objects become radio quiet in short timescales.",
"Although violent supernova explosions might have occurred in the past, the associated radio emission is likely smoothed out in the low potential well of NGC 205.",
"Also in other wavelength domains the detection of any supernova remnants will be challenging.",
"While the star formation is known to be active up to at least 60 Myr ago, a typical age $\\sim $ 10 Myr for a SNII progenitor and a SNR lifetime $\\sim $ 25000 years implies that it is difficult to detect any remaining evidence of supernova remnants in NGC 205."
],
[
"Environmental interactions",
"Galaxy harassment [99], starvation [78] and viscous stripping [103] are important transformation processes for galaxies in clusters, whereas tidal stirring [91] and galaxy threshing [3] are considered responsible for the formation of dwarf spheroidals and ultra-compact dwarfs in the low density environment of groups.",
"For the formation of NGC 205, a combination of ram pressure and tidal stripping would be more likely since those processes are capable of transforming gas-rich dwarf galaxies into dwarfs with a blue central core when passing through the halo of a galaxy of the same size as the Milky Way [92].",
"Tidal or gravitational interactions are also considered to be the main formation mechanism for spheroidal and lens-shaped galaxies in groups [73], [4], [120].",
"In a similar fashion, ram pressure stripping is found to form dwarf elliptical galaxies with blue nuclei in the Virgo cluster [15].",
"Several observations are indicative for the tidal influence of M31 on its companion NGC 205: a twist in the elliptical isophotes [60], [21], a stellar arc of blue metal-poor red giant branch stars northwest of M31 [64], [93], a tidal debris of C stars to the west of NGC 205 [30], a stripped Hi cloud 25$$ southwest of NGC 205 which overlaps in velocity with the galaxy [127], peculiar Hi morphology and kinematics [137], stars moving in the opposite direction with respect to the rotation of the main stellar body [48], [31] and an optical tidal tail extending at least 17$$ southwards from the galaxy's center [116].",
"This latter stellar tail coincides with the tentative dust tail detected in our SPIRE data (see Section REF ) and, thus, might be an indication for the removal of dust from NGC 205 as a result from the tidal influence of M31.",
"Besides the tidal features characterizing NGC 205, M31 is also affected by the tidal influence exerted by NGC 205.",
"While the origin of two off-centre spiral rings is probably attributable to a head-on collision with M32 [10], the warped structure in the outer Hi disk of M31 [114] and a distortion of the spiral structure in the disk [49] are likely caused to some extent by a two-body interaction between M31 and NGC 205.",
"Although those observations confirm the tidal influence of M31 on the outer regions of NGC 205, it does not provide an explanation for the removal of gas from the inner regions, where the last episode of star formation took place and we would expect to find the left-over gas reservoir.",
"Hydrodynamical simulations suggest that the gaseous component might be disrupted and partly removed even within the tidal radius (4$$ , [48]) without any indications for stellar bridges and tails at these radii [63], but whether it is the case for NGC 205 is difficult to inquire due to the uncertainties regarding its orbit around M31 [62].",
"However, in combination with supernova feedback expelling the gas from the inner regions, tidal interactions might strip the expelled gas from the outer regions.",
"Whether or not responsible for the removal of gas from the inner regions, a tidal encounter with a companion closer than 100-200 kpc is able to trigger SF through shocks in the disk of a typical dwarf galaxy when moving on a coplanar prograde orbit [19].",
"With a line-of-sight distance between the Andromeda galaxy and NGC 205 of $\\sim $ 39 kpc [94] and the likely prograde trajectory of NGC 205 toward its parent galaxy, M31 [48], [62], tidal interactions might have given the initial start for the last episode of star formation.",
"Furthermore, the most recent episodes of star formation during the last gigayear seem correlated with the orbit about M31 [26], which provide additional evidence for the tidal triggering of star formation in NGC 205.",
"Two other dwarf companions of M31, NGC 185 and NGC 147, are thought to have a star formation history comparable to NGC 205 based on their nearly identical optical appearances (Holmberg diameters, $B$ - $V$ colours, average surface brightnesses), mass-to-light ratios ($M$ /$L$ )$_{B}$ $\\sim $ 4 M$_{\\odot }$ /L$_{\\odot ,B}$ [31] and specific frequencies of C stars [27], implying similar fractions of gas and dust which have been turned into stars in the past.",
"Interestingly, both objects also feature a missing ISM mass problem [113].",
"In correspondence to the missing ISM mass problem in NGC 205, tidal interactions and supernova feedback are also believed to have influenced the ISM content in NGC 185 and NGC 147.",
"The tidal influence of M31 ($D$ = 785 $\\pm $ 25 kpc, [94]) is considered negligible at distances of $D$ = 616 $\\pm $ 26 and 675 $\\pm $ 27 kpc [94] for NGC 185 and NGC 147, respectively, compared to NGC 205 ($D$ = 824 $\\pm $ 27 kpc, [94]).",
"However, both galaxies are thought to form a gravitationally bound pair [129] and might have tidally interacted in the past.",
"Aside from the possible occurrence of tidal interactions, indications for supernova remnants are found in NGC 185 [43], [137], [80] and the current ISM content in this galaxy ( $\\sim $ 7.3 $\\times $ 10$^5$ M$_{\\odot }$ ) resembles the estimated mass returned to the ISM by planetary nebulae ($\\sim $ 8.4 $\\times $ 10$^5$ M$_{\\odot }$ , [113]).",
"However, the lack of ISM in NGC 147 remains a puzzling feature in this evolutionary framework."
],
[
"Conclusions",
"This work reports on Herschel dust continuum, [Cii] and [Oi] spectral line and JCMT CO(3-2) observations for NGC 205, the brightest early-type dwarf satellite of the Andromeda galaxy.",
"While direct gas observations (Hi+CO(1-0): 1.5 $\\times $ 10$^6$ M$_{\\odot }$ ) have proven to be inconsistent with theoretical predictions of the current gas content in NGC 205 ($>$ 10$^7$ M$_{\\odot }$ ), we could revise the missing ISM mass problem based on new gas mass estimates (CO(3-2), [Cii], [Oi]) and an indirect measurement of the ISM content probed through Herschel dust continuum observations taken in the frame of the VNGS and HELGA projects.",
"SED fitting to the FIR/submm fluxes results in a total dust mass $M_{\\text{d}}$ $\\sim $ 1.1-1.8 $\\times $ 10$^4$ M$_{\\odot }$ at an average temperature $T_{d}$ $\\sim $ 18-22 K. Based on Herschel data, we can also exclude the presence of a massive cold dust component ($M_{\\text{d}}$ $\\sim $ 5 $\\times $ 10$^5$ M$_{\\odot }$ , $T_{d}$ $\\sim $ 12 K), which was suggested based on millimeter observations from the inner 18.4$$ .",
"When assuming a metal abundance $Z$ $\\sim $ 0.3 Z$_{\\odot }$ and a corresponding gas-to-dust ratio $\\sim $ 400, a gas mass $M_{g}$ $\\sim $ 4-7 $\\times $ 10$^6$ M$_{\\odot }$ is probed indirectly through Herschel dust continuum observations.",
"The non-detection of [Oi] and the relatively low $L_{\\text{[CII]}}$ -to-$L_{\\text{CO(1-0)}}$ line intensity ratio ($\\sim $ 1850) imply that the molecular gas phase is well traced by CO molecules.",
"From CO(3-2) observations of the northern part of the galaxy, we infer a new molecular gas mass estimate $M_{\\text{H}_{2}}$ $\\sim $ 1.3 $\\times $ 10$^{5}$ M$_{\\odot }$ , implying that the molecular clouds in NGC 205 is mostly very diffuse.",
"New gas mass estimates from dust continuum and CO(3-2) line observations both confirm the missing ISM mass problem, i.e.",
"an inconsistency between theoretical predictions for the ISM mass and observations.",
"In an attempt to explain the deficiency in the ISM in the inner regions of NGC 205, we claim that efficient supernova feedback capable of expelling gas/dust from the inner, star-forming regions of NGC 205 to the outer regions and/or tidal interactions with M31 stripping the gas/dust component from the galaxy provide the best explanation for the removal of a significant amount of the ISM from NGC 205.",
"However, if supernova feedback is found to lack responsibility for the removal of gas/dust from the inner regions, we might have to reconsider the importance of tidal interactions on the gaseous component in a galaxy within the tidal radius and/or revise the parameters characterizing the orbit of NGC 205 in its approach toward M31."
],
[
"Acknowledgements",
"PACS has been developed by a consor- tium of institutes led by MPE (Germany) and in- cluding UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF- IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain).",
"This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain).",
"SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ.",
"Lethbridge (Canada); NAOC (China); CEA, OAMP (France); IFSI, Univ.",
"Padua (Italy); IAC (Spain); Stockholm Ob- servatory (Sweden); ISTFC and UKSA (UK); and Cal- tech/JPL, IPAC, Univ.",
"Colorado (USA).",
"This devel- opment has been supported by national funding agen- cies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); Stock- holm Observatory (Sweden); STFC (UK); and NASA (USA).",
"HIPE is a joint development by the Herschel Science Ground Segment Consortium, consist- ing of ESA, the NASA Herschel Science Center and the HIFI, PACS and SPIRE consortia.",
"This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada.",
"The research of C. D. W. is supported by grants from rom the Canadian Space Agency and the Natural Sciences and Engineering Research Council of Canada.",
"MB, JF, IDL and JV acknowledge the support of the Flemish Fund for Scientific Research (FWO-Vlaanderen), in the frame of the research projects no.",
"G.0130.08N and no.",
"G.0787.10N .",
"GG is a postdoctoral researcher of the FWO-Vlaanderen (Belgium)."
]
] | 1204.1264 |
[
[
"On the Relativistic Formulation of Matter"
],
[
"Abstract A critical analysis of the relativistic formulation of matter reveals some surprising inconsistencies and paradoxes.",
"Corrections are discovered which lead to the long-sought-after equality of the gravitational and inertial masses, which are otherwise different in general relativity.",
"Realizing the potentially great impact of the discovered corrections, an overview of the situation is provided resulting from the newly discovered crisis, amid the evidences defending the theory."
],
[
"Introduction",
"Modern theories of gravitation describe gravity not as a `force' in the usual sense but as a manifestation of the curvature of spacetime.",
"This renders the theories essentially geometrical in character with non-Euclidean geometries playing the central role.",
"Einstein's revolutionary discovery of the principle of general covariance then suggests that a relativistic theory of gravitation should be formulated using the language of tensors.",
"This implies that the matter, which is considered as the source of the curvature, should also be represented invariably by a tensor, leading to the usual relativistic hydrodynamic description of matter encoded in the famous energy-momentum-stress tensor (energy density-momentum-stress tensor, to be more precise) $T^{\\mu \\nu }$ ($\\mu ,\\nu =0,1,2,3$ ) which has become an integral part of all the relativistic theories of gravitation including the candidate theories of quantum gravity.",
"The simplest example of $T^{\\mu \\nu }$ belongs to the case of a perfect fluid (absence of viscosities and heat flow).",
"In a coordinate system $x^\\alpha $ , specifying the metric tensor $g_{\\mu \\nu }$ , the $T^{\\mu \\nu }$ of the perfect fluid is given by $T^{\\mu \\nu }=(\\rho +p) u^\\mu u^\\nu -p g^{\\mu \\nu },$ where $\\rho $ , the energy density of the fluid and $p$ , its isotropic pressure, are measured by an observer in a locally inertial reference frame which happens to be moving with the fluid at the instant of measurement.",
"In the considered coordinates $x^\\alpha $ , the fluid as a whole has 4-velocity $u^\\alpha =dx^\\alpha /ds$ .",
"It is important to note that the term $\\rho $ , appearing in (REF ), includes not only the rest mass of the individual particles of the fluid but also their kinetic energy, internal energy (for example, the energy of compression, energy of nuclear binding, etc.)",
"and all other sources of mass-energy [1] (excluding the energy of the gravitational field).",
"Modelling matter by $T^{\\mu \\nu }$ has revolutionized the way we used to think about the source of gravity.",
"As mass density is the source of gravity in Newtonian theory, we expect energy density to take over this role in the relativistic generalization of Poisson's equation.",
"To our surprise however, all the ten (independent) components of $T^{\\mu \\nu }$ become contributing source of gravity, making a revolutionary prediction that not only the energy density but pressure and momentum too have gravitational effects.",
"Though the momentum of the matter disappears from the scene once one uses a coordinate system comoving with the matter, however, it is not possible to avoid the pressure which becomes an integral part of the relativistic description of matter.",
"It is an undeniable fact that the standards of $T^{\\mu \\nu }$ , in terms of beauty, consistency and mathematical completeness, do not match the vibrant geometrical side of the modern theories of gravitation, perhaps not enough attention has been devoted to $T^{\\mu \\nu }$ as it deserves.",
"Einstein himself conceded this fact as he wrote about the standards of $T^{\\mu \\nu }$ in the general theory of relativity (GR): “GR is similar to a building, one wing (the geometry) of which is made of fine marble, but the other wing (the matter) of which is built of low grade wood\".",
"The mystery of the gravitational effects of the non-conventional sources (i.e., the components of $T^{\\mu \\nu }$ other than the energy density), is seldom addressed at a sufficiently foundational level.",
"It has not been realized, for example, that the formulation of matter in terms of $T^{\\mu \\nu }$ , modifies at the deepest level its well-known properties, so that the intuitive Newtonian knowledge needs revision upon closer investigation.",
"Let us consider, for example, the gravitational effects of pressure, which is the focal point of this paper.",
"Since pressure enters $T^{\\mu \\nu }$ as a scalar, just like the energy density, and since the dimensions of the pressure are those of the energy density, it implies that the pressure (somehow) contains energy density.",
"(To what degree it contains energy density, depends on the equation of state.)",
"This is an entirely new physics!",
"Albeit counter-intuitive, this is perfectly consistent with equation (REF ) wherein the pressure appears as being added to the energy density.",
"Further, this new physics is corroborated by many examples wherein pressure contributes to the energy density of the fluid.",
"One may mention some cases from GR, for example.",
"One may mention the pressure contributing to the active gravitational mass density $(\\rho +3p)/c^2$ in the Raychaudhuri equation $\\frac{3\\ddot{\\ell }}{\\ell }=2(\\omega ^2-\\sigma ^2)+\\dot{u}^\\alpha _{;\\alpha }-\\frac{4\\pi G}{c^2}(\\rho +3p),$ which is the relativistic analogue of Poisson's equation and determines the average contraction/expansion of a self-gravitating fluid in GR.",
"Here $\\ell $ represents the volume behaviour of the fluid and the kinematical quantities $\\omega $ , $\\sigma $ and $\\dot{u}^\\alpha $ respectively measure rotation, shear and acceleration of the fluid flow.",
"As another example, one can mention the pressure contributing to the passive gravitational mass density $(\\rho +p)/c^2$ of the fluid in the Tolman-Oppenheimer-Volkoff equation $\\frac{dp}{dr}+(\\rho +p) \\frac{d}{dr}\\ln (\\vert g_{00}|)^{1/2}=0,$ which represents the relativistic analogue of the classical hydrostatic equilibrium of a star.",
"Here $r$ is the radial coordinate and $g_{00}$ is the time-time component of the static spherically symmetric metric modelling the isotropic material of the star.",
"Let us note that the active and passive gravitational masses are not equal in the modern theories of gravitation.",
"The binding energy of the gravitational field is believed to be responsible for this.",
"However, why the contributions from the gravitational energy to the different masses are not equal, has remained a mystery in these theories.",
"Similar examples can be given from the alternative theories of gravity as well (one can see [2], for a review on the extended theories of gravity).",
"Notwithstanding strongly supported by various examples, the prediction that pressure contains energy, faces paradoxes and inconsistencies.",
"There is already a longstanding Tolman paradox [3] which can be described as follows."
],
[
"Tolman Paradox",
"Tolman [4] has derived a formula for the total energy $E$ , including gravitational energy, of a static system: $E=mc^2=\\int (\\mathcal {T}_0^0-\\mathcal {T}_1^1-\\mathcal {T}_2^2-\\mathcal {T}_3^3)~d^3x,$ where $\\mathcal {T}_\\mu ^\\nu =T_\\mu ^\\nu \\sqrt{-g}$ is the tensor-density corresponding to the energy density-stress tensor.",
"For the energy density-stress tensor (REF ), this formula reduces to $E=mc^2=\\int (\\rho +3p)\\sqrt{|g_{00}|}~dV,$ which applies to a wide class of cases including the one in which the matter is confined to some limited region.",
"Here $g_{00}$ is the time-time component of the metric (in an asymptotically flat coordinate system representing the spacetime at a sufficient distance from the material system of interest lying in the neighbourhood of the origin of the coordinate system) and $dV$ is the proper spatial volume element of the fluid sphere.",
"In modern terminology, $m$ is the ADM mass measured by a faraway inertial observer.",
"The Tolman paradox (related with the consequences of the gravitational effect of pressure) can be described as the following.",
"The matter ($p=0$ ) at rest in a container exhibits the total mass $m$ .",
"However, converting the matter inside the container into disordered radiation ($p=\\rho /3$ ) would double the total mass.",
"Or alternatively, a conversion of gamma rays of total energy $E=mc^2$ , enclosed in the container, into electron-positron pairs would lead to a total mass halfAs energy and momentum are conserved when two photons collide to give rise to an electron-positron pair, so, if all the photons pairs do not meet this requirement, only those will contribute to the paradox, which meet the requirement.",
"While their momentum will balance each others', their energy will be converted to the rest mass of the created electron-positron pairs.",
"However, the term $3p$ appearing in the Tolman integral (REF ) would remain unaccounted, and hence would contribute to the paradox.",
"of $m$ .",
"In both cases, the conservation of $E$ is seriously violated.",
"The Tolman paradox has not received much attention, as the formula (REF ) involves gravitational energy (ascertained by the presence of $\\sqrt{-g}$ in (REF )) which is notorious for violating its (local) conservation.",
"Thus one can readily claim that the gravitational energy is responsible for the above-mentioned violation of the conservation of $E$ in the Tolman paradox (though it is apparent that the paradox results from the term $p$ , and not from $\\sqrt{-g}$ in equation (REF )).",
"In addition to the Tolman paradox, there have also been claims for the violation of energy in the cosmological context (see, for example, [5]).",
"However, like the Tolman paradox, these claims have also not attracted much attention, as these usually take refuge, for a possible resolution, in the gravitational energy, which has been of an obscure nature and a controversial history.",
"So, is it possible to ascertain, beyond doubts, that there is really some fundamental problem in the relativistic formulation of matter, for example, given by the energy density-stress tensor (REF )?",
"If so, this must be done without implicating gravitational energy.",
"Let us try to do this in the following."
],
[
"New Paradoxes",
"In the following, we focus our attention on the relativistic formulation of matter given by the energy density-stress tensor $T^{\\mu \\nu }$ , without considering any particular theory of gravitation.",
"After all, all the modern theories of gravitation use this tensor to represent the matter source.",
"Usually, this tensor may have contributions from, or may be supplemented by, other sources as well, for example, the scalar fields (representing generalized `matter') or the geometric `matter' resulting from the modification of the effective Lagrangian of the gravitational field by adding higher-order terms in the curvature invariants (as is done in the extended theories of gravitation [2]).",
"To be more precise, we shall derive our results from the conservation of $T^{\\mu \\nu }$ , i.e., $T^{\\mu \\nu }_{ ~ ~ ;\\nu }=0$ which is identically satisfied, through the Bianchi identities, in the simple cases of the metric theories of gravitation, for example, GR.",
"There are however, other theories, for example, the Brans-Dicke theory, wherein this conservation does not follow from the divergence of the left hand side of the field equations.",
"However, it is a standard practice to assume $T^{\\mu \\nu }_{ ~ ~ ;\\nu }=0$ (as an extra degree of freedom) in this type of theories.",
"Hence, the results we shall obtain in the following, apply to almost all the theories of gravitation.",
"In the following we shall limit our analysis to the simplest case of $T^{\\mu \\nu }$ belonging to the perfect fluid, so that the physics does not get shrouded in the complexity of a more general $T^{\\mu \\nu }$ .",
"However, this compromise would not cost much since the perfect fluid represents/approximates a great many macroscopic physical systems, including the ordinary matter, the scalar fields, the dark energy candidates and perhaps the universe itself.",
"As the central point of our analysis in the following will be the energy density-stress tensor of a perfect fluid given by equation (REF ), it would not be out of place to appreciate its derivation.",
"The standard way to derive the tensor (REF ) is the following.",
"First, it is defined in the absence of gravity, i.e., in special relativity (SR).",
"Then, the general expression for the tensor in the presence of gravity, in coordinates, say $x^\\alpha $ (in which the metric tensor is identified as $g_{\\mu \\nu }$ ), is imported from SR through a coordinate transformation.",
"The bridge between the ideal case of SR and the actual case in the presence of gravity, is provided by a locally inertial coordinate system (LICS) which can always be found at any point of spacetime (thanks to the principle of equivalence).",
"Let us consider an LICS $x^\\alpha _\\star $ which, by definition, provides an infinitesimal spacetime neighbourhood around the selected point wherein the metric tensor $g_{\\mu \\nu }$ does not change significantly (and reduces to the Lorentzian metric $\\eta _{\\mu \\nu }$ ).",
"Hence $\\partial g_{\\mu \\nu }/\\partial x^\\alpha _\\star =0$ and the effects of gravity disappear in the neighbourhood (but not in a finite region).",
"This LICS is so chosen that a perfect fluid element is at rest in the neighbourhood of the position and time of interest.",
"As an observer at the selected point sees the fluid around him isotropic, the energy density-stress tensor $T^{\\mu \\nu }_\\star $ (in the coordinates $x^\\alpha _\\star $ ) takes the form characteristic of spherical symmetry: $T^{00}_\\star =\\rho , ~~T^{11}_\\star =T^{22}_\\star =T^{33}_\\star =p ~~\\text{and}~~ T^{\\mu \\nu }_\\star =0 ~~\\text{for} ~~\\mu \\ne \\nu ,$ where $\\rho $ and $p$ are, respectively, the energy density and pressure of the fluid measured by the considered observer.",
"As the fluid element is at rest in the above-mentioned neighbourhood, the spatial components of the 4-velocity vector then vanish, i.e.",
"$u_x\\equiv \\frac{dx}{d\\tau }=0, ~u_y\\equiv \\frac{dy}{d\\tau }=0, ~u_z\\equiv \\frac{dz}{d\\tau }=0 ~ ~{\\rm and}~~\\frac{dt}{d\\tau }=1,$ where we have specified $x^0_\\star \\equiv ct$ , $x^1_\\star \\equiv x$ , $x^2_\\star \\equiv y$ , $x^3_\\star \\equiv z$ ; and the proper time interval $d\\tau =ds/c$ .",
"In the general coordinate system $x^\\alpha $ , the energy density-stress tensor $T^{\\mu \\nu }$ is then given by $T^{\\mu \\nu }=\\frac{\\partial x^\\mu }{\\partial x^\\alpha _\\star } \\frac{\\partial x^\\nu }{\\partial x^\\beta _\\star } T^{\\alpha \\beta }_\\star ,$ which reduces to (REF ) by noting that $g^{\\mu \\nu }=\\eta ^{\\alpha \\beta } (\\partial x^\\mu /\\partial x^\\alpha _\\star )(\\partial x^\\nu /\\partial x^\\beta _\\star )$ and $dx^\\mu /ds=\\partial x^\\mu /\\partial x^0_\\star $ , owing to the relations in (REF ).",
"Equation (REF ) thus derived, is valid at all points, as an LICS can always be found at any point of spacetime so that the fluid element is at rest in an infinitesimal spacetime neighbourhood around the point.",
"After establishing the general expression for the energy density-stress tensor of the perfect fluid, let us study its divergence which is famous for describing the mechanical behaviour of the fluid.",
"As our main motive is to understand the mysterious gravitational effects of the pressure of the fluid without implicating the gravitational energy, let us consider the same LICS which has been used to derive (REF ) so that the subtleties of gravitation and the gravitational energy disappear locally in this coordinate system.",
"Let us reconsider the same fluid element, which is at rest in the chosen spacetime neighbourhood.",
"Though the spatial components of the 4-velocity vector vanish in this neighbourhood, the derivatives of the velocity will not be zero in general, except for its temporal component, which will vanish, as we see in the following: the general formula for the interval, $ds^2=g_{\\mu \\nu }~dx^\\mu _\\star ~dx^\\nu _\\star $ implies that $g_{00}\\left(c\\frac{dt}{ds}\\right)^2+g_{11}\\left(\\frac{dx}{ds}\\right)^2+....+2g_{01}c\\frac{dt}{ds}\\frac{dx}{ds}+....$ $+g_{33}\\left(\\frac{dz}{ds}\\right)^2=1,$ which on differentiation gives $\\frac{\\partial }{\\partial x^\\alpha _\\star }\\left(\\frac{dt}{ds}\\right)=0,$ by virtue of the relations in (REF ) and by recalling that $\\partial g_{\\mu \\nu }/\\partial x^\\alpha _\\star =0$ , $g_{\\mu \\nu }=\\eta _{\\mu \\nu }$ in the chosen neighbourhood.",
"Now we assume a vanishing divergence of (REF ) $T^{\\mu \\nu }_{~ ~;\\nu }=0,$ which, in the chosen coordinates, is given by $\\frac{\\partial T^{\\mu \\nu }}{\\partial x^\\nu _\\star }=0.$ The temporal component of equation (REF ) for the case $\\mu =0$ , can be written in accordance with (REF ) and (REF ) as $\\frac{d}{dt}(\\rho \\delta v)+p\\frac{d}{dt}(\\delta v)=0,$ where $\\delta v=\\delta x\\delta y\\delta z$ is the proper volume of the fluid element.",
"The usual interpretation of this equation is that the rate of change in the energy of the fluid element is given in terms of the work done against the external pressure.",
"This seems reasonable at the first sight, but cracks appear in it after a little reflection.",
"The first concern, as also noticed by Tolman [4], is that the fluid of a finite system can be divided into similar fluid elements and the same equation (REF ) can be applied to each of these elements, meaning that proper energy $(\\rho \\delta v)$ of every element is decreasing when the fluid is expanding or increasing when the fluid is contracting.",
"This leads to a paradoxical result that the sum of the proper energies of the fluid elements which make up an isolated system, is not constant!",
"Tolman overlooked this problem by assuming a possible role of the gravitational energy in it.",
"We however remind that there cannot be any role of the gravitational energy in equation (REF ) which has been derived in an LICS.",
"Another more important issue related with equation (REF ) is the following.",
"We have already mentioned that the term $\\rho $ includes in it all the possible sources of mass and energy (excluding the gravitational energy).",
"Hence, so is included in it the energy equivalent to the work done against the external pressure.",
"If we already know that energy (in the form of work done) is being supplied to the system or getting released from it (which can be calculated by checking by what amount $\\delta v$ is increasing or decreasing), why can't this too be taken care of by the term $\\rho $ ?",
"There is no natural law which dictates that $\\rho $ cannot include particular types of energies.",
"So, if $\\rho $ includes the energy equivalent to the work done against the external pressure, the additional work contained in the second term of equation (REF ) violates the conservation of energy!",
"Yet another paradox has been discovered recently [6] which demonstrates, more vividly, subtle inconsistencies in (REF ).",
"This can be derived for the spatial components of equation (REF ).",
"By the use of (REF ) and (REF ) in (REF ), we can write for the case $\\mu =1$ , the following equation $\\frac{\\partial p}{\\partial x}+\\frac{\\left(\\rho +p\\right)}{c^2}\\frac{du_x}{dt}=0,$ which has also been obtained by Tolman in some other context.",
"Here $du_x/dt=du_x/d\\tau =\\partial u_x/\\partial t$ (in the chosen coordinates) is the acceleration of the fluid element in the x-direction.",
"As any role of gravity is absent in this equation, it can be interpreted as the relativistic analogue of the Newtonian law of motion: the fluid element of unit volume, which moves under the action of the force $\\partial p/\\partial x$ , has got the inertial mass $(\\rho +p)/c^2$ .",
"But it is surprising that the inertial mass of the fluid element has got an additional contribution from $p$ , though without any apparent source!",
"Equation (REF ), taken at the face value, reveals that $p$ should be carrying some kind of energy (density) as $p/c^2$ contributes to the inertial mass (density) of the fluid element.",
"However, as the term $\\rho $ includes in it all the sources of energy-mass, so if $p$ `somehow' contains energy, it must be at the cost of violating the celebrated law of the conservation of energy!",
"Though equation (REF ) is not an energy conservation equation, but that does not allow it to defy the law of conservation of energy.",
"Finally, we derive the Tolman integral (REF ) in an LICS wherein it reduces to $E=mc^2=\\int (T_0^0-T_1^1-T_2^2-T_3^3)~d^3x=\\int (\\rho +3p)~dV,$ which does not contain any gravitational energy, but still pronounces the Tolman paradox, though the formula may be valid for a sufficiently small volume of the fluid.",
"From equationsEquation (REF ) can be written alternatively as $\\delta v ~d\\rho /dt+(\\rho +p)d(\\delta v)/dt=0.$ (REF ), (REF ) and (REF ), we notice something unexpected: we still encounter different (unequal) mass densities $(\\rho +3p)/c^2$ and $(\\rho +p)/c^2$ of the same fluid element though there remains no possibility, in the present situation, of any contribution from the gravitational energy to these masses to make them equal, as they are derived in an LICS!",
"Although the equality of the inertial and gravitational masses is the starting point of any metric theory of gravitation, however, the resulting theories have not achieved their own defining feature unequivocally.",
"The two gravitational masses, the active and the passive ones, have remained unequal and the equality of the inertial and the active gravitational masses have remained a dream.",
"The usual interpretation thereof seeks refuge in the gravitational energy.",
"However, the present analysis ensures that there is no possibility of any role of the gravitational energy in making the different masses equal.",
"Perhaps the origin of this problem is elsewhere.",
"It is thus established that the relativistic description of matter given by (REF ) suffers from some subtle inherent inconsistencies in its basic formulation.",
"The point to note is that there is no role of the notorious (pseudo) energy of the gravitational field in these problems."
],
[
"Corrections and Their Consequences",
"There already exists the resolution of the Tolman paradox.",
"Let us take tips from there to resolve the other paradoxes described above and study the consequences.",
"The standard resolution of the Tolman paradox, first presented by Misner & Putnam [7] in 1959, is provided through the following prescription: “If the pressure is confined by non-gravitational means, there must be tension (negative pressure) in the walls (to keep the fluid inside and the field static) which will compensate the additional energy causing the paradox.” In fact, the integrated value of the negative pressure ($-p=-T_i^i$ , $i=1, 2, 3$ , no summation over i) will contribute to (REF ) a term which just counterbalances the increased mass arising from the $3p=(T_1^1+T_2^2+T_3^3)$ term.",
"More recently this resolution has been confirmed by Ehlers et al.",
"[8] in a more refined version by giving a tensorial treatment to the surface tension of the walls and the trapped fluid sphere.",
"It should be noted that the key point of these resolutions $-$ the counterbalancing the pressure of the fluid by stresses in the wall $-$ does not emanate either from the derivation of $T^{\\mu \\nu }$ or from the underlying gravitational theory.",
"Rather it has been put by hand as a correction.",
"After getting corrected, equation (REF ) reduces to the classical result $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~E=\\int \\mathcal {T}_0^0~d^3x= \\int \\rho \\sqrt{|g_{00}|}~dV.\\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{(4$^\\prime $)}$ Let us apply this clue to the paradoxes represented by equations (REF ) and (REF ), for the case when the system is held by non-gravitational meansThe nature of the corrections is not clear when the system is held by gravitation.",
"Should one consider the systems held by the gravitation and those by non-gravitational means, equivalent?",
"Even in the situation when the system is held by gravitation, one would not get any contribution from the gravitational energy in equations (REF )$-$ (REF ) (as they are derived in an LICS), but still sufferring from the paradoxes.",
"This shows the equivalence of the cases held by gravitation and non-gravitational means, so far as the paradoxes are concerned.",
".",
"As the tension of the wall is transmitted to the fluid element, each of the $T_x^x$ , $T_y^y$ and $T_z^z$ present in the considered fluid element, is balanced by a -$T_i^i$ from the wall.",
"Hence the term $pd(\\delta v)/dt$ in equation (REF ) [which can also be written as $pd(\\delta v)/dt=\\delta y\\delta z~T_x^xd(\\delta x)/dt+\\delta z\\delta x~T_y^yd(\\delta y)/dt+\\delta x\\delta y~T_z^zd(\\delta z)/dt$ ] is balanced by $-\\delta y\\delta z~T_x^xd(\\delta x)/dt-\\delta z\\delta x~T_y^yd(\\delta y)/dt-\\delta x\\delta y~T_z^zd(\\delta z)/dt=-pd(\\delta v)/dt$ , modifying equation (REF ) to $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{d}{dt}(\\rho \\delta v)=0,\\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{(11$^\\prime $)}$ which is now free from the paradoxes.",
"We notice something very remarkable in these corrections.",
"The corrections made in equations (REF ) and (REF ) are effectively equivalent to replacing the active gravitational mass density $(\\rho +3p)/c^2$ as well as the inertial mass density $(\\rho +p)/c^2$ (which is also the passive gravitational mass density) by the simple (total) Newtonian mass density $\\rho /c^2$ .",
"Hence the long-sought-after validity of the principle of equivalence is achieved naturally here!",
"This modifies equation (REF ) to $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{\\partial p}{\\partial x}+\\frac{\\rho }{c^2}\\frac{du_x}{dt}=0,\\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{(12$^\\prime $)}$ which is equivalent to balancing the term $(p/c^2)(du_x/dt)$ of equation (REF ) by $-(p/c^2)(du_x/dt)$ .",
"We notice that there is only one contribution from $T_i^i$ s, viz.",
"from $T_x^x$ , to equation (REF ) whereas all the three $T_i^i$ s contribute to equations (REF ) and (REF ).",
"The reason is that equation (REF ) represents a balance of forces along the x-direction only.",
"Similar contributions from $T_y^y$ and $T_z^z$ would appear to the other two similar equations derived from equation (REF ) for the cases $\\mu =2$ and $\\mu =3$ respectively.",
"Obviously, equation (REF ) gets corrected to $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~E=\\int T_0^0~d^3x= \\int \\rho ~dV.\\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{(13$^\\prime $)}$ The corrected equations are convincing and free from the conceptual problems.",
"Additionally, they ascertain the equality of the gravitational and inertial masses, which is remarkable.",
"However, replacing the different general-relativistic energy densities, appearing in the equations of gravitation and cosmology, by the total energy density $\\rho $ , would be revolutionary."
],
[
"Any Observational Support for $T^{\\mu \\nu }$ ?",
"Does it then mean that Einstein's `wood' is not only low grade compared to the standards of his `marble' but it is also rotten and infected?",
"It should be noted that the relativistic description of the matter, in the form of the energy density-stress tensor (REF ), has never been tested in any direct experiment.",
"The classical tests of GR consider $\\rho = p = 0$ .",
"The same is true for the more precise tests of GR made through the observations of the radio pulsars, which are rapidly rotating strongly magnetized neutron stars.",
"The pulsar tests assume the neutron stars as point-like objects and look for the relativistic corrections in the post-Keplerian parameters by measuring the pulsar timing.",
"The test does not even require to know the exact nature of the matter that pulsars and other neutron stars are made of.",
"Hence, the celebrated tests of GR have remained limited to test only the geometric aspect of the theory.",
"Although the above-discussed theoretical crisis in the relativistic formulation of matter is too striking to be ignored, one should be cautious in pronouncing a judgment against a theory like GR, that has been agreed upon by experts for a century and is regarded a highly successful theory of gravitation in terms of its agreement with experimental results and its ability to predict new phenomena.",
"Although the experimental tests, which have verified the theory of GR, have been carried out in empty space, however the following two points may be considered in support of the energy density-stress tensor of the perfect fluid.",
"The general-relativistic formulation of the disordered radiation (which is a particular case of the energy density-stress tensor of the perfect fluid) is consistent with its formulation resulting from Maxwell's theory of classical electrodynamics.",
"Although the standard interpretation of the observations of the bending of starlight, when it passes past the Sun, is given in terms of the correct geometry resulting from the Einstein vacuum field equations $R^{\\mu \\nu }=0$ (Schwarzschild solution), however, the observed deflection, twice as much as predicted by a heuristic argument made in Newtonian gravity, appears roughly consistent with Tolman's formula (REF ) for the disordered radiation [4]Ironically a result derived by Tolman himself (page 250 of [4]) appears contradictory to this.",
"In a weak field, like that of the sun, where Newtonian gravitation can be regarded as a satisfactory approximation, equation (REF ) can be written as $E=\\int \\rho dV +(1/2c^2)\\int \\rho \\psi dV,$ where $\\psi $ is the Newtonian gravitational potential.",
"As $\\psi $ is negative, we note that the general relativistic active gravitational mass $E/c^2$ of the gravitating body is obviously less than its proper mass $(1/c^2)\\int \\rho dV$ and is expected to give a lower value for the gravitational deflection of light!."
],
[
"Conclusion",
"A critical analysis of the relativistic formulation of matter in terms of $T^{\\mu \\nu },$ reveals some surprising inconsistencies and paradoxes.",
"Corrections are discovered which lead to the long-sought-after equality of the gravitational and inertial masses, which are otherwise different in the metric theories of gravitation.",
"Although these obligatory corrections (which must be performed in order to rectify the relativistic formulation of matter) are important in their own right, they do not seem to be consistent with many cosmological observations.",
"For example, they render the standard cosmology altogether incompatible with an accelerating universe and hence with the supernovae Ia observations.",
"This indicates that perhaps the relativistic description of matter by the $T^{\\mu \\nu }$ is not compatible with the geometric description of the metric theories [9].",
"We have tried to provide an overview of the situation resulting from the newly discovered paradoxes, posing serious concern for the modern theories of gravitation, amid the evidences which defend them.",
"It seems likely that an entirely new paradigm is required to resolve this crisis which may completely revolutionize our understanding of the theory.",
"The resolution will certainly require additional ideas and a critical re-examination of the many simplifying assumptions underlying present scenarios."
],
[
"Appendix",
"It is only unto the experimental tests to decide the final validity of a theory.",
"While scientists are crafting experiments to measure already tested predictions of GR with ever-greater precision, it is surprising that not a single experiment has been devised so far to test the gravitational effect of pressure.",
"One such test is also warranted by the present crisis in the standard cosmology, resulting from the discovery of the mysterious `dark energy' (conjured up to explain the current observations), which poses a serious confrontation between fundamental physics and cosmology.",
"It may be mentioned that the most exotic property of the dark energy is its negative pressure, which is ultimately related with the gravitational effect of pressure.",
"In the following, we propose a simple experiment to test this novel prediction of the geometric theories of gravitation.",
"An experiment to test the gravitational effect of pressure: In order to test directly if pressure does carry energy density and hence makes gravitational contribution, we propose a simple Cavendish-type experiment in which the attractor masses, of the torsion balance, are replaced with hollow spherical containers containing some source material (say, $\\cal {S}$ ) which can be converted into gas when required.",
"The experiment is to be performed in two steps.",
"In the first step, one performs the experiment with $\\cal {S}$ in its solid or liquid state (when its pressure is zero).",
"In the second step, one repeats the experiment with $\\cal {S}$ converted into gas (which has a non-zero pressure).",
"If pressure does carry energy and hence makes a gravitational contribution, we should expect a difference between the readings of the two steps.",
"Of course, in either case, one would need to achieve a perfect absence of any heat transfer between the containers and the surroundings.",
"The recommendations for the material $\\cal {S}$ are (i) a material that sublimes from solid to gas, such as solid carbon dioxide; or (ii) two liquids or a liquid plus a solid that would react to produce a gas, analogous to the expansion of an automobile air bag.",
"For a volume $V$ of the container, we expect a gravitational contribution equivalent to an effective mass $pV/c^2$ from the pressure $p$ of the gas.",
"We can estimate this contribution by approximating the gas with an ideal gas.",
"If we consider $x$ kg of $\\cal {S}$ and produce a gas of molecular mass $M$ , the gas would contain $n=xN/M \\times 10^3$ molecules, where $N\\approx 6\\times 10^{23}$ is Avogadro's number.",
"Hence we should expect an energy contribution of $pV=nk_BT\\approx 25x/M\\times 10^5$ J at the room temperature ($k_B\\approx 1.38\\times 10^{-23}$ JK$^{-1}$ is the Boltzman constant) and hence a mass contribution of $pV/c^2\\approx 2.8 x/M\\times 10^{-11}$ kg.",
"Thus the fractional contribution to the mass of $\\cal {S}$ from the pressure of the gas $\\approx 2.8/M\\times 10^{-11}$ .",
"For a gas, like carbon dioxide ($M=44$ ), this comes out as $\\approx 10^{-12}$ (which is perhaps too small to be measured by the present means).",
"This feeble effect can be significantly enhanced by heating the gas (say, by some electrical appliance fitted with the containers).",
"Suppose we supply $y$ J of heat to the gas, this is imparted to the kinetic energy density of its molecules.",
"As the magnitude of the pressure of an ideal gas $=\\frac{2}{3}\\times {\\rm its ~kinetic ~energy ~density}$ , the fractional increase in the mass of the gas, from this effect, would be roughly $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{x\\times 10^{-12}+0.67y/c^2}{x+y/c^2},\\qquad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\text{(A.1)}$ where $c$ , the speed of light, is measured in meter per second.",
"Hence, it is clear that if we can increase $y$ significantly (higher than $10^5$ J), it would not be as difficult to detect (A.1) as is the meagre contribution of the order $10^{-12}$ .",
"This will test not only the gravitational effect of pressure but also the gravitational affirmation of the mass-energy equivalence which has remained just an extrapolation from special relativity to GR and needs to be tested.",
"Acknowledgement: The author thanks Vijay Rai for making available some important literature.",
"C. W. Misner, K. S. Thorn and J.",
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"R. C. Tolman, Relativity, Thermodynamics and Cosmology (Oxford University Press, 1934).",
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]
] | 1204.1553 |
[
[
"Counting Group Valued Graph Colorings"
],
[
"Abstract There are many variations on partition functions for graph homomorphisms or colorings.",
"The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group action so that the colors are identified with the elements of this group.",
"A Fourier transform is used to obtain an expansion for the numbers of colorings with terms indexed by isthmus free subgraphs of the domain.",
"The terms are products of a polynomial in the edge density a of the color graph and the number of colorings of the indexing subgraph of the domain into the complementary color graph.",
"The polynomial in a is independent of the color group and the term has order (1-a) to the r where r is the number of vertices minus the number of components in the indexing subgraph.",
"Thus if (1-a) is small there is a main term indexed by the empty subgraph which is a polynomial in a and the first dependence on the coloring group occurs in the lowest order corrections which are indexed by the shortest cycles in the graph and are of order (1-a) to the g-1 where g is the length of these shortest cycles.",
"The main theorem is stated as a reciprocity law.",
"Examples are given in which the coloring groups are long cycles and products of short cycles and adjacent vertices are required to have distant rather than distinct colors.",
"The chromatic polynomial of a graph corresponds to using any group and taking the allowed set to be the complement of the identity."
],
[
"Introduction",
"There are many variations on partition functions for graph homomorphisms or colorings [1, 2].",
"The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group action so that the colors are identified with the elements of this group.",
"Fix a finite set $V$ , a finite Abelian group ${F}$ and a subset ${A}=-{A}\\subseteq {F}$ .",
"The set $V$ will be the vertex set for the graphs; the elements of the group $F$ will be the colors applied to these vertices; and the subset $A$ will be the allowed differences between the two colors used for the ends of an edge.",
"Since $A$ is symmetric edge orientations can be ignored.",
"Write $f=|{F}|$ , $v=|V|$ , ${\\alpha }={|{A}|\\over f}$ , $\\overline{A}={F}-{A}$ and $\\overline{\\alpha } =1-{\\alpha }$ .",
"Write $P_V$ for the partially ordered set of isthmus-free simple graphs with vertex set $V$ .",
"Thus $P_V=(\\lbrace E\\subseteq {V\\atopwithdelims ()2}|c(E)=c(E-\\lbrace t\\rbrace )\\hbox{ for every }t\\in E\\rbrace ,\\subseteq )$ where $c(E)$ is the number of connected components of the graph with edge set $E$ and vertex set $V$ .",
"Write $\\delta =\\delta _E:{F}^V\\rightarrow {F}^E$ for the coboundary map for this graph.",
"If $P$ is a finite set, write $\\mathbb {C}^P$ for the $\\mathbb {C}$ -vector space with basis indexed by $P$ and with coordinates $[\\cdot ]_p:\\mathbb {C}^P\\rightarrow \\mathbb {C}$ and $GL(\\mathbb {C}^P)$ for the general linear group with matrix entries $[\\cdot ]^p_q:GL(\\mathbb {C}^P)\\rightarrow \\mathbb {C}$ .",
"The focus of this paper is on the probability that a uniformly chosen coloring of the vertices of a graph by elements of the group ${F}$ has all differences along edges in the set ${A}$ .",
"This is summarized in the vector $\\Gamma ^{{A}}\\in {\\mathbb {C}}^{P_V}$ with coordinates $[\\Gamma ^{{A}}]_E=f^{-v}|\\delta ^{-1}{A}^{E}|.$ This can also be viewed in terms of edge colorings since also $[\\Gamma ^{A}]_E=f^{c(E)-v}|{A}^{E}\\cap \\hbox{Im}(\\delta )|.$ This vector will be expanded using the linear operators $j$ and $r^e$ in $GL({\\mathbb {C}}^{P_V})$ .",
"The former is associated to the partial order and has entries $[j]^E_H=1$ if $E\\subseteq H$ and $[j]^E_H=0$ otherwise.",
"The latter is diagonal, associated to the linear extension of $P_V$ given by counting edges and has entries $[r^e]^E_E=r^{|E|}$ , where $r$ is any complex number.",
"Write $J_r=r^ej(r^{-1})^e$ and $M_r=J_{1-r}(-1)^eJ_r^{-1}$ .",
"These are matrices of polynomials in $r$ .",
"The main point is a reciprocity formula which can then be phrased as a formula for the probability of an allowed coloring in terms of the probability of totally disallowed ones for subgraphs or as a polynomial approximation independent of the coloring group.",
"Theorem.",
"$J_{{\\alpha }}^{-1}\\Gamma ^{{A}} = (-1)^eJ_{\\overline{\\alpha } }^{-1}\\Gamma ^{\\overline{A}}$ .",
"Corollary 1 $\\Gamma ^{{A}}=M_{\\overline{\\alpha } }\\Gamma ^{\\overline{A}}$ .",
"Write $g(E)$ for the girth or length of the shortest cycle of $E$ .",
"Corollary 2 $[\\Gamma ^{{A}}]_E= [M_{\\overline{\\alpha } }]^\\emptyset _E+O_{\\overline{\\alpha } \\rightarrow 0}(\\overline{\\alpha } ^{g(E)-1})$ .",
"Write $[\\chi (f)]_E=f^v[\\Gamma ^{{{F}}-\\lbrace 0\\rbrace }]_E$ for the chromatic polynomial of the graph $E$ and $[f^c]_E=f^{c(E)}$ .",
"Corollary 3 $\\chi (f)=M_{f^{-1}}f^c$ ."
],
[
"Proofs",
"Write $\\langle .,.\\rangle :\\hat{F}\\times {F}\\rightarrow S^1\\subseteq {\\mathbb {C}}$ for the canonical pairing between $F$ and its (isomorphic) Pontriajin dual and extend this to $\\langle .,.\\rangle _E:{\\hat{F}}^E\\times {F}^E\\rightarrow S^1\\subseteq {\\mathbb {C}}$ via $\\langle P,Q\\rangle _E=\\prod _{t\\in E}\\langle P(t),Q(t)\\rangle $ and similarly for $\\langle .,.\\rangle _V$ .",
"The coboundary map $\\delta _E$ has an $\\langle .,.\\rangle _E $ -adjoint boundary map $\\partial =\\partial _E:{\\hat{F}}^E\\rightarrow {\\hat{F}}^V$ , so that if $P\\in {\\hat{F}}^E$ , $Q\\in {F}^E$ and $X\\in {F}^V$ then $\\langle P,Q+\\delta X\\rangle _E=\\langle P,Q\\rangle _E\\langle P,\\delta X\\rangle _E=\\langle P,Q\\rangle _E\\langle \\partial P,X\\rangle _V$ and if ${\\bf 0}=0^V$ is the 0 vector in $\\hat{F}^V$ then $\\sum _{X\\in {F}^V}\\langle Y,X\\rangle _V={\\Bigg \\lbrace }\\begin{array}{ll} f^v & \\hbox{ if }Y={\\bf 0}\\\\ 0 &\\hbox{ otherwise.",
"}\\end{array}$ Combining these observations and using a Fourier transform gets from a sum over vertex colorings to a double sum over edge colorings.",
"Lemma 4 $[\\Gamma ^{{A}}]_E=f^{-|E|}\\sum _{P\\in \\partial ^{-1}({\\bf 0})}\\hspace{10.0pt}\\sum _{Q\\in {A}^{E}}\\langle P,Q\\rangle _{E}.$ Consider the Fourier expansion of a delta function: $d_{{A}}(x)={\\Bigg {\\lbrace }}\\begin{array}{ll} 1 &\\hbox{ if }x\\in {A}\\\\0 &\\hbox{ if }x\\in {\\overline{A}}\\end{array}{\\Bigg {\\rbrace }}=f^{-1}\\sum _{p\\in {\\hat{F}}}\\hspace{3.0pt}\\sum _{q\\in {A}}\\langle p,q-x\\rangle .$ Compute: $[\\Gamma ^{{A}}]_E=f^{-v}\\sum _{X\\in {F}^V}\\prod _{\\lbrace u,w\\rbrace \\in E}d_{{A}}([X]_u-[X]_w)$ $ =f^{-v}\\sum _{X\\in {F}^V}\\prod _{t\\in E}d_{{A}}([\\delta X]_t)$ $ =f^{-v}\\sum _{X\\in {F}^V}\\prod _{t\\in E}f^{-1}\\sum _{p\\in {\\hat{F}}}\\sum _{ q\\in {A}}\\langle p, q-[\\delta X]_t\\rangle $ $ =f^{-|E|-v}\\sum _{X\\in {F}^V}\\sum _{ P\\in {\\hat{F}}^E}\\sum _{ Q\\in {A}^E}\\langle P,Q-\\delta X\\rangle _E$ $ =f^{-|E|-v}\\sum _{X\\in {F}^V}\\sum _{ P\\in {\\hat{F}}^E}\\sum _{ Q\\in {A}^E}\\langle P,Q\\rangle _E\\langle P,-\\delta X\\rangle _E$ $ =f^{-|E|-v}\\sum _{X\\in {F}^V}\\sum _{ P\\in {\\hat{F}}^E}\\sum _{ Q\\in {A}^E}\\langle P,Q\\rangle _E\\langle \\partial P,-X\\rangle _V$ $ =f^{-|E|}\\sum _{P\\in \\partial ^{-1}({\\bf 0})}\\hspace{10.0pt}\\sum _{ Q\\in {A}^E}\\langle P,Q\\rangle _E.$ Write: $[\\Gamma ^{{A}}_+]_E=f^{-|E|}\\sum _{P\\in \\partial ^{-1} ({\\bf 0})\\cap ({\\hat{F}}-\\lbrace 0\\rbrace )^{E}}\\hspace{10.0pt}\\sum _{ Q\\in {A}^{E}}\\langle P,Q\\rangle _{E}.$ Lemma 5 $\\Gamma ^{{A}}=J_{{\\alpha }}\\Gamma ^{{A}}_+$ .",
"Write $E^{\\prime }$ for the support of $P$ and restrict $Q$ to $E^{\\prime }$ thereby losing a factor of $({\\alpha }f)^{|E-E^{\\prime }|}$ .",
"$f^{|E|}[\\Gamma ^{{A}}]_E=\\sum _{P\\in \\partial _E^{-1}({\\bf 0})}\\hspace{10.0pt}\\sum _{ Q\\in {A}^E}\\langle P,Q\\rangle _E$ $=f^{|E|}{\\alpha }^{|E|}\\sum _{E^{\\prime }\\le E} {\\alpha }^{-|E^{\\prime }|}f^{-|E^{\\prime }|} \\sum _{ P^{\\prime }\\in \\partial _{E^{\\prime }}^{-1}({\\bf 0})\\cap ({\\hat{F}}-\\lbrace 0\\rbrace )^{E^{\\prime }}}\\hspace{10.0pt}\\sum _{ Q^{\\prime }\\in {A}^{E^{\\prime }}}\\langle P^{\\prime },Q^{\\prime }\\rangle _{E^{\\prime }}$ $ =f^{|E|}[{\\alpha }^ej({\\alpha }^{-1})^e\\Gamma _+^{{A}}]_E.$ Lemma 6 $\\Gamma ^{{A}}_+=(-1)^e\\Gamma ^{\\overline{A}}_+$ .",
"If $p\\in {\\hat{F}}-\\lbrace 0\\rbrace $ then $\\sum _{q\\in {F}}\\langle p,q\\rangle =0$ so $\\sum _{q\\in {A}}\\langle p,q\\rangle =-\\sum _{q\\in {\\overline{A}}}\\langle p,q\\rangle $ .",
"Applying this observation $e$ times gives $\\sum _{Q\\in {A}^E}\\langle P,Q\\rangle _E=(-1)^e\\sum _{Q\\in {\\overline{A}^E}}\\langle P,Q\\rangle _E$ for any nowhere vanishing $P\\in ({\\hat{F}}-\\lbrace 0\\rbrace )^E$ .",
"The desired equation is the sum of this one over $P$ 's with trivial boundary.",
"The theorem now follows immediately and corollaries one and three are immediate consequences.",
"For corollary two note that $[\\Gamma ^{\\overline{A}}]_E\\le \\overline{\\alpha } ^{v-c(E)}$ with equality if $E$ is a forest.",
"Thus $[\\Gamma ^{\\overline{A}}]_\\emptyset =1$ gives the largest term if $\\overline{\\alpha } $ is small with the first corrections associated to the shortest cycles in $E$ ."
],
[
"Examples",
"Example 1.",
"($M$ for $v=3$ and $v=4$ ) It is straightforward to compute $M_{\\overline{\\alpha } }$ as a matrix of polynomials in $\\overline{\\alpha } $ .",
"If $v=3$ then $P_v$ has two elements (corresponding to the complete and empty graphs with three vertices) and $M_{\\overline{\\alpha } }=\\Big {[}\\begin{array}{cc} 1 & (1-\\overline{\\alpha } )^3\\\\ 0 & 1\\end{array}\\Big {]}\\Big {[}\\begin{array}{cc} -1 & 0\\\\ 0 & 1\\end{array}\\Big {]}\\Big {[}\\begin{array}{cc} 1 & -\\overline{\\alpha } ^3\\\\ 0 & 1\\end{array}\\Big {]}=\\Big {[}\\begin{array}{cc} -1 & 1-3\\overline{\\alpha } +3\\overline{\\alpha } ^2\\\\ 0 & 1\\end{array}\\Big {]}.$ If $v=4$ then $P_V$ has fifteen elements, falling into the five graph isomorphism classes: complete, complement of an edge, four cycle, three cycle and empty.",
"Use the notation ${{\\bf 1}}^a_b$ for the $a$ by $b$ matrix of 1s, $K^3_6=\\left[\\begin{array}{c}I_3\\\\I_3\\end{array}\\right]$ and $L^4_6=\\left[{\\tiny \\begin{array}{cccc} 1 & 1 & 0 & 0\\\\ 1 & 0 & 1 & 0 \\\\ 1 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 1 \\\\ 0 & 1 & 1 & 0\\end{array}}\\right]$ to express $M_{\\overline{\\alpha }}=J_{{\\alpha }}(-1)^eJ_{\\overline{\\alpha } }^{-1}$ in block form: $\\left[\\begin{array}{ccccc}1 & {\\alpha }{{\\bf 1}}^6_1 & {\\alpha }^2{{\\bf 1}}^3_1 & {\\alpha }^3{{\\bf 1}}^4_1 & {\\alpha }^6 \\\\0 & I_6 & {\\alpha }K^3_6 & {\\alpha }^2L^4_6 & {\\alpha }^5{{\\bf 1}}^1_6 \\\\0 & 0 & I_3 & 0 & {\\alpha }^3{{\\bf 1}}^1_3 \\\\0 & 0 & 0 & \\pm I_4 & c_9{{\\bf 1}}^1_4 \\\\0 & 0 & 0 & 0 & 1\\end{array}\\right]\\left[\\begin{array}{ccccc}1 & 0 & 0 & 0 & 0 \\\\0 & -I_6 & 0 & 0 & 0 \\\\0 & 0 & I_3 & 0 & 0 \\\\0 & 0 & 0 & -I_4 & 0 \\\\0 & 0 & 0 & 0 & 1\\end{array}\\right]\\left[\\begin{array}{ccccc}1 & -\\overline{\\alpha } {{\\bf 1}}^6_1 & \\overline{\\alpha } ^2{{\\bf 1}}^3_1 & 2\\overline{\\alpha } ^3{{\\bf 1}}^4_1 & -6\\overline{\\alpha } ^6 \\\\0 & I_6 & -\\overline{\\alpha } K^3_6 & -\\overline{\\alpha } ^2L^4_6 & 2\\overline{\\alpha } ^5{{\\bf 1}}^1_6 \\\\0 & 0 & I_3 & 0 & -\\overline{\\alpha } ^4{{\\bf 1}}^1_3 \\\\0 & 0 & 0 & I_4 & -\\overline{\\alpha } ^3{{\\bf 1}}^1_4 \\\\0 & 0 & 0 & 0 & 1\\end{array}\\right]$ $=\\left[\\begin{array}{ccccc}1 & -{{\\bf 1}}^6_1 & {{\\bf 1}}^3_1 & (-1+3\\overline{\\alpha } -\\overline{\\alpha } ^2+\\overline{\\alpha } ^3){{\\bf 1}}^4_1 & (1-6\\overline{\\alpha } +15\\overline{\\alpha } ^2-16\\overline{\\alpha } ^3) \\\\0 & -I_6 & K^3_6 & (-1+2\\overline{\\alpha } )L^4_6 & (1-5\\overline{\\alpha } +10\\overline{\\alpha } ^2-3\\overline{\\alpha } ^3){{\\bf 1}}^1_6 \\\\0 & 0 & I_3 & 0 & (1-4\\overline{\\alpha } +6\\overline{\\alpha } ^2-4\\overline{\\alpha } ^3){{\\bf 1}}^1_3 \\\\0 & 0 & 0 & -I_4 & (1-3\\overline{\\alpha } +3\\overline{\\alpha } ^2){{\\bf 1}}^1_4 \\\\0 & 0 & 0 & 0 & 1\\end{array}\\right].$ Next is a brief discussion of two infinite families of color groups.",
"The first is a single cycle and the second a product of short (length two) cycles.",
"In both cases the allowed sets are all elements far enough from the identity, with the latter using the Hamming metric.",
"Thus $[\\Gamma ^{{A}}]_E$ is the probability that a coloring has adjacent vertices colored with distant rather than distinct colors.",
"Example 2.",
"(Cyclic group) Consider ${F}={\\mathbb {Z}}\\slash f{\\mathbb {Z}}$ and ${A}=(k, f-k)$ so $k={\\overline{\\alpha } f-1\\over 2}$ .",
"If $E$ is fixed then $[\\Gamma ^{{A}}]_E$ is a piecewise polynomial in $\\overline{\\alpha } $ and $f^{-1}$ of degree at most $e$ in the first and $v$ in the second.",
"If $v=3$ then $[\\Gamma ^{{A}}]_\\emptyset =[\\Gamma ^{\\overline{A}}]_\\emptyset =1$ and $[\\Gamma ^{\\overline{A}}]_{K_3}=\\Bigg {\\lbrace }\\begin{array}{cl}1-3\\overline{\\alpha } +3\\overline{\\alpha } ^2 & \\hbox{ if }\\overline{\\alpha } >{2\\over 3} \\\\ {3\\over 4}\\overline{\\alpha } ^2 + {1\\over 4}f^{-2} & \\hbox{ otherwise}\\end{array}\\Bigg {\\rbrace }$ so $[\\Gamma ^{{A}}]_{K_3}=\\Bigg {\\lbrace }\\begin{array}{cl}0 & \\hbox{ if }\\overline{\\alpha } >{2\\over 3} \\\\ 1-3\\overline{\\alpha } +{9\\over 4}\\overline{\\alpha } ^2 - {1\\over 4}f^{-2} & \\hbox{ otherwise}\\end{array}\\Bigg {\\rbrace }.$ Example 3.",
"(Hamming) Consider ${F}=({\\mathbb {Z}}\\slash 2{\\mathbb {Z}})^n$ and ${A}=\\lbrace x\\in {F}||\\lbrace i|[x]_i=1\\rbrace | >k\\rbrace $ so $f=2^n$ and if $k={n\\over 2}+{r\\over 2}\\sqrt{n}$ then $\\lim _{n\\rightarrow \\infty }\\overline{\\alpha } ={1\\over \\sqrt{2\\pi }}\\int _{t=-\\infty }^{-r}e^{-t^2\\over 2}dt$ .",
"The central limit theorem also gives that the edge conditions become independent in the limit so that $\\lim _{n\\rightarrow \\infty }\\Gamma ^{{A}}=\\overline{\\alpha }^{-e}$ .",
"If $k=1$ then $\\overline{\\alpha }=(n+1)2^{-n}$ and $[\\Gamma ^{\\overline{A}}]_{K_3}=(3n+1)4^{-n}$ so $[\\Gamma ^{{A}}]_{K_3}=1-(3n+3)2^{-n}+(3n^2+3n)4^{-n}$ .",
"Authors: Eric Babson and Matthias Beck."
]
] | 1204.1283 |
[
[
"Multiple barriers in forced rupture of protein complexes"
],
[
"Abstract Curvatures in the most probable rupture force ($f^*$) versus log-loading rate ($\\log{r_f}$) observed in dynamic force spectroscopy (DFS) on biomolecular complexes are interpreted using a one-dimensional free energy profile with multiple barriers or a single barrier with force-dependent transition state.",
"Here, we provide a criterion to select one scenario over another.",
"If the rupture dynamics occurs by crossing a single barrier in a physical free energy profile describing unbinding, the exponent $\\nu$, from $(1- f^*/f_c)^{1/\\nu}\\sim(\\log r_f)$ with $f_c$ being a critical force in the absence of force, is restricted to $0.5 \\leq \\nu \\leq 1$.",
"For biotin-ligand complexes and leukocyte-associated antigen-1 bound to intercellular adhesion molecules, which display large curvature in the DFS data, fits to experimental data yield $\\nu<0.5$, suggesting that ligand unbinding is associated with multiple-barrier crossing."
],
[
"Single molecule pulling experiments have generated a wealth of data, which can be used to probe aspects of folding that were not previously possible [1], [2], [3].",
"In addition, DFS has been used to decipher the energy landscape of molecular complexes by measuring the rupture force ($f$ ) by linearly increasing load at a rate $r_f$ (= $df/dt$ ).",
"Because of the stochastic nature of the unbinding events, $f$ varies from one complex (or realization) to another, giving rise to an $r_f$ -dependent rupture force distribution ($P(f)$ ).",
"For a molecular complex obeying Bell's formula, $k(f) = k_{off}\\exp {(fx^{\\ddagger }/k_BT)}$ , Evans and Ritchie showed that the most probable force is $f^* = (k_BT/x^{\\ddagger })\\log (r_fx^{\\ddagger }/k_{off}k_BT)$ [4], where $x^{\\ddagger }(=x_{ts}-x_b)$ is the location of the transition state ($x_{ts}$ ) from the bound state ($x_b$ ) projected along the pulling coordinate and $k_{off}$ is the unbinding rate in the absence of force.",
"However, Bell's formula is applicable only if the molecular complexes are mechanically brittle or if the applied tension is sufficiently small that $x^{\\ddagger }$ does not shift upon application of force [5].",
"More generally, $f^*$ follows a $(\\log r_f)^{\\nu }$ dependence where $\\nu $ depends on the details of the assumed one dimensional (1D) model potential [6], [7], [8], [9], [10], [11], [12], [13].",
"The basic assumption in all these works is that a single free energy barrier along the pulling coordinate is sufficient to describe force-driven rupture of the bound complex.",
"Sometime ago Merkel et al.",
"used DFS to probe load dependent strength of biotin bound to ligands, streptavidin and avidin [14], showing that over six orders of variation in $r_f$ (from about $10^{-2}$ to $r_f$ in excess of $10^4$ pN/s) the plot of $f^*$ versus $\\log r_f$ ($[f^*,\\log r_f]$ plot) varies nonlinearly for both ligands.",
"We note parenthetically that it is also common to observe curvature in unfolding rates of proteins when the $r_f$ is varied [15].",
"By careful data analysis combined with molecular dynamics simulations they proposed an energy landscape for the complex, with multiple energy barriers [14].",
"A similar picture emerges in the rupture of intercellular adhesion molecules (ICAM-1 and ICAM-2) bound to leukocyte function-associated antigen-1 (LFA-1) upon application of force [16].",
"In principle, however, nonlinearity in $[f^*,\\log r_f]$ plot could also arise from load dependent variation in $x^{\\ddagger }$ [17] in a 1D energy landscape with a single barrier [5], [6], [7], [8], [9], [10], [12], [13], [17].",
"A theoretical model describing force-induced escape from a bound state with a single barrier in a cubic potential ($\\nu =2/3$ ) has been used to rationalize the biotin-ligand data by identifying various linear regimes demarcated by $r_f$ [9].",
"However, in the absence of easily discernible changes in the slopes in $[f^*, \\log r_f]$ plot it is difficult to justify such an analysis.",
"Here, we show by analyzing experimental data that the observed non-linearity in the DFS data of several protein complexes can be better accounted for with an energy landscape containing multiple sequential barriers, as originally demonstrated [14], [16].",
"Figure: Rupture characteristics obtained numerically using a potential with two barriers at constant loading rates.",
"(a) U(x)U(x) (magenta) and U(x)-f·xU(x)-f\\cdot x (cyan) with A=5A = 5 pN·nmpN\\cdot nm and f=50f = 50 pN.",
"Reflecting and absorbing boundary conditions are set at x=ax=a and x=bx=b, respectively.",
"(b) Rupture force distributions, P(f)=k(f)/r f ·exp-∫ 0 f df ' k(f ' )/r f P(f)=k(f)/r_f\\cdot \\exp {\\left[-\\int ^f_0df^{\\prime }k(f^{\\prime })/r_f\\right]}, at varying r f r_f were computed by using mean first passage time (MFPT),k -1 (f)=D -1 ∫ a b dye β(U(y)-f·y) ∫ a y dze -β(U(z)-f·z) k^{-1}(f)=D^{-1}\\int ^b_adye^{\\beta (U(y)-f\\cdot y)}\\int ^y_adze^{-\\beta (U(z)-f\\cdot z)}, starting from the first bound state at aa(=0 nm) to reach an absorbing boundary at bb(=5 nm).",
"MFPT expression is valid in the force regime where stationary flux approximation holds .The length was scaled by nm, and D=1.0×10 7 D=1.0\\times 10^7 nm 2 /snm^2/s was used for the diffusion constant.",
"(c) [f * ,logr f ][f^*,\\log r_f] plots at three AA values.",
"Fits of [f * ,logr f ][f^*,\\log r_f] to Eq.",
"yield ν≪0.5\\nu \\ll 0.5 for all AA values (ν=0.064\\nu =0.064, 0.0750.075, 0.0460.046 for A=4,5,6A=4, 5, 6 pN·nmpN\\cdot nm, respectively).In this case, the data should be divided into two regions and analyzed by the two linear fits as depicted using green lines on the curve with A=6pN·nmA=6pN\\cdot nm.",
"(d) loading rate dependent x ‡ (r f )(=x ts -x b )x^{\\ddagger }(r_f)(= x_{ts} -x_b), extracted from the slope of plot at each r f r_f in (c) with A=5A=5 pN·nmpN\\cdot nm, shows a sharp decrease from ∼\\sim 3 nm to <1<1 nm around r f ≈(e -3 -e 0 )r_f \\approx (e^{-3} - e^0) pN/s.To illustrate how steep curvatures in DFS data can arise naturally from a 1D free energy profile we calculated $P(f)$ and $[f^*,\\log r_f]$ of forced-escape kinetics of a quasiparticle from a potential with two barriers, $U(x)=Ax(x-1)[(x-2)(x-3)(x-4)(x-5)+1]$ with $A>0$ (Fig.REF ).",
"The distributions $P(f)$ are typical of what is observed in experiments (Fig.REF b).",
"For all values of $A$ , $[f^*, \\log r_f]$ plots are curved although one could discern a modest change in slope (Fig.REF c).",
"The loading rate dependent $x^{\\ddagger }(r_f)$ , calculated from the slope of the data $k_BT/x^{\\ddagger }(r_f)$ at each $r_f$ in Fig.REF c, changes from $\\sim $ 3 nm to $<1$ nm.",
"The precipitous change in $x^{\\ddagger }$ at $r_f\\approx (e^{-3} - e^0)$ pN/s reflects the transition of the confining barrier from outer to inner barrier with an increasing force (see Fig.REF a).",
"In contrast, gradual change of $x^{\\ddagger }$ in the range $0<\\log r_f <10$ is most likely due to the movement of the inner transition state (see Fig.5C in Ref.(5)).",
"Although it is straightforward to interpret that the two discrete slopes in Fig.REF c (or the precipitous transition of $x^{\\ddagger }$ in Fig.REF d) are due to crossing two barriers since the underlying potential is given in Fig.REF a, it is nontrivial to solve the inverse problem of unambiguously determining from $[f^*, \\log r_f]$ plots and decide whether the underlying free energy profile has a single barrier with a moving transition state as $r_f$ increases or multiple barriers.",
"Figure: The nn-dependent shape of G(x)G(x) (Eq.).",
"The potential with increasing n is associated with more brittle molecular complexes.",
"The yellow circle (x=x c x = x_c) denotes an inflection point and a cusp in each even and odd nn potential, respectively.To establish a criterion for ascertaining whether the energy landscapes for forced-ligand rupture from biotin and LFA-1 have multiple barriers, we study the range of applicability of DFS formalism based on a model potential with a single barrier.",
"Consider a Kramers' problem of barrier crossing in a free energy profile $G(x)$ in which a single barrier separates the bound and unbound states of a quasi-particle as in ligand bound in a pocket of a receptor: $G(x)=G(x_c)+f_c(x-x_c)+\\frac{(-1)^{n+1}M}{(n+1)!",
"}(x-x_c)^{n+1}$ with $M > 0$ .",
"In $G(x)$ , a 1D free energy profile with a single barrier, the shape of barrier and energy well is approximated using $n$ -th order polynomial with $n=1, 2, 3, \\cdots $ .",
"For odd $n$ , we assume that $G(x)=-\\infty $ for $x>x_c$ , so that the transition state of $G(x)$ is cusped.",
"In the absence of tension, the barrier height, $G^{\\ddagger }$ , and the location of transition state, $x^{\\ddagger }$ , are $G^{\\ddagger }=\\chi \\frac{n}{n+1} f_c(n!f_c/M)^{1/n}$ and $x^{\\ddagger }=\\chi (n!",
"f_c /M)^{1/n}$ , respectively, where $\\chi =1$ (for odd n), 2 (for even n).",
"Thus $f_c=\\frac{n+1}{n} G^{\\ddagger }/x^{\\ddagger }$ .",
"The form of $G(x)$ , an extension of the microscopic theories using harmonic-cusp or linear-cubic potential, accounts for the degree of plasticity (or ductility) or brittleness of the energy landscape [4] by changing $n$ (Fig.REF ) [5].",
"Under tension, $G_{eff}(x)=G(x)-f\\cdot x$ ; $f_c$ should be replaced with $f_c(1-f/f_c)=f_c\\varepsilon $ .",
"Therefore, $G^{\\ddagger }(f)=G^{\\ddagger }\\varepsilon ^{1+1/n}$ and $x^{\\ddagger }(f)=x^{\\ddagger }\\varepsilon ^{1/n}$ .",
"Although Eq.REF looks similar to the one Lin et al.",
"used to discuss rupture dynamics for $\\varepsilon \\ll 1$ where the barrier height is almost negligible [12], we did not impose any specific force condition on $G(x)$ .",
"Instead of attributing the movement of transition state to a large external tension [7], [8], [9], [10], [12], [13], we mapped the non-linearity in DFS data onto $G(x)$ that has the $n$ -dependent shape of transition barrier and bound state.",
"In $G(x)$ , increasing brittleness makes $x^{\\ddagger }(f)$ insensitive to applied tension, which is dictated by $n$ ; $x^{\\ddagger }(f)/x^{\\ddagger }=\\varepsilon ^{1/n}\\rightarrow 1$ .",
"For a generic free energy profile $F(x)$ with high curvatures at both $x=x_{ts}$ and $x_b$ , $x^{\\ddagger }(f)/x^{\\ddagger }=1-f/x^{\\ddagger }\\times (|F^{\\prime \\prime }(x_{ts})|^{-1}+|F^{\\prime \\prime }(x_b)|^{-1})\\rightarrow 1$ [5].",
"When free energy profile is associated with a brittle barrier, Bell's formula can be used to extract the feature of the underlying 1D profile from DFS data [5].",
"For general $n$ , the KramersÕ rate equation based on the Eq.REF under tension can be derived as: $k(\\varepsilon )=\\kappa \\varepsilon ^{\\alpha (n)}\\exp {(-\\beta G^{\\ddagger }\\varepsilon ^{(n+1)/n})}$ where $\\kappa $ is the prefactor in Kramers theory and $\\alpha (n)=\\chi (1-1/n)$ with $\\chi =1$ , 2 for odd and even $n$ , respectively.",
"For a given $k(f)$ , the most probable unbinding force is determined by $dP(f)/df|_{f=f^*}=0$ , resulting in a general equation for $f^*$ : $k^{\\prime }(f^*)=\\frac{1}{r_f}[k(f^*)]^2$ which leads to $\\varepsilon ^{\\frac{n+1}{n}}=\\frac{-1}{\\beta G^{\\ddagger }}\\log {\\left[\\frac{r_fx^{\\ddagger }}{\\kappa k_BT}\\varepsilon ^{1/n-\\alpha (n)}\\left(1-\\frac{1}{\\beta G^{\\ddagger }}\\frac{n\\alpha (n)}{n+1}\\varepsilon ^{-\\frac{n+1}{n}}\\right)\\right]}.$ Under the typical condition that rupture occurs by thermal activation, i.e., $f\\ll f_c (\\varepsilon \\approx 1)$ and $\\beta G^{\\ddagger }\\gg 1$ , the most probable unbinding force is approximated as: $f^*\\approx f_c\\left[1-\\left(-\\frac{k_BT}{G^{\\ddagger }}\\log \\frac{r_fx^{\\ddagger }}{\\kappa k_BT}\\right)^{\\nu }\\right]$ where $\\nu =\\frac{n}{n+1}$ .",
"In deriving Eq.REF using $G(x)$ , the large force $\\varepsilon (=1- f/f_c) \\ll 1$ or fast loading condition, an assumption made in obtaining the mean unbinding force expression similar to Eq.REF [6], [12], [13], is not needed.",
"Only the shape of the energy potential matters in deriving Eq.REF from Eq.REF .",
"The DFS data will have a larger curvature for smaller $n$ , namely when the energy landscape associated with a protein complex is more ductile (Fig.",
"2).",
"Because $n=1$ (harmonic cusp), 2 (linear cubic), $\\ldots $ , $\\infty $ (Bell), $\\nu $ must satisfy the bound, $1/2\\le \\nu \\le 1$ for an arbitrary 1D profile that suffices to describe rupture kinetics.",
"For forced-rupture of biotin-ligand complex, fits to the entire range of the data using Eq.REF give $\\nu $ in the disallowed range; $\\nu $ = 0.40 (biotin-streptavidin) and $\\nu $ = 0.070 (biotin-avidin) (see Fig.REF a).",
"Even in biotin-streptavidin case, the parameters extracted from the fits with $\\nu =0.40$ and $\\nu =0.5$ (fixed) are comparable; however, the fit with $\\nu =0.40$ is superior yielding both smaller relative error and reduced chi-square value, $\\chi _{red}^2$ , than with $\\nu =0.5$ , the lower bound of Eq.",
"REF , that gives the maximal curvature in the single-barrier picture (see Fig.REF (a) and Fig.REF in the SI).",
"For both biotin-ligand complexes, our criterion consistently suggests that the unbinding landscapes for the complexes involve more than one barrier, as was emphasized by Merkel et al.",
"[14].",
"Next, we analyzed the extensive data on LFA-1 expressed in Jurkat T cells whose binding affinity to ICAM-1 and ICAM-2 can be enhanced by treating the cells with phorbol myristate acetate (PMA) and the divalent counterion, Mg$^{2+}$ .",
"Under all conditions the exponents that best fit the DFS data are $\\nu <0.5$ (Fig.REF b).",
"As originally argued by entirely different method [16] rupture of ICAM-1 and ICAM-2 from LFA-1 is best described using a free energy profile with at least two barriers.",
"Taken together we arrive at a consistent conclusion that $\\nu <0.5$ implies that the underlying free energy landscape in protein-ligand complexes has multiple barriers.",
"Figure: Analysis of DFS data with large curvatures.",
"(a) The data obtained using biomembrane force probe (BFP) with force constant in the range 0.1-3 pN/nm were fitted to Eq.",
"(solid lines) with ν\\nu =0.40 for biotin-streptavidin (circle) and ν\\nu =0.070 for biotin-avidin (triangle).The x ‡ (r f )x^{\\ddagger }(r_f) at each r f r_f is calculated on the right using the slope of four successive data points of [f * ,logr f ][f^*, \\log r_f ] plot.Analyses of data using restricted ν\\nu values (ν=0.5\\nu =0.5 fit is in dashed line in Fig.a) are in the SI(b) Analysis of DFS data of LFA-1 and its ligands, ICAM-1 and ICAM-2 in Ref.",
".The fits in log-log scale are shown on the right.In all cases, ν<1/2\\nu <1/2 suggests that for these complexes as well the underlying free energy profiles must contain at least two barriers; thus multi-state fits are required by dividing the DFS data into multiple regions as was already surmised in .Mathematically the inequality (Eq.REF ) is not strict because it is possible to construct 1D profiles with $\\nu < 0.5$ for which $[f^*,\\log r_f]$ plots exhibit curvature like those observed in experiments.",
"However, such free energy profiles are physically pathological with non-existing first derivatives in the vicinity of the bound complex and fits to the data give manifestly unrealistic parameters (see Supporting Information for detailed calculations and analysis).",
"For the physical free energy profiles Eq.REF is rigorously satisfied.",
"In addition, there is no compelling reason to choose a special $n$ value even if 1D profile is deemed adequate, and thus $\\nu $ ought to be treated as a parameter.",
"Although Ref.",
"[11] used $\\nu $ as a free parameter, the validity range for $\\nu $ was not discussed.",
"If a global fit of $[f^*, \\log r_f]$ data using Eq.REF yields $\\nu < 0.5$ and the effect of probe stiffness [18] is absent in the DFS data (see below), we can conclude that a single barrier description of the energy landscape is inadequate.",
"In principle curvature in the DFS data could also arise due to probe stiffness.",
"Simple procedure of tiling free energy profile by the amount $-f\\cdot x$ under tension is widely used in analyzing single molecule force experiment.",
"However, more rigorous formulation for the effective free energy profile under load using a transducer with stiffness $k$ should read $G_{tot}(x,X_{tr})=G(x)+\\frac{1}{2}k_{eff}(x-X_{tr})^2$ where $x$ is the position of the end of molecule, $X_{tr}$ is the position of transducer, and $k_{eff}$ is the effective stiffness of molecular construct combining the transducer and the complex ($k_{eff}^{-1}=k_{tr}^{-1}+k_m^{-1}$ ).",
"In fact, $\\frac{1}{2}k_{eff}(x-X_{tr})^2=-f\\cdot x+\\frac{1}{2}k_{eff}x^2+\\frac{1}{2}k_{eff}X_{tr}^2$ with $f=k_{eff}X_{tr}$ .",
"Therefore, the effective free energy for the complex under tension should be written in general as $G_{eff}(x)=G(x)-[f-\\frac{1}{2}k_{eff}x]\\cdot x$ [19].",
"As long as $f\\gg \\frac{1}{2}k_{eff}x$ (or $X_{tr}\\gg x/2$ ) especially when $k_{eff}$ is small as in optical tweezers or BFP, one can approximate $G_{eff}(x)\\approx G(x)-f\\cdot x$ .",
"Otherwise, rebinding from transient capture well created by a large probe stiffness at near-equilibrium loading condition could give rise to the nonlinearity in the DFS data [18].",
"Therefore, the rupture force being measured should be replaced by $f^*\\rightarrow f^*-\\frac{1}{2}k_{eff}x^{\\ddagger }\\varepsilon ^{1/n}$ , and at low forces ($f\\ll f_c$ ) the most probable force measured by using a transducer with high stiffness such as AFM could be approximated as, $f^*\\approx \\underbrace{\\frac{1}{2}k_{eff}x^{\\ddagger }}_{=f_{pl}}+\\underbrace{f_c\\left[1-\\left(-\\frac{k_BT}{G^{\\ddagger }}\\log \\frac{r_fx^{\\ddagger }}{\\kappa k_BT}\\right)^{\\nu }\\right]}_{=f_{\\mathrm {DFS}}}.$ The effect of probe stiffness manifests itself as a non-vanishing plateau force, which could be as large as $f_{pl}\\approx (10-100)$ pN when $k_{eff}\\approx 100$ pN/nm and $x^{\\ddagger }=0.1-1$ nm, even when $r_f$ is small enough that $f_{\\mathrm {DFS}}=0$ [18].",
"The biotin-ligand complexes data preclude this possibility because the probe stiffness of BFP $k_{eff}=0.01-0.3$ pN/nm [14], which is 1-2 orders of magnitude smaller than the $k_{eff}$ value discussed in the literature [20], [21].",
"The value of $k_{eff}$ is $0.5-2.0$ pN/nm in the experiments involving LFA-1 [16].",
"Even the largest estimated $x^{\\ddagger }$ value for the outmost barrier ($x^{\\ddagger }\\approx 3$ nm) only yields $f_{pl}<1$ pN.",
"Furthermore, if the nonlinear curvature of DFS data is suspected to be due to the stiffness effect, this ought to be discerned from the curvature due to multiple barriers by producing DFS data at a reduced probe stiffness.",
"The curvature due to multiple barrier should persist in the DFS data even with a small $k_{eff}$ .",
"Thus, the curvature in the data in [14] can only be attributed to the presence of multiple barriers.",
"The condition (Eq.REF ) for single-barrier based 1D theories for DFS [7], [8], [9], [10], [11], [12], [13] provides a guideline to judge whether the curvature in DFS data is due to multiple barriers or single barrier with a ductile transition state.",
"Our work, which does not consider complications due to various multidimensional landscape scenarios [22], [23], shows that the extracted parameters from the data for the protein complexes with ligands using 1D profile with multiple barriers are physically reasonable [14], [16].",
"Additional justification for the use of such energy landscapes can only be made by studying the structures of the protein complexes in detail.",
"Acknowledgements: This work was funded by National Research Foundation of Korea (2010-0000602) (C.H.)",
"and National Institutes of Health (GM089685) (D.T.).",
"We thank Korea Institute for Advanced Study for providing computing resources.",
"SUPPORTING INFORMATION DFS theory for a free energy profile with non-integer $n$ .",
"It could be argued that the inequality $1/2\\le \\nu \\le 1$ (valid rigorously for integer $n$ ) that accounts for the curvature of DFS data is not mathematically required.",
"Here we show that it is possible to construct 1D free energy profiles for which $\\nu $ is clearly less than 0.5.",
"Indeed, these free energy profiles can even have nearly vanishing $\\nu $ .",
"However, such profiles are unphysical because near the bound state they have incorrect curvatures compared to the physical profiles discussed in the text and in the references cited therein.",
"More importantly, the first derivatives of these free energy profiles, which yield $\\nu < 0.5$ do not exist near the bound state i.e, they have a singularity.",
"For these and other reasons (see below) we reject these free energy profiles as plausible models for explaining the curvatures in the observed [$f^*, \\log {r_f}$ ] plots in a number of protein complexes discussed in the main text, which have all been explained using a two barrier model.",
"A free energy profile with $0<n<1$ (see Eq.REF ) can lead to $0<\\nu <1/2$ since $\\nu =\\frac{n}{n+1}$ .",
"To explore this possibility, we consider a free energy profile, $G(x)=\\sigma a x^{1/\\sigma }-b x$ with $\\sigma >1$ .",
"Here we may regard $n=1/\\sigma $ , and hence $n < 1$ .",
"The term $-bx$ is required for the potential to have a barrier at a finite value of $x^{\\ddagger }$ , the location of the transition state (TS).",
"The TS location and the associated barrier height are $x^{\\ddagger }&=\\left(\\frac{a}{b}\\right)^{\\frac{\\sigma }{\\sigma -1}}\\nonumber \\\\G^{\\ddagger }&=(\\sigma -1)a\\left(\\frac{a}{b}\\right)^{\\frac{1}{\\sigma -1}}.$ Under tension $f$ the potential becomes $G_{eff}=G(x)-fx=\\sigma a x^{1/\\sigma }-(b+f)x$ .",
"The $f$ -dependent TS location and the barrier height are $\\frac{x^{\\ddagger }(f)}{x^{\\ddagger }}&=\\left(\\frac{b}{b+f}\\right)^{\\frac{\\sigma }{\\sigma -1}}=\\left(\\frac{\\frac{G^{\\ddagger }/x^{\\ddagger }}{\\sigma -1}}{\\frac{G^{\\ddagger }/x^{\\ddagger }}{\\sigma -1}+f}\\right)^{\\frac{\\sigma }{\\sigma -1}}=\\eta ^{\\frac{\\sigma }{1-\\sigma }}\\nonumber \\\\\\frac{G^{\\ddagger }(f)}{G^{\\ddagger }}&=\\left(\\frac{b}{b+f}\\right)^{\\frac{1}{\\sigma -1}}=\\left(\\frac{\\frac{G^{\\ddagger }/x^{\\ddagger }}{\\sigma -1}}{\\frac{G^{\\ddagger }/x^{\\ddagger }}{\\sigma -1}+f}\\right)^{\\frac{1}{\\sigma -1}}=\\eta ^{\\frac{1}{1-\\sigma }}.$ where $\\eta \\equiv (1+f/f_{1/\\sigma })$ with $f_{1/\\sigma }\\equiv \\frac{1}{\\sigma -1}\\frac{G^{\\ddagger }}{x^{\\ddagger }}$ .",
"Therefore, one can rewrite Eq.REF ($G(x)$ ) and an effective free energy ($G_{eff}(x)$ ) under tension as $G(x)&=\\frac{\\sigma G^{\\ddagger }}{\\sigma -1}\\left(\\frac{x}{x^{\\ddagger }}\\right)^{1/\\sigma }-\\frac{G^{\\ddagger }}{\\sigma -1}\\left(\\frac{x}{x^{\\ddagger }}\\right),\\nonumber \\\\G_{eff}(x)&=\\frac{\\sigma G^{\\ddagger }}{\\sigma -1}\\left(\\frac{x}{x^{\\ddagger }}\\right)^{1/\\sigma }-(\\frac{G^{\\ddagger }}{\\sigma -1}+fx^{\\ddagger })\\left(\\frac{x}{x^{\\ddagger }}\\right).$ Given $G_{eff}(x)$ , it is possible to obtain the mean first passage time expression corresponding to the lifetime of the complex that can be measured in single molecule experiments.",
"It is given by, $k(f)^{-1}=\\frac{1}{D}\\int ^{\\infty }_0dx e^{\\beta G_{eff}(x)}\\int ^x_0dy e^{-\\beta G_{eff}(y)}.\\qquad \\mathrm {(S5)}$ The saddle-point approximation, $k(f)\\approx D\\left(\\int _{bound}dy e^{-\\beta G^{\\prime }_{eff}(0)y}\\right)^{-1}\\sqrt{\\frac{G^{\\prime \\prime }_{eff}(x^{\\ddagger }(f))}{2\\pi k_BT}}e^{-\\beta G^{\\ddagger }(f)}$ with Eq.A3, yields Kramers' equation for the escape rate: $k(f)=\\kappa \\eta ^{\\alpha (\\sigma )} \\exp \\left[{-\\beta G^{\\ddagger }\\eta ^{\\frac{1}{1-\\sigma }}}\\right]$ with $\\alpha (\\sigma )\\equiv \\frac{2\\sigma -1}{2(\\sigma -1)}$ .",
"Here, note that the $\\kappa $ , defined as the prefactor, contains the singular integral $\\left(\\int _{bound}dy e^{-\\beta G^{\\prime }_{eff}(0)y}\\right)^{-1}$ .",
"By using the relationship $k^{\\prime }(f^*)=[k(f^*)]^2/r_f$ to derive the most probable force, we get $\\eta ^{\\frac{1}{1-\\sigma }}=-\\frac{1}{\\beta G^{\\ddagger }}\\log {\\frac{r_fx^{\\ddagger }}{\\kappa k_BT}\\left[\\frac{(\\sigma -1)\\alpha (\\sigma )}{\\eta ^{\\alpha (\\sigma )+1}\\beta G^{\\ddagger }}+\\eta ^{\\frac{\\sigma }{1-\\sigma }}\\right]}.$ By assuming $\\beta G^{\\ddagger }\\gg 1$ and $f\\ll f_{1/\\sigma }$ , we can simplify the above equation into $\\eta ^{\\frac{1}{1-\\sigma }}\\approx -\\frac{1}{\\beta G^{\\ddagger }}\\log {\\frac{r_fx^{\\ddagger }}{\\kappa k_BT}}.$ Therefore, the most probable force ($f^*$ ) for the fractional potential (Eq.REF ) is $f^*&\\approx f_{1/\\sigma }\\left[\\left(-\\frac{1}{\\beta G^{\\ddagger }}\\log {\\frac{r_fx^{\\ddagger }}{\\kappa k_BT}}\\right)^{1-\\sigma }-1\\right]\\nonumber \\\\&=f_{1/\\sigma }\\left[\\left(-\\frac{1}{\\beta G^{\\ddagger }}\\log {\\frac{r_fx^{\\ddagger }}{\\kappa k_BT}}\\right)^{2-1/\\nu }-1\\right],$ where $\\sigma =\\frac{1-\\nu }{\\nu }$ was employed in the last line.",
"There are a few important comments about Eq.REF that need to be made.",
"(i) Note that the form of Eq.REF is very different from Eq.REF .",
"The difference is related to the aforementioned difficulties associated with $G_{eff}(x)$ .",
"Nevertheless, it is possible mathematically to construct model free energy profiles, without regard to physical considerations, for which [$f^*, \\log {r_f}$ ] plot has curvature that is reminiscent of what is observed in experiments.",
"(ii) Although Eq.REF can be used to obtain excellent fits to DFS data on protein complexes, it turns out that the extracted value of $\\kappa $ is extremely large and are clearly unphysical (see Fig.REF ).",
"The unphysical values are a consequence of the singularity of $G(x)$ (or $G_{eff}(x)$ ) at $x=0$ .",
"We conclude that the free energy profiles with a fractional power of $n$ , which mathematically creates singularity at bound state, is not suitable to be used to analyze experimental data.",
"Thus, the bound $1/2<\\nu <1$ in Eq.REF must hold for physical 1D free energy profiles with a single barrier.",
"Figure: (a) The blue curve is the bare (ff = 0) free energy profile of the form given in Eq.",
"(S1) and the green curve is the tilted form of G(x)G(x) in the presence of force.",
"By fitting the numerically computed (black circles) f * f^* as a function of r f r_f to Eq.",
"(red curve) we obtain the parameters shown below.",
"Although the features of original potential G(x)=100x 1/3 -10xG(x)=100 x^{1/3}-10 x are reasonably recovered (σ\\sigma is larger than the value in G(x)G(x)) by using Eq., the extracted value of κ\\kappa is unrealistically large.",
"(b) Eq.",
"was used to fit the DFS data of biotin-strepavidin (circles) and biotin-avidin (triangles).",
"Although the quality of fit is excellent, the unrealistically large value of κ\\kappa , due to the singularity of the hypothesized fractional potential at x=0x=0, suggests that the potential with a fractional power should not be used for the analysis.",
"In fact the κ\\kappa values are comparable to or much greater than the TST estimate k B T/hk_B T/h (≈6.2×10 12 \\approx 6.2\\times 10^{12} s -1 s^{-1}), which of course makes no sense.",
"Hence, we can rule out the free energy profiles of the form given in Eq.S1 to analyze DFS data on protein complexes.Figure: Analysis of DFS data using Eq.",
"for (a) biotin-streptavidin and (b) biotin-avidin.",
"For each ν\\nu , the fits, residuals (|f fit * -f exp * |/f exp * ×100|f^*_{fit}-f^*_{exp}|/f^*_{exp} \\times 100), and extracted parameters were summarized in the table on the right.We can draw some general conclusions from the fits.",
"For the biotin-streptavidin complex, fit with ν=0.397\\nu = 0.397 produces the smallest errors although at high loading rates the relative errors for ν=0.397\\nu = 0.397 and ν=0.5\\nu = 0.5 are comparable.",
"There are variations in other parameters (x ‡ x^{\\ddagger }, G ‡ G^{\\ddagger }, and κ\\kappa ) for all ν\\nu .",
"For the biotin-avidin complex the situation is far worse.",
"In particular, the relative errors in the fits are large even when ν\\nu is varied.",
"Similarly, the parameters extracted from the fits are not totally consistent.Taken together, the fits using a 1D free energy profile with a single barrier is not appropriate to describe the rupture kinetics of these two complexes."
]
] | 1204.1418 |
[
[
"Red and dead: The progenitor of SN 2012aw in M95"
],
[
"Abstract Core-collapse supernovae (SNe) are the spectacular finale to massive stellar evolution.",
"In this Letter, we identify a progenitor for the nearby core-collapse SN 2012aw in both ground based near-infrared, and space based optical pre-explosion imaging.",
"The SN itself appears to be a normal Type II Plateau event, reaching a bolometric luminosity of 10$^{42}$ erg s$^{-1}$ and photospheric velocities of $\\sim$11,000 \\kms\\ from the position of the H$\\beta$ P-Cygni minimum in the early SN spectra.",
"We use an adaptive optics image to show that the SN is coincident to within 27 mas with a faint, red source in pre-explosion HST+WFPC2, VLT+ISAAC and NTT+SOFI images.",
"The source has magnitudes $F555W$=26.70$\\pm$0.06, $F814W$=23.39$\\pm$0.02, $J$=21.1$\\pm$0.2, $K$=19.1$\\pm$0.4, which when compared to a grid of stellar models best matches a red supergiant.",
"Interestingly, the spectral energy distribution of the progenitor also implies an extinction of $A_V>$1.2 mag, whereas the SN itself does not appear to be significantly extinguished.",
"We interpret this as evidence for the destruction of dust in the SN explosion.",
"The progenitor candidate has a luminosity between 5.0 and 5.6 log L/\\lsun, corresponding to a ZAMS mass between 14 and 26 \\msun\\ (depending on $A_V$), which would make this one of the most massive progenitors found for a core-collapse SN to date."
],
[
"Introduction",
"That massive red supergiants explode as Type IIP (Plateau) supernovae (SNe) at the end of their lives has been clearly established (for example, [47], [41], [27], [12]), and confirmed by the disappearance of their progenitors ([28]).",
"What remains of great interest, however, is understanding how SN progenitor properties (eg.",
"mass, radius, metallicity) correlate with the explosion characteristics (kinetic energy, synthesised $^{56}$ Ni, luminosity etc).",
"The continued identification of progenitors of nearby SNe, coupled with follow-up observations of the SNe, can help shed light on this link.",
"In this Letter we present an analysis of the progenitor of SN 2012aw, which is the coolest, and probably the most massive progenitor of a Type IIP SN found thus far.",
"SN 2012aw in M95 was discovered independently by several amateur astronomers ([10]), with a first detection on 2012 Mar 16.9.",
"A non-detection at a limiting magnitude of $R$ $\\gtrsim $ 20.7 ([36]) on Mar 15.3 sets a rigorous constraint on the age of the SN at discovery of $<$ 1.6 days, and we adopt an explosion epoch of Mar 16.0 UT ($\\pm $ 0.8 d).",
"[30] obtained a spectrum of the SN on March 17.8, which showed a featureless blue continuum; subsequent spectra ([21], [39]) suggested the SN was a young Type IIP, although this classification is yet to be confirmed by the emergence of a plateau in the lightcurve at 3-4 weeks after explosion.",
"A candidate progenitor for SN 2012aw was identified in archival Hubble Space Telescope data by [8] and [13], who both suggested that a faint red supergiant was the likely precursor star.",
"The host galaxy of SN 2012aw, M95, is a barred and ringed spiral galaxy with an inclination of 55$$ .",
"[14] measured a Cepheid-based distance to M95 of 10 Mpc (corresponding to $=30.0\\pm 0.09$ mag), which we adopt in all of the the following.",
"This distance is consistent with that obtained from the Tip of the Red Giant Branch ([37]).",
"We also adopt a foreground (Milky Way) extinction of $E(B-V) = 0.028$ mag from [38].",
"Using the line strengths for the closest Hii region reported in [26] we calculate a metallicity of 12+log[O/H] = 8.8$\\pm $ 0.1 using the [Oiii]/[Nii] relation of [32].",
"From the recent study of the radial metallicity gradient in M95 by [33], at the position of SN 2012aw we estimate a metallicity of 12+ log [O/H] = 8.6$\\pm $ 0.2.",
"Hence we suggest that the metallicity of the progenitor of SN 2012aw is approximately solar (for comparison, Milky Way Hii regions have a typical metallicity of 12+ log [O/H] = 8.7$\\pm $ 0.3; [19]), albeit with the caveat that metallicity calibrations have significant uncertainties (as discussed in this context by [42]).",
"SN 2012aw was detected in X-rays with the Swift X-ray Telescope by [20], at a luminosity of $L_X = 9.2\\pm 2.5 \\times 10^{38}$ erg s$^{-1}$ .",
"This is at the upper end of the range of Type IIP X-ray luminosities ($0.16 - 4 \\times 10^{38}$ erg s$^{-1}$ ; [2]), although still much lower than the typical values seen for Type IIn SNe ($0.2 - 1.6 \\times 10^{41}$ erg s$^{-1}$ ; [11]) which are interacting with significant amounts of circumstellar material.",
"SN 2012aw was also detected in radio observations ([48]; [43]) with a flux at 22 GHz of 2$\\times 10^{25}$ erg s$^{-1}$ Hz$^{-1}$ at +1 week after explosion, rising to 4$\\times 10^{25}$ erg s$^{-1}$ Hz$^{-1}$ at +2 weeks.",
"These values are similar to those of the sample of Type IIP SNe presented by [2]."
],
[
"Supernova characterization and followup",
"Our collaborationhttp://graspa.oapd.inaf.it/index.php?option=com_content&vi ew=article&id=68&Itemid=93 commenced an intensive spectroscopic and photometric follow-up campaign for SN 2012aw immediately after the discovery.",
"Full coverage of the SN will be published in a future paper, here we present a limited set of optical observations from first two weeks after explosion to characterize the initial SN evolution.",
"Photometry was obtained for SN 2012aw with the Asiago Observatory Schmidt telescope in the Landolt $BVRI$ system.",
"The data were pipeline reduced, and PSF-fitting photometry was performed on the images using the QUBA pipeline ([46]).",
"A pseudo-bolometric (ie.",
"$BVRI$ ) lightcurve was constructed by integrating the observed SN flux over the optical filters.",
"The resulting lightcurve is shown in Fig.",
"REF , together with those of three other Type IIP SNe for comparison (SN 1999em, [9]; SN 1999gi, [24]; SN 2005cs, [31]).",
"The luminosity of SN 2012aw is comparable to that of the SN 1999em, placing it firmly in the continuum of normal Type IIP SNe, as opposed to sub-luminous events such as SN 2005cs.",
"Spectra of SN 2012aw were obtained from the Asiago 122cm telescope + Boller&Chivens Spectrograph + 300tr/mm, and the Nordic Optical Telescope + ALFOSC + Gr#4.",
"Spectra were reduced, extracted, and flux and wavelength calibrated with the QUBA pipeline.",
"The sequence of spectra from the first two weeks after explosion is presented in Fig.",
"REF .",
"Without any adjustment for either Milky Way or host galaxy extinction, the slope of the SN 2012aw spectrum (epoch 3.9 d) is almost identical to that of SN 1999em at +4 d [1], [18].",
"[1] found that a synthetic spectrum fit with $T_{\\rm BB}=11000-13000$ K and an extinction of $E(B-V)\\sim 0.05$ mag provides a good match to the continuum slope and Hi lines.",
"A similar result would necessarily be determined for SN 2012aw given its striking similarity to SN 1999em.",
"The H$\\alpha $ absorption minimum is at a velocity of $v_{phot}\\sim 13000$ km s$^{-1}$, again almost identical to the minimum for SNe 1999em and 1999gi.",
"Fig.",
"REF shows the early spectra of these three Type IIP SNe illustrating the similarity, also shown is the low energy, faint Type IIP SN 2005cs ([31]) which has a H$\\alpha $ minimum at a much lower velocity of 6500 km s$^{-1}$.",
"This clearly shows that SN 2012aw is not a low energy explosion.",
"A high resolution spectrum of SN 2012aw was obtained with the Telescopio Nazionale Galileo + Sarg on Mar 29.",
"The two components of the Nai D doublet were detected at 5907.998, 5914.004 Å , which is consistent with the recessional velocity of M95.",
"The equivalent width (EW) of the two components were 0.286 and 0.240 Å respectively.",
"From the empirical relation of [29], we estimate E(B-V) = 0.10$\\pm $ 0.05 mag for SN 2012aw.",
"We note however, that even if dust has been photo-evaporated in the explosion (see Sect.",
"), the Nai absorption should remain unchanged and hence these calibrations may systematically over-estimate the extinction."
],
[
"Archival data",
"We have searched both the Hubble and Spitzer Space Telescope archives, along with the publicly available archives of all the major ground based observing facilities, for pre-explosion images covering the site of SN 2012aw.",
"A log of all data used is given in Table .",
"The deepest, and highest resolution images of M95 found were from the Wide-Field and Planetary Camera 2 (WFPC2) onboard the Hubble Space Telescope (HST).",
"The site of SN 2012aw was observed on multiple occasions between 1994 Nov and 1995 Jan with the $F439W$ , $F555W$ and $F814W$ filters, and again in the $F555W$ filter in 1995 Dec, as part of the HST Key Project on the Extragalactic Distance Scale.",
"HST +WFPC2 observed M95 again in $F336W$ and $F658N$ on 2009 Jan 18.",
"M95 was also observed with the Subaru Telescope+ SuprimeCAM in $R$ on 1999 Jan 28, but as the seeing was $\\gtrsim $ 2.5\" in these images, they were of no use for constraining the progenitor.",
"NTT+SOFI ($K_S$ ) and VLT+ISAAC ($J_S$ ) images of M95 were obtained from the ESO archive.",
"A subset of the data used is shown in Fig.",
"REF .",
"All HST data were downloaded from the Multimission Archive at STScIhttp://archive.stsci.edu.",
"The individual WFPC2 $F814W$ images were combined using the drizzle algorithm within irafIRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.",
"([15]), although as the offsets between the individual exposures were not in an optimal subpixel dither pattern, we did not resample the image to a smaller pixel scale.",
"Small image shifts were removed using the tweakshifts task to improve the quality of the alignment.",
"The individual VLT+ISAAC images were reduced (bad-pixel masked, flat-fielded and sky-subtracted) using the ISAAC pipeline (version 6.0.6).",
"The individual frames from each night where the SN position was in the field of view were then aligned with sub-pixel shifts and combined within iraf.",
"The image from the second night of ISAAC observations (2000 Mar 27) was used for all of the subsequent analysis, as the total on source exposures are deeper, and the SN position is further from the edge of the chip.",
"The NTT+SOFI data from 2006 Mar 24 were not flat-fielded (as appropriate calibrations were not available), but were sky subtracted with the SOFI pipeline (version 1.5.4) before the individual on-source frames were combined.",
"A total of 600s was obtained on source, with a FWHM of 0.6.",
"For the SOFI data from 2002 Mar 25, the SOFI pipeline was unable to reduce the data, and so the iraf xdimsum package was used for the reduction.",
"The Spitzer Space Telescope + IRAC observed M95 as part of the SINGS survey ([23]), however the spatial resolution of these data (1.7 PSF FWHM in the shortest wavelength channel) is ill suited for progenitor identification.",
"We downloaded the IRAC data from the Spitzer Heritage Archive, but could not identify a point source at the SN coordinates.",
"We have not considered the IRAC data any further in the following."
],
[
"Progenitor identification",
"A deep, high resolution $K$ -band image of SN 2012aw was obtained on 2012 Mar 31 with Gemini+NIRI (Program: GN-2012A-Q-38).",
"To match the resolution of the pre-explosion WFPC2 images, we used the ALTAIR adaptive optics system to correct for the seeing; as the SN was sufficiently bright (mag$\\sim $ 13) at the time of the observations it was used as a natural guide star for ALTAIR.",
"The f14 camera was used, which has a 0.05 pixel scale over a 51$\\times $ 51 field of view.",
"As the region around SN 2012aw is not crowded, we used on source dithers of a few arcseconds, and median combined these to create the sky frame for each exposure.",
"The data were reduced with the iraf gemini package, yielding a final combined image with a PSF FWHM of 0.2.",
"40 sources were identified common to both the NIRI and the drizzled WFPC2 $F814W$ images.",
"The positions of these reference stars were measured in both images with iraf phot.",
"After rejecting outliers from the fit, which either had low signal to noise or centering errors, we used 22 sources to derive a geometric transformation between the pixel coordinate systems of the WFPC2 and NIRI images.",
"A general fit allowing for scaling, a shift in x and y, rotation and a skew term was used, with a root mean square error in the fit of 53 mas.",
"The pixel coordinates of SN 2012aw were measured in the NIRI image, and transformed to the pixel coordinates of the WFPC2 image.",
"The transformed position of the SN coincided with the same source identified by [8] and [13].",
"The position of this source was measured using the three different centering algorithms in phot, all of which agreed to within 14 mas.",
"The separation between the transformed SN position, and the progenitor candidate was 27 mas, which is well within the total (transformation + SN position + progenitor position) uncertainty of 55 mas.",
"Hence we formally identify the source as the progenitor candidate for SN 2012aw."
],
[
"Progenitor photometry",
"Photometry was performed on the WFPC2 images with the hstphot package ([4]), which is designed specifically for undersampled data from this instrument.",
"The pipeline reduced images were first masked for bad pixels, hot pixels and cosmic rays using the ancillary packages distributed with hstphot.",
"After masking, PSF-fitting photometry was performed on the individual images, which typically had exposure times of 1000 to 2000 s. The SN progenitor candidate was detected at a magnitude of $F555W$ =26.70$\\pm $ 0.06 and $F814W$ =23.39$\\pm $ 0.02 (in the photometric system as defined in [5]), but was not detected in either the $F336W$ or $F439W$ filter images.",
"Based on photometry of sources detected by hstphot in the $F439W$ image, we estimate a 5$\\sigma $ limiting magnitude of $F439W>$ 26.",
"The $F814W$ magnitude for the progenitor, as measured in the individual frames, appears to be unchanged within the uncertainties over the $\\sim $ 50 day period from the first to the last observation.",
"Unfortunately the progenitor is too faint to check for variability in the $F555W$ images.",
"The $F658N$ (H$\\alpha $ ) WFPC2 image was also examined, but there was no sign of emission at the SN position.",
"The SOFI $K_S$ and ISAAC $J_S$ images were aligned to an $F814W$ -filter mosaic of the four WFPC2 chips, and a counterpart to the optical progenitor candidate was found in both images.",
"PSF-fitting photometry was performed on the ISAAC $J_\\mathrm {s}$ and SOFI $K_\\mathrm {s}$ image with the snoopy package in iraf.",
"A magnitude was found for the progenitor of $J$ = 21.1$\\pm $ 0.2 (setting the photometric zeropoint from the $J$ magnitude of two 2MASS sources in the field, which is the dominant source of error).",
"As the bandpasses of the $J_\\mathrm {s}$ and $J$ filters are different, we calculated colour terms from synthetic photometry of MARCS model spectra between 3000 and 4000 K. Over this temperature range the colour term is $\\lesssim $ 0.02 mag, which is significantly less than our photometric error.",
"For the SOFI K$_\\mathrm {s}$ image from 2006, we measured a magnitude of $K$ = 19.3 $\\pm $ 0.4 with PSF-fitting photometry.",
"The uncertainty in the measurement consists of 0.36 mag from the progenitor photometry, and 0.21 uncertainty in the zero point, which was set from seven 2MASS sources in the field.",
"As a check of the PSF-fitting measurement, aperture photometry was performed on the same image with iraf phot, which returned a magnitude of $K$ = 19.1$\\pm $ 0.4.",
"Aperture photometry of the SOFI image from 2002 yielded a magnitude for the progenitor of 18.9$\\pm $ 0.3 mag.",
"We hence adopt an average value of $K$ =19.1 from the two epochs for the progenitor magnitude, with a conservative error of $\\pm $ 0.4 mag.",
"All NIR magnitudes are in the 2MASS system ([3])."
],
[
"Results and Discussion",
"We used our own Bayesian SED-fitting code (based on the Bayesian Inference-X nested sampling framework; Maund 2012 in prep; [40]) to compare the observed progenitor magnitudes to synthetic photometry of MARCS model spectra ([17]).",
"The models used were for 15 M$_{\\odot }$ stars with log g = 0 dex and solar metallicity, with temperatures between 3300 and 4400 K. The SEDs were rebinned by a factor of 10 prior to the calculation of synthetic photometry ([34]).",
"We fit the progenitor $VIJK$ magnitudes with flat priors on $T_\\mathrm {eff}$ and E(B-V), which was allowed to vary between $-0.5 <$ E(B-V) $< 2$ .",
"A match was found with the observed progenitor magnitudes along a banana-shaped region as shown in Fig.",
"REF .",
"The luminosity is ill constrained due to the uncertain extinction, and indeed varies within the 1$\\sigma $ contours between 5.0 dex at 3550 K to 5.6 dex for a progenitor with $T_\\mathrm {eff}$ =4450 K. The SED fit is poorer in $K$ than the $J$ or WFPC2 data, as can be seen in Fig.",
"REF .",
"Besides the larger photometric errors in $K$ data, the models are also quite sensitive to metalliciity at cooler temperatures, due to the TiO absorption which is present in the NIR spectra of cool stars.",
"Determining the luminosity of the progenitor from the $K$ -band magnitude only (as A$_K\\sim $ 0.1 A$_V$ , and the bolometric correction to $K$ -band only changes by 1.1 mag between 3300 and 4500 K) gives a value between 4.9 and 5.5 dex.",
"We also attempted to fit the progenitor SED with MARCS spectra of twice solar metallicity models, but found the fit to be poorer than for the solar metallicity models.",
"In Fig.",
"REF , we have also plotted stellar evolutionary tracks from the STARS code ([7]).",
"Comparing the luminosity of the progenitor to the endpoints of the tracks implies a ZAMS mass of between 14 and 26 M$_{\\odot }$ .",
"If we consider the hotter progenitors in Fig.",
"REF , the discrepancy in temperature with the end points of the STARS evolutionary tracks becomes more apparent.",
"However, increased mass loss, either in a binary or through rotation, would serve to bring the end points of these tracks over to hotter temperatures ([16]).",
"As the radius of the progenitor can be expressed as R=$\\sqrt{L/T_{eff}^4}$ , it is easy to calculate the expected radius for each point in the HR diagram.",
"We find that even for a temperature of 4500 K, and a luminosity of 5 dex, the radius of the progenitor is still $>$ 500 R$_{\\odot }$ .",
"This is sufficiently large that we may reasonably expect the progenitor to give rise to a Type IIP SN (e.g.. [35], [22]), and so cannot be used to further restrict the region of the HRD where the progenitor lies.",
"Within the 68% confidence contours, the solar metallicity SEDs favour an extinction which is greater than $E(B-V)>0.8$ mag, although within 95% (2$\\sigma $ ), there is a solution with a correspondingly poorer fit for $E(B-V)=0.4$ mag.",
"The implication of this is that a significant amount of dust could have been destroyed in the initial phases of the SN explosion (eg.",
"[45], [6]), and that the extinction towards a SN can not be taken as a proxy for the extinction towards the progenitor.",
"We stress, however, that regardless of extinction, SN 2012aw potentially arises from one of the highest mass progenitors found to date.",
"As was recently suggested by [44], significant amounts of circumstellar dust around SN progenitors is an appealing solution to the lack of high mass red supergiant progenitors identified by [42].",
"We caution however, that SN 2012aw is the reddest SN progenitor found thus far, and appears to suffer from extinction that is comparable to that of the most luminous Galactic red supergiants ([25]), but higher than is typical for a Type IIP progenitor (from the limited sample with colour information).",
"Hence it remains unclear whether circumstellar dust truly is the panacea for the “red supergiant problem”.",
"We acknowledge funding from STFC (MF), the ERC (SJS) and the Royal Society (JRM).",
"SB, MT, IJD and AP are partially supported by the PRIN-INAF 2009 with the project “Supernovae Variety and Nucleosynthesis Yields”.",
"Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership.",
"Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute.",
"STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.",
"Based on data obtained from the ESO Science Archive Facility.",
"Partially based on observations collected at Asiago observatory, Galileo 1.22 and Schmidt 67/92 telescopes operated by Padova University and INAF OAPd, and on observations made with the NOT, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.",
"We thank Ben Davies and Rolf Kudritzki for useful advice on metallicities.",
"Facilities: Gemini(NIRI+ALTAIR), HST(WFPC2), VLT(ISAAC), NTT(SOFI) llllc Pre-explosion images of the site of SN 2012aw 0pt Date Telescope Instrument Filter Exposure (s) 2009-01-18 HST WFPC2 F336W 4400 1994-11-29 2*- 2*- 2*F439W 2*5000 –1994-12-19 2009-01-18 - - F658N 1800 1994-11-29 2*- 2*- 2*F555W 2*34130 –1995-12-04 1994-11-29 2*- 2*- 2*F814W 2*9830 –1995-01-16 2000-03-26 VLT ISAAC J 2400 2000-03-27 - - - 4080 2002-03-25 NTT SOFI Ks 600 2006-03-24 - - - 600"
]
] | 1204.1523 |
[
[
"VLBI and Single Dish Monitoring of 3C84 in the Period of 2009-2011"
],
[
"Abstract The radio galaxy 3C 84 is a representative of gamma-ray-bright misaligned active galactic nuclei (AGNs) and one of the best laboratories to study the radio properties of the sub-pc jet in connection with the gamma-ray emission.",
"In order to identify possible radio counterparts of the gamma-ray emissions in 3C 84, we study the change in structure within the central 1 pc and the light curve of sub-pc-size components C1, C2, and C3.",
"We search for any correlation between changes in the radio components and the gamma-ray flares by making use of VLBI and single dish data.",
"Throughout the radio monitoring spanning over two GeV gamma-ray flares detected by the {\\it Fermi}-LAT and the MAGIC Cherenkov Telescope in the periods of 2009 April to May and 2010 June to August, total flux density in radio band increases on average.",
"This flux increase mostly originates in C3.",
"Although the gamma-ray flares span on the timescale of days to weeks, no clear correlation with the radio light curve on this timescale is found.",
"Any new prominent components and change in morphology associated with the gamma-ray flares are not found on the VLBI images."
],
[
"Introduction",
"With the Large Area Telescope (LAT) on board Fermi Gamma-ray Space Telescope, GeV $\\gamma $ -ray emission from 3C 84 (alias NGC 1275, z=0.0176) was detected for the first time (Abdo et al.",
"2009).",
"It was one of the two misaligned AGNs detected by Fermi after four-months operations.",
"More recently, $\\gamma $ -ray emission is detected in 12 misaligned AGNs (Abdo et al.",
"2010a, Brown & Adams 2012).",
"This gives new challenges to the AGN unification scenario.",
"Among these misaligned AGNs, 3C 84 exhibited strong variability in its $\\gamma $ -ray emission.",
"An averaged $\\gamma $ -ray flux during the first four months is $(2.10\\pm 0.23)\\times 10^{-7}$ ph cm$^{-2}$ s$^{-1}$ above 100 MeV.",
"This $\\gamma $ -ray flux is seven times brighter than the upper limit by EGRET.",
"It was claimed that the innermost jet of 3C 84 is the most likely source of $\\gamma $ -ray emission because of the time variation on the timescales shorter than years to decades (Abdo et al.",
"2009).",
"During the first 2 years, Fermi-LAT detected $\\gamma $ -ray emission from 3C 84, two episodes of increased $\\gamma $ -ray activity were observed: one occurred in 2009 Apr-May (Kataoka et al.",
"2010), and the other one occurred in 2010 June-August (Brown & Adams 2011).",
"In particular, the second flare was detected up to 102.5 GeV and very-high-energy $\\gamma $ -ray emission with a spectral cut-off around 500 GeV was found successively by MAGIC (Aleksić et al.",
"2011).",
"The radio source 3C 84 is known to be a bright radio source, and has been studied extensively since the early days of radio astronomy.",
"The radio flux has been monitored since 1960, and episodes of violent flux increase were reported (Kellermann & Pauliny-Toth.",
"1968; O'Dea et al.",
"1984).",
"In mid-1980s, the radio flux became exceptionally bright, more than 60 Jy at centimeter wavelengths, and then subsequently decreased such that, by the early 2000s, the radio flux decreased to $\\sim 10$ Jy (Nagai et al.",
"2010, hereafter Paper I).",
"This flux decrease can be ascribed to the adiabatic expansion of radio lobe in the central $\\sim 5$ pc (Asada et al.",
"2006).",
"The viewing angle of the jet is estimated to be 11-55$^{\\circ }$ to the line of sight (e.g., Asada et al.",
"2006; Lister et al.",
"2009).",
"Around 2005, it was reported that the radio flux started to increase again (Abdo et al.",
"2009).",
"The VLBI observations revealed that this flare originated from within the central pc-scale core, accompanying the ejection of a new jet component (Paper I).",
"This new component appeared from the south of the core around 2003 (Suzuki et al.",
"2012, hereafter Paper II), and was moving to the position angle $\\sim 160^{\\circ }$ steadily with slightly changing speed in both parallel and perpendicular directions.",
"In Paper I, we revealed that the central 1-pc structure mainly consists of three bright components C1, C2, and C3.",
"In Paper II, we argued that it is difficult to reconcile the observed GeV $\\gamma $ -ray emission with a one-zoned synchrotron-self Compton model in C3 because the observed spectral index of C3 between 22 and 43 GHz disagrees with that derived from the SED modeling.",
"In order to obtain an additional validation to this argument, this paper focuses on the comparison between radio light curve and GeV $\\gamma $ -ray flares which occurred in the period of 2009 April-May and 2010 June-August.",
"We argue which component is the most likely site of the observed $\\gamma $ -ray emission by comparing radio flux variability with the $\\gamma $ -ray flaring events.",
"We are also interested in the possible emergence of a new radio component associated with the $\\gamma $ -ray flares.",
"In this paper, we present the light curve of each component obtained by the Monitoring Of Jets in Active galactic nuclei with VLBA ExperimentsThe MOJAVE data archive is maintained at http://www.physics.purdue.edu/MOJAVE.",
"(MOJAVE: Lister et al.",
"2009) and detailed structural change obtained by the VLBI Exploration Radio Astrometry (VERA).",
"Moreover, the 37-GHz data from the Metsähovi have been used to complement the MOJAVE light curve.",
"Throughout this paper, we adopt $H_{0}=70.2 $ km sec$^{-1}$ Mpc$^{-1}$ , $\\Omega _{\\mathrm {M}}=0.27$ , and $\\Omega _{\\mathrm {\\Lambda }}=0.73$ (Komatsu et al.",
"2011; 1 mas=0.35 pc, 0.1 mas yr$^{-1}=0.11c$ ).",
"Observations were carried out between 2009 April and 2011 March for 8 epochs with four VERA stations at 43 GHz.",
"Each observation lasted 8 hours, consisting of a number of scans in which the length of each scan is 570 seconds, with a 30-seconds gap inserted between each scan.",
"Most of the scans were assigned for 3C 84 and a few scans were assigned for calibration.",
"The total on-source time for 3C 84 is about 7 hours for each observation.",
"The data were recorded at a rate of 128 Mbps for the first 6 epochs and 1024 Mbps for the remaining 2 epochs, with 16-MHz and 128-MHz bandwidth, respectively.",
"Data reduction was performed using the NRAO Astronomical Imaging Processing System (AIPS).",
"A standard $a$ $priori$ amplitude calibration was performed using the AIPS task APCAL based on the measurements of the system temperature during the observations and the aperture efficiency provided by each station for VERA data at 43 GHz.",
"Fringe fitting was done using the AIPS task FRING.",
"For the deconvolution of synthesized image, we used CLEAN and self-calibration technique.",
"Final images were obtained after a number of iterations with CLEAN and both phase and amplitude self-calibration using the DIFMAP software package (Shepherd 1997).",
"After the process described above, we found that the VERA 43-GHz was resolved and some portion of flux density within the central $\\sim 1$ pc region was missing.",
"Because of this missing flux, it was difficult to argue the flux change using VERA.",
"We do not use the VERA data for the argument of light curve, but they are used to study the structural change because the central region is less opaque at 43 GHz than at 15 GHz, i.e., the frequency of the MOJAVE data."
],
[
"The MOJAVE data",
"We imported the calibrated uv-data of the MOJAVE programme (15 GHz) into the NRAO AIPS package and we performed a few phase-only self-calibration iterations and flagging some data when the amplitudes were too high or too low with respect to the other visibilities.",
"Finally we produced high- and low-resolution images in order to study either the core structure or the extended emission.",
"The high-resolution image (FWHM=0.55 mas) was produced considering only the baselines longer than 50 M$\\lambda $ and using pure uniform weight, whereas the low-resolution image was obtained using the baseline shorter than 250 M$\\lambda $ and natural weight (FWHM=1 mas)."
],
[
"The Metsähovi data",
"The single dish monitoring data at 37 GHz is adopted from the Metsähovi quasar monitoring program.",
"The observations were carried out with the Metsähovi 14-m radio telescope.",
"A detailed description of the data reduction and analysis is given in Teräsranta et al.",
"(1998).",
"In order to compare the VLBI and single dish light curves, we selected the data from 2009 January 1 to 2011 November.",
"In order to investigate if there is any structural change, i.e., new component emergence, we deconvolved the VERA 43-GHz images by using the several components whose brightness distribution is elliptical Gaussian.",
"Throughout all epochs for VERA data at 43 GHz, overall structures were mostly well-represented by three major components C1, C2, and C3, which were the same components identified in Paper I and Paper II.",
"In addition to these three components, there was an additional emission bridging between C1 and C3.",
"We had a difficulty to model this emission: relative flux densities and positions of the Gaussian components in this region can be correlated with those of C1 and C3, and the position and size of C1 and C3 were often skewed by these components.",
"We operationally fitted this bridging emission by using one or two point sources (unresolved components) instead of Gaussians (see C4a and C4b in Figure REF ), in order to avoid them from such a skewing.",
"The choice of one or two point sources was somewhat arbitrary, and it was difficult to keep consistency across the epochs.",
"We always tried to maintain consistency of the three main components (C1, C2, and C3) even if the position and flux of the point sources C4a and C4b did not show smooth change across the epochs.",
"We do not use the result from this model fit for the evaluation of flux variability because of significant missing flux which is mentioned in section REF , but we use the VERA data for the discussion of structural change.",
"Figure REF shows the CLEAN images from the VERA and one example of model-fit image.",
"Note that the model-fit image and corresponding CLEAN image are essentially similar, demonstrating the validity of modeling procedure.",
"The analysis of the 15-GHz MOJAVE data was performed on the image domain instead of on the visibility domain.",
"The flux density, size and position of each component was derived by means of the AIPS task JMFIT, which performs a Gaussian fit to the components on the image plane.",
"This approach has been found preferable rather than the visibility-based approach due to the presence of extended structures.",
"Indeed, the minimum baseline of the VLBA at 15 GHz (6.5 M$\\lambda $ ) is sensitive to the extended flux density arising from both the northern and southern lobes located about 15 mas and 10 mas from the core, respectively (e.g., Asada et al.",
"2006).",
"Both components had complex sub-structures that make the analysis of the visibility data difficult to carry out.",
"On the other hand, the minimum baseline of VERA was too long to detect such extended structures, making the model fitting of the core components on the visibility data more reliable.",
"We use the result from MOJAVE data for the discussion of the light curve."
],
[
"Structure and Light curve",
"Figures REF and REF show the total intensity images of MOJAVE at 15 GHz and VERA at 43 GHz, respectively.",
"Two bright components C1 and C3 are separated by about 2 milli-arcsec in the north-south direction, while the fainter component C2 is located on the west of C3 (see Figure REF ).",
"The overall structure is very similar to that presented by Paper I and Paper II.",
"From the analysis of the data presented in this paper, no significant changes in the central structure of 3C 84 has been detected.",
"C3 is continuously moving to the south with a position angle $\\sim 170^{\\circ }$ with respect to C1, as it has already been found in previous papers.",
"The positional change of C2 is somewhat random rather than systematic, but it seems that C2 is also moving to the south slightly on average.",
"A detailed analysis of the kinematics of the central components of 3C 84 will be discussed in a forthcoming paper.",
"On average the total flux density increases with time, but some rise and fall in flux density on the timescales of a month is also seen in the Metsähovi data (Figure REF ).",
"Total flux increase shows similar trend with the sum of flux of C1, C2, and C3.",
"The most of flux increase originates in C3.",
"The increasing level is more than a factor of two throughout the monitoring (see Table REF ).",
"On the other hand, the flux variation of C2 and C1 is only about 10%.",
"In Paper II it was found that C3 showed moderate flux increase until 2008 and then both C1 and C3 showed a rapid flux increase after 2008.",
"Such a rapid flux increase of C1 is not seen anymore.",
"Surprisingly, any new component ejection accompanied by this flux increase is not detected with the exception of minor components C4a and C4b.",
"This is discussed in the section ."
],
[
"Comparison with $\\gamma $ -ray Events and radio light curve",
"In Figure REF , we present the radio data collected by Metsähovi and MOJAVE.",
"The pink vertical lines indicate the $\\gamma $ -ray flaring periods reported by Kataoka et al.",
"(2010) and Brown & Adams (2011).",
"The first one occurred in the period of 2009 April-May.",
"The flaring timescale is about 1 month.",
"The MOJAVE light curve shows that the flux density of C1 obtained in 15 days after the peak of this flare ($4.98\\pm 0.25$ Jy on 2009 May 28) increases slightly as compared to that obtained in $\\sim 4$ month before ($5.65\\pm 0.28$ Jy on 2009 January 30).",
"The flux density of C3 could be also increasing between 2009 January 30 ($5.56\\pm 0.28$ Jy) and 2009 May 28 ($5.84\\pm 0.29$ Jy), but the increasing level is not significant as compared to the flux calibration accuracy of VLBA ($\\sim 5$ %).",
"On the other hand, no obvious change in radio flux density is found from the Metsähovi 37-GHz data on the timescale of the $\\gamma $ -ray flaring event.",
"No structural change on the VLBI image is also seen before and after the flare despite our VERA observations pin down right beginning and end of the flare (2009 April 23 and 2009 May 24).",
"The second $\\gamma $ -ray flare occurred in the period of 2010 June-August.",
"Its e-folding rise time is about 1.5 days and a subsequent e-folding decay time is about 2.5 days.",
"This flare occurred during the increasing activity of total flux density at 37 GHz spanning from 2010 to 2011.",
"From Figure REF it is found that this radio flux increase mostly originates in C3.",
"However, by focusing on the timescale of days, the light curve of total flux does not show any signature of flare.",
"No obvious change in VLBI-scale structure before and after the flare is seen.",
"In the period of both $\\gamma $ -ray flares, no obvious change in radio flux density is detected by Metsähovi on the timescale of the $\\gamma $ -ray flaring events (days to weeks).",
"If the variability in the radio band had been of the same magnitude as in the $\\gamma $ -rays, where the flux doubled or tripled its value, the single dish observations would have detected such variation.",
"Thus, we may conclude that the magnitude of radio variability is smaller than that in $\\gamma $ -ray band or the size of the radio counterpart is so small that the radio variability of the flaring component is mitigated by the constant activity of the other components.",
"However, we must note that Metsähovi 37-GHz light curve shows a small rise and subsequent decay in the period from 2009 October to 2009 December (indicated by the oval of broken line in the small graph in Figure 3).",
"The timescale of this change is about one month.",
"Right after this rise and decay, a small increasing activity which occurs on the timescale of one week is detected again (indicated by the oval of broken dotted line in the small graph in Figure 3).",
"Either of these changes in the 37-GHz light curve is possibly related to the $\\gamma $ -ray flare with some time delay, as proposed in León-Tavares et al.",
"(2011).",
"More recently, an increasing of radio activity is observed from around 2011 June.",
"This probably originates in C3 because the flux density of C3 increases between 2011 June 24 and 2011 December 12 with the significance of 4.5$\\sigma $ (Table REF ).",
"This increasing activity seems to be difficult to relate to the $\\gamma $ -ray flare because the time difference between the two events is very large ($\\sim 1$ year).",
"Further multifrequency analysis to solve this puzzle will be presented in a forthcoming paper.",
"Where is the radio counterpart of the $\\gamma $ -ray flares?",
"One possibility is that the $\\gamma $ -ray flaring region is embedded in the self-absorbed core, i.e., C1 or further upstream of jet.",
"This seems quite natural because the $\\gamma $ -ray time variation is very fast, indicating a small emission region size.",
"In particular, the resultant emission region size for the case of second flare is estimated to be $\\sim 10^{-3}$ pc if we assume a Doppler factor ($\\delta $ ) of 2 (Brown & Adams 2010).",
"The size of $\\sim 10^{-3}$ pc is smaller than the beam size of VLBA at 15 GHz by two orders of magnitude.",
"Therefore, it is possible that time variation of radio counterpart is mitigated by the contamination from other radio emitting region.",
"However, a new jet component associated with the $\\gamma $ -ray flare is likely to be ejected, and such a component may be visible on the image some time after the $\\gamma $ -ray flare as a result of propagation down to optically thin region.",
"If the jet components are ejected from the core accompanied by the first and the second $\\gamma $ -ray flares with $\\delta =2$ , they should be visible on the image between C1 and C3 during our monitoring period.",
"As we mentioned in section , there is a smoothly distributed radio emission between C1 and C3.",
"This emission can be modeled with two point sources.",
"They might be jet components accompanied by the $\\gamma $ -ray flares.",
"It is worth pointing out for comparison that not all the $\\gamma $ -ray flare detected so far in blazars have a clear radio counterpart, as in the case of the GeV-TeV $\\gamma $ -ray flare in 3C 279 (Abdo et al.",
"2010b).",
"Another intriguing case in PKS 1510-089 is that some $\\gamma $ -ray flares seem to be related to changes in radio band (e.g., Marscher et al.",
"2010; Orienti et al.",
"2011) while others show no relation (e.g., D'Ammando et al.",
"2009).",
"Therefore, the lack of significant changes in radio band for 3C 84 after the detection of high $\\gamma $ -ray activity leaves the debate on the region responsible for the high-energy emission and its location still open.",
"Figure: (a)-((g) VERA 43-GHz images and model fit image.",
"The ellipse indicated at the bottom left corner in each image is the restoring beam.",
"The size of the restoring beam is (0.63×0.370.63\\times 0.37) mas at the position angle of -50 ∘ -50^{\\circ }, which is a typical beam size of all observing epochs.",
"The contours are plotted at the level of (-1, 1, 2, 4, 8, 16, 32) ×\\times 62.6 mJy/beam, which is the 3-times image noise rms on 2010 December 27 (f).",
"(h) Model-fit image on 2011 March 10.",
"Symbols indicated by blue lines are the model components obtained by the Gaussian model fit.",
"The plus signs are point sources (C4a and C4b) which are placed to model the emission between C1 and C3 (see text in §).Figure: Left:Low-resolution image of MOJAVE 15 GHz on 2011 December 12.",
"The size of restoring beam is (1×11\\times 1) mas.",
"The contours are plotted at the level of (-1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024)×\\times 3.8 mJy/beam, which is the 3-times image noise rms.",
"Right:High-resolution image of the central (8×88\\times 8) mas region on the same date.",
"The contour levels are same with those in low-resolution image.Figure: Single dish and VLBI light curves from Metsahovi at 37 GHz and MOJAVE at 15 GHz, respecctively.",
"Red squares, green circles, and blue triangles represent the MOJAVE flux densities of C1, C2, and C3, respectively.",
"The sum of flux densities of C1, C2, and C3 is indicated by black squares.",
"Single dish light curve by Metsähovi is indicated by cyan diamonds.",
"Two γ\\gamma -ray flaring events are indicated by pink vertical lines.",
"Vertical arrows represent the date of VERA observations.",
"Small graph inserted in overall plot is the Metsähovi light curve from 2009.8 to 2010 where the range indicated by the square of thin solid line (see text in §).",
"Note that Metsähovi flux density contains the emission from the entire 3C 84 source.Table: Flux densities of the central components from MOJAVE 15-GHz data"
],
[
"Acknowledgments",
"We thank Anthony Brown, the referee, for careful reading and helpful comments.",
"Part of this work was done with the contribution of the Italian Ministry of Foreign Affairs and University and Research for the collaboration project between Italy and Japan.",
"The VERA is operated by the National Astronomical Observatory of Japan.",
"This research has made use of data from the MOJAVE database that is maintained by the MOJAVE team (Lister et al.",
"2009).",
"The VLBA is operated by the US National Radio Astronomy Observatory (NRAO), a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"This work has made use of observations obtained with the 14 m Metsähovi Radio Observatory, a separate research institute of the Aalto University School of Electrical Engineering.",
"The Metsähovi team acknowledges the support from the Academy of Finland to our observing projects (numbers 212656, 210338, 121148, and others)."
]
] | 1204.1392 |
[
[
"Measurement of the elliptic anisotropy of charged particles produced in\n PbPb collisions at nucleon-nucleon center-of-mass energy = 2.76 TeV"
],
[
"Abstract The anisotropy of the azimuthal distributions of charged particles produced in PbPb collisions with a nucleon-nucleon center-of-mass energy of 2.76 TeV is studied with the CMS experiment at the LHC.",
"The elliptic anisotropy parameter defined as the second coefficient in a Fourier expansion of the particle invariant yields, is extracted using the event-plane method, two- and four-particle cumulants, and Lee--Yang zeros.",
"The anisotropy is presented as a function of transverse momentum (pt), pseudorapidity (eta) over a broad kinematic range: 0.3 < pt < 20 GeV, abs(eta) < 2.4, and in 12 classes of collision centrality from 0 to 80%.",
"The results are compared to those obtained at lower center-of-mass energies, and various scaling behaviors are examined.",
"When scaled by the geometric eccentricity of the collision zone, the elliptic anisotropy is found to obey a universal scaling with the transverse particle density for different collision systems and center-of-mass energies."
],
[
"Introduction",
"The azimuthal anisotropy of emitted charged particles is an important feature of the hot, dense medium produced in heavy-ion collisions, and has contributed to the suggestion of a strongly coupled quark-gluon plasma (sQGP) being produced in nucleus-nucleus collisions at RHIC [1], [2], [3], [4], [5].",
"In noncentral collisions, the beam direction and the impact parameter vector define a reaction plane for each event.",
"If the nucleon density within the nuclei is continuous, the initial nuclear overlap region is spatially asymmetric with an “almond-like” shape.",
"In this approximation, the impact parameter determines uniquely the initial geometry of the collision, as illustrated in Fig.",
"REF .",
"In a more realistic description, where the position of the individual nucleons that participate in inelastic interactions is considered, the overlap region has a more irregular shape and the event-by-event orientation of the almond fluctuates around the reaction plane.",
"Experimentally, the azimuthal distribution of the particles detected in the final state can be used to determine the “event plane” that contains both the beam direction and the azimuthal direction of maximum particle density.",
"Strong rescattering of the partons in the initial state may lead to local thermal equilibrium and the build up of anisotropic pressure gradients, which drive a collective anisotropic expansion.",
"The acceleration is greatest in the direction of the largest pressure gradient, i.e., along the short axis of the almond.",
"This results in an anisotropic azimuthal distribution of the final-state hadrons.",
"The anisotropy is quantified in terms of a Fourier expansion of the observed particle yields relative to the event-by-event orientation of the event plane [6]: $E\\frac{{^3 N}}{{^3 p}} = \\frac{{^3 N}}{{\\, \\, {}y\\, \\varphi }} = \\frac{1}{{2\\pi }}\\frac{{^2 N}}{{\\, \\, {}y}}\\left( {1 + \\sum \\limits _{n = 1}^\\infty {2v_n(,y) \\cos \\left[ {n\\left( {\\varphi - \\Psi } \\right)} \\right]} } \\right),$ Figure: A schematic diagram of a noncentral nucleus-nucleus collisionviewed in the plane orthogonal to the beam.The azimuthal angle ϕ\\varphi , the impact parameter vector b, and the reaction-plane angle Ψ R \\Psi _{\\textrm {R}} are shown.",
"Theevent-plane angle Ψ\\Psi , with respect to which the flow is measured, fluctuates around the reaction-plane angle.where $\\varphi $ , $E$ , $y$ , and are the particle's azimuthal angle, energy, rapidity, and transverse momentum, respectively, and $\\Psi $ is the event-plane angle.",
"The second coefficient of the expansion, often referred to as the “elliptic flow” strength, carries information about the early collision dynamics [7], [8].",
"The coefficients in the Fourier expansion may depend on , rapidity, and impact parameter.",
"Typically, the measurements are obtained for a particular class of events based on the centrality of the collisions, defined as a fraction of the total inelastic nucleus-nucleus cross section, with 0% denoting the most central collisions with impact parameter b = 0, and 100% the most peripheral collisions.",
"Expressions similar to Eq.",
"(REF ) can be written for the Fourier expansion of the yield integrated over or rapidity: $\\frac{{^2 N}}{{{}y\\, \\varphi }} = \\frac{1}{{2\\pi }}\\frac{{{}N}}{{{}y}}\\left( {1 + \\sum \\limits _{n = 1}^\\infty {2v_n(y) \\cos \\left[ {n\\left( {\\varphi - \\Psi } \\right)} \\right]} } \\right),$ and $\\frac{{^2 N}}{{\\, \\, \\varphi }} = \\frac{1}{{2\\pi }}\\frac{{{}N}}{{\\, }}\\left( {1 + \\sum \\limits _{n = 1}^\\infty {2v_n() \\cos \\left[ {n\\left( {\\varphi - \\Psi } \\right)} \\right]} } \\right).$ The Fourier coefficients $v_n(y)$ and $v_n()$ can be obtained by directly analyzing the yields integrated over or rapidity, or from the coefficients of the triple-differential invariant yield in Eq.",
"(REF ), $v_n(, y)$ , by performing the following yield-weighted average over the ranges of transverse momentum $\\Delta $ and rapidity $\\Delta y$ , from which the particles are taken: $v_n(y) = \\frac{ \\int _{\\Delta } \\, \\frac{{^2 N}}{{\\, \\, {}y}}v_n(,y)}{\\int _{\\Delta } \\, \\frac{{^2 N}}{{\\, \\,{}y}}}$ or $v_n() = \\frac{ \\int _{\\Delta y} {}y \\frac{{^2 N}}{{\\, \\, {}y}}v_n(,y)}{\\int _{\\Delta y} {}y \\frac{{^2 N}}{{\\, \\,{}y}}}.$ The coefficients of the Fourier expansion of the particles' invariant yield integrated over a broad rapidity and window, are often referred to as “integrated flow”: $\\frac{{{}N}}{{\\varphi }} = \\frac{1}{{2\\pi }}N\\left( {1 + \\sum \\limits _{n = 1}^\\infty {2v_n \\cos \\left[ {n\\left( {\\varphi - \\Psi } \\right)} \\right]} } \\right),$ where $v_n = \\frac{\\int _{\\Delta y} {}y \\int _{\\Delta } \\, \\frac{{^2 N}}{{\\, \\, {}y}}v_n(,y)}{\\int _{\\Delta y} {}y \\int _{\\Delta } \\, \\frac{{^2 N}}{{\\,\\, {}y}}}.$ In obtaining the Fourier coefficients, the absolute normalization in the particle yields is not important, as long as the particle detection efficiency is constant over the chosen transverse momentum and rapidity range.",
"However, if the efficiency varies, the appropriate efficiency corrections need to be applied.",
"This often leads to a two-step procedure in which first $v_n(,y)$ is obtained in a narrow phase-space window where the efficiency is constant, and then a yield-weighted average is performed using Eqs.",
"(REF ), (REF ), or (REF ), folding in the efficiency-corrected particle spectra.",
"When the particle mass is not determined in the measurement, the pseudorapidity $\\eta =-\\ln [\\tan (\\theta /2)]$ , with $\\theta $ being the polar angle, is used instead of the rapidity.",
"Elliptic flow has been measured at the AGS, SPS, and RHIC.",
"A notable feature in these measurements is that the elliptic flow measured as a function of transverse momentum, $v_2()$ , increases with the nucleon-nucleon center-of-mass energy ($\\sqrt{s_{NN}}$ ) up to about 22 [9], [10], and then saturates at a value compatible with predictions from ideal hydrodynamics [11], [12].",
"The most extensive experimental studies have been performed at the highest RHIC energy of $\\sqrt{s_{NN}} = 200$ in AuAu collisions [1], [2], [3].",
"First results from PbPb collisions at $\\sqrt{s_{NN}}= 2.76 $ from the Large Hadron Collider (LHC) [13], [14] indicate that there is little or no change in the transverse momentum dependence of the elliptic flow measured at the highest RHIC energy and the LHC, despite the approximately 14-fold increase in the center-of-mass energy.",
"Recent theoretical studies of elliptic flow have focused on quantifying the ratio of the shear viscosity to the entropy density of the produced medium assuming viscous hydrodynamics [15], [16], and taking into account a variety of possible initial conditions.",
"Based on experimental results and the corresponding theoretical descriptions of the data, the underlying physics processes that generate the elliptic anisotropy are thought to vary for different kinematic regions: Elliptic flow in the bulk system Hadrons produced in soft processes, carrying low transverse momentum ($\\lesssim 2$ for mesons, and $\\lesssim 3$ for baryons [17], [18], [19], [20]), exhibit azimuthal anisotropies that can be attributed to collective flow driven by the pressure gradients in the system.",
"The description of elliptic flow is amenable to hydrodynamic calculations [12], [21], [22], [15].",
"Comparisons to theory indicate that the flow is primarily generated during an early stage of the system evolution.",
"Recombination region At intermediate transverse particle momentum ($2\\lesssim \\lesssim 4$ ), the RHIC data show that the elliptic anisotropies for various identified hadron species approximately follow a common behavior when both the $v_2()$ value and the of the particle are divided by the number of valence quarks in the hadron [17], [18], [23].",
"This behavior is successfully reproduced by models invoking quark recombination as the dominant hadronization mechanism in this momentum range [24], [25], [26].",
"The quark-number scaling of the elliptic flow has been interpreted as evidence that quark degrees of freedom dominate in the early stages of heavy-ion collisions, when the collective flow develops [27].",
"The quark recombination may involve both thermally produced quarks and quarks originating in jet fragmentation.",
"Therefore, the elliptic anisotropy in the recombination region results from an interplay between the bulk flow of the system and the azimuthal anisotropy in hadron production induced by jet quenching.",
"Jet-fragmentation region At intermediate and high transverse momentum ($\\gtrsim 3$ ), where fragments from increasingly harder partons begin to contribute to the particle spectra, anisotropy in the azimuthal distributions may be generated from the stronger jet quenching in the direction of the long axis of the almond-shaped reaction zone [28], [29], [30], [31].",
"It is expected that this mechanism will dominate the elliptic anisotropy of hadrons with momenta in excess of 8.",
"Thus, measurements extending beyond this range carry information about the path-length dependence of energy loss in the produced medium.",
"The measurements presented in this paper span the transverse momentum range of $0.3<<20$ and provide the basis for comparisons to theoretical descriptions of the bulk properties of the system and the quenching of jets.",
"Such comparisons may give insight in determining the transport properties of sQGP, namely: the shear viscosity, and the opacity of the plasma.",
"The theoretical interpretation of the elliptic anisotropy in the recombination region requires identified-hadron measurements that are not included in this analysis.",
"In ideal hydrodynamics, the integrated elliptic flow is directly proportional to the initial spatial eccentricity of the overlap zone [32].",
"There are many factors that can change this behavior, including viscosity in the sQGP and the hadronic stages of the system evolution, incomplete thermalization in peripheral collisions, and variations in the equation-of-state.",
"Event-by-event fluctuations in the overlap zone [33], [34], [35], [36], [37], [38] could also influence the experimental results, depending on the method that is used to extract the $v_2$ signal.",
"Invoking multiple methods with different sensitivities to the initial-state fluctuations is important for disentangling the variety of physics processes that affect the centrality dependence of the elliptic flow.",
"Comparisons to results from lower-energy measurements and explorations of empirical scaling behaviors could provide additional insights into the nature of the matter produced in high-energy heavy-ion collisions.",
"The pseudorapidity dependence of the elliptic flow, $v_2(\\eta )$ , provides information on the initial state and the early-time development of the collision, constraining the theoretical models and the description of the longitudinal dynamics in the collisions [39], [40].",
"Longitudinal scaling in $v_2(\\eta )$ extending over several units of pseudorapidity (extended longitudinal scaling) has been reported at RHIC [41] for a broad range of collision energies ($\\sqrt{s_{NN}} = 19.6$ –200).",
"Studies of the evolution of $v_2(\\eta )$ from RHIC to LHC energies may have implications for the unified description of sQGP in both energy regimes.",
"The paper is organized as follows: Section gives the experimental details of the measurement including the Compact Muon Solenoid (CMS) detector, triggering and event selection, centrality determination, Glauber-model calculations, reconstruction of the charged-particle transverse momentum spectra, methods of measuring the elliptic anisotropy, and the studies of the systematic uncertainties.",
"In Section we present results of $v_2$ as a function of transverse momentum, centrality, and pseudorapidity, and the measurement of the charged-particle transverse momentum spectra.",
"The elliptic flow results are obtained with the event-plane method [6], two- and four-particle cumulants [42], and Lee–Yang zeros [43], [44].",
"The analysis is performed in 12 classes of collision centrality covering the most central 80% of inelastic collisions.",
"We study the eccentricity scaling of $v_2$ , and investigate the differences in the results obtained from different methods, taking into account their sensitivity to initial-state fluctuations.",
"Section is devoted to detailed comparisons with results obtained by other experiments at lower energies and the exploration of different scaling behaviors of the elliptic flow.",
"The results of our studies are summarized in Section .",
"The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diameter, providing a 3.8T field.",
"Within the field volume are a silicon tracker, a crystal electromagnetic calorimeter, and a brass/scintillator hadron calorimeter.",
"Muons are measured in gas-ionization detectors embedded in the steel return yoke.",
"In addition to these detectors, CMS has extensive forward calorimetry.",
"The inner tracker measures charged particles within the pseudorapidity range $|\\eta | < 2.5$ , and consists of silicon pixel and silicon strip detector modules.",
"The beam scintillation counters (BSC) are a series of scintillator tiles which are sensitive to almost the full PbPb interaction cross section.",
"These tiles are placed along the beam line at a distance of $\\pm 10.9$ m and $\\pm 14.4$ m from the interaction point, and can be used to provide minimum-bias triggers.",
"The forward hadron calorimeter (HF) consists of a steel absorber structure that is composed of grooved plates with quartz fibers inserted into these grooves.",
"The HF calorimeters have a cylindrical shape and are placed at a distance of 11.2m from the interaction point, covering the pseudorapidity range of $2.9<|\\eta |<5.2$ .",
"A more detailed description of the CMS detector can be found elsewhere [45].",
"The detector coordinate system has the origin centered at the nominal collision point inside the experiment, with the $y$ axis pointing vertically upward, the $x$ axis pointing radially inward towards the center of the LHC ring, and the $z$ axis pointing along the counterclockwise beam direction."
],
[
"Event selection",
"The measurements presented are performed by analyzing PbPb collision events recorded by the CMS detector in 2010.",
"From these data, the minimum-bias event sample is collected using coincidences between the trigger signals from both the $+z$ and $-z$ sides of either the BSC or HF.",
"The minimum-bias trigger used for this analysis is required to be in coincidence with the presence of both colliding ion bunches in the interaction region.",
"This requirement suppresses noncollision-related noise, cosmic rays, and beam backgrounds (beam halo and beam-gas collisions).",
"The total hadronic collision rate varied between 1 and 210Hz, depending on the number of colliding bunches and the bunch intensity.",
"In order to obtain a pure sample of inelastic hadronic collisions, several offline selections are applied to the triggered event sample.",
"These selections remove contamination from noncollision beam backgrounds and from ultraperipheral collisions (UPC) that lead to an electromagnetic breakup of one or both of the Pb nuclei [46].",
"First, beam-halo events are vetoed based on the BSC timing.",
"Then, to remove UPC and beam-gas events, an offline HF coincidence of at least three towers on each side of the interaction point is required, with a total deposited energy of at least 3in each tower.",
"A reconstructed primary vertex made of at least two tracks and consistent with the nominal interaction point position is required.",
"To further reject beam-gas and beam-halo events, the pixel clusters are required to have a length along the beam direction compatible with being produced by particles originating from the primary vertex, as for the study in [47].",
"Additionally, a small number of noisy events with uncharacteristic hadron calorimeter responses are removed.",
"The acceptance of the silicon tracker for $|\\eta | > 0.8$ is found to vary significantly with respect to the longitudinal position of the collision point relative to the geometric center of the detector.",
"This event-by-event variance in the tracking efficiency contributes to a systematic bias of the elliptic flow measurements at forward pseudorapidity.",
"In order to remove this bias, events in this analysis are required to have a longitudinal vertex position within 10cm of the geometric center of the detector.",
"After all selections, 22.6 million minimum-bias events, corresponding to an integrated luminosity of approximately 3, remain in the final sample."
],
[
"Centrality determination and Glauber model calculations",
"In this analysis, the observable used to determine centrality is the total energy in both HF calorimeters.",
"The distribution of this total energy is used to divide the event sample into 40 centrality bins, each representing 2.5% of the total nucleus-nucleus interaction cross section.",
"The events are then regrouped to form 12 centrality classes used in the analysis: 0–5% (most central), 5–10%, 10–15%, 15–20%, 20–25%, 25–30%, 30–35%, 35–40%, 40–50%, 50–60%, 60–70%, and 70–80% (see Table REF ).",
"Using Monte Carlo (MC) simulations, it is estimated that the minimum-bias trigger and the event selections include $(97\\pm 3)$ % of the total inelastic cross section.",
"For the events included in this analysis (0–80% centrality), the trigger is fully efficient.",
"For each group of events that comprises a centrality class, we evaluate a set of quantities that characterize the initial geometry of the collisions using a MC Glauber model.",
"The Glauber model is a multiple-collision model that treats a nucleus-nucleus collision as an independent sequence of nucleon-nucleon collisions (see Ref.",
"[48] and references therein).",
"The nucleons that participate in inelastic interactions are called “participants”.",
"A schematic view of a PbPb collision with an impact parameter $ \\text{b} = 6$ fm, as obtained from the Glauber model, is shown in Fig.",
"REF .",
"The direction and the magnitude of the impact parameter vector and the corresponding reaction-plane angle $\\Psi _{R}$ , are the same as in Fig.",
"REF .",
"However, the initial interaction zone as determined by the spatial distribution of the participants (filled circles) is no longer regular in shape and is not necessarily symmetric with respect to the reaction plane.",
"In each event one can evaluate the variances $\\sigma _{x}^{2}$ and $\\sigma _{y}^{2}$ , and the covariance $\\sigma _{xy}= \\langle xy \\rangle - \\langle x \\rangle \\langle y \\rangle $ of the participant distributions projected on the $x$ and $y$ axes.",
"One can then find a frame $x^{\\prime }$ -$y^{\\prime }$ that minimizes $\\sigma _{x^{\\prime }}$ , and define a “participant plane” using the beam direction and the $x^{\\prime }$ axis [34], [49].",
"In this frame, the covariance $\\sigma _{x^{\\prime }y^{\\prime }}$ of the participant spatial distribution vanishes.",
"To characterize the geometry of the initial state of the collision, we define [34], [49] the eccentricity of the participant zone $\\epsilon _\\mathrm {part}$ , its cumulant moments $\\epsilon \\lbrace 2\\rbrace $ and $\\epsilon \\lbrace 4\\rbrace $ , and the transverse overlap area of the two nuclei $S$ : $\\epsilon _{\\text{part}} \\equiv \\frac{\\sigma _{y^{\\prime }}^{2} - \\sigma _{x^{\\prime }}^{2}}{ \\sigma _{y^{\\prime }}^{2} + \\sigma _{x^{\\prime }}^{2} } = \\frac{ \\sqrt{ \\left( \\sigma _{y}^{2} - \\sigma _{x}^{2} \\right)^{2} + 4\\sigma _{xy}^{2} } }{ \\sigma _{y}^{2} + \\sigma _{x}^{2} },$ $\\epsilon \\lbrace 2\\rbrace ^{2} \\equiv \\langle \\epsilon _\\text{part}^{2} \\rangle ,$ $\\epsilon \\lbrace 4\\rbrace ^{4} \\equiv 2 \\langle \\epsilon _\\text{part}^{2} \\rangle ^{2} - \\langle \\epsilon _\\text{part}^{4} \\rangle , ~\\text{and}$ $S \\equiv \\pi \\sigma _{x^{\\prime }}\\sigma _{y^{\\prime }} = \\pi \\sqrt{\\sigma _{x}^{2}\\sigma _{y}^{2}-\\sigma _{xy}^{2}}.$ In Eqs.",
"(REF ) and (REF ), the average is taken over many events from the same centrality interval.",
"Figure: (Color online) A schematic view of a PbPb collision with an impactparameter b=6 b = 6fm as obtained from the Glauber model.",
"The nucleons that participate in inelastic interactionsare marked with filled circles.",
"The x and y coordinates represent the laboratory frame, while x' and y' represent theframe that is aligned with the axes of the ellipse in the participant zone.",
"The participant eccentricityϵ part \\epsilon _{\\text{part}} and the standard deviations of the participant spatialdistribution σ y ' \\sigma _{y^{\\prime }} and σ x ' \\sigma _{x^{\\prime }} from which the transverse overlap area of thetwo nuclei is calculated are also shown.",
"The angle Ψ R \\Psi _{\\textrm {R}} denotes theorientation of the reaction plane.Table: For the centrality bins used in the analysis, the average values of the number of participating nucleons,transverse overlap area of the two nuclei, participant eccentricity, and cumulant moments of the participant eccentricity,along with their systematic uncertainties from the Glaubermodel.To implement the Glauber model for PbPb collisions at $\\sqrt{s_{NN}} = 2.76$ , we utilize the foundation of a published Glauber MC software package TGlauberMC [50], which was developed for the PHOBOS Collaboration at RHIC.",
"Standard parameters of the Woods-Saxon function used for modeling the distribution of nucleons in the Pb nuclei are taken from Ref. [51].",
"The nucleon-nucleon inelastic cross section, which is used to determine how close the nucleon trajectories need to be in order for an interaction to occur, is taken as $64\\pm 5$ mb, based on a fit of the existing data for total and elastic cross sections in proton-proton and proton-antiproton collisions [52].",
"The uncertainties in the parameters involved in these calculations contribute to the systematic uncertainty in $N_\\text{part}$ , $S$ , and $\\epsilon _\\text{part}$ for a given centrality bin.",
"The connection between the experimentally defined centrality classes using the HF energy distribution and $N_\\text{part}$ from the Glauber model is obtained [53] from fully simulated and reconstructed MC events generated with the AMPT [54] event generator.",
"The calculated Glauber model variables for each centrality class are shown in Table REF ."
],
[
"Reconstruction of the charged-particle transverse momentum distributions and the mean\ntransverse momentum",
"To determine the transverse momentum distributions of the charged particles produced in the collisions, we first need to reconstruct the particles' trajectories (“tracks”) through the 3.8T solenoidal magnetic field.",
"The tracks are reconstructed by starting with a “seed” comprising two or three reconstructed signals (“hits”) in the inner layers of the silicon strip and pixel detectors that are compatible with a helical trajectory of some minimum and a selected region around the reconstructed primary vertex or nominal interaction point.",
"This seed is then propagated outward through subsequent layers using a combinatorial Kalman-filter algorithm.",
"Tracking is generally performed in multiple iterations, varying the layers used in the seeding and the parameters used in the pattern recognition, and removing duplicate tracks between iterations.",
"This algorithm is described in detail in Ref. [55].",
"The algorithm used in most of the CMS proton-proton analyses, as well as the tracking detector performance for the 2010 run, are described in Ref. [56].",
"The six-iteration process used in proton-proton collisions is computationally not feasible in the high-multiplicity environment of very central PbPb collisions.",
"In place of this, a simple two-iteration process is used.",
"The first iteration builds seeds from hits in some combination of three layers in the barrel and endcap pixel detectors compatible with a trajectory of $> 0.9$ and a distance of closest approach to the reconstructed vertex of no more than 0.1cm in the transverse plane and 0.2cm longitudinally.",
"These tracks are then filtered using selection criteria based on a minimum number of reconstructed hits, vertex compatibility along the longitudinal direction and in the transverse plane, and low relative uncertainty on the reconstructed momentum.",
"In the second iteration, seeding is also performed using three layers of the pixel detector, but the minimum transverse momentum requirement is relaxed to $> 0.2$ .",
"These tracks are not propagated through the silicon-strip detector, but simply refitted using the transverse position of the beam spot as an additional constraint.",
"These pixel-only tracks are then filtered using selection criteria of vertex compatibility along the longitudinal axis and statistical goodness of fit.",
"The tracks from both collections are checked for duplicate tracks using the number of hits in common between the two tracks, and duplicates are removed giving preference to the first-iteration tracks.",
"The tracking algorithm may sometimes misidentify tracks by combining silicon detector signals that do not originate from the same charged particle.",
"It is important to keep the proportion of misidentified tracks (referred to as “fake tracks”), or fake rate, as low as possible.",
"To create the final collection, first-iteration tracks with $> 1.5$ are combined with second-iteration pixel-only tracks with $< 1.8$ .",
"These limits were chosen to exclude kinematic regions where a given iteration has a high fake rate.",
"The efficiency and fake rate of this modified tracking collection are found both by using a full MC simulation of PbPb collisions based on the hydjet event generator [57] and by embedding simulated charged pions into PbPb data events.",
"The efficiency, fake rate, and momentum resolution of this final collection determined by the simulated events from the hydjet event generator are shown in Fig.",
"REF for events of five different centrality classes.",
"The abrupt change in efficiency, fake rate, and momentum resolution seen in the figure occurs at the transverse momentum where the two track collections are merged.",
"Figure: (Color online)Efficiency (top), fake rate (middle), and momentum resolution (bottom) of charged tracks obtained from hydjetsimulated events in four pseudorapidity regions: |η|<0.8|\\eta | < 0.8, 0.8<|η|<1.60.8 < |\\eta | < 1.6,1.6<|η|<2.01.6 < |\\eta | < 2.0, and 2.0<|η|<2.42.0 < |\\eta | < 2.4 displayed from left to right, and forthe five centrality classes given in the legend.The charged-particle transverse momentum distributions are corrected for the loss of acceptance, efficiency, and the contributions from fake tracks.",
"Each detected track is weighted by a factor $w_{tr}$ according to its centrality, transverse momentum, and pseudorapidity: $w_{tr} (\\text{centrality},, \\eta ) = \\frac{1-f}{e}.$ Here $f$ is the fraction of fake tracks that do not correspond to any charged particle, and $e$ is the absolute tracking efficiency accounting for both geometric detector acceptance and algorithmic tracking efficiency.",
"The proportion of multiply reconstructed particles and reconstructed particles from secondary decays is negligible and is not included in the correction factor.",
"The fully corrected transverse momentum distributions are measured in 12 centrality classes over the pseudorapidity range $|\\eta | < 2.4$ in bins of $\\Delta \\eta = 0.4$ , as discussed in Section REF .",
"These distributions are used in obtaining integrated $v_2$ values.",
"We also study the evolution of $\\langle \\rangle $ with pseudorapidity and centrality (Section REF ), and center-of-mass energy (Section ).",
"To evaluate $\\langle \\rangle $ , the spectra need to be extrapolated down to $= 0$ .",
"The extrapolation is performed using a Tsallis distribution [58], [59], [60]: $E \\frac{^3N_{\\mathrm {ch}}}{p^3} =\\frac{1}{2\\pi p_T} \\frac{E}{p} \\frac{^2N_{\\mathrm {ch}}}{\\eta p_T} =C\\left(1 + \\frac{E_T}{nT}\\right)^{-n} ,$ where $= \\sqrt{m^2 + ^2} - m$ , and $m$ is taken to be the charged pion mass.",
"The measured spectra are fitted in the range $0.3<<3.0$ , and the fit parameters $C$ , $n$ , and $T$ are determined.",
"The mean transverse momentum is then evaluated using the fit function in the extrapolation region of $0\\le <0.3 $ , and the data from the range $0.3\\le \\le 6.0 $ .",
"This method has been previously applied in CMS [47] in the measurement of $\\langle \\rangle $ in pp collisions at $\\sqrt{s}=7$ ."
],
[
"Methods for measuring the anisotropy parameter $v_{2}$",
"Anisotropic flow aims to measure the azimuthal correlations of the particles produced in heavy-ion collisions in relation to the initial geometry of the collisions.",
"Originally, the flow was defined as a correlation of the particle emission angles with the reaction plane [7], [6].",
"More recently, it was recognized [34], [49] that the initial geometry is better characterized by the positions of the individual nucleons that participate in inelastic interactions, and thus define a participant plane that fluctuates around the reaction plane on an event-by-event basis.",
"Neither the reaction plane nor the participant plane are directly measurable experimentally.",
"Instead, there are several experimental methods that have been developed to evaluate the anisotropic flow based on the final-state particle distributions.",
"In the present analysis, we use the event-plane method, two-particle and four-particle cumulants [42], and the Lee–Yang zeros method [43], [44] the last of which is based on correlations involving all the particles in the event.",
"The anisotropic flow measurements are affected by fluctuations that come from several sources.",
"Statistical fluctuations arise due to the fact that a finite number of particles is used to determine a reference plane for the flow and the multiplicity fluctuations within the chosen centrality interval.",
"The effect of these fluctuations is to reduce the measured flow signal and is largely compensated for by the resolution corrections described below.",
"Any remaining effects from statistical fluctuations on our measurements are included in the systematic uncertainties (see Section REF ).",
"Another more important source of fluctuations comes from the event-by-event fluctuations in the participant eccentricity that are present even at fixed $N_{\\text{part}}$ .",
"These dynamical fluctuations have been shown [35], [36], [37], [34], [38] to affect the various methods for estimating the flow differently, since each of them is based on a different moment of the final-particle momentum distribution.",
"For example, the two-particle cumulant method measures an r.m.s.",
"value of the flow that is higher than the mean value.",
"Conversely, the four-particle cumulant and other multiparticle correlation methods return a value that is lower than the mean value.",
"For the event-plane method, the results vary between the mean and the r.m.s.",
"value depending on the event-plane resolution, which varies in each centrality interval.",
"In addition, there exist other sources of correlations in azimuth, such as those from resonance decays, jets, and Bose–Einstein correlations between identical particles.",
"These correlations, which are not related to the participant plane, are called nonflow correlations.",
"The various methods proposed to estimate the magnitude of anisotropic flow have different sensitivities to the nonflow correlations, thus allowing systematic checks on the flow measurements.",
"The multiparticle correlation methods are least affected by nonflow correlations, but they do not work reliably when either the flow anisotropy ($v_2$ ) or the multiplicity in the selected phase-space window is small.",
"This happens in the most central and in the most peripheral events [42], [43].",
"Thus, the two-particle cumulant and the event-plane methods provide an extended centrality range, albeit with a larger nonflow contribution.",
"All four methods used here have been extensively studied and applied in different experiments.",
"Thus, we limit our description in the following subsections only to the features that are specific for our implementations of these methods."
],
[
"Event-plane method",
"The event-plane method estimates the magnitude of anisotropic flow by reconstructing an “event-plane” containing both the beam direction and the direction of the maximal flow determined from the azimuthal distributions of the final-state particles.",
"Under the assumption that the flow is driven by the initial-state asymmetry in the nuclear overlap zone, and that there are no other sources of azimuthal correlations in the final-state particles, the event-plane is expected to coincide with the participant plane [34] defined in Fig.",
"REF .",
"Recent theoretical calculations [61], [62], [38], [63] confirm that the event plane and the participant plane are strongly correlated event by event.",
"Since the event plane is determined using a finite number of particles, and detected with finite angular resolution, the measured event-plane angle fluctuates about its true value.",
"As a result, the observed particle azimuthal anisotropy with respect to the event-plane is smeared compared to its true value.",
"The true elliptic flow coefficient $v_2$ in the event-plane method is evaluated by dividing the observed $v^{\\mathrm {obs}}_2$ value by a resolution correction factor, $R$ , which accounts for the event-plane resolution.",
"To determine the event-plane resolution correction, a technique that sorts the particles from each event into three subevents based on their pseudorapidity values [6] is used.",
"For subevents $A$ , $B$ , $C$ in three different pseudorapidity windows, the event-plane resolution correction factor $R_A$ for subevent A is found as: $ R_A=\\sqrt{\\frac{\\langle \\cos [2 (\\Psi ^A-\\Psi ^B)]\\rangle \\langle \\cos [2 (\\Psi ^A-\\Psi ^C)]\\rangle }{\\langle \\cos [2(\\Psi ^B-\\Psi ^C)]\\rangle }},$ where $\\Psi ^A,\\Psi ^B$ , and $\\Psi ^C$ are the event-plane angles determined from the corresponding subevents, and the average is over all events in a selected centrality class used in the $v_{2}$ analysis.",
"In our implementation of the method, the event-plane angle determined for the subevent furthest in $\\eta $ from the track being used in the elliptic flow analysis is used, and the corresponding resolution correction is employed.",
"This selection minimizes the contributions of auto-correlations and other nonflow effects that arise if the particles used in the event-plane determination and those used in the flow analysis are close in phase space.",
"To achieve the largest pseudorapidity gap possible, two event planes are defined, with calorimeter data covering the pseudorapidity ranges of $-5 <\\eta <-3$ and $3<\\eta <5$ , labeled “HF-” and “HF+”, respectively.",
"These pseudorapidity ranges are primarily within the coverage of the HF calorimeters.",
"A third event plane, found using charged particles detected in the tracker in the pseudorapidity range $-0.8<\\eta <0.8$ , is also defined and used in the three-subevent technique for determining the resolution corrections for HF- and HF+.",
"The resulting resolution corrections are presented in Fig.",
"REF .",
"Particles detected in the tracker with $\\eta >0$ are then correlated with the HF- event plane, and those with $\\eta <0$ with the HF+ plane.",
"In this manner, the minimum pseudorapidity gap between particles used in the event-plane determination and those for which the $v_2$ signal is measured is 3 units.",
"A two-subevent technique based on HF- and HF+, with a resolution parameter defined as $ R_{A/B}=\\sqrt{\\langle \\cos [2 (\\Psi ^A-\\Psi ^B)]\\rangle }$ , is also implemented and used for systematic studies.",
"The values of $ R_{A/B}$ are shown for comparison in Fig.",
"REF .",
"A standard flattening procedure [64] using a Fourier decomposition of the distribution of the event-plane angles to 21st order is used to shift the event-by-event plane angle to correct for asymmetries in the event-plane distribution that arise from the detector acceptance and other instrumental effects.",
"Although most of these effects are already accounted for with a correction involving just the first four coefficients in the expansion, the larger order was used for data quality monitoring purposes.",
"For each centrality class in the analysis, the flattening parameters are calculated by grouping events according to the longitudinal location of their primary collision vertex in 5cm-wide bins.",
"Figure: (Color online) Event-plane resolution correction factors as a function of centrality for the two event-planes(HF- and HF+) used in determining the elliptic anisotropy parameter v 2 v_2.",
"The corrections determinedwith the three-subevent method used in the analysis are shown as open squares and star symbols.The results from a two-subevent method used in evaluatingthe systematic uncertainties are shown as filled circles,though they overlap the other points in all but the most peripheral bin."
],
[
"Cumulant method",
"The cumulant method measures flow utilizing a cumulant expansion of multiparticle azimuthal correlations, without determining the orientation of the event plane.",
"The idea is that if the particles are correlated with the event-plane orientation, then there also exist correlations between them.",
"In our analysis, we utilize two- and four-particle correlations.",
"To calculate the cumulants of these correlations, from which the flow coefficient is extracted, we use a generating function of the multiparticle correlations in a complex plane [42].",
"First, we evaluate the “integrated”, or reference, flow by constructing the corresponding generating function including all particles from a broad ($$ , $\\eta $ ) window, and averaging over the events in a given centrality class.",
"The reference flow may not be corrected for tracking efficiency and should not be equated with the fully corrected integrated flow of the events.",
"Then, the differential flow, i.e., the flow in a narrower phase-space window, either in or $\\eta $ , is measured with respect to the reference flow.",
"In the cumulant and Lee–Yang zeros methods, the reference flow serves the same purpose as the determination of the event-plane angle and the resolution correction factors in the event-plane method.",
"In our analysis of $v_{2}()$ , the and $\\eta $ ranges for the reference flow are $0.3<<3$ and $|\\eta |<0.8$ , respectively.",
"In the analysis of $v_{2}(\\eta )$ , the reference flow is obtained for the range $0.3<<3$ and $|\\eta |<2.4$ in order to maximize the resolution parameter, which improves as the charged hadron multiplicity $M$ in the selected phase-space window increases.",
"The transverse momentum restriction of $<3$ is imposed to limit the contributions from hadrons originating in jets, and thus reduce the nonflow correlations contributing to the measured elliptic anisotropy parameter.",
"To avoid auto-correlations, the particles used for determining differential flow are not included in evaluating the reference flow.",
"The generating function for the reference flow is calculated at several different points in the complex plane, and we then interpolate between these points.",
"We use three values for the radius parameter, $r_0$ , and seven values for the polar angle, as described in Ref. [42].",
"The radius parameters are determined according to the detected charged particle multiplicity and the number of events analyzed in each centrality class.",
"Each particle in the differential or $\\eta $ bin is correlated to the particles used for the reference flow through a differential generating function.",
"To account for the fact that the track reconstruction efficiency may vary across the chosen bin, we implement an efficiency correction that is applied as a track-by-track weight in the construction of the differential generating function."
],
[
" Lee–Yang zeros method",
"The Lee–Yang zeros (LYZ) method [43], [44] for directly measuring the flow is based on multiparticle correlations involving all particles in the event.",
"It uses the asymptotic behavior of the cumulant expansion to relate the location of the zeros of a complex function to the magnitude of the integrated flow in the system.",
"For a detector with a uniform detection efficiency for all particles in the chosen $\\eta $ and window, it is thus possible to obtain the integrated flow in a simple one-step procedure.",
"Since this is not the case for the CMS measurements presented here, we replace the term “integrated flow” that is used in the literature describing the method [43], [44] with “reference flow”.",
"In the generating function (i.e., $G^{\\theta }(ir)$ , where $r$ is the imaginary axis coordinate), the flow vector, constructed from all particles in the event, is projected onto a direction that makes an arbitrary angle $\\theta $ with respect to the $x$ axis.",
"We use five different projection angles and then average the results over events from the same centrality class to reduce the statistical uncertainties.",
"Subsequently, the minimum of the generating function is found, and the differential flow is determined with respect to the reference flow.",
"There are different ways to define the generating function that involve either a sum or a product of the individual particle contributions.",
"An example is given in Fig.",
"REF , showing how the minimum of the generating function is determined using both definitions.",
"The results presented here are based on the product generating function.",
"In our analysis of $v_{2}()$ , the and $\\eta $ ranges for the reference flow are $0.3<<12$ and $|\\eta |<0.8$ , respectively.",
"Since the Lee–Yang zeros method is less sensitive to jet-induced charged-particle correlations than the two-particle cumulant method, the range of the tracks included in the determination of the reference flow is not restricted to low .",
"In the analysis of $v_{2}(\\eta )$ , the pseudorapidity range for the particles included in the reference flow is extended to $|\\eta |<2.4$ .",
"The Lee–Yang zeros method is sensitive to multiplicity fluctuations.",
"To evaluate the effects of these fluctuations we perform a toy-model MC study.",
"Event ensembles are generated sampling the multiplicity for each event from Gaussian distributions with mean and r.m.s.",
"values comparable to the ones measured in the centrality bins used in the measurement.",
"For each particle, the corresponding , $\\eta $ , and $\\phi $ values are sampled from a realistic input $v_{2}(,\\eta )$ distribution.",
"These events are then analyzed using the same procedure as in the data and the resulting $v_{2}(,\\eta )$ values are compared to the input.",
"We find that the Lee–Yang zeros method tends to underestimate the results if the r.m.s.",
"of the multiplicity distribution is more than about 14% of the mean.",
"In order to keep the systematic uncertainties below 2%, the data are analyzed in 5%-wide centrality classes and then averaged and weighted with the charged-particle yield, to obtain results in wider centrality intervals needed for comparisons with other methods.",
"Efficiency corrections are implemented as a track-by-track weight in the differential generating function.",
"Figure: (Color online)An example of the modulus of the second harmonic Lee–Yang zero generating functionG θ (ir)G^{\\theta }(ir) as a function of theimaginary axis coordinate rr for θ\\theta = 0.",
"Both the sum and product generating functions are shown,calculated from events with centrality 15–20%20\\%, |η|<0.8|\\eta | < 0.8, and 0.3<<120.3 < < 12.An enlargement of |G θ (ir)||G^{\\theta }(ir)| around its first minimum is shown in the inset."
],
[
"Corrections to the anisotropy parameter $v_{2}$",
"In determining the $v_2()$ distributions for particles detected in the tracker, it is necessary to correct for the influence of misidentified (i.e., “fake”) tracks on the measurements.",
"As shown in Section REF , the fake-track contribution is particularly significant at low-values and for pseudorapidities $|\\eta |>1.6$ .",
"Of particular concern is the observation that the fake tracks can carry a $v_2$ signal at low similar to that of properly reconstructed (i.e., “real”) tracks at a higher .",
"Since the true $v_2$ signal is very small at low , but increases at higher , the fake tracks may contribute a significant fraction of the measured $v_2$ signal at low .",
"Studies using a full MC CMS simulation of PbPb collisions based on the hydjet event generator [57] indicate that the component of the $v_2$ signal due to fake tracks is relatively constant for $< 0.8$ , where the fraction of fake tracks is largest.",
"For higher , where the fraction of fake tracks is quite small, the value of $v_2$ is consistent with the measured value from correctly reconstructed tracks.",
"This suggests the following simple correction scheme.",
"Let $N_{\\mathrm {det}}()$ be the number of reconstructed tracks in a given bin, $f$ the fraction of these tracks that are “fake,” $N_{\\mathrm {true}}$ the number of “true” tracks in the bin, and $e$ the efficiency for reconstructing a true track in the bin.",
"Then $N_{\\mathrm {det}}-fN_{\\mathrm {det}} = e N_{\\mathrm {true}}$ .",
"The $fN_{\\mathrm {det}}$ fake tracks are characterized by a constant $v_2$ value given by $v_2^{\\mathrm {fake}}$ .",
"The $N_{\\mathrm {det}}-fN_{\\mathrm {det}}$ real tracks are characterized by $v_2^{\\mathrm {real}}$ .",
"Then the observed value $v_{2}^{\\mathrm {obs}}$ of $v_{2}$ will be $v_2^{\\mathrm {obs}}=(1-f)v_2^{\\mathrm {real}}+fv_2^{\\mathrm {fake}}$ and so $v_2^{\\mathrm {real}} = \\frac{{v_2^{\\mathrm {obs}} - fv_2^{\\mathrm {fake}}}}{{1 - f}}.$ This correction for the fake-track signal is only significant for values less than $\\approx $ 1.",
"In this range, an empirical correction that results in values of $v_2$ that are independent of the track selection requirements or fraction of fake tracks is applied using $v_2^{\\mathrm {fake}} = 1.3\\left\\langle {{v_2}} \\right\\rangle $ , where the yield-weighted average is performed over the transverse momentum range 0.3 to 3 , folding in the efficiency-corrected spectra.",
"This value for $v_2^{\\mathrm {fake}}$ is also supported by MC studies using hydjet."
],
[
"Systematic uncertainties in the measurements of $v_{2}$",
"The systematic uncertainties in the measurements of $v_{2}$ include those common to all methods, as well as method-specific uncertainties.",
"They are evaluated as relative uncertainties and are reported as percentages relative to the measured $v_2$ values.",
"Since we are reporting the results on $v_{2}$ for nonidentified charged particles, it is important to investigate the tracking efficiency as a function of particle species.",
"The tracking efficiencies for charged pions, kaons, protons, and antiprotons are determined using a full simulation of CMS.",
"Subsequently, the value of $v_2()$ for charged particles is obtained using different assumptions for the -dependence of $v_2$ and the transverse momentum spectra of each particle type, taking into account the corresponding reconstruction efficiencies.",
"The results are compared to those obtained with the assumption of a particle-species-independent efficiency.",
"The uncertainties in the charged particle $v_{2}$ results are estimated to be $\\lesssim 0.5\\%$ , independent of the , $\\eta $ , and centrality ranges.",
"This uncertainty is listed as “Part.",
"composition” in Tables REF –REF .",
"Since the $v_{2}$ value changes with centrality, an uncertainty in the centrality determination can lead to a shift in the $v_{2}$ measurement.",
"This uncertainty is evaluated by varying the value of the minimum-bias trigger efficiency to include ($97\\pm 3$ )% of the total inelastic cross section.",
"The resulting uncertainty on $v_{2}$ is of the order 1%, independent of the , $\\eta $ , and centrality ranges.",
"This uncertainty is listed as “Cent.",
"determination” in Tables REF –REF .",
"The kinematic requirements used to select tracks can affect the efficiency of track finding and the relative fraction of fake tracks in an event.",
"The requirements are varied from their default values to estimate the systematic uncertainty.",
"For each set of requirements, corresponding corrections are obtained for the fake-track contribution to the $v_2$ signal, as described in Section REF .",
"For a given centrality and range, the systematic uncertainty is estimated based on the stability of the $v_2$ value after corrections for fake tracks, independent of the track selection requirements.",
"The uncertainty is found to be directly related to the magnitude of the fake-track contribution, with the final results deemed unreliable when the fake-rate is higher than $\\approx $ 20%.",
"For the results presented in Section , the systematic uncertainties from this source remain below $4\\%$ over the entire range of , $\\eta $ , and centrality, and are significantly below this value for $> 0.5$ , centrality above $10\\%$ , and $|\\eta |<1.6$ .",
"The uncertainty in the efficiency corrections is evaluated by determining the efficiency based on the hydjet model, and by embedding simulated pions into PbPb events in data.",
"Although the two resulting efficiencies do have differences, the uncertainty on the $v_{2}$ value is small, at most 0.5%.",
"Variations in the $v_{2}$ results due to changing detector conditions throughout the data-taking period are studied by dividing the data into three subgroups and are found to be below 1% for all measurements.",
"The combined uncertainties from the efficiency corrections, fake-track corrections, and variations in detector conditions are listed under “Corrections” in Tables REF –REF .",
"Additional studies of the systematic uncertainty are conducted for each method.",
"In the event-plane method, flattening corrections are obtained using different procedures.",
"The vertex dependence of the flattening parameters is examined and different subevent $\\eta $ gaps are used in obtaining the resolution corrections.",
"The uncertainties from these sources are found to be negligible.",
"The resolution corrections are measured with the three-subevent and two-subevent methods and are found to be consistent within the statistical uncertainties.",
"The statistical uncertainties for the resolution correction factor are less than 1%, except for the most peripheral 70–80% centrality events, where the statistical uncertainty reaches a value of 2%.",
"We include the statistical uncertainties associated with the resolution-correction factors as part of the overall systematic uncertainty on tracking efficiency and fake-track corrections for the event-plane method.",
"In the cumulant method, we examine the numerical stability of the result when the radius parameter $r_{0}$ used in the interpolations of the generating function is increased or decreased by half of its central value.",
"The effects of multiplicity fluctuations are studied for the Lee–Yang zeros and the cumulant methods by analyzing the events in finer 2.5% centrality bins, and by using a fixed number of particles chosen at random from each event in a given centrality class.",
"The systematic uncertainties are smallest for the mid-rapidity region $|\\eta | < 0.8$ , $> 0.5$ , and in the mid-central events (10–40%), and range from 2.0 to 4.5% for the different methods.",
"At low for the most central events, and in the forward pseudorapidity region where the fake-track contributions are larger, the uncertainties increase to 2.4–6.8%.",
"Similarly, in the most peripheral events the uncertainties increase mostly due to multiplicity fluctuations and reach up to 3.2–7%, depending on the experimental method.",
"A summary of the systematic uncertainties is presented in Tables REF –REF .",
"Table: Systematic uncertainties in the measurement of v 2 ()v_{2}() for |η|<0.8|\\eta | < 0.8 with the event-plane methodfor different and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 (η)v_{2}(\\eta ) for 0.3<<30.3<<3with the event-plane method fordifferent η\\eta and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 ()v_{2}() for |η|<0.8|\\eta | < 0.8with the two-particle cumulant method for different andcentrality ranges.Table: Systematic uncertainties in the measurement of v 2 (η)v_{2}(\\eta ) for the range0.3<<30.3<<3with the two-particle cumulant methodfor different η\\eta and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 ()v_{2}() for |η|<0.8|\\eta | < 0.8with the four-particle cumulant method for different and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 (η)v_{2}(\\eta ) for the range0.3<<30.3<<3with the four-particle cumulant method for different η\\eta and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 ()v_{2}() for |η|<0.8|\\eta | < 0.8 with the Lee–Yang zeros method for different and centrality ranges.Table: Systematic uncertainties in the measurement of v 2 (η)v_{2}(\\eta ) for 0.3<<30.3<<3with the Lee–Yang zeros method for different η\\eta and centrality ranges."
],
[
"Systematic uncertainties in the measurements of the transverse momentum spectra and the mean\ntransverse momentum $\\langle \\rangle $ ",
"Several sources of systematic uncertainty are considered in obtaining the inclusive charged-particle transverse momentum distributions and their mean values, $\\langle \\rangle $ .",
"These include the uncertainties in the efficiency and fake-track correction factors, the particle-species-dependent efficiency, and the uncertainty in the minimum-bias trigger efficiency.",
"The effect on the overall normalization of the spectra is considered separately from the smaller effect on the shape of the spectra as a function of .",
"To obtain the mean transverse momentum, different functional forms are used to extrapolate the spectra down to $= 0$ , and the range over which the spectra are fitted is varied.",
"The combined point-to-point systematic uncertainties on the spectra are presented as a function of pseudorapidity in Table REF .",
"The total normalization uncertainty of the charged-particle spectra measured in each centrality interval is given in Table REF .",
"The systematic uncertainties in the measurement of $\\langle \\rangle $ in different pseudorapidity intervals are summarized in Table REF .",
"Table: Point-to-point systematic uncertainties in the measurement of the charged-particle spectrain different pseudorapidity intervals.Table: Normalization uncertainty in the measurement of the charged-particle spectrain different centrality intervals resulting from the uncertainty in the minimum-bias triggerefficiency.Table: Systematic uncertainty of the mean of charged particlesfrom each source and in total as a function of pseudorapidity."
],
[
"Results",
"The main results of the analysis using the four methods described above are as follows: $v_{2}()$ at mid-rapidity $|\\eta |<0.8$ .",
"Integrated $v_{2}$ at mid-rapidity $|\\eta |<0.8$ and $0.3 < < 3$ .",
"$v_{2}(\\eta )$ for $0.3 < < 3$ .",
"We also measure the charged-particle transverse momentum spectra and their mean for the centrality and pseudorapidity ranges in which the flow is studied.",
"The flow studies are performed in the 12 centrality classes listed in Table REF .",
"Using these results, we examine the scaling of the integrated $v_2$ with the participant eccentricity, as well as perform comparisons to measurements from other experiments.",
"Centrality classes are regrouped to perform these comparisons, i.e., the results of $v_2()$ , $v_2(\\eta )$ , or integrated $v_2$ obtained in the finer bins of centrality are averaged over wider bins, weighted using the corresponding $^{2}N/\\, {\\eta }$ spectra.",
"The evolution of the measured elliptic anisotropy as a function of centrality, center-of-mass energy, and transverse particle density is studied.",
"The scaling of $v_2(\\eta )$ in the longitudinal dimension is also examined through comparisons to RHIC data."
],
[
"Transverse momentum dependence of $v_{2}$",
"In Figs.",
"REF –REF , we present the measurement of $v_{2}$ for charged particles as a function of transverse momentum at mid-rapidity, obtained by each analysis method.",
"We use the notation $v_{2}\\lbrace \\mathrm {EP}\\rbrace $ to refer to the measurement of $v_{2}$ using the event-plane method, and $v_{2}\\lbrace 2\\rbrace $ , $v_{2}\\lbrace 4\\rbrace $ , $v_{2}\\lbrace \\mathrm {LYZ}\\rbrace $ to refer to those using the two-particle cumulant, four-particle cumulant, and Lee–Yang zeros methods, respectively.",
"Several trends can be observed and related to the physics processes dominating hadron production in different ranges.",
"The value of $v_{2}$ increases from central to peripheral collisions up to 40% centrality, as expected if the anisotropy is driven by the spatial anisotropy in the initial state [12], [21], [15].",
"The transverse momentum dependence shows a rise of $v_2$ up to $\\approx $ 3and then a decrease.",
"As a function of centrality, a tendency for the peak position of the $v_2()$ distribution to move to higher in more central collisions is observed, with the exception of the results from the most peripheral collisions in the two-particle cumulant and the event-plane methods.",
"In ideal hydrodynamics the azimuthal anisotropy continuously increases with increasing [12], [21].",
"The deviation of the theory from the RHIC data at $\\gtrsim 2$ –3 has been attributed to incomplete thermalization of the high-hadrons, and the effects of viscosity.",
"Indeed, viscous hydrodynamic calculations [65], [16], [15] show that the shear viscosity has the effect of reducing the anisotropy at high .",
"At $\\gtrsim 8$ , where hadron production is dominated by jet fragmentation, the collective-flow effects are expected to disappear [21], [15].",
"Instead, an asymmetry in the azimuthal distribution of hadron emission with respect to the reaction plane could be generated by path-length-dependent parton energy loss [28], [31], [30].",
"For events with similar charged-particle suppression [66], but different reaction-zone eccentricity, one might expect that the geometric information would be imprinted in the elliptic anisotropy signal.",
"The upper panels of Figs.",
"REF –REF (centrality 0–35%) show a trend that is consistent with this expectation.",
"In more central events, where the eccentricity is smaller, the elliptic anisotropy value is systematically lower.",
"In more peripheral collisions (centrality 35–80%), there is a complex interplay between the reduced energy loss and the increase in eccentricity that influence the $v_2()$ value in opposite directions.",
"The data presented here provide the basis for future detailed comparisons to theoretical models.",
"An important consideration in interpreting the $v_{2}()$ results is the contribution from nonflow correlations and initial-state eccentricity fluctuations.",
"To aid in assessing the magnitude of these effects and their evolution with the centrality of the collisions, the results of $v_{2}()$ obtained by all methods at mid-rapidity are compared in Fig.",
"REF for 12 centrality classes.",
"The four methods show differences as expected due to their sensitivities to nonflow contributions [42], [43], [37] and eccentricity fluctuations [35], [34], [38].",
"The method that is most affected by nonflow correlations is the two-particle cumulant, because of the fact that the reference and the differential flow signals are determined in the same pseudorapidity range.",
"The event-plane method is expected to be similarly affected if dedicated selections are not applied to reduce these contributions.",
"In our analysis, the particles used in the event-plane determination and the particles used to measure the flow are at least 3 units of pseudorapidity apart, which suppresses most nonflow correlations.",
"The differences between the two-particle cumulant and the event-plane methods are most pronounced at high and in peripheral collisions, where jet-induced correlations dominate over the collective flow.",
"In a collision where $M$ particles are produced, direct $k$ -particle correlations are typically of order $1/M^{k-1}$ , so that they become smaller as $k$ increases.",
"Therefore, the fourth-order cumulant and the Lee–Yang zeros methods are expected to be much less affected by nonflow contributions than the second-order cumulant method [42], [43], [37].",
"This trend is seen in our data.",
"Figure: (Color online)Comparison of the four different methods for determining v 2 v_{2} as a function of at mid-rapidity (|η|<0.8|\\eta |<0.8)for the 12 centrality classes given in the figures.",
"The error bars show the statistical uncertainties only."
],
[
"Centrality dependence of integrated $v_{2}$ and eccentricity scaling",
"To obtain the integrated $v_{2}$ values as a function of centrality at mid-rapidity, $v_2()$ measurements are averaged over , weighted by the corresponding charged-particle spectrum.",
"The integration range $0.3 < < 3 $ is limited to low to maximize the contribution from soft processes, which facilitates comparisons to hydrodynamic calculations.",
"The centrality dependence of the integrated $v_{2}$ at mid-rapidity $|\\eta |<0.8$ is presented in Fig.",
"REF , for the four methods.",
"The $v_{2}$ values increase from central to peripheral collisions, reaching a maximum in the 40–50% centrality range.",
"In the more peripheral collisions, a decrease in $v_{2}$ is observed in the event-plane and four-particle cumulant measurements, while the values obtained with the two-particle cumulant method remain constant within their uncertainties.",
"The results for $v_{2}\\lbrace 2\\rbrace $ are larger than those for $v_{2}\\lbrace \\mathrm {EP}\\rbrace $ , while the $v_{2}\\lbrace 4\\rbrace $ and $v_{2}\\lbrace \\mathrm {LYZ}\\rbrace $ values are smaller.",
"To facilitate a quantitative comparison between the methods, including their respective systematic uncertainties, the bottom panel of Fig.",
"REF shows the results from the cumulant and the Lee–Yang zeros methods divided by those obtained from the event-plane method.",
"The boxes represent the systematic uncertainty in the ratios, excluding sources of uncertainty common to all methods.",
"The ratios are relatively constant in the 10-60% centrality range, but the differences between the methods increase for the most central and the most peripheral collisions.",
"These findings are similar to results obtained by the STAR experiment at RHIC [67].",
"Below, we further investigate the differences in the $v_{2}$ values returned by each method.",
"Figure: (Color online)Top panel: Integrated v 2 v_{2} as a function of centralityat mid-rapidity |η|<0.8|\\eta |<0.8 for the four methods.",
"The boxes represent thesystematic uncertainties.",
"The magnitudes of the statistical uncertaintiesare smaller than the size of the symbols.",
"Bottom panel: The valuesfrom three of the methods are divided by the results from the event-plane method.",
"The boxes represent thesystematic uncertainties excluding the sources that are common to all methods.",
"The magnitudes of thestatistical uncertainties are smaller than the size of the symbols.The collective motion of the system, and therefore the anisotropy parameter, depend on the initial shape of the nucleus-nucleus collision area and the fluctuations in the positions of the interacting nucleons.",
"By dividing $v_2$ by the participant eccentricity, one may potentially remove this dependence across centralities, colliding species, and center-of-mass energies, enabling a comparison of results in terms of the underlying physics driving the flow.",
"In Fig.",
"REF , we examine the centrality dependence of the eccentricity-scaled anisotropy parameter obtained with the event-plane and cumulant methods at mid-rapidity, $|\\eta |<0.8$ .",
"The participant eccentricity and its cumulant moments are obtained from a Glauber-model simulation, as discussed in Section .",
"The statistical and systematic uncertainties in the integrated $v_{2}$ measurements are added in quadrature and represented by the error bars.",
"The dashed lines show the systematic uncertainties in the eccentricity determination.",
"In the left panel of Fig.",
"REF , the results from each method are divided by the participant eccentricity, $\\epsilon _{\\mathrm {part}}$ .",
"The data show a near-linear decrease from central to peripheral collisions, with differences between methods that were already observed in Fig.",
"REF .",
"In the right panel, the $v_{2}$ values for the cumulant measurements are scaled by their respective moments of the participant eccentricity, thus taking into account the corresponding eccentricity fluctuations [35], [34], [36], [37], [38].",
"With this scaling, the two-particle cumulant and the event-plane results become nearly identical, except for the most central and the most peripheral collisions, where the cumulant results are more affected by nonflow contributions.",
"This is expected [37] because in our application of the method there is no separation in rapidity between the particles used for the reference flow and those used in the differential flow measurement.",
"In the centrality range of 15–40%, the four-particle cumulant measurement of $v_{2}\\lbrace 4\\rbrace /\\epsilon \\lbrace 4\\rbrace $ is also in better agreement with the other two methods.",
"This indicates that the main difference in the results from the different methods could be attributed to their sensitivity to eccentricity fluctuations.",
"In the most central events, where the eccentricity $\\epsilon \\lbrace 4\\rbrace $ is very small, and in the most peripheral events, where the fluctuations are large, $v_{2}\\lbrace 4\\rbrace /\\epsilon _2\\lbrace 4\\rbrace $ deviates from the common scaling behavior.",
"For centralities above 50% the differences between the four-particle cumulant method and the other two methods do not seem to be accounted for by the initial-state fluctuations, as described in our implementation of the Glauber model.",
"In this centrality range, the event-by-event fluctuations in the eccentricity are non-Gaussian due to the underlying Poisson distributions from discrete nucleons [34], [38] and are more difficult to model.",
"It has also been suggested [37], [34] that when the event-plane resolution is smaller than $\\approx $ 0.6, as is the case for the peripheral collisions studied in CMS, the results from the event-plane method should be evaluated using the two-particle cumulant eccentricity $\\epsilon \\lbrace 2\\rbrace $ , rather than the participant eccentricity $\\epsilon _{\\mathrm {part}}$ .",
"We have used a common definition of eccentricity ($\\epsilon _{\\mathrm {part}}$ ) for all centrality classes studied in our event-plane analysis.",
"This would lower the measurements of $v_{2}\\lbrace \\mathrm {EP}\\rbrace /\\epsilon $ by about 10% in the most peripheral collisions, which is not sufficient to reconcile the differences between the event-plane and the four-particle cumulant results.",
"Another model of the initial state that has been used in the literature [39], [68], [69], [70], [71], [38], but has not been explored here, is the color glass condensate (CGC) model [72], which takes into account that at very high energies or small values of Bjorken $x$ , the gluon density becomes very large and saturates.",
"The CGC model predicts eccentricities that exceed the Glauber-model eccentricities by an approximately constant factor of around 1.2, with some deviation from this behavior in the most central and most peripheral collisions [70], [38].",
"The results presented here may give further insight into the nature of the initial-state fluctuations, especially in the regions where the eccentricity fluctuations become non-Gaussian.",
"Figure: (Color online)Left panel: Centrality dependence of the integrated v 2 v_{2} divided bythe participant eccentricity, ϵ part \\epsilon _{\\mathrm {part}}, obtained atmid-rapidity |η|<0.8|\\eta |<0.8 for the event-plane and two- and four-particle cumulantmethods.Right panel: The measurements of v 2 v_{2} as a function of centralityare the same as in the left panel, but here the two-particle and four-particlecumulant results are divided by their corresponding moments of theparticipant eccentricity, ϵ{2}\\epsilon \\lbrace 2\\rbrace and ϵ{4}\\epsilon \\lbrace 4\\rbrace .In both panels, the error bars show the sum in quadrature of the statisticaland systematic uncertainties in the v 2 v_{2} measurement, and the linesrepresent the systematic uncertainties in the eccentricitydetermination."
],
[
"Pseudorapidity dependence of $v_{2}$",
"The pseudorapidity dependence of the anisotropy parameter provides additional constraints on the system evolution in the longitudinal direction.",
"To obtain the $v_{2}(\\eta )$ distribution with the event-plane method, we first measure $v_{2}()$ in pseudorapidity bins of $\\Delta \\eta = 0.4$ , and then average the results over the range $0.3 < < 3 $ , weighting with the efficiency and fake-rate-corrected spectrum.",
"For the cumulant and Lee–Yang zeros methods, the measurements are done using all particles in the range $|\\eta |<2.4$ , and either $0.3 < < 3 $ or $0.3 < < 12$ in the generating function, to obtain the reference flow, and then extracting the pseudorapidity dependence in small pseudorapidity intervals of $\\Delta \\eta = 0.4$ .",
"Tracking efficiency and fake-rate corrections are applied using a track-by-track weight in forming the differential generating functions.",
"As a crosscheck, we have confirmed that at mid-rapidity the values obtained with this method agree with the ones obtained from a direct yield-weighted average of the $v_{2}()$ results from Figs.",
"REF –REF , within the stated systematic uncertainties.",
"As observed at mid-rapidity ($|\\eta |<0.8$ ) in Fig.",
"REF , the values of $v_{2}\\lbrace 4\\rbrace $ and $v_{2}\\lbrace \\mathrm {LYZ}\\rbrace $ are in agreement and are smaller than $v_{2}\\lbrace 2\\rbrace $ and $v_{2}\\lbrace \\mathrm {EP}\\rbrace $ .",
"This behavior persists at larger pseudorapidity, as shown in Fig.",
"REF , which suggests that similar nonflow correlations and eccentricity fluctuations affect the results over the full measured pseudorapidity range.",
"The results show that the value of $v_2(\\eta )$ is greatest at mid-rapidity and is constant or decreases very slowly at larger values of $|\\eta |$ .",
"This behavior is most pronounced in peripheral collisions and for the two-particle cumulant method, which is most affected by nonflow contributions.",
"Figure: (Color online)Pseudorapidity dependence of v 2 v_{2} for 0.3<<30.3 < < 3with all four methods in 12 centrality classes.The boxes give the systematic uncertainties.",
"The magnitudes of the statistical uncertaintiesare smaller than the size of the symbols.To assess whether the observed decrease in $v_{2}(\\eta )$ in the forward pseudorapidity region in peripheral PbPb collisions is due to a pseudorapidity dependence in the $v_{2}()$ distributions or in the underlying charged-particle spectra, in Fig.",
"REF we examine the values of $v_{2}()$ obtained with the event-plane method for several pseudorapidity intervals in each of the 12 centrality classes shown in Fig.",
"REF .",
"From the most central events up to 35–40% centrality there is no change in the $v_{2}()$ distributions with pseudorapidity within the statistical uncertainties.",
"Therefore, any change in the $v_{2}(\\eta )$ distribution can be attributed to changes in the underlying charged-particle transverse momentum spectra.",
"A gradual decrease is observed in the $v_{2}()$ values at forward pseudorapidity ($2.0<|\\eta |<2.4$ ) in more peripheral events.",
"For the 70–80% centrality class the values of $v_{2}()$ decrease by approximately 10% between the central pseudorapidity region $|\\eta |<0.4$ and the forward region $2.0<|\\eta |<2.4$ .",
"Thus, the pseudorapidity dependence in $v_{2}(\\eta )$ for peripheral collisions observed in Fig.",
"REF is caused by changes in the $v_{2}()$ distributions with pseudorapidity, as well as changes in the underlying transverse momentum spectra presented in Section REF .",
"Figure: (Color online)Results from the event-plane method for v 2 ()v_{2}() in three pseudorapidity regions and in 12 centralityclasses.",
"The error bars show the statistical uncertainties that in most cases have magnitudes smaller than the size of the symbol."
],
[
"Centrality and pseudorapidity dependence of the transverse momentum distributions ",
"Elliptic flow measures the azimuthal anisotropy in the invariant yield of the final-state particles.",
"Therefore, the charged-particle transverse momentum distributions influence the observed results.",
"The soft-particle-production mechanism and the evolution of the expanding nuclear medium are reflected in the low-range of the transverse momentum spectra.",
"In the hydrodynamics calculations, measurements of the pseudorapidity density of charged particles produced in collisions with different centrality constrain the description of the initial entropy and the energy density distribution in the collision zone, while the mean transverse momentum of the particle spectra constrains the final temperature and the radial-flow velocity of the system.",
"With additional input on the equation-of-state that is typically provided by lattice QCD, the hydrodynamics calculations then provide a description of the system evolution from some initial time, when local thermal equilibrium is achieved, to the “freeze-out”, when the particle interactions cease.",
"We have measured the charged-particle transverse momentum spectra for 12 centrality classes over the pseudorapidity range $|\\eta | < 2.4$ in bins of $\\Delta \\eta = 0.4$ .",
"Examples of these distributions for the mid-rapidity ($|\\eta | < 0.4$ ) and forward-rapidity ($2.0 <|\\eta | < 2.4$ ) regions are shown in Fig.",
"REF .",
"These results extend the measurements of charged-particle spectra previously reported by CMS [66] down to $= 0.3$ and to forward pseudorapidity.",
"The measurements presented here are in good agreement with the results in Ref.",
"[66] in their common ranges of , pseudorapidity, and centrality.",
"The evolution of the charged-particle spectra with centrality and pseudorapidity can be quantified in terms of the mean of the transverse momentum distributions.",
"The values of $\\langle \\rangle $ as a function of $N_{\\mathrm {part}}$ , which is derived from the centrality of the event, are shown in Fig.",
"REF .",
"In each pseudorapidity interval, the values of $\\langle \\rangle $ increase with $N_{\\mathrm {part}}$ up to $N_{\\mathrm {part}}\\approx $ 150 and then saturate, indicating that the freeze-out conditions of the produced system are similar over a broad range of collision centralities (0–35%).",
"This behavior is in contrast to the centrality dependence in the integrated $v_{2}$ values at mid-rapidity shown in Fig.",
"REF that vary strongly in this centrality range.",
"On the other hand, in more peripheral collisions (centrality greater than 35%) the $v_{2}()$ values do not vary much with centrality at low , as shown in the bottom panels of Figs.",
"REF –REF , while the spectral shapes change, as indicated by the $\\langle \\rangle $ measurement.",
"This behavior is qualitatively similar at forward pseudorapidities, as shown in Figs.",
"REF and REF .",
"These measurements taken together will help in understanding the early-time dynamics in the system evolution, which is reflected in the elliptic flow, and the overall evolution through the hadronic stage, which is reflected in the charged-particle spectra.",
"Figure: (Color online) Inclusive charged-particle spectra atmid-rapidity (left) and forward rapidity (right), for the 12 centrality classes given in the legend.The distributions are offset by arbitrary factors given in the legend for clarity.",
"The shaded bands represent thestatistical and systematic uncertainties added in quadrature, including the overall normalization uncertainties from the trigger efficiencyestimation.Figure: Mean transverse momentum of the charged-particle spectra as a function of N part N_{\\mathrm {part}} in three pseudorapidity intervals marked in the figure.The error bars represent the quadratic sum of the statistical and systematic uncertainties."
],
[
"Comparison with other measurements of $v_{2}$ at the LHC",
"Results on the elliptic anisotropy measured in PbPb collisions in $\\sqrt{s_{NN}}$ = 2.76 TeV have previously been reported by the ALICE [13] and ATLAS [14] experiments.",
"A comparison of $v_{2}\\lbrace 2\\rbrace $ and $v_{2}\\lbrace 4\\rbrace $ as a function of in the 40–50% centrality class for $|\\eta | < 0.8$ from CMS and ALICE [13] is shown in Fig.",
"REF .",
"The error bars give the statistical uncertainties, and the boxes represent the systematic uncertainties in the CMS measurements.",
"The two measurements are in good agreement over their common range.",
"Figure: (Color online) The values of v 2 {2}v_{2}\\lbrace 2\\rbrace and v 2 {4}v_{2}\\lbrace 4\\rbrace obtained with the cumulant method, as a function of from CMS (closedsymbols) and ALICE (open symbols) , measured in the range |η|<0.8|\\eta | < 0.8for the 40–50% centrality class.",
"The error bars show the statistical uncertainties, and the boxes give the systematicuncertainties in the CMS measurement.A comparison of $v_{2}()$ obtained with the event-plane method at mid-rapidity from CMS and ATLAS [14] is presented in Fig.",
"REF for the centrality ranges of the ATLAS measurement.",
"The error bars show the statistical and systematic uncertainties added in quadrature.The results are in good agreement within the statistical and systematic uncertainties.",
"Figure: (Color online) Comparison of results for v 2 ()v_{2}() obtained with the event-plane method from CMS (closed symbols) and ATLAS (open symbols) for the centrality classes marked in the figure.",
"The error bars show the statistical and systematic uncertainties added in quadrature."
],
[
"Discussion",
"We compare the CMS elliptic flow measurements presented in Section with results obtained at RHIC by the PHENIX, STAR, and PHOBOS experiments.",
"Since each method for measuring $v_2$ has a different sensitivity to nonflow correlations and initial-state fluctuations, we make these comparisons for measurements conducted with the same method and with similar kinematic requirements in the method's implementation.",
"In Fig.",
"REF the mid-rapidity measurement of $v_2()$ with the event-plane method in CMS is compared to results from PHENIX [73] for $\\sqrt{s_{NN}} = 200$ AuAu collisions.",
"For the PHENIX measurement, the event plane was determined at forward pseudorapidities, $|\\eta | =$ 3.1–3.9, while the $v_2()$ measurement was performed in the pseudorapidity interval $|\\eta |<0.35$ , thus providing a separation of at least 2.75 units of pseudorapidity between the charged particles used for the $v_2()$ analysis and the particles used in the event-plane determination.",
"This procedure is comparable to the CMS approach, where a separation of at least 3 units of pseudorapidity is used.",
"These large pseudorapidity gaps are expected [37] to suppress nonflow contributions in both measurements.",
"The pseudorapidity interval for the CMS measurement is wider, $|\\eta |< 0.8$ , but since the pseudorapidity dependence of $v_2(\\eta )$ was shown to be weak (see Fig.",
"REF ), this difference should not influence the comparison of the results.",
"The top panels in Fig.",
"REF show the measurements of $v_2()$ from CMS (closed symbols) and PHENIX (open symbols) for several centrality classes.",
"The error bars represent the statistical uncertainties only.",
"The shape of the $v_2()$ distributions and the magnitude of the signals are similar, in spite of the factor of $\\approx $ 14 increase in the center-of-mass energy.",
"To facilitate a quantitative comparison of these results, the CMS measurements are fitted with a combination of a fifth-order polynomial function (for $< 3.2$ ) and a Landau distribution (for $ 3 < < 7 $ ).",
"There is no physical significance attributed to these functional choices, other than an attempt to analytically describe the CMS $v_2()$ distributions, so that the value of $v_2$ can be easily compared to results from other experiments, which have been obtained with different binning.",
"The results from the fits are plotted as solid lines in the top panels of Fig.",
"REF .",
"In the bottom panels, the fit function is used to evaluate $v_2()$ at the values for each data set, and then to form the ratios between the CMS fit values and the PHENIX data, and the CMS fit to the CMS measurements.",
"The error bars represent the statistical uncertainties.",
"The systematic uncertainties from the CMS and PHENIX measurements are added in quadrature and plotted as shaded boxes.",
"The $v_2()$ values measured by CMS are systematically higher than those from PHENIX in all centrality classes and over the entire transverse momentum range measured by PHENIX.",
"The relative deviations of are the order 10%, except for the most peripheral collisions where they reach 15%.",
"A similar comparison is carried out for the two-particle and four-particle cumulant methods.",
"In Fig.",
"REF , the results from the STAR experiment [74] for AuAu collisions at $\\sqrt{s_{NN}} = 200$ in the 20–60% centrality range are compared with the CMS measurements.",
"The pseudorapidity interval for the STAR measurement is $|\\eta |< 1.3$ , compared to $|\\eta |< 0.8$ for CMS.",
"These $\\eta $ ranges are within a pseudorapidity region in which the $v_2(\\eta )$ values only weakly depend on the pseudorapidity.",
"The kinematic selections imposed on the charged particles used in determining the reference flow are also similar in the CMS and STAR measurements.",
"The top panels of Fig.",
"REF show the $v_2()$ distributions for the two-particle (left) and four-particle (right) cumulant method from both experiments, along with fits to the CMS data (lines).",
"The functional form used for the fit of the CMS $v_2()$ distributions is the same as the one used in Fig.",
"REF .",
"The bottom panels in Fig.",
"REF show the ratios of the fits to the CMS data to the actual measurements from CMS and STAR.",
"The error bars represent the statistical uncertainties.",
"The systematic uncertainties from the CMS and STAR measurements are added in quadrature and plotted as shaded boxes.",
"At low , the $v_2()$ values measured by CMS are larger than in the STAR data, but the relative deviations are smaller than 5% for the four-particle cumulant method, and are of the order 10–15% for the two-particle cumulant method.",
"Taken together, the comparisons to the RHIC results in Figs.",
"REF and REF indicate only a moderate increase in $v_2()$ at low from the highest RHIC energy to the LHC, despite the large increase in the center-of-mass energy.",
"Figure: (Color online)Top panels: Comparison of v 2 ()v_2() using the event-plane method asmeasured by CMS (solid circles) at s NN =2.76\\sqrt{s_{NN}}= 2.76, andPHENIX (open diamonds) at s NN =200\\sqrt{s_{NN}}= 200for mid-rapidity (|η|<0.8|\\eta |<0.8 and |η|<0.35|\\eta |<0.35,respectively).",
"The error bars represent the statistical uncertainties.",
"Thesolid line is a fit to the CMS data.",
"Bottom panels: Ratios of the CMSfit to the PHENIX data (open diamonds) and to the CMS data (opencircles).",
"The error bars show the statistical uncertainties, whilethe shaded boxes give the quadrature sum of the CMS and PHENIX systematic uncertainties.Figure: (Color online)Top panels: Comparison of v 2 ()v_2() using the two-particle (left)and the four-particle (right) cumulant method as measuredby CMS (solid circles) at s NN =2.76\\sqrt{s_{NN}}= 2.76, andSTAR (open stars) at s NN =200\\sqrt{s_{NN}}= 200atmid-rapidity (|η|<0.8|\\eta |<0.8 and |η|<1.3|\\eta |<1.3, respectively).",
"The errorbars represent the statistical uncertainties.",
"The line is a fit tothe CMS data.",
"Bottom panels: Ratios of the CMS fit values to the STAR data(open diamonds) and to the CMS data (open circles).",
"The error bars showthe statistical uncertainties, while the shaded boxes give the quadrature sum of the CMS and STAR systematicuncertainties.In Fig.",
"REF , we examine the $\\sqrt{s_{NN}}$ dependence of the integrated $v_2$ from mid-central collisions spanning $\\sqrt{s_{NN}}= 4.7$ GeV to $\\sqrt{s_{NN}}= 2.76$ TeV.",
"The CMS measurement is obtained with the event-plane method in the 20–30% centrality class by extrapolating the $v_2()$ and the charged-particle spectra down to $= 0$ .",
"In the extrapolation it is assumed that $v_2(0) = 0 $ , and the charged-particle yield is constrained to match the $N_{\\mathrm {ch}}/\\eta $ values measured by CMS [75].",
"The low-energy data are from Refs.",
"[20], [76], [23], [77], [78], [49], [79], [80], [81], as compiled in Ref.",
"[79] and tabulated in Ref. [76].",
"The error bars for the low-energy data represent the statistical uncertainties.",
"For the CMS data the error bar is the quadrature sum of the statistical and systematic uncertainties.",
"The integrated $v_2$ values increase approximately logarithmically with $\\sqrt{s_{NN}}$ over the full energy range, with a 20–30% increase from the highest RHIC energy to that of the LHC.",
"This has contributions from the increase in the mean of the charged-particle spectra with $\\sqrt{s_{NN}}$ , shown in Fig.",
"REF , and from the moderate increase in the $v_2()$ distributions at low , shown in Fig.",
"REF .",
"We note that the centrality selections, the collision species, and the methods employed in the integrated $v_2$ measurements are not identical in all experiments, so the comparison presented in Fig.",
"REF is only approximate.",
"Further comparisons to results from lower energies are presented in Figs.",
"REF –REF .",
"Figure: (Color online)The CMS integrated v 2 v_2 values for 20–30% centrality from the range |η|<0.8|\\eta | < 0.8 and 0<<30< < 3 obtained using the event-plane methodis compared as a function of s NN \\sqrt{s_{NN}} to results at mid-rapidity and similar centrality from ALICE ,STAR ,PHENIX ,PHOBOS , , ,NA49 ,E877 ,and CERES .The error bars for the lower-energy results represent statisticaluncertainties; for the CMS and ALICE measurements the statistical and systematic uncertainties are added in quadrature.Figure: Mean transverse momentum of the charged-particle spectra as a function of N part N_{\\mathrm {part}} measured by CMS in PbPb collisions ats NN =2.76\\sqrt{s_{NN}} = 2.76(closed circles) and by STAR in AuAu collisions at s NN =200\\sqrt{s_{NN}} = 200(open circles).",
"The error bars representthe quadratic sum of statistical and systematic uncertainties.Figure: (Color online) The CMS integrated v 2 v_{2} values from the event-plane method divided bythe participant eccentricity as a function of N part N_{\\mathrm {part}} with|η|<0.8|\\eta | < 0.8 and 0<<30< < 3 .",
"These results are comparedwith those from PHOBOS for different nuclear speciesand collision energies.",
"The PHOBOS v 2 v_{2} values are divided by the cumulant eccentricity ϵ{2}\\epsilon \\lbrace 2\\rbrace (see text).",
"The error bars give the statistical andsystematic uncertainties in the v 2 v_{2} measurements added in quadrature.",
"The dashed lines represent thesystematic uncertainties in the eccentricity determination.In ideal hydrodynamics, the eccentricity-scaled elliptic flow is constant over a broad range of impact parameters; deviations from this behavior are expected in peripheral collisions, in which the system freezes out before the elliptic flow fully builds up and saturates [32].",
"A weak centrality and beam-energy dependence is expected through variations in the equation-of-state.",
"In addition, the system is also affected by viscosity, both in the sQGP and hadronic stages [68], [83], [84], [22] of its evolution.",
"Therefore, the centrality and $\\sqrt{s_{NN}}$ dependence of $v_{2}/\\epsilon $ can be used to extract the ratio of the shear viscosity to the entropy density of the system.",
"In Fig.",
"REF , the integrated $v_2$ obtained from the event-plane method is divided by the eccentricity of the collisions and plotted as a function of $N_{\\mathrm {part}}$ , which is derived from the centrality of the event.",
"The result is compared to lower-energy AuAu and CuCu measurements from the PHOBOS experiment [34].",
"For the CMS measurement, the value of $v_2$ is divided by the participant eccentricity $\\epsilon _{\\mathrm {part}}$ since the event-plane resolution factor shown in Fig.",
"REF is greater than 0.6 for all but the most central and most peripheral event selections in our analysis.",
"It has been argued [34], [37] that for lower-resolution parameters, the event-plane method measures the r.m.s.",
"of the azimuthal anisotropy, rather than the mean, and therefore, the relevant eccentricity parameter in this case should be the second-order cumulant eccentricity $\\epsilon \\lbrace 2\\rbrace \\equiv \\sqrt{\\langle \\epsilon _\\mathrm {part}^{2} \\rangle }$ .",
"Thus, the comparison with the PHOBOS $v_2$ results, which were obtained with low event-plane resolution, is done by implementing this scaling using the data from Ref. [34].",
"An approximately 25% increase in the integrated $v_2$ scaled by the eccentricity between RHIC and LHC energies is observed, and with a similar $N_{\\mathrm {part}}$ dependence.",
"It was previously observed [85], [79], [34] that the $v_2/\\epsilon $ values obtained in different collision systems and varying beam energies scale with the charged-particle rapidity density per unit transverse overlap area $(1/S) (N_{\\mathrm {ch}}/y)$ , which is proportional to the initial entropy density.",
"In addition, it has been pointed out [69] that in this representation the sensitivity to the modeling of the initial conditions of the heavy-ion collisions is largely removed, thus enabling the extraction of the shear viscosity to the entropy density ratio from the data through the comparison with viscous hydrodynamics calculations.",
"With the factor of 2.1 increase in the charged-particle pseudorapidity density per participant pair, $(N_{\\mathrm {ch}}/\\eta )/(N_{\\mathrm {part}}/2)$ , from the highest RHIC energy to the LHC [86], [75], this scaling behavior can be tested over a much broader range of initial entropy densities.",
"In Fig.",
"REF , we compare the CMS results for $v_{2}/\\epsilon $ from the event-plane method to results from the PHOBOS experiment [34] for CuCu and AuAu collisions with $\\sqrt{s_{NN}} = 62.4$ and 200.",
"At lower energies, the scaling has been examined using the charged-particle rapidity density $N_{\\mathrm {ch}}/y$ [85], [79], [34].",
"However, since we do not identify the species of charged particles in this analysis, we perform the comparison using $(1/S) (N_{\\mathrm {ch}}/\\eta $ ) to avoid introducing uncertainties related to assumptions about the detailed behavior of the identified particle transverse momentum spectra that are needed to perform this conversion.",
"In Fig.",
"REF , the charged-particle pseudorapidity density $N_{\\mathrm {ch}}/\\eta $ measured by CMS [75] is used, and the value of the integrated $v_{2}$ for the ranges $0< < 3$ and $|\\eta | < 0.8$ .",
"The transverse nuclear-overlap area $S$ and the participant eccentricity are listed in Table REF .",
"The PHOBOS results from Ref.",
"[34] used $N_{\\mathrm {ch}}/y$ , and applied two factors to perform the Jacobian transformation from $N_{\\mathrm {ch}}/\\eta $ : the $x$ axis was scaled by a factor 1.15, and the $y$ axis by 0.9.",
"Both of these factors are reversed in order to compare the CMS and PHOBOS measurements in Fig.",
"REF .",
"As in Fig.",
"REF , the PHOBOS data are scaled by $\\epsilon \\lbrace 2\\rbrace $ , while the CMS data are scaled by the participant eccentricity $\\epsilon _{\\mathrm {part}}$ , taking into account the event-plane resolution factors in the two measurements.",
"The CMS result extends to very peripheral collisions (70–80% centrality), which allows for a significant overlap in the transverse charged-particle density measured at RHIC and LHC.",
"Despite the large systematic uncertainties quoted for the PHOBOS measurements, the data are in good agreement over the common $(1/S) (N_{\\mathrm {ch}}/\\eta )$ range.",
"A smooth increase in $v_{2}/\\epsilon $ proportional to the transverse particle density is observed over the entire measured range, except for a small decrease in the most central collisions in both the RHIC and LHC data.",
"The theoretical predictions [68], [83], [71] for the $\\sqrt{s_{NN}}$ dependence of the transverse-particle-density scaling of $v_{2}/\\epsilon $ differ, and do not generally predict a universal behavior.",
"The data presented here provide constraints on the model descriptions of the dynamical evolution of the system, and thus should aid the reliable extraction of the transport properties of the hot QCD medium from data.",
"Figure: (Color online) Eccentricity-scaled v 2 v_{2} as a function of the transverse charged-particle density from CMS andPHOBOS .",
"The error bars include both statistical and systematic uncertainties in v 2 v_{2}.The dashed lines represent the systematic uncertainties in the eccentricity determination.The $v_{2}(\\eta )$ results can be used to test theoretical descriptions of the longitudinal dynamics in the expanding system, as they have been shown to be sensitive to the choice of the initial conditions, the event-by-event fluctuations in the eccentricity, and the viscosity in the sQGP and the hadronic stages of the system evolution [39], [87].",
"The PHOBOS experiment observed [41] that the elliptic flow measured over a broad range of collision energies ($\\sqrt{s_{NN}}=19.6$ , 62.4, 130, and 200) exhibited longitudinal scaling extending over several units of pseudorapidity when viewed in the rest frame of one of the incident nuclei.",
"A similar phenomenon for soft-particle yields and spectra is known as limiting fragmentation [88].",
"Furthermore, with increasing $\\sqrt{s_{NN}}$ , this beam-energy independence of $v_2(\\eta )$ was found to extend over an increasingly wider pseudorapidity range [41].",
"To investigate the potential continuation of extended longitudinal scaling of the elliptic flow to LHC energies, in Fig.",
"REF we compare the pseudorapidity dependence of $v_2(\\eta )$ measured by CMS with that measured by PHOBOS at $\\sqrt{s_{NN}} = 200$ in three centrality intervals.",
"Neither the CMS nor the PHOBOS measurements are performed using identified particles.",
"The pseudorapidity $\\eta ^+$ ($\\eta ^-$ ) of the particles in the rest frame of the nuclei moving in the positive (negative) direction is approximated by $\\eta ^\\pm = \\eta \\pm y_{\\mathrm {beam}}$ , where $\\eta $ is the pseudorapidity of the particles in the center-of-mass frame, and $y_{\\mathrm {beam}} = \\textrm {arccosh} (E_{\\mathrm {lab}}/Am_N c^2) \\approx \\ln (\\sqrt{s_{NN}}~\\mathrm {[GeV]})$ , with $E_{\\mathrm {lab}}$ denoting the energy of the beam in the laboratory frame, $A$ the nuclear mass number, and $m_N$ the nucleon mass.",
"In Fig.",
"REF , the left (right) half of each plot depicts $v_2$ in the rest frame of the beam moving in the positive (negative) lab direction.",
"The PHOBOS $v_2(\\eta )$ results are from the hit-based analysis from Fig.",
"4 in Ref. [77].",
"The CMS results are obtained with the event-plane method in 0.4-unit-wide bins of pseudorapidity by averaging the corresponding $v_{2}()$ distributions over the range $0<<3$ and folding in the efficiency-corrected charged-particle spectra.",
"A comparable centrality interval (2.5–15% in CMS, and 3–15% in PHOBOS) is analyzed for the most central collisions, while for mid-central (15–25%) and more peripheral (25–50%) collisions, the centrality selections are the same in both experiments.",
"In Fig.",
"REF , the statistical uncertainties are shown as error bars, while the systematic ones are represented by the shaded boxes surrounding the points.",
"The CMS data cover 4.8 units of pseudorapidity, but do not overlap in pseudorapidity with the PHOBOS data when plotted in the rest frames of the colliding nuclei.",
"The CMS results show weaker pseudorapidity dependence than observed in the PHOBOS measurement.",
"The data suggest a nearly boost-invariant region that is several units wide for central events but considerably smaller for peripheral ones.",
"It has been noted [89] that if the QCD matter produced at mid-rapidity at RHIC is in local equilibrium, then deviations from the triangular shape of $v_{2}(\\eta )$ observed in the PHOBOS measurements [41] would be expected around mid-rapidity at LHC energies.",
"Detailed comparisons of theoretical calculations to the results presented here can give new insights into the nature of the matter produced at both RHIC and LHC energies.",
"Figure: (Color online) Measurements of v 2 v_2 as a function of the pseudorapidity of particles in the rest frame ofthe colliding nuclei η±y beam \\eta \\pm y_{\\mathrm {beam}} from CMS (closed symbols) and PHOBOS (open symbols) in three centrality intervals.",
"The error bars show the statistical uncertainties and the boxes give the systematic ones."
],
[
"Summary",
"Detailed measurements of the charged-particle azimuthal anisotropies in $\\sqrt{s_{NN}} = 2.76$ PbPb collisions and comparisons to lower collision energy results have been presented.",
"The results cover a broad kinematic range: $0.3 < < 20$ , $|\\eta | < 2.4$ , and 12 centrality classes from 0 to 80%.",
"The measurements employ four different methods that have different sensitivities to fluctuations in the initial conditions and nonflow correlations.",
"The systematic comparison between the methods provides the possibility to explore the underlying physics processes that cause these differences.",
"The elliptic anisotropy parameter $v_2()$ for $|\\eta |<0.8$ is found to increase with up to $\\approx 3$ , and then to decrease in the range $3 << 10$ .",
"For transverse momenta of $10 << 20$ , no strong dependence of $v_2$ on is observed.",
"The study of the high-azimuthal anisotropy in charged-particle production may constrain the theoretical descriptions of parton energy loss and its dependence on the path-length traveled through the medium.",
"The shapes of the $v_{2}()$ distributions are found to be similar to those measured at RHIC.",
"At low , only a moderate increase (5–15%) is observed in the comparison between results obtained at the highest RHIC energy and the LHC, despite the large increase in the center-of-mass energy.",
"The integrated $v_{2}$ at mid-rapidity and in mid-central collisions (20–30% centrality) increases approximately logarithmically with $\\sqrt{s_{NN}}$ .",
"An increase by 20–30% from the highest RHIC energy to that of the LHC is observed, which is mostly due to the increase in the mean of the underlying charged-particle spectra.",
"The integrated $v_{2}$ signal increases from the most central collisions to the 40–50% centrality range, after which a decrease is observed.",
"Conversely, the values of $\\langle \\rangle $ increase with $N_{\\mathrm {part}}$ up to $N_{\\mathrm {part}}\\approx $ 150 (from the most peripheral collisions up to centrality$\\approx $ 35%) and then saturate, indicating similar freeze-out conditions in the more central collisions.",
"The different methods of measuring $v_{2}$ give consistent results over a broad range of centrality, when scaled by their respective participant eccentricity moments.",
"Deviations from this scaling are observed in the most central collisions and in peripheral (centrality above 50%) collisions.",
"The eccentricity-scaled $v_{2}$ at mid-rapidity is measured to be approximately linear in the transverse particle density, and a universal scaling is observed in the comparison of results from different collision systems and center-of-mass energies measured at RHIC and the LHC.",
"The value of $v_{2}(\\eta )$ is found to be weakly dependent on pseudorapidity in central collisions; for peripheral collisions the values of $v_{2}(\\eta )$ gradually decrease as the pseudorapidity increases.",
"The results presented here provide further input to the theoretical models of relativistic nucleus-nucleus collisions and will aid in determining the initial conditions of the system, the degree of equilibration, and the transport properties of hot QCD matter produced in heavy-ion collisions."
],
[
"Acknowledgments",
"We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort.",
"In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses.",
"Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: the Austrian Federal Ministry of Science and Research; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education, Youth and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Recurrent financing contract SF0690030s09 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucléaire et de Physique des Particules / CNRS, and Commissariat à l'Énergie Atomique et aux Énergies Alternatives / CEA, France; the Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Science and Technological Development of Serbia; the Secretaría de Estado de Investigación, Desarrollo e Innovación and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Thailand Center of Excellence in Physics, the Institute for the Promotion of Teaching Science and Technology and National Electronics and Computer Technology Center; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.",
"Individuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of Czech Republic; the Council of Science and Industrial Research, India; the Compagnia di San Paolo (Torino); and the HOMING PLUS programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund."
],
[
"The CMS Collaboration ",
"Yerevan Physics Institute, Yerevan, Armenia S. Chatrchyan, V. Khachatryan, A.M. Sirunyan, A. Tumasyan Institut für Hochenergiephysik der OeAW, Wien, Austria W. Adam, T. Bergauer, M. Dragicevic, J. Erö, C. Fabjan, M. Friedl, R. Frühwirth, V.M.",
"Ghete, J. Hammer1, N. Hörmann, J. Hrubec, M. Jeitler, W. Kiesenhofer, M. Krammer, D. Liko, I. Mikulec, M. Pernicka$^{\\textrm {\\dag }}$ , B. Rahbaran, C. Rohringer, H. Rohringer, R. Schöfbeck, J. Strauss, A. Taurok, F. Teischinger, P. Wagner, W. Waltenberger, G. Walzel, E. Widl, C.-E. Wulz National Centre for Particle and High Energy Physics, Minsk, Belarus V. Mossolov, N. Shumeiko, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium S. Bansal, K. Cerny, T. Cornelis, E.A.",
"De Wolf, X. Janssen, S. Luyckx, T. Maes, L. Mucibello, S. Ochesanu, B. Roland, R. Rougny, M. Selvaggi, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel, A.",
"Van Spilbeeck Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, S. Blyweert, J.",
"D'Hondt, R. Gonzalez Suarez, A. Kalogeropoulos, M. Maes, A. Olbrechts, W. Van Doninck, P. Van Mulders, G.P.",
"Van Onsem, I. Villella Université Libre de Bruxelles, Bruxelles, Belgium O. Charaf, B. Clerbaux, G. De Lentdecker, V. Dero, A.P.R.",
"Gay, T. Hreus, A. Léonard, P.E.",
"Marage, T. Reis, L. Thomas, C. Vander Velde, P. Vanlaer Ghent University, Ghent, Belgium V. Adler, K. Beernaert, A. Cimmino, S. Costantini, G. Garcia, M. Grunewald, B. Klein, J. Lellouch, A. Marinov, J. Mccartin, A.A. Ocampo Rios, D. Ryckbosch, N. Strobbe, F. Thyssen, M. Tytgat, L. Vanelderen, P. Verwilligen, S. Walsh, E. Yazgan, N. Zaganidis Université Catholique de Louvain, Louvain-la-Neuve, Belgium S. Basegmez, G. Bruno, L. Ceard, C. Delaere, T. du Pree, D. Favart, L. Forthomme, A. Giammanco2, J. Hollar, V. Lemaitre, J. Liao, O. Militaru, C. Nuttens, D. Pagano, A. Pin, K. Piotrzkowski, N. Schul Université de Mons, Mons, Belgium N. Beliy, T. Caebergs, E. Daubie, G.H.",
"Hammad Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil G.A.",
"Alves, M. Correa Martins Junior, D. De Jesus Damiao, T. Martins, M.E.",
"Pol, M.H.G.",
"Souza Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil W.L.",
"Aldá Júnior, W. Carvalho, A. Custódio, E.M. Da Costa, C. De Oliveira Martins, S. Fonseca De Souza, D. Matos Figueiredo, L. Mundim, H. Nogima, V. Oguri, W.L.",
"Prado Da Silva, A. Santoro, S.M.",
"Silva Do Amaral, L. Soares Jorge, A. Sznajder Instituto de Fisica Teorica, Universidade Estadual Paulista, Sao Paulo, Brazil T.S.",
"Anjos3, C.A.",
"Bernardes3, F.A.",
"Dias4, T.R.",
"Fernandez Perez Tomei, E. M. Gregores3, C. Lagana, F. Marinho, P.G.",
"Mercadante3, S.F.",
"Novaes, Sandra S. Padula Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria V. Genchev1, P. Iaydjiev1, S. Piperov, M. Rodozov, S. Stoykova, G. Sultanov, V. Tcholakov, R. Trayanov, M. Vutova University of Sofia, Sofia, Bulgaria A. Dimitrov, R. Hadjiiska, V. Kozhuharov, L. Litov, B. Pavlov, P. Petkov Institute of High Energy Physics, Beijing, China J.G.",
"Bian, G.M.",
"Chen, H.S.",
"Chen, C.H.",
"Jiang, D. Liang, S. Liang, X. Meng, J. Tao, J. Wang, J. Wang, X. Wang, Z. Wang, H. Xiao, M. Xu, J. Zang, Z. Zhang State Key Lab.",
"of Nucl. Phys.",
"and Tech., Peking University, Beijing, China C. Asawatangtrakuldee, Y.",
"Ban, S. Guo, Y. Guo, W. Li, S. Liu, Y. Mao, S.J.",
"Qian, H. Teng, S. Wang, B. Zhu, W. Zou Universidad de Los Andes, Bogota, Colombia C. Avila, B. Gomez Moreno, A.F.",
"Osorio Oliveros, J.C. Sanabria Technical University of Split, Split, Croatia N. Godinovic, D. Lelas, R. Plestina5, D. Polic, I. Puljak1 University of Split, Split, Croatia Z. Antunovic, M. Dzelalija, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, S. Duric, K. Kadija, J. Luetic, S. Morovic University of Cyprus, Nicosia, Cyprus A. Attikis, M. Galanti, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A.",
"Razis Charles University, Prague, Czech Republic M. Finger, M. Finger Jr. Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt Y. Assran6, S. Elgammal, A. Ellithi Kamel7, S. Khalil8, M.A.",
"Mahmoud9, A. Radi8$^{, }$ 10 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia M. Kadastik, M. Müntel, M. Raidal, L. Rebane, A. Tiko Department of Physics, University of Helsinki, Helsinki, Finland V. Azzolini, P. Eerola, G. Fedi, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland J. Härkönen, A. Heikkinen, V. Karimäki, R. Kinnunen, M.J. Kortelainen, T. Lampén, K. Lassila-Perini, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, T. Peltola, E. Tuominen, J. Tuominiemi, E. Tuovinen, D. Ungaro, L. Wendland Lappeenranta University of Technology, Lappeenranta, Finland K. Banzuzi, A. Korpela, T. Tuuva DSM/IRFU, CEA/Saclay, Gif-sur-Yvette, France M. Besancon, S. Choudhury, M. Dejardin, D. Denegri, B. Fabbro, J.L.",
"Faure, F. Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, E. Locci, J. Malcles, L. Millischer, A. Nayak, J. Rander, A. Rosowsky, I. Shreyber, M. Titov Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France S. Baffioni, F. Beaudette, L. Benhabib, L. Bianchini, M. Bluj11, C. Broutin, P. Busson, C. Charlot, N. Daci, T. Dahms, L. Dobrzynski, R. Granier de Cassagnac, M. Haguenauer, P. Miné, C. Mironov, C. Ochando, P. Paganini, D. Sabes, R. Salerno, Y. Sirois, C. Veelken, A. Zabi Institut Pluridisciplinaire Hubert Curien, Université de Strasbourg, Université de Haute Alsace Mulhouse, CNRS/IN2P3, Strasbourg, France J.-L. Agram12, J. Andrea, D. Bloch, D. Bodin, J.-M. Brom, M. Cardaci, E.C.",
"Chabert, C. Collard, E. Conte12, F. Drouhin12, C. Ferro, J.-C. Fontaine12, D. Gelé, U. Goerlach, P. Juillot, M. Karim12, A.-C.",
"Le Bihan, P. Van Hove Centre de Calcul de l'Institut National de Physique Nucleaire et de Physique des Particules (IN2P3), Villeurbanne, France F. Fassi, D. Mercier Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, N. Beaupere, O. Bondu, G. Boudoul, H. Brun, J. Chasserat, R. Chierici1, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, S. Gascon, M. Gouzevitch, B. Ille, T. Kurca, M. Lethuillier, L. Mirabito, S. Perries, V. Sordini, S. Tosi, Y. Tschudi, P. Verdier, S. Viret Institute of High Energy Physics and Informatization, Tbilisi State University, Tbilisi, Georgia Z. Tsamalaidze13 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany G. Anagnostou, S. Beranek, M. Edelhoff, L. Feld, N. Heracleous, O. Hindrichs, R. Jussen, K. Klein, J. Merz, A. Ostapchuk, A. Perieanu, F. Raupach, J. Sammet, S. Schael, D. Sprenger, H. Weber, B. Wittmer, V. Zhukov14 RWTH Aachen University, III.",
"Physikalisches Institut A, Aachen, Germany M. Ata, J. Caudron, E. Dietz-Laursonn, D. Duchardt, M. Erdmann, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, T. Klimkovich, D. Klingebiel, P. Kreuzer, D. Lanske$^{\\textrm {\\dag }}$ , J. Lingemann, C. Magass, M. Merschmeyer, A. Meyer, M. Olschewski, P. Papacz, H. Pieta, H. Reithler, S.A. Schmitz, L. Sonnenschein, J. Steggemann, D. Teyssier, M. Weber RWTH Aachen University, III.",
"Physikalisches Institut B, Aachen, Germany M. Bontenackels, V. Cherepanov, M. Davids, G. Flügge, H. Geenen, M. Geisler, W. Haj Ahmad, F. Hoehle, B. Kargoll, T. Kress, Y. Kuessel, A. Linn, A. Nowack, L. Perchalla, O. Pooth, J. Rennefeld, P. Sauerland, A. Stahl Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, J. Behr, W. Behrenhoff, U. Behrens, M. Bergholz15, A. Bethani, K. Borras, A. Burgmeier, A. Cakir, L. Calligaris, A. Campbell, E. Castro, F. Costanza, D. Dammann, G. Eckerlin, D. Eckstein, D. Fischer, G. Flucke, A. Geiser, I. Glushkov, S. Habib, J. Hauk, H. Jung1, M. Kasemann, P. Katsas, C. Kleinwort, H. Kluge, A. Knutsson, M. Krämer, D. Krücker, E. Kuznetsova, W. Lange, W. Lohmann15, B. Lutz, R. Mankel, I. Marfin, M. Marienfeld, I.-A.",
"Melzer-Pellmann, A.B.",
"Meyer, J. Mnich, A. Mussgiller, S. Naumann-Emme, J. Olzem, H. Perrey, A. Petrukhin, D. Pitzl, A. Raspereza, P.M. Ribeiro Cipriano, C. Riedl, M. Rosin, J. Salfeld-Nebgen, R. Schmidt15, T. Schoerner-Sadenius, N. Sen, A. Spiridonov, M. Stein, R. Walsh, C. Wissing University of Hamburg, Hamburg, Germany C. Autermann, V. Blobel, S. Bobrovskyi, J. Draeger, H. Enderle, J. Erfle, U. Gebbert, M. Görner, T. Hermanns, R.S.",
"Höing, K. Kaschube, G. Kaussen, H. Kirschenmann, R. Klanner, J. Lange, B. Mura, F. Nowak, N. Pietsch, D. Rathjens, C. Sander, H. Schettler, P. Schleper, E. Schlieckau, A. Schmidt, M. Schröder, T. Schum, M. Seidel, H. Stadie, G. Steinbrück, J. Thomsen Institut für Experimentelle Kernphysik, Karlsruhe, Germany C. Barth, J. Berger, T. Chwalek, W. De Boer, A. Dierlamm, M. Feindt, M. Guthoff1, C. Hackstein, F. Hartmann, M. Heinrich, H. Held, K.H.",
"Hoffmann, S. Honc, U. Husemann, I. Katkov14, J.R. Komaragiri, D. Martschei, S. Mueller, Th.",
"Müller, M. Niegel, A. Nürnberg, O. Oberst, A. Oehler, J. Ott, T. Peiffer, G. Quast, K. Rabbertz, F. Ratnikov, N. Ratnikova, S. Röcker, C. Saout, A. Scheurer, F.-P. Schilling, M. Schmanau, G. Schott, H.J.",
"Simonis, F.M.",
"Stober, D. Troendle, R. Ulrich, J. Wagner-Kuhr, T. Weiler, M. Zeise, E.B.",
"Ziebarth Institute of Nuclear Physics \"Demokritos\", Aghia Paraskevi, Greece G. Daskalakis, T. Geralis, S. Kesisoglou, A. Kyriakis, D. Loukas, I. Manolakos, A. Markou, C. Markou, C. Mavrommatis, E. Ntomari University of Athens, Athens, Greece L. Gouskos, T.J. Mertzimekis, A. Panagiotou, N. Saoulidou University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas1, P. Kokkas, N. Manthos, I. Papadopoulos, V. Patras KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary G. Bencze, C. Hajdu1, P. Hidas, D. Horvath16, K. Krajczar17, B. Radics, F. Sikler1, V. Veszpremi, G. Vesztergombi17 Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Molnar, J. Palinkas, Z. Szillasi University of Debrecen, Debrecen, Hungary J. Karancsi, P. Raics, Z.L.",
"Trocsanyi, B. Ujvari Panjab University, Chandigarh, India S.B.",
"Beri, V. Bhatnagar, N. Dhingra, R. Gupta, M. Jindal, M. Kaur, J.M.",
"Kohli, M.Z.",
"Mehta, N. Nishu, L.K.",
"Saini, A. Sharma, J. Singh, S.P.",
"Singh University of Delhi, Delhi, India S. Ahuja, B.C.",
"Choudhary, A. Kumar, A. Kumar, S. Malhotra, M. Naimuddin, K. Ranjan, V. Sharma, R.K. Shivpuri Saha Institute of Nuclear Physics, Kolkata, India S. Banerjee, S. Bhattacharya, S. Dutta, B. Gomber, Sa.",
"Jain, Sh.",
"Jain, R. Khurana, S. Sarkar Bhabha Atomic Research Centre, Mumbai, India A. Abdulsalam, R.K. Choudhury, D. Dutta, S. Kailas, V. Kumar, A.K.",
"Mohanty1, L.M.",
"Pant, P. Shukla Tata Institute of Fundamental Research - EHEP, Mumbai, India T. Aziz, S. Ganguly, M. Guchait18, A. Gurtu19, M. Maity20, G. Majumder, K. Mazumdar, G.B.",
"Mohanty, B. Parida, K. Sudhakar, N. Wickramage Tata Institute of Fundamental Research - HECR, Mumbai, India S. Banerjee, S. Dugad Institute for Research in Fundamental Sciences (IPM), Tehran, Iran H. Arfaei, H. Bakhshiansohi21, S.M.",
"Etesami22, A. Fahim21, M. Hashemi, H. Hesari, A. Jafari21, M. Khakzad, A. Mohammadi23, M. Mohammadi Najafabadi, S. Paktinat Mehdiabadi, B. Safarzadeh24, M. Zeinali22 INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ , Bari, Italy M. Abbrescia$^{a}$$^{, }$$^{b}$ , L. Barbone$^{a}$$^{, }$$^{b}$ , C. Calabria$^{a}$$^{, }$$^{b}$$^{, }$ 1, S.S. Chhibra$^{a}$$^{, }$$^{b}$ , A. Colaleo$^{a}$ , D. Creanza$^{a}$$^{, }$$^{c}$ , N. De Filippis$^{a}$$^{, }$$^{c}$$^{, }$ 1, M. De Palma$^{a}$$^{, }$$^{b}$ , L. Fiore$^{a}$ , G. Iaselli$^{a}$$^{, }$$^{c}$ , L. Lusito$^{a}$$^{, }$$^{b}$ , G. Maggi$^{a}$$^{, }$$^{c}$ , M. Maggi$^{a}$ , B. Marangelli$^{a}$$^{, }$$^{b}$ , S. My$^{a}$$^{, }$$^{c}$ , S. Nuzzo$^{a}$$^{, }$$^{b}$ , N. Pacifico$^{a}$$^{, }$$^{b}$ , A. Pompili$^{a}$$^{, }$$^{b}$ , G. Pugliese$^{a}$$^{, }$$^{c}$ , G. Selvaggi$^{a}$$^{, }$$^{b}$ , L. Silvestris$^{a}$ , G. Singh$^{a}$$^{, }$$^{b}$ , G. Zito$^{a}$ INFN Sezione di Bologna $^{a}$ , Università di Bologna $^{b}$ , Bologna, Italy G. Abbiendi$^{a}$ , A.C. Benvenuti$^{a}$ , D. Bonacorsi$^{a}$$^{, }$$^{b}$ , S. Braibant-Giacomelli$^{a}$$^{, }$$^{b}$ , L. Brigliadori$^{a}$$^{, }$$^{b}$ , P. Capiluppi$^{a}$$^{, }$$^{b}$ , A. Castro$^{a}$$^{, }$$^{b}$ , F.R.",
"Cavallo$^{a}$ , M. Cuffiani$^{a}$$^{, }$$^{b}$ , G.M.",
"Dallavalle$^{a}$ , F. Fabbri$^{a}$ , A. Fanfani$^{a}$$^{, }$$^{b}$ , D. Fasanella$^{a}$$^{, }$$^{b}$$^{, }$ 1, P. Giacomelli$^{a}$ , C. Grandi$^{a}$ , L. Guiducci, S. Marcellini$^{a}$ , G. Masetti$^{a}$ , M. Meneghelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Montanari$^{a}$ , F.L.",
"Navarria$^{a}$$^{, }$$^{b}$ , F. Odorici$^{a}$ , A. Perrotta$^{a}$ , F. Primavera$^{a}$$^{, }$$^{b}$ , A.M. Rossi$^{a}$$^{, }$$^{b}$ , T. Rovelli$^{a}$$^{, }$$^{b}$ , G. Siroli$^{a}$$^{, }$$^{b}$ , R. Travaglini$^{a}$$^{, }$$^{b}$ INFN Sezione di Catania $^{a}$ , Università di Catania $^{b}$ , Catania, Italy S. Albergo$^{a}$$^{, }$$^{b}$ , G. Cappello$^{a}$$^{, }$$^{b}$ , M. Chiorboli$^{a}$$^{, }$$^{b}$ , S. Costa$^{a}$$^{, }$$^{b}$ , R. Potenza$^{a}$$^{, }$$^{b}$ , A. Tricomi$^{a}$$^{, }$$^{b}$ , C. Tuve$^{a}$$^{, }$$^{b}$ INFN Sezione di Firenze $^{a}$ , Università di Firenze $^{b}$ , Firenze, Italy G. Barbagli$^{a}$ , V. Ciulli$^{a}$$^{, }$$^{b}$ , C. Civinini$^{a}$ , R. D'Alessandro$^{a}$$^{, }$$^{b}$ , E. Focardi$^{a}$$^{, }$$^{b}$ , S. Frosali$^{a}$$^{, }$$^{b}$ , E. Gallo$^{a}$ , S. Gonzi$^{a}$$^{, }$$^{b}$ , M. Meschini$^{a}$ , S. Paoletti$^{a}$ , G. Sguazzoni$^{a}$ , A. Tropiano$^{a}$$^{, }$ 1 INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, S. Colafranceschi25, F. Fabbri, D. Piccolo INFN Sezione di Genova, Genova, Italy P. Fabbricatore, R. Musenich INFN Sezione di Milano-Bicocca $^{a}$ , Università di Milano-Bicocca $^{b}$ , Milano, Italy A. Benaglia$^{a}$$^{, }$$^{b}$$^{, }$ 1, F. De Guio$^{a}$$^{, }$$^{b}$ , L. Di Matteo$^{a}$$^{, }$$^{b}$$^{, }$ 1, S. Fiorendi$^{a}$$^{, }$$^{b}$ , S. Gennai$^{a}$$^{, }$ 1, A. Ghezzi$^{a}$$^{, }$$^{b}$ , S. Malvezzi$^{a}$ , R.A. Manzoni$^{a}$$^{, }$$^{b}$ , A. Martelli$^{a}$$^{, }$$^{b}$ , A. Massironi$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Menasce$^{a}$ , L. Moroni$^{a}$ , M. Paganoni$^{a}$$^{, }$$^{b}$ , D. Pedrini$^{a}$ , S. Ragazzi$^{a}$$^{, }$$^{b}$ , N. Redaelli$^{a}$ , S. Sala$^{a}$ , T. Tabarelli de Fatis$^{a}$$^{, }$$^{b}$ INFN Sezione di Napoli $^{a}$ , Università di Napoli \"Federico II\" $^{b}$ , Napoli, Italy S. Buontempo$^{a}$ , C.A.",
"Carrillo Montoya$^{a}$$^{, }$ 1, N. Cavallo$^{a}$$^{, }$ 26, A.",
"De Cosa$^{a}$$^{, }$$^{b}$ , O. Dogangun$^{a}$$^{, }$$^{b}$ , F. Fabozzi$^{a}$$^{, }$ 26, A.O.M.",
"Iorio$^{a}$$^{, }$ 1, L. Lista$^{a}$ , S. Meola$^{a}$$^{, }$ 27, M. Merola$^{a}$$^{, }$$^{b}$ , P. Paolucci$^{a}$ INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Università di Trento (Trento) $^{c}$ , Padova, Italy P. Azzi$^{a}$ , N. Bacchetta$^{a}$$^{, }$ 1, P. Bellan$^{a}$$^{, }$$^{b}$ , M. Biasotto$^{a}$$^{, }$ 28, D. Bisello$^{a}$$^{, }$$^{b}$ , A. Branca$^{a}$$^{, }$ 1, P. Checchia$^{a}$ , T. Dorigo$^{a}$ , U. Dosselli$^{a}$ , F. Gasparini$^{a}$$^{, }$$^{b}$ , A. Gozzelino$^{a}$ , M. Gulmini$^{a}$$^{, }$ 28, K. Kanishchev$^{a}$$^{, }$$^{c}$ , S. Lacaprara$^{a}$ , I. Lazzizzera$^{a}$$^{, }$$^{c}$ , M. Margoni$^{a}$$^{, }$$^{b}$ , G. Maron$^{a}$$^{, }$ 28, A.T. Meneguzzo$^{a}$$^{, }$$^{b}$ , L. Perrozzi$^{a}$ , N. Pozzobon$^{a}$$^{, }$$^{b}$ , P. Ronchese$^{a}$$^{, }$$^{b}$ , F. Simonetto$^{a}$$^{, }$$^{b}$ , E. Torassa$^{a}$ , M. Tosi$^{a}$$^{, }$$^{b}$$^{, }$ 1, S. Vanini$^{a}$$^{, }$$^{b}$ INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ , Pavia, Italy M. Gabusi$^{a}$$^{, }$$^{b}$ , S.P.",
"Ratti$^{a}$$^{, }$$^{b}$ , C. Riccardi$^{a}$$^{, }$$^{b}$ , P. Torre$^{a}$$^{, }$$^{b}$ , P. Vitulo$^{a}$$^{, }$$^{b}$ INFN Sezione di Perugia $^{a}$ , Università di Perugia $^{b}$ , Perugia, Italy G.M.",
"Bilei$^{a}$ , L. Fanò$^{a}$$^{, }$$^{b}$ , P. Lariccia$^{a}$$^{, }$$^{b}$ , A. Lucaroni$^{a}$$^{, }$$^{b}$$^{, }$ 1, G. Mantovani$^{a}$$^{, }$$^{b}$ , M. Menichelli$^{a}$ , A. Nappi$^{a}$$^{, }$$^{b}$ , F. Romeo$^{a}$$^{, }$$^{b}$ , A. Saha, A. Santocchia$^{a}$$^{, }$$^{b}$ , S. Taroni$^{a}$$^{, }$$^{b}$$^{, }$ 1 INFN Sezione di Pisa $^{a}$ , Università di Pisa $^{b}$ , Scuola Normale Superiore di Pisa $^{c}$ , Pisa, Italy P. Azzurri$^{a}$$^{, }$$^{c}$ , G. Bagliesi$^{a}$ , T. Boccali$^{a}$ , G. Broccolo$^{a}$$^{, }$$^{c}$ , R. Castaldi$^{a}$ , R.T. D'Agnolo$^{a}$$^{, }$$^{c}$ , R. Dell'Orso$^{a}$ , F. Fiori$^{a}$$^{, }$$^{b}$$^{, }$ 1, L. Foà$^{a}$$^{, }$$^{c}$ , A. Giassi$^{a}$ , A. Kraan$^{a}$ , F. Ligabue$^{a}$$^{, }$$^{c}$ , T. Lomtadze$^{a}$ , L. Martini$^{a}$$^{, }$ 29, A. Messineo$^{a}$$^{, }$$^{b}$ , F. Palla$^{a}$ , F. Palmonari$^{a}$ , A. Rizzi$^{a}$$^{, }$$^{b}$ , A.T. Serban$^{a}$$^{, }$ 30, P. Spagnolo$^{a}$ , P. Squillacioti1, R. Tenchini$^{a}$ , G. Tonelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Venturi$^{a}$$^{, }$ 1, P.G.",
"Verdini$^{a}$ INFN Sezione di Roma $^{a}$ , Università di Roma \"La Sapienza\" $^{b}$ , Roma, Italy L. Barone$^{a}$$^{, }$$^{b}$ , F. Cavallari$^{a}$ , D. Del Re$^{a}$$^{, }$$^{b}$$^{, }$ 1, M. Diemoz$^{a}$ , C. Fanelli$^{a}$$^{, }$$^{b}$ , M. Grassi$^{a}$$^{, }$ 1, E. Longo$^{a}$$^{, }$$^{b}$ , P. Meridiani$^{a}$$^{, }$ 1, F. Micheli$^{a}$$^{, }$$^{b}$ , S. Nourbakhsh$^{a}$ , G. Organtini$^{a}$$^{, }$$^{b}$ , F. Pandolfi$^{a}$$^{, }$$^{b}$ , R. Paramatti$^{a}$ , S. Rahatlou$^{a}$$^{, }$$^{b}$ , M. Sigamani$^{a}$ , L. Soffi$^{a}$$^{, }$$^{b}$ INFN Sezione di Torino $^{a}$ , Università di Torino $^{b}$ , Università del Piemonte Orientale (Novara) $^{c}$ , Torino, Italy N. Amapane$^{a}$$^{, }$$^{b}$ , R. Arcidiacono$^{a}$$^{, }$$^{c}$ , S. Argiro$^{a}$$^{, }$$^{b}$ , M. Arneodo$^{a}$$^{, }$$^{c}$ , C. Biino$^{a}$ , C. Botta$^{a}$$^{, }$$^{b}$ , N. Cartiglia$^{a}$ , R. Castello$^{a}$$^{, }$$^{b}$ , M. Costa$^{a}$$^{, }$$^{b}$ , N. Demaria$^{a}$ , A. Graziano$^{a}$$^{, }$$^{b}$ , C. Mariotti$^{a}$$^{, }$ 1, S. Maselli$^{a}$ , E. Migliore$^{a}$$^{, }$$^{b}$ , V. Monaco$^{a}$$^{, }$$^{b}$ , M. Musich$^{a}$$^{, }$ 1, M.M.",
"Obertino$^{a}$$^{, }$$^{c}$ , N. Pastrone$^{a}$ , M. Pelliccioni$^{a}$ , A. Potenza$^{a}$$^{, }$$^{b}$ , A. Romero$^{a}$$^{, }$$^{b}$ , M. Ruspa$^{a}$$^{, }$$^{c}$ , R. Sacchi$^{a}$$^{, }$$^{b}$ , V. Sola$^{a}$$^{, }$$^{b}$ , A. Solano$^{a}$$^{, }$$^{b}$ , A. Staiano$^{a}$ , A. Vilela Pereira$^{a}$ INFN Sezione di Trieste $^{a}$ , Università di Trieste $^{b}$ , Trieste, Italy S. Belforte$^{a}$ , F. Cossutti$^{a}$ , G. Della Ricca$^{a}$$^{, }$$^{b}$ , B. Gobbo$^{a}$ , M. Marone$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Montanino$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Penzo$^{a}$ , A. Schizzi$^{a}$$^{, }$$^{b}$ Kangwon National University, Chunchon, Korea S.G. Heo, T.Y.",
"Kim, S.K.",
"Nam Kyungpook National University, Daegu, Korea S. Chang, J. Chung, D.H. Kim, G.N.",
"Kim, D.J.",
"Kong, H. Park, S.R.",
"Ro, D.C.",
"Son, T. Son Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea J.Y.",
"Kim, Zero J. Kim, S. Song Konkuk University, Seoul, Korea H.Y.",
"Jo Korea University, Seoul, Korea S. Choi, D. Gyun, B. Hong, M. Jo, H. Kim, T.J. Kim, K.S.",
"Lee, D.H.",
"Moon, S.K.",
"Park, E. Seo University of Seoul, Seoul, Korea M. Choi, S. Kang, H. Kim, J.H.",
"Kim, C. Park, I.C.",
"Park, S. Park, G. Ryu Sungkyunkwan University, Suwon, Korea Y. Cho, Y. Choi, Y.K.",
"Choi, J. Goh, M.S.",
"Kim, E. Kwon, B. Lee, J. Lee, S. Lee, H. Seo, I. Yu Vilnius University, Vilnius, Lithuania M.J. Bilinskas, I. Grigelionis, M. Janulis, A. Juodagalvis Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-de La Cruz, R. Lopez-Fernandez, R. Magaña Villalba, J. Martínez-Ortega, A. Sánchez-Hernández, L.M.",
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"Zeuner Paul Scherrer Institut, Villigen, Switzerland W. Bertl, K. Deiters, W. Erdmann, K. Gabathuler, R. Horisberger, Q. Ingram, H.C. Kaestli, S. König, D. Kotlinski, U. Langenegger, F. Meier, D. Renker, T. Rohe, J. Sibille38 Institute for Particle Physics, ETH Zurich, Zurich, Switzerland L. Bäni, P. Bortignon, M.A.",
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"Ramirez Vargas, A. Zatserklyaniy Purdue University, West Lafayette, USA E. Alagoz, V.E.",
"Barnes, D. Benedetti, G. Bolla, D. Bortoletto, M. De Mattia, A. Everett, Z. Hu, M. Jones, O. Koybasi, M. Kress, A.T. Laasanen, N. Leonardo, V. Maroussov, P. Merkel, D.H. Miller, N. Neumeister, I. Shipsey, D. Silvers, A. Svyatkovskiy, M. Vidal Marono, H.D.",
"Yoo, J. Zablocki, Y. Zheng Purdue University Calumet, Hammond, USA S. Guragain, N. Parashar Rice University, Houston, USA A. Adair, C. Boulahouache, V. Cuplov, K.M.",
"Ecklund, F.J.M.",
"Geurts, B.P.",
"Padley, R. Redjimi, J. Roberts, J. Zabel University of Rochester, Rochester, USA B. Betchart, A. Bodek, Y.S.",
"Chung, R. Covarelli, P. de Barbaro, R. Demina, Y. Eshaq, A. Garcia-Bellido, P. Goldenzweig, Y. Gotra, J. Han, A. Harel, S. Korjenevski, D.C.",
"Miner, D. Vishnevskiy, M. Zielinski The Rockefeller University, New York, USA A. Bhatti, R. Ciesielski, L. Demortier, K. Goulianos, G. Lungu, S. Malik, C. Mesropian Rutgers, the State University of New Jersey, Piscataway, USA S. Arora, A. Barker, J.P. Chou, C. Contreras-Campana, E. Contreras-Campana, D. Duggan, D. Ferencek, Y. Gershtein, R. Gray, E. Halkiadakis, D. Hidas, D. Hits, A. Lath, S. Panwalkar, M. Park, R. Patel, V. Rekovic, A. Richards, J. Robles, K. Rose, S. Salur, S. Schnetzer, C. Seitz, S. Somalwar, R. Stone, S. Thomas University of Tennessee, Knoxville, USA G. Cerizza, M. Hollingsworth, S. Spanier, Z.C.",
"Yang, A. York Texas A&M University, College Station, USA R. Eusebi, W. Flanagan, J. Gilmore, T. Kamon56, V. Khotilovich, R. Montalvo, I. Osipenkov, Y. Pakhotin, A. Perloff, J. Roe, A. Safonov, T. Sakuma, S. Sengupta, I. Suarez, A. Tatarinov, D. Toback Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, P.R.",
"Dudero, C. Jeong, K. Kovitanggoon, S.W.",
"Lee, T. Libeiro, Y. Roh, I. Volobouev Vanderbilt University, Nashville, USA E. Appelt, D. Engh, C. Florez, S. Greene, A. Gurrola, W. Johns, P. Kurt, C. Maguire, A. Melo, P. Sheldon, B. Snook, S. Tuo, J. Velkovska University of Virginia, Charlottesville, USA M.W.",
"Arenton, M. Balazs, S. Boutle, B. Cox, B. Francis, J. Goodell, R. Hirosky, A. Ledovskoy, C. Lin, C. Neu, J.",
"Wood, R. Yohay Wayne State University, Detroit, USA S. Gollapinni, R. Harr, P.E.",
"Karchin, C. Kottachchi Kankanamge Don, P. Lamichhane, A. Sakharov University of Wisconsin, Madison, USA M. Anderson, M. Bachtis, D. Belknap, L. Borrello, D. Carlsmith, M. Cepeda, S. Dasu, L. Gray, K.S.",
"Grogg, M. Grothe, R. Hall-Wilton, M. Herndon, A. Hervé, P. Klabbers, J. Klukas, A. Lanaro, C. Lazaridis, J. Leonard, R. Loveless, A. Mohapatra, I. Ojalvo, G.A.",
"Pierro, I. Ross, A. Savin, W.H.",
"Smith, J. Swanson †: Deceased 1: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 2: Also at National Institute of Chemical Physics and Biophysics, Tallinn, Estonia 3: Also at Universidade Federal do ABC, Santo Andre, Brazil 4: Also at California Institute of Technology, Pasadena, USA 5: Also at Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France 6: Also at Suez Canal University, Suez, Egypt 7: Also at Cairo University, Cairo, Egypt 8: Also at British University, Cairo, Egypt 9: Also at Fayoum University, El-Fayoum, Egypt 10: Now at Ain Shams University, Cairo, Egypt 11: Also at Soltan Institute for Nuclear Studies, Warsaw, Poland 12: Also at Université de Haute-Alsace, Mulhouse, France 13: Now at Joint Institute for Nuclear Research, Dubna, Russia 14: Also at Moscow State University, Moscow, Russia 15: Also at Brandenburg University of Technology, Cottbus, Germany 16: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 17: Also at Eötvös Loránd University, Budapest, Hungary 18: Also at Tata Institute of Fundamental Research - HECR, Mumbai, India 19: Now at King Abdulaziz University, Jeddah, Saudi Arabia 20: Also at University of Visva-Bharati, Santiniketan, India 21: Also at Sharif University of Technology, Tehran, Iran 22: Also at Isfahan University of Technology, Isfahan, Iran 23: Also at Shiraz University, Shiraz, Iran 24: Also at Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Teheran, Iran 25: Also at Facoltà Ingegneria Università di Roma, Roma, Italy 26: Also at Università della Basilicata, Potenza, Italy 27: Also at Università degli Studi Guglielmo Marconi, Roma, Italy 28: Also at Laboratori Nazionali di Legnaro dell' INFN, Legnaro, Italy 29: Also at Università degli studi di Siena, Siena, Italy 30: Also at University of Bucharest, Bucuresti-Magurele, Romania 31: Also at Faculty of Physics of University of Belgrade, Belgrade, Serbia 32: Also at University of Florida, Gainesville, USA 33: Also at University of California, Los Angeles, Los Angeles, USA 34: Also at Scuola Normale e Sezione dell' INFN, Pisa, Italy 35: Also at INFN Sezione di Roma; Università di Roma \"La Sapienza\", Roma, Italy 36: Also at University of Athens, Athens, Greece 37: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 38: Also at The University of Kansas, Lawrence, USA 39: Also at Paul Scherrer Institut, Villigen, Switzerland 40: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia 41: Also at Gaziosmanpasa University, Tokat, Turkey 42: Also at Adiyaman University, Adiyaman, Turkey 43: Also at The University of Iowa, Iowa City, USA 44: Also at Mersin University, Mersin, Turkey 45: Also at Kafkas University, Kars, Turkey 46: Also at Suleyman Demirel University, Isparta, Turkey 47: Also at Ege University, Izmir, Turkey 48: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 49: Also at INFN Sezione di Perugia; Università di Perugia, Perugia, Italy 50: Also at University of Sydney, Sydney, Australia 51: Also at Utah Valley University, Orem, USA 52: Also at Institute for Nuclear Research, Moscow, Russia 53: Also at University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia 54: Also at Argonne National Laboratory, Argonne, USA 55: Also at Erzincan University, Erzincan, Turkey 56: Also at Kyungpook National University, Daegu, Korea"
]
] | 1204.1409 |
[
[
"Majorana-Like Modes of Light in a One-Dimensional Array of Nonlinear\n Cavities"
],
[
"Abstract The search for Majorana fermions in p-wave paired fermionic systems has recently moved to the forefront of condensed-matter research.",
"Here we propose an alternative route and show theoretically that Majorana-like modes can be realized and probed in a driven-dissipative system of strongly correlated photons consisting of a chain of tunnel-coupled cavities, where p-wave pairing effectively arises from the interplay between strong on-site interactions and two-photon parametric driving.",
"The nonlocal nature of these exotic modes could be demonstrated through cross-correlation measurements carried out at the ends of the chain---revealing a strong photon bunching signature---and their non-Abelian properties could be simulated through tunnel-braid operations."
],
[
"$P$ -wave pairing of strongly correlated photons",
"In this section, we demonstrate explicitly that $p$ -wave pairing generally emerges as a result of the interplay between two-photon parametric pumping and strong on-site photon-photon repulsion, and derive the effective drive Hamiltonian of Eq.",
"(2) in the main text.",
"Although multiple schemes can be envisioned to realize the key nonlinear pumping process that takes photons from a pump field and injects coherent photon pairs into the system, we will illustrate the general mechanism that leads to $p$ -wave pairing in a model based on a theoretical description of coupled cavity arrays that was developed in Ref. [1].",
"We start by assuming that the parametric pumps act locally in between the nearest-neighboring cavities corresponding to the sites $i$ and $i+1$ according to the standard Hamiltonian [2] $ H_{\\text{drive}, i} & = \\int \\mathrm {d}^3 r \\, \\chi ^{(2)} (\\mathbf {r}) E^{(+)}_{p, i} (\\mathbf {r}, t) E^{(-)}_{s, i} (\\mathbf {r}) E^{(-)}_{s, i} (\\mathbf {r}) + h.c.,$ where $E^{(+)}_{p, i} (\\mathbf {r}, t)$ is the positive-frequency part of the pump (index “p”) optical field, $E^{(-)}_{s, i} (\\mathbf {r})$ the negative-frequency part of the generated signal (index “s”) optical fields, and $\\chi ^{(2)} (\\mathbf {r})$ the effective second-order optical nonlinearity of the system (taking into account the polarization vectors of the three interacting fields).",
"We assume that the pump field can be treated as a classical monochromatic field of frequency $2 \\omega _{p, i}$ and complex amplitude $E^0_{p, i}$ , and that there is no significant depletion of the latter such that $E^0_{p, i}$ can be considered as constant.",
"Denoting the spatial mode function of the pump as $\\varphi _{p, i} (\\mathbf {r})$ , $E^{(+)}_{p, i} (\\mathbf {r}, t)$ then takes the form $ E^{(+)}_{p, i} (\\mathbf {r}, t) & = E^0_{p, i} \\varphi _{p, i} (\\mathbf {r}) e^{-2 \\mathrm {i} \\omega _p t}.$ The generated signal optical fields, on the other hand, can generally be expanded in terms of the Wannier modes $\\phi _i (\\mathbf {r})$ of the coupled cavity array.",
"Assuming that these modes decay sufficiently fast so that only nearest-neighboring modes have a non-vanishing overlap, we can write $ E^{(-)}_{s, i} (\\mathbf {r}) & = -\\mathrm {i} \\, \\sqrt{\\omega _c} \\left( \\phi ^*_i (\\mathbf {r}) b^\\dagger _i + \\phi ^*_{i+1} (\\mathbf {r}) b^\\dagger _{i+1} \\right),$ where $\\omega _c$ is the resonance frequency of the cavity (assumed to be the same for all cavities).",
"Plugging Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ), we thus obtain $H_{\\text{drive}, i} & = -E^0_{p, i} \\, \\omega _c \\int \\mathrm {d}^3 r \\, \\chi ^{(2)} (\\mathbf {r}) \\varphi _{p, i} (\\mathbf {r}) \\left( \\phi ^{* 2}_i (\\mathbf {r}) b^{\\dagger 2}_i + \\phi ^*_i (\\mathbf {r}) \\phi ^*_{i+1} (\\mathbf {r}) b^\\dagger _i b^\\dagger _{i+1} + \\phi ^{* 2}_{i+1} (\\mathbf {r}) b^{\\dagger 2}_{i+1} \\right) e^{-2 \\mathrm {i} \\omega _p t} + h.c. \\nonumber \\\\& \\approx -E^0_{p, i} \\, \\omega _c \\left( \\int \\mathrm {d}^3 r \\, \\chi ^{(2)} (\\mathbf {r}) \\varphi _{p, i} (\\mathbf {r}) \\phi ^*_i (\\mathbf {r}) \\phi ^*_{i+1} (\\mathbf {r}) \\right) b^\\dagger _i b^\\dagger _{i+1} e^{-2 \\mathrm {i} \\omega _p t} + h.c.,$ where we have invoked the presumably strong on-site repulsion between photons to neglect all terms of the form $b^{\\dagger 2}_i$ .",
"The strength of the degenerate parametric pumping therefore clearly appears to depend on the overlap of the interacting modes, as expected.",
"Assuming, for simplicity, that the pump field and the effective second-order optical nonlinearity are essentially constant over the volume $V_i$ between the cavities, and that the pump field vanishes outside this volume, we finally obtain $H_{\\text{drive}, i} & \\approx \\kappa _i \\left( \\int _{V_i} \\mathrm {d}^3 r \\, \\phi ^*_i (\\mathbf {r}) \\phi ^*_{i+1} (\\mathbf {r}) \\right) b^\\dagger _i b^\\dagger _{i+1} e^{-2 \\mathrm {i} \\omega _p t} + h.c. \\nonumber \\\\& \\equiv - \\Delta ^*_i \\, b^\\dagger _i b^\\dagger _{i+1} e^{-2 \\mathrm {i} \\omega _p t} + h.c.,$ where the complex coupling constant $\\kappa _i$ essentially depends (linearly) on $\\chi ^{(2)}$ and on the amplitude of the pump field $E^0_{p, i}$ .",
"This last result shows that the strength of the two-photon parametric drive (i.e., the $p$ -wave pairing strength) directly depends on the overlap of the Wannier modes corresponding to nearest-neighboring cavities, as argued in the main text."
],
[
"Steady-state second-order photon cross-correlations",
"In what follows, we give a brief derivation of Eq.",
"(11) from the main text.",
"In accordance with the discussion presented there, we start with the Lindblad master equation $ \\partial _t \\rho & = - \\mathrm {i} \\left[ H_{\\text{eff}}, \\rho \\right] + \\sum _{i=L,R} \\left( L_i \\rho \\, L^\\dagger _i - \\tfrac{1}{2} \\lbrace L^\\dagger _i L_i, \\rho \\rbrace \\right),$ where the Hamiltonian and the Lindblad operators are respectively given by $H_{\\text{eff}} & = \\delta _M \\sigma ^{z}_M - \\sqrt{2} \\tilde{J} (\\sigma ^{x}_L \\sigma ^{x}_M + \\sigma ^{x}_M \\sigma ^{x}_R), \\\\L_i & = \\sqrt{\\tfrac{\\tilde{\\Gamma }}{2}} \\sigma ^{-}_i.$ Regarding the system as a chain of 3 interacting spins and relabeling the sites as $L \\rightarrow 1$ , $M \\rightarrow 2$ , $R \\rightarrow 3$ , one can perform a Jordan-Wigner transformation and describe the problem in a basis of Majorana operators defined as ${3pt}\\begin{array}{ll}c_{2i-1} & = (\\prod _{j=1}^{i-1} \\sigma ^{z}_j) \\sigma ^{x}_i, \\\\c_{2i} & = (\\prod _{j=1}^{i-1} \\sigma ^{z}_j) \\sigma ^{y}_i.\\end{array}$ In this fermionic description, the Hamiltonian and the Lindblad operators take the form $H_{\\text{eff}} & = - \\mathrm {i} \\delta _M c_3 c_4 - \\mathrm {i} \\sqrt{2} \\tilde{J} (c_2 c_3 + c_4 c_5), \\\\L_1 & = \\sqrt{\\tfrac{\\tilde{\\Gamma }}{2}} (c_1 - \\mathrm {i} c_2), \\\\L_3 & = \\sqrt{\\tfrac{\\tilde{\\Gamma }}{2}} (c_5 - \\mathrm {i} c_6),$ and can conveniently be expressed as $H_{\\text{eff}} = \\tfrac{\\mathrm {i}}{2} \\mathbf {c}^T H \\mathbf {c}$ and $L_i = \\mathbf {l}_{i}^T \\mathbf {c}$ , where $\\mathbf {c}^T = (c_1, c_2, \\ldots , c_{6})$ .",
"Note that we have dropped the string-like operator of the form $\\prod _{j=1}^{3} \\sigma ^{z}_j$ that would normally appear in $L_3$ , since the latter does not affect Eq.",
"(REF ) (see, e.g., Ref. [3]).",
"The fact that Eq.",
"(REF ) is quadratic allows for a great simplification of the problem.",
"Assuming, without loss of generality, that the initial state of the system has a Gaussian form, one can focus exclusively on the time evolution of the correlation matrix $C_{i, j} = (\\mathrm {i} / 2) \\operatorname{tr} ([ c_i, c_j ] \\rho )$ which contains all information about the system.",
"As shown in Ref.",
"[4], the master equation of Eq.",
"(REF ) then reduces to a matrix equation of the form $\\partial _t C = X^T C + C X - Y,$ where $X = - 2 \\mathrm {i} H + 2 \\operatorname{Re} M$ and $Y = 4 \\operatorname{Im} M$ , with $M = \\sum _i \\mathbf {l}_{i} \\otimes \\mathbf {l}^\\dagger _i$ .",
"Here we find $X & =\\left(\\begin{array}{cccccc}\\tilde{\\Gamma } & 0 & 0 & 0 & 0 & 0 \\\\0 & \\tilde{\\Gamma } & 2 \\sqrt{2} \\tilde{J} & 0 & 0 & 0 \\\\0 & -2 \\sqrt{2} \\tilde{J} & 0 & -\\delta _M & 0 & 0 \\\\0 & 0 & \\delta _M & 0 & 2 \\sqrt{2} \\tilde{J} & 0 \\\\0 & 0 & 0 & -2 \\sqrt{2} \\tilde{J} & \\tilde{\\Gamma } & 0 \\\\0 & 0 & 0 & 0 & 0 & \\tilde{\\Gamma } \\\\\\end{array}\\right), \\quad Y & =\\left(\\begin{array}{cccccc}0 & 2 \\tilde{\\Gamma } & 0 & 0 & 0 & 0 \\\\-2 \\tilde{\\Gamma } & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 2 \\tilde{\\Gamma } \\\\0 & 0 & 0 & 0 & -2 \\tilde{\\Gamma } & 0 \\\\\\end{array}\\right),$ and the steady-state correlation matrix satisfies $ X^T C + C X = Y.$ In order to solve the above matrix equation, we follow the method of Ref.",
"[3] and make the ansatz $X = \\sum _{j = 1}^{6} \\alpha _{j, k} (\\mathbf {w}^T_j \\otimes \\mathbf {w}_k),$ where $\\mathbf {v}_j$ and $\\mathbf {w}_j$ are left and right eigenvectors of $X$ , respectively.",
"Introducing this ansatz into Eq.",
"(REF ) and assuming that $\\mathbf {v}_j$ , $\\mathbf {w}_j$ are normalized according to $\\mathbf {v}^\\dagger _j \\mathbf {v}_k = \\delta _{j, k}$ , we find $\\alpha _{j, k} = \\frac{1}{\\lambda _j + \\lambda _k} (\\mathbf {v}^\\dagger _j Y \\mathbf {v}^*_k),$ where $\\lambda _j$ are the eigenvalues corresponding to the right (or left) eigenvectors of $X$ .",
"Therefore, solving for the steady-state correlation matrix basically amounts to solving two eigenvalue problems: $X \\mathbf {w}_j = \\lambda _j \\mathbf {w}_j$ and $\\mathbf {v}_j X = \\lambda _j \\mathbf {v}_j$ .",
"Using the above explicit form of $X$ and $Y$ , straightforward algebra leads to $C & =\\left(\\begin{array}{cccccc}0 & \\frac{\\gamma ^2 \\left(\\gamma ^2+t^2+1\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} &\\frac{\\sqrt{2} \\gamma t \\left(\\gamma ^2+t^2\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} &-\\frac{\\sqrt{2} \\gamma ^2 t}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & -\\frac{\\gamma t^2}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 \\\\-\\frac{\\gamma ^2 \\left(\\gamma ^2+t^2+1\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 & 0 & 0 & 0& \\frac{\\gamma t^2}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} \\\\-\\frac{\\sqrt{2} \\gamma t \\left(\\gamma ^2+t^2\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 & 0 &0 & 0 & -\\frac{\\sqrt{2} \\gamma ^2 t}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} \\\\\\frac{\\sqrt{2} \\gamma ^2 t}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 & 0 & 0 & 0 & -\\frac{\\sqrt{2}\\gamma t \\left(\\gamma ^2+t^2\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} \\\\\\frac{\\gamma t^2}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 & 0 & 0 & 0 & \\frac{\\gamma ^2\\left(\\gamma ^2+t^2+1\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} \\\\0 & -\\frac{\\gamma t^2}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & \\frac{\\sqrt{2} \\gamma ^2 t}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & \\frac{\\sqrt{2} \\gamma t \\left(\\gamma ^2+t^2\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & -\\frac{\\gamma ^2 \\left(\\gamma ^2+t^2+1\\right)}{\\gamma ^2+\\left(\\gamma ^2+t^2\\right)^2} & 0 \\\\\\end{array}\\right),$ where we have defined the dimensionless quantities $t = \\tilde{J} / \\delta _M$ , $\\gamma = \\tilde{\\Gamma } / \\delta _M$ .",
"The steady-state second-order photon cross-correlations between the left and right probes then read $g^{(2)}_{LR} & = 1 + \\frac{\\langle \\sigma ^{z}_L \\sigma ^{z}_R\\rangle - \\langle \\sigma ^{z}_L\\rangle \\langle \\sigma ^{z}_R\\rangle }{(1 + \\langle \\sigma ^{z}_L\\rangle ) (1 + \\langle \\sigma ^{z}_R\\rangle )} \\nonumber \\\\& = 1 + \\frac{\\langle c_1 c_5\\rangle \\langle c_2 c_6\\rangle - \\langle c_1 c_6\\rangle \\langle c_2 c_5\\rangle }{(1 - \\mathrm {i} \\langle c_1 c_2\\rangle ) (1 -\\mathrm {i} \\langle c_5 c_6\\rangle )} \\nonumber \\\\& = 1 - \\frac{C_{1, 5} C_{2, 6} - C_{1, 6} C_{2, 5}}{(1 - C_{1, 2}) (1 - C_{5, 6})} \\nonumber \\\\& = 1 + \\frac{\\gamma ^2}{(t^2 + \\gamma ^2)^2},$ where we have used Wick's theorem in the second equality.",
"Similarly, the steady-state occupation of the probe cavities is given by $1 + \\langle \\sigma ^{z}_{L,R}\\rangle & = \\frac{t^2 (t^2 + \\gamma ^2)}{\\gamma ^2 + (t^2 + \\gamma ^2)^2} \\nonumber \\\\& = \\frac{t^2}{t^2 + \\gamma ^2} \\frac{1}{g^{(2)}_{LR}},$ and scales as $1/g^{(2)}_{LR}$ when $\\tilde{J} \\sim \\tilde{\\Gamma }$ ."
],
[
"Implementation in circuit QED",
"Our proposal for accessing Majorana physics in the optical context requires two crucial ingredients: strong coupling and (comparatively) weak photon losses, both of which are currently available in superconducting-circuit-based cavity quantum electrodynamics (circuit QED) (see Ref.",
"[5], e.g., for a recent review).",
"Superconducting qubits thus stand apart as an ideal platform for realizing our proposal—even more so because of the great flexibility in fabrication and control provided by state-of-the-art technologies.",
"We outline below a potential implementation of our proposal which we deem as closest to experimental realization, and give ballpark figures for the relevant parameters and energy scales.",
"The system that we typically envision consists of a chain of capacitively-coupled identical microwave resonators (playing the role of cavities), each incorporating two superconducting transmon qubits (playing the role of artificial atoms).",
"The first of these qubits is positioned at an antinode of the intracavity field, with a cavity-qubit coupling strength $g \\simeq \\omega _b \\sqrt{\\alpha }$ , where $\\omega _b$ is the bare-cavity resonance frequency and $\\alpha $ the fine-structure constant; the strength of the effective on-site interaction—or, equivalently, of the anharmonicity—associated with the lower polariton mode (with frequency $\\omega _c = \\omega _b - g$ ) is then given by $U = (2 - \\sqrt{2}) g$ .",
"The second qubit is located in between pairs of neighboring cavities, and is driven parametrically using strong microwave fields at frequency $2 \\omega _p$ so as to ensure the generation or annihilation of photon pairs at frequency $\\omega _b - g$ .",
"In the limit where the amplitude of the parametric drive (or “pump”) is weak as compared to $g$ , the only energetically allowed process is the generation (annihilation) of photon pairs in neighboring cavities, as discussed in the main text and in the first section above.",
"This results in an effective $p$ -wave pairing interaction with a phase determined by that of the microwave drive fields.",
"We emphasize that $g \\sim 0.1 \\omega _b \\sim 10^4 \\Gamma $ , where $\\Gamma $ is the cavity decay rate (see main text), is routinely obtained in state-of-the-art circuit-QED devices.",
"This implies that the tunneling amplitude $J$ between neighboring cavities and the amplitude $\\vert \\Delta \\vert $ of the parametric pumps can in principle be tailored to satisfy the relation $U \\sim g \\gg J, \\vert \\Delta \\vert \\gg \\Gamma $ which lies at the core of our proposal.",
"The first inequality ($U \\sim g \\gg J, \\vert \\Delta \\vert $ ) ensures that the system lies deep in the strong-coupling regime where photons effectively behave as fermions (or “fermionized” photons), and constitutes a sufficient condition for our model to map to that of Kitaev (see main text).",
"If one additionally adjust the detuning $\\mu = \\omega _p - \\omega _c$ between the cavities and the pumps so that $\\vert \\mu \\vert < 2J$ , the system is found in a topologically nontrivial regime and exhibits Majorana-like modes with an energy splitting $\\delta _M$ that depends on $J$ , $\\vert \\Delta \\vert $ , and $\\vert \\mu \\vert $ , and vanishes exponentially with the system size $N$ (see the explicit form below).",
"The second inequality ($J, \\vert \\Delta \\vert \\gg \\Gamma $ ), in contrast, determines the broadening of the Majorana levels, i.e., how well they can be resolved for a fixed energy splitting $\\delta _M$ .",
"For Majorana-like modes to be detectable using the scheme presented in the main text, this broadening must crucially be (i) much smaller than the energy splitting $\\delta _M$ between the Majorana levels, and (ii) much smaller than the gap $E_g$ of the system, so that Majorana levels do not overlap in energy with levels of the bulk.",
"In general, these two conditions depend on the system parameters in a nontrivial way given by the explicit form of $\\delta _M$ and $E_g$ [6], namely, $\\delta _M \\sim e^{-N / \\xi }$ , where $\\xi $ is the localisation length of the Majorana-like modes given by $\\xi ^{-1} = \\min { \\lbrace \\vert \\ln {\\vert x_{+} \\vert } \\vert , \\vert \\ln {\\vert x_{-} \\vert } \\vert \\rbrace }$ with $x_{\\pm } = \\tfrac{-\\mu \\pm \\sqrt{\\mu ^2 - 4J^2 + 4\\vert \\Delta \\vert ^2}}{2(J + \\vert \\Delta \\vert )}$ , $E_g = 2J - \\vert \\mu \\vert $ if $\\vert \\Delta \\vert \\ge J$ or if $\\vert \\Delta \\vert < J$ and $2J - \\vert \\mu \\vert < 2 \\vert \\Delta \\vert ^2 / J$ , and $E_g = \\vert \\Delta \\vert (4 - \\vert \\mu \\vert ^2 / (J^2 - \\vert \\Delta \\vert ^2))^{1/2}$ otherwise (in the topologically nontrivial regime with $2J - \\vert \\mu \\vert > 0$ ).",
"Away from the limit $2J - \\vert \\mu \\vert \\rightarrow 0$ where the gap closes (and where the topological phase transition occurs, in the limit $N \\rightarrow \\infty $ ), $\\delta _M \\gg \\Gamma $ is generally more restrictive than $E_g \\gg \\Gamma $ .",
"As long as the above inequality $J, \\vert \\Delta \\vert \\gg \\Gamma $ is ensured, both of these conditions can always be satisfied by tuning $\\mu $ well inside the topologically nontrivial region, as shown in the numerical results presented in the main text.",
"$J, \\vert \\Delta \\vert \\gg \\Gamma $ is therefore a sufficient condition for our detection scheme to work, up to the tuning of $\\mu $.",
"This condition is also necessary, since $J, \\vert \\Delta \\vert \\sim \\Gamma $ would make the broadening comparable to the gap.",
"Although this is not a requirement for our proposal, we finally remark that one typically expects to be able to work in a regime where $J \\sim \\vert \\Delta \\vert $ , since both quantities are essentially determined by the overlap of the Wannier modes of neighboring cavities.",
"We also emphasize that the fabrication of a linear chain of $\\sim 10$ almost identical cavities is technologically feasible, as suggested by the recent realization of a Kagome lattice of more than 200 nearly identical cavities [7], [5].",
"Photon correlation measurements on microwave cavities have also been recently demonstrated [8]."
],
[
"Simulating real-space braiding in 1D using tunnel-braid operations",
"The ability to exchange—or “braid”—Majorana modes in real space plays an important role in demonstrating the non-Abelian nature of the latter.",
"In the framework of the optical proposal presented in the main text, however, one is fundamentally restricted to one-dimensional systems where exchanging Majorana modes is physically impossible (Majorana modes would overlap and split in energy during the exchange, thereby hybridizing into complex fermionic modes and losing their exotic nature).",
"It thus appears crucial to find another way to perform non-Abelian operations in the degenerate ground-state subspace associated with Majorana modes while preserving their degeneracy.",
"Such a possibility was originally introduced in Ref.",
"[9], where it was shown that so-called “tunnel-braid” operations can simulate the braiding of Majorana modes in real space.",
"In what follows, we translate these ideas into the framework of our optical proposal, and give an explicit procedure to simulate the exchange of two Majorana-like modes of light tunnel-coupled to the same ancillary cavity.",
"Figure: Basic one-dimensional setup required to simulate non-Abelian operations such as braiding.",
"Two Kitaev chains—left (L) and right (R)—with pump phases φ L \\phi _L and φ R \\phi _R, respectively, exhibit Majorana-like modes γ L \\gamma _L and γ R \\gamma _R at their ends, as depicted.",
"Each chain tunnel-couples with amplitude J L J_L (J R J_R) to an intermediate nonlinear cavity (I) via its end cavity where γ L \\gamma _L (γ R \\gamma _R) is mostly localized.To illustrate the working principles of tunnel-braid operations, we consider a simple scenario where two Kitaev chains—left (L) and right (R)—are end-to-end tunnel-coupled through a single intermediate cavity (I) [Fig.",
"REF ].",
"We assume that the parametric pumps driving both chains have phases $\\phi _L$ and $\\phi _R$ , respectively, and the same frequency $\\omega _p$ .",
"The intermediate cavity is nonlinear—though not driven—and resonant with $\\omega _p$ , and both chains are in a topologically non-trivial phase with Majorana-like zero modes $\\gamma _L$ and $\\gamma _R$ located on the left and on the right of the intermediate cavity, respectively.",
"Photon tunneling occurs between the intermediate cavity and the end cavities of the Kitaev chains, as captured by the Hamiltonian $H_{\\text{t}} & = - 2 J_L (b^\\dagger _L b_I + h.c.) - 2 J_R (b^\\dagger _I b_R + h.c.),$ where $b_L$ , $b_I$ and $b_R$ are annihilation operators associated with photons in the respective cavities (with obvious notations), while $J_L$ and $J_R$ are positive amplitudes for tunneling into the left and right chains, respectively (factors of 2 are introduced for later convenience).",
"Transforming to the fermionic picture defined in the main text, we obtain $ H_{\\text{t}} & = - 2 J_L (a^\\dagger _L a_I + h.c.) - 2 J_R (a^\\dagger _I a_R + h.c.),$ with fermionic operators $a_L$ and $a_R$ that can be decomposed as $\\begin{split}a_L & = \\tfrac{1}{2} e^{- \\mathrm {i} \\phi _L / 2} (\\gamma ^{\\prime }_L + \\mathrm {i} \\gamma _L), \\\\a_R & = \\tfrac{1}{2} e^{- \\mathrm {i} \\phi _R / 2} (\\gamma _R + \\mathrm {i} \\gamma ^{\\prime }_R),\\end{split}$ where $\\gamma ^{\\prime }_L$ and $\\gamma ^{\\prime }_R$ are Majorana operators whose explicit form is not relevant here.",
"Since we are only interested in the low-energy physics associated with the Majorana-like zero modes $\\gamma _L$ and $\\gamma _R$ , we can make in Eq.",
"(REF ) the following replacement: $\\begin{split}a_L & \\rightarrow \\tfrac{\\mathrm {i}}{2} e^{- \\mathrm {i} \\phi _L / 2} \\gamma _L, \\\\a_R & \\rightarrow \\tfrac{1}{2} e^{- \\mathrm {i} \\phi _R / 2} \\gamma _R,\\end{split}$ thus obtaining the low-energy effective tunnel Hamiltonian $\\begin{split}H_{\\text{t,eff}} & = J_L \\left( (- \\mathrm {i} e^{- \\mathrm {i} \\phi _L / 2}) a^\\dagger _I - (- \\mathrm {i} e^{- \\mathrm {i} \\phi _L / 2})^* a_I \\right) \\gamma _L \\\\& + J_R \\left( (- e^{- \\mathrm {i} \\phi _R / 2}) a^\\dagger _I - (- e^{- \\mathrm {i} \\phi _R / 2})^* a_I \\right) \\gamma _R.\\end{split}$ This can be recast in a more convenient form by performing the gauge transformation $a_I \\rightarrow (- \\mathrm {i} e^{- \\mathrm {i} \\phi _L / 2}) a_I$ , leading to $H_{\\text{t,eff}} & = J_L (a^\\dagger _I - a_I ) \\gamma _L + J_R (e^{- \\mathrm {i} (\\phi _R - \\phi _L + \\pi ) / 2} a^\\dagger _I - e^{\\mathrm {i} (\\phi _R - \\phi _L + \\pi ) / 2} a_I) \\gamma _R.$ In this form, $H_{\\text{t,eff}}$ clearly appears asymmetric with respect to the interchange $\\gamma _L \\leftrightarrow \\gamma _R$ if the phase difference $\\Delta \\phi \\equiv \\phi _R - \\phi _L$ does not satisfy $\\Delta \\phi = (4n - 1) \\pi $ for some integer $n$ .",
"Away from these fine-tuned values of $\\Delta \\phi $ , the tunnel Hamiltonian thus provides a means of distinguishing between the different states of the Majorana qubit composed of $\\gamma _L$ and $\\gamma _R$ , i.e.",
"between different parity sectors of the full system.",
"In order for this information to remain unaccessible to the system—thereby restoring what is usually referred to as the “topological protection” or “parity protection” of the Majorana qubit—it is therefore necessary to fine-tune the phase difference between the Majorana-like modes to some appropriate value (which can easily be done in our optical setting by adjusting the relative phase of the parametric pumps of each Kitaev chain; see main text).",
"Assuming that this is satisfied—so that the even- and odd-parity sectors of the full system remain degenerate—the low-energy effective tunnel Hamiltonian reduces to $H_{\\text{t,eff}} & = (a^\\dagger _I - a_I) (J_L \\gamma _L + J_R \\gamma _R) \\nonumber \\\\& = J_{LR} (a^\\dagger _I - a_I) \\gamma _{LR},$ where $J_{LR} = \\sqrt{J^2_L + J^2_R}$ and $\\gamma _{LR} = (J_L \\gamma _L + J_R \\gamma _R) / J_{LR}$ .",
"In this parity-protected scenario, everything therefore happens as if a single Majorana mode $\\gamma _{LR}$ was tunnel-coupled to the intermediate cavity.",
"Taking into account the possible detuning $\\delta _I$ of the intermediate cavity with respect to the frequency $\\omega _p$ of the parametric pumps driving both Kitaev chains (see main text), the low-energy effective Hamiltonian describing the Majorana system tunnel-coupled to the intermediate cavity becomes $H_{\\text{eff}} & = \\delta _I a^\\dagger _I a_I + J_{LR} (a^\\dagger _I - a_I) \\gamma _{LR},$ where, of course, $\\vert \\delta _I \\vert , J_{LR} \\ll E_g$ must be assumed ($E_g$ being the energy gap of the Kitaev chains) in order for the low-energy description to be valid.",
"It was shown in Ref.",
"[9] that the above Hamiltonian allows for a variety of unitary “tunnel-braid” operations within the subspace associated with the Majorana modes $\\gamma _L$ and $\\gamma _R$ .",
"In particular, starting from an empty intermediate cavity with $\\delta _I > 0$ and $\\delta _I / J_{LR} \\gg 1$ , one can generate an adiabatic transition from an empty to an occupied intermediate cavity and a unitary rotation $U(J_L, J_R) = \\gamma _{LR}$ in the Majorana subspace by adiabatically tuning $\\delta _I / J_{LR}$ from large positive to large negative values (keeping $\\vert \\delta _I \\vert , J_{LR} \\ll E_g$ ).",
"The same unitary operation can be obtained if, starting from an occupied intermediate cavity with $\\delta _I < 0$ and $\\vert \\delta _I \\vert / J_{LR} \\gg 1$ , one adiabatically raises $\\delta _I / J_{LR}$ from large negative to large positive values.",
"In both cases, the intermediate cavity and the Majorana system become entangled, and an inversion of the occupation of the intermediate cavity as well as a rotation $\\gamma _{LR}$ in the Majorana subspace are obtained as the latter return to an unentangled state at the end of the process.",
"Using similar adiabatic processes, one can more generally perform non-Abelian operations within the degenerate ground-state subspace associated with $\\gamma _L$ and $\\gamma _R$ .",
"In particular, one can generate the operation $U_B = \\tfrac{1}{\\sqrt{2}} (1 + \\gamma _L \\gamma _R)$ and thus effectively simulate the braiding of $\\gamma _L$ and $\\gamma _R$ in real space (see, e.g., Ref.",
"[10]) following the procedure below: Start with $J_L = J_R = 0$ (no tunnel coupling) and the intermediate cavity far blue-detuned ($\\delta _I \\gg 0$ ), such that it is initially empty.",
"Switch on the tunnel coupling $J_L$ , raising it to a constant value $J \\ll \\delta _I$ .",
"At this point, the intermediate cavity and the Majorana system are still effectively decoupled.",
"Adiabatically red-shift the intermediate cavity, taking $\\delta / J_{LR}$ from large positive to large negative values.",
"In doing so, the intermediate cavity adiabatically changes from empty to occupied, and the unitary operation $U(J_L = J, J_R = 0) = \\gamma _L$ is performed in the Majorana subspace.",
"Switch on the tunnel coupling $J_R$ , raising to the constant value $J_L = J$ .",
"Adiabatically blue-shift the intermediate cavity, taking $\\delta / J_{LR}$ from large negative to large positive values.",
"In doing so, the intermediate cavity adiabatically changes from occupied to empty, and the unitary operation $U(J_L = J, J_R = J) = \\tfrac{1}{\\sqrt{2}} (\\gamma _L + \\gamma _R)$ is performed in the Majorana subspace.",
"Switch off both $J_L$ and $J_R$ .",
"At the end of the above cycle, the intermediate cavity and the Majorana system are left unentangled as they were initially.",
"While the intermediate cavity returns to its initial empty state, a non-trivial unitary operation $U_B = \\tfrac{1}{\\sqrt{2}} (\\gamma _L + \\gamma _R) \\gamma _L = \\tfrac{1}{\\sqrt{2}} (1 + \\gamma _L \\gamma _R)$ is obtained in the Majorana subspace, as if $\\gamma _L$ and $\\gamma _R$ had physically been exchanged.",
"This exemplifies the fact that one can simulate real-space braiding in purely one-dimensional systems.",
"We remark, however, that the tunnel-braid operations used in the above procedure are only parity-protected provided that the phase difference between the Majorana modes $\\gamma _L$ and $\\gamma _R$ satisfies $\\Delta \\phi = (4n - 1) \\pi $ (for some integer $n$ ).",
"Because of this fine-tuning requirement, tunnel-braid operations are not strictly speaking topologically protected.",
"Moreover, even if $\\Delta \\phi $ is fine-tuned so that the degeneracy of the Majorana modes is preserved, one must still be able to control the tunneling amplitudes $J_L$ and $J_R$ sufficiently well in order to ensure that the desired tunnel-braid operations are generated.",
"Indeed, different relative strengths $J_L / J_R$ generally lead to different operations (see Ref.",
"[9] for further details).",
"Although the lack of topological protection renders tunnel-braid operations somewhat unattractive for solid-state proposals where fermion parity conservation is almost perfect, it does not constitute an additional problem in our optical setting where parity is anyway only conserved on timescales much shorter than the lifetime of the photons.",
"If tunnel-braid operations can be completed within such timescales, the non-Abelian nature of Majorana-like modes of light can be simulated."
]
] | 1204.1238 |
[
[
"The Bose-Hubbard model with localized particle losses"
],
[
"Abstract We consider the Bose-Hubbard model with particle losses at one lattice site.",
"For the non-interacting case, we find that half of the bosons of an initially homogeneous particle distribution, are not affected by dissipation that only acts on one lattice site in the center of the lattice.",
"A physical interpretation of this result is that the surviving particles interfere destructively when they tunnel to the location of the dissipative defect and therefore never reach it.",
"Furthermore we find for a one-dimensional model that a fraction of the particles can propagate across the dissipative defect even if the rate of tunneling between adjacent lattice sites is much slower than the loss rate at the defect.",
"In the interacting case, the phase coherence is destroyed and all particles eventually decay.",
"We thus analyze the effect of small interactions and small deviations from the perfectly symmetric setting on the protection of the particles against the localized losses.",
"A possible experimental realization of our setup is provided by ultracold bosonic atoms in an optical lattice, where an electron beam on a single lattice site ionizes atoms that are then extracted by an electrostatic field."
],
[
"Introduction",
"Ultracold atoms in optical lattices have in recent years emerged as a very successful quantum simulator for many-particle and solid state physics.",
"The ability to tune parameters of the simulated Hamiltonians provides a unique tool to explore quantum phase transitions in this system [1].",
"Among the most studied models is the Bose-Hubbard Hamiltonian which describes interacting bosons in a lattice potential that can tunnel between neighboring lattice sites.",
"In its ground state, it can show two quantum phases, a superfluid regime for weak interactions and a Mott-insulator phase for strong interactions and commensurate filling [2].",
"Electron beams can provide means for locally probing ultracold quantum gasses since they can be focused onto spot sizes that are much smaller than the optical wavelengths of the trapping fields.",
"High-resolution scanning electron microscopy thus constitutes a method that combines single atom sensitivity and high spatial resolution [3], [4].",
"For this technique, a focused electron beam scans an optically trapped atomic gas.",
"The atoms are ionized by the electron impact and the produced ions are moved off the dipole trap by an electrostatic field.",
"With this method, single-site addressability of ultracold atomic gases loaded in optical lattices has been demonstrated in one and two dimensional structures [4], [5].",
"In addition, second and higher order correlation functions of a trapped gas of bosons have been extracted [6].",
"Besides providing a tool for local measurements, a focused electron beam can also be used as a method to introduce controlled localized dissipation.",
"Along these lines, the formation of solitons in a continuous system with a localized dissipative perturbation has been discussed employing a mean field approach [7] and corrections beyond the mean field level have also been considered [8].",
"Moreover, a three-site Bose-Hubbard model with dissipation in the central lattice site has been analyzed [9] and a numerical study of the dynamics during a finite time range that employed DMRG methods has been presented [10].",
"In this article we investigate the effect of a localized dissipative defect in a Bose-Hubbard model with analytical means.",
"This allows us to consider arbitrarily large lattices as well as the long time limit of the dynamics.",
"For the non-interacting case, we show that a certain fraction of the population of a uniform particle distribution is not affected by dissipation that is localized in the center of the trap.",
"In other words, we show the existence of a dissipation-free subspace for localized particle losses.",
"Furthermore we find for a one-dimensional model that a fraction of the particles can propagate across the dissipative defect even if the rate of tunneling between adjacent lattice sites is much slower than the loss rate at the defect.",
"Particles can however only tunnel through the dissipative defect at the expense of other particles being lost via the localized dissipation.",
"For the description of the system we employ a Born-Markov master equation in the standard Lindblad form [11], [12], [13], [14].",
"This equation represents the dynamics of the system under the influence of a mechanism that introduces one-by-one particle losses at one lattice site.",
"We start by investigating a non-interacting system with the defect located exactly at the center of the chain.",
"In this case, we show the existence of a dissipation-free subspace which consists of the half of the normal modes of the Hamiltonian.",
"Those are the modes that correspond to wave-functions which are anti-symmetric with respect to the trap center.",
"The bosons of those modes interfere destructively at the location of the defect and therefore never reach the lossy site.",
"Since their wavefunction is finite everywhere else, these bosons can tunnel between the right and the left sides of the system without appearing at the defect.",
"After this observation, we investigate cases that might present dynamical behavior similar to that of the non-interacting and symmetric one.",
"Most important candidates are cases that correspond to small deviations from the non-interacting and symmetric case.",
"One way to introduce such a deviation is to turn on small interactions while keeping the symmetric structure of the system.",
"Interactions destroy the phase coherence along the lattice since they can transform protected to unprotected modes.",
"Also in a non-interacting system with a slightly non-symmetric structure, we find that the destructive interference of particles that tunnel into the defect and therefore the dissipation free subspace are perturbed.",
"For those two types of small deviations from the non-interacting and symmetric case, namely small interactions and slightly non-symmetric structure, we show that there is a slow down of the losses for some fraction of the total population.",
"The remainder of the paper is organized as follows.",
"In section II we introduce our model and discuss general properties of particles in trapping potentials that are symmetric around a dissipative defect.",
"In section III, we present quantitative results for non-interacting bosons on a lattice with a dissipative defect in the central lattice site.",
"Section IV then discusses small deviations from the interaction-free and perfectly symmetric case and we finish the presentation with a summary and conclusions in section V."
],
[
"Localized losses in symmetrical potentials",
"Various quantum systems can feature states that are protected against certain types of dissipation.",
"Those states - often referred to as dark states - are well known in atomic physics and quantum optics [11], [12].",
"In many-body systems such states can appear as a property of a symmetry of a given Hamiltonian and be part of its eigenstates.",
"One of the symmetries that can generate such a dissipation-free subspace is the symmetry under reflection at the origin of the system.",
"In other words, when the potential is symmetric around a localized dissipative defect."
],
[
"Dissipation-free subspace of a single particle in a symmetrical potential",
"We consider a single particle in a symmetric potential.",
"Such a potential is described by an even function $U$ with $ U(x)=U(-x) $ .",
"The wavefunctions that correspond to eigenstates of such a potential must have a well defined parity, i.e.",
"some are even ($ \\psi _e(x)=\\psi _e(-x) $ ) and others are odd ($ \\psi _o(x)=-\\psi _o(-x) $ ).",
"These properties emerge since the Schrödinger equation is unchanged when the sign of a coordinate is reversed [15].",
"If one lists the eigenfunctions according to the number of nodes - increasing energy - then they are alternately even and odd.",
"The ground state of the system must be an even function since it contains no nodes.",
"Odd functions always vanish at the origin, since $\\psi _o(0)=-\\psi _o(0)=0.$ This means that whatever happens at this particular location in space does not affect the odd eigenfunctions.",
"A physical interpretation of this result is that the probability amplitudes for particles in odd eigenstates destructively interfere at the origin.",
"Now, let us assume that we locate a mechanism that introduces particles losses at the origin.",
"Then, if the particle is in a odd eigenfunction, it will never be caught by the loss mechanism.",
"This is to say, the odd eigenfunctions constitute a dissipation-free subspace.",
"The effect of the dissipation-free subspace may seem (almost) unphysical in a continuous description.",
"This is because we silently assumed that the loss mechanism has zero spatial width.",
"However, this wouldn't be problematic in a lattice setup.",
"If we assume that the relevant physics takes place within the lowest band of a tight binding Hamiltonian, i.e.",
"the energy of the system is low enough, then the lattice sites can be treated as points in space since their extension is much smaller than the wavelength of the mode-functions.",
"Therefore, the only requirement for the loss mechanism is to be restricted to one lattice site, specifically the central one - we assume odd number of sites.",
"Therefore the mechanism that generates the losses needs to be capable of addressing single lattice sites, see Fig.",
"(REF ).",
"Such mechanisms have recently been realized for ultracold atoms in optical lattices [1], [6], [7], [16], which motivates the study we present here.",
"Figure: (Color online) Illustration of the proposed setup for the realization of the dissipation-free subspace.",
"Introducing localized particle losses at the central site - assuming odd total number of lattice sites - leaves the odd modes - described by anti-symmetric functions - unaffected.",
"The loss mechanism consists of an electron beam focused onto the central site.",
"The atom-electron collisions leave the atoms ionized and by the use of an electrostatic field they can be extracted from the lattice.",
"This method provides single-site sensitivity for the control over the position of the defect and high precision control of the associated damping rate.We assume that particle losses occur one-by-one and can be described by a super-operator within the Born-Markov approximation [11], [12] which reads, $\\hat{\\mathcal {L}}_n \\left[ \\hat{\\rho }_S(t) \\right] = \\frac{\\gamma }{2} \\left( 2\\hat{a}_n \\hat{\\rho }_S(t) \\hat{a}^{\\dagger }_n - \\hat{a}^{\\dagger }_n \\hat{a}_n \\hat{\\rho }_S(t) - \\hat{\\rho }_S(t)\\hat{a}^{\\dagger }_n\\hat{a}_n \\right),$ where $ \\hat{\\rho }_S(t) $ is the (reduced) density matrix of the system and $ \\hat{a}^{\\dagger }_n $ and $ \\hat{a}_n $ are the creation and annihilation operators of one atom at site $ n $ .",
"For experimental realizations of local particle losses induced by a focused electron beam, our assumptions for the decay term are well justified.",
"Here the atoms are ionized by the electron impact and then extracted from the lattice with an electric field.",
"Therefore subsequent particle decays will be independent of each other which justifies the Markov assumption, c.f.",
"[7].",
"In an experimental setup with cold atoms in an optical lattice, the atoms are not only subject to the lattice potential but are also trapped by an additional confining potential that is usually well approximated by a harmonic trap.",
"In the central region of the lattice, the value of the confining potential typically only varies very little from one lattice site to another and we therefore approximate it by a constant.",
"In current experiments it is however not guaranteed that the bottom of the confining potential exactly overlaps with the site of the lattice where losses are induced.",
"To estimate the effect of this feature on our findings, we consider deviations from a lattice that is symmetric with respect to the dissipative site in section ."
],
[
"Non-interacting bosons in a lattice with particle losses at the central site",
"We start our quantitative study by considering a non-interacting gas of bosonic particles on a lattice with particle losses at the central site.",
"Assuming that the tight binding approximation holds, the model under consideration is the Bose-Hubbard Hamiltonian, $\\hat{H}= -J\\sum _{\\langle i,j \\rangle } \\left( \\hat{a}_{i}^{\\dagger }\\hat{a}_{j} + h.c. \\right) + \\frac{U}{2}\\sum _{i} \\hat{a}_{i}^{\\dagger }\\hat{a}_i \\left( \\hat{a}_{i}^{\\dagger }\\hat{a}_i -1 \\right),$ where $ J $ is the tunneling rate of the bosons between neighboring sites and $ U $ is the strength of the (pairwise) on-site particle interaction.",
"The operators $ \\hat{a}_i $ and $ \\hat{a}^{\\dagger }_i $ are the bosonic annihilation and creation operators, respectively.",
"They satisfy the commutation relations for bosons, $ [\\hat{a}_i,\\hat{a}^{\\dagger }_j]=\\delta _{ij} $ and $ [\\hat{a}_i,\\hat{a}_j]=[\\hat{a}_i^{\\dagger },\\hat{a}_j^{\\dagger }]=0 $ .",
"Since we are interested in the dynamics generated by the Hamiltonian (REF ) for a fixed number of particles, we omit a term related to a chemical potential.",
"Two quantum phases of this model are well known, namely the superfluid and the Mott insulator [2].",
"In general the phase of the sample depends on the magnitude of the ratio $ J/U $ .",
"An experimental realization of such a model is provided by ultracold bosonic atoms in an optical lattice [1].",
"Feshbach resonances provide a unique and efficient way to control interactions.",
"Based on this method, an ideal system of non-interacting atoms can be approximately implemented.",
"In this case Eq.",
"(REF ) is approximately written $\\hat{H}=-J\\sum _{\\langle i,j \\rangle } \\left( \\hat{a}_{i}^{\\dagger }\\hat{a}_{i} + h.c. \\right).$ This is the hopping Hamiltonian, and because of the absence of interactions the dynamics of each particle is unaffected by presence of the remaining particles."
],
[
"The dynamics in one dimension",
"For simplicity, we start with the dynamics in one dimension.",
"However, as we shall see, the generalization to higher dimensions is straight forward.",
"The Hamiltonian (REF ) in one dimension is written in the form $\\hat{H}=-J\\sum _{n=1}^{N-1} \\left( \\hat{a}_{n}^{\\dagger }\\hat{a}_{n+1} + h.c. \\right),$ where $ N $ is the total number of lattice sites.",
"We assume open boundary conditions, which imply that the wavefunction vanishes at the edges of the system.",
"The ladder operators for the normal modes, $ \\hat{b}_k $ , are in this case given by the following transform $\\begin{split}\\hat{a}_n=\\sqrt{\\frac{2}{N+1}}\\sum _{k}\\sin \\left(nk\\right) \\hat{b}_k, & \\\\\\hat{b}_k=\\sqrt{\\frac{2}{N+1}}\\sum _{n=1}^{N}\\sin \\left(nk\\right) \\hat{a}_n,\\end{split}$ where the quasi-momentum $ k $ is defined as $ k=\\frac{\\pi l}{N+1} $ , with $l=1,2,.....,N $ .",
"It can be easily shown that the operators $ \\hat{b}_k $ and $ \\hat{b}_k^{\\dagger } $ satisfy the commutation relations for bosons.",
"By substituting Eq.",
"(REF ) into Eq.",
"(REF ), we can write the Hamiltonian in the diagonal form $\\hat{H}=-2J\\sum _{k}\\cos \\left(k \\right) \\hat{b}_{k}^{\\dagger }\\hat{b}_k.$ This Hamiltonian corresponds to a set of $ N $ harmonic oscillators with frequencies given by the dispersion relation $ \\omega (k)=-2J\\cos \\left(k \\right) $ .",
"We note that even though the momentum modes (REF ) only diagonalize the Hamiltonian (REF ) in the absence of a confining potential, there is a different set of normal modes which diagonalizes the model with harmonic confining potential as there are no interactions.",
"Following our discussion in section half of those normal modes will nonetheless be protected against the localized dissipation."
],
[
"The Master Equation",
"The Born-Markov master equation that describes the evolution of the system under the influence of the dissipation at one lattice site is of the form, $\\frac{d}{dt}\\hat{\\rho }_S(t)= -i\\left[ \\hat{H},\\hat{\\rho }_S(t) \\right] + \\hat{\\mathcal {L}}_m \\left[ \\hat{\\rho }_S(t) \\right]$ where we have set $ \\hbar = 1$ .",
"The dissipator in the above master equation describes particle losses at site $ m $ , as defined in Eq (REF ).",
"In a first approximation, we shall assume that this site corresponds to the central site of the lattice.",
"This assumption of a symmetric structure is motivated by the fact that the lattice site on which the electron beam is focused can in experiments be selected with great accuracy [4], [5].",
"In terms of the mode operators given by Eq.",
"(REF ), the dissipator of the master equation reads $\\begin{split}\\mathcal {\\hat{L}}_m \\left[ \\hat{\\rho }_S(t) \\right] = & \\frac{\\gamma }{N+1} \\sum _{k,p} \\sin \\left( m k \\right) \\sin \\left( m p \\right) \\lbrace 2\\hat{b}_{k}\\hat{\\rho }_S(t)\\hat{b}_p^{\\dagger } \\\\& - \\hat{b}_k^{\\dagger }\\hat{b}_p\\hat{\\rho }_S(t) - \\hat{\\rho }_S(t)\\hat{b}_k^{\\dagger }\\hat{b}_p \\rbrace \\end{split} $ Under the assumption that the total number of sites is odd, the central site is numbered as $ m=(N+1)/2 $ and by using the definition of the quasi-momentum, the dissipator in Eq.",
"(REF ) is written $\\begin{split}\\mathcal {\\hat{L}}_m \\left[ \\hat{\\rho }_S(t) \\right] = & \\frac{\\gamma }{N+1} \\sum _{l,j=1}^{N} \\sin \\left( \\frac{\\pi l}{2} \\right) \\sin \\left( \\frac{\\pi j}{2} \\right) \\lbrace 2\\hat{b}_{l}\\hat{\\rho }_S(t)\\hat{b}_j^{\\dagger } \\\\& - \\hat{b}_l^{\\dagger }\\hat{b}_j\\hat{\\rho }_S(t) - \\hat{\\rho }_S(t)\\hat{b}_l^{\\dagger }\\hat{b}_j \\rbrace \\end{split}$ We immediately see that only odd values of $ l $ and $ j $ contribute to the summation.",
"That means that the dissipation affects only the modes that correspond to odd quasi-momentum labeling number.",
"Therefore, the modes with even $ l $ , constitute a dissipation-free subspace of the dissipator as given in Eq.",
"(REF ).",
"At first site, this result seems to be in contradiction with the discussion in the previous section.",
"There, we concluded that the dissipation-free subspace consists of the odd eigenfunctions.",
"However, this contradiction is a misconception since the odd eigenfunctions correspond to even quasi-momentum labeling number and vice versa.",
"The master equation, given by Eq.",
"(REF ) can now be written in the following form, $\\begin{split}\\frac{d}{dt}\\hat{\\rho }_S(t)= & -i\\left[ \\hat{H},\\hat{\\rho }_S(t) \\right] \\\\& + \\frac{\\gamma }{N+1} \\sum _{l,j=1}^{N} \\sin \\left( \\frac{\\pi l}{2} \\right) \\sin \\left( \\frac{\\pi j}{2} \\right) \\lbrace 2\\hat{b}_{l}\\hat{\\rho }_S(t)\\hat{b}_j^{\\dagger } \\\\& - \\hat{b}_l^{\\dagger }\\hat{b}_j\\hat{\\rho }_S(t) - \\hat{\\rho }_S(t)\\hat{b}_l^{\\dagger }\\hat{b}_j \\rbrace .\\end{split} $ Using the above equation, one can derive equations governing the evolution of the mean values of observables.",
"For the mean value of the number of bosons in odd modes (with even quasi-momentum labeling number) $ \\langle N(t)\\rangle _{Odd} = \\sum _{l^{\\prime }} \\langle \\hat{b}_{2l^{\\prime }}^{\\dagger } \\hat{b}_{2l^{\\prime }} \\rangle _{(t)} = \\sum _{l^{\\prime }} Tr \\lbrace \\hat{b}_{2l^{\\prime }}^{\\dagger } \\hat{b}_{2l^{\\prime }} \\hat{\\rho }_S(t) \\rbrace $ , where $ l^{\\prime }=1,2,....,N/2 $ , one gets, $\\frac{d}{dt} \\langle N(t)\\rangle _{Odd}=0.$ The obvious solution of this equation, $ \\langle N(t)\\rangle _{Odd} = \\langle N(0)\\rangle _{Odd}$ confirms that modes with an even quasi-momentum labeling number constitute a dissipation-free subspace.",
"On the other hand, for the mean value of the number of bosons in even modes (with odd quasi-momentum labeling number) $ \\langle N(t)\\rangle _{Even} = \\sum _{l^{\\prime }} \\langle \\hat{b}_{2l^{\\prime }-1}^{\\dagger } \\hat{b}_{2l^{\\prime }-1} \\rangle _{(t)} = \\sum _{l^{\\prime }} Tr \\lbrace \\hat{b}_{2l^{\\prime }-1}^{\\dagger } \\hat{b}_{2l^{\\prime }-1} \\hat{\\rho }_S(t) \\rbrace $ , where $ l^{\\prime }=1,2,....,(N+1)/2 $ , we have, $\\frac{d}{dt} \\langle N(t)\\rangle _{Even} = -\\frac{2 \\gamma }{N+1} \\sum _{l,j=1}^{(N+1)/2} \\langle \\hat{b}^{\\dagger }_{2j-1} \\hat{b}_{2l-1} \\rangle _{(t)}$ Figure: (Color online) The expectation value of the population of the even modes as a function of time.",
"All the bosons that populate these modes eventually decay.",
"The plot reveals also the rapid loss of the particles initially populating the defect.",
"A large proportion of the particles is annihilated almost instantly in the case where γ≫J \\gamma \\gg J .",
"This suggests that, in this case, the lossy site can be adiabatically eliminated from the description because of its fast rotating dynamics.",
"Parameters: γ=10 \\gamma =10 , J=0.1 J=0.1 , U=0 U=0 and N=5 N=5 .Hence, particles in these modes will eventually be lost from the system.",
"The numerical solution of the above equation is shown in Fig.",
"(REF ).",
"One sees that all bosons that populate the even modes eventually decay.",
"We now consider the case where initially one boson is located at a specific lattice site at one side of the dissipative defect.",
"The state of this localized particle is a coherent superposition of all the eigenmodes of the system [17].",
"Therefore, the fraction of the particle that occupies odd modes will be gradually lost due to the dissipation at the defect whereas the remaining part will be protected against the localized losses as it occupies even modes.",
"Since the even modes however show equal particle densities at both sides of the defect, a fraction of the initially localized particle is able to propagate through the defect into the other half of the lattice, see Fig.",
"(REF ).",
"In complete analogy, a fraction of a gas of non-interacting bosons that is initially located at one side of the dissipative defect, will propagate through the lossy site into the other half of the lattice at the expense of loosing half its initial number of particles, see Fig.",
"(REF ).",
"It is remarkable that this effect is independent of the rate of particle losses at the defect.",
"Hence even if the particles are lost from the defect a lot faster than they can tunnel into it, still a fraction of the particles ($1/4$ of them in our examples) is able to cross the defect and reach the other half of the lattice.",
"Figure: (Color online) The propagation across the defect of a particle that is initially localized at the first lattice site from the left.",
"Dashed line: The expectation value of the total population in lattice sites to the left of the defect.",
"Thin line: The expectation value of the total population in lattice sites to the right of the defect.",
"Thick line: The expectation value of the population in the defect.",
"Half of the initial number of particles survives the damping and ends up in a superposition of being located on the left and the right part of the lattice.",
"Parameters: γ=1 \\gamma =1 , J=0.1 J=0.1 , U=0 U=0 and N=5 N=5 ."
],
[
"Extension to higher dimensions",
"Since in our case the dissipation-free subspace is a consequence of the symmetry of our system, one expects to find an analogous behavior in higher dimensions.",
"To this end, we consider a two dimensional square lattice of size $ N \\times N $ and suppose that the electron beam focuses exactly on the center of the square lattice.",
"In this case, the Hamiltonian given by Eq.",
"(REF ) takes on the form, $\\hat{H}=-J \\sum _{i,j=1}^{N-1} \\left( \\hat{a}_{i,j}^{\\dagger } \\hat{a}_{i+1,j} + \\hat{a}_{i,j}^{\\dagger }\\hat{a}_{i,j+1} + h.c. \\right).$ By assuming again open boundary conditions, the ladder operators are transformed as, $\\begin{split}& \\hat{a}_{i,j}=\\frac{2}{N+1}\\sum _{k,p} \\sin \\left( ik \\right) \\sin \\left( jp \\right) \\hat{b}_{k,p}, \\\\& \\hat{b}_{k,p}=\\frac{2}{N+1}\\sum _{i,j=1}^{N} \\sin \\left( ik \\right) \\sin \\left( jp \\right) \\hat{a}_{i,j},\\end{split}$ where again the quasi-momenta take the values $ k,p=\\frac{\\pi l}{N+1} $ , with $ l=1,2,....,N $ .",
"In terms of the mode operators $ \\hat{b}_{k,p} $ and $ \\hat{b}_{k,p}^{\\dagger } $ , Hamiltonian (REF ) has the following diagonal form, $\\hat{H}=-2J\\sum _{k,p} \\left[ \\cos \\left( k \\right) + \\cos \\left( p \\right) \\right] \\hat{b}_{k,p}^{\\dagger } \\hat{b}_{k,p}.$ The dissipator that describes the particle losses at the central site reads, $\\begin{split}\\hat{\\mathcal {L}}_{m,m} \\left[ \\hat{\\rho }_S(t) \\right] = & \\frac{\\gamma }{2} \\left( 2\\hat{a}_{m,m}\\hat{\\rho }_S(t)\\hat{a}^{\\dagger }_{m,m} - \\hat{a}^{\\dagger }_{m,m}\\hat{a}_{m,m}\\hat{\\rho }_S(t) \\right.",
"\\\\& \\left.",
"- \\hat{\\rho }_S(t)\\hat{a}^{\\dagger }_{m,m}\\hat{a}_{m,m} \\right)\\end{split}$ where the coordinates of the lossy site are given by $ m=(N+1)/2 $ .",
"In terms of the eigenmode operators defined in Eq.",
"(REF ), the above dissipator reads, $\\begin{split}\\hat{\\mathcal {L}}_{m,m} \\left[ \\hat{\\rho }_S(t) \\right] & = \\frac{2\\gamma }{(N+1)^2}\\\\\\times & \\sum _{i,j,l,f=1}^{N} \\sin \\left( \\frac{\\pi i}{2} \\right)\\sin \\left(\\frac{\\pi j}{2}\\right) \\sin \\left(\\frac{\\pi l}{2}\\right)\\sin \\left(\\frac{\\pi f}{2}\\right) \\\\\\times &\\lbrace 2\\hat{b}_{i,j} \\hat{\\rho }_S(t) \\hat{b}^{\\dagger }_{l,f} - \\hat{b}^{\\dagger }_{i,j}\\hat{b}_{l,f}\\hat{\\rho }_S(t) - \\hat{\\rho }_S(t)\\hat{b}^{\\dagger }_{i,j}\\hat{b}_{l,f} \\rbrace .\\end{split}$ One immediately sees that the above dissipator is non-zero only when both the entries of the summations are odd (corresponding to even modes).",
"The eigenmodes that correspond to any other combination of quasi-momentum labeling numbers, constitute part of the dissipation-free subspace.",
"This can be demonstrated by computing the evolution of the total number of particles in modes corresponding to every such combination.",
"For the modes where at least one index of quasi-mometum labeling is an even number, one obtains, $\\frac{d}{dt}\\langle N(t) \\rangle _{O-O}=\\frac{d}{dt}\\langle N(t) \\rangle _{E-O} = 0.$ On the other hand for the modes where both indices are odd numbers (even modes) one gets, $\\frac{d}{dt} \\langle N(t) \\rangle _{E-E} = - \\frac{4\\gamma }{(N+1)^2}\\sum _{l,j,k,f=1}^{(N+1)/2} \\langle \\hat{b}^{\\dagger }_{2l-1,2j-1} \\hat{b}_{2k-1,2f-1} \\rangle _{(t)} .$ We therefore conclude that the dissipation-free subspace of a two dimensional non-interacting, symmetric system, constitutes the $3/4$ of all its eigenmodes.",
"In a completely analogous way, one finds that in a three-dimensional structure, the protected modes are $ 7/8 $ of all its eigenmodes."
],
[
"The case of large damping rate and the adiabatic elimination of the defect",
"Let us return to the one dimensional setup.",
"As illustrated in Fig.",
"(REF ), in the case where the damping rate $ \\gamma $ is much larger than the tunneling $ J $ , the loss of the particles that populate the defect, occurs almost instantly compared to the time-scale in which the tunneling takes place.",
"This suggests that one can assume the lossy site to be empty at all times and adiabatically eliminate its degrees of freedom to generate an effective picture for the description of the remainder of the system.",
"We now discuss this approach."
],
[
"The effective Markovian dissipator",
"We consider a system with three sites [9] with particle losses occurring at the central site.",
"By performing perturbative theory in the interaction picture [11], [12] and tracing out the degrees of freedom of the lossy site, we derive an expression for the evolution of the reduced density matrix of two outer sites $\\hat{\\rho }_{-m}$ , $\\hat{\\mathcal {L}}_{eff} \\left[\\hat{\\rho }_{-m}(t)\\right] = - \\int _{0}^{t} \\text{Tr}_{m}\\left[ \\hat{V}(t), \\left[ \\hat{V}(t-s),\\hat{\\rho }(s) \\right] \\right] ds.$ Since we trace out the degrees of freedom of the defect, we automatically place the lossy site in the status of a quantum environment.",
"Thus, the interaction Hamiltonian $ \\hat{V}(t) $ in Eq.",
"(REF ) corresponds to the tunneling between the defect and its neighbor sites, $\\hat{V}(t)=-J \\left( \\hat{a}^{\\dagger }_{m-1}(t) \\hat{a}_m(t) + \\hat{a}^{\\dagger }_{m+1}(t) \\hat{a}_m(t) + h.c. \\right).$ We now focus on the regime $ \\gamma \\gg J $ where we can assume that the lossy site practically remains empty at all times, as illustrated in Fig.",
"(REF ).",
"By imposing the Born approximation the equation (REF ) reads, $\\begin{split}\\hat{\\mathcal {L}}_{eff} \\left[\\hat{\\rho }_{-m}(t) \\right]= & - \\int _{0}^{t} \\text{Tr}_{m}\\left[ \\hat{V}(t), \\left[ \\hat{V}(t-s), \\right.\\right.",
"\\\\& \\left.\\left.",
"|0\\rangle \\langle 0| \\otimes \\hat{\\rho }_{-m}(s) \\right] \\right] ds,\\end{split}$ for this regime.",
"The integral kernel in the above equation contains the two-time correlation functions, $ \\langle 0| \\hat{a}_m(t)\\hat{a}^{\\dagger }_m(s) |0\\rangle $ and $ \\langle 0| \\hat{a}_m(s)\\hat{a}^{\\dagger }_m(t) |0\\rangle $ , of the defect.",
"These functions can be calculated using the dissipator given by Eq.",
"(REF ) which, in this case, yields the internal dynamics of the defect.",
"By doing so and using the quantum regression theorem [11], [12], one finds that the two-time correlations decay exponentially, $\\langle 0| \\hat{a}_m(t)\\hat{a}^{\\dagger }_m(t-s) |0\\rangle = \\langle 0| \\hat{a}_m(t-s)\\hat{a}^{\\dagger }_m(t) |0\\rangle = e^{-\\frac{\\gamma }{2}s}.$ The characteristic time-scale in which the correlations decay is $ \\tau =2/\\gamma $ .",
"In the case where the damping rate is large, the correlations decay rapidly, as illustrated in Fig.",
"(REF ).",
"Since the correlations are short-lived, the interactions occurs sharply at $ s=t $ .",
"Therefore, it is safe to replace $ s $ in $ \\hat{\\rho }_{-m}(s) $ in Eq.",
"(REF ), by $ t $ .",
"And by extending the upper limit of the integration to infinity, which concludes the Markov approximation, we can perform the integration with respect to $ s $ , obtaining, $\\begin{split}\\hat{\\mathcal {L}}_{eff}\\left[\\hat{\\rho }_{-m}(t)\\right] & = \\frac{2J^2}{\\gamma } \\left( 2 \\tilde{a}_{m} \\hat{\\rho }_{-m}(t) \\tilde{a}_{m}^{\\dagger } \\right.\\\\& \\left.",
"- \\tilde{a}_{m}^{\\dagger } \\tilde{a}_{m} \\hat{\\rho }_{-m}(t) - \\hat{\\rho }_{-m}(t) \\tilde{a}_{m}^{\\dagger } \\tilde{a}_{m} \\right).\\end{split}$ where $\\tilde{a}_{m} = \\hat{a}_{m-1}+\\hat{a}_{m+1}$ .",
"Hence, the dynamics of the lossy site is no longer part of this effective description and the losses are described as a property of certain type of states, namely symmetric superpositions between the neighbors of the defect.",
"Since the damping affects only symmetric superpositions, states that do not overlap with them are immune to particle losses.",
"Therefore, the anti-symmetric states of the from, $|\\psi \\rangle _A = \\frac{1}{\\sqrt{n!}}",
"\\left(\\frac{\\hat{a}^{\\dagger }_{m-1} - \\hat{a}^{\\dagger }_{m+1}}{\\sqrt{2}} \\right)^{n} |0\\rangle ,$ for any integer number $n$ have constant population.",
"It can be analytically shown that those anti-symmetric states identically equal to the odd modes that constitute the dissipation-free subspace.",
"Another implication of Eq.",
"(REF ) comes from the structure of the effective dumping rate $ 2J^2/\\gamma $ .",
"As the damping rate $ \\gamma $ increases, the actual losses decrease.",
"This behavior is a manifestation of the quantum Zeno effect [18], [19].",
"This result has been experimentally discovered for ultra-cold molecules by Syassen et al.",
"[20] and is predicted in[9], [10], see also [21], [22].",
"Although this effect is not directly obvious from the form of the dissipator given by Eq.",
"(REF ), the related phenomenon is of course not a consequence of the adiabatic elimination."
],
[
"Interactions and asymmetry: Two factors that induce losses to the dissipation-free subspace",
"All the above discussion was limited to the case of a symmetric structure (defect at the center of the lattice) and no interactions.",
"In this section we investigate how deviations from this ideal case can affect the protected modes.",
"First we shall consider the case of a symmetric structure but with small yet finite interactions.",
"Then we shall check the dynamics when the defect is not located at the center of the lattice but rather at a random position along it."
],
[
"Small interactions",
"Phenomena that depend on interference are typically affected by the presence of interactions, since the latter destroy the phase coherence.",
"This happens because the term in the Bose-Hubbard Hamiltonian (REF ), which is responsible for the interactions, is non-linear and gives rise to a non-linear term in the corresponding Schrödinger equation.",
"However, in the case of small enough interactions, one expects the system to behave similarly to the non-interacting case, at least for sort times.",
"In order to examine to what extent interactions affect our findings of the previous sections, we compare the strength of the interactions to the damping rate that appears in Eq.",
"(REF ) for an 1D lattice.",
"For a one-dimensional Bose-Hubbard system, the part of the Hamiltonian (REF ) that describes the on-site particle interaction, in terms of the mode operators defined in Eq.",
"(REF ), reads, $\\hat{H}_{int} = \\frac{U}{2} \\sum _{i,j,l,f} \\mathcal {T}(i,j,l,f) \\hat{b}^{\\dagger }_k\\hat{b}_p\\hat{b}^{\\dagger }_l\\hat{b}_f - \\frac{U}{2} \\sum _{i,j} \\mathcal {S}(i,j) \\hat{b}^{\\dagger }_i\\hat{b}_j,$ with $\\begin{split}\\mathcal {T}(k,p,l,f) = & \\frac{4}{(N+1)^2} \\sum _{n=1}^{N} \\sin \\left( \\frac{n\\pi i}{N+1} \\right)\\sin \\left( \\frac{n\\pi j}{N+1} \\right) \\\\& \\times \\sin \\left( \\frac{n\\pi l}{N+1} \\right)\\sin \\left( \\frac{n\\pi f}{N+1} \\right)\\end{split}$ and $\\mathcal {S}(i,j) = \\frac{2}{N+1} \\sum _{n=1}^{N} \\sin \\left( \\frac{n\\pi i}{N+1} \\right) \\sin \\left( \\frac{n\\pi j}{N+1} \\right).$ One can show that function $ \\mathcal {S} $ is identically equal to a delta function, $ \\mathcal {S}(i,j)= \\delta _{i,j } $ .",
"This means that the second term of the right-hand side of Eq.",
"(REF ) is diagonal and generates no hopping of particles between protected and unprotected modes.",
"On the contrary, the function $ \\mathcal {T} $ is not a delta function and therefore the non-linear term of the Hamiltonian (REF ) is responsible for hopping between the two types of modes.",
"For small enough interactions, the scattering of particles from the protected into the unprotected modes occurs a lot slower than the subsequent loss of particles from the unprotected modes.",
"Hence, one expects a that particles in the protected modes still survive a lot longer than the other particles.",
"To find a condition for when this scenario occurs, we consider the ratio between the scattering rate $ U \\mathcal {T} / 2$ and the effective damping rate $ \\Gamma = 2 \\gamma /(N+1) $ as given in Eq.",
"(REF ).",
"The scattering from protected to unprotected modes needs to be slower than the losses for all modes involved.",
"We therefore calculate when, $\\frac{U}{2 \\Gamma } \\, \\text{Max} [|\\mathcal {T}|] \\ll 1 .$ To this end, we numerically calculate the maximum value of $ T^{\\prime }(i,j,l,f) = \\frac{(N+1)^2}{4} \\mathcal {T}(i,j,l,f) $ as a function of the size of the system, $N$ , see Fig.",
"(REF ), and observe that $ \\max [T^{\\prime }] \\approx N/2 $ .",
"Figure: (Color online) The maximum value of T ' T^{\\prime } as a function of the size of the system - the total number of lattice sites NN.",
"The plot reveals the linear character of the function.",
"Specifically, Max[T ' ]∼N/2 \\text{Max}[T^{\\prime }]\\sim N/2 .Therefore, the ratio of the maximum value of the scattering over the effective damping is, $\\frac{U}{2 \\Gamma } \\, \\text{Max} [|\\mathcal {T}|] \\approx \\frac{UN}{2 \\gamma (N+1)} \\sim \\frac{U}{\\gamma }.$ We thus conclude that the slow-down takes place in the regime where $ \\gamma \\gg U $ .",
"An analogous treatment shows that this condition is also valid in two dimensions."
],
[
"Deviations from the symmetry",
"Up to now, we have assumed that particle losses occur at the center of the chain, see Eq.",
"(REF ).",
"Now, we shall place the losses not at the central site $ m $ but rather to a random position along the lattice.",
"Instead of subscript $ m $ , we use $ r $ which stands for \"random\".",
"From the structure of the normal modes in equation (REF ) one can see that for each location $r$ , those modes with vanishing amplitude at $r$ are protected against losses affecting the site $r$ only.",
"A general tendency to what degree the dissipation free subspace is degraded by deviations from the symmetry of the setup can be obtained from the following consideration.",
"We assume, without loss of generality, that the defect is located at a site on the left of the center of the lattice.",
"We can split the lattice into two parts, the symmetric part around the defect and the remaining part that consists of the sites that make the right side of the lattice larger - an illustration is given in Fig.",
"(REF ).",
"Figure: (Color online) Illustration of an asymmetric setup.",
"The defect is located at a random position along the lattice rather than being at the center.",
"For the description of the dynamics, we split the lattice into two parts, - each part is described by a different set of modes - the symmetric one that consists of the sites around the defect, obtaining in this way the structure of the symmetric setup, and the one consisting of the left-over sites.In this way, we can again retain modes that are symmetry-related.",
"Specifically, for the symmetric part of the lattice, namely for the first site from the left to site number $2r-1$ , we define the modes, $\\begin{split}& \\hat{a}_n =\\frac{1}{\\sqrt{r}} \\sum _{l=1}^{2r-1} \\sin \\left( \\frac{\\pi n l}{2r} \\right) \\hat{b}_l \\\\& \\hat{b}_l =\\frac{1}{\\sqrt{r}} \\sum _{n=1}^{2r-1} \\sin \\left( \\frac{\\pi n l}{2r} \\right) \\hat{a}_n.\\end{split}$ Accordingly, for the left-over sites, namely for site $ 2r $ to site $ N $ , we define the additional modes, $\\begin{split}& \\hat{a}_n =\\frac{2}{\\sqrt{N-2r}} \\sum _{l=2r}^{N} \\sin \\left( \\frac{\\pi n l}{N-2r} \\right) \\hat{c}_l \\\\& \\hat{c}_l =\\frac{2}{\\sqrt{N-2r}} \\sum _{n=2r}^{N} \\sin \\left( \\frac{\\pi n l}{N-2r} \\right) \\hat{a}_n.\\end{split}$ Since the lossy site $ r $ belongs to the symmetric part of the lattice the ladder operators $ \\hat{a}^{\\dagger }_r $ and $ \\hat{a}_r $ can be expanded in the mode operators defined in Eq.",
"(REF ).",
"Therefore, the dissipator given by Eq.",
"(REF ), reads in this case, $\\begin{split}\\hat{\\mathcal {L}}_{r} \\left[\\hat{\\rho }_S(t)\\right] = & \\frac{\\gamma }{2r} \\sum _{l,j=1}^{2r-1} \\sin \\left( \\frac{\\pi l}{2} \\right) \\sin \\left( \\frac{\\pi j}{2} \\right) \\lbrace 2\\hat{b}_l \\hat{\\rho }_S(t) \\hat{b}^{\\dagger }_j \\\\& - \\hat{b}^{\\dagger }_l \\hat{b}_j \\hat{\\rho }_S(t) - \\hat{\\rho }_S(t) \\hat{b}^{\\dagger }_l \\hat{b}_j \\rbrace .\\end{split}$ Similarly to the ideal case, we easily observe that the above dissipator affects only the modes which correspond to odd quasi-momentum labeling number.",
"That means that the modes with even quasi-momentum labeling number that correspond to the symmetric part and all the modes that correspond to the left-over part are not directly affected by the particle losses.",
"Since we consider again a non-interacting case, the unitary part of the evolution of the system is governed by the Hamiltonian given by Eq.",
"(REF ).",
"In order to write this Hamiltonian in terms of the mode operators (REF ) and (REF ), we split it into three parts, $\\begin{split}\\hat{H} = & -J \\sum _{n=1}^{2r-2} \\left( \\hat{a}^{\\dagger }_{n}\\hat{a}_{n+1} + h.c. \\right) - J \\sum _{n=2r}^{N-1} \\left( \\hat{a}^{\\dagger }_{n}\\hat{a}_{n+1} + h.c. \\right) \\\\& - J \\left( \\hat{a}^{\\dagger }_{2r-1}\\hat{a}_{2r} + h.c. \\right).\\end{split}$ The first part of the above Hamiltonian describes the symmetric part of the system, the second describes the left-over sites and the last one gives the tunneling between the last site of the symmetric part and the first one of the left-over part.",
"Now, in terms of the modes (REF ) and (REF ), the Hamiltonian reads, $\\begin{split}\\hat{H} = & -2J \\sum _{l=1}^{2r-1} \\cos \\left( \\frac{\\pi l}{2r} \\right) \\hat{b}^{\\dagger }_{l}\\hat{b}_{l} -2J \\sum _{j=2r}^{N} \\cos \\left( \\frac{\\pi j}{N-2r} \\right) \\hat{c}^{\\dagger }_{j}\\hat{c}_{j} \\\\& - J \\sum _{l=1}^{2r-1} \\sum _{j=2r}^{N} \\mathcal {R}(j,l) \\lbrace \\hat{b}^{\\dagger }_{l}\\hat{c}_{j} + h.c. \\rbrace ,\\end{split}$ where we have defined, $\\mathcal {R}(j,l) = \\frac{2}{\\sqrt{2r(N-2r)}} \\sin \\left( \\frac{\\pi (2r-1) l}{2r} \\right) \\sin \\left( \\frac{\\pi 2r j}{N-2r} \\right)$ The above Hamiltonian includes a tunneling term between the modes of the symmetric part and the left-over ones.",
"This tunneling process can transform protected to unprotected modes and destroys the effect of the dissipation-free subspace.",
"However, if the tunneling between both parts of the lattice occurs slowly compared to the rate of particle losses at the defect, then one expects a slow-down of particle losses for the modes with even quasi-momentum labeling number.",
"In such a case, the system approximately resembles the behavior of the ideal case described in the first section, at least for short times.",
"We therefore analyze this case in a similar way as the case of small interactions.",
"The tunneling of particle from modes with even quasi-momentum labeling number to lossy modes is slow compared to the losses $\\Gamma $ affecting the latter provided, $\\frac{J}{\\Gamma } \\, \\text{Max} [|\\mathcal {R}|] \\ll 1 .$ Inserting $\\Gamma = \\gamma /(2 r)$ and the values of $\\mathcal {R}$ , this ratio reads, $\\frac{J}{\\Gamma } \\, \\text{Max} [|\\mathcal {R}|] = \\frac{J}{\\gamma } \\sqrt{\\frac{2r}{N-2r}}.$ This ratio depends on the relative magnitudes of the parameters $ J $ and $ \\gamma $ but also on the size of the asymmetry.",
"In the case of small asymmetry, where $ N \\simeq 2r $ , the ratio is approximately written, $\\frac{J}{\\Gamma } \\, \\text{Max} [|\\mathcal {R}|] \\approx \\frac{J}{\\gamma } \\sqrt{N}.$ This indicates that there is no slow down of particle losses as long as $ \\gamma \\gg J \\sqrt{N} $ .",
"In the opposite case, where the asymmetry is large, i.e.",
"$ N > 2r $ , we approximately have, $\\frac{J}{\\Gamma } \\, \\text{Max} [|\\mathcal {R}|] \\approx \\frac{J}{\\gamma } \\sqrt{\\frac{2r}{N}}$ and since the ratio $ 2r /N $ is smaller than one, we have a slow-down if $ \\gamma > J $ ."
],
[
"conclusion",
"In this article we have considered the Bose-Hubbard model with particle losses at one lattice site.",
"For the non-interacting case, we found that particles in normal modes with vanishing amplitude at the dissipative defect are not affected by the localized losses.",
"For a one-dimensional model with the lossy site exactly in the center of the chain, half the modes thus form a dissipation free subspace.",
"This behavior can be attributed to a destructive interference of particles tunneling into the defect.",
"Furthermore a fraction of the particles can propagate across the dissipative defect even if the rate of tunneling between adjacent lattice sites is much slower than the loss rate at the defect.",
"To estimate the robustness of the features we predict, we have analyzed the effect of small particle interactions and deviations from a perfectly symmetric setup.",
"Our findings could be studied experimentally with ultracold bosonic atoms in an optical lattice, where an electron beam on a single lattice site ionizes atoms that are then extracted by an electrostatic field.",
"MJH acknowledges fruitful discussions with Herwig Ott.",
"This work is part of the Emmy Noether project HA 5593/1-1 and the CRC 631, both funded by the German Research Foundation, DFG."
]
] | 1204.0981 |
[
[
"What are the numbers in which spacetime?"
],
[
"Abstract Within an axiomatic framework, we investigate the possible structures of numbers (as physical quantities) in different theories of relativity."
],
[
"Introduction",
"Basically, we would like to investigate the following metaphysical question: What are the numbers in the physical world?",
"Without making this question more precise we can make the following two natural guesses which contradict each other: Obviously, the physical numbers are the real (or the complex) numbers since at least 99% of our physical theories are using these numbers.",
"Obviously, the set of physical numbers is a subset of the rational numbers (or even the integers) since the outcomes of the measurements have finite decimal representations.",
"Clearly, this informal level is too naive to meaningfully investigate our question.",
"However, that does not mean that it is impossible to scientifically investigate our question within some logical framework.",
"In this paper, we are going to reformulate and investigate this question (restricted to spacetime theories) within a rigorous logical framework.",
"First of all, what do numbers have to do with the geometry of spacetime?",
"The concepts related to numbers can be defined by the concepts of geometry by Hilbert's coordinatization, see, e.g., [12].",
"Moreover, purely geometrical statements can correspond to statements about the structure of numbers.",
"For example, in Cartesian planes over ordered fields, the statement “every line which contains a point from the interior of a circle intersects the circle” is equivalent to that “every positive number has a square root,” see, e.g., [13].",
"In the spirit of this example, here we investigate the question “How are some properties of spacetime reflected on the structure of numbers?” Among others, we will see axioms on observers also implying that positive numbers have square roots.",
"Ordered fields in which positive numbers have square roots are called Euclidean fields, which got their names after their role in Tarski's first-order logic axiomatization of Euclidean geometry [29].",
"Let $\\mathsf {Th}$ be a theory of space-time which contains the concept of numbers (as physical quantities) together with some algebraic operations on them, such as addition ($+$ ), multiplication ($\\cdot $ ) (or at least these concepts are definable in $\\mathsf {Th}$.).",
"In this case, we can introduce notation $\\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ for the class of the quantity parts (quantity structures) of the models of theory $\\mathsf {Th}$: $\\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})=\\lbrace \\text{The quantity parts of the models of \\leavevmode {\\color {axcolor}$\\mathsf {Th}$}}\\rbrace .$ We use the notation $\\mathfrak {Q}\\in \\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ for algebraic structure $\\mathfrak {Q}$ the same way as the model theoretic notation $\\mathfrak {Q}\\in Mod(\\leavevmode {\\color {axcolor}\\mathsf {AxField}})$ , e.g., $\\mathbb {Q}\\in \\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ means that $\\mathbb {Q}$ , the field of rational numbers, can be the structure of quantities (numbers) in $\\mathsf {Th}$.",
"Now we can scientifically investigate the question “What are the numbers in physical theory $\\mathsf {Th}$?” by studying what algebraic structures occur in $\\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ .",
"In this paper, we investigate this question only in the case when $\\mathsf {Th}$ is a theory of spacetimes.",
"However, this question can be investigated in any other physical theory the same way.",
"We will see that the answer to our question often depends on the dimension of spacetime.",
"Therefore, we will introduce notation $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ at page for the class of the possible quantity structures of theory $\\mathsf {Th}$ if all the investigated spacetimes are $n$ -dimensional.",
"In the logic language of Section , we will introduce several theories and axioms of relativity theory.",
"For example, our starting axiom system for special relativity (called $\\mathsf {SpecRel}$, see page REF ) captures the kinematics of special relativity perfectly, see Theorem REF and Corollary REF .",
"Furthermore, without any extra assumptions $\\mathsf {SpecRel}$ has a model over every ordered field, i.e., $\\textit {Num}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}})=\\lbrace \\,\\text{ordered fields} \\,\\rbrace ,$ see Remark REF .",
"Therefore, $\\mathsf {SpecRel}$ has a model over $\\mathbb {Q}$ , too.",
"However, if we assume that inertial observes can move with arbitrary speed less than that of light (in any direction every where), see $\\mathsf {AxThExp}$ at page , then every positive number has to have a square root if $n\\ge 3$ by Theorem REF , i.e., $\\textit {Num}_n(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})=\\lbrace \\text{\\,Euclidean fields\\,}\\rbrace .$ In particular, the number structure cannot be the field of rational numbers, but it can be the field of real algebraic numbers.",
"We will also see that our axiom system of special relativity has a model over $\\mathbb {Q}$ if we assume axiom $\\mathsf {AxThExp}$ only approximately (which is reasonable as we cannot be sure in anything perfectly accurately in physics), see Theorem REF , Corollary REF and Conjecture REF .",
"It is interesting that, if the spacetime dimension is 3, then we do not need the symmetry axiom of $\\mathsf {SpecRel}$ to prove that every positive number has a square root if $\\mathsf {AxThExp}$ is assumed, see Theorem REF .",
"However, in even dimensions, it is possible that some numbers do not have square roots, see Theorem REF and Questions REF and REF .",
"Moving toward general relativity we will see that our theory of accelerated observes ($\\mathsf {AccRel}$) requires the structure of quantities to be a real closed field, i.e., a Euclidean field in which every odd degree polynomial has a root, see Theorem REF .",
"However, any real closed field, e.g., the field of real algebraic numbers, can be the quantity structure of $\\mathsf {AccRel}$.",
"If we extend $\\mathsf {AccRel}$ by extra axiom $\\mathsf {Ax\\exists UnifOb}$ stating that there are uniformly accelerated observers, then the field of real algebraic numbers cannot be the structure of quantities any more if $n\\ge 3$ , see Theorem REF .",
"A surprising consequence of this result is that $\\textit {Num}_n(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}}+\\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ is not a first-order logic definable class of fields, see Remark REF .",
"In Section , we introduce an axiom system of general relativity $\\mathsf {GenRel}$ and investigate our question a bit for $\\mathsf {GenRel}$."
],
[
"The language of our theories",
"To investigate our reformulated question, we need an axiomatic theory of spacetimes.",
"The first important decision in writing up an axiom system is to choose the set of basic symbols of our logic language, i.e., what objects and relations between them we will use as basic concepts.",
"Here we will use the following two-sortedThat our theory is two-sorted means only that there are two types of basic objects (bodies and quantities) as opposed to, e.g., Zermelo–Fraenkel set theory where there is only one type of basic objects (sets).",
"language of first-order logic (FOL) parametrized by a natural number $d\\ge 2$ representing the dimension of spacetime: $\\lbrace \\, \\mathit {B},\\mathit {Q}\\,; \\mathsf {Ob}, \\mathsf {IOb}, \\mathsf {Ph},+,\\cdot ,\\le ,\\mathsf {W}\\,\\rbrace ,$ where $\\mathit {B}$ (bodiesBy bodies we mean anything which can move, e.g., test-particles, reference frames, electromagnetic waves, centers of mass, etc.)",
"and $\\mathit {Q}$ (quantities) are the two sorts, $\\mathsf {Ob}$ (observers), $\\mathsf {IOb}$ (inertial observers) and $\\mathsf {Ph}$ (light signals) are one-place relation symbols of sort $\\mathit {B}$ , $+$ and $\\cdot $ are two-place function symbols of sort $\\mathit {Q}$ , $\\le $ is a two-place relation symbol of sort $\\mathit {Q}$ , and $\\mathsf {W}$ (the worldview relation) is a $d+2$ -place relation symbol the first two arguments of which are of sort $\\mathit {B}$ and the rest are of sort $\\mathit {Q}$ .",
"Relations $\\mathsf {Ob}(o)$ , $\\mathsf {IOb}(m)$ and $\\mathsf {Ph}(p)$ are translated as “$o$ is an observer,” “$m$ is an inertial observer,” and “$p$ is a light signal,” respectively.",
"To speak about coordinatization of observers, we translate relation $\\mathsf {W}(k,b,x_1,x_2,\\ldots ,x_d)$ as “body $k$ coordinatizes body $b$ at space-time location $\\langle x_1,x_2,\\ldots ,x_d\\rangle $,” (i.e., at space location $\\langle x_2,\\ldots ,x_d\\rangle $ and instant $x_1$ ).",
"Quantity terms are the variables of sort $\\mathit {Q}$ and what can be built from them by using the two-place operations $+$ and $\\cdot $ , body terms are only the variables of sort $\\mathit {B}$ .",
"$\\mathsf {IOb}(m)$ , $\\mathsf {Ph}(p,b)$ , $\\mathsf {W}(m,b,x_1,\\ldots ,x_d)$ , $x=y$ , and $x\\le y$ where $m$ , $p$ , $b$ , $x$ , $y$ , $x_1$ , ..., $x_d$ are arbitrary terms of the respective sorts are so-called atomic formulas of our first-order logic language.",
"The formulas are built up from these atomic formulas by using the logical connectives not ($\\lnot $ ), and ($\\wedge $ ), or ($\\vee $ ), implies ($\\rightarrow $ ), if-and-only-if ($\\leftrightarrow $ ) and the quantifiers exists ($\\exists $ ) and for all ($\\forall $ ).",
"To make them easier to read, we omit the outermost universal quantifiers from the formalizations of our axioms, i.e., all the free variables are universally quantified.",
"We use the notation $\\mathit {Q}^n$ for the set of all $n$ -tuples of elements of $\\mathit {Q}$ .",
"If $\\bar{x}\\in \\mathit {Q}^n$ , we assume that $\\bar{x}=\\langle x_1,\\ldots ,x_n\\rangle $ , i.e., $x_i$ denotes the $i$ -th component of the $n$ -tuple $\\bar{x}$ .",
"Specially, we write $\\mathsf {W}(m,b,\\bar{x})$ in place of $\\mathsf {W}(m,b,x_1,\\dots ,x_d)$ , and we write $\\forall \\bar{x}$ in place of $\\forall x_1\\dots \\forall x_d$ , etc.",
"We use first-order logic set theory as a meta theory to speak about model theoretical terms, such as models, validity, etc.",
"The models of this language are of the form ${\\mathfrak {M}} = \\langle \\mathit {B}, \\mathit {Q};\\mathsf {Ob}_\\mathfrak {M}, \\mathsf {IOb}_\\mathfrak {M},\\mathsf {Ph}_\\mathfrak {M},+_\\mathfrak {M},\\cdot _\\mathfrak {M},\\le _\\mathfrak {M},\\mathsf {W}_\\mathfrak {M}\\rangle ,$ where $\\mathit {B}$ and $\\mathit {Q}$ are nonempty sets, $\\mathsf {Ob}_\\mathfrak {M}$ , $\\mathsf {IOb}_\\mathfrak {M}$ and $\\mathsf {Ph}_\\mathfrak {M}$ are subsets of $\\mathit {B}$ , $+_\\mathfrak {M}$ and $\\cdot _\\mathfrak {M}$ are binary functions and $\\le _\\mathfrak {M}$ is a binary relation on $\\mathit {Q}$ , and $\\mathsf {W}_\\mathfrak {M}$ is a subset of $\\mathit {B}\\times \\mathit {B}\\times \\mathit {Q}^d$ .",
"Formulas are interpreted in $\\mathfrak {M}$ in the usual way.",
"For the precise definition of the syntax and semantics of first-order logic, see, e.g., [7], [10]."
],
[
"Numbers required by special relativity",
"In this section, we will investigate our main question within special relativity.",
"To do so, first we formulate axioms for special relativity in the logic language of the previous section.",
"Since the language above contains the concept of quantities (and that of addition, multiplication and ordering), we can formulate statements about numbers directly.",
"In our first axiom, we state some basic properties of addition, multiplication and ordering true for real numbers.Using axiom $\\mathsf {AxOFiled}$ instead of assuming that the structure of quantities is the field of real numbers not just makes our theory more flexible, but also makes it possible to investigate our main question.",
"$\\mathsf {AxOField}$ The quantity part $\\langle \\mathit {Q},+,\\cdot ,\\le \\rangle $ is an ordered field, i.e., $\\langle \\mathit {Q},+,\\cdot \\rangle $ is a field in the sense of abstract algebra; and the relation $\\le $ is a linear ordering on $\\mathit {Q}$ such that i) $x \\le y\\rightarrow x + z \\le y + z$ and ii) $0 \\le x \\wedge 0 \\le y\\rightarrow 0 \\le xy$ holds.",
"$\\mathsf {AxOField}$ is a “mathematical\" axiom in spirit.",
"However, it has physical (even empirical) relevance.",
"Its physical relevance is that we can add and multiply the outcomes of our measurements and some basic rules apply to these operations.",
"Physicists use all properties of the real numbers tacitly, without stating explicitly which property is assumed and why.",
"The two properties of real numbers which are the most difficult to defend from empirical point of view are the Archimedean property, see [22], [23], [25],[24], and the supremum property,The supremum property (i.e., every nonempty and bounded subset of the numbers has a least upper bound) implies the Archimedean property.",
"So if we want to get ourselves free from the Archimedean property, we have to leave this one, too.",
"see the remark after the introduction of $\\mathsf {CONT}$ on page .",
"The rest of our axioms on special relativity will speak about the worldviews of inertial observers.",
"To formulate them, we use the following concepts.",
"The time difference of coordinate points $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ is defined as: $\\mathsf {time}(\\bar{x},\\bar{y}):=x_1-y_1.$ To speak about the spatial distance of any two coordinate points, we have to use squared distance since it is possible that the distance of two points is not amongst the quantities.",
"For example, the distance of points $\\langle 0,0\\rangle $ and $\\langle 1,1\\rangle $ is $\\sqrt{2}$ .",
"So in the field of rational numbers, $\\langle 0,0\\rangle $ and $\\langle 1,1\\rangle $ do not have distance but they have squared distance.",
"Therefore, we define the squared spatial distance of $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ as: $\\mathsf {space}^2(\\bar{x},\\bar{y}):=(x_2-y_2)^2+\\ldots +(x_d-y_d)^2.$ We denote the origin of $\\mathit {Q}^n$ by $\\bar{o}$ , i.e., $\\bar{o}:=\\langle 0,\\ldots ,0\\rangle $ .",
"The next axiom is the key axiom of our axiom system for special relativity, it has an immediate physical meaning.",
"This axiom is the outcome of the Michelson-Morley experiment.",
"It has been continuously tested ever since then.",
"Nowadays it is tested by GPS technology.",
"$\\mathsf {AxPh}$ For any inertial observer, the speed of light is the same everywhere and in every direction (and it is finite).",
"Furthermore, it is possible to send out a light signal in any direction (existing according to the coordinate system) everywhere: $\\mathsf {IOb}(m)\\rightarrow \\exists c_m\\Big [c_m>0\\wedge \\forall \\bar{x}\\bar{y}\\\\ \\big (\\exists p \\big [\\mathsf {Ph}(p)\\wedge \\mathsf {W}(m,p,\\bar{x})\\wedge \\mathsf {W}(m,p,\\bar{y})\\big ] \\leftrightarrow \\mathsf {space}^2(\\bar{x},\\bar{y})=c_m^2\\cdot \\mathsf {time}(\\bar{x},\\bar{y})^2\\big )\\Big ].$ Let us note here that $\\mathsf {AxPh}$ does not require (by itself) that the speed of light is the same for every inertial observer.",
"It requires only that the speed of light according to a fixed inertial observer is a positive quantity which does not depend on the direction or the location.",
"By $\\mathsf {AxPh}$, we can define the speed of light according to inertial observer $m$ as the following binary relation: $\\mathsf {c}(m,v)\\ \\stackrel{\\;def}{\\Longleftrightarrow }\\ v>0 \\wedge \\forall \\bar{x}\\bar{y}\\big (\\exists p \\big [\\mathsf {Ph}(p)\\wedge \\mathsf {W}(m,p,\\bar{x})\\wedge \\mathsf {W}(m,p,\\bar{y})\\big ]\\\\\\rightarrow \\mathsf {space}^2(\\bar{x},\\bar{y})= v^2\\cdot \\mathsf {time}(\\bar{x},\\bar{y})^2\\big ).$ By $\\mathsf {AxPh}$, there is one and only one speed $v$ for every inertial observer $m$ such that $\\mathsf {c}(m,v)$ holds.",
"From now on, we will denote this unique speed by $\\mathsf {c}_m$ .",
"Our next axiom connects the worldviews of different inertial observers by saying that all observers coordinatize the same “external\" reality (the same set of events).",
"By the event occurring for observer $m$ at point $\\bar{x}$ , we mean the set of bodies $m$ coordinatizes at $\\bar{x}$ : $\\mathsf {ev}_m(\\bar{x}):=\\lbrace b : \\mathsf {W}(m,b,\\bar{x})\\rbrace .$ $\\mathsf {AxEv}$ All inertial observers coordinatize the same set of events: $\\mathsf {IOb}(m)\\wedge \\mathsf {IOb}(k)\\rightarrow \\exists \\bar{y}\\, \\forall b\\big [\\mathsf {W}(m,b,\\bar{x})\\leftrightarrow \\mathsf {W}(k,b,\\bar{y})\\big ].$ From now on, we will use $\\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_k(\\bar{y})$ to abbreviate the subformula $\\forall b [\\mathsf {W}(m,b,\\bar{x})\\leftrightarrow \\mathsf {W}(k,b,\\bar{y})]$ of $\\mathsf {AxEv}$.",
"The next two axioms are only simplifying ones.",
"$\\mathsf {AxSelf}$ Any inertial observer is stationary relative to himself: $\\mathsf {IOb}(m)\\rightarrow \\forall \\bar{x}\\big [\\mathsf {W}(m,m,\\bar{x}) \\leftrightarrow x_2=\\ldots =x_d=0\\big ].$ Our last axiom on inertial observers is a symmetry axiom saying that they use the same units of measurement.",
"$\\mathsf {AxSymD}$ Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers: $\\mathsf {IOb}(m)\\wedge \\mathsf {IOb}(k) \\wedge x_1=y_1\\wedge x^{\\prime }_1=y^{\\prime }_1\\wedge \\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_k(\\bar{x}^{\\prime })\\\\ \\wedge \\mathsf {ev}_m(\\bar{y})=\\mathsf {ev}_k(\\bar{y}^{\\prime })\\rightarrow \\mathsf {space}^2(\\bar{x},\\bar{y})=\\mathsf {space}^2(\\bar{x}^{\\prime },\\bar{y}^{\\prime }),\\text{ and }\\\\\\mathsf {IOb}(m)\\rightarrow \\exists p\\big [\\mathsf {Ph}(p)\\wedge \\mathsf {W}(m,p,0,\\ldots ,0)\\wedge \\mathsf {W}(m,p,1,1,0,\\ldots ,0)\\big ].$ Let us introduce an axiom system for special relativity as the collection of the axioms above, if $d\\ge 3$ : $\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} :=\\lbrace \\leavevmode {\\color {axcolor}\\mathsf {AxOField}}, \\leavevmode {\\color {axcolor}\\mathsf {AxPh}}, \\leavevmode {\\color {axcolor}\\mathsf {AxEv}}, \\leavevmode {\\color {axcolor}\\mathsf {AxSelf}},\\leavevmode {\\color {axcolor}\\mathsf {AxSymD}}\\rbrace .$ In relativity theory, we are often interested in comparing the worldviews of two different observers.",
"To do so, we introduce the worldview transformation between observers $m$ and $k$ (in symbols, $\\mathsf {w}_{mk}$ ) as the binary relation on $\\mathit {Q}^d$ connecting the coordinate points where $m$ and $k$ coordinatize the same (nonempty) events: $\\mathsf {w}_{mk}(\\bar{x},\\bar{y})\\ \\stackrel{\\;def}{\\Longleftrightarrow }\\ \\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_k(\\bar{y})\\ne \\emptyset .$ Map $P:\\mathit {Q}^d\\rightarrow \\mathit {Q}^d$ is called a Poincaré transformation iff it is an affine bijection having the following property $\\mathsf {time}(\\bar{x},\\bar{y})^2-\\mathsf {space}^2(\\bar{x},\\bar{y})=\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2-\\mathsf {space}^2(\\bar{x}^{\\prime },\\bar{y}^{\\prime })$ for all $\\bar{x},\\bar{y},\\bar{x}^{\\prime },\\bar{y}^{\\prime }\\in \\mathit {Q}^d$ for which $P(\\bar{x})=\\bar{x}^{\\prime }$ and $P(\\bar{y})=\\bar{y}^{\\prime }$ .",
"Theorem REF shows that our streamlined axiom system $\\mathsf {SpecRel}$ perfectly captures the kinematics of special relativity since it implies that the worldview transformations between inertial observers are the same as in the standard non-axiomatic approaches.",
"Theorem 3.1 Let $d\\ge 3$ .",
"Assume $\\mathsf {SpecRel}$.",
"Then $\\mathsf {w}_{mk}$ is a Poincaré transformation if $m$ and $k$ are inertial observers.Actually, axioms $\\mathsf {AxOField}$, $\\mathsf {AxPh}$, $\\mathsf {AxEv}$, and $\\mathsf {AxSymD}$ are enough to prove this statement, see Theorem REF .",
"We postpone the proof of Theorem REF to Section , where we will prove a slightly stronger result, see Theorem REF .",
"For a similar result over Euclidean fields, see, e.g., [3], [4], [26].",
"The so-called worldline of body $b$ according to observer $m$ is defined as follows: $\\mathsf {wl}_m(b):=\\lbrace \\bar{x}: \\mathsf {W}(m,b,\\bar{x})\\rbrace .$ Corollary 3.2 Let $d\\ge 3$ .",
"Assume $\\mathsf {SpecRel}$.",
"The $\\mathsf {wl}_m(k)$ is a straight line if $m$ and $k$ are inertial observers.Axioms $\\mathsf {AxOField}$, $\\mathsf {AxPh}$, $\\mathsf {AxEv}$, and $\\mathsf {AxSelf}$ are enough to prove this statement since, by Theorem REF , axioms $\\mathsf {AxOField}$, $\\mathsf {AxPh}$, and $\\mathsf {AxEv}$ imply that the worldview transformations take lines to lines and $\\mathsf {w}_m(k)$ is the $\\mathsf {w}_{km}$ image of the time-axis by axiom $\\mathsf {AxSelf}$.",
"Let $m$ and $k$ be inertial observers.",
"The squared speed of $k$ according to $m$ is defined as follows: $\\mathsf {speed}^2(m,k,v) \\ \\stackrel{\\;def}{\\Longleftrightarrow }\\ \\\\ \\exists \\bar{x}\\bar{y}\\big [\\bar{x}\\ne \\bar{y}\\wedge \\mathsf {W}(m,k,\\bar{x})\\wedge \\mathsf {W}(m,k,\\bar{y})\\wedge \\mathsf {space}^2(\\bar{x},\\bar{y})=v\\cdot \\mathsf {time}(\\bar{x},\\bar{y})^2\\big ].$ By Corollary REF , $\\mathsf {SpecRel}$ implies that, for each $m,k\\in \\mathsf {IOb}$ , there is one and only one $v$ such that $\\mathsf {speed}^2(m,k,v)$ holds.",
"From now on let us denote this unique $v$ by $\\mathsf {speed}^2_m(k)$ .",
"Remark 3.3 Even if $\\langle \\mathit {Q},+,\\cdot ,\\le \\rangle $ is the ordered field of rational numbers, it is possible that the squared speed of an observer is 2.",
"For example, $\\mathsf {speed}^2_m(k)=2$ if $d=3$ and inertial observers $k$ goes trough points $\\langle 0,0,0\\rangle ,\\langle 1,1,1\\rangle \\in \\mathbb {Q}^3$ according to inertial observer $m$ .",
"However, some quantity cannot be the squared speed in some fields.",
"For example, the squared speed cannot be 3 if $\\langle \\mathit {Q},+,\\cdot ,\\le \\rangle $ is the ordered field of rational numbers and $d=3$ .",
"This is so, because the equation $x^2+y^2=3 z^2$ does not have a nonzero solution over the natural numbers (if $x$ , $y$ and $z$ are solutions, then $x$ , $y$ , and $z$ are divisible by $3^n$ for all natural numbers $n$ ; hence $x=y=z=0$ ).",
"Consequently, it does not have a nonzero solution over the field of rational numbers.",
"Corollary REF states basically that relatively moving inertial observers' clocks slow down by the Lorentz factor $\\gamma =(1-v^2/c^2)^{-1/2}$ where $v$ is the relative speed of the observers.",
"Corollary 3.4 Let $d\\ge 3$ .",
"Assume $\\mathsf {SpecRel}$.",
"Let $m,k\\in \\mathsf {IOb}$ and let $\\bar{x},\\bar{y},\\bar{x}^{\\prime },\\bar{y}^{\\prime }\\in \\mathit {Q}^d$ such that $\\bar{x},\\bar{y}\\in \\mathsf {wl}_k(k)$ , $\\mathsf {w}_{km}(\\bar{x})=\\bar{x}^{\\prime }$ and $\\mathsf {w}_{km}(\\bar{y})=\\bar{y}^{\\prime }$ .",
"Then $\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2 =\\frac{\\mathsf {time}(\\bar{x},\\bar{y})^2}{1-\\mathsf {speed}^2_m(k)}.$ Formula (REF ) is always defined since $\\mathsf {speed}^2_m(k)$ cannot be 1 by Theorem REF .",
"The case $\\bar{x}=\\bar{y}$ is trivial since, in this case, both $\\mathsf {time}(\\bar{x},\\bar{y})$ and $\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })$ are 0.",
"So let us assume that $\\bar{x}\\ne \\bar{y}$ .",
"Since $\\bar{x},\\bar{y}\\in \\mathsf {wl}_k(k)$ , we have that $\\mathsf {space}^2(\\bar{x},\\bar{y})=0$ by $\\mathsf {AxSelf}$.",
"By Theorem REF , $\\mathsf {w}_{km}$ is a Poincaré transformation.",
"Therefore, $\\mathsf {time}(\\bar{x},\\bar{y})^2=\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2-\\mathsf {space}^2(\\bar{x}^{\\prime },\\bar{y}^{\\prime }).$ Consequently, $\\mathsf {time}(\\bar{x},\\bar{y})^2=\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2\\left(1-\\frac{\\mathsf {space}^2(\\bar{x}^{\\prime },\\bar{y}^{\\prime })}{\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2}\\right).$ Hence, by the definition of $\\mathsf {speed}^2_m(k)$ , we get $\\mathsf {time}(\\bar{x},\\bar{y})^2=\\mathsf {time}(\\bar{x}^{\\prime },\\bar{y}^{\\prime })^2\\left(1-\\mathsf {speed}^2_m(k)\\right).$ since $\\mathsf {w}_{km}(\\bar{x})\\ne \\mathsf {w}_{km}(\\bar{y})$ and $\\mathsf {w}_{km}(\\bar{x}),\\mathsf {w}_{km}(\\bar{y})\\in \\mathsf {wl}_m(k)$ .",
"Theorem REF and its consequences show that $\\mathsf {SpecRel}$ captures special relativity well over every ordered field.",
"It is a natural question to ask what happens with these theorems if we assume less about the quantities.",
"This is one side of the question “what are the numbers?”, which is a whole research direction: Question 3.5 (Research direction) What remains from the theorems of $\\mathsf {SpecRel}$, if we replace ordered fields with other algebraic structures, e.g., with ordered rings?",
"Here we concentrate on the other side of our question; namely, “how can some physical assumptions implicitly enrich the structure of quantities?”.",
"To investigate this question, let us now introduce notation $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ for the class of the quantity parts of the models of theory $\\mathsf {Th}$ if $d=n$ : $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})=\\lbrace \\text{The quantity parts }\\\\ \\langle \\mathit {Q},+,\\cdot ,\\le \\rangle \\text{ of the models of \\leavevmode {\\color {axcolor}$\\mathsf {Th}$} if }d=n \\rbrace .$ The same way we use the notation $\\mathfrak {Q}\\in \\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {Th}})$ for ordered field $\\mathfrak {Q}$ as the model theoretic notation $\\mathfrak {Q}\\in Mod(\\leavevmode {\\color {axcolor}\\mathsf {AxField}})$ .",
"$\\mathsf {AxThExp}$ Inertial observers can move along any straight line with any speed less than the speed of light: $\\exists h \\mathsf {IOb}(h)\\wedge \\big (\\mathsf {IOb}(m)\\wedge \\mathsf {space}^2(\\bar{x},\\bar{y})<\\mathsf {c}_m^2\\cdot \\mathsf {time}(\\bar{x},\\bar{y})^2\\\\ \\rightarrow \\exists k \\big [\\mathsf {IOb}(k)\\wedge \\mathsf {W}(m,k,\\bar{x})\\wedge \\mathsf {W}(m,k,\\bar{y})\\big ]\\big ).$ Theorem REF below shows that axiom $\\mathsf {AxThExp}$ implies that positive numbers have square roots if $\\mathsf {SpecRel}$ is assumed.",
"Theorem 3.6 If $n\\ge 3$ , $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})=\\lbrace \\text{\\,Euclidean fields\\,}\\rbrace .$ By Theorem 3.8.7 of [2], we have that $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp}$ has a model over every Euclidean field.",
"Consequently, $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})\\supseteq \\lbrace \\text{\\,Euclideanfields\\,}\\rbrace .$ To show the converse inclusion, we have to prove that every positive quantity has a square root in every model of $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp}$.",
"To do so, let $x\\in Q$ be a positive quantity.",
"We have to show that $x$ has a square root in $\\mathit {Q}$ .",
"First we will prove that $1-v^2$ has a square root if $v\\in \\mathit {Q}$ and $0\\le v< 1$ .",
"To do so, let $v\\in \\mathit {Q}$ for which $0\\le v<1$ .",
"Let $\\bar{y}=\\langle 1, v,0,\\ldots ,0\\rangle $ .",
"By $\\mathsf {AxTheExp}$ there are inertial observers $m$ and $k$ such that $\\bar{o},\\bar{y}\\in \\mathsf {wl}_m(k)$ .",
"By Corollary REF , $\\mathsf {wl}_m(k)$ is a line.",
"Thus $\\mathsf {speed}^2_m(k)=v^2$ .",
"Therefore, there is a $z\\in \\mathit {Q}$ such that $1-v^2=z^2$ (i.e., $1-v^2$ has a square root in $\\mathit {Q}$ ) by $\\mathsf {AxField}$ and Corollary REF .",
"From $\\mathsf {AxField}$, it is easy to show that $x=\\left({\\frac{x+1}{2}}\\right)^2\\cdot \\left(1-\\left({\\frac{x-1}{x+1}}\\right)^2\\right)$ for all $x\\in \\mathit {Q}$ .",
"There is a $z\\in \\mathit {Q}$ such that $1-\\left(\\frac{x-1}{x+1}\\right)^2=z^2$ since $0\\le 1-\\left({\\frac{x-1}{x+1}}\\right)^2<1$ .",
"So there is a quantity, namely $\\frac{x+1}{2}\\cdot z$ , which is the square root of $x$ ; and that is what we wanted to prove.",
"Remark 3.7 Axiom $\\mathsf {AxThExp}$ cannot be omitted from Theorem REF since $\\mathsf {SpecRel}$ has a model over every ordered field, i.e., $\\textit {Num}_n(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}})=\\lbrace \\,\\text{ordered fields} \\,\\rbrace $ for all $n\\ge 2$ .",
"Moreover, it also has non trivial models in which there are several observers moving relative to each other.",
"We conjecture that there is a model of $\\mathsf {SpecRel}$ such that the possible speeds of observers are dense in interval $[0,1]$ , see Corollary REF and Conjecture REF at pages REF and REF .",
"In the proof of Theorem REF , axiom $\\mathsf {AxSymD}$ is strongly used since $\\mathsf {SpecRel}$ without $\\mathsf {AxSymD}$ does not imply the exact ratio of the slowing down of moving clocks; $\\mathsf {SpecRel}$ without $\\mathsf {AxSymD}$ only implies that at least one of two relatively moving inertial observers' clocks run slow according to the other, see [2].",
"So it is natural to investigate what remains of Theorem REF if we leave the symmetry axiom out.",
"It is surprising but, in the case of $d=3$ , Theorem REF remains valid even if we assume only $\\mathsf {\\mathsf {c}_m=1}$ from $\\mathsf {AxSymD}$, see Andréka–Madarász–Németi [2].",
"Now we will show that even the assumption $\\mathsf {\\mathsf {c}_m=1}$ is not necessary.",
"To do so, let us introduce the next axiom system $\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}}=\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}}-\\leavevmode {\\color {axcolor}\\mathsf {AxSymD}}.$ Theorem 3.8 $\\textit {Num}_{3}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})=\\lbrace \\text{\\,Euclideanfields\\,}\\rbrace $ By Theorem REF , $\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}}$ + $\\leavevmode {\\color {axcolor}\\mathsf {AxThExp}}$ has a model over every Euclidean field since even $\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}}$ + $\\leavevmode {\\color {axcolor}\\mathsf {AxThExp}}$ has one.",
"So $\\textit {Num}_{3}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}} +\\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})\\supseteq \\lbrace \\text{{Euclidean fields\\,}}\\rbrace .$ To prove the converse inclusion, we have to prove that the quantity structure of every model of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ is a Euclidean field if $d=3$ .",
"By Theorem 3.6.17 of [2], the quantity structures of the models of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ + $\\mathsf {\\mathsf {c}_m=1}$ are Euclidean fields if $d=3$ .",
"Therefore, it is enough to show that a model of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ + $\\mathsf {\\mathsf {c}_m=1}$ can be constructed from every model of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ without changing its quantity structure.",
"Let $\\mathfrak {M}$ be an arbitrary 3 dimensional model of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$.",
"Let $\\mathfrak {M}^+$ be the model which is constructed from $\\mathfrak {M}$ by rescaling the coordinatization of each inertial observer $m$ of $\\mathfrak {M}$ by the following map $\\bar{x}\\mapsto \\langle \\mathsf {c}_m x_1,x_2,\\ldots x_d\\rangle $ , i.e., rescaling the time of $m$ by the factor $\\mathsf {c}_m$ .",
"It is clear that the speed of light becomes 1 according to $m$ after the rescaling.",
"So $\\mathsf {\\mathsf {c}_m=1}$ holds in $\\mathfrak {M}^+$ .",
"It is also easy to see that this rescaling does not change the validity of $\\mathsf {AxThExp}$ and the other axioms of $\\mathsf {SpecRel_0}$.",
"Therefore, $\\mathfrak {M}^+$ is a model of axiom system $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ + $\\mathsf {\\mathsf {c}_m=1}$.",
"By the construction, the quantity parts of $\\mathfrak {M}^+$ and $\\mathfrak {M}$ are the same.",
"Consequently, the quantity part of $\\mathfrak {M}$ is a Euclidean field.",
"This completes our proof since $\\mathfrak {M}$ was an arbitrary model of axiom system $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$.",
"Until recently, it was unsolved whether Theorem REF is valid or not in any higher dimension (see [2]) when Hajnal Andréka has provided counterexamples in the even dimensions, i.e., the following is true: Theorem 3.9 $\\textit {Num}_{2k}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}} +\\leavevmode {\\color {axcolor}\\mathsf {AxThExp}} + \\mathsf {c}_m=1)\\supsetneqq \\lbrace \\text{{Euclidean fields\\,}}\\rbrace $ For the proof of Theorem REF , see [6].",
"The existence of models of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ over non Euclidean fields is a surprising result since it is natural to conjecture that a 3 dimensional model can be constructed from any $d\\ge 4$ dimensional model of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ without changing its quantity structure (by “cutting out” a 3 dimensional part).",
"Clearly, such a construction would imply Theorem REF in any dimension higher than 3, too.",
"It is interesting to note that this kind of construction works if the quantity structure is a Euclidean field.",
"Theorem REF only shows that there are models of $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ over some non-Euclidean fields.",
"However, the question “what are the fields over which $\\mathsf {SpecRel_0}$ + $\\mathsf {AxThExp}$ has a model?” is still unsolved even in 4 dimension: Question 3.10 Exactly which ordered fields are the elements of the class $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel_0}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp}})$ if $n\\ge 4$ .",
"Without adding extra axioms to $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp}$, it does not imply that the structure of numbers has to be a Euclidean field if $d=2$ .",
"One of the reasons for this fact is that, if $d=2$ , the axioms of $\\mathsf {SpecRel}$ do not imply that the world lines of inertial observers are straight lines.",
"So we have to add it as an extra axiom stating this ($\\mathsf {AxLine}$).",
"For a precise formulation of $\\mathsf {AxLine}$, see, e.g., [4].",
"Another reason is that, if $d=2$ , there are no two events which are simultaneous according to two relatively moving observers.",
"Therefore, $\\mathsf {AxSymD}$ states nothing if $d=2$ .",
"So we have to change this axiom.",
"For example, we may replace $\\mathsf {AxSymD}$ with the statement “moving observers see each others clock the same way and $\\mathsf {\\mathsf {c}_m=1}$” ($\\mathsf {AxSymT}$).",
"For a precise formulation of the first part of $\\mathsf {AxSymT}$, see, e.g., [3], [26].",
"Actually, $\\mathsf {AxSymT}$ is equivalent to $\\mathsf {AxSymD}$ if $\\mathsf {SpecRel_0}$ + $\\mathsf {\\mathsf {c}_m=1}$ is assumed and $d\\ge 3$ , see, e.g., [26].",
"Question 3.11 Does $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp}$ + $\\mathsf {AxLine}$ + $\\mathsf {AxSymT}$ imply that the quantities form a Euclidean field if $d=2$ ?",
"If not, what further natural axioms we have to assume to prove that the quantities form a Euclidean field?",
"Since our measurements have only finite accuracy, it is natural to assume $\\mathsf {AxThExp}$ only approximately.",
"To introduce an approximated version of $\\mathsf {AxThExp}$, we need some definitions.",
"The space component of coordinate point $\\bar{x}\\in \\mathit {Q}^d$ is defined as $ \\bar{x}_s:=\\langle x_2,\\ldots ,x_d\\rangle $ .",
"The squared Euclidean distance of $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ is defined as $\\mathsf {dist}^2(\\bar{x},\\bar{y}):=(x_1-y_1)^2+\\ldots +(x_d-y_d)^2$ and the difference of $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ is defined as $\\bar{x}-\\bar{y}:=\\langle x_1-y_1,\\ldots ,x_d-y_d\\rangle .$ Let the squared Euclidean length of $\\bar{x}\\in \\mathit {Q}^d$ be defined as $\\mathsf {length}^2(\\bar{x}):={x_1^2+\\ldots +x_d^2}.$ $\\mathsf {AxThExp^-}$ Inertial observers can move roughly with any speed less than the speed of light roughly in any direction: $\\exists h \\mathsf {IOb}(h) \\wedge \\Big (\\mathsf {IOb}(m)\\wedge \\varepsilon >0 \\wedge \\mathsf {length}^2(\\bar{v}_s)<\\mathsf {c}_m^2 \\\\\\wedge v_1=1 \\rightarrow \\exists \\bar{w}\\Big [\\mathsf {dist}^2(\\bar{w},\\bar{v})<\\varepsilon \\wedge \\forall \\bar{x}\\bar{y}\\,\\exists \\lambda \\big ( \\bar{x}-\\bar{y}=\\lambda \\bar{w}\\\\ \\rightarrow \\exists k\\big [ \\mathsf {IOb}(m)\\wedge \\mathsf {W}(m,k,\\bar{y})\\wedge \\mathsf {W}(m,k,\\bar{y})\\big ]\\big )\\Big ]\\Big ).$ A model of $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp^-}$ can be constructed over the field of rational numbers, i.e., the following is true: Theorem 3.12 $\\mathbb {Q}\\in \\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp^-}})$ For the proof of Theorem REF , see [16].",
"An ordered field is called Archimedean ordered field iff for all $a$ , there is a natural number $n$ such that $a<\\underbrace{1+\\ldots +1}_n$ holds.",
"By Pickert–Hion Theorem, every Archimedean ordered field is isomorphic to subfield of the field of real numbers, see, e.g., [11], [18].",
"Consequently, the field of rational numbers is dense in any Archimedean ordered field since it is dense in the field of real numbers.",
"Therefore, the following is a corollary of Theorem REF .",
"Corollary 3.13 $\\lbrace \\text{Archimedean orderedfields\\,}\\rbrace \\!\\subsetneqq \\!\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}}\\!",
"+\\!\\leavevmode {\\color {axcolor}\\mathsf {AxThExp^-}})$ By Lövenheim–Skolem Theorem it is clear that $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}}\\!",
"+\\!",
"\\leavevmode {\\color {axcolor}\\mathsf {AxThExp^-}})$ cannot be the class of Archimedean ordered fields since it has elements of arbitrarily large cardinality while Archimedean ordered fields are subsets of the field of real numbers by Pickert–Hion Theorem.",
"The question “exactly which ordered fields can be the quantity structures of theory $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp^-}$?” is open.",
"We conjecture that there is a model of $\\mathsf {SpecRel}$ + $\\mathsf {AxThExp^-}$ over every ordered field, i.e.",
": Conjecture 3.14 $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {SpecRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxThExp^-}})=\\lbrace \\,\\text{ordered fields\\,}\\rbrace $"
],
[
"Numbers implied by accelerated observers",
"Now we are going to investigate what happens with the possible structures of quantities if we extend our theory $\\mathsf {SpecRel}$ with accelerated observers.",
"To do so, let us recall our first-order logic axiom system of accelerated observers $\\mathsf {AccRel}$.",
"The key axiom of $\\mathsf {AccRel}$ is the following: $\\mathsf {AxCmv}$ At each moment of its worldline, each observer sees the nearby world for a short while as an inertial observer does.",
"For formalization of $\\mathsf {AxCmv}$, see [26].",
"In $\\mathsf {AccRel}$ we will also use the following localized version of axioms $\\mathsf {AxEv}$ and $\\mathsf {AxSelf}$ of $\\mathsf {SpecRel}$.",
"$\\mathsf {AxEv^-}$ Observers coordinatize all the events in which they participate: $\\mathsf {Ob}(k)\\wedge \\mathsf {W}(m,k,\\bar{x})\\rightarrow \\exists \\bar{y}\\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_k(\\bar{y}).$ $\\mathsf {AxSelf^-}$ In his own worldview, the worldline of any observer is an interval of the time-axis containing all the coordinate points of the time-axis where the observer sees something: $\\big [\\mathsf {W}(m,m,\\bar{x})\\rightarrow x_2=\\ldots =x_d=0\\big ]\\wedge \\\\ \\big [\\mathsf {W}(m,m,\\bar{y})\\wedge \\mathsf {W}(m,m,\\bar{z})\\wedge x_1<t<y_1\\rightarrow \\mathsf {W}(m,m,t,0,\\ldots ,0)\\big ] \\wedge \\\\ \\exists b \\big [\\mathsf {W}(m,b,t,0,\\ldots ,0) \\rightarrow \\mathsf {W}(m,m,t,0,\\ldots ,0)\\big ].$ Let us now introduce a promising theory of accelerated observers as $\\mathsf {SpecRel}$ extended with the three axioms above.",
"$\\leavevmode {\\color {axcolor}\\mathsf {AccRel_0}}= \\leavevmode {\\color {axcolor}\\mathsf {SpecRel}}\\cup \\lbrace \\leavevmode {\\color {axcolor}\\mathsf {AxCmv}},\\leavevmode {\\color {axcolor}\\mathsf {AxEv^-}},\\leavevmode {\\color {axcolor}\\mathsf {AxSf^-}}\\rbrace $ Since $\\mathsf {AxCmv}$ ties the behavior of accelerated observers to the inertial ones and $\\mathsf {SpecRel}$ captures the kinematics of special relativity perfectly by Theorem REF , it is quite natural to think that $\\mathsf {AccRel_0}$ is a strong enough theory of accelerated observers to prove the most fundamental results about accelerated observers.",
"However, $\\mathsf {AccRel_0}$ does not even imply the most basic predictions about accelerated observers such as the twin paradox or that stationary observers measure the same time between two events [15], [26].",
"Moreover, it can be proved that even if we add the whole firs-order logic theory of real numbers to $\\mathsf {AccRel_0}$ is not enough to get a theory that implies the twin paradox, see, e.g., [15], [26].",
"In the models of $\\mathsf {AccRel_0}$ in which $\\mathsf {TwP}$ is not true there are some definable gaps in the number line.",
"Our next assumption is an axiom scheme excluding these gaps.",
"$\\mathsf {CONT}$ Every parametrically definable, bounded and nonempty subset of $\\mathit {Q}$ has a supremum (i.e., least upper bound) with respect to $\\le $ .",
"In $\\mathsf {CONT}$ “definable” means “definable in the language of $\\mathsf {AccRel}$, parametrically.” For a precise formulation of $\\mathsf {CONT}$, see [15] or [26].",
"That $\\mathsf {CONT}$ requires the existence of supremum only for sets definable in the language of $\\mathsf {AccRel}$ instead of every set is important because it makes this postulate closer to the physical/empirical level.",
"This is true because $\\mathsf {CONT}$ does not speak about “any fancy subset” of the quantities, but just about those “physically meaningful” sets which can be defined in the language of our (physical) theory.",
"Our axiom scheme of continuity ($\\mathsf {CONT}$) is a “mathematical axiom\" in spirit.",
"It is Tarski's first-order logic version of Hilbert's continuity axiom in his axiomatization of geometry, see [12], fitted to the language of $\\mathsf {AccRel}$.",
"When $\\mathit {Q}$ is the ordered field of real numbers, $\\mathsf {CONT}$ is automatically true.",
"Let us introduce our axioms system $\\mathsf {AccRel}$ as the extension of $\\mathsf {AccRel_0}$ by axiom scheme $\\mathsf {CONT}$.",
"$\\leavevmode {\\color {axcolor}\\mathsf {AccRel}}=\\leavevmode {\\color {axcolor}\\mathsf {AccRel_0}} + \\leavevmode {\\color {axcolor}\\mathsf {CONT}}$ It can be proved that axiom system $\\mathsf {AccRel}$ implies the twin paradox, see [15], [26].",
"An ordered field is called real closed field if a first-order logic sentence of the language of ordered fields is true in it exactly when it is true in the field of real numbers, or equivalently if it is Euclidean and every polynomial of odd degree has a root in it, see, e.g., [28].",
"Theorem 4.1 $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}})=\\lbrace \\text{\\,real closed fields\\,}\\rbrace $ There is a model of $\\mathsf {AccRel}$ over every real closed field since every model of $\\mathsf {SpecRel}$ over a real closed field in which $\\mathit {B}=\\mathsf {Ph}\\cup \\mathsf {IOb}$ is a model of $\\mathsf {AccRel}$ and $\\mathsf {SpecRel}$ has a model even over every Euclidean ordered field by Theorem REF .",
"Axiom schema $\\mathsf {CONT}$ is stronger than the whole first-order logic theory of real numbers, see, e.g., [26].",
"Consequently, if axiom $\\mathsf {AxOField}$ is assumed, $\\mathsf {CONT}$ by itself implies that the quantities are real closed fields."
],
[
"Numbers implied by uniformly accelerated observers",
"We have seen that assuming existence of observers can ensure the existence of numbers.",
"So let us investigate another axiom of this kind.",
"The next axiom ensures the existence of uniformly accelerated observers.",
"To introduce it, let us define the life-curve $\\mathsf {lc}_{m}(k)$ of observer $k$ according to observer $m$ as the worldline of $k$ according to $m$ parametrized by the time measured by $k$, formally: $\\mathsf {lc}_{m}(k):=\\lbrace \\, \\langle t,\\bar{x}\\rangle \\in \\mathit {Q}\\times \\mathit {Q}^d \\::\\:\\exists \\bar{y}k\\in \\mathsf {ev}_k(\\bar{y})=\\mathsf {ev}_m(\\bar{x})\\wedge y_1=t\\,\\rbrace .$ $\\mathsf {Ax\\exists UnifOb}$ It is possible to accelerate an observer uniformly:In relativity theory, uniformly accelerated observers are moving along hyperbolas, see, e.g., [9], [19], [20].",
"$\\mathsf {IOb}(m)\\rightarrow \\exists k \\Big [ \\mathsf {Ob}(k) \\wedge \\mathsf {Dom}\\,\\mathsf {lc}_m(k)=\\mathit {Q}\\\\\\wedge \\forall \\bar{x}\\big [ \\bar{x}\\in \\mathsf {Ran}\\,\\mathsf {lc}_m(k) \\leftrightarrow x_2^2-x_1^2=a^2\\wedge x_3=\\ldots =x_d=0\\big ]\\Big ].$ Theorem 5.1 Let $d\\ge 3$ .",
"Assume $\\mathsf {AccRel}$ and $\\mathsf {Ax\\exists UnifOb}$.",
"Then there is a definable differentiable function $E:\\mathit {Q}\\rightarrow \\mathit {Q}$ such that $\\mathsf {Ran}\\,E=\\mathit {Q}^+=[0,\\infty )$ , $\\frac{dE}{dt}=E$ and $E(-t)=1/E(t)$ for all $t\\in \\mathit {Q}$ .",
"Let $\\bar{\\mathbb {Q}}\\cap \\mathbb {R}$ denote the ordered field of real algebraic numbers.",
"Theorem REF implies that the ordered field of algebraic real numbers cannot be the structure of quantities of theory $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$: Theorem 5.2 Let $n\\ge 3$ .",
"$\\bar{\\mathbb {Q}}\\cap \\mathbb {R}\\notin \\textit {Num}_n(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ See [27] for proofs and more details of Theorems REF and REF .",
"Remark 5.3 By Theorem REF , if $n\\ge 3$ , $\\textit {Num}_n(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} +\\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ is not an elementary class of ordered fields, i.e., it is not a first-order logic axiomatizable class in the language of ordered fields.",
"Of course, it is a pseudoelementary class, i.e., it is a reduct of an elementary class in a richer language.",
"By Theorem REF , we know that not every real closed field can be the quantity structure of $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$.",
"For example, the field of real algebraic numbers cannot be the quantity structure of $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$.",
"However, the problem that exactly which ordered fields can be the quantity structures of $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$ is still open: Question 5.4 Exactly which ordered fields are the elements of classes $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ and $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel_0}} + \\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ ?",
"Theorem REF suggests that the answer to Question REF may have something to do with ordered exponential fields, see, e.g., [8], [14]."
],
[
"Numbers required by general relativity",
"Let us now see some similar questions about the properties of numbers implied by axioms of general relativity.",
"To do so, let us recall our axiom system $\\mathsf {GenRel}$ of general relativity formulated in the same streamlined language as $\\mathsf {AccRel}$ and $\\mathsf {SpecRel}$.",
"$\\mathsf {GenRel}$ contains the localized versions of the axioms of $\\mathsf {SpecRel}$ and the postulate that the worldview transformations between observers are differentiable maps, which is the localized version of the theorem of $\\mathsf {SpecRel}$ stating that the worldview transformations between inertial observers are affine maps, see Theorem REF .",
"We have already introduced the localized versions of axioms $\\mathsf {AxEv}$ and $\\mathsf {AxSelf}$, see $\\mathsf {AxEv^-}$ and $\\mathsf {AxSelf^-}$ at page .",
"Now let us state the localized versions of $\\mathsf {AxPh}$ and $\\mathsf {AxSymD}$.For technical reasons, in $\\mathsf {GenRel}$ we use an equivalent version of $\\mathsf {AxSymD}$, and we introduce that the speed of light is 1 in $\\mathsf {AxPh}$ instead of in $\\mathsf {AxSym^-}$.",
"$\\mathsf {AxPh^-}$ The velocity of photons an observer “meets” is 1 when they meet, and it is possible to send out a photon in each direction where the observer stands.",
"$\\mathsf {AxSym^-}$ Meeting observers see each other's clocks slow down the same way.",
"$\\mathsf {AxDiff}$ The worldview transformations between observers are functions having linear approximations at each point of their domain (i.e., they are differentiable maps).",
"For a precise formulation of axioms $\\mathsf {AxPh^-}$, $\\mathsf {AxSym^-}$, and $\\mathsf {AxDiff}$, as well as a “derivation” of the axioms of $\\mathsf {GenRel}$ from that of $\\mathsf {SpecRel}$, see, e.g., [5], [26].",
"$\\leavevmode {\\color {axcolor}\\mathsf {GenRel}}:=\\lbrace \\leavevmode {\\color {axcolor}\\mathsf {AxOFiled}},\\leavevmode {\\color {axcolor}\\mathsf {AxPh^-}},\\leavevmode {\\color {axcolor}\\mathsf {AxEv^-}},\\leavevmode {\\color {axcolor}\\mathsf {AxSelf^-}},\\leavevmode {\\color {axcolor}\\mathsf {AxSym^-}},\\leavevmode {\\color {axcolor}\\mathsf {AxDiff}}\\rbrace \\cup \\leavevmode {\\color {axcolor}\\mathsf {CONT}}$ Axiom system $\\mathsf {GenRel}$ captures general relativity well since it is complete with respect the standard models of general relativity, i.e., with respect to Lorentzian manifolds, see, e.g., [5], [26].",
"We call the worldline of observer $m$ timelike geodesic, if each of its points has a neighborhood within which this observer “maximizes measured time\" between any two encountered events, see Figure REF for illustration and [5] for a formal definition of timelike geodesics in the language of $\\mathsf {GenRel}$.",
"According to the definition above, if there are only a few observers, then it is not a big deal that the worldline of $m$ is a timelike geodesic (it is easy to be maximal if there are only a few to be compared to).",
"To generate a real competition for the rank of having a timelike geodesic worldline, we postulate the existence of great many observers by the following axiom scheme of comprehension.",
"$\\mathsf {COMPR}$ For any parametrically definable timelike curve in any observer's worldview, there is another observer whose worldline is the range of this curve.",
"A precise formulation of $\\mathsf {COMPR}$ can be obtained from that of its analogue in [4].",
"Let us now show that $\\mathsf {COMPR}$ implies axiom $\\mathsf {Ax\\exists UnifOb}$, hence it requires at least as much properties of numbers.",
"Figure: Illustration for the definition of timelike geodesic in 𝖦𝖾𝗇𝖱𝖾𝗅\\mathsf {GenRel}Proposition 6.1 $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {COMPR}})\\subseteqq \\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {Ax\\exists UnifOb}})$ For all $a\\in \\mathit {Q}$ , the hyperbola (line if $a=0$ ) $\\lbrace \\bar{x}:x_2^2-x_1^2=a^2,x_3=\\ldots =x_d=0 \\rbrace $ can be parametrized by the definable timelike curve $\\lbrace \\langle x_1,\\bar{x}\\rangle : x_2^2-x_1^2=a^2,x_3=\\ldots =x_d=0\\rbrace .$ So by $\\mathsf {COMPR}$, there is an observer whose worldline is this set.",
"So $\\mathsf {COMPR}$ implies $\\mathsf {Ax\\exists UnifOb}$.",
"Therefore, every model of $\\mathsf {AccRel}$ + $\\mathsf {COMPR}$ is a model of of $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$.",
"Hence the possible quantity structures of $\\mathsf {AccRel}$ + $\\mathsf {COMPR}$ is a subset of the possible quantity structures of $\\mathsf {AccRel}$ + $\\mathsf {Ax\\exists UnifOb}$.",
"It is also quite easy to show that $\\mathsf {GenRel}$ does not require more properties of numbers than $\\mathsf {AccRel}$.",
"Proposition 6.2 $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxDiff}})\\subseteqq \\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {GenRel}})$ $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {AxDiff}} + \\leavevmode {\\color {axcolor}\\mathsf {COMPR}})\\subseteqq \\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {GenRel}} + \\leavevmode {\\color {axcolor}\\mathsf {COMPR}})$ To prove this statement it is enough to show that the models of $\\mathsf {AccRel}$ + $\\mathsf {AxDiff}$ are also models of $\\mathsf {GenRel}$.",
"Since $\\mathsf {AxPh^-}$ and $\\mathsf {AxSym^-}$ are the only two axioms of $\\mathsf {GenRel}$ which are not also contained in $\\mathsf {AccRel}$ + $\\mathsf {AxDiff}$, we only have to show that these two axioms are consequences of $\\mathsf {AccRel}$.",
"Axioms $\\mathsf {AxPh^-}$ and $\\mathsf {AxSym^-}$ follow from $\\mathsf {AccRel}$ since they are true for inertial observers in $\\mathsf {SpecRel}$ and by $\\mathsf {AxCmv}$ accelerated observers locally see the world the same way as their co-moving inertial observers.",
"Question 6.3 Exactly which ordered fields are the elements of classes $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {AccRel}} + \\leavevmode {\\color {axcolor}\\mathsf {COMPR}})$ and $\\textit {Num}_{n}(\\leavevmode {\\color {axcolor}\\mathsf {GenRel}} + \\leavevmode {\\color {axcolor}\\mathsf {COMPR}})$ ?",
"Maybe the ordered field reducts of differentially closed fields of Abraham Robinson, see [17], [21], have to do something with the answer to the question above."
],
[
"Proof of Theorem ",
"In this section, we are going to prove Theorem REF .",
"To do so, let us recall a version of Alexandrov–Zeeman theorem generalized over fields.",
"To state this theorem, we need some concepts.",
"Map $\\mathsf {q}:\\mathit {Q}^d\\rightarrow \\mathit {Q}$ is a quadratic form if $\\mathsf {q}(\\lambda \\bar{x})=\\lambda ^2\\mathsf {q}(\\bar{x})$ for all $\\lambda \\in \\mathit {Q}$ and $\\bar{x}\\in \\mathit {Q}^d$ , and $(\\bar{x},\\bar{y})_\\mathsf {q}:=\\mathsf {q}(\\bar{x}+\\bar{y})-\\mathsf {q}(vx)-\\mathsf {q}(\\bar{y})$ is a symmetric bilinear form.",
"Quadratic form $\\mathsf {q}$ is non-degenerate if $\\forall \\bar{x}\\bar{a}(\\bar{x},\\bar{a})_\\mathsf {q}=0 \\wedge \\mathsf {q}(\\bar{a})=0\\rightarrow \\bar{a}=\\bar{o}.$ A map $f:\\mathit {Q}^d\\rightarrow \\mathit {Q}^d$ is called a semilinear map iff there is a field automorphism $\\alpha $ such that $f(\\bar{x}+\\bar{y})=f(\\bar{x})+f(\\bar{y}) \\text{ and }f(\\lambda \\bar{x})=\\alpha (\\lambda )f(\\bar{x})$ for all $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ and $\\lambda \\in \\mathit {Q}$ .",
"Witt index of quadratic form $\\mathsf {q}$ is the maximal dimension of a subspace $X$ of $\\mathit {Q}^d$ with the property $\\mathsf {q}(\\bar{x})=0$ for all $\\bar{x}\\in X$ .",
"$\\mathsf {q}$ -null cone with vertex $\\bar{a}\\in \\mathit {Q}^d$ is defined as $C(\\bar{a})=\\lbrace \\bar{x}: \\mathsf {q}(\\bar{x}-\\bar{a})=0\\rbrace .$ Now we are ready to recall the version of Alexandrov-Zeeman theorem we need, see [30], [31]: Theorem 7.1 (Vroegindewey) Let $\\langle \\mathit {Q},+,\\cdot \\rangle $ be an commutative field.",
"Let $d\\ge 3$ and let $\\mathsf {q}$ be a non-degenerate quadratic form with Witt index 1.",
"Then every bijection of $\\mathit {Q}^d$ taking $\\mathsf {q}$ -null cones to $\\mathsf {q}$ -null cones is a composition of a translation and a semilinear map $f$ with the property $\\mathsf {q}\\big (f(\\bar{x})\\big )=\\lambda \\alpha \\big (\\mathsf {q}(x)\\big )$ for some $\\lambda \\ne 0$ and field automorphism $\\alpha $ .",
"We are going to apply Theorem REF to the worldview transformations of inertial observers in $\\mathsf {SpecRel}$.",
"To do so, we need several definitions and lemmas.",
"For all $c>0$ , let us define the $c\\,$-Minkowski quadratic form as $\\mu ^2_c(\\bar{x})=c\\cdot x_1^2-x_2^2-\\ldots -x_d^2.$ Lemma 7.2 Assume $\\mathsf {AxOField}$.",
"Let $\\bar{x}\\in \\mathit {Q}^d$ be such that $x_1=0$ and $\\mu ^2_c(\\bar{x})=0$ .",
"Then $\\bar{x}=\\bar{o}$ .",
"Since $x_1=0$ and $\\mu ^2_c(\\bar{x})=0$ , we have that $x_2^2+\\ldots +x_d^2=0$ .",
"This implies that $x_2=\\ldots = x_d=0$ in ordered fields.",
"Hence $\\bar{x}=\\bar{o}$ as stated.",
"Remark 7.3 Lemma REF is not valid in every field.",
"For example, in the field of complex numbers $\\bar{x}=\\langle 0,1, i\\rangle $ is a nonzero vector but $x_1=0$ and $\\mu ^2_1(\\bar{x})=0$ .",
"Lemma 7.4 Assume $\\mathsf {AxOField}$.",
"Let $c>0$ .",
"Then Minkowski quadratic form $\\mu ^2_c$ has Witt index 1.",
"Let $\\bar{x}$ and $\\bar{y}$ be vectors such that $\\mu ^2_c(\\alpha \\bar{x}+\\beta \\bar{y})=0$ for all $\\alpha ,\\beta \\in \\mathit {Q}$ .",
"Let $\\bar{z}=y_1\\bar{x}-x_1\\bar{y}$ .",
"Then $z_1=0$ and $\\mu ^2_c(\\bar{z})=0$ .",
"Hence, by Lemma REF , $\\bar{z}=\\bar{o}$ .",
"So $y_1\\bar{x}=x_1\\bar{y}$ .",
"Therefore, the subspace spanned by $\\bar{x}$ and $\\bar{y}$ is 1 dimensional.",
"Thus the Witt index of $\\mu ^2_c$ is 1 as stated.",
"The squared slope of line $l$ is defined as $\\mathsf {slope}^2(l)=\\frac{\\mathsf {space}^2(\\bar{x},\\bar{y})}{\\mathsf {time}(\\bar{x},\\bar{y})^2}$ for all $\\bar{x},\\bar{y}\\in l$ for wihc $x_1\\ne y_1$ .",
"Lemma 7.5 Assume $\\mathsf {AxOField}$.",
"Let $c>0$ .",
"There is no non-degenerate triangle whose every side is of squared slope $c$ .",
"Let $\\bar{x}$ , $\\bar{y}$ , and $\\bar{z}$ be the vertices of a triangle whose sides are of squared slope $c$ .",
"Then $c\\cdot \\mathsf {time}(\\bar{x},\\bar{y})^2=\\mathsf {space}^2(\\bar{x}, \\bar{y})$ , $c\\cdot \\mathsf {time}(\\bar{y},\\bar{z})^2=\\mathsf {space}^2(\\bar{y},\\bar{z})$ , and $c\\cdot \\mathsf {time}(\\bar{z},\\bar{x})^2=\\mathsf {space}^2(\\bar{z},\\bar{x})$ .",
"Let $\\bar{p}=\\bar{y}-\\bar{x}$ and $\\bar{q}=\\bar{z}-\\bar{y}$ .",
"Then $c p_1^2=p_2^2+\\ldots +p_d^2,$ $c q_1^2=q_2^2+\\ldots +q_d^2,\\text{ and}$ $c(p_1+q_1)^2=(p_2+q_2)^2+\\ldots +(p_d+q_d)^2.$ In other words $\\mu ^2_c(\\bar{p})=\\mu ^2_c(\\bar{q})=\\mu ^2_c(\\bar{p}+\\bar{q})=0$ .",
"By subtracting equations (REF ) and (REF ) from equation (REF ), we get $2cp_1 q_1=2p_2q_2+\\ldots +2p_dq_d.$ Let $\\alpha $ and $\\beta $ be arbitrary elements of $\\mathit {Q}$ .",
"Then $\\mu ^2_c(\\alpha \\bar{p}+\\beta \\bar{q})\\\\=\\alpha ^2\\mu ^2_c(\\bar{p})+2\\alpha \\beta (c p_1 q_1-p_2q_2-\\ldots -p_dq_d)+ \\beta ^2\\mu ^2_c(\\bar{q})$ for all $\\alpha ,\\beta \\in \\mathit {Q}$ .",
"Therefore, $\\mu ^2_c(\\alpha \\bar{p}+\\beta \\bar{q})=0$ .",
"By Lemma REF , $\\mu ^2_c$ has Witt index 1.",
"So $\\bar{p}$ and $\\bar{q}$ are in the same 1 dimensional subspace of $\\mathit {Q}^d$ .",
"Hence $\\bar{x}$ , $\\bar{y}=\\bar{x}+\\bar{p}$ , and $\\bar{z}=\\bar{x}+\\bar{p}+\\bar{q}$ are collinear.",
"The $f$ -image of set $H$ is defined as follows: $f[H]=\\left\\lbrace \\,b:\\exists a\\big [ a\\in H \\wedge f(a)=b\\big ]\\,\\right\\rbrace .$ Proposition 7.6 Assume $\\mathsf {AxOField}$, $\\mathsf {AxEv}$, and $\\mathsf {AxPh}$.",
"Let $m,k\\in \\mathsf {IOb}$ .",
"Then $\\mathsf {w}_{mk}$ is a bijection of $\\mathit {Q}^d$ taking lines of squared slope $\\mathsf {c}^2_m$ to lines of squared slope $\\mathsf {c}^2_k$ .",
"Let $m\\in \\mathsf {IOb}$ and let $\\bar{x}$ and $\\bar{y}$ be two distinct coordinate points.",
"Let $\\bar{v}:=\\langle 1, \\mathsf {c}_m,0,\\ldots , 0\\rangle $ and $\\bar{u}:=\\langle 1, -\\mathsf {c}_m,0,\\ldots ,0\\rangle $ .",
"By $\\mathsf {AxOField}$, at most one of the lines $l_1:=\\lbrace \\bar{x}+\\lambda \\cdot \\bar{v}:\\lambda \\in \\mathit {Q}\\rbrace \\text{ and }l_2:=\\lbrace \\bar{x}+\\lambda \\cdot \\bar{u}:\\lambda \\in \\mathit {Q}\\rbrace $ can contain $\\bar{y}$ since $a\\bar{v}=b\\bar{u}$ implies $a=b=0$ .",
"So, by $\\mathsf {AxPh}$, there is a light signal which is in $\\mathsf {ev}_m(\\bar{x})$ but not in $\\mathsf {ev}_m(\\bar{y})$ since $\\mathsf {slope}^2(l_1)=\\mathsf {slope}^2(l_2)=\\mathsf {c}^2_m$ .",
"Thus inertial observers see different events at different coordinate points, i.e., $\\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_m(\\bar{y})$ implies $\\bar{x}=\\bar{y}$ .",
"Therefore, binary relation $\\mathsf {w}_{mk}$ is an injective function for all $m,k\\in \\mathsf {IOb}$ .",
"Let $m,k\\in \\mathsf {IOb}$ .",
"By $\\mathsf {AxPh}$, every inertial observer sees a nonempty event in every coordinate point.",
"By $\\mathsf {AxEv}$, inertial observers coordinatize the same events.",
"Therefore, for all $\\bar{x}\\in \\mathit {Q}^d$ , there is a $\\bar{y}\\in \\mathit {Q}^d$ such that $\\mathsf {w}_{mk}(\\bar{x})=\\bar{y}$ .",
"So $\\mathsf {Dom}\\,\\mathsf {w}_{mk}=\\mathsf {Ran}\\,\\mathsf {w}_{km}=\\mathit {Q}^d$ .",
"Consequently, $\\mathsf {w}_{mk}$ is a bijection of $\\mathit {Q}^d$ for all $m,k\\in \\mathsf {IOb}$ .",
"Now we show that $\\mathsf {w}_{mk}$ takes lines of squared slope $\\mathsf {c}^2_m$ to lines of squared slope $\\mathsf {c}^2_k$ .",
"To do so, let $l$ be a line of squared slope $\\mathsf {c}^2_m$ and let $\\bar{x}$ , $\\bar{y}$ , $\\bar{z}$ be three distinct points of $l$ .",
"By $\\mathsf {AxPh}$, there are light signals $p_{xy}$ , $p_{yz}$ , and $p_{zx}$ such that $p_{xy},p_{zx}\\in \\mathsf {ev}_m(\\bar{x})$ , $p_{yz},p_{xy}\\in \\mathsf {ev}_m(\\bar{y})$ , and $p_{zx},p_{yz}\\in \\mathsf {ev}_m(\\bar{z})$ .",
"Since $\\mathsf {w}_{mk}$ is a bijection, $\\mathsf {w}_{mk}(\\bar{x})$ , $\\mathsf {w}_{mk}(\\bar{y})$ , and $\\mathsf {w}_{mk}(\\bar{z})$ are also distinct points.",
"By the definition of $\\mathsf {w}_{mk}$ , we have $p_{xy},p_{zx}\\in \\mathsf {ev}_k\\big (\\mathsf {w}_{mk}(\\bar{x})\\big )$ , $p_{yz},p_{xy}\\in \\mathsf {ev}_k\\big (\\mathsf {w}_{mk}(\\bar{y})\\big )$ , and $p_{zx},p_{yz}\\in \\mathsf {ev}_k\\big (\\mathsf {w}_{mk}(\\bar{z})\\big )$ .",
"So, by $\\mathsf {AxPh}$, coordinate points $\\mathsf {w}_{mk}(\\bar{x})$ , $\\mathsf {w}_{mk}(\\bar{y})$ , and $\\mathsf {w}_{mk}(\\bar{z})$ form a triangle such that all of its sides are of squared slope $\\mathsf {c}^2_k$ .",
"Therefore, by Lemma REF , they have to be on a line of squared slope $\\mathsf {c}^2_k$ .",
"So the $\\mathsf {w}_{mk}$ -image of $l$ is a subset of a line of $\\mathsf {c}^2_k$ .",
"Since $m$ and $k$ were arbitrary inertial observers we also have that the $\\mathsf {w}_{km}$ -image of the line containing $\\mathsf {w}_{mk}[l]$ is the subset of a line of squared slope $\\mathsf {c}^2_m$ .",
"Since $\\mathsf {w}_{mk}$ is a bijection and its inverse is $\\mathsf {w}_{km}$ , we have that $\\mathsf {w}_{km}\\big [\\mathsf {w}_{mk}[l]\\big ]=l$ .",
"Consequently, $\\mathsf {w}_{mk}[l]$ cannot be a proper subset of a line, but it has to be a whole line of squared slope $\\mathsf {c}^2_k$ .",
"This completes the proof of the proposition.",
"Corollary 7.7 Assume $\\mathsf {SpecRel}$.",
"Let $m$ and $k$ be inertial observers.",
"Then $\\mathsf {w}_{mk}$ is a bijection of $\\mathit {Q}^d$ taking lines of squared slope 1 to lines of squared slope 1.",
"$\\Box $ Let us call a liner bijection of $\\mathit {Q}^d$ almost Lorentz transformation iff there is a $\\lambda \\ne 0$ such that $\\mu ^2_1\\big (A(\\bar{x})\\big )=\\lambda \\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"We think of functions as special binary relations.",
"Hence we compose them as relations.",
"The composition of binary relations $R$ and $S$ is defined as: ${R \\fatsemi S}:=\\lbrace \\langle a,c\\rangle : \\exists bR(a,b)\\wedge S(b,c) \\rbrace .$ So $(g\\fatsemi f)(x)=f\\big (g(x)\\big )$ if $f$ and $g$ are functions.",
"We will also use the notation $x\\fatsemi g\\fatsemi f$ for $(g\\fatsemi f)(x)$ because the latter is easier to grasp.",
"In the same spirit, we will sometimes use the notation $x\\fatsemi f$ for $f(x)$ .",
"The inverse of $R$ is defined as: ${R^{-1}}:=\\lbrace \\langle a,b\\rangle : R(b,a) \\rbrace .$ Let us introduce, for all $c>0$ , the spatial distance and time rescaling maps as $S_c(\\bar{x})=\\langle x_1,cx_2,\\ldots , cx_d\\rangle \\text{ and }T_c(\\bar{x})=\\langle cx_1, x_2,\\ldots , x_d\\rangle $ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"It is clear that $T^{-1}_c=T_{1/c}$ and $S^{-1}_c=S_{1/c}$ .",
"Let $\\alpha $ be an automorphism of field $\\langle \\mathit {Q},\\cdot ,+\\rangle $ and let $\\tilde{\\alpha }$ be the map $\\tilde{\\alpha }(\\bar{x})=\\langle \\alpha (x_1),\\ldots , \\alpha (x_d) \\rangle $ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"A map from $\\mathit {Q}^d$ to $\\mathit {Q}^d$ is called automorphism-induced-map if it is the form $\\tilde{\\alpha }$ for some automorphism $\\alpha $ .Let us note that we have not required that $\\alpha $ is order preserving.",
"Theorem 7.8 Let $d\\ge 3$ .",
"Assume $\\mathsf {AxOField}$, $\\mathsf {AxEv}$, and $\\mathsf {AxPh}$.",
"Let $m,k\\in \\mathsf {IOb}$ .",
"Then $\\mathsf {w}_{mk}=S^{-1}_{\\mathsf {c}_m}\\fatsemi A\\fatsemi \\tilde{\\alpha }\\fatsemi T\\fatsemi S_{\\mathsf {c}_k}$ where $T$ is a translation, $A$ is an almost Lorentz transformation and $\\alpha $ is field automorphism.",
"$\\mathsf {w}_{mk}=T_{\\mathsf {c}_m} \\fatsemi A\\fatsemi \\tilde{\\alpha }\\fatsemi T\\fatsemi T^{-1}_{\\mathsf {c}_k}$ where $T$ is a translation, $A$ is an almost Lorentz transformation and $\\alpha $ is field automorphism.",
"By definitions, $S_c$ and $T^{-1}_c$ are linear bijections of $\\mathit {Q}^d$ taking lines of squared slope 1 to lines of squared slope $c^2$ .",
"Therefore, by Proposition REF , both maps $S_{\\mathsf {c}_m}\\fatsemi \\mathsf {w}_{mk}\\fatsemi S_{\\mathsf {c}_k}^{-1}$ and $T_{\\mathsf {c}_m}^{-1}\\fatsemi \\mathsf {w}_{mk}\\fatsemi T_{\\mathsf {c}_k}$ are bijections of $\\mathit {Q}^d$ taking lines of squared slope 1 to lines of squared slope 1.",
"Since the $\\mu ^2_1$ -null cone $C(\\bar{a})$ is the union of lines of squared slope 1 through $\\bar{a}$ , both $S_{\\mathsf {c}_m}\\fatsemi \\mathsf {w}_{mk}\\fatsemi S_{\\mathsf {c}_k}^{-1}$ and $T_{\\mathsf {c}_m}^{-1}\\fatsemi \\mathsf {w}_{mk}\\fatsemi T_{\\mathsf {c}_k}$ map $\\mu ^2_1$ -null cones to $\\mu ^2_1$ -null cones.",
"Therefore, by Theorem REF and Lemma REF , they are compositions of an almost Lorentz transformation $A$ , a field-automorphism-induced map $\\tilde{\\alpha }$ , and a translation $T$ .",
"Some of the following statements assume only that the quantity part is a field.",
"Therefore, let us introduce the following axiom: $\\mathsf {AxField}$ The quantity part $\\langle \\mathit {Q},+,\\cdot \\rangle $ is a (commutative) field.",
"Lemma 7.9 Assume $\\mathsf {AxField}$ and that $1+1\\ne 0$ .",
"Let $\\alpha $ and $\\beta $ be two automorphisms of $\\langle \\mathit {Q},+,\\cdot \\rangle $ such that $\\alpha (a)^2=\\beta (a)^2$ for all $a\\in \\mathit {Q}$ .",
"Then $\\alpha =\\beta $ .",
"For all $a\\in \\mathit {Q}$ , we have that $\\alpha (a)=\\beta (a)$ or $\\alpha (a)=-\\beta (a)$ .",
"Let $a\\in \\mathit {Q}$ such that $\\alpha (a)=-\\beta (a)$ .",
"Then $\\alpha (1+a)=1+\\alpha (a)=1-\\beta (a)$ .",
"Also $\\alpha (1+a)=\\beta (1+a)=1+\\beta (a)$ or $\\alpha (1+a)=-\\beta (1+a)=-1-\\beta (a)$ .",
"So $1-\\beta (a)=1+\\beta (a)$ or $1-\\beta (a)=-1-\\beta (a)$ .",
"Therefore, $\\beta (a)=0$ since $1+1\\ne 0$ .",
"Hence $a=0$ .",
"Thus $\\alpha (a)=\\beta (a)$ for all $a\\in \\mathit {Q}$ .",
"Let $\\mathsf {Id}_H$ denote the identity map from $H\\subseteq \\mathit {Q}^d$ to $H$ , i.e., $\\mathsf {Id}_H(\\bar{x})=\\bar{x}$ for all $\\bar{x}\\in H$ .",
"Remark 7.10 It is easy to see that Lemma REF is not valid if the field has characteristic 2, i.e., if $1+1=0$ .",
"For example, the 4 element field has two automorphisms $\\mathsf {Id}$ and $\\alpha $ ; and $\\alpha ^2=\\mathsf {Id}^2$ , but $\\alpha \\ne \\mathsf {Id}$ .",
"Lemma 7.11 Assume $\\mathsf {AxField}$.",
"Let $f:\\mathit {Q}^d\\rightarrow \\mathit {Q}^d$ be a semilinear transformation having the property $\\mu ^2_1\\big (f(\\bar{x})\\big )=\\lambda \\alpha \\big (\\mu ^2_1(\\bar{x})\\big )$ for some $\\lambda \\ne 0$ and field automorphism $\\alpha $ .",
"Then there are almost Lorentz transformations $A$ and $A^*$ such that $f=\\tilde{\\alpha }\\fatsemi A =A^*\\fatsemi \\tilde{\\alpha }$ .",
"Let $A$ be $\\tilde{\\alpha }^{-1}\\fatsemi f$ , i.e., $A(\\bar{x})=f\\big (\\tilde{\\alpha }^{-1}(\\bar{x})\\big )$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"$A$ is a bijection since both $\\tilde{\\alpha }^{-1}$ and $f$ are so.",
"$A$ is additive, i.e., $A(\\bar{x}+\\bar{y})=A(\\bar{x})+A(\\bar{y})$ for all $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ , since $\\tilde{\\alpha }^{-1}$ and $f$ are so.",
"Since $f$ is semilinear, there is a automorphism $\\beta $ such that $f(a\\bar{x})=\\beta (a)f(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ and $a\\in \\mathit {Q}$ .",
"Consequently, we have $\\mu ^2_1\\big (f(a\\bar{x})\\big )\\stackrel{(\\ref {eq-b})}{=}\\mu ^2_1\\big (\\beta (a)f(\\bar{x})\\big )\\stackrel{(\\ref {eq-q})}{=}\\beta (a)^2\\mu ^2_1\\big (f(\\bar{x})\\big )\\stackrel{(\\ref {eq-al})}{=}\\beta (a)^2\\lambda \\alpha \\big (\\mu ^2_1(\\bar{x})\\big )$ and $\\mu ^2_1\\big (f(a\\bar{x})\\big )\\stackrel{(\\ref {eq-al})}{=}\\lambda \\alpha \\big (\\mu ^2_1(a\\bar{x})\\big )\\stackrel{(\\ref {eq-q})}{=}\\lambda \\alpha \\big (a^2\\mu ^2_1(\\bar{x})\\big )=\\lambda \\alpha (a)^2\\alpha \\big (\\mu ^2_1(\\bar{x})\\big ).$ for all $a\\in \\mathit {Q}$ .",
"Consequently, $\\lambda \\beta (a)^2\\alpha \\big (\\mu ^2_1(\\bar{x})\\big )=\\lambda \\alpha (a)^2\\alpha \\big (\\mu ^2_1(\\bar{x})\\big )$ for all $a\\in \\mathit {Q}$ .",
"So $\\alpha (a)^2=\\beta (a)^2$ for all $a\\in \\mathit {Q}$ .",
"Therefore, by Lemma REF , $\\alpha =\\beta $ .",
"Consequently, equation (REF ) becomes $f(a\\bar{x})=\\alpha (a)f(\\bar{x}).$ Thus $A$ is a linear bijection since $A(a\\bar{x})\\stackrel{(\\ref {eq-A})}{=}f\\big (\\tilde{\\alpha }^{-1}(a \\bar{x})\\big )\\stackrel{(\\ref {eq-a})}{=}f\\big (\\alpha ^{-1}(a)\\tilde{\\alpha }^{-1}(\\bar{x})\\big )\\\\=\\alpha \\big (\\alpha ^{-1}(a)\\big )f\\big (\\tilde{\\alpha }^{-1}(\\bar{x})\\big )=af\\big (\\tilde{\\alpha }^{-1}(\\bar{x})\\big )\\stackrel{(\\ref {eq-A})}{=}a A(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ and $a\\in \\mathit {Q}$ .",
"Now we are going to show that $\\mu ^2_1\\big (A(\\bar{x})\\big )=\\lambda \\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"Let $\\bar{x}\\in \\mathit {Q}^d$ and let $\\bar{y}=\\tilde{\\alpha }^{-1}(\\bar{x})$ .",
"$\\mu ^2_1\\big (A(\\bar{x})\\big )\\stackrel{(\\ref {eq-A})}{=}\\mu ^2_1\\big (f\\big (\\tilde{\\alpha }^{-1}(\\bar{x})\\big )\\big )=\\mu ^2_1\\big (f(\\bar{y})\\big )\\\\\\stackrel{(\\ref {eq-al})}{=}\\lambda \\alpha \\big (\\mu ^2_1(\\bar{y})\\big )=\\lambda \\mu ^2_1\\big (\\tilde{\\alpha }(\\bar{y})\\big )=\\lambda \\mu ^2_1(\\bar{x}).$ This proves that $A$ is an almost Lorentz transformation; and $f=\\tilde{\\alpha }\\fatsemi A$ by the definition of $A$ .",
"We also have that $f=A^*\\fatsemi \\tilde{\\alpha }$ for almost Lorentz transformation $A^*=\\tilde{\\alpha }\\fatsemi A\\fatsemi \\tilde{\\alpha }^{-1}$ .",
"Vectors $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ are called orthogonal in the Euclidean sense, in symbols $\\bar{x}\\perp _e\\bar{y}$ , iff $x_1y_1+\\ldots +x_dy_d=0$ .",
"Vectors $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ are called Minkowski orthogonal, in symbols $\\bar{x}\\perp _\\mu \\bar{y}$ , iff $(\\bar{x},\\bar{y})_{\\mu ^2_1}=0$ , i.e., $x_1y_1=x_2y_2\\ldots +x_dy_d$ .",
"Lemma 7.12 Assume $\\mathsf {AxField}$.",
"Let $A$ be an almost Lorentz transformation.",
"Then $\\bar{x}\\perp _\\mu \\bar{y}$ iff $A(\\bar{x})\\perp _\\mu A(\\bar{y})$ for all $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ .",
"By definition, $\\bar{x}\\perp _\\mu \\bar{y}$ iff $(\\bar{x},\\bar{y})_{\\mu ^2_1}=0$ .",
"Also by definition $\\big (A(\\bar{x}),A(\\bar{y})\\big )_{\\mu ^2_1}=\\mu ^2_1\\big (A(\\bar{x})+A(\\bar{y})\\big )-\\mu ^2_1\\big (A(\\bar{x})\\big )-\\mu ^2\\big (A(\\bar{y})\\big )$ .",
"Since $A$ is an almost Lorentz transformation, $\\big (A(\\bar{x}),A(\\bar{y})\\big )_{\\mu ^2_1}=\\lambda \\cdot (\\bar{x},\\bar{y})_{\\mu ^2_1}$ for some $\\lambda \\ne 0$ .",
"Therefore, $\\big (A(\\bar{x}),A(\\bar{y})\\big )_{\\mu ^2_1}=0$ iff $(\\bar{x},\\bar{y})_{\\mu ^2_1}=0$ ; and this is what we wanted to prove.",
"Let us introduce the time unit vector as follows $\\bar{1}_t:=\\langle 1, 0,\\ldots ,0\\rangle $ .",
"Proposition 7.13 Assume $\\mathsf {AxField}$.",
"Let $A$ be an almost Lorentz transformation.",
"Then $y_1=0$ and $A(\\bar{y})_1=0$ iff $A(\\bar{1}_t)\\perp _eA(\\bar{y})$ and $A(\\bar{y})_1=0$ for all $\\bar{y}\\in \\mathit {Q}^d$ .",
"Let $\\bar{y}\\in \\mathit {Q}^d$ .",
"It is enough to show that $y_1=0$ is equivalent to $\\bar{1}_t\\perp _e A(\\bar{y})$ assuming that $A(\\bar{y})_1=0$ .",
"It is clear that $y_1=0$ iff $\\bar{1}_t\\perp _\\mu \\bar{y}$ .",
"By Lemma REF , $\\bar{1}_t\\perp _\\mu \\bar{y}$ iff $A(\\bar{1}_t)\\perp _\\mu A(\\bar{y})$ .",
"Since $A(\\bar{y})_1=0$ , we have $A(\\bar{1}_t)\\perp _\\mu A(\\bar{y})$ iff $A(\\bar{1}_t)\\perp _e A(\\bar{y})$ .",
"Therefore, $y_1=0$ iff $\\bar{1}_t\\perp _e A(\\bar{y})$ provided that $A(\\bar{y})_1=0$ .",
"Let $m$ and $k$ be inertial observers and let $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ .",
"Events $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are simultaneous for $k$ iff $x^{\\prime }_1=y^{\\prime }_1$ for all $\\bar{x}^{\\prime }$ and $\\bar{y}^{\\prime }$ for which $\\mathsf {ev}_m(\\bar{x})=\\mathsf {ev}_k(\\bar{x}^{\\prime })$ and $\\mathsf {ev}_m(\\bar{y})=\\mathsf {ev}_k(\\bar{y}^{\\prime })$ .",
"Events $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are separated orthogonally to the plane of motion of $k$ according to $m$ iff $x_1=y_1$ and $(\\bar{x}-\\bar{y})\\perp _e \\big (\\mathsf {w}_{km}(\\bar{1}_t)-\\mathsf {w}_{km}(\\bar{o})\\big )$ , see Figure REF .",
"Theorem 7.14 Let $d\\ge 3$ .",
"Assume $\\mathsf {AxOField}$, $\\mathsf {AxPh}$, and $\\mathsf {AxEv}$.",
"Let $m$ and $k$ be inertial observers and let $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ .",
"Events $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are simultaneous for both $m$ and $k$ iff $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are separated orthogonally to the plane of motion of $k$ according to $m$ .Specially, if $\\mathsf {speed}^2_m(k)=0$ , the same events are simultaneous for $m$ and $k$ .",
"Let $\\bar{x}^{\\prime }=\\mathsf {w}_{mk}(\\bar{x})$ , $\\bar{y}^{\\prime }=\\mathsf {w}_{mk}(\\bar{y})$ , and $\\bar{v}=\\bar{y}-\\bar{x}$ , see Figure REF .",
"Figure: Illustration for the proof of Theorem By Theorem REF , $\\mathsf {w}_{km}=S^{-1}_{\\mathsf {c}_k}\\fatsemi A\\fatsemi \\tilde{\\alpha }\\fatsemi T\\fatsemi S_{\\mathsf {c}_m}$ for some field automorphism $\\alpha $ , translation $T$ and almost Lorentz transformation $A$ .",
"Maps $S_c$ , $\\tilde{\\alpha }$ and $T$ do not change the facts whether $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are simultaneous for both $m$ and $k$ ; and whether they are separated orthogonally to the plane of motion of $k$ according to $m$ .",
"Therefore, we can assume, without loss of generality, that $\\mathsf {w}_{mk}$ is an almost Lorentz transformation.",
"Then $\\mathsf {w}_{km}(\\bar{o})=\\bar{o}$ .",
"Therefore, events $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are orthogonal to the plane of motion of $k$ according to $m$ iff $v_1=0$ and $\\bar{v}\\perp _e\\mathsf {w}_{km}(\\bar{1}_t)$ .",
"Let $\\bar{v}^{\\prime }=\\mathsf {w}_{mk}(\\bar{v})$ , then $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are orthogonal to the plane of motion iff $\\mathsf {w}_{km}(\\bar{v}^{\\prime })_1=0$ and $\\mathsf {w}_{km}(\\bar{v}^{\\prime })\\perp _e\\mathsf {w}_{km}(\\bar{1}_t)$ .",
"By Proposition REF , this is equivalent to $\\mathsf {w}_{km}(\\bar{v}^{\\prime })_1=0$ and $\\bar{v}^{\\prime }=0$ .",
"This means that $x_1=y_1$ and $x^{\\prime }_1=y^{\\prime }_1$ , i.e., that $\\mathsf {ev}_m(\\bar{x})$ and $\\mathsf {ev}_m(\\bar{y})$ are simultaneous both for $m$ and $k$ ; and that is what we wanted to prove.",
"Let $a\\in \\mathit {Q}$ such that $a\\ne 0$ .",
"Let us introduce dilation $D_a$ as the transformation mapping $\\bar{x}$ to $a\\bar{x}$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"It is clear that $D^{-1}_a=D_{1/a}$ .",
"Lemma 7.15 Assume $\\mathsf {AxField}$.",
"Let $A$ be an almost Lorentz transformation such that $\\mu ^2_1\\big (A(\\bar{x})\\big )=a^2\\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"There are a unique Lorentz transformation $L$ and a unique dilation $D$ such that $A=D\\fatsemi L=L \\fatsemi D$ .",
"Let $L$ be $D^{-1}_a\\fatsemi A$ .",
"$L$ is a Lorentz transformation since $\\mu ^2_1\\big (L(\\bar{x})\\big )=\\mu ^2_1\\left(\\frac{1}{a}A(\\bar{x})\\right)=\\frac{1}{a^2}\\mu ^2_1\\big (A(\\bar{x})\\big )=\\frac{1}{a^2}a^2\\mu ^2_1(\\bar{x})=\\mu ^2_1(\\bar{x}).$ Therefore, $A=D_a\\fatsemi L$ for Lorentz transformation $L$ and dilation $D_a$ .",
"Since $A$ is linear, $A=D^{-1}_a\\fatsemi A \\fatsemi D_a$ .",
"Thus $A=D^{-1}_a\\fatsemi D_a\\fatsemi L \\fatsemi D_a=L\\fatsemi D_a$ .",
"If $A= D\\fatsemi L$ for a Lorentz transformation $L$ and dilation $D$ , then $D$ has to be $D_a$ since $\\mu ^2_1\\big (L(\\bar{x})\\big )=\\mu ^2_1(\\bar{x})$ and $\\mu ^2_1\\big (A(\\bar{x})\\big )=a^2\\mu ^2_1(\\bar{x})$ .",
"Therefore, both $D$ and $L$ are unique in the decomposition of $A$ .",
"The same proof works when $A$ is decomposed as $A=L\\fatsemi D$ .",
"Lemma 7.16 Assume $\\mathsf {AxOField}$.",
"Let $\\bar{x},\\bar{y}\\in \\mathit {Q}^d$ such that $\\mu ^2_1(\\bar{x})>0$ and $(\\bar{x},\\bar{y})_{\\mu ^2_1}=0$ .",
"Then $\\mu ^2_1(\\bar{y})<0$ .",
"Assume indirectly that $\\mu ^2_1(\\bar{y})\\ge 0$ , i.e., $y^2_1\\ge y^2_2+\\ldots +y^2_d$ .",
"Since $x^2_1>x^2_2+\\ldots +x^2_d$ , we have that $x^2_1y^2_1>(x_2^2+\\ldots x_d^2)(y^2_2+\\ldots +y^2_d)$ .",
"By Cauchy–Schwarz inequalityFor a simple proof of Cauchy–Schwarz inequality that works also in ordered fields, see [1].",
"we have $(x_2^2+\\ldots x_d^2)(y^2_2+\\ldots +y^2_d)\\ge (x_2y_2+\\ldots + x_dy_d)^2$ .",
"Since $x_1y_1=x_2y_2+\\ldots + x_dy_d$ , we have that $x^2_1y^2_1>(x_1y_1)^2$ .",
"This contradiction proves that $\\mu ^2_1(\\bar{y})<0$ .",
"Proposition 7.17 Let $d\\ge 3$ .",
"Assume $\\mathsf {AxOField}$.",
"Let $A$ be an almost Lorentz transformation.",
"Then there is a $\\lambda >0$ such that $\\mu ^2_1\\big (A(\\bar{x})\\big )=\\lambda \\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"Since $A$ is an almost Lorentz transformation there is a $\\lambda \\ne 0$ such that $\\mu ^2_1\\big (A(\\bar{x})\\big )=\\lambda \\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"We are going to prove that this $\\lambda $ has to be positive.",
"Assume indirectly that $\\lambda <0$ .",
"Let $\\bar{y}=\\langle 0,1,0,\\ldots ,0\\rangle $ and $\\bar{z}=\\langle 0,0,1,0,\\ldots ,0\\rangle $ .",
"Then $\\mu ^2_1(\\bar{y})=\\mu ^2_1(\\bar{z})=-1$ and $(\\bar{y},\\bar{z})_{\\mu ^2_1}=0$ .",
"Let $\\bar{y}^{\\prime }=A(\\bar{y})$ and $\\bar{z}^{\\prime }=A(\\bar{z})$ .",
"Then $\\mu ^2_1(\\bar{y}^{\\prime })>0$ and $\\mu ^2_1(\\bar{z}^{\\prime })>0$ since $\\lambda <0$ ; and $(\\bar{y}^{\\prime },\\bar{z}^{\\prime })_{\\mu ^2_1}=0$ by Lemma REF .",
"These properties of $\\bar{y}^{\\prime }$ and $\\bar{z}^{\\prime }$ contradict Lemma REF .",
"Therefore, $\\lambda >0$ .",
"Remark 7.18 Proposition REF is not valid if $d=2$ since reflection $\\sigma _{tx}:\\langle t,x\\rangle \\mapsto \\langle x,t\\rangle $ is an almost Lorentz transformation and $\\mu ^2_1\\big (\\sigma _{tx}(\\bar{x})\\big )=-\\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^2$ .",
"Proposition 7.19 Let $d\\ge 3$ .",
"Assume that $\\langle \\mathit {Q}, + ,\\cdot \\rangle $ is a Euclidean field.",
"Then every almost Lorentz transformation is a composition of a Lorentz transformation and a dilation.$\\Box $ The statement follows from Lemma REF and Proposition REF since in Euclidean fields every positive number has a square root.",
"Remark 7.20 Proposition REF does not remain valid over arbitrary ordered fields.",
"To construct a counterexample, let $d=4$ , $\\langle \\mathit {Q},+, \\cdot , \\le \\rangle $ be the ordered field of rational numbers, and let $A$ be the following linear map $A(\\bar{x})=\\left\\langle \\frac{3x_1+x_2}{2}, \\frac{x_1+3x_2}{2}, x_3-x_4, x_3+x_4 \\right\\rangle $ for all $\\bar{x}\\in \\mathbb {Q}^4$ .",
"It is straightforward to check that $\\mu ^2_1\\big (A(\\bar{x})\\big )=2\\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathbb {Q}^4$ ; so $A$ is an almost Lorentz transformation.",
"However, $A$ cannot be the composition of a dilation $D$ and a Lorentz transformation $L$ over the field of rational numbers since then $A$ would also be the composition of $D$ and $L$ over the field of real numbers; and, by Lemma REF , the dilation in the unique decomposition of $A$ over the field of real numbers is $D_{\\sqrt{2}}$ , which does not map $\\mathbb {Q}^4$ to $\\mathbb {Q}^4$ .",
"Now we are ready to prove Theorem REF .",
"In Theorem REF we prove a slightly stronger result since we will not use axiom $\\mathsf {AxSelf}$.",
"Theorem 7.21 Let $d\\ge 3$ .",
"Assume $\\mathsf {AxOField}$, $\\mathsf {AxEv}$, $\\mathsf {AxPh}$, and $\\mathsf {AxSymD}$.",
"Let $m,k\\in \\mathsf {IOb}$ .",
"Then $\\mathsf {w}_{mk}$ is a Poincaré transformation.",
"Since, by $\\mathsf {AxSymD}$, the speed of light is 1 according to every inertial observer, $\\mathsf {w}_{mk}$ is a composition of an almost Lorentz transformation $A$ , a field-automorphism-induced map $\\tilde{\\alpha }$ and a translation $T$ by Theorem REF .",
"Specially, $\\mathsf {w}_{mk}$ maps lines to lines.",
"By $\\mathsf {AxOField}$, there is a line $l$ orthogonal to the plane of motion of $k$ according to $m$ .",
"By Theorem REF , both $l$ and $\\mathsf {w}_{mk}[l]$ are horizontal.",
"Therefore, by $\\mathsf {AxSymD}$, $\\mathsf {w}_{mk}$ maps $l$ to $\\mathsf {w}_{mk}[l]$ preserving the squared Euclidean distances of the points of $l$ .",
"Let $\\bar{v}$ be a direction vector of $l$ .That is, $\\bar{v}=\\bar{y}-\\bar{x}$ for two distinct points $\\bar{x}$ and $\\bar{y}$ of $l$ .",
"Then, by axiom $\\mathsf {AxSymD}$, we have that $\\mathsf {length}^2(x\\bar{v})=\\mathsf {length}^2\\big (\\tilde{\\alpha }(A(x\\bar{v}))\\big )$ for all $x\\in \\mathit {Q}$ since both $x\\bar{v}$ and $\\tilde{\\alpha }\\big ( A( x\\bar{v})\\big )$ are horizontal vectors.",
"Since both $l$ and $\\mathsf {w}_{mk}[l]$ are horizontal, we have that $\\mu ^2_1(\\bar{v})=\\mathsf {length}^2(\\bar{v})\\text{ and }\\mu ^2_1\\big (\\tilde{\\alpha }(A (\\bar{v}))\\big )=\\mathsf {length}^2\\big (\\tilde{\\alpha }(A(\\bar{v}))\\big ).$ Since $A$ is an almost Lorentz transformation, there is a $\\lambda \\ne 0$ such that $\\mu ^2_1\\big (A(\\bar{x})\\big )=\\lambda \\mu ^2_1(\\bar{x})$ for all $\\bar{x}\\in \\mathit {Q}^d$ .",
"Thus $\\mathsf {length}^2\\big (\\tilde{\\alpha }(A(\\bar{v}))\\big )\\stackrel{(\\ref {eq-x})}{=}\\mu ^2_1\\big (\\tilde{\\alpha }(A (\\bar{v}))\\big )=\\alpha \\big (\\mu ^2_1(A(\\bar{v}))\\big )\\stackrel{(\\ref {eq-y})}{=}\\alpha \\big (\\lambda \\mu ^2_1(\\bar{v})\\big )\\\\=\\alpha (\\lambda )\\alpha \\big (\\mu ^2_1(\\bar{v})\\big )\\stackrel{(\\ref {eq-x})}{=}\\alpha (\\lambda )\\alpha \\big (\\mathsf {length}^2(\\bar{v})\\big )$ Therefore, by the fact that that $\\mathsf {length}^2(a\\bar{y})=a^2\\mathsf {length}^2(\\bar{y})$ and Equations (REF ) and (REF ), we get $x^2\\mathsf {length}^2(\\bar{v})=\\alpha (\\lambda )\\alpha (x)^2\\alpha \\big (\\mathsf {length}^2(\\bar{v})\\big )$ for all $x\\in \\mathit {Q}$ .",
"Specially, $\\mathsf {length}^2(\\bar{v})=\\alpha (\\lambda )\\alpha \\big (\\mathsf {length}^2(\\bar{v})\\big )$ by choosing $x=1$ in equation (REF ).",
"Equations (REF ) and (REF ) imply that $x^2=\\alpha (x)^2$ for all $x\\in \\mathit {Q}$ .",
"Consequently, $\\alpha =\\mathsf {Id}_\\mathit {Q}$ by Lemma REF .",
"Thus $\\tilde{\\alpha }=\\mathsf {Id}_{\\mathit {Q}^d}$ and $1=\\alpha (\\lambda )$ by equation (REF ).",
"So $\\lambda =1$ , i.e., $A$ is a Lorentz transformation.",
"So $\\tilde{\\alpha }$ has to be the identity map and $A$ has to be a Lorentz transformation.",
"Thus $\\mathsf {w}_{mk}$ is a composition of a Lorentz transformation and a translation, i.e., it is a Poincaré transformation as it was stated."
],
[
"Concluding remarks",
"We have seen that the possible structures of quantities strongly depend on the other axioms of spacetime.",
"Typically, axioms requiring the existence of additional observers reduce the possible structures of quantities, see Theorems REF , REF , REF and Proposition REF .",
"We have proved several propositions about the connection between spacetime axioms and the possible structures of numbers.",
"However, there are still great many open questions in this research area, see Questions REF , REF , REF , REF , REF at pages REF , REF , REF , REF , REF , and Conjecture REF at page REF ."
],
[
"Acknowledgments",
"This research is supported by the Hungarian Scientific Research Fund for basic research grants No.",
"T81188 and No.",
"PD84093, as well as by a Bolyai grant for J. X. Madarász."
]
] | 1204.1350 |
[
[
"Nonlinear Damping in Graphene Resonators"
],
[
"Abstract Based on a continuum mechanical model for single-layer graphene we propose and analyze a microscopic mechanism for dissipation in nanoelectromechanical graphene resonators.",
"We find that coupling between flexural modes and in-plane phonons leads to linear and nonlinear damping of out-of-plane vibrations.",
"By tuning external parameters such as bias and ac voltages, one can cross over from a linear to a nonlinear-damping dominated regime.",
"We discuss the behavior of the effective quality factor in this context."
],
[
"Introduction",
"Advances in fabrication and detection techniques have enabled a wide range of experimental realizations of carbon-based nanoelectromechanical (NEM) resonators [1], [2], [3], [4].",
"However, to optimize their operation, an increased understanding of dissipation mechanisms is needed.",
"For NEM resonators in general, several processes leading to linear damping (LD) have been investigated [5], [6], [7], [8].",
"Specifically for graphene, at high temperatures, ohmic losses in the metallic gate and the graphene sheet have been argued to limit the quality factor [9].",
"Recently, the focus has shifted to study quantum aspects of mechanical motion [10], [11], such as mechanical cat states [12], which require a more detailed understanding of dissipation and decoherence mechanisms.",
"Since graphene-based resonators exhibit nonlinear behavior, one can expect the damping also to be amplitude dependent [13], [14], [15].",
"Nonlinear damping (NLD) was reported in recent experiments on graphene and carbon nanotube resonators [4].",
"However, little is known about the underlying physical mechanism, and typically phenomenological models are employed [13], [14], [15].",
"In these models, the resonator is coupled to a bath of harmonic oscillators.",
"For couplings that depend quadratically on the resonator amplitude, it is known that NLD emerges [16], [17], [13].",
"For carbon-based resonators such a coupling naturally arises if the strain couples linearly to the degrees of freedom of some subsystem, which can be regarded as a bath.",
"Two examples are the interaction between phonons and electrons[18], [19] and the coupling of mechanical modes.",
"The relative importance of the two mechanisms is a priori not known and will also depend on the details of the experimental realization.",
"In order to quantify the importance of the mechanical dissipation channel for NLD, we analyze the coupling between flexural modes and in-plane phonons.",
"We show that it leads to a quadratic coupling and, consequently, to both LD and NLD.",
"Whether LD or NLD dominates is determined by the ratio of vibrational amplitude and static deflection.",
"We give an estimate for the expected crossover between LD and NLD, which can be experimentally verified."
],
[
"Model and Method",
"We consider a graphene sheet of length $L$ and breadth $b$ , suspended over a trench of width $\\ell $ (cf.",
"Fig.",
"REF ).",
"The van der Waals attraction between the graphene and the substrate clamps down the sheet outside the suspended region [20], [21], [22].",
"The trench is modeled by allowing the sheet to freely displace vertically in this region.",
"Since out-of-plane displacement is accompanied by in-plane stretching or compression, flexural motion is converted into in-plane phonons in the suspended region.",
"The clamping constrains the out-of-plane motion over the substrate, but still allows for small in-plane displacements.",
"Consequently, in-plane phonons created in the suspended region transport energy away from this region.",
"In contrast to a phenomenological modeling approach we can relate dissipation to specific properties of the substrate and the graphene-substrate coupling.",
"These properties can be obtained independently by theoretical or experimental means.",
"Figure: (Color online) Schematic view of a suspended graphene membraneover a trenchin an insulating substrate.",
"A metallic gate is used for actuatingthe resonator.",
"In-plane phononsare created in thesuspended region and dissipate energy as they propagate away.The dynamics of graphene NEM-resonators are well described by the continuum theory of 2D-membranes [23].",
"For a resonator made from a sheet lying in the $xy$ -plane, this theory is conveniently formulated in terms of the in-plane displacement fields $u(x,y),v(x,y)$ in the $x-$ and $y-$ directions, respectively, and the displacement field in the $z-$ direction, $w(x,y)$ .",
"The equations of motion follow from the free energy ${\\cal F}=\\int dx dy\\, [{\\cal F}_b + {\\cal F}_s]$ where ${\\cal F}_b=\\frac{\\kappa }{2}|\\Delta w|^2$ is the free energy density associated with pure bending and ${\\cal F}_s=\\frac{1}{2}\\sum _{i,j}\\sigma _{ij}\\epsilon _{ij}$ is associated with stretching of the membrane.",
"The symmetric 2D strain and stress tensors are here defined as $\\epsilon _{xx}=u_{,x}+w_{,x}^2/2,\\quad 2\\epsilon _{xy}=(u_{,y}+v_{,x})+w_{,x}w_{,y},\\\\\\epsilon _{yy}=v_{,y}+w_{,y}^2/2\\;,$ and $\\sigma _{xx}=(\\lambda _{\\rm G}+2\\mu _{\\rm G})\\epsilon _{xx}+\\lambda _{\\rm G}\\epsilon _{yy},\\quad \\sigma _{xy}=2\\mu _{\\rm G}\\epsilon _{xy}, \\\\\\sigma _{yy}=(\\lambda _{\\rm G}+2\\mu _{\\rm G})\\epsilon _{yy}+\\lambda _{\\rm G}\\epsilon _{xx}\\;,$ respectively.",
"Spatial derivatives are denoted by subscripts, i.e., $u_{,x}=\\partial u/\\partial x$ .",
"The expression for the free energy, which is similar to that for large deflections of a plate [24], contains three material parameters, the bending energy $\\kappa \\approx 1.1 - 1.6$ eV, and the Lamé parameters, $\\mu _{\\rm G}\\approx 146$ N/m and $\\lambda _{\\rm G}\\approx 48$ N/m for graphene [25], [26], [27], [28].",
"To study qualitatively the effect of phonon radiation into the supporting substrate, we assume for simplicity a quasi 1D situation where variations in $y-$ direction are disregarded.",
"This would be valid for a wide sheet where deviations from this assumption is confined to the regions around the edges.",
"In this case we have only the displacement fields $u(x,t)$ and $w(x,t)$ .",
"In any realistic functioning device, there is some small amount of built in strain.",
"In practice, this implies that the energy contribution from the bending energy is always negligible for the lowest lying flexural modes [27].",
"Hence, to a good approximation we have for the quasi 1D graphene resonator attached to a substrate the free energy density $\\mathcal {F}(x,y) =\\frac{T_1}{2} \\left(u_{,x}^2 + u_{,x}w_{,x}^2 + \\frac{1}{4}w_{,x}^4\\right) \\\\+\\frac{1}{2} K(x) \\left( u - u_{\\rm S}\\right)^2+ \\mathcal {E}_{\\rm ext}[w]\\;,$ where we have defined $T_1 = \\lambda _{\\rm G}+2\\mu _{\\rm G}$ .",
"The potential $\\mathcal {E}_{\\rm ext}[w]$ accounts for interactions used to actuate the resonator.",
"The second to last term couples the graphene displacement to the substrate displacement $u_{\\rm S}(x,y)$ in a harmonic approximation [29], which largely allows us to obtain an analytical description.",
"The function $K(x)$ restricts this coupling to the supported region, i.e., $K(x)=K_0\\Theta (|x|-\\ell /2)$ with $\\Theta $ being the Heaviside step function.",
"The substrate is modeled as an elastic half-space and displacement at the surface, $\\vec{s}(\\vec{x},z=0,t) = (u_{\\rm S}, v_{\\rm S}, w_{\\rm S})$ , is given in terms of a response function [24], [30], [31], $s_{\\mu }(\\vec{x},z=0,\\omega ) = -\\sum _{\\nu }\\int \\frac{d^2 x^{\\prime }}{(2\\pi )^2} R_{\\mu \\nu }(\\vec{x}-\\vec{x}^{\\prime },\\omega ) \\\\\\times \\sigma _{\\nu z}(\\vec{x}^{\\prime },\\omega )\\;.$ Consistent with the 1D model of the graphene sheet, only $\\overline{u}_{\\rm S}(x)\\equiv \\int ^{b/2}_{-b/2} dy\\, u_{\\rm S}(x,y)$ is considered.",
"Within the harmonic approximation, $\\sigma _{xz}=K(x)\\left(u-u_{S}\\right)$ .",
"The free energy (REF ) leads to a coupling between flexural vibrations and in-plane motion via the coupling energy $\\mathcal {E}_{\\rm coup} = (T_1/2) u_{,x}w_{,x}^2$ , which is nonlinear in the flexural vibration amplitude.",
"This coupling leads to NLD of the flexural vibrations [16], [17], [13], [15]."
],
[
"Equations of motion",
"The equations of motion for the out-of-plane and in-plane vibrations resulting from Eq.",
"(REF ) are $\\rho _{\\rm G}\\ddot{w} - \\frac{T_1}{2} \\frac{d}{dx} \\left( 2 u_{,x} w_{,x} + w_{,x}^3 \\right)&= f_{\\rm dc} + f_{\\rm ac} \\cos (\\Omega t)\\;, \\\\\\rho _{\\rm G}\\ddot{u}- \\frac{T_1}{2} \\frac{d}{dx}\\left(2 u_{,x} + w_{,x}^2\\right)&=-K(x)\\left(u-\\overline{u}_{\\rm S}/b\\right), $ where $f_{\\rm dc}(x)$ and $f_{\\rm ac}(x) \\cos (\\Omega t)$ are the static and time dependent parts of the actuation force.",
"Typically, electrostatic actuation is used, resulting from a time dependent back-gate voltage of the form $V_{\\rm bg}(t)=V_{\\rm dc}+V_{\\rm ac}\\cos (\\Omega t)$ with $V_{\\rm dc}\\gg V_{\\rm ac}$ .",
"To simplify the analysis, we assume the equilibrium stress field resulting from $f_{\\rm dc}$ to be spatially uniform and equal to the tensile stress $T_0$ on the boundary [24].",
"Generally, at a given back-gate bias voltage, the resonance frequency $\\Omega _0(V_{\\rm dc})$ depends on initial stress and contains a shift due to electrostatic forces.",
"This so-called tuning behavior will be further discussed in Sec.",
"REF .",
"Since Eq.",
"() is linear in $u$ , the influence of the environment can be accounted for by a Green's function embedding technique.",
"The solution, $u(x,t) = \\int dx^{\\prime } \\int dt^{\\prime } G(x,x^{\\prime },t-t^{\\prime }) \\frac{c^2}{2} \\frac{d}{dx^{\\prime }} {w_{,x^{\\prime }}}^2(x^{\\prime },t^{\\prime })\\;,$ is given in terms of the in-plane response function $G$ , which contains information about the attachment to the substrate via Eq.",
"(REF ).",
"The speed of sound in graphene is denoted by $c=\\sqrt{T_1/\\rho _{\\rm G}}$ , where $\\rho _{\\rm G}$ is the mass density of graphene."
],
[
"Flexural mode dynamics",
"Next, we consider the fundamental flexural mode and set $w(x,t) = q(t) \\phi (x)$ for $|x| \\le \\ell /2$ and zero otherwise.",
"The mode shape $\\phi $ is normalized to the length of the resonator.",
"Upon projecting Eq.",
"(REF ) onto the fundamental mode, an ordinary differential equation for the vibration amplitude $q$ is obtained.",
"Further, moving to a rotating frame, we write $q(t)=\\left[q_0 +\\frac{1}{2}\\left(q_1(t)e^{i\\Omega t}+q_1^*(t)e^{-i\\Omega t}\\right)\\right]$ and $\\dot{q}(t)=\\frac{i\\Omega }{2}\\left[q_1(t)e^{i\\Omega t}-q_1^*(t)e^{-i\\Omega t}\\right]$ .",
"Inserting these expressions into the equation of motion and performing the averaging yields an equation for the slowly varying amplitude $q_1$ [dykr84], which contains memory terms related to linear and non-linear damping.",
"As the time-scales for flexural motion and in-plane phonons are well separated ($\\Omega _0\\ll c/\\ell $ ), the memory terms can be eliminated.",
"This procedure corresponds to a Markov approximation [13].",
"It is convenient to define new quantities $\\hat{\\chi }(\\Omega ) = \\frac{c^2}{2}\\int \\limits _{-l/2}^{l/2}dx\\int \\limits _{-l/2}^{l/2}dx^{\\prime } \\frac{d}{dx}\\left[\\phi _{,x}^2 \\hat{G}( x,x^{\\prime }, -\\Omega )\\right] \\\\\\times \\frac{d}{dx^{\\prime }}\\phi _{,x^{\\prime }}^2\\;,$ where $\\hat{G}( x,x^{\\prime }, \\omega ) = (2\\pi )^{-1}\\int d\\tau G(x,x^{\\prime },\\tau ) e^{i\\omega \\tau }$ is the Fourier transform of the in-plane response function.",
"We obtain an equation of motion for the complex envelope function $m \\dot{q_1} =\\left[im \\left(\\Omega _0-\\Omega \\right) q_1+ i\\frac{3}{8}\\frac{\\alpha }{\\Omega _0} |q_1|^2 q_1 \\right.\\\\\\left.-\\frac{1}{2} \\gamma q_1-\\frac{1}{8} \\eta |q_1|^2 q_1- \\frac{i}{2\\Omega _0} g\\right]\\;.$ For finite temperatures this equation has to be supplemented by noise forces, satisfying the fluctuation-dissipation relations.",
"The thermally induced vibrations can lead to an additional broadening of the response curves [13], [32].",
"In order to obtain a lower bound of LD and NLD we will work in the limit of zero temperature.",
"In Eq.",
"(REF ), the coefficients $m=\\rho _{\\rm G}\\ell b$ , $\\alpha $ , $\\gamma $ and $\\eta $ denote the suspended mass, the Duffing elastic constant, linear and non-linear damping, respectively.",
"They are given in terms of $\\hat{\\chi }$ as follows ${\\alpha } &={\\alpha _0} - \\frac{T_1 b}{2} \\frac{4}{3}{\\rm Re}\\;\\left(\\hat{\\chi }(0) + \\frac{1}{2}\\hat{\\chi }(2\\Omega )\\right) \\;,\\\\\\gamma &= -\\frac{T_1 b}{2\\Omega _0} q_0^2 \\,4 {\\rm Im}\\;\\hat{\\chi }(\\Omega ) \\;,\\\\\\eta &= -\\frac{T_1 b}{2\\Omega _0} \\,2 {\\rm Im}\\;\\hat{\\chi }(2\\Omega )\\;.$ Here, the bare Duffing constant is given by $\\alpha _0 = (T_1 b/2) \\int dx\\, \\phi _{,x}(x)^4$ .",
"The driving strength is $g = \\int dx\\, \\phi (x) f_{\\rm ac}(x)$ .",
"In accordance with our previous simplifications, we neglect the small polaronic shift of $\\Omega _0$ , which is proportional to ${\\rm Re}\\;\\hat{\\chi }$ , and an additional shift of $\\alpha $ due to the broken symmetry in the presence of static deflection.",
"Equation (REF ) is similar to the equations used to model NLD in micromechanical resonators [14], [15] and recent experiments on carbon-based resonators [4], the difference being the dependence of the damping coefficients in Eq.",
"(REF ) on the driving frequency.",
"In Eq.",
"(REF ) the prevailing damping mechanism is determined by the ratio $\\tilde{\\delta } \\equiv \\frac{\\eta |q_1|^2}{4\\gamma }\\approx \\frac{{{\\rm Im}\\;} \\hat{\\chi }(2\\Omega )}{8 {{\\rm Im}\\;}\\hat{\\chi }(\\Omega )}\\frac{|q^{\\rm max}_1|^2}{q_0^2}\\;.$ Here, $|q^{\\rm max}_1|$ denotes the maximum amplitude of the response for a given driving strength.",
"Thus, $\\tilde{\\delta }$ is determined by the ratio of the overlap integrals defined in Eq.",
"(REF ), which are purely geometrical quantities, and the ratio between the vibrational amplitude and the static deflection.",
"For a small static deflection, it is therefore expected that NLD dominates the damping caused by phonon radiation.",
"Similarly, the dimensionless ratio $\\tilde{\\eta } = \\frac{\\eta \\; \\Omega _0}{\\alpha }$ measures the relative importance of the two nonlinearities in Eq.",
"(REF ) [14].",
"For $\\tilde{\\eta }<\\sqrt{3}$ , the well-known bifurcation of the Duffing equation is present, while for $\\tilde{\\eta }>\\sqrt{3}$ this bifurcation vanishes.",
"The ratio $\\tilde{\\eta }$ is also a purely geometrical factor, apart from the weak dependence of $\\Omega _0$ on the static deformation of the graphene."
],
[
"Numerical method",
"To compute the overlap integrals (REF ) we first consider the Fourier transformed response of the substrate (REF ) $\\overline{u}_{\\rm S}(x, \\omega ) ={}&-\\int \\limits ^{L/2}_{-L/2} \\frac{d x^{\\prime }}{(2\\pi )^2}\\int \\limits ^{b/2}_{-b/2} dy^{\\prime }\\,\\int \\limits ^{b/2}_{-b/2} dy\\, \\\\& \\quad \\times R_{x x}(x-x^{\\prime },y-y^{\\prime },\\omega ) \\sigma _{x z}(x^{\\prime },y^{\\prime },\\omega ) \\\\\\approx {}&-\\int \\limits ^{L/2}_{-L/2} \\frac{d x^{\\prime }}{(2\\pi )^2}\\overline{R}_{xx}(x-x^{\\prime }, \\omega )\\, \\overline{\\sigma }_{x z}(x^{\\prime },\\omega )\\;.",
"$ In the second step, in order to get a purely 1D response function, we have approximated the $y^{\\prime }$ -dependence of $\\sigma _{x z}(x^{\\prime },y^{\\prime })$ by the mean value $\\frac{1}{b}\\overline{\\sigma }_{x z}$ and defined $\\overline{R}_{xx}(x-x^{\\prime },\\omega )\\equiv \\frac{1}{b}\\int ^{b/2}_{-b/2} dy^{\\prime }\\,\\int ^{b/2}_{-b/2} dy \\, R_{x x}(x-x^{\\prime },y-y^{\\prime },\\omega )$ We found that $\\overline{R}_{xx}(x-x^{\\prime },\\omega )$ is well approximated by the integral $\\int ^{b/2}_{-b/2} dy\\, R_{xx}(x-x^{\\prime },y,\\omega )$ ..",
"The response function $R_{\\mu \\nu }$ for an elastic half-space is known analytically [24], [30], [31] and mainly depends on the longitudinal and transversal sound velocities of the substrate (see Appendix ).",
"Evaluating Eq.",
"(REF ) at discrete positions $\\lbrace x_i\\rbrace ^N_1$ leads to the linear system $\\mathbb {K}\\,\\mathbf {u}_{\\rm S}(\\omega )= -\\left[ \\mathbb {I} - \\mathbb {K}\\,\\mathbb {R}(\\omega )\\right]^{-1}\\mathbb {K}\\,\\mathbb {R}(\\omega )\\mathbb {K}\\,\\mathbf {u}(\\omega )\\;,$ which can be solved for $\\overline{u}_{\\rm S}(x_i,\\omega )$ .",
"Here bold-face symbols denote vectors of length $N$ , e.g., $\\mathbf {u}=[u(x_1),\\ldots ,u(x_N)]$ and double struck symbols are $N\\times N$ matrices.",
"In particular, $\\mathbb {I}_{ij}=\\delta _{i,j}$ , $\\mathbb {K}_{ij}=K(x_i)\\delta _{i,j}$ and $\\mathbb {R}_{ij}=(2\\pi )^{-2}\\overline{R}_{xx}(x_i-x_j,\\omega )$ .",
"Using this result and the discretized version of the equation of motion () one obtains an equation for the in-plane response function $\\mathbb {G}_{ij}=\\hat{G}(x_i,x_j,\\omega )$ $\\left[-\\omega ^2 \\mathbb {I}- c^2 \\mathbb {L}+ \\frac{1}{\\rho _{\\rm G}} \\left[ \\mathbb {I} - \\mathbb {K}\\mathbb {R}(\\omega )\\right]^{-1}\\mathbb {K}\\right] \\mathbb {G}(\\omega )= \\mathbb {I}\\;,$ where $\\mathbb {L}$ is the discrete second derivative [34].",
"Approximating the integrations in Eq.",
"(REF ) by numerical quadratures, one finally obtains $\\hat{\\chi }(\\Omega ) = \\frac{c^2}{2}\\mbox{$\\Phi $}^{\\rm t} \\mathbb {G}(-\\Omega ) \\mbox{$\\Phi $}\\;$ with $\\mbox{$\\Phi $}_i = \\left.\\frac{d}{dx}\\phi _{,x}^2 \\right|_{x=x_i}$ , which allows the computation of $\\hat{\\chi }$ for a given geometry.",
"The parameters entering the equation of motion can then be calculated using Eqs.",
"(REF ).",
"Following Ref.",
"licr08, we set $\\tilde{\\gamma } = \\gamma /(m \\Omega _0)$ , $\\tilde{\\eta } = \\eta \\Omega _0/\\alpha $ , $\\tilde{g} = g \\sqrt{\\frac{\\alpha }{m^3}}/\\Omega _0^3$ , $\\tilde{\\Omega } = \\Omega /\\Omega _0$ , and $\\tilde{q} = q\\,\\sqrt{\\alpha /m \\Omega _0^2}$ .",
"In the limit of weak LD, $\\tilde{\\gamma }\\ll 1$ , the response of the resonator is determined solely by the dimensionless parameters $\\tilde{\\eta }$ , $\\tilde{g}$ and $\\tilde{\\Omega }$ , describing the nonlinear damping, the driving strength and the driving frequency.",
"To quantify the influence of LD and NLD, we consider the setup shown in Fig.",
"REF with a back-gate voltage $V_{\\rm bg} = V_{\\rm dc} + V_{\\rm ac}\\cos (\\Omega t)$ .",
"The fundamental-mode shape is taken to be $\\phi (x) = \\sqrt{2} \\cos (\\pi x/\\ell )$ , which gives $\\alpha _0 = 3 T_1 \\pi ^4 b/(4 \\ell ^3)$ .",
"Within a parallel plate model for electrostatic actuation, the force acting on the graphene sheet is given by $f(x) ={}& \\frac{\\partial }{\\partial w}\\frac{1}{2} C(w) V_{\\rm bg}^2 \\\\{}&\\approx -\\frac{\\epsilon _0 }{2(d + q(t)\\phi (x))^{2}} \\left( V_{\\rm dc}^2 + 2 V_{\\rm dc}V_{\\rm ac}\\cos (\\Omega t)\\right)\\;,$ where $C(w)=\\epsilon _0/(d+w)$ is the capacitance of a parallel plate capacitor with plates being separated by the distance $d+w$ and $\\epsilon _0$ is the vacuum permittivity.",
"The distance is determined by the depth $d$ of the trench and the flexural displacement $w$ of the resonator.",
"In the second line we further assumed $V_{\\rm dc}\\gg V_{\\rm ac}$ , which is typically found in experiments.",
"The force can be separated into a static and a time-dependent part, $f=f_{\\rm dc} + f_{\\rm ac}\\cos (\\Omega t)$ with $f_{\\rm dc}\\propto V_{\\rm dc}^2$ and $f_{\\rm ac}\\propto V_{\\rm dc}V_{\\rm ac}$ , respectively.",
"Since the displacement, which is on the order of a few nanometers, is much smaller than the trench depth, the force can be expanded in powers of $w$ .",
"Accordingly, the driving strength in Eq.",
"(REF ) becomes $g = 2 \\sqrt{2} \\ell b\\epsilon _0 V_{\\rm dc} V_{\\rm ac}/(\\pi d^2)$ .",
"Moreover, the static displacement can be found by solving Eqs.",
"(REF ) and () in the static limit (see Appendix ).",
"This yields $q_0 \\approx \\sqrt{2} \\ell ^2 \\epsilon _0 V_{\\rm dc}^2/(\\pi ^3 d^2 T_0)$ .",
"Note the dependence on the tensile stress $T_0$ ; $q_0$ becomes smaller for increasing tensile stress.",
"Table: Graphene and resonator parameters used for the calculations in Figs.",
"and .",
"Graphene and substrate parameters are taken from Refs.",
"lewe+08 and peue10.In the following, we consider a graphene resonator with dimensions and parameters as given in Tab.",
"REF .",
"We checked that the results do not change, for larger values of the total length $L$ .",
"Using Eqs.",
"(REF ) and (REF ) we obtain $\\alpha /\\alpha _0 \\approx 0.64$ and $\\tilde{\\eta }\\approx 7\\cdot 10^{-4}$ .",
"The latter implies bi-stable behavior of the resonator.",
"In general, these values depend sensitively on the geometry of the graphene sheet and on the substrate.",
"Our results provide a “best case” estimate, since the substrate is treated as a semi-infinite medium and the trench is modeled by the position dependent coupling $K(x)$ .",
"Lifting these restrictions will lead to a stronger response of the substrate, and more dissipation."
],
[
"Resonance frequency",
"As described in Sec.",
"REF the resonance frequency $\\Omega _0(V_{\\rm dc})$ depends on the initial stress and the bias voltage.",
"The dependence of $\\Omega _0$ on bias voltage, the so called tuning curve, is a characteristic feature of NEMS devices.",
"It is a result of the competition between softening (decreasing $\\Omega _0$ ) due to the electrostatic force [Eq.",
"(REF )], and stiffening (increasing $\\Omega _0$ ) due to the Duffing nonlinearity of the graphene sheet.",
"To obtain the tuning curve, we separate static and dynamic contributions to the displacement fields, $w(x,t)&=w_0(x) + \\delta w(x,t)\\;, \\\\u(x,t)&=u_0(x) + \\delta u(x,t)\\;$ and insert these expressions into the equations of motion given by Eqs.",
"(REF ).",
"The static solutions, $w_0$ and $u_0$ , are calculated in Appendix .",
"Further, we expand the static force $f_{\\rm dc}(x)$ up to first order in $\\delta w$ , $f_{\\rm dc}\\approx -\\frac{\\epsilon _0 V_{dc}^2}{2(d+w_0)^2}+\\frac{\\epsilon _0 V_{dc}^2}{(d+w_0)^3}\\delta w\\;.$ The resonance frequency is then obtained by collecting terms, which are linear in the vibration amplitude $\\delta w$ .",
"There are three such terms, which contribute to the resonance frequency, $\\Omega _0^2(V_{\\rm dc}) = \\Omega _0^2(0) + \\Delta \\Omega _{\\rm mech.",
"}^2 - \\Delta \\Omega _{\\rm el.",
"}^2\\;$ with $\\Omega _0^2(0) &= \\frac{T_0}{\\rho _{\\rm G}}\\frac{\\pi ^2}{\\ell ^2}\\;, \\\\\\Delta \\Omega _{\\rm mech.",
"}^2(V_{\\rm dc}) &= 2\\frac{T_1 \\pi ^4}{\\rho _{\\rm G}\\ell ^4} q_0^2=\\frac{8}{3 m}\\alpha _0 q_0^2\\;, \\\\\\Delta \\Omega _{\\rm el.",
"}^2(V_{\\rm dc}) &= \\frac{\\epsilon _0 V_{\\rm dc}^2}{d^3 \\rho _{\\rm G}}\\;.$ The three contributions are due to initial strain, mechanical stiffening and electrostatic softening, respectively.",
"Since the static deflection $q_0$ depends on the bias voltage $V_{\\rm dc}$ , the last two terms yield the voltage dependent tuning behavior.",
"Figure: (Color online) Resonance frequency Ω 0 \\Omega _0 vs. bias voltage.Symbols denote results of numerical calculation.The dashed (red) and dashed-dotted (blue) lines show the contributions ofmechanical stiffening and electrostatic softening for T 0 =10 -3 T 1 T_0=10^{-3} T_1, respectively.",
"Parameters are given in Tab.",
".Figure REF shows the tuning curve for the parameters given in in Tab.",
"REF .",
"For voltages, $V_{\\rm dc} > 10\\;\\text{V}$ , the resonance frequency (squared) is mainly determined by the mechanical stiffening, which scales with $V_{\\rm dc}^4$ while the softening term scales with $V_{\\rm dc}^2$ according Eqs.",
"(REF ).",
"Depending on the specific geometry and the initial stress, the resonance frequency of the resonator may be substantially tuned using the bias voltage.",
"Since the linear and nonlinear damping coefficients given by Eqs.",
"(REF ) depend on frequency, the magnitude of LD and NLD will, in principle, also be influenced by the tuning curve.",
"In order to disentangle the influence of $\\Omega _0(V_{\\rm dc})$ and the coupling to the in-plane phonons, we will only consider a constant resonance frequency $\\Omega _0=\\Omega _0(0) = \\sqrt{T_0/\\rho _{\\rm G}}(\\pi /\\ell )$ in the following discussions (see Appendix for the influence of the tuning on the quality factor)."
],
[
"Damping ratio",
"The relative importance of LD and NLD, which is quantified by $\\tilde{\\delta }$ defined in Eq.",
"(REF ), is determined by the ratios ${\\rm Im}\\;\\hat{\\chi }(2\\Omega )/(8{\\rm Im}\\;\\hat{\\chi }(\\Omega ))$ and $|q^{\\rm max}_1|/q_0$ .",
"The former weakly depends on the geometric details.",
"Figure: (Color online) Ratio δ ˜\\tilde{\\delta } of nonlinear (NLD) andlinear (LD) dampingterms according toEq.",
"(); a) bias voltage and b) ac voltage dependence.The thin dashed and dashed-dotted lines show the asymptotic behaviorfor strong LD and NLD.",
"A crossover between the two regimes is achieved bychanging the bias voltage.Parameters are given in Tab.",
".For small $\\Omega $ one can expand ${\\rm Im}\\;\\hat{\\chi }(\\Omega )$ in odd powers of $\\Omega $ .",
"As ${\\rm Im}\\;\\hat{\\chi }$ is proportional to the density of states of the substrate phonons, $D(\\Omega )\\propto \\Omega $ , we expect on symmetry-grounds for a quasi-1D geometry, that ${\\rm Im}\\;\\hat{\\chi }(\\Omega )\\propto \\Omega ^3$ .",
"Consistent with this expectation, we obtain numerically ${\\rm Im}\\;\\hat{\\chi }(2\\Omega )/(8{\\rm Im}\\;\\hat{\\chi }(\\Omega ))\\approx 0.93$ .",
"The maximum amplitude $q_1^{\\rm max}$ can easily be found from Eq.",
"(REF ) in the steady-state limit, which yields an implicit equation for the magnitude $|q_1|$ of the steady-state amplitude [14].",
"Sweeping the driving frequency, the maximum amplitude is attained when $d|q_1|/d\\Omega = 0$ , which results in the cubic equation $4 \\tilde{g} = |\\tilde{q}_1^{\\rm max}| (4 \\tilde{\\gamma } + \\tilde{\\eta }|\\tilde{q}_1^{\\rm max}|^2)\\;.$ Here, $\\tilde{\\gamma }$ and $\\tilde{g}$ depend on the bias voltage $V_{\\rm dc}$ via $q_0$ and $f_{\\rm ac}$ , respectively.",
"However, note that only $\\tilde{g}$ depends on the ac voltage.",
"Due to the different dependencies of $q_0$ and $|q_1^{\\rm max}|$ on the bias voltage, one can achieve a crossover from NLD to LD dominated behavior by increasing the bias voltage.",
"This is shown in Fig.",
"REF a.",
"In the limit of small $V_{\\rm dc}$ , $|\\tilde{q}_1^{\\rm max}| \\approx \\left( 4 \\tilde{g}/\\tilde{\\eta } \\right)^{1/3}\\propto V_{\\rm ac}^{1/3}V_{\\rm dc}^{1/3}$ and $\\tilde{\\delta }> 1$ , i.e., NLD dominates.",
"For large $V_{\\rm dc}$ , $|\\tilde{q}_1^{\\rm max}| \\approx \\tilde{g}/\\tilde{\\gamma }\\propto V_{\\rm ac}V_{\\rm dc}^{-3}$ and $\\tilde{\\delta }$ goes to zero with increasing $V_{\\rm dc}$ .",
"Since the static displacement is determined only by the geometry and the bias voltage, and the maximal amplitude additionally depends on the ac voltage, the crossover can also be realized by tuning $V_{\\rm ac}$ , which is shown in Fig.",
"REF b. Equating the expressions for $|\\tilde{q}_1^{\\rm max}|$ in the two limits gives an estimate for the crossover for both voltages.",
"Additionally, due to the dependencies of $q_0\\propto T_0^{-1}$ and $\\Omega _0\\propto \\sqrt{T_0}$ on the initial tension $T_0$ one finds that the damping ratio $\\tilde{\\delta }$ increases with increasing tension in both regimes ($\\tilde{\\delta } \\propto T_0^3$ and $\\tilde{\\delta }\\propto T_0$ in the LD and NLD regime, respectively).",
"Thus, the non-linear damping is enhanced for larger $T_0$ ."
],
[
"Quality factor",
"To quantify the energy loss we consider the quality factor $Q=\\Omega _0 \\langle E_\\perp \\rangle /\\langle \\dot{E}_\\perp \\rangle $ , which measures the time-averaged dissipated energy $\\langle \\dot{E}_\\perp \\rangle $ normalized to the average energy $\\langle E_\\perp \\rangle $ in the flexural modes.",
"The nonlinearities render $Q$ amplitude dependent.",
"To get a worst case estimate, we use the maximal amplitude.",
"In the slow envelope approximation we find $\\frac{1}{Q}\\approx \\frac{\\Omega _0 \\left(\\gamma + \\frac{1}{4} \\eta |q_1^{\\rm max}|^2\\right)}{m \\Omega _0^2 + \\frac{1}{2}\\frac{3}{8} \\alpha |q_1^{\\rm max}|^2} \\;.$ The nature of the damping influences $Q$ .",
"In the LD dominated regime, $\\tilde{\\delta }\\ll 1$ , $Q$ is independent of the vibrational amplitude, $Q_{\\rm LD}\\approx m \\Omega _0/\\gamma $ .",
"In contrast, for $\\tilde{\\delta }>1$ one gets $Q_{\\rm NLD}\\approx 4 m \\Omega _0/(\\eta |q_1^{\\rm max}|^2)$ for $\\tilde{\\eta }>1$ .",
"Thus, $Q$ increases with decreasing driving strength.",
"This agrees with the conclusions of Ref.",
"eimo+11.",
"Figure REF a shows the quality factor as a function of bias voltage for constant $V_{\\rm ac}$ .",
"As expected, $Q$ decreases with increasing bias and excitation voltages and its behavior with regard to applied voltage changes qualitatively at the crossover between LD and NLD regimes.",
"The asymptotic LD behavior limits the maximally attainable $Q$ -factor, which is indicated by the gray area.",
"We also compare to the case where the LD is additionally caused by a mechanism that does not depend on the bias voltage leading to $Q_0$ .",
"In this case the effective $Q$ -factor, $Q_{\\rm eff}^{-1}=Q^{-1} + Q_0^{-1}$ , has a cutoff for small $V_{\\rm dc}$ as shown in Fig.",
"REF b, which further limits the region of attainable $Q$ -factors.",
"The qualitative difference between LD and NLD is still present and should be experimentally observable.",
"Most importantly, by decreasing $V_{\\rm ac}$ the maximally attainable $Q$ -factor, which is determined by other damping mechanisms can be approached.",
"Figure: (Color online) Quality factor QQ vs. biasvoltage.",
"a) QQ calculated from Eq.",
"() and b)with additional voltage independent damping, Q eff -1 =Q -1 +Q 0 -1 Q_{\\rm eff}^{-1}=Q^{-1} + Q_0^{-1}with Q 0 =10 5 Q_0=10^5.",
"The gray area indicates the region ofattainable QQ-factors.",
"The dashed lines correspond to the behavior in a).Parameters are given in Tab.",
"."
],
[
"Conclusions",
"In conclusion, we have studied coupling between flexural vibrations and in-plane displacements as a physical mechanism for damping of flexural modes in graphene resonators.",
"A characteristic consequence, which influences the behavior of the dependence of the quality factor on bias and excitation voltages, is the competition between static deflection and vibrational amplitude.",
"We note that the same type of behavior would naturally occur for any dissipative process which couples linearly to the strain; for example, Ohmic dissipation induced by synthetic gauge fields [19].",
"The cross-over should allow for an experimental verification of this class of damping mechanisms.",
"We thank J. Atalaya for helpful discussions.",
"The research leading to these results has received funding [DM,AI] from the EU 7th framework program (FP7/2007-2013) RODIN (grant agreement no: 246026) and the Swedish Research Council [JK]."
],
[
"Response of an elastic half-space",
"The displacement response at the surface of an elastic half-space to a stress acting on the surface is given in terms of a response function by Eq.",
"(REF ).",
"If the stress is directed parallel to the $x$ -axis, the spatial Fourier transform of Eq.",
"(REF ) reads $u_{\\rm S}(\\vec{k},z=0,\\omega ) = - R_{xx}(\\vec{k},\\omega ) \\sigma _{x z}(\\vec{k},\\omega )\\;,$ where $\\vec{k}=(k_x,k_y)$ is the surface wave vector.",
"The response function $R_{xx}(\\vec{k},\\omega )$ for finite frequencies is explicitly given by[30], [31] $R_{xx}(\\vec{k}, \\omega )= -\\frac{i}{\\rho _{\\rm S} c_{\\rm T}^2}\\left(\\frac{p_{\\rm T}(\\omega , k)}{S(\\omega , k)} \\frac{\\omega ^2}{c_{\\rm T}^2}\\frac{k_x^2}{k^2}+\\frac{1}{p_{\\rm T}(\\omega , k)} \\frac{k_y^2}{k^2}\\right)$ with $p_{\\rm L,T}(\\omega , k) &= \\sqrt{\\left(\\frac{\\omega }{c_{\\rm L,T}}\\right)^2 +i\\varepsilon - k^2} \\;,\\\\S(\\omega , k) &= \\left[ \\left(\\frac{\\omega }{c_{\\rm L,T}}\\right)^2 - 2 k^2 \\right]^2+ 4 k^2 p_{\\rm L}(\\omega , k) p_{\\rm T}(\\omega , k)\\;,$ where $c_{\\rm L}$ and $c_{\\rm T}$ are the longitudinal and transversal speeds of sound, respectively, and the infinitesimal $\\varepsilon >0$ ensures causality.",
"Notice that $p_{\\rm L,T}$ and $S(\\omega , k)$ depend only on the modulus $k$ of the wave vector $\\vec{k}$ .",
"The response function in real space is then $R_{xx}(\\vec{x}, \\omega ) &= \\int d^2k R_{xx}(\\vec{k}, \\omega ) e^{i\\vec{k} \\cdot \\vec{x}} \\\\&= -\\frac{2\\pi i}{\\rho _{\\rm S} c_{\\rm T}^2}\\left(\\frac{\\partial }{\\partial x}I_{x}(x, y)+\\frac{\\partial }{\\partial y}I_{y}(x, y)\\right) \\;.$ Here, we defined $I_{x}(x, y) &= \\frac{x}{\\sqrt{x^2+y^2}} \\left(\\frac{\\omega }{c_{\\rm T}}\\right)^2 \\int dk\\;\\frac{p_{\\rm T}(\\omega , k)}{S(\\omega , k)} J_1( k \\sqrt{x^2+y^2}) \\;,\\\\I_{y}(x, y) &= \\frac{y}{\\sqrt{x^2+y^2}} \\int dk\\; \\frac{1}{p_{\\rm T}(\\omega , k)}J_1( k \\sqrt{x^2+y^2})\\;,$ where $J_1$ is a first order Bessel function of the first kind.",
"Note, that $I_{x}(x, -y) = I_{x}(x, y)\\,,\\quad I_{y}(x, -y) = -I_{y}(x, y)\\;.$ The expressions given in Eqs.",
"() are a very convenient starting point for the numerical evaluation of the response function used in Sec.",
"REF .",
"The zero-frequency response can be directly calculated in real space[24].",
"One finds $R_{xx}(\\vec{x}, \\omega =0)= \\frac{1}{4\\pi \\rho _{\\rm S}c_{\\rm T}^2}\\frac{2(c_{\\rm T}^2-c_{\\rm L}^2) x^2 - c_{\\rm L}^2 y^2}{(c_{\\rm L}^2-c_{\\rm T}^2)(x^2+y^2)^{3/2}}\\;.$"
],
[
"Static displacement",
"In the static limit, the equations for the in-plane and out-of-plane displacements (REF ) within the suspended region become $T_1 u_{,xx} + \\frac{T_1}{2}\\partial _x \\left(w_{,x}^2\\right)=&0\\;,\\\\-\\frac{T_1}{2}\\partial _x \\left[\\left(2 u_{,x}+w_{,x}^2\\right)w_{,x}\\right]=&f_{dc}\\;,$ with vanishing boundary conditions at $x=\\pm \\ell /2$ for the out-of-plane displacement.",
"To find the proper boundary conditions for the in-plane displacement, we need to consider the coupling to the substrate in the non suspended region.",
"Here, the equation for the in-plane displacement () is given by $T_1 u_{,xx}-K(x)(u(x)-\\overline{u}_S/b)=0\\;.$ Following the same line of reasoning as in the main text, the static substrate response can be written as $\\overline{u}_{\\rm S}(x) ={}& -\\int \\limits ^{L/2}_{-L/2} \\frac{d x^{\\prime }}{(2\\pi )^2}\\overline{R}_{xx}(x-x^{\\prime })\\,\\Theta (|x^{\\prime }|-\\ell /2) h(x^{\\prime })\\;.",
"$ with $h(x)=K_0(u-\\overline{u}_S/b)$ and $\\overline{R}_{xx}(x-x^{\\prime })$ being the static response function for an elastic half space given by Eq.",
"(REF ) integrated over $y$ .",
"To treat the problem analytically, we convert Eqs.",
"(REF ) and (REF ) into a local equation for the in-plane displacement.",
"In the limit of very strong coupling to the substrate, the spatial variation of $h(x)$ is small, in which case $-&\\int \\limits ^{L/2}_{-L/2} \\frac{d x^{\\prime }}{(2\\pi )^2}\\overline{R}_{xx}(x-x^{\\prime })\\, \\Theta (|x^{\\prime }|-\\ell /2)h(x^{\\prime })\\approx \\nonumber \\\\&-h(x)\\int \\limits ^{L/2}_{-L/2}\\frac{d x^{\\prime }}{(2\\pi )^2}\\overline{R}_{xx}(x-x^{\\prime }) \\Theta (|x^{\\prime }|-\\ell /2)\\;.$ This makes it possible to solve for $h(x)$ in terms of the in-plane displacement $u(x)$ .",
"One finds $h(x)=\\frac{K_0}{1-\\overline{R}_0(x)K_0}u(x)\\;,$ where $\\overline{R}_0(x)\\equiv (2\\pi )^{-2}\\int \\limits ^{L/2}_{-L/2} d x^{\\prime }\\; \\overline{R}_{xx}(x-x^{\\prime }) \\Theta (|x^{\\prime }|-\\ell /2)$ .",
"This expression is valid outside the suspended region and is approximately given by $h(x)\\approx -1/\\overline{R}_0(x)$ , which assumes $K_0 \\overline{R}_0(x)\\gg 1$ .",
"Consequently, the equation for the in-plane displacement, Eq.",
"(REF ), is modified to become $T_1 u_{,xx}+\\overline{R}_0(x)^{-1} u = 0$ for $|x|>\\ell /2$ .",
"Thus, the effect of the substrate is reduced to that of a spring with a spatially varying spring constant.",
"The displacement $u$ is expected to decay exponentially to zero in the clamped region with a decay length $\\lambda \\equiv \\sqrt{\\overline{R}_0(x) T_1}$ .",
"For the substrate parameters given in Table REF , this amounts to $\\lambda \\approx 100$ nm.",
"As a consequence, within a distance of 100 nm from the edge of the suspended region the in-plane displacement $u(x)$ is essentially zero.",
"To a good approximation, we therefore assume vanishing boundary conditions for in-plane displacement at $|x|=\\ell /2$ .",
"Setting $u(x)=(T_0/T_1)x+\\Delta u(x)$ , where the first terms accounts for initial strain in the graphene, the boundary conditions are $w(x=\\pm \\ell /2)=0$ and $\\Delta u(x=\\pm \\ell /2)=0$ .",
"Using the Ansatz $w(x)=q_0 \\phi (x)$ with $\\phi (x)=\\sqrt{2}\\cos \\pi x/\\ell $ , the in-plane equation (REF ) reads $\\Delta u_{,xx}=-\\frac{q_0^2}{2}\\partial _x \\left(\\phi _{,x}^2\\right)\\;.$ Consequently, the in-plane displacement will be given by $\\Delta u(x)= -q_0^2\\frac{\\pi ^2}{\\ell ^2}\\int \\limits ^{x}_{0} dx^{\\prime } \\sin ^2\\pi x^{\\prime }/\\ell + \\frac{\\pi ^2}{2\\ell ^2}q_0^2 x\\;.$ Inserting this expression into Eq.",
"() and we obtain $q_0\\left( \\frac{\\pi ^2}{\\ell ^2} T_0+ \\frac{\\pi ^4}{2 \\ell ^4} T_1 q_0^2\\right) =\\frac{2\\sqrt{2}}{\\pi }f_{dc}\\;.$ This is a purely algebraic equation for the static deflection.",
"In the limit $q_0\\ll \\frac{\\ell }{\\pi }\\sqrt{\\frac{T_0}{T_1}}\\approx 10$ nm for $\\ell =1\\mu $ m and $T_0/T_1=10^{-3}$ ,Êthe cubic term can be neglected and $q_0\\propto f_{\\rm dc}$ .",
"Figure: Static deflection q 0 q_0 vs. bias voltage for three values of initial tension.",
"The linear approximation [Eq.",
"()] is shown as dashed lines in the figure, while the squares and triangles correspond to the full numerical solution of the static problem.To compute $q_0$ , we need to consider the electrostatic interaction with the back gate.",
"The static force acting on the graphene is given by Eq.",
"(REF ).",
"Considering the limit $q_0\\ll d$ , we obtain for the static displacement $q_0=-\\sqrt{2}\\frac{\\ell ^2\\epsilon _0 V_{dc}^2}{\\pi ^3 T_0d^2}\\;,$ which is the expression given in Sec.",
".",
"In Fig.",
"REF the linear approximation (dashed line), given by Eq.",
"(REF ), is compared to the full numerical solution of Eq.",
"() (squares and triangles), which takes the substrate into account.",
"The linear approximation remains valid in the displayed interval for the two larger values of initial strain $T_0/T_1$ , while a more significant deviation is apparent for the lowest value of the strain."
],
[
"Influence of tuning and initial tension on the quality factor",
"In Sec.",
"REF we discussed the voltage dependence of the resonance frequency (tuning curve) and showed that the frequency can be substantially tuned by changing the bias voltage $V_{\\rm dc}$ .",
"Since the linear and nonlinear damping constants given by Eqs.",
"(REF ) depend on frequency, the quality factor will also depend on the tuning.",
"In order to quantify the influence of the voltage dependence of the resonance frequency on $Q$ , Fig.",
"REF shows the quality factor for constant $\\Omega _0=\\Omega _0(0)$ (dashed lines) and $\\Omega _0(V_{\\rm dc})$ (full lines).",
"One sees that deviations between these two cases appear only for larger voltages ($V_{\\rm dc} > 20\\;\\text{V}$ ).",
"Moreover, the qualitative behavior and the cross-over from NLD to LD behavior remains unchanged.",
"This confirms our statement in Sec.",
"REF , that the behavior of $Q$ is dominated by the damping coefficients $\\gamma $ and $\\eta $ rather than the voltage dependence of $\\Omega _0$ .",
"Additionally, Fig.",
"REF shows the quality factor for a smaller value of the initial tension.",
"In this case, the quality factor is decreased for all values of the static bias voltage.",
"In the limit of large LD, this is due the increased static deflection (see Eq.",
"(REF )).",
"In the opposite limit, the quality factor is independent of the static deflection, and the decrease in quality factor is instead a result of the decreasing resonance frequency $\\Omega _0(0)\\propto \\sqrt{T_0}$ .",
"Furthermore, as argued at the end of Sec.",
"REF , the cross-over between NLD and LD is shifted toward lower values of the bias voltage, signifying a decrease in the importance of NLD for lower tension."
]
] | 1204.0911 |
[
[
"Quantum teleportation between moving detectors in a quantum field"
],
[
"Abstract We consider the quantum teleportation of continuous variables modeled by Unruh-DeWitt detectors coupled to a common quantum field initially in the Minkowski vacuum.",
"An unknown coherent state of an Unruh-DeWitt detector is teleported from one inertial agent (Alice) to an almost uniformly accelerated agent (Rob, for relativistic motion), using a detector pair initially entangled and shared by these two agents.",
"The averaged physical fidelity of quantum teleportation, which is independent of the observer's frame, always drops below the best fidelity value from classical teleportation before the detector pair becomes disentangled with the measure of entanglement evaluated around the future lightcone of the joint measurement event by Alice.",
"The distortion of the quantum state of the entangled detector pair from the initial state can suppress the fidelity significantly even when the detectors are still strongly entangled around the lightcone.",
"We point out that the dynamics of entanglement of the detector pair observed in Minkowski frame or in quasi-Rindler frame are not directly related to the physical fidelity of quantum teleportation in our setup.",
"These results are useful as a guide to making judicious choices of states and parameter ranges and estimation of the efficiency of quantum teleportation in relativistic quantum systems under environmental influences."
],
[
"Introduction",
"Quantum teleportation is not only of practical values but also of theoretical interest because it contains many illuminating manifestations of quantum physics, clarifying fundamental issues such as quantum information and classical information, quantum nonlocality and relativistic locality, spacelike correlations and causality, and so on [1].",
"The first scheme of quantum teleportation is proposed by Bennett, Brassard, Crepeau, Jozsa, Peres, and Wootters (BBCJPW) [2], where an unknown state of a qubit $C$ is teleported from one spatially localized agent Alice to another agent Bob using an entangled pair of qubits $A$ and $B$ prepared in one of the Bell states and shared by Alice and Bob, respectively.",
"Such an idea is then adapted to the systems with continuous variables such as harmonic oscillators by Vaidman [3], who introduces an EPR state [4] for the shared entangled pair to teleport an unknown coherent state.",
"Braunstein and Kimble (BK) [5] generalized Vaidman's scheme from EPR states with exact correlations to squeezed coherent states.",
"In doing so the uncertainty of the measurable quantities has to be considered, which reduces the degree of entanglement of the $AB$ -pair as well as the fidelity of teleportation.",
"Since quantum teleportation can address the issues of nonlocality and causality, it is natural to consider quantum teleportation in fully relativistic systems and in non-inertial frames.",
"Alsing and Milburn made the first attempt of calculating the fidelity of quantum teleportation between two moving cavities in relativistic motions [6] – one is at rest (Alice), the other is uniformly accelerated (Rob, for relativistic motion) in the Minkowski frame – though their process of teleportation is not complete and the result is not quite reliable [7], [8].",
"Alternatively, one of us [9] has considered quantum teleportation in the Unruh-DeWitt (UD) detector theory [10], [11] with the agents in similar motions but based on the BK scheme: An unknown coherent state of an UD detector $C$ is teleported from Alice to Rob using an entangled pair of the UD detectors $A$ and $B$ initially in a two-mode squeezed state and held by Alice and Rob, respectively.",
"While both $A$ and $B$ are coupled with a common massless quantum field initially in the Minkowski vacuum, the detector $C$ is isolated from the environment to identify the best fidelity that $A$ and $B$ moving in a common environment can offer.",
"Unfortunately, the fidelity of quantum teleportation considered in [9] is not a physical one.",
"More careful consideration of the relativistic effects of quantum information associated with the quantum field is needed to get the correct results, which we do in this paper.",
"There are many subtle issues related to quantum teleportation in relativistic systems which play an important role but have hitherto been largely ignored.",
"Foremost, it is crucial that Bob or Rob has to have the full knowledge about the quantum state of the $AB$ -pair to achieve quantum teleportation perfectly.",
"For example, in the BBCJPW scheme, Alice performs a Bell measurement jointly on $C$ and $A$ , which collapses the $CA$ -pair into one of the Bell states, then sends the information to Bob classically about which Bell state the $CA$ -pair ends up in.",
"According to this classical information Bob can reproduce the initial state of $C$ by operating on $B$ .",
"Bob's “Manual\" in how to connect each kind of received information (the outcome of measurement on the $CA$ -pair) with some specific set of operations on $B$ is determined by the quantum state of the $AB$ -pair right before the joint measurement by Alice.",
"If the quantum state of the entangled $AB$ -pair is changed from their initial state, then Bob's “Manual\" must be changed accordingly to keep the fidelity of quantum teleportation perfect.",
"Second, categorically there seems to be a common misunderstanding or underestimation of relativistic effects in quantum information.",
"Many people may think of “relativistic\" as referring to fast motion of the atoms (qubits) or oscillators (detectors), which is not commonly encountered, but forget about the fact that a quantum field which is ever present in any setup is relativistic, namely that the laws of special relativity govern the field and hence enter in the interaction between the qubits or detectors even when there is no direct interaction amongst them.",
"This oversight is largely due to the focus of quantum foundational issues from the viewpoint of quantum mechanics, which is only an approximation – the nonrelativistic limit – to the consistent theory, which is quantum field theory.",
"Third, which is related to the second, is the neglect of environmental influences.",
"The presence of a quantum field is unavoidable and acts as an ubiquitous environment to the qubits or detectors in question ($A$ , $B$ , or $C$ described in the context of quantum teleportation above) whose effects need to be included in one's consideration.",
"They include: Each qubit or oscillator can be decohered by coupling to a quantum field, but decoherence can be lessened if the two qubits are placed in close range due to mutual influences mediated by the field;"
],
[
"Entanglement dynamics",
"The entanglement between two qubits or oscillators changes in time as their reduced state evolves;"
],
[
"Unruh effect",
"A pointlike object such as a UD detector coupled with a quantum field and uniformly accelerated in the Minkowski vacuum of that quantum field would experience a thermal bath of the field quanta at the Unruh temperature proportional to its proper acceleration.",
"Furthermore, objects in a relativistic system may behave differently when observed in different reference frames:"
],
[
"Frame dependence",
"Since quantum entanglement between two spatially localized degrees of freedom is a kind of spacelike correlation in a quantum state, which depends on reference frames, quantum entanglement of two localized objects but separated in space is frame-dependent;"
],
[
"Time dilation",
"For two moving objects localized in space and so parameterized by their proper times, the time dilations of them observed in the rest frame will naturally enter the dynamics of entanglement between them.",
"The above factors have been considered in some detail [14], [13], [12].",
"But there are new issues of foundational value which need be included in the consideration of quantum teleportation.",
"We mention two such issues related to open quantum systems below."
],
[
"Fidelity and entanglement in open quantum systems",
"As indicated in the BK scheme, fidelity of quantum teleportation could depend on (i) quantum entanglement of the $AB$ -pair, and (ii) the consistency of the quantum state of the $AB$ -pair with their initial state.",
"Both would be reduced by the coupling with an environment, the quantum field being an ubiquitous one.",
"We will compare the time evolution of the logarithmic negativity of the $AB$ -pair in a given reference frame and the fidelities of quantum teleportation in the same frame to distinguish the effects caused by these two factors.",
"For simplicity and for the purpose of comparing the degree of entanglement of the $AB$ -pair and the fidelity of quantum teleportation, one of us [9] considered the “pseudo-fidelity\" of teleportation by imagining that right at the moment the joint measurement on the $CA$ -pair was done by Alice, Rob gets the information of the outcome from Alice instantaneously and performs proper local operations on $B$ accordingly.",
"In reality classical information need some time to travel from Alice to Rob, and during the traveling time of the signal detector $B$ keeps evolving, so we expect the physical fidelity of teleportation will be further reduced.",
"Our calculation verifies this feature."
],
[
"Measurement in different frames",
"Quantum states make sense only in a given frame where a Hamiltonian is well defined [15].",
"Two quantum states of the same system with quantum fields in different frames are comparable only on those totally overlapping time-slices associated with certain moments in each frame.",
"By a measurement local in space, e.g.",
"on a point-like UD detector coupled with a quantum field, quantum states (of the combined system) in different frames can be interpreted as if they collapsed on different time-slices passing through the same measurement event.",
"Nevertheless, the post-measurement states will evolve to the same state up to a coordinate transformation when they are compared at some time-slice in the future [16].",
"Given the consistency of instantaneous measurement, we can further study the evolution of fidelities of teleportation in different frames.",
"Although they don't have to agree with each other if the time-slices of those different frames never overlap (except for the fiducial time-slice where the initial state is defined), we will show that the reduced state of the qubit or detector $B$ after wave functional collapse in different frames will become consistent once Rob enters the future lightcone of the joint measurement event by Alice.",
"To calculate the physical fidelity, we find it the most convenient to collapse the wave functional of the combined system almost on the future lightcone of the local measurement event by Alice so the continuous evolution during the moment of wave functional collapse and the local operation by Rob can be neglected in the cases with negligible mutual influences.",
"The paper is organized as follows.",
"In Section II we introduce the model we use for addressing these issues.",
"In Section III we review the definition of the averaged pseudo-fidelity and calculate the pseudo-fidelities in the ultraweak coupling limit in different frames; we then illustrate some results beyond the ultraweak coupling limit.",
"In Section IV we modify the setup and calculate the averaged physical fidelity for a more realistic case in the ultraweak coupling limit.",
"We then summarize our findings in Section V. In Appendix A we show explicitly that quantum entanglement of the $AB$ -pair is a necessary condition for the best fidelity of quantum teleportation beating the best classical fidelity in the ultraweak coupling limit of our model.",
"Finally a short discussion on nonlocality and causality is given in Appendix B.",
"To address the above issues, we consider three identical Unruh-DeWitt detectors $A$ , $B$ , and $C$ with mass $m=1$ and natural frequency $\\Omega $ , moving in a quantum field in (3+1)D Minkowski space.",
"The action of the combined system is given by [12] $S &=& -\\int d^4 x \\sqrt{-g} {1\\over 2}\\partial _\\mu \\Phi (x) \\partial ^\\mu \\Phi (x) +\\nonumber \\\\ & &\\sum _{{\\bf d}=A,B,C}\\int d\\tau _{\\bf d} \\left\\lbrace {1\\over 2}\\left[\\left(\\partial _{\\bf d}Q_{\\bf d}\\right)^2-\\Omega _{0}^2 Q_{\\bf d}^2\\right]+ \\lambda _0\\int d^4 xQ_{\\bf d}(\\tau _{\\bf d})\\Phi (x)\\delta ^4\\left(x^{\\mu }-z_{\\bf d}^{\\mu }(\\tau _{\\bf d})\\right)\\right\\rbrace ,$ where $\\mu , \\nu =0,1,2,3$ , $g_{\\mu \\nu } = {\\rm diag}(-1,1,1,1)$ , $\\partial ^{}_{\\bf d}\\equiv \\partial /\\partial \\tau ^{}_{\\bf d}$ , $\\tau ^{}_A$ , $\\tau ^{}_B$ and $\\tau ^{}_C$ are proper times for $Q_A$ , $Q_B$ , and $Q_C$ , respectively.",
"The scalar field $\\Phi $ is assumed to be massless, and $\\lambda _0$ is the coupling constant.",
"Detectors $A$ and $C$ are held by Alice, who is at rest in space with the worldline $z_A^\\mu =z_C^\\mu = (t, 1/b,0,0)$ , while $B$ is held by Rob, who is uniformly accelerated along the worldline $z_B^\\mu = (a^{-1}\\sinh a\\tau , a^{-1}\\cosh a\\tau ,0,0)$ , $0<a<b$ , where $\\tau $ is Rob's proper time, namely, $\\tau ^{}_A=\\tau ^{}_C=t$ and $\\tau ^{}_B=\\tau $ .",
"Suppose the initial state of the combined system at $t=\\tau =0$ is a product state $\\rho ^{}_{\\Phi _{\\bf x}}\\otimes \\rho ^{}_{AB}\\otimes \\rho _C^{(\\alpha )}$ of the Minkowski vacuum of the field $\\hat{\\rho }^{}_{\\Phi _{\\bf x}} = \\left| 0_M\\right>\\left< 0_M\\right|$ , a two-mode squeezed state of the detectors $A$ and $B$ , in the $(K, \\Delta )$ representation [17] (or the “Wigner characteristic function\" [18]), $& & \\rho ^{}_{AB}(K^A, K^B, \\Delta ^A, \\Delta ^B) = \\nonumber \\\\ & &\\exp -{1\\over 2\\hbar }\\left[ {e^{2r_1}\\over \\Omega }(K^A+K^B)^2 + \\Omega e^{-2r_1} (\\Delta ^A +\\Delta ^B)^2 +{e^{-2r_1}\\over \\Omega } (K^A - K^B)^2+ \\Omega e^{2r_1} (\\Delta ^A -\\Delta ^B)^2 \\right],$ and a coherent state of the detector $C$ , denoted $\\hat{\\rho }_C^{(\\alpha )} =\\left|\\alpha \\right>^{}_C\\left<\\alpha \\right|$ , or in the $(K, \\Delta )$ representation $(\\alpha =\\alpha ^{}_R + i\\alpha ^{}_I)$ , $\\rho _C^{(\\alpha )}(K^C, \\Delta ^C) = \\exp \\left[ -{1\\over 2\\hbar }\\left( {1\\over 2\\Omega } (K^C)^2+{\\Omega \\over 2}(\\Delta ^C)^2 \\right)+{i\\over \\hbar } \\left(\\sqrt{2\\hbar \\over \\Omega } \\alpha ^{}_R K^C - \\sqrt{2\\hbar \\Omega }\\alpha ^{}_I \\Delta ^C \\right) \\right].$ $\\rho _C^{(\\alpha )}$ is the quantum state to be teleported.",
"In general the factors in $\\rho _C^{(\\alpha )}$ will vary in time.",
"To concentrate on the best fidelity of quantum teleportation that the entangled $AB$ -pair can offer, however, we follow Ref.",
"[9] and assume the dynamics of $\\rho _C^{(\\alpha )}$ is frozen, or equivalently, assume $\\rho _C^{(\\alpha )}$ is created just before teleportation.",
"Note that, as $r_1\\rightarrow \\infty $ , $\\rho ^{}_{AB}$ goes to an EPR state with the correlations $\\left<\\right.",
"Q_A-Q_B \\left.\\right>= \\left<\\right.",
"P_A+P_B \\left.\\right>=0$ without uncertainty, while $Q_A+Q_B$ and $P_A-P_B$ are totally uncertain.",
"At $t=\\tau =0$ in the Minkowski frame, the detectors $A$ and $B$ start to couple with the field, while the detector $C$ is isolated from others.",
"By virtue of the linearity of the combined system $(\\ref {Stot1})$ , operators at some coordinate time $x^0$ after the initial moment are linear combinations of the operators defined at the initial moment [19]: $\\hat{Q}^{}_{\\bf d}(\\tau ^{}_{\\bf d}(x^0)) &=& \\sum _{{\\bf d}^{\\prime }}\\left[\\phi ^{{\\bf d}^{\\prime }}_{\\bf d}(\\tau ^{}_{\\bf d})\\hat{Q}^{[0]}_{{\\bf d}^{\\prime }} +f^{{\\bf d}^{\\prime }}_{\\bf d}(\\tau ^{}_{\\bf d})\\hat{P}^{[0]}_{{\\bf d}^{\\prime }} \\right]+\\int d^3y \\left[ \\phi ^{\\bf y}_{\\bf d}(\\tau ^{}_{\\bf d})\\hat{\\Phi }^{[0]}_{\\bf y} +f^{\\bf y}_{\\bf d}(\\tau ^{}_{\\bf d})\\hat{\\Pi }^{[0]}_{\\bf y} \\right], \\\\\\hat{\\Phi }^{}_{\\bf x}(x^0) &=& \\sum _{{\\bf d}^{\\prime }} \\left[\\phi ^{{\\bf d}^{\\prime }}_{\\bf x}(x^0)\\hat{Q}^{[0]}_{{\\bf d}^{\\prime }} +f^{{\\bf d}^{\\prime }}_{\\bf x}(x^0)\\hat{P}^{[0]}_{{\\bf d}^{\\prime }}\\right] + \\int d^3y \\left[\\phi ^{\\bf y}_{\\bf x}(x^0)\\hat{\\Phi }^{[0]}_{\\bf y} + f^{\\bf y}_{\\bf x}(x^0)\\hat{\\Pi }^{[0]}_{\\bf y} \\right], $ from which $\\hat{P}^{}_{\\bf d}(x^0)$ and $\\hat{\\Pi }^{}_{\\bf x}(x^0)$ can be derived.",
"Here $\\hat{\\cal O}_{\\zeta }^{[n]} \\equiv \\hat{\\cal O}_{\\zeta }(t_n)$ and all the “mode functions\" $\\phi ^{\\zeta }_\\xi (x^0)$ and $f^{\\zeta }_\\xi (x^0)$ are real functions of time ($\\zeta , \\xi = \\lbrace {\\bf d}\\rbrace \\cup \\lbrace {\\bf x}\\rbrace $ ), which can be related to those in $k$ -space in Ref.",
"[19].",
"Comparing the expansions $(\\ref {Qexp})$ and $(\\ref {Phiexp})$ of two equivalent continuous evolutions, one from $x^0_{\\bf 0}\\equiv 0$ to $x^0_{\\bf 1}$ then from $x^0_{\\bf 1}$ to $x^0_{\\bf 2}$ , the other from $x^0_{\\bf 0}$ all the way to $x^0_{\\bf 2}$ , one can see that the mode functions have the following identities, $& &\\phi ^{\\zeta [20]}_\\xi = \\sum _{{\\bf d}^{\\prime }}\\left[\\phi _\\xi ^{{\\bf d}^{\\prime }[21]}\\phi _{{\\bf d}^{\\prime }}^{\\zeta [10]} +f_\\xi ^{{\\bf d}^{\\prime }[21]} \\pi ^{\\zeta [10]}_{{\\bf d}^{\\prime }}\\right] +\\int d^3x^{\\prime } \\left[ \\phi ^{{\\bf x^{\\prime }}[21]}_\\xi \\phi ^{\\zeta [10]}_{\\bf x^{\\prime }} + f^{{\\bf x^{\\prime }}[21]}_\\xi \\pi ^{\\zeta [10]}_{\\bf x^{\\prime }}\\right], \\\\& & f^{\\zeta [20]}_\\xi = \\sum _{{\\bf d}^{\\prime }}\\left[\\phi _\\xi ^{{\\bf d}^{\\prime }[21]}f_{{\\bf d}^{\\prime }}^{\\zeta [10]} +f_\\xi ^{{\\bf d}^{\\prime }[21]} p^{\\zeta [10]}_{{\\bf d}^{\\prime }}\\right] +\\int d^3x^{\\prime } \\left[ \\phi ^{{\\bf x^{\\prime }}[21]}_\\xi f^{\\zeta [10]}_{\\bf x^{\\prime }} + f^{{\\bf x^{\\prime }}[21]}_\\xi p^{\\zeta [10]}_{\\bf x^{\\prime }}\\right], $ where $F^{[mn]} \\equiv F(x^0_m-x^0_n)$ , $\\pi ^{\\zeta }_{\\bf d}(\\tau ^{}_{\\bf d}(x^0)) \\equiv \\partial ^{}_{\\bf d}\\phi ^{\\zeta }_{{\\bf d}}(\\tau ^{}_{\\bf d}(x^0))$ , $\\pi ^{\\zeta }_{\\bf x}(x^0) \\equiv \\partial _0 \\phi ^{\\zeta }_{\\bf x}(x^0)$ , $p^{\\zeta }_{\\bf d}(\\tau ^{}_{\\bf d}(x^0)) \\equiv \\partial ^{}_{\\bf d}f^{\\zeta }_{{\\bf d}}(\\tau ^{}_{\\bf d}(x^0))$ , and $p^{\\zeta }_{\\bf x}(x^0) \\equiv \\partial _0 f^{\\zeta }_{\\bf x}(x^0)$ .",
"Similar identities for $\\pi ^\\zeta _\\xi $ and $p^\\zeta _\\xi $ can be derived straightforwardly from $(\\ref {id1})$ and $(\\ref {id2})$ .",
"Such identities can be interpreted as embodying the Huygens' principle of the mode functions, and can be verified by inserting particular solutions of the mode functions into the identities.",
"By virtue of the linearity of the combined system $(\\ref {Stot1})$ , the quantum state of the combined system started with a Gaussian state will always be Gaussian; therefore the reduced state of the three detectors is Gaussian for all times.",
"In the $(K,\\Delta )$ representation the reduced Wigner function at the coordinate time $x^0$ in the reference frame of some observer has the form $\\rho ^{}_{ABC}({\\bf K},{\\bf \\Delta };x^0) &=& \\exp \\left[ -{1\\over 2\\hbar ^2} \\left( K^{\\bf d} {\\cal Q}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) K^{\\bf d^{\\prime }}+\\Delta ^{\\bf d} {\\cal P}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) \\Delta ^{\\bf d^{\\prime }} - 2 K^{\\bf d} {\\cal R}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) \\Delta ^{\\bf d^{\\prime }}\\right)\\right.\\nonumber \\\\ & & \\left.+ {i\\over \\hbar } \\left( \\left<\\right.",
"\\hat{Q}_{\\bf d}(x^0)\\left.\\right> K^{\\bf d} -\\left<\\right.",
"\\hat{P}_{\\bf d}(x^0)\\left.\\right> \\Delta ^{\\bf d}\\right) \\right],$ where ${\\bf d}, {\\bf d^{\\prime }} = A,B,C$ , and the factors ${\\cal Q}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) &=& {\\hbar \\delta \\over i\\delta K^{\\bf d}} {\\hbar \\delta \\over i\\delta K^{\\bf d^{\\prime }}}\\rho ^{}_{ABC}({\\bf K},{\\bf \\Delta };x^0)|_{{\\bf K}={\\bf \\Delta }=0} =\\left<\\right.",
"\\delta Q_{\\bf d}(\\tau _{\\bf d}(x^0)), \\delta Q_{\\bf d^{\\prime }}(\\tau _{\\bf d^{\\prime }}(x^0)) \\left.\\right>,\\\\{\\cal P}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) &=& {i\\hbar \\delta \\over \\delta \\Delta ^{\\bf d}} {i\\hbar \\delta \\over \\delta \\Delta ^{\\bf d^{\\prime }}}\\rho ^{}_{ABC}({\\bf K},{\\bf \\Delta };x^0)|_{{\\bf K}={\\bf \\Delta }=0} =\\left<\\right.",
"\\delta P_{\\bf d}(\\tau _{\\bf d}(x^0)), \\delta P_{\\bf d^{\\prime }}(\\tau _{\\bf d^{\\prime }}(x^0)) \\left.\\right>,\\\\{\\cal R}_{{\\bf d}{\\bf d^{\\prime }}}(x^0) &=& {\\hbar \\delta \\over i\\delta K^{\\bf d}} {i\\hbar \\delta \\over \\delta \\Delta ^{\\bf d^{\\prime }}}\\rho ^{}_{ABC}({\\bf K},{\\bf \\Delta };x^0)|_{{\\bf K}={\\bf \\Delta }=0} =\\left<\\right.",
"\\delta Q_{\\bf d}(\\tau _{\\bf d}(x^0)), \\delta P_{\\bf d^{\\prime }}(\\tau _{\\bf d^{\\prime }}(x^0)) \\left.\\right>,$ are actually those symmetric two-point correlators of the detectors in their covariance matrices ($\\left<\\right.",
"{\\cal O}, {\\cal O}^{\\prime }\\left.\\right>\\equiv \\left<\\right.",
"{\\cal O}{\\cal O}^{\\prime }+{\\cal O}^{\\prime }{\\cal O}\\left.\\right>/2$ and $\\delta {\\cal O} \\equiv \\hat{\\cal O}-\\left<\\right.\\hat{\\cal O}\\left.\\right>$ ), which can be obtained in the Heisenberg picture by taking the expectation values of the evolving operators with respect to the initial state.",
"From $(\\ref {Qexp})$ and $(\\ref {Phiexp})$ , these correlators are combinations of the mode functions and the initial data, e.g., $\\left<\\right.",
"\\hat{Q}_A^2(\\tau ^{}_A)\\left.\\right> =\\phi _A^A(\\tau ^{}_A)\\phi _A^A(\\tau ^{}_A)\\left<\\right.",
"(\\hat{Q}_A^{[0]})^2\\left.\\right>_0 +\\int d^3x d^3y\\, \\phi _A^{\\bf x}(\\tau ^{}_A)\\phi _A^{\\bf y}(\\tau ^{}_A)\\left<\\right.",
"\\hat{\\Phi }_{\\bf x}^{[0]},\\hat{\\Phi }_{\\bf y}^{[0]}\\left.\\right>_0 + \\ldots ,$ where $\\left<\\right.",
"\\cdots \\left.\\right>_n$ denotes that the expectation values are taken from the quantum state right after $t=t_n$ ($t_0=0$ here.)",
"Suppose the reduced state of the three detectors continuously evolve to $\\rho ^{}_{ABC}({\\bf K},{\\bf \\Delta };t_1)$ in the Minkowski frame at some moment $t=t_1>0$ and $\\tau =\\tau _1 \\equiv a^{-1}\\sinh ^{-1} at_1$ , when a joint Gaussian measurement by Alice is performed locally in space on $A$ and $C$ so that the post-measurement state right after $t_1$ in the Minkowski frame becomes $\\tilde{\\rho }^{}_{ABC}({\\bf K},{\\bf \\Delta };t_1)= \\tilde{\\rho }^{(\\beta )}_{AC}(K^A, K^C,\\Delta ^A, \\Delta ^C)\\tilde{\\rho }_{B}(K^B, \\Delta ^B)$ , where we assume the quantum state of detectors $A$ and $C$ becomes another two-mode squeezed state The state $(\\ref {PMSAC})$ is chosen so that the analytic calculation is the simplest while the result is still interesting.",
"One may choose another state consistent with the EPR state as the squeeze parameter $r_2\\rightarrow \\infty $ instead, for example, $K^C$ and $\\Delta ^C$ are replaced by $(K^C-K^C)$ and $(\\Delta ^C+\\Delta ^A)$ , respectively.",
"Then $N_B$ and the $F_{av}$ will be more complicated and will depend on $\\alpha $ .",
"In practice the choice of the state may depend on the experimental setting.",
"$& &\\tilde{\\rho }^{(\\beta )}_{AC}(K^A, K^C,\\Delta ^A, \\Delta ^C) = \\nonumber \\\\ & & \\exp \\left[ -{1\\over 2\\hbar ^2}\\left( K^m \\tilde{\\cal Q}_{mn} K^n +\\Delta ^m \\tilde{\\cal P}_{mn} \\Delta ^n - 2 K^m \\tilde{\\cal R}_{mn} \\Delta ^n\\right) +{i\\over \\hbar } \\left( \\sqrt{2\\hbar \\over \\Omega }\\beta _R K^C -\\sqrt{2\\hbar \\Omega }\\beta _I\\Delta ^C\\right) \\right],$ with $m,n = A,C$ so that Alice gets the outcome $\\beta = \\beta _R + i\\beta _I$ .",
"Then $(\\ref {PMSAC})$ yields the reduced state of detector $B$ $\\tilde{\\rho }^{}_B(K^B,\\Delta ^B) &=& N_B \\int {dK^C d\\Delta ^C \\over 2\\pi \\hbar }{dK^A d\\Delta ^A \\over 2\\pi \\hbar }\\tilde{\\rho }_{AC}^{(\\beta ) *}(K^A, K^C, \\Delta ^A,\\Delta ^C)\\rho ^{}_{ABC}(K^A, K^B, K^C,\\Delta ^A,\\Delta ^B,\\Delta ^C; t_1),$ where $N_B$ is the normalization constant.",
"If we require ${\\rm Tr}^{}_B\\, \\tilde{\\rho }^{}_B= \\tilde{\\rho }^{}_B|_{K^B=\\Delta ^B=0}=1$ , then $N_B$ will depend on $\\beta $ .",
"Alternatively, following [9], we can require $N_B$ to be independent of $\\beta $ , then ${\\rm Tr}^{}_B\\, \\tilde{\\rho }^{}_B$ will be proportional to the probability $P(\\beta )$ of finding detectors $A$ and $C$ in the state $(\\ref {PMSAC})$ .",
"Let ${\\rm Tr}^{}_B\\, \\tilde{\\rho }^{}_B = P(\\beta )$ , then we have the normalization condition $1&=&\\int d^2\\beta P(\\beta )= \\int d\\beta ^{}_R d\\beta ^{}_I \\, \\tilde{\\rho }^{}_B(K^B=0,\\Delta ^B=0)\\nonumber \\\\&=& N_B\\int d\\beta _R d\\beta _I {dK^A d\\Delta ^A\\over 2\\pi \\hbar }{dK^C d\\Delta ^C\\over 2\\pi \\hbar }\\tilde{\\rho }_{AC}^{(\\beta ) *}(K^A, K^C, \\Delta ^A,\\Delta ^C)\\rho ^{}_{ABC}( K^A,0, K^C, \\Delta ^A,0,\\Delta ^C; t_1) \\nonumber \\\\&=& N_B \\int {dK^A d\\Delta ^A\\over 2\\pi \\hbar }{dK^C d\\Delta ^C\\over 2\\pi \\hbar } \\rho ^{}_{ABC}(K^A,0, K^C, \\Delta ^A,0,\\Delta ^C; t_1)2\\pi \\delta \\left(\\sqrt{2\\over \\hbar \\Omega }K^C\\right)2\\pi \\delta \\left(\\sqrt{2\\Omega \\over \\hbar }\\Delta ^C\\right)\\times \\nonumber \\\\ & & \\hspace{28.45274pt}\\exp \\left[ -{1\\over 2\\hbar ^2}\\left( K^m \\tilde{\\cal Q}_{mn} K^n +\\Delta ^m \\tilde{\\cal P}_{mn} \\Delta ^n - 2 K^m \\tilde{\\cal R}_{mn} \\Delta ^n\\right)\\right]\\nonumber \\\\&=& {N_B\\over 2\\hbar } \\int dK^A d\\Delta ^A \\exp {-1\\over 2\\hbar ^2}\\left[ \\left({\\cal Q}_{AA}^{[1]} + \\tilde{\\cal Q}_{AA}\\right) (K^A)^2 +\\left({\\cal P}_{AA}^{[1]} + \\tilde{\\cal P}_{AA}\\right) (\\Delta ^A)^2 -2 K^A \\left({\\cal R}_{AA}^{[1]} + \\tilde{\\cal R}_{AA}\\right) \\Delta ^A\\right],\\nonumber $ after inserting $(\\ref {rhoABC})$ and $(\\ref {PMSAC})$ into the integrand.",
"Here ${\\cal S}^{[n]}$ denotes the value of the factor ${\\cal S} = {\\cal Q}, {\\cal P}$ , or ${\\cal R}$ being taken at $t_n-\\epsilon $ with $\\epsilon \\rightarrow 0+$ .",
"Thus we have $N_B = {1\\over \\pi \\hbar }\\sqrt{ \\left({\\cal Q}_{AA}^{[1]} + \\tilde{\\cal Q}_{AA}\\right)\\left({\\cal P}_{AA}^{[1]} + \\tilde{\\cal P}_{AA}\\right)- \\left({\\cal R}_{AA}^{[1]} + \\tilde{\\cal R}_{AA}\\right)^2}.$"
],
[
"Pseudo-Fidelities and Entanglement in Different Frames",
"To compare the fidelity of quantum teleportation with quantum entanglement between detectors $A$ and $B$ at $t_1$ , which is defined on the same $t_1$ -slice in the Minkowski frame, we first imagine that Rob receives the outcome $\\beta $ of Alice's joint measurement on $A$ and $C$ and make the proper operation on detector $B$ instantaneously at $\\tau _1(t_1)$ when the worldline of $B$ intersects the $t_1$ -slice (see Fig.",
"REF ).",
"Physical situations with the classical signal from Alice traveling at the speed of light will be considered later in Section .",
"In the BK scheme [5], [9], according to the outcome $\\beta $ obtained by Alice, the operation that Rob should perform on detector $B$ is a displacement by $\\beta $ in phase space of $B$ from $\\tilde{\\rho }^{}_B$ to $\\rho ^{}_{out}$ , namely, $\\hat{\\tilde{\\rho }}^{}_{out} = \\hat{D}(\\beta )\\hat{\\rho }^{}_B$ , where $\\hat{D}(\\beta )$ is the displacement operator, or in the $(K,\\Delta )$ representation, $\\rho _{out}(K^B, \\Delta ^B) =\\tilde{\\rho }^{}_B(K^B, \\Delta ^B) \\exp {i\\over \\hbar }\\left( \\sqrt{2\\hbar \\over \\Omega }\\beta _R K^B-\\sqrt{2\\hbar \\Omega }\\beta _I \\Delta ^B\\right).$ The “pseudo-fidelity\" of quantum teleportation from $|\\alpha \\left.\\right>^{}_C$ to $|\\alpha \\left.\\right>^{}_B$ is then defined as $F(\\beta ) \\equiv {{}^{}_B\\left< \\right.\\alpha \\,|\\hat{\\rho }_{out}|\\,\\alpha \\left.\\right>^{}_B\\over {\\rm Tr}\\hat{\\rho }^{}_{out} }.$ Note that ${\\rm Tr}^{}_B\\hat{\\rho }^{}_{out} = \\rho ^{}_{out}(K^B=0, \\Delta ^B=0) ={\\rm Tr}^{}_B\\hat{\\rho }^{}_{B} = P(\\beta )$ .",
"A simpler quantity for calculation here is the averaged pseudo-fidelity, defined by $F_{av} &\\equiv & \\int d^2 \\beta P(\\beta ) F(\\beta ) = \\int d\\beta _R d\\beta _I \\,{}^{}_B\\left<\\right.\\alpha |\\hat{\\rho }_{out}|\\alpha \\left.\\right>^{}_B \\nonumber \\\\&=& \\int d\\beta _R d\\beta _I {dK^B d\\Delta ^B\\over 2\\pi \\hbar } \\rho _B^{(\\alpha ) *}(K^B, \\Delta ^B) \\rho _{out}(K^B,\\Delta ^B),$ where $\\hat{\\rho }_B^{(\\alpha )} =\\left|\\alpha \\right>^{}_B\\left<\\alpha \\right|$ .",
"From $(\\ref {rhoAl})$ and $(\\ref {rhoOut})$ , with the help of $(\\ref {rhoB})$ , $(\\ref {rhoABC})$ and $(\\ref {PMSAC})$ , we have $F_{av} &=& N_B \\int d\\beta _R d\\beta _I {\\prod _{\\bf d} dK^{\\bf d} d\\Delta ^{\\bf d} \\over (2\\pi \\hbar )^3} \\exp \\left\\lbrace {i\\over \\hbar }\\left[ \\sqrt{2\\hbar \\over \\Omega }(\\alpha ^{}_R-\\beta ^{}_R)(K^C-K^B)-\\sqrt{2\\hbar \\Omega }(\\alpha ^{}_I-\\beta ^{}_I)(\\Delta ^C-\\Delta ^B)\\right] +\\right.",
"\\nonumber \\\\& &\\left.-{1\\over 2\\hbar ^2}\\left[ {\\hbar \\over 2\\Omega } (K^B)^2+{\\hbar \\over 2}\\Omega (\\Delta ^B)^2 + K^m \\tilde{\\cal Q}_{mn} K^n +\\Delta ^m \\tilde{\\cal P}_{mn} \\Delta ^n - 2 K^m \\tilde{\\cal R}_{mn}\\Delta ^n\\right] \\right\\rbrace \\rho ^{}_{ABC}({\\bf K}, {\\bf \\Delta }; t_1) \\nonumber \\\\ &=&N_B \\int {\\prod _{\\bf d} dK^{\\bf d} d\\Delta ^{\\bf d}\\over (2\\pi \\hbar )^3} (2\\pi )^2 \\delta \\left(\\sqrt{2\\over \\hbar \\Omega }(K^C-K^B)\\right)\\delta \\left(\\sqrt{2\\Omega \\over \\hbar }(\\Delta ^C-\\Delta ^B)\\right) \\rho ^{}_{ABC}({\\bf K}, {\\bf \\Delta }; t_1)\\times \\nonumber \\\\ & &\\exp \\left\\lbrace -{1\\over 2\\hbar ^2}\\left[ {\\hbar \\over 2\\Omega } (K^B)^2+{\\hbar \\over 2}\\Omega (\\Delta ^B)^2 + K^m \\tilde{\\cal Q}_{mn} K^n +\\Delta ^m \\tilde{\\cal P}_{mn} \\Delta ^n - 2 K^m \\tilde{\\cal R}_{mn}\\Delta ^n\\right] \\right\\rbrace ,$ thus $F_{av} = {\\hbar ^2\\pi N_B \\over \\sqrt{\\det \\tilde{\\bf V}}} ,$ where $\\tilde{\\bf V} = \\left(\\begin{array}{cccc}{\\cal Q}_{AA}^{[1]}+\\tilde{\\cal Q}_{AA} & -{\\cal R}_{AA}^{[1]}-\\tilde{\\cal R}_{AA} &{\\cal Q}_{AB}^{[1]}+\\tilde{\\cal Q}_{AC} & -{\\cal R}_{AB}^{[1]}-\\tilde{\\cal R}_{AC} \\\\-{\\cal R}_{AA}^{[1]}-\\tilde{\\cal R}_{AA} & {\\cal P}_{AA}^{[1]}+\\tilde{\\cal P}_{AA} &-{\\cal R}_{BA}^{[1]}-\\tilde{\\cal R}_{CA} & {\\cal P}_{AB}^{[1]}+\\tilde{\\cal P}_{AC} \\\\{\\cal Q}_{AB}^{[1]}+\\tilde{\\cal Q}_{AC} & -{\\cal R}_{BA}^{[1]}-\\tilde{\\cal R}_{CA} &{\\cal Q}_{BB}^{[1]}+\\tilde{\\cal Q}_{CC}+ \\hbar \\Omega ^{-1} & -{\\cal R}_{BB}^{[1]}-\\tilde{\\cal R}_{CC} \\\\-{\\cal R}_{AB}^{[1]}-\\tilde{\\cal R}_{AC} & {\\cal P}_{AB}^{[1]}+\\tilde{\\cal P}_{AC} &-{\\cal R}_{BB}^{[1]}-\\tilde{\\cal R}_{CC} & {\\cal P}_{BB}^{[1]}+\\tilde{\\cal P}_{CC}+\\hbar \\Omega \\end{array}\\right).$ Note that $F_{av}$ in $(\\ref {Favformula})$ is independent of $\\alpha $ because of the choice of the state $(\\ref {PMSAC})$ .",
"Below we consider the cases with the factors in the two-mode squeezed state $(\\ref {PMSAC})$ of detectors $A$ and $C$ right after the joint measurement given by: $\\tilde{\\cal Q}_{AA}=\\tilde{\\cal Q}_{CC}= {\\hbar \\over 2\\Omega }\\cosh 2r_2$ , $\\tilde{\\cal Q}_{AC}={\\hbar \\over 2\\Omega }\\sinh 2r_2$ , $\\tilde{\\cal P}_{AA}=\\tilde{\\cal P}_{CC}={\\hbar \\over 2}\\Omega \\cosh 2 r_2$ , $\\tilde{\\cal P}_{AC}= -{\\hbar \\over 2}\\Omega \\sinh 2r_2$ with squeezed parameter $r_2$ , and $\\tilde{\\cal R}_{mn}=0$ .",
"If the joint measurement on detectors $A$ and $C$ is done perfectly such that $r_2\\rightarrow \\infty $ , then from $(\\ref {Favformula})$ , $(\\ref {tildeV})$ , and $(\\ref {NormB})$ , we have $F_{av}(t_1,\\tau _1) \\rightarrow \\left[ \\left( {1\\over \\Omega }+{1\\over \\hbar }\\left<\\right.Q_-^2\\left.\\right>\\right)\\left(\\Omega +{1\\over \\hbar }\\left<\\right.P_+^2\\left.\\right>\\right)-\\left(\\left<\\right.Q_-,P_+\\left.\\right>\\right)^2\\right]^{-1/2},$ where $Q_- \\equiv Q_A(t_1)-Q_B(\\tau _1)$ and $P_+ \\equiv P_A(t_1)+P_B(\\tau _1)$ .",
"For $t_1 = t_0=0$ , the initial state $\\rho ^{}_{AB}$ of detectors $A$ and $B$ in $(\\ref {rhoABI})$ without coupling to the field gives $F_{av}(0,0) = {1\\over 1+e^{-2r_1}},$ which implies $F_{av}\\rightarrow 1$ as $r_1 \\rightarrow \\infty $ when $\\rho ^{}_{AB}$ is nearly an EPR state, while $F_{av}\\rightarrow 1/2$ as $r_1 \\rightarrow 0$ when $\\rho ^{}_{AB}$ is almost the coherent state of free detectors.",
"In the latter case $F_{av}=1/2$ is known as the best fidelity of “classical\" teleportation using coherent states [5], without considering the coupling of the UD detectors with the environment.",
"This does not imply that $F_{av}$ of quantum teleportation must be greater than $1/2$ .",
"In our result, if we start with the state with $r_1=0$ , then $F_{av}$ will always be less than $1/2$ after the detectors are coupled to the field, and detectors $A$ and $B$ are always separable, too.",
"Once the correlations such as $\\left<Q_-\\right>=0$ needed in the protocol of quantum teleportation becomes more uncertain than the minimum quantum uncertainty, $F_{av}-1/2$ will become negative."
],
[
"ultraweak coupling limit",
"In the ultraweak coupling limit, $\\gamma $ is so small that $\\gamma \\Lambda _1 \\ll a, \\Omega $ .",
"Inserting the expressions for the correlators in this limit [12], one obtains $(\\ref {QAAwc})$ -$(\\ref {RAAwc})$ and $\\tilde{\\bf V} \\approx \\left( \\begin{array}{cccc}{\\hbar \\over 2\\Omega } {\\cal A}(t_1) & 0 & {\\hbar \\over 2\\Omega } {\\cal X}(t_1,\\tau _1)& {\\hbar \\over 2} {\\cal Y}(t_1,\\tau _1) \\\\0 & {\\hbar \\over 2} \\Omega {\\cal A}(t_1)+\\upsilon &{\\hbar \\over 2} {\\cal Y}(t_1,\\tau _1) & -{\\hbar \\over 2}\\Omega {\\cal X}(t_1,\\tau _1)\\\\ {\\hbar \\over 2\\Omega } {\\cal X}(t_1,\\tau _1) & {\\hbar \\over 2} {\\cal Y}(t_1,\\tau _1) & {\\hbar \\over 2\\Omega } {\\cal B}(\\tau _1) & 0 \\\\{\\hbar \\over 2} {\\cal Y}(t_1,\\tau _1) & -{\\hbar \\over 2}\\Omega {\\cal X}(t_1,\\tau _1) & 0 &{\\hbar \\over 2} \\Omega {\\cal B}(\\tau _1)+\\upsilon \\end{array}\\right)$ by writing $\\upsilon \\equiv 2\\hbar \\gamma \\Lambda _1/\\pi $ .",
"Here $t^{}_1(x_{\\bf 1}^0)$ and $\\tau ^{}_1(x_{\\bf 1}^0)$ are the proper times of detectors $A$ and $B$ , respectively, when Alice performs the joint measurement on detectors $A$ and $C$ at coordinate time $x_{\\bf 1}^0$ observed in some reference frame, and ${\\cal A}(t_1) &\\equiv & C_2 +e^{-2\\gamma t_1} C_1 +1-e^{-2\\gamma t_1},\\\\{\\cal B}(\\tau _1) &\\equiv & 2+C_2 + e^{-2\\gamma \\tau _1}C_1 + \\left(1-e^{-2\\gamma \\tau _1}\\right)\\coth {\\pi \\Omega \\over a},\\\\{\\cal X}(t_1,\\tau _1) &\\equiv & S_2 + e^{-\\gamma (t_1+\\tau _1)} \\cos \\Omega (t_1+\\tau _1)\\, S_1,\\\\{\\cal Y}(t_1,\\tau _1) &\\equiv & e^{-\\gamma (t_1+\\tau _1)}\\sin \\Omega (t_1+\\tau _1)\\,S_1 ,$ with $C_n \\equiv \\cosh 2r_n$ and $S_n\\equiv \\sinh 2r_n$ .",
"So the averaged pseudo-fidelity in the ultraweak coupling limit can be written in a simple form, $F_{av}(t_1, \\tau _1) = {2 {\\cal A}\\over {\\cal AB}-({\\cal X}^2+{\\cal Y}^2)} + O(\\upsilon ), $ where ${\\cal X}^2+{\\cal Y}^2 = S_2^2+ S_1^2\\, e^{-2\\gamma (t_1+\\tau _1)}+2S_1 S_2 \\, e^{-\\gamma (t_1+\\tau _1)}\\cos \\Omega (t_1+\\tau _1)$ is oscillating in time due to the natural squeeze-antisqueeze oscillation of the two-mode squeezed state of detectors $A$ and $B$ .",
"The maximum (minimum) values of $F_{av}$ , denoted by $F^+_{av}$ ($F^-_{av}$ ), occur at $\\cos \\Omega (t_1+\\tau _1)\\approx 1$ ($-1$ ), when ${\\cal Y}=0$ and $F_{av}^{\\pm }(t_1, \\tau _1) \\approx {2{\\cal A}\\over {\\cal AB} - \\left[S_2\\pm S_1 \\, e^{-\\gamma (t_1+\\tau _1)}\\right]^2}.$ In the BBCJPW scheme, Alice and Rob have full knowledge of the entangled AB-pair.",
"In the BK scheme for continuous variables, we may assume the same thing: while the two-mode squeezed state of $A$ and $B$ squeezes and antisqueezes alternatingly in time, Alice completely knows when the AB-pair will give the best correlation needed and thus the best fidelity of quantum teleportation with peak values $F_{av}^+$ , so she always performs the joint measurement on $A$ and $C$ at one of those moments.",
"This actually requires that Alice has had the full knowledge about how Rob moves to guarantee that $\\tau (t_1)$ would make $\\cos \\Omega (t_1+\\tau (t_1))\\approx 1$ .",
"In Appendix A we show that, in the ultraweak coupling limit of our model with the initial state $(\\ref {rhoABI})$ and the post-measurement state $(\\ref {PMSAC})$ , whenever detectors $A$ and $B$ are separable in some frame, the averaged pseudo-fidelity of quantum teleportation $F_{av}$ must have been less than the best classical fidelity $1/2$ in that frame.",
"In other words, quantum entanglement between detectors $A$ and $B$ is necessary to provide the advantage of quantum teleportation, at least in the ultraweak coupling limit of our model.",
"Results in the Minkowski frame, where Alice performs the joint measurement at $x^0_{\\bf 1}=t_1$ and Rob's proper time $\\tau _1 = \\tau (t_1)= a^{-1}\\sinh ^{-1}at_1$ , are shown in Figs.",
"REF , REF (left), and REF (left).",
"In Figs.",
"REF and REF one can see that the averaged pseudo-fidelity $F_{av}$ oscillates in a time-varying frequency due to the $\\cos \\Omega (t_1+\\tau (t_1))$ term in $(\\ref {X2Y2})$ .",
"$F_{av}$ is larger as $\\left<\\right.",
"Q_-^2 \\left.\\right>$ (and $\\left<\\right.",
"P_+^2 \\left.\\right>$ ) gets smaller, when the two-mode squeezed state of $A$ and $B$ looks closer to the EPR state so the BK scheme is closer to the idealized case given by Vaidman [5], [3].",
"In more than half of the period in an oscillation, however, $F_{av}$ is less than $1/2$ because the squeezing of the $AB$ -state has oscillated to the orthogonal direction, so that $Q_-$ and $P_+$ are uncertain.",
"One may improve the teleportation by initiating the $AB$ -pair as a rotating squeezed state and using a local oscillator to track its phase angle [1], or simply switching to an alternative protocol using $Q_+$ and $P_-$ instead of $Q_-$ and $P_+$ in Vaidman's scheme whenever $\\left<Q_-^2\\right> > \\left<Q_+^2\\right>$ and $\\left<P_+^2\\right> >\\left<P_-^2\\right>$ , then $F_{av}$ could be greater than $1/2$ during most of the early times, though the peak values will never exceed $F_{av}^+$ .",
"The positions of the peaks of $F_{av}$ at early times are different in the cases with different proper accelerations $a$ of detector $B$ .",
"This is because in this setup different degrees of time-dilation of Rob seen by an observer at rest in Minkowski frame will shift the oscillations $\\cos \\Omega (t_1+\\tau (t_1))$ in $(\\ref {X2Y2})$ in different ways.",
"As mentioned earlier, since $t$ and $\\tau (t)$ depend on the motion of detectors $A$ and $B$ , the position of the peaks of $F_{av}$ also depends on how the detectors move.",
"When $t_1$ gets larger, time dilation of detector $B$ becomes more significant in our setup and so detector $B$ appears to change extremely slowly in the Minkowski frame.",
"Thus $F_{av}$ oscillates approximately in frequency $\\Omega $ at late times.",
"The peak values $F^+_{av}$ fall below the best fidelity of classical teleportation $1/2$ at some time much earlier than the disentanglement time $t_{dE}$ when the logarithmic negativity $E_{\\cal N}$ become zero.",
"One can estimate the moment $t_{1/2}$ when $F_{av}^+$ touches $1/2$ if $a$ is not too small in the ultraweak coupling limit.",
"For large $t_1$ with $\\gamma t_1\\sim O(1)$ , $\\tau (t_1)\\ll t_1$ , so $e^{-\\gamma (t_1+\\tau (t_1))}\\approx e^{-\\gamma t_1}$ , and $B(t) \\approx 2+C_2 + C_1$ .",
"From $(\\ref {maxminFav})$ , one has $F_{av}^+(t_1) \\approx {2{\\cal A}(t_1)\\over (2+C_2+C_1){\\cal A}(t_1)-(S_2+e^{-\\gamma t_1}S_1)^2}$ where ${\\cal A}(t_1)$ has been given in $(\\ref {Aoft})$ , which is independent of $a$ in this limit.",
"So one can see that, while the moments when $F_{av}(t_1)$ reaches a local extremum depend on the proper acceleration $a$ of detector $B$ , $F^+_{av}(t_1)$ in the ultraweak coupling limit is insensitive to $a$ in the Minkowski frame, just like the degree of entanglement $\\Sigma $ or $E_{\\cal N}$ in this case.",
"From $(\\ref {FavpMwc})$ , one obtains $t_{1/2}$ by solving the equation $(C_1-1)(C_2-3)e^{-2\\gamma t_{1/2}}-2S_1 S_2 e^{-\\gamma t_{1/2}} - (C_1 -1)(C_2+1) = 0,$ such that $F_{av}^+(t_{1/2}) \\approx 1/2$ .",
"Unlike the disentanglement time in this case [12], $t_{dE}\\approx (2\\gamma )^{-1}\\ln (\\pi \\Omega /\\gamma \\Lambda _1)$ , which is almost independent of the initial state of the entangled detectors, the moment $t_{1/2}$ when $F_{av}^+\\approx 1/2$ strongly depends on the squeezed parameter $r_1$ of the initial state of detectors $A$ and $B$ , as well as the squeezed parameter $r_2$ introduced by the joint measurement on $A$ and $C$ , though $t_{1/2}$ is still insensitive to $a$ .",
"Figure: A comparison of the averaged pseudo-fidelity F av (t 1 )F_{av}(t_1) (solid curve) in the Minkowski frame in the ultraweakcoupling limit, the evolution of the correlator Q - 2 /20\\left<\\right.Q_-^2\\left.\\right>/20 (dotted curve), Q - ≡Q A -Q B Q_-\\equiv Q_A - Q_Band the logarithmic negativity E 𝒩 /3.5E_{\\cal N}/3.5 (dot-dashed curve) at early times.",
"One can see that F av F_{av} reaches a maximum wheneverQ - 2 \\left<\\right.Q_-^2\\left.\\right> reaches a minimum, while the logarithmic negativity E 𝒩 E_{\\cal N} evolves smoothly remaining always well above zero during the same time interval.So we can see that the oscillation of F av F_{av} is due to the natural oscillation of the initial two-mode squeezed state,and what causes F av F_{av} to drop below 1/21/2 here comes from the distortion of the quantum states of detectors AA and BB from their initial state; it is not an indication of disentanglement of the detectors.",
"Here γ=0.0002\\gamma = 0.0002, Ω=2.3\\Omega = 2.3, m=ℏ=1m=\\hbar =1, r 1 =1.1r_1=1.1, r 2 =1.2r_2=1.2,Λ 0 =Λ 1 =20\\Lambda _0=\\Lambda _1=20, a=1/4a=1/4, and b=2.01ab=2.01 a.Figure: Averaged pseudo-fidelity F av F_{av} in the ultraweak coupling limit in the Minkowski frame (left) and in the quasi-Rindler frame(right) at early times for different aa.",
"Here the parameters are the same as those in Fig.",
"except thatthe proper accelerations are a=1/4a=1/4 (solid), a=1a=1 (dotted) and a=4a=4 (gray).",
"Dashed lines are F av + F^+_{av} (top), 1/21/2 (middle),and F av - F^-_{av} (bottom), where F av ± F^{\\pm }_{av} assume the approximated values obtained from ()(\\ref {maxminFav}).",
"One can see thatthe position of the peaks at early times are different for different aa.",
"This is because different degreesof time-dilation of detector BB (detector AA) seen by an observer at rest in Minkowski (quasi-Rindler) frame will shift theoscillations cosΩ(t 1 +τ(t 1 ))\\cos \\Omega (t_1+\\tau (t_1)) (cosΩ(t(τ 1 )+τ 1 )\\cos \\Omega (t(\\tau _1)+\\tau _1)) in ()(\\ref {X2Y2}) in different ways.Figure: (Left) F av + (t 1 )-1/2F^+_{av}(t_1) -1/2 (solid curves) and E 𝒩 (t 1 )/10E_{\\cal N}(t_1)/10 (dashed curves) in the Minkowski frame.The parameters are the same as those in Fig.",
".The curves for a=1/4a=1/4, a=1a=1, and a=4a=4 are almost indistinguishable in the left plot since both F av + F^+_{av}and E 𝒩 E_{\\cal N} are insensitive to aa in the Minkowski frame in the ultraweak coupling limit.One can see that the peak-values of averaged pseudo-fidelity of quantum teleportation, namely F av + F^+_{av}, become less than 1/21/2 fort 1 >t 1/2 ≈0.57/γt_1 > t_{1/2} \\approx 0.57/\\gamma when entanglement between AA and BB are still quite strong right before the joint measurementin the Minkowski frame [in the sense that the value of the logarithmic negativity E 𝒩 (t 1 )E_{\\cal N}(t_1) is well above zero].The disentanglement time is t dE ≈3.75/γt_{dE}\\approx 3.75/\\gamma according to Ref.",
", much later than the momentt 1/2 t_{1/2} that quantum teleportation loses its advantage.",
"(Right) F av + (t 1 )-1/2F^+_{av}(t_1) -1/2 (solid curves) and -E 𝒩 (t 1 )/10-E_{\\cal N}(t_1)/10 (dashed curves) in the quasi-Rindler frame.There are actually six curves in each set: from right to left they correspond to the caseswith a=1/4a=1/4, 1, 2 (all are light gray curves, indistinguishable in this plot), 4 (gray), 8 (dark gray), and 16 (black),respectively, in quasi-Rindler frame.",
"Other parameters are the same as before.",
"One can see thatF av + F^+_{av} here still fall below the best classical fidelity 1/21/2 earlier than the disentanglement time whenE 𝒩 E_{\\cal N} touches zero.",
"As illustrated here, the larger the proper acceleration aa, the earlier the value of F av + -1/2F^+_{av}-1/2becomes negative in the quasi-Rindler frame.",
"For aa large enough, that moment will be quite close to, but no later than the disentanglementtime τ dE ≈πΩ/γa\\tau _{dE} \\approx \\pi \\Omega /\\gamma a ."
],
[
"Quasi-Rindler frame",
"By a quasi-Rindler frame we refer to the coordinate system in which each time-slice almost overlaps a Rindler time-slice in the R-wedge but the part in the L-wedge has been bent to the region with positive $t$ to make the whole time-slice located after the initial time-slice for the Minkowski observer, as illustrated in Fig.",
"REF .",
"Results in the quasi-Rindler frame in the ultraweak coupling limit, where Alice performs the joint measurement at $x_{\\bf 1}^0 = \\tau _1$ such that $t_1 = t(\\tau _1)=b^{-1}\\tanh a\\tau _1$ , are shown in Figs.",
"REF (right) and REF (right).",
"In Fig.",
"REF (right) there are similar oscillations to those in Fig.",
"REF (left) because of the same $\\cos \\Omega (t_1(\\tau _1)+\\tau _1)$ term in $(\\ref {X2Y2})$ , but here the shift of the peaks at early times is due to the time-dilation of detector $A$ seen by the Rindler observer.",
"When $\\tau _1$ gets larger, the frequency of the oscillation also approaches $\\Omega $ , since detector $A$ looks frozen in the quasi-Rindler frame.",
"In contrast to the case in the Minkowski frame, $F^+_{av}(\\tau _1)$ in the quasi-Rindler frame is sensitive to the proper acceleration $a$ .",
"Indeed, if $a$ and $b$ are not extremely small, for large $\\tau _1$ , one has $t_1 = t(\\tau _1) \\approx 1/b$ , $F_{av}^+(\\tau _1) \\approx {2\\left[C_2+1+e^{-2\\gamma /b}(C_1-1)\\right]\\over \\left[C_2+1+e^{-2\\gamma /b}(C_1-1)\\right]{\\cal B}(\\tau _1)-(S_2+e^{-\\gamma (\\tau _1+1/b)}S_1)^2},$ where ${\\cal B}(\\tau _1)$ given in $(\\ref {BofT})$ depends on $a$ .",
"Again, the moment $\\tau _{1/2}$ when $F_{av}^+(\\tau _{1/2})=1/2$ depends on $r_1$ in the initial state of the detectors $A$ and $B$ as well as $r_2$ from the joint measurement on $A$ and $C$ .",
"In Fig.",
"REF (right) one can see that the larger proper acceleration $a$ , the earlier the value of $F^+_{av}-1/2$ touches zero, while the value of $\\tau _{1/2}$ is always less than $\\tau _{dE}$ ($\\approx \\pi \\Omega /\\gamma a$ for large $a$ [12]).",
"Indeed, the conclusion in Appendix A is valid for the Rindler observer as well as the Minkowski observer: it implies that the peak values of the averaged pseudo-fidelity of quantum teleportation $F^+_{av}$ must have been less than the best averaged fidelity $1/2$ of classical teleportation at the disentanglement time $\\tau _{dE}$ when $E_{\\cal N}$ touches zero, so $\\tau _{1/2} \\le \\tau _{dE}$ .",
"For large $a$ , this can be easily seen by inserting $\\tau _1=\\tau _{dE}\\approx \\pi \\Omega /\\gamma a$ into $(\\ref {FavRindwc})$ , which implies that $F_{av}^+ (\\tau _{dE}) \\ge 1/2$ if $0 &\\ge & (C_1+1)(C_2+1)e^{-2\\pi \\Omega /a} - 2 e^{-\\gamma /b}S_1 S_2 e^{-\\pi \\Omega /a} + e^{-2\\gamma /b}(C_1-1)(C_2-1) \\nonumber \\\\&=& (C_1+1)(C_2+1)\\left( e^{-\\pi \\Omega /a} - e^{-\\gamma /b} \\tanh r_1 \\tanh r_2\\right)^2.$ But the right hand side of the above equation is positive definite.",
"Thus for all parameters $r_1$ , $r_2$ , and $\\Omega $ , $F_{av}^+(\\tau _{dE})$ is always less than $1/2$ , unless the parameters happen to satisfy the equality ${\\pi \\Omega \\over a} = {\\gamma \\over b} - \\ln (\\tanh r_1 \\tanh r_2)$ such that $\\tau _{1/2}\\approx \\tau _{dE}$ in this particular case.",
"Beyond the ultraweak coupling limit, both $F_{av}$ and $E_{\\cal N}$ are strongly affected by the environment.",
"The calculation can be more complicated if the mutual influences between detectors $A$ and $B$ are strong.",
"For simplicity, we consider the cases with $b>2a$ with $a$ not extremely large so that only the first and second order corrections from the mutual influences are needed while they are still small compared with the zeroth order [12].",
"In Fig.",
"REF one can see that quantum entanglement disappears quickly both in the Minkowski frame and the quasi-Rindler frame due to strong interplays with the environment.",
"The averaged pseudo-fidelity $F_{av}$ drops below $1/2$ even quicker, and the peak values of $F_{av}$ never exceed $1/2$ again once they dropped below this level in these examples.",
"Figure: Beyond the ultraweak coupling limit, quantum entanglement disappears quickly both in the Minkowski frame (upper left)and in the quasi-Rindler frame (upper right) as witnessed by E 𝒩 E_{\\cal N} (solid curves) becoming zero quickly.The averaged pseudo-fidelity F av F_{av} drops below 1/21/2 even quicker, both in the Minkowski frame (lower left)and in the quasi-Rindler frame (lower right).Here γ=0.1\\gamma =0.1, Ω=2.3\\Omega =2.3, (a,b)=(0.2,0.401)(a, b)=(0.2,0.401), r 1 =1.1r_1=1.1, r 2 =1.2r_2=1.2, and Λ 0 =Λ 1 =20\\Lambda _0=\\Lambda _1=20."
],
[
"Physical fidelity in a more realistic setup",
"Suppose Rob stops accelerating at his proper time $\\tau _2$ when $t_2 = a^{-1}\\sinh a\\tau _2$ in Minkowski time, after this moment Rob moves with constant velocity along the worldline $( (\\tau -\\tau _2)\\cosh a \\tau _2+a^{-1}\\sinh a\\tau _2,(\\tau -\\tau _2)\\sinh a \\tau _2 + a^{-1}\\cosh a\\tau _2, 0,0)$ in Minkowski coordinate for $\\tau ^{}_B=\\tau >\\tau _2$ , while Alice stays at $(t, 1/b, 0,0)$ and performs the joint measurement on A and C at $t_1$ (see Fig.",
"REF ).",
"In this setup the classical information about the outcome at the very moment of the measurement can always reach Rob if the signal is traveling at the speed of light.",
"Assume Alice sends out the information right after $t_1$ when the joint measurement on $A$ and $C$ is done, then Rob will receive the message at his proper time $\\tau ^{adv}_1 =\\left\\lbrace \\begin{array}{lll}-a^{-1}\\ln a\\left( b^{-1} - t_1\\right) & \\,\\,\\,{\\rm if}\\,\\,\\, t_1 < b^{-1}-a^{-1}e^{-a\\tau _2}, \\\\\\left(t_1 - b^{-1}\\right)e^{a\\tau _2} + a^{-1} + \\tau _2 & \\,\\,\\,{\\rm otherwise.}",
"\\end{array} \\right.$ Figure: Setup for quantum teleportation from Alice (thick dotted line) at rest to Rob (thick solid curve)accelerated constantly from 0 to τ 2 \\tau _2 in his proper time then turning to inertial motion.The gray solid curve represents the τ 1 ' \\tau _1^{\\prime }-slice in the quasi-Rindler frame, and the gray dotted horizontal linerepresents the t 1 t_1-slices in the Minkowski frame.The shaded region represents the future lightcone of the joint measurement event on AA and CC by Alice, andthe hypersurface t=x 1 t=x^1 is the event horizon of Rob for τ 2 →∞\\tau _2\\rightarrow \\infty .Suppose Rob performs the local operation on $B$ at some moment $\\tau ^{}_P > \\tau ^{adv}_1$ according to the received information.",
"Then the averaged physical fidelity should be given by $F_{av} = \\int d^2\\beta P(\\beta )\\times {}^{}_B\\left<\\alpha \\right| \\hat{\\rho }^{}_{out}(\\tau ^{}_P)\\left|\\alpha \\right>^{}_B,$ where $\\rho ^{}_{out}(\\tau ^{}_P)$ has the same form as Eq.",
"($\\ref {rhoOut}$ ) but $\\tilde{\\rho }^{}_B$ defined at $t_1$ there is replaced by $\\tilde{\\rho }^{}_B(\\tau ^{}_P)$ which started with the initial state $\\tilde{\\rho }^{}_B(\\tau _1)$ with $\\tau _1 = a^{-1}\\sinh ^{-1}a t_1$ and evolves from $\\tau _1$ to $\\tau ^{}_P$ according to the Schrödinger equation.",
"During the continuous evolution mutual influences from detector $A$ will start to affect $B$ at $\\tau _1^{adv}$ .",
"This makes the calculation more complicated.",
"Fortunately, if the classical signal travels at lightspeed and Rob performs the local operation right after he receives the signal, namely, $\\tau ^{}_P = \\tau ^{adv}_1 + \\epsilon $ with $\\epsilon \\rightarrow 0+$ , and if mutual influences are weak enough and negligible (this is easy to satisfy in the weak coupling and large distance limit, see [20]), the calculation can be greatly simplified."
],
[
"Reduced state of a detector with its entangled partner being measured",
"In Ref.",
"[16] one of us has shown that a quantum state of a Raine-Sciama-Grove detector-field system in (1+1)D Minkowski space started with the same initial state defined on the same fiducial time-slice, then collapsed by a spatially local measurement on the detector at some moment, will evolve to the same quantum state on the same final time-slice (up to a coordinate transformation), independent of which frame is used by the observer or which time-slice the wave function collapsed on between the initial and the final time-slices.",
"This implies that the reduced state of the detector $B$ at the final time is coordinate-independent.",
"For the Unruh-DeWitt detector theory in (3+1)D Minkowski space considered here, the argument is similar.",
"Right after the local measurement on detectors $A$ and $C$ at $t_1$ (for a simpler case with the local measurement only on detector $A$ , see Ref.",
"[21]), the quantum state at $t_1$ collapses to $\\tilde{\\rho }^{}_{AC} \\otimes \\tilde{\\rho }^{}_{B\\Phi _{\\bf x}}$ on $t_1$ -slice in the Minkowski frame or $\\tau _1^{\\prime }$ -slice in quasi-Rindler frame (see Fig.",
"REF ), or whatever time-slice depending on the observer's frame.",
"Similar to $(\\ref {rhoB})$ , here the post-measurement state $\\tilde{\\rho }^{}_{B\\Phi _{\\bf x}}$ of detector $B$ and the field $\\Phi _{\\bf x}$ is obtained by $\\tilde{\\rho }^{}_{B\\Phi _{\\bf x}}(K^{\\bar{\\sigma }},\\Delta ^{\\bar{\\sigma }}) &=& N \\int {dK^C d\\Delta ^C \\over 2\\pi \\hbar }{dK^A d\\Delta ^A \\over 2\\pi \\hbar }\\tilde{\\rho }_{AC}^*(K^A, K^C, \\Delta ^A,\\Delta ^C)\\rho ({\\bf K}, K^{\\bf x}, {\\bf \\Delta },\\Delta ^{\\bf x}; t_1)$ where $\\rho $ is the quantum state of the combined system evolved from $t_0\\equiv 0$ to $t_1$ and ${\\bar{\\sigma }}=\\lbrace B\\rbrace \\cup \\lbrace {\\bf x} \\rbrace $ .",
"Since $\\tilde{\\rho }^{}_{AC}$ is Gaussian, a straightforward calculation shows that the post-measurement state of detector $B$ and the field has the form: $\\rho ^{}_{B\\Phi _{\\bf x}}(K^{\\bar{\\sigma }},\\Delta ^{\\bar{\\sigma }})&=& \\exp \\left[ {i\\over \\hbar } \\left( {\\cal J}^{(0)}_{\\bar{\\zeta }} K^{\\bar{\\zeta }} -{\\cal M}^{(0)}_{\\bar{\\zeta }}\\Delta ^{\\bar{\\zeta }}\\right)-{1\\over 2\\hbar ^2} \\left( K^{\\bar{\\zeta }} {\\cal Q}_{{\\bar{\\zeta }}{\\bar{\\xi }}} K^{\\bar{\\xi }} +\\Delta ^{\\bar{\\zeta }} {\\cal P}_{{\\bar{\\zeta }}{\\bar{\\xi }}} \\Delta ^{\\bar{\\xi }}-2 K^{\\bar{\\zeta }} {\\cal R}_{{\\bar{\\zeta }}{\\bar{\\xi }}} \\Delta ^{\\bar{\\xi }} \\right) \\right.\\nonumber \\\\& & \\hspace{14.22636pt} \\left.",
"+{1\\over 2\\hbar ^2}\\sum _{n=1}^4 {1\\over {\\cal W}^{(n)}}\\left( K^{\\bar{\\zeta }}{\\cal J}^{(n)}_{\\bar{\\zeta }}- \\Delta ^{\\bar{\\zeta }}{\\cal M}^{(n)}_{\\bar{\\zeta }} \\right)\\left( {\\cal J}^{(n)}_{\\bar{\\xi }} K^{\\bar{\\xi }}- {\\cal M}^{(n)}_{\\bar{\\xi }}\\Delta ^{\\bar{\\xi }}\\right) \\right].$ Here we use the Einstein-DeWitt notation for $\\bar{\\zeta }, \\bar{\\xi } =\\lbrace B\\rbrace \\cup \\lbrace {\\bf x} \\rbrace $ , which run over the degrees of freedom of detector $B$ and the field defined at ${\\bf x}$ on the whole time-slice, $n$ runs from 1 to 4 corresponding to the four dimensional Gaussian integrals in $(\\ref {rhoBPhi0})$ , ${\\cal W}^{(n)}$ depends only on the two-point correlators of detectors $A$ and $C$ at the moment of measurement, while ${\\cal J}^{(i)}_{\\bar{\\zeta }}(\\hat{\\Phi }_{\\bar{\\zeta }})$ and ${\\cal M}^{(i)}_{\\bar{\\zeta }}(\\hat{\\Pi }_{\\bar{\\zeta }})$ are linear combinations of the terms with a cross correlator between detector $A$ or $C$ and the operators $\\hat{\\Phi }_{\\bar{\\zeta }}$ or $\\hat{\\Pi }_{\\bar{\\zeta }}$ ($\\hat{\\Phi }_{B}\\equiv \\hat{Q}_B$ and $\\hat{\\Pi }_{B}\\equiv \\hat{P}_B$ ), respectively, multiplied by a few correlators of $A$ and/or $C$ , all of which are the correlators of the operators evolved from $t_0$ to $t_1$ with respect to the initial state given at $t_0$ .",
"This implies that the two-point correlators right after the wave functional collapse become $\\left<\\right.",
"\\hat{\\Phi }^{[1]}_{\\bar{\\zeta }},\\hat{\\Phi }^{[1]}_{\\bar{\\xi }}\\left.\\right>_1&=& \\left<\\right.\\hat{\\Phi }^{[10]}_{\\bar{\\zeta }}, \\hat{\\Phi }^{[10]}_{\\bar{\\xi }}\\left.\\right>_0 -\\sum _{n=1}^4 {{\\cal J}^{(n)}_{\\bar{\\zeta }}(\\hat{\\Phi }^{[10]}_{\\bar{\\zeta }}){\\cal J}^{(n)}_{\\bar{\\xi }}(\\hat{\\Phi }^{[10]}_{\\bar{\\xi }}) \\over {\\cal W}^{(n)}}, \\\\\\left<\\right.",
"\\hat{\\Pi }^{[1]}_{\\bar{\\zeta }},\\hat{\\Pi }^{[1]}_{\\bar{\\xi }}\\left.\\right>_1&=& \\left<\\right.\\hat{\\Pi }^{[10]}_{\\bar{\\zeta }}, \\hat{\\Pi }^{[10]}_{\\bar{\\xi }}\\left.\\right>_0 -\\sum _{n=1}^4 {{\\cal M}^{(n)}_{\\bar{\\zeta }}(\\hat{\\Pi }^{[10]}_{\\bar{\\zeta }}){\\cal M}^{(n)}_{\\bar{\\xi }}(\\hat{\\Pi }^{[10]}_{\\bar{\\xi }}) \\over {\\cal W}^{(n)}}, \\\\\\left<\\right.",
"\\hat{\\Phi }^{[1]}_{\\bar{\\zeta }},\\hat{\\Pi }^{[1]}_{\\bar{\\xi }}\\left.\\right>_1&=& \\left<\\right.\\hat{\\Phi }^{[10]}_{\\bar{\\zeta }}, \\hat{\\Pi }^{[10]}_{\\bar{\\xi }}\\left.\\right>_0 -\\sum _{n=1}^4 {{\\cal J}^{(n)}_{\\bar{\\zeta }}(\\hat{\\Phi }^{[10]}_{\\bar{\\zeta }}){\\cal M}^{(n)}_{\\bar{\\xi }}(\\hat{\\Pi }^{[10]}_{\\bar{\\xi }}) \\over {\\cal W}^{(n)}}.",
"$ For example, $\\left<\\right.",
"(\\hat{Q}_B^{[1]})^2\\left.\\right>_1= {\\cal Q}_{BB}(t_1) -\\sum _{n=1}^4 [{\\cal J}^{(n)}_{B}(\\hat{Q}_B^{[10]}) {\\cal J}^{(n)}_{B}(\\hat{Q}_B^{[10]})/{\\cal W}^{(n)}]$ where ${\\cal Q}_{BB}(t_1) =\\left<\\right.",
"(\\hat{Q}_B^{[10]})^2\\left.\\right>_0$ .",
"Here $\\hat{\\cal O}_B^{[1]}$ refers to the operator $\\hat{\\cal O}_B$ defined at $t_1$ and $\\hat{\\cal O}_B^{[10]}$ refers to the operator $\\hat{\\cal O}_B(t_1-t_0)$ in the Heisenberg picture.",
"At some moment $t_M$ in the Minkowski frame before the detector $B$ enters the future lightcone of the measurement event on $A$ , namely, when $\\tau ^{}_B=\\tau (t_M) < \\tau ^{adv}_1$ , the two-point correlators of the detector $B$ is either in the original, uncollapsed form, e.g.",
"$\\left<\\right.",
"\\hat{Q}_B^2(t_M-t_0) \\left.\\right>_0$ , if the wave functional collapse does not happen yet in some observers' frames, or in the collapsed form evolved from the post-measurement state, e.g., $\\left<\\right.",
"\\hat{Q}_B^2(t_M) \\left.\\right> &=&\\left< \\left[ \\sum _{{\\bf d}} \\left(\\phi _B^{{\\bf d}[M1]}\\hat{Q}_{\\bf d}^{[1]}+f_B^{{\\bf d}[M1]}\\hat{P}_{\\bf d}^{[1]}\\right)+\\int dx \\left(\\phi _B^{x[M1]}\\hat{\\Phi }_x^{[1]}+f_B^{x[M1]}\\hat{\\Pi }_x^{[1]}\\right) \\right]^2 \\right>_1 \\nonumber \\\\& & = \\left<\\right.",
"(\\tilde{\\Upsilon }_B^{[M0]})^2 \\left.",
"\\right>_0 -\\sum _{n=1}^4{{\\cal I}^{(n)} [\\tilde{\\Upsilon }_B^{[M0]}, \\tilde{\\Upsilon }_B^{[M0]} ]\\over {\\cal W}^{(n)}} ,$ in other observers' frames.",
"Here we have used the Huygens' principles $(\\ref {id1})$ and $(\\ref {id2})$ , and defined $\\tilde{\\Upsilon }^{[M0]}_B &\\equiv & \\hat{\\Phi }_\\zeta ^{[0]}\\left[\\phi _B^{\\zeta [M0]}-\\phi _B^{A[M1]}\\phi _A^{\\zeta [10]}-f_B^{A[M1]}\\pi _A^{\\zeta [10]}\\right] + \\hat{\\Pi }_\\zeta ^{[0]}\\left[f_B^{\\zeta [M0]}-\\phi _B^{A[M1]}f_A^{\\zeta [10]}-f_B^{A[M1]}p_A^{\\zeta [10]}\\right]$ with $\\hat{\\Phi }_{A,C}\\equiv \\hat{Q}_{A,C}$ and $\\hat{\\Pi }_{A,C}\\equiv \\hat{P}_{A,C}$ , while ${\\cal I}^{(n)}$ is derived from those ${\\cal J}^{(n)}_{\\bar{\\zeta }}$ and ${\\cal J}^{(n)}_{\\bar{\\xi }}$ pairs in $(\\ref {Q2PM})$ -$(\\ref {PQPM})$ .",
"Note that before the detector $B$ enters the lightcone, $\\phi _B^{A[M1]} = f_B^{A[M1]}=0$ , such that $\\tilde{\\Upsilon }^{[M0]}_B$ reduces to $\\hat{Q}^{[M0]}_B$ .",
"So at the moment $t_M$ the correlators of detector $B$ do not depend on the data on $t_1$ -slice except those right at the local measurement event on detector $A$ and $C$ .",
"This means that once we discover the reduced state of detector $B$ has been collapsed, the form of the reduced state of $B$ will be independent of the moment when the collapse occurs in the history of detector $B$ (e.g.",
"$\\tau ^{}_B=\\tau _1$ or $\\tau ^{\\prime }_1$ in Fig.",
"REF if $\\tau _2 > \\tau ^{\\prime }_1$ there), namely, the moment where the worldline of detector $B$ intersects the time-slice that the wave functional collapsed on.",
"No matter in which frame the system is observed, the correlators in the reduced state of detector $B$ must have become the collapsed ones like $(\\ref {QB2clpsed})$ exactly when detector $B$ is entering the future lightcone of the measurement event by Alice, namely, $\\tau ^{}_B = \\tau ^{adv}_1$ , after which the reduced states of detector $B$ observed in different frames become consistent.",
"Also after this moment the retarded mutual influences will reach $B$ such that $\\phi _B^{A[M1]}$ and $f_B^{A[M1]}$ would become nonzero and get involved in the correlators of $B$ .",
"In fact, some information of measurement has entered the correlators of $B$ via the correlators of $A$ and $C$ at $t_1$ at the position of Alice in ${\\cal J}^{(n)}$ , ${\\cal M}^{(n)}$ and ${\\cal W}^{(n)}$ much earlier.",
"Nevertheless, that information cannot be recognized by Rob before he has causal contact with Alice.",
"A short discussion on this point is given in Appendix B.",
"Thus we are allowed to choose $t_M$ in $(\\ref {QB2clpsed})$ so that $\\tau ^{}_M\\equiv \\tau ^{}_B(t_M) = \\tau ^{adv}_1-\\epsilon $ and collapse or project the wave functional right before $\\tau ^{}_M$ , namely, collapse on a time-slice almost overlapping the future lightcone of the measurement event by Alice.",
"It is guaranteed that there exists some coordinate system having such a time slice which intersects the worldline of Alice at $\\tau ^{}_A = t_1$ and the worldline of Rob at $\\tau ^{}_B = \\tau ^{adv}_1-\\epsilon $ .",
"If we further assume that mutual influences are negligible and Rob performs the local operation right after the classical information from Alice is received, namely, $\\tau ^{}_P = \\tau ^{adv}_1+\\epsilon $ with $\\epsilon \\rightarrow 0+$ , then the continuous evolution of the reduced state of detector $B$ from $\\tau ^{}_M$ to $\\tau ^{}_P$ is negligible.",
"In this case we can still calculate the averaged fidelity of quantum teleportation using $(\\ref {Favformula})$ with $(\\ref {NormB})$ and $(\\ref {tildeV})$ by inserting $\\tau ^{adv}_1$ into $\\tau _1$ there, then working out the correlators ${\\cal S}^{[1]}_{BB}$ , ${\\cal S}^{[1]}_{AB}$ , and ${\\cal S}^{[1]}_{BA}$ (${\\cal S}= {\\cal Q}, {\\cal P}, {\\cal R}$ ) accordingly."
],
[
"Correlators of non-uniformly accelerated detector",
"To guarantee the classical information from Alice can always reach Rob, we have assumed Rob stops accelerating at the moment $\\tau _2$ .",
"This means the acceleration of detector $B$ is not uniform.",
"The dynamics of the correlators of non-uniformly accelerated detectors have been studied in Ref.",
"[20].",
"In the weak coupling limit the behavior of such a detector is similar to a harmonic oscillator in contact with a heat bath with a time-varying “temperature\" corresponding to the proper acceleration of the detector.",
"From Ref.",
"[20], the dynamics of entanglement will be dominated by the zeroth order results of the a-parts of the self and cross correlators and the v-parts of the self correlators of detectors $A$ and $B$ .",
"The deviation of the v-parts of the self correlators of detector $B$ from those of a uniformly accelerated detector (an inertial detector in [20]) and higher-order corrections from mutual influences are negligible in the weak coupling limit with large initial entanglement and large spatial separation between the detectors.",
"For larger initial accelerations of detector $B$ , the changes of the v-part of its self correlators during and after the transition of the proper acceleration of detector $B$ from $a$ to 0 are more significant.",
"Suppose the changing rate of the proper acceleration of detector $B$ from a finite $a$ to 0 is fast enough so that we can approximate the proper acceleration of detector $B$ as a step function of time, but not too fast to produce significant non-adiabatic oscillation on top of the smooth variation.",
"According to the results in [20] and [22], for $\\tau _2$ sufficiently large, the correlators of detector $B$ behave roughly like $\\left<\\right.Q_B^2(\\tau )\\left.\\right>_{\\rm v} &\\approx &\\left.\\left<\\right.Q_B^2(\\tau )\\right|_{a_\\mu a^\\mu = a^2}\\left.\\right>_{\\rm v} +\\theta (\\tau -\\tau _2)\\times \\nonumber \\\\ & & \\left[\\left( \\left.\\left<\\right.Q_B^2(\\infty )\\right|_0 \\left.\\right>_{\\rm v}-\\left.\\left<\\right.Q_B^2(\\infty )\\right|_{a^2}\\left.\\right>_{\\rm v}\\right)\\left(1-e^{-2\\gamma (\\tau -\\tau _2)}\\right)-{\\gamma \\hbar a^2 e^{-2\\gamma (\\tau -\\tau _2)}\\over 6\\pi m_0(\\gamma ^2+\\Omega ^2)^2}\\right], \\\\\\left<\\right.P_B^2(\\tau )\\left.\\right>_{\\rm v} &\\approx &\\left.\\left<\\right.P_B^2(\\tau )\\right|_{a^2}\\left.\\right>_{\\rm v} +\\theta (\\tau -\\tau _2)\\left[\\left( \\left.\\left<\\right.P_B^2(\\infty )\\right|_0\\left.\\right>_{\\rm v}-\\left.\\left<\\right.P_B^2(\\infty )\\right|_{a^2}\\left.\\right>_{\\rm v}\\right)\\left(1-e^{-2\\gamma (\\tau -\\tau _2)}\\right)\\right],$ where $\\left<\\right.Q_B^2(\\infty )\\left.\\right>_{\\rm v}$ and $\\left<\\right.P_B^2(\\infty )\\left.\\right>_{\\rm v}$ are the correlators in steady state at late times.",
"These approximated bahaviors have been verified by numerical calculations (see Figs.",
"3(right) and 4(right) in [22]).",
"Note that the last term of $(\\ref {QB2NUAD})$ is actually $O(\\gamma )$ and negligible in the ultraweak coupling limit.",
"Also $\\left<Q_A^2\\right>_{\\rm v}$ and $\\left<P_A^2\\right>_{\\rm v}$ behave as the approximated form in $(\\ref {tildeVwc})$ , and other $\\left<\\right.",
"\\cdots \\left.\\right>_{\\rm v}$ are $O(\\gamma )$ and negligible in the this limit.",
"Below we apply these approximations to calculate the averaged fidelity of quantum teleportation in the ultraweak coupling limit.",
"Figure: Averaged physical fidelity F av F_{av} against the moment t 1 t_1 of the joint measurement on AA and CC in the ultraweak couplinglimit in the more realistic cases where the detector BB stops accelerating at τ 2 =10\\tau _2=10, and the local operation on detector BB is performedright after τ 1 adv \\tau ^{adv}_1.",
"Before this moment the wave functional was collapsed either on Minkowski time-slice then evolve toτ 1 adv \\tau _1^{adv} (left) or almost on the future lightcone of the joint measurement event by Alice (middle), namely,the hypersurface intersecting the worldlines of Alice and Rob at t 1 t_1 and τ 1 adv \\tau _1^{adv}.Other parameters are the same as those in Fig.",
".The difference between the results in the left and the middle plotsis shown in the right plot.",
"One can see that it is within O(γ)O(\\gamma ) (here γ=0.0002\\gamma =0.0002).Compare the left plots with Fig.",
"(left) one can see the hugedifference from the averaged pseudo-fidelity there.",
"The difference is due to the natural oscillations of thedetector BB from t 1 t_1 to the much later moment τ 1 adv \\tau _1^{adv}."
],
[
"Averaged physical fidelity of quantum teleportation in ultraweak coupling limit",
"Replacing $\\tau _1$ in $(\\ref {tildeVwc})$ by $\\tau ^{adv}_1$ in $(\\ref {tau1adv})$ , while inserting $(\\ref {QB2NUAD})$ and $(\\ref {PB2NUAD})$ into the v-parts of the self correlators in ${\\cal Q}_{BB}^{[1]}$ and ${\\cal P}_{BB}^{[1]}$ , respectively, we obtain the results in Figs.",
"REF (middle) and REF .",
"In Fig.",
"REF (right) one can see that the differences between the results with wave functional collapsed on Minkowski time-slice then evolving the system to $\\tau _1^{adv}$ (left), and those collapsed almost on the future lightcone of the measurement event (middle), are $O(\\gamma )$ , which is within the error of the two-point correlators in the ultraweak coupling limit so they should be considered negligible.",
"In all the plots of Fig.",
"REF the number of peaks of the physical $F_{av}$ in the same duration of $t_1$ in this more realistic case is much more than the one for the averaged pseudo-fidelity.",
"This is because it takes a long time from $\\tau _1$ to the moment $\\tau _1^{adv}$ when the classical signal from Alice reaches Rob, during which detector $B$ has oscillated many times.",
"In Fig.",
"REF we see that the moment $t_1=t_{1/2}$ when the best averaged physical fidelity of quantum teleportation $F_{av}^+$ drops to $1/2$ is again earlier than any $F_{av}^+$ of pseudo-fidelity has.",
"So Alice must perform the joint measurement on detectors $A$ and $C$ much earlier than what was estimated from the pseudo-fidelities to achieve successful quantum teleportation.",
"The larger $a\\tau _2$ , the later $\\tau ^{adv}_1$ Rob has by $(\\ref {tau1adv})$ , and so the lower value of the physical $F_{av}^+$ at that time due to the longer time of coupling with the environment.",
"When $a\\tau _2$ is large enough, $\\tau _1^{adv}$ is so large that $t_{1/2}\\approx b^{-1}$ , which is the moment that Alice enters the event horizon of Rob for $\\tau _2\\rightarrow \\infty $ (see Fig.",
"REF (upper-right), (lower-middle), and (lower-right)).",
"From the same argument in Appendix with the proper time of detector $B$ substituted by $\\tau _1^{adv}$ (actually $\\tau _1^{adv}\\pm \\epsilon $ as $\\epsilon \\rightarrow 0+$ ), one can see that quantum entanglement of $AB$ -pair evaluated almost on the future lightcone of the measurement event by Alice is still a necessary condition of the best averaged physical fidelity of quantum teleportation beating the classical one in the ultraweak coupling limit of our model.",
"Figure: F av + -1/2F_{av}^+-1/2 (solid curves) and E 𝒩 /10E_{\\cal N}/10 (dashed curves) in the Minkowski frame in the more realistic caseas a function of the moment of the joint measurement t 1 t_1.The black curves are those physical ones with the wave functionals almost collapsed on the lightcone,while the gray curves are those pseudo-fidelities with the wave functionals collapsed on the t 1 t_1-slice in the Minkowski frame.Here the parameters are the same as those in Fig.",
"except aa, bb and τ 2 \\tau _2.One can see that the moment that the peak values of the averaged physical fidelity F av + F_{av}^+ becomes less than 1/21/2 is always earlierthan the pseudo-F av + F_{av}^+ has, and both happens earlier than the disentanglement times evaluated on the correspondinghypersurfaces of wave functional collapse.The lower-right plot is a close up of the lower-middle plot at very early times.",
"If we increasethe value of aτ 2 a\\tau _2 further we get the upper-right plot, wherethe solid black curve indicates the case with a=4a=4, b=8.04b=8.04, τ 2 =10\\tau _2=10,the dotted curve indicates the case with a=2a=2, b=4.02b=4.02, τ 2 =10\\tau _2=10,and the dot-dashed curve indicates the case with a=1a=1, b=2.01b=2.01, τ 2 =20\\tau _2=20.All these three curves become negative at about t 1 =1/bt_1=1/b."
],
[
"Summary",
"Quantum entanglement of two localized but spatially separated objects is a kind of spacelike correlations while the physical fidelity of quantum teleportation, is a kind of timelike correlations.",
"In general they are incommensurate.",
"We define the pseudo-fidelity of quantum teleportation on the same time-slice that the joint measurement occurs in some reference frame to compare with the quantum entanglement of the $AB$ -pair on that time-slice.",
"In the more realistic cases assuming that Rob stops accelerating after some $\\tau _2 > 0$ , the classical signal from Alice can always reach Rob, and the reduced state of detector $B$ collapsed on different time-slices by the same joint measurement by Alice in different frames will become consistent when Rob is entering the future lightcone of the measurement event.",
"Thus we are allowed to perform the projection of the wave functional almost on that future lightcone, and right after that, let Rob perform the local operation.",
"The physical fidelity of quantum teleportation obtained in this way can be compared with the quantum entanglement of the $AB$ -pair evaluated right before the wave functional collapsed almost on the future lightcone of the joint measurement event by Alice.",
"In the ultraweak coupling limit of our model, when detectors $A$ and $B$ are separable on a time-slice, the averaged pseudo-fidelity $F_{av}$ of quantum teleportation obtained on that time-slice must be less than or equal to $1/2$ , which is the best possible fidelity of classical teleportation.",
"Thus quantum entanglement between detectors $A$ and $B$ is a necessary condition that the averaged pseudo-fidelity of quantum teleportation has advantage over the best classical one.",
"We have a similar observation in our result beyond the ultraweak coupling limit.",
"Similarly, if we assume that the classical signal from Alice travels with lightspeed and Rob performs the local operation right after he received the signal, then in the ultraweak coupling limit the entanglement “on the lightcone\", which is evaluated right before Rob enters the future lightcone of the measurement event by Alice, will also be a necessary condition for a physical fidelity of quantum teleportation beating the classical ones.",
"We have seen that the logarithmic negativity $E_{\\cal N}$ evolves quite smoothly while the averaged pseudo-fidelity evaluated in whatever reference frame or the averaged physical fidelity of quantum teleportation oscillates in $t_1$ , which is the moment when Alice performs the joint measurement on detectors $A$ and $C$ .",
"Even at very early times $F_{av}$ drops below $1/2$ frequently.",
"It is clear that the oscillation of the averaged fidelities are mainly due to the distortion of the quantum state of the $AB$ -pair from their initial state (caused by the alternating squeeze-antisqueeze natural oscillations of their quantum state) rather than the time evolution of quantum entanglement between them.",
"In all cases considered in this paper different values of $a$ have different ways of time-dilation, which causes different shifts of the peaks of $F_{av}$ at early times in $t_1$ , though the $a$ -dependence of the peak values of $F_{av}$ is not significant at this stage in all cases in the ultraweak coupling limit.",
"In a longer time scale, while the peak values of the averaged pseudo-fidelity in the Minkowski frame are insensitive to the proper acceleration $a$ , the ones in the quasi-Rindler frame as well as the averaged physical fidelity in the more realistic cases do depend on $a$ significantly: the larger $a$ is, the quicker the best fidelities $F_{av}^+ (t_1)$ drops below $1/2$ .",
"Finally, the best averaged physical fidelity becomes less than $1/2$ always earlier (in $t_1$ ) than any best averaged pseudo-fidelity we considered does, while each of these moments is earlier than the disentanglement time evaluated on the corresponding hypersurfaces which the wave functional collapses on.",
"In our more realistic cases, the later Rob turns into inertial motion (i.e.",
"the larger $a \\tau _2$ ), the earlier the moment in $t_1$ that the best averaged physical fidelity drops below $1/2$ , and the earlier the corresponding disentanglement time in $t_1$ .",
"For $a \\tau _2$ large enough, detectors $A$ and $B$ become separable “on the lightcone\" right after $t_1=1/b$ in the ultraweak coupling limit, meaning that quantum teleportation loses advantage right after Alice goes beyond the event horizon of Rob in the limiting case $\\tau _2 \\rightarrow \\infty $ in our setup.",
"Part of this work was done while BLH visited the National Center for Theoretical Sciences (South) and the Department of Physics of National Cheng-Kung University, Tainan, Taiwan, the Center for Quantum Information and Security at Macquarie University, Sydney, the Center for Quantum Information and Technology at the University of Queensland, Brisbane, Australia in January-March, 2011 and the National Changhua University of Education, Taiwan in January 2012.",
"He wishes to thank the hosts of these institutions for their warm hospitality.",
"This work is supported by the National Science Council of Taiwan under grant NSC 100-2112-M-006-007, NSC 99-2112-M-018-001-MY3, and in part by the National Center for Theoretical Sciences, Taiwan, and by USA NSF PHY-0801368 to the University of Maryland."
],
[
"Averaged fidelity and entanglement in ultraweak coupling limit",
"In our model in the ultraweak coupling limit, it is straightforward to show to the leading order that quantum entanglement between detectors $A$ and $B$ is a necessary condition for the corresponding averaged fidelity of quantum teleportation to be better than the best classical ones.",
"Suppose in the post-measurement state $\\tilde{\\cal Q}_{AA}=\\tilde{\\cal Q}_{CC}= \\hbar C_2/2\\Omega $ , $\\tilde{\\cal Q}_{AC}=\\hbar S_2/ 2\\Omega $ , $\\tilde{\\cal P}_{AA}=\\tilde{\\cal P}_{CC}=\\hbar \\Omega C_2/2$ , $\\tilde{\\cal P}_{AC}=\\hbar \\Omega S_2/2$ , and $\\tilde{\\cal R}_{mn}=0$ (with $C_n \\equiv \\cosh 2r_n$ and $S_n\\equiv \\sinh 2r_n$ , see Section REF ).",
"Also from Eqs.$(28)$ , $(29)$ , $(32)$ , $(33)$ , and $(B2)$ -$(B8)$ in Ref.",
"[12] with $\\alpha ^2 = (\\hbar /\\Omega )e^{-2r_1}$ and $\\beta ^2 = \\hbar \\Omega e^{-2r_1}$ there (not the complex numbers $\\alpha $ and $\\beta $ in this paper), writing $\\upsilon \\equiv 2\\hbar \\gamma \\Lambda _1/\\pi $ , with $1\\gg \\upsilon \\gg \\gamma \\gg \\upsilon ^2$ , one has ${\\cal Q}_{AA} &\\approx &{\\hbar \\over 2\\Omega }\\left[ C_1 e^{-2\\gamma t_1} + 1-e^{-2\\gamma t_1}\\right]+O(\\gamma ),\\\\{\\cal Q}_{BB} &\\approx & {\\hbar \\over 2\\Omega }\\left[ C_1 e^{-2\\gamma \\tau _1} + 1-e^{-2\\gamma \\tau _1}\\right] +\\delta \\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v}+O(\\gamma ),\\\\{\\cal Q}_{AB} &=& {\\hbar \\over 2\\Omega } S_1 e^{-\\gamma (t_1+\\tau _1)}\\cos \\Omega (t_1+\\tau _1)+O(\\gamma ),\\\\{\\cal P}_{AA} &\\approx & {\\hbar \\Omega \\over 2}\\left[ C_1 e^{-2\\gamma t_1} + 1-e^{-2\\gamma t_1}\\right] + \\upsilon +O(\\gamma ),\\\\{\\cal P}_{BB} &\\approx & {\\hbar \\Omega \\over 2}\\left[ C_1 e^{-2\\gamma \\tau _1} + 1-e^{-2\\gamma \\tau _1}\\right] + \\upsilon +\\delta \\left<\\right.",
"\\hat{P}_B^2(\\tau _1)\\left.\\right>_{\\rm v}+O(\\gamma ),\\\\{\\cal P}_{AB} &\\approx & -\\Omega ^2 {\\cal Q}_{AB}+O(\\gamma ), \\\\{\\cal R}_{AB} &\\approx & {\\cal R}_{BA}\\approx {\\cal R} \\equiv -{\\hbar \\over 2}S_1 e^{-\\gamma (t_1+\\tau _1)}\\sin \\Omega (t_1+\\tau _1)+O(\\gamma ), \\\\{\\cal R}_{AA} &\\approx & {\\cal R}_{BB} \\approx 0+O(\\gamma ), $ for the initial state $(\\ref {rhoABI})$ in the ultraweak coupling limit.",
"Here $\\delta \\left<\\right.",
"\\hat{P}_B^2(\\tau _1)\\left.\\right>_{\\rm v}\\approx \\Omega ^2 \\delta \\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v} \\approx (\\hbar /2\\Omega )(\\coth (\\pi \\Omega /a)-1)(1-e^{-2\\gamma \\tau _1})$ if Rob is uniformly accelerated ($\\tau _2\\rightarrow \\infty $ ) with proper acceleration $a$ , and $\\delta \\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v} \\equiv \\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v}-\\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)|_{a_\\mu a^\\mu \\rightarrow 0}\\left.\\right>_{\\rm v}$ and $\\delta \\left<\\right.",
"\\hat{P}_B^2(\\tau _1)\\left.\\right>_{\\rm v}\\equiv \\left<\\right.",
"\\hat{P}_B^2(\\tau _1)\\left.\\right>_{\\rm v}-\\left<\\right.",
"\\hat{P}_B^2(\\tau _1)|_{a_\\mu a^\\mu \\rightarrow 0}\\left.\\right>_{\\rm v}$ $(\\approx \\Omega ^2 \\delta \\left<\\right.",
"\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v}$ , too) are given in $(\\ref {QB2NUAD})$ and $(\\ref {PB2NUAD})$ in the more realistic case, so ${\\cal P}_{AA}\\approx \\Omega ^2 {\\cal Q}_{AA}+\\upsilon + O(\\gamma ), \\hspace{14.22636pt}{\\cal P}_{BB}\\approx \\Omega ^2 {\\cal Q}_{BB}+\\upsilon + O(\\gamma ).$ Then on the time-slices passing through the worldlines of detectors $A$ and $B$ at $t_1$ and $\\tau _1$ , respectively, $\\Sigma &\\approx & {\\cal Z}^2 -{\\hbar ^2\\over 4}\\Omega ^2\\left({\\cal Q}_{AA}+{\\cal Q}_{BB}\\right)^2 +\\upsilon ({\\cal Q}_{AA}+{\\cal Q}_{BB} ) \\left({\\cal Z}-{\\hbar ^2\\over 2}\\right)+O(\\gamma )\\\\\\hbar ^2\\pi N_B &\\approx & \\hbar \\sqrt{ \\left(\\Omega {\\cal Q}_{AA}+{\\hbar \\over 2}C_2\\right)^2 + \\upsilon \\left({\\cal Q}_{AA}+{\\hbar \\over 2\\Omega }C_2\\right)}+O(\\gamma ), \\\\\\det \\tilde{V} &\\approx & {\\cal F}^2+ \\upsilon \\left[ {\\cal Q}_{AA}+{\\cal Q}_{BB}+ {\\hbar \\over \\Omega }(1+C_2)\\right]{\\cal F} + O(\\gamma ),$ where $\\Sigma $ defined in [12] indicates the degree of entanglement between detectors $A$ and $B$ , ${\\cal Z}$ and ${\\cal F}$ are given by ${\\cal Z} &\\equiv & {\\hbar ^2\\over 4} + \\Omega ^2{\\cal Q}_{AA}{\\cal Q}_{BB} -\\Omega ^2 {\\cal Q}_{AB}^2-{\\cal R}^2,\\\\{\\cal F} &\\equiv & {\\cal Z} + \\hbar \\Omega {\\cal Q}_{AA} - \\hbar S_2 \\Omega {\\cal Q}_{AB}+{\\hbar \\over 2} C_2\\left[\\hbar + \\Omega ({\\cal Q}_{AA}+{\\cal Q}_{BB})\\right].$ Detectors $A$ and $B$ are separable if and only if $\\Sigma \\ge 0$ , or ${\\cal Z} \\ge {\\hbar \\over 2}\\Omega \\left({\\cal Q}_{AA}+{\\cal Q}_{BB}\\right) -{\\upsilon \\over 2}\\left({\\cal Q}_{AA}+{\\cal Q}_{BB}-{\\hbar \\over \\Omega }\\right) + O(\\gamma ).$ This implies $& &\\det \\tilde{V}-4\\left(\\hbar ^2\\pi N_B\\right)^2 \\ge {\\cal J}^2 - \\hbar ^2\\left(2\\Omega {\\cal Q}_{AA}+\\hbar C_2\\right)^2 +\\nonumber \\\\ & & \\,\\,\\,\\,\\,\\upsilon {\\cal J}\\left[ {3\\over 16}\\left({\\cal Q}_{AA}+{\\cal Q}_{BB}\\right)+{\\hbar \\over \\Omega }\\left({5+4C_2\\over 16}\\right)\\right]-2\\upsilon \\hbar ^2\\left(2{\\cal Q}_{AA} + {\\hbar \\over \\Omega } C_2\\right)+ O(\\gamma ), $ where ${\\cal J} \\equiv {\\hbar \\over 2}\\Omega (1+C_2)\\left({\\cal Q}_{AA}+{\\cal Q}_{BB}\\right) +\\hbar \\Omega {\\cal Q}_{AA}-\\hbar S_2 \\Omega {\\cal Q}_{AB}+ {\\hbar ^2\\over 2} C_2.$ Now one can see $& & {\\cal J} - 2\\hbar \\left(\\Omega {\\cal Q}_{AA}+{\\hbar \\over 2}C_2\\right)\\nonumber \\\\&=& \\hbar \\Omega \\left[ \\left({C_2-1\\over 2}\\right)\\left( {\\cal Q}_{AA}-{\\hbar \\over 2\\Omega }\\right) +\\left({C_2+1\\over 2}\\right)\\left( {\\cal Q}_{BB}-{\\hbar \\over 2\\Omega }\\right)-S_2 {\\cal Q}_{AB}\\right] \\nonumber \\\\&\\ge & \\hbar ^2 \\left(\\sinh r_1 \\sinh r_2 e^{-\\gamma t_1}+\\cosh r_1 \\cosh r_2 e^{-\\gamma \\tau _1} \\right)^2 +\\hbar \\Omega \\,\\,\\delta \\left<\\right.\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v} \\cosh ^2 r_2>0,$ after the approximated expressions for the correlators were inserted.",
"Notice that $\\delta \\left<\\right.\\hat{Q}_B^2(\\tau _1)\\left.\\right>_{\\rm v}$ is always positive here.",
"This implies that the $O(\\upsilon ^0)$ terms of $\\det \\tilde{V}-4(\\hbar ^2\\pi N_B)$ in $(\\ref {Favineq})$ is positive whenever $\\Sigma \\ge 0$ , and the $O(\\upsilon )$ terms in $(\\ref {Favineq})$ in this case must be greater than $8\\upsilon \\hbar \\left(\\Omega {\\cal Q}_{AA}+{\\hbar \\over 2}C_2\\right)\\left[ {3\\over 16}\\left({\\cal Q}_{AA}+{\\cal Q}_{BB}\\right)+ {\\hbar \\over \\Omega }\\left({1+4C_2\\over 16}\\right)\\right] >0.",
"\\nonumber $ Therefore, for the entangled pair of the UD detectors initially in the state $(\\ref {rhoABI})$ in the ultraweak coupling limit, if $\\Sigma \\ge 0$ , then $\\det \\tilde{V} > 4(h^2\\pi N_B)^2$ up to $O(\\hbar \\gamma \\Lambda _1)$ and so $F_{av} = h^2\\pi N_B / \\sqrt{\\det \\tilde{V}}$ must be less than $1/2$ to this order.",
"In other words, once quantum entanglement between $A$ and $B$ disappears, the corresponding averaged pseudo-fidelity of quantum teleportation must have been less than the best classical fidelity $1/2$ in the ultraweak coupling limit of our model."
],
[
"Entanglement, quantum nonlocality, and causality",
"Suppose $\\tau _2\\rightarrow \\infty $ so Rob has an event horizon at the hypersurface $t=x$ .",
"Suppose at the moment $t_1$ before Alice goes beyond the event horizon of Rob (Fig.",
"REF ) Alice performs a joint measurement on $A$ and $C$ .",
"Then the wave functional of the combined system (including the quantum fields that these UD detectors are coupled with) will collapse at that moment either on the Minkowski time-slice (gray dashed horizontal line in Fig.",
"REF ), or on the quasi-Rindler time-slice (the gray solid curve in the same plot), or whatever time-slice may be passing through the same measurement event in some observer's frame.",
"All post-measurement states of the combined system in different frames will evolve to the same state when compared on the same time-slice after the measurement [16].",
"The Gaussian reduced state of the detector $B$ consists of the two-point correlators of $Q_B$ and $P_B$ .",
"It has a sudden change from the uncollapsed to the collapsed one at $\\tau ^{}_B=\\tau _1$ in Fig.",
"REF , as observed in the conventional Minkowski frame, or at $\\tau ^{\\prime }_1$ , as observed in the quasi-Rindler frame.",
"Such a sudden change occurs at different spacetime points for different observers.",
"In other words, when observed at some moment $\\tau ^{}_B < \\tau ^{adv}_1$ before Rob enters the future lightcone of the joint measurement event, those correlators of $B$ may either be in the uncollapsed form or the collapsed form, with the two-point correlators look like $(\\ref {QA2examp})$ or $(\\ref {QB2clpsed})$ , depending on the observer [21].",
"In both cases all the correlators of $B$ are independent of the data on the time-slice that the wave functional collapsed on except those localized right at the position of the detector $A$ and $C$ .",
"If Rob never performs any further measurement on $B$ before entering the future lightcone of the joint measurement event done by Alice, certainly he will have no idea that $B$ is in the uncollapsed or collapsed state.",
"But right before the moment $\\tau ^{}_B =\\tau ^{adv}_1$ when Rob is entering the future lightcone of the joint measurement event, these different reduced states of $B$ in different frames must have become the same collapsed one, with the same combination of the mode functions depending only on the data on the initial time-slice.",
"So quantum teleportation after Rob receives the classical information from Alice will give a definite result consistent in all frames.",
"Suppose Rob performs a measurement on $B$ before entering the future lightcone of the joint measurement event to see which state $B$ is in.",
"In some observers' frames $B$ 's reduced state at the moment of Rob's measurement is in the collapsed state, then the outcome will have some dependence on the outcome of $A$ since $A$ and $B$ were entangled initially.",
"In other observers' frames, $B$ is still in the uncollapsed state before the measurement, then this measurement will result in a wave functional collapse of the combined system, later in these frames the outcome of Alice's joint measurement will have the same correlation with the outcome of $B$ .",
"Both kinds of histories interpreted by different observers will be consistent a posteriori, but both could not help Rob to conclude $B$ was in the uncollapsed or collapsed state right before the measurement because the quantum state is only sampled by Rob in one single measurement.",
"If Rob and Alice share an ensemble of many copies of the entangled pairs, in some observers' frames Rob performs the measurement first, then using the outcomes Rob can recognize the reduced state of $B$ as uncollapsed by quantum state tomography.",
"In other observers' frames, Rob performs the measurement after Alice.",
"For these observers the exact collapsed state of one copy of the entangled pair may be different from another after the measurements by Alice and not predictable (either by Rob or by Alice), while the distribution of the collapsed state is determined by the uncollapsed state right before Alice's measurement.",
"So the reduced state of detector $B$ recognized by Rob from his measurement on this ensemble of the collapsed states will have no difference from the uncollapsed one.",
"Rob still cannot determine whether the reduced state of the single detector $B$ before his measurement is collapsed or not.",
"If the joint measurement on $A$ and $C$ is performed after Alice went beyond the event horizon $t=x$ of Rob, the mutual influences ($\\phi _B^A$ and $f_B^A$ in $(\\ref {Updef})$ ) will never reach Rob, though it appears that some information of measurement could enter the collapsed reduced state of $B$ [21].",
"Similar to the previous case with Rob still outside the future lightcone of the joint measurement, the functional form of the correlators in the reduced state of $B$ will be suddenly changed at the moment of the projective measurement, and different observers will recognize different spacetime points where this change occurs.",
"But again Rob will never know whether the state of $B$ is in the uncollapsed or collapsed state.",
"These reduced states in different frames will not be identical until $\\tau ^{}_B$ goes to infinity or some moment Rob performs a measurement on $B$ ."
]
] | 1204.1525 |
[
[
"Tuning entanglement and ergodicity in two-dimensional spin systems using\n impurities and anisotropy"
],
[
"Abstract We consider the entanglement in a two dimensional XY model in an external magnetic field h. The model consists of a set of 7 localized spin-1/2 particles in a two dimensional triangular lattice coupled through nearest neighbor exchange interaction J.",
"We examine the effect of single and double impurities in the system as well as the degree of anisotropy on the nearest neighbor entanglement and ergodicity of the system.",
"We have found that the entanglement of the system at the different degrees of anisotropy mimics that of the one dimensional spin systems at the extremely small and large values of the parameter h/J.",
"The entanglement of the Ising and partially anisotropic system show phase transition in the vicinity of h/J = 2 while the entanglement of the isotropic system suddenly vanishes there.",
"Also we investigate the dynamic response of the system containing single and double impurities to an external exponential magnetic field at different degrees of anisotropy.",
"We have demonstrated that the ergodicity of the system can be controlled by varying the strength and location of the impurities as well as the degree of anisotropy of the coupling."
],
[
"Introduction",
"Quantum entanglement is a corner stone in the structure of quantum theory with no classical analog [1].",
"Entanglement is a nonlocal correlation between two (or more) quantum systems such that the description of their states has to be done with reference to each other even if they are spatially well separated.",
"Particular fields where entanglement is considered as a crucial resource are quantum teleportation, cryptography and quantum computation [2], [3], where it provides the physical basis for manipulating the linear superposition of the quantum states used to implement the different computational algorithms.",
"On the other hand, many questions regarding the behavior of the complex quantum systems significantly rely on a deep understanding and a good quantification of the entanglement [4], [5], [6], [7], [8], [9].",
"Particularly, entanglement is considered as the physical resource responsible for the long range correlations taking place in many-body systems during quantum phase transitions.",
"There has been great interest in studying the different sources of errors in quantum computing and their effect on quantum gate operations [10], [11].",
"Different approaches have been proposed for protecting quantum systems during the computational implementation of algorithms such as quantum error correction [12] and decoherence-free subspace [13], [14].",
"Nevertheless, realizing a practical protection against the different types of induced decoherence is still a hard task.",
"Therefore, studying the effect of naturally existing sources of errors such as impurities and lack of isotropy in coupling between the quantum systems implementing the quantum computing algorithms is a must.",
"Furthermore, considerable efforts should be devoted to utilizing such sources to tune the entanglement rather than eliminating them.",
"The effect of impurities and anisotropy of coupling between neighbor spins in a one dimensional spin system has been investigated [15].",
"It was demonstrated that the entanglement can be tuned in a class of one-dimensional systems by varying the anisotropy of the coupling parameter as well as by introducing impurities into the spin system.",
"For a physical quantity to be eligible for an equilibrium statistical mechanical description it has to be ergodic, which means that its time average coincides with its ensemble average.",
"To test ergodicity for a physical quantity one has to compare the time evolution of its physical state to the corresponding equilibrium state.",
"There has been an intensive efforts to investigate ergodicity in one-dimensional spin chains where it was demonstrated that the entanglement, magnetization, spin-spin correlation functions are non-ergodic in Ising and XY spin chains for finite number of spins as well as at the thermodynamic limit [16], [17], [18], [8].",
"Studying quantum entanglement in two-dimensional systems face more obstacles in comparison to the one dimensional case, particularly the rapid increase in the dimension of the Hilbert spaces which lead to much larger scale calculations relying mainly on the numerical methods.",
"The existence of exact solutions has contributed enormously to the understanding of the entanglement for 1D systems [19], [20], [18], [8].",
"In a previous work, the entanglement in a 19-site two-dimensional transverse Ising model at zero temperature [21] was studied.",
"The spin-$1/2$ particles are coupled through an exchange interaction $J$ and subject to an external time-independent magnetic field $h$ .",
"It was demonstrated that for such a class of systems the entanglement can be tuned by varying the parameter $\\lambda =h/J$ and also by introducing impurities into the system, which showed a quantum phase transition at a critical value of the parameter $\\lambda $ in the vicinity of 2.",
"Recently, we have investigated the time evolution of entanglement in a two dimensional triangular transverse Ising system with seven spins in an external magnetic field [22].",
"Different time dependent forms of the magnetic field were applied.",
"The system have demonstrated different responses based on the type of applied field, where for a smoothly changing magnetic field the system entanglement follows the profile of the field very closely.",
"In this paper, we consider the entanglement in a two-dimensional $XY $ triangular spin system, where the nearest neighbor spins are coupled through an exchange interaction $J$ and subject to an external magnetic field $h$ .",
"We consider the system at different degrees of anisotropy to test its effect on the system entanglement and dynamics.",
"The number of spins in the system is 7 with a number of impurities existing.",
"We consider two different cases of impurities, the first case is a single impurity existing either at the border of the system or at the center with the coupling strength between the impurity spin and its neighbors different from that between the rest of the spins.",
"The second case is double impurities, existing both at the border or one at the border and one at the center.",
"We consider the coupling between the two impurities as $J^{\\prime }$ , which is different from the coupling $J^{\\prime \\prime }$ between each one of them and its neighbors, while the interaction among the other spins is $J$ .",
"We show that the entanglement profile of the system at different degrees of anisotropy has great resemblance to that of the one dimensional spin systems as the parameter $\\lambda \\rightarrow 0$ and $\\infty $ .",
"On the other hand, both the Ising and the partially anisotropic systems show phase transition behavior in the vicinity of $\\lambda =2$ but the isotropic system show sharp step variations in the same region before suddenly vanishing.",
"Examining the effect of an external exponential magnetic field on the time evolution of the entanglement showed that the ergodicity of the system can be tuned by varying the strength and location of the impurities and the degree of anisotropy in the system.",
"This paper is organized as follows.",
"In the next section we present our model and quantification of entanglement.",
"In sec.",
"III we consider the case of a single impurity.",
"In sec IV we study the system with a double impurity.",
"We conclude in sec V."
],
[
"Model and quantification of entanglement",
"We consider a set of 7 localized spin-$\\frac{1}{2}$ particles in a two dimensional triangular lattice coupled through exchange interaction $J$ and subject to an external time-dependent magnetic field of strength $h(t)$ .",
"All the particles are identical except one (or two) of them which are considered impurities.",
"The Hamiltonian for such a system is given by $H=-\\frac{(1+\\gamma )}{2}\\sum _{<i,j>}J_{i,j}\\sigma _{i}^x\\sigma _{j}^x -\\frac{(1-\\gamma )}{2}\\sum _{<i,j>}J_{i,j}\\sigma _{i}^y\\sigma _{j}^y - h(t) \\sum _{i} \\sigma _{i}^z,$ Figure: The two dimensional triangular spin lattice in presence of an external transverse magnetic field.where $<i,j>$ is a pair of nearest-neighbors sites on the lattice, $J_{i,j}=J$ for all sites except the sites nearest to an impurity site.",
"For a single impurity, the coupling between the impurity and its neighbors $J_{i,j}=J^{\\prime }=(\\alpha +1)J$ , where $\\alpha $ measures the strength of the impurity.",
"For double impurities $J_{i,j}=J^{\\prime }=(\\alpha _1+1)J$ is the coupling between the two impurities and $J_{i,j}=J^{\\prime \\prime }=(\\alpha _2+1)J$ is the coupling between any one of the two impurities and its neighbors while the coupling is just $J$ between the rest of the spins.",
"For this model it is convenient to set $J=1$ .",
"For a system of 7 spins, its Hilbert space is huge with $2^7$ dimensions, yet it is exactly diagonalizable using the standard computational techniques.",
"Exactly solving Schrodinger equation of the Hamiltonian (REF ), yielding the system energy eigenvalues ${E_i}$ and eigenfunctions ${\\psi _i}$ .",
"The density matrix of the system is defined by $\\rho = |\\psi _0 \\rangle \\langle \\psi _0 | \\; ,$ where $|\\psi _0\\rangle $ is the ground state energy of the entire spin system.",
"We confine our interest to the entanglement between two spins, at any sites $i$ and $j$ [23].",
"All the information needed in this case, at any moment $t$ , is contained in the reduced density matrix $\\rho _{i, j}(t)$ which can be obtained from the entire system density matrix by integrating out all the spins states except $i$ and $j$ .",
"We adopt the entanglement of formation, as a well known measure of entanglement where Wootters [24] has shown that, for a pair of binary qubits, the concurrence $C$ , which goes from 0 to 1, can be taken as a measure of entanglement.",
"The concurrence between two sites $i$ and $j$ is defined as $C(\\rho )=max\\lbrace 0,\\epsilon _1-\\epsilon _2-\\epsilon _3-\\epsilon _4\\rbrace ,$ where the $\\epsilon _i$ 's are the eigenvalues of the Hermitian matrix $R\\equiv \\sqrt{\\sqrt{\\rho }\\tilde{\\rho }\\sqrt{\\rho }}$ with $\\tilde{\\rho }=(\\sigma ^y \\otimes \\sigma ^y)\\rho ^*(\\sigma ^y\\otimes \\sigma ^y)$ and $\\sigma ^y$ is the Pauli matrix of the spin in y direction.",
"For a pair of qubits the entanglement can be written as, $E(\\rho )=\\epsilon (C(\\rho )),$ where $\\epsilon $ is a function of the “concurrence” $C$ $\\epsilon (C)=h\\left(\\frac{1-\\sqrt{1-C^2}}{2}\\right),$ where $h$ is the binary entropy function $h(x)=-x\\log _{2}x-(1-x)log_{2}(1-x).$ In this case, the entanglement of formation is given in terms of another entanglement measure, the concurrence $C$ .",
"The dynamics of entanglement is evaluated using the same techniques applied in our previous work [22].",
"Specifically, we apply the step-by-step time-evolution projection technique, which was proved to give the same exact result as the matrix transformation technique, where both techniques were introduced in [22], but 20 times faster.",
"In this technique we assume that our system is initially, at $t_0$ , in the ground state at zero temperature $|\\phi \\rangle $ with energy, say, $\\varepsilon $ in an external magnetic field with strength $a$ .",
"The magnetic field is turned to a new value $b$ and the system Hamiltonian becomes $H$ with $N$ eigenpairs $E_i$ and $|\\psi _{i}\\rangle $ .",
"The original state $|\\phi \\rangle $ can be expanded in the basis $\\lbrace |\\psi _{i}\\rangle \\rbrace $ : $|\\phi \\rangle =c_{1}|\\psi _{1}\\rangle +c_{2}|\\psi _{2}\\rangle +...+c_{N}|\\psi _{N}\\rangle ,$ where $c_i=\\langle \\psi _i|\\phi \\rangle .$ When $H$ is independent of time between $t$ and $t_0$ then we can write $U(t,\\, t_{0})\\,|\\psi _{i,t_0}\\rangle =e^{-iH(t>t_0)(t-t_{0})/\\hbar }|\\psi _{i,t_0}\\rangle =e^{-iE_{i}(t-t_{0})/\\hbar }|\\psi _{i,t_0}\\rangle ,$ where $U(t,\\, t_{0})$ is the time evolution operator.",
"The ground state will evolve with time as $|\\phi (t)\\rangle &=& c_{1}|\\psi _{1}\\rangle e^{-iE_{1}(t-t_0)}+c_{2}|\\psi _{2}\\rangle e^{-iE_{2}(t-t_0)}+...+c_{N}|\\psi _{N}\\rangle e^{-iE_{N}(t-t_0)}\\nonumber \\\\&=& \\sum _{i=1}^{N}c_{i}|\\psi _{i}\\rangle e^{-iE_{i}(t-t_0)}.$ and the pure state density matrix becomes $\\rho (t)=|\\phi (t)\\rangle \\langle \\phi (t)|.$ Simply any complicated function can be treated as a collection of step functions.",
"When the state evolves to the next step just repeat the procedure to get the next step results.",
"Of course the lack of smoothness in the magnetic field function imposes a challenging obstacle in the calculations but this can be overcome by choosing a proper small enough time step.",
"Because the size of our 7-site system is still manageable, in our actual calculations, we included all the $2^7=128$ states in every step, without any truncation of the higher energy eigenstates.",
"This ensures us no approximation in this step.",
"But the method itself is aiming at larger size system, like 19 sites XY model.",
"By then, due to the computation limit, cutting off higher energy eigenstates might be a necessary action."
],
[
"Static system with border impurity",
"We define a dimensionless coupling parameter $\\lambda =h/J$ and we set $J=1$ throughout this paper for convenience.",
"We start by considering the effect of a single impurity located at the border site 1.",
"The concurrence between the impurity site 1 and site 2, $C(1,2)$ , versus the parameter $\\lambda $ for the three different models, Ising ($\\gamma =1$ ), partially anisotropic ($\\gamma =0.5$ ) and isotropic XY ($\\gamma =0$ ) at different impurity strengths ($\\alpha = -0.5, 0, 0.5, 1$ ) is in fig.",
"REF .",
"Figure: (Color online) The concurrence C(1,2)C(1,2) versus the parameter λ\\lambda with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,0.5,1\\alpha = -0.5, 0, 0.5, 1 for different degrees of anisotropy γ=1,0.5,0\\gamma = 1, 0.5, 0 as shown in the subfigures.",
"The legend for all subfigures is as shown in subfigure (a).Firstly, the impurity paramter $\\alpha $ is set to zero.",
"For the corresponding Ising model, the concurrence $C(1,2)$ , in fig.",
"REF (a), demonstrates the usual phase transition behavior where it starts at zero value and increases gradually as $\\lambda $ increases reaching a maximum at $\\lambda \\approx 2$ then decays as $\\lambda $ increases further.",
"As the degree of anisotropy decreases the behavior of the entanglement changes, where it starts with a finite value at $\\lambda =0$ and then shows a step profile for the small values of $\\lambda $ .",
"For the partially anisotropic case, the step profile is smooth and the entanglement mimics the Ising case as $\\lambda $ increases but with smaller magnitude.",
"The entanglement of isotropic XY system shows a sharp step behavior then suddenly vanishes before reaching $\\lambda =2$ .",
"Interestingly, the entanglement behavior of the two-dimensional spin system at the different degrees of anisotropy mimics the behavior of the one-dimensional spin system at the same degrees of anisotropy at the extreme values of the parameter $\\lambda $ .",
"The ground state of the one-dimensional Ising model is characterized by a quantum phase transition that takes place at the critical value $h/J = 1$ [5], [8] which corresponds to a maximum entanglement in the system.",
"The order parameter is the magnetization $\\langle \\sigma ^x \\rangle $ which is different from zero for $J \\ge h$ and zero otherwise.",
"The ground state is paramagnetic when $J/h \\rightarrow 0$ where the spins get aligned in the magnetic field direction, the $z$ -direction.",
"It is ferromagnetic when $J/h\\rightarrow \\infty $ where the spins are aligned in the $x$ -direction.",
"Both cases cause zero entanglement.",
"Comparing the entanglement behavior in the two-dimensional Ising spin system with the one-dimensional system, one can see a great resemblance except that the critical value becomes $h/J \\approx 2$ in the two dimensional case as shown in fig.",
"REF .",
"On the other hand, for the partially anisotropic and isotropic $XY$ systems, the entanglements of the two-dimensional and one-dimensional system agrees at the extreme values of $\\lambda $ where it vanishes for $h >> J$ and reaches a finite value for $h << J$ .",
"The former case corresponds to an alignment of the spins in the $z$ -direction, paramagnetic state, while the latter case corresponds to alignment in the $x$ and $y$ -directions which a ferromagnetic state.",
"The effect of a weak impurity ($J^{\\prime }<J$ ), $\\alpha =-0.5$ , is shown in fig.",
"REF (b) where the entanglement behavior is the same as before except that the entanglement magnitude is reduced compared with the pure case.",
"On the other hand, considering the effect of a strong impurity ($J^{\\prime }> J$ ), where $\\alpha =0.5$ and 1, as shown in fig.",
"REF (c) and fig.",
"REF (d) respectively, one can see that the entanglement profile for $\\gamma =1$ and 0.5 have the same overall behavior as in the pure and weak impurity cases except that the entanglement magnitude becomes higher as the impurity gets stronger and the peaks shift toward higher $\\lambda $ values.",
"Nevertheless, the isotropic $XY$ system behaves differently from the previous cases where it starts to increase first in a step profile before suddenly dropping to zero again, which will be explained latter.",
"Figure: (Color online) The concurrence C(2,4)C(2,4) versus the parameter λ\\lambda with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,0.5,1\\alpha = -0.5, 0, 0.5, 1 for different degrees of anisotropy γ=1,0.5,0\\gamma = 1, 0.5, 0 as shown in the subfigures.",
"The legend for all subfigures is as shown in subfigure (a).To study the entanglement between two sites, none of them is impurity, we consider $C(2,4)$ which is depicted in fig.",
"REF .",
"There are two main differences between the behavior of $C(2,4)$ and $C(1,2)$ .",
"Firstly, the magnitude of the entanglement envelope is higher for $C(2,4)$ for $\\gamma =0.5$ and 0 (but not $\\gamma =1$ ) when $\\alpha =0$ , while $C(2,4)$ is greater than $C(1,2)$ for all $\\gamma $ values for the weak impurity case, $\\alpha =0$ .",
"This is an interesting result as internal sites entanglement should be smaller in value than the edge sites.",
"Secondly, the entanglement of the isotropic XY case increases in a multi-step profile for all values of $\\alpha $ before suddenly dropping to zero."
],
[
"Static system with center impurity",
"To explore the effect of the impurity location we investigate the case of a single impurity spin located at site 4, instead of site 2, where we plot the concurrences $C(1,2)$ and $C(1,4)$ in figs.",
"REF and REF respectively.",
"Figure: (Color online) The concurrence C(1,2)C(1,2) versus the parameter λ\\lambda with a single impurity at the central site 4 with different impurity coupling strengths α=-0.5,0,0.5,1\\alpha = -0.5, 0, 0.5, 1 for different degrees of anisotropy γ=1,0.5,0\\gamma = 1, 0.5, 0 as shown in the subfigures.",
"The legend for all subfigures is as shown in subfigure (a).Figure: (Color online) The concurrence C(1,4)C(1,4) versus the parameter λ\\lambda with a single impurity at the border site 4 with different impurity coupling strengths α=-0.5,0,0.5,1\\alpha = -0.5, 0, 0.5, 1 for different degrees of anisotropy γ=1,0.5,0\\gamma = 1, 0.5, 0 as shown in the subfigures.",
"The legend for all subfigures is as shown in subfigure (a).Interestingly, while changing the impurity location has almost no effect on the behavior of the entanglement $C(1,2)$ of the partially anisotropic and isotropic XY systems, it has a great impact on that of the Ising system where the peak value of the entanglement increases significantly in the weak impurity case and decreases as the impurity gets stronger as shown in fig.",
"REF .",
"Now considering the entanglement between the central impurity site 4 and the edge site 1, and comparing with the results in fig.",
"REF of the entanglement between the edge site 2 and central site 4, one can see that the entanglement $C(1,4)$ profile for all degrees of anisotropy is very close to the $C(2,4)$ .",
"Nevertheless the entanglement $C(1,4)$ magnitude is lower for weak impurity case and higher for the strong impurity which means that the central impurity made a significant change to the entanglement magnitude."
],
[
"Effect of system energy gap on entanglement",
"To explain the distinct behavior of the entanglement corresponding to the different degrees of anisotropy $\\gamma $ we depict the lowest few energy eigenvalues of the system at the different $\\gamma $ values for the two cases of border and central impurities in fig.",
"REF and fig.",
"REF respectively.",
"As can be noticed in fig.",
"REF (a), the energies of the ground state and the first excited state of the Ising system coincide at the beginning at the small values of $\\lambda $ until a specific value where they deviate from each other.",
"This is corresponding to the transition from the degenerated ground state to non-degenerated one, from paramagnetic to ferromagnetic order, by breaking the $Z_2$ symmetry, which explains the phase transition curve observed in the Ising case.",
"Figure: (Color online) The energy spectrum versus the parameter λ\\lambda with a single impurity at the central site 4 with impurity coupling strength α=1\\alpha = 1 for different degrees of anisotropy γ=1,0.5,0\\gamma = 1, 0.5, 0 as shown in the subfigures.",
"The legend for all subfigures is as shown in subfigure (a).The energy spectrum of the partially anisotropic $XY$ system is a little bit different at the small values of $\\lambda $ where the ground state and the first excited state coincide at the beginning but then deviates slightly from each other before recombining again and at last separate from each other completely, this behavior is repeated quite few times depending on the impurity strength, as illustrated in fig.",
"REF (b).",
"This energy spectrum behavior explains the roughness in the ascending part of the entanglement curves of the partially anisotropic $XY$ system corresponding to subsequent transitions between the ground state and the first excited state taking place before reaching the maximum entanglement point.",
"In fig.",
"REF (c), the energy spectrum of the isotropic $XY$ system is explored where clearly the deviations and recombination between the ground and first excited state energies become sharper and more frequent compared with the partially anisotropic system.",
"This distinct behavior of the energy spectrum corresponding to $\\gamma = 0$ is the reason for the sharp step behavior of the entanglement as was shown in figs.",
"REF and REF .",
"Figure: (Color online) (a) The concurrence C 14 C_{14} versus λ\\lambda ; (b) the first derivative of the concurrence C 14 C_{14} with respect to λ\\lambda versus λ\\lambda ; (c) the energy gap between the ground state and first excited state versus λ\\lambda ; (d) the first derivative (in units of JJ) and second derivative (in units of J 2 J^2) of the energy gap with respect λ\\lambda versus λ\\lambda for the pure Ising system (γ=1\\gamma =1 and α=0\\alpha =0).Figure: (Color online) (a) The concurrence C 14 C_{14} versus λ\\lambda ; (b) the first derivative of the concurrence C 14 C_{14} with respect to λ\\lambda versus λ\\lambda ; (c) the energy gap between the ground state and first excited state versus λ\\lambda ; (d) the first derivative (in units of JJ) and second derivative (in units of J 2 J^2) of the energy gap with respect λ\\lambda versus λ\\lambda for the pure partially anisotropic system (γ=0.5\\gamma =0.5 and α=0\\alpha =0).",
"Notice that the first derivative of energy gap is enlarged 10 times its actual scale for clearness.Figure: (Color online) (a) The concurrence C 14 C_{14} versus λ\\lambda ; (b) the first derivative of the concurrence C 14 C_{14} with respect to λ\\lambda versus λ\\lambda ; (c) the energy gap between the ground state and first excited state versus λ\\lambda ; (d) the first derivative (in units of JJ) and second derivative (in units of J 2 J^2) of the energy gap with respect λ\\lambda versus λ\\lambda for the pure isotropic system (γ=0\\gamma =0 and α=0\\alpha =0).",
"Notice that the first derivative of energy gap is enlarged 20 times its actual scale for clearness.Critical quantum behavior in a many body system happens either when an actual crossing takes place between the excited state and the ground state or a limiting avoided level-crossing between them exists, i.e.",
"an energy gab between the two states that vanishes in the infinite system size limit at the critical point [20].",
"When a many body system crosses a critical point, significant changes in both its wave function and ground state energy takes place, which are manifested in the behavior of the entanglement function.",
"The entanglement in one dimensional infinite spin systems, Ising and $XY$ , was shown to demonstrate scaling behavior in the vicinity of critical points [23].",
"The change in the entanglement across the critical point was quantified by considering the derivative of the concurrence with respect to the parameter $\\lambda $ .",
"This derivative was explored versus $\\lambda $ for different system sizes and although it didn't show divergence for finite system sizes, it showed clear anomalies which developed to a singularity at the thermodynamic limit.",
"The ground state of the Heisenberg spin model is known to have a double degeneracy for an odd number of spins which is never achieved unless the thermodynamic limit is reached [20].",
"Particularly, the Ising 1D spin chain in an external transverse magnetic field has doubly degenerate ground state in a ferromagnetic phase that is gapped from the excitation spectrum by $2 J (1-h/J)$ , which is removed at the critical point and the system becomes in a paramagnetic phase.",
"Now let us first consider our two-dimensional finite size Ising spin system.",
"The concurrence $C_{14}$ and its first derivative are depicted versus $\\lambda $ in figs.",
"REF (a) and (b) respectively.",
"As one can see, the derivative of the concurrence shows strong tendency of being singular at $\\lambda _c = 1.64$ .",
"The characteristics of the energy gap between the ground state and the first excited state as a function of $\\lambda $ are explored in fig.",
"REF (c).",
"The system shows strict double degeneracy, zero energy gap, only at $\\lambda =0$ i.e.",
"at zero magnetic field, but once the magnetic field is on the degeneracy is lifted and an extremely small energy gap develops, which increase very slowly for small magnetic field values but increases abruptly at certain $\\lambda $ value.",
"It is important to emphasis here that at $\\lambda =0$ , regardless of which one of the double ground states is selected for evaluating the entanglement, the same value is obtained.",
"The critical point of a phase transition should be characterized by a singularity in the ground state energy, and an abrupt change in the energy gap of the system as a function of the system parameter as it crosses the critical point.",
"To better understand the behavior of the energy gap across the prospective critical point and identify it, we plot the first and second derivatives of the energy gap as a function of $\\lambda $ in fig.",
"REF (d).",
"Interestingly, the first derivative $d\\Delta E / d \\lambda $ which represents the rate of change of the energy gap as a function of $\\lambda $ starts with a zero value at $\\lambda =0$ and then increase very slowly before it shows a great rate of change and finally reaches a saturation value.",
"This behavior is best represented by the second derivative $d^2 \\Delta E / d \\lambda ^2$ , which shows strong tendency of being singular at $\\lambda _c=1.8$ , which indicates the highest rate of change the energy gap as a function of $\\lambda $ .",
"The reason for the small discrepancy between the two values of the $\\lambda _c$ extracted from the $dC / d \\lambda $ plot and the one of $d^2 \\Delta E / d \\lambda ^2$ is that the concurrence $C_{14}$ is only between two sites and does not represent the whole system in contrary to the energy gap.",
"One can conclude that the rate of change of the energy gap as a function of the system parameter, $\\lambda $ in our case, should be maximum across the critical point.",
"Turning to the case of the partially anisotropic spin system, $\\gamma =0.5$ , presented in fig.",
"REF , one can notice from fig.",
"REF (a) that the concurrence shows few sharp changes, which is reflected in the energy gap plot as an equal number of minima as shown in fig.",
"REF (b).",
"Nevertheless, again there is only one strict double degeneracy at $\\lambda =0$ while the other three energy gap minima are non-zero and in the order of $10^{-5}$ .",
"It is interesting to notice that the anomalies in both $dC / d \\lambda $ and $d^2 \\Delta E / d \\lambda ^2$ are much stronger and sharper compared with the Ising case as shown in figs.",
"REF (c) and (d).",
"Finally the isotropic system which is depicted in fig.",
"REF , shows even sharper energy gap changes as a result of the sharp changes in the concurrence and the anomalies in the derivatives $dC / d \\lambda $ and $d^2 \\Delta E / d \\lambda ^2$ are even much stronger than the previous two cases."
],
[
"System dynamics with impurity",
"Now we turn to the dynamics of the two dimensional spin system under the effect of a single impurity and different degrees of anisotropy.",
"We investigate the dynamical reaction of the system to an applied time-dependent magnetic field with exponential form $h(t)= b + (a-b) e^{-w \\; t}$ for $t > 0$ and $h(t)=a$ for $t \\le 0$ .",
"We start by considering the Ising system, $\\gamma =1$ with a single impurity at the border site 1, which is explored in fig.",
"REF , where we set $a=1$ , $b=3.5$ and $\\omega =0.1$ .",
"For the pure case, $\\alpha =0$ shown in fig.",
"REF (a), the results confirms the ergodic behavior of the system that was demonstrated in our previous work [22], where the asymptotic value of the entanglement coincide with the equilibrium state value at $h(t)=b$ .",
"As can be noticed from figs.",
"REF (b), REF (c) and REF (d) neither the weak nor strong impurities have effect on the ergodicity of the Ising system.",
"Nevertheless, there is a clear effect on the asymptotic value of entanglements $C(1,2)$ and $C(1,4)$ but not on $C(2,4)$ which relates two regular sites.",
"The weak impurity, $\\alpha =-0.5$ reduces the asymptotic value of $C(1,2)$ and $C(1,4)$ while the strong impurities, $\\alpha = 1, 2$ raise it compared to the pure case.",
"Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional Ising lattice (γ=1\\gamma = 1) under the effect of an exponential magnetic field with parameters values a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (a).In fig.",
"REF , we consider the same system but under the effect of a weaker exponential magnetic field with set of parameters $a =1$ , $b=1.5$ and $\\omega =0.1$ .",
"As can be noticed, the entanglement of the Ising system is still showing an ergodic behavior at all impurity strengths.",
"This means that the Ising system with a single border impurity is always ergodic under the effect of different impurity strengths and different magnetic field parameters.",
"Interestingly, the different impurity strengths have different effects on the asymptotic value of the entanglements compared to the previous case under the effect of the new magnetic field.",
"The weak impurity, as shown in fig.",
"REF (b), raises the asymptotic value of $C(2,4)$ and splits those of $C(1,2)$ and $C(1,4)$ from each other.",
"On the other hand, the strong impurity effects are depicted in fig.",
"REF (c) and fig.",
"REF (d) which show that the asymptotic values of all concurrences are reduced significantly as the impurity strength increases, in contrary to the previous case.",
"This emphasis the important role that the magnetic filed parameters play beside the impurity strength in controlling the entanglement behavior.",
"Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(1,5)C(1,2), C(1,4), C(1,5) with a single impurity at the central site 4 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional Ising lattice (γ=1\\gamma = 1) under the effect of an exponential magnetic field with parameters values a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (a).It is of great interest to examine the effect of the impurity location, which we investigate in fig.",
"REF where a single impurity is located at the central site 4 instead of the border site 1 with exponential magnetic field parameters $a =1$ , $b=1.5$ and $\\omega =0.1$ .",
"Very interestingly, the entanglement behavior changes significantly as a result of changing the impurity location.",
"Though the pure Ising system is still ergodic as shown in fig.",
"REF (a), the system with weak and strong impurity becomes non-ergodic which is illustrated in figs.",
"REF (b), REF (c) and REF (d) respectively.",
"Again the weak impurity raises the asymptotic values wheres the strong impurities reduces them significantly.",
"Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) under the effect of an exponential magnetic field with parameters values a=1, b=3.5 and ω=0.1\\omega =0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=3.5h=3.5.",
"The legend for all subfigures is as shown in subfigure (b).The dynamics of the partially anisotropic XY system under the effect exponential magnetic field with parameters $a =1$ , $b=3.5$ and $\\omega = 0.1$ , is explored in fig.",
"REF .",
"It is remarkable to see that while for both the pure and weak impurity cases, $\\alpha =0$ and $-0.5$ , the system is nonergodic as shown in figs.",
"REF (a) and REF (b), and it is ergodic in the strong impurity cases $\\alpha = 1$ and 2 as illustrated in figs.",
"REF (c) and REF (d).",
"Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) under the effect of an exponential magnetic field with parameters values a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (a).Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the central site 4 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) under the effect of an exponential magnetic field with parameters values a=1,b=1.5a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (a).Clearly the asymptotic values of the entanglement are higher in the pure and weak impurity cases compared with the strong impurities.",
"Changing the magnetic field parameter value $b$ to $1.5$ , one can observe the great impact in fig.",
"REF , where only the pure system becomes ergodic while the system with any impurity strength is nonergodic.",
"This means that the magnetic field parameters control ergodicity as well.",
"In fig.",
"REF , we study the same system with a single impurity at the central site 4 with magnetic field parameters $a =1$ , $b=1.5$ and $\\omega = 0.1$ .",
"As can be seen from the different subfigures, the system is ergodic only in the pure case and the asymptotic values are higher for the pure and weak impurity systems compared with the strong impurities.",
"Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the border site 1 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional XY lattice (γ=0\\gamma = 0) under the effect of an exponential magnetic field with parameters values a=1,b=1.5a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight (thicker) lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (a).Figure: (Color online) Dynamics of the concurrences C(1,2),C(1,4),C(2,4)C(1,2), C(1,4), C(2,4) with a single impurity at the central site 4 with different impurity coupling strengths α=-0.5,0,1,2\\alpha = -0.5, 0, 1, 2 for the two dimensional XY lattice (γ=0\\gamma = 0) under the effect of an exponential magnetic field with parameters values a=1,b=1.5a=1, b=1.5 and ω=0.1\\omega =0.1.",
"The straight (thicker) lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.5h=1.5.",
"The legend for all subfigures is as shown in subfigure (b).The complete isotropic $XY$ system with a single border impurity at site 1 under the effect of an exponential magnetic field with parameter values $a =1$ , $b=1.5$ and $\\omega = 0.1$ , is investigated in fig.",
"REF .",
"The trivial effect of the magnetic field, similar to the one dimensional case results [8], is clear where the entanglement assumes a constant value for all pair of spins.",
"This trivial effect is the result of the fact that for $\\gamma =0$ the exchange coupling terms in the Hamiltonian commute with the magnetic field term.",
"Nevertheless one still can see an effect of the impurity on the ergodicity of the system where for $\\alpha =0$ and 1, the system is nonergodic wile for $\\alpha =-0.5$ and 2 it is ergodic as shown in fig.",
"REF .",
"In fact, testing a wide range of $\\alpha $ values indicates that for the values approximately in the range $-0.4 \\ge \\alpha \\le 1.9$ the system is nonergodic, otherwise it is ergodic i.e.",
"for small absolute values of the impurity.",
"Examining the same system under the effect of the same magnetic field but with a single central impurity, for wide range of $\\alpha $ , demonstrates that the system becomes nonergodic at all values of $\\alpha $ which is illustrated in fig.",
"REF ."
],
[
"Static system with impurities",
"In this section we study the effect of double impurity, where we start with two located at the border sites 1 and 2.",
"We set the coupling strength between the two impurities as $J^{\\prime } = (1+\\alpha _1) J$ , between any one of the impurities and its regular nearest neighbors as $J^{\\prime \\prime } = (1+\\alpha _2) J$ and between the rest of the nearest neighbor sites on the lattice as $J$ .",
"The effect of the impurities strength on the concurrence between different pairs of sites for the Ising lattice is shown in fig.",
"REF .",
"In fig.",
"REF (a) we consider the entanglement between the two impurity sites 1 and 2 under a constant external magnetic field $h=2$ .",
"The concurrence $C(1,2)$ takes a large value when the impurity strengths $\\alpha _1$ , controlling the coupling between the impurity sites, is large and when $\\alpha _2$ , controlling coupling between impurities and their nearest neighbors, is weak.",
"As $\\alpha _1$ decreases and $\\alpha _2$ increases, $C(1,2)$ decreases monotonically until it vanishes.",
"As one can conclude, $\\alpha _1$ is more effective than $\\alpha _2$ in controlling the entanglement in this case.",
"On the other hand, the entanglement between the impurity site 1 and the regular central site 4 is illustrated in fig.",
"REF (b) which behaves completely different from C(1,2).",
"Figure: (Color online) The concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(4,5)C(4,5) versus the impurity coupling strengths α 1 \\alpha _1 and α 2 \\alpha _2 with double impurities at sites 1 and 2 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) in an external magnetic field h=2.Figure: (Color online) The concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(4,5)C(4,5) versus the impurity coupling strengths α 1 \\alpha _1 and α 2 \\alpha _2 with double impurities at sites 1 and 2 for the two dimensional XY lattice (γ=0\\gamma = 0) in an external magnetic field h=1.The concurrence C(1,4) is mainly controlled by the impurity strength $\\alpha _2$ where it starts with a very small value when the impurity is very weak and increases monotonically until it reaches a maximum value at $\\alpha _2=0$ , i.e.",
"with no impurity, and decays again as the impurity strength increases.",
"The effect of $\\alpha _1$ in that case is less significant and makes the concurrence slowly decreases as $\\alpha _1$ increases which is expected since as the coupling between the two border sites 1 and 2 increases the entanglement between 1 and 4 decreases.",
"It is important to note that in general $C(1,2)$ is much larger than $C(1,4)$ since the border entanglement is always higher than the central one as the entanglement is shared by many sites.",
"The entanglement between two regular sites is shown in fig.",
"REF (c) where the concurrence C(4,5) is depicted against $\\alpha _1$ and $\\alpha _2$ , the entanglement decays gradually as $\\alpha _2$ increases while $\\alpha _1$ has a very small effect on the entanglement, which slightly decreases as $\\alpha _1$ increases as shown.",
"Interestingly, the behavior of the energy gap between the ground state and the first excited state of the Ising system $\\Delta E$ versus the impurity strengths $\\alpha _1$ and $\\alpha _2$ , which is explored in fig.",
"REF (d) has a strong resemblance to that of the concurrence $C(4,5)$ except that the decay of $\\Delta E$ against $\\alpha _2$ is more rapid.",
"The effect of changing the location of the impurities is considered in fig.",
"REF where the two impurities exist at sites 1 and 4 in the Ising system.",
"The behavior of the concurrences are very much the same as in the previous case except that the profiles of $C(1,2)$ and $C(1,4)$ have been exchanged as C(1,4) now represents the concurrence between the two impurity sites.",
"Figure: (Color online) Dynamics of the concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(5,7)C(5,7) with double impurities at sites 1 and 2 for the two dimensional Ising lattice (γ=1\\gamma = 1) in an exponential magnetic field where a=1, b=2, and w=0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=2h=2.",
"The legend for all subfigures is as shown in subfigure (a).Figure: (Color online) Dynamics of the concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(5,7)C(5,7) with double impurities at sites 1 and 2 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) in an exponential magnetic field where a=1, b=2, and w=0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=2h=2.",
"The legend for all subfigures is as shown in subfigure (a).The partially anisotropic system, $\\gamma =0.5$ , with double impurity at sites 1 and 2 and under the effect of the external magnetic field $h=2$ is explored in fig.",
"REF .",
"As one can see, the overall behavior specially at the border values of the impurity strengths is the same as observed in the Ising case except that the concurrences suffer a local minimum within a small range of the impurity strength $\\alpha _2$ between 0 and 1 while corresponding to the whole $\\alpha _1$ range.",
"The change of the entanglement around this local minimum takes a step-like profile which is very clear in the case of the concurrence $C(1,4)$ shown in fig.",
"REF (b).",
"Remarkably, the local minima in the plotted concurrences coincide with the line of vanishing energy gap as shown in fig.",
"REF (d), which means that these minima correspond to a transition between a ground state and another one which takes place as the system parameters change.",
"The anisotropic $XY$ model with two impurities at sites 1 and 2 in an external magnetic field $h=1.8$ is explored in fig.",
"REF .",
"The entanglement for this system shows much sharper changes as a function of the impurity strengths and the sharp step changes take place in a narrow region of both $\\alpha _1$ and $\\alpha _2$ specifically for $-1 \\le \\alpha _1 \\le 1$ and $-1 \\le \\alpha _2 \\le 0$ .",
"It is interesting to note that again the sharp step changes in the entanglement are corresponding to the line of minimum energy gap as shown in fig.",
"REF (d) where this line varies continuously between a very small value and zero which explains the many steps appearing in the different concurrences and particularly $C(1,4)$ depicted in fig.",
"REF (b)."
],
[
"System dynamics with impurities",
"Now we turn to the dynamics of the two dimensional spin system with double impurity under the effect of an external exponential magnetic field to test the ergodicity of the system as we vary the degree of anisotropy or the location of the impurities.",
"In fig.",
"REF we consider the dynamics of the Ising system with two impurities at the sites 1 and 2 under the effect of an exponential magnetic field with parameters $a=1$ , $b=2$ and $\\omega =0.1$ .",
"As one can see the system shows an ergodic behavior for the different values of the impurities strengths $\\alpha _1, \\alpha _2= (0,0), (0,1), (1,0)$ and $(1,1)$ .",
"The Ising system sustains its ergodicity for all shown values of impurities strengths and other tested values.",
"Figure: (Color online) Dynamics of the concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(5,7)C(5,7) with double impurities at sites 1 and 4 for the two dimensional partially anisotropic lattice (γ=0.5\\gamma = 0.5) in an exponential magnetic field where a=1, b=2, and w=0.1.",
"The straight lines represent the equilibrium concurrences corresponding to constant magnetic field h=2h=2.",
"The legend for all subfigures is as shown in subfigure (a).Figure: (Color online) Dynamics of the concurrence C(1,2)C(1,2), C(1,4)C(1,4), C(5,7)C(5,7) with double impurities at sites 1 and 4 for the two dimensional isotropic XY lattice (γ=0\\gamma = 0) in an exponential magnetic field where a=1, b=1.8, and w=0.1.",
"The straight (thicker) lines represent the equilibrium concurrences corresponding to constant magnetic field h=1.8h=1.8.",
"The legend for all subfigures is as shown in subfigure (a).The partially anisotropic system under the same condition behaves differently.",
"Its pure case, $\\alpha _1,\\,\\alpha _2 = (0,1)$ case and $\\alpha _1,\\,\\alpha _2 = (1,1)$ case are all nonergodic as dipicted in figs.",
"REF (a) and REF (b).",
"Nevertheless, the system with impurity strengths $\\alpha _1=1$ and $\\alpha _2 = 0$ show ergodic behavior which means that the nonergodicity of the partially anisotropic system is sensitive for the strength and location of impurities.",
"The isotropic system is explored in fig.",
"REF which behaves nonergodically for all impurity strengths.",
"Testing the effect of the impurity location we consider the same system with impurities at the sites 1 and 4.",
"While the Ising system shows ergodicty at all impurity strengths as shown in fig.",
"REF , the partially and isotropic $XY$ systems are nonergodic at the different impurity strengths as plotted in figs.",
"REF and REF respectively."
],
[
"Conclusion and future directions",
"We have investigated the nearest neighbor entanglement and ergodicity of a two-dimensional $XY$ spin lattice in an external magnetic field $h$ .",
"The spins are coupled to each other through nearest neighbor exchange interaction $J$ .",
"The number of spins in the lattice are 7 where we may consider one or two of them as impurities.",
"We have found that the completely anisotropic (the Ising), the partially anisotropic and isotropic systems behave in a very similar fashion to that of the one dimensional spin systems at the extreme, small and large, values of the parameter $\\lambda =h/J$ but may deviate at the intermediate values.",
"The first two systems show phase transition in the vicinity of the parameter critical value $\\lambda =2$ and their entanglement vanishes as $\\lambda $ increases.",
"The entanglement of the isotropic system changes in a sharp step profile before suddenly vanishing in the vicinity of $\\lambda =2$ .",
"The entanglement dynamics of the system with impurities was investigated under the effect of an external time-dependent magnetic field of exponential form.",
"It was found that the ergodicity of the system can be tuned using the strength and location of the impurities as well as the degree of anisotropy of the coupling between the spins.",
"It is interesting in future to investigate the same systems coupled to a dissipative environment and to examine the effect of impurity to tune the decoherence in the spin system and to investigate the ergodicity status under coupling to the environment.",
"Furthermore we would like to investigate the same system with larger number of sites to test the system size effect and to clarify the critical value of the parameter $\\lambda $ using finite size scaling [25], [26].",
"Previously, the 19 sites triangular static Ising lattice was treated exactly using the the trace minimization algorithm [22].",
"The dynamics of entanglement in the 19-site XY system is currently under consideration, and by taking advantage of parallel computing we can reach 34 spins by far."
],
[
"Acknowledgments",
"We are grateful to the Saudi NPST for support (project no.",
"11-MAT1492-02) and the deanship of scientific research, King Saud University.",
"We are also grateful to the USA Army research office for partial support of this work at Purdue."
]
] | 1204.0759 |
[
[
"Calculating error bars for neutrino mixing parameters"
],
[
"Abstract One goal of contemporary particle physics is to determine the mixing angles and mass-squared differences that constitute the phenomenological constants that describe neutrino oscillations.",
"Of great interest are not only the best fit values of these constants but also their errors.",
"Some of the neutrino oscillation data is statistically poor and cannot be treated by normal (Gaussian) statistics.",
"To extract confidence intervals when the statistics are not normal, one should not utilize the value for chisquare versus confidence level taken from normal statistics.",
"Instead, we propose that one should use the normalized likelihood function as a probability distribution; the relationship between the correct chisquare and a given confidence level can be computed by integrating over the likelihood function.",
"This allows for a definition of confidence level independent of the functional form of the !2 function; it is particularly useful for cases in which the minimum of the !2 function is near a boundary.",
"We present two pedagogic examples and find that the proposed method yields confidence intervals that can differ significantly from those obtained by using the value of chisquare from normal statistics.",
"For example, we find that for the first data release of the T2K experiment the probability that chisquare is not zero, as defined by the maximum confidence level at which the value of zero is not allowed, is 92%.",
"Using the value of chisquare at zero and assigning a confidence level from normal statistics, a common practice, gives the over estimation of 99.5%."
],
[
"ACKNOWLEDGMENTS",
"The work of B. K. C. is supported, in part, by US Department of Education Grant P200A090275, the work of D. J. E. is supported, in part, by US Department of Energy Grant DE-FG02-96ER40975; the work of J. E-R. is supported, in part, by CONACyT, Mexico."
]
] | 1204.1354 |
[
[
"Magnetic adatom induced skyrmion-like spin texture in surface electron\n waves"
],
[
"Abstract When a foreign atom is placed on a surface of a metal, the surrounding sea of electrons responds screening the additional charge leading to oscillations or ripples.",
"On surfaces, those electrons are sometimes confined to two-dimensional surface states, whose spin-degeneracy is lifted due to the Rashba effect arising from the spin-orbit interaction of electrons and the inversion asymmetric environment.",
"It is believed that at least for a single adatom scanning tunneling microscopy measurements are insensitive to the Rashba splitting i.e.",
"no signatures in the charge oscillations will be observed.",
"Resting on scattering theory, we demonstrate that, if magnetic, one single adatom is enough to visualize the presence of the Rashba effect in terms of an induced spin-magnetization of the surrounding electrons exhibiting a twisted spin texture described as superposition of two skyrmionic waves of opposite chirality."
],
[
"Magnetic adatom induced skyrmion-like spin texture in surface electron waves Samir [email protected] Andreas Bringer Stefan Blügel Peter Grünberg Institut & Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany When a foreign atom is placed on a surface of a metal, the surrounding sea of electrons responds screening the additional charge leading to oscillations or ripples.",
"On surfaces, those electrons are sometimes confined to two-dimensional surface states, whose spin-degeneracy is lifted due to the Rashba effect arising from the spin-orbit interaction of electrons and the inversion asymmetric environment.",
"It is believed that at least for a single adatom scanning tunneling microscopy measurements are insensitive to the Rashba splitting i.e.",
"no signatures in the charge oscillations will be observed.",
"Resting on scattering theory, we demonstrate that, if magnetic, one single adatom is enough to visualize the presence of the Rashba effect in terms of an induced spin-magnetization of the surrounding electrons exhibiting a twisted spin texture described as superposition of two skyrmionic waves of opposite chirality.",
"The lack of spatial inversion symmetry is the triggering ingredient for a wide range of new phenomena that are accessible with state of the art experimental techniques [1], [2], [3], [4], [5], [6].",
"Angle-resolved photoemission spectroscopy (ARPES) was the first tool used by LaShell and coworkers [7] to discover a small energy splitting in the $sp$ surface state band of the Au(111) surface.",
"This splitting has been interpreted by the same authors as a realization of the interaction between the spin and orbital angular momentum, which promoted the renaissance of the Rashba physics.",
"The surge of interest in similar effects involving the spin-orbit interaction in a structural asymmetric environment is the incitement of many additional sophisticated measurements and theoretical simulations on relevant surfaces with [8], [9], [10] and without [11], [12], [13], [5], [14], [15] topologically protected surface states.",
"Since charge oscillations (Friedel oscillations) induced by scattering of the surface-state electrons at a single adatom are blind with respect to such spin-orbit effects [16], an alternative [17] has been proposed on the basis of a multiple scattering study consisting on probing with the scanning tunneling microscope (STM) charge oscillations confined within a corral of adatoms.",
"Theoretical calculations [18] outline the possibility of very complex magnetic structures.",
"We pursue a different route and investigate the possibility of grasping information on the spin-orbit interaction at surfaces exploiting the break of time inversion symmetry introduced by a magnetic adatom.",
"As shown and discussed later, we discover an intriguing spin-texture in the spin-polarized electron gas surrounding the magnetic adatom (Fig.",
"REF ).",
"This new magnetic behavior can be very similar to the topological twists, called skyrmions [20], [1], [2], [3], [21].",
"Parts of our predictions have recently been verified in the interferences produced by the scattering at a MnPc molecule of the complex surface states of Bi(110) surface [15].",
"Figure: Skyrmionic-like spin-texture at the Fermi energy of Au(111) surface.",
"(a) Spatial visualization of the induced local density of states (LDOS) andthe local magnetization direction of Au surface electronsafter scattering with an Fe adatom.",
"The spin texture found in (a)can be decomposed into a linear combination of two smoothly rotatingskyrmionic magnetic waves with opposite vector chirality shown in (b).The wavelengths of LDOS oscillation, the left and right skyrmionic waves are ∼18.7\\sim 18.7 Å, ∼17.3\\sim 17.3 Å and 20.320.3 Å, respectively.Our investigation is based on a Rashba Hamiltonian [19] that describes a two-dimensional gas of free electrons confined in the ($xy$ ) plane of a metallic surface: $\\hat{H} =\\frac{\\hat{p}_x^2 + \\hat{p}_y^2}{2m^*} - \\frac{\\alpha _R}{\\hbar }(\\sigma _x\\hat{p}_y -\\sigma _y\\hat{p}_x)$ considered with respect to a zero-energy reference defined by the bottom of the dispersion curve in absence of the spin-orbit coupling.",
"$m^*$ is the effective mass of the electron and $\\alpha _R$ is the effective Rashba parameter, describing the strength of the effect, whose value is determined in principle by the atomic spin-orbit strength as well as by the degree of asymmetry of the wavefunction imposed by the presence of the surface [16].",
"Here, however, $\\alpha _R$ is chosen to model the experimental dispersion relation of the suface state.",
"Quite generally the gradient of the potential at the surface acts as an electric field $\\vec{E}$ normal to the surface in the lab frame of the sample.",
"Electrons propagating with momentum $\\vec{k}$ across the surface experience this field in their local frame of reference as an effective magnetic field, $\\vec{B}=\\frac{\\hbar }{m^*c}\\vec{k}\\times \\vec{E}$ , which is the origin of the functional form for the Rashba Hamiltonian and defines a spin-quantization axis $\\hat{n}(\\vec{k})$ to be located in the surface plane normal to the wavevector $\\vec{k}=(k_x,\\,k_y)=k(\\cos \\phi ,\\,\\sin \\phi )\\perp \\hat{n}(\\vec{k})$ , and $c$ is the speed of light.",
"The eigenvectors $\\vert \\psi _{1(2)}\\rangle $ of the Hamiltonian (REF ), associated with the wavevectors $\\vec{k}_1$ and $\\vec{k}_2$ , are spin-up and spin-down states, respectively, with respect to $\\hat{n}$ , but are also a coherent superposition of spin-up and -down states, $\\vert \\!\\psi _{1(2)}(\\vec{k})\\rangle =\\frac{e^{ik_{1(2)}}}{\\sqrt{2}}(\\vert \\uparrow \\rangle -(+) i e^{i\\phi }\\vert \\downarrow \\rangle )$ , expressed by the spin functions $\\vert \\uparrow \\rangle $ and $\\vert \\downarrow \\rangle $ , when measured with respect to the surface normal ($z$ -direction).",
"The eigenvalue spectrum $E_{1(2)}(\\vec{k})=\\frac{\\hbar ^2}{2m^*}(k_{1(2)}^2-k_{\\mathrm {so}}^2)$ is a two spin-split cone-shaped parabolic energy dispersion curve by which the origins of the parabola $E_{1(2)}$ are shifted with respect to $k=0$ by $k_{\\mathrm {so}}=\\pm m^*\\alpha _{R}/\\hbar ^2$ , i.e.",
"$k_{1(2)}=k+(-)k_{\\mathrm {so}}$ (see Fig.",
"REF ).",
"When $k_2$ changes sign from positive to negative, the two branches of the dispersion curves cross.",
"Unless stated otherwise, we shall consider in the upcoming text only positive $k_2$ .",
"Figure: Energy dispersion of spin-split surface-state electrons.",
"When plotted with respect to the two components of thetwo-dimensional wave vector k →\\vec{k}, the energy dispersion are cone-shaped.",
"Due to spin-orbit interaction, two spin-split parabolas are obtained and are centered around ±k so \\pm k_{so} (see text).Arrows indicate thevector fields of the spin-quantization axes (or the patterns of the spin) atthe constant energy contour.",
"For every energy, two opposite spins have different wave vectors leading to two concentric circle with opposite spins.",
"The effective B-fieldfelt by the electrons is perpendicular to the propagation direction defined by k →\\vec{k}.The scattering of the surface-state Rashba electrons on a magnetic adatom deposited on a substrate is investigated using scattering theory, which has been successfully applied in the description of electron scattering at adatoms [22], [23], [24].",
"This theory involves the calculation of Green functions, which allow an elegant treatment of the electronic properties including the embedded adatom by solving the Dyson equation, $G= G_0 + G_0tG_0$ , where all quantities are site, spin and energy dependent matrices.",
"We note that the electronic properties of the adatom are inscribed into the scattering matrix $t$ , the amplitude of the electron-wave scattering at the adatom.",
"$G_0$ is the Green function of the pure two-dimensional electron gas corresponding to the Hamiltonian (REF ) that is constructed using the eigenvectors $|\\psi _{1(2)}\\rangle $ : $G_0 =\\begin{bmatrix}G_\\mathrm {D} & e^{-i\\phi } G_\\mathrm {ND}\\\\-e^{i\\phi } G_\\mathrm {ND} & G_\\mathrm {D}\\end{bmatrix} .$ In case the Rashba effect vanishes, the off-diagonal part of the Green function, $G_\\mathrm {ND}$ , is zero and the diagonal part, $G_\\mathrm {D}$ , reverts to the Green function of the free electron surface states.",
"From $G$ , quantities related to those measured by STM, such as the local density of states (LDOS) [25] $n(\\vec{r};E)= -1/\\pi \\, \\mathrm {Im Tr}_{\\sigma } G(\\vec{r},\\vec{r}; E)$ or measured by the spin-polarized STM (SP-STM) as the local magnetization density of states (LMDOS) [26] $\\vec{M}(\\vec{r};E)=-1/\\pi \\,\\mathrm {Im Tr}_{\\sigma } \\vec{\\sigma }G(\\vec{r},\\vec{r}; E)$ can be calculated, where $\\vec{\\sigma }$ is the vector of Pauli matrices and Tr$_{\\sigma }$ is a trace in spin space.",
"It is convenient to express all quantities in cylindrical coordinates as it turns out that they depend only on the radial distance $R$ measured from the position of the adatom.",
"The LMDOS is expressed by two components, $\\vec{M}(R)=(M_z,M_r)$ , the radial component, $M_r$ , and the one normal to the surface, $M_z$ .",
"The azimuthal component, $M_\\phi $ , vanishes.",
"The obtained results are general, but to strengthen our point we consider as an application the system of an Fe adatom on the Au(111) surface, since the system has been proven experimentally accessible by low-temperature STM and for Au the presence of a large Rashba effect has been shown experimentally [7].",
"On such a surface, Jamneala and coworkers [27] found that contrary to Ti, Co, and Ni, no Kondo peak was observed for V, Cr, Mn, and Fe adatoms meaning, that for the latter elements indeed a local magnetic moment exists and it is not screened by the conduction electrons [28].",
"We treat Fe as an adatom with a magnetic moment pointing along the $z$ -direction perpendicular to the surface since our first-principles calculations [29] (see supplementary material) predict this to be the easy axis with a magnetic anisotropy energy of about 12 meV.",
"Hence, such a case imposes a scattering matrix that is diagonal in spin-space, $t &=&\\begin{bmatrix}t_{\\uparrow \\uparrow }& 0 \\\\0& t_{\\downarrow \\downarrow }\\end{bmatrix},$ which can be related to the phase-shift $\\delta (E)$ experienced by the incoming electron waves scattering at the adatom ($t = \\hbar ^2/m^*[\\exp (2i\\delta (E))-1]$ ).",
"Note that a generalization to an arbitrary rotation angle of the magnetic moment will be straightforward using standard unitary transformations.",
"For Fe, as for Au, all majority-spin states are fully occupied and the scattering of majority electrons in the vicinity of the Fermi energy ($E_{\\mathrm {F}}$ ) is practically zero, i.e.",
"the phase-shift vanishes ($t_{\\uparrow \\uparrow }=0$ ).",
"The minority-spin LDOS shows, on the contrary to Au, high values around the Fermi energy and therefore a large scattering.",
"Thus, we assume a phase-shift of $\\pi /2$ ($t_{\\downarrow \\downarrow } = -2\\hbar ^2/m^*$ ).",
"The parameters describing the Au(111) surface state are identical to those chosen by Walls and Heller [17]: $m^* = 0.26\\, m_e$ , $E_F = 0.41$ eV and $\\alpha _{R} = 0.4$ eVÅ that correspond to a Fermi wavelength $\\lambda _{\\mathrm {F}}=\\frac{2\\pi }{k} \\sim 37.4$ Å and a spin rotation length $\\lambda _{\\mathrm {so}}=\\frac{\\pi }{k_{\\mathrm {so}}}$ of 230.5 Å.",
"After solving the Dyson equation, the induced circular LDOS oscillations, $\\Delta n(R;E)$ , emanating from the adatom located at the center ($R=0$ ) is given by $\\Delta n=-1/\\pi \\, {\\mathrm {Im}} [(G_\\mathrm {D}G_\\mathrm {D}+G_\\mathrm {ND}G_\\mathrm {ND})(t_{\\uparrow \\uparrow }+t_{\\downarrow \\downarrow })]$ and is plotted in Fig.",
"REF (a) for all states at $E=E_{\\mathrm {F}}$ .",
"At large distances $R$ , the circular induced standing wave undulations, $\\Delta n(R;E)$ , can be expressed as $\\sim \\frac{m^*}{\\pi ^2\\hbar ^2k^2R}\\sqrt{k_1k_2}\\cos (2kR)$ .",
"When the spin-orbit coupling is negligible, i.e.",
"$k_1 \\sim k_2$ , we recover the conventional form of the adatom induced energy dependent charge density oscillations $\\sim \\frac{\\cos (2kR)}{2kR}$ [23], [22].",
"Regardless of the spin-orbit interaction, there is only one wave vector, $2k= k_1+k_2$ , that describes the oscillations, with the corresponding wavelength at $E_F$ given by $\\lambda _{F}/2 \\sim 18.7$ Å .",
"Consistent to the work of Petersen and Hedegård [16] and confirmed by Walls and Heller [17], no signature or information on the Rashba effect or spin-orbit interaction of the surface atoms can be extracted from the LDOS.",
"This picture changes fundamentally when we look at the LMDOS.",
"Due to the presence of a magnetic adatom ($t_{\\uparrow \\uparrow }\\ne t_{\\downarrow \\downarrow }$ ), the magnetization density perpendicular to the surface and in the surface plane are non-zero.",
"We find concentric rings $M(R)$ of equal size magnetization density surrounding the Fe adatom, with magnetization densities emanated at the center of the Fe atom wrangling in the $(M_z(R),\\, M_r(R))$ plane (Fig.",
"REF (a)).",
"Contrary to the LDOS, the magnetization density is non-trivially modulated by the Rashba effect.",
"Analysing the asymptotic behavior of $M_z(R) \\sim \\frac{m^*}{2\\pi ^2 \\hbar ^2 k^2 R}(k_1 \\cos (2k_1 R) + k_2 \\cos (2k_2 R))$ as calculated from $M_z=-1/\\pi \\, {\\mathrm {Im}} [(G_\\mathrm {D}G_\\mathrm {D}-G_\\mathrm {ND}G_\\mathrm {ND})(t_{\\uparrow \\uparrow }-t_{\\downarrow \\downarrow })]$ ) as well as of $M_r(R) \\sim \\frac{m^*}{2\\pi ^2\\hbar ^2k^2R}(k_1\\sin (2k_1R)-k_2\\sin (2k_2R))$ as calculated from $M_{r}(R) = - 2/\\pi \\, {\\mathrm {Im}} [G_\\mathrm {D}(t_{\\uparrow \\uparrow }-t_{\\downarrow \\downarrow })G_\\mathrm {ND}]$ , one finds that both wave vectors $k_1$ and $k_2$ enter nontrivially.",
"This observation implies the possibility of using a SP-STM with an out-of-plane magnetized tip to probe the $z$ -projected Rashba induced interferences.",
"Any magnetic signal detected by a SP-STM with an in-plane magnetized tip would be a clear fingerprint of the Rashba effect.",
"If there is no Rashba effect, i.e.",
"$G_\\mathrm {ND}=0$ , the quantization axis is only determined by the adatom and the scattering electrons are not a coherent superposition of spin-up and down-electrons that that led to a finite in-plane spin-component.",
"Such spin-textures are a reminder of skyrmionic magnetic configurations but are completely different.",
"Indeed, with skyrmions, the magnetic moments can rotate smoothly from one direction at the center of the structure to the opposite direction.",
"Here, however, the magnetization ripples experience phase-switch as well as beating as sketched in the top-right of Fig.",
"REF .",
"In order to simplify our analysis, the induced LDOS and the $z$ - and radial components of the magnetization passing by the adatom position are additionally illustrated in the top-left of the same figure.",
"Surprisingly, the amplitude of the $M_r$ and $M_z$ oscillations are of the same size with a phase-shift between them that depends on the strength of the Rashba coupling term.",
"This can be better understood when looking at the asymptotic behavior.",
"For instance, $M_z$ can be rewritten in terms of $k_{\\mathrm {so}}$ and $k$ as $\\sim \\frac{m^*}{\\pi ^2 \\hbar ^2k^2 R} (k\\cos (2kR)\\cos (2k_{\\mathrm {so}}R) - k_{so} \\sin (2kR)\\sin (2k_{\\mathrm {so}}R))$ .",
"At the Fermi energy of gold, $k_{\\mathrm {so}}$ is expected to be very small compared to $k$ .",
"At distances much smaller than the spin-rotation length but long enough that the previous asymptotic behavior holds, $M_z$ simplifies more to $\\frac{m^*}{\\pi ^2 \\hbar ^2kR}\\cos (2kR)$ and oscillates similarly to the LDOS with or without spin-orbit interaction.",
"This explains the wavelengths of the induced initial ripples being, as expected, twice the Fermi wavevector.",
"In fact one has to go far beyond the vicinity of the adatom, at distances of the order of the spin-rotation length, to see an effect achieved by the intriguing phase switch observed around 60 Å.",
"Indeed, the new spin-rotation length corresponding to a full rotation from 0o to 180o occurs around 115 Å, which means that the change of sign occurs at half this value.",
"Another substrate with a stronger Rashba coupling parameter would impose a more important phase-shift at smaller distances.",
"$M_{r}$ , which is proportional to $-\\frac{m^*}{\\pi ^2 \\hbar ^2 k^2R}(k\\cos (2kR)\\sin (2k_{\\mathrm {so}}R)+k_{\\mathrm {so}}\\sin {2kR}\\cos (2k_{\\mathrm {so}}R))$ can be further simplified – for rather short distances – to $-\\frac{2m^*k_{\\mathrm {so}}}{\\pi ^2\\hbar ^2k}\\cos (2kR)$ , similar to the cosine behavior that characterizes $M_z$ .",
"Although the spin-texture looks very complex at first sight it can be understood as a linear combination of two skyrmionic waves, $\\vec{M}=\\vec{M}_{k_1}+\\vec{M}_{k_2}$ , with $\\vec{M}_{k_1}(R)\\propto k_1(\\cos (2k_1R),-\\sin (2k_1R))$ and $\\vec{M}_{k_2}(R)\\propto k_2(\\cos (2k_2R),\\sin (2k_2R))$ of opposite vector chirality defined as $\\vec{c}=\\vec{M}(R)\\times \\vec{M}(R+dR)$ .",
"The chirality $\\vec{c}_{k_2}= \\sin (2k_2dR)\\hat{e}_\\phi $ points in the $\\phi $ -direction and forms with the directions $\\hat{e}_z$ and $\\hat{e}_r$ a right handed coordinate system, while $\\vec{c}_{k_1}= -\\sin (2k_1dR)\\hat{e}_\\phi $ points in the $-\\phi $ -direction and forms with $\\hat{e}_z$ and $\\hat{e}_r$ a left-handed coordinate system.",
"Accordingly, we call $\\vec{M}_{k_1}$ and $\\vec{M}_{k_2}$ a left- and right-rotating skyrmionic wave, respectively.",
"Our definition of a skyrmion in the actual work designates well-defined magnetic waves which are (i) centered around the adatoms, and have (ii) fixed rotation sense [1].",
"One great virtue of skyrmions is their topological nature that is protected under the presence of reasonable size magnetic fields.",
"This multi-skyrmionic magnetic waves can be destroyed or manipulated by modifying the spin-orbit interaction.",
"By switching it off, only the magnetization along the $z$ -direction would be finite.",
"One way to tune the Rashba effect is to change the substrate nature gradually [30], [31].",
"Such an experiment is very difficult to realize but we propose another tuning method: Instead of the Fermi energy, one could probe at different energy varying the bias voltage between tip and sample.",
"For example, by decreasing the value of the energy probed, the wavelength corresponding to $2k$ decreases which is accompanied by a decrease in the number of nodes.",
"This is observable in the two additional examples depicted in Fig.",
"REF calculated at 140 meV and 20 meV.",
"The final multi-skyrmionic waves have a different texture but share the common feature in the phase switch at 60 Å.",
"One should bear in mind that, as for the LDOS, two different regimes have to be expected for the magnetization components depending on the sign of $k_2$ (see supplementary material).",
"This suggests that the scanning tunneling spectroscopy (STS) experiment of Ast et al.",
"[14], but spin-polarized, would be also a way to verify our predictions.",
"To summarize, we have revealed a new type of spin-texture (see Fig.",
"REF ) induced in a confined electron gas subject to the Rashba effect by a magnetic atom.",
"These structures can be understood as a kind of combination of skyrmionic-like waves of opposite chirality.",
"By tuning the energy level or the spin-orbit strength, the observation of the final spin-texture is possible because of the large magnitude of the induced magnetization and the creation and exploration of new complex configurations that manageable with state of the art experiments.",
"Such non-trivial magnetic Friedel oscillations have an impact on the magnetic interactions between adatoms and other nanostructures and consequently on their magnetic behavior.",
"Simple asymptotic expressions for the induced magnetizations are derived which offer a simple understanding of new types of experiments involving the manipulation and construction of adatom based spin-nanostructures.",
"Figure: Comparison of the induced LDOS and LMDOS components.Due to the Rashba effect the Fe magnetic adatom induces non-trivial spin interferences: onthe left panels are shown the radial dependence of the LDOS and LMDOS at different energies while on the right panelsthe corresponding magnetization unit vectors are depicted.Figure: Non-collinear spin-configuration of the electron-gas.Spectroscopic skyrmionic-like spin-textures of theelectron-magnetization surrounding the adatoms atdifferent energies (410 meV, 140 meV and 20 meV).",
"The number of nodesdiminishes when decreasing the energy leading to smoother rotatingspherical magnetic waves.We thank Prof. K. Scharnberg for stimulating discussions at the initial stage of the work, and Drs. P.",
"Lazic and H. Schumacher for their assistance in preparing some of the figures.",
"SL acknowledges the support of the HGF-YIG Programme VH-NG-717 (Functional nanoscale structure and probe simulation laboratory, Funsilab)."
]
] | 1204.0999 |
[
[
"On the Change of the Inner Boundary of an Optically Thick Accretion Disk\n around White Dwarfs Using the Dwarf Nova SS Cyg as an Example"
],
[
"Abstract We present the results of our studies of the aperiodic optical flux variability for SS Cyg, an accreting binary systemwith a white dwarf.",
"The main set of observational data presented here was obtained with the ANDOR/iXon DU-888 photometer mounted on the RTT-150 telescope, which allowed a record(for CCD photometers) time resolution up to 8 ms to be achieved.",
"The power spectra of the source's flux variability have revealed that the aperiodic variability contains information about the inner boundary of the optically thick flow in the binary system.",
"We show that the inner boundary of the optically thick accretion disk comes close to the white dwarf surface at the maximum of the source's bolometric light curve, i.e., at the peak of the instantaneous accretion rate onto the white dwarf, while the optically thick accretion disk is truncated at distances 8.5e9 cm ~10 R_{WD} in the low state.",
"We suggest that the location of the inner boundary of the accretion disk in the binary can be traced by studying the parameters of the power spectra for accreting white dwarfs.",
"In particular, this allows the mass of the accreting object to be estimated."
],
[
"INTRODUCTION",
"Compact objects in binary systems (white dwarfs, neutron stars, black holes) form accretion disks around themselves from the matter coming into the region of dominance of their gravity (see, e.g., Frank et al.",
"2002).",
"Moving toward the compact object, the matter in the accretion disk releases gravitational energy and heats up to high temperatures, radiating in the optical, ultraviolet, and X-ray bands (see, e.g., [38]).",
"It has long been pointed out that the accretion flow around compact objects can be inhomogeneous and consist of several regions with significantly differing physical properties, for example, optically thick and optically thin/corona regions [39], [15], [33], [10].",
"The so-called dwarf novae, accreting white dwarfs in binary systems with low-mass Roche-lobe-filling companion stars, are among the best-studied systems with accretion disks.",
"An ever-increasing set of measurements exists for them, suggesting that the accretion disk in some states of the binary system is not optically thick over its entire extent [32], [22], [16], [12], [3], [20].",
"In such states, the optically thick accretion disk ends at a certain distance from the white dwarf within which the flow becomes optically thin.",
"Both the interaction with the white dwarf magnetosphere, if the white dwarf has a sufficiently strong magnetic field [31], [3], and gradual evaporation of the accretion disk [26] can be responsible for this transition.",
"However, the current status of the accretion disk models does not allow the radius at which the optically thick flow evaporates to be predicted with confidence (see, e.g., [27]).",
"Thus, the question about the truncation (evaporation) of the inner parts of the accretion disks has not been ultimately resolved.",
"An additional reliable measurement of the inner radii for the accretion disks of accreting systems in various states would be an important confirmation of the validity of the existing model for the accretion flow in binary systems.",
"Recently, it has been shown that the inner boundary of the optically thick disk can be estimated by analysing the power spectrum of aperiodic noise in the accreting source [34], [35].",
"In this paper, we applied this method to determine the radius at which the accretion disk in SS Cyg, a binary system with an accreting white dwarf, is truncated."
],
[
"APERIODIC NOISE OF ACCRETING\nSOURCES FOR DIAGNOSING THE GEOMETRY OF ACCRETION FLOWS",
"It has been known almost since the discovery of accreting binary systems that, as a rule, they exhibit aperiodic flux variability in a wide range of time scales–flickering (see, e.g., [19], [6]).",
"As recent studies have shown, a modulation of the instantaneous accretion rate at different distances from the central object and their subsequent multiplicative addition (the so-called model of propagating fluctuations [24], [9]) are the most probable mechanism of flickering in the light curves of accreting systems.",
"In this model, a variable emission is generated in the central regions of the accretion flow (which is confirmed, for example, by the curves of the variability amplitude in eclipsing systems (see [7], [2]) for accreting white dwarfs and is shown by the fast (up to time scales of the order of tens of milliseconds) variability of the large fraction of their X-ray flux for accreting neutron stars and black holes).",
"However, the accretion flow itself is modulated at various (including) large distances from the compact object as a result of the stochastic nature of viscosity in accretion disks (see, e.g., [1], [5], [14]), with the variations on shorter time scales that emerge in the inner disk regions modulating the accretion rate of the matter coming into this region from the outer regions.",
"In this model, the emerging broadband power spectrum must have features in the range of frequencies corresponding to the characteristic time scales at the accretion disk edges.",
"In particular, the truncation of the accretion disk in its central part in the model of propagating fluctuations must lead to a break in the variability power spectrum: the source's flux variability must be suppressed at higher frequencies relative to the situation where the accretion disk is not truncated.",
"As the inner disk boundary moves toward the compact object, the break frequency must, accordingly, increase.",
"An observational confirmation of this prediction was given previously [34].",
"In particular, it was shown that accreting magnetized neutron stars (X-ray pulsars) whose radius of the magnetosphere (which determines the inner boundary of the optically thick accretion disk) depends on the current accretion rate exhibit a change of the break frequency in the variability power spectrum during bursts of activity, i.e., in periods of a significant increase in the accretion rate.",
"The dependence of the break frequency on the object's X-ray luminosity (accretion rate) agrees well with the theory of a dipole neutron star magnetosphere compressed by a Keplerian accretion disk.",
"If a similar situation with a change in the accretion flow geometry occurs in accreting white dwarfs, then one might expect a similar change in the properties of their aperiodic variability.",
"However, the number of white dwarfs with a significant magnetic field (i.e., with a field that is capable of destroying the accretion disk at great distances from the white dwarf surface) exhibiting bursts of the accretion rate is very small, the bursts are rare and irregular, and, consequently, it is very difficult to observe such systems in the low and high states.",
"Such periods of an increase in the instantaneous accretion rate are much more often observed for systems with a weak magnetic field.",
"As the object most suitable for testing this prediction, we chose the binary system SS Cyg – a well known dwarf nova, i.e., a binary system in which the periods of a low accretion rate onto the compact object are replaced by outbursts, the periods of active accretion (see, e.g., the review by [29]).",
"It follows from the existing model of accretion disks around white dwarfs (see, e.g., [16], [18]) that, despite the absence of a significant magnetic field on the white dwarf in the low state, the optically thick accretion disk in this binary system still ends at a considerable distance from the white dwarf ($>5-10R_{WD}$ ), but not through the interaction with the stellar magnetosphere but through its evaporation and transformation into an optically thin flow [26], [21], [27].",
"In the periods of peak bolometric luminosity, when the current accretion rate onto the white dwarf increases considerably, the optically thick accretion disk comes close to the white dwarf.",
"Since the X-ray flux from this system is fairly low (several mCrab, i.e., only a few photons per second per 1000 cm), it is very difficult to obtain a high quality power spectrum in the required frequency range (up to 1-5 Hz) from X-ray light curves.",
"To measure the properties of the object's aperiodic variability, we used its optical observations.",
"Although the optically emitting regions are considerably larger in size than the X-ray-emitting ones, their sizes nevertheless do not exceed $\\sim 10^{10}$ cm and, hence, the smoothing of the variability due to the finite time it takes for light to traverse the emitting region plays no major role up to frequencies of the order of several Hz.",
"It is in this frequency range that we carried out the studies presented here."
],
[
"Influence of the Atmosphere on Photometric\nMeasurements",
"The flux variations in the photometric series obtained with ground-based telescopes are determined by a number of factors.",
"First of all, these include the object's Poissonian photon counting noise.",
"However, the influence of this noise on the power spectrum of the source's flux variability is fairly easy to take into account by subtracting the constant level ($P(f)\\propto const$ ) produced by this noise from it.",
"In the case of the CCD array used in our measurements (EM CCD), additional noise of the photometric signal is produced by the electronic amplification cascade in the reception channel.",
"This is because the signal amplification itself in the CCD array becomes noisy due to the Poissonian variations of the amplification cascade electrons (multiplicative noise).",
"However, this amplification is a random variable whose values are uncorrelated in neighbouring exposures.",
"Therefore, it actually leads to a slightly changed value of the constant level in the derived power spectra.",
"The most significant factor that makes it difficult to measure the aperiodic variability parameters for sources is the influence of turbulence in the atmosphere.",
"Because of this turbulence, chaotic variations of the refractive index always exist in the atmosphere.",
"This, in turn, leads to changes in the shape of the stellar image at the telescope's focus, their jitter, and scintillations.",
"To give an idea of what additional variability is produced by the atmosphere above the RTT-150 telescope, we present the variability power spectrum for a nonvariable star taken on September 2, 2010, in Fig.",
"1.",
"Unfortunately, the properties of the atmosphere producing this additional (with respect to the Poissonian one) noise is not strictly fixed during the night, i.e., a correction for the atmosphere can be made by analysing the variability power spectrum for the reference star photographed at a slightly different time than that for the source being investigated only with a low accuracy.",
"Differential photometry is a popular method for combatting the influence of these factors.",
"In this method, the flux from the star being investigated is measured not directly but in comparison with one or more nonvariable field stars.",
"Using the ratio of the fluxes from the investigated and (nearby) reference stars allows the influence of the changing atmosphere to be taken into account, at least to a first approximation (see, e.g., the power spectra of the photometric series obtained with the same RTT-150 telescope; [8]).",
"However, a necessary condition for this method to be efficient is the requirement that the field stars be at least brighter than the investigated star.",
"Otherwise, the statistical uncertainty in measuring the brightness of the investigated star will be determined not by the star itself but by the noise of the comparison star.",
"The optical brightness of SS Cyg in the low state is about 12 magnitudes.",
"This allows USNO B.1 1335-0436095 at a distance of 2.06 arcmin from SS Cyg with a brightness $R\\sim 11$ to be used as a reference star.",
"However, during outbursts (i.e., in the periods of a high accretion rate in the binary system), the system's brightness increases to $R\\sim 8.5$ , which makes the use of USNO B.1 1335–0436095 as a reference star nonoptimal.",
"In this case, the star BD+42 4190 at a distance of $\\sim $ 4.7 arcmin with a brightness $R\\sim 8.0$ should be used.",
"Here, we used both the first and second reference stars for the application of differential photometry."
],
[
"RTT-150 Observations",
"The observations of SS Cyg during its outburst were performed with the Russian-Turkish 1.5-m telescope RTT-150 using a CCD photometer mounted at the Cassegrain focus f = 1/7.7.",
"A log of observations is presented in Table 1.",
"The photometric measurements were made with the ANDOR iXon DU-888 CCD array.",
"The iXon DU-888 back-illuminated CCD approximately 4x4 arcmin in size is divided into 1024x1024 pixels.",
"The CCD is equipped with electronic multiplication (EMCCD), which allows one to reduce considerably the readout noise effect at very short exposure times and, consequently, to use it for measuring the brightness of faint objects.",
"TheCCDis cooled electronically to a temperature of -60C.",
"The entire field with the readout of all CCD pixels (1024x1024) can be imaged eight times per second; when reducing the readout region and binning the readout rows, the exposure time can be reduced to $\\sim $ 1-3 ms. Table: Observations of SS Cyg on telescope RTT150 in 2010-2011, used in our workIn our case, we used two approaches: When using USNO B.1 1335-0436095 at a distance of 2.06 arcmin from SS Cyg as a reference star, we recorded the image of the sky around SS Cyg with a width of about 14 arcsec (see Fig.",
"2) summed in the vertical direction into a one-dimensional strip.",
"This allowed us to to obtain images with a frequency of 123.46 Hz; the length of one exposure in our observations is 1.36 ms.",
"The photometric measurements were made in a one-dimensional strip with a fixed center and the aperture width determined from the summed (over the interval of observations) one-dimensional brightness profile on the CCD.",
"The CCD background illumination outside bright sources was approximated by a linear function at distances up to 100 pixels from the sources.",
"As the aperture width, we took a value that was a factor of 8 larger than the full width at half maximum of the stellar profile.",
"In our case, the choice of such aperture (with a fixed value of a wide aperture) stellar photometry is related primarily to the fact that we needed to gather the stellar flux as completely as possible to reduce the parasitic noise level of the atmosphere.",
"Reducing the aperture width or using the aperture width determined for each specific frame, despite the fact that the shape of the stellar image on the detector changes in a chaotic way unknown to us – the image centroid moves over the detector, the fraction of the flux in the wings of the stellar image changes under the atmospheric effect, will cause the stellar flux variability amplitude to increase through atmospheric jitter.",
"Figure: Region of the sky around SS Cyg observed with the high-speed ANDOR iXon DU-888 CCD during its high state.The comparison star (USNO B1 1335–0436095, mR = 11.0) is located in the left part of the image; SS Cyg is located in itsright part.",
"The image with the original CCD resolution is shown at the top; the one-dimensional profile obtained by the imageaddition along the vertical axis is shown at the bottom.",
"The scale of the plot along the Y axis indicates the number of countsrecorded at the points of the one-dimensional profile in 100 s. We see that the brightness of the reference star is several timeslower than that of the investigated one.",
"In 2010, we also used the method described above during our observations of SS Cyg in the high state, which is nonoptimal from the standpoint of maximizing the signal-to-noise ratio for the source due to the comparative faintness of the reference star.",
"To maximize the signal-to-noise ratio in our photometric measurements of SS Cyg in the July 2011 observation of the high state, we changed the method and used BD+42 4190 as a reference star.",
"Since the CCD size is about 4 arcmin, in this case, we cannot use the technique described above.",
"In this case, we placed the reference star and the investigated source SS Cyg along the CCD diagonal and read out the entire CCD field by binning its pixels in 2x2 (see Fig.",
"3).",
"In such a case, the time resolution was 62.5 ms.",
"The photometric measurements were made with a fixed (circular) aperture.",
"The count measurements in adjacent circles were used as the CCD background illumination measurements.",
"Figure: Map of the sky region around SS Cyg duringits outburst in July 2011 with the reference starBD+42 4190.",
"The image orientation corresponds to theCCD orientation during our observation."
],
[
"The Low State",
"First of all, we checked whether the variability power spectrum for SS Cyg in quiescence was consistent with the predictions of the model of propagating fluctuations in an accretion disk with an inner boundary, i.e., whether there was a break in the power spectrum similar to that observed in accreting systems with disks truncated by the magnetospheres of compact objects [34], [35].",
"For this purpose, we used both RTT-150 data (see the table) and moderate-time-resolution (about 10–30 s) observations with telescopes at the Crimean Astrophysical Observatory (CrAO) (see [42]).",
"Figure: Power spectrum of the optical flux variability for SS Cyg in the low state obtained from the RTT-150 observations in2010 (see the text) and from the CrAO observations .",
"The dashed curve indicates the model fit to thepower spectrum by a function of the form f×P(f)∝(1+(f/f 0 ) 4 ) -0.25 f\\times P(f)\\propto (1+(f/f_0)^4)^{-0.25}, where f 0 =2.1×10 -3 f_0=2.1\\times 10^{-3} Hz.The power spectrum of the optical flux variability for SS Cyg obtained in these observations is presented in Fig.",
"4.",
"Since the source's flux variability is undetectable at high frequencies, Fig.",
"4 does not show the frequency range above $\\sim $ 0.1 Hz.",
"We clearly see that the variability power spectrum has two characteristic regions – the region where the variability power behaves approximately as $P\\propto f^{-1}$ (the flat part in Fig.",
"4) and the region where the variability decreases with frequency as $P\\propto f^{-2}$ , similar to what we observed for intermediate polars (see Revnivtsev et al.",
"2010).",
"Note that a similar change in the pattern of behavior of the power spectra for accreting nonmagnetic white dwarfs was demonstrated previously by Kraicheva et al.",
"(1999) and Pandel et al.",
"(2003).",
"Figure: Lightcurves of SS Cyg according to data of association of astronomers AAVSO (www.aavso.org) during periods Aug.-Sept. 2010 and June-July 2011.",
"Wide black stripes denote the dates of RTT150 observations.If we fit the power spectrum for SS Cyg by the analytical model $f\\times P(f) \\propto [1 + (f/f_0)^4]^{-0.25}$ used in [34], then the break frequency will be $(2.1 \\pm 0.5) \\times 10^{-3}$ Hz.",
"For a white dwarf with a mass of $0.81M_\\odot $ [4], this corresponds to the Keplerian rotation frequency of the matter at a distance of $(8.5 \\pm 1.4) \\times 10^9$ cm.",
"The inner disk radius estimated from the break frequency in the power spectrum agrees satisfactorily with the estimates based on other physical effects.",
"In particular, the inner radius of the optically thick disk in the intermediate polar EX Hya estimated from the break frequency in the power spectrum of its flux variability, $1.9 \\times 10^{9}$ cm (Revnivtsev et al.",
"2011), agrees, to within about 30%, with the results of the measurements made by analyzing emission line profiles, $\\sim 1-2 \\times 10^9$ cm (Hellier et al.",
"1987), and analyzing eclipses at different phases of the white dwarf pulsations, $\\sim 1.5 \\times 10^9$ cm (Siegel et al.",
"1989).",
"Interestingly, the inner boundary of the optically thick disk determined in this way from the source?s variability power spectrum agrees qualitatively with the predictions of the model of evaporating accretion disks around nonmagnetic white dwarfs (see, e.g., Liu et al.",
"1997)."
],
[
"The High State",
"In the high state of SS Cyg, we made its photometric observations in two outbursts.",
"We managed to obtain photometric measurements at the very peak of the system's optical brightness in the September 2010 outburst (see Fig.",
"5a) and at the optical brightness decline in the July 2011 outburst (see Fig.",
"5b).",
"In the September 2010 outburst, we made our measurements using USNO B.1 1335-0436095 as a reference star.",
"As a result, the signal-to-noise ratio of the photometric series obtained by the method of differential photometry is much lower than the best one (the brightness of the reference star was lower than that of SS Cyg by a factor of 60-70).",
"For this reason, the photometric series is essentially insensitive to the possible residual atmospheric jitter effects on the variability.",
"Consequently, the power spectrum of the signal from SS Cyg at frequencies above approximately 0.5-1 Hz reaches a constant that represents the statistical measurement errors (see also the power spectra of nonvariable stars obtained by the method of differential photometry with RTT 150 by [8]).",
"Figure: Power spectra of the optical flux variability for SS Cyg in periods with different accretion rates in the low state (a),in the period of an intermediate accretion rate (the July 11, 2011 observation) (b), and in the period of peak accretion rate(the September 2, 2010 observation) (c).",
"The dashed curve indicates the model used to describe the power spectrum in thelow state (see Fig.",
"4).",
"The dotted curve indicates a similar model in which 0.091 Hz was substituted as the break frequency;this corresponds to the Keplerian rotation frequency of matter in a circular orbit near the surface of a white dwarf with massM=0.81M ⊙ M = 0.81M_\\odot (the white dwarf radius was calculated using a formula from Nauenberg (1972).During the July 2011 outburst, we used the brighter star BD+42 4190.",
"This allowed us (1) to obtain a higher-quality variability power spectrum for the objects and (2) to detect the residual atmospheric jitter effects.",
"Figure 6 presents the power spectrum for SS Cyg in the July 11, 2011 observation obtained by the method of differential photometry.",
"We clearly see the flux variability with a shape of the power spectrum resembling that of the atmospheric variability power spectrum at frequencies above approximately 0.05 Hz (see Fig.",
"1).",
"Thus, we may conclude that the method of differential photometry allows us to take into account the atmospheric effect up to a level of about 0.3% of the source's flux (the exact level of the residual atmospheric jitter effect depends on the conditions in each specific observation and on the distance between the object and the reference star).",
"We took into account the influence of this residual variability by subtracting the components in a power spectrum of the form $P \\propto A(1 + (f/f_1)^2)^\\beta $ .",
"In the observation under consideration, $A = 5.52 \\times 10^{-5}$ , $f1 = 0.557$ , and $\\beta = -0.192$ .",
"The described model is indicated in Fig.",
"6 by the dashed curve.",
"The power spectra of the flux variability for SS Cyg in the light curves obtained by the method of differential photometry during different stages of its outburst state are presented in Fig.",
"7.",
"It should be noted that in the September 2, 2011 observations, apart from aperiodic noise (see Fig.",
"8), we detected a transient appearance of the so-called dwarf nova oscillations (DNOs) commonly observed in SS Cyg (see, e.g., [25]).",
"For example, in the time interval MJD 55441.7644– 55441.9581, we observed oscillations with a period of 6.735 s and a relative amplitude of $3.2\\times 10^{-4}$ .",
"However, their presence in no way affects our results on the investigation of aperiodic variability, because the amplitude of these oscillations is more than an order of magnitude lower than that of the observed aperiodic variations."
],
[
"CHANGES OF THE INNER DISK BOUNDARY",
"The pattern of change in the aperiodic flux variability of SS Cyg is clearly seen from Fig.",
"7.",
"As the brightness (accretion rate) of the source rose, the fraction of the fast variability clearly increased at frequencies up to 0.01 Hz in the July 11, 2011 observation (i.e., during the source's intermediate brightness) and up to 0.1 Hz in the observations early in September 2010 (i.e., during the source's peak optical brightness).",
"It is in this frequency range that we should expect the appearance of an additional noise component due to the appearance of parts of the accretion disk near the white dwarf surface: the rotation frequency of matter in a Keplerian orbit near the surface of a white dwarf with mass $M = 0.81M_\\odot $ is about 0.091 Hz.",
"A similar appearance of the high-frequency component in the source's flux noise was observed in magnetic systems (in particular, accreting magnetized neutron stars) in which the inner radius of the accretion disk changed during the period of activity because of the change in the balance of matter pressures in the disk and the neutron star magnetosphere (see Revnivtsev et al.",
"2009).",
"Thus, we may conclude that the results of our observations show that the accretion flow around a nonmagnetic white dwarf (in our case, the white dwarf in the binary system SS Cyg) is clearly divided into two regions the location of the boundary between which depends on the current accretion rate in the inner part of the accretion flow (where the main energy release takes place).",
"In the model of an evaporating accretion disk [26], this boundary separates the regions of optically thick and optically thin flows.",
"Interestingly, a distinct feature like a quasiperiodic oscillation (QPO)with a low Q ($Q = f/\\Delta f \\sim 1$ ) is observed near the break in the variability power spectrum for SS Cyg.",
"The appearance of QPO near the break in the variability power spectrum is not unique to the object considered but is a rather common phenomenon in systems in which matter is accreted through two regions with different physical properties.",
"For example, in magnetized neutron stars, in whichmatter fromthe optically thick disk penetrates into the magnetosphere (see, e.g., [34]); in accreting black holes in a hard spectral state, in which matter from the optically thick disk transforms into an optically thin coronal flow [43].",
"However, in the mentioned cases, the QPO Q factor is, as a rule, higher.",
"It can be assumed that the appearance of QPO in the case of SS Cyg is associated with instabilities whenmatter penetrates fromthe optically thick accretion disk into the coronal flow.",
"The detection of a break in the variability power spectrum for nonmagnetic white dwarfs opens up possibilities for an independent estimation of their masses.",
"Indeed, the inner radius of the accretion disk, which must exceed the white dwarf radius, can be estimated by measuring the break frequency in the object's variability power spectrum at the peak of its optical-ultraviolet brightness.",
"Next, given the equation of state for the white dwarf (i.e., its mass– radius relation), we can estimate its mass.",
"Unfortunately, in our case, the statistical quality of the observational data at the peak accretion rate onto the white dwarf (September 2010) is too low to independently estimate the white dwarf mass by this method.",
"New, higher-quality observations are needed for this purpose."
],
[
"CONCLUSIONS",
"We analysed the pattern of aperiodic optical flux variability for the accreting nonmagnetic white dwarf SS Cyg.",
"This work is a pilot project whose goal is to test the hypothesis that the accretion disk around the nonmagnetic white dwarf in SS Cyg is truncated at a certain distance from the white dwarf in the low state and approaches the white dwarf in the high state.",
"For this purpose, we carried out special observations of the source in the low and high states with a record (for CCD photometers) time resolution up to 123 Hz.",
"We showed the following.",
"In the low state, the power spectrum of the optical flux variability for SS Cyg is similar to the variability power spectrum for intermediate polars with a break in the range of frequencies $\\sim 2\\times 10^{-3}$ Hz.",
"This suggests that the optically thick accretion disk in SS Cyg is truncated/evaporated (Meyer and Meyer- Hofmeister 1994; Liu et al.",
"1997) at a distance of about $(8.5 \\pm 1.4) \\times 10^9$ cm, or $R \\sim 10R_{WD}$ .",
"In the high state, the flux variability amplitude for SS Cyg in the frequency range $10^{-3}-1$ Hz is lower than that in the low one, 1.3%.",
"However, it contains a much larger fraction of fast variability, up to frequencies of $\\sim 0.1$ Hz that roughly correspond to the rotation frequency of matter in a Keplerian orbit near the white dwarf surface.",
"In the state with the source's maximum optical brightness, the break frequency is maximal, about 0.1 Hz; in the state with an intermediate brightness, the break frequency is about 0.01 Hz.",
"To further study the behaviour of the inner boundary of the optically thick accretion disk at various stages of the burst of the accretion rate, we need to carry out additional observations with a bright reference star in the instrument's field of view and to try to compare more quantitatively the inner boundary of the accretion disk for nonmagnetic white dwarfs with the predictions of various theoretical models.",
"We thank the Excellence Cluster Universe of the Technische Universität München for the opportunity to work with the ANDOR/iXon CCD array at the RTT-150 telescope.",
"M.G.",
"Revnivtsev expresses special gratitude to Andreas Müller for his great help in acquiring the ANDOR/iXon CCD array.",
"We used the source's light curves measured by the American Association of Variable Star Observers (AAVSO).",
"M.G.",
"Revnivtsev is grateful to I.B.",
"Voloshina who provided the light curves of SS Cyg in the low state.",
"We thank the TÜBÏTAK National Observatory (TUG, Turkey), the Space Research Institute of the Russian Academy of Sciences, and the Kazan State University for support in using the Russian–Turkish 1.5-m telescope (RTT-150).",
"This work was supported by the Russian Foundation for Basic Research (project nos.",
"07-02-01004, 08-02-00974, 09-02-12384-ofi-m, 10-02-01442, 10-02-01145, 10-02-00492, 10-02-91223-ST-a), the Program for Support of Leading Scientific Schools of the Russian Federation (NSh-5069.2010.2), the Programs of the Russian Academy of Sciences P-19 and OPhN- 16, The ”Dynasty” Foundation for Noncommercial Programs, and the TÜBÏTAK Programs 209T055 and 10BRTT150-25-0.",
"Translated by V. Astakhov"
]
] | 1204.0902 |
[
[
"Distortion of the HBT image by mean field interaction"
],
[
"Abstract We study effects of a mean-field interaction on the spacetime geometry of the hadron source measured by utilizing the Hanbury Brown and Twiss (HBT) interferometry in the ultrarelativistic heavy-ion collisions.",
"We show how a modification of a pion amplitude, caused in the freeze-out process, is incorporated into the correlation function of the interferometry within a semiclassical method.",
"Profiles of the distorted images are illustrated.",
"To make a quantitative estimate of the effects, we construct a mean-field-interaction model on the basis of the pion-pion scattering amplitude, and then investigate to what extent the effects of the mean-field interaction acts to efficiently modify the HBT radii."
],
[
"Introduction",
"In prior to the RHIC experiments, the dynamics of the collective expansion was simulated with hydrodynamical models assuming a formation of the quark-gluon plasma.",
"Their results indicated a prolonged lifetime of the matter, and a considerable difference in the transverse HBT radii, $R_{\\rm out}/R_{\\rm side}\\simeq 2$ , which has been supported by sophisticated models[1].",
"While these models have precisely reproduced the single-particle spectra, especially the elliptic flow, of the RHIC data, it was unexpectedly found that there were systematic discrepancies between the theoretically predicted and experimentally measured HBT radii.",
"It has been known as the “RHIC HBT puzzle\".",
"This contribution is based on the works [4], [5] originally motivated by solving this puzzle.",
"We investigate effects of a mean-field interaction, sketched in Fig.",
"REF , as a possible origin of the discrepancies.",
"Effects of mean-field interactions have been examined independently in Ref.",
"[2] and [3] to the end of solving the puzzle.",
"We thank H. Fujii for calling our attention to those works.. We have shown how the modification of a pion amplitude due to a mean-field interaction reflects in a distorted HBT image[4], within a semiclassical framework.",
"We also found a possible improvement of the discrepancies, owing to cooperative effects of attractive and absorptive mean-field interactions.",
"To pursue this possibility, we have constructed a more realistic mean-field interaction on the basis of the pion-pion forward scattering amplitudes[5].",
"There, we actually obtained attractive and absorptive mean-field interactions following from a strong attraction and attenuation around the momentum regime of the $\\rho $ -meson resonance.",
"Based on those frameworks, we showed how and to what extent the mean-field interaction efficiently modify the HBT radii."
],
[
"Modification of the correlation function",
"We show how the effects of the mean-field interaction, encoded in the phase and attenuation factors of the amplitude, reflect in the interference pattern.",
"By using a semiclassical method to evaluate the amplitude, a modified correlation function has been obtained in a time-independent case[4], and then the framework has been extended to a time-dependent case to take into account the expansion dynamics of the hadron source[5].",
"The correlation function $C$ in the HBT interferometry is defined by $C(\\mathbf {k}_1,\\mathbf {k}_2) = \\frac{ P_2(\\mathbf {k}_1,\\mathbf {k}_2) }{ P_1(\\mathbf {k_1}) P_1(\\mathbf {k_2}) } \\ ,$ where $\\mathbf {k}_1$ and $\\mathbf {k}_2$ are the momenta of a particle pair.",
"$P_1(\\mathbf {k})$ and $P_2(\\mathbf {k}_1,\\mathbf {k}_2)$ are probabilities of detecting a particle and identical particle pair, respectively.",
"Without any interaction after the emission, the correlation function is simply given by the Fourier transform of the source function $S(x, {\\mathbf {k}}) $ as, $C({\\mathbf {k}}, {\\mathbf {q}}) = 1 + \\eta ^2({\\mathbf {k}})\\left| \\int \\!\\!d^4x \\ S(x, {\\mathbf {k}}) \\ e^{i q x } \\right|^2 \\ .$ The correlation function is measured as a function of the average and relative momenta, ${\\mathbf {k}}$ and ${\\mathbf {q}}$ .",
"The product of the single-particle spectra appears as a normalization factor, $\\eta ^{-2}({\\mathbf {k}}) \\simeq P_1^2({\\mathbf {k}}) = \\left[ \\ \\int \\!\\!d^4x \\ S(x, {\\mathbf {k}}) \\ \\right] ^2\\ .$ In this expression, we have approximated the momenta as $\\mathbf {k}_1 \\simeq \\mathbf {k}_2 \\simeq \\mathbf {k}$ for the small relative momentum $( |{\\mathbf {q}}| \\ll |{\\mathbf {k}}|)$ , which is a relevant regime for the HBT interferometry.",
"Once we incorporate effects of the mean-field interaction, they distort the kernel of the Fourier transform in Eq.",
"(REF ).",
"In terms of the classical action of a pion propagating in the mean field, a modified correlation function is obtained as, $C ({\\mathbf {k}}, {\\mathbf {q}}) = 1 + \\eta ^{2} ({\\mathbf {k}})\\left|\\int \\!\\!d^4x \\ S(x, {\\mathbf {p}}_0 (x,{\\mathbf {k}}) ) \\ e^{-2\\gamma (x,{\\mathbf {k}})} \\ e^{ i q_{\\mu } \\left( x^{\\mu } + \\partial _k ^{\\mu } \\delta \\!",
"S^{c\\ell }_{{\\mathbf {k}}} (x) \\right) }\\right| ^2\\ , $ where $x^{\\mu }=(t,{\\mathbf {x}})$ is the emission point, and the derivative $\\partial _k^\\mu $ operates on the four momentum $k^\\mu $ .",
"The mean-field interaction acts to accelerate the pion emitted with an initial momentum ${\\mathbf {p}}_0 (x,{\\mathbf {k}})$ so that it becomes the asymptotic one ${\\mathbf {k}}$ .",
"An attenuation factor is given by the coefficient $\\gamma (x,^bk)$ which is related to an imaginary part of pion self-energy (see Sec.)",
"We extended the semiclassical framework to take into account the relativity in an appendix in Ref.",
"[5]..",
"The deviation of the classical action from that of the free propagation, $\\delta \\!",
"S^{c\\ell }_{{\\mathbf {k}}}= S^{c\\ell }_{{\\mathbf {k}}} - k^\\mu x_\\mu $ , appears as the phase distortion.",
"Reflecting the acceleration and attenuation in the mean field, the single-particle spectrum is also modified as, $\\eta ^{-2} ({\\mathbf {k}}) = \\left[ \\int \\!\\!d^4x \\ S(x, {\\mathbf {p}}_0 (x,{\\mathbf {k}}) ) \\ e^{-2\\gamma (x,{\\mathbf {k}})} \\right] ^2 $ .",
"In Fig.",
"REF , we illustrate the distorted HBT images due to the mean-field interactions (see Ref.",
"[4] for details).",
"A repulsive mean-field interaction induces an image elongated along the outward axis, whereas an attractive one induces a stretched image with a longer extension along the sideward axis.",
"Effects of an attenuation also acts to stretch the sideward extension, because it cuts off a propagation of the pion emitted on the opposite side to the detection point.",
"Figure: Phenomenological pion self-energy: a propagating pion is scattered by other evaporating pions represented with a loop.Blob indicates the forward scattering amplitude shown in Fig.",
".Figure: Distorted HBT images at 𝐤 ⊥ =100{\\mathbf {k}}_\\perp = 100 MeV:contour plots show how an isotropic Gaussian profile, with its variance σ=5 fm \\sigma = 5 \\ {\\rm fm}, is apparently distorted due to effects of repulsive and attractive interactions,and cooperative effects of attenuation, as indicated above."
],
[
"Phenomenological mean-field interaction",
"In the isospin-symmetric limit, an in-medium pion self-energy depicted in Fig.",
"REF is written as, (t, x, p) = - d4p(2)4 2(p2-m2) T(s) f(t, x,p) pf(t, x,p) = S(t, x,p) with the isospin averaged scattering amplitude $T(s)$ and distribution of the medium pion $f(t, {\\mathbf {x}},{\\mathbf {p}}^\\prime )$ .",
"A Mandelstam variable $s = (p+q)^2$ is the squared center-of-mass energy of a two-body scattering of on-shell pions.",
"Eq.",
"() describes the time-evolution of the distribution function in the presence of a source term $S(t, {\\mathbf {x}},{\\mathbf {p}}^\\prime )$ which take into account the pion emission from the hadron source.",
"We take the source function being common to the one in Eq.",
"(REF ) as, $S(t, {\\mathbf {x}}, {\\mathbf {p}}) = \\frac{1}{(2\\pi )^3}\\int f_{eq}(x^{\\prime }, p)\\ \\delta ^{_(4)}(x-x^{\\prime }) \\ p^{ \\mu } d\\sigma _{\\mu }(x^\\prime ) \\ , $ where $x^\\prime $ is the emission point on the hypersurface, of which normal vector is given by $d\\sigma _{\\mu }(x^\\prime )$ .",
"The thermal distribution function $f_{eq}(x, p) = \\left[\\exp \\lbrace (p^{ \\nu } u_{\\nu }-\\mu )/T \\rbrace -1 \\right]^{-1}$ is specified with the macro variables: temperature $T$ , pion chemical potential $\\mu $ and the flow vector $u^{\\mu } \\propto \\left( \\frac{t}{\\tau }, \\mathbf {v}_{\\perp }, \\frac{z}{\\tau } \\right)$ with the normalization $u^\\mu u_\\mu = 1$ .",
"As a simple model, we take $ T=130 \\ {\\rm MeV}$ and $\\mu = 30 \\ {\\rm MeV}$ , and assume Bjorken flow and the transverse flow profile proportional to the transverse coordinate $r$ as $\\mathbf {v}_\\perp = 0.06 r \\ {\\it c}^{-1}$ (see Ref [5] for details of the model).",
"The forward pion-pion scattering amplitude () is given by averaging over the isospin channels as $T(s) = 3 \\left( \\frac{1}{9} T_0 (s) + \\frac{3}{9}T_1 (s) + \\frac{5}{9}T_2 (s) \\right) $ .",
"The overall factor comes from the contributions of the pion spices, $\\pi ^\\pm $ and $\\pi ^0$ , assuming the same emission rate (REF ) for each.",
"Panels in Fig.",
"REF show the real and imaginary parts of the isospin-dependent scattering amplitudes and their average, respectively.",
"A strong attraction and attenuation around the $\\rho $ -meson resonance efficiently modify a pion amplitude.",
"Because effects of the mean-field interaction () depend on a path and momentum of a pion propagation, pions acquire phase shifts and attenuations in different magnitudes, depending on their motions after the emissions.",
"Therefore, they cannot maintain the interference compared with the case without the mean-field interaction.",
"Figure: Real and imaginary parts of the forward pion-pion scattering amplitude.Effects of the ρ\\rho -meson resonance manifest in the mean-field interaction as an attraction and attenuation."
],
[
"Results",
"In Fig.",
"REF , we show effects of the mean-field interaction on the transverse HBT radii and single-pion spectrum.",
"Those effects appear as displacements from dotted curves (see caption) which imitate typical results of the hydrodynamical simulations.",
"We find that the mean-field interaction mostly modify the sideward radius and that it acts to reduce a discrepancy between the dotted curve and the experimental data which is seen in typical results of conventional hydrodynamical simulations.",
"Owing to this improvement, a deviation in the ratio of the transverse radii is also improved in a low momentum regime.",
"The effect of the real part do not cause a considerable shift of the single-pion spectrum, and an amount of attenuated pions noticed by the blue curve would be totally complemented if we consistently take into account a decay of the $\\rho $ -meson resonance as an inverse reaction of the attenuation.",
"This is a good point for a consistent description of the transverse HBT radii and single-pion spectrum in a sense of improving the discrepancies found in the hydrodynamic picture: otherwise precisely reproduced single-pion spectrum would be sacrificed.",
"Figure: Transverse HBT radii and single-pion spectrum:red curves show these quantities in the presence of the real part of the self-energy,while blue curves show cooperative effects of the real and imaginary parts of the self-energy.Dotted curves show the quantities obtained without the mean-field interaction."
],
[
"Concluding remarks",
"We would like to remark on a prospect toward a consistent description of the hadron phase.",
"It would be worthwhile to investigate the effects of the mean-field interaction throughout the hadron phase by more elaborate descriptions, because the interaction between an emitted pion and a rather dense hadron source would be more effective, and because the magnitudes of the effects obtained in our work are comparable to those of the individual effects obtained with several upgrades of hydrodynamical modeling[6].",
"An analytic study of a mean-field interaction in terms of kinetic theory will be found in Ref.",
"[7]."
],
[
"Acknowledgments",
"The author thanks the organizers and especially S. Pratt as the convener of the session for giving him an opportunity to talk in the conference.",
"A large part of this contribution is based on his Ph.D. thesis submitted in 2010.",
"He is grateful to his supervisor, Prof. T. Matsui, for his encouraging instruction."
]
] | 1204.0779 |
[
[
"Class I methanol masers in low-mass star formation regions"
],
[
"Abstract Four Class I maser sources were detected at 44, 84, and 95 GHz toward chemically rich outflows in the regions of low-mass star formation NGC 1333I4A, NGC 1333I2A, HH25, and L1157.",
"One more maser was found at 36 GHz toward a similar outflow, NGC 2023.",
"Flux densities of the newly detected masers are no more than 18 Jy, being much lower than those of strong masers in regions of high-mass star formation.",
"The brightness temperatures of the strongest peaks in NGC 1333I4A, HH25, and L1157 at 44 GHz are higher than 2000 K, whereas that of the peak in NGC 1333I2A is only 176 K. However, rotational diagram analysis showed that the latter source is also a maser.",
"The main properties of the newly detected masers are similar to those of Class I methanol masers in regions of massive star formation.",
"The former masers are likely to be an extension of the latter maser population toward low luminosities of both the masers and the corresponding YSOs."
],
[
"Introduction",
"In spite of a number of observations and theoretical works, the nature of Class I methanol masers is still unknown.",
"This is partly because until recently these masers have been observed only in regions of massive star formation, which are typically distant (2–3 kpc from the Sun or farther) and highly obscured at optical and even NIR wavelengths.",
"In addition, high mass stars usually form in clusters.",
"These properties make it difficult to resolve maser spots and to associate masers with other objects in these regions.",
"In contrast, regions of low-mass star formation are much more widespread and many of them are only 200–300 pc from the Sun; they are less heavily obscured than regions of high-mass star formation, and there are many isolated low-mass protostars.",
"Therefore, the study of masers in these regions might be more straightforward compared to that of high-mass regions, and hence, the detection of Class I masers there might have a strong impact on maser exploration.",
"Bearing this in mind, we undertook a search for Class I methanol masers in regions of low-mass star formation.",
"Since the most common viewpoint is that these masers arise in postshock gas in the wings of bipolar outflows ([9]; [4]) our source list was composed of these objects.",
"The naive expectation is to find methanol masers towards bright thermal sources of methanol; therefore the basis of our source list consists of “chemically rich outflows”, where methanol abundance is significantly enhanced relative to that in quiescent gas.",
"Because methanol enhancement has been detected in young, well-collimated outflows from Class 0 and I sources, we included several such objects in our list regardless of whether methanol enhancement had been previously found there.",
"A subsample of our list consisted of YSOs with known outflows and/or H$_2$ O masers located in Bok globules.",
"Like other objects from our list, these YSOs are typically isolated objects of low or intermediate mass, located in nearby ($<$ 500 pc) small and relatively simple molecular clouds.",
"In total, our source list consisted of 37 regions which harbor 46 known outflows driven by Class 0 and I low-mass protostars, taken from the literature.",
"All of them were observed in the $7_0-6_1A^+$ transition at 44 GHz, where the strongest Class I masers have been found so far.",
"In addition to the $7_0-6_1A^+$ transition, most sources were observed in other Class I maser lines, namely, in the $4_{-1}-3_0E$ line at 36 GHz, in the $5_{-1}-4_0E$ line at 84 GHz, and in the $8_0-7_1A^+$ line at 95 GHz, as well as the “purely thermal” $2_K-1_K$ lines at 96 GHz.",
"Figure: Spectra of the newly detected masers."
],
[
"Observations and results",
"Single-dish observations.",
"The single-dish observations are described in detail by Kalenskii et al.",
"(2006, 2010a).",
"They were carried out with the 20-m radio telescope of the Onsala Space Observatory (OSO) during several observing sessions in 2004–2011.",
"As a result, we detected maser candidates at 44 GHz towards NGC 1333I2A, NGC 1333I4A, HH 25 and L1157.",
"Toward NGC 1333I4A and HH 25, narrow features were also found at 95 and 84 GHz.",
"In addition, a narrow line was detected at 36 GHz toward the blue lobe of an extremely high-velocity outflow in the vicinity of the bright reflection nebula NGC 2023.",
"The source spectra are shown in Fig.",
"REF .",
"VLA/EVLA observations.",
"To check whether the newly detected sources are really masers we observed them with the NRAOThe National Radio Astronomy Observatory is operated by Associated Universities, Inc., under contract with the National Science Foundation.",
"VLA/EVLA array in the D configuration, which provides an angular resolution about $1.^{\\prime \\prime }5$ at 44 GHz.",
"L1157 was observed with the VLA on March 17, 2007; the other sources were observed with the EVLA on August 08, 2010.",
"The data were reduced using the NRAO Astronomical Image Processing System (AIPS) package.",
"The source parameters are presented in Table REF ."
],
[
"Are the newly detected sources really masers?",
"The small sizes and high brightness temperatures at 44 GHz indicate that the newly detected sources are masers.",
"The exceptions are NGC 2023, which was not found at 44 GHz, and NGC 1333I2A, with a line brightness temperature of only 170 K (Table REF ).",
"The nature of the 36 GHz line in the blue lobe of the bipolar outflow in NGC 2023 is unclear.",
"On the one hand, the line is fairly narrow, and offset measurements showed that the source is compact at least with respect to the 105-arcsec Onsala beam.",
"These properties suggest that the source is a maser.",
"This assumption has further support in the fact that the line LSR velocity, $\\approx 6.5$ km s$^{-1}$ , is less than the systemic velocity of about 10 km s$^{-1}$ .",
"On the other hand, the line has no counterpart at 44 GHz, which is more typical for thermal emission.",
"Note, however, that there are known masers at 36 GHz without 44-GHz counterparts; in particular, no 44 GHz emission was found at the velocity of a fairly strong 36-GHz maser detected $3^{\\prime }$ north of DR21(OH) by [10].",
"Therefore, we tentatively conclude that the narrow line in NGC 2023 is a maser.",
"The fairly low brightness temperature and finite sizes (Table REF ) of NGC 1333I2A (M1 and M2) suggest that they are thermal sources.",
"However, a rotational diagram analysis (Kalenskii et al., in prep.)",
"shows that they are low-gain masers or a cluster of weak masers.",
"Table: Parameters of maser sources determined by the VLA observations."
],
[
"Properties of the new masers",
"Association with chemically rich outflows.",
"New masers were found towards the lobes of outflows in NGC 1333I4A, NGC 1333I2A, NGC 2023, HH25, and L1157.",
"These outflows are known to be chemically rich outflows with enhanced methanol abundances.",
"Comparison of the VLA maps with high-resolution maps of thermal methanol and other molecules shows that the masers coincide with chemically rich gas clumps, where the abundances of methanol and other molecules are enhanced (e.g., [5]; [1]; [3]).",
"In L1157, the masers are located in gas clumps, which, according to chemical modeling of [13], probably pre-existed the outflow.",
"LSR velocities and intensities.",
"Comparison of the maser LSR velocities with those of thermal methanol lines, observed in the same directions, show that these velocities coincide within 0.5 km/s.",
"This coincidence occurs even when the LSR velocities of some other molecular lines toward the maser positions are significantly different.",
"The LSR velocities of both maser and thermal methanol lines are usually close to the systemic velocities.",
"An exception is the 36-GHz maser in the EHV outflow NGC 2023.",
"Its radial velocity is less than the systemic velocity by about 3.5 km $^{-1}$ .",
"Note that Voronkov (this volume) has detected a high-velocity Class I maser just at 36 GHz.",
"Maser intensities.",
"The new masers are weaker than the bright masers typical in regions of massive star formation.",
"However, they obey the same relationship between the maser and YSO luminosities as reported by [2] for masers in regions of high- and intermediate-mass star formation, thus extending this relationship toward low luminosities (Kalenskii et al., in prep).",
"Variability.",
"Several sessions of repeated observations of NGC 1333I4A, HH25, and L1157 at 44 GHz were performed in 2008–2011.",
"No notable variations were found.",
"Slight changes in line intensities can be attributed to poor signal-to-noise ratios and calibration uncertainties.",
"However, further monitoring of these sources is desirable in order to search for flares similar to that which occurred in DR 21(OH).",
"To summarize, the main properties of the newly detected masers are similar to those of Class I methanol masers in regions of massive star formation.",
"The former masers are likely to be an extension of the latter maser population toward lower luminosities of both the masers and the corresponding YSOs."
],
[
"Maser models",
"The fact that the maser LSR velocities coincide with the systemic velocities allows us to conclude that the masers appear in dense clumps of gas, probably pre-existing the outflows.",
"However, the exact nature of the masers remains unknown.",
"[11] suggested that compact maser spots arise in extended, turbulent clumps because in a turbulent velocity field the coherence lengths along some directions are larger than the mean coherence length, resulting in a random increase of the optical depth absolute values along certain sight lines in a clump.",
"According to [8] such a model can easily explain the observed brightness of the maser lines, but within the framework of this model it is difficult to explain why single peaks dominate the maser emission in the L1157 clumps.",
"However, natural additional assumptions, such as the existence of shocks or centrally condensed clumps, makes it possible to explain the observational data.",
"An examination of the maser spectra in L1157 may lead to another interpretation of our results.",
"Both 44 GHz masers detected in this source have double line profiles.",
"It is known that a double thermal line with a “blue asymmetry” may be a signature of collapse [12].",
"Contrary to this, the masers in L1157 exhibit a “red asymmetry”.",
"However, just such an asymmetry is what one would expect for Class I masers arising in a collapsing clump.",
"This model is discussed in more detail by [8].",
"Note that this model, if correct, is specific for the masers in L1157; no other maser in our sample exhibits a double line profile.",
"The work was financially supported by RFBR (grants No.",
"04-02-17547, 07-02-00248, and 10-02-00147-a), and Federal National Scientific and Educational Program (project number 16.740.11.0155).",
"P.H.",
"acknowledges partial support from NSF grant AST 0908901.",
"S.Kurtz acknowledges support from UNAM DGAPA grant IN101310.",
"The Onsala Space Observatory is the Swedish National Facility for Radio Astronomy and is operated by Chalmers University of Technology, Göteborg, Sweden, with financial support from the Swedish Research Council and the Swedish Board for Technical Development."
]
] | 1204.0950 |
[
[
"An exact solution of the moving boundary problem for the relativistic\n plasma expansion in a dipole magnetic field"
],
[
"Abstract An exact analytic solution is obtained for a uniformly expanding, neutral, highly conducting plasma sphere in an ambient dipole magnetic field with an arbitrary orientation of the dipole moment in the space.",
"Based on this solution the electrodynamical aspects related to the emission and transformation of energy have been considered.",
"In order to highlight the effect of the orientation of the dipole moment in the space we compare our results obtained for parallel orientation with those for transversal orientation.",
"The results obtained can be used to treat qualitatively experimental and simulation data, and several phenomena of astrophysical and laboratory significance."
],
[
"Introduction",
"There are many processes in physics which require the solution of boundary and initial value problems.",
"When the boundary is immobile the standard techniques for the solution of the problem are well known.",
"However, the situation becomes very complicated introducing a moving boundary into the problem.",
"Usually this excludes the achievement of an exact solution of the problem and the development of some approximate methods is desirable [1], [2], [3].",
"The simplest case often allowing an analytical solution is the moving plane boundary.",
"With the help of a time-dependent transformation it is possible to immobilize the boundary but increasing essentially the complexity of the underlying dynamical equations.",
"The purpose of this work is to present an example of an exactly solvable moving boundary and initial value problem and generalize the results of our previous paper [4].",
"The problems with moving boundary are very important in fundamental mathematical physics but they have also important applications in many area of physics.",
"For instance, an important example is the expansion of a plasma cloud with a sharp boundary across an ambient magnetic field.",
"Such kind of processes are particularly of interest for many space and laboratory investigations (see [5], [6], [7], [8], [9] and references therein).",
"This is a topic of intense interest across a wide variety of disciplines within plasma physics, with applications to laser generated plasmas [9], solar [10] and magnetospheric [11], [12] physics, astrophysics [13], and pellet injection for tokamak refueling [14].",
"The dynamics of the plasma cloud has been studied both numerically [5], [7], [8], [6] and analytically [15], [16], [17], [18], [19].",
"Usually the plasma is treated as a highly conducting media shielded from the penetration of the magnetic field.",
"Thus the magnetic field is zero inside.",
"An exact analytic solution for a uniformly expanding plasma sphere in an external uniform magnetic field has been obtained in [18], [19] both for non-relativistic [18] and relativistic [19] expansions.",
"Within one-dimensional geometry a similar problem has been considered in [15].",
"In the previous papers [4] and [17] we obtained the exact solutions for the uniform relativistic expansion of the plasma sphere and cylinder in the presence of a dipole and homogeneous magnetic fields, respectively.",
"However, in Ref.",
"[4] we have considered a somewhat simplified situation when the dipole moment is directed to the centre of the plasma sphere thus providing an azimuthal symmetry to the problem.",
"In the present paper we generalize the solution obtained in [4] considering a similar problem of the expansion of the plasma sphere in the presence of a dipole magnetic field but with an arbitrary orientation of the dipole moment in the space."
],
[
"Moving boundary and initial value problem",
"We consider the moving boundary and initial value problem of the relativistic ($v\\lesssim c$ , where $v$ is the radial velocity of the sphere) expansion of the plasma sphere in the vacuum in the presence of a dipole magnetic field.",
"Consider the dipole with the magnetic moment $\\mathbf {p}$ and a neutral infinitely conducting plasma sphere with radius $R(t)$ located at the origin of the coordinate system.",
"The dipole is placed in the position $\\mathbf {r}_{0}$ from the centre of the sphere ($R(t)<r_{0}$ ).",
"The orientation of the dipole moment is given by the angle $\\vartheta $ between the vectors $\\mathbf {p}$ and $\\mathbf {r}_{0}$ .",
"In contrast to our previous paper [4] we consider here an arbitrary orientation of the dipole moment in space with arbitrary $\\vartheta $ .",
"The plasma sphere has expanded at $t=0$ to its present state from a point source located at the point $\\mathbf {r}=0$ .",
"It is convenient to introduce the vector potential of the induced and the dipole magnetic fields.",
"The magnetic field of the dipole is given by $\\mathbf {H}_{0}=\\mathbf {\\nabla }\\times \\mathbf {A}_{0}$ , where the vector potential $\\mathbf {A}_{0}$ is $\\mathbf {A}_{0}=\\frac{\\mathbf {p}\\times \\left( \\mathbf {r}-\\mathbf {r}_{0}\\right) }{\\left|\\mathbf {r}-\\mathbf {r}_{0}\\right|^{3}}.$ As the spherical plasma cloud expands it both perturbs the external magnetic field and generates an electric field.",
"Within the spherical plasma region there is neither an electric field nor a magnetic field.",
"We shall obtain an analytic solution of the electromagnetic field configuration for an arbitrary $\\vartheta $ .",
"When an infinitely conducting plasma sphere is introduced into a background magnetic field the plasma expands and excludes the background magnetic field to form a magnetic cavity.",
"The magnetic energy of the dipole in the excluded volume, i.e., in the volume of the conducting sphere is calculated as $\\int _{\\Omega _{R}}\\frac{H_{0}^{2}(\\mathbf {r}) }{8\\pi } d\\mathbf {r} \\equiv Q(\\xi )=Q_{\\parallel }(\\xi )\\cos ^{2} \\vartheta +Q_{\\perp }(\\xi ) \\sin ^{2} \\vartheta ,$ where $\\xi =R/r_{0}<1$ , $\\Omega _{R}$ is the volume of the plasma sphere, $Q_{\\parallel }(\\xi ) =(p^{2}/2r_{0}^{3})\\xi f_{1}(\\xi )$ , $Q_{\\perp }(\\xi ) =(p^{2}/4r_{0}^{3}) \\xi f_{2}(\\xi )$ and the functions $f_{1}(x)$ and $f_{2}(x)$ are determined in Appendix A by Eqs.",
"(A.3) and (A.4), respectively.",
"Here $Q_{\\parallel }(\\xi )$ and $Q_{\\perp }(\\xi )$ are the magnetic energies escaped from the plasma volume at parallel ($\\vartheta =0$ ) and transversal ($\\vartheta =\\pi /2$ ) orientations of the dipole, respectively.",
"The energy $Q(\\xi )$ increases with decreasing $\\vartheta $ and reach its maximum value at $\\vartheta =0$ or $\\vartheta =\\pi $ , that is the magnetic moment $\\mathbf {p}$ is parallel or antiparallel to the symmetry axis $\\mathbf {r}_{0}$ .",
"We now turn to solve the boundary problem and calculate the induced electromagnetic fields which arise near the surface of the plasma sphere due to the dipole magnetic field.",
"Since the sphere is highly-conducting the electromagnetic fields vanish inside the sphere.",
"In addition, the normal component of the magnetic field $H_{r}$ vanishes on the surface of the sphere.",
"To solve the boundary problem we introduce the spherical coordinate system $r,\\theta ,\\varphi $ with the $z$ -axis along the vector $\\mathbf {r}_{0}$ and the azimuthal angle $\\varphi $ is counted from the plane ($xz$ -plane) containing the vectors $\\mathbf {r}_{0}$ and $\\mathbf {p}$ .",
"The spherical coordinate $\\theta $ is the angle between the radius vector $\\mathbf {r}$ and $\\mathbf {r}_{0}$ .",
"The vector potential (REF ) at $r<r_{0}$ can alternatively be represented by the sum of Legendre polynomials.",
"Using the summation formulas derived in Appendix A of Ref.",
"[4] one obtaines: $&&A_{0r}=\\frac{p}{r_{0}^{2}}\\sin \\vartheta \\sin \\varphi \\sum _{\\ell =1}^{\\infty }\\left( \\frac{r}{r_{0}}\\right) ^{\\ell -1}P_{\\ell }^{1}\\left(\\cos \\theta \\right) , \\\\&&A_{0\\theta }=\\frac{p}{r_{0}^{2}}\\sin \\vartheta \\sin \\varphi \\sum _{\\ell =1}^{\\infty }\\ell \\left( \\frac{r}{r_{0}}\\right) ^{\\ell -1}P_{\\ell }\\left(\\cos \\theta \\right) , \\\\&&A_{0\\varphi }=\\frac{p}{r_{0}^{2}}\\left[ \\cos \\vartheta \\sum _{\\ell =1}^{\\infty }\\left( \\frac{r}{r_{0}}\\right) ^{\\ell }P_{\\ell }^{1}\\left( \\cos \\theta \\right) \\right.",
"\\\\&&\\left.+\\sin \\vartheta \\cos \\varphi \\sum _{\\ell =0}^{\\infty }\\left(\\ell +1\\right) \\left( \\frac{r}{r_{0}}\\right) ^{\\ell }P_{\\ell }\\left( \\cos \\theta \\right) \\right] , \\nonumber $ where $P_{\\ell }^{\\nu }(x)$ is the generalized Legendre polynomials with $\\nu =0,1$ and $P_{\\ell }(x)=P_{\\ell }^{0}(x)$ .",
"Since the external region of the plasma sphere is devoid of free charge density, a suitable gauge $\\nabla \\cdot \\mathbf {A} =0$ allows the electric and magnetic fields to be derived from the vector potential $\\mathbf {A}$ .",
"Having in mind the symmetry of the problem with respect to the $xz$ -plane it is sufficient to choose the vector potential in the form $A_{r}=A_{0r}$ , and $&&A_{\\theta } =A_{0\\theta } +\\frac{p}{r_{0}^{2}}\\sin \\vartheta \\sin \\varphi \\sum _{\\ell =1}^{\\infty }\\mathcal {A}_{\\ell }(r,t) P_{\\ell }^{\\prime }(\\cos \\theta ) , \\\\&&A_{\\varphi } =A_{0\\varphi } +\\frac{p}{r_{0}^{2}}\\sum _{\\ell =1}^{\\infty }\\bigg [\\cos \\vartheta \\mathcal {B}_{\\ell }(r,t) P_{\\ell }^{1}(\\cos \\theta ) \\\\&&+\\sin \\vartheta \\cos \\varphi \\mathcal {C}_{\\ell }(r,t) \\frac{\\partial }{\\partial \\theta }P_{\\ell }^{1}(\\cos \\theta ) \\bigg ] , \\nonumber $ and the components of the electromagnetic field are given by $E_{r}=0$ and $&&H_{r}=\\frac{1}{r\\sin \\theta }\\left[ \\frac{\\partial }{\\partial \\theta }\\left( A_{\\varphi }\\sin \\theta \\right) -\\frac{\\partial A_{\\theta }}{\\partial \\varphi }\\right] , \\\\&&H_{\\theta }=\\frac{1}{r\\sin \\theta }\\frac{\\partial A_{r}}{\\partial \\varphi }-\\frac{1}{r}\\frac{\\partial \\left( rA_{\\varphi }\\right) }{\\partial r}, \\\\&&H_{\\varphi }=\\frac{1}{r}\\frac{\\partial \\left( rA_{\\theta }\\right) }{\\partial r}-\\frac{1}{r}\\frac{\\partial A_{r}}{\\partial \\theta }, \\\\&&E_{\\theta }=-\\frac{1}{c}\\frac{\\partial A_{\\theta }}{\\partial t}, \\quad E_{\\varphi }=-\\frac{1}{c}\\frac{\\partial A_{\\varphi }}{\\partial t} .",
"$ In Eqs.",
"(REF ) and () the prime indicates the derivative with respect to the argument and $\\mathcal {A}_{\\ell }(r,t)$ , $\\mathcal {B}_{\\ell }(r,t)$ and $\\mathcal {C}_{\\ell }(r,t)$ are the expansion coefficients which are determined from the boundary and initial conditions.",
"It is straightforward to show that the gauge $\\nabla \\cdot \\mathbf {A} =0$ is satisfied automatically if $\\mathcal {C}_{\\ell }(r,t) =\\mathcal {A}_{\\ell }(r,t)$ .",
"Previously in Ref.",
"[4] we have considered the case of the parallel or antiparallel orientation of the dipole when $\\vartheta =0$ or $\\vartheta =\\pi $ .",
"In this case Eqs.",
"(REF )–() are essentially simplified because $A_{r}=A_{0r}=0$ , $A_{\\theta }=A_{0\\theta }=0$ and $H_{\\varphi } =E_{r}=E_{\\theta }=0$ .",
"Then the problem is reduced to the evaluation of the vector potential $A_{\\varphi }$ which involves only the coefficient $\\mathcal {B}_{\\ell }(r,t)$ .",
"For arbitrary $\\vartheta $ the equations for the expansion coefficients $\\mathcal {A}_{\\ell }(r,t)$ and $\\mathcal {B}_{\\ell }(r,t)$ are obtained from the Maxwell's equations $\\frac{\\partial ^{2}\\mathcal {A}_{\\ell }}{\\partial r^{2}}+\\frac{2}{r}\\frac{\\partial \\mathcal {A}_{\\ell }}{\\partial r}-\\frac{\\ell (\\ell +1)}{r^{2}}\\mathcal {A}_{\\ell }-\\frac{1}{c^{2}}\\frac{\\partial ^{2}\\mathcal {A}_{\\ell }}{\\partial t^{2}}=0 .$ Similar equation is obtained for the quantity $\\mathcal {B}_{\\ell }(r,t)$ .",
"This equation is to be solved in the external region $r>R(t)$ subject to the boundary and initial conditions.",
"Here $R(t)$ is the plasma sphere radius at the time $t$ .",
"The initial conditions at $t=0$ are $\\mathcal {A}_{\\ell }(r,0)=\\mathcal {B}_{\\ell }(r,0)=0 , \\quad \\frac{\\partial \\mathcal {A}_{\\ell }(r,0)}{\\partial t}=\\frac{\\partial \\mathcal {B}_{\\ell }(r,0)}{\\partial t}=0 .$ The first initial condition states that the initial value of $\\mathbf {A}$ is that of a dipole magnetic field, $\\mathbf {A}(\\mathbf {r},0)=\\mathbf {A}_{0}(\\mathbf {r})$ .",
"The second initial condition states that there is no initial electric field.",
"Boundary conditions should be imposed at the spherical surface $r=R(t)$ and at infinity.",
"Because of the finite propagation velocity of the perturbed electromagnetic field the magnetic field at infinity will remain undisturbed for all finite times.",
"Thus, for all finite times $\\mathcal {A}_{\\ell }(r,t)\\rightarrow 0$ and $\\mathcal {B}_{\\ell }(r,t)\\rightarrow 0$ at $r\\rightarrow \\infty $ .",
"The boundary condition at the expanding spherical surface is $H_{r}=0$ or, alternatively, $\\mathcal {A}_{\\ell }(R,t)=-\\frac{1}{\\ell +1}\\left( \\frac{R}{r_{0}}\\right) ^{\\ell } , \\quad \\mathcal {B}_{\\ell }(R,t)=-\\left( \\frac{R}{r_{0}}\\right) ^{\\ell } .$ We consider the case of the uniform expansion of the plasma sphere $R(t)=vt$ with a constant expansion velocity $v$ .",
"This special case of the uniform expansion falls within the conical flow techniques which has been applied previously in Refs.",
"[19], [17].",
"From symmetry considerations one seeks a solution for the total (i.e., the unperturbed potential $\\mathbf {A}_{0}(\\mathbf {r})$ plus the induced one) vector potential of the form $\\mathcal {A}_{\\ell }(r,t)=r^{\\nu }\\Psi _{\\ell }(\\zeta )$ and $\\mathcal {B}_{\\ell }(r,t)=r^{\\nu }\\Phi _{\\ell }(\\zeta )$ with $\\zeta =r/ct$ , where $c$ is the velocity of light.",
"Here $\\Psi _{\\ell }(\\zeta )$ and $\\Phi _{\\ell }(\\zeta )$ are some unknown functions and $\\nu >0$ .",
"Having in mind the symmetry of the unperturbed magnetic field and also the boundary conditions (REF ) it is sufficient to choose the parameter $\\nu $ as $\\nu =\\ell $ .",
"The equation for the vector potential $\\mathbf {A}(\\mathbf {r},t)$ is obtained from the Maxwell's equations which for the unknown functions $\\Psi _{\\ell }(\\zeta )$ and $\\Phi _{\\ell }(\\zeta )$ yields the same ordinary differential equation $\\zeta (1-\\zeta ^{2})\\Psi _{\\ell }^{\\prime \\prime }(\\zeta )+2(\\ell +1-\\zeta ^{2})\\Psi _{\\ell }^{\\prime }(\\zeta )=0 .$ This equation is to be solved in the external region $r>R(t)$ subject to the boundary and initial conditions.",
"The boundary condition at the expanding spherical surface is $H_{r}=0$ which is equivalent to the relations $\\Psi _{\\ell }(\\beta )=-\\frac{1}{\\ell +1} (\\frac{1}{r_{0}})^{\\ell }$ and $\\Phi _{\\ell }(\\beta )=-(\\frac{1}{r_{0}})^{\\ell }$ (see Eq.",
"(REF )) with $\\beta =v/c<1$ .",
"In addition, imposing that $\\mathbf {A}(\\mathbf {r},t)=\\mathbf {A}_{0}(\\mathbf {r})$ at $r\\geqslant ct$ we obtain another boundary conditions $\\Psi _{\\ell }(1)=\\Phi _{\\ell }(1)=0$ .",
"Thus, the solution of Eq.",
"(REF ) subject to the initial and boundary conditions may be finally written in the form at $vt<r<ct$ $\\mathcal {A}_{\\ell }(r,t)=\\frac{1}{\\ell +1} \\mathcal {B}_{\\ell }(r,t) , \\quad \\mathcal {B}_{\\ell }(r,t)=-\\left( \\frac{r}{r_{0}}\\right) ^{\\ell }\\frac{p_{\\ell }(1/\\zeta )}{p_{\\ell }(1/\\beta )} ,$ $\\mathbf {A}(\\mathbf {r},t)=\\mathbf {A}_{0}(\\mathbf {r})$ at $r\\geqslant ct$ and $\\mathbf {A}(\\mathbf {r},t)=0$ at $r\\leqslant vt$ .",
"Here $p_{\\ell }(z)=2^{\\ell }\\ell !",
"(z^{2}-1)^{\\frac{\\ell +1}{2}}P_{\\ell }^{-\\ell -1}(z)=\\int _{1}^{z}(\\tau ^{2}-1)^{\\ell } d\\tau $ and $P_{\\mu }^{\\nu }(z)$ are the generalized Legendre functions with $z>1$ , $\\mu =\\ell $ , and $\\nu =-\\ell -1$ .",
"The electromagnetic field components are determined according to Eqs.",
"(REF )–().",
"From Eqs.",
"(REF )–() and (REF ) it is straightforward to show that the boundary conditions on the moving surface (see, e.g., [3], [20] for details), $\\mathbf {E}(R)=-\\frac{1}{c}[\\mathbf {v}\\times \\mathbf {H}(R)]$ (or $E_{\\theta }(R)=\\beta H_{\\varphi }(R)$ and $E_{\\varphi }(R)=-\\beta H_{\\theta }(R)$ ), are satisfied automatically.",
"It should be emphasized that all above results are valid only for $R(t)<r_{0}$ or $t<r_{0}/v$ .",
"At the time $t=r_{0}/v$ the plasma sphere reaches to the dipole which will be completely shielded by the plasma.",
"Therefore at $t\\geqslant r_{0}/v$ the total electromagnetic field vanishes.",
"Consider now briefly the non-relativistic limit of Eqs.",
"(REF ) and () with the expansion coefficients (REF ) recalling that $\\mathcal {C}_{\\ell }(r,t) =\\mathcal {A}_{\\ell }(r,t)$ .",
"This limit can be obtained using at $\\zeta \\rightarrow 0$ and $\\beta \\rightarrow 0$ the asymptotic expression $p_{\\ell } (1/\\zeta )/p_{\\ell }(1/\\beta )=(\\beta /\\zeta )^{2\\ell +1} =(R/r)^{2\\ell +1}$ (see, e.g., Ref.",
"[21] for the asymptotic expansion of the generalized Legendre functions $P_{\\mu }^{\\nu } (z)$ with $z>1$ ) which yields $&&A_{\\theta }(\\mathbf {r},t) =A_{0\\theta }(\\mathbf {r})-\\frac{p}{r_{0}R}\\sin \\vartheta \\sin \\varphi \\\\&&\\times \\sum _{\\ell =1}^{\\infty }\\frac{1}{\\ell +1}\\left( \\frac{r_{\\ast }}{r}\\right) ^{\\ell +1}P_{\\ell }^{\\prime }(\\cos \\theta ), \\nonumber \\\\&&A_{\\varphi }(\\mathbf {r},t) =A_{0\\varphi }(\\mathbf {r})-\\frac{p}{r_{0}R}\\sum _{\\ell =1}^{\\infty }\\left( \\frac{r_{\\ast }}{r}\\right) ^{\\ell +1} \\\\&&\\times \\left[\\cos \\vartheta P_{\\ell }^{1}(\\cos \\theta )+\\sin \\vartheta \\cos \\varphi \\frac{1}{\\ell +1}\\frac{\\partial }{\\partial \\theta }P_{\\ell }^{1}(\\cos \\theta )\\right] , \\nonumber $ where $r_{\\ast }=R^{2}/r_{0} <R$ .",
"The remaining $\\ell $ -summations in these expressions can be performed using the summation formulas derived in Ref. [4].",
"As has been shown in [4] the resulting induced vector potential in the non-relativistic limit represents a sum of the dipole and the quadrupole terms.",
"These fictitious dipole and quadrupole ('image' dipole or quadrupole) are located in the $xz$ -plane inside the plasma sphere at the distance $\\mathbf {r}_{\\ast }= (R^{2}/r^{2}_{0}) \\mathbf {r}_{0}$ from the centre.",
"Finally, in the lowest order with respect to the factor $\\beta $ the components of the electric field are given by $E_{\\theta } =-\\beta \\partial A_{\\theta }/\\partial R$ , $E_{\\varphi } =-\\beta \\partial A_{\\varphi }/\\partial R$ , where $A_{\\theta }$ and $A_{\\varphi }$ are determined by Eqs.",
"(REF ) and (), respectively."
],
[
"Energy balance",
"In the problem of the plasma expansion in an ambient magnetic field it is important to study the fraction of energy emitted and lost in the form of electromagnetic pulse propagating outward of the expanding plasma [15], [18], [4], [17], [16].",
"In this section we consider the energy balance during the plasma sphere expansion in the presence of the magnetic dipole with arbitrary orientation thus generalizing our previous treatment [4] with specific orientations $\\vartheta =0,\\,\\pi $ .",
"It is assumed that the plasma sphere of the zero initial radius is created at $t=0$ .",
"When it starts expanding, ambient dipole magnetic field $\\mathbf {H}_{0}$ is perturbed by the electromagnetic pulse, $\\mathbf {H}^{\\prime } =\\mathbf {H}-\\mathbf {H}_{0}$ , $\\mathbf {E}$ , propagating outward with the speed of light.",
"The tail of this pulse coincides with the moving plasma boundary $r=R(t)$ while the leading edge is at the information sphere, $r=ct$ .",
"Ahead of the leading edge, the magnetic field is not perturbed and equals $\\mathbf {H}_{0}$ while the electric field is absent.",
"For the evaluation of the energy balance we employ the Poynting equation $\\mathbf {\\nabla }\\cdot \\mathbf {S}=-\\mathbf {j}\\cdot \\mathbf {E}-\\frac{\\partial }{\\partial t}\\frac{E^{2}+H^{2}}{8\\pi } ,$ where $\\mathbf {S}=\\frac{c}{4\\pi }[\\mathbf {E}\\times \\mathbf {H}]$ is the Poynting vector and $\\mathbf {j}=j_{\\theta }\\mathbf {e}_{\\theta }+j_{\\varphi }\\mathbf {e}_{\\varphi }$ (with $|\\mathbf {e}_{\\theta }|=|\\mathbf {e}_{\\varphi }|=1$ ) is the surface current density.",
"The energy components involved in Eq.",
"(REF ) have been evaluated in detail in [4].",
"We recall here some results for completeness.",
"In order to calculate the energy emitted to infinity we should integrate the Poynting vector over time and over the surface $S_{c}$ of the sphere with radius $r_{c}<r_{0}$ (control sphere) and the volume $\\Omega _{c}$ enclosing the plasma sphere ($r_{c}>R$ or $0\\leqslant t<r_{c}/v$ ).",
"Integrating over time and over the volume $\\Omega _{c}$ Eq.",
"(REF ) is represented as $W_{S}(t)=W_{J}(t)+\\Delta W_{\\mathrm {EM}}(t) ,$ where $&&W_{S}(t)=r_{c}^{2}\\int _{0}^{t} d t^{\\prime }\\int _{0}^{\\pi }\\sin \\theta d\\theta \\int _{0}^{2\\pi }S_{r} d\\varphi , \\nonumber \\\\&&W_{J}(t)=-\\int _{0}^{t} d t^{\\prime }\\int _{\\Omega _{c}}\\mathbf {j}\\cdot \\mathbf {E} d\\mathbf {r} .",
"$ Here $S_{r}=\\frac{c}{4\\pi }(E_{\\theta }H_{\\varphi }-E_{\\varphi }H_{\\theta })$ is the radial component of the Poynting vector.",
"Also $W_{\\mathrm {EM}}(t)$ and $\\Delta W_{\\mathrm {EM}}(t)=W_{\\mathrm {EM}}(0)-W_{\\mathrm {EM}}(t)$ are the total electromagnetic energy and its change (with minus sign) in a volume $\\Omega _{c}$ , respectively.",
"$W_{J}(t)$ is the energy transferred from plasma sphere to electromagnetic field and is the mechanical work with minus sign performed by the plasma on the external electromagnetic pressure.",
"At $t=0$ the electromagnetic fields are given by $\\mathbf {H}(\\mathbf {r},0)=\\mathbf {H}_{0}(\\mathbf {r})$ and $\\mathbf {E}(\\mathbf {r},0)=0$ .",
"Hence $W_{\\mathrm {EM}}(0) $ is the energy of the dipole magnetic field in a volume $\\Omega _{c}$ and can be calculated from Eq.",
"(REF ) by replacing $R$ by $r_{c}$ .",
"Thus $W_{\\mathrm {EM}}(0) =Q(u)$ , where $u=r_{c}/ r_{0}<1$ .",
"Taking into account that $\\mathbf {H}=\\mathbf {E}=0$ in a plasma sphere, the change of the electromagnetic energy $\\Delta W_{\\mathrm {EM}}(t)$ in a volume $\\Omega _{c}$ can be represented in the form [4] $\\Delta W_{\\mathrm {EM}}(t)=Q(u)-\\int _{\\Omega _{c}^{\\prime }}\\frac{E^{2}+H^{2}}{8\\pi } d\\mathbf {r} ,$ where $\\Omega _{c}^{\\prime }$ is the volume of the control sphere excluding the volume of the plasma sphere.",
"Hence the total energy flux, $W_{S}(t)$ given by Eq.",
"(REF ) is calculated as a sum of the energy loss by plasma due to the external electromagnetic pressure and the decrease of the electromagnetic energy in a control volume $\\Omega _{c}$ .",
"Consider now explicitly each energy component $W_{S}(t)$ , $W_{J}(t)$ and $\\Delta W_{\\mathrm {EM}}(t)$ .",
"$W_{S}(t)$ is evaluated from Eq.",
"(REF ).",
"In the first expression of Eq.",
"(REF ) the $t^{\\prime }$ -integral must be performed at $\\frac{r_{c}}{c}\\leqslant t^{\\prime }\\leqslant t$ ($t<\\frac{r_{c}}{v}$ ) since at $0\\leqslant t^{\\prime }<\\frac{r_{c}}{c}$ the electromagnetic pulse does not reach to the control surface yet and $S_{r}(r_{c}) =0$ .",
"Using the summation formulas derived in Appendix from Eqs.",
"(REF )–(), (REF ) and (REF ) we obtain $&&W_{S}(t)=Q(u)+\\frac{p^{2}}{2r_{0}^{3}}\\sum _{\\ell =1}^{\\infty }\\frac{\\ell a_{\\ell }(\\vartheta )}{2\\ell +1}u^{2\\ell +1} \\nonumber \\\\&&\\times \\left\\lbrace \\frac{(1/\\eta ^{2}-1)^{2\\ell +1}}{(2\\ell +1)p_{\\ell }^{2}(1/\\beta )}-(\\ell +1)\\left[ \\frac{p_{\\ell }(1/\\eta )}{p_{\\ell }(1/\\beta )}-1\\right] ^{2}\\right\\rbrace , $ where $\\eta =r_{c}/ct<1$ , and $a_{\\ell }(\\vartheta )=(\\ell +1)\\cos ^{2}\\vartheta + \\frac{\\ell }{2}\\sin ^{2}\\vartheta $ is the orientation factor of the dipole magnetic field.",
"In the non-relativistic limit, $\\beta \\rightarrow 0$ , using the asymptotic expression (see, e.g., Ref.",
"[21]) $p_{\\ell }(z)=z^{2\\ell +1}/(2\\ell +1)$ at $z\\rightarrow \\infty $ , as well as the summation formulas of Appendix from Eq.",
"(REF ) we obtain $&&W_{S}(t)=2Q(\\xi )-Q(\\kappa ) \\nonumber \\\\&&+\\frac{p^{2}}{r_{0}^{3}}\\frac{\\kappa ^{3}}{(1-\\kappa ^{2})^{3}}\\left( \\cos ^{2}\\vartheta +\\frac{1+\\kappa ^{2}}{4}\\sin ^{2}\\vartheta \\right) $ with $\\kappa =R^{2}/r_{0}r_{c}$ .",
"In Eq.",
"(REF ) $Q(\\kappa )$ represents the magnetic energy of the dipole field in a sphere having the radius $R_{\\ast }=R^{2}/r_{c}<R$ and enclosed in the plasma sphere.",
"Figure: (Color online) The energy components W S (t)W_{S}(t) (the lines without symbols) and W J (t)W_{J}(t) (the lines with symbols)normalized to the quantity p 2 /r 0 3 p^{2}/ r^{3}_{0} for two values of the factor β\\beta as a function of ct/r 0 ct/r_{0}calculated from expressions () and () with r c =0.5r 0 r_{c}=0.5r_{0}.",
"The solid and dashed linescorrespond to the dipole orientations ϑ=0\\vartheta =0 and ϑ=π/2\\vartheta =\\pi /2, respectively.Next, we evaluate the energy loss $W_{J}(t)$ by the plasma which is determined by the surface current density, $\\mathbf {j}$ localized within thin skin layer, $R-\\delta <r<R+\\delta $ with $\\delta \\rightarrow 0$ , near plasma boundary.",
"Therefore in Eq.",
"(REF ) the volume $\\Omega _{c}$ is in fact replaced by the volume $\\Omega _{\\delta }\\sim R^{2}\\delta $ of the shell with $R-\\delta <r< R+\\delta $ .",
"Within the skin layer we take into account that $\\mathbf {E}=-\\frac{1}{c}[\\mathbf {v}\\times \\mathbf {H}]$ and $H_{r}(R)=0$ .",
"Also it should be mentioned that the boundary of the volume $\\Omega _{\\delta }$ moves with a velocity $v$ and the electrical field has a jump across the plasma surface.",
"The detailed calculations are given in [4] and we provide here only the final result which reads $Q_{J}(t)=-\\int _{\\Omega _{\\delta }}\\mathbf {j}\\cdot \\mathbf {E} d\\mathbf {r}=\\frac{v}{\\gamma ^{2}}\\int \\nolimits _{S_{R}}\\frac{H^{2}(R)}{8\\pi } dS .$ Here $\\gamma ^{-2}=1-\\beta ^{2}$ and $S_{R}$ are the relativistic factor and the surface of the expanding plasma, respectively.",
"Note that the moving boundary modifies the surface current which is now proportional to the factor $\\gamma ^{-2}$ [3], [20].",
"Equation (REF ) shows that the energy loss by the plasma per unit time is equal to the work performed by the plasma on the external electromagnetic pressure.",
"As shown in Ref.",
"[4] this external pressure is formed by the difference between magnetic and electric pressures, i.e., the induced electric field tends to decrease the force acting on the expanding plasma surface.",
"The total energy loss by the plasma sphere is calculated as $W_{J}(t) =\\int _{0}^{t}Q_{J}\\left( t^{\\prime }\\right) d t^{\\prime } =\\frac{p^{2}}{2r_{0}^{3}}\\sum _{\\ell =1}^{\\infty }\\frac{\\ell a_{\\ell }(\\vartheta )}{(2\\ell +1)^{2}} \\nonumber \\\\\\times \\left( \\frac{\\xi }{\\beta ^{2}\\gamma ^{2}}\\right) ^{2\\ell +1}\\frac{1}{p_{\\ell }^{2}(1/\\beta )} , $ where $\\xi =R/r_{0}$ .",
"In a non-relativistic case Eq.",
"(REF ) yields (see Appendix): $W_{J}(t)=\\frac{p^{2}}{r_{0}^{3}}\\frac{\\xi ^{3}}{\\left( 1-\\xi ^{2}\\right) ^{3}}\\left( \\cos ^{2}\\vartheta +\\frac{1+\\xi ^{2}}{4}\\sin ^{2}\\vartheta \\right) .$ The change of the electromagnetic energy in a control sphere is calculated from Eq.",
"(REF ).",
"At $R<r_{c}<ct$ (the electromagnetic pulse fills the whole control sphere) it is straightforward to show that $\\Delta W_{\\mathrm {EM}}(t)=W_{S}(t) -W_{J}(t)$ as predicted by the energy balance equation (REF ).",
"The non-relativistic limit of $\\Delta W_{\\mathrm {EM}}(t)$ can be evaluated from Eqs.",
"(REF ) and (REF ).",
"Also it should be emphasized that in Eqs.",
"(REF ), (REF ) as well as in $\\Delta W_{\\mathrm {EM}}(t)$ the dependence on the orientation $\\vartheta $ of the dipole moment is determined by the factor $a_{\\ell } (\\vartheta )$ given by (REF ).",
"At parallel (or antiparallel) orientation with $\\vartheta =0$ (or $\\vartheta =\\pi $ ) this factor is $a_{\\ell } =\\ell +1$ and Eqs.",
"(REF ), (REF ) as well as $\\Delta W_{\\mathrm {EM}}(t)$ coincide with the corresponding formulas derived in Ref. [4].",
"As in Eq.",
"(REF ) it is convenient to represent the energy components $W_{S}(t)$ and $W_{J}(t)$ in the form $W_{S,J}(t) =W_{S,J\\parallel }(t) \\cos ^{2} \\vartheta +W_{S,J\\perp }(t) \\sin ^{2} \\vartheta $ , where $W_{\\parallel }(t)$ and $W_{\\perp }(t)$ are the energy components at parallel ($\\vartheta =0$ ) and transversal ($\\vartheta =\\pi /2$ ) orientations of the dipole moment, respectively.",
"As an example in Figure REF we show the results of model calculations for the time evolution of the energy components $W_{S}(t)$ and $W_{J}(t)$ for two extreme orientations of the dipole magnetic field with $\\vartheta =0$ and $\\vartheta =\\pi /2$ .",
"We recall that at $0\\leqslant t\\leqslant r_{c}/c$ , i.e.",
"the electromagnetic pulse does not yet reach to the surface of the control sphere, the energy flux vanishes and $W_{S}(t)=0$ .",
"Thus, for a given radius $r_{c}$ of the control sphere the energy flux occurs within the time interval $\\Delta t= (r_{c}/c) (1/\\beta -1)$ .",
"Note that this interval decreases with the expansion velocity.",
"As expected in Figure REF it is seen that both components of the energy $W_{S}(t)$ and $W_{J}(t)$ increase monotonically with time and strongly depend on the orientation $\\vartheta $ of the magnetic dipole.",
"In addition, since the dipole magnetic field at the location of the plasma cloud is stronger at smaller orientation angles $\\vartheta $ the energy flux $W_{S}(t)$ and the energy loss $W_{J}(t)$ are maximal for the parallel (or antiparallel) orientation of $\\mathbf {p}$ .",
"It is also noteworthy the dependence of the energy components on the relativistic factor $\\beta $ of the plasma expansion.",
"In the case of the non-relativistic expansion the energy flux $W_{S}(t)$ is systematically larger than the energy loss $W_{J}(t)$ (see Figure REF , left panel) at least in the wide interval of the expansion time (for comparison see Figure REF below).",
"Furthermore, both energies increase with the factor $\\beta $ but now $W_{J}(t)$ exceeds $W_{S}(t)$ in the wide interval of the expansion time (Figure REF , right panel).",
"Figure: (Color online) The ratios Γ S (t)\\Gamma _{S}(t) (the lines without symbols) and Γ J (t)\\Gamma _{J}(t) (the lines with symbols)for four values of the factor β\\beta as a function of ct/r 0 ct/r_{0} calculated from expressions () and() with r c =0.5r 0 r_{c}=0.5r_{0}.",
"The solid and dashed lines correspond to the dipole orientations ϑ=0\\vartheta =0and ϑ=π/2\\vartheta =\\pi /2, respectively.To gain more insight in Figure REF we show also the time evolution of the ratios $\\Gamma _{S}(t)=W_{S}(t)/Q_{0}(t)$ and $\\Gamma _{J}(t)=W_{J}(t)/Q_{0}(t)$ for $\\vartheta =0$ and $\\vartheta =\\pi /2$ .",
"The relativistic factor $\\beta $ varies from the non-relativistic ($\\beta =0.01$ ) to the relativistic ($\\beta =0.9$ ) values.",
"Here $Q_{0}(t)=Q(\\xi )$ is the dipole magnetic energy escaped from the plasma sphere.",
"Previously it has been found [15] that for the non-relativistic ($\\beta \\ll 1$ ) expansion of a one-dimensional plasma slab and for uniform external magnetic field approximately the half of the outgoing energy is gained from the plasma, while the other half is gained from the magnetic energy, $W_{S}\\simeq 2W_{J}\\simeq 2\\Delta W_{\\mathrm {EM}}$ .",
"In the case of the expansion of a spherical plasma with $\\beta \\ll 1$ in the uniform magnetic field, $W_{J}=1.5Q_{0}$ , $\\Delta W_{\\mathrm {EM}}=0.5Q_{0}$ , and $W_{S}=2Q_{0}$ , where $Q_{0}=H_{0}^{2}R^{3}/6$ [18].",
"Therefore in this case the released electromagnetic energy is mainly gained from the plasma.",
"In contrast to the cases with uniform magnetic field discussed above in the present context there are no simple relations between the energy components $W_{S}(t)$ , $W_{J}(t)$ and $Q_{0}(t)$ .",
"However, at the initial stage ($t\\ll r_{c}/v$ ) of non-relativistic expansion the dipole field at large distances can be treated as uniform and the energies $W_{S}(t) $ and $W_{J}(t)$ are close to the values $2Q_{0}(t)$ and $1.5Q_{0}(t)$ (Figure REF ), respectively.",
"It is noteworthy that for arbitrary $\\beta $ the ratios $\\Gamma _{S}(t)$ and $\\Gamma _{J}(t)$ are only weakly sensitive to the orientation of the dipole magnetic field although the quantities $W_{S}(t)$ , $W_{J}(t)$ and $Q_{0}(t)$ strongly depend on $\\vartheta $ (see Figure REF and Eq.",
"(REF ) with (A.3), (A.4)).",
"Moreover, for any $\\beta $ the ratio $\\Gamma _{J}(t)$ is almost constant and may be approximated as $\\Gamma _{J}(t)\\simeq \\Gamma _{J}(0)$ or alternatively $W_{J}(t)\\simeq 1.5CQ_{0}(t)$ , where $C=\\gamma ^{-6}(1-\\beta )^{-4}(1+2\\beta )^{-2}$ is some kinematic factor which is independent on the dipole orientation $\\vartheta $ .",
"These features merely indicate that for arbitrary $\\beta $ the energy loss $W_{J}(t)$ increases proportionally to the escaped energy $Q_{0}(t)$ while the parallel ($W_{S\\parallel }(t)$ ) and the transversal ($W_{S\\perp }(t)$ ) components of the emitted energy increase proportionally to the parallel ($Q_{0\\parallel }(t)$ ) and transversal ($Q_{0\\perp }(t)$ ) components of the escaped energy, respectively, with $W_{S\\parallel }(t)/Q_{0\\parallel }(t) \\simeq W_{S\\perp }(t)/Q_{0\\perp }(t)\\simeq \\Gamma _{S}(t)$ .",
"Finally, in Figures REF and REF it is seen that for non-relativistic expansion the emitted energy is gained from both the plasma cloud and electromagnetic energy in the control sphere.",
"However, at the final stage ($t=r_{c}/v$ ) of the relativistic expansion (with $\\beta \\sim 1$ ) $W_{S}\\simeq W_{J}$ and hence in this case the emitted energy $W_{S}$ is mainly gained from the plasma."
],
[
"Conclusion",
"An exact solution of the uniform radial expansion of a neutral, infinitely conducting plasma sphere in the presence of a dipole magnetic field has been obtained for an arbitrary orientation of the dipole moment in the space.",
"Thus we have generalized our result obtained in Ref.",
"[4] for a parallel (or antiparallel) orientation of the dipole moment when $\\mathbf {p} \\parallel \\mathbf {r}_{0}$ .",
"The electromagnetic fields are derived by using the appropriate initial and boundary conditions, Eqs.",
"(REF ) and (REF ), respectively.",
"As expected the electromagnetic fields are perturbed only within the domain located between the surfaces of the expanding plasma ($r=R$ ) and the information ($r=ct$ ) spheres.",
"Outside the sphere $r=ct$ the magnetic field is not perturbed and coincides with the dipole magnetic field.",
"In addition, since the dipole magnetic field at the location of the plasma cloud is stronger for parallel (or antiparallel) orientation the induced electromagnetic fields are maximal for $\\mathbf {p} \\parallel \\mathbf {r}_{0}$ .",
"We have also studied the energy balance during the plasma sphere expansion with an arbitrary orientation of the dipole.",
"The model calculations demonstrate that the energy components increase monotonically with time and strongly depend on the orientation of the dipole moment (Figure REF ), being maximal at $\\vartheta =0,\\pi $ .",
"On the other hand, the ratios $\\Gamma _{S}(t)$ and $\\Gamma _{J}(t)$ are only weakly sensitive to the variation of $\\vartheta $ , the latter being a slowly varying function of time (Figure REF ).",
"This is because the energy loss $W_{J}(t)$ varies almost linearly with the escaped energy $Q_{0}(t)$ while the parallel ($W_{S\\parallel }(t)$ ) and the transversal ($W_{S\\perp } (t)$ ) components of the emitted energy vary linearly with the parallel ($Q_{0\\parallel }(t)$ ) and transversal ($Q_{0\\perp }(t)$ ) components of the escaped energy, respectively.",
"The calculations also show that for non-relativistic expansion the emitted energy is gained from both the plasma cloud and perturbations of the electromagnetic energy in the control sphere.",
"For relativistic expansion $W_{S}\\simeq W_{J}$ and the emitted energy is practically gained only from the plasma sphere.",
"Lastly, to solve the boundary value problem we have applied the conical flow technique which is only valid for the uniform expansion of the plasma cloud with constant $v$ .",
"In principle the solution for the arbitrary time-dependent velocity $v(t)$ can be obtained by the Laplace transformation technique used, for instance, in [4].",
"This yields an integral equation for the vector potential which allows a general analytical solution only for a one–dimensional expansion.",
"For two– and three–dimensional expansions the general solutions are not known and these cases require separate investigations.",
"This work has been supported by the State Committee of Science of Armenian Ministry of Higher Education and Science (Project No.",
"11–1c317)."
],
[
"Appendix: Some summation formulas",
"Using the known relation [21] $\\chi (x) =\\sum _{\\ell =1}^{\\infty }x^{2\\ell }=\\frac{x^{2}}{1-x^{2}} , \\qquad \\mathrm {(A.1)}$ where $x<1$ , one can derive some summation formulas which are used in the main text of the paper.",
"The first relation is evaluated as $f(x) =\\sum _{\\ell =1}^{\\infty }\\frac{\\ell (\\ell +1) }{2\\ell +1}x^{2\\ell +1}= \\frac{1}{4} \\left\\lbrace x\\frac{\\partial }{\\partial x} [x\\chi (x)] -\\int _{0}^{x}\\chi (t) dt \\right\\rbrace \\nonumber \\\\=\\frac{1}{8}\\left[ \\frac{x(1+x^{2})}{(1-x^{2})^{2}}- \\frac{1}{2}\\ln \\frac{1+x}{1-x}\\right] .",
"\\qquad \\mathrm {(A.2)}$ In (A.2) we have used the boundary condition $f(0)=0$ .",
"Similarly one obtains: $f_{1}(x) =\\sum _{\\ell =1}^{\\infty }\\frac{\\ell (\\ell +1) ^{2}}{2\\ell +1}x^{2\\ell }=\\frac{1}{2x}\\frac{\\partial }{\\partial x}\\left[ xf(x) \\right] \\nonumber \\\\=\\frac{1}{8}\\left[ \\frac{8x^{2}-x^{4}+1}{(1-x^{2})^{3}}-\\frac{1}{2x}\\ln \\frac{1+x}{1-x}\\right] , \\\\f_{2}(x) =\\sum _{\\ell =1}^{\\infty }\\frac{\\ell ^{2}(\\ell +1)}{2\\ell +1}x^{2\\ell }=\\frac{x}{2}\\frac{\\partial }{\\partial x}\\left[ \\frac{1}{x}f(x) \\right] \\nonumber \\\\=\\frac{1}{8}\\left[ \\frac{x^{4}+8x^{2}-1}{(1-x^{2}) ^{3}}+\\frac{1}{2x}\\ln \\frac{1+x}{1-x}\\right] .",
"\\qquad \\mathrm {(A.3,A.4)}$"
]
] | 1204.0874 |
[
[
"Cosmological neutrino entropy changes due to flavor statistical mixing"
],
[
"Abstract Entropy changes due to delocalization and decoherence effects should modify the predictions for the cosmological neutrino background (C$\\nu$B) temperature when one treats neutrino flavors in the framework of composite quantum systems.",
"Assuming that the final stage of neutrino interactions with the $\\gamma e^{-}e^{+}$ radiation plasma before decoupling works as a measurement scheme that projects neutrinos into flavor quantum states, the resulting free-streaming neutrinos can be described as a statistical ensemble of flavor-mixed neutrinos.",
"Even not corresponding to an electronic-flavor pure state, after decoupling the statistical ensemble is described by a density matrix that evolves in time with the full Hamiltonian accounting for flavor mixing, momentum delocalization and, in case of an open quantum system approach, decoherence effects.",
"Since the statistical weights, $w$, shall follow the electron elastic scattering cross section rapport given by $0.16\\,w_{e} = w_{\\mu} = w_{\\tau}$, the von-Neumann entropy will deserve some special attention.",
"Depending on the quantum measurement scheme used for quantifying the entropy, mixing associated to dissipative effects can lead to an increasing of the flavor associated von-Neumann entropy for free-streaming neutrinos.",
"The production of von-Neumann entropy mitigates the constraints on the predictions for energy densities and temperatures of a cosmologically evolving isentropic fluid, in this case, the cosmological neutrino background.",
"The effects of entropy changes on the cosmological neutrino temperature are quantified, and the {\\em constraint} involving the number of neutrino species, $N_{\\nu} \\approx 3$, in the phenomenological confront with Big Bang nucleosynthesis parameters is consistently relieved."
],
[
"Cosmological neutrino entropy changes due to flavor statistical mixing A. E. Bernardini [email protected] [On leave of absence from] Departamento de Física, Universidade Federal de São Carlos, PO Box 676, 13565-905, São Carlos, SP, Brasil.",
"Departamento de Física e Astronomia, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal.",
"Entropy changes due to delocalization and decoherence effects should modify the predictions for the cosmological neutrino background (C$\\nu $ B) temperature when one treats neutrino flavors in the framework of composite quantum systems.",
"Assuming that the final stage of neutrino interactions with the $\\gamma e^{-}e^{+}$ radiation plasma before decoupling works as a measurement scheme that projects neutrinos into flavor quantum states, the resulting free-streaming neutrinos can be described as a statistical ensemble of flavor-mixed neutrinos.",
"Even not corresponding to an electronic-flavor pure state, after decoupling the statistical ensemble is described by a density matrix that evolves in time with the full Hamiltonian accounting for flavor mixing, momentum delocalization and, in case of an open quantum system approach, decoherence effects.",
"Since the statistical weights, $w$ , shall follow the electron elastic scattering cross section rapport given by $0.16\\,w_{e} = w_{\\mu } = w_{\\tau }$ , the von-Neumann entropy will deserve some special attention.",
"Depending on the quantum measurement scheme used for quantifying the entropy, mixing associated to dissipative effects can lead to an increasing of the flavor associated von-Neumann entropy for free-streaming neutrinos.",
"The production of von-Neumann entropy mitigates the constraints on the predictions for energy densities and temperatures of a cosmologically evolving isentropic fluid, in this case, the cosmological neutrino background.",
"Our results states that the quantum mixing associated to decoherence effects are fundamental for producing an additive quantum entropy contribution to the cosmological neutrino thermal history.",
"According to our framework, it does not modify the predictions for the number of neutrino species, $N_{\\nu } \\approx 3$ .",
"It can only relieve the constraints between $N_{\\nu }$ and the neutrino to radiation temperature ratio, $T_{\\nu }/T_{\\gamma }$ , by introducing a novel ingredient to re-direct the interpretation of some recent tantalizing evidence than $N_{\\nu }$ is significantly larger than by more than 3.",
"03.65.Ta, 14.60.Pq, 98.80.-k flavor mixing and oscillation, neutrino, cosmology, density matrix The question of late-time entropy production that leads to changes on the cosmological neutrino background (C$\\nu $ B) temperature [1], [2], [3], [4] has been recently posed as theoretical puzzle on the foundations of the cosmological standard model.",
"The textbook literature sets that, due to the electron-positron ($e^+e^-$ ) annihilations which heated the background radiation after neutrino-radiation decoupling, the C$\\nu $ B temperature should be lower by a factor $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ when compared to the cosmological microwave background (CMB) temperature, where $T_{\\nu \\mbox{\\tiny $0$}}$ and $T_{\\gamma \\mbox{\\tiny $0$}}$ are, respectively, the C$\\nu $ B and CMB temperatures at present.",
"Speculative factors which may cause some slight departure from the standard value of $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ produce an overall increasing of the neutrino energy density of an order of $1\\%$ , i. e. a tiny effect when included in practical calculations.",
"It is originated either from corrections due to finite-temperature quantum field theories, which lead to an additional slight heating of the neutrinos [5], or from a secondary heating due to $e^+e^-$ annihilation prior to decoupling [2].",
"The questions posed in this letter are therefore how flavor mixing effects could introduce additional corrections to $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}}$ , which become effective some time after decoupling, deep inside the free-streaming regime, and how it affects the constraints over the number of neutrino species.",
"Flavor related von-Neumann entropies are therefore relevant in the context of describing neutrino flavor oscillations in the framework of composite quantum system [6].",
"Any relative contribution to the entropy changes of a cosmologically evolving isentropic fluid, namely the C$\\nu $ B, could modify the predictions for its corresponding temperature which would affect the rapport to the energy densities.",
"Since the von-Neumann entropy for a composite quantum system of well-defined flavor quantum numbers is obtained in the framework of the generalized theory of quantum measurement, we extend such a discussion to the arena of cosmological neutrinos and we provide a general and functional characterization of the corresponding flavor associated entropies.",
"Let us recapitulate that the main difference between $T_{\\nu }$ and $T_{\\gamma }$ must arise from $e^+e^-$ annihilations after neutrino decoupling from the $\\gamma e^+e^-$ radiation plasma.",
"Prior to annihilations, the total entropy density that includes all the ultra-relativistic fermionic and bosonic species is given by [7] $s_i (a) = \\frac{2\\pi ^2}{45} \\, T_r^{3} \\left[ g_{\\gamma } + \\frac{7}{8} \\left(g_{e} + g_{\\nu }\\right)\\right],$ where $g_{\\gamma } = 2$ , $g_{e} = 4$ , and $g_{\\nu } = 6$ are the respective numbers of degrees of freedom for photon, electron/positron and neutrino/antineutrino according to the Standard Model, and $T_r$ is the temperature of the radiation plasma.",
"After annihilations, the electrons and positrons have gone away and the photon and neutrino temperatures are no longer identical.",
"The total entropy density is thus given in terms of the above defined temperatures, $T_{\\nu }$ and $T_{\\gamma }$ , by $s_f (a) = \\frac{2\\pi ^2}{45} \\, T_{\\nu }^{3} \\left[ g_{\\gamma }\\, \\frac{ T_{\\gamma }^{3}}{ T_{\\nu }^{3}} + \\frac{7}{8} g_{\\nu }\\right].$ Assuming the isentropic evolution over Eqs.",
"(REF ) and (REF ), one sets $a^{3} \\, s_i(a) = a^{3} \\, s_f(a)$ and easily obtains the temperature ratio $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ , which can be substituted into the energy and particle number density definitions, $\\rho $ and $n$ , in order to give $\\rho _{\\nu }/\\rho _{\\gamma } = N_{\\nu } (7/8) (4/11)^{4/3}$ and $n_{\\nu }/n_{\\gamma } = 3/11$ , where $N_{\\nu }$ is the number of neutrino species [7].",
"One should notice that the value of $g_{\\nu } = 6$ is introduced in the absence of right-handed neutrinos.",
"As the right handed neutrinos are not produced in the early universe (the masses are so small as to be irrelevant) so only one spin state is produced and $g = 6$ Indeed $g = 12$ is already completely ruled out by cosmological data..",
"Meanwhile, it is straightforward to verify that the number of $\\nu $ degrees of freedom do not affect the rapport between $T_{\\nu }$ and $T_{\\gamma }$ .",
"Moreover, as usually noticed in the literature, the temperature ratio of radiation and neutrinos is assumed to be constant.",
"Once neutrinos are massive they may enter the non-relativistic regime below a scale factor $a$ of about $10^{-4}$ .",
"The reference [8] provides us with a suitable explanation for the consequence of finite neutrino thermal speed: the initial phase-space density for massive neutrinos is a relativistic Fermi-Dirac distribution, preserved from the time when the neutrinos decoupled in the early universe.",
"Decreasing the temperature with time is compensated by relating proper momentum to comoving momentum.",
"Therefore, ignoring perturbations, the present-day distribution for massive neutrinos is the relativistic Fermi-Dirac - not the equilibrium nonrelativistic distribution - because the phase space-distribution was preserved after neutrino decoupling.",
"The neutrinos distribution function is similar to that for a massless particle and neutrinos temperature scales as $a^{-1}$ .",
"Assuming that free-streaming flavor oscillating neutrinos evolve isentropically straightforward to the era of low temperatures, the maximal free streaming neutrino entropy may change due to decoherence effects.",
"It is parameterized by $\\delta S_{\\mbox{\\tiny VN}}$ into the following relation for neutrino temperatures, $g_{\\nu } \\left(\\frac{7}{8}\\right) \\frac{2\\pi ^2}{45}\\, T^{3}_{\\nu } = g^{\\prime }_{\\nu } \\left[\\left(\\frac{7}{8}\\right) \\frac{2\\pi ^2}{45}\\, T^{\\prime \\,3}_{\\nu } \\pm n_{\\nu }\\, \\delta S_{\\mbox{\\tiny VN}}\\right].$ By observing that $n_{\\nu } = (9/5)\\,\\pi ^{-2}\\, T^{\\prime 3}_{\\nu }$ , and assuming that $g^{\\prime }_{\\nu } = g_{\\nu }$ , i. e. that the number of degrees of freedom is conserved, Eq.",
"(REF ) results in $T_{\\nu } = T^{\\prime }_{\\nu } \\left(1 \\pm \\frac{324}{7 \\pi ^{4}}\\,\\delta S_{\\mbox{\\tiny VN}} \\right)^{\\frac{1}{3}}$ where, as we shall notice, $\\delta S_{\\mbox{\\tiny VN}}$ stands for the associated von-Neumann entropy per flavor quantum ensemble in a volume $dp^{3}\\,dq^{3}$ of the phase-space, so that $n_{\\nu } \\,\\delta S_{\\mbox{\\tiny VN}}$ corresponds to the total von-Neumann entropy density.",
"The modifications introduced by Eq.",
"(REF ) are guaranteed by the subadditivity of entropies associated to density matrices that describe independent quantum systems $A$ and $B$ , i. e. systems with completely or partially uncorrelated quantum properties.",
"For cosmological neutrino ensembles, the quantum property related to the system $A$ is approximated by the momentum, $p$ , which is read as a quantum phase-space element through its relation to the classical momentum established by the distribution function, $f(p)$ , utilized for computing averaged quantities like $\\rho _{\\nu }$ and $n_{\\nu }$ [7].",
"The quantum property related to the system $B$ is obviously the flavor quantum number.",
"The triangle inequality given by $|S_A -S_B|\\,\\le \\, S_{AB} \\, \\le \\, |S_A + S_B|,$ sets upper and lower limits for the total entropy, $S_{AB}$ , that describes a composite quantum system with the abovementioned characteristics.",
"While according to the Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts [9], in quantum mechanics the triangle inequality sets that the entropy of the joint system can be less than the sum of the entropy of its components due to the possibility of entanglement.",
"It happens when the localization character introduced by the momentum distribution function, $f(p)$ , changes the pattern of flavor oscillations, as an intrinsic decoherence mechanism promoted by some dynamics of the delocalization effect similar to those caused by the space-time evolution of mass-eigenstate wave-packets.",
"In parallel, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are completely uncorrelated, i. e. when the entanglement disappears.",
"Turning back to the main point of our analysis, let us report about some foundations on quantum statistics in order to quantify $\\delta S_{\\mbox{\\tiny VN}}$ into Eq.",
"(REF ).",
"The density matrix representation of a composite quantum system of three flavor species, namely $e$ , $\\mu $ and $\\tau $ , is given by $\\rho \\mbox{\\footnotesize $(t)$} \\equiv \\rho = \\sum _{\\alpha = e,\\mu ,\\tau }w_{\\alpha } \\, M^{\\alpha }_{(t)} ~~\\mbox{with} ~~ \\sum _{\\alpha = e,\\mu ,\\tau } w_{\\alpha } = 1,$ where $w_{\\alpha }$ are the statistical weights, and $M^{\\alpha }_{(t)}$ are the $\\alpha $ -flavor projection operators which depend on the flavor associated mixing angles and are constrained by the unitarity condition $\\sum _{\\alpha } M^{\\alpha }_{(t)} = \\mathbf {1}$ .",
"The von-Neumann entropy provides an important functional defined in terms of the density matrix as $S\\mbox{\\footnotesize $(\\rho )$} = - \\mbox{Tr}\\lbrace \\rho \\, \\ln \\mbox{\\footnotesize $(\\rho )$}\\rbrace ,$ where the Boltzmann constant, $k_{B}$ , was set equal to unity.",
"The entropy $S\\mbox{\\footnotesize $(\\rho )$}$ quantifies the departure of a composite quantum system from a pure state, i. e. it implicitly measures the entanglement of an ensemble of flavor states describing a given finite system.",
"As one can expect, quantum measurements induce modifications on the the von-Neumann entropy of the system.",
"The entropy change due to a non-selective measurement scheme described by operations parameterized by the projection operators $M^{\\alpha }_{(0)}$ [9] is given by $\\Delta S = S\\mbox{\\footnotesize $(\\rho ^{\\prime })$} - S\\mbox{\\footnotesize $(\\rho )$} \\ge 0,$ where $S\\mbox{\\footnotesize $(\\rho ^{\\prime })$} =S\\left(\\sum _{\\alpha } {P^{\\alpha }_{(t)} \\rho _{\\alpha }}\\right),$ with $\\rho _{\\alpha } = \\left(P^{\\alpha }_{(t)}\\right)^{-1} \\, M^{\\alpha }_{(0)}\\, \\rho \\, M^{\\alpha }_{(0)},$ and where $P^{\\alpha }_{(t)}$ are the probabilities of measuring $\\alpha $ -flavor eigenstates at time $t$ .",
"In terms of the density matrix, one has $P^{\\alpha }_{(t)} = \\mbox{Tr}\\lbrace M^{\\alpha }_{(0)}\\, \\rho \\rbrace &=& \\sum _{\\beta = e,\\mu ,\\tau }{w_{\\beta } \\, \\mbox{Tr}\\lbrace M^{\\alpha }_{(0)}\\,M^{\\beta }_{(t)}\\rbrace } = \\sum _{\\beta = e,\\mu ,\\tau }{w_{\\beta } \\,\\mathcal {P}_{\\alpha \\rightarrow \\beta }\\mbox{\\footnotesize $(t)$}},$ with $\\mathcal {P}_{\\alpha \\rightarrow \\beta }\\mbox{\\footnotesize $(t)$} = |\\langle \\nu ^{\\beta }_{(0)}\\mid \\nu ^{\\alpha }_{(t)} \\rangle |^{\\mbox{\\tiny $2$}}$ describing the $\\alpha $ - to $\\beta $ -flavor conversion probabilities in the single-particle quantum mechanics framework.",
"We assume that $\\delta S_{\\mbox{\\tiny VN}}$ quantifies the level of flavor-mixing during the evolution of cosmological neutrino ensembles either from pure states to statistical mixtures, or from statistical mixtures to maximal statistical mixtures.",
"It results into a kind of late-time entropy production.",
"The above conceptions related to the von-Neumann entropy bring up important insights into the scope of distinguishing measurement procedures [9] and quantifying the degree of mixture of statistical ensembles.",
"In order to distinguish possible decoherence effects, i. e. those caused by dissipative mechanisms (extrinsic decoherence) from those due to delocalization characteristics (intrinsic decoherence), we shall parameterize $\\delta S_{\\mbox{\\tiny VN}}$ by two different ways.",
"Both effects lead to increasing entropy density values after neutrino decoupling.",
"The cosmological standard model prescription for neutrino decoupling in the early universe sets that the three neutrino species ($e$ , $\\mu $ , $\\tau $ ) are kept in thermal contact with the radiation plasma through the elastic scattering process with background electrons(positrons).",
"Different flavor neutrinos ($e$ , $\\mu $ , $\\tau $ ) coexist with the same averaged temperature: neutrinos corresponding to the same volume of the phase space (that is constrained by some momentum distribution), reach the thermal equilibrium through measurement schemes produced by the elastic scattering.",
"The proportion between the corresponding cross sections, $\\sigma _{\\nu }$ , is given by $\\sigma _e \\,:\\,\\sigma _{\\mu }\\,:\\, \\sigma _{\\tau } \\Leftrightarrow 1\\,:\\,0.16\\,:\\,0.16$ After scattering ends up, one should have an averaged statistical ensemble described by $1\\,:\\,0.16\\,:\\,0.16 \\Leftrightarrow w_e \\,:\\,w_{\\mu }\\,:\\,w_{\\tau }$ where we have introduced the statistical weights $w_{\\alpha }$ , with $\\alpha = $ $e$ , $\\mu $ and $\\tau $ Since it does not correspond to the maximal statistical mixture, $w_e = w_{\\mu } = w_{\\tau }$ , decoherence effects may lead the free streaming flavor ensemble to the maximal entropy configuration.",
"The composite quantum system of neutrino flavors certainly reaches the configuration of a maximal statistical mixture corresponding to $S = \\ln (3)$ before entering the non-relativistic regime.",
"Due to some extrinsic decoherence mechanism, one identifies the variation of the von-Neumann entropy as a deviation from the entropy of such a maximal statistical mixing through $\\delta S_{\\mbox{\\tiny VN}} = \\ln (3) - S(\\rho _{(t=0)})$ , for which $t=0$ is defined as the time of neutrino decoupling and $S(\\rho _{(t=0)})$ is computed in terms of the statistical weights $w_{\\alpha }$ .",
"The alternative way of quantifying such an entropy increasing after decoupling is through the parametrization of $\\delta S_{\\mbox{\\tiny VN}}$ by $\\delta S^{DL}_{\\mbox{\\tiny VN}} = S(\\langle \\rho \\rangle _{\\rm time}) - S(\\rho _{(t=0)})$ Assuming that some intrinsic (dynamical) decoherence mechanism suppresses the time dependence of the non-diagonal elements of the density matrix, the delocalization (DL) effects can be reproduced by time-averaging the density matrix: $\\langle \\rho \\rangle _{\\rm time}$ , Both parameterizations can be conceptually modified by adding to $\\delta S^{DL}_{\\mbox{\\tiny VN}}$ the entropy change due to a non-selective measurement scheme given by $\\Delta S (t = 0)$ from Eq.",
"(REF ).",
"As one can depict from Fig.",
"REF , the non-selective measurements do not change $\\delta S_{\\mbox{\\tiny VN}}$ at $t = 0$ .",
"By assuming the next to standard phenomenological values for the neutrino flavor mixing angles, namely the tri-bimaximal approximation $\\theta _{12} = \\arcsin {(1/\\sqrt{3})}$ , $\\theta _{23} = \\pi /4$ , and $\\theta _{13} = 0$ , one can depict from Fig.",
"REF the modifications on the averaged ratio $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ due to entropy changes produced by flavor statistical mixing as functions of the electronic statistical weight, $w_e$ .",
"Besides the unitarity given by Eq.",
"(REF ), we have set $w_{\\mu } = w_{\\tau }$ in order to plot the curves.",
"Otherwise, by replacing $\\delta S_{\\mbox{\\tiny VN}} = \\ln (3) - S(\\rho _{(t=0)})$ by $\\delta S^{DL}_{\\mbox{\\tiny VN}} = S(\\langle \\rho \\rangle _{\\rm time}) - S(\\rho _{(t=0)})$ into Eq.",
"(REF ), one quantifies the degree of mixture of the flavor definite statistical ensemble as function of its mixing properties since the decoherence is exclusively due to delocalization effects, i. e. an intrinsic decoherence mechanism.",
"The differences due to flavor mixing properties can be depicted from the Fig.",
"REF for which we have considered three particular cases: i) the next to standard phenomenological values for the neutrino flavor mixing angles, namely the tri-bimaximal mixing approximation for which $\\theta _{12} = \\arcsin {1/\\sqrt{3}}$ , $\\theta _{23} = \\pi /4$ , and $\\theta _{13} 0$ ii) the maximal mixing for which $\\theta _{12} =\\theta _{23} = \\theta _{13}= \\pi /4$ ; and iii) no mixing for which $\\theta _{12} =\\theta _{23} = \\theta _{13}= 0$ .",
"In case of analyzing $\\delta S_{\\mbox{\\tiny VN}}$ (c. f. Fig.",
"REF ) the entropy change ($\\Delta S$ ) due to a non-selective measurement intervention is explicit.",
"To sum up the relevant aspects of the above properties, Fig.",
"REF show the upper and lower limits for the modifications on the rate $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ due to the corrections obtained from Eq.",
"(REF ) for the cases where $\\delta S_{\\mbox{\\tiny VN}} = \\ln (3) - S(\\rho ^{\\prime })$ (red lines) and $\\delta S_{\\mbox{\\tiny VN}} = \\ln (3) - S(\\rho )$ (blue lines).",
"Fig.",
"REF follows the same correspondence.",
"As expected, Figs.",
"REF and REF reveals that the maximal decoherence effects occur for $w_e = 1$ which correspond to a pure state configuration at time of neutrino decoupling.",
"One also notices that quantum mixing is fundamental for introducing the additive quantum entropy.",
"It is convenient to notice that the entropy change due to a non-selective measurement performed over a maximal statistical mixture is null, i. e. $S(\\rho ) = S(\\rho ^{\\prime })$ .",
"For the values corresponding to the rapport from Eq.",
"(REF ), one should have $w_{e}\\sim 0.68$ .",
"It leads to corrections of the order of 3-$4\\%$ on the maximal bounds.",
"Figure: Upper (thick lines) and lower (thin line) maximal limits for modifications on the averaged ratio T ν0 /T γ0 =(4/11) 1/3 T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3} due to maximal entropy changes δS VN \\delta S_{\\mbox{\\tiny VN}}.The flavor statistical mixing is parameterized by the electronic statistical weight, w e w_e.We have considered a three-level composite quantum system with neutrino mixing angles approximated by the tri-bimaximal parameters, i. e. θ 12 =arcsin1/3\\theta _{12} = \\arcsin {1/\\sqrt{3}}, θ 23 =π/4\\theta _{23} = \\pi /4, and θ 13 =0\\theta _{13} = 0.The unitarity condition, w e +w μ +w τ =1w_{e} + w_{\\mu } + w_{\\tau } = 1, with w μ =w τ w_{\\mu } = w_{\\tau }, reduces the dependence of the entropies on the statistical weights to the one-degree of freedom dependence on w e w_{e}.",
"Blue and red lines describing the constraint of δS VN \\delta S_{\\mbox{\\tiny VN}} respectively with S(ρ)S(\\rho ) and S(ρ ' )S(\\rho ^{\\prime }) are coincident.Figure: Upper (thick lines) and lower (thin lines) limits for the modifications on the averaged rate T ν0 /T γ0 =(4/11) 1/3 T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3} due to late-time entropy changes δS VN DL \\delta S^{DL}_{\\mbox{\\tiny VN}} producedby intrinsic decoherence (delocalization) effects.The mixing properties are defined in terms of the mixing angles.In the first plot we have assumed the tri-bimaximal mixing with θ 12 =arcsin1/3\\theta _{12} = \\arcsin {1/\\sqrt{3}}, θ 23 =π/4\\theta _{23} = \\pi /4, and θ 13 =0\\theta _{13}= 0.In the second one we have set θ 12 =θ 23 =θ 13 =π/4\\theta _{12} =\\theta _{23} = \\theta _{13}= \\pi /4.And in the third one obviously no mixing is considered.Red lines are obtained through the non-selective quantum measuremet scheme.Finally, one should observe that the mixing entropy assumes its maximum value which, in case of an $n$ -level system, corresponds to $\\delta S_{m} = \\ln \\mbox{\\footnotesize $(n)$}$ .",
"For a three flavor system of neutrinos, it corresponds to $\\delta S_{\\mbox{\\tiny VN}} = 0$ .",
"For maximal statistical mixing, the non-selective measurement does not change neither the energy nor the entropy of the system while the selective measurement changes the entropy [9].",
"It means that the von-Neumann entropy and the above-related quantities gain relevance in the study of the measurement procedures which take into account the flavor eigenstate correspondence to measurable energies.",
"Observing that the entropy increasing follows the level of mixing of the systems, the maximal variation for the ratio $T_{\\nu \\mbox{\\tiny $0$}}/T_{\\gamma \\mbox{\\tiny $0$}} = (4/11)^{1/3}$ can be set through the reading of Eq.",
"(REF ) by means of Eq.",
"(REF ) that results in $\\left|\\left(1 - \\frac{324}{7 \\pi ^{4}}\\,\\ln {3} \\right)^{\\frac{1}{3}} \\right|\\lesssim \\frac{T_{\\nu \\mbox{\\tiny $0$}}}{T^{\\prime }_{\\nu \\mbox{\\tiny $0$}}}\\lesssim \\left|\\left(1 + \\frac{324}{7 \\pi ^{4}}\\,\\ln {3} \\right)^{\\frac{1}{3}} \\right|~~\\Rightarrow ~~0.86 \\lesssim \\frac{T^{\\prime }_{\\nu \\mbox{\\tiny $0$}}}{T_{\\nu \\mbox{\\tiny $0$}}}\\lesssim 1.28$ in case of realistic values given by the tri-bimaximal mixing.",
"The above values can be mitigated if one considers the $\\delta S^{DL}_{\\mbox{\\tiny VN}}$ in place of $\\delta S_{\\mbox{\\tiny VN}}$ .",
"In the same fashion of some eventual indirect exotic coupling of neutrinos to electrons or photons that could have kept neutrinos longer in equilibrium with photons, entropy changes due to flavor mixing introduce a novel ingredient that suggests that the $\\nu - \\gamma $ number density ratio could not be diluted by $4/11$ .",
"The possibility of attenuating the constraints on the late-time entropy production from the large scale structure and CMB anisotropies has already been considered [10], [11], [12], [13].",
"Herein the referred entropy modifications can change the rapport between the effective number of neutrino families, $N_{\\nu }$ , and any parameter phenomenologically depicted from the pattern of large scale structures and CMB anisotropies.",
"To clear up this point, let us assume that the standard value for the $\\nu -\\gamma $ energy density ratio, $7/8$ , leads to the following relation between the red-shift of matter-radiation equality and $N_{\\nu }$ , $1 + z_{eq} \\propto \\left[1 + \\frac{7}{8} \\left(\\frac{4}{11}\\right)^{4/3}N_{\\nu }\\right]^{-1}.$ To keep the ratio $\\rho _{\\nu }/\\rho _{\\gamma } = N_{\\nu } (7/8) (4/11)^{4/3}$ consistent with the phenomenology, the modifications introduced by the growing von-Neumann entropy discussed above introduce the upper and lower bounds to $N_{\\nu }$ , through the modified parameter $N^{\\prime }_{\\nu } \\sim N_{\\nu } \\left|\\left(1 \\pm \\frac{324}{7 \\pi ^{4}}\\,\\delta S_{\\mbox{\\tiny VN}(w_e = 1)} \\right)^{-\\frac{4}{3}}\\right|$ that, from this point, has to be interpreted as novel phenomenological parameter, and not as the number of neutrino species.",
"For $N_{\\nu } \\sim 3$ , one should have the bounds $1.7 \\lesssim N^{\\prime }_{\\nu } \\lesssim 8.1,$ where the bounds are for neutrino ensembles being produced as pure states after decoupling and evolving to a maximal statistical mixture in the free-streaming regime.",
"Essentially, we are not modifying the predictions for the number of neutrino species, $N_{\\nu }$, which was already accurately computed, for instance, at [17], where the distortions in the $\\nu _e$ and $\\nu _{\\mu }/\\nu _{\\tau }$ phase-space distribution that arise in the standard cosmology due to electron-positron annihilations have been considered.",
"Our results just relieve the constraints on the value of the parameter $N^{\\prime }_{\\nu }$ that re-enters into the expression derived from matter-radiation equality (c. f. Eq.",
"(REF )) and that could lead to some phenomenological tension [2], [20].",
"Our result enlarges the range of phenomenological agreement for tantalizing cosmological and terrestrial evidences that suggest the number of light neutrinos may be greater than three [21], [22].",
"A recent re-examination of cosmological bounds on extra light species have been performed [22] in order to consider the cosmological scenario with two sterile neutrinos and explore whether partial thermalization of the sterile states can mitigate the conflict between apparently ambiguous cosmological constraints on the number of neutrino species.",
"Accurately computed values of Helium abundance depicted from the Big Bang nucleosynthesis (BBN)formalism constrains the number of relativistic neutrino species present during nucleosynthesis, while measurements of the CMB angular power spectrum constrains the values of effective energy density of relativistic neutrinos and photons.",
"Therefore, scenarios where new sterile neutrino species may have different contributions to $N_{\\nu }^{(eff)}$ , respectively from BBN and CMB data, can be reconciled through the entropy corrections to C$\\nu $ B computed through the approach that we have introduced.",
"The same argument can be reported on the analysis of increasing the effective number of neutrino species, $N_{\\nu }$ , in the early universe, focussed on introducing extra relativistic species (hot dark matter or dark radiation [23]).",
"The above results does not change the significance and the magnitude of the finite-temperature electromagnetic corrections to the energy density of the $\\gamma e^+ e^-$ radiation plasma [17], [18], [19] or of the finite temperature QCD corrections [16].",
"They are of the same order of magnitude of flavor mixing corrections upon the averaged temperature of decoupling for different neutrino species, which also depend on the mixing parameters [16].",
"Although the above limits relieve the constraints on the possible values for $N_{\\nu }$ , neutrino heating/freezing corrections can be a little larger than those predicted by the finite temperature quantum field theories, for instance, late-time entropy production due to some weakly interacting scalar field decay [1], [2].",
"To conclude, one should observe that it is commonly assumed that ultra-relativistic thermal relics are in perfect equilibrium state even after decoupling.",
"For photons in cosmic microwave background (CMB) this has been established with a very high degree of accuracy.",
"Thus, the same assumption has been made about free-streaming neutrinos.",
"Our line of reasoning calls into attention the premise of coherent flavor eigenstates at the epoch of neutrino decoupling from the radiation background.",
"It has been assumed that the free evolution of flavor ensembles leads to some spontaneous lowering of the coherence interference effects, associated with the destruction of the oscillation pattern and with the vanishing of the 3-partite quantum entanglement.",
"Some previous studies have addressed to the issue of the decoherence history of the cosmological neutrinos and its implications on probing best values for neutrino masses and on modifying the power spectrum of large scale structures [14], [15], [16].",
"In the single-particle quantum mechanics framework, it has been supposed that flavor wave-packets could have spatial extents that would be comparable to the space-time curvature scale of the universe itself.",
"The delocalization effects lead to decoherence in the same way as we have quoted in this letter (c. f. Fig.",
"REF .)",
"Likewise, it could also result from some suitable prescription for some extrinsic dissipative mechanism that results into the wave function collapse, for instance, as a consequence of describing the flavor ensemble as an open quantum system.",
"Assuming that some kind of decoherence mechanism results in increasing the level of mixing for neutrino flavor ensembles in the cosmological background, the analysis developed here would not have only addressed to the entropy issues related to neutrinos, but would also suggest some new insights into the role of quantum coherence and decoherence in the history of these relic particles.",
"Finally, our results states that the flavor quantum mixing of neutrino mass eigenstates associated to decoherence effects are fundamental for producing an additive contribution of quantum entropy to the cosmological neutrino thermal history.",
"According to our framework, it does not modify the predictions for the number of neutrino species, $N_{\\nu } \\approx 3$ .",
"It can only relieve the constraints between $N_{\\nu }$ and the neutrino to radiation temperature ratio, $T_{\\nu }/T_{\\gamma }$ , by introducing a novel ingredient to re-direct the interpretation of some recent tantalizing evidence than $N_{\\nu }$ is significantly larger than by more than 3.",
"Obtaining the neutrino entropy changes well inside the free-streaming propagation regime is therefore a relevant aspect that has to be considered while computing cosmological neutrino properties, namely the cosmic energy density, the constraints related to the precise number of neutrino species, $N_{\\nu }$ , the specific entropy itself, and eventually, the neutrino mass values [6].",
"Acknowledgments - This work has been supported by the Brazilian Agencies FAPESP (grant 12/03561-0) and CNPq (grant 300233/2010-8)."
]
] | 1204.1504 |
[
[
"Conformational Collapse of Surface-Bound Helical Filaments"
],
[
"Abstract Chiral polymers are ubiquitous in nature and in the cellular context they are often found in association with membranes.",
"Here we show that surface bound polymers with an intrinsic twist and anisotropic bending stiffness can exhibit a sharp continuous phase transition between states with very different effective persistence lengths as the binding affinity is increased.",
"Above a critical value for the binding strength, determined solely by the torsional modulus and intrinsic twist rate, the filament can exist in a zero twist, surface bound state with a homogeneous stiffness.",
"Below the critical binding strength, twist walls proliferate and function as weak or floppy joints that sharply reduce the effective persistence length that is measurable on long lengthscales.",
"The existence of such dramatically different conformational states has implications for both biopolymer function {\\it in vivo} and for experimental observations of such filaments {\\it in vitro}."
],
[
"Introduction",
"Both prokaryotic and eukaryotic cells possess a rich variety of filamentous proteins with a wide range of compositions, sizes and flexibility [1].",
"These biopolymers play a vital role in a number of critical cellular functions including maintaining structural integrity, serving as a template for cell growth, cell adhesion and motility, mechanical signal transduction and cell division.",
"In many of these roles, the association of the polymers with membranes is critical and necessary for function.",
"For example, the actin rich cell cortex in eukaryotic cells is a major determinant of the cell's structural integrity and mediates adhesion and signal transduction.",
"A number of different actin binding proteins link the cortex with the membrane both directly, and indirectly through other proteins and complexes [2].",
"In the axons and dendrites of nerve cells, members of the MAP1 family link microtubules, which are critical for vesicle transport and mechanotransduction, to the plasma membrane [3].",
"In red blood cells, the remarkably flexible spectrin-actin network, responsible for cell shape regulation, is tightly associated with the membrane by a number of proteins such as ankyrin and dematin [4].",
"Similarly, the structural support lamin networks inside the nucleus are attached to the nuclear envelope by LAP family proteins [5].",
"During cell division, forces required for pinching in the division plane are generated by shrinking actin contractile rings attached to the membrane [6].",
"A similar phenomena occurs in bacterial cell division where the bacterial filamentous protein FtsZ makes up the contractile ring [7] which can be linked to the membrane by FtsA, ZipA and ZapA [8], [9].",
"Another bacterial protein MreB which provides structural support and guides cell wall synthesis is attached to the membrane via RodZ, MreC/D [10], [9] or even directly [11].",
"While these and several more examples illustrate the common nature of filamentous proteins binding to membranes in vivo, it is also known that many filamentous proteins can show direct interactions with membrane components in vitro typically through electrostatic or hydrophobic interactions [4].",
"This is of fundamental importance when considers that many experimental setups that study such filaments in vitro have them in close proximity or immobilized/fixed on a surface.",
"Clearly, understanding the association of filamentous proteins with membranes and surfaces is important from a scientific and technological perspective.",
"To date, there has a been a large amount of work studying polymers at interfaces and surfaces [12], [13].",
"However, many of these studies tend to neglect key features of real biopolymers that could have qualitative effects on their structure and their interactions with surfaces.",
"Firstly, a vast majority of biopolymers are chiral with an intrinsically helical structure.",
"This helicity can manifest itself in a number of fascinating ways such as the cholesteric liquid crystal and the blue phases of concentrated DNA solutions and the self-limiting bundle size of aggregates of generic chiral biopolymers such as actin [14], [15], [16], [17].",
"Furthermore, classical models of polymers such as the worm-like chain model do not take anisotropies in elastic moduli or finite widths of the biopolymers into account.",
"A number of recent efforts have therefore focused on the statistical mechanics of elastic “ribbons” with a finite aspect ratio as a more realistic model for typical biopolymers or other protein aggregates [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].",
"The ribbon-like nature of the polymers can lead to interesting effects including layering transitions in highly anisotropic condensates and the existence of an underlying helical structure even in the absence of twist [29], [30].",
"Thus the introduction of helicity and anisotropy can result in qualitative changes in the structure and dynamics of both single filaments and their aggregates.",
"Given the ubiquity of membrane and surface adsorbed biopolymers, in this paper, we take the first step toward understanding how helicity and anisotropy can influence the structure and dynamics of polymers that are bound to surfaces.",
"Rather than study a polymer with more detailed resolution, for example a full atomistic representation, we focus on the generic effects of intrinsic helicity on polymer adsorption using a minimal, coarse-grained description of molecular structure.",
"We show that the interplay between helicity and surface interactions can have profound effects on the structural and thermodynamic properties of the polymer.",
"We thus probe the consequences of helical structure on the large-length scale properties of absorbed chains, which is of potential relevance to a broad class of biomacromolecules.",
"We first build a general model that allows us to prescribe the thermodynamics of a helical and anisotropic filament binding to a surface of arbitrary curvature.",
"For the rest of the paper, we then treat only planar surfaces.",
"We first consider the infinitely stiff limit where the anisotropy becomes irrelevant and only the helicity plays a role.",
"The key conclusion from this analysis, is that the structure and conformational fluctuations of a helical polymer are critically sensitive to the attractive interactions with a surface.",
"We find that the filament undergoes a rapid and continuous phase transition from a strongly bound untwisted state to a weakly bound state for a critical value of the binding strength.The transition occurs via a proliferation of “twist-walls”, regions where the chain unbinds locally from the surface and rapidly twists a single turn.",
"We then consider how this transition alters the conformations and fluctuations of an anisotropic flexible filament on a planar surface.",
"The coupling between the twist and bend degrees of freedom introduced by the anisotropy and binding results in surface bound conformations with highly heterogeneous bending stiffness below the critical binding strength.",
"The transition between strongly and weakly bound states is then accompanied by a sharp drop in the effective bending rigidity of the polymer.",
"We demonstrate these conclusions, first, through an exact, “zero-temperature\" analysis of the continuum mechanical model of the filament that ignores thermal fluctuations of twist.",
"We then extend our analysis to finite-temperature by way of Monte Carlo simulations, which demonstrate that the sharp transition between the strongly-bound and weakly-bound states of helical filaments persists in the presence of both bending and torsion fluctuations of the chain backbone.",
"Thus, we predict that strong coupling of the twist degrees of freedom of a helical polymer to surface interactions, not only restructures the torsional state of the chain, but it also leads to a dramatic transition in the conformational fluctuations of the entire chain backbone.",
"This suggests that experimentally measured properties of biopolymers absorbed to a surface such as the persistence length and end-to-end size are crucially sensitive to the strength of surface interactions.",
"This paper is organized as follows.",
"In Section.REF , we build the general model for a helical and anisotropic filament binding to a surface of arbitrary curvature and in Section.REF , we describe our Monte Carlo simulation procedure.",
"In Section.REF , we analyze the model for planar surfaces in the infinitely stiff filament limit.",
"We specifically study the transition between the strongly bound and weakly bound states of the filament.",
"In Section.REF , we study how the transition influences the in-plane bending fluctuations of an anisotropic surface adsorbed filament.",
"We conclude with a short discussion in Section..",
"Here we introduce to a continuum model of a helical filament bound to an attractive surface.",
"For the purposes of generality, we consider the filament bound to a cylindrical surface of radius, $r$ .",
"This is a natural geometry to describe the binding the cylindrical cell wall of a bacterium, as in [9] , though we will specialize in the next section to the case of planar surfaces, $r \\rightarrow \\infty $ .",
"The filament is modelled by a helical strip, whose cross-section is rectangular with a thickness, $t$ and width, $w$ .",
"In the absence of external forces or interactions with the surface, the wide axis of the filament has a natural rotation rate, $\\omega _0$ , along the long axis of the filament (see Fig.",
"REF ).",
"Both thermal energy and surface interactions distort the filament geometry, leading to elastic costs which we described with the Kirchoff-Love of elastic beams [31].",
"The deformations of the filament are described by rotations of the material frame along the arc-length, $s$ , of the filament backbone.",
"We choose $\\hat{{\\bf e}}_1$ ($\\hat{{\\bf e}}_2$ ) to describe the local orientation of the wide (thin) axis of the filament, and $\\hat{{\\bf e}}_3 = \\hat{{\\bf t}}$ to describe the tangent.",
"Figure: Diagram defining the coordinates used to describe an intrinsically helical filament.",
"(a) A section of an intrinsically twisted filament which is forming a helix in a cylindrical coordinate system, where 𝐭 ^\\hat{{\\bf t}}, 𝐞 ^ 1 \\hat{{\\bf e}}_1 and 𝐞 ^ 2 \\hat{{\\bf e}}_2 are the material frame basis vectors.",
"The grey curve is the center line of the filament.",
"(b) The projection of the tangent vector, 𝐭 ^\\hat{{\\bf t}}, along z ^\\hat{z} and φ ^\\hat{\\phi } defines of the bending angle θ(s)\\theta (s).",
"(c) The rotation of the material coordinates 𝐞 ^ 1 \\hat{{\\bf e}}_1 and 𝐞 ^ 2 \\hat{{\\bf e}}_2 defines of the twist angle ψ(s)\\psi (s).Figure: Schematic of a helical filament with an anisotropic cross section.",
"On the right, we show a series of helical configurations for increasing strength of surface interactions, which show progressive unwinding of the helical twist.Here, we compute the mechanical energy of the filament in terms of an angle, $\\theta (s)$ , describing in-plane orientation of $\\hat{{\\bf t}}$ and an angle, $\\psi (s)$ , describing the orientation of the wide axis of the filament with respect to the surface normal.",
"If the center line of the filament, ${\\bf r}(s)$ , is confined to a helical surface (with long axis centered on the $x=y=0$ axis), then the tangent vector can be described in terms of the angle $\\theta (s)$ , where $\\partial _s {\\bf r} = \\hat{{\\bf t}}= \\cos \\theta (s) \\hat{z}+ \\sin \\theta (s) \\hat{\\phi },$ where $ \\hat{\\phi }= \\hat{z}\\times {\\bf r}(s)/r$ is the local, azimuthal direction on the cylinder.",
"The curvature of the filament center is determined by $\\partial _s \\hat{{\\bf t}}$ , which is non-zero when $\\theta ^{\\prime } \\ne 0$ and the filament curves in the surfaces of the cylinder, or when $\\sin \\theta \\ne 0$ , and the filament, following a straight, or geodesdic, path on the cylinder is deflected normal to the cylinder.",
"That is, the $\\hat{\\phi }$ direction changes along the arc-length as the curve is convected around the cylinder axis as $\\partial _s \\hat{\\phi }=-\\frac{\\sin \\theta }{r} \\hat{r}$ where the radial direction is given by $ \\hat{r}= [{\\bf r}- \\hat{z}({\\bf r}\\cdot \\hat{z})]/r$ .",
"Hence, $\\partial _s \\hat{{\\bf t}}= - \\frac{ \\sin ^2 \\theta }{r} \\hat{r}+ \\theta ^{\\prime } (\\hat{{\\bf t}}\\times \\hat{r}),$ where $\\hat{{\\bf t}}\\times \\hat{r}= \\sin \\theta \\hat{z}- \\cos \\theta \\hat{\\phi }$ .",
"To describe the local anisotropy of the filament cross-section we define the material directions, perpendicular to $\\hat{{\\bf t}}$ , $\\hat{{\\bf e}}_1 = \\sin \\psi \\hat{r}- \\cos \\psi (\\hat{{\\bf t}}\\times \\hat{r}) ,$ and $\\hat{{\\bf e}}_2 = \\cos \\psi \\hat{r}+ \\sin \\psi (\\hat{{\\bf t}}\\times \\hat{r}) .$ Defining $\\kappa _i = \\hat{{\\bf e}}_i \\cdot \\partial _s \\hat{{\\bf t}}$ as the rate of back bone bending into the direction $\\hat{{\\bf e}}_i$ , we have, $\\kappa _1 = \\kappa \\sin (\\psi -\\delta ), \\ \\kappa _2 = \\kappa \\cos (\\psi - \\delta ),$ where $\\kappa ^2 = (\\theta ^{\\prime })^2 + \\frac{ \\sin ^4 \\theta }{r^2},$ is the square curvature of the central axis and $\\tan \\delta = r \\theta ^{\\prime }/\\sin ^2 \\theta $ .",
"Finally, we extract the rate of cross section twist, $\\omega $ , from the rate of rotation of $\\hat{{\\bf e}}_1$ and $\\hat{{\\bf e}}_2$ around the tangent, $\\omega = \\hat{{\\bf e}}_2 \\cdot \\partial _s \\hat{{\\bf e}}_1= \\psi ^{\\prime } - \\frac{\\sin (2 \\theta ) }{2 r}.$ From these quantities we define the following elastic energy for the helical filament, $E_{\\rm mech} = \\frac{1}{2} \\int ds \\Big [ C_1 \\kappa _1^2 + C_2 \\kappa _2^2 + K (\\omega - \\omega _0)^2 \\Big ] ,$ where $C_1$ and $C_2$ are bend moduli for bending around the respective wide and thin axes of the filament cross section and $K$ is the twist modulus.",
"Modeling the filament interior as an isotropic, elastic medium these three constants are related by $K = \\frac{ 3\\mu (w t)^3}{\\pi ^2( w^2+t^2)} ; C_1 = \\frac{ E w^3 t}{12} ; C_2 = \\frac{E w t^3}{12}$ where $E$ and $\\mu $ are the Young's and shear moduli of the of the elastic medium We have used approximation of the torsional modulus of a rectangular beam, $K = \\mu A^4/ 4 \\pi ^2 I$ , where $A$ and $I$ and the area and second-moment of the cross section, respectively.",
"See e.g.",
"A. E. Love A Treatise on the Mathematical Theory of Elasticity (New York, Dover, 1944), 4th ed., Chap.",
"14.. Next we introduce an effective potential to describe the thermodynamics of interactions between the helical filament and the attractive wall.",
"We assume that filament maintains a stronger state of binding when the surface is in close contact with the wide axis of the filament, that is, the minimal energy configuration aligns the $\\hat{{\\bf e}}_2$ with the surface normal, ${\\bf r}$ .",
"Thus, we consider a simple energy gain per unit length proportional to $(\\hat{{\\bf e}}_2 \\cdot {\\bf r})^2$ , $E_{\\rm int} = \\int ds V \\sin ^2(\\psi ) ,$ where $V$ describes the binding affinity difference (per unit length) between “face on\" and “edge on\" binding of the filament.",
"We note that similar models have been applied to study other aspects of biopolymer behavior, such a base-pair stacking interactions between bound pairs of DNA [32], [33].",
"However, to our knowledge, our study is the first to use an intrinsically-twisted elastic filament model to study the frustrations between surface-interactions and mechanics of helical polymers, and the first to study structural transitions driven by these interactions.",
"In the remainder of this article, we focus on the case of a filament bound to a planar surface ($r \\rightarrow \\infty $ ).",
"The lack of surface curvature greatly simplifies the coupling between $\\theta $ and $\\psi $ , $E(r \\rightarrow \\infty ) = \\int ds \\bigg [ \\Big (\\frac{C_1}{2} \\cos ^2 \\psi + \\frac{C_2}{2} \\sin ^2 \\psi \\Big ) |\\theta ^{\\prime }|^2 + \\frac{K}{2} (\\psi ^{\\prime }- \\omega _0)^2 +V \\sin ^2\\psi \\Big ] .$ However, due to the anisotropy of bending response, an important coupling remains between in-plane backbone fluctuations and rotations of cross section.",
"It is easier to bend the filament about the thin axis, and hence, rotation of $\\psi $ along the backbone correlates to variations stiffness to bending in the plane.",
"We find below this coupling the exists significant consequences for the sensitivity of contour fluctuations of the filament backbone to the state of filament binding.",
"We now describe our Monte Carlo simulations that serve as an independent test of our analytic results."
],
[
"Numerical Simulations",
"In this section we describe our numerical setup to study an adsorbed filament on a flat surface.",
"In the subsequent section, we will study the analytical solution to the equation of mechanical equilibrium for absorbed helical filaments.",
"While this provides an exact description of the minimal-energy states of the model, it neglects thermally-induced fluctuations of the torsion as well as the backbone bending.",
"It is reasonable to expect that at finite temperature, the presence of both types of fluctuations modify the “zero-temperature\" analysis of the theory.",
"Thus, we carry out our numerical simulations of the model to quantify the contributions of finite-temperature fluctuations absent from the analytic solutions of the theory.",
"Monte Carlo simulations were performed to find equilibrium configurations of a $2d$ chain of length $L$ that is parsed into $N$ discrete segments or plaquettes.",
"Each plaquette is of unit length and in general can undergo bending and torsional deformations.",
"Plaquette $i$ has associated with it a pair of angles ($\\theta _i$ , $\\psi _i$ ), which represent the local orientation and twist of the chain respectively.",
"A discrete version of the energy given by eq.",
"(REF ) is implemented to evaluate the energy of each chain configuration, $E=\\frac{1}{2}\\sum _{i}^{N}\\big (C_1 \\cos ^2(\\bar{\\psi }_{i,i+1}) + C_2\\sin ^2(\\bar{\\psi }_{i,i+1})\\big )\\big (\\Delta \\theta _{i,i+1}\\big )^2 + K(\\Delta \\psi _{i,i+1}-\\omega _0)^2 + V\\sin ^2(\\psi _i).$ Here $i$ refers to the segment index, $\\bar{\\psi }_{i,i+1}$ refers to the average twist between nearest neighbor plaquettes, $\\Delta $ refers to finite differences between neighbors along the chain and elastic parameters are in units of $k_B T$ .",
"Configurations of the chain were accepted (rejected) upon comparing the trial configuration energy to the previous configuration energy using a METROPOLIS algorithm with the usual Boltzmann weight ($\\exp (-\\Delta E)$ ) .",
"Specifically we look for configurations $z^{(t+1)}$ , where $t$ denotes the Monte Carlo step, that represent equilibrium configurations of the chain.",
"Trial configurations of the chain are accepted/reject by the following criterion, $z^{(t+1)} = \\left\\lbrace \\begin{array}{ll}z^\\prime & \\mbox{\\text{with probability} $r(z,z^\\prime ) = \\text{min}\\big (1,\\frac{p(z^\\prime )}{p(z)}\\big )$} \\\\z & \\mbox{if $r_0 > r(z,z^\\prime )$} \\\\\\end{array}\\right.",
"\\\\.$ Here $z$ is the previous configuration at step $t$ and $p(z^\\prime )$ is the Boltzmann weight of the trial configuration at step $t+1$ .",
"Also, $r_0$ is a random number chosen from a uniform distribution between 0 and 1.",
"Updates of either degree of freedom ($\\theta _i$ or $\\psi _i$ ) per plaquette were done independently with equal probability for each Monte Carlo step.",
"Update acceptance for trail configurations of the chain were between $30\\%$ and $40\\%$ of the total amount of Monte Carlo steps ($10^7$ steps per chain).",
"Data was averaged over $\\sim 10^5$ equilibrium configurations of the chain.",
"Equilibration of the chain was determined by measuring the rotational diffusion of the end to end vector using the condition, $\\vec{R}(t) \\cdot \\vec{R}(0)=0,$ where $t$ is the Monte Carlo step.",
"Usual equilibration times were of the order $10^6$ steps.",
"Chain lengths of $N=32$ were used in all numerical results unless otherwise stated."
],
[
"Thermodynamics of surface-induced untwisting",
"In this section, we analyze the thermodynamics of surface binding of the helical filament, initially ignoring the coupling between in-plane orientation and filament twist described in eq.",
"(REF ).",
"This can be justified in the limit of stiff filaments, where stiffness is characterized by the 2D persistence length, $\\bar{\\ell }_p = 2 \\frac{ \\sqrt{C_1 C_2} }{k_B T} ,$ which is the effective persistence length of a planar helical filament twisting at a constant rate Notice this persistence length differs by a factor of 2 from the 3D persistence length due to the surface confinement [36] In the limit $L \\ll \\bar{\\ell }_p$ , the backbone remains effectively rod-like, and we may assume $\\theta ^{\\prime } \\simeq 0$ .",
"With this approximation in mind, the remaining degrees of freedom describe filament twist, and the contact between the wide edge of the filament and surface, the final two terms in eq.",
"(REF ).",
"The underlying frustration of helical filament binding is straightforward to understand from these terms.",
"While the twist elastic energy is minimized by a constant rate of rotation, $\\psi = \\omega _0 s$ , the binding energy is minimized by locking-in to constant orientations, $\\psi = n \\pi $ .",
"The equation of motion describing the minimal energy configurations of $\\psi (s)$ , is well-known to the analyses of 1D models of incommensurate solids [34] and rigid pendula, $|\\psi ^{\\prime }|^2 = \\lambda ^{-2} \\big ( \\sin ^2 \\psi + \\epsilon \\big ) ,$ where $\\epsilon $ is a positive-valued “constant of motion\" and $\\lambda ^2 = \\frac{K}{2 V}$ is a length scale defined by the ratio of twist modulus to strength of surface binding.",
"The limit, $\\epsilon \\rightarrow 0$ , describes “lock in\" to an untwisted state of face-on binding, $\\psi = n \\pi $ .",
"For small, but finite, values of $\\epsilon $ , the solution becomes inhomogeneous.",
"The filament locks into near perfect surface registry, with $\\psi \\simeq n \\pi $ , over spans of arc length much greater than $\\lambda $ .",
"These nearly “commensurate\" domains are separated by rapid jumps, twist walls, where $\\psi $ rapidly jumps by $\\pi $ over an arc-distance of order $\\lambda $ .",
"For large $\\epsilon $ , the filament twists homogeneously.",
"The transition from locked-in, untwisted filaments, to helical filaments has a highly non-linear, and critical, sensitivity to $V$ and the elastic cost (per unit length) to untwist the filament, proportional to $K \\omega _0^2$ .",
"To analyze this, we follow the analysis of the Emery and Bak [35] to derive an effective theory for filament binding in terms of the length-averaged rate of twist, $\\langle \\psi ^{\\prime } \\rangle = 2 \\pi / P$ , where $P$ is the pitch of rotation.",
"This pitch is twice the distance between twist walls and can be computed from eq.",
"(REF ) as function of $\\epsilon $ , $P( \\epsilon ) = \\lambda \\int _0^{2\\pi } \\frac{d \\psi }{\\sqrt{\\sin ^2 (\\psi ) + \\epsilon }} = 4 \\lambda \\epsilon ^{-1/2} K(-\\epsilon ^{-1}) ,$ where $K(k)$ is the complete elliptic function of the first kind.",
"The asymptotic behavior near the lock-in transition has the form $P(\\epsilon \\rightarrow 0) \\simeq - 2\\lambda \\ln (\\epsilon /16)$ , from which we express the constant, as the approximate of function of pitch, $\\epsilon (P) \\simeq 16 e^{- P/2 \\lambda }$ .",
"Substituting the solution, eq.",
"(REF ), into eq.",
"(REF ) and integrating over one helical turn, we may derive an expression for the effective elastic energy per unit length as a function of $P$ , $E(P)/P \\simeq \\big ( 8 V \\lambda - 2 \\pi K \\omega _0 )/P+ 4 V \\lambda e^{-P / 2 \\lambda }/P + C_0.$ Note that the density of twist walls is $2/P$ , so that the first term in parentheses is proportional the energy per twist wall.",
"The cost of the wall corresponds an increase in potential energy over a length of order $\\lambda $ , while a twist wall relaxes the torsional strain on the untwisted filament, leading the negative contribution to twist wall energy.",
"The second term can be thought of as the cost of exponentially screened repulsive interactions between twist walls.",
"It is straightforward to show that when the elastic energy gain is smaller then the potential energy cost of a twist wall, the untwisted, locked-in state ($P\\rightarrow \\infty $ ) is stable.",
"This occurs when $V>V_c$ , where $V_c = \\frac{K \\pi ^2 \\omega _0^2}{8} .$ However, when $V<V_c$ , the energy per twist wall is negative and the filament begins unbind and wind through the incorporation of twist walls.",
"For small $|V_c-V|$ , at the unbinding threshold, the equilibrium pitch grows rapidly as $P \\sim \\ln |V_c-V |.$ This indicates that the filament undergoes a rapid and continuous phase transition from the untwisted and strongly-bound state to the helical and unbound state at $V=V_c$ .",
"The dependence of the mean-twist, $2\\pi /P$ , can be calculated using results summarized in the Appendix.",
"The profile of $\\langle \\psi ^{\\prime } \\rangle = 2 \\pi /P$ vs. $V$ as calculated in absence of thermal fluctuations is shown in Fig.",
"REF (a).",
"In Fig.",
"REF we show examples of torsional state of a helical filament with a rectangular cross section as the unwinding transitions is approached from below.",
"Simulation data is plotted on top of Fig.",
"REF (a) which is in excellent agreement with this unwinding transition.",
"Figure: Both figures together highlight mechanically what is taking place for both degrees of freedom as the filament becomes strongly bound to the substrate.",
"In (a), the solid curve shows the predicted dependence of the net filament twist on the strength of surface interaction.",
"Solid points with error bars display simulation results at finite temperature.",
"(Red) w/t=2.0w/t=2.0, (Purple) w/t=4.0w/t=4.0 and (Blue) w/t=8.0w/t=8.0.",
"In (b), a plot of the angular diffusion, for w/t=4.0w/t=4.0, associated in-plane orientational fluctuations (eq.",
"() of the filament tangent along its backbone for fixed profiles of equilibrium twist ψ(s)\\psi (s) at various levels of surface binding.",
"Here, ψ(0)=0\\psi (0)=0 for each configurations."
],
[
"Conformational collapse of desorbing helical filament",
"In this section, we consider the effect of the transition between the strongly-bound, untwisted state to the helical state on the backbone fluctuations of the surface confined filament.",
"Clearly, when the filament is strongly-bound in the face-on configuration in-plane bending, around the wide edge of the filament cross section, is suppressed.",
"As described above, just below the binding threshold, a low density of twist walls populate the length of the filament.",
"As the wide axis is normal to the surface in these regions, they act as localized weak spots for in-plane bending, whose bending stiffness is reduce by a factor of $(t/w)^2$ relative to the face-on spans.",
"Hence, in the weakly-bound, helical state, the bending stiffness becomes highly heterogeneous.",
"This for sufficiently anisotropic filament cross sections, this heterogeneity will have a profound impact on the bending fluctuations of the filament backbone in the plane.",
"These fluctuations are observable in the tangent-tangent correlations, $\\langle \\hat{{\\bf t}}(s_1) \\cdot \\hat{{\\bf t}}(s_2) \\rangle _{\\psi } = \\exp \\big [-\\langle |\\theta (s_1,s_2)|^2 \\rangle _{\\psi } /2 \\big ] .$ where the notation $\\langle \\cdot \\rangle _{\\psi }$ denotes an average for a fixed twist profile, $\\psi (s)$ .",
"From the mechanical energy for bending we may derive the following simple result for the “diffusion\" of $\\theta $ along the contour length, $\\langle |\\theta (s_1,s_2)|^2 \\rangle _\\psi = 2 \\int _{s_1}^{s_2} ds ^{\\prime }\\frac{1}{\\ell _1 \\cos ^2\\psi (s^{\\prime }) + \\ell _2 \\sin ^2 \\psi (s^{\\prime }) } ,$ where $\\ell _1 =2 C_1/k_B T$ and $\\ell _2 = 2 C_2/k_B T$ , are 2D persistence lengths corresponding to bending around, respective, thin and wide directions of the cross section.",
"The experimentally measurable tangent-tangent correlation function, $C_{\\hat{{\\bf t}}} (s_2-s_1)=\\langle \\cos [ \\theta (s_2) - \\theta (s_1) ] \\rangle $ , is obtained by averaging $\\langle \\hat{{\\bf t}}(s_1) \\cdot \\hat{{\\bf t}}(s_2) \\rangle _{\\psi } $ over all torsional states.",
"According to the analysis above, distinct configurations of $\\psi (s)$ at a given binding energy are periodic functions of $s$ (modulo shifts by $\\pi $ ), differing only by a uniform translation along the backbone (modulo $P/2$ ).",
"We may perform the averaging over torsional configurations by fixing the twist angle at a reference point, say $\\psi (s=0) = 0$ , and averaging $\\langle \\hat{{\\bf t}}(s_0) \\cdot \\hat{{\\bf t}}(s_0+ s) \\rangle _{\\psi }$ over a span $P/2$ along the backbone for a given $s$ $C_{\\hat{{\\bf t}}}(s) = \\frac{2}{P} \\int _0^{P/2} ds_0\\langle \\hat{{\\bf t}}(s_0) \\cdot \\hat{{\\bf t}}(s_0+s) \\rangle _{\\psi } .$ Hence, the zero-energy torsional fluctuations associated with sliding the twist-wall array along the backbone distribute the weak spots uniformly along the chain and lead to translationally-invariant tangent correlations.",
"Figure: In (a), a plot of the (logarithm) of tangent-tangent correlation function of helical filament with bending anisotropy C 1 /C 2 =16C_1/C_2=16 and ω 0 ℓ 1 =1\\omega _0 \\ell _1 = 1.",
"Three different surface binding strengths are shown: (blue) vanishing binding strength; (red) near-threshold binding strength; and (yellow) above-threshold binding strength.",
"In (b), a plot of the (logarithm) of the ensemble average tangent-tangent correlation along the chain contour nn for 3 different binding strengths, here C 1 /C 2 =64C_1/C_2=64.",
"As we can see the oscillations about and average effective persistence length is present in good agreement with analytical results.In Fig.",
"REF (b) we plot angular excursion, $\\langle |\\theta (0,s)|^2 \\rangle _\\psi $ , for filaments with a bending aspect ratio, $C_1/C_2 = 16$ and $\\psi (0)=0$ , for weak and strong surface binding.",
"Above the absorption threshold, we find that angular diffusion grows linearly with arc-length, with the relative small slope of $1/\\ell _1$ .",
"Just below the critical binding strength, the signature of twist walls appear, where the local persistence length drops to $\\ell _2=\\ell _1/4$ over a relativity short length span of roughly $\\lambda $ , corresponding to “weak spots\" of easy bending around the thin direction.",
"As the binding strength drops to negligible levels, the filament adopts its native twist, $\\omega _0$ , and the local persistence length, as measured by the inverse slope of $\\langle |\\theta (0,s)|^2 \\rangle $ , oscillates between $\\ell _1$ and $\\ell _2$ over equivalent quarter-pitch spans of arc-length.",
"Fig.",
"REF (a) shows $-\\ln C_{\\hat{{\\bf t}}} (s)$ for a range of binding strengths, for the case $\\ell _1 \\omega =1$ and $C_1/C_2=16$ as in Fig.",
"REF (b).",
"Despite the translation invariance of $C_{\\hat{{\\bf t}}} (s)$ regularly spaced intervals of relatively flexible backbone are still apparent in the tangent-tangent correlation functions below the threshold binding strength.",
"Though the angular diffusion is not strictly a linear of function of arc-length over short distances, we may extract the long-distance, effective persistence length from the fall of tangent-tangent correlations over a span of $P/2$ corresponding to a period span of the arc-length modulation, $\\ell _{\\rm eff}^{-1} = -\\frac{\\ln C_{\\hat{{\\bf t}}} (P/2)}{ P/2} = \\langle |\\theta (0,P/2)|^2 \\rangle _\\psi /P .$ In Fig.",
"REF we plot the dependence of $\\ell _{\\rm eff}$ on the binding strength, for three values of cross-section anisotropy.",
"Clearly, in the locked-in, untwisted state, the persistence length is maximal, $\\ell _{\\rm eff} = \\ell _1$ , as in-plane bending occurs only around wide axis of the filament.",
"In the opposite limit of vanishing binding strength, the uniform helical rotation of the filament, $\\psi = \\omega _0 s$ , samples bending around all cross-sectional directions evenly, and from eq.",
"(REF ) we find that $\\ell _{\\rm eff}= ( \\ell _1 \\ell _2)^{1/2}$ .",
"Hence, going from strong binding to weak binding of a helical filament we observe a $(\\ell _1/\\ell _2)^{1/2} = w/t$ -fold drop in the persistence length of the backbone.",
"From Fig.",
"REF we see that most of this drop occurs over a narrow region near to the untwisting transition, $V \\lesssim V_c$ .",
"However, it is significant to note from Fig.",
"REF that even far from this transition, at weak binding strengths we predict a weaker, and roughly linear, dependence of $\\ell _{\\rm eff}$ on $V$ .",
"This demonstrates that the apparent persistence length of a surface absorbed helical filament is always sensitive to anisotropic effects of surface binding.",
"Figure: Plots of the effective persistence lengths as functions of surface binding strength for filaments of different in-plane anisotropy.",
"For a continuum model of elasticity, a given width and thickness ratio implies a ratio of bending moduli, C 1 =(w/t) 2 C 2 C_1= (w/t)^2 C_2.",
"Monte Carlo simulation data with vertical error bars are plotted on top of analytical result and show good agreement with theoretical predictions.We know look at actual conformations of the polymer as we cross this transition.",
"Monte Carlo simulations revealed an excellent agreement with analytic results as seen in Fig.",
"REF and Fig.",
"REF .",
"Moreover the presence of twist walls (Fig.",
"REF ) can be seen in our simulation results as well.",
"In Fig.",
"REF we can see two chain conformations which characterize qualitatively the interplay of average twist and effective persistence length.",
"Fig.",
"REF (a) is a snapshot of a collapsed chain configuration when $V=0$ .",
"The chain twists at its natural twist rate and the effective persistence length is leading to a collapsed chain state.",
"In contrast Fig.",
"REF (b) shows a situation when the binding potential is above the critical potential.",
"Here the chain is almost flat as predicted and the effective persistence length is large leading to a swollen chain configuration.",
"Figure: Typical Monte Carlo chain configurations acquired during simulation runtime at two values of binding potential.",
"(a) For V=0V=0 the chain twist with its natural twist rate ω 0 \\omega _0 and is collapsed with a characteristic persistence length given by eq.",
"().",
"(b) For V>V c V>V_c we see that twist walls are being expelled and the chain is essentially flat with ℓ 1 \\ell _1 as the characteristic persistence length.",
"Here C 1 /C 2 =2C_1/C_2 = 2 and L=100 subunits.Figure: Simulation data for the end to end displacement normalized by the square of the polymer contour length.",
"Each data set if for a fixed contour length but varying aspect ratio.",
"As the aspect ratio of the polymer is increase from w/t=2w/t=2 (red) to w/t=8w/t=8 (blue) the magnitude of the transition to constant persistence length (V>V c V>V_c) is less dramatic.",
"Dashed lines are generated using the ℓ eff \\ell _{\\rm eff} data from figure and eq.",
"().To quantify the collapse transition in a more experientially accessible way, we now focus on the squared end to end displacement, $\\langle R^2\\rangle \\rangle /L^2$ normalized by the contour length of the polymer.",
"For binding potentials $V<V_c$ we can see that $\\langle R^2\\rangle \\rangle /L^2$ increases as the potential is increased from zero.",
"This behavior should be expected from the behavior of the $\\ell _{\\rm eff}$ data (Fig.",
"REF ).",
"Once $V>V_c$ the polymer untwists and only the larger of the two bending stiffness is accessible when $\\langle \\psi ^\\prime \\rangle =0$ and hence the fluctuations of the polymer should be indicative of that length scale.",
"This picture is consistent with the data in Fig.",
"REF in the $V>V_c$ regime where $\\langle R^2\\rangle /L^2$ is constant.",
"An expression for the squared end to end displacement for a fixed contour length in 2d is [36], $\\frac{\\langle R^2\\rangle }{L^2} = \\frac{4\\ell _{p}}{L} \\big ( 1-\\frac{2 \\ell _{p}}{L}( 1-e^{( -\\frac{L}{2\\ell _{p}} )} \\big )\\big )$ In Fig.",
"REF the dashed lines were generated using the extracted persistence lengths from the data in Fig.",
"REF .",
"As we can see the $\\langle R^2\\rangle /L^2$ data is in very good agreement with eq.",
"(REF )."
],
[
"Conclusion",
"Structural and mechanical properties of biopolymers adsorbed to a surface are of great importance in biology.",
"In vivo there are many examples of membrane bound polymers which are essential for cellular functions such as cell shape maintenance, motility, mechanical sensing and the cleaving and contraction of the cell membrane during division.",
"It is therefore imperative to understand if and how biopolymers change their mechanical and structural properties when adsorbed/bound to a membrane surface.",
"This question becomes particularly interesting when one considers that real biopolymers typically have helical structures and can have anisotropic bending stiffnesses.",
"The interplay between the intrinsic helicity, the binding affinity to the surface and the bending stiffness anisotropy lead to non-trivial conformational and mechanical changes upon filament binding, which in turn can be tuned or regulated by the filament's affinity for the surface.",
"Here we have studied a continuum model for a ribbon polymer which has an anisotropic bending stiffness and a native helical twist around it's long axis.",
"While the anisotropy in a real biopolymer can originate in the detailed chemical structure, here we choose to model the biopolymer as uniformly elastic ribbon with a width $w$ and thickness $t$ , thus the structurally the anisotropic bending stiffness arises when aspect ratio $w/t \\ne 1$ .",
"The helical twist implies that the filament's affinity for the surface will be modulated on a period set by the native twist rate when the filament is in its naturally twisted state.",
"In the context of the ribbon model we assume that the wider side with more surface area also has a higher binding affinity for the surface and the twist therefore determines which side is in contact with the surface.",
"Though this assumption is follows intuitively from the fact that face-on binding of the wider edge of the cross-section allows for greater filament-surface contact, the detailed orientation dependence of filament-surface interactions need not conform to this simplified picture.",
"It is quite easy to see that, however, that any sufficiently strong orientation dependence of interactions (say, a strong preference edge-on binding) will give rise to the unwinding transition described here.",
"We first considered an infinitely stiff limit ($\\theta ^\\prime \\simeq 0$ ) where the larger surface area is in contact with the binding surface most of the time.",
"For surface potential strengths above the critical value $V_c$ (eq.",
"REF ), the net twist vanishes and the wider face of the ribbon maintains contact with the surface.",
"Once below the critical value of the potential ($V\\lesssim V_c$ ), a sharp continuous transition occurs to a configuration where twist walls proliferate and form a periodic array (in the absence of thermal fluctuations) which are are separated by a distance $P/2$ .",
"Due to the highly heterogeneous structure of the twist in this transition from the weakly-bound to strongly-bound regime, we found that spectrum of in-plane bending fluctuations is highly sensitive to surface interactions near to this transition.",
"In the weak-binding limit, thermal bending of the backbone is enhanced in the “softer\" regions where the filament has an edge-on orientation.",
"Here, excursions of the in-plane angle $\\theta $ oscillate around a purely diffusive dependence in the contour length, a signature of the appearance of twist walls which function as floppy joints with the period of the oscillations being set by the average distance between twist walls .",
"The effective persistence length has a highly nonlinear increase with increased potential for $V< V_c$ .",
"As the magnitude of the potential approaches the critical potential from below we find that the amplitude of the oscillations diminishes to zero as the period continuously approaches infinity.",
"Mechanically this is due to the larger flat side constituting more and more of the bound side and thus the effective persistence length approaches that of the stiffer bending direction, leading to a homogeneous bending stiffness corresponding to the bending about the wide filament direction.",
"We propose that one could use the strong dependence of the apparent persistence length on surface interactions to address in important and previously unresolved biomechanical property of biological filaments experimentally.",
"Though the bending stiffness of helical biofilaments like F-actin is well characterized on long lengthscales, the are no measurements to our knowledge of the local anisotropy of bending stiffness.",
"By modulating the strength of filament-surface interactions and monitoring fluctuations of the backbone orientation, say by modulating surface charge or osmotic pressure of the solution, the dependence of effective persistence length on binding strength, $\\ell _{\\rm eff} (V)$ can be measured by standard microscopy techniques.",
"Extrapolating these data to the limits of low- and high-surface binding one has a direct access to stiffness anisotropy via the relation, $\\ell _{\\rm eff}(V \\rightarrow \\infty )/\\ell _{\\rm eff}(V \\rightarrow 0) = \\sqrt{C_1/C_2}$ .",
"To assess the physiological relevance and experimental accessibility of surface-induced untwisting helical biopolymers, we now discuss known values of the physical parameters affecting this behavior.",
"For simplicity we focus on electrostatic interactions between the polymer and surface, which is easily accessible in vitro.",
"We are interested then in the surface charge density, $\\sigma $ , required to attain the critical binding strength $V_c$ say for F-actin whose long-wavelength mechanical properties are well-characterized.",
"F-actin is negatively charged with linear charge density of $\\rho \\simeq 4 ~{\\rm nm}^{-1}$ (in units of $e$ ) [37], and thus in solution would bind to an oppositely charged surface.",
"Given the equation for the critical binding strength (eqn.REF ) we can calculate the necessary charge density.",
"First we quantify the critical potential per unit-length that is required to untwist F-actin, we use a native twist of 1.6 turns per micron ($10~{\\rm \\mu m}^{-1}$ ) [38] and a twist stiffness in the range of $8 \\times 10^{-2}~{\\rm pN}~{\\rm \\mu m}^2$ This provides us with a scale of critical potential (per-unit length), $V_c \\approx 10~{\\rm pN}$ To quantify the critical surface charge we assume that the solution of F-actin is dilute such that the distance between neighboring adsorbed filaments is very large compared to their length.",
"Assuming a screened electrostatics, the electrostatic free energy gained per unit charge brought to an oppositely charged surface is roughly $k_B T\\sigma \\ell _B/\\kappa $ , where $\\ell _B$ is the Bjerrum length and $\\kappa ^{-1}$ is the Debye screening length of the solution.",
"From this we may crudely estimate the value of the surface-binding potential to be of order $V/k_B T \\approx \\frac{\\sigma \\rho \\ell _B}{\\kappa }.$ From this we estimate the value of $\\sigma $ for which we expect surface interactions to be sufficiently strong to drive the unwinding transtion of F-actin, $\\sigma \\approx V_c \\kappa / k_B T \\rho \\ell _B$ , at room temperature in $1\\hspace*{3.0pt}\\mu M$ monovalent salt, where $\\ell _B /\\kappa =0.21~{\\rm nm}^2$ .",
"The critical surface charge to fully untwist the filament is, $\\sigma _c =3.e~{\\rm nm}^{-2}$ These values are consistent with physiological values for the surface charge density on cell membranes indicating that such conformational transitions and the associated changes in mechanical properties of surface bound filaments may in fact be exploited in vivo.",
"The modest scale $\\sigma _c$ required also suggest that such the a careful regulation of electrostatically-induced surface binding can be used to indirectly probe the anisotropic mechanical properties of helical helical biofilaments like F-actin and bacterial homologs like MreB.",
"It is important to take these effects into account when backing out elastic properties of twisted filaments adsorbed on surfaces.",
"For instance, there have been recent observations that DNA can form abnormally high bending deformations when adsorbed to a surface [39].",
"DNA is also intrinsically twisted (1 revolution per 3.4 nm) on length scales where anomalous elasticity is observed ($\\sim 5$ nm).",
"In the context of our model we have presented here it would be interesting to understand how local denatured “weak spots\" could affect the formation of twist walls and how the overall observed persistence length is determined by the interplay of intrinsic twist and adsorption onto a surface.",
"The authors would like to thank the Kavli Institute for Theoretical Physics (supported by NSF PHY11-25915) where some of this work was done.",
"GG was supported by the NSF Career program under DMR Grant 09-55760.",
"AG would also like to acknowledge support from a James S. McDonnell Foundation Award, NSF grant DBI-0960480, NSF grant EF-1038697, a UC MEXUS grant, and a George E. Brown, Jr. Award."
],
[
"Twist profile solutions",
"Given the constant of integration, $\\epsilon $ , the relationship between $\\psi $ and arc-length follows the solution to eq.",
"(REF ), $s = \\lambda \\int _0^\\psi d \\psi ^{\\prime } \\frac{1}{\\sqrt{\\sin ^2 \\psi ^{\\prime } + \\epsilon }} = \\lambda \\epsilon ^{-1/2} F(\\psi ,-\\epsilon ^{-1}),$ where $F(x, k)$ is the incomplete elliptic integral of the first kind and we have set $\\psi (s=0) = 0$ .",
"This equation can be inverted to given $\\psi (s)$ in terms of the Jacobi elliptic amplitude function, $\\psi (s) = {\\rm am} \\Big (\\frac{s\\sqrt{\\epsilon } }{\\lambda }, -\\epsilon ^{-1} \\Big ) .$ Using the solution to Euler-Lagrange equation for $\\psi (s)$ , we may derive an expression for the energy of a single pitch of the bound helix as function of the parameter $\\epsilon $ , $E(\\epsilon ) = \\int _0^{P(\\epsilon )} ds \\Big [ \\frac{K}{2} |\\psi ^{\\prime }|^2 + V \\sin ^2\\psi \\Big ] - 2 \\pi K \\omega _0 + \\frac{K}{2} \\omega _0^2 P(\\epsilon ) .$ Using eq.",
"(REF ) we may the integral above as, $\\int _0^{P(\\epsilon )} ds \\Big [ \\frac{K}{2} |\\psi ^{\\prime }|^2 + V \\sin ^2\\psi \\Big ] = 2 V \\lambda \\int _0^{2 \\pi } dp \\sqrt{ \\sin ^2 p + \\epsilon } - V \\epsilon P(\\epsilon ) .$ From this, we have the energy density, $\\frac{ E(\\epsilon ) }{ P(\\epsilon )} = \\frac{2 V \\lambda }{P(\\epsilon )} \\int _0^{2 \\pi } d \\psi \\sqrt{ \\sin ^2 \\psi + \\epsilon } - V\\epsilon - \\frac{2 \\pi K \\omega _0}{ P(\\epsilon )} ,$ which we minimize with respect to $\\epsilon $ to derive the equation of state, effectively relating mean pitch to the binding potential and twist-elastic cost for $V \\le V_c$ , $\\frac{ \\pi K \\omega _0}{ V \\lambda } = \\int _0^{2 \\pi } d \\psi \\sqrt{ \\sin ^2 \\psi + \\epsilon } = 4\\epsilon ^{1/2} E( -\\epsilon ^{-1}),$ where $E(k)$ is the complete elliptic function of the second kind.",
"For a given $\\epsilon $ this relation can easily be solved to binding strength, $V(\\epsilon )=\\frac{ V_c }{\\epsilon ^{1/2} E( -\\epsilon ^{-1}) } .$ Noting that $\\lambda = \\lambda _c (V/V_c)^{-1/2}$ , where $\\lambda _c^{-1} = \\pi \\omega _0 /2$ , eqs.",
"(REF ) we may rewrite the equilibrium pitch in terms of complete elliptic integrals, $P(\\epsilon ) = \\lambda _c E( -\\epsilon ^{-1}) K( -\\epsilon ^{-1}) .$ Using these expressions, $V(\\epsilon )$ and $P(\\epsilon )$ , we describe the relationship between equilibrium pitch of the helical filament and binding strength through the parametric dependence on $\\epsilon $ , plotted in Fig.",
"REF (b)."
]
] | 1204.1115 |
[
[
"A study of Wigner functions for discrete-time quantum walks"
],
[
"Abstract We perform a systematic study of the discrete time Quantum Walk on one dimension using Wigner functions, which are generalized to include the chirality (or coin) degree of freedom.",
"In particular, we analyze the evolution of the negative volume in phase space, as a function of time, for different initial states.",
"This negativity can be used to quantify the degree of departure of the system from a classical state.",
"We also relate this quantity to the entanglement between the coin and walker subspaces."
],
[
"introduction",
"Quantum Walks (QW) are considered a as a piece of potential importance in the design of quantum algorithms [1], [2], [3], [4], [5], as it is the case of classical random walks in traditional computer science.",
"As in the case of random walks, QW's can appear both under its discrete-time [6] or continuous-time [7] form.",
"Moreover, it has been shown that any quantum algorithm can be recast under the form of a QW on a certain graph: QWs can be used for universal quantum computation, this being provable for both the continuous [8] and the discrete version [9].",
"Experiments have been designed or already performed to implement the QW [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].",
"In this paper, we concentrate on the discrete-time QW on a line.",
"We perform a systematic study making use of Wigner functions, which are defined for this problem.",
"Wigner functions [20], [21] were introduced as an alternative description of quantum states.",
"They play an important role in quantum mechanics, having been widely used in quantum optics to visualize light states.",
"From the experimental point of view, they provide a way for quantum state reconstruction via tomography and inverse Radon transformation [22].",
"Wigner functions are quasi-probability distributions in phase space, meaning that they cannot be interpreted as a probability measure in momentum and space configurations.",
"This is an obvious fact for any quantum description, and only marginal distributions can be associated to probabilities in position or momentum (or any linear combination, i.e.",
"any quadrature).",
"In fact, Wigner functions can take negative values, thus invalidating a direct link to a probability distribution.",
"This caveat, however, turns out to be a potential advantage, for it can be used to identify “true” quantum states.",
"More precisely, the volume of the negative part of the Wigner function, its negativity, has been suggested as a figure of merit to quantify the degree of non-classicality [23].",
"This idea has been recently exploited [24] to directly estimating nonclassicality of a state by measuring its distance from the closest one with a positive Wigner function.",
"When dealing with the discrete QW, one has to account for the extra degree of freedom (in addition to the spatial motion): the coin.",
"We consider the simplest case of a two level coin.",
"Therefore, the Wigner function has to incorporate this extra index and, with the prescription we use, it turns into a matrix.",
"We will propose a rather straightforward extended definition of negativity for this Wigner “function”.",
"Then the question arises, what kind of states of the QW are nonclassical?",
"Does this quantumness increase in time, as the QW evolves through its unitary evolution?",
"We want to explore these questions using the Wigner function.",
"A different topic, although it is closely related to the previous one, is whether this nonclassicality can be related to the entanglement between the walker and the coin, since this quantity is also evolving during the QW evolution.",
"This paper is organized as follows.",
"In Sect.",
"II we review the main definitions pertinent to the QW on a line.",
"Sect.",
"III introduces the Wigner function for our problem and the main associated properties.",
"We present some examples showing our numerical results for the Wigner function evolution in Sect.",
"IV.",
"In Sect.",
"V we define an extension of the negativity to the QW, based on the proposal made in [23] for a scalar function.",
"We end in Sect.",
"VI by summarizing our main results and conclusions."
],
[
"Discrete-time QW walk on a lattice",
"The discrete-time QW on a line is defined as the evolution of a one-dimensional quantum system following a direction which depends on an additional degree of freedom, the coin (or chirality), with two possible states: “left” $|L\\rangle $ or “right” $|R\\rangle $ .",
"The total Hilbert space of the system is the tensor product $H_{s}\\otimes H_{c}$ , where $H_{s}$ is the Hilbert space associated to the lattice, and $H_{c}$ is the coin Hilbert space.",
"Let us call $T_{-}$ ($T_{+}$ ) the operators in $H_{s}$ that move the walker one site to the left (right), and $|L\\rangle \\langle L|$ , $|R\\rangle \\langle R|$ the chirality projector operators in $H_{c}$ .",
"The QW is defined by $U(\\theta )=T_{-}\\otimes |L\\rangle \\langle L|\\text{ }C(\\theta )+T_{+}\\otimes |R\\rangle \\langle R|\\text{ }C(\\theta ),$ where $C(\\theta )=\\sigma _{z}\\cos \\theta +\\sigma _{x}\\sin \\theta $ , and $\\sigma _{z}$ , $\\sigma _{x}$ are Pauli matrices acting on $H_{c}$ .",
"For $\\theta =\\pi /4$ the operator $C(\\theta )$ becomes the Hadamard transformation.",
"The unitary operator $U(\\theta )$ transforms the state in one time step as $|\\psi (t+1)\\rangle =U(\\theta )|\\psi (t)\\rangle .$ The state at time $t$ can be expressed as the spinor $|\\psi (t)\\rangle =\\sum \\limits _{n=-\\infty }^{\\infty }\\left[\\begin{array}{c}a_{n}(t)\\\\b_{n}(t)\\end{array}\\right]|n\\rangle ,$ where the upper (lower) component is associated to the right (left) chirality, and $\\left\\lbrace \\mid n\\rangle /n\\in \\mathbb {Z}\\right\\rbrace $ is a basis of position states on the lattice.",
"A basis in the whole Hilbert space can be constructed as the set of states $\\left\\lbrace \\mid n,\\alpha \\rangle =\\mid n\\rangle \\otimes \\mid \\alpha \\rangle /n\\in \\mathbb {Z};\\alpha =L,R\\right\\rbrace $ ."
],
[
"Wigner functions for the quantum walk",
"An important tool in some fields related to quantum physics is the use of quasi-probability distributions.",
"Wigner functions constitute the major example, although other functions as the Glauber-Sudarshan $P$ function [25], [26] or the Husimi $Q$ function [27] are commonly used in quantum optics.",
"For the case of a one-dimensional system with continuous position $x$ and conjugate momentum $p$ , the Wigner function is defined as [20]: $W(x,p)=\\frac{1}{\\pi }_{-\\infty }^{\\infty }\\psi ^{*}(x-y)\\psi (x+y)e^{-2ipy}dy,$ where $\\psi (x)$ is the wave function for the system in a state $|\\psi \\rangle $ , and we are using units such that $\\hbar =1$ .",
"A number of properties can be derived from the definition, the most important ones giving the probability in position (momentum) space as the marginal distributions obtained by integration over momentum (position), respectively.",
"We refer the interested reader to references [21], [22] for an overview about these properties.",
"We are interested in describing the QW with the help of Wigner functions.",
"The case of a finite lattice, with periodic conditions, has been widely studied [28], [29], [30], [31] .",
"Here we want to describe the QW on an infinite lattice of equally-spaced positions.",
"A proposal for the Wigner function to study this problem will be published elsewhere [32].",
"As discussed in this reference, due to the discreteness of the phase space, one needs to double the phase space in order to fulfill the necessary properties of the Wigner function.",
"This doubling feature is a characteristic of discrete Wigner functions [33], [30], and we will return to this point later.",
"Secondly, we need to incorporate the chiral (or spin) degree of freedom.",
"To do this, we consider the walker as a spin 1/2 particle moving on the lattice.",
"This analogy allows us to make the connection to the extensive literature of Wigner functions describing particles with spin.",
"They have been extended to relativistic particles [34] and widely used in kinetic theory [35], nuclear physics [36] or to describe neutrino propagation in matter [37].",
"Within this approach, wave functions as defined in [32] are to be replaced by (Dirac) spinors.",
"Since we are interested in a nonrelativistic description, we simply use 2-dimensional spinors and define: $W_{\\alpha \\beta }(n,k,t)=\\frac{1}{\\pi }e^{ikn}\\sum _{l\\in \\mathbb {Z}}\\langle l,\\alpha \\mid \\psi (t)\\rangle \\langle \\psi (t)\\mid n-l,\\beta \\rangle e^{-2ikl}.$ In the latter equation, $|\\psi (t)\\rangle $ represents the state of the QW at time $t$ .",
"This definition can be extended to the case when the state of the system (walker plus coin) is described by the density matrix $\\rho (t).$ In such a case, we would have $W_{\\alpha \\beta }(n,k,t)=\\frac{1}{\\pi }e^{ikn}\\sum _{l\\in \\mathbb {Z}}\\langle l,\\alpha \\mid \\rho (t)\\mid n-l,\\beta \\rangle e^{-2ikl}.$ In what follows, we will omit the chirality subindices, so that $W(n,k,t)$ will represent an hermitian matrix in chiral space.",
"Also, it will be understood that summations with natural indices are performed over all integers in $\\mathbb {Z}$ .",
"Substitution of Eq.",
"(REF ) into Eq.",
"(REF ) immediately gives $W(n,k,t) & = & \\frac{1}{\\pi }e^{ikn}\\sum _{l}e^{-2ikl}\\left(\\begin{array}{cc}a_{l}(t)a_{n-l}^{*}(t) & a_{l}(t)b_{n-l}^{*}(t)\\\\b_{l}(t)a_{n-l}^{*}(t) & b_{l}(t)b_{n-l}^{*}(t)\\end{array}\\right).$ Notice that the Wigner function is defined over the phase space associated to the Hilbert space of the lattice.",
"This phase space is given by pairs $(n,k)$ , with $n\\in \\mathbb {Z}$ and $k\\in [-\\pi ,\\pi [$ .",
"These coordinates are not to be confused with the position and momentum of the real system, although they are associated to them.",
"To avoid ambiguity, we will refer to $n$ and $k$ as the phase space spatial and momentum coordinates.",
"Using the above definitions, there are many properties that can be proven in a straightforward way.",
"The most important ones are given below.",
"By performing the integration over $k$ one readily obtains, for even values of the spatial phase space coordinate, $_{-\\pi }^{\\pi }W(2n,k,t)dk=2\\left(\\begin{array}{cc}\\mid a_{n}(t)\\mid ^{2} & a_{n}(t)b_{n}^{*}(t)\\\\b_{n}(t)a_{n}^{*}(t) & \\mid b_{n}(t)\\mid ^{2}\\end{array}\\right),$ while, for odd values $_{-\\pi }^{\\pi }W(2n+1,k,t)dk=0.$ This result is a consequence of the above mentioned doubling feature.",
"From here, the spatial probability distribution can be recovered by performing the trace over coin variables: $\\frac{1}{2}Tr\\left(_{-\\pi }^{\\pi }W(2n,k,t)dk\\right)=\\mid a_{n}(t)\\mid ^{2}+\\mid b_{n}(t)\\mid ^{2}=P(n,t),$ where $P(n,t)$ stands for the probability of detecting the walker at position $n$ , regardless of the coin state.",
"To obtain the distribution in momentum space we start from Eq.",
"(REF ) and introduce the quasi-momentum basis $\\left\\lbrace \\mid k\\rangle /k\\in [-\\pi ,\\pi [\\right\\rbrace $ (restricted to the first Brillouin zone), which is related to the spatial basis $\\left\\lbrace \\mid n\\rangle /n\\in \\mathbb {Z}\\right\\rbrace $ via Fourier discrete transformation, i.e.",
": $\\langle n\\mid k\\rangle =\\frac{1}{\\sqrt{2\\pi }}e^{ink}.$ After projecting over a given $|k\\rangle $ one obtains $\\langle k\\mid \\psi (t)\\rangle =\\left(\\begin{array}{c}\\tilde{a}_{k}(t)\\\\\\tilde{b}_{k}(t)\\end{array}\\right),$ where $\\tilde{a}_{k}(t)\\equiv \\frac{1}{\\sqrt{2\\pi }}\\sum _{n}e^{-ink}a_{n}(t);\\,\\,\\,\\,\\tilde{b}_{k}(t)\\equiv \\frac{1}{\\sqrt{2\\pi }}\\sum _{n}e^{-ink}b_{n}(t)$ are the chirality components of the wave function in momentum space.",
"By introducing the closure relation for the basis of states $\\lbrace \\mid k\\rangle \\rbrace $ , one can relate the Wigner function to the matrix elements of $\\rho (t)$ in momentum space: $W_{\\alpha \\beta }(n,k,t)=\\frac{1}{\\pi }\\int _{-\\pi }^{\\pi }e^{in(q-k)}\\langle q,\\alpha \\mid \\rho (t)\\mid 2k-q,\\beta \\rangle dq,$ with $\\mid k,\\alpha \\rangle =\\mid k\\rangle \\otimes \\mid \\alpha \\rangle $ .",
"Summation over $n$ leads to $M(k,t)\\equiv \\sum _{n}W(n,k,t)=2\\langle k\\mid \\rho \\mid k\\rangle ,$ where use was made of the equation $\\sum _{n}e^{ink}=2\\pi \\delta (k,2\\pi )$ , with $\\delta (k,2\\pi )\\equiv \\sum _{m}\\delta (k+2\\pi m)$ the “Dirac comb” function.",
"Obviously, $M(k,t)$ is also a $2\\times 2$ matrix.",
"For a pure state we have, using Eq.",
"(REF ), $M(k,t)=2\\langle k\\mid \\psi (t)\\rangle \\langle \\psi (t)\\mid k\\rangle =2\\left(\\begin{array}{cc}\\mid \\tilde{a}_{k}(t)\\mid ^{2} & \\tilde{a}_{k}(t)\\tilde{b}_{k}^{*}(t)\\\\\\tilde{b}_{k}(t)\\tilde{a}_{k}^{*}(t) & \\mid \\tilde{b}_{k}(t)\\mid ^{2}\\end{array}\\right).$ The diagonal components of this matrix give (up to a factor 2) the probability in momentum space, when chirality is specified, whereas non-diagonal components correspond to coherences between different chiralities.",
"Now, the trace over chirality provides the probability in momentum space, when chirality is not measured: $\\frac{1}{2}Tr\\lbrace M(k,t))\\rbrace =\\tilde{a}_{k}^{2}(t)+\\tilde{b}_{k}^{2}(t)\\equiv P(k).$ Since the evolution in momentum space is diagonal in $k$ and given by a unitary transformation (see, for example [38]), it can be easily proven that the magnitude $P(k)$ defined above remains constant with time.",
"Another important property that can be derived for the Wigner function of the QW is a recursion formula relating $W(n,k,t+1)$ to other components of this function at time $t$ .",
"Using Eq.",
"(REF ) one obtains, after some algebra: spanish $W(n,k,t+1) & =M_{R}W(n-2,k,t)M_{R}^{\\dagger }+e^{-2ik}M_{R}W(n,k,t)M_{L}^{\\dagger }\\nonumber \\\\+ & e^{2ik}M_{L}W(n,k,t)M_{R}^{\\dagger }+M_{L}W(n+2,k,t)M_{L}^{\\dagger },$ englishwhere $M_{L}=(|L\\rangle \\langle L|)C(\\theta )$ and $M_{L}=(|R\\rangle \\langle R|)C(\\theta )$ .",
"An immediate consequence of this recursion formula is that sites with even $n$ evolve independently of those with odd $n$ ."
],
[
"Numerical results",
"In what follows, we will discuss a couple of examples showing the main features of the Wigner function.",
"We have numerically simulated the QW evolution for various initial states, and explicitly computed the Wigner function.",
"We take the lattice large enough, so that boundaries do not need to be considered.",
"In practice, this is equivalent to assuming an infinite lattice.",
"In the cases we will consider here, the initial state is such that at $t=0$ the Wigner function is non-vanishing only at phase space points with even $n$ .",
"Then, Eq.",
"(REF ) warrants that $W(2s+1,k,t)=0$ , $\\forall s\\in \\mathbb {Z}$ , at any time.",
"Therefore, we only plot the Wigner function over the part of the phase space with even spatial coordinate.",
"Fig REF shows $W_{RR}(n,k,t)$ for $t=500$ , with initially localized conditions, spanish $\\mid \\psi (0)\\rangle =\\frac{1}{\\sqrt{2}}(\\mid R\\rangle +i\\mid L\\rangle )\\otimes \\mid 0\\rangle $ englishThe initial Wigner function for this state can be easily evaluated, with the resultspanish $W(n,k,0)=\\frac{1}{2\\pi }\\delta _{n,0}\\left(\\begin{array}{cc}1 & -i\\\\i & 1\\end{array}\\right).$ Since the matrix defining the Wigner function is Hermitian, this component has no imaginary part.",
"The evolved component $W_{RR}$ after 500 iterations is shown in Fig.",
"REF , and the time evolution of this component can be seen on Fig.",
"REF .",
"One can observe an intricate structure, arising from interference effects.",
"Notice, for example, the similarity with the threads mentioned in [30].",
"It is interesting to mention that, although the Wigner function expands in space, as the walker distribution broadens, it keeps the same structure.",
"The rest of components of the Wigner function show a similar appearance.",
"As an example, we have represented in Fig.",
"REF the real part of the off-diagonal component $W_{RL}$ for $t=500$ , starting from the localized state Eq.",
"(REF ).",
"Figure: (Color online): W RR W_{RR} component of the Wigner function with initialconditions given as in (), after 500 iterations.Figure: (Color online): Contour plots showing the time evolution of the W RR W_{RR}component of the Wigner function starting from the localized state().",
"From left to right, the sub figures correspondto t=0t=0, t=100t=100 and t=500t=500, respectively.Figure: (Color online): Real part of the W RL W_{RL} component of the Wignerfunction with initial conditions given as in (),after 500 iterations.The momentum distribution is obtained from Eq.",
"(REF ).",
"We show in Fig.",
"REF , as an example, the RR component of $M(k,t)$ plotted for $t=50,100,200$ and the same initial condition (REF ) as before.",
"Since the moment $k$ is bounded, the distribution becomes more intricate as the QW evolves.",
"At later times, the figure shows more oscillations, although the envelop remains constant, in accordance to the self-resemblance of the Wigner function as time increases.",
"Figure: M RR (k,t)M_{RR}(k,t) component with the same initial conditions as before,corresponding to times t=50t=50, t=100t=100 and t=200t=200 (left to right).For comparison, we investigate the Wigner function for a different initial condition.",
"It corresponds to a “Schrödinger cat” in two positions $|\\pm a\\rangle $ which are entangled with two chiralities, i.e.",
": $\\mid \\psi (0)\\rangle =\\frac{1}{\\sqrt{2}}(\\mid a,R\\rangle +i\\mid -a,L\\rangle ).$ The initial Wigner function for this state is now $W(n,k,0)=\\frac{1}{2\\pi }\\left(\\begin{array}{cc}\\delta _{n,2a} & -ie^{-2ika}\\delta _{n,0}\\\\ie^{2ika}\\delta _{n,0} & \\delta _{n,-2a}\\end{array}\\right).$ Notice that $a=0$ reproduces the localized state described above.",
"The evolved component $W_{RR}$ of the Wigner function is shown in Fig.",
"REF .",
"Fig.",
"REF reveals the time evolution of this component.",
"As compared to the previous case, it shows an even more complicated structure, thus suggesting a less classical state.",
"This suggestion will be confirmed in the next section by comparing the negativity for both cases.",
"Figure: (Color online): W RR W_{RR} component of the Wigner function with initialconditions given as in (), and a=10a=10, after 500iterations.Figure: (Color online): Contour plots showing the time evolution of the W RR W_{RR}component of the Wigner function starting from the localized state() with a=10a=10.",
"From left to right, the sub figurescorrespond to t=0t=0, t=100t=100 and t=500t=500."
],
[
"Negativity of the wigner function in the quantum walk",
"Consider the Wigner function, as defined in Eq.",
"(REF ).",
"As already mentioned, the fact that this function can take negative values implies that it cannot be considered as a classical probability distribution.",
"Therefore, non positivity of the Wigner function can be interpreted as a measure of the non-classicality of the system.",
"In quantum optics, this is interpreted as a signature for non-classical states of light, caused by a quantum interference phenomenon.",
"In the context of finite dimensional systems, this idea is exploited in spanish[39] to establish a criterion for entanglement in a system of two spin 1/2 particles.",
"In spanish[40], a connection is found between entanglement and the negativity of the Wigner function for hyperradial s-waves.",
"Such states, in $D=2d$ dimensions, can be interpreted as the wave function of two entangled particles in $d$ dimensions.",
"A quantitative measure of non-classicality is given by the negativity, as defined in [23].",
"In the continuous case, with variables $x$ and $p$ , this volume can be written as: $\\delta (W)=\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }[\\mid W(x,p)\\mid -W(x,p)]dpdx=\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }\\mid W(x,p)\\mid dpdx-1.$ In deriving the latter equality, we made use of the fact that the total probability is normalized to one, so that $\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }W(x,p)dpdx=1$ .",
"We will use $\\delta (W)$ , adopted to the discrete case, to compute the negativity of the Wigner function, and to explore whether we can relate the deviations from a classical behavior of the QW to the entanglement between the walker and the coin.",
"In our case, the Wigner function is a 2x2 matrix, and variables $x$ , $p$ are to be replaced by $n$ and $k$ , as defined in Sect.",
"III.",
"We propose, as a possible generalization of the above equation [32]: $\\delta (W)=\\sum _{n}\\int _{-\\pi /2}^{\\pi /2}[\\mid \\mid W(n,k)\\mid \\mid -Tr(W(n,k))]dk=\\sum _{n}\\int _{-\\pi /2}^{\\pi /2}\\mid \\mid W(n,k)\\mid \\mid dk-1,$ and where we omitted the dependence on time to simplify the notation.",
"Again, the latter equality arises as a consequence of normalization.",
"As a measure of the norm $||A||$ of a matrix $A$ , we adopt the trace norm, defined as: $\\mid \\mid A\\mid \\mid \\equiv Tr\\sqrt{A^{\\dagger }A},$ where $A^{\\dagger }$ denotes the hermitian conjugate of $A$ .",
"If $\\lambda _{1}(n,k)$ , $\\lambda _{2}(n,k)$ are the eigenvalues of $W(n,k)$ for a given $n$ and $k$ , one obtains $||W||=\\mid \\lambda _{1}(n,k)\\mid +\\mid \\lambda _{2}(n,k)\\mid $ .",
"In this way, Eq.",
"(REF ) adopts the following form: $\\delta (W)=\\int _{-\\pi /2}^{\\pi /2}\\sum _{n}\\left(\\mid \\lambda _{1}(n,k)\\mid +\\mid \\lambda _{2}(n,k)\\mid -\\lambda _{1}(n,k)-\\lambda _{2}(n,k)\\right)dk,$ which can be regarded as a natural generalization of (REF ).",
"We have numerically calculated the negativity (REF ) as a function of time, for the same initial conditions considered in the previous section: the localized state and two different cat states (Fig.",
"REF ).",
"Figure: (Color online): Negativity, as a function of time, for the localizedstate a=0a=0 (blue, bottom curve) and for the Schrödinger cat statewith a=4a=4 (red, middle curve) and a=30a=30 (green, top curve).We immediately observe that one obtains a higher degree of “quantumness” for the Schrödinger cat states, as compared to the localized state.",
"This was expected from the higher degree of complexity observed in the Wigner function for the cat state, and implies a larger amount of interference effects.",
"Higher values of the initial separation yield a larger negativity.",
"The question arises whether this higher degree of quantumness will also imply a higher entanglement between the walker and the coin.",
"To this end, we use the entropy of entanglement as a quantity to characterize this property.",
"To be more precise, we compute the quantity $S(t)=-Tr\\lbrace \\rho _{c}(t)\\log _{2}\\rho _{c}(t)\\rbrace ,$ where $\\rho _{c}(t)=Tr_{S}\\lbrace \\rho (t)\\rbrace $ is the density matrix for the coin, which is obtained by tracing out the spatial degrees of freedom.",
"Fig.",
"REF shows $S(t)$ for the localized state ($a=0$ ) and for the cat state with different values of the (half) separation $a$ .",
"The entanglement entropy is bounded by the (log of the) dimension of the coin space: $S(t)\\le 1$ , and we can see that the time scale to reach this maximum is shorter for larger values of $a$ , which also correspond to higher degrees of negativity, as seen in previous figures.",
"Figure: Entropy of the localized state a=0a=0 (blue, bottom curve) and forthe Schrödinger cat state with a=4a=4 (red, middle curve) and a=30a=30(green, top curve)."
],
[
"Conclusions",
"In this paper, we have studied the discrete-time quantum walk on a one-dimensional lattice using the Wigner function.",
"We have explored the potential of this phase-space representation to characterize the dynamics of the system.",
"Differently to the case of a scalar wave function, we now have a $2\\times 2$ matrix, defined over a phase space $(n,k)$ .",
"We have examined two cases, which correspond to different initial conditions.",
"The first case starts with an initially localized state at $n=0$ , as considered in many works in the literature.",
"Its Wigner function shows a quite intricate structure, which is build up by interference effects on the lattice.",
"We have also considered Schrödinger cat states, in which space and coin are initially entangled.",
"It is apparent, from the Wigner function plots, that such states are capable of building up higher interference effects, and thus describe states that are even “less classical” than the previous one.",
"In order to quantify this effect, we have computed the negativity associated to the Wigner function [23], as defined by the negative volume of the function in phase space.",
"This definition has been extended to our case by the use of the trace norm of the Wigner matrix.",
"As expected, the cat states give rise to a larger extent of negativity, which is even larger when the initial separation of the cat state is increased.",
"In accordance to these ideas, one also obtains that the coin-walker entanglement evolves faster for the latter states.",
"Altogether, the time evolution of the QW translates into an evolving state which separates from classicality.",
"This separation, as time goes on, was expected from the fact that the walker distribution expands faster (with a quadratic deviation $\\sigma \\sim t$ ) than its classical counterpart (the random walk, for which $\\sigma \\sim \\sqrt{t}$ ).",
"This work was supported by the Spanish Grants FPA2011-23897, 'Generalitat Valenciana' grant PROMETEO/2009/128, and by the DFG through the SFB 631, and the Forschergruppe 635."
]
] | 1204.1050 |
[
[
"Hindered proton collectivity in 28S: Possible magic number at Z=16"
],
[
"Abstract The reduced transition probability B(E2;0 ->2+) for 28S was obtained experimentally using Coulomb excitation at 53 MeV/nucleon.",
"The resultant B(E2) value 181(31) e2fm4 is smaller than the expectation based on empirical B(E2) systematics.",
"The double ratio |M_n/M_p|/(N/Z) of the 0+ ->2+ transition in 28S was determined to be 1.9(2) by evaluating the M_n value from the known B(E2) value of the mirror nucleus 28Mg, showing the hindrance of proton collectivity relative to that of neutrons.",
"These results indicate the emergence of the magic number Z=16 in the |T_z|=2 nucleus 28S."
],
[
"APS/123-QED Hindered proton collectivity in $^{28}_{16}$ S$^{}_{12}$ : Possible magic number at $Z=16$ Y. Togano Present address: EMMI, GSI, D-64291 Darmstadt, Germany RIKEN Nishina Center, Saitama 351-0198, Japan Department of Physics, Rikkyo University, Tokyo 171-8501, JapanY.",
"Yamada Department of Physics, Rikkyo University, Tokyo 171-8501, JapanN.",
"Iwasa Department of Physics, Tohoku University, Miyagi 980-8578, JapanK.",
"Yamada RIKEN Nishina Center, Saitama 351-0198, JapanT.",
"Motobayashi RIKEN Nishina Center, Saitama 351-0198, JapanN.",
"Aoi RIKEN Nishina Center, Saitama 351-0198, JapanH.",
"Baba RIKEN Nishina Center, Saitama 351-0198, JapanS.",
"Bishop RIKEN Nishina Center, Saitama 351-0198, JapanX.",
"Cai Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, ChinaP.",
"Doornenbal RIKEN Nishina Center, Saitama 351-0198, JapanD.",
"Fang Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, ChinaT.",
"Furukawa RIKEN Nishina Center, Saitama 351-0198, JapanK.",
"Ieki Department of Physics, Rikkyo University, Tokyo 171-8501, JapanT.",
"Kawabata Center for Nuclear Study, University of Tokyo, Saitama 351-0198, JapanS.",
"Kanno RIKEN Nishina Center, Saitama 351-0198, JapanN.",
"Kobayashi Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan Y. Kondo RIKEN Nishina Center, Saitama 351-0198, JapanT.",
"Kuboki Department of Physics, Saitama University, Saitama 338-8570, JapanN.",
"Kume Department of Physics, Tohoku University, Miyagi 980-8578, Japan K. Kurita Department of Physics, Rikkyo University, Tokyo 171-8501, JapanM.",
"Kurokawa RIKEN Nishina Center, Saitama 351-0198, JapanY. G.",
"Ma Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, ChinaY.",
"Matsuo RIKEN Nishina Center, Saitama 351-0198, JapanH.",
"Murakami RIKEN Nishina Center, Saitama 351-0198, JapanM.",
"Matsushita Department of Physics, Rikkyo University, Tokyo 171-8501, JapanT.",
"Nakamura Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan K. Okada Department of Physics, Rikkyo University, Tokyo 171-8501, JapanS.",
"Ota Center for Nuclear Study, University of Tokyo, Saitama 351-0198, Japan Y. Satou Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan S. Shimoura Center for Nuclear Study, University of Tokyo, Saitama 351-0198, Japan R. Shioda Department of Physics, Rikkyo University, Tokyo 171-8501, JapanK. N.",
"Tanaka Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan S. Takeuchi RIKEN Nishina Center, Saitama 351-0198, JapanW.",
"Tian Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, ChinaH.",
"Wang Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, ChinaJ.",
"Wang Institute of Modern Physics, Chinese Academy of Science, Lanzhou 730000, ChinaK.",
"Yoneda RIKEN Nishina Center, Saitama 351-0198, Japan The reduced transition probability $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ) for $^{28}$ S was obtained experimentally using Coulomb excitation at 53 MeV/nucleon.",
"The resultant $B$ (E2) value 181(31) e$^2$ fm$^4$ is smaller than the expectation based on empirical $B$ (E2) systematics.",
"The double ratio $|M_n/M_p|/(N/Z)$ of the $0^+_{gs} \\rightarrow 2^+_1$ transition in $^{28}$ S was determined to be 1.9(2) by evaluating the $M_n$ value from the known $B$ (E2) value of the mirror nucleus $^{28}$ Mg, showing the hindrance of proton collectivity relative to that of neutrons.",
"These results indicate the emergence of the magic number $Z=16$ in the $|T_z|=2$ nucleus $^{28}$ S. 23.20.Js, 25.60.-t, 25.70.De Magic numbers characterize the shell structure of fermionic quantum system such as atoms, metallic clusters [1] and nuclei [2].",
"A unique feature of the nuclear system is the fact that it comprises two types of fermions, the protons and neutrons, and hence the magic numbers appear both for protons and neutrons.",
"Most of the recent studies regarding the magic numbers are for neutron-rich nuclei.",
"Disappearance of the conventional magic numbers of $N$ =8, 20 and 28 [3], [4], [5] or the appearance of the new magic number $N=16$ [6], [7], [8] has been shown.",
"They are associated with nuclear collectivity, which is enhanced, for instance, in the neutron-rich $N=20$ nucleus $^{32}$ Mg caused by disappearance of the magic number [9], [10].",
"The new neutron magic number $N=16$ has been confirmed experimentally for $^{27}$ Na ($|T_z|=5/2$ ) and more neutron-rich isotones [6], [8], [7], [11], [12].",
"Its appearance can be theoretically interpreted as a result of a large gap between the neutron $d_{3/2}$ and $s_{1/2}$ orbitals caused by the low binding energy [6] and/or the spin-isospin dependent part of the residual nucleon-nucleon interaction [13].",
"In analogy to the magic number $N=16$ , the proton magic number $Z=16$ must also exist in proton-rich nuclei.",
"However, it has not been identified experimentally in the proton-rich sulfur isotopes.",
"The present Letter reports on a study of the magic number $Z=16$ at the most proton-rich even-even isotope $^{28}$ S with $|T_z|=2$ through a measurement of the reduced transition probability $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ).",
"The $B$ (E2) value is directly related to the amount of quadrupole collectivity of protons.",
"The relative contribution of the proton- and neutron-collectivities can be evaluated using the ratio of the neutron transition matrix element to the proton one (the $M_n/M_p$ ratio) for $0^+_{gs} \\rightarrow 2^+_1$ transitions [14], [15].",
"$M_p$ is related to $B$ (E2) by $e^2 M_p^2 = B({\\rm E2};0^+_{gs} \\rightarrow 2^+_1)$ .",
"The $M_n$ value can be deduced from the $M_p$ value in the mirror nucleus, where the numbers of protons and neutrons are interchanged.",
"If collective motions of protons and neutrons have the same amplitudes, the double ratio $|M_n/M_p|/(N/Z)$ is, therefore, expected to be unity.",
"Deviation from $|M_n/M_p|/(N/Z) = 1$ corresponds to a proton/neutron dominant excitation and should indicate a difference in the motions of protons and neutrons.",
"Such a difference appears typically for the singly-magic nuclei [14], [16].",
"For proton singly-magic nuclei, the proton collectivity is hindered by the magicity, leading to $|M_n/M_p|/(N/Z) > 1$ .",
"For example, the singly-magic nucleus $^{20}$ O has a large double ratio of $1.7 \\sim 2.2$ for the $0^+_{gs} \\rightarrow 2^+_1$ transition [17], [18], [19].",
"We used Coulomb excitation at an intermediate energy to extract the $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ) value of the proton-rich nucleus $^{28}$ S. Intermediate-energy Coulomb excitation is a powerful tool to obtain $B$ (E2) with relatively low intensity beams because a thick target is available [10], [20].",
"The double ratio $|M_n/M_p|/(N/Z)$ of the $0^+_{gs} \\rightarrow 2^+_1$ transition is obtained by combining the $B$ (E2) values of $^{28}$ S and the mirror nucleus $^{28}$ Mg.",
"The experiment was performed using the RIBF (Radioactive Isotope Beam Factory) accelerator complex operated by RIKEN Nishina Center and Center for Nuclear Study, University of Tokyo.",
"A $^{28}$ S beam was produced via projectile fragmentation of a 115-MeV/nucleon $^{36}$ Ar beam from the $K=540$ MeV RIKEN Ring Cyclotron incident on a 531 mg/cm$^2$ thick Be target.",
"The secondary beam was obtained by the RIKEN Projectile-fragment separator (RIPS) [21] using an aluminum energy degrader with a thickness of 221 mg/cm$^2$ and a wedge angle of 1.46 mrad placed at the first dispersive focus.",
"The momentum acceptance was set to be $\\pm $ 1%.",
"A RF deflector system [22] was placed at the second focal plane of RIPS to purify the $^{28}$ S in the beam with intense contaminants (mostly of $^{27}$ P, $^{26}$ Si and $^{24}$ Mg) that could not be removed only by the energy loss in the degrader.",
"Particle identification for the secondary beam was performed event-by-event by measuring time of flight (TOF), energy loss ($\\Delta E$ ), and the magnetic rigidity of each nucleus.",
"TOF was measured by using a RF signal from the cyclotron and a 0.1 mm thick plastic scintillator located 103 cm upstream of the third focal plane.",
"$\\Delta E$ was obtained by a 0.1 mm thick silicon detector placed 117 cm upstream of the third focal plane.",
"The average $^{28}$ S beam intensity was 120 s$^{-1}$ , which corresponded to approximately 1.9% of the total intensity of the secondary beam.",
"The secondary target was a 348 mg/cm$^2$ -thick lead sheet which was set at the third focal plane.",
"The average beam energy at the center of the lead target was 53 MeV/nucleon.",
"Three sets of PPACs [23] were placed 155.6 cm, 125.6 cm, and 66.2 cm upstream of the secondary target, respectively, to obtain the beam trajectory on the secondary target.",
"An array of 160 NaI(Tl) scintillator crystals, DALI2 [24], was placed around the target to measure de-excitation $\\gamma $ rays from ejectiles.",
"The measured full energy peak efficiency was 30 % at 0.662 MeV, in agreement with a Monte-Carlo simulation made by the GEANT4 code, and the energy resolution was 9.5 % (FWHM).",
"The full-energy-peak efficiency for 1.5 MeV $\\gamma $ rays emitted from the ejectile with the velocity of $0.32c$ was evaluated to be 16% by the Monte-Carlo simulation.",
"The scattering angle, energy loss ($\\Delta E$ ), and total energy ($E$ ) of the ejectiles from the lead target were obtained by a detector telescope located 62 cm downstream of the target.",
"It consisted of four layers of silicon detectors arranged in a $5 \\times 5$ matrix without 4 detectors at the corners for the first two layers, and a $3 \\times 3$ matrix for the third and fourth layers.",
"The silicon detectors in the four layers had an effective area of $50 \\times 50$ mm$^2$ and a thickness of 500, 500, 325, and 500 $\\mu $ m, respectively.",
"The detectors in the first and second layers had 5-mm-wide strip electrodes on one side to determine the hit position of the ejectiles.",
"The $\\Delta E$ -$E$ method was employed to identify $^{28}$ S. The mass number resolution for sulfur isotopes was 0.35 (1$\\sigma $ ).",
"The angle of the ejectile was obtained from the hit position on the telescope and the beam angle and position on the target measured by the PPACs.",
"The scattering angle resolution was 0.82 degree.",
"The Doppler-shift corrected $\\gamma $ -ray energy-spectrum measured in coincidence with inelastically scattered $^{28}$ S is shown in Fig.",
"REF .",
"A peak is clearly seen at 1.5 MeV.",
"Figure: Doppler-shift corrected γ\\gamma -rayenergy-spectrum in the Pb( 28 ^{28}S, 28 ^{28}S γ\\gamma )Pb reaction.The fit by the response function (dashed curve) and the exponentialbackground (dotted curve) is shown by the solid curve.The spectrum was fitted by a detector response obtained by the Monte-Carlo simulation and an exponential background.",
"The peak energy was obtained to be 1.497(11) MeV, which was consistent with the previous measurement, 1.512(8) MeV, by the two neutron removal reaction on $^{30}$ S [25].",
"This peak has been assigned to the transition from the $2^+_1$ state to the $0^+$ ground state [25].",
"In extracting the inelastic cross section, transitions feeding the $2^{+}_1$ state were not accounted for, because the proton separation energy of 2.46(3) MeV is relatively low and no higher excited states were seen in the present spectrum and the two-neutron removal reaction on $^{30}$ S [25].",
"This was supported by the location of the second excited state in the mirror nucleus $^{28}$ Mg of 3.86 MeV.",
"The angular distribution of the scattered $^{28}$ S excited to its 1.5 MeV state is shown in Fig.",
"REF (a).",
"Figure REF (b) shows the angle-dependence of the detection efficiency for scattered $^{28}$ S obtained by a Monte-Carlo simulation.",
"It took into account the spacial and angular distributions of the $^{28}$ S beam, the size of the silicon detectors, and effect of multiple scattering in the target.",
"The cross section integrated up to 8 degree was obtained to be 99(16) mb by taking into account the angle-dependent detection efficiency.",
"The error was nearly all attributed to the statistical uncertainties, while the systematic errors of the $\\gamma $ -ray detection efficiency and the angle-dependence of the detection efficiency were also included (3%).",
"Figure: (a) Angular distribution for thePb( 28 ^{28}S, 28 ^{28}S γ\\gamma )Pb reaction exciting the 1.5 MeV state in 28 ^{28}S. The solid curve represents the best fit with ECIScalculation assuming ΔL=2\\Delta L=2.The dashed and dotted curves show the Coulomb and nuclear contributions,respectively.",
"(b) Detection efficiency calculated by the Monte-Carlo simulation.The distribution was fitted by that for an angular momentum transfer of $\\Delta L=2$ , calculated by the coupled-channel code ECIS97 [26] taking into account the scattering angle resolution.",
"As seen in the figure, the $\\Delta L=2$ distribution well reproduced the experimental one, supporting the $2^+$ assignment for the 1.5 MeV state.",
"The ECIS calculation is almost equivalent to the distorted wave born approximation, since higher-order processes are negligible in the present experimental conditions.",
"The optical potential parameters were taken from the study of the $^{17}$ O + $^{208}$ Pb elastic scattering at 84 MeV/nucleon [27].",
"The collective deformation model was employed to obtain a form factor for nuclear excitation.",
"The Coulomb- and nuclear-deformation parameters $\\beta _C$ and $\\beta _N$ were employed to obtain the $B$ (E2) value as $B({\\rm E2})=(3ZeR^2/4\\pi )^2 \\beta _C^2$ .",
"$\\beta _N$ is related to $\\beta _C$ by a Bernstein prescription [14], $\\frac{\\beta _N}{\\beta _C} =\\frac{1+(b_n^F/b_p^F)(M_n/M_p)}{1+(b_n^F/b_p^F)(N/Z)},$ where $b_{n(p)}^F$ is the interaction strength of a probe $F$ with neutrons (protons) in the nucleus.",
"$b_n^F/b_p^F$ is estimated to be 0.81 for the inelastic scattering on Pb at around 50 MeV/nucleon [19].",
"The $M_n$ was deduced from the adopted $B$ (E2) value of the mirror nucleus $^{28}$ Mg [28].",
"The $B$ (E2) value for $^{28}$ S was obtained by adjusting $\\beta _C$ and hence $M_p$ with $\\beta _N$ calculated by eq.",
"(REF ) to reproduce the experimental angular distribution.",
"The dashed and dotted curves in Fig.",
"REF shows the Coulomb and nuclear contributions, respectively.",
"The use of the optical potential determined from the $^{40}$ Ar + $^{208}$ Pb scattering [29] gave a 5.5% smaller $B$ (E2) value.",
"By taking the average of the results with the two optical potentials, the $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ) value was determined to be 181(31) e$^2$ fm$^4$ .",
"The associated error included the uncertainty of the measured cross section and the systematic error due to the choice of optical potentials.",
"The $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ) value for the $^{24}$ Mg, a contaminant of the secondary beam, was obtained to be 444(66) e$^2$ fm$^4$ by the same analysis.",
"This agreed with the adopted value of 432(11) e$^2$ fm$^4$ [28], exhibiting the reliability of the present analysis for $^{28}$ S. Figure: Plot of the BB(E2;0 gs + →2 1 + 0^+_{gs} \\rightarrow 2^+_1) values (a),the excitation energies of 2 1 + 2^+_1 states (b), andthe double ratio |M n /M p |/(N/Z)|M_n/M_p|/(N/Z) (c) for sulfur (Z=16Z=16) isotopes.The shell model predictions with the USDB interaction are shown by the dotted curves for each quantity.The shaded region represents the BB(E2) predictions by the empiricalBB(E2) systematics .The present result is represented by the filled circles.The $B$ (E2) and $E_x(2^+_1)$ values for $Z=16$ isotopes are plotted in Fig.",
"REF (a) and (b), respectively.",
"The filled circles show the present results.",
"The open triangles for $B$ (E2) and $E_x(2^+_1)$ represent known values for the $Z=16$ isotopes up to $A=40$ [28].",
"The $B$ (E2) value increases from $^{36}$ S, the neutron singly-magic nucleus, to $^{30}$ S, and decreases at $^{28}$ S. On the other hand, the $2^+_1$ energy of $^{28}$ S is smaller than those of $^{30-36}$ S. These features contradict the empirical systematics.",
"For example, Raman proposed the relation $B({\\rm E2}) = (25.7 \\pm 4.5) E_x(2^+_1)^{-1} Z^2 A^{-2/3}$ which is obtained by a global fit to $E_x(2^+_1)$ and $B$ (E2) in a wide range of nuclei [28].",
"The shaded band in Fig.",
"REF (a) represents the $B$ (E2) values calculated by this formula.",
"As clearly seen, the present data for $^{28}$ S is much smaller than the expectation of 472(83) e$^2$ fm$^4$ .",
"An explanation of these small $B$ (E2) and $E_x(2^+_1)$ is given by the hindered proton collectivity and the neutron dominance in the $0^+_{gs} \\rightarrow 2^+_1$ transition.",
"A similar mechanism is proposed for $^{16}$ C [31], [32], [33] and $^{136}$ Te [34], [35], where small $B$ (E2) and $E_x(2^+_1)$ values in comparison with neighboring isotopes are observed.",
"Figure REF (c) shows the double ratio $|M_n/M_p|/(N/Z)$ of the $Z=16$ isotopes.",
"The filled circle and open triangles show the present result and the known values, respectively.",
"They are obtained by the $B$ (E2) values of the mirror pairs.",
"The open squares represent the double ratios obtained by the combinations of $B$ (E2) and the result of $(p,p^{\\prime })$ on the nuclei of interest [36].",
"The ratio for $^{28}$ S amounts to 1.9(2) by taking the present result and adopted $B$ (E2) of 350(50) e$^2$ fm$^4$ for the mirror nucleus $^{28}$ Mg [28].",
"The double ratio of 1.9(2) is significantly larger than unity indicating again the hindered proton collectivity relative to neutron and the neutron dominance in the $0^+_{gs} \\rightarrow 2^+_1$ transition in $^{28}$ S. This hindrance can be understood if $^{28}$ S is the proton singly-magic nucleus by the $Z=16$ magicity.",
"This picture is supported by the larger $B$ (E2) value and $|M_n/M_p|/(N/Z) \\sim 1$ of the neighboring $N=12$ isotones: 356 e$^2$ fm$^4$ and 1.05(6) for $^{26}$ Si [28], [37], and 432(11) e$^2$ fm$^4$ and 0.95(8) for $^{24}$ Mg [28], [38].",
"The double ratios of $^{30-36}$ S are close to unity, as seen in the figure, indicating that the hindrance of the proton collectivity does not appear in these nuclei.",
"The large double ratios for $^{38, 40}$ S can be explained by the neutron skin effect caused by the $Z=16$ sub-shell closure [36], [39].",
"The dotted lines in Fig.",
"REF (a)-(c) show shell model predictions with the USDB effective interaction using the effective charges of $e_p=1.36$ and $e_n=0.45$ [30], [40].",
"The calculation shows excellent agreement with the experimental $E_x(2^+_1)$ values.",
"The overall tendencies of the $B$ (E2) and $|M_n/M_p|/(N/Z)$ are reasonably reproduced.",
"Especially the sudden decrease of $B$ (E2) and increase of $|M_n/M_p|/(N/Z)$ at $^{28}$ S are mostly predicted.",
"It indicates that the shell model calculation with the USDB interaction accounts for the phenomena observed in the present study.",
"It should be note that the model interprets the $N=16$ magicity in neutron-rich nuclei with the large $s_{1/2}$ -$d_{3/2}$ gap, and hence the $Z=16$ magicity in proton-rich nuclei is inherent in the model reflecting the isospin symmetry.",
"Slight difference remaining between the predictions and the experimental data may require further development of the theory.",
"In summary, the $B$ (E2;$0^+_{gs} \\rightarrow 2^+_1$ ) value for the proton-rich nucleus $^{28}$ S was measured using Coulomb excitation at 53 MeV/nucleon.",
"The resultant $B$ (E2) value is determined to be 181(31) e$^2$ fm$^4$ .",
"The double ratio $|M_n/M_p|/(N/Z)$ for the $0^+_{gs} \\rightarrow 2^+_1$ transition in $^{28}$ S is obtained to be 1.9(2), by evaluating the $M_n$ value from the known $B$ (E2) value of the mirror nucleus $^{28}$ Mg.",
"These results show a hindered proton collectivity relative to that of neutrons in $^{28}$ S. It indicate the emergence of $Z=16$ magicity in the $|T_z|=2$ nucleus $^{28}$ S. The systematics of the $|M_n/M_p|/(N/Z)$ values for the $Z=16$ isotopes indicates that the hindrance of proton collectivity in proton-rich region appears only at $^{28}$ S. The authors thank the staff of RIKEN Nishina Center for their work of the beam operation during the experiment.",
"One of the authors (Y.T.)",
"is grateful for the support of the Special Postdoctoral Researcher Program at RIKEN and Research Center for Measurement in Advanced Science at Rikkyo University."
]
] | 1204.0920 |
[
[
"Minimal model of associative learning for cross-situational lexicon\n acquisition"
],
[
"Abstract An explanation for the acquisition of word-object mappings is the associative learning in a cross-situational scenario.",
"Here we present analytical results of the performance of a simple associative learning algorithm for acquiring a one-to-one mapping between $N$ objects and $N$ words based solely on the co-occurrence between objects and words.",
"In particular, a learning trial in our learning scenario consists of the presentation of $C + 1 < N$ objects together with a target word, which refers to one of the objects in the context.",
"We find that the learning times are distributed exponentially and the learning rates are given by $\\ln{[\\frac{N(N-1)}{C + (N-1)^{2}}]}$ in the case the $N$ target words are sampled randomly and by $\\frac{1}{N} \\ln [\\frac{N-1}{C}] $ in the case they follow a deterministic presentation sequence.",
"This learning performance is much superior to those exhibited by humans and more realistic learning algorithms in cross-situational experiments.",
"We show that introduction of discrimination limitations using Weber's law and forgetting reduce the performance of the associative algorithm to the human level."
],
[
"Introduction",
"Early word-learning or lexicon acquisition by children, in which the child learns a fixed and coherent lexicon from language-proficient adults, is still a polemic problem in developmental psychology [1].",
"The classical associationist viewpoint, which can be traced back to empiricist philosophers such as Hume and Locke, contends that the mechanism of word learning is sensitivity to covariation – if two events occur at the same time, they become associated – being part of humans' domain-general learning capability.",
"An alternative viewpoint, dubbed social-pragmatic theory, claims that the child makes the connections between words and their referents by understanding the referential intentions of others.",
"This idea, which seems to be originally due to Augustine, implies that children use their intuitive psychology or theory of mind [2] to ‘read’ the adults' minds.",
"Although a variety of experiments with infants demonstrate that they exhibit a remarkable statistical learning capacity [3], the findings that the word-object mappings are generated both fast and errorless by children are difficult to account for by any form of statistical learning.",
"We refer the reader to the book by Bloom [1] for a review of this most controversial and fascinating theme.",
"Regardless of the mechanisms children use to learn a lexicon, the issue of how good humans are at acquiring a new lexicon using statistical learning in controlled experiments has been tackled recently [4], [5], [6], [7], [8], [9].",
"In addition, it has been conjectured that statistical learning may be the principal mechanism in the development of pidgin [10].",
"In this context (pidgin), however, it is necessary to assume that the agents are endowed with some capacity to grasp the intentions of the others as well as to understand nonlinguistic cues, otherwise one cannot circumvent the referential uncertainty inherent in a word-object mapping [11].",
"The statistical learning scenario we consider here is termed cross-situational or observational learning, and it is based on the intuitive idea that one way that a learner can determine the meaning of a word is to find something in common across all observed uses of that word [12], [13], [14].",
"Hence learning takes place through the statistical sampling of the contexts in which a word appears.",
"There are two competing theories about word learning mechanism within the cross-situational scenario, namely, hypothesis testing and associative learning (see [9] for a review).",
"The former mechanism assumes that the learner builds coherent hypotheses about the meaning of a word which is then confirmed or disconfirmed by evidence [15], [16], [17], [18], whereas the latter is based essentially on the counting of co-occurrences of word-object statistics [19], [20].",
"Albeit associative learning can be made much more sophisticated than the mere counting of contingencies [9], in this contribution we focus on the simplistic interpretation of that learning mechanism, which allows the derivation of explicit mathematical expressions to characterize the learner's performance.",
"Although cross-situational associative learning has been a very popular lexicon acquisition scenario since it can be easily implemented and studied through numerical simulations (see, e.g., [10], [21], [22], [23]), there were only a few attempts to study analytically this learning strategy [24], [25].",
"These works considered a minimal model of cross-situational learning, in which the one-to-one mapping between $N$ objects and $N$ words must be inferred through the repeated presentation of $C + 1 < N$ objects (the context) together with a target word, which refers to one of the objects in the context.",
"The co-occurrences between objects and words are stored in a confidence matrix, whose integer entries count how many times an object has co-occurred with a given word during the learning process.",
"The meaning of a particular word is then obtained by picking the object corresponding to the greatest confidence value associated to that word, i.e., the object that has co-occurred more frequently with that word.",
"In this paper, we expand on the work of Smith et al.",
"[24] and offer analytical expressions for the learning rates of this minimal associative algorithm for different word sampling schemes, see Eqs.",
"(REF ), (REF ) and (REF ).",
"To assess the relevance of our findings to the efforts on understanding how humans perform on cross-situational learning tasks, we use Monte Carlo simulations to compare the performance of the minimal associative algorithm with the performance of humans for short learning times [6] and with the performance of a more elaborated learning algorithm for long times [7].",
"Our finding that the accuracy of the minimal associative algorithm is much higher than that observed in the experiments is imputed to the illimited storage and discrimination capability of the algorithm.",
"In fact, introduction of errors in the discrimination of confidence values according to Weber's law reduces the performance to a level below that of humans.",
"Somewhat surprisingly, introduction of forgetting acts synergistically with our prescription for Weber's law resulting in an increase of performance that eventually matches the experimental results.",
"The rest of this paper is organized as follows.",
"In Sect.",
"we describe the learning scenario and in Sect.",
"we introduce and study analytically the simplest associative learning scheme for counting co-occurrences of words and objects, in which the words are learned independently.",
"We consider first the problem of learning a single word and then investigate the effect of using different word sampling schemes for learning the complete $N$ -word lexicon.",
"In Sect.",
"we compare the performance of the minimal associative algorithm with the performance exhibited by adult subjects.",
"To understand the high efficiency of the algorithm we introduce constraints on its storage and discrimination capabilities and show how the constraint parameters can be tunned to describe the experimental results.",
"Finally, in Sect.",
"we discuss our findings and present some concluding remarks."
],
[
"Cross-situational learning scenario",
"We assume that there are $N$ objects, $N$ words and a one-to-one mapping between words and objects.",
"To describe the one-to-one word-object mapping, we use the index $i= 1, \\ldots , N$ to represent the $N$ distinct objects and the index $h = 1, \\ldots , N$ to represent the $N$ distinct words.",
"Without loss of generality, we define the correct mapping as that for which the object represented by $i=1$ is named by the word represented by $h=1$ , object represented by $i=2$ by word represented by $h=2$ , and so on.",
"Henceforth we will refer to the integers $i$ and $h$ as objects and words, respectively, but we should keep in mind that they are actually labels to those complex entities.",
"At each learning event, a target word, say word $h=1$ , is selected and then $C+1$ distinct objects are selected from the list of $N$ objects.",
"This set of $C+1$ objects forms a context for the selected word.",
"The correct object ($i=1$ , in this case) must be present in the context.",
"The learner's task is to guess which of the $C+1$ objects the word refers to.",
"This is then an ambiguous word learning scenario in which there are multiple object candidates for any word.",
"The parameter $C$ is a measure of the ambiguity (and so of the difficulty) of the learning task.",
"In particular, in the case $C=N-1$ the word-object mapping is unlearnable.",
"At first sight one could expect that learning would be trivial for $C=0$ since there is no ambiguity, but the learning complexity depends also on the manner the objects are selected to compose the contexts.",
"Typically, the objects are chosen randomly and without replacement from the list of $N$ objects (see, e.g., [23], [24], [25]), which for $C=0$ results in a learning error (i.e., the fraction of wrong word-object associations) that decreases exponentially with learning rate $-\\ln \\left( 1 - 1/N \\right)$ as the number of learning trials $t$ increases.",
"This is so because there is a non-vanishing probability that some words are not selected in the $t$ trials [25].",
"In order to avoid testing subjects on the meaning of words they never heard, most experimental studies on word-learning mechanisms use a deterministic word selection procedure which guarantees that all words are uttered before the testing stage, although some words may be spoken more frequently than others [4], [5], [6], [7].",
"Hence we consider here, in addition to the random selection procedure, a deterministic selection procedure which guarantees that all $N$ words are selected in $t=N$ trials.",
"For this procedure the case $C=0$ is trivial and the learning error becomes zero at $t=N$ .",
"However, since encountering words whose meaning is unknown is not a rare event in the real world (hence the utility of dictionaries), a non-uniform Zipfian random selection of words is likely to be a more realistic sampling scheme for learning natural word-referent associations (see, e.g., [25])."
],
[
"Minimal Associative Learning Algorithm",
"Here we consider one of the earliest mathematical learning models – the linear learning model [26].",
"The basic assumption of this model is that learning can be modeled as a change in the confidence with which the learner associates the target word to a certain object in the context.",
"More to the point, this confidence is represented by a matrix whose non-negative integer entries $p_{hi}$ yield a value for the confidence with which word $h$ is associated to object $i$ .",
"We assume that at the outset ($t=0$ ) all confidences are set to zero, i.e., $p_{hi} = 0$ with $i,h = 1, \\ldots ,N$ and whenever object $i^*$ appear in a context in companion with target word $h^*$ the confidence $p_{i^* h^*}$ increases by one unit.",
"Hence at each learning trial, $C+1$ confidences are updated.",
"Note that this learning algorithm considers reinforcement only.",
"To determine which object corresponds to word $h$ the learner simply chooses the object index $i$ for which $p_{hi}$ is maximum.",
"In the case of ties, the learner selects one object at random among those that maximize the confidence $p_{hi}$ .",
"Recalling our definition of the correct word-object mapping in the previous section, the learning algorithm achieves a perfect performance when $p_{hh} > p_{hi}$ for all $h$ and $i \\ne h$ .",
"The learning error $E$ at a given trial $t$ is then given by the fraction of wrong word-object associations.",
"Note that we have $p_{hi} \\le p_{hh}$ with $ i \\ne h$ since object $i=h$ must appear in the contexts of all learning events in which the target word is $h$ (see Sect.",
").",
"In this case, the learning error of any single word, say $h$ , which we denote by $\\epsilon _{sw}$ , is the reciprocal of the number of objects for which $p_{hi} = p_{hh}$ with $i \\ne h$ .",
"Interestingly, it can easily be shown that this very simple and general learning algorithm is identical to the algorithm presented in [24] which is based on detecting the intersections of context realizations in order to single out the set of confounder objects at a given trial $t$ .",
"This equivalence has already been noted in the literature [27] (see also [8]).",
"The minimal associative learning algorithm can be immediately adapted to incorporate more realistic features, such as finite memory and imprecision in the comparison of magnitudes, whereas the confounder reducing algorithm is restricted to an ideal learning scenario.",
"A salient feature of the minimal associative learning algorithm which allows the analytical study of its performance is the fact that words are learned independently.",
"This is easily seen by noting that the confidences $p_{hi}, i=1, \\ldots , N$ are updated only when the target word $h$ is selected.",
"This means that, aside from a trivial rescaling of the learning time, our scenario is equivalent to the experimental settings (see Sect. )",
"in which $C+1$ target words are presented together with a context exhibiting $C+1$ objects, with each object associated to one of the target words [4], [5], [6], [7].",
"Taking advantage of this feature, we will first solve a simplified version of the cross-situational learning in which a given target word $h$ (and its associated object $i=h$ ) appears in all learning trials whereas the $C$ other objects (the confounders) that make up the rest of the context vary in each learning trial.",
"Once the problem of learning a single word is solved (see Sect.",
"REF ), we can easily work out the generalization to learning the whole lexicon (see Sects.",
"REF and REF ).",
"We will use $\\tau $ to measure the time of the learning trials in the case of single-word learning and $t$ in the whole lexicon learning case."
],
[
"Learning a single word",
"Before any learning event has taken place, the target word may be associated to any one of the $N$ objects, so the initial state of the learning error is always equal to $\\left(N-1\\right)/N$ .",
"When the first learning event takes place, the target word may be incorrectly assigned to the $C$ other confounder objects shown in the context, so the probability of error at the first trial is always equal to $C/ \\left( C+1\\right)$ .",
"In the second trial, there are two possibilities: the probability of error is unchanged because the same context is chosen or the probability of error decreases to the value $n/ \\left( n+1\\right)$ l with $n < C$ because $n$ confounder objects of the first context appeared again in the second trial.",
"The same reasoning allows us to describe the probability of error in any trial given that this probability is known in the previous trial as described next.",
"As pointed out, the possible error values are $n/ \\left( n+1\\right)$ with $n=0,1,...,C$ .",
"Labeling these values by the index $n$ , the probability of error at trial $\\tau $ can be written as $\\mathbf {W} \\left( \\tau \\right) = \\left(w_{C}\\left( \\tau \\right), w_{C-1}\\left( \\tau \\right), \\cdots , w_{1}\\left( \\tau \\right) ,w_{0}\\left( \\tau \\right) \\right) .$ The time evolution of $\\mathbf {W} \\left( \\tau \\right)$ is given by the Markov chain $\\mathbf {W} \\left( \\tau + 1 \\right) = \\mathbf {W} \\left( \\tau \\right) T ,$ where $T$ is a $\\left( C + 1 \\right) \\times \\left( C + 1 \\right) $ transition matrix whose entries $T_{m n}$ yield the probability that the error at a certain trial is $n/\\left( n + 1 \\right)$ given that the error was $m/\\left( m + 1 \\right)$ in the previous trial.",
"Clearly, $T_{m n } = 0$ for $m < n$ since the error cannot increase during the learning stage in the absence of noise.",
"It is a simple matter to derive $T_{m n }$ for $m \\ge n$ [24].",
"In fact, it is given by the probability that in $C$ choices one selects exactly $n$ of the $m$ confounder objects from the list of $N-1$ objects.",
"(We recall that the object associated to the target word is picked with certainty and so the list comprises $N-1$ objects, rather than $N$ , and the number of selections is $C$ rather than $C+1$ .)",
"This is given by the hyper-geometric distribution [28] $T_{m n} = \\frac{\\dbinom{m}{n}\\dbinom{N-1-m}{C-n}}{\\dbinom{N-1}{C}}$ for $m \\ge n$ and $T_{m n} = 0$ for $m < n$ .",
"Since the transition matrix is triangular, its eigenvalues $\\lambda _{n}$ with $n=0,1,...,C$ are the elements of the main diagonal that correspond to transitions that leave the learning error unchanged, i.e., $\\lambda _{n} = T_{n n} = \\frac{\\dbinom{N-1-n}{C-n}}{\\dbinom{N-1}{C}} .$ Note that $\\lambda _0 = 1 > \\lambda _{n \\ne 1} > 0$ as expected for eigenvalues of a transition matrix.",
"In addition, since $\\lambda _n/\\lambda _{n+1} = \\left( N-1 -n \\right)/\\left( C -n \\right) > 1$ the eigenvalues are ordered such that $\\lambda _0 > \\lambda _1 > \\ldots > \\lambda _{N-1}$ .",
"Recalling that the probability vector is known at $\\tau = 1$ , namely, $\\mathbf {W}_{1} = \\left( 1, 0, \\ldots , 0 \\right)$ we can write $\\mathbf {W} \\left( \\tau \\right) = \\mathbf {W} \\left( \\tau =1 \\right) T^{\\tau - 1} .$ Although it is a simple matter to write $T^{\\tau -1}$ in terms of the right and left eigenvectors of $T$ , this procedure does not produce an explicit analytical expression for $W_n \\left( \\tau \\right)$ in terms of the two parameters of the model $C$ and $N$ , since we are not able to find analytical expressions for the eigenvectors.",
"However, Smith et al.",
"[24] have succeeded in deriving a closed analytical expression for $W_n \\left( \\tau \\right)$ using the inclusion-exclusion principle of combinatorics [29], $W_n \\left( \\tau \\right) = \\binom{C}{n} \\sum _{i=n}^C \\left( -1 \\right)^{i-n} \\binom{C-n}{i-n} \\lambda _i^{\\tau - 1},$ where $\\lambda _i$ , given by Eq.",
"(REF ), is the probability that a particular set of $i$ members of the $C$ confounders in the first learning episode $\\tau = 1$ appear in any subsequent episode.",
"Although the spectral decomposition of $T$ plays no role in the derivation of Eq.",
"(REF ) we choose to maintain the notation $\\lambda _i$ for the above mentioned probability.",
"Figure: (Color online) The expected single-word learning error ϵ sw \\epsilon _{sw} as a function of the number of learning trials τ\\tau .",
"The solid curves are the results of Eq.",
"() and thefilled circles the results of Monte Carlo simulations.",
"The upper panel shows the results for C=2C=2 and (left to right) NN = 100, 50, 30 and 20, and the lower panel for N=20N=20 and (left to right) CC = 5, 10, 13, 15 and 16.Recalling that a situation described by $n$ corresponds to the learning error $n/\\left( n + 1 \\right)$ we can immediately write the average learning error for a single word as $ \\epsilon _{sw} \\left( \\tau \\right) = \\sum _{n=0}^C \\frac{n}{n+1} W_n \\left( \\tau \\right) ,$ which is valid for $\\tau > 0$ only.",
"For $\\tau = 0$ one has $\\epsilon _{sw} \\left( 0 \\right) = 1 - 1/N$ .",
"The dependence of $\\epsilon _{sw}$ on the number of learning trials $\\tau $ for different values of $N$ and $C$ is illustrated in Fig.",
"REF using a semi-logarithmic scale.",
"Except for very small $\\tau $ , the learning error exhibits a neat exponential decay which is revealed by considering only the leading non-vanishing contribution to $W_n$ for large $\\tau $ , namely, $\\epsilon _{sw} \\left( \\tau \\right) \\sim \\frac{C}{2} \\lambda _1^{\\tau -1} = \\frac{N-1}{2} \\exp \\left[ -\\tau \\ln \\left( \\frac{N -1}{C} \\right) \\right] .$ Hence the learning rate for single-word learning is $\\alpha _{sw} = \\ln \\left[ \\left( N-1 \\right) / C \\right]$ which is zero in the case $C=N-1$ , i.e., all objects appear in the context and so learning is impossible.",
"In the case $C=0$ , the learning rate diverges so that $\\epsilon _{sw} = 0$ at the first learning trial $\\tau = 1$ already.",
"Most interestingly, the learning rate increases with increasing $N$ (see Fig.",
"REF ) indicating that the larger the number of objects, the faster the learning of a single word.",
"This apparently counterintuitive result has a simple explanation: a large list of objects to select from actually decreases the chances of choosing the same confounding object during the learning events."
],
[
"Learning the whole lexicon with random sampling",
"We turn now to the original learning problem in which the learner has to acquire the one-to-one mapping between the $N$ words and the $N$ objects.",
"In this section we focus in the case the target word at each learning trial is chosen randomly from the list of $N$ words.",
"Since all words have the same probability of being chosen, the probability of choosing a particular word is $1/N$ .",
"At trial $t$ we assume that word 1 appeared $k_1$ times, word 2 appeared $k_2$ times, and so on with $k_1 + k_2 + \\ldots + k_N = t$ .",
"The integers $k_i = 0, \\ldots , t$ are random variables distributed by the multinomial $P \\left(k_{1}, \\ldots , k_{N} \\right) = N^{-t} \\frac{t!}{k_{1}!",
"\\cdots k_{N}!}",
"\\delta _{t, k_1 + \\ldots + k_N} .$ Figure: (Color online) The expected learning error E r E_r in the case the NN words are sampled randomly as a function of the number of learning trials tt.The solid curves are the results of Eq.",
"() and thefilled circles the results of Monte Carlo simulations.",
"The upper panel shows the results for C=2C=2 and (left to right) N=10,20,...,80N = 10, 20, \\ldots , 80 andthe lower panel the results for N=20N=20 and (left to right) C=1,2,...,10C = 1, 2, \\ldots , 10 .Clearly, if word $i$ appeared $k_i$ times in the course of $t$ trials then the expected error associated to it is $\\epsilon _{sw} \\left( k_i \\right)$ with the (word independent) single word error given by Eq.",
"(REF ) for $k_i > 0$ .",
"With this observation in mind, we can immediately write the expected learning error in the case the $N$ words are sampled randomly, $E_{r} \\left( t \\right) & = & \\sum _{k_1, \\ldots , k_N} P \\left(k_{1}, \\ldots , k_{N} \\right) \\frac{1}{N} \\sum _{i=1}^N \\epsilon _{sw} \\left( k_i \\right) \\nonumber \\\\& = & \\sum _{k=0} ^t \\dbinom{t}{k} \\left( \\frac{1}{N} \\right)^k \\left( 1 - \\frac{1}{N} \\right)^{t-k} \\epsilon _{sw} \\left( k \\right) .$ The sum over $k$ can be easily carried out provided we take into account the fact that $\\epsilon _{sw} \\left( k \\right) $ has different prescriptions for the cases $k=0$ and $k > 0$ .",
"We find $E_{r} \\left( t \\right) & = & \\sum _{n=0}^C \\frac{n}{n+1} \\dbinom{C}{n} \\sum _{i=n}^C \\dbinom{C-n}{i-n} \\frac{\\left( -1 \\right)^{i-n}}{\\lambda _i} \\times \\nonumber \\\\& & \\left[ \\left( \\frac{\\lambda _i + N -1}{N} \\right)^t - \\left( \\frac{ N -1}{N} \\right)^t \\right] \\nonumber \\\\& & + \\left( \\frac{N-1}{N} \\right)^{t+1}$ with $\\lambda _i$ given by Eq.",
"(REF ).",
"This is a formidable expression which can be evaluated numerically for $C$ not too large and in Fig.",
"REF we exhibit the dependence of $E_r$ on the number of learning trials for a selection of values of $N$ and $C$ .",
"To obtain the asymptotic time dependence of $E_r$ we need to keep in the double sum only the leading order term.",
"Since the summand in Eq.",
"(REF ) vanishes for $n=0$ , the largest eigenvalue that appears in that expression is $\\lambda _1$ , corresponding to the term $i=n=1$ , and so this is the term that dominates the sum in the limit $t \\rightarrow \\infty $ .",
"Hence $E_r$ exhibits the exponential decay $E_r \\sim \\frac{C}{2 \\lambda _1} \\left( \\frac{\\lambda _1 + N - 1}{N} \\right) ^t = \\frac{N-1}{2} \\exp \\left[ - t \\alpha _r \\left( C,N \\right) \\right]$ where $\\alpha _r \\left( C,N \\right) = \\ln \\left[ \\frac{ N \\left( N-1 \\right)}{C + \\left( N-1 \\right)^2} \\right]$ is the learning rate of our algorithm in the case the $N$ words are sampled randomly.",
"As already mentioned, it is interesting that the unambiguous learning scenario $C=0$ results in the finite learning rate $ - \\ln \\left( 1 - 1/N \\right)$ simply because some words may never be chosen in the course of the $t$ learning trials.",
"Interestingly, the learning rate $\\alpha _r$ exhibits a non-monotone dependence on $N$ for fixed $C$ : for $N > 2C+1$ , it decreases with increasing $N$ (this is the parameter selection used to draw the upper panel of Fig.",
"REF ), and it increases with increasing $N$ otherwise.",
"Recalling that for fixed $C$ the minimum value of $N$ is $N =C + 1$ at which $\\alpha _r = 0$ , increasing $N$ from this minimal value must result in an increase of $\\alpha _r$ .",
"The fact that $\\alpha _r$ decreases for large $N$ – an effect of sampling – implies that there is an optimal value $N^* = 2C + 1$ that maximizes the learning speed for fixed $C$ .",
"Of course, for fixed $N$ the learning speed is maximized by $C=0$ ."
],
[
"Learning the whole lexicon with deterministic sampling",
"To better understand the effects of the random sampling of the $N$ words we consider here a deterministic sampling scheme in which every word is guaranteed to be chosen in the course of $N$ learning trials.",
"Let us begin with the first $N$ learning trials and recall that at time $t=0$ all words have error $\\epsilon _{sw} \\left( 0 \\right) = \\left(N-1\\right)/N$ .",
"Then during the learning process for $t= 1, \\ldots , N$ there will be $t$ words with error $\\epsilon _{sw} \\left( 1 \\right) = C/ \\left(C+1 \\right)$ and $N-t$ with error $\\epsilon _{sw} \\left( 0 \\right) $ so that the total learning error for the deterministic sampling is $E_d \\left( t \\right) = \\frac{1}{N} \\left[ t \\epsilon _{sw} \\left( 1 \\right) +\\left( N-t \\right) \\epsilon _{sw} \\left( 0 \\right) \\right], \\qquad t \\le N .$ This expression can be easily extended for general $t$ by introducing the single-word learning time $\\tau = \\lfloor t/N \\rfloor $ , $E_d \\left( t \\right) = \\frac{1}{N} \\left[ \\left( t - N \\tau \\right) \\epsilon _{sw} \\left( \\tau + 1 \\right) +\\left( N \\tau + N - t \\right) \\epsilon _{sw} \\left( \\tau \\right) \\right]$ where $\\lfloor x\\rfloor $ is the largest integer not greater than $x$ .",
"The time-dependence of the learning error for the deterministic sampling of the $N$ words is shown in Fig.",
"REF .",
"For $t \\gg N$ , $\\tau $ becomes a continuous variable for any practical purpose, and then we can see that $E_d$ decreases exponentially with increasing $t$ .",
"Clearly, the learning rate is determined by the single-word learning error [see Eq.",
"(REF )] and so replacing $\\tau $ by $t/N$ in that equation we obtain the learning rate for the deterministic sampling case $\\alpha _d \\left( C,N \\right) = \\frac{1}{N} \\ln \\left[ \\frac{ N-1 }{C} \\right] .$ As in the single-word learning case, the learning rate diverges for $C=0$ in accordance with our intuition that in the absence of ambiguity, the learning task should be completed in $N$ steps.",
"In fact, the learning error decreases linearly with $t$ as given by Eq.",
"(REF ).",
"Similarly to our findings for the random sampling, $\\alpha _d$ exhibits a non-monotonic dependence on $N$ : beginning from $\\alpha _d = 0$ at $N=C+1$ , it increases until reaching a maximum at $N^* \\approx e C$ and then decreases towards zero again as the size of the lexicon further increases.",
"It is interesting to compare the learning rates for the two sampling schemes, Eqs.",
"(REF ) and (REF ).",
"In the leading non-vanishing order for large $N$ and $C \\ll N$ , we find $\\alpha _r \\approx C/N^2$ whereas $\\alpha _d \\approx \\left( \\ln N \\right)/N$ .",
"In the more realistic situation in which the context size grows linearly with the lexicon size, i.e., $C = \\gamma N$ with $\\gamma \\in \\left[0,1 \\right]$ , for large $N$ we find $\\alpha _r \\approx \\left( 1 - \\gamma \\right)/N$ and $\\alpha _d \\approx - \\left( \\ln \\gamma \\right)/N$ .",
"Hence for small $C$ or $\\gamma \\approx 0$ , the deterministic sampling of words results in much faster learning than the random sampling.",
"For large $C$ or $\\gamma \\approx 1$ , however, the two sampling schemes produce equivalent results."
],
[
"Effects of imperfect memory and discriminability",
"The simplicity of the minimal associative learning algorithm analyzed in the previous section is deceiving.",
"In fact, the algorithm contains two assumptions that make it extremely powerful.",
"The first assumption is illimited memory, since the algorithm stores the confidence values from the very first to the last learning episode, regardless of the number of learning episodes.",
"The second is perfect discriminability, since it always identifies the largest confidence regardless of the closeness to, say, the second-largest one.",
"The scheme we use to relax the perfect discriminability assumption is inspired by Weber's law, which asserts that the discriminability of two perceived magnitudes is determined by the ratio of the objective magnitudes.",
"Accordingly, we assume that the probability that the algorithm selects object $i$ as the referent of any given word $h$ is simply $p_{hi}/\\sum _j p_{hj}$ , so that referents with similar confidence values have similar probabilities of being selected.",
"This differs from the original minimal algorithm for which the referent selection probability is either one or zero, except in the case of ties when the probability is divided equally among the referents with identical confidence values.",
"Forgetting or decaying of the confidence values is implemented by subtracting a fixed factor $\\beta \\in \\left[ 0, 1 \\right]$ from the confidences $p_{hi}, i=1,\\ldots ,N $ whenever word $h$ is absent from a learning episode.",
"The problem with this procedure is that the confidence values may become negative and when this happens we reset them to zero.",
"Another difficulty that may rise is when $p_{hi}= 0$ for all $i=1,\\ldots ,N $ and in this case we reset $p_{hi}= 1/N$ for all $i=1,\\ldots ,N $ .",
"These resetting procedures are responsible for the discontinuities observed in the performance of the algorithm as we will see next.",
"As in the minimal algorithm, we add 1 to the confidences associated to the target word and the objects exhibited in the context.",
"Relaxation of the perfect memory assumption makes the forgetting parameter $\\beta $ dependent on the sampling scheme of words, which precludes an analytical approach to this problem.",
"As we have to resort to simulations to study the performance of the modified algorithm anyway, in this section we consider a very specific sampling scheme used in experiments with adult subjects to test the effect of varying the frequency of presentation of the target words on their learning performances [6].",
"More importantly, use of this sampling scheme allows us to compare quantitatively the performance of the minimal as well as of the modified associative learning algorithms with the performances of the adult subjects.",
"The experiment we consider here aims at evaluating the performance of the associative algorithms in learning a mapping between $N=18$ words and $N=18$ objects after 27 training episodes [6].",
"Each episode comprises the presentation of 4 objects together with their corresponding words.",
"Following Ref.",
"[6], we investigate two conditions.",
"In the two frequency condition, the 18 words are divided into two subsets of 9 words each.",
"In the first subset the 9 words appear 9 times and in the second only 3 times (see Fig.",
"REF ).",
"In the three frequency condition, the 18 words are divided in three subsets of 6 words each.",
"In the first subset, the 6 words appear 3 times, in the second, 6 times and in the third, 9 times (see Fig.",
"REF ).",
"In these two conditions, the same word was not allowed to appear in two consecutive learning episodes.",
"Once the cross-situational learning scenario is defined, we carry out $10^4$ runs of the modified associative learning algorithm for a fixed value of the forgetting parameter.",
"The results are shown in terms of the average accuracy $1 - \\langle \\epsilon \\rangle $ as function of $\\beta $ in Figs.",
"REF and REF .",
"The horizontal straight lines and the shaded zones around them represent the means and standard deviations of the results of experiments carried out with 33 adult subjects [6].",
"Before discussing the interesting dependence of the accuracy on the forgetting parameter exhibited in Figs.",
"REF and REF , a word is in order about the performance of the original minimal algorithm that is not shown in those figures.",
"In the two frequency condition, the mean accuracy is $0.99$ for words in the 9-repetition subset and $0.90$ for those in the 3-repetition subset.",
"In the three frequency condition, the mean accuracy is $0.99$ for words in the 9- and 6-repetition subsets, and $0.91$ for those in the 3-repetition subset.",
"These accuracy values are well above those exhibited in Figs.",
"REF and REF .",
"Moreover, adding the forgetting factor to the minimal associative algorithm does not affect its performance, since subtracting the same quantity from all confidence values $p_{hi}$ for a fixed word $h$ does not alter the rank order of these confidences.",
"Figure: (Color online) Expected accuracy for the two frequency condition as function of the forgetting parameter β\\beta at learning trial t=27t=27.The curves show the accuracy of the set of words sampled 9 and 3 times as indicated in the figure.",
"The horizontal lines and the shaded zones arethe experimental results .",
"For β≈0.16\\beta \\approx 0.16 we get an excellent agreement between the model and experiments.Figure: (Color online) Expected accuracy for the three frequency condition as function of the forgetting parameter β\\beta at learning trial t=27t=27.The curves show the accuracy of the set of words sampled 9, 6 and 3 times as indicated in the figure.",
"The horizontal lines and the shaded zones arethe experimental results .",
"For β≈0.08\\beta \\approx 0.08 we get an excellent agreement between the model and experiments.Although we intuitively expect that words that appear more frequently would be learned better, this outcome actually depends on the value of the forgetting parameter as shown in Figs.",
"REF and REF .",
"This counterintuitive finding was first observed in the three frequency condition experiment on adult subjects [6].",
"In fact, the results of those experiments (i.e., the expected accuracies) can be described very well by choosing $\\beta = 0.16$ in the two frequency condition and $\\beta = 0.08$ in the three frequency condition.",
"It is interesting that the choice of a moderate value for the forgetting parameter $\\beta $ may result in a considerable improvement of the performance of the algorithm.",
"This is a direct consequence of Weber's law prescription for the discrimination of the confidence values and so there is a synergy between discrimination and memory in our algorithm.",
"To see this we note that at a given learning trial the ratio between the probabilities of selecting referent $i=1$ and referent $i=2$ for a word $h$ is $r = p_{h1}/p_{h2}$ .",
"If word $h$ does not appear in the next trial then this ratio becomes $r^{\\prime } = \\frac{p_{h1} - \\beta }{ p_{h2} - \\beta } \\approx r \\left[ 1 + \\frac{\\beta }{p_{h1}p_{h2}} \\left( p_{h1} -p_{h2} \\right) \\right]$ so that $r^{\\prime } > r $ if $p_{h1} > p_{h2}$ , thus implying that the forgetting parameter helps the discrimination of the largest confidence.",
"Of course, too large values of $\\beta $ deteriorate the performance of the algorithm as shown in the figures.",
"We note that the dents and jumps in the learning curves are not statistical fluctuations but consequences of the discontinuities introduced by the ad hoc regularization procedures discussed before.",
"The above analysis, summarized in part by Figs.",
"REF and REF , evinces the better performance of the associative algorithm with perfect storage and discrimination capabilities when compared with humans' performance for a finite number of learning trials ($t=27$ , in the case).",
"In addition, it shows that introduction of imprecision in the discrimination of confidence values following Weber's law prescription together with forgetting brings that performance down to the human level.",
"For the sake of completeness, it would be interesting to compare the performance of the minimal associative algorithm with humans' performance in the limit of very long learning times, which was in fact the main focus of Sect.",
".",
"As there are no such experiments – we guess it would be nearly impossible to keep the subjects' attention focused on such boring tasks for too long – next we compare the performance of the minimal algorithm with the performance of a rather sophisticated learning algorithm which, among other things, models the attention of the learners to regular and novel words [7].",
"The algorithm is described briefly as follows.",
"At any given trial, the confidence values $p_{hi}$ are adjusted according to the update rule $p_{hi}^{\\prime } = \\hat{\\beta } p_{hi} + \\chi \\frac{ p_{hi} \\exp \\left[ \\lambda \\left( H_h + H_i \\right) \\right]}{\\sum _{hi} p_{hi} \\exp \\left[ \\lambda \\left( H_h + H_i \\right) \\right]}$ where $H_h = - \\sum _i \\Lambda _{hi} \\ln \\Lambda _{hi}$ with $\\Lambda _{hi} = p_{hi}/\\sum _i p_{hi}$ , and similarly for $H_i$ with the indexes of the sums running over the set of words [7].",
"In this equation the entropies $H_h$ and $H_i$ are used as measures of the novelty of word $h$ and object $i$ at the current learning episode.",
"The parameter $\\hat{\\beta }$ governs forgetting, $\\chi $ is the weight distributed among the potential associations in the trial, and $\\lambda $ weights the uncertainty (entropies) and prior knowledge ($p_{hi}$ ).",
"We refer the reader to Ref.",
"[7] for a detailed explanation of the algorithm as well as for a comparison with experimental results for short learning times.",
"Here we present its performance in acquiring the word-object mapping in the simplified scenario of Sect.",
"(i.e., one target word and $C+1$ objects in the context) for randomly sampled words.",
"Figure: (Color online) Expected learning error for N=10N=10 and C=2C=2 as function of the number of learning trials tt in the case words are sampled randomly.The open circles are results of the minimal associative algorithm whereas the filled symbols are the results of the algorithm proposed by Karchergis et al.",
":diamonds (χ=3.01,λ=1.39,β ^=0.64\\chi =3.01, \\lambda =1.39, \\hat{\\beta } =0.64), circles (χ=0.31,λ=2.34,β ^=0.91\\chi =0.31, \\lambda =2.34, \\hat{\\beta } =0.91), and squares (χ=0.20,λ=0.88,β ^=0.96\\chi =0.20, \\lambda =0.88, \\hat{\\beta } =0.96).Figure REF summarizes our findings for $N=10$ , $C=2$ and three selection of the parameter set ($\\chi , \\lambda , \\hat{\\beta }$ ) used by Karchergis et al.",
"to reproduce the experimental results [7].",
"The symbols in this figure represent an average over $10^4$ independent samples.",
"The expected learning error decreases exponentially with increasing $t$ and the rate of learning (the slope of the learning curves for large $t$ in the semi-log scale) is roughly insensitive to the choice of the parameters of the algorithm.",
"As expected from our previous analysis of short learning times, the minimal associative learning algorithm performs much better than the more realistic algorithm.",
"These conclusions hold true for a vast variety of different selections of $N$ and $C$ , as well as for the deterministic word sampling scheme."
],
[
"Discussion",
"As the problem of learning a lexicon within a cross-situational scenario was studied rather extensively by Smith et al.",
"[24], it is appropriate that we highlight our original contributions to the subject in this concluding section.",
"Although we have borrowed from that work a key result for the problem of learning a single word, namely, Eq.",
"(REF ), even in this case the focal points of our studies deviate substantially.",
"In fact, throughout the paper our main goal was the determination of the learning rates in several learning scenarios, whereas the main interest of Smith et al.",
"was in quantifying the number of learning trials required to learn a word with a fixed given probability [24].",
"In addition, those authors addressed the problem of the random sampling of words using various approximations, leading to inexact results from where the learning rate $\\alpha _r$ , see Eq.",
"(REF ), cannot be recovered.",
"As a result, the interesting non-monotonic dependence of $\\alpha _r$ (and $\\alpha _d$ , as well) on the size $N$ of the lexicon passed unnoticed.",
"The study of the deterministic sampling of words and the introduction and analysis of the effects of limited storage and discrimination capabilities on the original minimal associative algorithm are original contributions of our paper.",
"We note that in the cross-situational scenarios studied previously [24], [25] the set of objects that can be associated to a given word is word-dependent, rather than constant as considered here.",
"In other words, if the target word is $h$ then the elements of the context in a learning episode are drawn from a fixed subset of $N_h \\le N $ objects.",
"These subsets can freely overlap with each other.",
"Here we have assumed $N_h = N$ for $h =1, \\ldots , N$ .",
"Of course, this generalization does not affect the analysis of the single-word learning, except that $\\epsilon _{sw}$ becomes word-dependent since the parameter $N$ is replaced by $N_h$ [see Eq.",
"(REF )] and similarly for the learning rate $ \\alpha _{sw}$ [see Eq.",
"(REF )].",
"More importantly, since words are learned independently by the minimal associative algorithm, the single-word learning errors contribute additively to the total lexicon learning error regardless of the sampling procedure [see Eqs.",
"(REF ) and (REF )].",
"Hence the asymptotic behavior of the total error is determined by the word that takes the longest to be acquired, i.e., the word with the lowest learning rate or equivalently with the smallest subset cardinality $N_h$ .",
"With this in mind we can easily obtain the learning rates for this more general situation, namely, $\\alpha _r = \\ln \\left\\lbrace N \\left( N_m - 1 \\right) /\\left[ C + \\left( N_m - 1 \\right) \\left( N - 1 \\right) \\right] \\right\\rbrace $ and $\\alpha _d = \\ln \\left[ \\left( N_m - 1 \\right)/C \\right]/N$ where $N_m = \\min _h \\left\\lbrace N_h, h=1, \\ldots , N \\right\\rbrace $ .",
"As expected, in the case $N_m = N$ these expressions reduce to Eqs.",
"(REF ) and (REF ).",
"The cross-situational learning scenario considered here, as well as those used in experimental studies, does not account for the presence of external noise, such as the effect of out-of-context target words.",
"This situation can be modeled by introducing a probability $\\gamma \\in \\left[ 0, 1 \\right]$ that the correct object is not part of the context so the target word can be said to be out of context.",
"Since we have assumed that learning is based on the perception of differences in the co-occurrence of objects and target words, in the case all $N$ objects have the same probability of being selected to form the contexts regardless of the target word, such a purely observational learning is clearly unattainable.",
"To determine the critical value of the noise parameter $\\gamma _c$ at which this situation occurs we simply equate the probability of selecting the correct object with the probability of selecting any given confounding object to compose the context in a learning episode, $1 - \\gamma _{c} = \\frac{\\left( 1 - \\gamma _{c}\\right) C}{N-1} + \\frac{\\gamma _{c} \\left(C+1\\right)}{N-1},$ from where we get $\\gamma _{c} = 1 -\\frac{C+1}{N}.$ Since in this case all objects and all words are equivalent, in the sense they have the same probability of co-occurrence, the average single-word learning error, as wells as the total error regardless of the sampling scheme, is simply $\\epsilon _{sw} = 1 - 1/N$ .",
"We refer the reader to Ref.",
"[30] for a detailed study of the behavior of the minimal associative learning algorithm near the critical noise parameter using statistical mechanics techniques.",
"Here we emphasize that the existence of $\\gamma _c$ is not dependent on the algorithm used to learn the word-object mapping.",
"Rather, it is a limitation of cross-situational learning in general.",
"The simplifying feature of our model that allowed an analytical approach, as well as extremely efficient Monte Carlo simulations (in all graphs the error bars were smaller than the symbol sizes), is the fact that words are learned independently from each other.",
"In this context, the minimal associative algorithm considered here corresponds to the optimal learning strategy.",
"Moreover, the fact that the minimal associative algorithm exhibits effectively illimited storage and discrimination capabilities makes its learning performance much superior to that of adult subjects in controlled experiments [6] and to that of sophisticated algorithms designed to capture the strategies used by humans in the observational learning task [7].",
"Interestingly, introduction of errors in the discrimination of the confidence values using Weber's law reduced the performance of the minimal algorithm to the level reported in the experiments.",
"Perhaps, sophisticated learning strategies such as the mutual exclusivity constraint [15], which directs children to map novel words to unnamed referents, have evolved to compensate the limitations imposed by Weber's law to evaluate the frequency of co-occurrence of words and referents."
],
[
"Acknowledgments",
"This research was supported by The Southern Office of Aerospace Research and Development (SOARD), Grant No.",
"FA9550-10-1-0006, and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).",
"P.F.C.T.",
"was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)."
]
] | 1204.1564 |
[
[
"Modeling and forecasting exchange rate volatility in time-frequency\n domain"
],
[
"Abstract This paper proposes an enhanced approach to modeling and forecasting volatility using high frequency data.",
"Using a forecasting model based on Realized GARCH with multiple time-frequency decomposed realized volatility measures, we study the influence of different timescales on volatility forecasts.",
"The decomposition of volatility into several timescales approximates the behaviour of traders at corresponding investment horizons.",
"The proposed methodology is moreover able to account for impact of jumps due to a recently proposed jump wavelet two scale realized volatility estimator.",
"We propose a realized Jump-GARCH models estimated in two versions using maximum likelihood as well as observation-driven estimation framework of generalized autoregressive score.",
"We compare forecasts using several popular realized volatility measures on foreign exchange rate futures data covering the recent financial crisis.",
"Our results indicate that disentangling jump variation from the integrated variation is important for forecasting performance.",
"An interesting insight into the volatility process is also provided by its multiscale decomposition.",
"We find that most of the information for future volatility comes from high frequency part of the spectra representing very short investment horizons.",
"Our newly proposed models outperform statistically the popular as well conventional models in both one-day and multi-period-ahead forecasting."
],
[
"Introduction",
"Much of the recent popularity of realized volatility is mainly due to its two distinct implications for practical estimation and forecasting.",
"The first relates to the nonparametric estimation of latent volatility process without the need for any assumptions about the explicit model.",
"The second brings the possibility of forecasting volatility directly through standard time series econometrics with discretely sampled daily data, while effectively extracting information from intraday high-frequency data.",
"In contrast to the popular framework of a generalized autoregressive conditional heteroscedasticity (GARCH) model, volatility is directly observed and can be used for forecasting when we apply realized volatility concept.",
"While [20] argue that GARCH(1,1) can hardly be beaten by any other model, recent active research shows that with help of high frequency measures, we can improve the forecasts significantly.",
"[24] for example utilize intra-day data and show that we can obtain forecasts superior to forecasts from GARCH(1,1).",
"[23] assesses the informational content of alternative realized volatility estimators using Realized GARCH in Value-at-Risk prediction.",
"We aim to extend this line of research by investigating the importance of different realized measures in recently developed framework combining appeal of GARCH(1,1) model and high frequency data.",
"Moreover, we employ recently developed estimators which allow to decompose volatility into several investment horizons and thus we attempt to study the influence of intraday investment horizons on daily volatility forecasts.",
"A simple nonparametric estimate of price variability over a given time interval has been formalized by [2], [8].",
"While these authors provide a unified framework for modeling, [31] was one of the first to provide a formal assessment of the relationship between cumulative squared intraday returns and the underlying return variance.",
"A vast quantity of literature on several aspects of estimating volatility has emerged in the wake of these fundamental contributions.",
"Our work builds on this popular realized volatility approach.",
"While most time series models are set in the time domain, we enrich the analysis by the frequency domain.",
"This is enabled by the use of the wavelet transform.",
"It is a logical step to take, as the stock markets are believed to be driven by heterogeneous investment horizons.",
"In our work, we ask if wavelet decomposition can improve our understanding of volatility series.",
"One very appealing feature of wavelets is that they can be embedded into stochastic processes, as shown by [5].",
"Thus we can conveniently use them to extend the theory of realized measures as shown by [17], or [12].",
"One of the common issues with the interpretation of wavelets in economic applications is that they are filter, thus they can hardly be used for forecasting in econometrics.",
"Models based on wavelets are often outperformed by simple benchmark models, as shown by [18].",
"Rather, they can provide a useful “lens\" into the time series.",
"Our wavelet-based estimator of realized volatility uses wavelets only to decompose the daily variation of the returns using intraday information, hence this is no longer an issue.",
"As wavelets are used to measure realized volatility at different investment horizons, we can construct a forecasting model based on the wavelet decomposed volatility conveniently.",
"Several attempts to use wavelets in the estimation of realized variation have emerged in the past few years.",
"[21] were the first to suggest a wavelet estimator of realized variance.",
"[15], for example, proposes to use a wavelet transform as a comparable estimator of quadratic variation.",
"[28] uses wavelets to decompose volatility into a multi-horizon scale.",
"One exception is the work of [17], who were the first to use the wavelet-based realized variance estimator together with the methodology for estimation of jumps.",
"In [12], we revisit and extend this work and using large monte carlo study we show that this estimator improves forecasting of the volatility substantially when compared to other estimators.",
"Moreover, in [12] we attempt to use the estimators to decompose stock market volatility into several investment horizons in a non-parametric way.",
"Motivated by previous results, this paper focuses on proposing a model which will improve the forecasting of volatility.",
"Similarly to [22] and [4], we use the decomposition of the quadratic variation with the intention of building a more accurate forecasting model.",
"Our approach is very different though, as we use wavelets to decompose the integrated volatility into several investment horizons and jumps.",
"Moreover, we employ recently proposed realized GARCH framework of [19].",
"Realized GARCH allows to model jointly returns and realized measures of volatility, while key feature is a measurement equation that relates the realized measure to the conditional variance of returns.",
"We investigate several measures of realized volatility, namely realized volatility estimator proposed by [3], the bipower variation estimator of [10], the two-scale realized volatility of [30], the realized kernel of [6] and finally jump wavelet two-scale realized variance (JWTSRV) estimator of [12] in the framework of Realized GARCH, and we find significant differences in volatility forecasts, while our JWTSRV estimator brings the largest improvement.",
"The main contribution of the paper thus lies in investigating how the forecasts of volatility vary with different realized measures and mainly testing the time-frequency realized measures in forecasting.",
"We show that the most important information influencing the future volatility is carried by the high frequency part of the spectra representing very short investment horizons of 10 minutes.",
"This decomposition gives us an interesting insight into the volatility process.",
"Moreover, we utilize jumps estimated by the JWTSRV estimator to build a Realized Jump-GARCH model, which outperforms significantly other models based on different realized measures.",
"The paper is organized in sections as follows.",
"After the introduction, the second section introduces theoretical framework for time-frequency decomposition of volatility, the third section introduces a Realized GARCH and Jump-GARCH models, the fourth section applies the presented theory, decomposes the empirical volatility of forex futures and finally uses the decomposition for forecasting."
],
[
"Theoretical framework for time-frequency decomposition of realized volatility",
"While most time series models are naturally set in the time domain, wavelet transform help us to enrich the analysis of quadratic variation by the frequency domain.",
"It is a logical step to take, as the stock markets are believed to be driven by heterogeneous investment horizons, so volatility dynamics should be understood not only in time but at investment horizons as well.",
"In this section, we introduce an estimator that is able to separate the continuous part of the price process containing noise from the jump variation.",
"We will briefly introduce general ideas of constructing the estimator here, while for the details necessary to understand the derivation of the estimator using wavelet theory, we refer to [12].",
"In the analysis, we assume that the latent logarithmic asset price follows a standard jump-diffusion process contamined with microstructure noise.",
"Let $y_t$ be the observed logarithmic prices evolving over $0 \\le t \\le T$ , which will have two components; the latent, so-called “true log-price process\", $dp_t=\\mu _t dt+\\sigma _t dW_t+\\xi _t dq_t$ , and zero mean $i.i.d.$ microstructure noise, $\\epsilon _t$ , with variance $\\eta ^2$ .",
"In a latent process, $q_t$ is a Poisson process uncorrelated with $W_t$ , and the magnitude of the jump, denoted as $J_l$ , is controlled by factor $\\xi _t \\sim N(\\bar{\\xi },\\sigma ^2_{\\xi })$ .",
"Thus, the price process is $y_t=p_t+\\epsilon _t$ .",
"The quadratic return variation over one day $[t-1,t]$ , associated with the price process $y_t$ can be naturally decomposed into two parts: integrated variance of the latent price process, $IV_t$ and jump variation $JV_t$ $QV_t=\\underbrace{\\int _{t-1}^t \\sigma _s^2 ds}_{\\mbox{$IV_t$}}+ \\underbrace{\\sum _{t-1 \\le l \\le t} J_l^2}_{\\mbox{$JV_t$}}$ As detailed by [2] and [8], quadratic variation is a natural measure of variability in the logarithmic price.",
"A simple consistent estimator of the overall quadratic variation under the assumption of zero noise contamination in the price process is provided by the well-known realized variance, introduced by [1].",
"The realized variance over $\\left[t-1,t\\right]$ can be estimated as $\\widehat{QV}^{(RV)}_t=\\sum _{k=1}^N r_{k,t}^2\\hspace{2.84526pt},$ where $r_{k,t}$ is the $k$ -th intraday return in the $\\left[t-1,t\\right]$ and $N$ is the number of intraday observations.",
"The estimator in Eq.",
"(REF ) converges in probability to $IV_{t}+JV_{t}$ as $N\\rightarrow \\infty $ [1], [2], [3], [7], [8], [9].",
"While the observed price process $y_t$ is contamined with noise and jumps in real data, we need to account for this, as the main object of interest is the $IV_t$ part of quadratic variation.",
"[30] propose solution to the noise contamination by introducing the so-called two-scale realized volatility (TSRV henceforth) estimator.",
"They adopt a methodology for estimation of the quadratic variation utilizing all of the available data using an idea of precise bias estimation.",
"The two-scale realized variation over $\\left[t-1,t\\right]$ is measured by $\\widehat{QV}_{t}^{(TSRV)}=\\widehat{QV}_{t}^{(average)}-\\frac{\\bar{N}}{N} \\widehat{QV}_{t}^{(all)},$ where $\\widehat{QV}_{t}^{(all)}$ is computed as in Eq.",
"(REF ) on all available data and $\\widehat{QV}_{t}^{(average)}$ is constructed by averaging the estimators $\\widehat{QV}_{t}^{(g)}$ obtained on $G$ grids of average size $\\bar{N}=N/G$ as: $\\widehat{QV}_{t}^{(average)}=\\frac{1}{G}\\sum _{g=1}^G \\widehat{QV}_{t}^{(g)}.$ where the original grid of observation times, $M=\\lbrace t_1,\\dots ,t_N\\rbrace $ is subsampled to $M^{(g)}$ , $g=1,\\dots ,G$ , where $N/G\\rightarrow \\infty $ as $N\\rightarrow \\infty $ .",
"The estimator in Eq.",
"(REF ) provides the first consistent and asymptotic estimator of the quadratic variation of $p_t$ with rate of convergence $N^{-1/6}$ .",
"[30] also provide the theory for optimal choice of $G$ grids, $G^*=c N^{2/3}$ , where the constant $c$ can be set to minimize the total asymptotic variance.",
"A different approach to deal with noise is realized kernels (RK) due to [6].",
"In our study, we use the non-negative realized kernel with Parzen weights which guarantees a positive estimate.The realized kernel estimator is computed without accounting for end effects, for more information see [6] Since we are interested in decomposing quadratic variation into the integrated variance and jump variation component, we have use a methodology for jump detection as well.",
"[10], [11] develop bipower variation estimator (BV), which can detect the presence of jumps in high-frequency data.",
"The main idea of the BV estimator is to compare two measures of the integrated variance, one containing the jump variation and the other being robust to jumps and hence containing only the integrated variation part.",
"In our work, we use the [4] adjustment of the original [10] estimator, which helps render it robust to certain types of microstructure noise."
],
[
"Estimation of quadratic variation using Wavelets ",
"[17] use a different approach to realized volatility measurement.",
"They use wavelets in order to separate jump variation from the price process, as well as for estimation of the integrated variance on the jump–adjusted data.",
"In addition, wavelet methodology offers decomposition of the estimated volatility into scales representing investment horizons.",
"Therefore, we can observe how particular investment horizon contributes to the total variance.",
"In the empirical section, we aim to study information content of investment horizons for volatility forecasting, thus we describe the wavelet jump detection and then introduce the wavelet estimator of integrated variance of [12], which allows to decompose the volatility into several investment horizons.",
"Assume the sample path of the price process $y_t$ has a finite number of jumps.",
"Following results of [29] on the wavelet jump detection of the deterministic functions with $i.i.d.$ additive noise, we use a special form of a discrete wavelet transform, the MODWT, which unlike the ordinary discrete wavelet transform, is not restricted to a dyadic sample length.",
"Jumps locations are detected with use of the first level wavelet coefficients obtained on the process $y_t$ over $[t-1,t]$ , $\\mathcal {W}_{1,k,t}$ .",
"Since we use the MODWT, we have $k$ wavelet coefficients at the first scale, which corresponds to number of intraday observations, i.e., $k=1,\\dots ,N$ .",
"In case the value of the wavelet coefficient $\\mathcal {W}_{1,k,t}$ is greaterUsing the MODWT filters, we need to slightly correct the position of the wavelet coefficients to get the precise jump position, see [26].",
"than the universal threshold $d_t\\sqrt{2 \\log N}$ [16], than a jump with size $\\Delta J_{k,t}$ is detected as $\\Delta J_{k,t} =(y_{k,t} - y_{k-1,t}) \\mathbb {1}_{\\lbrace |\\mathcal {W}_{1,k,t}| >d_t\\sqrt{2 \\log N} \\rbrace } \\hspace{14.22636pt} k\\in [1,N],$ where $d_t=\\sqrt{2}median\\lbrace |\\mathcal {W}_{1,k,t}|\\rbrace /0.6745$ for $k\\in [1,N]$ denotes the intraday median absolute deviation estimator [27], .",
"Following [17], the jump variation over $[t-1,t]$ in the discrete time is estimated as the sum of squares of all the estimated jump sizes, $\\widehat{JV}_t=\\sum _{k=1}^N \\left(\\Delta J_{k,t}\\right)^2.$ [17] prove that using (REF ), we are able to estimate the jump variation from the process consistently with the convergence rate of $N^{-1/4}$ .",
"Having precisely detected jumps, we proceed to jump adjustment of the observed price process $y_t$ over $\\left[t-1,t\\right]$ .",
"We adjust the data for jumps by subtracting the intraday jumps from the price process as: $y_{k,t}^{(J)}=y_{k,t}-\\Delta J_{k,t}, \\hspace{14.22636pt} k=1,\\dots N,$ where $N$ is the number of intraday observations.",
"Finally, the volatility can be computed using the jump-adjusted wavelet two-scale realized variance (JWTSRV) estimator on the jump adjusted data $y_{k,t}^{(J)}$ .",
"The JWTSRV is an estimator that is able to estimate integrated variance from the process under the assumption of data containing noise as well as jumps.",
"The estimator utilizes the TSRV approach of [30] as well as the wavelet jump detection method.",
"Another advantage of the estimator is, that it decomposes the integrated variance into $J^m+1$ components, therefore we are able to study the dynamics of volatility at various investment horizons.",
"Following [12], we define the JWTSRV estimator over $\\left[t-1,t\\right]$ , on the jump-adjusted data as: $\\widehat{IV}_{t}^{(JWTSRV)}=\\sum _{j=1}^{J^m+1}\\widehat{IV}_{j,t}^{(JWTSRV)}=\\sum _{j=1}^{J^m+1}\\left(\\widehat{IV}_{j,t}^{(average)}-\\frac{\\bar{N}}{N} \\widehat{IV}_{j,t}^{(all)}\\right),$ where $\\widehat{IV}_{j,t}^{(average)}=\\frac{1}{G} \\sum _{g=1}^G \\sum _{k=1}^N \\mathcal {W}_{j,k,t}^2 $ is obtained from wavelet coefficient estimates on a grid of size $\\bar{N}=N/G$ , and $\\widehat{IV}_{j,t}^{(all)}=\\sum _{k=1}^N \\mathcal {W}_{j,k,t}^2$ is the wavelet realized variance estimator at a scale $j$ on all the jump-adjusted observed data, $y_{k,t}^{(J)}$ .",
"$\\mathcal {W}_{j,k,t}^2$ denotes the MODWT wavelet coefficient at scale $j$ with position $k$ obtained over $\\left[t-1,t\\right]$ .",
"[12] show that the JWTSRV is consistent estimator of the integrated variance as it converges in probability to the integrated variance of the process $p_t$ , and they test the small sample performance of the estimator in a large Monte Carlo study.",
"The JWTSRV is found to be able to recover true integrated variance from the noisy process with jumps very precisely.",
"Moreover, the JWTSRV estimator is also tested in forecasting exercise, which confirms to improve forecasting of the integrated variance substantially."
],
[
"A forecasting model based on decomposed integrated volatilities ",
"Similarly to [22] and [4], we use the decomposition of the quadratic variation with the intention of building a more accurate forecasting model.",
"Our approach is very different though, as we use wavelets to decompose the integrated volatility into several investment horizons and jumps first.",
"Then, we employ recently proposed Realized GARCH framework of [19].",
"Realized GARCH allows to model jointly returns and realized measures of volatility, while key feature is a measurement equation that relates the realized measure to the conditional variance of returns.",
"We use the decomposed realized measures in the Realized GARCH, and expect that our modification will result in better in-sample fits of the data as well as out-of-sample forecasts.",
"For comparison, we also use other estimators and study how they improve the forecasting ability of Realized GARCH."
],
[
"Realized GARCH framework for forecasting",
"The key object of interest in GARCH family is the conditional variance, $h_t=var(r_t|\\mathcal {F}_{t-1})$ , where $r_t$ is a time series of returns.",
"While in a standard GARCH(1,1) model the conditional variance, $h_t$ is dependent on its past $h_{t-1}$ and $r^2_{t-1}$ , [19] propose to utilize realized measures of volatility and make $h_t$ dependent on them as well.",
"The authors introduce so-called measurement equation which ties the realized measure to latent volatility.",
"The general framework of Realized GARCH$(p,q)$ models is well connected to existing literature in [19].",
"Here, we restrict ourselves to the simple log-linear specification of Realized GARCH$(1,1)$ with Gaussian innovations which we will use to build our model.",
"A simple log-linear Realized GARCH$(1,1)$ model is given by $r_t &=&\\sqrt{h_t} z_t, \\\\\\log (h_t)&=& \\omega +\\beta \\log (h_{t-1}) +\\gamma \\log (x_{t-1})\\\\\\log (x_t) &=& \\xi +\\psi \\log (h_t)+\\tau _1 z_t+\\tau _2 z^2_t+u_t,$ where $r_t$ is the return, $x_t$ a realized measure of volatility, $z_t \\sim i.i.d (0,1)$ and $u_t \\sim i.i.d (0,\\sigma _u^2)$ with $z_t$ and $u_t$ being mutually independent, $h_t=var(r_t|\\mathcal {F}_{t-1})$ with $\\mathcal {F}_{t}=\\sigma (r_t,x_t,r_{t-1},x_{t-1},\\ldots )$ and $\\tau (z)=\\tau _1 z_t+\\tau _2 z^2_t$ is called leverage function.",
"It is worth noting that while we use only this specific version of Realized GARCH, [19] introduces a general family of models which generalized a GARCH models as it can nest any GARCH specification.",
"Assumption on innovations is not essential, and can be changed to other common assumptions as Student's $t$ for example.",
"[19] provide the asymptotic properties of the quasi-maximum likelihood estimator (QMLE henceforth), and propose to use it for the parameter estimation.",
"The structure of the QMLE is very similar to that of the standard GARCH model, although one needs to accommodate possible error from realized measures in the estimation.",
"The log-likelihood function is thus given by $\\log L(\\lbrace r_t,x_t\\rbrace _{t=1}^T;\\theta )=\\sum _{t=1}^T \\log f(r_t,x_t|\\mathcal {F}_{t-1}).$ Standard GARCH models do not have realized measure $x_t$ , so joint conditional density needs to be factorized $f(r_t,x_t|\\mathcal {F}_{t-1})=f(r_t|\\mathcal {F}_{t-1})f(x_t|r_t,\\mathcal {F}_{t-1}).$ When comparing the fits to a standard GARCH, the partial log-likelihood, $\\ell (r)=\\sum _{t=1}^T \\log f(r_t|\\mathcal {F}_{t-1})$ can be used conveniently.",
"For the Gaussian specification of $z_t$ and $u_t$ , the joint likelihood is then split into the sum $\\ell (r,x)=\\underbrace{-0.5 \\sum _{t=1}^T \\left( \\log (2\\pi )+\\log (h_t)+r^2_t/h_t\\right)}_{ \\ell (r)}+\\underbrace{-0.5 \\sum _{t=1}^T\\left(\\log (2\\pi )+\\log (\\sigma _u^2)+u_t^2/\\sigma _u^2 \\right)}_{\\ell (x|r)}.$ Realized GARCH framework is rather general, as it allows to accommodate different realized measures.",
"In our analysis, we will estimate Realized GARCH(1,1) models using different $x_t$ , namely RV, BV, TSRV, RK and JWTSRV from previous sections and compare its performance.",
"Moreover, we will use the decomposition of the volatility from JWTSRV to study which investment horizon has greatest effect on future volatility."
],
[
"Realized Jump-GARCH(1,1) on decomposed volatilities",
"By estimating different Realized GARCH models using various realized measures, we will see which measure carries the best information for forecasting of volatility.",
"In addition, we would like to utilize estimated jumps as well as decomposition of JWTSRV to see which investment horizon has impact on the future volatility as well.",
"By addition of estimated jumps into the variance equation, we obtain Realized Jump-GARCH(1,1) model (Realized J-GARCH) given by $r_t &=&\\sqrt{h_t} z_t, \\\\\\log (h_t)&=& \\omega +\\beta \\log (h_{t-1})+\\gamma \\log (x_{t-1})+\\gamma _J \\log (1+JV_{t-1}),\\\\\\log (x_t)&=& \\xi +\\psi \\log (h_t)+\\tau _1 z_t+\\tau _2 z^2_t+u_t,$ where $x_t$ and $JV_t$ is estimated using Eq.",
"(REF ), and Eq.",
"(REF ) by our $\\widehat{IV}_t^{(JWTSRV)}$ and $\\widehat{JV}_t$ respectively and $z_t$ and $u_t$ come from Gaussian normal distribution and are mutually independent.",
"This model is logical step in generalizing the Realized GARCH structure as $\\widehat{IV}_t^{(JWTSRV)}$ and $\\widehat{JV}_t$ add up to a quadratic variation of underlying price process which is not biased by noise.",
"If jumps have a significant impact on volatility forecasts, $\\gamma _J$ coefficient should be significantly different from zero.",
"Finally, we utilize a wavelet decomposition of integrated volatility to different investment horizons and estimate the model where $x_{j,t}$ will represent $\\widehat{IV}_{j,t}^{(JWTSRV)}$ at all estimated investment horizons $j=1,...,J^m+1$ .",
"Thus second equation for $\\log (h_t)$ will be $\\log (h_t)&=& \\omega +\\beta \\log (h_{t-1})+ \\gamma _{W_{j}} \\log (x_{j,t-1})+\\gamma _J \\log (1+JV_{t-1}),$ where $x_{j,t}$ is estimated using Eq.",
"(REF ) by $\\widehat{IV}_{j,t}^{(JWTSRV)}$ .",
"Our last model is motivated by the decomposition of realized volatility into several investment horizons.",
"$\\gamma _{W_j}$ will provide a good guide for significance of various investment horizons.",
"All the models are estimated by QMLE and can be easily generalized by assuming different distributions of $z_t$ and $u_t$ .",
"We have also tried to incorporate different distributionsResults for other cases are available from authors upon request.",
"but the results did not change qualitatively and to keep the number of estimated models under control, we report the results for the Gaussian case only."
],
[
"Forecast evaluation using different realized variance measures",
"To analyze the forecast efficiency and information content of different volatility estimators in the Realized GARCH framework, we employ the popular approach of [25] regressions.",
"The regression takes the form: $V_{t+1}^{(m)}=\\alpha +\\beta V_{t}^{RG-(k)}+\\epsilon _t,$ with $V_{t+1}^{(m)}$ being the integrated volatility estimated using the square root of the $m$ -th estimator, namely, RV, BV, TSRV, RK and JWTSRV, respectively.",
"$V_{t}^{RG-(k)}$ denotes the 1-day ahead forecast of $V_{t+1}^{(m)}$ using the $k$ -th estimator based on Realized GARCH(1,1), namely RV, BV, TSRV, RK, JWTSRV and finally Realized J-GARCH(1,1).",
"We report in-sample as well as rolling out-of-sample results.",
"After testing the forecasting efficiency of the different volatility models we would also like to test the information content of the wavelet decomposition of the realized volatility.",
"For this purpose, we separately estimate Realized J-GARCH$(1,1)$ for all components $\\widehat{IV}_{j,t}^{(JWTSRV)}$ for $j=1,\\dots ,5$ of the realized volatility.",
"Finally, we use Heteroskedasticity-adjusted Mean Square Error (HMSE) of [14] and QLIKE of [13]."
],
[
"Data description",
"Foreign exchange future contracts are traded on the Chicago Mercantile Exchange (CME) on a 24-hour basis.",
"As these markets are among the most liquid, they are suitable for analysis of high-frequency data.",
"We will estimate the realized volatility of British pound (GBP), Swiss franc (CHF) and euro (EUR) futures.",
"All contracts are quoted in the unit value of the foreign currency in US dollars.",
"It is advantageous to use currency futures data for the analysis instead of spot currency prices, as they embed interest rate differentials and do not suffer from additional microstructure noise coming from over-the-counter trading.",
"The cleaned data are available from Tick Data, Inc.http://www.tickdata.com/ It is important to look first at the changes in the trading system before we proceed with the estimation on the data.",
"In August 2003, for example, the CME launched the Globex trading platform, and for the first time ever in a single month, the trading volume on the electronic trading platform exceeded 1 million contracts every day.",
"On Monday, December 18, 2006, the CME Globex(R) electronic trading platform started offering nearly continuous trading.",
"More precisely, the trading cycle became 23 hours a day (from 5:00 pm on the previous day until 4:00 pm on current day, with a one-hour break in continuous trading), from 5:00 pm on Sunday until 4:00 pm on Friday.",
"These changes certainly had a dramatic impact on trading activity and the amount of information available, resulting in difficulties in comparing the estimators on the pre-2003 data, the 2003–2006 data and the post–2006 data.",
"For this reason, we restrict our analysis to a sample period extending from January 5, 2007 through November 17, 2010, which contains the most recent financial crisis.",
"The futures contracts we use are automatically rolled over to provide continuous price records, so we do not have to deal with different maturities.",
"The tick-by-tick transactions are recorded in Chicago Time, referred to as Central Standard Time (CST).",
"Therefore, in a given day, trading activity starts at 5:00 pm CST in Asia, continues in Europe followed by North America, and finally closes at 4:00 pm in Australia.",
"To exclude potential jumps due to the one-hour gap in trading, we redefine the day in accordance with the electronic trading system.",
"Moreover, we eliminate transactions executed on Saturdays and Sundays, US federal holidays, December 24 to 26, and December 31 to January 2, because of the low activity on these days, which could lead to estimation bias.",
"Finally, we are left with 944 days in the sample.",
"Looking more deeply at higher frequencies, we find a large amount of multiple transactions happening exactly at the same time stamp.",
"We use the arithmetic average for all observations with the same time stamp.",
"Table REF presents the summary statistics for the daily log-returns of GBP, CHF and EUR futures over the sample period, $t=1,\\dots ,944$ , i.e., January 5, 2007 to November 17, 2010.",
"The summary statistics display an average return very close to zero, skewness, and excess kurtosis which is consistent with the large empirical literature.",
"Having prepared the data, we can estimate the integrated volatility using different estimators and use them within proposed forecasting framework.",
"For each futures contract, the daily integrated volatility is estimated using the square root of realized variance estimator of [3], the bipower variation estimator of [10], the two-scale realized volatility of [30], the realized kernel of [6].",
"Finally, we utilize our jump wavelet two-scale realized variance estimator defined by Eq.",
"(REF ).",
"All the estimators are adjusted for small sample bias.",
"For convenience, we refer to the estimators in the description of the results as RV, BV, TSRV, RK and JWTSRV, respectively.",
"The RV, BV, TSRV, and the JWTSRV are estimated on 5-minute log-returns.",
"The decomposition of volatility into the so-called continuous and jump part is depicted by Figure REF , which provide the returns, estimated jumps and finally integrated variances using JWTSRV estimator for all three futures pairs.",
"Figure REF shows the further decomposition into several investment horizons.",
"For better illustration, we annualize the square root of the integrated variance in order to get the annualized volatility and we compute the components of the volatility on several investment horizons.",
"Figure REF (a) to (e) show the investment horizons of 10 minutes, 20 minutes, 40 minutes, 80 minutes and up to 1 day, respectively.",
"It is very interesting that most of the volatility (around 50%) comes from the 10-minute investment horizon which is a new insight.",
"Moreover, the longer the investment horizon, the lower the contribution of the variance to the total variation."
],
[
"Forecasting results",
"The main results of estimation and forecasting are presented in this section.",
"The estimation strategy is as follows.",
"For each of three forex futures considered, namely GBP, CHF and EUR, we first estimate benchmark GARCH(1,1) model.",
"Then, we estimate the Realized GARCH of [19] with several realized volatility measures, namely RV, BV, RK, TSRV and JWTSRV to find out the influence of realized measure on the final forecasts.",
"Finally, we add our Realized Jump-GARCH model, and estimate it with JWTSRV components.",
"We use the period from January 5, 2007 to February 2, 2010 for estimation of all the models.",
"Thus, we refer to this period as the in-sample period.",
"The rest of the year 2010 is saved for comparison of the out-of-sample forecasts on a rolling basis.",
"We use open-to-close returns as well as open-to-close realized measures in the analysis.",
"Tables REF , REF and REF contain all results for GBP futures, CHF futures and EUR futures respectively.",
"By observing partial log-likelihood $\\ell (r)$ , we can see immediately that all the Realized GARCH models reported by the second, third, fourth, fifth and sixth columns bring significant improvement to the GARCH(1,1) model reported by the first column (in testing significance of the difference, we restrict ourselves to use simple log-likelihood ratio test).",
"When we focus on comparison of Realized GARCH(1,1) with different realized measures $x_t$ , we observe further significant differences.",
"This points to importance of usage proper realized measure.",
"While the simplest measure RV is contammined with noise and jumps, we expect the worst performance for the model which uses it as a realized measure proxy.",
"BV is robust to jumps and RK with TSRV are robust to noise.",
"Finally, our JWTSRV estimator is robust to both jumps and noise in the realized variance so we expect the best performance of model which uses JWTSRV.",
"Looking at the results, all the parameter estimates for the different realized measures are similar to each other, while log-likelihoods $\\ell (r,x)$ uncover rather large differences between the models.",
"In all three currency futures used in this study, Realized GARCH(1,1) model with JWTSRV realized measure performs significantly better than in RV, BV, RK and TSRV cases.",
"Its log-likelihood brings the largest improvement to all other models.",
"Models with RV, BV and TSRV are more or less on the similar levels of the log-likelihood, while surprisingly the model with RK measure of realized variance is far worst in all cases.",
"Figure REF compares the latent volatility $h_t$ and measured volatility $x_t$ from all models.",
"It brings further insight into the various fits and it confirms our findings.",
"When compared to RV, BV, RK and TSRV measures, we can see that relationship between $h_t$ and $x_t$ is strongest for our last three models based on the JWTSRV measure.",
"Moreover the plot for RK explains why it performs so badly in comparison to other estimators.",
"Figure REF shows the scattered plots of residuals $z_t$ and $u_t$ and confirms a good specification of all models.",
"Knowing that Realized GARCH(1,1) with the JWTSRV measure performs far best in all cases and improves the log-likelihood supports our further modifications.",
"Motivated by these results, we study if inclusion of jumps in the model improves the fits in our newly proposed Realized Jump-GARCH(1,1) model denoted as Realized J-GARCH in the Tables.",
"Realized J-GARCH model brings further significant improvement in the log-likelihood in all cases, while $\\gamma _J$ coefficient is significantly different from zero in the case of CHF and EUR, but can not be statistically distinguished from zero in the case of GBP.",
"The only reason we can see is that in case of GBP futures the estimated jump variation is lowest in comparison to other currencies used so it does not play significant role in forecasts.",
"Still, we can conclude that jumps bring significant improvement in the modeling and Realized Jump-GARCH(1,1) using JWTSRV outperforms other models.",
"As the last step, we would like to utilize the realized variance decomposition of JWTSRV as we expect that it will further improve the forecasts.",
"Our motivation is straightforward.",
"We would like to find out if the different investment horizons bring improvement to the volatility forecasts.",
"For this, we utilize Realized Jump-GARCH(1,1) model again, where we include different components of JWTSRV and jumps.",
"Thus the resting five models in the Tables REF , REF and REF are Realized Jump-GARCH(1,1) on JWTSRV decompositions separately.",
"Specifically, we use $\\widehat{IV}_{j,t}^{(JWTSRV)}$ for $x_t$ to see the contribution of different decompositions.",
"Results for all three currencies are strikingly conclusive.",
"The first scale $j=1$ , representing volatility on investment horizon of 10 minutes brings statistically same results as the best model with full JWTSRV without decomposition.",
"Looking at the coefficients together with the full and partial log-likelihoods, and out-of-sample measures, the replacement of full volatility with volatility from the first scale makes no difference.",
"This means that the most information for future volatility is carried over in this very short investment horizon.",
"Replacing the realized measure with other components of volatility brings slight deterioration of fits as well as forecasts.",
"This points us to the conclusion that the most of the information can be found in the high frequency part of the spetral density of volatility.",
"Until now, we have been focusing on in-sample results.",
"Turning our attention to the out-of-sample results we can see that they confirm our findings from the in-sample estimation.",
"Realized J-GARCH model with JWTSRV measure improves out-of-sample forecasts in terms of $R^2$ from the Mincer-Zarnowitz regression substantially in comparison to all other models, while HMSE and QLIKE confirms this result.",
"We also note that out-of-sample forecasts are very accurate as $\\beta $ from the Mincer-Zarnowitz regressions is very close to 1.",
"It is worth noting that using other realized measures this is not always the case.",
"Still, $\\alpha $ shows some bias in the forecasts, especially in case of GBP and CHF currencies.",
"This bias can be contributed to the period we choose for forecasting.",
"Thus the forecasting exercise brings an important result, that we need to rely on proper realized measure when forecasting volatility."
],
[
"Conclusion",
"In this paper, we investigate how the decomposed integrated volatilities and jumps influence the future volatility using realized GARCH framework.",
"Utilizing a jump wavelet two scale realized volatility estimator, which measures volatility in the time-frequency domain, we study the influence of intra-day investment horizons on daily volatility forecasts.",
"After the introduction of wavelet-based estimation of quadratic variation together with forecasting model, we compare our estimators to several most popular estimators, namely, realized variance, bipower variation, two-scale realized volatility and realized kernels in the forecasting exercise.",
"The wavelet-based estimator proves to bring significant improvement in the volatility forecasts.",
"Model incorporating jumps improves forecasting ability even more.",
"Concluding the empirical findings, we show that our wavelet-based estimators bring a significant improvement to the volatility estimation and forecasting.",
"It also offers a new method of time-frequency modeling of realized volatility which helps us to better understand the dynamics of stock market behavior.",
"Specifically, it uncovers that most of the volatility is created on higher frequencies.",
"3pt Appendix:Tables Table: The table summarizes the daily log-return distributions of GBP, CHF and EUR futures.",
"The sample period extends from January 5, 2007 through November 17, 2010, accounting for a total of 944 observations.Table: Results for the GBP futures: in-sample fits of GARCH(1,1), Realized GARCH(1,1) with RV, BV, RK, TSRV, JWTSRV, Realized Jump-GARCH denoted as Realized J-GARCH, and Realized Jump-GARCH on IV ^ j,t (JWTSRV) \\widehat{IV}_{j,t}^{(JWTSRV)} decompositions.",
"Robust standard errors are reported in parentheses.Table: Results for the CHF futures: in-sample fits of GARCH(1,1), Realized GARCH(1,1) with RV, BV, RK, TSRV, JWTSRV, Realized Jump-GARCH denoted as Realized J-GARCH, and Realized Jump-GARCH on IV ^ j,t (JWTSRV) \\widehat{IV}_{j,t}^{(JWTSRV)} decompositions.",
"Robust standard errors are reported in parentheses.Table: Results for the EUR futures: in-sample fits of GARCH(1,1), Realized GARCH(1,1) with RV, BV, RK, TSRV, JWTSRV, Realized Jump-GARCH denoted as Realized J-GARCH, and Realized Jump-GARCH on IV ^ j,t (JWTSRV) \\widehat{IV}_{j,t}^{(JWTSRV)} decompositions.",
"Robust standard errors are reported in parentheses.",
"Appendix: Figures Figure: Daily returns, estimated jump variation and IV t IV_t estimated by JWTSRV for (a) GBP, (b) CHF and (c) EUR futures.Figure: Decomposed annualized volatility (by 252 days) of GBP, CHF and EUR futures using JWTSRV, (a) volatility on investment horizon of 10 minutes, (b) volatility on investment horizon of 20 minutes, (c) volatility on investment horizon of 40 minutes, (d) volatility on investment horizon of 80 minutes, (e) volatility on investment horizon up to 1 day.",
"Note that sum of components (a), (b), (c), (d) and (e) give total volatility.Figure: Scattered plot of h t h_t on x t x_t mapped into probability integral transform (PITs) for all different models.",
"Rows contain estimates of GBP, CHF and EUR futures, while columns contain GARCH(1,1), Realized GARCH(1,1) estimates using RV, BV, RK, TSRV, JWTSRV denoted as RG-RV, RG-BV, RG-RK, RG-TSRV, RG-JWTSRV, and finally Realized Jump-GARCH(1,1), and Realized Jump-GARCH(1,1) with j=1j=1 component denoted as RJ-G and RWJ-G respectively.Figure: Scattered plot of z t z_t on u t u_t residuals mapped into probability integral transform (PITs) obtained for different models.",
"Rows contain estimates of GBP, CHF and EUR futures, while columns contain Realized GARCH(1,1) estimates using RV, BV, RK, TSRV, JWTSRV denoted as RG-RV, RG-BV, RG-RK, RG-TSRV, RG-JWTSRV, and finally Realized Jump-GARCH(1,1), and Realized Jump-GARCH(1,1) with j=1j=1 component denoted as RJ-G and RWJ-G respectively."
]
] | 1204.1452 |
[
[
"On the NLO QCD corrections to the production of the heaviest neutral\n Higgs scalar in the MSSM"
],
[
"Abstract We present a calculation of the two-loop top-stop-gluino contributions to Higgs production via gluon fusion in the MSSM.",
"By means of an asymptotic expansion in the heavy particle masses, we obtain explicit and compact analytic formulae that are valid when the Higgs and the top quark are lighter than stops and gluino, without assuming a specific hierarchy between the Higgs mass and the top mass.",
"Being applicable to the heaviest Higgs scalar in a significant region of the MSSM parameter space, our results complement earlier ones obtained with a Taylor expansion in the Higgs mass, and can be easily implemented in computer codes to provide an efficient and accurate determination of the Higgs production cross section."
],
[
"Introduction",
"With the coming into operation of the Large Hadron Collider (LHC), a new era has begun in the search for the Higgs boson(s).",
"This search requires an accurate control of all the Higgs production and decay mechanisms, including the effects due to radiative corrections [1].",
"At the LHC the main production mechanism for the Standard Model (SM) Higgs boson, $H_{\\scriptscriptstyle {\\rm SM}}$ , is the loop-induced gluon fusion mechanism [2], $gg \\rightarrow H_{\\scriptscriptstyle {\\rm SM}}$ , where the coupling of the gluons to the Higgs is mediated by loops of colored fermions, primarily the top quark.",
"The knowledge of this process in the SM includes the full next-to-leading order (NLO) QCD corrections [3], [4], [5], [6], [7], [8]; the next-to-next-to-leading order (NNLO) QCD corrections [9] including finite top mass effects [10]; soft-gluon resummation effects [11]; the first-order electroweak (EW) corrections [12], [13], [14]; estimates of the next-to-next-to-next-to-leading order (NNNLO) QCD corrections [15] and of the mixed QCD-EW corrections [16].",
"The Higgs sector of the Minimal Supersymmetric Standard Model (MSSM) consists of two $SU(2)$ doublets, $H_1$ and $H_2$ , whose relative contribution to electroweak symmetry breaking is determined by the ratio of vacuum expectation values of their neutral components, $\\tan \\beta \\equiv v_2/v_1$ .",
"The spectrum of physical Higgs bosons is richer than in the SM, consisting of two neutral CP-even bosons, $h$ and $H$ , one neutral CP-odd boson, $A$ , and two charged bosons, $H^\\pm $ .",
"The couplings of the MSSM Higgs bosons to matter fermions differ from those of the SM Higgs, and they can be considerably enhanced (or suppressed) depending on $\\tan \\beta $ .",
"As in the SM, gluon fusion is one of the most important production mechanisms for the neutral Higgs bosons, whose couplings to the gluons are mediated by top and bottom quarks and their supersymmetric partners, the stop and sbottom squarks.",
"In the MSSM, the cross section for Higgs boson production in gluon fusion is currently known at the NLO.",
"The contributions arising from diagrams with quarks and gluons, with full dependence on the Higgs and quark masses, can be obtained from the corresponding SM results [4], [5], [6], [7], [8] with an appropriate rescaling of the Higgs-quark couplings.",
"The contributions arising from diagrams with squarks and gluons were first computed under the approximation of vanishing Higgs mass in ref.",
"[17], and the full Higgs-mass dependence was included in later calculations [6], [7], [8], [18].",
"The contributions of two-loop diagrams involving top, stop and gluino to both scalar and pseudoscalar Higgs production were computed in the vanishing-Higgs-mass limit (VHML) in refs.",
"[19], [20], whose results were later confirmed and cast in a compact analytic form in refs.",
"[21], [22].",
"Finally, first results for the NNLO contributions in the limit of vanishing Higgs mass and degenerate stop and gluino masses were presented in ref. [23].",
"The VHML can provide reasonably accurate results as long as the Higgs mass is well below the threshold for creation of the massive particles running in the loops.",
"For the production of the lightest scalar Higgs, this condition does apply to the two-loop diagrams involving top, stop and gluino, but it obviously does not apply to the corresponding diagrams involving the bottom quark, whose contribution can be relevant for large values of $\\tan \\beta $ .",
"In turn, the masses of the heaviest scalar $H$ and of the pseudoscalar $A$ might very well approach (or exceed) the threshold for creation of top quarks or even of squarks.",
"Unfortunately, retaining the full dependence on the Higgs mass in the quark-squark-gluino contributions has proved a rather daunting task.",
"A calculation based on a combination of analytic and numerical methods was presented in ref.",
"[24] (see also ref.",
"[25]), but neither explicit analytic results nor a public computer code have been made available so far.",
"However, results from the first year of supersymmetry (SUSY) searches at the LHC (see, e.g., ref.",
"[26]) set preliminary lower bounds on the squark and gluino masses of the order of the TeV, albeit for specific models of SUSY breaking.",
"This suggests that – if the MSSM is actually realized in nature – there might be wide regions of its parameter space in which all three of the neutral Higgs bosons are somewhat lighter than the squarks and the gluino.",
"Approximate analytic results for the quark-squark-gluino contributions can be derived in this case.",
"In particular, ref.",
"[27] presented an approximate evaluation of the bottom-sbottom-gluino contributions to scalar production, based on an asymptotic expansion in the large supersymmetric masses that is valid up to and including terms of ${\\cal O}(m_b^2/m_\\phi ^2)$ , ${\\cal O}(m_b/M)$ and ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ , where $m_\\phi $ denotes a Higgs boson mass and $M$ denotes a generic superparticle mass.",
"An independent calculation of the quark-squark-gluino contributions to scalar production, restricted to the limit of zero squark mixing and degenerate superparticle masses, was also presented in ref.",
"[28], confirming the results of ref.",
"[27] for the bottom contributions.",
"More recently, ref.",
"[22] presented an evaluation of the quark-squark-gluino contributions to pseudoscalar production that is also based on an asymptotic expansion in the large supersymmetric masses, but does not assume any hierarchy between the pseudoscalar mass and the quark mass, thus covering both the top-stop-gluino and bottom-sbottom-gluino cases.",
"Exploiting the asymptotic-expansion techniques developed in refs.",
"[27] and [22], we provide in this paper an evaluation of the two-loop top-stop-gluino contributions to Higgs-scalar production valid when the Higgs and the top quark are lighter than stops and gluino, without assuming a specific hierarchy between the Higgs mass and the top mass.",
"In particular, we provide explicit and compact analytic formulae which include terms up to ${\\cal O}(m_\\phi ^2/M^2)$ , ${\\cal O}(m_t^2/M^2)$ and ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ .",
"The results presented in this paper complement the earlier ones of ref.",
"[21], which, being obtained via a Taylor expansion in the Higgs mass, are not accurate for a Higgs mass comparable to (or greater than) the top mass, as might well be the case for the heaviest Higgs scalar of the MSSM.",
"Our formulae can be easily implemented in computer codesAn implementation of the MSSM gluon-fusion cross section in the POWHEG framework was presented in ref.",
"[29]., allowing for an efficient and accurate determination of the Higgs-boson production cross section in the MSSM.",
"The paper is organized as follows: in section we summarize general results on the form factors for Higgs boson production via gluon fusion in the MSSM.",
"Section contains our explicit results for the contributions arising from two-loop top-stop-gluino diagrams, as well as a discussion of the renormalization conditions for the parameters in the top/stop sector.",
"In section we compare numerically the results of our asymptotic expansion in the heavy masses with the results of a Taylor expansion in the Higgs mass, up to and including terms of ${\\cal O}(m_\\phi ^2/m_t^2)$ and ${\\cal O}(m_\\phi ^2/M^2)$ , discussing the regions of applicability of the two different expansions and the effect of different renormalization conditions.",
"Finally, in the last section we present our conclusions."
],
[
"Higgs boson production via gluon fusion in the MSSM",
"In this section we recall for completeness some general results on Higgs boson production via gluon fusion in the MSSM.",
"The leading-order (LO) partonic cross section for the $gg\\rightarrow \\phi $ process (with $\\phi = h,H$ ) reads $\\sigma ^{(0)} =\\frac{G_\\mu \\,\\alpha _s^2 (\\mu _{\\scriptscriptstyle R}) }{128\\, \\sqrt{2} \\, \\pi }\\,\\left|{\\cal H}^{1\\ell }_\\phi \\right|^2~,$ where $G_\\mu $ is the muon decay constant, $\\alpha _s(\\mu _{\\scriptscriptstyle R})$ is the strong gauge coupling expressed in the $\\overline{\\rm MS}$ renormalization scheme at the scale $\\mu _{\\scriptscriptstyle R}$ , and ${\\cal H}_\\phi $ is the form factor for the coupling of the CP-even Higgs boson $\\phi $ with two gluons, which we decompose in one- and two-loop parts as ${\\mathcal {H}}_\\phi ~=~ {\\mathcal {H}}_\\phi ^{1\\ell }~+~ \\frac{\\alpha _s}{\\pi } \\, {\\mathcal {H}}_\\phi ^{2\\ell }~+~{\\cal O}(\\alpha _s^2)~.$ The form factors for the lightest and heaviest Higgs mass eigenstates can be decomposed as ${\\cal H}_{h} ~=~ T_F\\,\\left(-\\sin \\alpha \\,{\\mathcal {H}}_1 +\\cos \\alpha \\,{\\mathcal {H}}_2 \\right)~,~~~~~~{\\cal H}_{H} ~=~ T_F\\,\\left(\\cos \\alpha \\,{\\mathcal {H}}_1 +\\sin \\alpha \\,{\\mathcal {H}}_2 \\right)~,$ where $T_F = 1/2$ is a color factor, $\\alpha $ is the mixing angle in the CP-even Higgs sector of the MSSM and ${\\mathcal {H}}_i$ ($i = 1,2$ ) are the form factors for the coupling of the neutral, CP-even component of the Higgs doublet $H_i$ with two gluons.",
"Focusing on the contributions involving the third-generation quarks and squarks, and exploiting the structure of the Higgs-quark-quark and Higgs-squark-squark couplings, we can write to all orders in the strong interactions [21] ${\\mathcal {H}}_1 & = & \\lambda _t \\,\\left[m_t\\,\\mu \\,s_{2\\theta _t}\\,F_t\\,+ m_{\\scriptscriptstyle Z}^2 \\,s_{2\\beta }\\,D_t \\right] \\;+\\lambda _b \\,\\left[m_b\\,A_b\\,s_{2\\theta _b}\\,F_b \\,+ 2\\,m_b^2\\,G_b \\,+2\\, m_{\\scriptscriptstyle Z}^2 \\,c_\\beta ^2 \\,D_b\\right]\\,, \\\\{\\mathcal {H}}_2 & = & \\lambda _b\\,\\left[m_b\\,\\mu \\,s_{2\\theta _b}\\,F_b\\,-m_{\\scriptscriptstyle Z}^2 \\,s_{2\\beta }\\,D_b \\right] +\\lambda _t\\, \\left[m_t\\,A_t\\,s_{2\\theta _t}\\,F_t \\,+ 2\\,m_t^2\\,G_t \\,-2\\, m_{\\scriptscriptstyle Z}^2 \\,s_\\beta ^2 \\,D_t\\right]~.$ In the equations above $\\lambda _t = 1/\\sin \\beta $ and $\\lambda _b =1/\\cos \\beta $ .",
"Also, $\\mu $ is the higgsino mass parameter in the MSSM superpotential, $A_q$ (for $q=t,b$ ) are the soft SUSY-breaking Higgs-squark-squark couplings and $\\theta _q$ are the left-right squark mixing angles (here and thereafter we use the notation $s_\\varphi \\equiv \\sin \\varphi , \\, c_\\varphi \\equiv \\cos \\varphi $ for a generic angle $\\varphi $ ).",
"The functions $F_q$ and $G_q$ appearing in eqs.",
"(REF ) and () denote the contributions controlled by the third-generation Yukawa couplings, while $D_q$ denotes the contribution controlled by the electroweak, D-term-induced Higgs-squark-squark couplings.",
"The latter can be decomposed as $D_q = \\frac{I_{3q}}{2} \\, \\widetilde{G}_q+ c_{2\\theta _{\\tilde{q}}} \\,\\left(\\frac{I_{3q}}{2} - Q_q \\,s^2_{\\theta _{\\scriptscriptstyle W}}\\right) \\,\\widetilde{F}_q \\, , $ where $I_{3q}$ denotes the third component of the electroweak isospin of the quark $q$ , $Q_q$ is the electric charge and $\\theta _{\\scriptscriptstyle W}$ is the Weinberg angle.",
"The form factors ${\\mathcal {H}}_i$ can in turn be decomposed in one- and two-loop parts as in eq.",
"(REF ).",
"The one-loop parts, ${\\cal H}_i^{1\\ell }$ , contain contributions from diagrams involving quarks ($q$ ) or squarks ($\\tilde{q}_i$ ).",
"The functions entering ${\\mathcal {H}}_i^{1\\ell }$ are $F_q^{1\\ell } ~=~ \\widetilde{F}_q^{1\\ell }& =& \\frac{1}{2}\\,\\left[\\frac{1}{m^2_{\\tilde{q}_{1}}} {\\mathcal {G}}^{1\\ell }_{0}(\\tau _{\\tilde{q}_{1}}) -\\frac{1}{m^2_{\\tilde{q}_{2}}} {\\mathcal {G}}^{1\\ell }_{0}(\\tau _{\\tilde{q}_{2}})\\right]\\, , \\\\G_q^{1\\ell } & =& \\frac{1}{2}\\,\\left[\\frac{1}{m^2_{\\tilde{q}_{1}}} {\\mathcal {G}}^{1\\ell }_{0}(\\tau _{\\tilde{q}_{1}}) +\\frac{1}{m^2_{\\tilde{q}_{2}}} {\\mathcal {G}}^{1\\ell }_{0} (\\tau _{\\tilde{q}_{2}}) +\\frac{1}{m_q^2} {\\mathcal {G}}^{1\\ell }_{1/2} (\\tau _q)\\right]~, \\\\\\widetilde{G}_q^{1\\ell } & =& \\frac{1}{2}\\,\\left[\\frac{1}{m^2_{\\tilde{q}_{1}}} {\\mathcal {G}}^{1\\ell }_{0} (\\tau _{\\tilde{q}_{1}}) +\\frac{1}{m^2_{\\tilde{q}_{2}}} {\\mathcal {G}}^{1\\ell }_{0} (\\tau _{\\tilde{q}_{2}})\\right]~,$ where $\\tau _k \\equiv 4\\,m_k^2/m_h^2$ , and the functions ${\\mathcal {G}}^{1\\ell }_{0}$ and ${\\mathcal {G}}^{1\\ell }_{1/2}$ read ${\\mathcal {G}}^{1\\ell }_{0} (\\tau ) & =& ~~~~\\,\\tau \\!\\left[ 1 + \\frac{\\tau }{4}\\,\\ln ^2 \\left(\\frac{\\sqrt{1- \\tau } - 1}{\\sqrt{1- \\tau } + 1}\\right) \\right]\\,, \\\\{\\mathcal {G}}^{1\\ell }_{1/2} (\\tau ) & = & - 2\\,\\tau \\left[ 1 - \\frac{ 1 -\\tau }{4} \\,\\ln ^2\\left(\\frac{\\sqrt{1-\\tau } - 1}{\\sqrt{1-\\tau } + 1} \\right) \\right] \\,.$ The analytic continuations are obtained with the replacement $m_h^2\\rightarrow m_h^2 + i \\epsilon $ .",
"We remark that in the limit in which the Higgs boson mass is much smaller than the mass of the particle running in the loop, i.e.",
"$\\tau \\gg 1$ , the functions ${\\mathcal {G}}^{1\\ell }_{0}$ and ${\\mathcal {G}}^{1\\ell }_{1/2}$ behave as ${\\mathcal {G}}^{1\\ell }_{0} \\rightarrow -\\frac{1}{3} -\\frac{8}{45\\,\\tau }~+~{\\cal O}(\\tau ^{-2})~,~~~~~~~~~~{\\mathcal {G}}^{1\\ell }_{1/2} \\rightarrow -\\frac{4}{3}-\\frac{14}{45\\,\\tau }~+~{\\cal O}(\\tau ^{-2})~.$ The two-loop parts of the form factors, ${\\cal H}_i^{2\\ell }$ , contain contributions from diagrams involving quarks, squarks, gluons and gluinos.",
"We point the reader to, e.g., section 2 of ref.",
"[21] for explicit formulae showing how ${\\cal H}_i^{2\\ell }$ (or, equivalently, ${\\cal H}_\\phi ^{2\\ell }$ ) enter the total NLO cross section for Higgs boson production in hadronic collisions.",
"In the next section we present our new evaluation of the top/stop contributions to ${\\cal H}_i^{2\\ell }$ , based on an asymptotic expansion in the stop and gluino masses."
],
[
"Two-loop contributions to the Higgs-production form factors",
"In the case of the lightest Higgs boson $h$ , the top/stop contributions to the two-loop form factor ${\\cal H}_h^{2\\ell }$ are well under control.",
"Typically, the mass ratios between the lightest Higgs and the particles running in the loops allow for the evaluation of the relevant diagrams via a Taylor expansion in the Higgs mass, with the zero-order term in the series – for which ref.",
"[21] provides explicit analytic formulae – already a good approximation to the full result.",
"In the case of the heaviest Higgs boson $H$ , on the other hand, the assumption that it is much lighter than the particles running in the loops is valid only in a limited portion of the MSSM parameter space.",
"In particular, $m_{\\scriptscriptstyle H}$ might very well sit around or above the threshold for the creation of a real top-quark pair in the loops, in which case – as found in ref.",
"[22] for the pseudoscalar – a Taylor expansion in $m_{\\scriptscriptstyle H}^2$ would certainly fail to approximate the correct result for the Higgs-production form factor.",
"To address this possibility, we present in this section explicit analytic results for the two-loop top/stop contributions to the form factors ${\\cal H}_i^{2\\ell }$ that include terms up to ${\\cal O}(m_\\phi ^2/M^2)$ , ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ and ${\\cal O}(m_t^2/M^2)$ , without assuming a specific hierarchy between the Higgs mass and the top mass.",
"Figure: Examples of two-loop diagrams for gg→φgg\\rightarrow \\phi that donot involve gluinos.Figure: Examples of two-loop diagrams for gg→φgg\\rightarrow \\phi involving gluinos.The top/stop contributions to ${\\cal H}_i^{2\\ell }$ come from two-loop diagrams such as the ones depicted in figs.",
"REF and REF .",
"In analogy to what was done in refs.",
"[21], [27], we can decompose the functions $F_t^{2\\ell },\\,G_t^{2\\ell },\\,\\widetilde{F}_t^{2\\ell }$ and $\\widetilde{G}_t^{2\\ell }$ entering the two-loop parts of eqs.",
"(REF ) and () as $F_t^{2\\ell } &=& Y_{\\tilde{t}_1} - Y_{\\tilde{t}_2} -\\frac{4\\,c_{2\\theta _t}^2}{m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2}\\, Y_{c_{2\\theta _t}^2}\\,,\\\\G_t^{2\\ell } &=& Y_{\\tilde{t}_1} + Y_{\\tilde{t}_2} + Y_t\\,,\\\\\\widetilde{F}_t^{2\\ell } &=& Y_{\\tilde{t}_1} - Y_{\\tilde{t}_2}+\\frac{4\\,s_{2\\theta _t}^2}{m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2}\\,Y_{ c_{2\\theta _t}^2}\\,, \\\\\\widetilde{G}_t^{2\\ell } &=& Y_{\\tilde{t}_1} + Y_{\\tilde{t}_2}\\,.$ The various terms in eqs.",
"(REF )–() can be split in the contributions coming from diagrams with (s)top (s)quarks and gluons ($g$ , figs.",
"REF a and REF b); with a quartic stop coupling ($4\\tilde{t}$ , fig.",
"REF c); with top quarks, stop squarks and gluinos ($\\tilde{g}$ , figs.",
"REF a and REF b): $Y_x = Y_x^{g} + Y_x^{4\\tilde{t}} + Y_x^{\\tilde{g}}~~~~~~~(x=t,\\tilde{t}_1,\\tilde{t}_2,c_{2\\theta _t}^2)~.$ Furthermore, we remark that the term $Y_t$ entering eq.",
"() contains only contributions from diagrams with a Higgs-top coupling, figs.",
"REF a and REF a, therefore $Y_t^{4\\tilde{t}} =0$ .",
"On the other hand, the terms $Y_{\\tilde{t}_1},\\, Y_{\\tilde{t}_2}$ and $Y_{c_{2\\theta _t}^2}$ in eqs.",
"(REF )–() contain only contributions from diagrams with a Higgs-stop coupling, figs.",
"REF b, REF c, and REF b."
],
[
"Top-gluon, stop-gluon and four-stop contributions",
"The top-gluon, stop-gluon and four-stop contributions to the terms $Y_x$ in eq.",
"(REF ) can be extracted from the existing literature, and we collect them in this section for completeness.",
"We assume that the parameters entering the one-loop parts of the form factors ${\\cal H}_i$ in eqs.",
"(REF ) and () are expressed in the $\\overline{\\rm DR}$ renormalization scheme at the scale $Q$ .",
"The contribution to the term $Y_t$ arising from two-loop diagrams with top quarks and gluons (fig.",
"REF a) must be computed for arbitrary values of $\\tau _t \\equiv 4m_t^2/m_\\phi ^2\\,$ .",
"It reads: $2\\,m_t^2\\,Y^{g}_t ~=~ C_F\\,\\left[{\\cal F}^{(2\\ell ,a)}_{1/2}(x_t) ~+~{\\cal F}^{(2\\ell ,b)}_{1/2}(x_t)\\,\\left(\\ln \\frac{m_t^2}{Q^2}-\\frac{1}{3}\\right)\\right]~+~ C_A\\; {\\cal G}^{(2\\ell ,C_A)}_{1/2}(x_t)~,$ where $C_F=4/3$ and $C_A=3$ are color factors, and exact expressions for ${\\cal F}^{(2\\ell ,a)}_{1/2}$ , ${\\cal F}^{(2\\ell ,b)}_{1/2}$ and ${\\cal G}^{(2\\ell ,C_A)}_{1/2}\\,$ as functions of $x_t \\equiv (\\sqrt{1-\\tau _t}-1)/(\\sqrt{1-\\tau _t}+1)$ are given in eqs.",
"(2.12), (2.13) and (3.8) of ref.",
"[7], respectively.",
"The contributions to the terms $Y_{\\tilde{t}_1},\\, Y_{\\tilde{t}_2}$ and $Y_{c_{2\\theta _t}^2}$ arising from two-loop diagrams with stop squarks and gluons (fig.",
"REF b) and from diagrams with a quartic stop coupling (fig.",
"REF c) can, to the accuracy required by our expansion, be computed in the limit of vanishing $m_\\phi $ .",
"They read [21] $Y_{\\tilde{t}_1}^{g} &=& -\\frac{1}{2\\,m_{\\tilde{t}_1}^2}\\,\\left(\\frac{3\\,C_F}{4} + \\frac{C_A}{6}\\right)~,\\\\Y_{\\tilde{t}_1}^{4 \\tilde{t}}&=&-\\frac{C_F}{24}\\,\\left[\\frac{c_{2\\theta _t}^2\\,m_{\\tilde{t}_1}^2+s_{2\\theta _t}^2\\,m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_1}^4}+\\frac{s_{2\\theta _t}^2}{m_{\\tilde{t}_1}^4\\,m_{\\tilde{t}_2}^2}\\,\\left(m_{\\tilde{t}_1}^4\\,\\ln \\frac{m_{\\tilde{t}_1}^2}{Q^2}-m_{\\tilde{t}_2}^4\\,\\ln \\frac{m_{\\tilde{t}_2}^2}{Q^2}\\right)\\right]~,\\\\Y_{c_{2\\theta _t}^2}^{4 \\tilde{t}}&=&-\\frac{C_F}{24}\\,\\left[\\frac{(m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2)^2}{m_{\\tilde{t}_1}^2\\,m_{\\tilde{t}_2}^2}-\\frac{m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_2}^2}\\,\\ln \\frac{m_{\\tilde{t}_1}^2}{Q^2}-\\frac{m_{\\tilde{t}_2}^2-m_{\\tilde{t}_1}^2}{m_{\\tilde{t}_1}^2}\\,\\ln \\frac{m_{\\tilde{t}_2}^2}{Q^2}\\right]~.$ The term $Y_{c_{2\\theta _t}^2}^{g}$ is zero, while the terms $Y_{\\tilde{t}_2}^{g}$ and $Y_{\\tilde{t}_2}^{4 \\tilde{t}}$ can be obtained by performing the substitutions $\\tilde{t}_1\\leftrightarrow \\tilde{t}_2$ in eqs.",
"(REF ) and (), respectively."
],
[
"Top-stop-gluino contributions",
"In this section we present our original results for the asymptotic expansion of the top-stop-gluino contributions in the stop and gluino masses.",
"We retain in our formulae only terms that contribute to the form factors ${\\cal H}_i$ up to ${\\cal O}(m_t^2/M^2)$ , ${\\cal O}(m_\\phi ^2/M^2)$ or ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ , where $M$ denotes a generic superparticle mass.",
"Again, we assume that the one-loop parts of ${\\cal H}_i$ in eqs.",
"(REF ) and () are expressed in terms of $\\overline{\\rm DR}$ -renormalized parameters evaluated at the scale $Q$ .",
"The top-stop-gluino contributions to the term $Y_t$ , arising from diagrams with a Higgs-top coupling (fig.",
"REF a), read $2\\,m_t^2\\,Y_t^{\\tilde{g}} &=&\\frac{4}{3}\\, {\\cal F}^{(2\\ell ,b)}_{1/2}(\\tau _t) \\,\\frac{\\delta m_t}{m_t}^{\\!\\scriptscriptstyle SUSY}- ~\\frac{C_F}{4} \\, {\\mathcal {G}}^{1\\ell }_{1/2} (\\tau _t)\\,\\frac{m_{\\tilde{g}}}{m_t}\\,s_{2\\theta _t}\\,\\left(\\frac{x_1}{1-x_1} \\ln x_1-\\frac{x_2}{1-x_2}\\ln x_2\\right)\\nonumber \\\\&&+~ s_{2\\theta _t}\\frac{m_t}{m_{\\tilde{g}}} \\,{\\cal R}_1 ~+~ \\frac{m_t^2}{m_{\\tilde{g}}^2}\\,{\\cal R}_2~,$ where $x_i = m_{\\tilde{t}_i}^2/m_{\\tilde{g}}^2\\,$ ($i=1,2$ ), and $\\delta m_t^{\\scriptscriptstyle SUSY}$ denotes the SUSY contribution to the top self-energy, in units of $\\alpha _s/\\pi $ , expanded in powers of $m_t$ up to terms of ${\\cal O}(m_t^3)$ $\\delta m_t^{\\scriptscriptstyle SUSY} &=&-\\frac{C_F}{4}\\,m_t\\,\\left[s_{2\\theta _t}\\,\\frac{m_{\\tilde{g}}}{m_t}\\,\\frac{x_{1}}{1-x_{1}} \\ln x_{1}+ \\frac{1}{2} \\ln \\frac{m_{\\tilde{g}}^2}{Q^2} +\\frac{x_1 - 3}{4\\, (1-x_1)} + \\frac{x_1\\,(x_1-2)}{2\\,(1-x_1)^2} \\,\\ln x_1\\right.\\nonumber \\\\&+&\\!\\left.\\frac{s_{2\\theta _t}\\,m_t}{2\\,m_{\\tilde{g}}\\,(1-x_1)^3} \\left( 1-x_1^2 + 2\\,x_1 \\ln x_1 \\right)+\\frac{m_t^2}{6\\,m_{\\tilde{g}}^2\\,(1-x_1)^3}\\left(x_1^2-5 x_1-2 -\\frac{6\\,x_1}{1-x_1} \\ln x_1 \\right)\\right]\\nonumber \\\\&+&\\!\\biggr (x_1\\longrightarrow x_2\\,,~~s_{2\\theta _t}\\longrightarrow -s_{2\\theta _t}\\biggr )~.$ The terms ${\\cal R}_1$ and ${\\cal R}_2$ in eq.",
"(REF ) collect contributions suppressed by $m_t/M$ and $m_t^2/M^2\\,$ , respectively: ${\\cal R}_1 &=&\\frac{C_A}{6 \\left(1-x_1\\right)^2} \\,\\left[3\\, \\left(1-x_1+x_1 \\ln x_1\\right) \\left(\\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}-\\mathcal {B}(\\tau _t)-\\frac{1}{2}\\,{\\mathcal {K}^{1\\ell }(\\tau _t)}+2\\right) \\right.",
"\\nonumber \\\\& & ~~~~~~~~~~~~~~~~~+ \\left.",
"\\vphantom{\\frac{\\mathcal {K}^{1\\ell }}{2}}6\\, x_1\\, {\\rm Li}_2\\left(1-x_1\\right)+2\\, x_1+2\\, x_1\\left(1+x_1\\right)\\, \\ln x_1-2\\right] \\nonumber \\\\&-&\\frac{C_F}{6\\, x_1 \\,\\left(1-x_1\\right)^3}\\left[3\\, \\left(x_1-x_1^3+2\\, x_1^2 \\ln x_1\\right)\\left(\\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}-\\mathcal {B}(\\tau _t)-\\frac{1}{4}\\, {\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}-\\frac{1}{2}\\, {\\mathcal {K}^{1\\ell }(\\tau _t)}+2\\right) \\right.",
"\\nonumber \\\\& & \\left.",
"~~~~~~~~~~~+\\left(1-x_1\\right)^3\\, \\ln \\frac{m_{\\tilde{g}}^2}{Q^2}+12\\, x_1^2\\, {\\rm Li}_2\\left(1-x_1\\right)+5\\, x_1^3-5\\, x_1^2+x_1 -1 +2\\, \\left(x_1^3+2\\, x_1^2\\right) \\ln x_1\\right] \\nonumber \\\\&- & \\biggr (x_1 \\longrightarrow x_2\\biggr )~,\\\\&&\\nonumber \\\\{\\cal R}_2&=&-\\frac{C_A}{12 \\left(1-x_1\\right)^3}\\left[3\\, \\left(1-x_1^2+2\\, x_1 \\ln x_1\\right)\\left(2\\, \\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}-\\mathcal {B}(\\tau _t)-\\frac{1}{2}\\,{\\mathcal {K}^{1\\ell }(\\tau _t)}+2\\right) \\right.",
"\\nonumber \\\\&& \\left.",
"\\vphantom{\\frac{\\mathcal {K}^{1\\ell }}{2}}~~~~~~~~~~~~~~~~~~~~~+24\\, x_1 {\\rm Li}_2\\left(1-x_1\\right) +1-x_1^2+2\\, x_1\\, \\left(3\\, x_1+10\\right)\\, \\ln x_1\\right] \\nonumber \\\\&+& \\frac{C_F}{18\\, x_1\\, \\left(1-x_1\\right)^4}\\left\\lbrace \\vphantom{\\frac{\\mathcal {K}^{1\\ell }}{2}}3\\,x_1 \\biggr [\\left(1-x_1\\right)\\,(5\\,x_1 - x_1^2+2)+6\\, x_1 \\ln x_1\\biggr ] \\times \\right.",
"\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~~\\times \\left(2\\, \\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}-\\mathcal {B}(\\tau _t) -\\frac{1}{2}\\,{\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}-\\frac{1}{2}\\,{\\mathcal {K}^{1\\ell }(\\tau _t)}+2\\right)+6\\, \\left(1-x_1\\right)^4 \\ln \\frac{m_{\\tilde{g}}^2}{Q^2} \\nonumber \\\\&& ~~~~~~~~~~~\\left.",
"\\vphantom{\\frac{\\mathcal {K}^{1\\ell }}{2}}+72\\, x_1^2 \\, {\\rm Li}_2 (1-x_1) -x_1\\, (1-x_1)^2\\,(11\\, x_1-26)-6\\, \\left(1-x_1\\right)+6\\, x_1^2 \\left(2\\, x_1+9\\right)\\, \\ln x_1\\right\\rbrace \\nonumber \\\\&+& \\biggr (x_1 \\longrightarrow x_2\\biggr )~.$ We recall that the function $\\mathcal {G}^{1\\ell }_{1/2}(\\tau )$ is defined in eq.",
"(), while $\\mathcal {B}(\\tau )$ and $\\mathcal {K}^{1\\ell }(\\tau )$ are defined as ${\\cal B}(\\tau ) ~=~2+\\sqrt{1-\\tau }\\,\\ln \\left(\\frac{\\sqrt{1- \\tau }-1}{\\sqrt{1-\\tau }+1}\\right)~,~~~~~~~~~~{\\cal K}^{1\\ell }(\\tau ) ~=~\\frac{\\tau }{2}\\,\\ln ^2\\left(\\frac{\\sqrt{1- \\tau }-1}{\\sqrt{1-\\tau }+1}\\right)~.$ Finally, ${\\cal F}^{(2\\ell ,b)}_{1/2}$ can be expressed directly as a function of $\\tau $ in terms of the other three functions: ${\\cal F}^{(2\\ell ,b)}_{1/2}(\\tau ) ~=~ -\\frac{3}{2}\\,\\left[2 \\,\\mathcal {G}^{1\\ell }_{1/2}(\\tau ) + \\tau \\,{\\cal B}(\\tau )-{\\cal K}^{1\\ell }(\\tau )\\right].$ The top-stop-gluino contributions to the terms $Y_{\\tilde{t}_1}$ and $Y_{c_{2\\theta _t}^2}$ , arising from diagrams with a Higgs-stop coupling (fig.",
"REF b), read $Y^{\\tilde{g}}_{\\tilde{t}_1} &=&\\left(\\frac{C_F}{4}\\,\\frac{s_{2\\theta _t}}{m_t\\,m_{\\tilde{g}}}\\,\\mathcal {G}^{1\\ell }_{1/2}(\\tau _t)- \\frac{2\\,C_F+C_A}{12\\,m_{\\tilde{g}}^2}\\right)\\,\\left(\\frac{1}{1-x_1}+\\frac{1}{\\left(1-x_1\\right)^2}\\ln x_1\\right)\\nonumber \\\\&+&\\frac{C_F}{24\\,m_{\\tilde{g}}^2\\,x_1^2\\,(1-x_1)^3}\\,\\left\\lbrace 4\\,(1-x_1)^3\\,\\left(1-\\ln \\frac{m_{\\tilde{g}}^2}{Q^2}\\right)-3 \\,x_1^2\\,\\mathcal {G}^{1\\ell }_{1/2}(\\tau _t)\\,\\biggr [(1-x_1)(3-x_1) + 2\\ln x_1\\biggr ]\\right\\rbrace \\nonumber \\\\&+&\\frac{C_F\\,s_{2\\theta _t}\\,m_t}{6\\,m_{\\tilde{g}}^3\\, x_1^2\\, \\left(1-x_1\\right)^4}\\left\\lbrace 3 \\,x_1^2\\, \\biggr [\\left(1-x_1\\right)\\left(x_1+5\\right)+2\\,\\left(2 \\,x_1+1\\right)\\, \\ln x_1\\biggr ]\\left(\\frac{1}{4}\\,{\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}-\\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}\\right) \\right.",
"\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~~~~~~+\\left(1-x_1\\right)^4 \\ln \\frac{m_{\\tilde{g}}^2}{Q^2}-12\\, x_1^2\\, \\left(2 \\,x_1+1\\right) \\,{\\rm Li}_2\\left(1-x_1\\right) \\nonumber \\\\&& \\left.\\phantom{\\frac{1}{4}{\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}}~~~~~~~~~~~~~~-\\left(1-x_1\\right) \\,\\left(14\\,x_1^2-3\\,x_1+1\\right)-2\\, x_1^2 \\left(x_1^2+18\\, x_1+5\\right) \\ln x_1\\right\\rbrace \\nonumber \\\\&+&\\frac{C_A\\,s_{2\\theta _t}\\,m_t}{6\\,m_{\\tilde{g}}^3\\,\\left(1-x_1\\right)^3}\\left\\lbrace 3\\,\\biggr [2-2\\, x_1+\\left(x_1+1\\right)\\, \\ln x_1\\biggr ]\\left(1+ \\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}\\right)\\right.",
"\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~ \\left.",
"\\vphantom{\\frac{m_t^2}{m_{\\tilde{g}}^2}}+6\\, \\left(1+x_1\\right)\\, {\\rm Li}_2\\left(1-x_1\\right)+2\\,x_1\\,\\left(1-x_1\\right)+2\\,\\left(6\\, x_1+1\\right)\\,\\ln x_1\\right\\rbrace \\nonumber \\\\&+&\\frac{C_F\\,s_{2\\theta _t}\\,m_\\phi ^2\\, \\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}{48\\,m_t\\,m_{\\tilde{g}}^3\\, x_1\\, \\left(1-x_1\\right)^4}\\, \\biggr [\\left(1-x_1\\right) \\left(x_1^2-5\\, x_1-2\\right)-6\\, x_1 \\ln x_1\\biggr ]~,\\\\&&\\nonumber \\\\&&\\nonumber \\\\Y^{\\tilde{g}}_{c_{2\\theta _t}^2} &=&-\\frac{C_F\\,m_{\\tilde{g}}}{8\\,s_{2\\theta _t}\\,m_t}\\,\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)\\,\\left(\\frac{x_1}{1-x_1}\\,\\ln x_1-\\frac{x_2}{1-x_2}\\,\\ln x_2\\right)\\nonumber \\\\&+&\\!\\!\\left\\lbrace \\frac{C_F\\,m_t}{12\\,s_{2\\theta _t}\\,m_{\\tilde{g}}\\,x_1\\left(1-x_1\\right)^3}\\left[\\,3\\,x_1 \\left(x_1^2-2\\, x_1 \\,\\ln x_1-1\\right) \\!\\left(\\frac{1}{4}\\,{\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}-\\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}\\right)+\\left(1-x_1\\right)^3 \\ln \\frac{m_{\\tilde{g}}^2}{Q^2} \\right.",
"\\right.\\nonumber \\\\&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\left.",
"\\vphantom{\\frac{m_t^2}{m_{\\tilde{g}}^2}}+12 \\,x_1^2\\, {\\rm Li}_2\\left(1-x_1\\right)-\\left(1-2 x_1\\right) \\left(1-x_1\\right)^2+2 \\,x_1^2\\, \\left(x_1+5\\right) \\ln x_1\\right]\\nonumber \\\\&&+~ \\frac{C_A\\,m_t\\,x_1}{12\\,s_{2\\theta _t}\\,m_{\\tilde{g}}\\, \\left(1-x_1\\right)^2}\\left[\\left(x_1 - 1 - \\ln x_1\\right)\\left(3\\, \\ln \\frac{m_t^2}{m_{\\tilde{g}}^2}+1\\right)-6\\, {\\rm Li}_2\\left(1-x_1\\right)-2\\, \\left(x_1+2\\right) \\ln x_1\\right]\\nonumber \\\\&&+~\\frac{C_F\\,m_\\phi ^2\\,\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t)}{32\\,s_{2\\theta _t}\\,m_{\\tilde{g}}\\,m_t\\left(1-x_1\\right)^2}\\,\\left[\\frac{\\left(1-x_1\\right) \\left(x_1+x_2-2 x_1 x_2\\right)}{\\left(1-x_2\\right) \\left(x_1-x_2\\right)}+\\frac{2 x_1 \\left(x_1^2+x_1 x_2-2 x_2\\right) \\ln x_1}{\\left(x_1-x_2\\right)^2}\\right] \\nonumber \\\\&&\\nonumber \\\\&&-~\\biggr (x_1\\longleftrightarrow x_2\\biggr )~\\biggr \\rbrace ~,$ while $Y^{\\tilde{g}}_{\\tilde{t}_2}$ can be obtained by performing the substitutions $x_1\\rightarrow x_2$ and $s_{2\\theta _t}\\rightarrow -s_{2\\theta _t}$ in eq.",
"(REF ).",
"We remark that some care is required in order to properly include in the form factors ${\\cal H}_i$ only terms up to ${\\cal O}(m_\\phi ^2/M^2)$ , ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ and ${\\cal O}(m_t^2/M^2)$ .",
"In particular, in the calculation of the function $F_t^{2\\ell }$ , eq.",
"(REF ), we must use the full formulae for $Y_{\\tilde{t}_1}^{\\tilde{g}}$ and $Y_{c_{2\\theta _t}^2}^{\\tilde{g}}$ in eqs.",
"(REF ) and (), respectively.",
"On the other hand, in the calculation of the functions $G_t^{2\\ell },\\,\\widetilde{F}_t^{2\\ell }$ and $\\widetilde{G}_t^{2\\ell }$ , eqs.",
"()–(), we must retain only the terms in the first two lines of eq.",
"(REF ) for $Y_{\\tilde{t}_1}^{\\tilde{g}}$ , and only the term in the first line of eq.",
"() for $Y_{c_{2\\theta _t}^2}^{\\tilde{g}}$ .",
"As a first check of the correctness of our calculation, we verified that by taking the VHML (i.e., taking $m_\\phi \\rightarrow 0$ ) in the formulae presented in this section, which implies $\\mathcal {G}_{1/2}^{1\\ell }(\\tau _t) \\rightarrow -4/3,~\\mathcal {K}^{1\\ell }(\\tau _t) \\rightarrow -2,~\\mathcal {B}(\\tau _t)\\rightarrow 0$ and ${\\cal F}^{(2\\ell ,b)}_{1/2}\\rightarrow 0$ , we obtain for the top-stop-gluino contributions the same result that we would obtain by expanding the VHML results of ref.",
"[21] in powers of $m_t$ up to and including ${\\cal O}(m_t^2/M^2)$ .",
"It is also straightforward to check that, by performing the trivial replacement $t\\rightarrow b$ in the formulae presented in this section and then dropping all terms that contribute to the form factors beyond ${\\cal O}(m_b^2/m_\\phi ^2)$ , ${\\cal O}(m_b/M)$ and ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ , we recover the results of ref.",
"[27] for the bottom-sbottom-gluino contributions.",
"To this effect it must be kept in mind that $\\mathcal {G}_{1/2}^{1\\ell }(\\tau _b)$ and $\\mathcal {K}^{1\\ell }(\\tau _b)$ are of ${\\cal O}(m_b^2/m_\\phi ^2)$ , while $\\mathcal {B}(\\tau _b) = 2 - \\ln (-m_\\phi ^2/m_b^2) + {\\cal O}(m_b^2/m_\\phi ^2)$ ."
],
[
"On-shell renormalization scheme for the stop parameters",
"If the parameters entering the one-loop part of the form factors are expressed in a renormalization scheme different from $\\overline{\\rm DR}$ , the two-loop results presented in the previous section must be shifted in a way analogous to that described in section 3.2 of ref.",
"[21], to which we point the reader for details.",
"Indeed, in our “on-shell” (OS) scheme we adopt the same prescriptions as in ref.",
"[21] for the input parameters that are subject to ${\\cal O}(\\alpha _s)$ corrections: the top and stop masses are defined as the poles of the corresponding propagators; the counterterm of the stop mixing angle $\\theta _t$ is chosen as to cancel the anti-hermitian part of the stop wave-function renormalization matrix; the trilinear coupling $A_t$ is treated as a derived quantity, related to the other parameters in the top/stop sector by $s_{2\\theta _t}= \\frac{2\\,m_t\\,(A_t + \\mu \\cot \\beta )}{m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2}~.$ Some differences with respect to the treatment in ref.",
"[21] arise, however, due to the fact that the results presented in that paper were obtained in the VHML for arbitrary values of the top mass, while the results presented here are valid up to and including terms of ${\\cal O}(m_\\phi ^2/M^2)$ and ${\\cal O}(m_t^2/M^2)$ , without assuming a hierarchy between $m_\\phi $ and $m_t$ .",
"Defining the $\\overline{\\rm DR}$ –OS shift for a generic parameter $x$ according to $x^{\\overline{\\rm DR}} = x^{\\rm OS}+ (\\alpha _s/\\pi )\\,\\delta x$ , we need here to expand the various shifts in powers of $m_t$ .",
"Up to the order relevant to our calculation, the explicit expressions for the shifts in the stop masses and mixing angle read $\\delta m_{\\tilde{t}_1}^2&=&\\frac{C_F}{4}\\,m_{\\tilde{t}_1}^2\\,\\left\\lbrace 3\\, \\ln \\frac{m_{\\tilde{t}_1}^2}{Q^2} - 3 -c_{2\\theta _t}^2 \\left( \\ln \\frac{m_{\\tilde{t}_1}^2}{Q^2} -1 \\right)-s_{2\\theta _t}^2 \\frac{m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_1}^2} \\left( \\ln \\frac{m_{\\tilde{t}_2}^2}{Q^2} -1 \\right)\\right.",
"\\nonumber \\\\& & ~~~~~~~~~~~~~ -6 \\frac{m_{\\tilde{g}}^2}{m_{\\tilde{t}_1}^2}- 2 \\left( 1 - 2 \\frac{m_{\\tilde{g}}^2}{m_{\\tilde{t}_1}^2} \\right) \\ln \\frac{m_{\\tilde{g}}^2}{Q^2}- 2 \\left( 1-\\frac{m_{\\tilde{g}}^2}{m_{\\tilde{t}_1}^2} \\right)^2\\ln \\left| 1-\\frac{m_{\\tilde{t}_1}^2}{m_{\\tilde{g}}^2} \\right|\\nonumber \\\\& & ~~~~~~~~~~~~~\\left.",
"-\\frac{4\\,s_{2\\theta _t}\\,m_t\\,m_{\\tilde{g}}}{m_{\\tilde{t}_1}^2}\\left[ \\ln \\frac{m_{\\tilde{g}}^2}{Q^2} + \\left( 1-\\frac{m_{\\tilde{g}}^2}{m_{\\tilde{t}_1}^2} \\right)\\ln \\left| 1-\\frac{m_{\\tilde{t}_1}^2}{m_{\\tilde{g}}^2} \\right| - 2 \\right]\\,\\right\\rbrace ~,\\\\&&\\nonumber \\\\\\delta \\theta _t &=&\\frac{C_F}{4}\\frac{c_{2\\theta _t}\\,s_{2\\theta _t}}{(m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2)}\\,\\left\\lbrace m_{\\tilde{t}_1}^2\\,\\left(\\ln \\frac{m_{\\tilde{t}_1}^2}{Q^2}-1\\right)-\\frac{2\\,m_t\\,m_{\\tilde{g}}}{s_{2\\theta _t}}\\,\\left[\\ln \\frac{m_{\\tilde{g}}^2}{Q^2} + \\left( 1-\\frac{m_{\\tilde{g}}^2}{m_{\\tilde{t}_1}^2} \\right)\\ln \\left| 1-\\frac{m_{\\tilde{t}_1}^2}{m_{\\tilde{g}}^2} \\right| - 2 \\right]\\,\\right\\rbrace \\nonumber \\\\&&\\nonumber \\\\&+&\\biggr (\\tilde{t}_1\\longleftrightarrow \\tilde{t}_2\\,,~~s_{2\\theta _t}\\longrightarrow -s_{2\\theta _t}\\,,~~c_{2\\theta _t}\\longrightarrow -c_{2\\theta _t}\\biggr )~,$ where the analogous expression for $\\delta m_{\\tilde{t}_2}^2$ can be obtained by performing the substitutions $\\tilde{t}_1\\leftrightarrow \\tilde{t}_2$ and $s_{2\\theta _t}\\rightarrow -s_{2\\theta _t}$ in eq.",
"(REF ).",
"We also define $\\delta s_{2\\theta _t}=2\\, c_{2\\theta _t}\\, \\delta \\theta _t$ and $\\delta c_{2\\theta _t}= - 2\\, s_{2\\theta _t}\\, \\delta \\theta _t$ .",
"The shift for the top mass reads $\\delta m_t=\\frac{C_F}{4}\\,m_t\\,\\left(3 \\ln \\frac{m_t^2}{Q^2} - 5\\right)+ \\delta m_t^{\\scriptscriptstyle SUSY},$ where the SUSY contribution $\\delta m_t^{\\scriptscriptstyle SUSY}$ was given in eq.",
"(REF ).",
"Finally, the shift for the trilinear coupling $A_t$ can be expressed in terms of the other shifts according to $\\delta A_t ~=~ \\left(\\frac{\\delta m_{\\tilde{t}_1}^2-\\delta m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_1}^2-m_{\\tilde{t}_2}^2}+\\frac{\\delta s_{2\\theta _t}}{s_{2\\theta _t}}-\\frac{\\delta m_t}{m_t}\\right)(A_t+\\mu \\,\\cot \\beta )~.$ If the one-loop form factors are evaluated in terms of OS parameters, the two-loop functions in eqs.",
"(REF )–() must be replaced by $F_t^{2\\ell } & \\longrightarrow & F_t^{2\\ell } ~+~\\frac{1}{6}\\,\\left[\\frac{\\delta m_{\\tilde{t}_1}^2}{m_{\\tilde{t}_1}^4}-\\frac{\\delta m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_2}^4}-\\left(\\frac{\\delta m_t}{m_t}+\\frac{\\delta s_{2\\theta _t}}{s_{2\\theta _t}}\\right)\\,\\left(\\frac{1}{m_{\\tilde{t}_1}^2}-\\frac{1}{m_{\\tilde{t}_2}^2}\\right)- \\frac{2 \\,m_\\phi ^2}{15}\\,\\frac{\\delta m_t}{m_t}\\left(\\frac{1}{m_{\\tilde{t}_1}^4}-\\frac{1}{m_{\\tilde{t}_2}^4}\\right)\\right]\\,,\\nonumber \\\\&&\\\\G_t^{2\\ell } & \\longrightarrow & G_t^{2\\ell } ~+~\\frac{1}{6}\\,\\left[\\frac{\\delta m_{\\tilde{t}_1}^2}{m_{\\tilde{t}_1}^4}+\\frac{\\delta m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_2}^4}-2\\,\\frac{\\delta m_t}{m_t}\\,\\left(\\frac{1}{m_{\\tilde{t}_1}^2}+\\frac{1}{m_{\\tilde{t}_2}^2}\\right)\\right]~-~\\frac{2}{3}\\, {\\cal F}^{(2\\ell ,b)}_{1/2}(\\tau _t) \\,\\frac{\\delta m_t}{m_t^3}~,\\\\&&\\nonumber \\\\\\widetilde{F}_t^{2\\ell } & \\longrightarrow & \\widetilde{F}_t^{2\\ell } ~+~\\frac{1}{6}\\,\\left[\\frac{\\delta m_{\\tilde{t}_1}^2}{m_{\\tilde{t}_1}^4}-\\frac{\\delta m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_2}^4}- \\frac{\\delta c_{2\\theta _t}}{c_{2\\theta _t}}\\,\\left(\\frac{1}{m_{\\tilde{t}_1}^2}-\\frac{1}{m_{\\tilde{t}_2}^2}\\right)\\right]~,\\\\&&\\nonumber \\\\\\widetilde{G}_t^{2\\ell } & \\longrightarrow & \\widetilde{G}_t^{2\\ell } ~+~\\frac{1}{6}\\,\\left[\\frac{\\delta m_{\\tilde{t}_1}^2}{m_{\\tilde{t}_1}^4}+\\frac{\\delta m_{\\tilde{t}_2}^2}{m_{\\tilde{t}_2}^4}\\right]~.$ In addition, the two-loop form factor ${\\cal H}_2^{2\\ell }$ gets contributions originating from the shift in $A_t$ : ${\\cal H}_2^{2\\ell } \\longrightarrow {\\cal H}_2^{2\\ell } ~-~\\frac{m_t\\,s_{2\\theta _t}}{6\\,s_\\beta }\\,\\left[\\delta A_t \\,\\left(\\frac{1}{m_{\\tilde{t}_1}^2}-\\frac{1}{m_{\\tilde{t}_2}^2}\\right)+ \\frac{2 \\,m_\\phi ^2}{15}\\,\\delta A_t \\,\\left(\\frac{1}{m_{\\tilde{t}_1}^4}-\\frac{1}{m_{\\tilde{t}_2}^4}\\right)\\right]~.$ Once again, care is required in order to properly include in the form factors ${\\cal H}_i$ only terms up to ${\\cal O}(m_\\phi ^2/M^2)$ , ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ and ${\\cal O}(m_t^2/M^2)$ .",
"In particular: the shifts $\\delta m_{\\tilde{t}_1}^2$ , $\\delta m_{\\tilde{t}_2}^2$ and $\\delta \\theta _t$ must be computed up to ${\\cal O}(m_t)$ in $F_t^{2\\ell }$ , eq.",
"(REF ), while they must be truncated at order zero in $m_t$ in $G_t^{2\\ell }$ , $\\widetilde{F}_t^{2\\ell }$ and $\\widetilde{G}_t^{2\\ell }$ , eqs.",
"()–(); in $F_t^{2\\ell }$ , eq.",
"(REF ), the first occurrence of $\\delta m_t$ must be computed up to ${\\cal O}(m_t^2)$ , while the second occurrence, in the term proportional to $m_\\phi ^2$ , must be truncated at order zero in $m_t$ ; in $G_t^{2\\ell }$ , eq.",
"(), the first occurrence of $\\delta m_t$ must be truncated at ${\\cal O}(m_t)$ , while the second occurrence, in the term proportional to ${\\cal F}^{(2\\ell ,b)}_{1/2}(\\tau _t)$ , must be computed up to ${\\cal O}(m_t^3)$ ; finally, in eq.",
"(REF ) the first occurrence of $\\delta A_t$ must be computed up to ${\\cal O}(m_t)$ by means of eq.",
"(REF ), while the second occurrence, in the term proportional to $m_\\phi ^2$ , must be truncated at ${\\cal O}(m_t^{-1})$ .",
"Ref.",
"[28] provides formulae for the two-loop SUSY contributions to the form factors for scalar production in gluon fusion, in the OS renormalization scheme, also based on an asymptotic expansion in the superparticle masses but restricted to the limit of zero squark mixing and degenerate superparticle masses.",
"We checked that our OS results agree with those of ref.",
"[28] in the simplified limit considered in that paper, after taking into account a difference in the overall normalization factor and the fact that ref.",
"[28] employs the opposite convention for the sign of $\\mu $ with respect to our eq.",
"(REF )."
],
[
"A numerical example",
"We will now discuss a numerical evaluation of the two-loop SUSY contributions to the form factors for scalar Higgs production in a representative region of the MSSM parameter space.",
"The SM parameters entering our calculation include the $Z$ boson mass $m_{\\scriptscriptstyle Z}= 91.1876$ GeV, the $W$ boson mass $m_{\\scriptscriptstyle W}= 80.399$ GeV and the strong coupling constant $\\alpha _s(m_{\\scriptscriptstyle Z}) = 0.118$ [30].",
"For the pole mass of the top quark we take $M_t = 173.2$ GeV [31].",
"For the relevant SUSY parameters we choose $m_Q=m_U=\\mu ~=~ 1~{\\rm TeV}\\,,~~~A_t ~=~ 2~{\\rm TeV}\\,,~~~m_{\\tilde{g}}~=~ 800~{\\rm GeV}\\,,~~~\\tan \\beta ~=~ 5~,$ where $m_Q$ and $m_U$ are the soft SUSY-breaking masses for the left and right stops, respectively.",
"For a given value of the pseudoscalar mass $m_{\\scriptscriptstyle A}$ , the scalar masses $m_h$ and $m_{\\scriptscriptstyle H}$ and the mixing angle $\\alpha $ are computed including the leading one-loop corrections of ${\\cal O}(\\alpha _t)$ and the leading two-loop corrections of ${\\cal O}(\\alpha _s \\alpha _t)$ [32].",
"Figure: Real part of the SUSY contributions to ℋ H 2ℓ {\\cal H}_{\\scriptscriptstyle H}^{2\\ell }, plotted as afunction of m H m_{\\scriptscriptstyle H}.",
"The choice of SUSY parameters and the meaning ofthe different curves are explained in the text.",
"The plot on the leftrefers to the DR ¯\\overline{\\rm DR} scheme, while the plot on the right refers tothe OS scheme.In fig.",
"REF we show the real part of the SUSY (i.e., all except top-gluon) contributions to the two-loop form factor for heaviest-Higgs production, ${\\cal H}_{\\scriptscriptstyle H}^{2\\ell }$ , as a function of $m_{\\scriptscriptstyle H}$ .",
"Since, as mentioned above, $m_{\\scriptscriptstyle H}$ is not a free parameter in our calculation, its variation is obtained by varying $m_{\\scriptscriptstyle A}$ between 100 GeV and 500 GeV.",
"For simplicity, in the computation of the form factor we neglected the small D-term-induced electroweak contributions.",
"The left plot in figure REF is obtained assuming that the parameters $m_t,\\,m_{\\tilde{t}_1},\\,m_{\\tilde{t}_2}$ and $\\theta _t$ entering the one-loop part of the form factor, ${\\cal H}_{\\scriptscriptstyle H}^{1\\ell }$ , are expressed in the $\\overline{\\rm DR}$ renormalization scheme at the scale $Q=1$ TeV.",
"In this case we extract the $\\overline{\\rm DR}$ top mass $m_t(Q)$ from the input value for the pole mass $M_t$ by means of eq.",
"(B2) of ref.",
"[32], and we interpret the input parameters $m_Q,\\,m_U$ and $A_t$ in eq.",
"(REF ) directly as running parameters evaluated at the scale $Q$ .",
"The right plot, on the other hand, is obtained assuming that the parameters $m_t,\\,m_{\\tilde{t}_1},\\,m_{\\tilde{t}_2}$ and $\\theta _t$ entering ${\\cal H}_{\\scriptscriptstyle H}^{1\\ell }$ are expressed in the OS scheme described in section 3.2 of ref. [21].",
"In this case we identify $m_t$ directly with the pole mass $M_t$ , and we interpret the input parameters $m_Q,\\,m_U$ and $A_t$ in eq.",
"(REF ) as the parameters that can be obtained by rotating the diagonal matrix of the physical stop masses by the “physical” angle $\\theta _t$ , defined through eq.",
"(37) of ref. [21].",
"In each plot, the dashed (blue) line represents the result obtained in the VHML, as given in ref.",
"[21], while the solid (red) line represents the result computed at the first order of a Taylor expansion in $m_{\\scriptscriptstyle H}^2$ , i.e.",
"it includes the effect of terms of ${\\cal O}(m_{\\scriptscriptstyle H}^2/m_t^2)$ and ${\\cal O}(m_{\\scriptscriptstyle H}^2/M^2)$ which were also computed in ref.",
"[21] but proved too lengthy to be presented in analytic form.",
"The dot-dashed (black) line represents instead the result of the asymptotic expansion in the superparticle masses derived in this paper.",
"The latter is applicable when both $m_t$ and $m_{\\scriptscriptstyle H}$ are smaller than the generic superparticle mass $M$ , as is indeed the case here since $M\\approx 1$ TeV, but it does not require any specific hierarchy between $m_{\\scriptscriptstyle H}$ and $m_t$ .",
"The comparison between the dashed and solid lines shows that, as $m_{\\scriptscriptstyle H}$ increases, the effect of the terms of ${\\cal O}(m_{\\scriptscriptstyle H}^2/m_t^2)$ and ${\\cal O}(m_{\\scriptscriptstyle H}^2/M^2)$ becomes more and more relevant, and the VHML does not provide an accurate approximation to ${\\cal H}_{\\scriptscriptstyle H}^{2\\ell }$ .",
"Furthermore, the comparison between the dot-dashed and solid lines shows that, even if the inclusion of the first-order terms pushes the validity of the Taylor expansion up to larger values of $m_{\\scriptscriptstyle H}$ , the Taylor expansion fails anyway when $m_{\\scriptscriptstyle H}$ gets close to the threshold for the production of a real top-quark pair in the loops.",
"In that case one can use the result of our asymptotic expansion in $M$ , provided that the latter is still considerably larger than $m_{\\scriptscriptstyle H}$ .",
"A few additional comments are in order concerning the comparison between the left ($\\overline{\\rm DR}$ ) and right (OS) plots in fig.",
"REF .",
"There is no reason to expect the plots to look similar to each other, first of all because the difference between the values of ${\\cal H}_{\\scriptscriptstyle H}^{2\\ell }$ in the two schemes is compensated for, up to higher-order terms, by a shift in the value of the one-loop form factor, ${\\cal H}_{\\scriptscriptstyle H}^{1\\ell }$ , and also because the different interpretation of the input parameters in the two schemes means that, by using the numerical inputs in eq.",
"(REF ) for both schemes, we are in fact considering two different points of the MSSM parameter space.",
"This said, a striking difference between the two schemes is visible in the behavior of the asymptotic expansion (i.e., the dot-dashed line) around the threshold for the production of a real top-quark pair in the loops.",
"The fact that in the $\\overline{\\rm DR}$ plot the threshold is located at a lower value of $m_{\\scriptscriptstyle H}$ than in the OS plot is an artifact, due to lower value of the MSSM running top mass with respect to the pole top mass (indeed, for our choice of parameters $m_t(Q) = 144.3$ GeV).",
"The much sharper behavior around the threshold of the dot-dashed line in the $\\overline{\\rm DR}$ plot, on the other hand, can be traced back to the contribution of the first term in the right-hand side of eq.",
"(REF ) for $Y_t^{\\tilde{g}}$ .",
"That term reflects the fact that the running top mass of the MSSM (i.e., including the stop-gluino contribution) is used in the top-quark contribution to ${\\cal H}_{\\scriptscriptstyle H}^{1\\ell }$ , and it is canceled out by the last term of eq.",
"() if the pole top mass (or, for that matter, the running top mass of the SM) is used instead.",
"Indeed, we checked that, in a “mixed” renormalization scheme in which the stop contributions to ${\\cal H}_{\\scriptscriptstyle H}^{1\\ell }$ are expressed in term of running parameters (including the MSSM running top mass) but the top-quark contribution is expressed in terms of the pole top mass, the qualitative behavior of the dot-dashed line around the threshold would be similar to the one in the OS plot.",
"To conclude this discussion, we show in fig.",
"REF the real part of the SUSY contributions to the two-loop form factor for lightest-Higgs production, ${\\cal H}_h^{2\\ell }$ , as a function of the pseudoscalar mass $m_{\\scriptscriptstyle A}$ , which is varied in the same range used to produce fig.",
"REF .",
"The meaning of the different curves is the same as in fig.",
"REF , and again the left plot is obtained assuming that the parameters entering the one-loop form factor ${\\cal H}_h^{1\\ell }$ are expressed in the $\\overline{\\rm DR}$ scheme, while the right plot is obtained assuming that they are expressed in the OS scheme.",
"In the MSSM the mass of the lightest Higgs scalar $h$ is bounded from above, and for large enough values of the pseudoscalar mass it becomes independent of $m_{\\scriptscriptstyle A}$ , as do the couplings of $h$ to the top quark and to the stops.",
"Indeed, for the choice of SUSY parameters in eq.",
"(REF ) our crude ${\\cal O}(\\alpha _t+\\alpha _t\\alpha _s)$ calculation of the Higgs mass yields $m_h<123.8$ GeV in the $\\overline{\\rm DR}$ plot and $m_h<122.5$ GeV in the OS plot, and all the curves in fig.",
"REF become essentially flat for $m_{\\scriptscriptstyle A}> 250$ GeV.",
"Due to the relative smallness of $m_h$ no real-particle threshold is crossed, thus the result of the asymptotic expansion (dot-dashed line) is rather close to the result of the Taylor expansion at the first order in $m_h^2$ (solid line).",
"However, a comparison between the left and right plots of fig.",
"REF shows that in the $\\overline{\\rm DR}$ calculation the VHML result (dashed line) provides a less-than-perfect approximation to ${\\cal H}_h^{2\\ell }$ , while in the OS calculation the effect of the terms proportional to $m_h^2$ is small, and the VHML result essentially overlaps with the other two results.",
"This difference between the two schemes can again be traced to the contribution of the first term in the right-hand side of eq.",
"(REF ), i.e.",
"to the choice of renormalization scheme for the top mass entering the top-quark contribution to ${\\cal H}_h^{1\\ell }$ .",
"Even in this case we checked that, in a “mixed” scheme in which the top-quark contribution to ${\\cal H}_h^{1\\ell }$ is expressed in terms of the pole top mass while the stop contributions are expressed in terms of running parameters, the VHML would provide as good an approximation to ${\\cal H}_h^{2\\ell }$ as it does in the full OS scheme.",
"Figure: Real part of the SUSY contributions to ℋ h 2ℓ {\\cal H}_h^{2\\ell }, plotted as afunction of m A m_{\\scriptscriptstyle A}.",
"The choice of SUSY parameters and the meaning ofthe different curves are explained in the text.",
"The plot on the leftrefers to the DR ¯\\overline{\\rm DR} scheme, while the plot on the right refers tothe OS scheme."
],
[
"Conclusions",
"The calculation of the production cross section for the MSSM Higgs bosons is not quite as advanced as in the SM.",
"Indeed, a full computation of the two-loop quark-squark-gluino contributions, valid for arbitrary values of all the relevant particle masses, has not been made publicly available so far.",
"Moreover, the complexity of such a computation is going to be reflected in results that will probably be too lengthy and computer-time-consuming to be efficiently implemented in event generators.",
"An alternative approach consists in deriving approximate analytic results that can be easily implemented in computer codes, and that are valid in specific regions of the MSSM parameter space such as, e.g., when the Higgs bosons are somewhat lighter than the squarks and the gluino.",
"In this paper we presented a new calculation of the two-loop top-stop-gluino contributions to the form factors for Higgs scalar production in gluon fusion.",
"We exploited techniques developed in our earlier computations of the production cross section for the MSSM Higgs bosons [27], [22] to obtain explicit and compact analytic results based on an asymptotic expansion in the heavy particle masses, up to and including terms of ${\\cal O}(m_\\phi ^2/M^2)$ , ${\\cal O}(m_{\\scriptscriptstyle Z}^2/M^2)$ and ${\\cal O}(m_t^2/M^2)$ .",
"We compared our new results with the VHML results of ref.",
"[21], as well as with the results of a Taylor expansion in the Higgs mass, up to and including terms of ${\\cal O}(m_\\phi ^2/m_t^2)$ and ${\\cal O}(m_\\phi ^2/M^2)$ , and we discussed the regions of applicability of the different expansions.",
"We also discussed the effect of choosing different renormalization schemes for the parameters in the top/stop sector.",
"From the example presented in section it appears that, in the case of the heaviest Higgs boson $H$ , the use of our asymptotic expansion becomes mandatory when $m_{\\scriptscriptstyle H}$ approaches (or crosses) the threshold for the production of a real top-quark pair in the loops.",
"It also appears that choosing the OS scheme for the parameters in the top/stop sector leads to a milder behavior of the two-loop form factor ${\\cal H}_{\\scriptscriptstyle H}^{2\\ell }$ around the threshold.",
"In the case of the lightest Higgs boson $h$ , whose mass is bounded from above in the MSSM, we need not worry about thresholds.",
"However, our discussion showed that in the $\\overline{\\rm DR}$ scheme the VHML provides a worse approximation to ${\\cal H}_h^{2\\ell }$ than it does in the OS scheme.",
"Finally, we remark that the results derived in this paper for the production cross section can be straightforwardly adapted to the NLO computation of the gluonic and photonic decay widths of the MSSM Higgs bosons, in analogy to what described in section 5 of ref.",
"[21]."
],
[
"Acknowledgments",
"This work was partially supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet)."
]
] | 1204.1016 |
[
[
"Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels"
],
[
"Abstract Paragrassmann algebras are given a sesquilinear form for which one subalgebra becomes a Hilbert space known as the Segal-Bargmann space.",
"This Hilbert space as well as the ambient space of the paragrassmann algebra itself are shown to have reproducing kernels.",
"These algebras are not isomorphic to algebras of functions so some care must be taken in defining what \"evaluation at a point\" corresponds to in this context.",
"The reproducing kernel in the Segal-Bargmann space is shown to have most, though not all, of the standard properties.",
"These quantum spaces provide non-trivial examples of spaces which have a reproducing kernel but which are not spaces of functions."
],
[
"Introduction",
"This paper is inspired in large measure by the work in [2] on paragrassmann algebras.",
"We begin in Sections 1-4 by reviewing some of the material in [2], though sometimes re-working that presentation by using our own notation and sometimes by making mild generalizations.",
"We also prove some basic propositions for later use and explain in detail the conjugation we will be using.",
"This makes the paper more self-contained logically.",
"However, see [2] for references to previous works on this topic in mathematics and physics.",
"We note that the deformation parameter $q$ in this paper is non-zero and complex, while in [2] it lies on the unit circle in the complex plane.",
"But more importantly, the conjugation used here is different from that in [2].",
"So strictly speaking this paper treats topics not discussed in [2], though there are ideas in common.",
"The core material of the paper starts in Section 5 where reproducing kernels are defined and discussed in the context of a Segal-Bargmann space that we define as a subalgebra of a paragrassmann algebra.",
"The Segal-Bargmann (or coherent state) transform is introduced in Section 6, and its relation to the reproducing kernel is proved.",
"We follow in Section 7 with a proof of the existence of the reproducing kernel in the full space of paragrassmann variables.",
"This might seem to be a rather surprising result since this is a non-commutative algebra in general unlike the Segal-Bargmann space, which is a commutative algebra though also not isomorphic to an algebra of functions.",
"However, reproducing kernels in the finite dimensional case are quite common and, as we shall see, this is even the generic case in some sense to be specified later.",
"In the last section we discuss some possible avenues for future research.",
"One of these possibilities, the definition and study of Toeplitz operators in this context, will be the topic of a forthcoming paper [8]."
],
[
"Preliminaries",
"Throughout this article we take $l$ to be an integer with $l \\ge 2$ .",
"We put $q_l = e^{2 \\pi i / l}$ , a primitive $l$ -th root of unity in the complex plane $\\mathbb {C}$ .",
"(N.B.",
"Our parameter $l$ corresponds to $k^\\prime $ in [2].)",
"We take the set $\\lbrace \\theta , \\overline{\\theta } \\rbrace $ of two elements and consider the free algebra over the field of complex numbers $\\mathbb {C}$ generated by this set.",
"It is denoted by $\\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace $ .",
"It is also called the algebra of complex polynomials in the two non-commuting variables $ \\theta , \\overline{\\theta } $ which satisfy no relation whatsoever.",
"In this paper all spaces are vector spaces over the field $\\mathbb {C}$ , and all algebras are unital, that is, have an identity element.",
"Moreover, algebra morphisms map the identity element in the domain to the identity element in the codomain.",
"As in [2] we define the paragrassmann algebra associated to $l$ to be the quotient algebra $PG_l = PG_l(\\theta , \\overline{\\theta } ) := \\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace / \\langle \\theta ^l,\\overline{\\theta }{}^l, \\theta \\overline{\\theta } - q_l \\overline{\\theta } \\theta \\rangle .$ Here as usual the notation $\\langle \\, \\cdot \\, \\rangle $ means the two-sided ideal generated by the elements listed inside the braces.",
"We let $ \\theta , \\overline{\\theta } $ also denote the quotients (i.e., equivalence classes) of these two elements in $PG_l$ .",
"Seen this way $ \\theta $ and $\\overline{\\theta } $ are nilpotent elements in $PG_l$ , each having order of nilpotency $l$ .",
"They do not commute, since $ \\theta \\overline{\\theta } = q_l \\overline{\\theta } \\theta $ in $PG_l$ and $q_l \\ne 1$ .",
"(The case $l=1$ has been excluded not so much because $q_l =1$ in that case, but really because $ \\theta = \\overline{\\theta } =0 $ in that case and so $PG_1 = \\mathbb {C}$ , a trivial case we wish to exclude.)",
"However, in the case $l=2$ we have $q_2 = -1$ and so $ \\theta $ and $\\overline{\\theta } $ anti-commute.",
"Also, $ \\theta ^2 = \\overline{\\theta }{}^2 =0$ .",
"So the case $l=2$ corresponds to two grassmann variables in the standard definition of this term.",
"In [2] the paragrassmann algebra studied is $PG_l$ .",
"Notice that in this algebra the order of nilpotency of the variables $\\theta $ and $\\overline{\\theta }$ determines the parameter $q_l$ in the commutation relation, and conversely.",
"However, we would like to generalize slightly this concept in the following definition.",
"Definition 2.1 Let $l \\ge 2$ be an integer and $q \\in \\mathbb {C}$ .",
"The paragrassmann algebra $PG_{l,q}$ with paragrassmann variables $\\theta $ and $\\overline{\\theta }$ is defined by $PG_{l,q} =PG_{l,q}(\\theta , \\overline{\\theta }) := \\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace / \\langle \\theta ^l, \\overline{\\theta }{}^l, \\theta \\overline{\\theta }- q \\overline{\\theta } \\theta \\rangle $ Clearly, $PG_l$ is the special case of $PG_{l,q} $ when $q = q_l$ .",
"We will be studying $PG_{l,q} $ .",
"The equation $\\theta \\overline{\\theta } - q \\overline{\\theta } \\theta = 0$ in $PG_{l,q}$ is called the $q$ -commutation relation, while $\\theta ^l =0$ and $ \\overline{\\theta }{}^l =0$ in $PG_{l,q}$ are called the nilpotency conditions.",
"We note that $PG_{l,q}(\\theta , \\overline{\\theta }) = PG_{l,q^{-1}}(\\overline{\\theta }, \\theta )$ for $ q \\ne 0$ .",
"This is an equality (not just an isomorphism) of sets, of vector spaces and also of algebras.",
"Notice that the order of the paragrassmann variables is different on the two sides of this equality.",
"Since neither element in the pair of paragrassmann variables $\\theta , \\overline{\\theta }$ is more fundamental than the other (each being the conjugate of the other in a conjugation defined later) we also have an isomorphism $PG_{l,q}(\\theta , \\overline{\\theta }) \\cong PG_{l,q^{-1}}(\\theta , \\overline{\\theta })$ of algebras, where the isomorphism on the generators maps $\\theta \\mapsto \\overline{\\theta }$ and $\\overline{\\theta } \\mapsto \\theta $ .",
"This is then extended multiplicatively to the basis $AW$ , defined below, and then linearly to $PG_{l,q}(\\theta , \\overline{\\theta })$ .",
"Notice that the order of the paragrassmann variables is the same on the two sides of this isomorphism.",
"One can combine the equality (REF ) and the isomorphism (REF ) to get other identifications.",
"The moral of these basic facts is that at this stage one can not distinguish between the creation and annihilation elements, where one of these should be $\\theta $ while the other should be $\\overline{\\theta }$ .",
"It is only after quantization that such a distinction can be made by examining the quantizations of $\\theta $ and $\\overline{\\theta }$ .",
"We will see this in detail in [8] in the quantization given by the Toeplitz operators.",
"So the algebra $PG_{l,q} $ could be viewed as `classical' object in some sense even though it is also a `quantum' object, that is, it is not commutative.",
"The case $q=0$ is different as would be expected.",
"In any quantization scheme (and there are many) this `classical' algebra for $q=0$ , namely $PG_{l,0} $ , gives rise to what could be called a `quantum' free probability theory of paragrassmann variables.",
"We will be using the following index set throughout: $I_l = \\lbrace 0, 1, \\dots , l-1 \\rbrace .$ When an index, say $i$ , is given without an explicit index set, we assume $i \\in I_l$ .",
"The Segal-Bargmann (or holomorphic) space is defined to be $\\mathcal {B}_H = \\mathcal {B}_H(\\theta ) := \\mathrm {span}_{\\mathbb {C}} \\, \\lbrace \\theta ^i \\, \\vert \\, i \\in I_l \\rbrace .$ Similarly, the anti-Segal-Bargmann (or anti-holomorphic) space is defined to be $\\mathcal {B}_{AH} =\\mathcal {B}_{AH}(\\overline{\\theta }) := \\mathrm {span}_{\\mathbb {C}} \\, \\lbrace \\overline{\\theta }{}^i \\, \\vert \\, i \\in I_l \\rbrace .$ The Segal-Bargmann space is not only a vector subspace of $PG_{l,q}(\\theta , \\overline{\\theta })$ ; it is also a subalgebra.",
"Actually, it is a commutative subalgebra isomorphic to the truncated polynomial algebra $\\mathbb {C} [\\theta ] \\, / \\langle \\theta ^l \\rangle $ .",
"Similarly, the anti-Segal-Bargmann space is a commutative subalgebra isomorphic to the exact same truncated polynomial algebra, although this is usually written as $\\mathbb {C} [\\overline{\\theta }] \\, / \\langle \\overline{\\theta }{}^l \\rangle $ .",
"The subspace $\\mathcal {S} = \\mathcal {B}_H + \\mathcal {B}_{AH}$ of $PG_{l,q} $ plays a special role.",
"Note that this is not a subalgebra.",
"Also this is not a direct sum but nearly so, since $\\mathcal {B}_H \\cap \\mathcal {B}_{AH} = \\mathbb {C}1$ .",
"We define a conjugation operation in $\\mathcal {S}$ by $(\\theta ^i)^* := \\overline{\\theta }{}^i$ and $(\\overline{\\theta }{}^i)^* := \\theta ^i$ and extend anti-linearly to all of $\\mathcal {S}$ .",
"For the time being we do not introduce a conjugation operation in $PG_{l,q}$ though we will do this later on by extending this rather natural conjugation in $\\mathcal {S}$ to $PG_{l,q}$ .",
"We have two canonical bases of $PG_{l,q}$ provided that $q \\ne 0$ .",
"First, there is the Wick basis $W = \\lbrace \\overline{\\theta }{}^i \\theta ^j \\, \\vert \\, i,j \\in I_l \\rbrace ,$ which is also a basis when $q=0$ .",
"Second, there is the anti-Wick basis $AW = \\lbrace \\theta ^i \\overline{\\theta }{}^j \\, \\vert \\, i,j \\in I_l \\rbrace $ for the case $q \\ne 0$ .",
"Of course, we are using here the usual convention that $\\theta ^0 =1$ and $\\overline{\\theta }{}^0 = 1$ .",
"Notice that the elements of these two bases are not the same, except when $q=1$ .",
"For $q \\ne 1$ their only common element is the identity element 1.",
"Here we follow [2] by saying that an expression with all factors of $\\theta $ to the right (respectively, left) of all factors of $\\overline{\\theta } $ is in Wick (respectively, anti-Wick) order.",
"In physics (e.g., see [6]) the original definition is that an expression with all annihilation operators to the right of all creation operators is in Wick order.",
"Since we have no way for identifying at this level which variable corresponds to annihilation, our present definition is a rather arbitrary choice whose only virtue is that it agrees with [2].",
"Clearly, we have $\\mathrm {dim}_{\\mathbb {C}} \\, PG_{l,q} = l^2$ .",
"In the rest of this article, we will consider the case $q \\in \\mathbb {C} \\setminus \\lbrace 0 \\rbrace $ .",
"The results in this paper do not depend directly on the specific value of the parameter $q$ .",
"It seems that the principle role of the $q$ -commutation relation (in conjunction with the nilpotency conditions) is to force the space under consideration to have finite dimension $l^2$ with the very specific standard basis $AW$ .",
"The integral is a linear functional $PG_{l,q} \\rightarrow \\mathbb {C}$ defined on the basis $AW$ by $\\int \\!",
"\\!",
"\\!",
"\\int d \\theta \\, \\, \\theta ^i \\overline{\\theta }{}^j \\, \\, d \\overline{\\theta } := \\delta _{i,l-1} \\delta _{j,l-1},$ where $\\delta _{ab}$ is the Kronecker delta of the integers $a,b$ .",
"This is a Berezin type integral, by which is meant that only the highest non-zero power element $\\theta ^{l-1} \\overline{\\theta }{}^{l-1}$ has a non-zero integral."
],
[
"Conjugation",
"We next introduce a conjugation (or $*$ -operation) in $PG_{l,q}$ by expanding an arbitrary $f \\in PG_{l,q}$ in the basis $AW$ as $f = \\sum _{i,j} f_{ij} \\theta ^i \\overline{\\theta }{}^j,$ where the coefficients $f_{ij} \\in \\mathbb {C}$ are uniquely determined.",
"Then we define the conjugation of $f$ by $f^* := \\sum _{i,j} f_{ij}^* \\theta ^j \\overline{\\theta }{}^i.$ (The usual complex conjugate of $\\lambda \\in \\mathbb {C}$ is denoted by $\\lambda ^*$ .)",
"This gives the expansion of $f^*$ in the same basis $AW$ .",
"This is an anti-linear operation.",
"It also immediately follows that $f^{**} = f$ , that is, this operation is an involution.",
"Note that the conjugation being an involution depends on $\\theta $ and $\\overline{\\theta }$ having the same order of nilpotency.",
"Actually, as another immediate consequence of the definition (REF ) we also have that $( \\theta ^i \\overline{\\theta }{}^j )^* = \\theta ^j \\overline{\\theta }{}^i $ and in particular the relations $(\\theta ^i)^* = \\overline{\\theta }{}^i \\quad \\mathrm {and} \\quad (\\overline{\\theta }{}^j)^* = \\theta ^j \\!$ as promised earlier in Section 2.",
"The action of the conjugation on elements in the basis $W$ is given by $( \\overline{\\theta }{}^i \\theta ^j )^* = ( q^{-i j} \\, \\theta ^j \\overline{\\theta }{}^i )^* =( q^{-i j} )^* \\theta ^i \\overline{\\theta }{}^j= (q^{-ij})^* q^{ i j} \\overline{\\theta }{}^j \\, \\theta ^i= \\left( \\dfrac{q}{q^*} \\right)^{ij} \\overline{\\theta }{}^j \\theta ^i .$ So for $q \\in \\mathbb {R} \\setminus \\lbrace 0 \\rbrace $ the two bases $W$ and $AW$ are treated in a similar manner by the conjugation.",
"But the expression on the far right of (REF ) has no limit when $ q \\rightarrow 0$ and $q$ is complex.",
"We note that the subspace $\\mathcal {S}$ is invariant under the conjugation and that this operation interchanges its subalgebras $\\mathcal {B}_H$ and $\\mathcal {B}_{AH}$ .",
"As noted earlier, these two subalgebras are isomorphic as algebras under the $\\mathbb {C}$ -linear map $\\mathcal {B}_H \\rightarrow \\mathcal {B}_{AH}$ induced by $ \\theta ^i \\mapsto \\overline{\\theta }{}^i$ for all $i \\in I_l$ .",
"Moreover, the conjugation gives us an explicit anti-isomorphism (and its inverse) between these subalgebras, because of the relations (REF ).",
"An anti-isomorphism between algebras $A$ and $A^\\prime $ over $\\mathbb {C}$ is an anti-linear bijection $\\alpha : A \\rightarrow A^\\prime $ that also satisfies $\\alpha (ab) = \\alpha (b) \\alpha (a)$ for all $a,b \\in A$ .",
"For example, the next result immediately implies that the conjugation is an anti-isomorphism of the algebra $PG_{l,q}$ with itself for non-zero real $q$ .",
"Proposition 3.1 The algebra $PG_{l,q}$ is a $*$ -algebra, that is we have $(fg)^* = g^*f^*$ for all $f,g \\in PG_{l,q}$ , if and only if $q \\in \\mathbb {R} \\setminus \\lbrace 0 \\rbrace $ .",
"Proof: First take $q \\ne 0$ and real.",
"We then consider the special case $f = \\theta ^i \\overline{\\theta }{}^j$ and $g = \\theta ^k \\overline{\\theta }{}^m$ , where $i,j,k,m \\in I_l$ .",
"Then on one hand we have $(f g)^* = (\\theta ^i \\overline{\\theta }{}^j \\theta ^k \\overline{\\theta }{}^m)^* =(\\theta ^i q^{-jk} \\theta ^k \\overline{\\theta }{}^j \\overline{\\theta }{}^m)^* =q^{-jk} (\\theta ^{i+k} \\overline{\\theta }{}^{j+m} )^*= q^{-jk} \\theta ^{j+m} \\overline{\\theta }{}^{i+k} ,$ while on the other we get $g^* f^* = (\\theta ^k \\overline{\\theta }{}^m)^* (\\theta ^i \\overline{\\theta }{}^j)^*= \\theta ^m \\overline{\\theta }{}^k \\theta ^j \\overline{\\theta }{}^i= q^{-jk} \\theta ^{j+m} \\overline{\\theta }{}^{i+k}.$ So the result holds in this special case.",
"But since these are an arbitrary pair of elements in the basis $AW$ , we get $(f g)^* = g^* f^*$ for all $f, g \\in PG_{l,q}$ .",
"For $q \\in \\mathbb {C} \\setminus \\mathbb {R}$ equation (REF ) shows that $PG_{l,q}$ is not a $*$ -algebra.",
"$\\quad \\blacksquare $ Remarks: In earlier preprint versions of this paper, I assumed that the conjugation made $PG_{l,q}$ into a $*$ -algebra.",
"So I only considered the case $q \\in \\mathbb {R} \\setminus \\lbrace 0 \\rbrace $ .",
"This is an unnecessary restriction.",
"Here we consider the more general case $q \\in \\mathbb {C} \\setminus \\lbrace 0 \\rbrace $ .",
"The only way that the conjugation enters into the subsequent theory is through the definition of the sesquilinear form given in the next section.",
"And the properties that we will use of that sesquilinear form do not require that the conjugation gives $PG_{l,q}$ a $*$ -algebra structure.",
"I thank R. Fresneda [4] for clarifying for me that the definition of the conjugation used in [2] is the anti-linear extension of $( \\theta ^i \\overline{\\theta }{}^j )^* = \\overline{\\theta }{}^i \\theta ^j $ .",
"(We note that this does not give a $*$ -algebra.)",
"The fact that we will be using the above definition (REF ) of the conjugation means that we are considering structures that are strictly speaking distinct from those discussed in [2].",
"Nonetheless, there will still be things in common with the approach in [2].",
"We note that what is behind these various definitions of conjugation are different ways of dealing with an ordering problem.",
"The conjugation (REF ) is nicely related to the integral.",
"The proof is left to the reader.",
"Proposition 3.2 $\\int \\!",
"\\!",
"\\!",
"\\int d \\theta \\, f(\\theta , \\overline{\\theta })^* \\, d \\overline{\\theta } =\\left( \\int \\!",
"\\!",
"\\!",
"\\int d \\theta \\, f(\\theta , \\overline{\\theta }) \\, d \\overline{\\theta } \\right)^*$ for all $f(\\theta , \\overline{\\theta }) \\in PG_{q,l}(\\theta , \\overline{\\theta })$ .",
"Here the conjugation on the right side is complex conjugation of a complex number.",
"We would also like to note that there are other reasonable definitions for a conjugation.",
"One of these for any $q \\in \\mathbb {C} \\setminus \\lbrace 0 \\rbrace $ is the anti-linear extension of the following definition on elements of $AW$ : $(\\theta ^i \\overline{\\theta }{}^j)^{*} := (q^*/q)^{i j/2} \\, \\theta ^j \\overline{\\theta }{}^i$ for a fixed choice of the square root of the phase factor $q^*/q$ .",
"This extends our definition for $q \\in \\mathbb {R} \\setminus \\lbrace 0 \\rbrace $ provided that we take $1^{1/2}$ to be 1.",
"Note that this conjugation is an involution, that is $f^{**} =f$ , and satisfies equations (REF ).",
"However for $q \\notin \\mathbb {R}$ it does not satisfy $ (fg)^* = g^* f^*$ for all $f,g$ .",
"It has the virtue of acting on the basis $W$ in a similar way to its action on $AW$ .",
"Note that $ \\theta $ and $\\overline{\\theta } $ are a pair of conjugate complex variables, that is, $ \\theta ^* = \\overline{\\theta } $ and $\\overline{\\theta }{}^* = \\theta $ and the intersection of the two subalgebras generated by $ \\theta $ and by $\\overline{\\theta } $ , respectively, is simply the smallest it could possibly be: $\\mathbb {C} 1$ .",
"This is in close analogy with the pair of conjugate complex variables $z$ and $\\overline{z}$ (which are functions $\\mathbb {C} \\rightarrow \\mathbb {C}$ and not points in $\\mathbb {C}$ ) as studied in complex analysis, where $z$ generates the algebra of holomorphic functions on $\\mathbb {C}$ while $\\overline{z}$ generates the algebra of anti-holomorphic functions.",
"Also the intersection of these algebras consists of the constant functions.",
"Notice how the non-commutative geometry approach of viewing elements of algebras (in this case functions) as the primary objects of study clarifies a common confusion even in this commutative example where one might not otherwise understand how the complex plane (whose complex dimension is one) can support two independent complex variables, neither of which is more `fundamental' than the other.",
"This short discussion motivates the definition of a variable as any element in a unital algebra that is not a scalar multiple of the identity element.",
"And then a pair of complex variables in any unital $*$ -algebra is defined as any pair of conjugate variables which generate subalgebras with intersection $\\mathbb {C} 1$ ."
],
[
"Sesquilinear form",
"Much of the material in this section comes from the paper [2], though we define a more general sesquilinear form.",
"We want to introduce a sesquilinear form on $PG_{l,q}$ in order to turn it into something like an $L^2$ space.",
"We start with any element (a `positive weight') in $PG_{l,q}$ of the form $w = w(\\theta , \\overline{\\theta }) = \\sum _{m \\in I_l} w_{l-1-m} \\theta ^m \\overline{\\theta }{}^m,$ where $w_n > 0$ for all $n \\in I_l$ .",
"The strange looking way of writing the sub-index on the right side of this equation will be justified later on.",
"In [2] the authors take $w_n = [n]_q!$ which is a $q$ -deformed factorial of the integer $n$ .",
"(See [2] for their definitions.)",
"In any case, this couples the `weight' factors $w_n$ with the deformation parameter $q$ , which itself is coupled in [2] with the nilpotency power $l$ .",
"We have preferred to keep all of these parameters decoupled from one and other.",
"Take $f = f(\\theta , \\overline{\\theta })$ and $ g = g(\\theta , \\overline{\\theta })$ in $PG_{l,q}$ .",
"Informally, we would like to define the sesquilinear form or inner product as in [2] by $\\langle f , g \\rangle _w := \\int \\!\\!\\!",
"\\int d \\theta : f(\\theta , \\overline{\\theta })^* g(\\theta , \\overline{\\theta })w(\\theta , \\overline{\\theta }) : d \\overline{\\theta },$ where $ : \\, \\, : $ is the anti-Wick (or anti-normal) ordering, that is, put all $\\theta $ 's to the left and all $\\overline{\\theta }$ 's to the right.",
"However, the anti-Wick ordering is only well defined on the space $\\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace $ and does not pass to its quotient space $PG_{l,q}(\\theta , \\overline{\\theta })$ .",
"By the formal expression (REF ) we really mean $\\langle f , g \\rangle _w := \\sum _{m \\in I_l} w_{l-1-m} \\int \\!\\!\\!\\int d \\theta \\,\\, \\theta ^m: f(\\theta , \\overline{\\theta })^* : \\, : g(\\theta , \\overline{\\theta }) :\\overline{\\theta }{}^m \\,\\, d \\overline{\\theta },$ where the anti-Wick product $\\, : \\cdot : \\, : \\cdot : \\,$ is defined as the $\\mathbb {C}$ -bilinear extension of this defining formula for pairs of basis elements in $AW$ : $: \\theta ^a \\overline{\\theta }{}^b : \\, : \\theta ^c \\overline{\\theta }{}^d : \\,\\,\\, \\equiv \\,\\,\\, \\theta ^{a+c} \\overline{\\theta }{}^{b+d} .$ So the anti-Wick product is a well defined mapping from $PG_{l,q} \\times PG_{l,q}$ to $PG_{l,q}$ .",
"Clearly the expression in (REF ) is anti-linear in $f$ and linear in $g$ .",
"Also, by the definition of the integral on the right side of (REF ), we have that $\\langle f , g \\rangle _w $ is a complex number.",
"Even though the multiplication in $PG_{l,q}$ is not commutative we have the following result.",
"Proposition 4.1 The sesquilinear form (REF ) is complex symmetric, that is we have that $ \\langle f , g \\rangle _w^* = \\langle g , f \\rangle _w$ for all $f,g \\in PG_{l,q}$ .",
"Proof: The point here is that the anti-Wick product is commutative.",
"A consequence is that $\\langle \\theta ^a \\overline{\\theta }{}^b , \\theta ^c \\overline{\\theta }{}^d \\rangle _w =\\langle \\theta ^c \\overline{\\theta }{}^d , \\theta ^a \\overline{\\theta }{}^b \\rangle _w.$ Also this is a real number, as we will show momentarily below.",
"Then the result follows immediately by expanding $f$ and $g$ in the basis $AW$ .",
"$\\quad \\blacksquare $ We also define $\\vert \\vert f \\vert \\vert ^2_w := \\langle f , f \\rangle _w.$ Note that the previous proposition implies that $\\langle f , f \\rangle _w$ is real.",
"As we will subsequently see, this real number can be negative.",
"So, in general, we are not defining $\\vert \\vert f \\vert \\vert _w$ .",
"Let $ \\theta ^a \\overline{\\theta }{}^b$ and $ \\theta ^c \\overline{\\theta }{}^d$ be arbitrary elements in the basis $AW$ , for $a, b, c, d \\in I_l$ .",
"We now compute their inner product: $\\langle \\theta ^a \\overline{\\theta }{}^b , \\theta ^c \\overline{\\theta }{}^d \\rangle _w &=&\\int \\!",
"\\!",
"\\!",
"\\int d \\theta : (\\theta ^a \\overline{\\theta }{}^b)^* \\theta ^c \\overline{\\theta }{}^d w(\\theta , \\overline{\\theta }) :d \\overline{\\theta } \\nonumber \\\\&=& \\sum _{n \\in I_l} w_{l-1-n}\\int \\!",
"\\!",
"\\!",
"\\int d \\theta \\,\\, \\theta ^{n} : \\theta ^b \\overline{\\theta }{}^a: \\, : \\theta ^c \\overline{\\theta }{}^d :\\overline{\\theta }{}^{n} \\,\\,d \\overline{\\theta } \\nonumber \\\\&=& \\sum _n w_{l-1-n} \\int \\!",
"\\!",
"\\!",
"\\int d \\theta \\, \\theta ^{b+c+n}\\overline{\\theta }{}^{a+d+n} d \\overline{\\theta } \\nonumber \\\\&=& \\sum _n w_{l-1-n} \\, \\delta _{b+c+n,l-1} \\, \\delta _{a+d+n,l-1} \\quad \\mathrm {(Kronecker~deltas)} \\nonumber \\\\&=& \\Big \\lbrace \\begin{array}{cc} w_{b+c} = w_{a+d} & \\mathrm {if~} b+c = a+d \\le l -1 \\\\0 & \\mathrm {otherwise} \\nonumber \\end{array} \\Big \\rbrace \\\\&=& \\delta _{a+d, b+c} \\, \\chi _l (a+d) \\, w_{a+d},$ where $\\chi _l$ is the characteristic function of $I_l$ , that is for every integer $k$ we put $\\chi _l(k) = 1$ if $k \\in I_l$ and $\\chi _l(k) = 0$ if $k \\notin I_l$ .",
"As noted above, () is a real number.",
"Strictly speaking, we should define $w_n$ here for $n \\ge l$ .",
"But the exact definition is unimportant since the $\\chi _l$ factor is zero in that case.",
"Having established the closed formula () for the inner product of any pair of elements in the basis $AW$ , we will never again have need to calculate a Berezin integral.",
"Another way of saying this is that we could have taken () as the definition of the inner product (extending anti-linearly in the first argument and linearly in the second) and dispensed with the Berezin integral entirely.",
"Now in general there are pairs such that $(a,b) \\ne (c,d)$ but $b+c = a+d \\le l-1$ , so that there are distinct basis elements in $AW$ which are not orthogonal.",
"(For example, take $a=1, b=l -1, c=0, d=l -2$ .",
"These all lie in $I_l$ since $l \\ge 2$ .)",
"That is to say, the basis $AW$ is not orthogonal.",
"Similar calculations show that the basis $W$ also is not orthogonal.",
"In particular, $\\langle \\theta ^a \\overline{\\theta }{}^b, \\theta ^c \\overline{\\theta }{}^d \\rangle _w = 0$ if $a-b \\ne c -d$ .",
"For example, taking $b=d=0$ and $a \\ne c$ , we see that $\\langle \\theta ^a, \\theta ^c \\rangle _w = 0$ .",
"Similarly, taking $a=c=0$ and $b \\ne d$ , gives us $\\langle \\overline{\\theta }{}^b, \\overline{\\theta }{}^d \\rangle _w = 0$ .",
"Finally, taking $b=c=0$ and $a \\ge 1$ and $d\\ge 1$ we have that $\\langle \\theta ^a, \\overline{\\theta }{}^d \\rangle _w =0$ .",
"All of this shows that the following set is orthogonal: $\\lbrace 1, \\theta , \\theta ^2 , \\dots , \\theta ^{l -1}, \\overline{\\theta }, \\overline{\\theta }{}^{2} , \\dots , \\overline{\\theta }{}^{l -1} \\rbrace .$ Also, from (REF ) we immediately get $|| \\theta ^a \\overline{\\theta }{}^b ||_w^2 = \\Big \\lbrace \\begin{array}{cc} w_{a+b} & \\mathrm {if~} a+b \\le l -1 \\\\0 & \\mathrm {otherwise}\\end{array} \\Big \\rbrace = \\chi _l(a+b) \\, w_{a+b}.$ This already shows that there are non-zero elements $f \\in PG_{l,q}$ such that $|| f ||_w^2 =0$ .",
"One simply takes $f = \\theta ^a \\overline{\\theta }{}^b$ with $a,b \\in I_l$ and $a+b \\ge l$ .",
"(For example, $a=b=l-1$ will do, since $l \\ge 2$ .)",
"Taking $b=0$ (with $a \\in I_l$ ) and $a=0$ (with $b \\in I_l$ ) respectively in equation (REF ) we get the formulas $|| \\theta ^a ||_w^2 = w_{a} \\quad \\mathrm {and} \\quad || \\overline{\\theta }{}^b ||_w^2 = w_{b}.$ The `nice' subindices of the $w$ 's in these identities are the reason we took the unusual looking convention for the subindices in the definition (REF ) of $w(\\theta , \\overline{\\theta })$ .",
"There are also always elements $f \\in PG_{l,q}$ such that $|| f ||_w^2 < 0$ .",
"To show this explicitly we note that we have these formulas for any $l \\ge 2$ : $\\langle 1, 1 \\rangle _w = w_0, \\quad \\quad \\langle \\theta ^{l-1} \\overline{\\theta }{}^{l-1} , \\theta ^{l-1} \\overline{\\theta }{}^{l-1} \\rangle _w = 0,\\\\\\langle 1, \\theta ^{l-1} \\overline{\\theta }{}^{l-1} \\rangle _w =\\langle \\theta ^{l-1} \\overline{\\theta }{}^{l-1} , 1 \\rangle _w = w_{l-1}.$ Consequently, taking $f= \\alpha 1 + \\beta \\theta ^{l-1} \\overline{\\theta }{}^{l-1} $ for any $\\alpha , \\beta \\in \\mathbb {C}$ , we have that $\\langle f ,f \\rangle _w =\\langle \\alpha 1 + \\beta \\theta ^{l-1} \\overline{\\theta }{}^{l-1} , \\alpha 1 + \\beta \\theta ^{l-1} \\overline{\\theta }{}^{l-1}\\rangle _w =| \\alpha |^2 w_0 + (\\alpha ^* \\beta + \\alpha \\beta ^*) w_{l-1}\\\\= | \\alpha |^2 w_0 + ( | \\alpha + \\beta |^2 - | \\alpha |^2 - | \\beta |^2 ) w_{l-1}.$ Since $w_{l-1} > 0$ it is not difficult to find $\\alpha $ and $\\beta $ making this last expression strictly less than zero.",
"For example, $\\alpha = 1$ and $\\beta < - w_0/ (2 w_{l-1})$ will do.",
"We now define $\\phi _n(\\theta , \\overline{\\theta }) := w_n^{-1/2} \\theta ^n$ for every $n \\in I_l$ .",
"(Recall that $w_n > 0$ .",
"So we take the positive square root of $w_n$ .)",
"We also write $\\phi _n(\\theta ) := w_n^{-1/2} \\theta ^n$ , since this element does not `depend' on $\\overline{\\theta }$ , that is, it lies in the subalgebra generated by $\\theta $ alone.",
"These vectors clearly form an orthonormal basis of $\\mathcal {B}_{H}$ .",
"But given the above calculations we can immediately say more.",
"Proposition 4.2 $\\mathcal {B}_{H}$ , $\\mathcal {B}_{AH}$ and $\\mathcal {S}$ are Hilbert spaces with respect to $\\langle \\cdot , \\cdot \\rangle _w $ ., which is a positive definite inner product when restricted to any of these subspaces of $PG_{l,q}$ .",
"Moreover, with respect to this inner product we have the following statements: $\\lbrace \\phi _n \\, \\vert \\, n \\in I_l \\rbrace $ is an orthonormal basis of $\\mathcal {B}_{H}$ and $\\mathrm {dim}_{\\mathbb {C}} \\, \\mathcal {B}_{H} = l$ .",
"$\\lbrace \\phi _n^* \\, \\vert \\, n \\in I_l \\rbrace $ is an orthonormal basis of $\\mathcal {B}_{AH}$ and $\\mathrm {dim}_{\\mathbb {C}} \\, \\mathcal {B}_{AH} = l$ .",
"$\\lbrace \\phi _n \\, \\vert \\, n \\in I_l \\rbrace \\cup \\lbrace \\phi _n^* \\, \\vert \\, n \\in I_l \\setminus \\lbrace 0\\rbrace \\rbrace $ is an orthonormal basis of $\\mathcal {S}$ and $\\mathrm {dim}_{\\mathbb {C}} \\, \\mathcal {S} = 2l - 1$ .",
"Note that $\\phi _n^*(\\theta , \\overline{\\theta }) = ( \\phi _n(\\theta , \\overline{\\theta }) )^* = w_n^{-1/2} \\overline{\\theta }{}^n$ using $w_n > 0$ .",
"We also write this element as $\\phi _n^*( \\overline{\\theta })$ ."
],
[
"Reproducing kernel for the Segal-Bargmann space",
"It is not possible to find an algebra of complex valued functions which is an isomorphic copy of the commutative algebra $\\mathcal {B}_H$ .",
"We can see this is so by simply noting that $\\theta \\in \\mathcal {B}_H$ satisfies $\\theta \\ne 0$ since $l \\ge 2$ and is nilpotent, namely, $\\theta ^l =0$ .",
"But no non-zero complex valued function is nilpotent.",
"Similarly, the commutative algebra $\\mathcal {B}_{AH}$ is not isomorphic to an algebra of functions.",
"Nonetheless $\\mathcal {B}_H$ and $\\mathcal {B}_{AH}$ are reproducing kernel Hilbert spaces, properly understood.",
"The classical theory of reproducing kernel Hilbert spaces whose elements are functions goes back to the seminal work of Bergman in the 20th century.",
"(See [3] for example.)",
"Here we start with $\\mathcal {B}_H$ , the Segal-Bargmann space, which we will now write as $\\mathcal {B}_H(\\theta )$ to indicate that the paragrassmann variable in this space is $\\theta $ .",
"First, let us note that when we write an arbitrary element in this Segal-Bargmann space uniquely as $f(\\theta ) = \\sum _{j \\in I_l} \\lambda _j \\theta ^j,$ where $\\lambda _j \\in \\mathbb {C}$ for all $j \\in I_l$ , this really can be interpreted as a function of $\\theta $ .",
"In fact, if we let $f(x) = \\sum _{j =0 }^N \\beta _j x^j \\in \\mathbb {C}[x]$ be an arbitrary polynomial in $x$ , an indeterminant, then we can define a functional calculus (where the `functions' are polynomials) of any element $a \\in \\mathcal {B}_H(\\theta )$ precisely by defining $ f(a)$ to be $\\sum _{j =0 }^N \\beta _j a^j$ .",
"With no loss of generality, we can take $N \\ge l-1$ .",
"If we now take $\\beta _j = \\lambda _j$ for all $j \\in I_l$ and any value whatsoever for $\\beta _j$ for $j \\ge l$ , then $f(\\theta )$ is simply the arbitrary element $\\sum _j \\lambda _j \\theta ^j$ considered above.",
"The mapping from $\\mathbb {C}[x] $ , the algebra of polynomials $f(x)$ in $x$ , to $\\mathcal {B}_H(\\theta )$ given by $f(x) \\mapsto f(\\theta )$ is clearly an algebra morphism that is surjective.",
"This is standard material, but it aids us in considering $f(\\theta ) = \\sum _{j \\in I_l} \\lambda _j \\theta ^j$ in $ \\mathcal {B}_H(\\theta )$ and its corresponding element $f(\\eta ) = \\sum _{j \\in I_l} \\lambda _j \\eta ^j$ in $ \\mathcal {B}_H(\\eta )$ for another paragrassmann variable $\\eta $ .",
"We provide $PG_{l,q}(\\eta , \\overline{\\eta })$ and its subspace $ \\mathcal {B}_H(\\eta )$ with essentially the same inner product as above, simply replacing $\\theta , \\overline{\\theta }$ with $\\eta , \\overline{\\eta }$ everywhere.",
"Also, we wish to emphasize that this functional calculus is what replaces in this context the concept of `evaluation at a point' in the usual theory of reproducing kernel Hilbert spaces of functions.",
"We would like to establish for every $f(x) \\in \\mathbb {C}[x]$ the reproducing formula $f(\\theta ) = \\langle K(\\theta , \\eta ) , f(\\eta ) \\rangle _w,$ where the inner product here is roughly speaking taken with respect to the variable $\\eta $ (that is, basically in $\\mathcal {B}_H(\\eta )$ ; more on this later) and where $K(\\theta , \\eta ) \\in \\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_H(\\eta ).$ This last condition says that the reproducing kernel $K(\\theta , \\eta )$ is holomorphic in $\\eta $ and anti-holomorphic in $\\theta $ .",
"This condition is in analogy with the theory of reproducing kernel functions in holomorphic function spaces.",
"(See [7], for example.)",
"The reader should note that we are using the notation $K(\\theta , \\eta )$ , which is analogous to the notation in the classical theory of reproducing kernel Hilbert spaces.",
"If we had been consistent with our own conventions, we would have denoted this as $K(\\overline{\\theta }, \\eta )$ .",
"Now the unknown we have to solve for is the kernel $K(\\theta , \\eta )$ .",
"We write $K(\\theta , \\eta ) = \\sum _{i,j} a_{ij} \\overline{\\theta }{}^i \\otimes \\eta ^j,$ an arbitrary element in $ \\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_H(\\eta )$ , and see what are the conditions that the reproducing formula (REF ) imposes on the coefficients $a_{ij} \\in \\mathbb {C}$ .",
"We take an arbitrary element $f(\\theta ) = \\sum _k \\lambda _k \\theta ^k \\in \\mathcal {B}_H (\\theta )$ .",
"So we have the corresponding element $f(\\eta ) = \\sum _k \\lambda _k \\eta ^k \\in \\mathcal {B}_H (\\eta )$ .",
"We then calculate out the right side of equation (REF ) and get $\\langle K(\\theta , \\eta ) , f(\\eta ) \\rangle _w =\\langle \\sum _{i,j} a_{ij} \\, \\overline{\\theta }{}^i \\otimes \\eta ^j , f(\\eta ) \\rangle _w= \\sum _{i,j} a_{ij}^* \\, \\langle \\eta ^j , f(\\eta ) \\rangle _w \\, \\theta ^i\\\\= \\sum _{i,j} a_{ij}^* \\, \\langle \\eta ^j ,\\sum _k \\lambda _k \\eta ^k \\rangle _w \\, \\theta ^i= \\sum _{i,j} a_{ij}^* \\, \\sum _k \\lambda _k \\langle \\eta ^j ,\\eta ^k \\rangle _w \\, \\theta ^i\\\\= \\sum _{i,j} a_{ij}^* \\, \\sum _k \\lambda _k \\delta _{j,k} w_j \\, \\theta ^i= \\sum _{i,j} a_{ij}^* \\, \\lambda _j w_j \\, \\theta ^i= \\sum _i \\left( \\sum _j w_j a_{ij}^* \\, \\lambda _j \\right) \\theta ^i .$ (Note that the second equality is nothing other than the promised, rigorous definition of the inner product in (REF ).)",
"Now we want this to be equal to $f(\\theta ) = \\sum _i \\lambda _i \\theta ^i$ for all $f(\\theta )$ in $\\mathcal {B}_H (\\theta )$ , that is, for all vectors $\\lbrace \\lambda _i | i \\in I_l \\rbrace $ in $\\mathbb {C}^l$ .",
"So the matrix $(w_j a_{ij}^*)$ has to act as the identity on $\\mathbb {C}^l$ and thus has to be the identity matrix $(\\delta _{ij})$ , where $\\delta _{ij}$ is the Kronecker delta.",
"The upshot is that $a_{ij} = \\delta _{ij} / w_j$ does the job, and nothing else does.",
"So, substituting in equation (REF ) we see that $K(\\theta , \\eta ) = \\sum _{j} \\dfrac{1}{w_j} \\overline{\\theta }{}^j \\otimes \\eta ^j$ is the unique reproducing kernel `function.'",
"And, as one might expect, an abstract argument also shows that reproducing kernels are unique.",
"For suppose that $K_1(\\theta , \\eta ) \\in \\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_{H}(\\eta )$ is also a reproducing kernel.",
"Then the standard argument makes sense in this context, namely, $K_1(\\rho , \\eta ) = \\langle K(\\eta , \\theta ) , K_1(\\rho , \\theta ) \\rangle _w =\\langle K_1(\\rho , \\theta ) , K(\\eta , \\theta ) \\rangle _w^* \\nonumber \\\\= K(\\eta , \\rho ) ^* = K(\\rho , \\eta ),$ where $\\rho $ is another paragrassmann variable.",
"The astute reader will have realized that these innocent looking formulas require a bit of justification, including a rigorous definition of the inner product in this context.",
"We leave most of these details to the reader.",
"But, for example, in the last equality we are using the standard natural isomorphisms $\\left( \\mathcal {B}_{AH}(\\overline{\\eta }) \\otimes \\mathcal {B}_H(\\rho ) \\right)^*\\cong \\mathcal {B}_{H}(\\eta ) \\otimes \\mathcal {B}_{AH}(\\overline{\\rho })\\cong \\mathcal {B}_{AH}(\\overline{\\rho }) \\otimes \\mathcal {B}_{H}(\\eta ).$ The relation $K(\\eta , \\rho ) ^* = K(\\rho , \\eta )$ is also well known in the classical theory.",
"For example, Eq.",
"(1.9a) in [1] is an analogous result.",
"Also, by putting $K_1$ equal to $K$ in the first equality of (REF ) we get another result that is analogous to a result in the classical case.",
"This is $K(\\rho , \\eta ) = \\langle K(\\eta , \\theta ) , K(\\rho , \\theta ) \\rangle _w,$ which is usually read as saying that two `evaluations' of the reproducing kernel (with the holomorphic variable, here $\\theta $ , being the same in the two) have an inner product with respect to that holomorphic variable that is itself an `evaluation' of the reproducing kernel.",
"See [7], Theorem 2.3, part 4, for the corresponding identity in the context of holomorphic function spaces.",
"Next we put $\\eta = \\rho $ to get $K(\\rho , \\rho ) = \\langle K(\\rho , \\theta ) , K(\\rho , \\theta ) \\rangle _w = || K(\\rho , \\theta ) ||_w^2 = || K(\\rho , \\cdot ) ||_w^2.$ And again this last formula is analogous to a result in the classical theory.",
"Later on, we will discuss the positivity of $K(\\rho , \\rho )$ .",
"We note that this theory is consistent with many expectations coming from the usual theory of reproducing kernels.",
"As we have seen, $K(\\theta , \\eta )$ is the only element in the appropriate space with the reproducing property.",
"Also, we clearly have the following well known relation with the elements of the standard orthonormal basis, namely that $K(\\theta , \\eta ) = \\sum _{j} \\phi _j (\\theta )^* \\otimes \\phi _j (\\eta ).$ This follows from (REF ) and the definition of $\\phi _j$ .",
"Suppose that $\\psi _j(\\eta )$ for $j \\in I_l$ is another orthonormal basis of $\\mathcal {B}_{H}(\\eta )$ .",
"It then follows from $\\langle f^*, g^* \\rangle _w = \\langle f, g \\rangle _w^*$ for all $f,g \\in PG_{l,q}$ (which we leave to the reader as another exercise) that $\\psi _j(\\theta )^*$ is an orthonormal basis of $\\mathcal {B}_{AH}(\\theta )$ .",
"Then by a standard argument in linear algebra we obtain from (REF ) that $K(\\theta , \\eta ) = \\sum _{j} \\psi _j (\\theta )^* \\otimes \\psi _j (\\eta ).$ This formula is the analogue of an identity in the classical theory.",
"(For example see [1], Eq.",
"(1.9b) or [7], Theorem 2.4.)",
"One result from the classical theory of reproducing kernel Hilbert spaces of functions seems to have no analogue in this context.",
"That result is the point-wise estimate that one gets by applying the Cauchy-Schwarz inequality to the reproducing formula.",
"(For example, see [7], Theorem 2.3, part 5 for this result in the holomorphic function setting.)",
"What happens in the classical case is that one takes the absolute value of an expression such as $f(z)$ which is a complex number (being the value of a function $f$ at the point $z$ ) and then applies Cauchy-Schwarz to the inner product on the right side of the reproducing formula, which gives a bound by the product of two Hilbert space norms: one of the reproducing kernel with respect to its holomorphic variable and the other of $f$ .",
"So there are two semi-norms of $f$ in this formulation.",
"The first is $|f(z)|$ and the second (actually a norm) is $|| f ||$ in the appropriate Hilbert space norm.",
"But in the context of this paper the expression $f(\\theta )$ on the left side of (REF ) is an element in the Hilbert space $\\mathcal {B}_H(\\theta )$ with dimension greater than one; in particular it is not a complex number.",
"Moreover, the expression $f(\\eta )$ on the right side of (REF ) is an element in the isomorphic Hilbert space $\\mathcal {B}_H(\\eta )$ and as such is the isomorphic copy of $f(\\theta )$ in $\\mathcal {B}_H(\\theta )$ .",
"So $f(\\theta )$ and $f(\\eta )$ are essentially the same object, and it seems that we have available only one semi-norm to measure each of them, namely the norm in the corresponding Hilbert space.",
"Of course, by definition of the norms $|| f(\\theta ) ||_{\\mathcal {B}_H(\\theta )} = || f(\\eta ) ||_{\\mathcal {B}_H(\\eta )}$ holds.",
"One could weaken this to an estimate $|| f(\\theta ) ||_{\\mathcal {B}_H(\\theta )} \\le || f(\\eta ) ||_{\\mathcal {B}_H(\\eta )}$ and say that this is the point-wise estimate in this context.",
"Naturally, saying such a thing is rather ridiculous, albeit true.",
"So, while the discussion of this paragraph does not preclude the possibility of another semi-norm that gives a non-trivial `point-wise' estimate, it does indicate what is behind this issue.",
"However, I have not been able to find such a suitable semi-norm.",
"The reproducing formula in the usual theory of reproducing kernel Hilbert spaces is interpreted as saying that the Dirac delta function is realized as integration against a smooth kernel function.",
"Since the inner product in the reproducing formula (REF ) is a Berezin integral, we can say that the Dirac delta function in this context is realized as Berezin integration against the `smooth function' $K(\\theta , \\eta )$ .",
"But what is the Dirac delta function in this context?",
"We recall that $f(\\theta )$ in this paper is merely convenient notation for an element in a Hilbert space.",
"We are not evaluating $f$ at a point $\\theta $ in its domain.",
"It seems that the simplest interpretation for the Dirac delta $\\delta _{\\eta \\rightarrow \\theta }$ in the present context is that it acts on $f(\\eta )$ to produce $f(\\theta )$ , namely that it is a substitution operator.",
"We use a slightly different notation for this Dirac delta in part because it is different from the usual Dirac delta and also to distinguish it from the Kronecker delta function that we have been using.",
"An appropriate definition and notation would be $\\delta _{\\eta \\rightarrow \\theta } [ f(\\eta )] := f(\\theta )$ so that $ \\delta _{\\eta \\rightarrow \\theta } : \\mathcal {B}_H (\\eta ) \\rightarrow \\mathcal {B}_H (\\theta )$ is an isomorphism of Hilbert spaces and of algebras.",
"In particular, with this way of defining the Dirac delta we do not get a functional acting on a space of test functions.",
"However, the left side of equation (REF ) is $\\delta _{\\eta \\rightarrow \\theta } [ f(\\eta )] $ .",
"Notice that in this approach $ \\delta _{\\eta \\rightarrow \\theta } \\in \\mathrm {Hom}_{\\mathrm {Vect}_\\mathbb {C}} ( \\mathcal {B}_H (\\eta ) , \\mathcal {B}_H (\\theta ) ) \\cong \\mathcal {B}_H (\\theta ) \\otimes \\mathcal {B}_H (\\eta )^\\prime \\cong \\mathcal {B}_H (\\theta ) \\otimes \\mathcal {B}_{AH} (\\overline{\\eta })$ using standard notation from category theory and viewing the space $\\mathcal {B}_{AH} (\\overline{\\eta })$ as the dual space $ \\mathcal {B}_H (\\eta )^\\prime $ .",
"This does agree with our previous analysis where we had that $K(\\theta , \\eta ) \\in \\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_H(\\eta )$ , because the inner product in equation (REF ) is anti-linear in its first argument.",
"So that previous analysis simply identifies which element in $\\mathcal {B}_{H} (\\theta ) \\otimes \\mathcal {B}_{AH} (\\overline{\\eta }) $ is the Dirac delta, namely $\\delta _{\\eta \\rightarrow \\theta } = \\sum _{j} \\dfrac{1}{w_j} \\theta ^j \\otimes \\overline{\\eta }{}^j = \\sum _{j} \\phi _j (\\theta )\\otimes \\phi _j^* (\\overline{\\eta }),$ which is analogous to a standard formula for the Dirac delta.",
"We now try to see to what extent this generalized notion of a reproducing kernel has the positivity properties of a usual reproducing kernel.",
"First, we `evaluate' the diagonal `elements' $K(\\theta , \\theta )$ in $\\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_{H}(\\theta )\\cong \\mathcal {B}_{H}(\\theta )^\\prime \\otimes \\mathcal {B}_{H}(\\theta )$ getting $K(\\theta , \\theta ) = \\sum _j \\dfrac{1}{w_j} \\overline{\\theta }{}^j \\otimes \\theta ^j= \\sum _j \\dfrac{1}{w_j} (\\theta ^j)^* \\otimes \\theta ^j.$ This is positive element by using the usual definition of a positive element in a $*$ -algebra, provided we adequately define the $*$ -operation in $ \\mathcal {B}_{AH}(\\overline{\\theta }) \\otimes \\mathcal {B}_{H}(\\theta )\\cong \\mathcal {B}_{H}(\\theta )^\\prime \\otimes \\mathcal {B}_{H}(\\theta ) $ .",
"Given that this exercise has been done and without going into further details we merely comment that we can identify $\\mathcal {B}_{H}(\\theta )^\\prime \\otimes \\mathcal {B}_{H}(\\theta )$ with $\\mathcal {L} (\\mathcal {B}_{H}(\\theta ) )$ , the vector space (and $*$ -algebra) of all of the linear operators from the Hilbert space $\\mathcal {B}_{H}(\\theta )$ to itself.",
"Under this identification the positive elements of $\\mathcal {B}_{H}(\\theta )^\\prime \\otimes \\mathcal {B}_{H}(\\theta )$ correspond exactly to the positive operators in $\\mathcal {L} (\\mathcal {B}_{H}(\\theta ) )$ , and we have that $K(\\theta , \\theta ) =\\sum _j \\dfrac{1}{w_j} | \\theta ^j \\rangle \\langle \\theta ^j | = \\sum _j | \\phi _j (\\theta ) \\rangle \\langle \\phi _j (\\theta ) |\\in \\mathcal {L} (\\mathcal {B}_{H}(\\theta ) ),$ using the Dirac bra and ket notation.",
"But this is clearly a positive linear operator since $\\sum _j | \\phi _j (\\theta ) \\rangle \\langle \\phi _j (\\theta ) | = I_{ \\mathcal {B}_{H}(\\theta ) } \\equiv I \\ge 0 ,$ the identity operator on $\\mathcal {B}_{H}(\\theta )$ .",
"The upshot is that $ K(\\theta , \\theta ) = I$ .",
"But $|| K( \\theta , \\cdot ) ||_w^2 = K( \\theta , \\theta )$ as we showed in (REF ).",
"So, $|| K( \\theta , \\cdot ) ||_w^2 = I$ as well.",
"Next, we take a finite number of pairs of paragrassmann variables $\\theta _n, \\overline{\\theta }_n$ and a finite sequence of complex numbers $\\lambda _n$ , where $n = 1, \\dots , N$ .",
"Then we investigate the positivity of the usual expression, that is, we consider $\\sum _{n,m=1}^N \\lambda _n^* \\lambda _m K(\\theta _n , \\theta _m) =\\sum _{n,m=1}^N \\lambda _n^* \\lambda _m \\sum _{j \\in I_l} \\dfrac{1}{w_j} \\overline{\\theta }{}^j_n \\otimes \\theta _m^j\\\\= \\sum _{j \\in I_l} \\dfrac{1}{w_j} \\sum _{n,m=1}^N \\lambda _n^* \\lambda _m (\\theta _n^j)^*\\otimes \\theta _m^j=\\sum _{j \\in I_l} \\dfrac{1}{w_j} \\left( \\sum _{n=1}^N \\lambda _n \\theta _n^j \\right)^*\\otimes \\left( \\sum _{m=1}^N \\lambda _m \\theta _m^j\\right),$ which is a positive element in the appropriate $*$ -algebra and therefore also corresponds to a positive linear map.",
"A detail here is that one has to define a $*$ -algebra where the sums $\\sum _{n} \\lambda _n \\theta _n^j$ makes sense for all $j \\in I_l$ .",
"But this is a straightforward exercise left to the reader.",
"It now is natural to ask whether this procedure can be reversed, as we know is the case with the usual theory of reproducing kernel functions.",
"That is to say, can we start with a mathematical object, call it $K$ , that has the properties (in particular, the positivity) of a reproducing kernel in this context and produce from it a reproducing kernel Hilbert space that has that given object $K$ as its reproducing kernel?",
"This seems not to be possible, at least not using an argument based on the identity (REF ) as is done in the classical case.",
"It turns out that (REF ) is only analogous to the identity in the classical case, since it says something decidedly different given that the left side of it is not a complex number.",
"In the classical theory the operation of evaluation at a point gives a complex number.",
"But in this context the evaluation of a function at a variable in an algebra gives another element in that same algebra.",
"In the argument in the classical case, one uses the analogue of (REF ) to define an inner product (on a set of functions) having available only the candidate mathematical object $K$ (in that case a function of two variables).",
"But one can not use (REF ) directly to define in this context a complex valued inner product.",
"Perhaps an inverse procedure can be found, but it will have to differ somewhat from the procedure in the classical case.",
"We now come back to the question of finding an analogy to a point-wise bound for $f (\\theta ) \\in \\mathcal {B}_H(\\theta )$ .",
"Suppose that $f (\\theta ) \\ne 0$ and put $u_0 := || f(\\theta ) ||_w^{-1} f(\\theta )$ , a unit vector in $ \\mathcal {B}_H(\\theta )$ .",
"Extend this to an orthonormal basis $u_j$ of $ \\mathcal {B}_H(\\theta )$ for $j \\in I_l$ .",
"So we get the operator inequality $ | u_0 \\rangle \\langle u_0 | \\le \\sum _{j} | u_j \\rangle \\langle u_j | = I_{\\mathcal {B}_H(\\theta )}$ .",
"Then we obtain $| f(\\theta ) \\rangle \\langle f(\\theta ) | = || f(\\theta ) ||_w^2 \\, | u_0 \\rangle \\langle u_0 |\\le || f(\\theta ) ||_w^2 \\, I_{\\mathcal {B}_H(\\theta )} =|| f(\\theta ) ||_w^2 \\, || K( \\theta , \\cdot ) ||_w^2$ which is an operator inequality involving positive operators.",
"For $f(\\theta ) = 0$ this inequality is trivially true (and is actually an equality).",
"The point here is that (REF ) has some resemblance to the point-wise estimate in the classical case.",
"(See [7].)",
"The left side of (REF ) can be considered a type of `outer product' of $f(\\theta )$ with itself.",
"An entirely analogous argument shows that $\\mathcal {B}_{AH}$ is a reproducing kernel Hilbert space with reproducing kernel $K_{AH}$ given by $K_{AH}(\\theta , \\eta ) = K(\\theta , \\eta )^* = \\sum _{j} \\phi _j (\\theta ) \\otimes \\phi _j (\\eta )^* =\\sum _{j} \\dfrac{1}{w_j} \\theta ^j \\otimes \\overline{\\eta }{}^j.$ The theory for this reproducing kernel is really the same as the material presented in this section.",
"Finally, we note that the results of this section depend on the weight $w(\\theta , \\overline{\\theta })$ and are independent of the value of the parameter $q \\in \\mathbb {C} \\setminus \\lbrace 0 \\rbrace $ ."
],
[
"Coherent States and the Segal-Bargmann Transform",
"We now introduce the coherent state quantization of Gazeau.",
"(See [5] for example.)",
"We let $\\mathcal {H}$ be any complex Hilbert space of dimension $l$ and choose any orthonormal basis of $\\mathcal {H}$ , which we will denote as $e_n$ for $n \\in I_l$ .",
"While $\\theta $ is a complex variable, it does not `run' over a domain of values, say in some phase space.",
"So the coherent state $| \\theta \\rangle $ we are about to define is one object, not a parameterized family of objects.",
"Actually, we define two coherent states corresponding to the variable $\\theta $ : $| \\theta \\rangle := \\sum _{n \\in I_l} \\phi _n (\\theta ) \\otimes e_n \\in \\mathcal {B}_{H} (\\theta )\\otimes \\mathcal {H},$ $\\langle \\theta | := \\sum _{n \\in I_l} \\phi _n^* (\\overline{\\theta }) \\otimes e_n^\\prime \\in \\mathcal {B}_{AH} (\\theta )\\otimes \\mathcal {H}^\\prime .$ Here $ \\mathcal {H}^\\prime $ denotes the dual space of all linear functionals on $ \\mathcal {H}$ , and $e_n^\\prime $ is its orthonormal basis that is dual to $e_n$ , where $n \\in I_l$ .",
"We refrain from using the super-script $*$ for the dual objects in order to avoid confusion in general with the conjugation.",
"We are following Gazeau's conventions here.",
"(See [5].)",
"As noted in [2] these objects give a resolution of the identity using the Berezin integration theory, and so this justifies calling them coherent states.",
"Now we have $\\langle \\, \\theta \\, | \\, \\eta \\, \\rangle = \\sum _{j,k} \\phi _j^* (\\overline{\\theta }) \\otimes \\phi _k (\\eta ) \\,\\langle e_j^\\prime , e_k \\rangle =\\sum _{j,k} \\phi _j^* (\\overline{\\theta }) \\otimes \\phi _k (\\eta ) \\, \\delta _{j,k}\\\\= \\sum _{j} \\phi _j^* (\\overline{\\theta }) \\otimes \\phi _j (\\eta ) = K(\\theta ,\\eta ).$ Here the inner product (or pairing) of the coherent states $\\langle \\, \\theta \\, | $ and $| \\, \\eta \\, \\rangle $ is simply defined by the first equality, which is a quite natural definition.",
"So the inner product of two coherent states gives us the reproducing kernel.",
"This is a result that already appears in the classical theory.",
"(See [1], Eq.",
"(1.9a) for a formula of this type.)",
"Since we have coherent states, we can define the corresponding Segal-Bargmann (or coherent state) transform in the usual way.",
"Essentially the same transform was discussed in [2], where it is denoted by $\\mathcal {W}$ .",
"This will be a unitary isomorphism $C: \\mathcal {H} \\rightarrow \\mathcal {B}_{H}(\\theta )$ .",
"(Recall that we are considering an abstract Hilbert space $\\mathcal {H}$ of dimension $l$ with orthonormal basis $e_n$ .)",
"We define $C$ for all $\\psi \\in \\mathcal {H}$ by $C \\psi (\\theta ) := \\langle \\, \\langle \\theta |, \\psi \\rangle .$ Note that $ \\langle \\theta | \\in \\mathcal {B}_{AH}(\\theta ) \\otimes \\mathcal {H}^\\prime $ .",
"So the outer bracket $\\langle \\cdot , \\cdot \\rangle $ here refers to the pairing of the dual space $\\mathcal {H}^\\prime $ with $\\mathcal {H}$ .",
"As noted earlier $ \\langle \\theta |$ is one object, not a parameterized family of objects.",
"So $ C \\psi (\\theta )$ is one element in the Hilbert space $\\mathcal {B}_{H}(\\theta )$ .",
"And as usual the $\\theta $ in the notation is a convenience for reminding us what the variable is, and it is not a point at which we are evaluating $C \\psi $ .",
"Actually, in this context where it is understood that the variable under consideration is $\\theta $ , $ C \\psi (\\theta )$ and $ C \\psi $ are two notations for one and the same object.",
"Substituting the definition for $\\langle \\theta |$ gives $C\\psi (\\theta ) = \\langle \\, \\langle \\theta |, \\psi \\rangle = \\left\\langle \\sum _{n \\in I_l} \\phi _n^* (\\overline{\\theta }) \\otimes e_n^\\prime , \\psi \\right\\rangle \\\\= \\sum _{n \\in I_l} \\langle \\phi _n^* (\\overline{\\theta }) \\otimes e_n^\\prime , \\psi \\rangle = \\sum _{n \\in I_l} \\langle e_n^\\prime , \\psi \\rangle \\, \\phi _n (\\theta ) .$ So by taking $\\psi $ to be $e_j$ we see that the Segal-Bargmann transform $C: \\mathcal {H} \\rightarrow \\mathcal {B}$ is the (unique) linear transformation mapping the orthonormal basis $e_j$ to the orthonormal basis $\\phi _j(\\theta )$ .",
"This shows that $C$ is indeed a unitary isomorphism.",
"Also the above shows that this is an `integral kernel' operator with kernel given by the coherent state $ \\langle \\theta | = \\sum _j \\phi _j^*(\\overline{\\theta }) \\otimes e_j^\\prime $ .",
"This is also a well-known relation in the Segal-Bargmann theory.",
"Equation (2.10b) in [1] is this sort of formula.",
"The Segal-Bargmann transform of the coherent state $| \\, \\eta \\, \\rangle $ is given by $C | \\, \\eta \\, \\rangle (\\theta ) = \\langle \\, \\langle \\theta |, | \\, \\eta \\, \\rangle \\rangle =\\langle \\theta \\, | \\, \\eta \\, \\rangle = K(\\theta , \\eta ).$ So, this says that the Segal-Bargmann transform of a coherent state is the reproducing kernel.",
"This is also a type of relation known in the classical context of [1], where it appears as Eq.",
"(2.8).",
"There are also two coherent states corresponding to the other variable $\\overline{\\theta }$ , which also give a resolution of the identity.",
"And $\\langle \\, \\overline{\\theta } \\, | \\, \\overline{\\eta } \\, \\rangle $ gives the reproducing kernel for the anti-holomorphic Segal-Bargmann space, and so forth."
],
[
"Reproducing kernel for the Paragrassmann space",
"Consider $PG_{l,q}(\\theta , \\overline{\\theta })$ with $l=2$ , a `fermionic' case.",
"This is a non-commutative algebra of dimension 4.",
"The basis $AW$ is $\\lbrace 1, \\theta , \\overline{\\theta } , \\theta \\overline{\\theta } \\rbrace $ .",
"The weight `function' in this case is $w(\\theta , \\overline{\\theta }) = w_1 + w_0 \\theta \\overline{\\theta }.$ (We are using our convention for the sub-indices.",
"See equation (REF ).)",
"So if this space has a reproducing kernel $K_{PG} (\\theta , \\overline{\\theta }, \\eta , \\overline{\\eta } )$ , it must satisfy $f (\\theta , \\overline{\\theta } ) = \\langle K_{PG} (\\theta , \\overline{\\theta }, \\eta , \\overline{\\eta } ) , f (\\eta , \\overline{\\eta } ) \\rangle _w$ for all $f = f (\\theta , \\overline{\\theta } ) \\in PG_{2,q}$ .",
"(Notice that here, and throughout this section, we use a functional calculus of a pair of non-commuting variables in place of evaluation at a point.",
"Actually, we can define a functional calculus for all $f \\in \\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace $ .)",
"This last equation in turn is equivalent to satisfying $\\theta ^i \\overline{\\theta }{}^j =\\langle K_{PG} (\\theta , \\overline{\\theta }, \\eta , \\overline{\\eta } ) , \\eta ^i \\overline{\\eta }^j \\rangle _w$ for all $i,j \\in I_2 = \\lbrace 0, 1 \\rbrace $ .",
"Substituting the general element $K_{PG} (\\theta , \\overline{\\theta }, \\eta , \\overline{\\eta } ) = \\sum _{abcd} k_{abcd} \\theta ^a \\overline{\\theta }{}^b \\otimes \\eta ^c \\overline{\\eta }{}^d$ (which lies in a space of dimension $2^4$ ) into the previous equation gives $\\theta ^i \\overline{\\theta }{}^j =\\sum _{abcd} k_{abcd}^* \\theta ^b \\overline{\\theta }{}^a \\langle \\eta ^c \\overline{\\eta }{}^d , \\eta ^i \\overline{\\eta }^j \\rangle _w=\\sum _{ab} \\left( \\sum _{cd} k_{abcd}^* G_{cdij} \\right) \\theta ^b \\overline{\\theta }{}^a ,$ where $ G_{cdij} = \\langle \\eta ^c \\overline{\\eta }{}^d , \\eta ^i \\overline{\\eta }^j \\rangle _w$ .",
"So the unknown coefficients $ k_{abcd} \\in \\mathbb {C}$ must satisfy $\\sum _{cd} k_{abcd}^* G_{cdij} = \\delta _{(b,a),(i,j)},$ where this Kronecker delta is 1 if the ordered pair $(b,a)$ is equal to the ordered pair $(i,j)$ and otherwise is 0.",
"So everything comes down to showing the invertibility of the matrix $ G = ( G_{cdij} )$ , whose rows (resp., columns) are labelled by $\\eta ^c \\overline{\\eta }{}^d$ (resp., $\\eta ^i \\overline{\\eta }^j \\rangle _w$ ) where the ordered pairs $(c,d), (i,j) \\in I_2 \\times I_2$ .",
"So $G$ is a $4 \\times 4$ matrix.",
"To calculate it we note that the pertinent identities are as follows: $\\langle 1 ,1\\rangle _w = w_0,\\\\\\langle 1 , \\eta \\overline{\\eta } \\rangle _w = \\langle \\eta \\overline{\\eta } , 1\\rangle _w = w_1,\\\\\\langle \\eta , \\eta \\rangle _w = \\langle \\overline{\\eta } , \\overline{\\eta } \\rangle _w = w_1,\\\\\\langle \\eta \\overline{\\eta } , \\eta \\overline{\\eta } \\rangle _w = 0.$ All other inner products of pairs of elements in $AW$ are zero.",
"The matrix $G$ we are dealing with here in the case $l=2$ is $G = \\left(\\begin{array}{cccc}w_0 & 0 & 0 & w_1 \\\\0 & w_1 & 0 & 0 \\\\0 & 0 & w_1 & 0 \\\\w_1 & 0 & 0 & 0\\end{array}\\right)$ with respect to the ordered basis $\\lbrace 1 , \\eta , \\overline{\\eta }, \\eta \\overline{\\eta } \\rbrace $ .",
"Then $\\det G = - (w_1)^4 \\ne 0$ and so $G^{-1} = \\left(\\begin{array}{cccc}0 & 0 & 0 & 1/w_1 \\\\0 & 1/w_1 & 0 & 0 \\\\0 & 0 & 1/w_1 & 0 \\\\1/w_1 & 0 & 0 & -w_0/w_1^2\\end{array}\\right)$ by using standard linear algebra.",
"It follows that the (unique!)",
"reproducing kernel for $PG_{2,q}$ is given by $K_{PG} (\\theta , \\overline{\\theta }, \\eta , \\overline{\\eta } ) = \\dfrac{1}{w_1} \\theta \\overline{\\theta } \\otimes 1+ \\dfrac{1}{w_1} \\overline{\\theta } \\otimes \\eta + \\dfrac{1}{w_1} \\theta \\otimes \\overline{\\eta } +\\dfrac{1}{w_1} 1 \\otimes \\eta \\overline{\\eta }- \\dfrac{w_0}{w_1^2} \\theta \\overline{\\theta } \\otimes \\eta \\overline{\\eta } .$ (This is also works when $w_0 \\le 0$ or $w_1 < 0$ .)",
"Even though the reproducing kernel in this example lies in a space of dimension 16, only 5 terms in the standard basis have non-zero coefficients.",
"Actually, the method in the previous paragraph is the systematic way to arrive at a formula for the reproducing kernel for $PG_{l,q}$ in general.",
"Everything comes down to showing the invertibility of the matrix $G$ , where $G_{cdij} = \\langle \\eta ^c \\overline{\\eta }{}^d , \\eta ^i \\overline{\\eta }^j \\rangle _w$ for $(c,d),(i, j) \\in I_l \\times I_l$ and then finding the inverse matrix.",
"Again, we label the rows and columns of $G$ by the elements in $AW$ .",
"As is well known, invertibility is a generic property of $G$ (that is, true for an open, dense set of matrices $G$ ).",
"So there are many, many examples of sesquilinear forms (including positive definite inner products) defined on the non-commutative algebra $PG_{l,q}$ , making it into a reproducing kernel space.",
"Then it becomes clear that it is straightforward to give any finite dimensional algebra $\\mathcal {A}$ , commutative or not (but such that every element in $\\mathcal {A}$ is in the image of some functional calculus), an inner product so that $\\mathcal {A}$ has a reproducing kernel.",
"The infinite dimensional case will require more care due to the usual technical details.",
"But is the matrix $G$ associated to the sesquilinear form (REF ) invertible?",
"Theorem 7.1 Taking $G$ to be the matrix (REF ) associated to the sesquilinear form defined by (REF ), we have that $\\det G = \\pm (w_{l-1})^{l^2} \\ne 0$ for every $l \\ge 2$ .",
"Proof: We will argue by induction on $l$ for $l \\ge 4$ .",
"We have shown above that $\\det G = - (w_1)^4 = - (w_1)^{2^2} \\ne 0$ when $l=2$ .",
"So we must establish this result for $l=3$ as well.",
"We claim in that case that $\\det G = (w_2)^9 \\ne 0$ .",
"First we calculate the matrix entries of $G$ , which is a $9 \\times 9$ matrix, and get $G = \\left( \\begin{array}{ccccccccc}w_0 & 0 & 0 & w_1 & 0 & 0 & 0 & 0 & w_2 \\\\0 & w_1 & 0 & 0 & 0 & 0 & 0 & w_2 & 0 \\\\0 & 0 & w_1 & 0 & 0 & 0 & w_2 & 0 & 0 \\\\w_1 & 0 & 0 & w_2 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0& w_2 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & w_2 & 0& 0 & 0 & 0 \\\\0 & 0 & w_2 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & w_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\w_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\\end{array}\\right)$ with respect to the ordered basis $\\lbrace 1, \\theta , \\overline{\\theta }, \\theta \\overline{\\theta } , \\theta ^2, \\overline{\\theta }{}^2,\\theta \\overline{\\theta }{}^2 ,\\theta ^2 \\overline{\\theta } ,\\theta ^2 \\overline{\\theta }{}^2 \\rbrace $ .",
"Now the last 5 columns have all entries equal to zero, except for one entry equal to $w_2$ .",
"Similarly, the last 5 rows have all entries equal to zero, except for one entry equal to $w_2$ .",
"We calculate the determinant by expanding successively along each of the 5 last columns, thereby obtaining 5 factors of $w_2$ and a sign (either plus or minus).",
"With each expansion the corresponding row is also eliminated and this will eliminate the first 2 of the last 5 rows, but leave the remaining 3 rows.",
"So we now expand along these remaining last 3 rows, getting 3 more factors of $w_2$ as well as a sign.",
"We then have remaining a $1 \\times 1$ matrix whose entry comes from the 4th row and 4th column of the above matrix.",
"And that entry is again $w_2$ .",
"So the determinant of $G$ in the case $l=3$ is a sign times 9 factors of $w_2$ .",
"We leave it to the reader to check that the overall sign is positive and so we get $ \\det G = (w_2)^9 = (w_2)^{3^2}$ as claimed.",
"Now we assume $l \\ge 4$ and prove this case by induction.",
"The induction hypothesis that we will use is that in the case for $l-2$ the matrix $G$ (which is an $(l-2)^2 \\times (l-2)^2$ matrix) has determinant $\\pm (w_{l-3})^{(l-2)^2}$ .",
"This is why we started this argument by proving separately the cases $l=2$ and $l=3$ .",
"We start by using the same argument of expansion of the determinant as used above when $l=3$ .",
"The matrix $G$ is an $l^2 \\times l^2$ matrix.",
"We consider this matrix in a basis made by ordering the basis $AW$ in such a way that the last elements are all of the form $\\theta ^{l-1} \\overline{\\theta }{}^k$ or of the form $\\theta ^{k} \\overline{\\theta }{}^{l-1}$ , where $k \\in I_l$ .",
"Notice that the element $\\theta ^{l-1} \\overline{\\theta }{}^{l-1}$ is the only basis element in $AW$ that has both of these forms.",
"So, there are $2 \\, \\mathrm {card} (I_l) -1 = 2l -1$ such elements.",
"We claim that each of these elements has exactly one non-zero inner product with the elements in the ordered basis $AW$ and that the value of that inner product is $w_{l-1}$ .",
"Starting with $\\theta ^{l-1} \\overline{\\theta }{}^k$ we note that $\\langle \\theta ^i \\overline{\\theta }{}^j, \\theta ^{l-1} \\overline{\\theta }{}^k \\rangle _w = \\delta _{i+k, j+l-1} w_{j+l-1} \\chi _l (j+l-1).$ This is zero if $j > 0$ because of the $\\chi _l$ factor.",
"But for $j = 0$ we have $\\langle \\theta ^i , \\theta ^{l-1} \\overline{\\theta }{}^k \\rangle _w = \\delta _{i+k, l-1} w_{l-1}$ which is only non-zero for $i = l -1 - k$ , in which case we have $\\langle \\theta ^{l-1-k} , \\theta ^{l-1} \\overline{\\theta }{}^k \\rangle _w = w_{l-1}.$ A similar calculation shows that $\\langle \\overline{\\theta }{}^{l-1-k} , \\theta ^{k} \\overline{\\theta }{}^{l-1} \\rangle _w = w_{l-1},$ while all other elements in $AW$ have zero inner product with $ \\theta ^{k} \\overline{\\theta }{}^{l-1}$ .",
"So we expand the determinant of $G$ along the last $2l-1$ columns, obtaining $2l-1$ factors of $w_{l-1}$ and some sign, either plus or minus.",
"The corresponding rows that are eliminated, according to the above, are labelled by all the powers of $\\theta $ alone or the powers of $\\overline{\\theta }$ alone.",
"Of these powers, only the two powers $\\theta ^{l-1}$ and $\\overline{\\theta }{}^{l-1}$ label one of the last $2l - 1$ rows.",
"So we proceed by expanding along the remaining $2l -3$ last rows, thereby obtaining $2l-3$ more factors of $w_{l-1}$ and some sign.",
"These $(2l -1) + (2l -3) = 4l -4 $ factors of $w_{l-1}$ as well as the sign multiply the determinant of a square matrix which has $ l^2 - ( 4l -4 ) = (l-2)^2 $ rows and the same number of columns.",
"Now, we claim that we can calculate the determinant of this remaining $ (l-2)^2 \\times (l-2)^2 $ matrix, call it $M$ , using the induction hypothesis.",
"However, $M$ is not the matrix $G$ for the case $l-2$ , but is related to it as we shall see.",
"From the labeling of the rows and columns of $G$ for the case $l$ , the matrix $M$ inherits a labeling, namely its rows and columns are labeled by the basis elements $ \\theta ^i \\overline{\\theta }{}^j$ of $AW$ for $ 1 \\le i,j \\le l-2$ .",
"This is clear by recalling the labeling of the columns and rows which were eliminated in the above expansions.",
"But for any $a,b,c,d \\in I_l$ we have that the entries of $G$ are $\\langle \\theta ^a \\overline{\\theta }{}^b , \\theta ^c \\overline{\\theta }{}^d \\rangle _w = \\delta _{a+d, b+c} w_{a+d} \\chi _l (a+d).$ In particular, this holds for $a,b,c,d \\in I_l \\setminus \\lbrace 0, l-1 \\rbrace $ in which case we can write $\\langle \\theta ^a \\overline{\\theta }{}^b , \\theta ^c \\overline{\\theta }{}^d \\rangle _w = \\delta _{a+d-2, b+c-2} w_{a+d} \\chi _{l-2} (a+d-2),$ and so these are the entries in the matrix $M$ .",
"By changing to new variables $a^\\prime , b^\\prime , c^\\prime , d^\\prime \\in I_{l-2}$ where $a^\\prime = a-1 , b^\\prime = b-1, c^\\prime = c-1 , d^\\prime = d-1$ we see that the matrix entries of $M$ are $\\delta _{a^\\prime +d^\\prime , b^\\prime +c^\\prime } w_{a^\\prime +d^\\prime +2} \\chi _{l-2} (a^\\prime +d^\\prime ).$ But the entries for the matrix $G$ in the case $l-2$ are $\\langle \\theta ^{a^\\prime } \\overline{\\theta }{}^{b^\\prime } , \\theta ^{c^\\prime } \\overline{\\theta }{}^{d^\\prime } \\rangle _w =\\delta _{a^\\prime +d^\\prime , b^\\prime +c^\\prime } w_{a^\\prime +d^\\prime } \\chi _{l-2} (a^\\prime +d^\\prime )$ for all $a^\\prime , b^\\prime , c^\\prime , d^\\prime \\in I_{l-2}$ .",
"So, except for a shift by $+2$ in the sub-indices of the weights, the entries in $M$ correspond to the entries in $G$ for the case $l-2$ .",
"Consequently, by the induction hypothesis as stated above, we have that $ \\det M = \\pm (w_{(l-3)+2})^{(l-2)^2} = \\pm (w_{l-1})^{(l-2)^2}$ .",
"Putting all this together, we have that $\\det G = \\pm (w_{l-1})^{(4l -4)} \\det M = \\pm (w_{l-1})^{(4l -4)} (w_{l-1})^{(l-2)^2} = \\pm (w_{l-1})^{l^2}$ which proves our result.",
"$\\quad \\blacksquare $ We leave it to the interested reader to track down the correct sign in the previous result.",
"We also note that $w_{l-1}$ is the coefficient of 1 in the definition of the weight in (REF ), according to our convention.",
"Since $w_{l-1} \\ne 0$ , the main result of this section now follows immediately.",
"Here it is: Theorem 7.2 For all $l \\ge 2$ and for all $q \\in \\mathbb {R} \\setminus \\lbrace 0 \\rbrace $ the paragrassmann space $PG_{l,q}$ has a unique reproducing kernel with respect to the inner product $\\langle \\cdot , \\cdot \\rangle _w$ defined in equation (REF ).",
"Remark: $PG_{l,q}$ is a non-commutative algebra and so it is a quantum space in the broadest interpretation of that terminology, that is, it is not isomorphic to an algebra of functions defined on some set (`classical' space).",
"Also it is not even a Hilbert space with respect to the sesquilinear form that we are using on it.",
"Nonetheless, contrary to what one might expect from studying the theory of reproducing kernel Hilbert spaces, this space does have a reproducing kernel.",
"Also, we would like to comment that the existence of the reproducing kernel for $PG_{l,q}$ is a consequence of the definition of the sesquilinear form and is not dependent on the parameter $q$ ."
],
[
"Concluding Remarks",
"In this paper we have introduced ideas from the analysis of reproducing kernel Hilbert spaces of functions to the study of the non-commutative space $PG_{l,q}$ of paragrassmann variables.",
"We are rather confident that other ideas from analysis will find application to $PG_{l,q}$ , and this will be one direction for future research.",
"Even more important could be the application of these ideas from analysis to other classes of non-commutative spaces.",
"We expect that there could be many such applications.",
"Another possible direction for generalizing these results is to eliminate the nilpotency condition and simply work in the infinite dimensional algebra $\\mathbb {C} \\lbrace \\theta , \\overline{\\theta } \\rbrace / \\langle \\theta \\overline{\\theta } - q \\overline{\\theta } \\theta \\rangle $ , which is also known as the quantum plane.",
"Greater care must be taken in this case and possibly some of our results will not hold in complete generality.",
"The reproducing kernel for the Segal-Bargmann space $\\mathcal {B}_H$ will be used in a subsequent paper [8] to define Toeplitz operators in that space with symbols in the space $PG_{q,l}$ .",
"Acknowledgments This paper was inspired by a talk based on [2] given by Jean-Pierre Gazeau during my sabbatical stay at the Laboratoire APC, Université Paris Diderot (Paris 7) in the spring of 2011.",
"Jean-Pierre was my academic host for that stay, and so I thank him not only for stirring my curiosity in this subject but also for his most kind hospitality which was, as the saying goes, above and beyond the call of duty.",
"(I won't go into the details, but it really was.",
"Way beyond, actually.)",
"Merci beaucoup, Jean-Pierre!",
"Also my thanks go to Rodrigo Fresneda for very useful comments as well as for being my most gracious host at the UFABC in São Paulo, Brazil in April, 2012 where work on this paper continued.",
"Muito obrigado, Rodrigo!"
]
] | 1204.1033 |
[
[
"Can type II Semi-local cosmic strings form?"
],
[
"Abstract We present the simplest possible model for a semi-local string defect in which a U(1) gauged subgroup of an otherwise global SU(2) is broken to produce local cosmic strings endowed with current-carrying properties.",
"Restricting attention to type II vortices for which the non current-carrying state is unstable, we show that a condensate must form microscopically and macroscopically evolve towards a chiral configuration.",
"It has been suggested that such configurations could potentially exist in a stable state, thereby inducing large cosmological consequences based on equilibrium angular momentum supported loop configurations (vortons).",
"Here we show that the current itself induces a macroscopic (longitudinal) instability: we conclude that type II semi-local cosmic strings cannot form in a cosmological context."
],
[
"Introduction",
"Cosmic strings have long lost their status of plausible competitors to the inflation paradigm [1].",
"However, from the point of view of particle physics and high energy models thereof, the opposite should be true: even though it is not immediately obvious to build consistent models of inflation based on the most natural extensions of the standard model such as supersymmetric Grand Unified Theories (GUT) or strings, those naturally predict vortex-like objects, i.e.",
"linear topological defects [2] (see however Ref.",
"[3]).",
"Thus, constraints provided by cosmic string network simulations are very much still of current interest, would it be only to understand why and how one can construct an inflation model without strings.",
"Assuming strings to form however is not yet the end of the story.",
"In practice, most research has been made under the assumption that the vortices were not endowed with any particular structure, and hence that the spacelike two-dimensional worldsheet they described was well modeled by a Nambu-Goto Lorentz invariant action, i.e.",
"the area spanned by the worldsheet.",
"That such a model attracted attention makes full sense since it turns out that any more complicated model would be essentially intractable by means of the currently available technology.",
"Besides, it was also shown that any Lorentz symmetry-breaking current on the vortices could lead to centrifugally-supported equilibrium states, dubbed vortons [4], [5], whose existence merely rules out the string scenario altogether [6], provided they are sufficiently long-lived.",
"Structureless Nambu-Goto strings, on the other hand, are very difficult to produce in almost any reasonable high energy theory.",
"Indeed, and unless one assumes a special sector put by hand to generate the strings themselves, which comes very short of the original idea to describe the high energy phenomena in a unified and consistent way, the string-forming Higgs field present in most GUT model must couple to scalars, fermions or gauge fields in such a way as to produce currents.",
"Even the cosmic strings present in the superstring framework do not escape this conclusion, as they must couple to moduli, at least the volume of the compact extra dimensions.",
"Thus, one expects cosmic strings to be of the current-carrying kind, as originally introduced by Witten in 1985 [7].",
"Many models have since been discussed and investigated by numerous authors, with the general conclusion that the equation of state of the strings is highly non trivial, with specific properties such as the existence of a maximal spacelike current, a phase frequency threshold for timelike current above which there is no bound state anymore, and the possibility, in all known models, to build a lightlike current which ought to be absolutely stable, thus enhancing the vorton excess problem [8], [9], [10].",
"Solutions have been proposed, most of them based on the instabilities of current-carrying loop configurations that would dissipate most of the large loops before they have time to evolve into cosmologically dangerous vortons.",
"The present work, although not directly concerned with this problem, suggests yet another possibility, namely that the current could form directly in a configuration that would be unstable with respect to longitudinal (soundlike along the string) modes.",
"Our model can be seen as the next-to-simple one after the neutral Witten bosonic model, consisting of a global U(1) condensate in a local U(1) vortex.",
"Here we still assume the vortex to be produced by a gauged U(1) symmetry breaking, but instead of adding extra symmetries, we embed this local U(1) into an otherwise global SU(2).",
"Non-current carrying strings in this model have been investigated in [11], [12], [13], [14], while the current-carrying case has been discussed in [15], [16].",
"This is merely the limit of the usual would-be semi-local strings found in the standard electroweak model; except that the measured parameters of this model preclude their actual stability.",
"In fact, the stability of non-current carrying semi-local strings does not follow from the topology of the vacuum manifold (as it does for the U(1) case), but from dynamical arguments.",
"The ratio between the gauge and Higgs boson masses governs the stability of semi-local strings: for Higgs boson mass larger (smaller) than the gauge boson mass semi-local strings are unstable (stable) and in the BPS limit a degenerate one-parameter family of stable solutions exists [12].",
"The parameter corresponds roughly to the width of the strings and as such semi-local strings of arbitrary width have the same energy in the BPS limit.",
"Whenever this zero mode gets excited it leads to the growth of the string core [17].",
"As such these non-current carrying semi-local strings have been studied in the context of cosmological applications regarding the formation and evolution of string networks [18], [19], [20], [21] as well as implications for the CMB [22].",
"The stability of the current-carrying counterparts has been discussed in [23] using linear perturbation theory; there it was also found that these embedded type II vortices have a single unstable mode, and so it has been suggested that the current-carrying ones, being less energetic, could be stable.",
"We show that this is not the case because some other instability develops.",
"In a sense, the category of this model is more natural than the Witten-kind of models because one expects a large GUT group to be partially broken to yield the low energy particle physics currently tested at the LHC, so the strings, if present, once formed, are expected to be embedded in a larger structure.",
"It is obviously mostly a parameter dependent question to know whether the strings here described will form rather than the Witten kind of strings.",
"Finally, such a model permits to embed a cosmic string in a non abelian framework in a tractable way, contrary to what happens in the case of a pure non abelian current-carrying situation [24], [25].",
"As already mentioned above, if the ratio between the Higgs and gauge boson masses is large, the corresponding type II vortices are unstable.",
"In Ref.",
"[15] and [16], it was shown that a current could build along such vortices, and that the resulting current-carrying state was less energetic than the structureless one.",
"A stability analysis [23] then showed that even though long wavelength perturbations tend to grow exponentially, there was a limit below which the current-carrying string state could be stable; this could imply important cosmological consequences whenever small loops form.",
"The purpose of the current article is to close this window of stability by performing a global analysis showing the current-carrying configurations will also develop a short wavelength instability, the so-called longitudinal instability introduced by Carter [26], [27], [28].",
"The paper is organized as follows: in the following section , we set up the actual model and discuss the stringlike solutions that can be expected.",
"We then move on, in Sec.",
"to evaluating the currents that could condense in a string core, summarizing a stability analysis first discussed in [12].",
"These currents are examined thoroughly in Sec.",
"REF and it is shown that the lightlike current limit is defined as the endpoint of the state parameter space in this case, with the phase frequency threshold being at the null point.",
"Finally, Sec.",
"shows that the corresponding equation of state leads to the longitudinal loop instabilities: right after a condensate has formed, it should evolve towards the chiral limit [29], thereby destroying many would-be vortons [30] through emission of high energy particles [31], [32].",
"We conclude that type II vortices cannot form at all in such models."
],
[
"Partly gauged SU(2) string model",
"The simplest embedded current-carrying string model is provided by the partly ungauged version of the electroweak theory in which the SU(2) coupling constant is made to vanish, while the equivalent to electromagnetism U(1) remains gauged.",
"In practice, this amounts to starting with the following Lagrangian $\\mathcal {L}= -g^{\\mu \\nu } (D_\\mu \\mathbf {\\Phi })^\\dagger \\cdot D_\\nu \\mathbf {\\Phi }-\\frac{1}{4} F_{\\mu \\nu } F^{\\mu \\nu } - V(\\mathbf {\\Phi }),$ where the U(1) covariant derivative acting on the SU(2) Higgs doublet $\\mathbf {\\Phi }$ is $D_\\mu \\mathbf {\\Phi } \\equiv \\left( \\partial _\\mu - i e A_\\mu \\right) \\mathbf {\\Phi }$ , $F_{\\mu \\nu } \\equiv \\partial _{\\mu }A_{\\nu } -\\partial _{\\nu }A_{\\mu }$ is the Faraday tensor of the U(1) gauge field, and finally the scalar field potential $V$ is taken to be of the symmetry-breaking kind $V(\\mathbf {\\Phi }) = \\frac{\\lambda }{2} \\left( \\mathbf {\\Phi }^\\dagger \\cdot \\mathbf {\\Phi }-\\eta ^2 \\right)^2,$ so the self coupling $\\lambda $ combines with the vacuum expectation value (vev) $\\eta $ of $\\mathbf {\\Phi }$ to provide the scalar field excitation mass as $m_\\phi = \\sqrt{2\\lambda } \\eta $ .",
"The vector field also acquires a mass $m_A = \\sqrt{2} e\\eta $ , and the mass ratio is thus defined as $2 \\beta \\equiv m_\\phi ^2 / m_A = \\lambda /e^2$ .",
"(Note our definition of $\\beta $ differs by a factor of 2 with that of Ref. [15].)",
"The lowest energy configuration, having $\\mathbf {\\Phi }^\\dagger \\cdot \\mathbf {\\Phi } = \\eta ^2$ admits vortex defects of the local U(1) kind: fixing the SU(2) gauge in which $\\mathbf {\\Phi }_0 = \\begin{pmatrix} \\Phi _0 \\\\ 0\\end{pmatrix},$ there remains a local U(1) gauge to be fixed through the phase of $\\Phi _0$ ; if it takes the form of a non vanishing winding, i.e.",
"if $\\Phi _0 \\propto \\mathrm {e}^{in\\theta }$ with index $n\\in \\mathbb {Z}\\ne 0$ and $\\theta $ a local coordinate angle, then $\\Phi _0\\rightarrow 0$ defines a string around which the phase winds.",
"One can then locally set the string to be aligned along a $z-$ axis around which one defines the cylindrical coordinates $r$ and $\\theta $ , and the non vanishing component of the Higgs field becomes $\\Phi _0 = \\varphi (r)\\mathrm {e}^{in\\theta }$ , where $\\lim _{r\\rightarrow \\infty } \\varphi (r)=\\eta $ and $\\varphi (0)=0$ .",
"The question then arises as to the actual stability of the above configuration.",
"An analysis similar to that in [7] is carried out below showing that one does indeed expect a current of the kind we discussed in the following sections.",
"From the Lagrangian (REF ), one obtains the general equations of motion for the gauge field $A_\\mu $ as $\\frac{1}{\\sqrt{-g}}\\partial _\\nu \\left(\\sqrt{-g}F^{\\nu \\mu }\\right) =2 e^2 \\mathbf {\\Phi }^\\dagger \\cdot \\mathbf {\\Phi } A^\\mu + i e \\mathbf {\\Phi }\\stackrel{\\leftrightarrow }{\\partial }\\!",
"{}^\\mu \\mathbf {\\Phi },$ and for the Higgs scalar $\\frac{1}{\\sqrt{-g}} \\partial _\\mu \\left(\\sqrt{-g}g^{\\mu \\nu }D_\\nu \\mathbf {\\Phi }\\right) = i e A^\\nu D_\\nu \\mathbf {\\Phi } +\\mathbf {\\Phi }\\frac{\\mathrm {d}V(\\mathbf {\\Phi })}{\\mathrm {d}(\\mathbf {\\Phi }^\\dagger \\cdot \\mathbf {\\Phi } )},$ with the hermitian conjugate equation applying for $\\mathbf {\\Phi }^\\dagger $ .",
"These give, for the background configuration (REF ) with the potential (REF ), $\\frac{\\mathrm {d}^2\\varphi }{\\mathrm {d}r^2}+\\frac{1}{r}\\frac{\\mathrm {d}\\varphi }{\\mathrm {d}r}= \\left[ \\frac{Q^2}{r^2} +\\lambda \\left( \\varphi ^2-\\eta ^2\\right)\\right] \\varphi ,$ and $\\frac{\\mathrm {d}^2 Q}{\\mathrm {d}r^2}-\\frac{1}{r}\\frac{\\mathrm {d}Q}{\\mathrm {d}r}= 2e^2 \\varphi ^2 Q,$ after setting $Q=n-eA_\\theta $ to account for the winding number.",
"We now assume – see the following sections – that we have (numerical) solutions for the functions $\\varphi (r)$ and $Q(r)$ .",
"Because the Higgs doublet is coupled with itself, and even though finite energy solutions of Eqs.",
"(REF ) and (REF ) exist, one needs verify that these are stable.",
"Following Witten [7], we set an arbitrary perturbation $\\mathbf {\\Phi } = \\mathbf {\\Phi }_0 +\\delta \\mathbf {\\Phi }$ with $\\delta \\mathbf {\\Phi } = \\begin{pmatrix} 0 \\\\ \\sigma \\mathrm {e}^{i\\omega t}\\end{pmatrix},$ where $\\sigma =\\sigma (r)$ depends on the radial coordinate only.",
"Plugging Eq.",
"(REF ) into (REF ) and keeping only first order terms, one gets the Schrödinger-like equation $-\\Delta _2 \\sigma + \\mathcal {V}(r) \\sigma = \\omega ^2 \\sigma ,$ where $\\Delta _2 = \\partial _x^2 + \\partial _y^2=\\partial ^2_r + r^{-1} \\partial _r+r^{-2} \\partial _\\theta ^2$ is the two-dimensional laplacian and the effective potential $\\mathcal {V}$ reads $\\mathcal {V}(r) = \\frac{\\left[n-Q(r)\\right]^2}{r^2}+ \\lambda \\left[\\varphi ^2(r) -\\eta ^2\\right].$ This potential is shown on Fig.",
"REF for different values of the parameter $\\beta \\equiv \\lambda /(2e^2)$ .",
"One expects from the figure that there could be bound states provided $\\beta $ is large enough.",
"Figure: The potential (), rescaled so asto be dimensionless, appearing in the Schrödinger equation() for various values of the parameter β=λ/(2e 2 )\\beta =\\lambda /(2e^2) as a function of the dimensionless distanceto the string core ρ=2eηr\\rho =\\sqrt{2} e\\eta r(see Sec.",
"for details on the numerics).Since $\\lim _{r\\rightarrow 0}\\mathcal {V}(r) = -\\lambda \\eta ^2$ is negative and $\\mathcal {V} \\sim n^2/r^2$ asymptotically, the potential satisfies the usual quantum mechanical conditions for having a bound state: a range of values for the parameter $\\beta $ can be found for which there exist solutions of Eq.",
"(REF ) with $\\omega ^2 <0$ , and hence an instability of the background solution (REF ) should develop.",
"With the non linear terms taken fully into account, the instability translates into a condensate that can carry a current.",
"Comparison with Ref.",
"[12] shows that for $\\beta >\\frac{1}{2}$ , i.e.",
"$\\lambda >e^2$ , one expects a condensate to form: according to the usual classification, this means that type I vortices are absolutely stable (no condensate) while type II vortices spontaneously form a current-carrying state.",
"Note also that since type II vortices are energetically favored to occur with unit winding number, we shall for now on restrict attention to the case $n=1$ .",
"The question now is whether or not these current-carrying solutions can lead to the stable enough configurations (for cosmological purposes) discussed in [23].",
"It should be remarked at this stage that the mere existence of an instability does not guarantee that it has an endpoint which one then identifies with the current-carrying state.",
"The numerical solutions obtained below show that it does, and because the field equations stem from minimizing the energy per unit length to be minimized, they provide more stable configurations satisfying the boundary conditions.",
"As we shall see, these solutions will turn out to initiate another instability."
],
[
"The current-carrier condensate",
"For now on, we follow [23] and assume a condensate did form and we write the Higgs doublet as $\\mathbf {\\Phi } =\\left[ \\begin{matrix} \\varphi (r) \\mathrm {e}^{in\\theta + i\\psi (z,t)} \\\\\\sigma (r)\\mathrm {e}^{im\\theta + i \\xi (z,t)}\\end{matrix} \\right],$ where $n\\in \\mathbb {Z}$ is the winding number of the string, $m\\in \\mathbb {Z}$ leaves the possibility for the perturbation to wind as well, and the phases $\\psi $ and $\\xi $ only depend on the internal string coordinates.",
"This field can then source $A_\\theta $ , $A_z$ and $A_t$ , all being functions of the radius $r$ only in order for the worldsheet to be localized.",
"Note that the form (REF ) assumes no modes are present in the transverse direction, i.e.",
"the phases $\\psi $ and $\\xi $ do not depend on $r$ , so we do consider neither ingoing nor outgoing waves: the field configuration we are investigating is at equilibrium, hence may only have excitations along the worldsheet.",
"We shall also occasionally use a latin index to denote worldsheet coordinates $\\lbrace z,t\\rbrace $ collectively."
],
[
"State parameters",
"With the ansatz (REF ), the field equations now read $A^{\\prime \\prime }_a+\\frac{1}{r}A^{\\prime }_a +2 e \\left[ (\\partial _a \\psi -eA_a ) \\varphi ^2+ (\\partial _a \\xi -eA_a ) \\sigma ^2 \\right] = 0,$ for the internal gauge fields, $Q^{\\prime \\prime }-\\frac{1}{r}Q^{\\prime }=2 e^2 \\left[ Q \\varphi ^2+ (Q+m-n) \\sigma ^2 \\right] ,$ with the same convention as before that $Q=n-eA_\\theta $ , $\\varphi ^{\\prime \\prime }+\\frac{1}{r}\\varphi ^{\\prime } = \\left[ P_\\psi ^2 +\\frac{Q^2}{r^2} +\\lambda \\left( \\varphi ^2+\\sigma ^2-\\eta ^2\\right)\\right]\\varphi ,$ with $P_\\psi ^2 = (\\partial _z\\psi -eA_z)^2 - (\\partial _t\\psi -eA_t)^2$ , $\\sigma ^{\\prime \\prime }+\\frac{1}{r}\\sigma ^{\\prime } = \\left[ P_\\xi ^2 +\\frac{(Q+m-n)^2}{r^2} +\\lambda \\left( \\varphi ^2+\\sigma ^2-\\eta ^2\\right)\\right]\\sigma ,$ where $P_\\xi ^2$ is defined in a similar fashion as $P_\\psi $ , namely $P_\\xi ^2 = (\\partial _z\\xi -eA_z)^2 - (\\partial _t\\xi -eA_t)^2$ .",
"Finally, the phases represent massless modes propagating along the string, as is clear from their equations of motion $(\\partial ^2_t - \\partial ^2_z)\\psi = \\gamma ^{ab}\\partial _a \\partial _b\\psi =0 = \\gamma ^{ab}\\partial _a \\partial _b \\xi .$ In Eqs.",
"(REF ) to (REF ), we have set a prime to denote a derivative with respect to the radial distance $r$ .",
"One now needs to look at the boundary conditions to restrict attention to the physically meaningful cases.",
"In particular, noting that $\\lim _{r\\rightarrow 0}Q(r)=n$ and assuming $P_\\xi ^2$ to be regular at the string core location, Eq.",
"(REF ) implies the following: setting $\\sigma \\simeq \\sigma _0 + \\sigma _0^{\\prime } r +\\frac{1}{2}\\sigma _0^{\\prime \\prime } r^2 +\\cdots $ , the expansion $\\sigma _0^{\\prime \\prime } \\left( 2-\\frac{m^2}{2}\\right) + \\frac{\\sigma _0^{\\prime }}{r}\\left( 1-m^2 \\right) + \\frac{m^2 \\sigma _0}{r^2} P_\\xi ^2(0)\\sigma _0+ \\mathcal {O}\\left( r\\right)=0$ should hold.",
"In order for the $r^{-2}$ term to be regular, one must impose either $m=0$ or demand that $\\sigma _0=0$ .",
"In the latter case, assuming $m\\ne 0$ , one finds that $m^2=1$ and $m^2=4$ simultaneously, which is self-contradictory.",
"Hence, we must set $m=0$ and $\\lim _{r\\rightarrow 0} \\sigma ^{\\prime }(r)=0$ .",
"Moreover, asymptotically, i.e.",
"when $Q\\rightarrow 0$ , $\\sigma \\rightarrow 0$ and $\\varphi \\rightarrow \\eta $ , Eq.",
"(REF ) becomes $A_a^{\\prime \\prime }+\\frac{1}{r} A_a^{\\prime } +2e\\eta ^2 (\\partial _a\\psi - eA_a)=0,$ the solution of which can only be made to vanish – i.e.",
"we demand $\\lim _{r\\rightarrow \\infty }A_a(r)=0$ in order for the total energy of the configuration to be finite – only provided $\\partial _a\\psi =0$ .",
"As $\\psi $ must now be a constant, it can, without lacking generality, be set to zero by means of a global SU(2) gauge transformation (which can also remove any constant part that would be present in $\\xi $ as well).",
"The general solution of (REF ) then reads $\\xi = \\xi _-(z-t) + \\xi _+(z+t) + kz-\\omega t,$ where $\\xi _\\pm $ represent the left and right massless modes moving along the string and the last term represents a coherent mode, that can, in the usual case, be built as a superposition of left and right movers.",
"If a string segment is considered, the left and right moving modes are responsible for the leaking out of the current; again, following [23], we shall in what follows consider a $z-$ independent string (approximating a closed loop when setting periodic boundary conditions), assuming it can somehow be formed in the first place and thus neglect these modes; we shall accordingly set $\\xi _\\pm \\rightarrow 0$ in what follows.",
"Because of Eq.",
"(REF ), the last term of Eq.",
"(REF ) is a constant.",
"This implies that the two functions $P_a \\equiv eA_a-\\partial _a\\xi $ satisfy the same linear equation and hence are merely proportional to one another for all values of $r$ .",
"One then has $P_z \\propto P_t$ , the proportionality constant being found by taking the asymptotic limit of this relation for which we want the gauge field $A_a$ to vanish.",
"This yields $P_z=-k P_t /\\omega $ , and thus $A_z=-k A_t /\\omega $ .",
"We are now in a position to define the relevant degree of freedom as $A_z^2 - A_t^2 = \\left( \\frac{k^2}{\\omega ^2}-1\\right) A_t^2=\\left( 1-\\frac{\\omega ^2}{k^2}\\right) A_z^2 \\equiv w P^2,$ with $w$ the state parameter, and the function $P$ is dimensionless.",
"The fields $A_z$ and $A_t$ are then related to $P$ through $A_t = \\omega P \\sqrt{\\frac{w}{k^2-\\omega ^2}}, \\ \\ \\ \\ \\hbox{and}\\ \\ \\ \\ A_z = -k P \\sqrt{\\frac{w}{k^2-\\omega ^2}};$ note that $w$ has dimensions of a squared mass.",
"In view of this, one needs to complement the system with yet another independent – and dimensionless – parameter $b$ , representing the bias between the gauge fields and the phase gradient, through $k^2 - \\omega ^2 = w b^2.$ The sign of $w$ determines that of the phase gradient, so the current is described by two positive parameters and a sign.",
"For $w>0$ (resp.",
"$w<0$ ), the current is spacelike (resp.",
"timelike), and the equation of motion for $P$ is $P^{\\prime \\prime }+\\frac{1}{r} P^{\\prime } = 2e^2 P \\left( \\varphi ^2 + \\sigma ^2 \\right) +2eb \\sigma ^2,$ where we assume $b>0$ .",
"Having constructed the current-carrying configuration and taken account of all the symmetries, we now turn to the range of parameters that one should investigate to fully describe such strings."
],
[
"The lightlike current limit",
"The ordinary – neutral [33] or charged [34] – current-carrying cosmic string is known to have a maximum charge density (timelike current) above which it is energetically favored for the condensed particles to form ingoing and outgoing massive radial modes.",
"In the model here discussed, such a phase frequency threshold is also acting, and as it turns out, it prevents the timelike currents to form altogether.",
"With the degrees of freedom as obtained in the previous section, we can rewrite Eq.",
"(REF ) as $\\sigma ^{\\prime \\prime }+\\frac{1}{r} \\sigma ^{\\prime } = \\left[ P_\\xi ^2+ \\frac{(Q-n)^2}{r^2}+\\lambda \\left(\\varphi ^2+\\sigma ^2-\\eta ^2\\right)\\right]\\sigma ,$ where now $P_\\xi ^2 = w\\left( b+e P \\right)^2$ .",
"In the asymptotic regime, one is left with $\\sigma ^{\\prime \\prime }+\\frac{1}{r}\\sigma ^{\\prime } \\sim \\left( w b^2 +\\frac{n^2}{r^2}\\right)\\sigma ,$ as $\\sigma $ decreases to vanishingly small values.",
"The general solution for this Bessel equation is $\\sigma \\sim a_\\mathrm {I} I_n(b\\sqrt{w} r) + a_\\mathrm {K}K_n(b\\sqrt{w} r)$ for constant $a_\\mathrm {I}$ and $a_\\mathrm {K}$ , with $I_n$ and $K_n$ the modified Bessel functions of order $n$ .",
"For $w>0$ , the field is a condensate provided we set $a_\\mathrm {I}=0$ .",
"The energy contained in this solution converges exponentially fast far from the string core provided $w>0$ : for $w<0$ , instead the general solution is a combination of oscillatory Bessel functions.",
"In the usual Witten current-carrying case [33], [27], there is a similar transition for a given, nonzero negative value $w_\\mathrm {th}$ of $w$ that leads to a logarithmic divergence in the equation of state in the limit $w\\rightarrow w_\\mathrm {th}$ .",
"Here however, the threshold would be for a lightlike current, with $w_\\mathrm {th}=0$ : the would-be divergence is regularized by the $w$ prefactor that enters into the definition of the energy per unit length and tension (see next Sec. )",
"and the result is perfectly finite: there is no phase frequency thresholdRather, one could say that there is a frequency threshold as in the usual case, but the asymptotic mass of the current-carrier vanishes since it is akin to a Goldstone mode here, so the threshold does not imply a divergent behavior of either the energy per unit length or the tension.",
"in this case, the current can, from a spacelike configuration, smoothly evolve towards an almost lightlike situation.",
"In fact, Eq.",
"(REF ) also gives the behavior of $\\sigma $ with $w$ in the limit $w\\rightarrow 0$ .",
"First, setting $w=0$ into Eq.",
"(REF ) yields $\\sigma \\sim A r^{-n} + B r^{n}$ , with $A$ and $B$ unknown constants; a necessary condition for the condensate to be localized on the vortex is that $B=0$ .",
"On the other hand, taking directly the solution (REF ) with $a_\\mathrm {I}=0$ and expanding the Bessel function $K_n$ in the neighborhood of $w\\sim 0$ (we assume an analytic continuation with $n\\rightarrow n+\\epsilon $ and take afterwards the limit $\\epsilon \\rightarrow 0$ to handle the singularity), one obtains $K_n(b\\sqrt{w} r) \\sim 2^{-1-n} b^n r^n w^{n/2} \\Gamma (-n)+2^{-1+n} b^{-n} r^{-n} w^{-n/2} \\Gamma (n),$ so that, providing $w^{n/2}$ converges to zero faster than the pole in the $\\Gamma $ function, one can identify $a_\\mathrm {K} = \\frac{\\left(b\\sqrt{w}\\right)^n}{2{n-1}(n-1)!}",
"A,$ where $A$ , although arbitrary at this stage, is independent of $w$ as it comes from the solution for $w=0$ .",
"Therefore, in the small (but finite) $w$ limit, we have that $\\sigma \\propto w^{n/2} K_n(b\\sqrt{w} r)$ whose asymptotic behavior gives $\\sigma \\propto w^{n/2-1/4} \\mathrm {e}^{-b\\sqrt{w} r} /\\sqrt{r}$ .",
"It is this behavior that implies the chiral current limit to be well defined.",
"We now move on to evaluating the integrated quantities leading to this equation of state."
],
[
"Integrated quantities",
"In order to describe the network of strings that will be generated by the single strings here considered, one needs to integrate over the transverse directions in order to be able to approximate each defect by means of an actually zero thickness object.",
"This means we should derive the current and stress energy tensor associated with the solutions obtained above.",
"As we will then show the string to be unstable with respect to longitudinal perturbations, the worldsheet these integrated quantities suppose will not actually last; assuming its presence is however necessary for calculation purposes."
],
[
"Current",
"Among the integrated quantities of interest, the current, defined as $J^\\mu \\equiv \\frac{1}{2e} \\frac{\\delta \\mathcal {L}}{\\delta A_\\mu },$ provides two independent ways to verify that the following configurations obtained numerically are indeed solutions and not mere artifacts.",
"With the framework of model (REF ), this is $J^\\mu = -\\frac{i}{2} \\left[ \\mathbf {\\Phi }^\\dagger \\cdot \\left(\\partial ^\\mu \\mathbf {\\Phi }\\right) -\\left( \\partial ^\\mu \\mathbf {\\Phi }^\\dagger \\right) \\cdot \\mathbf {\\Phi }\\right]-e A^\\mu \\mathbf {\\Phi }^\\dagger \\cdot \\mathbf {\\Phi },$ which gives, using the explicit form (REF ) in terms of the components of $\\mathbf {\\Phi }$ $J_r = 0 \\ \\ \\ \\hbox{and} \\ \\ \\ \\ \\ J_\\theta = Q\\varphi ^2 + (Q-n) \\sigma ^2,$ for the transverse components, and $J_a = \\sigma ^2 \\left(\\partial _a \\xi -eA_a\\right) -2e\\varphi ^2 A_a,$ with $a\\in \\lbrace z,t\\rbrace $ for the longitudinal, worldsheet components.",
"Integration over the transverse degrees of freedom yield two macroscopically defined quantities, namely the rotational current flux around the string $I_\\theta \\equiv \\int \\mathrm {d}^2x^\\perp J_\\theta = 2\\pi \\int \\left[ Q \\varphi ^2+(Q-n)\\sigma ^2\\right]\\,r\\,\\mathrm {d}r = \\frac{2\\pi n}{e^2},$ when the field equation (REF ) with $m=0$ is used, and the Lorentz-invariant current scalar $J$ along the worldsheet defined through $J^2 \\equiv \\left( \\int \\mathrm {d}^2x^\\perp J_z \\right)^2 -\\left( \\int \\mathrm {d}^2x^\\perp J_t \\right)^2 ,$ which is readily evaluated in terms of the underlying field solution previously derives as $J=2\\pi \\sqrt{w}\\int \\left[ eP\\varphi ^2 +\\left( b+eP\\right) \\sigma ^2 \\right] \\,r\\,\\mathrm {d}r,$ because the difference of the integrals is itself a squared integral, as expected for Lorentz symmetry reasons along the string.",
"Making use of the field equation (REF ) then yields $J=0$ , so this definition cannot account for a conserved current along the worldsheet.",
"This stems from the fact that the current is now supported by both components of the doublet, whereas in the usual Witten situation, there is only one field that carries the current.",
"Although mostly useless for physical purposes, the current components (REF ) and (REF ) can be used as a measure of the validity of the numerical calculation: once the fields are calculated, evaluating the integrals should reproduce the analytic results above.",
"An alternative way to define the current is obtained by recalling that it physically comes from the phase gradient along the string.",
"In other words, what really matters is the current-carrying phase instead of the field itself, so that a suitable worldsheet covariant – but not SU(2) covariant – definition is $\\mathcal {J}_a = -\\frac{1}{2} \\eta _{ab}\\frac{\\delta \\mathcal {L}}{\\delta \\partial _b\\xi },$ where $\\eta _{ab}\\equiv \\mathrm {diag}\\, (-1,1)$ is the internal Minkowski metric in the string.",
"Since the action only depends on the phase gradient and not on the phase itself, this current is automatically conserved.",
"With the definition (REF ), one can construct an integrated current $I$ which is merely one part of that given in (REF ), namely one finds, using the same integration procedure as in (REF ) (with the replacements $J\\rightarrow I$ and $J_a\\rightarrow \\mathcal {J}_a$ ) $I=2\\pi \\sqrt{w}\\int \\left( b+eP\\right) \\sigma ^2 \\,r \\,\\mathrm {d}r.$ The nonzero value of this quantity also explains the difference between the spacelike and timelike eigenvalues of the stress energy tensor to which we now turn."
],
[
"Worldsheet stress-energy tensor",
"From the Lagrangian (REF ), one also derives the stress energy tensor $T_{\\mu \\nu } = -2\\frac{\\delta \\mathcal {L}}{\\delta g^{\\mu \\nu }}+g_{\\mu \\nu } \\mathcal {L},$ leading to the worldsheet components $T_{tt} &=&2e^2 \\varphi ^2 A_t^2 + 2\\sigma ^2 \\left(\\partial _t\\xi -e A_t \\right)^2 + A_t^{\\prime 2}-\\mathcal {L}(\\varphi ,\\sigma ,Q,P),\\cr && \\\\T_{zz} &=& 2e^2 \\varphi ^2 A_z^2 + 2\\sigma ^2 \\left(\\partial _z\\xi -e A_z \\right)^2+A_z^{\\prime 2}+\\mathcal {L}(\\varphi ,\\sigma ,Q,P),\\cr && \\\\T_{zt} &=& 2e^2 \\varphi ^2 A_z A_t + 2\\sigma ^2 \\left( \\partial _z \\xi -e A_z \\right) \\left( \\partial _t -e A_t \\right) + A_z^{\\prime }A^{\\prime }_t,\\cr && $ where we have made use of the symmetries discussed in the previous sections, and the Lorentz-invariant part stems from the background Lagrangian $\\mathcal {L}(\\varphi ,\\sigma ,Q,P)=-\\varphi ^{\\prime 2}-\\sigma ^{\\prime 2}-\\frac{Q^2 \\varphi ^2}{r^2}-\\frac{(Q-n)^2 \\sigma ^2}{r^2}- w\\left[ e^2\\varphi ^2 P^2 + (b+eP)^2 \\sigma ^2\\right]-\\frac{1}{2}\\left( w P^{\\prime 2}+\\frac{Q^{\\prime 2}}{e^2 r^2}\\right)-\\frac{\\lambda }{2}\\left( \\varphi ^2 + \\sigma ^2 -\\eta ^2\\right)^2.$ We assume the other components, i.e.",
"in the transverse direction, to vanish once integrated along the radial coordinates for the on-shell solution [35].",
"Following [36], we write $T_{ab} = \\begin{pmatrix} \\mathcal {A}+\\mathcal {B} & \\mathcal {C} \\cr \\mathcal {C} & -\\mathcal {A}+\\mathcal {B} \\end{pmatrix},$ where $\\mathcal {A}=-\\mathcal {L}(\\varphi ,\\sigma ,Q,P;w\\rightarrow 0)$ , i.e.",
"that part of $\\mathcal {L}$ of Eq.",
"(REF ) without the variations along the vortex, and $\\mathcal {B} = & &\\hspace{-8.53581pt}\\varphi ^2 e^2 (A_z^2+A_t^2) +\\sigma ^2 \\left[ \\left( k-eA_z\\right)^2+\\left(\\omega +eA_t\\right)^2 \\right]\\cr & &\\hspace{-8.53581pt}+\\frac{1}{2} (A_z^{\\prime 2} +A_t^{\\prime 2}),$ and the non diagonal component reads $\\mathcal {C} = 2\\varphi ^2 e^2 A_z A_t - 2\\sigma ^2 (k-e A_z) (\\omega +e A_t) + A_z^{\\prime } A_t^{\\prime } .$ Diagonalization of $T_{ab}$ with respect to $\\eta _{ab} = \\mathrm {diag}\\, (-1,1)$ the two-dimensional Minkowski metric yields the eigenvalues $E_\\pm $ .",
"Those are $E_\\pm &\\equiv & \\mathcal {A}\\pm \\sqrt{\\mathcal {B}^2-\\mathcal {C}^2}\\cr &=& \\mathcal {A} \\pm w \\left[\\frac{1}{2} P^{\\prime 2}+e^2 P^2 \\varphi ^2 +\\left(b+e P\\right)^2\\sigma ^2 \\right],$ from which one derives the energy per unit length $U$ and tension $T$ by integration over the transverse degrees of freedom, namely $U = 2\\pi \\int E_+(r) \\, r\\,\\mathrm {d}r \\ \\ \\ \\ \\hbox{and} \\ \\ \\ \\ T = 2\\pi \\int E_- (r) \\, r\\,\\mathrm {d}r.$ Note at this point that since the quantity appearing in the diagonalizing solution Eq.",
"(REF ) is a perfect square, the integration and diagonalization procedures commute, just as in the case of the current for which (REF ) could be straightforwardly derived, so the resulting macroscopic quantities are really defined in an unambiguous way.",
"Figure: Rescaled fields around the vortex: X(ρ)X(\\rho ) – full line –and Y(ρ)Y(\\rho ) – dashed –are the Higgs field components in units of the Higgs VEV η\\eta , whilethe vector field fluxQ(ρ)Q(\\rho ) – dotted – renders the vortex local and P(ρ)P(\\rho ) – dot-dashed –condenses in such away as to support the current otherwise induced by condensation ofYY.",
"This figureis obtained for parameter values α=1\\alpha =1, β=3\\beta =3 andw ˜=0.1β/α 2 \\tilde{w}=0.1 \\beta /\\alpha ^2.In order to evaluate the actual behavior of the equation of state relating the energy per unit length and the tension, and in particular the stability of the resulting current-carrying string, we now discuss the numerical solutions."
],
[
"Numerics",
"Solving numerically the system of equations (REF ), (REF ), (REF ) and (REF ), requires that we cancel out the dimensions of the relevant quantities.",
"Setting $\\rho =\\sqrt{2} e\\eta r$ the radius in units of the gauge vector mass, and rescaling the fields and state parameter through $\\varphi = \\eta X(\\rho ), \\ \\ \\ \\ \\sigma = \\eta Y(\\rho ) \\ \\ \\ \\ \\hbox{and}\\ \\ \\ \\ w = 2 \\eta ^2 \\tilde{w},$ we obtain the dimensionless equations of motion in the form $\\ddot{X} + \\frac{1}{\\rho }\\dot{X} &=&\\left[\\tilde{w}P^2+\\frac{Q^2}{\\rho ^2}+\\beta \\left(X^2+Y^2-1\\right) \\right] X,\\\\\\ddot{Q} - \\frac{1}{\\rho }\\dot{Q} &=& Q X^2+(Q-n) Y^2,\\\\\\ddot{Y} + \\frac{1}{\\rho }\\dot{Y} &=&\\left[\\tilde{w}(\\alpha +P)^2+\\frac{(Q-n)^2}{\\rho ^2}+\\beta \\left(X^2+Y^2-1\\right) \\right] Y,\\cr & &\\\\\\ddot{P} + \\frac{1}{\\rho }\\dot{P} &=& P\\left( X^2+Y^2\\right) +\\alpha Y^2,$ where a dot denotes differentiation with respect to the rescaled radius $\\rho $ and the constants are defined by $\\alpha \\equiv b/e$ and $\\beta \\equiv \\lambda /(2e^2)$ .",
"Figure: Variation of the internal current eI/ηeI/\\eta as a function of the rescaled state parameterw ˜\\tilde{w} for α=1\\alpha =1 and various values of β\\beta (same as onFig.",
").A point worth discussing in relation with these equations concerns the evolution of the condensate as the state parameter increases.",
"Expanding the field functions around the string core as $X\\propto \\rho ^m + \\cdots $ , $Y\\sim Y_0 + \\frac{1}{2} \\ddot{X}_0 \\rho ^2 +\\cdots $ , $Q\\sim n+ \\frac{1}{2} \\ddot{Q}_0 \\rho ^2 +\\cdots $ and $P\\sim P_0 + \\frac{1}{2}\\ddot{P}_0 \\rho ^2 +\\cdots $ , where we have taken into account the regular boundary conditions, the zeroth order expansion of Eqs.",
"(REF ) to (), one gets that $\\ddot{P}_0 = \\frac{Y_0^2}{2}\\left(\\alpha +P_0\\right),$ implying that $-\\alpha \\le P_0\\le 0$ : if $P_0> 0$ , then (REF ) implies that $\\ddot{P}_0>0$ , and hence $P$ should be a growing and positive function of $\\rho $ , which is inconsistent with the requirement that $\\lim _{\\rho \\rightarrow \\infty } P= 0$ (we assume, following the figures, that the functions are monotonic).",
"If $P_0< -\\alpha $ , then $\\ddot{P}_0 <0$ , the same argument applies with a negative and decreasing function.",
"Eq.",
"(REF ) tells us that $m=n$ , as usual, while Eq.",
"() is trivially satisfied at the lowest order with the given expansion.",
"However, Eq.",
"() translates into $\\ddot{Y}_0 = \\frac{Y_0}{2}\\left[ \\tilde{w} \\left(\\alpha +P_0\\right)^2+\\beta \\left( Y_0^2 - 1\\right) \\right],$ so that, demanding $\\ddot{Y}_0 Y_0<0$ for the reasons just discussed for $P$ , one finds that $Y_0^2 \\le 1-\\frac{\\tilde{w}}{\\beta }\\left(\\alpha +P_0\\right)^2,$ indicating that for large values of $\\tilde{w}$ , assuming $P_0$ to depend only mildly on $\\tilde{w}$ (indeed, $P_0\\rightarrow 0$ in this limit), the available range for $Y_0$ abruptly shrinks to zero when $\\tilde{w} \\ge \\tilde{w}_\\mathrm {max} \\equiv \\beta /\\alpha ^2$ , or in other words for $w\\ge w_\\mathrm {max} \\equiv \\lambda \\eta ^2 /b$ : the range of variations for the state parameter is automatically constrained, as in the ordinary Witten case [33].",
"The finite range of variation of the state parameter can be understood in the following way.",
"Imagine a region along the string network where a statistical fluctuation on the phase gradient implies the condensate should form with a very large value of $w$ .",
"This gives the would-be condensate enough momentum to pass over the potential barrier (REF ), and hence blocks the instability to effectively take place until the fluctuation goes to a more reasonable value below the maximum $(\\partial \\xi )^2 \\le w_\\mathrm {max}$ .",
"These equations are derivable from the dimensionless action $\\mathcal {S}_+$ , where $\\mathcal {S}_\\pm = \\int \\left\\lbrace \\dot{X}^2 + \\dot{Y}^2 +\\tilde{w} \\dot{P}^2+\\frac{\\dot{Q}^2}{\\rho ^2} \\pm \\tilde{w}\\left[X^2P^2 +\\left(\\alpha +P\\right)^2 Y^2\\right] +\\frac{Q^2 X^2 + \\left(Q-n\\right)^2 Y^2}{\\rho ^2} + \\frac{1}{2} \\beta \\left(X^2+Y^2-1\\right)^2\\right\\rbrace \\,\\rho \\,\\mathrm {d}\\rho ,$ which is used to produce the numerical solutions shown on Fig.",
"REF that are discussed below.",
"The quantities $\\mathcal {S}_\\pm $ serve to define the energy per unit length and tension through $U=2\\pi \\eta ^2 \\mathcal {S}_+ \\ \\ \\ \\ \\hbox{and} \\ \\ \\ \\ \\ T=2\\pi \\eta ^2 \\mathcal {S}_-.$ We also derive the currents in terms of dimensionless variables as $I=\\frac{\\eta }{e} \\pi \\sqrt{2\\tilde{w}} \\int \\left(\\alpha + P\\right) Y^2 \\rho \\,\\mathrm {d}\\rho ,$ It is shown on Fig.",
"REF as functions of $\\tilde{w}$ .",
"The limit provided by Eq.",
"(REF ) compares with our numerical calculations in the sense that the would-be current (REF ) obtained in Sec.",
"REF abruptly vanishes when $\\tilde{w}$ exceeds the critical value above which the condensate does not form at all.",
"The other currents, i.e., the constraints stemming from Eqs.",
"(REF ) and (REF ), are numerically verified to hold, hence ensuring our field functions to solve their equations of motion.",
"Eqs.",
"(REF ) and (REF ) permit to show explicitely, using the asymptotic behaviors derived above for $\\sigma $ , that the energy and tension are both well behaved at the would-be phase frequency threshold $w\\rightarrow 0$ .",
"In terms of dimensionless variables, we have, for $\\rho \\gg 1$ , that $Y(\\rho )$ behaves as $Y\\sim f(\\tilde{w}) \\tilde{w}^{n/2-1/4}\\mathrm {e}^{-\\alpha \\sqrt{\\tilde{w}}\\rho }/\\sqrt{\\alpha \\rho }$ , where $f(\\tilde{w})$ is an unknown function of $\\tilde{w}$ whose behavior for small values of the state parameter $\\lim _{\\tilde{w}\\rightarrow 0} f(\\tilde{w})$ is a constant.",
"Figure: Values of the condensate function P 0 (0)P_0(0) and Y 0 (0)Y_0(0)in the string core (ρ=0\\rho =0) as functions of the rescaled state parameterw ˜\\tilde{w} for α=1\\alpha =1 and various values of β\\beta (same as onFig.",
").Now, in this small $w$ regime, it is a simple matter to evaluate the leading behavior of the integrated quantities, as most of the field hardly depend on $w$ : as shown on Fig.",
"REF , the condensate value at the string core and the current gauge function $P$ , as well as the background fields $X$ and $Q$ , are essentially independent of $w$ .",
"The only term that really matters for the variation of the integrals with $w$ is the asymptotic behavior of the current carrier $\\sigma $ : as in the ordinary Witten case, the condensate tends to spread around the string around the phase frequency threshold, i.e.",
"here in the almost chiral case.",
"Thus, assuming the asymptotic behavior to hold from a distance $\\rho _\\mathrm {M}$ on, the dominant contribution $\\Delta $ comes from the $Y$ terms in Eq.",
"(REF ), namely $\\Delta _\\pm &=& \\int _{\\rho _\\mathrm {M}}^\\infty \\left[\\dot{Y}^2 \\pm \\tilde{w} \\left( \\alpha +P\\right)^2 Y^2+\\frac{\\left(Q-n\\right)^2}{\\rho ^2} Y^2 \\right.",
"\\cr && \\left.",
"\\hspace{28.45274pt}+\\beta \\left(X^2-1\\right) Y^2+\\frac{1}{2} \\beta Y^4\\right] \\rho \\,\\mathrm {d}\\rho .\\nonumber $ For $\\rho > \\rho _\\mathrm {M}$ , one can further make the assumption that the other fields have reached their asymptotic regime, namely we can set $(P,Q)\\rightarrow 0$ and $X\\rightarrow 1$ , so the only important contributions end up being $\\Delta = \\int _{\\rho _\\mathrm {M}}^\\infty \\left[\\dot{Y}^2 \\pm \\tilde{w} \\alpha ^2 Y^2+\\frac{n^2}{\\rho ^2} Y^2+\\frac{1}{2} \\beta Y^4\\right] \\rho \\,\\mathrm {d}\\rho ,$ which can be explicitly calculated.",
"Neglecting irrelevant constant terms and keeping only the leading contributions, this gives, for $n=1$ (the general case leads to similar conclusions but is merely more involved and, as discussed above, not relevant to the current discussion since the type II vortices here considered are unstable for $n>1$ , splitting into $n$ unit winding vortices) $\\Delta \\sim A \\pm B \\tilde{w} + C \\sqrt{\\tilde{w}}+ D \\tilde{w} \\ln \\tilde{w},$ where $A$ , $B$ , $C$ and $D$ can be evaluated as asymptotic integrals over the fields that do not depend on $\\tilde{w}$ .",
"Figure: Energy per unit length UU (full lines) and tensionTT (dashed lines) of the semi-local string in units of thesquared Higgs vev η 2 \\eta ^2for α=1\\alpha =1 as on Fig.",
", and different valuesof β\\beta as indicated on the curves.This shows explicitly the absence of a phase frequency thresholdat w ˜=0\\tilde{w}=0, i.e.",
"the null current limit is perfectly regular.",
"Thefunctions end abruptly for a maximum value of w ˜\\tilde{w}, asindicated in Eq.",
"() after which the condensate identicallyvanished.",
"It is also seen that UU and TT vary in the same waywith ww for all values of ww, so that dT/dU>0\\mathrm {d}T/\\mathrm {d}U >0, and hencethe longitudinal perturbation velocity c L 2 c_{_\\mathrm {L}}^2 is always negative,signaling an unstable behavior of the string seen as a macroscopicobject; the relevant string evolution presumably leads to a chiralbehavior independently of the initial value of the cosmological wwdistribution at the string network formation time.What makes $U$ different from $T$ as functions of $\\tilde{w}$ is, in the above expression, the second term involving $B$ .",
"In the limit $\\tilde{w}\\rightarrow 0$ , this term rapidly becomes negligible, and the dominant contribution thus implies that $U$ and $T$ evolve in similar ways with respect to $\\tilde{w}$ , the unique parameter describing the string state.",
"As a result, variations of the tension with the energy per unit length are always positive, so the longitudinal perturbation velocity $c^2_{_\\mathrm {L}}\\equiv - \\frac{\\mathrm {d}T}{\\mathrm {d}U} \\le 0,$ is negative in the limit $\\tilde{w}\\rightarrow 0$ .",
"Numerical calculation shown in Fig.",
"REF for the full range of available variations of $\\tilde{w}$ shows that in fact, Eq.",
"(REF ) is valid for all possible states attainable by the strings under scrutiny here."
],
[
"Discussion and conclusion",
"We have investigated a specific model of embedded type II gauged vortices coming from the gauging of a U(1) subgroup of an otherwise global SU(2).",
"When the U(1) symmetry is broken through a Higgs doublet acquiring a nonvanishing vacuum expectation value, another component of the same doublet can be excited because of a well-known condensate instability.",
"This leads to possible current-carrying string states as the phase gradient of the carrier part of the doublet varies along the string: at least at the time when the condensate forms, variations from one point to another are subject to fluctuations over distances larger than the correlation length, i.e.",
"the inverse mass of the Higgs field.",
"Because type II vortices exhibit only a single unstable mode, it was suggested in [23] that those thus formed could be stable provided they appear as sufficiently small loops so that the unstable long wavelength microscopic perturbations do not take over the dynamics.",
"It remained to understand whether these loops could be macroscopically stable, and this requires that we solve the internal string structure in order to be able to integrate over the irrelevant degrees of freedom.",
"This is achieved by means of a numerical integration of the field equations and a calculation of the relevant integrated quantities forming the stress-energy tensor, to be later coupled to gravity, and the currents.",
"We found that contrary to the original U(1)$\\times $ U(1) Witten model [7], [33], [34] for which a large region of stability with timelike, lightlike and spacelike currents could be identified, here only a spacelike current could be constructed.",
"This relies on the fact that the condensate is essentially a massless Goldstone mode, so that any timelike excitation would be energetically favored to move away from the string.",
"The lightlike limit, however, appears to be reasonably well-defined.",
"We obtained another crucial difference with the usual current-carrying string models: the spacelike current configurations happen to be unstable with respect to longitudinal (sound-wave like) perturbations.",
"As a result, our investigation closes the window of possible stability zones opened in Ref.",
"[23], and we are led to the definite conclusion that type II vortices cannot form, or if they do, they will spontaneously decay in such a way that their cosmological relevance is vanishing.",
"Let us finally point out that the non-current carrying semi-local strings share some features with BPS D-term string solutions [37], in particular the latter possess a zero mode - very similar to semi-local strings.",
"While the zero mode can also be excited [38] one might wonder whether any of the results obtained in our paper would also be valid in this case and whether this could lead to any consequence on inflationary models rooted in String Theory.",
"BH gratefully acknowledges support within the framework of the DFG Research Training Group 1620 Models of gravity.",
"PP would like to thank support from Jacobs University (Bremen – Germany).",
"This research was supported in part by Perimeter Institute for Theoretical Physics (Waterloo, ON – Canada)."
]
] | 1204.1270 |
[
[
"Power of earthquake cluster detection tests"
],
[
"Abstract Testing the global earthquake catalogue for indications of non-Poissonian attributes has been an area of intense research, especially since the 2011 Tohoku earthquake.",
"The usual approach is to test statistically for the hypothesis that the global earthquake catalogue is well explained by a Poissonian process.",
"In this paper we analyse one aspect of this problem which has been disregarded by the literature: the power of such tests to detect non-Poissonian features if they existed; that is, the probability of type II statistical errors.",
"We argue that the low frequency of large events and the brevity of our earthquake catalogues reduces the power of the statistical tests so that an unequivocal answer for this question is not granted.",
"We do this by providing a counter example of a stochastic process that is clustered by construction and by analysing the resulting distribution of p-values given by the current tests."
],
[
"Introduction",
"Several investigators have proposed the presence of two temporal clusters of very large earthquakes during the past century, e.g.",
"[2], [1].",
"The first cluster occurred in the middle of last century and included the 1952 Mw 9.0 Kamchatka earthquake, the 1960 Mw 9.5 Chile earthquake and the Mw 9.2 Alaska earthquake ([2]).",
"The second aparent cluster began with the occurrence of the Mw 9.15 Sumatra earthquake of 26 December 2004 and has continued with the Mw 8.8 Chile earthquake on 27 February 2010 and the Mw 9.0 Tohoku earthquake on 11 March 2011 [3], [1].This recent cluster has given rise to debate about whether the observed temporal clustering of these very large earthquakes has some physical cause or has occurred by random chance [4].",
"[5] used three statistical tests to conclude that the global clustering can be explained by the random variability in a Poisson process.",
"His first test was an analysis of inter-event times using a one-sided Kolmogorov-Smirnov test.",
"The second test showed that the occurrence of very large earthquakes is not correlated with the occurrence of smaller events.",
"The third test demonstrated that temporal clustering in seismic moment release occurs in about $50\\%$ of the samples when the number of events is drawn from a Poisson distribution and is not constrained as in the modeling of [2], [3].",
"In another article, [6] reach the same conclusions testing for the Poissonian hypothesis using a different set of statistical quantities.",
"The purpose of this paper is to discuss the power of traditional statistical tests to establish unequivocally the existence or not of earthquake clusters for catalogues with small numbers of events and not amenable to experimental repeatability.",
"In general, to study the power of statistical tests we need to enunciate an alternative hypothesis and calculate the probability of correctly rejecting a false hypothesis [7]; this is not the case for most studies of earthquake clusters since, to our knowledge, no stochastic process other than Poisson has been widely hypothesized and tested for in the earthquake catalogue.",
"The objective of our study is to determine the probability with which a random sample of a contrived non-Poissonian process is rejected in a test in which the null hypothesis is a Poisson process.",
"To aid the discussion we have devised a stochastic process which is clustered by construction and whose samples play the role of earthquake catalogues with a given magnitude threshold and de-clustered to remove aftershocks.",
"To each one of these samples we apply a specific statistical test and use the set of p-values obtained in this way to calculate their probability distribution.",
"This distribution will inform us the probability that any random sample of this process will pass or fail a test for Poissonian statistics.",
"To justify the merits of our analysis, we start by observing that Poisson is the unique discrete stochastic process that satisfies two conditions: lack of memory (Markov assumption) and a constant probability of event occurrence through time [8].",
"The exceptional character of this process makes it a valuable tool in the natural sciences since lack of memory and time independence can be inferred either a priori or a posteriori - in this case by showing that the observed data fits well a Poisson distribution.",
"Statistical inference of this sort is usually obtained through consensus of a large number of independent experiments, sometimes aided by theoretical models One example is photo counting experiments of stable laser light, in which a Poisson distribution is derived from first principles quantum mechanics and experimentally verified to an large degree of confidence ($ \\ll 1\\% $ ) On the other hand, there is an infinite number of processes not satisfying one or both conditions.",
"This becomes relevant when the available data is limited and a consensus view cannot be established since short data series could be explained adequately by more than one stochastic process.",
"It is granted that, even in such occasions a Poisson distribution can be postulated on arguments of simplicity and plausibility, which, while scientific valid does not constitute objectively an explanation for a phenomenon.",
"Otherwise, a Poisson process should be regarded as only one among many possible explanations.",
"One obvious question that arises from these considerations regards how much data is enough so that an inference exercise can assert beyond reasonable doubt which model explains the data observed.",
"The answer to this question lies in the scale of the stochastic process as compared with the length of the observed, which is illustrated by study."
],
[
"Description of the process",
"The stochastic process that we use to assess the skill of Poissonian properties tests was devised as a theoretical artefact and not as a statistical model for the earthquake catalogue or associated with any particular physical reality.",
"It was designed to convey in a synthetic manner the features of clustered data in which clusters may occur randomly and at relative low frequencies.",
"This process is constructed by generating a Poisson series at low event rates (from 2 to 3 cluster per century) equivalent to 110 years of observations.",
"A cluster is a period of increased rate of event occurrence; we express this by inserting in each cluster occurrence a Poisson sample with a 10-fold increase in frequency (3 to 4 events per decade) and a duration of 15 years.",
"Clusters are not allowed to overlap, but can neighbour each other to form mega-clusters of $\\approx 30 $ years.",
"Each particular choice of parameter will give rise to a particular distribution of the 110-years average event rate.",
"We have chosen the parameters above to coincide with the general scale of the observed global earthquake catalogue: average event frequencies will range from 1 to 2 evens per decade, with a large variability for the averages derived from any single 110 years sample (standard deviation $\\sigma \\approx 0.32$ ).",
"As a reference, the global earthquake catalogue for a cut-off magnitude of 8.3 is approximately 2 events per decade.",
"The samples generated by this process will produce clusters which are aperiodic and could be interpreted either as due to self sustained triggering (one event increases the probability of another) or as an overall increase in event rates due to a single underlying cause.",
"In either case, the samples are, on average, clustered enough so that the the p-value distribution is that of a non-uniform distribution and skewed to the left.",
"To best represent the variability in the genesis of earthquakes both clusters and events within clusters are subjected to full variability of Poisson process - this means a non-zero probability of entire centuries with no clusters.",
"We do not consider the problem of cluster detection for catalogues where a true Poisson noise or another independent clustered process was added to the background, nor of a periodic cyclic process - we assume it to be self evident that this entails a greater similarity with a Poissonian process.",
"The approach we take here is conservative insofar as the process we envisage produces samples which are more clustered than a true Poissonian process."
],
[
"p-Values distribution",
"P-values distribution were obtained by generating 10,000 independent samples of our process and by performing three different statistical tests in each independent sample to obtain the corresponding p-value.",
"The tests we have chosen are three: (a) Kolmogorov-Smirnov test on inter-event time distribution (b) Pearson $\\chi $ -square test on event count and (c) same same test on multiple event inter-time distribution.",
"Test (a) was performed under the null hypothesis that inter-event times follow an exponential distribution characteristic of a Poisson process $P(t)=\\exp (-\\lambda t)$ where $\\lambda $ is the average event rate per year and $t$ is the time measured in years.",
"The annual event rate $\\lambda $ is re-calculated for each sample, simulating our ignorance of the true event rate.",
"Test (b) is the usual Pearson $\\chi $ -square that tests for similarities in the histogram distribution between the samples and a Poisson distribution.",
"Test (c) performs the same test as (b) on the inter-event time distribution for multiple events using the corresponding Poisson distribution for null hypothesis.",
"These computations were performed using Mathematica© software package [10].",
"Our procedure was tested by performing these same tests with Poisson generated samples, which correctly output uniform distributions of p-values."
],
[
"results",
"A sample result is the p-value distribution shown in Figure REF .",
"It represents the probability of measuring a given p-value for test (a) for a single 110 years clustered sample.",
"We have adopted the most common view of rejecting a hypothesis for p-values smaller than 5% – a criterion that we return to in later discussions.",
"We have repeated the same process by varying the parameters of our process to estimate the power of detection of test (a) as the frequency of clusters and of events within clusters vary.",
"The process parameters for the result of Figure REF are 3 clusters per century and 4 events per decade over a 15 year cluster period – or 3-by-4 in short.",
"With these parameters, the average annual frequency of events is 0.12 events/year, with a 70th and 90th percentiles above median of 0.15 and 0.2 events/year approximately.",
"In the histogram of Figure REF bins are plotted in intervals of 5% and we can see that the probability of a p-value smaller than $5\\%$ is $\\approx 40\\%$ .",
"If the 5% significance level is strictly adopted, this is the chance that an observer would correctly reject the hypothesis and implies a type-II error probability of 60%.",
"We performed test (a) varying the parameters of the generating Poisson processes and show in Figure REF the probabilities of obtaining a p-value smaller than 5%.",
"As expected, for low frequency of events per cluster the process looks more like a Poisson process and is less frequently rejected; the samples of this process are maximally non-Poissonian for high cluster and event frequencies, with probability of a correct \"reject\" above 70%.",
"Average event frequencies for these values vary from 2 to 1 events per decade in the 5-by-5 and 3-by-4 cases respectively (see Figure REF ).",
"We will not discuss test (b), which has shown to be the less skilled of all, with probability of rejection on the order of 20%.",
"The most successful test among those we studied is (c).",
"In it, we generated the null hypothesis distribution from the inter-$n$ -event time distribution from samples of a Poisson process, against which the clustered samples were tested.",
"In this test, the Poisson hypothesis distribution is assumed to have a known average event frequency given by the long term mean over all samples.",
"A more sensible approach is to take into consideration the probability that a single 110 years average event rate will be that of the long term average.",
"This can be done, brute force, by generating one null hypothesis for each sample to be tested against all samples, thus accounting for the chance that a particular 110-years and the long term average event rate are the same.",
"In the interest of focusing on the essential points, we show the distributions for this test for the cases where the Poisson frequency is the long term average (over all samples), the 70% and 90% quantiles above median for the same parameters as those shown in Figure REF (3 clusters per century 4 events per decade within clusters).",
"The results are explained in Figures REF from (a) through (c).",
"In it we see that, if we pick a sample whose value is the same as the average event rate, the test will detect correctly the non-Poissonian nature of this process 80% of times.",
"For samples whose average event rate is above the 90% quantile, the probability of correctly rejecting the null hypothesis drops to about 70%.",
"The true value of the power for this test will depend on the degree of confidence in the estimation of the average event frequency and our belief of how accurately a single 110-years sample will inform on the \"true\" long term sample.",
"This reflects the fact that this process unfolds on time-scales greater than 110 years."
],
[
"Discussion",
"We stress that our presentation is not a claim that the stochastic process we devised intends to be a realistic model of the genesis of mega-earthquakes on a global scale.",
"The results we have show are solely an illustration of the pitfalls of statistical tests and of type II errors.",
"At the heart of these issues lies the statistical variability of the process we used, which can be plainly expressed by saying that some of the samples are more Poissonian than others.",
"It is the assessment of differences between trajectories enables us to determine the falsehood of the Poissonian hypothesis.",
"It remains to be seen the results of a similar study using plausible non-Poissonian processes, and the effects of the introduction of a Poissonian background.",
"As we noted before, arguments of plausibility and simplicity based on Markovian and time-independence assumptions provide solid grounds for hypothesising a Poisson process as a likely candidate to explain the global earthquake catalogue.",
"However, when viewed only on the merits of the observed data, the probability of type II statistical error, such as those we computed here, must be taken into account.",
"Our degree of belief in a given premise is explicitly manifest in Bayesian inference through the assignment of prior probabilities [9]; any argument that deems to inquire the data alone should state clearly its prior, whether equal probabilities or weighted towards a Poisson distribution.",
"[Such argument can be made not only as a matter of scientific clarity but as as an aid to scientific imagination.]",
"Another aspect we discuss here regards the levels of significance commonly used in statistics.",
"We have used 5% as the par excellence standard in rejecting an hypothesis.",
"Economics can provide the basis for a rational approach to choosing levels of significance by considering the costs of taking an erroneous decision based on a failed test [7].",
"Such type of argument in the case of earthquake clusters is not straightforward nor scientifically objective.",
"Regarding the unequivocal establishment of a scientific statement, the setting of a level of significance should take into consideration the probabilities such as those we have derived here.",
"In Figure REF , for example, the probability of a p-value above 20% is non-trivial ($>10\\%$ ).",
"In light of this discussion, we can consider the recent results of earthquake cluster detection.",
"Test (a) corresponds to the first test of [5] which was applied to the global catalogue at a large cut-off magnitude of $M_w=9$ , which corresponds to a frequency of approximately 0.04 event/year, and he reports p-values as low as 0.12.",
"We have shown that the same test would not be accurate event at much lower cut-off implying an event frequency of 0.1-0.2 event/year (corresponding magnitude thresholds between 8.4 and 8.3).",
"From [6], we are mostly interested in their multinomial test as it is equivalent to our case (c), which we assessed as the most powerful; in their work, the p-values reported for a $M_w>8$ magnitude cut-off range between 35 to 25% depending on the de-clustering undertaken.",
"The average event rate for these magnitudes is well above any we have analysed here (0.8-0.7 event/year).",
"More relevant to this discussion is [6] assertion that: \"(...) the null hypothesis that times of large earthquakes follow a homogeneous Poisson process would not be rejected by any of the tests\".",
"Based on our discussion, the criteria to accept (or reject) a hypothesis is not a clear-cut line.",
"These considerations go beyond the specifics of cluster detection (e.g.",
"[11] This work does not suggest that clustering is a real phenomenon – considering that our test-process is highly contrived.",
"This is a tentative way to introduce some objectivity into assertions such as \"random variability explains earthquake catalogue\": what would really be meant by \"explains\"?",
"The fact that a given series of events has a \"reasonable\" probability according to such a process (and we have yet to define what we mean by reasonable) – at most we could say it is consistent when favouring some prior, but as far a such limited data set is available such strong conclusions must not be taken for granted."
],
[
"acknowledgments",
"The author acknowledges the contributions of Profs Paul Somerville, Rob Van den Honert and John McAneney to this paper, and the financial support of Lloyd’s of London."
]
] | 1204.1131 |
[
[
"Delaunay Hodge Star"
],
[
"Abstract We define signed dual volumes at all dimensions for circumcentric dual meshes.",
"We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive.",
"This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes.",
"This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal.",
"A correct definition requires that all entries be positive.",
"DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing.",
"Our result allows the use of DEC with a much larger class of meshes than was previously considered possible."
],
[
"Introduction",
"Discrete Exterior Calculus (DEC) is a framework for numerical solution of partial differential equations on simplicial meshes and for geometry processing tasks [8], [4].",
"DEC has had considerable impact in computer graphics and scientific computing.",
"It is related to finite element exterior calculus and differs from it in how inner products are defined.",
"The main objects in DEC are $p$ -cochains, which for the purpose of this paper may be considered to be a vector of real values with one entry for each $p$ -dimensional simplex in the mesh.",
"For $p$ -cochains $a$ and $b$ their inner product in DEC is $a^T \\operatorname{\\ast }_p b$ where $\\operatorname{\\ast }_p$ is a diagonal discrete Hodge star operator.",
"This is a diagonal matrix of order equal to the number of $p$ -simplices and with entries that are ratios of volumes of $p$ -simplices and their $(n-p)$ -dimensional circumcentric dual cells.",
"For this to define a genuine inner product the entries have to be positive.",
"Simply taking absolute values or considering all volumes to be unsigned does not lead to correct solutions of partial differential equations.",
"See Figure REF to see the spectacular failure when unsigned volumes are used in solving Poisson's equation in mixed form.",
"That figure also shows the importance of the Delaunay property and of our boundary assumptions and the success of the signed dual volumes for such meshes that we describe in this paper.",
"When DEC was invented it was known that completely well-centered meshes were sufficient but perhaps not necessary for defining the Hodge star operator.",
"(A completely well-centered mesh is one in which the circumcenters are contained within the corresponding simplices at all dimensions.",
"Examples are acute-angled triangle meshes and tetrahedral meshes in which each triangle is acute and each tetrahedron contains its circumcenter.)",
"For such meshes, the volumes of circumcentric dual cells are obviously positive.",
"For some years now there has been numerical evidence that for codimension 1 duality some (but not all) pairwise Delaunay meshes yield positive dual volumes if volumes are given a sign based on some simple rules.",
"Moreover, these seem to yield correct numerical solutions for a simple partial differential equation.",
"A pairwise Delaunay mesh (of dimension $n$ embedded in $\\mathbb {R}^N$ , $N\\ge n$ ) is one in which each pair of adjacent $n$ -simplices sharing a face of dimension $n-1$ is Delaunay when embedded in $\\mathbb {R}^n$ .",
"(Imagine a pair of triangles with a hinge at the shared edge and lay the pair flat on a table.)",
"This generalizes the Delaunay condition to triangle mesh surfaces embedded in three dimensions and analogous higher dimensional settings.",
"For planar triangle meshes and for tetrahedral meshes in three dimensions pairwise Delaunay is same as Delaunay.",
"We give a sign convention for dual cells and a mild assumption on boundary simplices.",
"With these in hand, for pairwise Delaunay meshes it is easy to see that the codimension 1 dual lengths are positive in the most general case (dimension $n$ mesh embedded in $\\mathbb {R}^N$ ).",
"In addition, we prove that such triangle meshes embedded in two or three dimensions have positive vertex duals and that the duals of vertices and edges of tetrahedral meshes in three dimensions are positive.",
"This settles the question of positivity for all dimensions of duality for simplicial meshes used in physical and graphics applications and opens up the possibility of using DEC with a much larger class of meshes.",
"Note that the results of this paper are relevant for the assembly of the duals from elementary duals (defined in Section ).",
"Thus it is important for algorithms which compute these dual volumes piece by piece from the elementary duals (such as are used in the software PyDEC [2]).",
"It may be possible to have alternative formulas which bypass the assembly process and give the dual volumes directly.",
"Such formulas are hinted at in [9].",
"Zobel [11] has also hinted at the positivity of the duals in some cases but without proofs or details.",
"Figure: Solution of Poisson's equation -Δu=f-\\Delta u = f in mixedform.",
"In mixed form this equation is the system σ=-gradu\\sigma = -\\operatorname{grad}u and divσ=f\\operatorname{div}\\sigma = f. The boundary condition is constantinflux on left and outflux on right.",
"The correct solution islinear uu which varies only along x-direction and a constanthorizontal σ\\sigma .",
"The top row shows uu and bottom row showsσ\\sigma .",
"The first column shows the correct solution using theresults of this paper on a Delaunay mesh with correct boundarysimplices.",
"The next three columns show various failure modes ofalternatives.",
"Second column is for unsigned duals using the samemesh as first column.",
"The third column has a single bad (i.e., notone-sided – see Section ) boundary triangle shownshaded in a Delaunay mesh.",
"The fourth column is a non Delaunaymesh.There are alternatives to the discrete diagonal Hodge star.",
"In finite element exterior calculus, the role of discrete diagonal Hodge star is played by the mass matrix corresponding to polynomial differential forms [1].",
"This matrix is in general not diagonal.",
"Sometimes these are referred to as the Galerkin Hodge stars.",
"Our results do not apply to that case directly.",
"For finite elements, there are other quality measures that may be important.",
"See for instance [10].",
"Notation: We will sometimes write the dimension of a simplex as a superscript.",
"The notation $\\tau \\prec \\sigma $ means that $\\tau $ is a face of $\\sigma $ .",
"Circumcenter of a simplex $\\tau $ is denoted $c_\\tau $ and the circumcentric dual of $\\tau $ as $\\operatorname{\\star }\\tau $ .",
"Some figure labels like $\\operatorname{\\star }p$ stand for dual of a $p$ -simplex."
],
[
"Signed Circumcentric Dual Cells",
"The dual mesh of a (primal) simplicial mesh is often defined using a barycentric subdivision.",
"For every $p$ -dimensional simplex, there is a dual $(n - p)$ -dimensional cell where $n$ is the dimension of the simplicial complex.",
"In DEC circumcenters are used instead of barycenters because the resulting orthogonality between the primal and dual is an integral part of the definition of some of the operators of DEC [8].",
"The circumcentric dual cell of a $p$ -dimensional primal simplex $\\tau $ is constructed from a set of simplices incident to the circumcenter of $\\tau $ .",
"These are called elementary dual simplices with vertices being a sequence of circumcenters of primal simplices incident to $\\tau $ .",
"The sequence begins with the circumcenter of $\\tau $ , moves through circumcenters of higher-dimensional simplices $\\sigma ^i$ and ends with the circumcenter of a top-dimensional simplex $\\sigma ^n$ such that $\\tau \\prec \\sigma ^{p + 1} \\prec \\cdots \\prec \\sigma ^n$ .",
"Taking each of the possibilities for $\\sigma ^i$ at each dimension $i$ yields the full dual cell.",
"In the past in the software PyDEC the volume of the dual cells has been taken to be the sum of unsigned volumes of the elementary dual simplices.",
"Instead we define the volume of the dual cells as the sum of signed volumes of its elementary dual simplices.",
"Our contribution is in defining the sign convention and describing with proofs the class of meshes for which the dual volumes and hence Hodge star entries are positive.",
"If the primal complex is completely well centered every elementary dual has a positive volume.",
"In general, the sign of the volume of an elementary dual simplex is defined as follows.",
"Start from the circumcenter of $\\tau $ .",
"Let $v_p$ be the vertex such that $v_p * \\tau $ is the simplex $\\sigma ^{p + 1}$ formed by the vertices of $\\tau $ together with $v_p$ .",
"Similarly, for $p + 1\\le i \\le n - 1$ , let $v_i$ be the vertex such that $v_i * \\sigma ^i$ is the simplex $\\sigma ^{i + 1}$ .",
"If the circumcenter of $\\sigma ^{p +1}$ is in the same half space of $\\sigma ^{p+1}$ as $v_p$ relative to $\\tau $ , let $s_p = +1$ , otherwise, $s_p = -1$ .",
"Likewise, for $p + 1\\le i \\le n - 1$ , if the circumcenter of $\\sigma ^{i + 1}$ is in the same half space as $v_i$ relative to $\\sigma ^i$ , let $s_i = +1$ , otherwise, $s_i = -1$ .",
"Then, the sign $s$ of the elementary dual simplex is the product of the signs at each dimension, that is, $s =s_p \\, s_{p + 1} \\cdots s_n$ .",
"For illustration of this sign rule, we now consider various cases in two and three dimensions.",
"The first example is the dual of an edge $ab$ in a triangle $abc$ .",
"From the midpoint of $ab$ – its circumcenter – we move to the circumcenter of triangle $abc$ .",
"If this move is towards vertex $c$ , then the sign is $s = s_1 = +1$ , but if it is away from vertex $c$ , as it will be if the angle at vertex $c$ is obtuse, then the sign is $s = s_1 = -1$ .",
"The next example is the dual of vertex $a$ in triangle $abc$ .",
"We will consider the simplex formed from the circumcenter of $a$ , the circumcenter of $ac$ , and the circumcenter of $abc$ .",
"The move from $a$ to the midpoint of $ac$ gives $s_1 = +1$ , since vertex $c$ and the midpoint of $ac$ are in the same direction from $a$ .",
"The move from the midpoint of $ac$ to the circumcenter of $abc$ gives $s_2 = +1$ if we go towards $b$ and $s_2 = -1$ if we move away from $b$ .",
"The sign of the volume of this contribution to the dual of vertex $a$ is $s = s_1 \\, s_2 = s_2$ .",
"For a tetrahedron $abcd$ we can expand on the cases for triangle $abc$ .",
"For the dual to face $abc$ , we move from the circumcenter of $abc$ to the circumcenter of $abcd$ .",
"If the circumcenter of $abcd$ is in the same half space as vertex $d$ relative to $abc$ , this move is towards $d$ , the sign is $s = s_1 = +1$ , and the signed length (volume) is positive; otherwise, it is negative.",
"Of the two contributions to the dual of edge $ab$ , we focus on the simplex formed from the circumcenter of $ab$ , the circumcenter of $abc$ and the circumcenter of $abcd$ .",
"The sign $s_1$ is determined as it was for the dual of edge $ab$ in triangle $abc$ .",
"The sign $s_2$ is $+1$ if vertex $d$ and the circumcenter of $abcd$ are in the same half space relative to $abc$ .",
"Thus for the dual of edge $ab$ , the sign of the volume is $s = s_1 \\, s_2$ , and both $s_1$ and $s_2$ can be either positive or negative.",
"As a final example, consider the simplex formed from vertex $a$ , the circumcenter of $ac$ , the circumcenter of $abc$ , and the circumcenter of $abcd$ .",
"This simplex contributes to the dual of vertex $a$ .",
"Signs $s_1$ and $s_2$ are the same as they were for the dual of vertex $a$ in triangle $abc$ .",
"Sign $s_3$ is $-1$ if triangle $abc$ separates vertex $d$ from the circumcenter of tetrahedron $abcd$ .",
"The sign of this elementary volume then is $s =s_1 \\, s_2 \\, s_3$ .",
"Figure: Examples of sign rule application in 2d.",
"The dot marks thecircumcenter and green and red are used to denote positive andnegative volumes respectively.The significance of the sign rule defined above is that it orients the elementary dual simplices in a particular way with respect to the dual orientation for a completely well-centered simplex.",
"Consider two $n$ -dimensional simplices $\\sigma $ and $\\sigma _w$ which have the same orientation but such that $\\sigma _w$ is well-centered.",
"We are given a bijection between the vertices of these two simplices such that the resulting simplicial map is orientation preserving.",
"This vertex map induces a bijection between faces of the two simplices and between their elementary duals.",
"Let $\\tau $ and $\\tau _w$ be two corresponding $p$ -dimensional faces in the two simplices and consider their duals $\\operatorname{\\star }\\tau $ and $\\operatorname{\\star }\\tau _w$ .",
"If we consider two corresponding elementary duals in $\\operatorname{\\star }\\tau $ and $\\operatorname{\\star }\\tau _w$ we can affinely map these such that the first vertex (the circumcenter of $\\tau $ or $\\tau _w$ ) is mapped to the origin and the others are mapped to $+1$ or $-1$ along a coordinate axis.",
"For the elementary dual in $\\operatorname{\\star }\\tau _w$ we always choose $+1$ for all $n-p$ coordinate axes.",
"For $\\operatorname{\\star }\\tau $ we choose $+1$ if the sign along that direction of the elementary dual is positive according to the sign rule described above and $-1$ otherwise.",
"It is clear (and is easy to show using determinants) that the orientation of the corresponding elementary duals will be same if an even number of $-1$ directions are used for the elementary dual in $\\operatorname{\\star }\\tau $ and the orientations will be opposite otherwise.",
"Thus we have shown the following result.",
"With $\\sigma $ , $\\sigma _w$ , $\\tau $ , and $\\tau _w$ as above, the orientation of $\\operatorname{\\star }\\tau $ is same as that of $\\operatorname{\\star }\\tau _w$ if an even number of $-1$ signs appear according to sign rule and is opposite otherwise.",
"If the orientation of $\\operatorname{\\star }\\tau $ is same as $\\operatorname{\\star }\\tau _w$ we will assign a positive volume to $\\operatorname{\\star }\\tau $ and otherwise a negative volume."
],
[
"Signed Dual of a Delaunay Triangulation",
"We first consider the codimension 1 case in the most general setting of a simplicial complex of arbitrary dimension $n$ embedded in dimension $N \\ge n$ .",
"After that we consider cases other than codimension 1 but in more restricted settings.",
"For these latter cases we restrict ourselves to the physically most useful cases of triangle meshes embedded in two or three dimensions ($n = 2$ and $N = 2$ or 3) and tetrahedral meshes embedded in three dimensions ($n = N = 3$ ).",
"We conjecture that these results can be extended to the more general setting of arbitrary $n$ and $N\\ge n$ but those cases are not as important for physical applications and we leave those for future work.",
"For the general codimension 1 case we first prove the following basic fact about circumcenter ordering for Delaunay pairs.",
"[Circumcenter Order] Let $\\tau $ be an $(n - 1)$ -dimensional simplex in $\\mathbb {R}^n$ .",
"Let $L$ and $R$ be points such that $\\lambda = L * \\tau $ and $\\rho = R * \\tau $ form a non-degenerate Delaunay pair of $n$ -dimensional simplices with circumcenters $c_{\\lambda }$ and $c_{\\rho }$ , respectively.",
"Then, $c_{\\lambda }$ and $c_{\\rho }$ have the same relative ordering with respect to $\\tau $ as $L$ and $R$ .",
"Consider the collection of $(n-1)$ -dimensional spheres containing the vertices of $\\tau $ .",
"Since $\\lambda $ and $\\rho $ are a non-degenerate Delaunay pair, their circumspheres are empty and belong to this collection.",
"It is then easy to see that $c_{\\lambda }$ and $c_{\\rho }$ will be in the same order as $\\lambda $ and $\\rho $ .",
"See Figure REF .",
"For an alternative, more algebraic and detailed proof, see Appendix .",
"(In fact there we show the stronger result that the correctness of the circumcenter ordering is equivalent to the simplices being a non-degenerate Delaunay pair.)",
"Figure: For a Delaunay pair the ordering of the circumcenters isthe same as that of the top dimensional simplices.",
"SeeLemma .The above lemma can now be used to show easily that the codimension 1 duals always have positive net length.",
"This is the content of the next result.",
"[Codimension 1] Let $\\tau $ be a codimension 1 shared face of two $n$ -dimensional simplices embedded in $\\mathbb {R}^N$ , $N \\ge n$ forming a Delaunay pair.",
"Then the signed length $\\operatorname{\\star }\\tau $ is positive.",
"When $N = n$ , the results directly follows from Lemma since the circumcenters are in the correct order.",
"For $N > n$ , we can isometrically embed the simplices in $\\mathbb {R}^n$ in which case, the circumcenters are again in the correct order and the result follows.",
"In the $N > n$ case, the signs of the elementary dual edges of $\\operatorname{\\star }\\tau $ are assigned in the affine spaces of the corresponding $n$ -dimensional simplices.",
"For example, consider a pair of triangles embedded in $\\mathbb {R}^3$ and meeting at a shared edge at an angle other than $\\pi $ .",
"In this case, the signed length of the dual edge of the shared edge is determined as the sum of the two elementary dual edges which are measured in the planes of the two triangles individually."
],
[
"Dual of a Vertex in Triangle Mesh Surface",
"Now we show that the area of the dual of an internal vertex in a pairwise Delaunay triangle mesh is always positive.",
"We prove this below by showing that the net dual area corresponding to a pair of triangles is positive.",
"Let $\\tau $ be an internal vertex in a pairwise Delaunay triangle mesh embedded in $\\mathbb {R}^N$ , $N = 2, 3$ .",
"Then the signed area of $\\operatorname{\\star }\\tau $ is a positive.",
"$\\operatorname{\\star }\\tau $ is the Voronoi cell of vertex $\\tau $ in the pairwise Delaunay mesh.",
"Consider a pair of triangles sharing a common edge incident to $\\tau $ and if they are embedded in $\\mathbb {R}^3$ , isometrically project to $\\mathbb {R}^2$ (i.e., treat the shared edge as a hinge, and flatten the pair.)",
"The circumcenters of these two triangles are in correct order by Lemma and there are three possible cases as shown in Figure REF .",
"Thus the net area of the two elementary dual simplices is positive when the signs are assigned using the rule described in Section .",
"Summing over all edges containing $\\tau $ yields the full $\\operatorname{\\star }\\tau $ as a positive area.",
"Figure: Elementary dual simplices of a vertex in a pair oftriangles sharing an edge.",
"The cases shown correspond to variouspositions of the circumcenters of the shared edge and the twotriangles."
],
[
"Dual of an Edge in Tetrahedral Mesh",
"Let $\\tau $ be an internal edge in a tetrahedral Delaunay triangulation embedded in $\\mathbb {R}^3$ .",
"Then $\\operatorname{\\star }\\tau $ is a simple, planar, convex polygon whose signed area is positive.",
"$\\operatorname{\\star }\\tau $ of an internal edge $\\tau $ in a Delaunay triangulation may or may not intersect $\\tau $ .",
"The vertices of $\\operatorname{\\star }\\tau $ are circumcenters of tetrahedra incident to $\\tau $ and the boundary edges of $\\operatorname{\\star }\\tau $ are dual edges of triangles incident to $\\tau $ .",
"Note that $\\operatorname{\\star }\\tau $ is the interface between the Voronoi cells corresponding to the two vertices of $\\tau $ and thus is a bounding face of both Voronoi cells.",
"Since the Voronoi cell of a vertex is a convex polyhedron [6], $\\operatorname{\\star }\\tau $ is simple, planar and convex.",
"Suppose $\\tau $ intersects $\\operatorname{\\star }\\tau $ .",
"Then the tetrahedra incident to $\\tau $ and the edges of $\\operatorname{\\star }\\tau $ have to be in a configuration shown in left part of Figure REF .",
"A configuration in which the triangles incident to $\\tau $ are reflected about $\\tau $ is impossible due to Lemma .",
"Now, to see that the signed area of $\\operatorname{\\star }\\tau $ is positive, consider two elementary dual simplices of $\\operatorname{\\star }\\tau $ incident to a shared face $\\sigma $ of two tetrahedra in the fan of tetrahedra incident to $\\tau $ .",
"These two elementary dual simplices can be in one of the two configurations as shown in Figure REF .",
"In both cases, $c_{\\tau }$ is the circumcenter of the edge $\\tau $ , $c_{\\sigma }$ is the circumcenter of the shared face $\\sigma $ , and $c_{\\rho }$ and $c_{\\lambda }$ are the circumcenters of the two tetrahedra.",
"Also, in both cases, using the sign rule of Section the sum of the signed areas of the elementary dual simplices is positive, and hence, the signed area of $\\operatorname{\\star }\\tau $ composed of these elementary dual simplices is positive.",
"Next consider the case in which $\\tau $ does not intersect $\\operatorname{\\star }\\tau $ as shown in right part of Figure REF .",
"A boundary edge of $\\operatorname{\\star }\\tau $ is called near side if it is visible from the midpoint of $\\tau $ , otherwise, it is called a far side edge.",
"Figure REF shows the net dual simplices of a near side and far side boundary edge of $\\operatorname{\\star }\\tau $ .",
"By the sign rule of Section , far side elementary dual simplices have a net positive signed area while near side elementary dual simplices have a net negative signed area.",
"The negative areas of the near side dual simplices are covered by the positive areas of the far side dual simplices.",
"Thus, the sum of all these elementary dual simplices which is the signed area of $\\operatorname{\\star }\\tau $ is positive.",
"Figure: An internal edge τ\\tau of a tetrahedral mesh may or may notintersect errorτ\\operatorname{\\star }\\tau .",
"The views here are along τ\\tau whichappears as a point.",
"The short lines are half-planes of the trianglesincident to τ\\tau .",
"The tetrahedra are labeled aa, bb, etc.",
"Eachboundary edge of errorτ\\operatorname{\\star }\\tau corresponds to the triangle indicatedby the coloring.",
"The half planes could potentially be a reflectionabout τ\\tau but that is impossible in a Delaunay mesh due toLemma .Figure: Representative elementary dual simplices of errorτ\\operatorname{\\star }\\tau when it intersects τ\\tau (left side) and does not intersectτ\\tau (right side) corresponding to the two cases shown inFigure ."
],
[
"Dual of a Vertex in Tetrahedral Mesh",
"Let $\\tau $ be an internal vertex of a tetrahedra Delaunay mesh embedded in $\\mathbb {R}^3$ .",
"Then the volume of $\\operatorname{\\star }\\tau $ is positive.",
"$\\operatorname{\\star }\\tau $ of a vertex $\\tau $ in a Delaunay tetrahedral mesh is a convex polyhedron that is the Voronoi dual cell of $\\tau $ [6] and thus $\\tau $ is inside $\\operatorname{\\star }\\tau $ .",
"The faces of $\\operatorname{\\star }\\tau $ are duals of edges incident to $\\tau $ .",
"By Theorem REF all these faces have a positive signed area.",
"The direction corresponding to traversal from $\\tau $ to an edge center always has a positive sign.",
"Thus each pyramid formed by $\\tau $ and a boundary face of $\\star \\tau $ has positive volume.",
"Thus, the volume of $\\star \\tau $ is positive."
],
[
"Requirements on Boundary Simplices",
"In the previous section we have only considered internal simplices in a pairwise Delaunay mesh.",
"For simplices lying in the boundary of a domain we require an assumption to ensure positive duals.",
"We call a simplex $\\sigma $ one-sided with respect to a codimension 1 face $\\tau $ if its circumcenter $c_{\\sigma }$ lies in the same half space as the apex with respect to $\\tau $ in the affine space of $\\sigma $ .",
"We show below that the only assumption then needed is that a top dimensional simplex with a codimension 1 face in the domain boundary should be one-sided with respect to the boundary face.",
"Consider a pairwise Delaunay mesh of dimension $n$ embedded in $\\mathbb {R}^N$ , $N \\ge n$ .",
"Assume that $\\tau $ is an $(n - 1)$ -dimensional face appearing in domain boundary and $\\tau \\prec \\sigma ^n$ such that $\\sigma ^n$ is one-sided with respect to $\\tau $ .",
"For a mesh such as above, a dual of codimension 1 faces has positive length.",
"For $n = 2$ and $N = 2$ or 3, and for $n = N = 3$ , duals of all simplices at all dimensions have positive areas or volumes.",
"The codimension 1 dual of $\\tau $ in all cases has positive length using our sign rule since $\\sigma ^n$ is one-sided with respect to $\\tau $ .",
"As a result, for a surface triangle mesh, that is $n = 2$ and $N = 2$ or 3, it easily follows from our sign rule in Section that the dual of a vertex on the boundary also has a positive area.",
"For $n = N = 3$ , one configuration for the dual of an edge $\\tau $ incident to the boundary is shown in Figure REF .",
"In this figure, the plane containing the codimension 1 faces incident to $\\tau $ are shown as short line segments, and the coloring of boundary edges of $\\star \\tau $ show the corresponding codimension 1 face they are dual to.",
"The other configuration in which the planes containing the faces incident to $\\tau $ are mirror images of ones shown is not possible since then the circumcenters of tetrahedra will not be in the correct order as in Lemma .",
"Thus, by our sign rule, all elementary dual simplices of $\\star \\tau $ are positive and hence the signed area of $\\star \\tau $ is positive.",
"Finally, it follows from our sign rule that the dual of a vertex on the boundary is also positive since each of the elementary dual pyramids will have a positive volume.",
"Figure: Dual of an edge τ\\tau lying in the boundary of a Delaunaytetrahedral mesh.",
"The meaning of colors and small lines is as inFigure ."
],
[
"Conclusions and Outlook",
"For planar triangle meshes and for tetrahedral meshes in three dimensional space the condition of being pairwise Delaunay is equivalent to being Delaunay.",
"Thus most commercial and freely available meshing software can generate such meshes.",
"In our experience, several codes for planar meshing also generate meshes for which the one-sidedness condition on the boundary is satisfied.",
"For example, the popular meshing code called Triangle has an option for conforming Delaunay triangulation which generates Delaunay meshes with one-sided boundary triangles.",
"For tetrahedral meshes with acute input angles this property may be harder to achieve.",
"In general however, algorithms for creating tetrahedral meshes with one-sided boundary tetrahedra do exist [5], [3], [7].",
"Note that one-sidedness is equivalent to an “oriented” Gabriel property (using diametral half-balls) for the boundary faces.",
"The pairwise Delaunay condition also appears to be more natural for DEC than other conditions that are used in place of Delaunay in the case of surfaces.",
"For example, some researchers require that the equatorial balls of triangles not contain another vertex.",
"This disqualifies surfaces with many folds or sharp turns.",
"Another alternative is to define intrinsic Delaunay condition based on geodesics on the triangle mesh but algorithms for such surfaces can be complicated to implement.",
"Yet another alternative is to use Hodge-optimized triangulations [9].",
"But creation of these requires an additional optimization step.",
"On the other hand Hodge-optimized triangulation is a very interesting generalization of Voronoi-Delaunay duality with many applications.",
"The invention of algorithms that generate pairwise Delaunay surface meshes is left for future work.",
"So is the proof of our conjecture that the case of codimension other than 1 has positive volume for general dimension and embedding space for pairwise Delaunay meshes with one-sided boundary simplices.",
"Nevertheless, the practically important cases have all been settled by this paper.",
"Acknowledgement: ANH and KK were supported by NSF Grant DMS-0645604.",
"We thank the anonymous referees for pointing out some important references and for their other suggestions."
],
[
"Circumcenter Order",
"In fact here we prove a stronger result than Lemma .",
"We will show that the circumcenters are in the correct order if and only if the pair of simplices is a non-degenerate Delaunay pair.",
"Let $\\tau $ be an $(n - 1)$ -dimensional simplex in $\\mathbb {R}^n$ .",
"Let $L$ and $R$ be points separated by $\\tau $ .",
"Let $c_{\\lambda }$ and $c_{\\rho }$ be the circumcenters of the $n$ -dimensional simplices $\\lambda = L *\\tau $ and $\\rho = R * \\tau $ , respectively.",
"Then, $c_{\\lambda }$ and $c_{\\rho }$ have the same relative ordering with respect to $\\tau $ as $L$ and $R$ if and only if $\\lambda $ and $\\rho $ are a pair of non-degenerate Delaunay simplices.",
"Since $\\lambda $ and $\\rho $ are a Delaunay pair, the affine space of $\\tau $ separates $L$ and $R$ .",
"See Figure REF .",
"Let $c_\\tau $ and $r_\\tau $ be the circumcenter and the circumradius of $\\tau $ , respectively.",
"Now, $c_\\lambda $ and $c_\\rho $ lie on a line $\\ell $ that passes through $c_\\tau $ and is orthogonal to the affine space of $\\tau $ .",
"Let $h_\\lambda $ be the signed distance along $\\ell $ from $c_\\lambda $ to $c_\\tau $ .",
"Similarly, let $h_\\rho $ be the signed distance from $c_\\rho $ to $c_\\tau $ .",
"For now, it is sufficient that these distances be signed and whether the positive direction is along $L$ or $R$ is not important.",
"Next, orthogonally project $R$ onto $\\ell $ , and let $r_R$ be the (positive) distance from $R$ to its projection onto $\\ell $ .",
"Finally, let $h_R$ be the signed distance (along $\\ell $ ) from the projection of $R$ onto $\\ell $ to $c_\\tau $ .",
"Notice that $h_R$ is necessarily either negative or positive depending on the choice of positive direction to be either along $L$ or $R$ , respectively.",
"By elementary geometry, the squared circumradius of $\\lambda $ is $h_\\lambda ^2 + r_\\tau ^2$ and the squared circumradius of $\\rho $ is $h_\\rho ^2 + r_\\tau ^2$ .",
"Similarly, the squared distance from $c_\\lambda $ to $R$ is $(h_\\lambda - h_R)^2 + r_R^2$ .",
"Since $\\lambda $ and $\\rho $ form a Delaunay pair, $R$ lies outside the circumsphere of $\\lambda $ .",
"Thus, the squared circumradius of $\\lambda $ is less than the squared distance from $c_\\lambda $ to $R$ : $&& h_\\lambda ^2 + r_\\tau ^2 & < (h_\\lambda - h_R)^2 + r_R^2 \\, , \\\\\\Rightarrow && r_\\tau ^2 & < r_R^2 + h_R^2 - 2\\, h_R \\, h_\\lambda \\, .$ Also, since $R$ lies on the circumsphere of $\\rho $ , the distance from $c_\\rho $ to $R$ is the same as the distance from $c_\\rho $ to a vertex of $\\tau $ .",
"Thus, we have: $&& h_\\rho ^2 + r_\\tau ^2 & = (h_\\rho - h_R)^2 + r_R^2 \\, , \\\\\\Rightarrow && r_\\tau ^2 & = r_R^2 + h_R^2 - 2 \\, h_R \\, h_\\rho \\, .$ Using this in the previous inequality, we obtain: $&& r_R^2 + h_R^2 - 2 \\, h_R \\, h_\\rho & < r_R^2 + h_R^2 - 2 \\, h_R \\,h_\\lambda \\, , \\\\\\Rightarrow && h_R \\, h_\\rho & > h_R \\, h_\\lambda \\, .$ Finally, we choose a coordinate direction along $\\ell $ to fix signs in the signed distances along $\\ell $ .",
"If we choose the direction towards the half space containing $R$ to be positive, $h_R$ is positive.",
"(We will call this the positive $R$ -direction.)",
"As a result, the last inequality above simplifies to $h_\\rho > h_\\lambda $ .",
"This means that $h_\\rho $ is larger along the positive $R$ -direction.",
"If we choose the direction along $L$ to be positive, $h_R$ is negative and we obtain $h_\\rho < h_\\lambda $ .",
"In this case, $h_\\lambda $ is larger along the positive $L$ -direction.",
"Conversely, if $\\lambda $ and $\\rho $ are not a Delaunay pair, then the distance from $c_\\lambda $ to $R$ is less than the circumradius of $\\lambda $ .",
"Thus, all inequalities will reverse directions and therefore the circumradii will be in the wrong order."
]
] | 1204.0747 |
[
[
"On the Roman bondage number of a graph"
],
[
"Abstract A Roman dominating function on a graph $G=(V,E)$ is a function $f:V\\rightarrow\\{0,1,2\\}$ such that every vertex $v\\in V$ with $f(v)=0$ has at least one neighbor $u\\in V$ with $f(u)=2$.",
"The weight of a Roman dominating function is the value $f(V(G))=\\sum_{u\\in V(G)}f(u)$.",
"The minimum weight of a Roman dominating function on a graph $G$ is called the Roman domination number, denoted by $\\gamma_{R}(G)$.",
"The Roman bondage number $b_{R}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $E'\\subseteq E(G)$ for which $\\gamma_{R}(G-E')>\\gamma_R(G)$.",
"In this paper, we first show that the decision problem for determining $b_{\\rm R}(G)$ is NP-hard even for bipartite graphs and then we establish some sharp bounds for $b_{\\rm R}(G)$ and characterizes all graphs attaining some of these bounds."
],
[
"Introduction",
"For terminology and notation on graph theory not given here, the reader is referred to [13], [14], [35].",
"In this paper, $G$ is a simple graph with vertex set $V=V(G)$ and edge set $E=E(G)$ .",
"The order $|V|$ of $G$ is denoted by $n=n(G)$ .",
"For every vertex $v\\in V$ , the open neighborhood $N(v)$ is the set $\\lbrace u\\in V\\mid uv\\in E\\rbrace $ and the closed neighborhood of $v$ is the set $N[v] = N(v) \\cup \\lbrace v\\rbrace $ .",
"The degree of a vertex $v\\in V$ is $\\deg _G(v)=\\deg (v)=|N(v)|$ .",
"The minimum and maximum degree of a graph $G$ are denoted by $\\delta =\\delta (G)$ and $\\Delta =\\Delta (G)$ , respectively.",
"The open neighborhood of a set $S\\subseteq V$ is the set $N(S)=\\cup _{v\\in S}N(v)$ , and the closed neighborhood of $S$ is the set $N[S]=N(S)\\cup S$ .",
"The complement $\\overline{G}$ of $G$ is the simple graph whose vertex set is $V$ and whose edges are the pairs of nonadjacent vertices of $G$ .",
"We write $K_n$ for the complete graph of order $n$ and $C_n$ for a cycle of length $n$ .",
"For two disjoint nonempty sets $S,T\\subset V(G)$ , $E_G(S,T)=E(S,T)$ denotes the set of edges between $S$ and $T$ .",
"A subset $S$ of vertices of $G$ is a dominating set if $|N(v)\\cap S|\\ge 1$ for every $v\\in V-S$ .",
"The domination number $\\gamma (G)$ is the minimum cardinality of a dominating set of $G$ .",
"To measure the vulnerability or the stability of the domination in an interconnection network under edge failure, Fink et at.",
"[10] proposed the concept of the bondage number in 1990.",
"The bondage number, denoted by $b(G)$ , of $G$ is the minimum number of edges whose removal from $G$ results in a graph with larger domination number.",
"An edge set $B$ for which $\\gamma (G-B)>\\gamma (G)$ is called a bondage set.",
"A $b(G)$ -set is a bondage set of $G$ of size $b(G)$ .",
"If $B$ is a $b(G)$ -set, then obviously $\\gamma (G-B)=\\gamma (G)+1.$ A Roman dominating function on a graph $G$ is a labeling $f:V\\rightarrow \\lbrace 0, 1, 2\\rbrace $ such that every vertex with label 0 has at least one neighbor with label 2.",
"The weight of a Roman dominating function is the value $f(V(G))=\\sum _{u\\in V(G)}f(u)$ , denoted by $f(G)$ .",
"The minimum weight of a Roman dominating function on a graph $G$ is called the Roman domination number, denoted by $\\gamma _{R}(G)$ .",
"A $\\gamma _{R}(G)$ -function is a Roman dominating function on $G$ with weight $\\gamma _{R}(G)$ .",
"A Roman dominating function $f : V\\rightarrow \\lbrace 0, 1, 2\\rbrace $ can be represented by the ordered partition $(V_0,V_1, V_2)$ (or $(V_{0}^{f},V_{1}^{f},V_{2}^{f})$ to refer to $f$ ) of $V$ , where $V_i=\\lbrace v\\in V\\mid f(v) = i\\rbrace $ .",
"In this representation, its weight is $\\omega (f)=|V_1|+2|V_2|$ .",
"It is clear that $V_1^f\\cup V_2^f$ is a dominating set of $G$ , called the Roman dominating set, denoted by $D^f_{\\rm R}=(V_1,V_2)$ .",
"Since $V_1^f\\cup V^f_2$ is a dominating set when $f$ is an RDF, and since placing weight 2 at the vertices of a dominating set yields an RDF, in [4], it was observed that $\\gamma (G)\\le \\gamma _{R}(G)\\le 2\\gamma (G).$ A graph $G$ is called to be Roman if $\\gamma _{\\rm R}(G)=2\\gamma (G)$ .",
"The definition of the Roman dominating function was given implicitly by Stewart [26] and ReVelle and Rosing [25].",
"Cockayne, Dreyer Jr., Hedetniemi and Hedetniemi [4] as well as Chambers, Kinnersley, Prince and West [3] have given a lot of results on Roman domination.",
"For more information on Roman domination we refer the reader to [3], [4], [5], [9], [11], [16], [17], [18], [21], [22], [27], [28], [29], [30], [33].",
"Let $G$ be a graph with maximum degree at least two.",
"The Roman bondage number $b_{R}(G)$ of $G$ is the minimum cardinality of all sets $E^{\\prime }\\subseteq E$ for which $\\gamma _{R}(G-E^{\\prime })>\\gamma _{R}(G)$ .",
"Since in the study of Roman bondage number the assumption $\\Delta (G)\\ge 2$ is necessary, we always assume that when we discuss $b_R(G)$ , all graphs involved satisfy $\\Delta (G)\\ge 2$ .",
"The Roman bondage number $b_R(G)$ was introduced by Jafari Rad and Volkmann in [23], and has been further studied for example in [1], [6], [7], [8], , [24].",
"An edge set $B$ that $\\gamma _{\\rm R}(G-B)>\\gamma _{\\rm R}(G)$ is called the Roman bondage set.",
"A $b_R(G)$ -set is a Roman bondage set of $G$ of size $b_R(G)$ .",
"If $B$ is a $b_R(G)$ -set, then clearly $\\gamma _{\\rm R}(G-B)=\\gamma _{\\rm R}(G)+1.$ In this paper, we first show that the decision problem for determining $b_{\\rm R}(G)$ is NP-hard even for bipartite graphs and then we establish some sharp bounds for $b_{\\rm R}(G)$ and characterizes all graphs attaining some of these bounds.",
"We make use of the following results in this paper.",
"Proposition A (Chambers et al.",
"[3]) If G is a graph of order $n$ , then $\\gamma _R(G)\\le n-\\Delta (G) + 1$ .",
"Proposition B (Cockayne et al.",
"[4]) For a grid graph $P_2\\times P_n$ , $\\gamma _{\\rm R}(P_2\\times P_n)=n+1.$ Proposition C (Cockayne et al.",
"[4]) For any graph $G$ , $\\gamma (G)\\le \\gamma _{\\rm R}(G)\\le 2\\gamma (G)$ .",
"Proposition D (Cockayne et al.",
"[4]) For any graph $G$ of order n, $\\gamma (G)=\\gamma _{\\rm R}(G)$ if and only if $G=\\bar{K_n}$ .",
"Proposition E (Cockayne et al.",
"[4]) If $G$ is a connected graph of order n, then $\\gamma _{\\rm R}(G) =\\gamma (G)+1$ if and only if there is a vertex $v\\in V(G)$ of degree $n-\\gamma (G)$ .",
"Proposition F (Hu and Xu [20]) If $G=K_{3,3,\\ldots ,3}$ is the complete $t$ -partite graph of order $n\\ge 9$ , then $b_{\\rm R}(G)=n-1$ .",
"Proposition G (Jafari Rad and Volkmann [23]) If $G$ is a connected graph of order $n\\ge 3$ , then $b_{\\rm R}(G)\\le \\delta (G)+2\\Delta (G)-3$ .",
"Proposition H (Fink et al.",
"[10], Rad and Volkmann [23]) For a cycle $C_n$ of order $n$ , $b(C_n)= \\left\\lbrace \\begin{array}{ll}3, & {\\rm if}\\ n=1\\, ({\\rm mod}\\, 3);\\\\2, & {\\rm otherwise}.\\end{array} \\right.$ $b_{\\rm R}(C_n)= \\left\\lbrace \\begin{array}{ll}3, & {\\rm if}\\ n=2\\, ({\\rm mod}\\, 3);\\\\2, & {\\rm otherwise}.\\end{array} \\right.$ Observation 1 Let $G$ be a connected graph of order $n\\ge 3$ .",
"Then $\\gamma _{\\rm R}(G)=2$ if and only if $\\Delta (G)=n-1$ .",
"Observation 2 Let $G$ be a graph of order $n$ with maximum degree at least two.",
"Assume that $H$ is a spanning subgraph of $G$ with $\\gamma _{\\rm R}(H)=\\gamma _{\\rm R}(G)$ .",
"If $K=E(G)-E(H)$ , then $b_{\\rm R}(H)\\le b_{\\rm R}(G)\\le b_{\\rm R}(H)+|K|$ .",
"Proposition I Let $G$ be a nonempty graph of order $n\\ge 3$ , then $\\gamma _{\\rm R}(G)=3$ if and only if $\\Delta (G)=n-2$ .",
"Let $\\Delta (G)=n-2$ .",
"Assume that $u$ is a vertex of degree $n-2$ and $v$ is the unique vertex not adjacent to $u$ in $G$ .",
"By Observation REF , $\\gamma _{\\rm R}(G)\\ge 3$ and clearly $f=(V(G)-\\lbrace u,v\\rbrace ,\\lbrace v\\rbrace ,\\lbrace u\\rbrace )$ is a Roman dominating set of $G$ with $f(G)=3$ .",
"Thus, $\\gamma _{\\rm R}(G)=3$ .",
"Conversely, assume $\\gamma _{\\rm R}(G)=3$ .",
"Then $\\Delta (G)\\le n-2$ by Proposition REF .",
"Let $f=(V_0,V_1,V_2)$ be a $\\gamma _{\\rm R}$ -function of $G$ .",
"If $V_2=\\emptyset $ , then $f(v)=1$ for each vertex $v\\in V(G)$ , and hence $n=3$ .",
"Sine $G$ is nonempty and $\\Delta (G)\\le n-2=1$ , we have $\\Delta (G)=n-2=1$ .",
"Let $V_2\\ne \\emptyset $ .",
"Since $\\gamma _{\\rm R}(G)=3$ , we deduce that $|V_1|=|V_2|=1$ .",
"Suppose $V_1=\\lbrace v\\rbrace $ and $V_2=\\lbrace u\\rbrace $ .",
"Then other $n-2$ vertices assigned 0 are must be adjacent to $u$ .",
"Thus, $\\Delta (G)\\ge d_G(u)\\ge n-2$ and hence $\\Delta (G)=n-2$ ."
],
[
"Complexity of Roman bondage number",
"In this section, we will show that the Roman bondage number problem is NP-hard and the Roman domination number problem is NP-complete even for bipartite graphs.",
"We first state the problem as the following decision problem.",
"Roman bondage number problem (RBN): Instance: A nonempty bipartite graph $G$ and a positive integer $k$ .",
"Question: Is $b_{\\rm R}(G)\\le k$ ?",
"Roman domination number problem (RDN): Instance: A nonempty bipartite graph $G$ and a positive integer $k$ .",
"Question: Is $\\gamma _{\\rm R}(G)\\le k$ ?",
"Following Garey and Johnson's techniques for proving NP-completeness given in [12], we prove our results by describing a polynomial transformation from the known-well NP-complete problem: 3SAT.",
"To state 3SAT, we recall some terms.",
"Let $U$ be a set of Boolean variables.",
"A truth assignment for $U$ is a mapping $t: U\\rightarrow \\lbrace T,F\\rbrace $ .",
"If $t(u)=T$ , then $u$ is said to be “ true\" under $t$ ; If $t(u)=F$ , then $u$ is said to be “ false\" under $t$ .",
"If $u$ is a variable in $U$ , then $u$ and $\\bar{u}$ are literals over $U$ .",
"The literal $u$ is true under $t$ if and only if the variable $u$ is true under $t$ ; the literal $\\bar{u}$ is true if and only if the variable $u$ is false.",
"A clause over $U$ is a set of literals over $U$ .",
"It represents the disjunction of these literals and is satisfied by a truth assignment if and only if at least one of its members is true under that assignment.",
"A collection $C$ of clauses over $U$ is satisfiable if and only if there exists some truth assignment for $U$ that simultaneously satisfies all the clauses in $C$ .",
"Such a truth assignment is called a satisfying truth assignment for $C$ .",
"The 3SAT is specified as follows.",
"3-satisfiability problem (3SAT): Instance: A collection ${C}=\\lbrace C_1,C_2,\\ldots ,C_m\\rbrace $ of clauses over a finite set $U$ of variables such that $|C_j| =3$ for $j=1, 2,\\ldots ,m$ .",
"Question: Is there a truth assignment for $U$ that satisfies all the clauses in ${C}$ ?",
"Theorem 3 (Theorem 3.1 in [12]) 3SAT is NP-complete.",
"Theorem 4 RBN is NP-hard even for bipartite graphs.",
"The transformation is from 3SAT.",
"Let $U=\\lbrace u_1,u_2,\\ldots ,u_n\\rbrace $ and ${C}=\\lbrace C_1,C_2,\\ldots ,$ $C_m\\rbrace $ be an arbitrary instance of 3SAT.",
"We will construct a bipartite graph $G$ and choose an integer $k$ such that ${C}$ is satisfiable if and only if $b_{\\rm R}(G)\\le k$ .",
"We construct such a graph $G$ as follows.",
"For each $i=1,2,\\ldots ,n$ , corresponding to the variable $u_i\\in U$ , associate a graph $H_i$ with vertex set $V(H_i)=\\lbrace u_i,\\bar{u}_i,v_i,v_i^{\\prime },x_i,y_i,z_i,w_i\\rbrace $ and edge set $E(H_i)=\\lbrace u_iv_i,u_iz_i,\\bar{u}_iv_i^{\\prime },\\\\\\bar{u}_iz_i,y_iv_i,y_iv_i^{\\prime },y_iz_i,w_iv_i,w_iv_i^{\\prime },w_iz_i,x_iv_i,x_iv_i^{\\prime }\\rbrace $ .",
"For each $j=1,2,\\ldots ,m$ , corresponding to the clause $C_j=\\lbrace p_j,q_j,r_j\\rbrace \\in {C}$ , associate a single vertex $c_j$ and add edge set $E_j=\\lbrace c_jp_j, c_jq_j,c_jr_j\\rbrace $ , $1\\le j\\le m$ .",
"Finally, add a path $P=s_1s_2s_3$ , join $s_1$ and $s_3$ to each vertex $c_j$ with $1\\le j\\le m$ and set $k=1$ .",
"Figure REF shows an example of the graph obtained when $U=\\lbrace u_1,u_2,u_3,u_4\\rbrace $ and ${C}=\\lbrace C_1,C_2,C_3\\rbrace $ , where $C_1=\\lbrace u_1,u_2,\\bar{u}_3\\rbrace , C_2=\\lbrace \\bar{u}_1,u_2,u_4\\rbrace ,C_3=\\lbrace \\bar{u}_2,u_3,u_4\\rbrace $ .",
"Figure: An instance of the Roman bondagenumber problem resulting from an instance of 3SAT.",
"Here k=1k=1 andγ R (G)=18\\gamma _{\\rm R}(G)=18, where the bold vertex ww means a Romandominating function with f(w)=2f(w)=2.To prove that this is indeed a transformation, we only need to show that $b_{\\rm R}(G)=1$ if and only if there is a truth assignment for $U$ that satisfies all clauses in ${C}$ .",
"This aim can be obtained by proving the following four claims.",
"Claim 4.1 $\\gamma _{\\rm R}(G)\\ge 4n+2$ .",
"Moreover, if $\\gamma _{\\rm R}(G)=4n+2$ , then for any $\\gamma _{\\rm R}$ -function $f$ on $G$ , $f(H_i)=4$ and at most one of $f(u_i)$ and $f(\\bar{u}_i)$ is 2 for each $i$ , $f(c_j)=0$ for each $j$ and $f(s_2)=2$ .",
"Let $f$ be a $\\gamma _{\\rm R}$ -function of $G$ , and let $H_i^{\\prime }=H_i-u_i-\\bar{u}_i$ .",
"If $f(u_i)=2$ and $f(\\bar{u}_i)=2$ , then $f(H_i)\\ge 4$ .",
"Assume either $f(u_i)=2$ or $f(\\bar{u}_i)=2$ , if $f(x_i)=0$ or $f(y_i)=0$ , then there is at least one vertex $t$ in $\\lbrace v_i,v_i^{\\prime },z_i\\rbrace $ such that $f(t)=2$ .",
"And hence $f(H_i^{\\prime })\\ge 2$ .",
"Thus, $f(H_i)\\ge 4$ .",
"If $f(u_i)\\ne 2$ and $f(\\bar{u}_i)\\ne 2$ , let $f^{\\prime }$ be a restriction of $f$ on $H_i^{\\prime }$ , then $f^{\\prime }$ is a Roman dominating function of $H_i^{\\prime }$ , and $f^{\\prime }(H_i^{\\prime })\\ge \\gamma _{\\rm R}(H_i^{\\prime })$ .",
"Since the maximum degree of $H_i^{\\prime }$ is $V(H_i^{\\prime })-3$ , by Lemma REF , $\\gamma _{\\rm R}(H_i^{\\prime })>3$ and hence $f^{\\prime }(H_i^{\\prime })\\ge 4$ and $f(H_i)\\ge 4$ .",
"If $f(s_1)=0$ or $f(s_3)=0$ , then there is at least one vertex $t$ in $\\lbrace c_1, \\cdots ,c_m, s_2\\rbrace $ such that $f(t)=2$ .",
"Then $f(N_G[V(P)])\\ge 2$ , and hence $\\gamma _{\\rm R}(G)\\ge 4n+2$ .",
"Suppose that $\\gamma _{\\rm R}(G)=4n+2$ , then $f(H_i)=4$ and since $f(N_G[x_i])\\ge 1$ , at most one of $f(u_i)$ and $f(\\bar{u}_i)$ is 2 for each $i=1,2,\\ldots ,n$ , while $f(N_G[V(P)])=2$ .",
"It follows that $f(s_2)=2$ since $f(N_G[s_2])\\ge 1$ .",
"Consequently, $f(c_j)=0$ for each $j=1,2,\\ldots ,m$ .",
"Claim 4.2 $\\gamma _{\\rm R}(G)=4n+2$ if and only if ${C}$ is satisfiable.",
"Suppose that $\\gamma _{\\rm R}(G)=4n+2$ and let $f$ be a $\\gamma _{\\rm R}$ -function of $G$ .",
"By Claim 4.1, at most one of $f(u_i)$ and $f(\\bar{u}_i)$ is 2 for each $i=1,2,\\ldots ,n$ .",
"Define a mapping $t: U\\rightarrow \\lbrace T,F\\rbrace $ by $t(u_i)=\\left\\lbrace \\begin{array}{l}T \\ \\ {\\rm if}\\ f(u_i)=2\\ {\\rm or}\\ f(u_i)\\ne 2\\ {\\rm and} f(\\bar{u}_i)\\ne 2, \\\\F \\ \\ {\\rm if}\\ f(\\bar{u}_i)=2.\\end{array}\\right.\\ i=1,2,\\ldots ,n.$ We now show that $t$ is a satisfying truth assignment for ${C}$ .",
"It is sufficient to show that every clause in ${C}$ is satisfied by $t$ .",
"To this end, we arbitrarily choose a clause $C_j\\in {C}$ with $1\\le j\\le m$ .",
"By Claim 4.1, $f(c_j)=f(s_1)=f(s_3)=0$ .",
"There exists some $i$ with $1\\le i\\le n$ such that $f(u_i)=2$ or $f(\\bar{u}_i)=2$ where $c_j$ is adjacent to $u_i$ or $\\bar{u}_i$ .",
"Suppose that $c_j$ is adjacent to $u_i$ where $f(u_i)=2$ .",
"Since $u_i$ is adjacent to $c_j$ in $G$ , the literal $u_i$ is in the clause $C_j$ by the construction of $G$ .",
"Since $f(u_i)=2$ , it follows that $t(u_i)=T$ by (REF ), which implies that the clause $C_j$ is satisfied by $t$ .",
"Suppose that $c_j$ is adjacent to $\\bar{u}_i$ where $f(\\bar{u}_i)=2$ .",
"Since $\\bar{u}_i$ is adjacent to $c_j$ in $G$ , the literal $\\bar{u}_i$ is in the clause $C_j$ .",
"Since $f(\\bar{u}_i)=2$ , it follows that $t(u_i)=F$ by (REF ).",
"Thus, $t$ assigns $\\bar{u}_i$ the truth value $T$ , that is, $t$ satisfies the clause $C_j$ .",
"By the arbitrariness of $j$ with $1\\le j\\le m$ , we show that $t$ satisfies all the clauses in ${C}$ , that is, ${C}$ is satisfiable.",
"Conversely, suppose that ${C}$ is satisfiable, and let $t:U\\rightarrow \\lbrace T,F\\rbrace $ be a satisfying truth assignment for ${C}$ .",
"Create a function $f$ on $V(G)$ as follows: if $t(u_i)=T$ , then let $f(u_i)=f(v_i^{\\prime })=2$ , and if $t(u_i)=F$ , then let $f(\\bar{u}_i)=f(v_i)=2$ .",
"Let $f(s_2)=2$ .",
"Clearly, $f(G)=4n+2$ .",
"Since $t$ is a satisfying truth assignment for ${C}$ , for each $j=1,2,\\ldots ,m$ , at least one of literals in $C_j$ is true under the assignment $t$ .",
"It follows that the corresponding vertex $c_j$ in $G$ is adjacent to at least one vertex $w$ with $f(w)=2$ since $c_j$ is adjacent to each literal in $C_j$ by the construction of $G$ .",
"Thus $f$ is a Roman dominating function of $G$ , and so $\\gamma _{\\rm R}(G)\\le f(G)= 4n+2$ .",
"By Claim 4.1, $\\gamma _{\\rm R}(G)\\ge 4n+2$ , and so $\\gamma _{\\rm R}(G)=4n+2$ .",
"Claim 4.3 $\\gamma _{\\rm R}(G-e)\\le 4n+3$ for any $e\\in E(G)$ .",
"For any edge $e\\in E(G)$ , it is sufficient to construct a Roman dominating function $f$ on $G-e$ with weight $4n+3$ .",
"We first assume $e\\in E_G(s_1)$ or $e\\in E_G(s_3)$ or $e\\in E_G(c_j)$ for some $j=1,2,\\ldots ,m$ , without loss of generality let $e\\in E_G(s_1)$ or $e=c_ju_i$ or $e=c_j\\bar{u}_i$ .",
"Let $f(s_3)=2,f(s_1)=1$ and $f(u_i)=f(v_i^{\\prime })=2$ for each $i=1,2,\\ldots ,n$ .",
"For the edge $e\\notin E_G(u_i)$ and $e\\notin E_G(v_i^{\\prime })$ , let $f(s_1)=2, f(s_3)=1$ and $f(u_i)=f(v_i^{\\prime })=2$ .",
"For the edge $e\\notin E(\\bar{u}_i)$ and $e\\notin E(v_i)$ , let $f(s_1)=2, f(s_3)=1$ and $f(\\bar{u}_i)=f(v_i)=2$ .",
"If $e=u_iv_i$ or $e=\\bar{u}_iv_i^{\\prime }$ , let $f(s_1)=2, f(s_3)=1$ and $f(x_i)=f(z_i)=2$ .",
"Then $f$ is a Roman dominating function of $G-e$ with $f(G-e)=4n+3$ and hence $\\gamma _{\\rm R}(G-e)\\le 4n+3$ .",
"Claim 4.4 $\\gamma _{\\rm R}(G)=4n+2$ if and only if $b_{\\rm R}(G)=1$ .",
"Assume $\\gamma _{\\rm R}(G)=4n+2$ and consider the edge $e=s_1s_2$ .",
"Suppose $\\gamma _{\\rm R}(G)=\\gamma _{\\rm R}(G-e)$ .",
"Let $f^{\\prime }$ be a $\\gamma _{\\rm R}$ -function of $G-e$ .",
"It is clear that $f^{\\prime }$ is also a $\\gamma _{\\rm R}$ -function on $G$ .",
"By Claim 4.1 we have $f^{\\prime }(c_j)=0$ for each $j=1,2,\\ldots ,m$ and $f^{\\prime }(s_2)=2$ .",
"But then $f^{\\prime }(N_{G-e}[s_1])=0$ , a contradiction.",
"Hence, $\\gamma _{\\rm R}(G)<\\gamma _{\\rm R}(G-e)$ , and so $b_{\\rm R}(G)=1$ .",
"Now, assume $b_{\\rm R}(G)=1$ .",
"By Claim 4.1, we have $\\gamma _{\\rm R}(G)\\ge 4n+2$ .",
"Let $e^{\\prime }$ be an edge such that $\\gamma _{\\rm R}(G)<\\gamma _{\\rm R}(G-e^{\\prime })$ .",
"By Claim 4.3, we have that $\\gamma _{\\rm R}(G-e^{\\prime })\\le 4n+3$ .",
"Thus, $4n + 2\\le \\gamma _{\\rm R}(G)< \\gamma _{\\rm R}(G-e^{\\prime })\\le 4n+3$ , which yields $\\gamma _{\\rm R}(G)=4n+2$ .",
"By Claim 4.2 and Claim 4.4, we prove that $b_{\\rm R}(G)=1$ if and only if there is a truth assignment for $U$ that satisfies all clauses in ${C}$ .",
"Since the construction of the Roman bondage number instance is straightforward from a 3-satisfiability instance, the size of the Roman bondage number instance is bounded above by a polynomial function of the size of 3-satisfiability instance.",
"It follows that this is a polynomial reduction and the proof is complete.",
"Corollary 5 Roman domination number problem is NP-complete even for bipartite graphs.",
"It is easy to see that the Roman domination problem is in NP since a nondeterministic algorithm need only guess a vertex set pair $(V_1,V_2)$ with $|V_1|+2|V_2|\\le k$ and check in polynomial time whether that for any vertex $u\\in V\\setminus (V_1\\cup V_2)$ whether there is a vertex in $V_2$ adjacent to $u$ for a given nonempty graph $G$ .",
"We use the same method as Theorem REF to prove this conclusion.",
"We construct the same graph $G$ but does not contain the path $P$ .",
"We set $k=4n$ , then use the same methods as Claim 4.1 and 4.2, we have that $\\gamma _{\\rm R}(G)=4n$ if and only if ${C}$ is satisfiable."
],
[
"General bounds",
"Lemma 6 Let $G$ be a connected graph of order $n\\ge 3$ such that $\\gamma _{\\rm R}(G)=\\gamma (G)+1$ .",
"If there is a set $B$ of edges with $\\gamma _{\\rm R}(G-B)=\\gamma _{\\rm R}(G)$ , then $\\Delta (G)=\\Delta (G-B)$ .",
"Since $G$ is connected and $n\\ge 3$ , $\\gamma _{\\rm R}(G)=\\gamma (G)+1\\le n-1$ .",
"Since $\\gamma _{\\rm R}(G-B)=\\gamma _{\\rm R}(G)\\le n-1$ , $G-B$ is nonempty.",
"It follows from Propositions REF and REF that $\\gamma _{\\rm R}(G-B)\\ge \\gamma (G-B)+1$ .",
"Since $\\gamma _{\\rm R}(G-B)=\\gamma _{\\rm R}(G)=\\gamma (G)+1\\le \\gamma (G-B)+1,$ we have $\\gamma _{\\rm R}(G-B)=\\gamma (G-B)+1$ , and then $\\gamma (G-B)=\\gamma (G)$ .",
"If $G-B$ is connected, then by Proposition REF , $\\Delta (G-B)=n-\\gamma (G-B)=n-\\gamma (G)=\\Delta (G).$ If $G-B$ is disconnected, then let $G_1$ be a nonempty connected component of $G-B$ .",
"By Propositions REF and REF , $\\gamma _{\\rm R}(G_1)\\ge \\gamma (G_1)+1$ .",
"Then $\\begin{array}{ccc}\\gamma (G)+1 &= &\\gamma _{\\rm R}(G-B)\\hfill \\\\&= & \\gamma _{\\rm R}(G_1)+\\gamma _{\\rm R}(G-G_1)\\\\&\\ge &\\gamma (G_1)+1+\\gamma (G-G_1)\\hfill \\\\&\\ge & \\gamma (G)+1,\\hfill \\end{array}$ and hence $\\gamma _{\\rm R}(G_1)=\\gamma (G_1)+1$ , $\\gamma _{\\rm R}(G-G_1)=\\gamma (G-G_1)$ and $\\gamma (G)=\\gamma (G_1)+\\gamma (G-G_1)$ .",
"By Proposition REF , $G-G_1$ is empty and hence $\\gamma (G-G_1)=|V(G-G_1)|$ .",
"By Proposition REF , $\\Delta (G_1)&=|V(G_1)|-\\gamma (G_1)\\\\&=n-|V(G-G_1)|-\\gamma (G_1)\\\\&=n-\\gamma (G-G_1)-\\gamma (G_1)\\\\&=n-\\gamma (G)=\\Delta (G)$ as desirable.",
"Theorem 7 Let $G$ be a connected graph of order $n\\ge 3$ with $\\gamma _{\\rm R}(G)=\\gamma (G)+1$ .",
"Then $b_{\\rm R}(G)\\le \\min \\lbrace b(G),n_{\\Delta }\\rbrace ,$ where $n_{\\Delta }$ is the number of vertices with maximum degree $\\Delta $ in $G$ .",
"Since $n\\ge 3$ and $G$ is connected, we have $\\Delta (G)\\ge 2$ and hence $\\gamma (G)\\le n-2$ .",
"Let $B$ be a $b(G)$ - set.",
"By (REF ), $\\gamma (G-B)=\\gamma (G)+1\\le n-1$ and so $G-B$ is nonempty.",
"It follows from Propositions REF and REF that $\\gamma _{\\rm R}(G-B)\\ge \\gamma (G-B)+1>\\gamma (G)+1=\\gamma _{\\rm R}(G)$ and hence $B$ is a Roman bondage set of $G$ .",
"Thus, $b_{\\rm R}(G)\\le b(G)$ .",
"We now prove that $b_{\\rm R}(G)\\le n_{\\Delta }$ .",
"It follows from Propositions REF , REF and the fact $\\gamma _{\\rm R}(G) =\\gamma (G)+1$ that $\\Delta (G)=n-\\gamma (G)$ .",
"Let $\\lbrace v_1,\\ldots ,v_{n_{\\Delta }}\\rbrace $ be the set consists of all vertices of degree $\\Delta $ and let $e_i$ be an edge adjacent to $v_i$ for each $1\\le i\\le n_{\\Delta }$ .",
"Suppose $B^{\\prime }=\\lbrace e_1,\\ldots ,e_{n_{\\Delta }}\\rbrace $ .",
"Clearly, $\\Delta (G-B^{\\prime })<\\Delta (G)=n-\\gamma (G)$ and $G-B^{\\prime }$ is nonempty.",
"Since $G-B^{\\prime }$ is nonempty, it follows from Propositions REF and REF that $\\gamma _{\\rm R}(G-B^{\\prime })\\ge \\gamma (G-B^{\\prime })+1$ .",
"We claim that $\\gamma _{\\rm R}(G-B^{\\prime })>\\gamma _{\\rm R}(G)$ .",
"Assume to the contrary that $\\gamma _{\\rm R}(G-B^{\\prime })=\\gamma _{\\rm R}(G)$ .",
"We deduce from Lemma REF that $\\Delta (G-B^{\\prime })=\\Delta (G)=n-\\gamma (G)$ , a contradiction.",
"Hence $b_{\\rm R}(G)\\le |B^{\\prime }|\\le n_{\\Delta }$ .",
"This completes the proof.",
"Theorem 8 For every Roman graph $G$ , $b_{\\rm R}(G) \\ge b(G).$ The bound is sharp for cycles on $n$ vertices where $n\\equiv 0\\;({\\rm mod}\\;3)$ .",
"Let $B$ be a $b_R(G)$ - set.",
"Then by (REF ) we have $2\\gamma (G-B)\\ge \\gamma _{\\rm R}(G-B)>\\gamma _{\\rm R}(G)=2\\gamma (G).$ Thus $\\gamma (G-B)>\\gamma (G)$ and hence $b_{\\rm R}(G) \\ge b(G)$ .",
"By Proposition REF , we have $b_{\\rm R}(C_n) \\ge b(C_n)=2$ when $n\\equiv 0\\;({\\rm mod}\\;3)$ .",
"The strict inequality in Theorem REF can hold, for example, $b(C_{3k+2})=2<3=b_{\\rm R}(C_{3k+2})$ by Proposition REF .",
"A graph $G$ is called to be vertex domination-critical ( vc-graph for short) if $\\gamma (G-x) < \\gamma (G)$ for any vertex $x$ in $G$ .",
"We call a graph $G$ to be vertex Roman domination-critical (vrc-graph for short) if $\\gamma _{\\rm R}(G-x)<\\gamma _{\\rm R}(G)$ for every vertex $x$ in $G$ .",
"The vertex covering number $\\beta (G)$ of $G$ is the minimum number of vertices that are incident with all edges in $G$ .",
"If $G$ has no isolated vertices, then $\\gamma _{\\rm R}(G)\\le 2\\gamma (G)\\le 2\\beta (G)$ .",
"If $\\gamma _{\\rm R}(G)=2\\beta (G)$ , then $\\gamma _{\\rm R}(G)=2\\gamma (G)$ and hence $G$ is a Roman graph.",
"In [31], Volkmann gave a lot of graphs with $\\gamma (G)=\\beta (G)$ .",
"Theorem 9 Let $G$ be a graph with $\\gamma _{\\rm R}(G)=2\\beta (G)$ .",
"Then (1) $b_{\\rm R}(G)\\ge \\delta (G)$ ; (2) $b_{\\rm R}(G)\\ge \\delta (G)+1$ if $G$ is a vrc-graph.",
"Let $G$ be a graph such that $\\gamma _{\\rm R}(G)=2\\beta (G)$ .",
"(1) If $\\delta (G)=1$ , then the result is immediate.",
"Assume $\\delta (G)\\ge 2$ .",
"Let $B\\subseteq E(G)$ and $|B|\\le \\delta (G)-1$ .",
"Then $\\delta (G-B)\\ge 1$ and so $\\gamma _{\\rm R}(G)\\le \\gamma _{\\rm R}(G-B) \\le 2\\beta (G-B) \\le 2\\beta (G)=\\gamma _{\\rm R}(G)$ .",
"Thus, $B$ is not a Roman bondage set of $G$ , and hence $b_{\\rm R}(G)\\ge \\delta (G)$ .",
"(2)Let $B$ be a Roman bondage set of $G$ .",
"An argument similar to that described in the proof of (1), shows that $B$ must contain all edges incident with some vertex of $G$ , say $x$ .",
"Hence, $G-B$ has an isolated vertex.",
"On the other hand, since $G$ is a vrc-graph, $\\gamma _{\\rm R}(G-x) <\\gamma _{\\rm R}(G)$ which implies that the removal of all edges incident to $x$ can not increase the Roman domination number.",
"Hence, $b_{\\rm R}(G)\\ge \\delta (G)+1$ .",
"The cartesian product $G=G_1\\times G_2$ of two disjoint graphs $G_1$ and $G_2$ has $V(G)=V(G_1)\\times V(G_2)$ , and two vertices $(u_1,u_2)$ and $(v_1,v_2)$ of $G$ are adjacent if and only if either $u_1=v_1$ and $u_2v_2\\in E(G_2)$ or $u_2=v_2$ and $u_1v_1\\in E(G_1)$ .",
"The cartesian product of two paths $P_r=x_1x_2\\ldots x_r$ and $P_t=y_1y_2\\ldots y_t$ is called a grid.",
"Let $G_{r,s}=P_r\\times P_t$ is a grid, and let $V(G_{r,s})=\\lbrace u_{i,j}=(x_i,y_j)|1\\le i\\le r\\,\\,{\\rm and}\\,\\,1\\le j\\le t\\rbrace $ be the vertex set of $G$ .",
"Next we determine Roman bondage number of grids.",
"Theorem 10 For $n\\ge 2$ , $b_{\\rm R}(G_{2,n})=2$ .",
"By Proposition REF , we have $\\gamma _{\\rm R}(G_{2,n})=n+1$ .",
"Since $\\gamma _{\\rm R}(G_{2,n}-u_{1,1}u_{1,2}-u_{2,1}u_{2,2})=2+\\gamma _{\\rm R}(G_{2,n-1})=n+2,$ we deduce that $b_{\\rm R}(G_{2,n})\\le 2$ .",
"Now we show that $\\gamma _{\\rm R}(G_{2,n}-e)= \\gamma _{\\rm R}(G_{2,n})$ for any edge $e\\in E(G_{2,n})$ .",
"Consider two cases.",
"Case 1 $n$ is odd.",
"For $i=1,2,3,4$ , define $f_i:V(G_{2,n})\\rightarrow \\lbrace 0,1,2\\rbrace $ as follows: $f_1(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 1\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 3\\;({\\rm mod}\\;4)\\\\0 & {\\rm if} & {\\rm otherwise},\\hfill \\end{array}\\right.$ $f_2(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 3\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 1\\;({\\rm mod}\\;4)\\\\0 & {\\rm if} & {\\rm otherwise},\\hfill \\end{array}\\right.$ and if $n\\equiv 1\\;({\\rm mod}\\;4) $ , then $f_3(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 0\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 2\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=j=1 \\;\\;{\\rm or}\\;\\;i=2 \\;{\\rm and}\\; j=n \\hfill \\\\0 & {\\rm if} & {\\rm otherwise}.\\hfill \\end{array}\\right.$ and if $n\\equiv 3\\;({\\rm mod}\\;4) $ , then $f_4(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 2\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 0\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=2 \\;{\\rm and}\\; j=1\\;\\;{\\rm or}\\;\\;i=2 \\;{\\rm and}\\; j=n\\hfill \\\\0 & {\\rm if} & {\\rm otherwise}.\\hfill \\end{array}\\right.$ Obviously, $f_i$ is a $\\gamma _R(G_{2,n})$ -function for each $i=1,2,3$ when $n\\equiv 1\\;({\\rm mod}\\;4)$ and $f_i$ is a $\\gamma _R(G_{2,n})$ -function for each $i=1,2,4$ when $n\\equiv 3\\;({\\rm mod}\\;4)$ .",
"Let $e\\in E(G)$ be an arbitrary edge of $G$ .",
"Then clearly , $f_1$ or $f_2$ or $f_3$ is a Roman dominating function of $G-e$ if $n\\equiv 1\\;({\\rm mod}\\;4) $ and $f_1$ or $f_2$ or $f_3$ is a Roman dominating function of $G-e$ if $n\\equiv 3\\;({\\rm mod}\\;4) $ .",
"Hence $b_R(G_{2,n})\\ge 2$ .",
"Case 2 $n$ is even.",
"For $i=1,2,3,4$ , define $f_i:V(G_{2,n})\\rightarrow \\lbrace 0,1,2\\rbrace $ as follows: $f_1(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 0\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 2\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=j=1\\hfill \\\\0 & {\\rm if} & {\\rm otherwise},\\hfill \\end{array}\\right.$ $f_2(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 2\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 0\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=2 \\;{\\rm and}\\; j=1\\hfill \\\\0 & {\\rm if} & {\\rm otherwise}.\\hfill \\end{array}\\right.$ and if $n\\equiv 0\\;({\\rm mod}\\;4)$ , then $f_3(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 1\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 3\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=1 \\;{\\rm and}\\; j=n\\hfill \\\\0 & {\\rm if} & {\\rm otherwise},\\hfill \\end{array}\\right.$ and if $n\\equiv 2\\;({\\rm mod}\\;4)$ , then $f_4(u_{i,j})=\\left\\lbrace \\begin{array}{ccc}2 & {\\rm if} & i=1 \\;{\\rm and}\\; j\\equiv 1\\;({\\rm mod}\\;4)\\;\\;{\\rm or}\\;\\; i=2 \\;{\\rm and}\\; j\\equiv 3\\;({\\rm mod}\\;4)\\\\1 & {\\rm if} & i=2 \\;{\\rm and}\\; j=n\\hfill \\\\0 & {\\rm if} & {\\rm otherwise},\\hfill \\end{array}\\right.$ Obviously, $f_i$ is a $\\gamma _R(G_{2,n})$ -function for each $i=1,2,3$ when $n\\equiv 0\\;({\\rm mod}\\;4)$ and $f_i$ is a $\\gamma _R(G_{2,n})$ -function for each $i=1,2,4$ when $n\\equiv 2\\;({\\rm mod}\\;4)$ .",
"Let $e\\in E(G)$ be an arbitrary edge of $G$ .",
"Then clearly , $f_1$ or $f_2$ or $f_3$ is a Roman dominating function of $G-e$ if $n\\equiv 0\\;({\\rm mod}\\;4) $ and $f_1$ or $f_2$ or $f_4$ is a Roman dominating function of $G-e$ if $n\\equiv 2\\;({\\rm mod}\\;4) $ .",
"Hence $b_R(G_{2,n})\\ge 2$ .",
"This completes the proof."
],
[
"Roman bondage number of graphs with small Roman domination number",
"Dehgardi, Sheikholeslami and Volkmann [7] posed the following problem: If $G$ is a connected graph of order $n\\ge 4$ with Roman domination number $\\gamma _R(G)\\ge 3$ , then $b_{R}(G)\\le (\\gamma _R(G)-2)\\Delta (G).$ Theorem REF shows that the inequality (REF ) holds if $\\gamma _R(G)\\ge 5$ .",
"Thus the bound in (REF ) is of interest only when $\\gamma _R(G)$ is 3 or 4.",
"In this section we prove (REF ) for all graphs $G$ of order $n\\ge 4$ with $\\gamma _R(G)=3,4$ , improving Proposition REF .",
"Theorem 11 If $G$ is a connected graph of order $n\\ge 4$ with $\\gamma _R(G)=3$ , then $b_{R}(G)\\le \\Delta (G)=n-2.$ Let $\\gamma _R(G)=3$ .",
"Then $\\Delta (G)=n-2$ by Proposition REF .",
"Let $M$ be maximum matching of $G$ and let $U$ be the set consisting of unsaturated vertices.",
"Since $G$ is connected and $\\gamma _R(G)=3$ , we deduce that $|M|\\ge 2$ .",
"If $U=\\emptyset $ , then $G-M$ has no vertex of degree $n-2$ and it follows from Proposition REF that $\\gamma _R(G-M)\\ge 4$ .",
"Thus $b_R(G)\\le |M|\\le \\frac{n}{2}\\le n-2=\\Delta (G).$ Assume now that $U\\ne \\emptyset $ .",
"Clearly $U$ is an independent set.",
"Since $G$ is connected and $M$ is maximum, there exist a set $J$ of $|U|$ edges such that each vertex of $U$ is incident with exactly one edge of $J$ .",
"Then $|J|=|U|=n-2|M|$ .",
"Now let $F=J\\cup M$ .",
"Obviously, $G-F$ has no vertex of degree $n-2$ , and it follows from Proposition REF that $\\gamma _R(G_F)\\ge 4$ .",
"This implies that $b_R(G)\\le |M|+|U|=n-|M|\\le n-2=\\Delta (G).$ This completes the proof.",
"Next we characterize all graphs that achieve the bound in Theorem REF .",
"Theorem 12 If equality holds in Theorem REF , then $G$ is regular.",
"Let $\\gamma _R(G)=3$ and $b_R(G)=\\Delta (G)=n-2$ .",
"If $G$ has a perfect matching $M$ , then it follows from (REF ) that $\\frac{n}{2}=n-2$ and hence $n=4$ .",
"This implies that $b_R(G)=|M|=2=\\Delta (G)$ .",
"Since $b_R(P_4)=1$ , we have $G=C_4$ as desired.",
"Let $G$ does not have a perfect matching and let $M$ be a maximum matching of $G$ .",
"It follows from (REF ) that $|M|=2$ .",
"Let $X$ be the independent set of $M$ -unsaturated vertices.",
"We consider two cases.",
"Case 1.",
"$|X|=1$ .",
"Then $n=5$ .",
"Let $V(G)=\\lbrace v_1,\\ldots ,v_5\\rbrace $ .",
"Since $\\gamma _R(G)=3$ , $\\Delta (G)=n-2=3$ by Proposition REF .",
"Since $n$ is odd, $G$ has a vertex of even degree 2.",
"Let $\\deg (v_1)=2$ and let $v_1v_2, v_1v_3\\in E(G)$ .",
"Since $b_R(G)=3>\\deg (v_1)$ , we have $\\gamma _R(G-v_1)=\\gamma _R(G)-1=2$ .",
"By Observation REF , $\\Delta (G-v_1)=3$ .",
"Since $\\gamma _R(G)=3$ , we may assume without loss of generality that $\\deg (v_4)=3$ and $\\lbrace v_4v_2,v_4v_3,v_4v_5\\rbrace \\subseteq E(G)$ .",
"Let $F=\\lbrace v_1v_2, v_3v_4\\rbrace $ .",
"Since $b_R(G)=3>|F|$ , we have $\\gamma _R(G-F)=3$ .",
"It follows from Proposition REF and the fact $\\gamma _R(G-F)=3$ that $\\deg _{G-F}(v_5)=3$ .",
"This implies that $\\lbrace v_5v_2,v_5v_3,v_5v_4\\rbrace \\subseteq E(G)$ .",
"Thus $E(G)=\\lbrace v_1v_2,v_1v_3,v_2v_4,v_2v_5,v_3v_4,v_3v_5,v_4v_5\\rbrace $ .",
"Now we have $G-\\lbrace v_2v_4,v_3v_5\\rbrace \\simeq C_5$ and hence $\\gamma _R(G-\\lbrace v_2v_4,v_3v_5\\rbrace )=4$ .",
"This implies that $b_R(G)\\le 2$ a contradiction.",
"Case 2.",
"$|X|\\ge 2$ .",
"Then $n\\ge 6$ .",
"Let $M=\\lbrace u_1v_1, u_2v_2\\rbrace $ be a maximum matching of $G$ .",
"If $y$ and $z$ are vertices of $X$ and $yu_i\\in E(G)$ , then since the matching $M$ is maximum, $zv_i\\notin E(G)$ .",
"Therefore, we may assume without loss of generality that $N_G(X)\\subseteq \\lbrace u_1,u_2\\rbrace $ .",
"So $\\deg (y)+\\deg (z)\\le 4$ for every pair of distinct vertices $y$ and $z$ in $X$ .",
"Let $y,z\\in X$ and $F$ be the set of edges incident with $y$ or $z$ .",
"Then $y,z$ are isolated vertices in $G-F$ and hence $\\gamma _R(G-F)\\ge 4$ .",
"If $|F|\\le 3$ , then $n-2=b_R(G)\\le 3$ which leads to a contradiction.",
"Therefore, $|F|=4$ .",
"It follows that $n-2=b_R(G)\\le 4$ and hence $n= 6$ .",
"Let $V(G)=\\lbrace u_1,u_2,v_1,v_2,y,z\\rbrace $ .",
"Then $\\deg (y)=\\deg (z)=2$ and $\\deg (u_1),\\deg (u_2)\\ge 3$ .",
"If $v_1v_2\\in E(G)$ , then $\\lbrace yu_1,xu_2,v_1v_2\\rbrace $ is a matching of $G$ which is a contradiction.",
"Thus $\\deg (v_1),\\deg (v_2)\\le 2$ .",
"Since $\\gamma _R(G)=3$ , $\\Delta (G)=n-2=4$ by Proposition REF .",
"We distinguish two subcases.",
"Subcase 2.1 $\\delta (G)=1$ .",
"Assume without loss of generality that $\\deg (v_1)=1$ .",
"Let $F$ be the set of edges incident with $y$ or $v_1$ .",
"Then $|F|=3$ and $y,v_1$ are isolated vertices in $G-F$ and hence $\\gamma _R(G-F)\\ge 4$ .",
"Thus $n-2=b_R(G)\\le 3$ , a contradiction.",
"Subcase 2.2 $\\delta (G)=2$ .",
"Then we must have $\\deg (v_1)=\\deg (v_2)=2$ and $v_1u_2,v_2u_1\\in E(G)$ .",
"Let $F=\\lbrace yu_1,zu_2\\rbrace $ .",
"Clearly $\\Delta (G-F)=3=n-3$ and it follows from Proposition REF that $\\gamma _R(G-F)\\ge 4$ .",
"Hence $b_R(G)\\le 2$ , which is a contradiction.",
"This completes the proof.",
"Proposition J The complete graph $K_{2r}$ is 1-factorable.",
"According to Theorem REF , Theorem REF , Proposition REF and Proposition REF , we prove the next result.",
"Theorem 13 Let $G$ be a connected graph of order $n\\ge 4$ with $\\gamma _R(G)=3$ .",
"Then $b_{R}(G)=\\Delta (G)= n-2$ if and only if $G\\simeq C_4$ .",
"Let $G$ be a connected graph of order $n\\ge 4$ with $\\gamma _R(G)=3$ .",
"It follows from Theorem REF that $b_{R}(G)\\le n-2$ .",
"If $G\\simeq C_4$ , then obviously $b_R(G)=2=n-2$ .",
"Conversely, assume that $b_R(G)=n-2$ .",
"It follows from Proposition REF and Theorem REF that $G$ is $(n-2)$ -regular.",
"This implies that $n$ is even and hence $G=K_n-M$ where $M$ is a perfect matching in $K_n$ .",
"By Proposition REF , $G$ is 1-factorable.",
"Let $M_1$ be a perfect matching in $G$ .",
"Now $G-M_1$ is an $(n-3)$ -regular and it follows from Proposition REF that $\\gamma _R(G-M_1)\\ge 4$ .",
"Thus $n-2=b_R(G)\\le \\frac{n}{2}$ which implies that $n=4$ and hence $G=C_4$ .",
"Theorem 14 If $G$ is a connected graph of order $n\\ge 4$ with $\\gamma _R(G)=4$ , then $b_{R}(G)\\le \\Delta (G)+\\delta (G)-1.$ Obviously $\\Delta (G)\\ge 2$ .",
"Let $u$ be a vertex of minimum degree $\\delta (G)$ .",
"If $b_R(G)\\le \\deg (u)$ , then we are done.",
"Suppose $b_{R}(G)> \\deg (u)$ .",
"Then $\\gamma _R(G-u)=\\gamma _R(G)-1=3$ .",
"By Theorem REF , $b_R(G-u)\\le \\Delta (G-u)$ .",
"If $b_R(G-u)=\\Delta (G-u)$ , then $G-u=C_4$ by Theorem REF and since $G$ is connected, we deduce that $\\gamma _R(G)=3$ , a contradiction.",
"Thus $b_R(G-u)\\le \\Delta (G-u)-1$ .",
"It follows from Observation REF that $b_R(G)\\le b_R(G-u)+\\deg (u)\\le \\Delta (G-u)-1+\\deg (u) \\le \\Delta (G)+\\delta (G)-1,$ as desired.",
"This completes the proof.",
"Dehgardi et al.",
"[7] proved that for any connected graph $G$ of order $n\\ge 3$ , $b_{\\rm R}(G)\\le n-1$ and posed the following problems.",
"Problem 1.",
"Prove or disprove: For any connected graph $G$ of order $n\\ge 3$ , $b_{\\rm R}(G)=n-1$ if and only if $G\\cong K_3$ .",
"Problem 2.",
"Prove or disprove: If $G$ is a connected graph of order $n\\ge 3$ , then $b_{\\rm R}(G)\\le n-\\gamma _{\\rm R}(G)+1.$ Since $\\gamma _R(K_{3,3,\\ldots ,3})=4$ , Proposition REF shows that Problems 1 and 2 are false.",
"Recently Akbari and Qajar [1] proved that: Proposition K If $G$ is a connected graph of order $n\\ge 3$ , then $b_{R}(G)\\le n-\\gamma _R(G)+5.$ We conclude this paper with the following revised problems.",
"Problem 3.",
"Characterize all connected graphs $G$ of order $n\\ge 3$ for which $b_R(G)=n-1$ .",
"Problem 4.",
"Prove or disprove: If $G$ is a connected graph of order $n\\ge 3$ , then $b_{R}(G)\\le n-\\gamma _R(G)+3.$"
]
] | 1204.1438 |
[
[
"Hawking radiation from dynamical horizons"
],
[
"Abstract In completely local settings, we establish that a dynamically evolving black hole horizon can be assigned a Hawking temperature.",
"Moreover, we calculate the Hawking flux and show that the radius of the horizon shrinks."
],
[
"Hawking radiation from dynamical horizons Ayan [email protected] Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India.",
"Bhramar [email protected] Amit [email protected] Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064, INDIA.",
"In completely local settings, we establish that a dynamically evolving black hole horizon can be assigned a Hawking temperature.",
"Moreover, we calculate the Hawking flux and show that the radius of the horizon shrinks.",
"The laws of black hole mechanics in general relativity are remarkably analogous to the laws of thermodynamics [1].",
"This analogy is exact when quantum effects are taken into account.",
"Indeed, Hawking's semiclassical analysis establishes that quantum mechanically, a stationary black hole with surface gravity $\\kappa $ radiates particles to infinity with a perfect black body spectrum at temperature $\\kappa /2\\pi $ [2].",
"Consequently, asymptotic observers perceive a thermal state and assign a physical temperature to the black hole.",
"The precise match to thermodynamics is complete when the thermodynamic entropy of the black hole is identified with a quarter of its area [3].",
"The original calculation of Hawking is independent of the gravitational field equations.",
"It relies only on the behavior of quantum fields in a specific spacetime geometry describing a stationary black hole formed due to a gravitational collapse.",
"Over the years, several other techniques have been developed to study spontaneous particle emission and the Hawking temperature for more general spacetimes.",
"For example, the Hartle-Hawking proposal [5] and the Euclidean approach [6] have been extensively used to associate thermal states to spacetimes with bifurcate Killing horizons.",
"In fact, it has been established that in any globally hyperbolic spacetime with bifurcate Killing horizon, there can exist a vacuum thermal state at temperature $\\kappa /2\\pi $ which remains invariant under the isometries generating the horizon [7].",
"Although these constructions are elegant, they are quite restrictive, inapplicable even for spacetimes with superradiance [7].",
"These formulations also do not indicate how such a thermal state may arise as a result of some version of physical process.",
"In addition, their existence requires knowledge of global structure of spacetime.",
"As a result, they do not appear very useful to study thermal properties of local horizons.",
"On the other hand, the laws of black hole mechanics apply equally well to black hole horizons which can been proved using only local geometrical properties of null surfaces, without any assumptions on the global development of the spacetime in which the horizon is embedded [8], [9], [10], [11].",
"It has also been established that such horizons can be assigned an entropy proportional to the area of the local horizon [12], [13].",
"Thus, it seems to be a reasonable physical expectation that even with a local definition of black hole horizon one should be able to establish the analogy to thermodynamics.",
"More precisely, such horizons should have a temperature of $\\kappa /2\\pi $ .",
"Incidentally, this question has been investigated in a semiclassical approach which treats Hawking radiation as a quantum tunneling phenomenon [14]-[18].",
"The method involves calculating the imaginary part of the action for the (classically forbidden) process of s-wave emission, from inside and through the horizon (see [19] for more details).",
"Using the WKB-approximation the tunnelling probability for such a classically forbidden trajectory is calculated to be, $\\Gamma = e^{-2{\\rm Im} S}$ where, $S$ is the classical action of the trajectory to leading order in $\\hbar $ .",
"This is equated to the Boltzmann factor $ e^{-{\\beta }E}$ to extract the inverse Hawking temperature $\\beta $ .",
"The main advantage of this formalism is that the calculations involve only the local geometry and hence can be applied to any local horizon.",
"Indeed, tunnelling method has been applied to local dynamical black hole horizons and the temperature is found to be $\\kappa /2\\pi $ where $\\kappa $ is the dynamical surface gravity [20], [21].",
"Still there are some problems with the method itself and some issues which have not been addressed in this treatment of dynamical horizons.",
"First, the approach depends heavily on the semiclassical approximation and though it is argued that this remains valid near the horizon, it would be better to devise a more general formalism which does not rely on WKB-like approximations.",
"Secondly, in calculating the imaginary part of the semiclassical action $S$ from the Hamilton-Jacobi equation, a singular integral appears with a pole at the horizon.",
"While for the static case the result is standard, for the dynamical horizon it is not clear how the integration is to be performed since the position of the horizon changes in a dynamical process.",
"Lastly, in all these treatments of radiation from dynamical horizons the evolution of the horizon itself is never addressed.",
"In other words, it is not clear how the horizon loses area due to emission of a flux of radiation.",
"The local formalisms of black hole horizon should be able to address these issues.",
"In this paper, a formalism is developed to establish two basic issues.",
"First, that one can associate a temperature to local dynamical horizons without the need for any WKB-like approximation schemes.",
"Second, that there exists a precise relation between the radiation emitted by the horizon and area loss, i.e., flux of outgoing radiation through the horizon in between two partial Cauchy slices exactly equals the difference of radii of the sphere that foliates the horizon at those two instances.",
"We elucidate our arguments as follows.",
"To calculate temperature for local dynamical horizons, we begin by considering the Kodama vector field [22].",
"For dynamical spacetimes, this vector field provides a preferred timelike direction and is parallel to the Killing vector at spatial infinity which we assume to be flat.",
"We can construct well-behaved positive frequency field modes on both sides of the horizon by considering the Kodama vector field but the outgoing modes exhibit logarithmic singularities on the horizon under some approximation.",
"However, if considered as distribution valued, these modes can be interpreted as horizon crossing and the probability current for these modes remain well defined.",
"The Hawking temperature is determined if one equates the conditional probability, that modes incident on one side is emitted to the other side, to the Boltzmann factor [23], [24], $P_{(emission|incident)} = \\frac{P_{(emission{\\cap }incident)}}{P_{(incident)}} = e^{-{\\beta }E}.$ Since this method does not depend on the entire evolution of the field modes in the spacetime, it is ideally suited for our purpose.",
"To evaluate the Hawking flux, we recall that there are two well known (and related) definitions of local black hole horizon, the future outward trapping horizon (FOTH) [8], [9] and the Dynamical Horizon [25], [26] (or its equilibrium version called the isolated horizon).",
"In these local settings, black hole horizons are a stack of apparent horizons which, under suitable energy conditions, are either null or spacelike.",
"As such, energy flux can only remain on the surface or flow into the horizon.",
"In order that matter fields flow out off such a surface requires that the surface must be timelike in some affine interval.",
"However, to achieve a timelike evolution of the horizon, some energy conditions need to be violated.",
"This is only natural since Hawking radiation necessarily associates, with the thermal emission of particles, a positive flux of energy flowing to infinity (we shall assume that the spacetime is asymptotically flat) and a corresponding flux of negative energy flowing into the black hole (this negative energy flux can also be motivated by the fact that the expectation values of stress energy tensor of quantum fields generically violate energy conditions).",
"In this process the horizon looses area and energy.",
"The plan of the paper is as follows: First, we will discuss the geometrical setup which is based on future outer trapping horizon (FOTH).",
"Next, we show that how the Hawking temperature is proportional to the dynamical surface gravity associated with the Kodama vector.",
"Finally, we will calculate the flux of energy radiated in a dynamical process.",
"We begin with definitions.",
"We follow the conventions of [9].",
"Consider a four dimensional spacetime $\\mathfs {M}$ with signature $(-, +,+,+)$ .",
"A three-dimensional submanifold $\\Delta $ in $\\mathfs {M}$ is said to be a future outer trapping horizon (FOTH) if 1) It is foliated by a preferred family of topological two-spheres such that, on each leaf $S$ , the expansion $\\theta _+$ of a null normal $l^{a}_+$ vanishes and the expansion $\\theta _-$ of the other null normal $l^{a}_-$ is negative definite, 2) The directional derivative of $\\theta _+$ along the null normal $l^{a}_-$ (i.e., $\\mathfs {L}_{l_-}\\theta _+$ ) is negative definite.",
"Thus, $\\Delta $ is foliated by marginally trapped two-spheres.",
"According to a theorem due to Hawking, the topology of $S$ is necessarily spherical in order that matter or gravitational flux across $\\Delta $ is non-zero.",
"If these fluxes are identically zero then $\\Delta $ becomes a Killing or isolated horizon.",
"Even though our arguments will remain local, for definiteness, we choose a spherically symmetric background metric $ds^2=-2e^{-f}dx^+dx^-+r^2(d\\theta ^2+\\sin ^2\\!\\theta d\\phi ^2)$ where both $f$ and $r$ are smooth functions of $x^\\pm $ .",
"The expansions of the two null normals are $\\theta _\\pm =(2/r)\\,\\partial _\\pm r$ respectively where $\\partial _\\pm =\\partial /\\partial x^{\\pm }$ .",
"In this coordinate system, the second requirement for FOTH translates to $\\partial _-\\theta _+<0$ on $\\Delta $ .",
"Let the vector field $t^{a}=l^{a}_+ + h\\,l^{a}_-$ be tangential to the FOTH for some smooth function $h$ .",
"Then the Raychaudhuri equation for $l^{a}_+$ and the Einstein equation implies $\\partial _+\\theta _+=-h\\partial _-\\theta _+=-8\\pi \\, T_{++}.$ where $T_{++}=T_{ab}\\,l^{a}_+ l^{b}_+$ and $T_{ab}$ is the energy momentum tensor.",
"Several consequences follow from this equation.",
"First, the FOTH is degenerate (or null) if and only if $T_{++}=0$ on $\\Delta $ .",
"In that case, the FOTH is generated by $l^{a}_+$ .",
"Degenerate FOTH is not interesting for Hawking radiation because this implies $\\partial _+r=0$ .",
"As a consequence, the area, $A=4\\pi r^2$ of $S$ , and the Misner-Sharp energy for this spacetime, given by $E={\\textstyle {\\frac{1}{2}}}r$ , also remains unchanged.",
"Secondly, since $t^2=-2h\\,e^{-f}$ , a FOTH becomes spacelike if and only if $T_{++}>0$ and is timelike if and only if $T_{++}<0$ .",
"For a timelike FOTH, several consequences follow.",
"Here, $\\mathfs {L}_tr <0$ , and hence, $\\Delta $ is timelike if and only if the area $A$ and the Misner-Sharp energy $E$ decreases along the horizon.",
"This is also expected on general grounds since the horizon receives an incoming flux of negative energy, $T_{++}<0$ .",
"As we have emphasized before, in the dynamical spacetime (REF ) the Kodama vector field plays the analog role of the Killing vector.",
"For this spacetime, it is given by $K^{a}=e^f\\,(\\partial _-r) \\,\\partial ^a_+-e^f\\,(\\partial _+r)\\,\\partial ^a_-.$ The surface gravity is defined through $K^a\\nabla _{[b} K_{a]}=\\kappa \\, K_b$ and is $k=-e^f\\,\\partial _-\\partial _+r$ .",
"The FOTH condition $\\partial _-\\theta _+<0$ implies $k>0$ .",
"Let us now determine the positive frequency modes of the Kodama vector.",
"It is easy to see that any smooth function of $r$ is a zero-mode of the Kodama vector.",
"Once, a zero-mode is obtained, other positive frequency eigenmodes are evaluated using $iK\\,Z_\\omega =\\omega \\,Z_\\omega $ Here, $Z_\\omega $ are the eigenfunctions corresponding to the positive frequency $\\omega $ .",
"For simplification, let us introduce new coordinates, $y=x^-$ and $r$ and two new functions, $\\bar{Z}_\\omega (y,r)=Z_\\omega (x^+,x^-)$ and $G(y,r)=e^f\\,(\\partial _+r)$ .",
"As a result, the eigenvalue equation (REF ) reduces to $G\\,\\partial _y\\bar{Z}_\\omega =i\\omega \\,\\bar{Z}_\\omega .$ Integrating and transforming back to old coordinates, the above equation gives $Z_\\omega =F(r)\\exp \\Big (i\\omega \\int _r\\frac{dx^-}{e^f\\partial _+r}\\Big )$ where $F(r)$ is an arbitrary smooth function of $r$ and the subscript $r$ under the integral sign denotes that while doing the integration $r$ is kept fixed.",
"To evaluate the integral in (REF ), we multiply the numerator and the denominator by $(\\partial _-\\theta _+)$ and use the fact that for any fixed $r$ surface, $e^f\\,(\\partial _-\\theta _+)=-2k/r$ , (although the strict interpretation of $k$ as the surface gravity holds only for surfaces with $\\theta _+ =0$ , it exists as a function in any neighbourhood of the horizon).",
"Thus, in some neighbourhood of the horizon we get $\\int _r\\frac{dx^-\\,\\partial _-\\theta _+}{e^f\\,\\partial _+r \\partial _-\\theta _+}=-\\int _r\\frac{d\\theta _+}{k\\theta _+}$ We now assume (this is the only assumption we make in this calculation) that during the dynamical evolution $k$ is a slowly varying function in some small neighbourhood of the horizon (the zeroth law takes care of it on the horizon, but we also assume it to hold in a small neighbourhood of the horizon).",
"This gives $Z_\\omega =F(r){\\left\\lbrace \\begin{array}{ll} \\theta _+^{-\\frac{i\\omega }{k}} &\\text{for}\\;\\theta _+>0\\\\(-|\\theta _+|)^{-\\frac{i\\omega }{k}} & {\\rm for}\\;\\theta _+<0.\\end{array}\\right.",
"}$ where the spheres are not trapped `outside the trapping horizon' ($\\theta _+>0$ ) and fully trapped `inside' ($\\theta _+<0$ ).",
"These are precisely the modes which are defined outside and inside the dynamical horizon respectively but not on the horizon.",
"Now we have to keep in mind the modes (REF ) are not ordinary functions, but are distribution-valued.",
"Comparing with the spherically symmetric static case [24], we find for $\\epsilon \\rightarrow 0^+$ $(\\theta _++i\\epsilon )^\\lambda ={\\left\\lbrace \\begin{array}{ll} \\theta _+^\\lambda &\\text{for}\\;\\theta _+>0\\\\|\\theta _+|^\\lambda e^{i\\lambda \\pi } &\\text{for}\\;\\theta _+<0\\end{array}\\right.",
"}$ for the choice $\\lambda =-i\\omega /k$ .",
"The distribution (REF ) is well-defined for all values of $\\theta _+$ and $\\lambda $ , and it is differentiable to all orders.",
"The modes $Z^*_\\omega $ are given by the complex conjugate distribution.",
"We wish to calculate the probability density in a single particle Hilbert space for positive frequency solutions across the dynamical horizon $\\varrho (\\omega )= -\\frac{i}{2}\\Big [Z^*_\\omega KZ_\\omega -KZ^*_\\omega Z_\\omega \\Big ]=\\omega Z_\\omega ^*Z_\\omega .$ A straightforward calculation gives, apart from a positive function of $r$ , $\\varrho (\\omega )&=\\omega (\\theta _++i\\epsilon )^{-\\frac{i\\omega }{k}}(\\theta _+-i\\epsilon )^ {\\frac{i\\omega }{k}}.\\nonumber \\\\&={\\left\\lbrace \\begin{array}{ll} \\omega & \\text{for}\\;\\theta _+>0\\\\\\omega e^{\\frac{2\\pi \\omega }{k}} & \\text{for}\\;\\theta _+<0.\\end{array}\\right.",
"}$ The conditional probability that a particle emits when it is incident on the horizon from inside is, $P_{(emission|incident)} = e^{-\\frac{2\\pi \\omega }{k}}$ This gives the correct Boltzmann weight with the temperature $k/2\\pi $ , which is the desired value.",
"We now show that as the horizon evolves, the radius of the 2-sphere foliating the horizon shrinks in precise accordance with the amount of flux radiated by the horizon.",
"To study the flux equation, consider new coordinates, ($x^+,x^-)\\mapsto (\\theta _+,\\tilde{x}^-)$ where $\\tilde{x}^-=x^-$ .",
"On FOTH, $(\\partial _-\\theta _+)/(\\partial _+\\theta _+)$ is equal to $-(\\partial _-\\partial _+r)/(4\\pi r\\,T_{++})$ and negative definite.",
"As a result, the derivatives are related to each other by $\\tilde{\\partial }_-=\\partial _-+\\left(\\frac{\\partial _-\\partial _+r}{4\\pi r\\,T_{++}}\\right)\\,\\partial _+.$ It is not difficult to show that $\\tilde{\\partial }_-$ is proportional to the tangent vector $t^a$ to the FOTH.",
"Observe that the normal one-form to $\\Delta $ must be proportional to $(dr-2\\,dE)$ , which on the horizon is equal to the one-form $(8\\pi e^fr^2\\,T_{++}-2re^f\\partial _-\\partial _+r)\\,\\partial _-r \\, dx^-.$ In arriving at the above identity we have made use of two Einstein's equations [9] $r\\,\\partial _-\\partial _+r+\\partial _+r\\,\\partial _-r+{\\textstyle {\\frac{1}{2}}}e^{-f}&=&4\\pi r^2\\,T_{-+},\\\\ \\nonumber \\partial _+^2r+\\partial _+f\\,\\partial _+r&=&-4\\pi r\\,T_{++},$ and energy equations $\\partial _\\pm E=2\\pi e^f r^3(T_{-+}\\,\\theta _\\pm -T_{\\pm \\pm }\\,\\theta _{\\mp }).$ As a result, the normal vector $n^a$ is proportional to $\\partial _+-\\left(\\frac{4\\pi r\\,T_{++}}{\\partial _-\\partial _+r}\\right)\\partial _-=\\partial _+-h\\partial _-,$ so that the tangent vector $t^a=\\partial ^a_++h\\partial ^a_-$ , which is clearly proportional to (REF ).",
"So $\\tilde{x}^-,\\theta ,\\phi $ are natural coordinates on FOTH.",
"The line-element (REF ) induces a line-element on $\\Delta $ $ds^2=-2e^{-f}h^{-1}(d\\tilde{x}^-)^2+r^2(d\\theta ^2+\\sin ^2\\!\\theta d\\phi ^2).$ Consequently, the volume element on $\\Delta $ is given by $d\\mu =\\sqrt{2e^{-f}h^{-1}}r^2\\sin \\theta \\,d\\tilde{x}^-d\\theta d\\phi $ .",
"We can now calculate the flux of matter energy that crosses the dynamical horizon—it is an integral on a slice of horizon bounded by two spherical sections $S_1$ and $S_2$ $\\mathfs {F}=\\int d\\mu \\;T_{ab}\\hat{n}^a K^b$ where $\\hat{n}^a$ is the unit normal vector $\\hat{n}^a=\\frac{1}{\\sqrt{2he^{-f}}}(\\partial _+^a-h\\partial _-^a)$ and $K^a$ is the Kodama vector.",
"Using spherical symmetry, eqn.",
"(REF ) and eqn.",
"(REF ), we get $ \\mathfs {F}&=\\int d\\tilde{x}^-\\;4\\pi r^2(\\frac{1}{h}T_{++}-T_{+-})e^f\\partial _-r\\nonumber \\\\&=\\int d\\tilde{x}^-\\;4\\pi r^2(\\frac{1}{4\\pi r}\\partial _+\\partial _-r-T_{+-})e^f\\partial _-r.$ Making use of the Einstein equation (REF ) on the horizon and (REF ), we get $ \\mathfs {F}&=-\\int d\\tilde{x}^-\\;\\frac{1}{2}\\partial _-r=-\\int d\\tilde{x}^-\\;\\frac{1}{2}\\tilde{\\partial }_-r\\nonumber \\\\&=-\\frac{1}{2}(r_2-r_1) $ where $r_1,r_2$ are respectively the two radii of $S_1,S_2$ .",
"Since the area is decreasing along the horizon, $r_2<r_1$ where $S_2$ lies in the future of $S_1$ .",
"As a result, the outgoing flux of matter energy radiated by the dynamical horizon is positive definite (and the ingoing flux of matter energy is negative definite).",
"The flux formula (REF ) differs from that given in [25].",
"Since the Kodama vector field provides a timelike direction and is null on the horizon, it seems more appropriate to use $K^a$ for the dynamical horizon.",
"The derivation of Hawking temperature and the flux law depends on two assumptions.",
"First, that the Kodama vector exists in the spacetime.",
"For spherically symmetric spacetimes, the Kodama vector field exists unambiguously and the Misner-Sharp energy is well defined.",
"For more general spacetimes, a Kodama-like vector field is not known, however, one can still define some mass for such cases that reduces to the Misner-Sharp energy in the spherical limit [27].",
"The second assumption, the existence of a slowly varying $k$ can also be motivated for large black holes.",
"In such cases, the horizon evolves slowly enough so that the surface gravity function should vary slowly in some small neighbourhood of the horizon.",
"Alternatively, we can conclude that the Hawking temperature for a dynamically evolving large black hole is $k/2\\pi $ if the dynamical surface gravity is slowly varying in the vicinity of the horizon.",
"The set-up described in this paper can be further developed to model dynamically evaporating black hole horizons through Hawking radiation, analytically as well as numerically.",
"Over the years, several models have been constructed which study radiating black holes, formed in a gravitational collapse, based on the imploding Vaidya metric with a negative energy-momentum tensor, show that a timelike apparent horizon forms due to violation of energy conditions [28].",
"However, such models are based on global considerations of event horizons, while local structures like that used in [29] might be useful for a better understanding of Hawking radiation and computations of quantum field theoretic effects (see also [30]).",
"It is also interesting to speculate on the extension of the present method for other diffeomorphism invariant theories of gravity.",
"While the zeroth and the first law hold for any arbitrary such theory, the second law has only been proved for a class of such theories [31].",
"If the present formalism can be extended to other theories of gravity, it will lend a support to the existence of the area increase theorem for such theories.",
"While more interesting and deeper issues can only be understood in a full quantum theory of gravity, the present framework can elucidate the suggestions of [32] and provide a better understanding of the Hawking radiation process."
]
] | 1204.1530 |
[
[
"Diagrammatic approach to coherent backscattering of laser light by cold\n atoms: Double scattering revisited"
],
[
"Abstract We present a diagrammatic derivation of the coherent backscattering spectrum from two two-level atoms using the pump-probe approach, wherein the multiple scattering signal is deduced from single-atom responses, and provide a physical interpretation of the single-atom building blocks."
],
[
"Introduction",
"Coherent backscattering (CBS) of light emerges due to the constructive interference of multiply scattered counter-propagating waves surviving the disorder average [1].",
"With optical waves, CBS was successfully observed using classical – e.g., polysterene particles [2] – and quantum – dilute cold atoms [3] – scatterers alike.",
"A remarkable property of cold atoms is their nonlinear inelastic scattering induced by a powerful resonant laser field.",
"Atomic saturation and inelastic scattering processes accompanying it were shown to affect the CBS interference from cold Sr (strontium) [4] and Rb (rubidium) [5] atoms, but a corresponding theory of CBS of intense laser light from cold atomic clouds is still lacking.",
"The main challenge one has to deal with when describing CBS from cold saturated atoms can be briefly summarized as follows.",
"A multiple scattering signal must be built on the basis of the accurately described responses of individual scatterers to a strong laser field.",
"These responses can be found by solving the optical Bloch equations (OBE) [6].",
"However, a standard generalization of the OBE to the $N$ -atom case leads to a Lehmberg-type master equation governing the evolution of the reduced density operator of all atoms which are laser-driven and dipole-dipole interacting [7], [8], with the number of equations for the atomic averages growing exponentially with the number of scatterers.",
"So far, such a master equation in the context of CBS has been solved only for $N=2$ atoms [9], [10].",
"Recently, we have initiated an alternative method of generalizing the OBE to the many-atom case which we call the diagrammatic approach to CBS [11], [12], [13].",
"In its framework, the double scattering signal can be obtained from the solutions of the OBE for an atom subjected to a bichromatic classical driving.",
"One component thereof represents the laser field and another one the field scattered from the second atom.",
"Single-atom responses to a bichromatic field were evaluated non-perturbatively in the laser field and perturbatively – up to a second order – in the scattered field amplitude.",
"Furthermore, by self-consistently combining single-atom responses (to which we will also refer as `building blocks'), we were able to derive analytical expressions for the background and interference spectra of double scattering.",
"The spectra thus evaluated were rigorously shown to be equivalent to that deduced on the basis of the two-atom master equation [14].",
"The motivation of the present contribution is twofold.",
"First, previously we focused on a discussion of the inelastic building blocks and their self-consistent combination into double scattering diagrams [12], [13].",
"These blocks determine the CBS signal only in the case of a very strong laser driving.",
"In the present contribution we will consider a general case and present a diagrammatic derivation of the CBS spectra for arbitrary intensity of the laser field.",
"Second, the single-atom building blocks have not been given a physical interpretation.",
"Here, we furnish the single atom building blocks with a physical interpretation and establish a close connection with the results of Mollow [15].",
"The paper is organized as follows.",
"In the next section we present the building blocks contributing to the double scattering background and interference spectra.",
"Thereafter it is shown how these blocks can be evaluated by solving the OBE under bichromatic driving.",
"Thereby we establish the connection to the method used by Mollow in [15].",
"In Sec.",
"we formulate rules for combining single-atom building blocks into double scattering diagrams, present the full set of diagrams contributing to the elastic and inelastic spectra, and give the explicit expressions thereof.",
"We conclude our work in Sec. .",
"To be self-contained, we will briefly outline here the main idea of the diagrammatic approach to CBS of laser light from two two-level atoms which was presented in detail in [12], [13].",
"The fundamental double scattering processes surviving the disorder averaging and contributing to the CBS background and interference signals are shown in Fig.",
"REF .",
"Figure: Diagrammatic representation of the double scatteringprocesses contributing to the background (a) and interference (b)spectra of CBS.",
"The laser wave (thick arrows) of frequency ω L \\omega _L isscattered by the atoms (gray circles) into waves whose frequenciesω ' \\omega ^{\\prime }, ω '' \\omega ^{\\prime \\prime } and ω D \\omega _D may differ from ω L \\omega _L.We assume that the laser field may be sufficiently strong to saturate the atomic transitions.",
"Accordingly, the frequencies $\\omega ^{\\prime }$ , $\\omega ^{\\prime \\prime }$ and $\\omega _D$ of the waves scattered by the atoms towards each other and a detector can, but need not, differ from the laser frequency $\\omega _L$ , i.e., correspond to inelastic scattering.",
"The co-propagating positive and negative frequency amplitudes (solid and dashed arrows, respectively) contribute to the background double scattering intensity, see Fig.",
"REF (a), which is independent from the observation direction, whereas the counter-propagating ones contribute to the CBS interference, see Fig.",
"REF (b).",
"We will be interested in finding the frequency distributions of the background and interference spectra (intensity vs. $\\omega _D$ ) as a function of the driving field parameters (such as the Rabi frequency and the offset from the atomic transition frequency).",
"In the framework of the diagrammatic approach [11], [12], [13], these frequency distributions can be derived on the basis of single-atom building blocks.",
"The latter represent spectral responses of an atom subjected to a classical bichromatic driving field.",
"The first component thereof corresponds to a laser field of arbitrary strength, whereas a second, weak, component describes the far-field scattered by the other atom.",
"Following the nomenclature used in laser spectroscopy [6], [16], we will refer to the laser and weak field components as the pump and probe fields, respectively.",
"While the classical description of the laser field is common, applying the same description to the atomic radiation exhibiting photon antibunching [17] is, in general, wrong.",
"However, in the dilute regime, when the double scattering originates from exchange of a single photon, the validity of the semiclassical treatment of the atom-probe field interaction does not contradict the nonclassical character of the scattered field [12], [13], which manifests itself only in correlations between at least two photons.",
"Furthermore, the validity of the classical ansatz for the probe field was proven analytically by establishing the equivalence between the results of the diagrammatic and master equation calculations [14].",
"Figure: Decomposition of scattering processes at each of the atoms(gray circles) from Fig.",
"as a sum of elastic(open circles) and inelastic (hatched squares) building blocks.",
"Processescontributing to (a) background, and (b) interference intensities.",
"Frequenciesof the probe and scattered fields will be defined after the expressions for the building blocks on the right hand side, as well as the rules for combining the building blocks are known.The classical ansatz for the fields exchanged between the atoms allows us to consider two atoms with their incoming and outgoing fields separately from each other.",
"The left hand sides in Fig.",
"REF (a) and (b) show the decomposition of the background and interference contributions to CBS from Fig.",
"REF (a) and (b), respectively, into single-atom blocks.",
"In order not to overburden diagrams, in Fig.",
"REF only fields scattered by the atoms are depicted, but here and henceforth one should remember that the atoms are also laser-driven and, furthermore, the effect of the laser field on atoms is accounted for non-perturbatively.",
"Note that the arrows are not labeled by their frequencies in Fig.",
"REF .",
"The latter will be defined in the course of the subsequent analysis.",
"In general, for any finite value of the saturation parameter, there are non-zero cross-sections for elastic and inelastic scattering of photons by a single laser-driven atom [6].",
"The same also holds in presence of additional probe field(s).",
"In fact, one of the purposes of the diagrammatic approach is to calculate elastic and inelastic responses of a single laser-driven atom subject to probe fields.",
"Accordingly, the right hand sides of Fig.",
"REF represent decompositions of the total single-atom responses into elastic (blank circles) and inelastic (hatched squares) building blocks.",
"Different shapes of the blocks (i.e., circles or squares) emphasize that the corresponding expressions result from different equations of motion (for the atomic dipole averages and temporal correlation functions, see ) describing the elastic and inelastic scattering, respectively.",
"Different colours refer to photons emitted at the laser frequency (blank), or at the frequency different from the laser frequency (hatched), as will be explained in Sec.",
"REF .",
"There is an important difference between the diagrams in Figs.",
"REF and REF .",
"While the former diagrams depict general (background and interference) double scattering processes for a random configuration of atoms, the latter ones show single-atom blocks which, by construction, only produce those background and interference contributions which automatically survive the disorder average.",
"In accordance with this, all arrows in Fig.",
"REF are for convenience oriented along the horizontal line.",
"Prior to presenting the explicit expressions for the building blocks on the right hand side of Fig.",
"REF , we will show that the elastic blocks can be decomposed further.",
"Each of the building blocks in Fig.",
"REF contains two outgoing arrows.",
"We will now express all elastic blocks (circles) as combinations of the blocks which each contain only one outgoing arrow.",
"We will refer to such building blocks with one outgoing arrow as `elementary' ones.",
"In the case when there are no incoming arrows, see Fig.",
"REF (a), the outgoing arrows correspond to the elastic intensity of light scattered by the laser-driven atom.",
"It is well-known [6] that the elastic intensity is proportional to $\\langle \\sigma ^+\\sigma ^-\\rangle ^{(0,{\\rm el})}=\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}$ , that is, given by the product of the expectation values of the atomic dipole operators for an atom driven only by the laser field (we will supply the expectation values in this case by the superscript $(0)$ ).",
"We represent this equation graphically in Fig.",
"REF (a).",
"There, a circle with the outgoing solid (dashed) arrow corresponds to $\\langle \\sigma ^-\\rangle ^{(0)}$ ($\\langle \\sigma ^+\\rangle ^{(0)}$ ), respectively, whereas the product of these two amplitudes is depicted by symbol `$\\times $ '.",
"It is easy to generalize this result to the case when there are incoming probe-field amplitudes.",
"In this case, the elastic building blocks can be expanded into sums of products of two elementary amplitudes, with the number of terms in this expansion being equal to the number of ways in which the incoming arrows can be distributed among the two circles (see Fig.",
"REF (b-d)).",
"Figure: Decomposition of the elastic scattering processes intoelementary processes.",
"(a) the elastic intensity of light scattered by atwo-level atom is equal to (a1): the product of the amplitudes for theelastic scattering of positive and negative frequency amplitudesdescribed by circles with outgoing solid and dashed arrows; (b) incase of two incoming probe fields there are four ways, (b1)-(b4),to distribute the probe fields among the irreducible blocks; (c),(d) decomposition of the elastic responses in case of one incomingprobe field into (c1), (c2) and (d1), (d2), two elementary blocks, respectively."
],
[
"`Inelastic' building blocks",
"Concerning the inelastic building blocks of Fig.",
"REF (hatched squares), they result from the non-factorizable part of the atomic response, i.e., $\\langle \\sigma ^+\\sigma ^-\\rangle -\\langle \\sigma ^+\\rangle \\langle \\sigma ^-\\rangle $ , see equation (REF ) below.",
"Unlike the elastic blocks, these blocks, which exhibit two outgoing arrows corresponding to $\\sigma ^+$ and $\\sigma ^-$ , respectively, cannot be factorized into a product of blocks with only one outgoing arrow.",
"In Sec.",
"REF we presented the disorder averaged elastic and inelastic single-atom building blocks which contribute to the double scattering background and interference spectra of CBS.",
"All these blocks represent a single, two-level, laser-driven atom which additionally receives none, one or two classical probe-field amplitudes.",
"We will next show how to evaluate these blocks using the OBE under bichromatic driving."
],
[
"Single-atom optical Bloch equations under bichromatic driving",
"A classical bichromatic driving field reads $ {\\cal E}(t)={\\cal E}_L^*e^{i\\omega _L t}+ {\\cal E}_Le^{-i\\omega _Lt}+\\varepsilon ^*e^{i\\omega _p t}+\\varepsilon e^{-i\\omega _pt},$ where ${\\cal E}_L$ , $\\varepsilon $ are complex amplitudes, and $\\omega _L$ , $\\omega _p$ the frequencies of the pump and probe field, respectively.",
"It is easy to show that the optical Bloch vector $\\langle \\vec{\\sigma }\\rangle =(\\langle \\sigma ^-\\rangle ,\\langle \\sigma ^+\\rangle ,\\langle \\sigma ^z\\rangle )$ , where $\\sigma ^-=|0\\rangle \\langle 1|$ , $\\sigma ^+=|1\\rangle \\langle 0|$ , $\\sigma ^z=|1\\rangle \\langle 1|-|0\\rangle \\langle 0|$ , and $|0\\rangle $ , $|1\\rangle $ the ground and excited states of the atom, obeys the following equation of motion written in the frame rotating at the laser frequency: $\\langle \\dot{\\vec{\\sigma }}\\rangle =M\\langle \\vec{\\sigma }\\rangle +\\vec{L}+v\\;e^{i\\omega t}\\Delta ^{(+)} \\langle \\vec{\\sigma }\\rangle +v^*e^{-i\\omega t}\\Delta ^{(-)}\\langle \\vec{\\sigma }\\rangle ,$ where $M$ denotes the optical Bloch matrix for an atom driven by the laser field: $M=\\left(\\begin{array}{ccc}-\\gamma +i\\delta &0&-i\\Omega /2\\\\0&-\\gamma -i\\delta &i\\Omega ^*/2\\\\-i\\Omega ^*&i\\Omega &-2\\gamma \\end{array}\\right),$ with $\\delta =\\omega _L-\\omega _0$ the laser detuning from the atomic transition frequency $\\omega _0$ , and $\\gamma $ half the spontaneous decay rate.",
"$\\Omega =2{\\cal E}_L/\\hbar $ and $v=2\\varepsilon d/\\hbar $ are the Rabi frequencies of the pump and probe fields, respectively, with $d$ the (real) matrix element of the atomic dipole transition.",
"$\\omega =\\omega _p-\\omega _L$ is the detuning between the probe and pump frequencies.",
"Finally, the matrices $\\Delta ^{(-)}=\\left(\\begin{array}{ccc}0&0&-i/2\\\\0&0&0\\\\0&i&0\\end{array}\\right),\\;\\;\\Delta ^{(+)}=\\left(\\begin{array}{ccc}0&0&0\\\\0&0&i/2\\\\-i&0&0\\end{array}\\right) $ describe the coupling of the atom to the positive and negative frequency components of the scattered field, respectively, and $\\vec{L}=(0,0,-2\\gamma )^T $ is a constant vector."
],
[
"Elastic building blocks",
"It is straightforward to establish the correspondence between the elementary elastic building blocks of Fig.",
"REF and solutions of equation (REF ).",
"We will start by noting that if $v=v^*=0$ in equation (REF ), it reduces to the standard OBE having the steady-state solution $\\langle \\vec{\\sigma }\\rangle ^{(0)}$ .",
"For non-zero probe fields, there appear corrections to the unperturbed components of the Bloch vector as well as oscillations at the frequencies $\\pm \\omega $ (see ).",
"Depending on whether a single incoming probe field represents a positive- or negative-frequency amplitude at frequency $\\omega $ , the corrections will be denoted $\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(-)}$ and $\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(+)}$ , respectively.",
"A second-order correction, which corresponds to one solid and one dashed incoming arrow at the frequency $\\omega $ , will be denoted as $\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}$ .",
"Finally, we recall that the emission of the positive-(negative-) frequency amplitudes is associated with the expectation values of the atomic lowering (raising) operators [6].",
"Summarizing the above, we express the elementary elastic building blocks in terms of the solutions of equation (REF ) as shown in Fig.",
"REF .",
"From now on we use the convention not to label the incoming (outgoing) waves which are exactly on resonance with the laser frequency.",
"Furthermore, we will denote elastic building blocks with outgoing wave at the laser frequency by a blank circle, whereas a hatched circle corresponds to an outgoing wave at a frequency $\\omega \\ne \\omega _L$ (as shown below, the latter requires the presence of an incoming wave with frequency $\\omega $ or $-\\omega $ ).",
"Figure: Graphical representation of the elementary diagrams forelastic scattering, and their mathematical expressions in terms ofsolutions of equation () for the optical Bloch vector.",
"Waves at the laser frequencyare not labelled.",
"Left andright columns describe positive- and negative-frequency amplitudes,respectively, of: (a) and (b) – elastic scattering in absence ofthe probe field; (c) and (d) – amplitudes of stimulated emission at the probefield frequency ω\\omega by a laser-driven atom; (e) and (f) –phase conjugation via nonlinear wave mixing of the pump andprobe fields; (g) and (h) – amplitudes of stimulated emission at the laser frequencyby an atom subjected to a positive- and negative-frequency probe fieldat the frequency ω\\omega .Diagrams (a) and (b) of Fig.",
"REF describe positive and negative frequency amplitudes, respectively, of the elastic scattering by a laser-driven atom.",
"Clearly, their product is nothing but the diagram (a1) in Fig.",
"REF .",
"Diagrams (c)-(h) include one or two incoming arrows at the frequency $\\omega $ .",
"Atomic response functions depicted by these diagrams were studied by Mollow [15] in the context of the weak probe absorption or amplification by a laser-pumped two-level atom (see ).",
"As follows from the analysis presented in , the processes (c) and (d) are correspond to the amplitudes of stimulated emission at the probe field frequency, and hence, their outgoing arrows are labeled by the same frequency values as the incoming ones.",
"In accord with the convention adopted above, these building blocks are hatched, since the frequency of the incoming wave – and therefore also of the elastically scattered outgoing wave – differs from the laser frequency.",
"Further on, the processes (e) and (f) correspond to a nonlinear phase conjugation whereupon the positive- (negative-)frequency amplitude at the frequency $\\omega $ is scattered into a negative- (positive-)frequency, that is, conjugated, amplitude at the frequency $-\\omega $ , via a nonlinear mixing with the laser wave described by the nonlinear atomic susceptibility.",
"Using the nonlinear optics terminology, one can regard $\\langle \\sigma ^-(\\omega )\\rangle ^{(-)}$ , $\\langle \\sigma ^+(\\omega )\\rangle ^{(-)}$ (and their complex conjugate quantities) as, respectively, the effective (that is, dependent on the laser pump) linear and nonlinear susceptibilities of a two-level system with respect to the probe field [18].",
"We note that the relevance of nonlinear susceptibilities for CBS from saturated atoms was mentioned before [19].",
"Finally, the processes (g) and (h) describe stimulated emission of the amplitudes at the pump field frequency in the presence of the probe field.",
"We conclude this part by a short remark.",
"The diagrams of Fig.",
"REF describe all possible elementary building blocks that appear in Fig.",
"REF .",
"Consequently, solutions of equation (REF ) for the Bloch vector to second order in the probe-field amplitude allow us to evaluate completely the `elastic' responses on the right hand side of Fig.",
"REF ."
],
[
"Inelastic building blocks",
"Let us now address the inelastic building blocks.",
"They are deduced from the solutions of equation (REF ) for the atomic dipole temporal correlation function [13], [12] (see also ), $g(t_1,t_2)=\\langle \\sigma ^+(t_1)\\sigma ^-(t_2)\\rangle -\\langle \\sigma ^+(t_1)\\rangle \\langle \\sigma ^-(t_2)\\rangle , $ in the steady-state limit when $g(t_1,t_2)=g(t_1-t_2)$ , and subsequent Laplace transformation to the frequency domain.",
"Figure: Graphical representation of the inelastic building blocksand their mathematical expressions (see equation ()).",
"(a) P (0) (ω)P^{(0)}(\\omega ) – the Mollow triplet; (b)P (2) (ω,ν)P^{(2)}(\\omega ,\\nu ) – correction tothe Mollow triplet due to the probe field at the frequency ω\\omega ; (c)P (+) (ω,ν)P^{(+)}(\\omega ,\\nu ) and (d) P (-) (ω,ν)P^{(-)}(\\omega ,\\nu ) – inelastic wave mixingof the pump and probe fields resulting in scattering of the waves neither of which is in resonance with the laser or probe waves.The Laplace transformed solutions of equation (REF ), with the probe field amplitudes $v$ and $v^*$ set equal to zero, together with the linear ($\\propto v, v^*$ ) and quadratic ($\\propto |v|^2$ ) corrections, yield the inelastic building blocks shown in Fig.",
"REF .",
"The physical interpretation of the functions $P^{(0)}(\\omega )$ , $P^{(\\pm )}(\\omega ,\\nu )$ and $P^{(2)}(\\omega ,\\nu )$ is analogous to the interpretation of the building blocks of Fig.",
"REF , and merely generalizes it to include the multi-photon scattering of the laser photons.",
"Diagram (a) in Fig.",
"REF represents the inelastic spectrum of a two-level atom subject to monochromatic driving.",
"That is, the function $P^{(0)}(\\omega )$ is the resonance fluorescence spectrum known, for $\\Omega \\gg \\gamma $ , as the Mollow triplet [20].",
"Diagram (b) represents the correction to the Mollow triplet $P^{(2)}(\\omega ,\\nu )$ proportional to the intensity of the probe field at the frequency $\\omega $ .",
"This diagram can be regarded as a generalization of diagrams (g) and (h) of Fig.",
"REF , which accounts for the inelastic scattering.",
"Finally, the functions $P^{(\\pm )}(\\omega ,\\nu )$ (see Fig.",
"REF (c,d)) represent inelastic wave mixing processes resulting in the emission of two amplitudes which are off-resonant with either the laser or the probe fields.",
"These diagrams should be contrasted with the diagrams (c,d) in Fig.",
"REF .",
"Indeed, in the latter diagrams, the two outgoing arrows are on-resonant with the probe and laser fields."
],
[
"Rules",
"In the previous Section we defined all single-atom building blocks.",
"We recall that the elastic and inelastic building blocks from which the background and interference contributions are constructed are shown on the right hand sides of Fig.",
"REF (a) and (b), respectively.",
"The four elastic blocks in Fig.",
"REF can be further decomposed according to Fig.",
"REF , with the elementary diagrams defined in Fig.",
"REF , whereas the inelastic blocks and their corresponding spectral response functions are shown in Fig.",
"REF .",
"So, all diagrams on the right hand side of Fig.",
"REF are defined, and we need to introduce rules for combining them.",
"These rules can be formulated as follows: The ladder and crossed contributions are obtained by combining the building blocks of atom 1 with that of atom 2, on the right hand side of Fig.",
"REF (a) and (b); each combined two-atom diagram is proportional to the factor $g^2$ , where $g=(k_L\\ell )^{-1}$ scales as the far-field dipole-dipole coupling between the two atoms separated by the mean distance $\\ell $ .",
"The elastic building blocks for atoms 1 and 2 are expanded according to Fig.",
"REF .",
"Each outgoing solid (dashed) arrow of atom 1 is merged with each incoming solid (dashed) arrow of atom 2.",
"As a result, one obtains sums of diagrams containing two intermediate and two outgoing arrows.",
"The intermediate arrows' frequencies are defined self-consistently: the frequencies of the incoming and outgoing arrows must coincide and, moreover, respect the form of the response functions presented in Figs.",
"REF , REF .",
"As a result, for example, there is only one independent inelastic frequency describing intermediate amplitudes.",
"The response functions associated with the single atom building blocks in the resulting double scattering diagrams must be multiplied.",
"The symbolic expressions for these functions are given in Figs.",
"REF and REF .",
"If an intermediate arrow representing the inelastic probe field (say, at frequency $\\omega $ ) changes its frequency then the expression corresponding to that diagram is integrated over this frequency as $\\int _{-\\infty }^{\\infty }d\\omega /(2\\pi )$ (note that $\\omega =\\omega _p-\\omega _L$ , and can take on negative values).",
"Diagrams containing blocks connected by intermediate counter-propagating arrows with no outgoing arrows must be excluded.",
"Such diagrams correspond to a photon cycling between the atoms without contributing to the detected signal.",
"Exactly the same rule exists in the nonlinear diagrammatic theory of CBS in a Kerr medium [21].",
"Using the above rules, we can represent the background and the interference contributions as sums of the diagrams shown in Figs.",
"REF and REF , respectively.",
"When writing down the explicit expressions for different contributions, we will omit the overall factor $g^2$ .",
"Figure: Background contributions to double scattering.",
"(a) Fullyelastic contribution originating from both atoms scatteringelastically; (b) inelastic contribution due to second atomscattering inelastically; (c) mixed contribution includingdiagrams (c1), (c2) corresponding to elastic scattering,and (c3), (c4) corresponding to inelastic scattering;(d) fully inelastic contribution.By definition, the elastic background spectrum $L_{\\rm el}(\\nu )=L_{\\rm el}\\delta (\\nu )$ exhibits a single $\\delta $ -peak precisely at the laser frequency, corresponding to $\\nu =0$ in the rotating frame.",
"Hence, it is given by the sum of all diagrams with unlabeled signal (outgoing) lines in Fig.",
"REF , that is, diagrams (a), (c1), and (c2).",
"Note that diagram (a) is to be unfolded into a sum of products of the following diagrams of Fig.",
"REF : (a1)(b1), (a1)(b2), (a1)(b3), and (a1)(b4).",
"Taking the expressions for the building blocks from Figs.",
"REF , REF , and applying the above rules for combining these building blocks, the elastic background intensity reads: $L_{\\rm el}&=\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+(0)\\rangle ^{(2)}\\nonumber \\\\&+\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-(0)\\rangle ^{(2)}\\nonumber \\\\&+\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+(0)\\rangle ^{(+)}\\langle \\sigma ^-(0)\\rangle ^{(-)}\\nonumber \\\\&+\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^-(0)\\rangle ^{(+)}\\langle \\sigma ^+(0)\\rangle ^{(-)}\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }P^{(0)}(\\omega )\\langle \\sigma ^+(\\omega )\\rangle ^{(2)}\\langle \\sigma ^-\\rangle ^{(0)}\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }P^{(0)}(\\omega )\\langle \\sigma ^-(\\omega )\\rangle ^{(2)}\\langle \\sigma ^+\\rangle ^{(0)},$ where equations (REF ), () and () correspond to diagrams (a), (c1) and (c2) in Fig.",
"REF , respectively (see for the explicit evaluation of the right hand side of equation (REF ))."
],
[
"Inelastic spectrum",
"As evident from Fig.",
"REF , its remaining diagrams describe the inelastic spectrum $L_{\\rm inel}(\\nu )&=\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}P^{(2)}(0,\\nu )\\\\&+P^{(0)}(\\nu )\\langle \\sigma ^-(\\nu )\\rangle ^{(-)}\\langle \\sigma ^+(\\nu )\\rangle ^{(+)}\\\\&+P^{(0)}(\\nu )\\langle \\sigma ^+(\\nu )\\rangle ^{(-)}\\langle \\sigma ^-(\\nu )\\rangle ^{(+)}\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }P^{(0)}(\\omega )P^{(2)}(\\omega ,\\nu ),$ where equations (REF ), (), (), and () correspond to diagrams (b), (c3), (c4) and (d) of Fig.",
"REF , and the (signal) frequency $\\omega $ was for convenience relabeled in lines 2 and 3 of equation (REF ) to $\\nu $ .",
"Figure: Interference contributions to double scattering.",
"(a) Fullyelastic contribution; (b) and (c) mixed contributions includingdiagrams (b1), (c1) corresponding to elastic scattering, and(b2), (c2) corresponding to inelastic scattering; (d) fully inelasticcontribution."
],
[
"Elastic spectrum",
"By the same argument as in the case of the background diagrams, we identify the diagrams (a), (b1), and (c1) of Fig.",
"REF as contributing to the elastic spectrum.",
"In the present case, the fully elastic contribution represented by diagram (a) is explicitly decomposed into diagrams (a1), (a2), and (a3) corresponding to a combination of diagrams (c1)(d2), (c2)(d1), and (c2)(d2) of Fig.",
"REF , respectively.",
"The product of diagrams Fig.",
"REF (c1) and (d1) does not contribute to Fig.",
"REF (a) because it corresponds to a photon cycling between the atoms (see Sec.",
"REF ).",
"Note that the squares which are hatched in Fig.",
"REF (b), (c) become blank in diagrams (b1) and (c1).",
"This is precisely the consequence of the self-consistent combination of diagrams: the connection of the phase conjugation diagram Fig.",
"REF (e), for example, to Fig.",
"REF (d) enforces the occurrence of two frequencies $\\omega $ and $-\\omega $ , and therefore $\\nu =0$ , in the building block Fig.",
"REF (d).",
"Thus, a blank square denotes an inelastic process, where the frequency of one of the outgoing arrows equals the laser frequency.",
"After these remarks, we present the explicit expression for the elastic interference spectrum: $ C_{\\rm el}&=\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-(0)\\rangle ^{(+)}\\langle \\sigma ^-(0)\\rangle ^{(-)}\\\\&+\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+(0)\\rangle ^{(-)}\\langle \\sigma ^+(0)\\rangle ^{(+)}\\\\&+\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-(0)\\rangle ^{(-)}\\langle \\sigma ^+(0)\\rangle ^{(+)}\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-(-\\omega )\\rangle ^{(+)}P^{(-)}(\\omega ,0)\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+(-\\omega )\\rangle ^{(-)}P^{(+)}(\\omega ,0),$ where the subsequent lines of equation (REF ) correspond to diagrams (a1), (a2), (a3), (b1), and (c1) of Fig.",
"REF , respectively."
],
[
"Inelastic spectrum",
"The three contributions to the inelastic spectrum are depicted by diagrams (b2), (c2) and (d) in Fig.",
"REF .",
"In diagram (d), both the solid and dashed lines corresponding to intermediate amplitudes that change their frequencies.",
"To write down the expression for this diagram, we note that the outgoing inelastic photon frequency is $\\nu $ .",
"Since the frequencies of the intermediate amplitudes are correlated, e.g., $\\omega $ and $\\nu -\\omega $ in Fig.",
"REF (d), there is only one independent probe field frequency $\\omega $ to be integrated over at fixed detected frequency $\\nu $ .",
"Denoting the signal frequency $\\nu $ in diagrams (b2) and (c2) of Fig.",
"REF , finally we obtain the following result for the inelastic interference spectrum: $C_{\\rm inel}(\\nu )&=\\langle \\sigma ^-\\rangle ^{(0)}\\langle \\sigma ^+(\\nu )\\rangle ^{(+)}P^{(-)}(0,\\nu )\\\\&+\\langle \\sigma ^+\\rangle ^{(0)}\\langle \\sigma ^-(\\nu )\\rangle ^{(-)}P^{(+)}(0,\\nu )\\\\&+\\int _{-\\infty }^{\\infty }\\frac{d\\omega }{2\\pi }P^{(+)}(\\omega ,\\nu )P^{(-)}(\\nu -\\omega ,\\nu ),$ where equations (REF ), (), and () correspond to diagrams (b2), (c2), and (d) in Fig.",
"REF ."
],
[
"Conclusion",
"We presented a diagrammatic derivation of the double scattering background and interference spectral distributions of coherent backscattering from two two-level atoms.",
"Although these distributions were known before [11], [12], [13], here we re-derived them in an intuitive way which, we believe, makes the diagrammatic approach to CBS of intense laser light from cold atoms more accessible and attractive to the reader.",
"It is not only the diagrams that render our present approach intuitive, but also a physical interpretation of the building blocks which is given in this work.",
"In particular, we established a connection of the `elastic' building blocks with the response functions used to calculate the weak probe absorption spectra by a laser-driven atom [15].",
"Furthermore, the `inelastic' building blocks include the Mollow triplet, a modification thereof proportional to the intensity of the weak probe field, and inelastic response functions which are linear in the probe fields, and which emerge due to mixing of the laser and probe waves.",
"Among the single-atom spectral response functions relevant to CBS, only the resonance fluorescence and probe absorption spectra were observed experimentally [22], [23].",
"It remains to be seen whether the other functions can also be measured directly in experiments with single quantum scatterers interacting with one strong and one weak laser field.",
"Perhaps, strongly driven quantum dots [16] are most suitable candidates for such observations.",
"The possibility of obtaining the double scattering signal diagrammatically from single-atom building blocks suggests two directions of future research, in the field of CBS of intense laser light from cold atoms.",
"First, the diagrammatic techniques presented in this work will be further developed to assess higher scattering orders of CBS.",
"Second, a generalization of this method to realistic atoms having degenerate dipole transitions will be another useful application.",
"Partial financial support by the DFG Research Grant 760 is gratefully acknowledged.",
"V.S.",
"acknowledges financial support through DFG grant BU-1337/9-1."
],
[
"Expressions for building blocks",
"Here we present expressions for the elementary elastic and inelastic building blocks (Figs.",
"REF and REF , respectively).",
"A detailed derivation thereof can be found, for example, in [13]."
],
[
"Elastic blocks",
"The expressions associated with the building blocks of Fig.",
"REF follow from the perturbative (in the probe field amplitude) solutions of the optical Bloch equation (REF ).",
"They are given by the first or second entries of the following vectors: $\\langle \\vec{\\sigma }\\rangle ^{(0)}&=G\\vec{L},\\\\\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(+)}&=G(i\\omega )\\Delta ^{(+)}\\langle \\vec{\\sigma }\\rangle ^{(0)},\\\\\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(-)}&=G(-i\\omega )\\Delta ^{(-)}\\langle \\vec{\\sigma }\\rangle ^{(0)},\\\\\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}&=G\\Delta ^{(+)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(-)}+ G\\Delta ^{(-)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(+)},$ where $G(x)$ is the free propagator given by $ G(x)=(x-M)^{-1},$ where $M$ is the Bloch matrix (see equation (REF )), $G\\equiv G(0)$ , and the matrices $\\Delta ^{(\\pm )}$ describe the coupling of the atomic dipole to the weak probe fields (see equation (REF )).",
"For the Bloch vector $\\langle \\vec{\\sigma }\\rangle =(\\langle \\sigma ^-\\rangle ,\\langle \\sigma ^+\\rangle ,\\langle \\sigma ^z\\rangle )$ , the zeroth-order elastic scattering amplitude $\\langle \\sigma ^-\\rangle ^{(0)}$ is given by $(G\\vec{L})_1$ .",
"Likewise, using (REF ), one finds all of the amplitudes of Fig.",
"REF ."
],
[
"Inelastic blocks",
"The expressions for the four elementary inelastic building blocks of Fig.",
"REF can be found by solving the equations of motions for the two-time correlation vectors: $\\vec{g}_1(t_1,t_2)&=\\langle \\vec{\\sigma }(t_1)\\sigma ^-(t_2)\\rangle -\\langle \\vec{\\sigma }(t_1)\\rangle \\langle \\sigma ^-(t_2)\\rangle ,\\; t_1>t_2,\\\\\\vec{g}_2(t_1,t_2)&=\\langle \\sigma ^+(t_1)\\vec{\\sigma }(t_2)\\rangle -\\langle \\sigma ^+(t_1)\\rangle \\langle \\vec{\\sigma }(t_2)\\rangle ,\\; t_1<t_2.$ By virtue of the quantum regression theorem the vectors $\\vec{g}_i(t_1,t_2)$ of equation (REF ) satisfy equations of motion similar to the OBE (REF ), which can be solved by Laplace transformation.",
"The resulting expressions for the inelastic building blocks are given as a sum of two terms corresponding to equation (REF a) and (REF b), respectively: $P^{(0)}(\\omega )&=[\\vec{\\tilde{g}}^{(0)}_2(-i\\omega )]_1+[\\vec{\\tilde{g}}^{(0)}_1(i\\omega )]_2,\\\\P^{(-)}(\\omega ,\\nu )&=[\\vec{\\tilde{g}}^{(-)}_2(\\omega ,i\\omega -i\\nu )]_1+[\\vec{\\tilde{g}}^{(-)}_1(\\omega ,i\\nu )]_2,\\\\P^{(+)}(\\omega ,\\nu )&=[\\vec{\\tilde{g}}^{(+)}_2(\\omega ,-i\\nu )]_1+[\\vec{\\tilde{g}}^{(+)}_1(\\omega ,i\\nu -i\\omega )]_2,\\\\P^{(2)}(\\omega ,\\nu )&=[\\vec{\\tilde{g}}^{(2)}_2(\\omega ,-i\\nu )]_1+[\\vec{\\tilde{g}}^{(2)}_1(\\omega ,i\\nu )]_2,$ where the Laplace transform solutions read: $\\vec{\\tilde{g}}_{1,2}(z)^{(0)}&=G(z)\\vec{g}^{(0)}_{1,2}(0),\\\\\\vec{\\tilde{g}}_{1,2}(\\omega ,z)^{(+)}&=G(z+i\\omega )[\\Delta ^{(+)}\\vec{\\tilde{g}}^{(0)}_{1,2}(z)+\\vec{g}^{(+)}_{1,2}(\\omega ,0)],\\\\\\vec{\\tilde{g}}_{1,2}(\\omega ,z)^{(-)}&=G(z-i\\omega )[\\Delta ^{(-)}\\vec{\\tilde{g}}^{(0)}_{1,2}(z)+\\vec{g}^{(-)}_{1,2}(\\omega ,0)],\\\\\\vec{\\tilde{g}}_{1,2}(\\omega ,z)^{(2)}&=G(z)[\\Delta ^{(+)}\\vec{\\tilde{g}}^{(-)}_{1,2}(\\omega ,z)+\\Delta ^{(-)}\\vec{\\tilde{g}}^{(+)}_{1,2}(\\omega ,z)\\nonumber \\\\&+\\vec{g}^{(2)}_{1,2}(\\omega ,0)].$ The initial conditions in the right hand side of equation (REF ) result from setting $t_1=t_2$ in (REF ): $\\vec{g}^{(0)}_{1}(0)&=-i\\Delta ^{(+)}\\langle \\vec{\\sigma }\\rangle ^{(0)}+\\vec{L}_1-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^-\\rangle ^{(0)},\\\\\\vec{g}^{(0)}_{2}(0)&=i\\Delta ^{(-)}\\langle \\vec{\\sigma }\\rangle ^{(0)}+\\vec{L}_2-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^+\\rangle ^{(0)},\\\\\\vec{g}_1^{(\\pm )}(\\omega ,0)&=-i\\Delta ^{(+)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(\\pm )}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(\\pm )}\\langle \\sigma ^-\\rangle ^{(0)}\\nonumber \\\\&-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^-(\\omega )\\rangle ^{(\\pm )},\\\\\\vec{g}_2^{(\\pm )}(\\omega ,0)&=i\\Delta ^{(-)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(\\pm )}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(\\pm )}\\langle \\sigma ^+\\rangle ^{(0)}\\nonumber \\\\&-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^+(\\omega )\\rangle ^{(\\pm )},\\\\\\vec{g}_1^{(2)}(\\omega ,0)&=-i\\Delta ^{(+)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}\\langle \\sigma ^-\\rangle ^{(0)}\\nonumber \\\\&-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^-(\\omega )\\rangle ^{(2)}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(+)}\\langle \\sigma ^-(\\omega )\\rangle ^{(-)}\\nonumber \\\\&-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(-)}\\langle \\sigma ^-(\\omega )\\rangle ^{(+)},\\\\\\vec{g}_2^{(2)}(\\omega ,0)&=i\\Delta ^{(-)}\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(2)}\\langle \\sigma ^+\\rangle ^{(0)}\\nonumber \\\\&-\\langle \\vec{\\sigma }\\rangle ^{(0)}\\langle \\sigma ^+(\\omega )\\rangle ^{(2)}-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(-)}\\langle \\sigma ^+(\\omega )\\rangle ^{(+)}\\nonumber \\\\&-\\langle \\vec{\\sigma }(\\omega )\\rangle ^{(+)}\\langle \\sigma ^+(\\omega )\\rangle ^{(-)},$ with $\\vec{L}_1=(0,1/2,0)^T$ , and $\\vec{L}_2=(1/2,0,0)^T$ ."
],
[
"Relation of the building blocks to the results of Mollow",
"Apart from notation, the elastic building blocks are equivalent to the results obtained by Mollow, in connection with the study of stimulated emission and absorption of a weak probe field by a coherently pumped two-level atom [15].",
"Since Mollow's derivation and definitions differ from ours, we here establish the relation between his and our results.",
"Upon the replacements $\\kappa &\\rightarrow & 2\\gamma , \\; \\kappa ^\\prime \\rightarrow \\gamma ,\\; \\Delta \\omega \\rightarrow \\Delta ,\\; \\nonumber \\\\\\Delta \\nu &\\rightarrow & \\omega , \\;2|\\lambda \\mathcal { E}_0|\\rightarrow \\Omega ,\\; \\lambda ^\\prime \\mathcal { E}^\\prime _0\\rightarrow v,\\nonumber $ where the left and right hand sides of each substitution correspond to the results of [15] and of our present work, respectively, we obtain the following identities: $\\bar{\\alpha }&\\equiv \\langle \\sigma ^-\\rangle ^{(0)}, & (\\bar{\\alpha })^*&\\equiv \\langle \\sigma ^+\\rangle ^{(0)},\\\\\\delta \\alpha _+&\\equiv v\\langle \\sigma ^-(\\omega )\\rangle ^{(-)}, &(\\delta \\alpha _+)^*&\\equiv v^*\\langle \\sigma ^+(\\omega )\\rangle ^{(+)},\\\\\\delta \\alpha _-&\\equiv v\\langle \\sigma ^+(\\omega )\\rangle ^{(-)},&(\\delta \\alpha _-)^*&\\equiv v^*\\langle \\sigma ^-(\\omega )\\rangle ^{(+)},\\\\\\delta \\alpha _0&\\equiv 4v^2\\langle \\sigma ^-(\\omega )\\rangle ^{(2)}, &(\\delta \\alpha _0)^*&\\equiv 4(v^*)^2\\langle \\sigma ^+(\\omega )\\rangle ^{(2)}.$ We note that the right hand side of each identity is equal, up to a prefactor, to the elementary building block from Fig.",
"REF .",
"For completeness, we also provide the expressions on which the elementary building blocks depend implicitly: $\\bar{n}-\\bar{m}&\\equiv \\langle \\sigma ^z\\rangle ^{(0)},& \\eta &\\equiv 2v\\langle \\sigma ^z(\\omega )\\rangle ^{(-)},\\\\\\delta \\bar{n}&\\equiv 2 |v|^2\\langle \\sigma ^z(\\omega )\\rangle ^{(2)},$ where, as in equations (REF )-(), the left and right hand sides of each identity refer to our present results and those of [15], respectively.",
"As shown in [15], the rate $\\mathcal { W}^{\\prime }$ of absorption/stimulated emission of the probe field is given by $\\mathcal { W}^{\\prime }=-i\\lambda ^\\prime \\mathcal { E}_0^{\\prime *}\\delta \\alpha _++i\\lambda ^\\prime \\mathcal { E}_0^{\\prime }(\\delta \\alpha _+)^*.$ Hence, the identities () imply our interpretation of the building blocks in Fig.",
"REF (c,d) as describing a stimulated emission of the probe field in presence of the pump field (whether the field is actually emitted or absorbed is defined by the sign of $\\mathcal { W}^{\\prime }$ [15]).",
"Analogously, the rate $\\delta \\mathcal { W}$ of absorption/stimulated emission of the pump field reads $\\delta \\mathcal { W}=-i\\lambda ^* \\mathcal {E}_0^*\\delta \\alpha _0+i\\lambda \\mathcal {E}_0(\\delta \\alpha _0)^*.$ Therefore, due to the identities (), we interpret the amplitudes $\\langle \\sigma ^\\pm (\\omega )\\rangle ^{(2)}$ (see Fig.",
"REF (g,h)) as describing stimulated emission of the pump field in presence of the probe field.",
"The physical interpretation of the function $\\delta \\alpha _-$ ($(\\delta \\alpha _-)^*$ ) was not considered by Mollow.",
"But, by analogy with the above results, and using the fact it represents a correction to the negative- (positive-)frequency scattering amplitude for a pumped atom probed by a positive- (negative-)frequency field, we deduce that $\\delta \\alpha _-$ ($(\\delta \\alpha _-)^*$ ) and, consequently, $\\langle \\sigma ^+(\\omega )\\rangle ^{(-)}$ ($\\langle \\sigma ^-(\\omega )\\rangle ^{(+)}$ ), see Fig.",
"REF (e,f), describes a phase-conjugation due to non-linear mixing of the pump and probe fields."
]
] | 1204.1512 |
[
[
"Towards extremely dense matter on the lattice"
],
[
"Abstract QCD is expected to have a rich phase structure.",
"It is empirically known to be difficult to access low temperature and nonzero chemical potential $\\mu$ regions in lattice QCD simulations.",
"We address this issue in a lattice QCD with the use of a dimensional reduction formula of the fermion determinant.",
"We investigate spectral properties of a reduced matrix of the reduction formula.",
"Lattice simulations with different lattice sizes show that the eigenvalues of the reduced matrix follow a scaling law for the temporal size $N_t$.",
"The properties of the fermion determinant are examined using the reduction formula.",
"We find that as a consequence of the $N_t$ scaling law, the fermion determinant becomes insensitive to $\\mu$ as $T$ decreases, and $\\mu$-independent at T=0 for $\\mu<m_\\pi/2$.",
"The $N_t$ scaling law provides two types of the low temperature limit of the fermion determinant: (i) for low density and (ii) for high-density.",
"The fermion determinant becomes real and the theory is free from the sign problem in both cases.",
"In case of (ii), QCD approaches to a theory, where quarks interact only in spatial directions, and gluons interact via the ordinary Yang-Mills action.",
"The partition function becomes exactly $Z_3$ invariant even in the presence of dynamical quarks because of the absence of the temporal interaction of quarks.",
"The reduction formula is also applied to the canonical formalism and Lee-Yang zero theorem.",
"We find characteristic temperature dependences of the canonical distribution and of Lee-Yang zero trajectory.",
"Using an assumption on the canonical partition function, we discuss physical meaning of those temperature dependences and show that the change of the canonical distribution and Lee-Yang zero trajectory are related to the existence/absence of $\\mu$-induced phase transitions."
],
[
"Introduction",
"QCD, first principle of strong interaction, has two important features; confinement and chiral symmetry breaking.",
"They change its nature depending on circumstances, e.g., temperature ($T$ ) and quark chemical potential ($\\mu $ ), which leads to rich structure in the QCD phase diagram [1], [2], [3].",
"Hadrons and nuclei are formed at ordinary temperatures and chemical potentials.",
"The quark-gluon plasma (QGP) is formed at high $T$ , which is expected to be created in the early universe and also in experiments, RHIC, LHC, etc.",
"Several possibilities have been proposed for high density states of matter, which are realized in the core of neutron stars.",
"Finite temperature properties of QCD have been uncovered by lattice QCD simulations [4], [5], [6], [7], [8], [9], which is a computational approach implemented with Monte Carlo methods.",
"For instance, recent simulations on finer lattices show that the deconfinement transition is crossover [4], and occurs at $T_{\\rm pc}=150-170$ MeV depending on observables [5], [6], [7].",
"In contrast, the properties of QCD at nonzero $\\mu $ have been difficult to study because of the notorious sign problem [10].",
"The sign problem spoils the importance sampling at $\\mu (\\ne 0)$ , and makes it unfeasible to generate gauge ensemble.",
"Several approaches have been developed to avoid the sign problem and study nonzero-$\\mu $ systems in lattice QCD simulations, see e.g., Refs.",
"[11], [12], [13], [14], [15].",
"The consistency between different approaches are shown for small $\\mu /T$ .",
"For instance, it was shown [16], [17] that several approaches of finite density lattice studies are consistent up to $\\mu /T\\sim 1$ , for staggered fermions.",
"For Wilson fermions, we showed the consistency of the Taylor expansion, multi-parameter reweighting (MPR) and imaginary chemical potential methods [18].",
"For status of lattice studies of the QCD phase diagram, see e.g, Refs.",
"[1], [19].",
"Extensive studies have been made for the finite density properties of QCD in particular for the location of the QCD critical end point (CEP).",
"The QCD phase diagram contains rich physics also at low temperatures [2], [1], [3].",
"One expectation is that towards large density region, QCD changes its state from hadron gas, nuclear liquid and color superconducting state.",
"In addition, the discovery of a pulser with twice solar mass [20] calls the reliable equation of state based on QCD at low temperature and high density.",
"The study of the low temperature and finite density states would be an important challenge for lattice QCD simulations.",
"There are expectations of the existence of sign free regions at low temperatures.",
"The finite density lattice QCD at low temperatures had been extensively investigated in the Glasgow method, see e.g.",
"Ref.",
"[21], [22], [23], [24], where it was shown the onset of the baryon density is given by $m_\\pi /2$ .",
"Recently, the phase at low temperatures was investigated in a lattice simulation [25].",
"It is empirically expected that the fermion determinant is independent of $\\mu $ at $T=0$ up to $\\mu <M_N/3$ .",
"However, the inclusion of chemical potentials can generally change the eigen spectrum of the Dirac operator and hadron spectrum.",
"The $\\mu $ -independence at $T=0$ was, therefore, considered as a small puzzle and called “Silver Blaze” problem [26].",
"Cohen showed the $\\mu $ -independence at $T=0$ for isospin chemical potential $\\mu _I$ case and Adams [27] for quark chemical potential case.",
"The orbifold equivalence for baryon chemical potential and isospin chemical potential also suggested [28], [29] that the phase of the fermion determinant is small up to $m_\\pi /2$ .",
"The same result was also obtained by Splittorff and Verbaarschot by using the chiral perturbation theory (ChPT).",
"On the other hand, the mechanism to extend the Silver Blaze region from $\\mu =m_\\pi /2$ to $\\mu =M_N/3$ is not yet understood.",
"A sign-free region was also suggested for high density.",
"Hong derived a high density effective theory (HDET) of low energy modes in dense QCD $\\mu \\gg \\Lambda _{\\rm QCD}$ [30].",
"The positivity of the fermion action in HDET was later discussed by Hong and Hsu [31], [32].",
"HDET was also studied in e.g., Refs.",
"[33], [34].",
"In this paper, we study the property of QCD at low temperature and finite density, using lattice QCD.",
"Particularly, the properties of the fermion determinant at low temperature and nonzero $\\mu $ will be clarified.",
"The key idea to tackle this issue is a reduction formula for the fermion determinant.",
"The formula is obtained by performing the temporal part of the determinant of a fermion matrix.",
"The technique is analogous to the transfer matrix method in condensed matter physics, and provides several advantages in lattice simulations of QCD at finite density.",
"The formula produces a matrix with a reduced dimension, which is called the reduced matrix.",
"We will see that the reduced matrix plays an important role in the properties of the fermion determinant at low temperature and nonzero $\\mu $ .",
"Some issues are discussed in the present paper.",
"In § , we focus on the reduced matrix, and discuss its derivation, interpretation and spectral property, where the connection with important dynamics of QCD is considered.",
"In particular, we will find a clear indication from lattice QCD simulations that the eigenvalues of the reduced matrix follow a scaling law of the temporal size $N_t$ .",
"This section is partly a review of the reduction formula.",
"In § , the property of the fermion determinant is investigated in detail.",
"We show that the quark determinant is insensitive to $\\mu $ at low temperatures for small $\\mu $ .",
"Using the relation between the eigenvalues of the reduced matrix and pion mass, we will show that the $\\mu $ -independence continues up to $\\mu =m_\\pi /2$ .",
"We will see that the $N_t$ scaling law leads to two expression of the low temperature limit of the fermion determinant both for low density and for high density.",
"The low density expression corresponds to the $\\mu $ -independence at small $\\mu $ .",
"Hence the $\\mu $ -independence at $T=0$ for $\\mu <m_\\pi /2$ is a consequence of the $N_t$ -scaling law of the eigenvalues of the reduced matrix.",
"The fermion determinant becomes real and the theory is free from the sign problem both cases.",
"In the case of high density and low temperature limit, QCD approaches to a theory, where quarks interact only in spatial direction, and the gluon action is given by the ordinary Yang-Mills action.",
"The corresponding partition function is exactly $Z_3$ invariant even in the presence of dynamical quarks because of the absence of the temporal interaction of quarks.",
"Property, possible application and numerical feasibility are discussed.",
"In § , the reduction formula is applied to the canonical formalism and Lee-Yang zero theorem.",
"These methods are often used in analysis to find CEP and first order phase transition line at low temperatures.",
"The sign problem causes difficulties in these methods [35], [16].",
"For instance, Ejiri pointed out that the sign problem causes a fictitious signal for the finite size scaling analysis of the Lee-Yang zero closest to the positive real axis, which makes it difficult to distinguish crossover and first order transition.",
"We consider the fugacity expansion of the grand partition function, where $Z_{GC}(\\mu )$ is expressed via the canonical partition functions $Z_n$ with quark number $n$ .",
"We show that both the $n$ -dependence of $Z_n$ and the Lee-Yang zero trajectory show characteristic change from high to low temperatures in a correlated manner.",
"These behavior can be used to qualitatively distinguish the crossover and first order phase transition."
],
[
"Reduction Formula",
"In this section, we study the reduction formula of the fermion determinant.",
"A basic idea of the reduction formula is to analytically carry out the temporal part of the fermion determinant, which reduces the rank of the determinant.",
"Since the chemical potential $\\mu $ couples only to the temporal link variables, the reduction formula rearranges the fermion determinant regarding $\\mu $ , which enables us to separate out $\\mu $ from link variables.",
"These characters offer some advantages in finite density simulations.",
"The reduction formula is not only a technical tool but also has physical interpretation.",
"The formula produces a matrix with a reduced dimension, which we refer to as the reduced matrix.",
"Its eigenvalues characterize the $\\mu $ -dependence of the fermion determinant.",
"The reduced matrix physically corresponds to a transfer matrix or a generalized Polyakov line, and its eigenvalues are related to a free energy of dynamical quarks.",
"In this section, the spectral properties of the reduced matrix are investigated in detail.",
"The formulation is given in § REF .",
"The interpretation and overview of the reduced matrix are presented in § REF .",
"Simulation setup is explained in § REF .",
"Some important and fundamental properties of the reduced matrix are shown in § REF .",
"In § REF , we will find a clear indication of the $N_t$ -scaling law of the eigenvalues of the reduced matrix.",
"In § REF , we discuss the relation between the eigenvalues of the reduced matrix and pion mass."
],
[
"Formulation",
"The reduction formula was derived by Gibbs [36], for staggered fermions.",
"Alternative derivation was found by Hasenfratz and Toussaint [37].",
"Later it has been applied to various studies, e.g.",
"Glasgow method [21], [22], [23], [24], [38], multi-parameter reweighting [39], canonical formalism [40], [16], [41], [42].",
"The derivation for Wilson fermions needs a delicate treatment because the Wilson terms $(r\\pm \\gamma _4)$ are singular for $r=1$ .",
"Adams derived the formula using zeta-regularization [43].",
"Borici derived the formula by introducing a permutation matrix [44].",
"Borici's method was further studied by Alexandru and Wenger [45] and two of present authors [46].",
"The formula for the Wilson fermion was later applied to studies of CEP [47] and thermodynamical quantities [18].",
"The reduction formula in continuum theory was found by Adams using zeta-regularization, which was used to show the $\\mu $ -independence of the fermion determinant at $T=0$ [27].",
"Throughout the present paper, we consider the Wilson fermions.",
"For the derivation for the staggered fermions, see e.g.",
"Ref. [12].",
"The grand partition function of $N_f$ -flavor QCD at a temperature $T$ and quark chemical potential $\\mu $ is given by ZGC(, T) = DU [()]Nf e-SG.",
"In our simulations, a renormalization group(RG)-improved action is used for $S_G$ .",
"The clover-improved Wilson fermion is employed for the fermion action () = x, x -i=13 [ (r-i) Ui(x) x, x+i + (r+i) Ui(x) x, x-i] -[ e+a (r-4) U4(x) x, x+4 +e-a (r+4) U4(x) x, x-4] - CSW x, x F, where $\\kappa $ and $r$ are the hopping parameter and Wilson parameter, respectively.",
"The quark chemical potential $\\mu $ couples to the fourth current $\\bar{\\psi }\\gamma _4\\psi $ in Euclidean path-integral, then a temporal hop accompanies a factor $e^{\\pm \\mu a}$ [48].",
"$C_{SW}$ is a coefficient of the clover term.",
"Note that the clover term is diagonal in the coordinate space, hence it does not cause any problem in the derivation of the reduction formula.",
"The fermion determinant satisfies a $\\gamma _5$ -hermiticity relation (())* = (-*).",
"It ensures that $\\det \\Delta (0)\\in \\mathbb {R}$ .",
"In the presence of nonzero real $\\mu $ , the fermion determinant is in general complex, which causes the sign problem.",
"The $\\det \\Delta $ is real if $\\mu $ is pure imaginary.",
"This property has been used in the study of the QCD phase diagram in lattice QCD simulations [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62] and also in phenomenological studies, see e.g.",
"Refs.",
"[63], [64].",
"For the preparation of the reduction formula, we define the spatial part of the Wilson fermion matrix as $B$ , B = x, x -i=13 [ (r-i) Ui(x) x, x+i + (r+i) Ui(x) x, x-i] - CSW x, x F, and two block-matrices i = ab, (x, y, ti) = c- Bab, (x, y, ti) r- -2 c+ r+ ab (x-y), i = ab, (x, y, ti) = c+ Bac,(x, y, ti) r+ U4cb(y, ti) -2 c- r- (x-y) U4ab(y, ti).",
"Here $r_\\pm = (r \\pm \\gamma _4)/2$ , which are projection operators in case of $r=1$ .",
"This property is used in the derivation of the formula.",
"$c_{\\pm }$ are arbitrary scalar except for zero, which is introduced in so called permutation matrix $P = (c_- r_- + c_+ r_+ V e^{\\mu a})$ .",
"$\\alpha _i$ describes a spatial hopping on a fixed time plane $t_i$ , while $\\beta _i$ describes a spatial hopping at $t_i$ and a temporal hopping to the next time slice.",
"They are independent of $\\mu $ .",
"A temporal matrix representation of the Wilson fermion matrix contains block-elements proportional to $r_\\pm $ , which are singular.",
"This fictitious singularity is avoided by calculating $\\det (P\\Delta )$ [44].",
"Then the determinant is obtained from $\\det (P \\Delta )/\\det P$ .",
"Using the block matrices, the reduction formula is given by [46] () = (c+ c- )-N/2 -Nred/2 C0 ( + Q ), with Q = (1-1 1) (Nt-1 Nt), C0 = (i = 1Nt (i ) ), where $\\xi =\\exp (-\\mu /T)$ is the fugacity.",
"$N=4N_c N_s^3 N_t$ and $N_{\\rm red}= N/N_t$ are the dimension of $\\Delta $ and $Q$ , respectively.",
"The rank of $\\alpha _i$ and $Q$ is reduced by $1/N_t$ compared to $N$ .",
"Instead, $Q$ is a dense matrix.",
"$Q$ and $C_0$ are independent of $\\mu $ .",
"Hence, the reduction formula separates the chemical potential from the gauge parts.",
"To obtain $\\det \\Delta $ , we need to evaluate $\\det (Q+\\xi )$ .",
"Here we calculate the eigenvalues $\\lambda $ for $|Q-\\lambda I|=0$ .",
"Once we obtain $\\lambda $ , the quark determinant is the analytic function of $\\mu $ .",
"Then, values of $\\det \\Delta (\\mu )$ are obtained for arbitrary values of $\\mu $ , which is an advantage, although the eigen problem requires large numerical cost.",
"Alternative methods such as LU decomposition of $Q+\\xi $ are available instead of solving the eigenvalue problem for $Q$ .",
"In this case, we need to perform the LU decomposition for each value of $\\mu $ .",
"Once having the eigenvalues of $Q$ , we obtain () = C0 -Nred/2n=1Nred (n + ) = C0 n=0Nredcn n-Nred/2 = C0 n=-Nred/2Nred/2 cn n, where we set $c_\\pm =1$ for simplicity.",
"In the second line, we redefine the index $n$ from the first expression to the second one.",
"The fermion determinant is described in two forms: a product form Eq.",
"(REF ), and a summation form Eq.",
"(REF ).",
"The second one is nothing but a fugacity or winding number expansion of the quark determinant, where fugacity coefficients $c_n$ are polynomials of $\\lambda _n$ [46].",
"The consistency between the reduction formula for the Wilson fermions and staggered fermions can be found by considering the fugacity expansion form.",
"The fugacity expansion of the fermion determinant is also obtained in other approaches, such as a domain decomposition technique [65].",
"Here, we summarize some features of the reduction formula.",
"The formula offers several advantages; The dimension of the determinant is reduced by $1/N_t$ , which reduces the numerical cost of the direct calculation of the fermion determinant.",
"$\\det \\Delta (\\mu )$ is the analytic function of $\\mu $ , which suppresses the numerical cost to evaluate values of $\\det \\Delta (\\mu )$ for many values of $\\mu $ and also provides an insight into the $\\mu $ -dependence of $\\det \\Delta (\\mu )$ .",
"Increase of the numerical cost for low temperature(large $N_t$ ) is also suppressed.",
"On the other hand, it has an applicable range; A direct method is used for the eigenproblem of the reduced matrix.",
"This requires large numerical cost.",
"In particular, it is difficult to apply the reduction formula to large lattice size because of the limitation of a memory."
],
[
"Physical interpretation",
"In the reduced matrix $Q$ , block elements $(\\alpha _i^{-1} \\beta _i)$ describe propagations of a quark from $t=t_i$ to $t=t_{i+1}$ .",
"$Q$ is the product of the block elements, and therefore it describes propagations from the initial $t=t_1$ to final time $t=t_{N_t}$ (depicted in the right panel of Fig.",
"REF ).",
"Accordingly, the reduction formula is analogous to the transfer matrix method, where block elements is interpreted as transfer matrices [36], [44], [45].",
"The matrix $Q$ is also understood as a generalized Polyakov line.",
"If we pick up the temporal links in $Q$ , it is written as $Q = \\cdots U_4(t_1) \\cdots U_4(t_2) \\cdots U_4(t_{N_t})$ .",
"If the trace is taken, $\\mbox{tr}Q$ is similar to the Polyakov loop.",
"It is known that the Polyakov loop describes the free energy of static quarks $\\langle P \\rangle \\sim e^{-F/T}$ with $P=\\mbox{tr}\\prod _{i=1}^{N_t} U_4(t_i)$ .",
"However, $Q$ contains spatial hopping terms denoted as the dots, where quarks are not static.",
"This suggests that the reduced matrix is related to the free energy of dynamical quarks.",
"$C_0$ consists of the spatial loops, where each loop is in a fixed time slice, which accompanies no temporal hops, see the left panel of Fig.",
"REF and Fig.",
"REF .",
"Thus the spatial quark loops $C_0$ are separated from temporal hoppings and do not contribute to the $\\mu $ -dependence of the fermion determinant.",
"Here, we summarize properties of the reduction formula.",
"From $\\gamma _5$ -hermiticity, it follows ; $C_0$ is real.",
"Two eigenvalues form a pair n, 1/n*.",
"This is because of a symmetry of $Q$ , see e.g., Ref. [45].",
"We give a simple proof of this relation in Appendix.",
"REF .",
"The coefficients of the positive and negative winding terms satisfy c-n=cn*.",
"The product of all the eigenvalues is unity (followed from $c_{-n}=c_n^*$ ); Q = n=1Nred n = 1 2 Nred = 1.",
"For the case $c_\\pm = 1$ , it is straightforward to show i = i.",
"From Eq.",
"(REF ) and Eq.",
"(REF ), we can separate all eigenvalues into two sets ($\\lbrace \\lambda _k | |\\lambda _k|>1\\rbrace $ and $\\lbrace \\lambda _k | |\\lambda _k|<1\\rbrace $ ).",
"Then, the product of the normalized eigenvalues in each set is $\\pm 1$ , k=1Nred/2 k|k| = 1.",
"For later convenience, we introduce a notation $\\lambda _L$ for $|\\lambda |>1$ and $\\lambda _S$ for $|\\lambda |<1$ .",
"These properties are satisfied for configuration by configuration, independent of temperature.",
"Although it is non-trivial to clarify the correspondence between the reduced matrix $Q$ and the original matrix $\\Delta $ , properties of QCD can be seen in the spectral property of $Q$ ; e.g., Confinement: As we have mentioned, the matrix $Q$ is related to the Polyakov line.",
"A correlation between the Polyakov loop and the eigenvalues of the reduced matrix was found in Ref. [45].",
"We can find the confinement property is realized in the angular distribution of $\\lambda $ .",
"Chiral symmetry breaking : It is known that the distribution of the eigenvalues form a gap near $|\\lambda |=1$ .",
"Gibbs pointed out [36] that the gap of the eigenvalue distribution is related to the pion mass.",
"Hence, the behavior of the eigenvalues near the unit circle is related to the chiral symmetry breaking.",
"Hadron Spectroscopy: Since the reduced matrix is related to the free energy of dynamical quarks, its eigenvalues are related to hadron spectrum.",
"Fodor, Szabo and Tóth proposed to extract hadron masses from the eigenvalues of the reduced matrix based on thermodynamical approach [66].",
"Low temperature behavior : Because $Q$ is a product of $N_t$ block-matrices, then it is expected [46] that the eigenvalues of $Q$ follow a scaling law for the temporal lattice size $N_t$ .",
"The scaling law is useful to study the low $T$ behavior of the fermion determinant.",
"For instance, we will see that the scaling law explains the $\\mu $ -independence of the fermion determinant at $T=0$ for $\\mu <m_\\pi /2$ ."
],
[
"Simulation setup",
"Before proceed to numerical simulations, simulation details are given here.",
"Simulations were performed on four lattice setups; $(N_s^3\\times N_t, m_{\\rm PS}/m_{\\rm V})$ = (i) $(8^3\\times 4, 0.8)$ (ii) $(8^4, 0.8)$ , (iii) $(10^3\\times 4, 0.8)$ and (iv) $(8^3\\times 4, 0.6)$ .",
"The wide range of temperatures were covered by using (i) and (ii).",
"The setup (iii) and (iv) were used to investigate the finite size effect and quark mass dependence.",
"The detail of these simulations is as follows.",
"(i) We investigated 29 values of $\\beta $ in the interval $1.5\\le \\beta \\le 2.4$ , which covers the temperature range $T/T_{\\rm pc}= 0.76 - 3$ .",
"The data were used in our previous studies on imaginary chemical potential approach [62] and thermodynamical quantities [18].",
"(ii) This simulation was performed to examine lower temperature.",
"We investigated 16 values of $\\beta $ in the interval $1.79 \\le \\beta \\le 1.94$ corresponding to the temperature range $T/T_{\\rm pc}=0.460-0.587$ .",
"(iii) This setup was performed to study the finite size effects.",
"We investigated 29 values of $\\beta $ in the interval $1.79\\le \\beta \\le 1.94$ , which covers the temperature range $T/T_{\\rm pc}= 0.93 - 1.2$ .",
"(iv) This simulation was performed to investigate the quark mass dependence.",
"We investigated four values of $\\beta =1.6, 1.7, 1.95$ and $2.0$ , which correspond to $T/T_{\\rm pc}=0.86, 0.94, 1.48$ and $1.67$ , respectively.",
"For all the four setup, the value of the hopping parameter $\\kappa $ was determined for each $\\beta $ by following lines of constant physics (LCP) in Ref. [67].",
"The clover coefficient $C_{\\rm SW}$ was determined by using a result obtained in the one-loop perturbation theory : $C_{\\rm SW}= ( 1- 0.8412 \\beta ^{-1})^{-3/4}$ .",
"We also used the data in Ref.",
"[67] to obtain the values of the temperature from $\\beta $ .",
"Gauge configurations were generated at $\\mu =0$ with the hybrid Monte Carlo simulations.",
"The size of the molecular dynamics step was $\\delta \\tau =0.02$ for $m_{\\rm PS}/m_{\\rm V}=0.8$ and $\\delta \\tau = 0.01$ for $m_{\\rm PS}/m_{\\rm V}=0.6$ with unit length.",
"The acceptance ratio was more than 90 % for $N_s=8$ and 80 % for $N_s=10$ .",
"HMC simulations were carried out for 11, 000 trajectories for each parameter set for $m_{\\rm PS}/m_{\\rm V}=0.8$ , and 4, 000 trajectories $m_{\\rm PS}/m_{\\rm V}=0.6$ .",
"It should be noted that the statistics are small for $m_{\\rm PS}/m_{\\rm V}=0.6$ .",
"The measurements were performed for each 20 HMC trajectories after thermalization.",
"The eigenvalue calculations were performed for 400 configurations for heavy quark case, and 50 configurations for light quark case.",
"Computational details of the eigenvalue calculation were explained in Ref.",
"[18]."
],
[
"Pair nature, $Z_{3}$ symmetry and temperature dependence",
"Let us study the spectral properties of the eigenvalues of the reduced matrix.",
"Figure: Eigenvalue distribution on the complex λ\\lambda planefor three temperatures on the 8 3 ×48^3\\times 4 lattice with m PS /m V =0.8m_{\\rm PS}/m_{\\rm V}=0.8.The top panels show the distribution of whole the eigenvalues,while the bottom panels magnify the region near the origin of the top panels.Figure: The spectral density of the larger half of the eigenvalues λ n \\lambda _nfor three temperatures.",
"The result for the smaller half of the eigenvalues isobtained from the pair nature.",
"Left : absolute value, right : phase of λ n \\lambda _n.In the right panel, thick lines are obtained from the three-Gaussian function :ρ(θ)=∑ i=1,2,3 a i exp(-(θ-c i ) 2 /(2c i 2 ))\\rho (\\theta )=\\sum _{i=1, 2, 3} a_i \\exp ( - (\\theta -c_i)^2/(2c_i^2)), (θ=arg(λ))(\\theta =\\arg (\\lambda )).",
"The vertical lines denote arg(λ)=±2π/3\\arg (\\lambda )=\\pm 2\\pi /3.We look at typical behaviors of the eigenvalue distribution on the complex $\\lambda $ plane in Fig.",
"REF , where results are shown for three temperatures.",
"Each result is obtained from one configuration on the $8^3\\times 4$ lattice.",
"For each temperature, the eigenvalues are shown in two different scales; the top panels show the distribution of whole the eigenvalues in the complex plane, and bottom panels enlarge the region near the origin.",
"These panels show two features of the reduced matrix, the pair nature and $Z_{3}$ -like property.",
"First, we consider the angular (phase) distribution of the eigenvalues of the reduced matrix.",
"The histogram of the angular distribution is shown in the right panel of Fig.",
"REF .",
"We find that the histogram of the angular distribution is well fitted with the three Gaussian function $\\rho (\\theta )=\\sum _{i=1, 2, 3} a_i \\exp ( - (\\theta -c_i)^2/(2c_i^2))$ , where $\\theta =\\arg (\\lambda )$ .",
"Three peaks are located at $\\theta = 0, \\pm 2\\pi /3$ at $T\\le T_{\\rm pc}$ , and two of them ($\\theta = \\pm 2\\pi /3$ ) shift towards $\\theta = 0$ as $T$ increases.",
"Although $Z_3$ is explicitly broken in the presence of the quarks, the eigenvalue distribution is correlated to the Polyakov loop, as we have mentioned in the previous subsection.",
"Indeed, Alexandru and Wenger found a correlation between the argument of the Polyakov loop and the location of the blank region of the eigenvalue distribution at high $T$ [45].",
"The confinement property of QCD may be seen in the angular distribution of the eigenvalues of the reduced matrix.",
"The left panel of Fig.",
"REF shows the distribution of the magnitude of the large eigenvalues $\\lambda _L$ .",
"The distribution of $|\\lambda _L|$ broadens at high temperatures.",
"Next, we consider a gap of the eigenvalues.",
"According to the pair nature, the eigenvalues are split into two regions.",
"The half of the eigenvalues are distributed in a region $|\\lambda |<1$ , and the other half in a region $|\\lambda |>1$ .",
"There is the gap between the two regions where no eigenvalue exists, see the bottom panels of Fig.",
"REF .",
"As we will see later, the gap is related to the pion mass.",
"Let us consider a physical meaning of the two types of the eigenvalues: larger half ($|\\lambda |>1$ ) and smaller half ($|\\lambda |<1$ ).",
"The pair nature of the eigenvalues is a consequence of $\\gamma _5$ -hermiticity, and therefore of the symmetry between the quark and anti-quark.",
"As we have mentioned, $\\mbox{tr}\\; Q$ is related to a generalized Polyakov loop and to free energies of an quark and anti-quark.",
"Qualitatively, the relation can be written as $\\lambda \\sim e^{-F/T}$ with a free energy $F$ .",
"Obviously $|\\lambda |<1$ if $F>0$ , while $|\\lambda |>1$ if $F<0$ .",
"Hence, the smaller half of the eigenvalues correspond to quarks and the larger half correspond to anti-quarks.",
"Figure: Finite size effect on the spectral densities of the larger half of theeigenvalues.",
"Left : absolute value, right: phase.In each panel, the blue-solid and red-dotted lines denote the results for8 3 ×48^3\\times 4 and 10 3 ×410^3\\times 4.",
"Note V s (N s =10)/V s (N s =8)∼2V_s(N_s=10)/V_s(N_s=8)\\sim 2.The volume dependence of the eigenvalues is shown in Fig.",
"REF .",
"It turns out that the spectral density of the eigenvalues is almost insensitive to the spatial lattice size $N_s$ .",
"The effect of increasing $N_s$ is small even for the gap and tail part of the distribution.",
"This insensitivity may be a consequence that long-range correlations are suppressed due to the heavy quark mass used in this work.",
"The volume dependence should be investigated in future simulations both with small quark masses and large lattice sizes.",
"Although the histogram of eigenvalues of $Q$ is insensitive to $N_s$ , the fermion determinant depends on $N_s$ .",
"The fermion determinant can be written as $\\det \\Delta (\\mu )\\sim \\exp ( \\sum _{n=1}^{N_{\\rm red}} \\ln (\\lambda _n + \\xi ))$ , or spectral representation, () = C0 ( 2Nc Vs /T + 4 Nc Vs d() (+) ), where $V_s=N_s^3$ .",
"$\\rho (\\lambda )$ is the spectral density on the complex $\\lambda $ plane.",
"The term $\\ln (\\lambda + \\xi )$ is complex, and generates the phase of $\\det \\Delta (\\mu )$ .",
"As we have shown in Fig.",
"REF , $\\rho (\\lambda )$ is insensitive to $N_s$ , and therefore the $\\lambda $ -integral is also insensitive to $N_s$ .",
"The imaginary part of the $\\lambda $ -integral, $\\int d\\lambda \\rho (\\lambda ) \\ln (\\lambda +\\xi )$ , is also insensitive to $N_s$ .",
"This means the phase of $\\det \\Delta (\\mu )$ is proportional to $V_s$ , which shows the well-known fact [10] that the sign problem becomes severe for large lattice volume.",
"Note that $N_t$ dependence is included in $\\lambda $ itself, but does not appear in the overall factor of the exponent.",
"Hence, the increase of $N_t$ does not increase the phase of the determinant.",
"This suggests that the sign problem is not necessarily severe at low temperatures.",
"We will see later, the increase of $N_t$ makes the sign problem milder.",
"Figure: The quark mass dependence of the spectral density.Here we consider the absolute values of the larger eigenvalues |λ|>1|\\lambda |>1.The parameter is β=1.7\\beta =1.7 for m ps /m V =0.6m_{\\rm ps}/m_{\\rm V}=0.6 and β=1.8\\beta =1.8 for m ps /m V =0.8m_{\\rm ps}/m_{\\rm V}=0.8.The decrease of the quark mass narrows the width of the distribution.Figure: The quark mass dependence of the scatter plot of theeigenvalues.",
"The eigenvalues near the unit circle, whichis related to the pion mass, is shown.",
"Eigenvalues approach tothe unit circle for small quark mass.The quark mass dependence of the eigenvalues of the reduced matrix is shown in Figs.",
"REF and REF .",
"The eigenvalues $\\lambda $ depend on the quark mass for both the tail part and gap part.",
"It is shown in Fig.",
"REF that the decrease of the quark mass narrows the eigenvalue distribution.",
"Figure REF shows the quark mass dependence of eigenvalues near the unit circle.",
"Qualitatively, eigenvalues approach to the unit circle as the quark mass becomes smaller.",
"Gibbs pointed out [36] for staggered fermions that eigenvalues outside the unit circle move away from the unit circle as the quark mass increases.",
"The results in Figs.",
"REF and REF are consistent with the result in Ref. [36].",
"However, the quark mass dependence is considered without the ensemble average.",
"We will consider the quark mass dependence of the average of the gap later."
],
[
"Low-$T$ behavior and {{formula:4fdc594b-8e90-458c-8696-37dbf79bd6ea}} -scaling law",
"Next, we investigate the $N_t$ dependence and the properties at lower temperature.",
"The left panel of Fig.",
"REF shows the eigenvalue distribution for the $8^4$ lattice with $\\beta =1.8$ , which corresponds to $T/T_{\\rm pc}\\sim 0.5$ .",
"Since $T=1/(a N_t)$ , the temperature in Fig.",
"REF is almost half compared to the case of Fig REF .",
"A major difference between $N_t=4$ and $N_t=8$ is the magnitude of the eigenvalues.",
"As the temperature decreases, the smaller half of the eigenvalues become smaller and larger half of the eigenvalues become larger.",
"The right panel of Fig.",
"REF shows the histogram of the absolute value of $\\lambda $ scaled by $N_t$ .",
"The results for $N_t=4$ and $N_t=8$ agree well.",
"This agreement implies that the eigenvalue distribution as a function of a variable $\\ln |\\lambda |/N_t$ is almost independent of $N_t$ , which leads to the scaling law $|\\lambda |\\sim l^{N_t}$ with a quantity $\\ln l\\equiv \\ln |\\lambda |/N_t$ .",
"Note that $l$ depends on the lattice spacing $a$ .",
"We have pointed out this scaling law in the previous study [46].",
"The matrix $Q$ is given as the product of $A_i = \\alpha _i^{-1} \\beta _i,(i=1, \\cdots N_t)$ .",
"Let us express $A_i = \\bar{A} + \\delta A_i$ , where $\\bar{A}$ is a matrix independent of time.",
"$A_i$ is expected to depend on time moderately in equilibrium, i.e., $|A_i| > | \\delta A_i|$ .",
"An additional cancellation would occur in $\\sum _i \\delta A_i$ , since $\\delta A_i$ is a fluctuation from $\\bar{A}$ .",
"If this argument holds, $Q$ is parameterize as $Q\\sim \\bar{A}^{N_t}$ .",
"The agreement observed in Fig.",
"REF clearly indicates this scaling behavior.",
"The right panel of Fig.",
"REF also shows the gap and pair nature of the eigenvalues.",
"We focus on the gap in the next subsection."
],
[
"Gap and pion mass",
"As we have mentioned, the reduced matrix describes the generalized Polyakov line and is related to the free energy of a dynamical quark.",
"Combining several Polyakov lines, it is possible to construct states having quantum numbers of hadrons.",
"Thus it is natural to expect that the eigenvalues of the reduced matrix have something to do with the hadron spectrum.",
"Indeed, Gibbs pointed out [36] a relation at $T=0$ between the pion mass and the eigenvalue gap.",
"Later, Fodor, Szabo and Tóth related the hadron spectrum to the eigenvalues of the reduced matrix, based on thermodynamical consideration [66].",
"In this subsection, we study the pion mass using an eigenvalue close to $|\\lambda |=1$ , which controls the gap.",
"Figure: Left panel: Eigenvalues near the unit circle |λ|=1|\\lambda |=1.The parameters for each result are (β,N t )=(1.8,8),(1.9,8),(1.8,4),(\\beta , N_t)=(1.8, 8), (1.9, 8), (1.8, 4),and (1.9,4)(1.9, 4) for T/T pc =0.47,0.54,0.93,T/T_{\\rm pc}=0.47, 0.54, 0.93, and 1.081.08, respectively.Each result is obtained from one configuration.Right panel : The largest eigenvalue in smaller half max |λ n |<1 |λ n |\\max _{|\\lambda _n|<1} |\\lambda _n|,which is the eigenvalue closest to the unity |λ|=1|\\lambda |=1.",
"The gap is defined asthe deviation between the data and one.The temperature dependence of the gap is shown Fig.",
"REF .",
"The left panel shows twenty eigenvalues close to unity, where the result for each temperature is obtained from one configuration.",
"It is shown that the gap shrinks as the temperature increases.",
"To take into account the ensemble average, we focus on the maximum eigenvalue among the smaller half: $\\max _{|\\lambda |<1} \\lambda $ .",
"The gap is given by the difference between unity and this quantity.",
"The result is shown in the right panel, where the gap is large at low $T$ and decreases as $T$ increases.",
"In case of $m_{\\rm ps}/m_{\\rm V}=0.8$ , the gap is saturated at high $T$ .",
"The right panel of Fig.",
"REF contains the results for two values of $m_{\\rm ps}/m_{\\rm V}$ .",
"We find that the average value of $\\max _{|\\lambda |<1} |\\lambda |$ is larger for smaller quark mass.",
"This implies that the gap shrinks for smaller quark mass, which is consistent with Ref. [36].",
"However, the simulations for $m_{\\rm ps}/m_{\\rm V}=0.6$ were done only for four temperatures and for the small number of HMC trajectories.",
"Further simulations are needed to reveal the quark mass dependence of the eigenvalues of the reduced matrix.",
"Figure: Pion mass obtained from Gibbs's definition.At low TT, the curve is well fitted with c 1 (T/T pc ) -1 c_1 (T/T_{\\rm pc})^{-1} (theblue dashed line).",
"The pion mass is extracted from the coefficientc 1 c_1 of the 1/T1/T fit at low temperatures.According to Gibbs [36], the pion mass is given by a m= - 1Nt |k|<1 |k|2, which is expected to be valid at $T=0$ [22].",
"On the other hand, Fodor, Szabo and Tóth derived a modified expression [66], a m= Nt ( -1Nt | k |2 ).",
"The pion mass in Eq.",
"(REF ) is shown in Fig.",
"REF as a function of $T/T_{\\rm pc}$ .",
"The results are shown for $m_{\\rm ps}/m_{\\rm V}=0.6$ and $0.8$ .",
"In case of $m_{\\rm ps}/m_{\\rm V}=0.8$ , $m_{\\pi }/T$ is well fitted with $c_1(T/T_{pc})^{-1}$ at $T/T_{\\rm pc}=0.5$ with $c_1=4.060(6)$ .",
"The pion mass is extracted from the low temperature behavior as $m_{\\pi } = c_1 T_{\\rm pc}$ , which is approximately $4 T_{\\rm pc}$ .",
"This large value is because of the present simulation setup.",
"At high $T$ , $m_\\pi $ is almost linearly proportional to $T$ , which is probably due to thermal effects.",
"The decreasing $m_{\\rm ps}/m_{\\rm V}$ makes $m_\\pi /T$ smaller.",
"However, simulation data are not sufficient to obtain $c_1$ .",
"Hadron masses are extracted through the exponential damping in an usual method.",
"In the present approach, the pion mass is extracted through $1/T$ behavior.",
"It is important to note that the $1/T$ behavior is already obtained at $N_t=8$ .",
"Hence, this approach may be an alternative method for hadron spectroscopy, which was discussed in detail in Ref.",
"[66]."
],
[
"Low temperature limit of QCD",
"In this section, we consider the behavior of the fermion determinant at low temperatures.",
"For small $\\mu $ , the fermion determinant becomes insensitive to $\\mu $ as $T$ decreases, and becomes almost independent of $\\mu $ at enough low $T$ .",
"We explain this behavior by using the properties of the eigenvalues of the reduced matrix.",
"We first consider $T$ - and $\\mu $ -dependence of a reweighting factor.",
"Then, we consider the low temperature limit with the use of the $N_t$ -scaling law of the eigenvalues of the reduced matrix.",
"We will derive two expressions for low temperature limit of the quark determinant; one is for small $\\mu $ and the other is for large $\\mu $ .",
"The low density expression shows the $\\mu $ -independence of the fermion determinant at $T=0$ for $\\mu <m_\\pi /2$ .",
"We will conclude that the $\\mu $ -independence at $T=0$ for small $\\mu $ is the consequence of the $N_t$ scaling law of the eigenvalues of the reduced matrix.",
"The other expression is its high-density counter part.",
"We discuss the physical meaning and criterion for these limits.",
"We discuss the high density limit of the fermion determinant.",
"In low temperature and high density limit, QCD approaches to a theory, where quarks interact in spatial directions with the ordinary Yang-Mills type of the gauge action.",
"The corresponding partition function is $Z_3$ invariant even in the presence of dynamical quarks.",
"The fermion determinant becomes real and the theory is free from the sign problem."
],
[
"Fluctuation of the fermion determinant",
"In this subsection, we consider the quark determinant at low temperature and finite $\\mu $ .",
"Figure: The scatter plot of the reweighting factor on the complex plane.Left panel : hadron phase (T/T pc =0.93T/T_{\\rm pc}=0.93).",
"Right panel : QGP phase(T/T pc =1.08T/T_{\\rm pc}=1.08).The figures are taken from Ref.",
".The results are obtained for 8 3 ×48^3\\times 4 lattice with m ps /m V =0.8m_{\\rm ps}/m_{\\rm V}=0.8.Figure REF is the scatter plot of $\\ln R(\\mu , \\beta _0)_{(0,\\beta _0)}=N_f\\ln \\det \\Delta (\\mu )/\\det \\Delta (0)$ , which is so called reweighting factor.",
"This shows how the quark determinant develops as $\\mu $ is varied from a simulation point $(\\mu =0,\\beta _0)$ , where gauge configurations are generated.",
"Here we fix a temperature ($\\beta =\\beta _0$ ).",
"The left and right panels show the results for $T/T_{\\rm pc}\\sim 0.93 (\\beta _0=1.8)$ and $1.08(1.9)$ , respectively.",
"The horizontal and vertical axes are the real and imaginary parts of $\\ln R$ , i.e., the exponent of $|R|$ and the phase of $R$ .",
"Figure: Left : the average of ln|R|=N f ln|detΔ(μ)/detΔ(0)|\\ln |R|=N_f\\ln |\\det \\Delta (\\mu )/\\det \\Delta (0)|, right : the averagephase factor defined by 〈cosθ〉\\langle \\cos \\theta \\rangle , where θ=arg(R)\\theta = \\arg (R).The results are obtained for 8 3 ×48^3\\times 4 lattice with m ps /m V =0.8m_{\\rm ps}/m_{\\rm V}=0.8.The results are shown in Fig.",
"REF now as functions of $T/T_{\\rm pc}$ for three values of $\\mu $ .",
"The magnitude of the reweighting factor is very small at low temperature, and rapidly increases near $T_{\\rm pc}$ .",
"As $T$ decreases, the average of $\\ln |R|$ approaches to zero at least up to $\\mu a = 0.25$ .",
"This behavior means $|\\det \\Delta (\\mu ) / \\det \\Delta (0)|\\sim 1$ , and therefore the fermion determinant is insensitive to $\\mu $ at low temperatures $T/T_{\\rm pc}\\sim 0.5$ .",
"In the right panel, we plot the average phase factor $\\langle \\cos \\theta \\rangle $ with $\\theta = \\arg (R)$ .",
"The average phase factor is close to one at high and low temperatures, and has minimum at slightly below $T_{\\rm pc}$ .",
"The sign problem is very severe in the vicinity of $T_{\\rm pc}$ .",
"At high temperatures, the average phase factor increases with increasing $\\mu $ .",
"On the other hand, the $\\mu $ -dependence of the average phase factor is small at low temperatures.",
"The average phase factor approaches to one for the three values of $\\mu $ .",
"Figure: The quark chemical potential(μ\\mu ) dependence of the average ofln|R|=N f ln|detΔ(μ)/detΔ(0)|\\ln |R|=N_f\\ln |\\det \\Delta (\\mu )/\\det \\Delta (0)|.We found that the fermion determinant is insensitive to $\\mu $ at low temperatures.",
"However, the low-$T$ behavior of the fermion determinant depends on $\\mu $ .",
"Next, we consider the reweighting factor as a function of $\\mu $ in Fig.",
"REF .",
"At high $T$ ($T\\ge 0.76 T_{\\rm pc}$ ), the average value of $\\ln |R|$ smoothly increases as $\\mu $ goes to larger.",
"A discontinuous change is found at $\\mu a=0.5\\sim 0.6$ at low $T$ ($T\\le 0.54 T_{\\rm pc}$ ).",
"$|R|$ is approximately unity for small $\\mu $ , and starts to increase at $\\mu a = 0.5\\sim 0.6$ , see the right panel.",
"The onset of the $\\mu $ -dependence of the fermion determinant is about $\\mu a = 0.5 \\sim 0.6(\\mu /T = 4.0 \\sim 4.8)$ .",
"Using the value of the pion mass obtained in the previous section, it corresponds to $\\mu = 0.5 m_\\pi \\sim 0.6 m_\\pi $ .",
"This is consistent with a well known result that the fermion determinant is independent of $\\mu $ for $\\mu <m_\\pi /2$ at $T=0$ , see e.g.",
"Refs [23], [26], [27], [68].",
"Phenomenologically, it is expected that the $\\mu $ -independence at $T=0$ continues up to $\\mu =M_N/3$ , where $M_N$ is the mass of the nucleon.",
"This problem was raised long time ago in the Glasgow method, see e.g.",
"Ref. [23].",
"There would be several reasons for the discrepancy between the critical value of $\\mu $ and $M_N/3$ .",
"For instance, the importance sampling at $\\mu =0$ may cause this discrepancy, because the quark chemical potential is equivalent to the isospin chemical potential at $\\mu =0$ .",
"Figure: The fluctuation of the reweighting factor as a function of T/T pc T/T_{\\rm pc}.σ 2 =1 n∑ i (x i -x ¯) 2 ,x ¯=1 n∑ i x i \\sigma ^2= \\frac{1}{n}\\sum _i (x_i-\\bar{x})^2, \\bar{x}=\\frac{1}{n}\\sum _i x_i,where x= Re [lnR(μ,β 0 ) (0,β 0 ) ]x={\\rm Re}[\\ln R(\\mu , \\beta _0)_{(0,\\beta _0)}] for left panel and x= Im [lnR(μ,β 0 ) (0,β 0 ) ]x={\\rm Im}[\\ln R(\\mu , \\beta _0)_{(0,\\beta _0)}]for right panel.",
"The results for N t =4N_t=4 are taken fromRef.",
".The deviation of the reweighting factor, $\\sigma ^2$ , is shown in Fig.",
"REF .",
"The deviation of the reweighting factor reaches the maximum near the crossover transition point $T_{\\rm pc}$ both in the magnitude (left panel) and the phase (right panel), and decreases as the temperature is away from $T_{\\rm pc}$ .",
"The peak becomes prominent as $\\mu $ increases.",
"In Fig.",
"REF , we showed the reweighting factor for low $T$ ($T/T_{\\rm pc}=0.93$ ) and high $T$ ($1.08$ ).",
"In the vicinity of $T_{\\rm pc}$ , gauge configurations visit low-$T$ states and high-$T$ states, which results in the peak of the fluctuation.",
"The real part of $\\ln |R|$ reaches the maximum at $T_{\\rm pc}$ , while the imaginary part reaches the maximum at slightly below $T_{\\rm pc}$ .",
"Hence the sign problem is most severe at slightly below $T_{\\rm pc}$ , see also the right panel of Fig.",
"REF .",
"Figure: The fluctuation of the quark determinant as a function of T/T pc T/T_{\\rm pc} andthe Gaussian fit.",
"See also Fig.",
".In Fig.",
"REF , we focus on the low temperature region.",
"As $T$ decreases, the deviation rapidly decreases for both real and imaginary parts.",
"We find that the low temperature behavior is well fitted with the Gaussian function, which is shown in the solid lines in Fig.",
"REF .",
"This implies that fluctuation of the quark determinant is exponentially suppressed as the temperature decreases for small $\\mu $ .",
"We have seen that the fermion determinant is insensitive to $\\mu $ at low temperatures for $\\mu <m_\\pi /2$ .",
"As we have discussed in § REF , the eigenvalues follow the $N_t$ -scaling law.",
"Let us consider an large eigenvalue $\\lambda $ ($|\\lambda |>1$ ) and describe its counterpart as $1/\\lambda ^*$ .",
"As increasing $N_t$ or decreasing $T$ , the large eigenvalue $\\lambda $ becomes lager and the smaller eigenvalue $1/\\lambda ^*$ smaller.",
"If $\\mu $ is fixed at a small value, $\\xi $ is $O(1)$ .",
"This leads to the scale separation $|1/\\lambda ^*| \\ll \\xi \\ll |\\lambda |$ .",
"The contribution of each eigenvalue to the fermion determinant is approximated as $\\xi + \\lambda \\sim \\lambda $ and $\\xi + 1/\\lambda ^* \\sim \\xi $ .",
"This causes the $\\mu $ -independence of the fermion determinant at low temperatures.",
"We will turn back to this point in the next subsection.",
"The average phase factor $\\langle \\cos \\theta \\rangle $ is close to one at low temperatures as well as high temperatures.",
"Does it mean the feasibility of MC simulations at low $T$ ?",
"At high $T$ , although the fluctuation is small, the $\\mu $ dependence of the fermion determinant is non vanishing, while the $\\mu $ -dependence almost disappears at low $T$ .",
"It is already known that the neglecting phase leads to the phase transition at $\\mu =m_\\pi /2$ .",
"The phase of the determinant plays an important role to go beyond $m_\\pi /2$ .",
"The careful analysis would be required to evaluate the phase of the determinant, which may cause another difficulty at low temperature simulations.",
"Figure: The fluctuation of the quark determinant on 10 3 ×410^3\\times 4.See also the results for N s =8N_s=8 in Fig.",
".σ\\sigma is given in the caption of Fig.",
".Note that the above results are obtained for fixed spatial lattice size $N_s$ .",
"As we have mentioned, the fermion determinant depends on $N_s$ , although the eigenvalues of the reduced matrix are insensitive to $N_s$ .",
"Finally in this subsection, we consider the volume dependence of the reweighting factor.",
"The result for $N_s=10$ is shown in Fig.",
"REF .",
"See also the result for $N_s=8$ in Fig.",
"REF .",
"Increasing $N_s$ from 8 to 10, the maximum value of the deviation is almost twice both in the left and right panels, which is approximately equal to the volume ratio $V_s(N_s=10)/V_s(N_s=8)=10^3/8^3 \\sim 2$ .",
"A question arises on the phase of the fermion determinant at the low temperature limit $N_t\\rightarrow \\infty $ and thermodynamical limit $N_s\\rightarrow \\infty $ .",
"As we have shown, decreasing temperature suppresses the phase, while increasing the spatial volume enhances the phase.",
"What happens for the phase at the low temperature limit and thermodynamical limit ?",
"The result depends in general on the order of taking these two limits.",
"In this paper, we gave only a result in Fig.",
"REF which was obtained for a fixed lattice volume.",
"We will extend the present study for various sets of lattice sizes $N_s$ and $N_t$ in the next step Splittorff and Verbaarschot investigated the average phase factor at low temperature in chiral perturbation theory [68], and discussed this problem.."
],
[
"Low temperature limit of the fermion determinant",
"According to the $N_t$ -scaling law, we parameterize the larger half of the eigenvalues as $\\lambda _n = l_n^{N_t}e^{i\\theta _n}$ .",
"Using the pair nature, the smaller half of the eigenvalues are described as $1/\\lambda _n^*=l_n^{-N_t}e^{i\\theta _n}$ .",
"Now, the quark determinant is parameterize as = C0-Nred/2 n=1Nred/2( + lnNt ein ) n=1Nred/2( + ln-Nt ein), where we set $c_\\pm =1$ for simplicity.",
"In Eq.",
"(REF ), we divide the product into two parts and replace the smaller eigenvalues with $1/\\lambda _n^*$ .",
"In the low temperature limit $T= 1/(a N_t)\\rightarrow 0, (N_t\\rightarrow \\infty )$ , large and small eigenvalues follow lnNt, (ln)-Nt 0.",
"Since the fugacity $\\xi = \\exp (-\\mu a N_t)$ also decreases with the same exponent as the small eigenvalues, $\\det \\Delta (\\mu )$ in the low-$T$ limit depends on $\\mu $ .",
"I) Fixed $\\mu /T$ .",
"In this case, the fugacity $\\xi $ is constant, and the smaller eigenvalues decrease faster than $\\xi $ : $\\xi \\gg l_n^{-N_t}$ .",
"Then, the quark determinant is reduced to = C0 n=1Nred/2n, where the product is taken over the larger half of the eigenvalues $|\\lambda _n|>1$ .",
"$\\det \\Delta (\\mu )$ is independent of $\\mu $ , and therefore there is no sign problem in this limit.",
"II) Fixed $\\mu a$ .",
"In this case, the fugacity decreases with the same exponent as the smaller eigenvalues.",
"This case is further classified into two cases corresponding to the magnitude relation of $ \\exp (\\mu a)$ and $l_n$ .",
"We introduce a typical magnitude $\\bar{l}$ , which is used to compare the eigenvalues and $\\exp (\\mu a)$ .",
"This may be the average of $l_n$ or smallest one of $l_n$ depending on the distribution of eigenvalues.",
"We will turn back to this point later.",
"a) $\\exp (\\mu a) < \\bar{l}$ .",
"In this case, the smaller eigenvalues decrease faster than the fugacity; $\\xi \\gg (l_n)^{-N_t}$ .",
"We obtain = C0 n=1Nred/2n.",
"This is equal to Eq.",
"(REF ), which implies that the quark determinant remains unchanged up to $\\mu a < \\ln \\bar{l}$ in the low temperature limit.",
"Namely, $\\det \\Delta (\\mu )$ is independent of $\\mu $ at low temperature and small $\\mu $ regions.",
"This is nothing but the Silver Blaze phenomena discussed above [27].",
"This is analogous to a situation in Fermi statistics.",
"If $T$ is smaller than a lowest excitation energy of a system, then the inclusion of small $\\mu $ can excite no quark in excited energy levels, the system remains to stay at the lowest energy state.",
"This can lead the system to be independent of $\\mu $ .",
"We discuss this point in Ref. [69].",
"b) $\\exp (\\mu a) > \\bar{l}$ .",
"In this case, the fugacity decreases faster than the smaller eigenvalues; $\\xi \\ll (l_n^*)^{-N_t}$ .",
"Then, we obtain = -Nred/2 C0 Q = -Nred/2 i=1Nt (i) = -Nred/2 i=1Nt (Bac,(x, y, ti) r+ -2 r- (x-y) ).",
"where we first use Eq.",
"(REF ), then substitute Eq.",
"(REF ).",
"In the last line, we use $\\det U =1$ .",
"In Eq.",
"(REF ), $B(t_i)$ contains spatial hops in the $i$ -th time slice, but does not contain any temporal hopping terms.",
"The $\\mu $ dependence comes only from the overall factor $\\xi ^{-N_{\\rm red}/2}=\\exp (2 N_c N_s^3 \\mu /T)$ .",
"This is the highest order term in the fugacity expansion and means that all the states are occupied by quarks.",
"As we have discussed, $\\det Q=1$ and $C_0$ is real, therefore Eq.",
"(REF ) is real and free from the sign problem.",
"The fermion determinant is real in the low temperature limit both for large and small $\\mu $ .",
"However, the fermion determinant is given by the different expressions at large and small $\\mu $ , which may suggest that two different states exist at $T=0$ .",
"One may think to determine the critical value of $\\mu _0 a=\\ln \\bar{l}$ .",
"However the determination of $\\bar{l}$ is nontrivial task because of the finite width of the distribution of the eigenvalues.",
"If we employ the eigenvalue closest to 1 for $\\bar{l}$ , then $\\bar{l}^{-N_t} = \\max _{|\\lambda _n|<1} |\\lambda _n|$ .",
"Using Eq.",
"(REF ), we obtain l = a = a m/2.",
"The fermion determinant is independent of $\\mu $ for $\\mu <m_\\pi /2$ .",
"This value of $\\bar{l}$ corresponds to the phase transition point for a pion condensation observed in the phase quench simulations.",
"If we employ the average value of the larger half of $\\lambda _n$ for $\\bar{l}$ , we approximately obtain $\\bar{l}\\sim (200)^{1/4}$ .",
"We obtain $\\mu a \\sim \\ln \\bar{l} = \\frac{1}{4} \\ln 200 \\sim 1.32$ which is much larger than $m_\\pi /2$ and beyond the lattice cutoff $\\mu a = 1$ .",
"Figure: Schematic figure for the low temperature limit.The solid line denotes the histogram of ln|λ|/N t \\ln |\\lambda |/N_t,see also Fig.",
".Dotted vertical lines denote the behavior of -μ/T=lnξ-\\mu /T = \\ln \\xi with increasing μ\\mu .The criterion $\\bar{l}$ may be different for the low density limit and high density limit, depending on the eigenvalue distribution at $T=0$ .",
"The situation of the low temperature limit is shown in a schematic figure Fig.",
"REF .",
"If $\\mu $ is smaller than $m_\\pi /2$ , the fugacity is located within the gap.",
"Taking $N_t\\rightarrow \\infty $ leads to $|1/\\lambda ^*| \\ll \\xi \\ll |\\lambda |$ , (for $|\\lambda |>1$ ), which corresponds to (II-a).",
"The fermion determinant is independent of $\\mu $ in this case.",
"Increasing $\\mu $ , the fugacity becomes comparable to the smaller half of the eigenvalues, which causes the $\\mu $ dependence.",
"If $\\mu $ goes beyond a certain value, which is probably given by the minimum eigenvalue, then the low temperature limit (II-b) is obtained.",
"The fermion determinant has a trivial $\\mu $ -dependence in this case.",
"According to the above discussion, the criterion would be given by > ||<1 ||, for (II-a), < ||, for (II-b).",
"The criterion for (II-a) is related to the pion mass.",
"The criterion for (II-b) may also have a similar interpretation.",
"As we have discussed in § REF , the eigenvalues of the reduced matrix is related to the free energy of quarks.",
"According to the discussion there, the minimum eigenvalue is related to the highest energy state of a quark.",
"Hence, the low temperature limit (II-b) is obtained if $\\mu $ is sufficiently larger than the highest energy state of a quark.",
"A question arises if the high density limit reflects a real physics of QCD or just a consequence of lattice artifacts, because the highest energy state probably depends on the lattice spacing $a$ and is a consequence of UV-cutoff.",
"The $a$ -dependence of the minimum eigenvalue is understood from the left panel of Fig.",
"REF .",
"The increase of $T$ means the decrease of $a$ , which follows from $T=(a N_t)^{-1}$ .",
"It turns out that the maximum eigenvalue $\\lambda _{\\rm max}$ becomes larger as $a$ decreases.",
"This means that the minimum eigenvalue $\\lambda _{\\rm min}$ becomes smaller with decreasing $a$ because of the relation $\\lambda _{\\rm min}= 1/\\lambda _{\\rm max}^*$ .",
"The minimum eigenvalue is described as $\\lambda _{\\rm min} \\sim e^{-F/T}$ assuming the eigenvalues correspond to free energies of a quark.",
"With decreasing $a$ , $\\lambda _{\\rm min}$ becomes smaller, which means $F$ becomes larger.",
"Thus, the highest energy state depends on the lattice spacing $a$ .",
"Investigations of the eigenvalue distribution for finer lattices would clarify if the limit (II-b) corresponds to the low temperature and high density limit of QCD.",
"It is also important to consider the dependence of $\\lambda _{\\rm max}$ and $\\lambda _{\\rm min}$ on the quark mass and lattice volume as well as the lattice spacing.",
"In the present work, the value of $\\lambda _{\\rm max}$ depends on the quark mass (Fig.",
"REF ), while it is insensitive to the lattice volume $N_s^3$ (Fig.",
"REF )."
],
[
"QCD at low temperature and high density",
"In this subsection, we study further the low temperature and high density limit (II-b).",
"Even though the limit (II-b) may be the consequence of the lattice cutoff, it is meaningful to consider this case.",
"For instance, it can be used to generate gauge configurations at high density regions.",
"So far, there is no case where direct MC simulations are feasible and the quark number density is high.",
"If gauge configurations are generated at high density regions by direct MC simulations, they may provide valuable information, e.g., for multi-ensemble reweighting.",
"Using Eq.",
"(REF ), we obtain the low-temperature and high-density limit of the QCD partition function T 0 ZGC(, T) = e2 Nf Nc Ns3 /T DU (()|T0)Nf e-SG, (, T)|T0 = i=1Nt (Bac,(x, y, ti) r+ -2 r- (x-y)), where the definitions of $B$ , $r_\\pm $ etc were given in § REF .",
"In Eq.",
"(REF ), the gluon part remains unchanged and is given by the ordinary Yang-Mills action, while the fermionic part is different from the ordinary QCD action.",
"Quarks interact only in spatial directions, where no interaction exists in the temporal direction.",
"Equation (REF ) is also different from the naive spatial fermion matrix $B$ , but contains a constant term $2\\kappa $ with the projection operator $r_\\pm = (r \\pm \\gamma _4)/2$ .",
"The quark chemical potential appears only in the bulk factor $\\exp (2 N_f N_c N_s^3 \\mu /T)$ , which gives the quark number density, n = 2 Nf Nc , (lattice unit) 2 Nf Nc 3, (physical unit).",
"Since Eq.",
"(REF ) contains no temporal hopping term, the partition function is $Z_{3}$ symmetric even in the presence of dynamical quarks.",
"Naively, it is expected that a deconfinement transition occurs if baryon number density is large enough to cause the overlap of baryon's wave functions, where effective degrees of freedom would be quarks rather than hadrons.",
"If this is the case, $Z_{3}$ symmetry would be an exact symmetry of QCD in extremely high-dense matter.",
"Another high density limit was proposed in Ref.",
"[70] with an approximation that the quark mass and chemical potential are simultaneously made large.",
"In the approximation, the quark mass depends on the chemical potential.",
"In the present case, the fine tune of the quark mass is not needed in taking the low-$T$ and high-$\\mu $ limit.",
"Equation (REF ) is realized at extremely large $\\mu $ .",
"It would be very difficult to access such a high density region in experiments.",
"Nevertheless there are several interests in the high density limit (II-b).",
"Theoretically, it is interesting to consider the nature of QCD at high density regions : confinement, chiral symmetry and color superconductivity.",
"In lattice QCD simulations, the knowledge on important configurations at high density regions would be valuable information.",
"For instance, such configurations can be used in multi-ensemble reweighting method, or they may be used for a reweighting from high density regions in order to find a QCD phase transition line at low temperatures.",
"It is important to consider the numerical feasibility.",
"As we have mentioned, whole the determinant is real, and therefore sign free.",
"Although it is free from the sign problem, it needs large $N_T$ to take the low temperature limit, which requires a large numerical cost.",
"This increase of the computational time may be suppressed by using the property of the fermion determinant in the low temperature limit.",
"In the low-$T$ and high-$\\mu $ limit, the fermion determinant is expressed as the product of the $N_t$ block determinants.",
"If each block determinant is real, it may be possible to evaluate the block determinant by the Gaussian integral with the pseudo-fermion field with smaller dimension.",
"Instead, the Gaussian integral appears $N_t$ times.",
"The real positivity of the block determinant is necessary to implement this idea.",
"We leave the proof of this expectation in future works."
],
[
"The partition function on the complex plane",
"The determination of the confinement/deconfinement phase boundary is an important issue in the study of the QCD phase diagram.",
"A canonical formalism and Lee-Yang zero analysis are useful approaches to identify a phase transition point, and have been investigated in Refs.",
"[37], [71], [41], [16], [42], [72], [73], [74], [75], [76], [77], [45], [47], [78].",
"However, some difficulties were pointed out [16], [35].",
"For instance, the sign problem causes a fictitious signal in Lee-Yang zero analysis, which makes it difficult to distinguish a physical phase transition and fictitious signal [35].",
"In this section, we consider the canonical formalism and Lee-Yang zero theorem, where the reduction formula is useful.",
"The purpose of this section is to propose a method to identify a $\\mu $ -induced phase transition by using a temperature dependence of canonical partition functions and of Lee-Yang zero trajectories.",
"The method provides a qualitative way to distinguish a crossover and first order phase transition, and would be useful in case that the sign problem causes a difficulty in ordinary methods such as finite size scaling of the Lee-Yang zero near a positive real axis.",
"Canonical partition functions $Z_n$ with the quark number $n$ are obtained from the fugacity expansion form of the fermion determinant.",
"Then, we consider the temperature dependence of two quantities: the canonical distribution $Z_n$ and trajectory of Lee-Yang zeros.",
"They show characteristic changes as the temperature decreases from high $T$ to low $T$ .",
"We discuss the relation between their temperature dependences and the existence of a $\\mu $ -induced phase transition.",
"In § REF , we give a brief overview of the canonical formalism and Lee-Yang zero analysis.",
"Fugacity coefficients of the fermion determinant are calculated in § REF .",
"Canonical partition functions are calculated in § REF , where we employ the Glasgow method [23].",
"Lee-Yang zeros are calculated in § REF ."
],
[
"Brief overview",
"According to statistical mechanics, a grand canonical partition function is described as a superposition of canonical partition functions, Z(T,) = n=1N Zn(T) n, where $Z_n$ describes a canonical partition function with a fixed particle number $n$ , and $N$ is the maximum number of the particle.",
"In thermodynamical limit, $N\\rightarrow \\infty $ .",
"Several methods are available for the determination of phase transition points.",
"For instance, a phase transition point is identified by the convergence radius of Eq.",
"(REF ), or the finite size scaling analysis of a Lee-Yang zero $Z(\\mu )=0$ , the Maxwell construction for an S-shape in a $\\mu $ -$n$ diagram, etc.",
"The grand partition function Eq.",
"(REF ) can be considered as a polynomial of $\\xi $ at a given temperature.",
"Phase transitions result from discontinuities of derivatives of the free energy.",
"In general, $Z(\\xi )$ contains the $N$ -roots on the complex $\\xi $ plane.",
"Since the canonical partition function $Z_n$ is real positive for all $n$ , no root exists for real positive values of $\\xi $ .",
"Hence, all the roots are distributed somewhere on the complex $\\xi $ plane except for the positive real axis.",
"Lee and Yang showed [79], [80] that in the case a phase transition occurs, zeros of the grand partition function approach to the positive real axis in the thermodynamical limit $V\\rightarrow \\infty $ .",
"Thus, the phase transition is described by zeros of the grand partition function, which are called the Lee-Yang zeros.",
"The order of the phase transition is distinguished by considering the trajectory formed by zeros near the positive real axis [81].",
"In Ref.",
"[81] the application to non-equilibrium systems was also discussed.",
"Stephanov investigated the properties of Lee-Yang zero of QCD by using the scaling and universality [82].",
"In lattice QCD simulations, the thermodynamical singularities of Lee-Yang zeros were investigated in e.g.",
"Refs.",
"[24], [39], [83], [35], [84]."
],
[
"Fugacity expansion of the fermion determinant",
"We start from the fugacity expansion of $\\det \\Delta (\\mu )$ , ()= C0 n=-Nred/2Nred/2 cn n, which is defined in Eq.",
"(REF ).",
"Before seeing numerical results, we discuss the numerical procedure for the calculation of $c_n$ .",
"There are two difficulties in the determination of $c_n$ ; numerical precision and overflow/underflow due to the applicable range of the double precision.",
"First, the expansion of Eq.",
"(REF ) requires an enormous amount of calculations because of the larger number $N_{\\rm red}$ , which easily causes a loss of precision.",
"A recursive algorithm with a recurrence relation was used to expand Eq.",
"(REF ).",
"A Vandermonde matrix approach was not effective because of the large dimension and of the existence of close eigenvalues.",
"Divide and conquer algorithms are best in respect to the number of the calculation steps.",
"Second, $c_n$ span a wide range of magnitude, from order $O(1)$ to $O(e^{4 N_c N_s^3})$ .",
"A huge number such as $O(e^{4 N_c N_s^3})$ exceeds the maximum value of double precision, where an ordinary double precision variable is not applicable.",
"Numerical libraries such as FMLib [85] are available for this calculation.",
"Other libraries have been also used in literature, see e.g., Ref. [45].",
"However, the use of libraries may become an obstructive factor of fast computation.",
"Therefore, we developed a new variable [46].",
"These procedures to calculate $c_n$ are explained in Appendix. .",
"Figure: The magnitude of all the fugacity coefficients for varioustemperature.",
"The lattice size is 8 3 ×48^3\\times 4.",
"Data is taken from oneconfiguration for each temperature.Figure: The argument of the fugacity coefficients for varioustemperature.",
"Data are shown for small quark number sector.The lattice size is 8 3 ×48^3\\times 4.",
"Data is taken from oneconfiguration.Now we proceed to the numerical results of $c_n$ .",
"Figures REF and REF show the modulus and argument of the fugacity coefficients $c_n$ for a gauge configuration.",
"The simulation setup was given in the section REF .",
"The magnitude $|c_n|$ spans over the wide range of order from $O(1)$ to $O(e^{10^4})$ .",
"At $\\mu =0$ , $\\det \\Delta (\\mu )$ is dominated by several $c_n$ near $n=0$ .",
"The argument of $c_n$ shows complicated $n$ dependence, as shown in Fig.",
"REF , where $\\arg (c_n)$ is defined for $-\\pi \\le \\arg (c_n) \\le \\pi $ .",
"Qualitatively, $\\arg (c_n)$ tends to oscillate with higher frequency as the temperature becomes lower.",
"Calculating $c_n$ for 400 configurations, we found that $|c_n|$ was stable for the change of configurations.",
"On the other hand, $\\arg (c_n)$ rapidly changes for configuration by configuration, which leads to the cancellation of $c_n$ in the ensemble average.",
"This cancellation becomes more severe at low temperatures.",
"We will see this point in the next subsection.",
"The fugacity coefficients $c_n$ satisfy the relation $c_{-n}^* = c_n$ , as a consequence of the $\\gamma _5$ hermiticity.",
"Hence, positive and negative $n$ describe the winding number of a quark and an anti-quark around the temporal cylinder, respectively.",
"This is realized as the reflection symmetry $|c_{-n}|=|c_{n}|$ with respect to $n=0$ for the absolute value, which is well satisfied see Fig.",
"REF .",
"The relation for the phase, which is given by $\\arg (c_{-n}) = - \\arg (c_n)$ , was numerically satisfied for several hundreds $c_n$ near $n \\sim 0$ and near $n\\sim \\pm N_{\\rm red}/2$ .",
"For instance, in the left panel of Fig.",
"REF , $\\arg (c_{-n}) = - \\arg (c_n)$ for $n=0\\sim 300$ , while $\\arg (c_{-n}) \\ne - \\arg (c_n)$ for $n\\sim 500$ .",
"Fortunately, the quark determinant is dominated by coefficients near $n=0$ for $\\mu =0$ ."
],
[
"Canonical partition function",
"The grand canonical partition function of QCD with $N_f$ flavors can be also expanded in powers of the fugacity, ZGC(, T) = n=-NqNq ZC(n, T) n, where $N_q=N_f N_{\\rm red}/2 = 2N_f N_c N_s^3$ is the maximum quark number which can be put on the $N_s^3$ lattice.",
"$Z_C(n, T)$ is a canonical partition function with a fixed quark number $n$ .",
"Using the Fourier transformation [41], [16], [42], the canonical partition function is obtained ZC(n) = 12/3 -/3/3 de-in ZGC(=iI), where $\\phi =\\mu _I/T$ .",
"We have used the Roberge-Weiss periodicity.",
"In this work, we employ the Glasgow method, which is based on the reweighting in $\\mu $ and reduction formula, for the calculation of the canonical partition functions.",
"It was pointed out that it suffers from the overlap problem [11].",
"In this work, we focus on the properties of the canonical partition functions and Lee-Yang zeros as a first step, and leave the improvement of the overlap for future works.",
"Substituting the reduction formula into Eq.",
"(REF ), the canonical partition function is given by Zn ZC(n) = C02 dn((0))20.",
"Here $d_n$ are the fugacity coefficients of the two-flavor determinant, and $\\langle \\cdot \\rangle _0$ denotes an ensemble average for gauge configurations generated at $\\mu =0$ .",
"$d_n$ is determined by using the recursive algorithm for $\\left(\\xi ^{-N_{\\rm red}/2}\\prod _{n=1}^{N_{\\rm red}}(\\xi +\\lambda _n)\\right)^{N_f} = \\sum _{n=0}^{N_f N_{\\rm red}} d_n \\xi ^n$ .",
"Although it is possible to obtain $d_n$ in terms of $c_n$ , this method was slower than the above procedure.",
"Figure: The canonical partition functions Z n Z_n for three temperatures.Left : real part, right: imaginary part.In the left panel, the solid lines are exp(-a L |n|)\\exp (- a_L |n|) forT/T pc =0.93T/T_{\\rm pc}=0.93 and exp(-a H n 2 )\\exp ( - a_H n^2) for T/T pc =1T/T_{\\rm pc}=1 and 1.351.35.The results are obtained by using the Glasgow method with the use of 400sets of the eigenvalues of the reduced matrix.The canonical partition functions $Z_n$ are shown up to $|n|=30$ for three temperatures in Fig.",
"REF .",
"$Z_n$ must be real positive, i.e., ${\\rm Im}[Z_n]=0$ .",
"The results for high temperatures satisfy this condition.",
"Even at low temperature, ${\\rm Im}[Z_n]$ is zero within errorbars for most $n$ .",
"However, the errorbars are large, which is caused by the fluctuation of the phase of the fugacity coefficients $c_n$ for configuration by configuration, which is the sign problem in canonical approaches [40].",
"The signal-to-noise ratio for ${\\rm Re}[Z_n]$ becomes small for large $n$ .",
"This is probably because of the importance sampling at $\\mu =0$ .",
"The average value of the quark number density is zero at $\\mu =0$ , and therefore configurations generated at $\\mu =0$ have less overlap with large quark number sectors.",
"The canonical partition functions are obtained up to about $|n|=30$ in the present simulation setup.",
"The calculation of $Z_n$ for larger $n$ needs an improvement of overlap or increase of statistics.",
"However, it should be noted that the real part of the canonical partition functions show a characteristic temperature dependence even for $|n|<30$ .",
"The canonical partition function ${\\rm Re}[Z_n]$ exponentially decreases as $|n|$ increases for all the three temperatures.",
"The width of the distribution of ${\\rm Re}[Z_n]$ becomes broad at high temperature, which is consistent with the increase of the effective degrees of freedom at high $T$ .",
"The $n$ -dependence of ${\\rm Re}[Z_n]$ changes at high temperatures ($T/T_{\\rm pc}=1, 1.35$ ) and low temperature $(T/T_{\\rm pc}=0.93)$ .",
"Qualitatively, ${\\rm Re}[Z_n]$ is approximated by a Gaussian function $\\exp (-a_H n^2)$ for $T\\ge T_{\\rm pc}$ , and to a function $\\exp (-a_L |n|)$ for $T< T_{\\rm pc}$ , where $a_{H,L}$ are parameters.",
"To be precise, the result agrees with the Gaussian function for $T/T_{\\rm pc}=1.35$ up to $|n|=30$ , while there is a deviation from the Gaussian function for $|n|=20\\sim 30$ for $T/T_{\\rm pc}=1$ .",
"There is also deviation from the function $\\exp (-a_L |n|)$ for $T/T_{\\rm pc}=0.93$ .",
"In case of $\\exp (-a_L |n|)$ , the partition function becomes a geometric series.",
"Triality nonzero terms ${\\rm mod}(n,3)\\ne 0$ do not vanish, which is a consequence of the importance sampling at $\\mu =0$ .",
"Those terms can be eliminated by using the Roberge-Weiss periodicity [49].",
"We discuss this point in Appendix. .",
"The triality nonzero terms change neither the trajectory of Lee-Yang zeros nor the convergence radius of the fugacity polynomial as long as $Z_n$ with zero- and nonzero-triality is described by the same function of $n$ .",
"Triality nonzero terms change the density of Lee-Yang zeros on its trajectory.",
"However, this effect vanishes if thermodynamical limit is taken.",
"Hence, we keep the triality nonzero terms, which does not cause any problem in the following discussion.",
"Here it is interesting to note that the low-temperature high density limit of QCD may suggest the importance of the other $Z_3$ sectors.",
"If all the $Z_3$ sectors are visited in MC simulations, the triality nonzero terms would disappear.",
"The $n$ -dependence of the canonical partition functions change from high temperatures to low temperatures.",
"Now, we discuss the relation between this change of the $n$ -dependence of ${\\rm Re}[Z_n]$ and a finite density phase transition.",
"We found that ${\\rm Re}[Z_n]$ is approximately given by Re[ZC(n)] { e- aL |n|, (T<Tpc), e- aH n2, (TTpc).",
".",
"Assuming that these distributions hold for larger $n$ and that the imaginary part is sufficiently small, the convergence radius $r$ of Eq.",
"(REF ) is given by r-1 =n | Z(n+1)Z(n) | = { e - aL, (T< Tpc), 0 , (T Tpc).",
".",
"The Gaussian function for $T\\ge T_{\\rm pc}$ suggests no phase transition, while the function $e^{-a|n|}$ for $T<T_{\\rm pc}$ suggests the existence of a phase transition at finite $\\mu $ .",
"Note that the result for $T=T_{\\rm pc}$ shows the Gaussian behavior, which is consistent with the absence of a $\\mu $ -induced phase transition at $T=T_{\\rm pc}$ .",
"Thus, the shape change of the canonical distribution is related to exisitence or absence of a $\\mu $ -induced phase transition.",
"The deviation from Eq.",
"(REF ) observed for $T=T_{\\rm pc}$ and $T=0.93T_{\\rm pc}$ may be important in the study of CEP.",
"The shape of the canonical distribution including the large-$n$ part and the deviation from Eq.",
"(REF ) can contribute to higher-order moments, such as skewness and kurtosis.",
"In the present analysis, we employed the one-parameter reweighting, where the overlap problem becomes severe for large $n$ parts of $Z_n$ .",
"The improvement of the overlap is needed in order to apply the above discussion to cases where the tail part of the canonical distribution is important."
],
[
"Lee-Yang zeros",
"In this subsection, we consider the Lee-Yang zeros.",
"Several methods are available for the calculation of Lee-Yang zeros.",
"For instance, it is possible to search zeros of the left hand side of Eq.",
"(REF ).",
"It is also possible to solve roots of the fugacity polynomial, which is the right hand side of Eq.",
"(REF ).",
"Here, we adopt the latter approach by using the canonical partition functions obtained in the previous subsection.",
"In order to calculate roots of the fugacity polynomial, We consider the truncation of the fugacity polynomial in order to calculate roots of the right hand side of Eq.",
"(REF ), because the order of the fugacity polynomial $N_q$ is large.",
"In general, it is not allowed to truncate the polynomial in the vicinity of phase transition points.",
"However, the truncated polynomial can reproduce the trajectory of Lee-Yang zeros in the case of the geometric series.",
"First, we discuss this point.",
"We divide the fugacity polynomial into small $n$ and large $n$ parts ZGC() = |n|M ZC(n) n + |n|M ZC(n) n where $M$ is an integer.",
"Although ${\\rm Re}[Z_n]$ rapidly decreases as $|n|$ increases, the sum of the higher-order terms significantly contribute to $Z_{GC}(\\mu )$ in the vicinity of the convergence radius, i.e., phase transition points.",
"At high $T$ , the higher-order terms do not affect the convergence property because of the Gaussian shape.",
"At low $T$ , the $e^{-a |n|}$ -shape provides the nonzero convergence radius, where the sum of higher-order terms is significant near transition points.",
"However, if ${\\rm Re}[Z_n]=\\exp (-a |n|)$ even for large $n$ , the trajectory formed by Lee-Yang zeros does not depend on the maximum order of the fugacity polynomial.",
"This is a consequence of the geometric series.",
"For instance, considering $1+x+x^2+\\cdots + x^N=0$ , its roots lie on the unit circle.",
"The order $N$ changes the density of the roots on the circle, but does not change the trajectory formed by roots.",
"In fact, the fugacity polynomial with ${\\rm Re}[Z_n]=\\exp (-a |n|)$ is different from a naive geometric series, because it contains both the quark and anti-quark components.",
"However, as we will show below, the trajectory of the roots for the quark and anti-quark sectors are separated from inside and outside of $|\\xi |=1$ because of the symmetry between quarks and anti-quarks.",
"Figure: The distribution of Lee-Yang zeros, which are roots ofthe fugacity Polynomial shown in Fig..The roots are obtained for the leading order terms for M=24M=24.Solid lines are the unit circle.Accordingly, the Lee-Yang zero trajectory can be reproduced via the truncated polynomial on the assumption on $Z_n$ .",
"Now, we consider the numerical result.",
"We employ the data of $Z_n$ up to $M=24$ , because the signal-to-noise ratio becomes large for $n>M$ at $T/T_{\\rm pc}=0.93$ .",
"Although this value of $M$ is small, it corresponds to the baryon density $\\sim $ 2 [fm$^{-3}$ ].",
"IMSL Library was used for the calculation of the roots of the fugacity polynomial.",
"The result is shown in Fig.",
"REF .",
"Corresponding to the change of $Z_n$ , the Lee-Yang zero trajectory also changes its shape from high temperatures to low temperatures.",
"The zeros are approximately distributed on two circles at $T/T_{\\rm pc}=0.93$ and one circle with two branches at $T/T_{\\rm pc}=1$ and $1.35$ .",
"The trajectory is symmetry with regard to the unit circle because of the charge conjugation symmetry between the quark and anti-quark.",
"The results for the Lee-Yang zero trajectories were obtained by using the data of $Z_n$ shown in Fig.",
"REF , where both the real and imaginary parts were considered.",
"We did not use Eq.",
"(REF ) to obtain Fig.",
"REF .",
"Hence, the trajectories in Fig.",
"REF contain the deviation of $Z_n$ from Eq.",
"(REF ).",
"The two-circle trajectory at low $T$ is similar to a typical behavior of geometric series with positive and negative $n$ components.",
"If the polynomial is an ordinary geometric series with positive $n$ components, then the roots lie on a circle.",
"Adding the negative $n$ components with the charge conjugation symmetry, then the roots locate on two circle.",
"Similarly, the two-branch trajectory is a typical behavior of the Gaussian distribution.",
"Therefore, the trajectories obtained from the data of $Z_n$ qualitatively agree with the trajectories obtained from Eq.",
"(REF ), which suggests that Eq.",
"(REF ) is a good approximation for $Z_n$ at least for small $n$ .",
"Figure: The distribution of Lee-Yang zeros on the complex μ/T\\mu /T plane.In Fig.",
"REF , we have shown the Lee-Yang zero distribution for $T/T_{\\rm pc}=0.84$ and $0.93$ in the complex $\\mu /T$ plane.",
"The results are invariant under $\\mu /T\\leftrightarrow -\\mu /T$ .",
"In the Lee-Yang zero theorem, the phase transition point is obtained from the finite size scaling analysis of the zero nearest the positive real axis.",
"In the present approach, we have made the assumption on $Z_n$ .",
"If the assumption is valid, then Lee-Yang zeros are on the same trajectories.",
"Then, the zeros would approach to the positive real axis as the volume increases in case of a phase transition.",
"The phase transition point can be estimated in the canonical formalism, by using the Maxwell construction for the S-shape of the $\\mu $ -$n$ diagram [16], [47].",
"Figure: A conjectured phase boundary for the first order phase transition.The data are obtained from Fig.",
"() as the intersection ofthe linear fit and positive real axis with the assumption on Z n Z_n.Following the assumption Eq.",
"(REF ), we estimate a phase transition point.",
"Here we should note that the overlap problem causes a loss of reliability.",
"From the linear fit ${\\rm Re}[\\mu /T]=$ const., we estimate the location of the intercept of the trajectory and real positive axis, and obtain c/T={ 0.97(3) T/Tpc=0.84 0.70(2) T/Tpc=0.93 .",
"The result is mapped onto the QCD phase diagram in Fig.",
"REF .",
"Here the errorbars for $\\mu _c/T$ were obtained from $\\chi ^2$ fit.",
"We also estimated the errorbars for $T/T_{\\rm pc}$ , which is taken from [67].",
"The result is almost consistent with previous studies with staggered fermions [16] at $\\mu /T=0.7$ , and undershoots those results at $\\mu /T=0.97$ .",
"Here, it is important to consider the applicable limit of the Glasgow method.",
"This can be done by using the imaginary chemical potential.",
"The pure imaginary chemical potential region is located on the unit circle on the complex fugacity plane.",
"The Roberge-Weiss (RW) endpoint should be located on ${\\rm Im}[\\mu /T]=\\pi /3$ .",
"It turns out that the singularity on the unit circle appears about ${\\rm Im}[\\mu /T]\\sim \\pi /4$ , which is smaller than the value expected from the RW endpoint.",
"This would imply that the overlap problem becomes severe for $|\\mu /T| >\\pi /4$ .",
"The phase transition points in Fig.",
"REF are almost corresponds to this limit.",
"In order to determine the phase transition point, the configurations should be improved to obtain a better overlap.",
"In this section, we have presented an approach for the study of the QCD phase boundary.",
"Although our analysis is at fundamental one, we found that the canonical distribution and Lee-Yang zero trajectory distinguish the crossover behavior at $T_{\\rm pc}$ and first order behavior at low $T$ .",
"It is useful to consider the canonical partition function together with standard techniques for the phase transition, which may provides complementary information to identify the phase transition point.",
"Figure: Volume dependence of the canonical partition functions (left panel)and Lee-Yang zero distribution (right panel).The assumption used in the present analysis should be examined further.",
"In particular, the improvement of the overlap and the finite size effect on $Z_n$ are important task.",
"We showed the results for $N_s=10$ in Fig.",
"REF .",
"The increasing $N_s$ causes the broadening of the canonical partition functions.",
"Although the eigenvalue distribution of the reduced matrix is insensitive to $N_s$ , canonical partition functions are sensitive to $N_s$ .",
"Since $N_q$ for $N_s=10$ is twice as larger as that for $N_s=8$ , we employed $M=48$ for $N_s=10$ .",
"The trajectory of the Lee-Yang zero was not affected largely by the increase of $N_s$ .",
"However, the lattice volume is still small for $N_s=10$ , and the finite size effect may appear for larger lattices.",
"The finite size effect should be investigated for larger lattices.",
"Note that in Fig.",
"REF , the result for $N_s=8$ and 10 were obtained with the same number of statistics.",
"Since the sign problem becomes more severe for lager lattices, it is also important to increase the number of the statistics to study the finite size effect."
],
[
"Summary",
"We have studied QCD at nonzero chemical potential and temperature in the lattice QCD simulations.",
"We particularly focused on the low temperature regions of the QCD phase diagram, and studied several issues with the use of the dimensional reduction formula of the fermion determinant.",
"In § , we studied the reduction formula of the Wilson fermion determinant and showed several properties of the reduced matrix.",
"The reduced matrix is interpreted as the transfer matrices or the generalized Polyakov line, and the eigenvalues of the reduced matrix is related to the free energy of dynamical quarks.",
"The angular distribution of the eigenvalues manifests the $Z_3$ or confinement properties of QCD.",
"The eigenvalues form the gap, which is related to the pion mass and therefore chiral symmetry breaking.",
"We found an indication that the eigenvalues of the reduced matrix is scaled by the temporal size as $\\lambda \\sim l^{N_t}$ for $|\\lambda |>1$ and $\\lambda \\sim l^{-N_t}$ for $|\\lambda |<1$ .",
"The $N_t$ scaling law controls the temperature dependence of the fermion determinant, and therefore is the important finding.",
"In section , we studied the property of the fermion determinant at low $T$ and finite $\\mu $ .",
"We showed from the lattice simulations that at $T/T_{\\rm pc}\\sim 0.5$ and for $\\mu <m_\\pi /2$ , the fermion determinant is insensitive to $\\mu $ and the average phase factor $\\langle \\cos \\theta \\rangle $ approaches to one.",
"The fluctuation of the reweighting factor, the ratio of the determinant at $\\mu =0$ and $\\mu \\ne 0$ , exponentially decreases as the temperature decreases.",
"Using the $N_t$ -scaling law, we also derived the $\\mu $ -independence of the fermion determinant at $T=0$ for $\\mu <m_\\pi /2$ .",
"Hence we concluded that the fermion determinant is $\\mu $ -independence at low $T$ and for $\\mu <m_\\pi /2$ , which is the consequence of the $N_t$ -scaling law of the eigenvalues of the reduced matrix.",
"Extending the low temperature studies further, we have considered the low-temperature limit of the fermion determinant and of QCD, and obtained two expressions of the quark determinant; one is for low density and the other is for high density.",
"Low density expression corresponds to $\\mu $ -independence for $\\mu <m_\\pi /2$ .",
"The other expression is its high-density counter part.",
"We discussed the nature of the high density and low temperature limit of the QCD partition function.",
"In the case, QCD approaches to a theory, where quarks interacts only in the spatial direction with the ordinary Yang-Mills type of the gauge action.",
"The corresponding partition function is $Z_3$ invariant even in the presence of dynamical quarks.",
"Furthermore, the fermion determinant becomes real and the theory is free from the sign problem.",
"In § , we studied the canonical formalism and Lee-Yang zero theorem.",
"We have shown that the canonical distribution and Lee-Yang zero trajectory show characteristic changes when $T$ is varied from high to low temperatures.",
"The canonical distribution is similar to a Gaussian function of the quark number $n$ at high $T$ , and to a geometric series with negative $n$ components.",
"The Lee-Yang zero trajectory is a circle with two-branch at high $T$ and two circle at low $T$ .",
"The eigenvalue distribution was almost independent of the lattice size in our simulations on $8^3$ and $10^3$ , which may suggest that the canonical partition function is insensitive to the spatial volume.",
"Assuming the obtained $n$ -dependence holds for larger $n$ , we have shown that the $T$ -dependence of the canonical distribution and Lee-Yang zero trajectory are related to the phase transition.",
"The canonical distribution and Lee-Yang zero trajectory distinguish the crossover and first order phase transition.",
"Hence the investigation of the canonical partition function and the Lee-Yang zeros may provides a way to distinguish the phase transition and fictitious signals caused from sign problem.",
"It would be interesting to combine the present approach and the ordinary techniques such as the finite size scaling of the Lee-Yang zero, Maxwell construction of the canonical formalism, etc.",
"For confirmation of the results found in the present work, several points should be clarified further.",
"Our analysis was performed on the small and coarse lattices with heavy quark mass.",
"It is important to remove these lattice artifacts on the eigenvalues of the reduced matrix.",
"The gap of the eigenvalues is sensitive to the quark mass.",
"The tail of the eigenvalue distribution is sensitive to the quark mass and lattice spacing.",
"These quantities are related to the pion mass and highest energy level, which determine the critical values of $\\mu $ for the low temperature limits.",
"It is also important to examine the $N_t$ -scaling law for larger temporal lattice size.",
"It is also important to investigate the lattice artifacts on the canonical partition function.",
"In addition, the finite-size scaling and the improvement of the overlap are necessary task to identify the phase transition point.",
"In particular, the large quark number sector of the canonical partition function plays an important role in the phase transition.",
"For the improvement of the overlap, a multi-ensemble reweighting or histogram method may be useful [86], [87].",
"With further confirmations, it will be interesting to study the thermodynamical properties of QCD at low temperatures."
],
[
"Acknowledgements",
"We thank S. Ejiri, Ph.",
"de Forcrand, S. Hashimoto, M. Hanada and H. Matsufuru for valuable comments and stimulating discussions.",
"Parts of this work were done during the stay at the workshop “New Type of Fermions on Lattice” held at YITP.",
"We thank to A. Ohinishi, T. Misumi, D. Adams, C. Hoelbling for stimulating discussions and valuable information.",
"KN specially thanks to T. Hatsuda for the hospitality and encouragement during the stay at Tokyo University.",
"AN acknowledges the hospitality and discussions to M. Yahiro at Kyushu university.",
"This work was supported by Grants-in-Aid for Scientific Research 20340055, 20105003, 23654092 and 20105005.",
"The simulation was performed on NEC SX-8R at RCNP, NEC SX-9 at CMC, Osaka University, and System A and System B at KEK."
],
[
"Pair nature of the eigenvalues",
"We show a proof of a pair nature of the eigenvalues of the reduced matrix.",
"The pair nature is a consequence of the symmetry of the reduced matrix, which was shown in Ref. [45].",
"Here we present a brief proof.",
"Substituting Eq.",
"(REF ) into the $\\gamma _5$ -hermiticity relation leads to (*)-Nred2 n=1Nred(n* + *) = (*)Nred2n=1Nred(n + (*)-1).",
"Since the $\\gamma _5$ hermiticity holds for $\\forall \\mu \\in \\mathbb {C}$ , Eq.",
"(REF ) also holds for $\\forall \\xi \\in \\mathbb {C}$ .",
"Let $\\xi _0$ a value of fugacity which set l.h.s of Eq.",
"(REF ) to be zero, i.e., $\\xi _0 = -\\lambda _n$ .",
"If l.h.s is zero, then r.h.s also must be zero.",
"Hence, an eigenvalue for l.h.s $=0$ must exist for $\\xi =\\xi _0$ .",
"This is satisfied by $\\lambda + (\\xi _0^*)^{-1}=0$ .",
"Thus, two eigenvalues n, 1n* appears at the same time.",
"This procedure is applied to all the eigenvalues $\\lambda _n, (n=1, \\cdots N_{\\rm red})$ .",
"Thus, the pair nature of the eigenvalues of the transfer matrix is proved."
],
[
"Z$_3$ Properties of Fugacity coefficients",
"Next, we consider the property of the fugacity coefficients $c_n$ under center transformation $Z_{3}$ , U4(x,ti) U4(x,ti), ti, x, where $\\omega = \\exp ( 2\\pi i k / 3), \\;\\; (k = \\pm 1)$ is an element of $Z_3$ .",
"In Eq.",
"(REF ), $C_0$ is invariant under $Z_{3}$ , since it does not contain temporal link variables.",
"From Eqs.",
"(REF ) and (REF ) $Q$ transforms as $Q\\rightarrow \\omega Q$ under $Z_3$ .",
"The fermion determinant is transformed as () = C0 -Nred/2 ( Q + ) , = C0 -Nred/2 ( Q + -1), = C0 -Nred/2 n=0Nred cn -n n, = C0 n=-Nred/2Nred/2 cn -n n Then, the fugacity coefficients $c_n$ transforms $c_n \\rightarrow c_n \\omega ^{-n}$ under the $Z_3$ transformations.",
"$c_n$ is invariant for the triality sector $n=3m(m\\in \\mathbb {Z})$ and covariant for the triality nonzero sector $n= 3m + 1, 3m + 2$ ."
],
[
"RW invariance and Triality",
"Consider the Roberge-Weiss transformation [49].",
"Temporal links are transformed under $Z_3$ U4(x, ti) U4(x, ti) = U4 (x, ti), ti, x, and the chemical potential is shifted in the imaginary direction as T T = T - 2i k3, which acts on the fugacity as the rotation $(\\xi \\rightarrow \\xi \\omega )$ .",
"It is obvious from Eq.",
"(REF ) that the transformations for the link variables and chemical potential cancel.",
"The fermion determinant is invariant under the RW transformation, (, {U}) (, {U}) =(, {U}) The grand partition function is also invariant under the RW transformation, Z() Z() = DU(, {U}) e-SG(U), = DU (, {U}) e-SG(U) , = Z().",
"An important property on the canonical partition function is deduced from this invariance [24], [71].",
"For simplicity we set $\\mu $ to be pure imaginary.",
"Expressing the fermion determinant as the fugacity polynomial, the grand partition function is given as ZGC(iI) = DU n=-MM cn n eSg, and ZGC(i IT+ i 2k3) = DU n=-MM cn n n eSg.",
"Here note that the maximum quark number $M$ is proportional to the spatial volume $V$ and diverges at thermodynamical limit, which may spoils this discussion near phase transition points.",
"Assuming the analiticity, Eq.",
"(REF ) leads to ( 1 - ( i 2n k3 ) ) DU cn eSg =0 .",
"Thus, we obtain Z(n)=DU cn eSg =0 (mod(n,3)0).",
"Hence the canonical partition function must vanish for the triality nonzero sector.",
"The above argument is applied to arbitrary number of flavors, which can be obtained by changing $M$ ."
],
[
"Canonical partition function for triality nonzero sectors",
"We have seen the properties of the canonical partition function Eq.",
"(REF ).",
"However, we showed that the canonical partition functions do not vanish for the triality nonzero sector, see Fig.",
"REF .",
"This small paradox is caused by the importance sampling at $\\mu =0$ , where configurations for one $Z_3$ sector are collected in the presence of the quarks.",
"Effects of the non-zero triality sector was investigated in Ref. [42].",
"Here, we consider this point.",
"First, we classify the configuration space into three regions according to the location of the Polyakov loop $(L)$ on the complex $L$ plane; R1={U | -/3 (L) < /3}, R2={U | /3 (L) < }, R3={U |-(L) < -/3}.",
"The grand partition function is written as Z() = (R1 + R2 + R3) DU () e-Sg As we have already seen, whole the $Z(\\mu )$ is invariant under the Roberge-Weiss transformation.",
"Let us consider the transformation property of each component.",
"Let $I_j$ to be one of $Z_3$ component of the partition function, Ij = Rj DU () e-Sg.",
"Under the Roberge-Weiss transformation, the Polyakov loop $L$ is rotated in the complex $L$ plane, then the configurations $\\lbrace U\\rbrace $ move from one of $R_i$ sector to another $R_{i+1}$ .",
"Therefore, $I_j$ transforms as Ij() = Ij+1(), (I4 = I1).",
"Therefore, each $I_j$ is not invariant under the Roberge-Weiss transformation.",
"In the importance sampling at $\\mu =0$ in the presence of the quarks, configurations for one $Z_3$ sector are extensively collected, which cover a part of the configuration space, e.g.",
"$R_1$ .",
"This causes the non vanishing of the triality nonzero sectors in the canonical partition function.",
"In order to satisfy the triality is to include other $Z_3$ sectors by using the RW transformation, Z() = I1 () + I2() + I3() = I1 () + I1(+2i T /3) + I1(+ 4i T /3) = R1 DU n cn n (1 + -n + -2n) e-Sg = R1 DU n,mod(n,3)=0 3 cn n e-Sg, which contains only ${\\rm mod}(n,3)=0$ terms."
],
[
"Calculation of coefficients $c_n$",
"The fugacity coefficients $c_n$ of the fermion determinant can be obtained in the reduction formula, $\\det \\Delta (\\mu ) = C_0 \\xi ^{-N_{red}/2} \\prod (\\lambda _n + \\xi )= C_0 \\sum _{n=-N_{red}/2}^{N_{red}/2} c_n \\xi ^n .$ Their values vary from order one to order $10^{900}$ even on the small $4^4$ lattice.",
"They cannot be handled in the double precision.",
"The simplest way is to use an arbitrary accuracy libraries.",
"This is more than necessary.",
"To express $c_k$ , we need wide range of floating numbers, but we do not need very high precision.",
"In other words, we need wide range of the exponent, but we do not need very huge significant numbers.",
"We express each real and imaginary parts of $c_k$ in a form of $a \\times b^L ,$ where $1 \\le |a| < b ,$ and $a$ is a double precision real and $L$ is an integer.",
"We employ the “module” in Fortran 90, which allows us to define a new type of data and mathematical operation among them.",
"The base $b$ can be any number, and we set it to be 8.",
"There are several way to get $c_n$ in Eq.",
"(REF ).",
"The simplest way is to use in a recursive way: $\\sum _{k=0}^M C_k^{\\prime }\\xi ^k=(B_0 + B_1 \\xi ) \\sum _{k=0}^{M-1} C_k\\xi ^k$ and $C_0^{\\prime } &=& B_0 C_0\\nonumber \\\\C_k^{\\prime } &=& B_{k-1} C_k + B_k C_{k-1}\\quad (k = 1, 2, \\cdots , M-1)\\nonumber \\\\C_M^{\\prime } &=& B_1 C_{M-1}$ We calculate several cases by this method, and by a high accuracy library, FMLIB[85].",
"We got the same results.",
"A smarter way is a divide-and-conquer method.",
"See ().",
"Here for simplicity we assume $N=2^M$ , but this can be loosened $c \\leftarrow c1 \\times c2$ is an operation to determine $c$ from $c_1$ and $c_2$ where $&&(c(0) + c(1)*x + ... + c(N_3)*x^N_3)\\leftarrow \\nonumber \\\\&&(c_1(0) + c_1(1)*x + ... + c_1(N_1)*x^{N_1})\\\\&& \\times (c_2(0) + c_2(1)*x + ... + c_2(N_2)*x^{N_2})$ and $N_3=N_1+N_2$ .",
"[h] Divide-Conquer calculation for the coefficients $c_n$ for $\\prod _{i=1}^{N} (a(i)+x) = \\sum _{k=0}^N c(k)*x^k$ Input $a$ , Output $c$ ) Set N read a CALL DandQ ( a, c, N) Recursive SUBROUTINE DandQ ( a, c, N) IF N corresponds the Bottom, RETURN CALL DandQ (a(1), c1, N/2) CALL DandQ (a(N/2+1), c2, N/2) $c \\leftarrow c1 \\times c2$ RETURN END SUBROUTINE When $N$ is large (i.e., we are near to the “top” of the recursive level) , Eq.",
"(REF ) is easy to vectorize or parallelize.",
"For small $N$ (i.e., we are near to the “bottom” of the recursive level) , we handle many calculations of the type of Eq.",
"(REF ) , then it is also easy to vectorize or parallelize."
]
] | 1204.1412 |
[
[
"Scattering theory for Schr\\\"odinger operators on steplike, almost\n periodic infinite-gap backgrounds"
],
[
"Abstract We develop direct scattering theory for one-dimensional Schr\\\"odinger operators with steplike potentials, which are asymptotically close to different Bohr almost periodic infinite-gap potentials on different half-axes."
],
[
"Introduction",
"One of the main tools for solving various Cauchy problems, since the seminal work of Gardner, Green, Kruskal, and Miura [10] in 1967, is the inverse scattering transform and therefore, since then, a large number of articles has been devoted to direct and inverse scattering theory.",
"Given two (in general different) one-dimensional background Schrödinger operators $L_\\pm $ with real finite-gap potentials $p_\\pm (x)$ , i.e.",
"$L_\\pm =-\\frac{d^2}{dx^2}+p_\\pm (x), \\quad x\\in \\mathbb {R},$ one can consider the perturbed one-dimensional Schrödinger operator $L=-\\frac{d^2}{dx^2}+p(x),\\quad x\\in \\mathbb {R},$ where $p(x)$ satisfies a second moment condition, i.e.",
"$\\pm \\int _0^{\\pm \\infty } (1+x^2)\\vert p(x)-p_\\pm (x)\\vert dx<\\infty .$ Then one of the main tools when considering the scattering problem for the Schrödinger operator $L$ , are the transformation operators which map the background Weyl solutions of the operators $L_\\pm $ to the Jost solutions of $L$ .",
"In particular, if the background operators are well-understood, the transformation operators enable us to perform the direct scattering step, which means to characterize the scattering data and to derive the Gel'fand-Levitan-Marchenko equation.",
"The starting point for the inverse scattering step is the Gel'fand-Levitan-Marchenko equation together with the scattering data, from which one deduces the kernels of the transformation operators and recovers the potential $p(x)$ .",
"In much detail the scattering problem has been studied in the case where $p(x)$ is asymptotically close to $p_\\pm (x)=0$ .",
"For a complete investigation and discussions on the history of this problem we refer to the monographs of Levitan [17] and Marchenko [21].",
"Taking this as a starting point, two natural extension have been considered.",
"On the one hand the case of steplike constant asymptotics $p_\\pm (x)=c_\\pm $ , where $c_+\\ne c_-$ denote some constants, has been investigated by Buslaev and Fomin [2], Cohen and Kappeler [4], and Davies and Simon [5].",
"On the other hand Firsova [9] studied the case of equal periodic, finite-gap potentials $p_+(x)=p_-(x)$ .",
"Rather recently, the combination of these two cases, namely the case that the initial condition is asymptotically close to steplike, quasi-periodic, finite-gap potentials $p_-(x)\\ne p_+(x)$ , has been investigated by Boutet de Monvel, Egorova, and Teschl [1].",
"Trace formulas in the case of one periodic background were given by Mikikits-Leitner and Teschl [22] and a Paley–Wiener theorem in Egorova and Teschl [6].",
"Of course the inverse scattering theory is also the main ingredient for solving the Cauchy problem of the Korteweg–de Vries (KdV) equation via the inverse scattering transform [21].",
"Moreover, scattering theory is also the basic ingredient for setting up the associated Riemann–Hilbert problem from which the long-time asymptotics can be derived via the nonlinear steepest descent analysis (see [14] for an overview).",
"In the case of finite-gap backgrounds the Cauchy problem was solved by Grunert, Egorova, and Teschl [13], [8].",
"Note that the analogous Cauchy problem for the modified KdV equations can be obtained via the Miura transform [7].",
"The long-time asymptotics in case of one finite-gap background were recently derived by Mikikits-Leitner and Teschl [23].",
"Of much interest is also the case of asymptotically periodic solutions, which has been first considered by Firsova [9].",
"In the present work we propose a complete investigation of the direct scattering theory for Bohr almost periodic infinite-gap backgrounds, which belong to the so–called Levitan class.",
"It should be noticed, that this class, as a special case, includes the set of smooth, periodic infinite–gap operators.",
"To set the stage, we need: Hypothesis H. 1.1 Let $\\nonumber 0\\le E_0^\\pm < E_1^\\pm < \\dots <E_n^\\pm < \\dots $ be two increasing sequences of points on the real axis which satisfy the following conditions: for a certain $l^\\pm >1$ , $\\sum _{n=1}^\\infty (E_{2n-1}^\\pm )^{l^\\pm } (E_{2n}^\\pm -E_{2n-1}^\\pm ) <\\infty $ and $E_{2n+1}^\\pm -E_{2n-1}^\\pm > C^\\pm n^{\\alpha ^\\pm }$ , where $C^\\pm $ and $\\alpha ^\\pm $ are some fixed, positive constants.",
"We will call, in what follows, the intervals $(E_{2j-1}^\\pm ,E_{2j}^\\pm )$ for $j=1,2,\\dots $ gaps.",
"In each closed gap $[E_{2j-1}^\\pm ,E_{2j}^\\pm ]$ , $j=1,2,\\dots $ , we choose a point $\\mu _j^\\pm $ and an arbitrary sign $\\sigma _j^\\pm \\in \\lbrace -1,1\\rbrace $ .",
"Next consider the system of differential equations for the functions $\\mu _j^\\pm (x)$ , $\\sigma _j^\\pm (x)$ , $j=1,2,...$ , which is an infinite analogue of the well-known Dubrovin equations, given by $\\frac{d \\mu _j^\\pm (x)}{d x}=& -2\\sigma _j^\\pm (x)\\sqrt{-(\\mu _j^\\pm (x)-E_0^\\pm )}\\sqrt{\\mu _j^\\pm (x)-E_{2j-1}^\\pm }\\sqrt{\\mu _j^\\pm (x)-E_{2j}^\\pm } \\\\ \\nonumber &\\times \\prod _{k=1, k\\ne j}^\\infty \\frac{\\sqrt{\\mu _j^\\pm (x)-E_{2k-1}^\\pm }\\sqrt{\\mu _j^\\pm (x)-E_{2k}^\\pm }}{\\mu _j^\\pm (x)-\\mu _k^\\pm (x)}$ with initial conditions $\\mu _j^\\pm (0)=\\mu _j^\\pm $ and $\\sigma _j^\\pm (0)=\\sigma _j^\\pm $ , $j=1,2,\\dots $ We will use the standard branch cut of the square root in the domain $\\mathbb {C}\\setminus \\mathbb {R}_+$ with $\\mathop {\\rm Im}\\sqrt{z}>0$ .. Levitan [17], [18], and [19], proved, that this system of differential equations is uniquely solvable, that the solutions $\\mu _j^\\pm (x)$ , $j=1,2,\\dots $ are continuously differentiable and satisfy $\\mu _j^\\pm (x)\\in [E_{2j-1}^\\pm , E_{2j}^\\pm ]$ for all $x\\in \\mathbb {R}$ .",
"Moreover, these functions $\\mu _j^\\pm (x)$ , $j=1,2,\\dots $ are Bohr almost periodic For informations about almost periodic functions we refer to [20]..",
"Using the trace formula (see for example [17]) $p_\\pm (x)=E_0^\\pm +\\sum _{j=1}^\\infty (E_{2j-1}^\\pm +E_{2j}^\\pm -2\\mu _j^\\pm (x)),$ we see that also $p_\\pm (x)$ are real Bohr almost periodic.",
"The operators $L_\\pm := -\\frac{d^2}{dx^2}+p_\\pm (x),\\quad \\text{dom}(L_\\pm )=H^2(\\mathbb {R}),$ in $L^2(\\mathbb {R})$ , are then called almost periodic infinite-gap Schrödinger operators of the Levitan class.",
"The spectra of $L_\\pm $ are purely absolutely continuous and of the form $\\nonumber \\sigma _\\pm =[E_0^\\pm ,E_1^\\pm ]\\cup \\dots \\cup [E_{2j}^\\pm ,E_{2j+1}^\\pm ] \\cup \\dots ,$ and have spectral properties analogous to the quasi-periodic finite-gap Schrödinger operator.",
"In particular, they are completely defined by the series $\\sum _{j=1}^\\infty (\\mu _j^\\pm ,\\sigma _j^\\pm )$ , which we call the Dirichlet divisor.",
"These divisors are associated to Riemann surfaces of infinite genus, which are connected with the functions $Y_\\pm ^{1/2}(z)$ , where $Y_\\pm (z)=-(z-E_0^\\pm )\\prod _{j=1}^\\infty \\frac{(z-E_{2j-1}^\\pm )}{E_{2j-1}^\\pm }\\frac{(z-E_{2j}^\\pm )}{E_{2j-1}^\\pm },$ and where the branch cuts are taken along the spectrum.",
"It is known, that the Schrödinger equations $\\Big (-\\frac{d^2}{dx^2}+p_\\pm (x)\\Big )y(x)=z y(x)$ with any continuous, bounded potential $p_\\pm (x)$ have two Weyl solutions $\\psi _\\pm (z,x)$ and $\\breve{\\psi }_\\pm (z,x)$ , which satisfy $\\nonumber \\psi _\\pm (z,.",
")\\in L^2(\\mathbb {R}_\\pm ), \\quad \\text{ resp. }",
"\\quad \\breve{\\psi }_\\pm (z,.",
")\\in L^2(\\mathbb {R}_\\mp ),$ for $z\\in \\sigma _\\pm $ and which are normalized by $\\psi _\\pm (z,0)=\\breve{\\psi }_\\pm (z,0)=1$ .",
"In our case of Bohr almost periodic potentials of the Levitan class, these solutions have complementary properties similar to the properties of the Baker-Akhiezer functions in the finite-gap case.",
"We will briefly discuss them in the next section.",
"The object of interest, for us, is the one-dimensional Schrödinger operator $L$ in $L^2(\\mathbb {R})$ $L:=-\\frac{d^2}{dx^2}+q(x), \\quad \\text{dom}(L)=H^2(\\mathbb {R}),$ with the real potential $q(x)\\in C(\\mathbb {R})$ satisfying the following condition $\\pm \\int _0^{\\pm \\infty } (1+\\vert x\\vert ^2 ) \\vert q(x)-p_\\pm (x)\\vert dx<\\infty ,$ for which we will characterize the corresponding scattering data and derive the Gel'fand-Levitan-Marchenko equation with the help of the transformation operator, which has been investigated in [12]."
],
[
"The Weyl solutions of the background operators",
"In this section we want to summarize some facts for the background Schrödinger operators $L_\\pm $ of Levitan class.",
"We present these results, obtained in [12], [17], [25], and [26], in a form, similar to the finite-gap case used in [1] and [11].",
"Let $L_\\pm $ be the quasi-periodic one-dimensional Schrödinger operators associated with the potentials $p_\\pm (x)$ .",
"Let $s_\\pm (z,x)$ , $c_\\pm (z,x)$ be sine- and cosine-type solutions of the corresponding equation $\\left(-\\frac{d^2}{dx^2}+p_\\pm (x)\\right)y(x)=zy(x), \\quad z\\in $ associated with the initial conditions $\\nonumber s_\\pm (z,0)=c^\\prime _\\pm (z,0)=0, \\quad c_\\pm (z,0)=s^\\prime _\\pm (z,0)=1,$ where prime denotes the derivative with respect to $x$ .",
"Then $c_\\pm (z,x)$ , $c^\\prime _\\pm (z,x)$ , $s_\\pm (z,x)$ , and $s^\\prime _\\pm (z,x)$ are entire with respect to $z$ .",
"Moreover, they can be represented in the following form $c_\\pm (z,x)&=\\cos (\\sqrt{z}x)+\\int _0^x\\frac{\\sin (\\sqrt{z}(x-y))}{\\sqrt{z}}p_\\pm (y)c_\\pm (z,y)dy,\\\\s_\\pm (z,x)&=\\frac{\\sin (\\sqrt{z}x)}{\\sqrt{z}}+\\int _0^x \\frac{\\sin (\\sqrt{z}(x-y))}{\\sqrt{z}}p_\\pm (y)s_\\pm (z,y)dy$ The background Weyl solutions are given by $\\psi _\\pm (z,x) &= c_\\pm (z,x) + m_\\pm (z,0) s_\\pm (z,x), \\\\\\text{resp.",
"}\\breve{\\psi }_\\pm (z,x) & = c_\\pm (z,x) +\\breve{m}_\\pm (z,0) s_\\pm (z,x),$ where $m_\\pm (z,x)=\\frac{H_\\pm (z,x)\\pm Y_\\pm ^{1/2}(z)}{G_\\pm (z,x)}, \\quad \\breve{m}_\\pm (z,x)=\\frac{H_\\pm (z,x)\\mp Y_\\pm ^{1/2}(z)}{G_\\pm (z,x)},$ are the Weyl functions of $L_\\pm $ (cf [17]), where $Y_\\pm (z)$ are defined by (REF ), $G_\\pm (z,x)=\\prod _{j=1}^\\infty \\frac{z-\\mu _j^\\pm (x)}{E_{2j-1}^\\pm }, \\quad \\text{ and } \\quad H_\\pm (z,x)=\\frac{1}{2}\\frac{d}{d x}G_\\pm (z,x).$ Using (REF ) and (REF ), we have $H_\\pm (z,x)=\\frac{1}{2}\\frac{d}{d x}G_\\pm (z,x)= G_\\pm (z,x) \\sum _{j=1}^\\infty \\frac{\\sigma _j^\\pm (x) Y_\\pm ^{1/2}(\\mu _j^\\pm (x))}{\\frac{d}{dz} G_\\pm (\\mu _j^\\pm (x),x)(z-\\mu _j^\\pm (x))}.$ The Weyl functions $m_\\pm (z,x)$ and $\\breve{m}_\\pm (z,x)$ are Bohr almost periodic.",
"Lemma 2.1 The background Weyl solutions, for $z\\in , can be represented in the following form{\\begin{@align}{1}{-1}\\psi _{\\pm }(z,x) =\\exp \\left(\\int _0^x m_{\\pm }(z,y)dy\\right)= \\left( \\frac{G_\\pm (z,x)}{G_\\pm (z,0)}\\right)^{1/2}\\exp \\left( \\pm \\int _0^x \\frac{Y_\\pm ^{1/2}(z)}{G_\\pm (z,y)}dy \\right),\\end{@align}}and{\\begin{@align*}{1}{-1}\\breve{\\psi }_\\pm (z,x)=\\exp \\left(\\int _0^x \\breve{m}_\\pm (z,y)dy\\right)=\\left(\\frac{G_\\pm (z,x)}{G_\\pm (z,0)}\\right)^{1/2}\\exp \\left(\\mp \\int _0^x \\frac{Y_\\pm ^{1/2}(z)}{G_\\pm (z,y)}dy\\right).\\end{@align*}}If for some $ >0$, $ z-j(x)> $ for all $ jN$ and $ xR$, then the following holds:For any $ 1>>0$ there exists an $ R>0$ such that\\begin{equation*}\\vert \\psi _{\\pm }(z,x)\\vert \\le \\mathrm {e}^{\\mp (1-\\delta )x\\mathop {\\rm Im}(\\sqrt{z})}\\Big (1+\\frac{D_R}{\\vert z\\vert }\\Big ), \\text{ for any } \\vert z\\vert \\ge R,\\quad \\pm x>0,\\end{equation*}and\\begin{equation*}\\vert \\breve{\\psi }_\\pm (z,x)\\vert \\le \\mathrm {e}^{\\pm (1-\\delta )x\\mathop {\\rm Im}(\\sqrt{z})}\\Big (1+\\frac{D_R}{\\vert z\\vert }\\Big ), \\text{ for any } \\vert z\\vert \\ge R,\\quad \\pm x<0,\\end{equation*}where $ DR$ denotes some constant dependent on $ R$.$ As the spectra $\\sigma _\\pm $ consist of infinitely many bands, let us cut the complex plane along the spectrum $\\sigma _\\pm $ and denote the upper and lower sides of the cuts by $\\sigma _\\pm ^{\\mathrm {u}}$ and $\\sigma _\\pm ^{\\mathrm {l}}$ .",
"The corresponding points on these cuts will be denoted by $\\lambda ^{\\mathrm {u}}$ and $\\lambda ^{\\mathrm {l}}$ , respectively.",
"In particular, this means $f(\\lambda ^{\\mathrm {u}}) := \\lim _{\\varepsilon \\downarrow 0} f(\\lambda +I\\varepsilon ),\\qquad f(\\lambda ^{\\mathrm {l}}) := \\lim _{\\varepsilon \\downarrow 0}f(\\lambda -I\\varepsilon ), \\qquad \\lambda \\in \\sigma _\\pm .$ Defining $g_\\pm (\\lambda )= -\\frac{G_\\pm (\\lambda ,0)}{2Y_\\pm ^{1/2}(\\lambda )},$ where the branch of the square root is chosen in such a way that $\\frac{1}{I} g_\\pm (\\lambda ^{\\mathrm {u}}) = \\mathop {\\rm Im}(g_\\pm (\\lambda ^{\\mathrm {u}})) >0 \\quad \\mbox{for}\\quad \\lambda \\in \\sigma _\\pm ,$ it follows from Lemma REF that $W(\\breve{\\psi }_\\pm (z), \\psi _\\pm (z))=m_\\pm (z)-\\breve{m}_\\pm (z)=\\mp g_\\pm (z)^{-1},$ where $W(f,g)(x)=f(x)g^\\prime (x)-f^\\prime (x)g(x)$ denotes the usual Wronskian determinant.",
"For every Dirichlet eigenvalue $\\mu _j^\\pm =\\mu _j^\\pm (0)$ , the Weyl functions $m_\\pm (z)$ and $\\breve{m}_\\pm (z)$ might have poles.",
"If $\\mu ^\\pm _j$ is in the interior of its gap, precisely one Weyl function $m_\\pm $ or $\\breve{m}_\\pm $ will have a simple pole.",
"Otherwise, if $\\mu ^\\pm _j$ sits at an edge, both will have a square root singularity.",
"Hence we divide the set of poles accordingly: $M_\\pm &=\\lbrace \\mu ^\\pm _j\\mid \\mu ^\\pm _j \\in (E_{2j-1}^\\pm ,E_{2j}^\\pm ) \\text{ and } m_\\pm \\text{ has a simple pole}\\rbrace ,\\\\\\breve{M}_\\pm &=\\lbrace \\mu ^\\pm _j\\mid \\mu ^\\pm _j \\in (E_{2j-1}^\\pm ,E_{2j}^\\pm ) \\text{ and } \\breve{m}_\\pm \\text{ has a simple pole}\\rbrace ,\\\\\\hat{M}_\\pm &=\\lbrace \\mu ^\\pm _j\\mid \\mu ^\\pm _j \\in \\lbrace E_{2j-1}^\\pm ,E_{2j}^\\pm \\rbrace \\rbrace ,$ and we set $M_{r,\\pm }=M_\\pm \\cup \\breve{M}_\\pm \\cup \\hat{M}_\\pm $ .",
"In particular, we obtain the following properties of the Weyl solutions (see, e.g.",
"[3], [12], [17], [27], and [28]): Lemma 2.2 The Weyl solutions have the following properties: The function $\\psi _{\\pm }(z,x)$ (resp.",
"$\\breve{\\psi }_\\pm (z,x)$ ) is holomorphic as a function of $z$ in the domain $\\mathbb {C}\\setminus (\\sigma _\\pm \\cup M_{\\pm })$ (resp.",
"$(\\sigma _\\pm \\cup \\breve{M}_\\pm )$ ), real valued on the set $\\mathbb {R}\\setminus \\sigma _\\pm $ , and have simple poles at the points of the set $M_{\\pm }$ (resp.",
"$\\breve{M}_\\pm $ ).",
"Moreover, they are continuous up to the boundary $\\sigma _\\pm ^u\\cup \\sigma _\\pm ^l$ except at the points from $\\hat{M}_\\pm $ and $\\psi _\\pm (\\lambda ^{\\mathrm {u}}) =\\breve{\\psi }_\\pm (\\lambda ^{\\mathrm {l}}) =\\overline{\\psi _\\pm (\\lambda ^{\\mathrm {l}})},\\quad \\lambda \\in \\sigma _\\pm .$ For $E \\in \\hat{M}_\\pm $ the Weyl solutions satisfy $\\psi _{\\pm }(z,x)=O\\left(\\frac{1}{\\sqrt{z-E}}\\right), \\quad \\breve{\\psi }_{\\pm }(z,x)=O\\left(\\frac{1}{\\sqrt{z-E}}\\right),\\quad \\mbox{as } z\\rightarrow E\\in \\hat{M}_\\pm ,$ where the $O((z-E)^{-1/2})$ -term is independent of $x$ .",
"The same applies to $\\psi ^{\\prime }_{\\pm }(z,x)$ and $\\breve{\\psi }^\\prime _{\\pm }(z,x)$ .",
"At the edges of the spectrum the Weyl solutions satisfy $\\nonumber \\lim _{z\\rightarrow E}\\psi _\\pm (z,x)-\\breve{\\psi }_\\pm (z,x)=0 \\quad \\text{ for }\\quad E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm ,$ and $\\nonumber \\psi _\\pm (z,x)+\\breve{\\psi }_\\pm (z,x)=O(1) \\quad \\text{ for } \\quad z \\text{ near }E\\in \\hat{M}_\\pm ,$ where the $O(1)$ -term depends on $x$ .",
"The functions $\\psi _{\\pm }(z,x)$ and $\\breve{\\psi }_{\\pm }(z,x)$ form an orthonormal basis on the spectrum with respect to the weight $d\\rho _\\pm (z)=\\frac{1}{2\\pi I}g_\\pm (z) dz,$ and any $f(x)\\in L^2(\\mathbb {R})$ can be expressed through $f(x)=\\oint _{\\sigma _\\pm } \\left(\\int _\\mathbb {R}f(y)\\psi _\\pm (z,y)dy\\right)\\breve{\\psi }_\\pm (z,x)d\\rho (z).$ Here we use the notation $\\nonumber \\oint _{\\sigma _\\pm }f(z)d\\rho _\\pm (z) := \\int _{\\sigma ^u_\\pm } f(z)d\\rho _\\pm (z)- \\int _{\\sigma ^l_\\pm } f(z)d\\rho _\\pm (z).$ For a proof of (i) and (iii) we refer to [12].",
"(ii) We only prove the claim for the $+$ case (the $-$ case can be handled in the same way) and drop the $+$ in what follows.",
"In [12] we showed that $\\lim _{z\\rightarrow E}\\exp \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right)={\\left\\lbrace \\begin{array}{ll}\\pm 1, & \\mu _j(0)\\ne E, \\mu _j(x)\\ne E, \\\\\\pm 1, & \\mu _j(0)=E, \\mu _j(x)=E, \\\\\\pm I, & \\mu _j(0)=E, \\mu _j(x)\\ne E, \\\\\\pm I, & \\mu _j(0)\\ne E, \\mu _j(x)=E, \\end{array}\\right.",
"}$ for any $E\\in \\partial \\sigma $ .",
"Thus assuming that $\\mu _j(0)\\ne E$ , we can write along the spectrum $\\psi (z,x)-\\breve{\\psi }(z,x)=2i\\left(\\frac{G(z,x)}{G(z,0)}\\right)^{1/2}\\sin \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right).$ If in addition $\\mu _j(x)\\ne E$ , then by (REF ) we have $\\lim _{z\\rightarrow E}\\sin \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right)=0$ and $\\lim _{z\\rightarrow E}\\frac{G(z,x)}{G(z,0)}$ exists.",
"Thus we end up with $\\lim _{z\\rightarrow E} (\\psi (z,x)-\\breve{\\psi }(z,x)) =0.$ If $\\mu _j(x)=E$ , then by (REF ) we get $\\lim _{z\\rightarrow E}\\sin \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right)=\\pm 1$ and $\\lim _{z\\rightarrow E} \\frac{G(z,x)}{G(z,0)}=0$ .",
"Hence $\\lim _{z\\rightarrow E}(\\psi (z,x)-\\breve{\\psi }(z,x))=0.$ To prove the second claim, assume that $\\mu _j(0)=E$ and write $\\psi (z,x)-\\breve{\\psi }(z,x)=2\\left(\\frac{G(z,x)}{G(z,0)}\\right)^{1/2}\\cos \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right).$ If $\\mu _j(x)=E$ , then by (REF ) we get $\\lim _{z\\rightarrow E}\\cos \\left(\\int _0^x\\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right)=\\pm 1$ and $\\lim _{z\\rightarrow E} \\frac{G(z,x)}{G(z,0)}$ exists.",
"Therefore $\\lim _{z\\rightarrow E}(\\psi (z,x)-\\breve{\\psi }(z,x))$ exists and especially $\\psi (z,x)-\\breve{\\psi }(z,x)=O(1) \\quad \\text{ for }\\quad z\\text{ near } E=\\mu _j(0).$ If $\\mu _j(x)\\ne E$ , we cannot conclude as before, because $\\lim _{z\\rightarrow E}\\frac{G(z,x)}{G(z,0)}$ does not exist.",
"Assume that $E=E_{2j}$ (the case $E=E_{2j-1}$ can be handled in a similar way).",
"Then we can seperate for fixed $x\\in \\mathbb {R}$ the interval $[0,x]$ into smaller intervals $[x_0,x_1]\\cup [x_1,x_2]\\cup \\dots \\cup [x_{2l},x]$ such that $x_0=0$ , $\\mu _j(x_k)\\in \\lbrace E_{2j-1}, E_{2j}\\rbrace $ for $k=0,1,2, \\dots ,2l$ , $\\mu _j(x)$ is monotone increading or decreasing on every interval $[x_k, x_{k+1}]$ and $\\mu _j(x_0)=E_{2j}=\\mu _j(x_{2l})$ .",
"Folowing the proof of [12], one obtains for $E=E_{2j}$ that $\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau & =I\\sigma _j2(l+1) \\arctan \\left(\\frac{\\sqrt{E_{2j}-E_{2j-1}}}{\\sqrt{z-E_{2j}}}\\right)\\\\ \\nonumber &\\quad +I\\sigma _j\\arctan \\left(\\frac{\\sqrt{E_{2j}-\\mu _j(x)}}{\\sqrt{z-E_{2j}}}\\right)+IO(\\sqrt{z-E_{2j}}),$ where the $O(\\sqrt{z-E_{2j}})$ term depends on $x$ .",
"Using now that $\\arctan (x)=\\frac{\\pi }{2}+O\\left(\\frac{1}{x}\\right)$ for $x\\rightarrow \\infty $ , we have $\\arctan \\left(\\frac{\\sqrt{E_{2j}-E_{2j-1}}}{\\sqrt{z-E_{2j}}}\\right)=\\frac{\\pi }{2}+O\\left(\\frac{\\sqrt{z-E_{2j}}}{\\sqrt{E_{2j}-E_{2j-1}}}\\right),$ and $\\arctan \\left(\\frac{\\sqrt{E_{2j}-\\mu _j(x)}}{\\sqrt{z-E_{2j}}}\\right)=\\frac{\\pi }{2}+O\\left(\\frac{\\sqrt{z-E_{2j}}}{\\sqrt{E_{2j}-\\mu _j(x)}}\\right),$ which gives $\\cos \\left(\\int _0^x \\frac{Y^{1/2}(z)}{G(z,\\tau )}d\\tau \\right)=O(\\sqrt{z-E_{2j}}),$ where the $O(\\sqrt{z-E_{2j}})$ term depends on $x$ .",
"Plugging this into (REF ) yields $\\psi (z,x)-\\breve{\\psi }(z,x)=O(1)\\quad \\text{ for } \\quad z\\text{ near } E=\\mu _j(0),$ where the $O(1)$ -term depends on $x$ ."
],
[
"The direct scattering problem",
"Consider the Schrödinger equation $\\left(-\\frac{d^2}{dx^2}+q(x)\\right) y(x)=zy(x), \\quad z\\in $ with a potential $q(x)$ satisfying the following condition $\\pm \\int _0^{\\pm \\infty } (1+x^2)\\vert q(x)-p_\\pm (x)\\vert dx <\\infty .$ Then there exist two solutions, the so-called Jost solutions $\\phi _\\pm (z,x)$ , which are asymptotically close to the background Weyl solutions $\\psi _\\pm (z,x)$ of equation (REF ) as $x\\rightarrow \\pm \\infty $ and they can be represented as $\\phi _\\pm (z,x)=\\psi _\\pm (z,x)\\pm \\int _x^{\\pm \\infty } K_\\pm (x,y)\\psi _\\pm (z,y)dy.$ Here $K_\\pm (x,y)$ are real-valued functions, which are continuously differentiable with respect to both parameters and satisfy the estimate $\\vert K_\\pm (x,y) \\vert \\le C_\\pm (x)Q_\\pm (x+y)=\\pm C_\\pm (x)\\int _{\\frac{x+y}{2}}^{\\pm \\infty }\\vert q(t)-p_\\pm (t)\\vert dt,$ where $C_\\pm (x)$ are continuous, positive, monotonically decreasing functions, and therefore bounded as $x\\rightarrow \\pm \\infty $ .",
"Furthermore, $\\left|\\frac{d K_\\pm (x,y)}{dx}\\right|+\\left|\\frac{d K_\\pm (x,y)}{dy}\\right|\\le C_\\pm (x)\\left(\\left|q_\\pm \\left(\\frac{x+y}{2}\\right)\\right|+Q_\\pm (x+y)\\right)$ and $\\pm \\int _a^{\\pm \\infty }(1+x^2)\\left|\\frac{d}{dx}K_\\pm (x,x)\\right|dx<\\infty , \\quad \\forall a\\in \\mathbb {R}.$ For more information we refer to [12].",
"Moreover, for $\\lambda \\in \\sigma _\\pm ^u\\cup \\sigma _\\pm ^l$ a second pair of solutions of (REF ) is given by $\\overline{\\phi _\\pm (\\lambda ,x)}=\\breve{\\psi }_\\pm (\\lambda ,x)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,y)\\breve{\\psi }_\\pm (\\lambda ,y)dy, \\quad \\lambda \\in \\sigma _\\pm ^u\\cup \\sigma _\\pm ^l.$ Note $\\breve{\\psi }_\\pm (\\lambda ,x)=\\overline{\\psi _\\pm (\\lambda ,x)}$ for $\\lambda \\in \\sigma _\\pm $ .",
"Unlike the Jost solutions $\\phi _\\pm (z,x)$ , these solutions only exist on the upper and lower cuts of the spectrum and cannot be continued to the whole complex plane.",
"Combining (REF ), (REF ), (REF ), and (REF ), one obtains $W(\\phi _\\pm (\\lambda ),\\overline{\\phi _\\pm (\\lambda )})=\\pm g(\\lambda )^{-1}.$ In the next lemma we want to point out, which properties of the background Weyl solutions are also inherited by the Jost solutions.",
"Lemma 3.1 The Jost solutions $\\phi _\\pm (z,x)$ have the following properties: The function $\\phi _\\pm (z,x)$ considered as a function of $z$ , is holomorphic in the domain $(\\sigma _\\pm \\cup M_\\pm )$ , and has simple poles at the points of the set $M_\\pm $ .",
"It is continuous up to the boundary $\\sigma _\\pm ^u \\cup \\sigma _\\pm ^l$ except at the points from $\\hat{M}_\\pm $ .",
"Moreover, we have $\\nonumber \\phi _\\pm (z,x)\\in L^2(\\mathbb {R}_\\pm ), \\quad z\\in \\sigma _\\pm $ For $E\\in \\hat{M}_\\pm $ they satisfy $\\nonumber \\phi _\\pm (z,x)=O\\left(\\frac{1}{\\sqrt{z-E}}\\right), \\quad \\text{ as } z\\rightarrow E\\in \\hat{M}_\\pm ,$ where the $O((z-E)^{-1/2})$ -term depends on $x$ .",
"At the band edges of the spectrum we have the following behavior: $\\lim _{z\\rightarrow E}\\phi _\\pm (z,x)-\\overline{\\phi _\\pm (z,x)}=0\\quad \\text{ for } \\quad E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm ,$ and $\\phi _\\pm (z,x)+\\overline{\\phi _\\pm (z,x)}=O(1) \\quad \\text{ for } \\quad z \\text{ near }E\\in \\hat{M}_\\pm ,$ where the $O(1)$ -term depends on $x$ .",
"Everything follows from the fact that these properties are only dependent on $z$ and therefore the transformation operator does not influence them.",
"Now we want to characterize the spectrum of our operator $L$ , which consists of an (absolutely) continuous part, $\\sigma =\\sigma _+\\cup \\sigma _-$ and an at most countable number of discrete eigenvalues, which are situated in the gaps, $\\sigma _d\\subset \\mathbb {R}\\backslash \\sigma $ .",
"In particular every gap can only contain a finite number of discrete eigenvalues (cf.",
"[15], [16], and [24]) and thus they cannot cluster.",
"For our purposes it will be convenient to write $\\sigma =\\sigma _-^{(1)}\\cup \\sigma _+^{(1)}\\cup \\sigma ^{(2)},$ with $\\sigma ^{(2)}:=\\sigma _-\\cap \\sigma _+, \\quad \\sigma _\\pm ^{(1)}=\\mathop {\\rm clos}(\\sigma _\\pm \\backslash \\sigma ^{(2)}).$ It is well-known that a point $\\lambda \\in \\mathbb {R}\\backslash \\sigma $ corresponds to the discrete spectrum if and only if the two Jost solutions are linearly dependent, which implies that we should investigate $W(z):=W(\\phi _-(z,.",
"), \\phi _+(z,.",
")),$ the Wronskian of the Jost solutions.",
"This is a meromorphic function in the domain $\\sigma $ , with possible poles at the points $M_+\\cup M_- \\cup (\\hat{M}_+\\cap \\hat{M}_-)$ and possible square root singularities at the points $(\\hat{M}_+\\cup \\hat{M}_-)\\backslash (\\hat{M}_+\\cap \\hat{M}_-)$ .",
"For investigating the function $W(z)$ in more detail, we will multiply the possible poles and square root singularities away.",
"Thus we define locally in a small neighborhood $U_j^\\pm $ of the j'th gap $[E_{2j-1}^\\pm ,E_{2j}^\\pm ]$ , where $j=1,2,\\dots $ $\\tilde{\\phi }_{j,\\pm }(z,x)=\\delta _{j,\\pm }(z)\\phi _\\pm (z,x),$ where $\\delta _{j,\\pm }(z)={\\left\\lbrace \\begin{array}{ll}z-\\mu _j^\\pm , & \\text{ if } \\mu _j^\\pm \\in M_\\pm ,\\\\1, & \\text{else}\\end{array}\\right.",
"}$ and $\\hat{\\phi }_{j,\\pm }(z,x)=\\hat{\\delta }_{j,\\pm }(z)\\phi _\\pm (z,x),$ where $\\hat{\\delta }_{j,\\pm }(z)={\\left\\lbrace \\begin{array}{ll}z-\\mu _j^\\pm , & \\text{ if } \\mu _j^\\pm \\in M_\\pm ,\\\\\\sqrt{z-\\mu _j^\\pm }, & \\text{ if } \\mu _j^\\pm \\in \\hat{M}_\\pm ,\\\\1, & \\text{ else}.",
"\\end{array}\\right.",
"}$ Correspondingly, we set $\\tilde{W}(z)=W(\\tilde{\\phi }_-(z,.",
"),\\tilde{\\phi }_+(z,.",
")), \\quad \\hat{W}(z)=W(\\hat{\\phi }_-(z,.",
"),\\hat{\\phi }_+(z,.",
")).$ Here we use the definitions $\\tilde{\\phi }_\\pm (z,x)={\\left\\lbrace \\begin{array}{ll}\\tilde{\\phi }_{j,\\pm }(z,x), & \\text{ for } z\\in U_j^\\pm , j=1,2,\\dots ,\\\\\\phi _\\pm (z,x), & \\text{ else },\\end{array}\\right.",
"}$ $\\hat{\\phi }_\\pm (z,x)={\\left\\lbrace \\begin{array}{ll}\\hat{\\phi }_{j,\\pm }(z,x), & \\text{ for } z\\in U_j^\\pm , j=1,2,\\dots ,\\\\\\phi _\\pm (z,x), & \\text{ else }.",
"\\end{array}\\right.",
"}$ and we will choose $U_j^+=U_m^-$ , if $[E_{2j-1}^+, E_{2j}^+]\\cap [E_{2m-1}^-,E_{2m}^-]\\ne \\emptyset $ .",
"Analogously, one can define $\\delta _\\pm (z)$ and $\\hat{\\delta }_\\pm (z)$ .",
"Note that the function $ \\hat{W}(z)$ is holomorphic in the domain $U_j^\\pm \\cap (\\sigma )$ and continuous up to the boundary.",
"But unlike the functions $W(z)$ and $\\tilde{W}(z)$ it may not take real values on the set $\\mathbb {R}\\backslash \\sigma $ and complex conjugated values on the different sides of the spectrum $\\sigma ^u \\cup \\sigma ^l$ inside the domains $U_j^\\pm $ .",
"That is why we will characterize the spectral properties of our operator $L$ in terms of the function $\\tilde{W}(z)$ which can have poles at the band edges.",
"Since the discrete spectrum of our operator $L$ is at most countable, we can write it as $\\sigma _d=\\bigcup _{n=1}^\\infty \\sigma _n\\subset \\mathbb {R}\\backslash \\sigma ,$ where $\\sigma _n=\\lbrace \\lambda _{n,1}, \\dots ,\\lambda _{n,k(n)}\\rbrace , \\quad n\\in \\mathbb {N}$ and $k(n)$ denotes the number of eigenvalues in the n'th gap of $\\sigma $ .",
"For every eigenvalue $\\lambda _{n,m}$ we can introduce the corresponding norming constants $(\\gamma _{n,m}^\\pm )^{-2}=\\int _{\\mathbb {R}}\\tilde{\\phi }_{\\pm }^2(\\lambda _{n,m},x)dx.$ Now we begin with the study of the properties of the scattering data.",
"Therefore we introduce the scattering relations $T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)=\\overline{\\phi _\\pm (\\lambda ,x)}+R_\\pm (\\lambda )\\phi _\\pm (\\lambda ,x), \\quad \\lambda \\in \\sigma _\\pm ^{u,l},$ where the transmission and reflection coefficients are defined as usual, $T_\\pm (\\lambda ):=\\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\pm (\\lambda ))}{W(\\phi _\\mp (\\lambda ),\\phi _\\pm (\\lambda ))}, \\quad R_\\pm (\\lambda ):=-\\frac{W(\\phi _\\mp (\\lambda ),\\overline{\\phi _\\pm (\\lambda ))}}{W(\\phi _\\mp (\\lambda ),\\phi _\\pm (\\lambda ))}, \\quad \\lambda \\in \\sigma _\\pm ^{u,l}$ Theorem 3.2 For the scattering matrix the following properties are valid: $T_\\pm (\\lambda ^u)=\\overline{T_\\pm (\\lambda ^l)}$ and $R_\\pm (\\lambda ^u)=\\overline{R_\\pm (\\lambda ^l)}$ for $\\lambda \\in \\sigma _\\pm $ .",
"$\\dfrac{T_\\pm (\\lambda )}{\\overline{T_\\pm (\\lambda )}}=R_\\pm (\\lambda )$ for $\\lambda \\in \\sigma _\\pm ^{(1)}$ .",
"$1-\\vert R_\\pm (\\lambda )\\vert ^2=\\dfrac{g_\\pm (\\lambda )}{g_\\mp (\\lambda )}\\vert T_\\pm (\\lambda )\\vert ^2$ for $\\lambda \\in \\sigma ^{(2)}$ .",
"$\\overline{R_\\pm (\\lambda )}T_\\pm (\\lambda )+R_\\mp (\\lambda )\\overline{T_\\pm (\\lambda )}=0$ for $\\lambda \\in \\sigma ^{(2)}$ .",
"(i) and (iv) follow from (REF ), (REF ), (REF ), and Lemma REF For showing (ii) observe that $\\tilde{\\phi }_\\mp (\\lambda ,x)\\in \\mathbb {R}$ as $\\lambda \\in \\mathop {\\rm int}(\\sigma _\\pm ^{(1)})$ , which implies (ii).",
"To show (iii), assume $\\lambda \\in \\mathop {\\rm int}\\sigma ^{(2)}$ , then by (REF ) $\\nonumber \\vert T_\\pm \\vert ^2W(\\phi _\\mp ,\\overline{\\phi _\\mp })=(\\vert R_\\pm \\vert ^2-1)W(\\phi _\\pm ,\\overline{\\phi _\\pm }).$ Thus using (REF ) finishes the proof.",
"Theorem 3.3 The transmission and reflection coefficients have the following asymptotic behavior, as $\\lambda \\rightarrow \\infty $ for $\\lambda \\in \\sigma ^{(2)}$ outside a small $\\varepsilon $ neighborhood of the band edges of $\\sigma ^{(2)}$ : $R_\\pm (\\lambda )&=O(\\vert \\lambda \\vert ^{-1/2}),\\\\T_\\pm (\\lambda )&=1+O(\\vert \\lambda \\vert ^{-1/2}).$ The asymptotics can only be valid for $\\lambda \\in \\sigma ^{(2)}$ outside an $\\varepsilon $ neighborhood of the band edges, because the Jost solutions $\\phi _\\pm $ might have square root singularities there.",
"At first we will investigate $W(\\phi _-(\\lambda ,0),\\phi _+(\\lambda ,0))$ : $\\phi _-(\\lambda ,0)\\phi _+^\\prime (\\lambda ,0)=& \\left(1+\\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy\\right)\\\\ \\nonumber & \\times \\left(m_+(\\lambda )-K_+(0,0)+\\int _0^\\infty K_{+,x}(0,y)\\psi _+(\\lambda ,y)dy\\right).$ Using (cf.",
"()) $\\psi _\\pm ^\\prime (\\lambda ,x)=m_\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,x),$ we can write $\\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy= \\int _{-\\infty }^0 \\frac{K_-(0,y)}{m_-(\\lambda ,y)}\\psi _-^\\prime (\\lambda , y)dy.$ Hence $\\nonumber \\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy=\\frac{K_-(0,0)}{m_-(\\lambda )}+I_1(\\lambda ),$ $I_1(\\lambda )=-\\int _{-\\infty }^0\\left(K_{-,y}(0,y)\\frac{\\psi _-(\\lambda ,y)}{m_-(\\lambda ,y)}-K_-(0,y)\\psi _-(\\lambda ,y)\\frac{m_-^\\prime (\\lambda ,y)}{m_-(\\lambda ,y)^2}\\right)dy.$ Here it should be noticed that $m_\\pm (\\lambda ,y)^{-1}$ has no pole, because (see e.g.",
"[17]) $G_\\pm (\\lambda ,y)N_\\pm (\\lambda ,y)+H_\\pm (\\lambda ,y)^2=Y_\\pm (\\lambda ),$ where $N_\\pm (\\lambda ,y)=-(\\lambda -\\nu _0^\\pm (y))\\prod _{j=1}^\\infty \\frac{\\lambda -\\nu _j^\\pm (y)}{E_{2j-1}^\\pm },$ with $\\nu _0^\\pm (y)\\in (-\\infty ,E_0^\\pm ]$ and $\\nu _j^\\pm (y)\\in [E_{2j-1}^\\pm ,E_{2j}^\\pm ]$ .",
"Thus we obtain $m_\\pm (\\lambda ,y)^{-1}=\\frac{G_\\pm (\\lambda ,y)}{H_\\pm (\\lambda ,y)\\pm Y_\\pm (\\lambda )^{1/2}}=-\\frac{H_\\pm (\\lambda ,y)\\mp Y_\\pm (\\lambda )^{1/2}}{N_\\pm (\\lambda ,y)},$ and therefore $\\frac{K_-(0,0)}{m_-(\\lambda )}=O(\\frac{1}{\\sqrt{\\lambda }})$ .",
"Moreover $I_1(\\lambda )=O\\big (\\frac{1}{\\sqrt{\\lambda }}\\big )$ as the following estimates show: $\\vert I_1(\\lambda )& \\vert \\le \\int _{-\\infty }^0 \\vert K_{-,y}(0,y)\\frac{\\psi _-(\\lambda ,y)}{m_-(\\lambda ,y)}\\vert dy +\\int _{-\\infty }^0\\vert K_-(0,y)\\psi _-(\\lambda ,y)\\frac{m_-^\\prime (\\lambda ,y)}{m_-(\\lambda ,y)^2}\\vert dy \\\\ \\nonumber &\\le \\frac{C}{\\sqrt{\\lambda }}\\int _{-\\infty }^0(\\vert q(y)-p_-(y) \\vert +Q_-(y))dy,$ where we used that $\\vert \\psi _\\pm (\\lambda ,y)\\vert =\\vert \\frac{G _\\pm (\\lambda ,y)}{G_\\pm (\\lambda ,0)}\\vert =O(1)$ and $m_\\pm ^{-1}(\\lambda ,y)=O\\left(\\frac{1}{ \\sqrt{\\lambda }}\\right)$ for all $y$ by the quasi-periodicity, together with (REF ) and $\\psi _\\pm ^{\\prime \\prime }(\\lambda ,x)=m_\\pm (\\lambda ,x)^2\\psi _\\pm (\\lambda ,x)+m_\\pm ^\\prime (\\lambda ,x)\\psi _\\pm (\\lambda ,x).$ Making the same conclusions as before, one obtains $\\nonumber \\int _0^\\infty K_{+,x}(0,y)\\psi _+(\\lambda ,y)dy=O(1).$ In a similar manner one can investigate $\\nonumber \\phi _-^\\prime (\\lambda ,0)\\phi _+(\\lambda ,0)=&\\left(m_-(\\lambda )+K_-(0,0)+\\int _{-\\infty }^0 K_{-,x}(0,y)\\psi _-(\\lambda ,y)dy\\right) \\\\&\\times \\left(1+\\int _0^\\infty K_+(0,y)\\psi _+(\\lambda ,y)dy\\right),$ where $\\int _{-\\infty }^0 K_{-,x}(0,y)\\psi _-(\\lambda ,y)dy=O(1),$ $\\int _0^\\infty K_+(0,y)\\psi _+(\\lambda ,y)dy=-\\frac{K_+(0,0)}{m_+(\\lambda )}+I_2(\\lambda ),$ $I_2(\\lambda )=-\\int _0^\\infty \\left(K_{+,y}(0,y)\\frac{\\psi _+(\\lambda ,y)}{m_+(\\lambda ,y)}-K_+(0,y)\\psi _+(\\lambda ,y)\\frac{m_+^\\prime (\\lambda ,y)}{m_+(\\lambda ,y)^2}\\right)dy,$ and $I_2(\\lambda )=O\\big (\\frac{1}{\\sqrt{\\lambda }}\\big )$ .",
"Thus combining all the informations we obtained so far yields $\\nonumber W(\\phi _-(\\lambda ),\\phi _+(\\lambda ))& = m_+(\\lambda )-m_-(\\lambda )+ K_-(0,0)\\left(\\frac{m_+(\\lambda )-m_-(\\lambda )}{m_-(\\lambda )}\\right)\\\\ \\nonumber &+ K_+(0,0)\\left(\\frac{m_-(\\lambda )-m_+(\\lambda )}{m_+(\\lambda )}\\right)+O(1).$ and therefore, using (REF ), $T_\\pm (\\lambda )=1+O\\left(\\frac{1}{\\sqrt{\\lambda }}\\right).$ Analogously one can investigate the behavior of $W(\\phi _\\mp (\\lambda ),\\overline{\\phi _\\pm (\\lambda )}$ to obtain $R_\\pm (\\lambda )=O\\Big (\\frac{1}{\\sqrt{\\lambda }}\\Big )$ .",
"Theorem 3.4 The functions $T_\\pm (\\lambda )$ can be extended analytically to the domain $(\\sigma \\cup M_\\pm \\cup \\breve{M}_\\pm )$ and satisfy $\\frac{-1}{T_+(z)g_+(z)}=\\frac{-1}{T_-(z)g_-(z)}=:W(z),$ where $W(z)$ possesses the following properties: The function $\\tilde{W}$ is holomorphic in the domain $U_j^\\pm \\cap (\\sigma )$ , with simple zeros at the points $\\lambda _k$ , where $\\left( \\frac{d\\tilde{W}}{dz}(\\lambda _k)\\right)^2=(\\gamma _{n,k}^+ \\gamma _{n,k}^-)^{-2}.$ Besides it satisfies $\\overline{\\tilde{W}(\\lambda ^u)}=\\tilde{W}(\\lambda ^l), \\quad \\lambda \\in U_j^\\pm \\cap \\sigma \\quad \\text{and} \\quad \\tilde{W}(\\lambda )\\in \\mathbb {R}, \\quad \\lambda \\in U_j^\\pm \\cap (\\mathbb {R}\\backslash \\sigma ).$ The function $\\hat{W}(z)$ is continuous on the set $U_j^\\pm \\cap \\sigma $ up to the boundary $\\sigma ^l\\cup \\sigma ^u$ .",
"It can have zeros on the set $\\partial \\sigma \\cup (\\partial \\sigma _+^{(1)} \\cap \\partial \\sigma _-^{(1)}) $ and does not vanish at any other points of $\\sigma $ .",
"If $\\hat{W}(E)=0$ as $E\\in \\partial \\sigma \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)})$ , then $\\hat{W}(z)=\\sqrt{z-E}(C(E)+o(1))$ , $C(E)\\ne 0$ .",
"Except for (REF ) everything follows from the corresponding properties of $\\phi _\\pm (z,x)$ .",
"Therefore assume $\\hat{W}(\\lambda _0)=0$ for some $\\lambda _0\\in \\sigma $ , then $\\tilde{\\phi }_\\pm (\\lambda _0,x)=c_\\pm \\tilde{\\phi }_\\mp (\\lambda _0,x),$ for some constants $c_\\pm $ , which satisfy $c_-c_+=1$ .",
"Moreover, every zero of $\\tilde{W}$ (or $\\hat{W}$ ) outside the continuous spectrum, is a point of the discrete spectrum of $L$ and vice versa.",
"Denote by $\\gamma _\\pm $ the corresponding norming constants defined in (REF ) for some fixed point $\\lambda _0$ of the discrete spectrum.",
"Proceeding as in [21] one obtains $W\\big (\\tilde{\\phi }_\\pm (\\lambda _0,0), \\frac{d}{d\\lambda }\\tilde{\\phi }_\\pm (\\lambda _0,0)\\big )=\\int _0^{\\pm \\infty } \\tilde{\\phi }_\\pm ^2(\\lambda _0,x)dx.$ Thus using (REF ) and (REF ) yields $\\gamma _\\pm ^{-2}& =\\mp c_\\pm ^2\\int _0^{\\mp \\infty }\\tilde{\\phi }_\\mp ^2(\\lambda _0,x)dx \\pm \\int _0^{\\pm \\infty }\\tilde{\\phi }_\\pm ^2(\\lambda _0,x)dx\\\\ \\nonumber & =\\mp c_\\pm ^2 W\\big ( \\tilde{\\phi }_\\mp (\\lambda _0,0),\\frac{d}{d\\lambda }\\tilde{\\phi }_\\mp (\\lambda _0,0)\\big )\\pm W\\big (\\tilde{\\phi }_\\pm (\\lambda _0,0), \\frac{d}{d\\lambda }\\tilde{\\phi }_\\pm (\\lambda _0,0)\\big ) \\\\ \\nonumber & =c_\\pm \\frac{d}{d\\lambda }W(\\tilde{\\phi }_-(\\lambda _0),\\tilde{\\phi }_+(\\lambda _0)).$ Applying now $c_-c_+=1$ , we obtain (REF ).",
"The continuity of $\\hat{W}(z)$ up to the boundary follows immediately from the corresponding properties of $\\hat{\\phi }_\\pm (z,x)$ .",
"Now we will investigate the possible zeros of $\\hat{W}(\\lambda )$ for $\\lambda \\in \\sigma $ .",
"Assume $W(\\lambda _0)=0$ for some $\\lambda _0\\in \\mathop {\\rm int}(\\sigma ^{(2)})$ .",
"Then $\\phi _+(\\lambda _0,x)=c\\phi _-(\\lambda _0,x)$ and $\\overline{\\phi _+(\\lambda _0,x)}=\\overline{c}\\overline{\\phi _-(\\lambda _0,x)}$ .",
"Thus $W(\\phi _+,\\overline{\\phi _+})=\\vert c\\vert ^2 W(\\phi _-, \\overline{\\phi _-})$ and therefore $\\mathop {\\rm sign}g_+(\\lambda _0)=-\\mathop {\\rm sign}g_-(\\lambda _0)$ by (REF ), contradicting (REF ).",
"Next let $\\lambda _0\\in \\mathop {\\rm int}(\\sigma _\\pm ^{(1)})$ and $\\tilde{W}(\\lambda _0)=0$ , then $\\phi _\\pm (\\lambda _0,x)$ and $\\overline{\\phi _\\pm (\\lambda _0,x)}$ are linearly independent and bounded, moreover $\\tilde{\\phi }_\\mp (\\lambda _0,x)\\in \\mathbb {R}$ .",
"Therefore $\\tilde{W}(\\lambda _0)=0$ implies that $\\tilde{\\phi }_\\mp =c_1^\\pm \\phi _\\pm =c_2^\\pm \\overline{\\phi _\\pm }$ and thus $W(\\phi _\\pm ,\\overline{\\phi _\\pm })=0$ , which is impossible by (REF ).",
"Note that in this case $\\lambda _0$ can coincide with a pole $\\mu \\in M_\\mp $ .",
"Since $\\hat{W}(\\lambda )\\ne 0$ for $\\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)})\\cup \\mathop {\\rm int}(\\sigma ^{(1)}_+)\\cup \\mathop {\\rm int}(\\sigma ^{(1)}_-)$ , it is left to investigate the behavior at the band edges of $\\sigma _+$ and $\\sigma _-$ .",
"Therefore introduce the local parameter $\\tau =\\sqrt{z-E}$ in a small neighborhood of each point $E\\in \\partial \\sigma _\\pm $ and define $\\dot{y}(z,x)=\\frac{d}{d\\tau }y(z,x)$ .",
"A simple calculation shows that $\\frac{dz}{d\\tau }(E)=0$ , hence for every solution $y(z,x)$ of (REF ), its derivative $\\dot{y}(E,x)$ is again a solution of (REF ).",
"Therefore, the Wronskian $W(y(E),\\dot{y}(E))$ is independent of $x$ .",
"For each $x\\in \\mathbb {R}$ in a small neighborhood of a fixed point $E\\in \\partial \\sigma _\\pm $ we introduce the function $\\hat{\\psi }_{\\pm ,E}(z,x)={\\left\\lbrace \\begin{array}{ll}\\psi _\\pm (z,x), & E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm , \\\\ \\nonumber \\tau \\psi _\\pm (z,x), & E\\in \\hat{M}_\\pm .\\end{array}\\right.",
"}$ Proceeding as in [1] one obtains $W\\Big (\\hat{\\psi }_{\\pm ,E}(E), \\frac{d}{d\\tau }\\hat{\\psi }_{\\pm ,E}(E)\\Big )=\\pm \\lim _{z\\rightarrow E}\\frac{\\alpha \\tau ^\\alpha }{2g_\\pm (z)},$ where $\\alpha =-1$ if $E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm $ and $\\alpha =1$ if $E\\in \\hat{M}_\\pm $ .",
"Using representation () for $\\psi _\\pm (z,x)$ one can show (cf [12]), $\\nonumber \\psi _\\pm (E,x)=\\left(\\frac{G_\\pm (E,x)}{G_\\pm (E,0)}\\right)^{1/2}\\exp \\left(\\pm \\lim _{z\\rightarrow E}\\int _0^x\\frac{Y_\\pm (z)^{1/2}}{G_\\pm (z,\\tau )}d\\tau \\right), \\quad E\\in \\partial \\sigma $ where $\\nonumber \\exp \\left(\\pm \\lim _{z\\rightarrow E}\\int _0^x\\frac{Y_\\pm (z)^{1/2}}{G_\\pm (z,\\tau )}d\\tau \\right)={\\left\\lbrace \\begin{array}{ll}I^{2s+1}, & \\mu _j\\ne E, \\mu _j(x)=E,\\\\I^{2s+1}, & \\mu _j=E, \\mu _j(x)\\ne E,\\\\I^{2s}, & \\mu _j=E, \\mu _j(x)=E,\\\\I^{2s}, & \\mu _j\\ne E, \\mu _j(x)\\ne E,\\end{array}\\right.",
"}$ for $s\\in \\lbrace 0,1\\rbrace $ .",
"Defining $\\nonumber \\hat{\\phi }_{\\pm ,E}(\\lambda ,x)={\\left\\lbrace \\begin{array}{ll}\\phi _\\pm (\\lambda ,x), & E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm ,\\\\\\tau \\phi _\\pm (\\lambda ,x), & E\\in \\hat{M}_\\pm ,\\end{array}\\right.",
"}$ we can conclude using (REF ) that $\\overline{\\phi _\\pm (E,x)}=\\phi _\\pm (E,x), \\quad \\text{ for } E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm .$ Moreover, for $E\\in \\hat{M}_\\pm $ , ${\\left\\lbrace \\begin{array}{ll}\\overline{\\hat{\\phi }_{\\pm ,E}(E,x)}=-\\hat{\\phi }_{\\pm ,E}(E,x), & \\text{ a left band edge from } \\sigma _\\pm ,\\\\\\overline{\\hat{\\phi }_{\\pm ,E}(E,x)}=\\hat{\\phi }_{\\pm ,E}(E,x), & \\text{ a right band edge from } \\sigma _\\pm .\\end{array}\\right.",
"}$ If $\\lambda _0=E\\in \\partial \\sigma ^{(2)}\\cap \\mathop {\\rm int}(\\sigma _\\pm )\\subset \\mathop {\\rm int}(\\sigma _\\pm )$ , then $\\hat{W}(E)=0$ if and only if $W(\\psi _\\pm ,\\hat{\\psi }_{\\mp ,E})(E)=0$ .",
"Therefore, as $\\hat{\\phi }_{\\mp ,E}(E,.",
")$ are either pure real or pure imaginary, $W(\\overline{\\phi _\\pm },\\hat{\\phi }_{\\mp ,E})(E)=0$ , which implies that $\\overline{\\phi _\\pm }(E,x)$ and $\\phi _\\pm (E,x)$ are linearly dependent, a contradiction.",
"Thus the function $\\hat{W}(z)$ can only be zero at points $E$ of the set $\\partial \\sigma \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)})$ .",
"We will now compute the order of the zero.",
"First of all note that the function $\\hat{W}(\\lambda )$ is continuously differentiable with respect to the local parameter $\\tau $ .",
"Since $\\frac{d}{d\\tau }(\\delta _+\\delta _-)(E)=0$ , the function $W(\\hat{\\phi }_{+,E}, \\hat{\\phi }_{-,E})$ has the same order of zero at $E$ as $\\hat{W}(\\lambda )$ .",
"Moreover, if $\\hat{\\delta }_\\pm (E)\\ne 0$ ,then $\\frac{d}{d\\tau }\\hat{\\delta }_\\pm (E)=0$ and if $\\hat{\\delta }_-(E)=\\hat{\\delta }_+(E)=0$ , then $\\frac{d}{d\\tau }(\\tau ^{-2}\\hat{\\delta }_+\\hat{\\delta }_-)(E)=0$ .",
"Hence $\\frac{d}{d\\tau }\\hat{W}(E)=0$ if and only if $\\frac{d}{d\\tau }W(\\hat{\\phi }_{+,E}, \\hat{\\phi }_{-,E})=0$ .",
"Combining now all the informations we obtained so far, we can conclude as follows: if $\\hat{W}(E)=0$ , then $\\hat{\\phi }_{\\pm ,E}(E,.",
")=c_\\pm \\hat{\\phi }_{\\mp ,E}(E,.",
")$ , with $c_-c_+=1$ .",
"Furthermore we can write $\\nonumber \\dot{W}(\\hat{\\phi }_{+,E},\\hat{\\phi }_{-,E})(E)& =W(\\frac{d}{d\\tau }\\hat{\\phi }_{+,E},\\hat{\\phi }_{-,E})(E)-W(\\frac{d}{d\\tau }\\hat{\\phi }_{-,E}, \\hat{\\phi }_{+,E})(E)\\\\ \\nonumber & = c_-W(\\frac{d}{d\\tau }\\hat{\\phi }_{+,E},\\hat{\\phi }_{+,E})(E)-c_+W(\\frac{d}{d\\tau }\\hat{\\phi }_{-,E}, \\hat{\\phi }_{-,E})(E)\\\\ \\nonumber & = c_-W(\\frac{d}{d\\tau }\\hat{\\psi }_{+,E},\\hat{\\psi }_{+,E})(E)-c_+W(\\frac{d}{d\\tau }\\hat{\\psi }_{-,E}, \\hat{\\psi }_{-,E})(E).$ Using (REF ), (REF ), (REF ), and distinguishing several cases as in [1] finishes the proof.",
"Theorem 3.5 The reflection coefficient $R_\\pm (\\lambda )$ satisfies: The reflection coefficient $R_\\pm (\\lambda )$ is a continuous function on the set $\\mathop {\\rm int}(\\sigma _\\pm ^{u,l})$ .",
"If $E\\in \\partial \\sigma _+\\cap \\partial \\sigma _-$ and $\\hat{W}(E)\\ne 0$ , then the function $R_\\pm (\\lambda )$ is also continuous at $E$ .",
"Moreover, $\\nonumber R_\\pm (E)={\\left\\lbrace \\begin{array}{ll}-1 & \\text{ for } E\\notin \\hat{M}_\\pm ,\\\\1 & \\text{ for } E\\in \\hat{M}_\\pm .\\end{array}\\right.",
"}$ At first it should be noted that by Lemma REF the reflection coefficient is bounded, as $\\frac{g_\\pm (\\lambda )}{g_\\mp (\\lambda )}>0$ for $\\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)})$ .",
"Thus, using the corresponding properties of $\\phi _\\pm (z,x)$ , finishes the first part.",
"We proceed as in the proof of [1].",
"By (REF ) the reflection coefficient can be represented in the following form: $R_\\pm (\\lambda )=-\\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\mp (\\lambda ))}{W(\\phi _\\pm (\\lambda ),\\phi _\\mp (\\lambda ))}=\\pm \\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\mp (\\lambda ))}{W(\\lambda )},$ and is therefore continuous on both sides of the set $\\mathop {\\rm int}(\\sigma _\\pm )\\backslash (M_\\mp \\cup \\hat{M}_\\mp )$ .",
"Moreover, $\\nonumber \\vert R_\\pm (\\lambda )\\vert =\\left|\\frac{W(\\overline{\\hat{\\phi }_\\pm (\\lambda )},\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )}\\right|,$ where the denominator does not vanish, by assumption and hence $R_\\pm (\\lambda )$ is continuous on both sides of the spectrum in a small neighborhood of the band edges under consideration.",
"Next, let $E\\in \\lbrace E_{2j-1}^\\pm , E_{2j}^\\pm \\rbrace $ with $\\hat{W}(E)\\ne 0$ .",
"Then, if $E\\notin \\hat{M}_\\pm $ , we can write $\\nonumber R_\\pm (\\lambda )=-1\\mp \\frac{\\hat{\\delta }_{j,\\pm }(\\lambda )W(\\phi _\\pm (\\lambda )-\\overline{\\phi _\\pm (\\lambda )},\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )},$ which implies $R_\\pm (\\lambda )\\rightarrow -1$ , since $\\phi _\\pm (\\lambda )-\\overline{\\phi _\\pm (\\lambda )}\\rightarrow 0$ by Lemma REF as $\\lambda \\rightarrow E$ .",
"Thus we proved the first case.",
"If $E\\in \\hat{M}_\\pm $ with $\\hat{W}(E)\\ne 0$ , we use (REF ) in the form $\\nonumber R_\\pm (\\lambda )=1\\pm \\frac{\\hat{\\delta }_{j,\\pm }(\\lambda )W(\\phi _\\pm (\\lambda )+\\overline{\\phi }_\\pm (\\lambda ),\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )},$ which yields $R_\\pm (\\lambda )\\rightarrow 1$ , since $\\hat{\\delta }_{j,\\pm }(\\lambda )\\rightarrow 0$ and $\\phi _\\pm (\\lambda )+\\overline{\\phi _\\pm }(\\lambda )=O(1)$ by Lemma REF as $\\lambda \\rightarrow E$ .",
"This settles the second case."
],
[
"The Gel'fand-Levitan-Marchenko Equation",
"The aim of this section is to derive the Gel'fand-Levitan-Marchenko (GLM) equation, which is also called the inverse scattering problem equation and to obtain some additional properties of the scattering data, as a consequence of the GLM equation.",
"Therefore consider the function $G_\\pm (z,x,y) &= T_\\pm (z)\\phi _\\mp (z,x)\\psi _\\pm (z,y)g_\\pm (z)-\\breve{\\psi }_\\pm (z,x)\\psi _\\pm (z,y)g_\\pm (z)\\\\ \\nonumber & := G^\\prime _\\pm (z,x,y)+G^{\\prime \\prime }_\\pm (z,x,y), \\quad \\pm y >\\pm x,$ where $x$ and $y$ are considered as fixed parameters.",
"As a function of $z$ it is meromorphic in the domain $\\sigma $ with simple poles at the points $\\lambda _k$ of the discrete spectrum.",
"It is continuous up to the boundary $\\sigma ^u\\cup \\sigma ^l$ , except for the points of the set, which consists of the band edges of the background spectra $\\partial \\sigma _+$ and $\\partial \\sigma _-$ , where $G_\\pm (z,x,y)=O((z-E)^{-1/2}) \\quad \\text{ as } \\quad E\\in \\partial \\sigma _+\\cup \\partial \\sigma _-.$ Outside a small neighborhood of the gaps of $\\sigma _+$ and $\\sigma _-$ , the following asymptotics as $z\\rightarrow \\infty $ are valid: $\\nonumber \\phi _\\mp (z,x)&=\\mathrm {e}^{\\mp I\\sqrt{z}x(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1/2})\\right), \\quad g_\\pm (z)=\\frac{-1}{2I\\sqrt{z}}+O(z^{-1}),\\\\ \\nonumber \\breve{\\psi }_\\pm (z,x)& =\\mathrm {e}^{\\mp I\\sqrt{z}x(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1})\\right), \\quad T_\\pm (z)=1+O(z^{-1/2}),\\\\ \\nonumber \\psi _\\pm (z,y) & = \\mathrm {e}^{\\pm I\\sqrt{z}y(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1})\\right),$ and the leading term of $\\phi _\\mp (z,x)$ and $\\breve{\\psi }_\\pm (z,x)$ are equal, thus $G_\\pm (z,x,y)=\\mathrm {e}^{\\pm I\\sqrt{z}(y-x)(1+O(\\frac{1}{z}))}O(z^{-1}), \\quad \\pm y > \\pm x.$ Figure: Contours Γ ε,n \\Gamma _{\\varepsilon ,n}Consider the following sequence of contours $\\Gamma _{\\varepsilon ,n,\\pm }$ , where $\\Gamma _{\\varepsilon ,n,\\pm }$ consists of two parts for every $n\\in \\mathbb {N}$ and $\\varepsilon \\ge 0$ : $C_{\\varepsilon ,n,\\pm }$ consists of a part of a circle which is centered at the origin and has as radii the distance from the origin to the midpoint of the largest band of $[E_{2n}^\\pm ,E_{2n+1}^\\pm ]$ , which lies inside $\\sigma ^{(2)}$ , together with a part wrapping around the corresponding band of $\\sigma $ at a small distance, which is at most $\\varepsilon $ , as indicated by figure 1.",
"Each band of the spectrum $\\sigma $ , which is fully contained in $C_{\\varepsilon ,n,\\pm }$ , is surrounded by a small loop at a small distance from $\\sigma $ not bigger than $\\varepsilon $ .",
"W.l.o.g.",
"we can assume that all the contours are non-intersecting.",
"Using the Cauchy theorem, we obtain $\\nonumber \\frac{1}{2\\pi I}\\oint _{\\Gamma _{\\varepsilon ,n,\\pm }} G_\\pm (z,x,y)dz=\\sum _{\\lambda _k \\in \\mathop {\\rm int}(\\Gamma _{\\varepsilon ,n,\\pm })}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm (z,x,y), \\quad \\varepsilon >0.$ By (REF ) the limit value of $G_\\pm (z,x,y)$ as $\\varepsilon \\rightarrow 0$ is integrable on $\\sigma $ , and the function $G_\\pm ^{\\prime \\prime }(z,x,y)$ has no poles at the points of the discrete spectrum, thus we arrive at $\\frac{1}{2\\pi I}\\oint _{\\Gamma _{0,n,\\pm }}G_\\pm (z,x,y)dz=\\sum _{\\lambda _k \\in \\mathop {\\rm int}(\\Gamma _{0,n,\\pm })}\\mathop {\\rm Res}_{\\lambda _k}G^\\prime _\\pm (z,x,y),\\quad \\pm y > \\pm x.$ Estimate (REF ) allows us now to apply Jordan's lemma, when letting $n\\rightarrow \\infty $ , and we therefore arrive, up to that point only formally, at $\\frac{1}{2\\pi I}\\oint _{\\sigma }G_\\pm (\\lambda ,x,y)d\\lambda =\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y), \\quad \\pm y>\\pm x.$ Next, note that the function $G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)$ does not contribute to the left part of (REF ), since $G_\\pm ^{\\prime \\prime }(\\lambda ^u,x,y)=G_\\pm ^{\\prime \\prime }(\\lambda ^l,x,y)$ for $\\lambda \\in \\sigma _\\mp ^{(1)}$ and, hence $\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)d\\lambda =0$ .",
"In addition, $\\oint _{\\sigma _\\pm }G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)d\\lambda =0$ for $x\\ne y$ by Lemma REF (iv).",
"Therefore we arrive at the following equation, $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\pm }G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y), \\quad \\pm y>\\pm x.$ To make our argument rigorous we have to show that the series of contour integrals along the parts of the spectrum contained in $C_{0,n,\\pm }$ on the left hand side of (REF ) converges as $n\\rightarrow \\infty $ and that the contribution of the integrals along the circles $C_{0,n,\\pm }$ converges against zero as $n\\rightarrow \\infty $ , by applying Jordan's lemma.",
"This will be done next.",
"Using (REF ), (REF ), (REF ), and (REF ), we obtain $\\nonumber \\frac{1}{2\\pi I} & \\oint _{\\sigma _\\pm } G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\oint _{\\sigma _\\pm }T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & = \\oint _{\\sigma _\\pm } \\Big (R_\\pm (\\lambda )\\phi _\\pm (\\lambda ,x)+\\overline{\\phi _\\pm (\\lambda ,x)}\\Big )\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & =\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )+\\oint _{\\sigma _\\pm } \\breve{\\psi }_\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ) \\\\ \\nonumber & \\quad \\pm \\int _x^{\\pm \\infty } dt K_\\pm (x,t)\\Big (\\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,t)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )+\\delta (t-y)\\Big )\\\\ \\nonumber & = F_{r,\\pm }(x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{r,\\pm }(t,y)dt+K_\\pm (x,y),$ where $F_{r,\\pm }(x,y)=\\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ).$ Now properties (ii) and (iii) from Lemma REF imply that $\\vert R_\\pm (\\lambda )\\vert <1 \\quad \\text{for} \\quad \\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)}), \\quad \\vert R_\\pm (\\lambda )\\vert =1 \\quad \\text{for} \\quad \\lambda \\in \\sigma _\\pm ^{(1)}.$ and by () we can write $\\nonumber F_{r,\\pm } (x,y) & =\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & = -\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\frac{(G_\\pm (\\lambda ,x)G_\\pm (\\lambda ,y))^{1/2}}{2Y_\\pm (\\lambda )^{1/2}}\\exp (\\eta _\\pm (\\lambda ,x)+\\eta _\\pm (\\lambda ,y))d\\lambda ,$ with $\\nonumber \\eta _\\pm (\\lambda ,x):=\\pm \\int _0^x \\frac{Y_\\pm (\\lambda )^{1/2}}{G_\\pm (\\lambda ,\\tau )}d\\tau \\in I\\mathbb {R}.$ We are now ready to prove the following lemma.",
"Lemma 4.1 The sequence of functions $F_{r,n,\\pm }(x,y)=\\oint _{\\sigma _\\pm \\cap \\Gamma _{0,n,\\pm }} R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y) d\\rho _\\pm (\\lambda ),$ is uniformly bounded with respect to $x$ and $y$ , that means for all $n\\in \\mathbb {N}$ , $\\vert F_{r,n,\\pm }(x,y)\\vert \\le C$ .",
"Moreover, $F_{r,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{r,\\pm }(x,y) = \\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ),$ which is again uniformly bounded with respect to $x$ and $y$ .",
"In particular, $F_{r,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"For $\\lambda \\in \\sigma _\\pm $ as $\\lambda \\rightarrow \\infty $ we have the following asymptotic behavior in a small neighborhood $V_{n}^\\pm $ of $E=E_{n}^\\pm $ $\\nonumber \\vert R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )\\vert =O\\Big (\\frac{\\sqrt{E_{2j}^\\pm -E_{2j-1}^\\pm }}{\\sqrt{\\lambda (\\lambda -E)}}\\Big ),$ in a small neighborhood $W_n^\\pm $ of $E=E_n^\\mp $ , if $E\\in \\sigma _\\pm $ $\\nonumber R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=O(\\frac{1}{\\sqrt{\\lambda }}),$ and for $\\lambda \\in \\sigma _\\pm \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_{n}^\\pm \\cup W_{n}^\\pm )$ $R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=\\exp (\\pm I\\sqrt{\\lambda }(\\vert x\\vert +\\vert y\\vert )(1+O(\\frac{1}{\\lambda })))\\Big (\\frac{C}{\\lambda }+O\\Big (\\frac{1}{\\lambda ^{3/2}}\\Big )\\Big ).$ These estimates are good enough to show that $F_{r,\\pm }(x,y)$ exists, if we choose $V_n^\\pm $ and $W_n^\\pm $ in the following way: We choose $V_n^\\pm \\subset \\sigma _\\pm ^{(1)}\\cup \\sigma ^{(2)}$ , if $E_n^\\pm $ is a band edge of $\\sigma _\\pm ^{(1)}$ , such that $V_n^\\pm $ consists of the corresponding band of $\\sigma _\\pm ^{(1)}$ together with the following part of $\\sigma _\\pm ^{(2)}$ with length $E_n^\\pm -E_{n-1}^\\pm $ , if $n$ is even and $E_{n+1}^\\pm -E_n^\\pm $ , if $n$ is odd.",
"If $E_n^+$ is a band edge of $\\sigma ^{(2)}$ , we choose $V_n^\\pm \\subset \\sigma ^{(2)}$ , where the length of $V_n^\\pm $ is equal to the length of the gap pf $\\sigma _\\pm $ next to it.",
"We set $W_n^\\pm \\subset \\sigma ^{(2)}$ with length $3(E_n^\\mp -E_{n-1}^\\mp )$ , if $n$ is even and $3(E_{n+1}^\\mp -E_n^\\mp )$ , if $n$ is odd, centered at the midpoint of the corresponding gap in $\\sigma _\\mp $ .",
"As we are working in the Levitan class and we therefore know that $\\sum _{n=1}^\\infty (E_{2n-1}^\\pm )^{l^\\pm }(E_{2n}^\\pm -E_{2n-1}^\\pm )<\\infty $ for some $l^\\pm >1$ , we obtain that the sequences belonging to $V_n^\\pm $ and $W_n^\\pm $ converge.",
"As far as the behavior along the spectrum away from the band edges of $\\sigma _+$ and $\\sigma _-$ is concerned observe that $\\vert \\exp (\\pm I\\sqrt{\\lambda }(x+y)O(\\frac{1}{\\lambda }))\\vert \\le 1+ (x+y)O(\\frac{1}{\\sqrt{\\lambda }}),\\quad \\lambda \\rightarrow \\infty .$ and therefore $R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=\\exp (\\pm I\\sqrt{\\lambda }(x+y))\\Big (\\frac{C}{\\lambda }+(1+x+y)O\\Big (\\frac{1}{\\lambda ^{3/2}}\\Big )\\Big ).$ To show the convergence of the series $F_{r,\\pm }(x,y)$ for fixed $x$ and $y$ , we split the integral along the spectrum $\\sigma $ up into three integrals along $\\bigcup _{n\\in \\mathbb {N}} V_n^\\pm $ , $\\bigcup _{n\\in \\mathbb {N}} W_n^\\pm $ , and $\\sigma \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_n^\\pm \\cup W_n^\\pm )$ respectively.",
"As far as the integral along $\\bigcup _{n\\in \\mathbb {N}} V_n^pm$ is concerned observe that the integrand has a square root singularity at the boundary and is therefore integrable along $V_n^\\pm $ for all $n\\in \\mathbb {N}$ .",
"Since we are working within the Levitan class the sum over all $n\\in \\mathbb {N}$ converges.",
"The integrand can be uniformly bounded for all $\\lambda \\in W_n^\\pm $ such that $\\lambda \\ge 1$ .",
"Since there are only finitely many $n\\in \\mathbb {N}$ such that $W_n^\\pm \\subset [0,1]$ , the corresponding series converges by the definition of the Levitan class.",
"Thus it is left to consider the integral along $\\sigma \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_n^\\pm \\cup W_n^\\pm )$ :=I.",
"Direct computation yields $\\nonumber \\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{C}{\\lambda }d\\lambda & =\\pm \\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{2C}{I\\lambda ^{1/2}(x+y)}\\vert _a^b\\\\ \\nonumber & \\pm \\int _a^b\\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{C}{I\\lambda ^{3/2}(x+y)}d\\lambda ,$ which is finite since by assumption $\\pm x\\le \\pm y$ .",
"Hence one possibility to see that the corresponding series of integrals converges is to integrate first the function describing the asymptotic behavior along $[E_0^\\pm ,\\infty ]$ and substract from it the series of integrals corresponding to the $[E_0^\\pm ,\\infty ]\\cap I^c$ .",
"Since every interval belonging to the complement belongs to a neighborhood of the gaps of $\\sigma ^{(2)}$ and the integrand can be uniformly bounded, the definition of the Levitan class implies that this series converges.",
"Similarly we conclude $\\nonumber \\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))(x+y)O(\\frac{1}{\\lambda ^{3/2}})d\\lambda & =\\exp (\\pm I\\sqrt{\\lambda }(x+y))O(\\frac{1}{\\lambda })\\vert _a^b\\\\ \\nonumber & +\\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))O(\\frac{1}{\\lambda ^{2}})d\\lambda $ Note that since we are working within the Levitan class all estimates are independent of $x$ and $y$ .",
"For investigating the other terms, we will need the following lemma, which is taken from [9]: Lemma 4.2 Suppose in an integral equation of the form $f_\\pm (x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)f_\\pm (t,y)dt=g_\\pm (x,y), \\quad \\pm y>\\pm x,$ the kernel $K_\\pm (x,y)$ and the function $g_\\pm (x,y)$ are continuous for $\\pm y>\\pm x$ , $\\nonumber \\vert K_\\pm (x,y) \\vert \\le C_\\pm (x) Q_\\pm (x+y),$ and for $g_\\pm (x,y)$ one of the following estimates hold $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)Q_\\pm (x+y), \\quad \\text{ or }$ $\\vert g_\\pm (x,y) \\vert \\le C_\\pm (x)(1+\\max (0,\\pm x)).$ Furthermore assume that $\\pm \\int _0^{\\pm \\infty } (1+\\vert x \\vert ^2)\\vert q(x)-p_\\pm (x)\\vert dx<\\infty .$ Then (REF ) is uniquely solvable for $f_\\pm (x,y)$ .",
"The solution $f_\\pm (x,y)$ is also continuous in the half-plane $\\pm y >\\pm x$ , and for it the estimate (REF ) respectively (REF ) is reproduced.",
"Moreover, if a sequence $g_{n,\\pm }(x,y)$ satisfies (REF ) or (REF ) uniformly with respect to $n$ and pointwise $g_{n,\\pm }(x,y)\\rightarrow 0$ , for $\\pm y>\\pm x$ , then the same is true for the corresponding sequence of solutions $f_{n,\\pm }(x,y)$ of (REF ).",
"For a proof we refer to [9].",
"Remark 4.3 An immediate consequence of this lemma is the following.",
"If $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ denotes a bounded function, then $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)(1+\\max (0,\\pm x))$ and therefore $\\vert f_\\pm (x,y)\\vert \\le C_\\pm (x)(1+\\max (0,\\pm x))$ .",
"Rewriting this integral equation as follows $f_\\pm (x,y)=g_\\pm (x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)f_\\pm (t,y)dy,$ we obtain that the absolute value of the right hand side is smaller than a bounded function $\\tilde{C}_\\pm (x)$ by using (REF ) and (REF ), and hence the same is true for the left hand side.",
"In particular if $C_\\pm (x)$ is a decreasing function the same will be true for $\\tilde{C}_\\pm (x)$ .",
"We will now continue the investigation of our integral equation.",
"Lemma 4.4 The sequence of functions $F_{h,n,\\pm }(x,y)=\\int _{\\sigma _\\pm ^{(1),u}\\cap \\Gamma _{0,n,\\pm }} \\vert T_\\mp (\\lambda )\\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda )$ is uniformly bounded, that means for all $n\\in \\mathbb {N}$ , $\\vert F_{h,n,\\pm }(x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ are monotonically decrasing functions as $x\\rightarrow \\pm \\infty $ .",
"Moerover, $F_{h,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{h,\\pm }(x,y) =\\int _{\\sigma _\\mp ^{(1),u}} \\vert T_\\mp (\\lambda ) \\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y) d\\rho _\\mp (\\lambda ),$ which is again bounded by some monotonically increasing function.",
"In particular, $F_{h,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"On the set $\\sigma _\\mp ^{(1)}$ both the numerator and the denominator of the function $G_\\pm ^\\prime (\\lambda ,x,y)$ have poles (resp.",
"square root singularities) at the points of the set $\\sigma _\\mp ^{(1)}\\cap (M_\\pm \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)}))$ (resp.",
"$\\sigma _\\mp ^{(1)}\\cap (M_\\mp \\backslash (M_\\mp \\cap M_\\pm ))$ , but multiplying them, if necessary away, we can avoid singularities.",
"Hence, w.l.o.g., we can suppose $\\sigma _\\mp ^{(1)}\\cap (M_{r,+}\\cup M_{r,-})=\\emptyset $ .",
"Thus we can write $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda .$ For investigating this integral we will consider, using (REF ), $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\mp (\\lambda ) &\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\mp (\\lambda )d\\lambda \\\\ \\nonumber & = \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}\\phi _\\mp (\\lambda ,x)\\Big (\\overline{\\phi _\\mp (\\lambda ,y)}+R_\\mp (\\lambda )\\phi _\\mp (\\lambda ,y)\\Big )g_\\mp (\\lambda )d\\lambda .$ First of all note that the integrand, because of the representation on the right hand side, can only have square root singularities at the boundary $\\partial \\sigma _\\mp ^{(1)}$ and we therefore have $\\nonumber \\int _{\\sigma _\\mp ^{(1)}\\cap [E_{2n-1}^\\pm ,E_{2n}^\\pm ]}& \\vert \\phi _\\mp (\\lambda ,x) \\Big (\\overline{\\phi _\\mp (\\lambda ,y)}+ R_\\mp (\\lambda )\\phi _\\mp (\\lambda ,y)\\Big )g_\\mp (\\lambda )\\vert d\\lambda \\\\ \\nonumber & \\le 2\\int _{\\sigma _\\mp ^{(1)}\\cap [E_{2n-1}^\\pm , E_{2n}^\\pm ]}\\vert \\phi _\\mp (\\lambda ,x)\\phi _\\mp (\\lambda ,y)g_\\mp (\\lambda )\\vert d\\lambda \\\\ \\nonumber & \\le C_\\pm (y)C_\\pm (x)\\left( \\frac{(E_{2n}^\\pm -E_{2n-1}^\\pm )}{\\sqrt{\\lambda -E_0^\\mp }}+\\frac{\\sqrt{E_{2n}^\\pm -E_{2n-1}^\\pm }}{\\sqrt{\\lambda -E_0^\\mp }}\\right),$ where $E_{2n-1}^\\pm $ and $E_{2n}^\\pm $ denote the edges of the gap of $\\sigma _\\pm $ in which the corresponding part of $\\sigma _\\mp ^{(1)}$ lies and $C_\\pm (x)$ denote monotonically decreasing functions from now on.",
"Therefore as we are working in the Levitan class and by separating $\\sigma _\\mp ^{(1)}$ into the different parts, one obtains that $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (y)C_\\pm (x).$ Thus we can now apply Lemma REF , and hence $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (y)C_\\pm (x)(1+\\max (0,\\pm y)).$ Note that we especially have, because of (REF ), $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (x)$ Therefore we can conclude that for fixed $x$ and $y$ the left hands side of (REF ) exists and satisfies $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (x),$ and hence $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G^{\\prime }_\\pm (z,x,y)dz\\vert \\le C_\\pm (x).$ We will now rewrite the integrand in a form suitable for our further purposes.",
"Namely, since $\\psi _\\pm (\\lambda ,x)\\in \\mathbb {R}$ as $\\lambda \\in \\sigma _\\mp ^{(1)}$ , we have $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\frac{1}{2\\pi I}\\int _{\\sigma _\\mp ^{(1),u}}\\psi _\\pm (\\lambda ,y)\\Big (\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}-\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}\\Big )d\\lambda $ Moreover, (REF ) and Lemma REF (ii) imply $\\nonumber \\overline{\\phi _\\mp (\\lambda ,x)}=T_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)-\\frac{T_\\mp (\\lambda )}{\\overline{T_\\mp (\\lambda )}}.$ Therefore, $\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}-\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}& =\\phi _\\mp (\\lambda ,x)\\Big (\\frac{1}{W(\\lambda )}+\\frac{T_\\mp (\\lambda )}{\\overline{T_\\mp (\\lambda )W(\\lambda )}}\\Big )-\\frac{T_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)}{\\overline{W(\\lambda )}}\\\\ \\nonumber & =\\phi _\\mp (\\lambda ,x)\\frac{2\\mathop {\\rm Re}(T_\\mp ^{-1}(\\lambda )\\overline{W(\\lambda )})T_\\mp (\\lambda )}{\\vert W(\\lambda )\\vert ^2}-\\frac{T_\\mp (\\lambda )\\phi _\\pm (\\lambda )}{\\overline{W(\\lambda )}}.$ But by (REF ) $T_\\mp ^{-1}(\\lambda )\\overline{W(\\lambda )}=\\vert W(\\lambda )\\vert ^2g_\\mp (\\lambda )\\in I\\mathbb {R}, \\quad \\text{ for }\\lambda \\in \\sigma _\\mp ^{(1)},$ and therefore the first summand of (REF ) vanishes.",
"Using now $\\overline{W}=(\\overline{T_\\mp }g_\\mp )^{-1}$ we arrive at $\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}-\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}=\\vert T_\\mp (\\lambda )\\vert ^2g_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)$ and hence $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =F_{h,\\pm }(x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{h,\\pm }(t.y)dt,$ where $F_{h,\\pm }(x,y)=\\int _{\\sigma _\\pm ^{(1),u}}\\vert T_\\mp (\\lambda )\\vert ^2\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda ),$ and $\\vert F_{h,\\pm }(x,y)\\vert \\le C_\\pm (x)C_\\pm (y)$ by Lemma REF .",
"The partial sums $F_{h,n,\\pm }(x,y)$ can be investigated similarly We will now investigate the r.h.s.",
"of (REF ) and (REF ).",
"Therefore we consider first the question of the existence of the right hand side: To prove the boundedness of the corresponding series on the left hand side, it is left to investigate the series, which correspond to the circles.",
"We will derive the necessary estimates only for the part of the n'th circle $K_{R_{n,\\pm }}$ , where $R_{n,\\pm }$ denotes the radius, in the upper half plane as the part in the lower half plane can be considered similarly.",
"We have $\\nonumber \\vert \\int _{K_{R_{n},\\pm }}G_\\pm (z,x,y)dz\\vert & \\le \\int _0^\\pi C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )\\sin (\\theta /2)}d\\theta \\\\ \\nonumber & \\le \\int _0^{\\pi /2} C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )\\sin (\\eta )} d\\eta \\\\ \\nonumber & \\le \\int _0^{\\pi /2} C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu ) 2\\frac{\\eta }{\\pi }}d\\eta \\\\ \\nonumber &\\le C\\frac{1}{\\sqrt{R}(x-y)(1-\\nu )}\\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )2\\frac{\\eta }{\\pi }}\\vert _0^{\\frac{\\pi }{2}},$ where $C$ and $\\nu $ denote some constant, which are dependent on the radius (cf.",
"Lemma REF ).",
"Therefore as already mentioned the part belonging to the circles converges against zero and hence the same is true for the corresponding series, by Jordan's lemma.",
"Thus we obtain that the sequence of partial sums on the right hand side of (REF ) and (REF ) is uniformly bounded and we are therefore ready to prove the following result.",
"Lemma 4.5 The sequence of functions $F_{d,n,\\pm }(x,y)=\\sum _{\\lambda _k\\in \\sigma _d\\cap \\Gamma _{0,n,\\pm }}(\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y)$ is uniformly bounded, that means for all $n\\in \\mathbb {N}$ , $\\vert F_{d,n,\\pm }(x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ are monotonically increasing functions.",
"Moreover, $F_{d,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{d,\\pm }(x,y) = \\sum _{\\lambda _k\\in \\sigma _d}(\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y),$ which is again bounded by some monotonically increasing function.",
"In particular, $F_{d,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"Applying (REF ), (REF ), (REF ), (REF ), and (REF )to the right hand side of (REF ), yields $\\nonumber \\sum _{\\lambda _k\\in \\sigma _d} \\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y)&=-\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}\\frac{\\tilde{\\phi }_\\mp (\\lambda ,x)\\tilde{\\psi }_\\pm (\\lambda ,y)}{\\tilde{W}(\\lambda )} \\\\ \\nonumber & = -\\sum _{\\lambda _k\\in \\sigma _d}\\frac{\\tilde{\\phi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y)}{\\tilde{W}^\\prime (\\lambda _k)c_{k,\\pm }}\\\\ \\nonumber & =-\\sum _{\\lambda _k\\in \\sigma _d}(\\gamma _k^\\pm )^2\\tilde{\\phi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y) \\\\ \\nonumber &= -F_{d,\\pm }(x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{d,\\pm }(t,y)dt,$ where $F_{d,\\pm }(x,y):=\\sum _{\\lambda _k\\in \\sigma _d} (\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y).$ Thus we obtained the following integral equation, $F_{d,\\pm }(x,y)& = -K_\\pm (x,y)-F_{c,\\pm }(x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{c,\\pm }(t,y)dt\\\\ \\nonumber & \\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{d,\\pm }(t,y)dt,$ which we can now solve for $F_{d,\\pm }(x,y)$ using again Lemma REF and hence $F_{d,\\pm }(x,y)$ exists and satisfies the given estimates.",
"The corresponding partial sums can be investigated analogously using the considerations from above.",
"Putting everything together, we see that we have obtained the GLM equation.",
"Theorem 4.6 The GLM equation has the form $K_\\pm (x,y)+F_\\pm (x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_\\pm (t,y)dt=0, \\quad \\pm (y-x)>0,$ where $F_\\pm (x,y)& = \\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ) \\\\ \\nonumber & +\\int _{\\sigma _\\mp ^{(1),u}} \\vert T_\\mp (\\lambda )\\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda ) \\\\ \\nonumber & +\\sum _{k=1}^\\infty (\\gamma ^\\pm _k)^2 \\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y).$ Moreover, we have Lemma 4.7 The function $F_\\pm (x,y)$ is continuously differentiable with respect to both variables and there exists a real-valued function $q_\\pm (x)$ , $x\\in \\mathbb {R}$ with $\\pm \\int _a^{\\pm \\infty }(1+x^2)\\vert q_\\pm (x)\\vert dx<\\infty , \\quad \\text{ for all } a\\in \\mathbb {R},$ such that $\\vert F_\\pm (x,y)\\vert \\le \\tilde{C}_\\pm (x)Q_\\pm (x+y),$ $\\left|\\frac{d}{dx}F_\\pm (x,y)\\right|\\le \\tilde{C}_\\pm (x)\\left(\\left|q_\\pm \\left(\\frac{x+y}{2}\\right)\\right|+Q_\\pm (x+y)\\right),$ $\\pm \\int _a^{\\pm \\infty }\\left|\\frac{d}{dx}F_\\pm (x,x)\\right|(1+x^2)dx<\\infty ,$ where $Q_\\pm (x)=\\pm \\int _{\\frac{x}{2}}^{\\pm \\infty }\\vert q_\\pm (t) \\vert dt,$ and $\\tilde{C}_\\pm (x)>0$ is a continuous function, which decreases monotonically as $x\\rightarrow \\pm \\infty $ .",
"Applying once more Lemma REF , one obtains (REF ).",
"Now, for simplicity, we will restrict our considerations to the + case and omit + whenever possible.",
"Set $Q_1(u)=\\int _u^\\infty Q(t)dt$ .",
"Then, using (REF ), the functions $Q(x)$ and $Q_1(x)$ satisfy $\\int _a^\\infty Q_1(t)dt<\\infty , \\quad \\int _a^\\infty Q(t)(1+\\vert t\\vert )dt<\\infty .$ Differentiating (REF ) with respect to $x$ and $y$ yields $\\vert F_x(x,y)\\vert \\le \\vert K_x(x,y)\\vert +\\vert K(x,x)F(x,y)\\vert +\\int _x^\\infty \\vert K_x(x,t)F(t,y)\\vert dt,$ $F_y(x,y)+K_y(x,y)+\\int _x^\\infty K(x,t)F_y(t,y)dt=0.$ We already know that the functions $Q(x)$ , $Q_1(x)$ , $C(x)$ , and $\\tilde{C}(x)$ are monotonically decreasing and positive.",
"Moreover, $\\int _x^\\infty \\Big (\\left|q_+\\Big (\\frac{x+t}{2}\\Big )\\right|+Q(x+t)\\Big )Q(t+y)dt\\le (Q(2x)+Q_1(2x))Q(x+y),$ thus we can estimate $F_x(x,y)$ and $F_y(x,y)$ can be estimates using (REF ) and the method of successive approximation.",
"It is left to prove (REF ).",
"Therefore consider (REF ) for $x=y$ and differentiate it with respect to $x$ : $\\frac{d F(x,x)}{dx}+\\frac{d K(x,x)}{dx}-K(x,x)F(x,x)+\\int _x^\\infty (K_x(x,tF(t,x)+K(x,t)F_y(t,x))dt=0.$ Next (REF ) and (REF ) imply $\\vert K(x,y)F(x,x)\\vert \\le \\tilde{C}(a)C(a)Q^2(2x), \\quad \\text{ for } x>a,$ where $\\int _a^\\infty (1+x^2)Q^2(2x)dx<\\infty $ .",
"Moreover, by (REF ) and (REF ) $\\nonumber \\left|K_x^\\prime (x,t) F(t,x)\\right| + \\left|K(x,t) F_y^\\prime (t,x)\\right|\\le 4\\tilde{C}(a)\\hat{C}(a)\\Bigl \\lbrace \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert Q(x+t) +Q^2(x+t)\\Bigr \\rbrace ,$ together with the estimates $& \\int _a^\\infty d x\\,x^2\\int _x^\\infty Q^2(x+t)d t\\le \\int _a^\\infty |x| Q(2x)d x\\ \\sup _{x\\ge a}\\int _x^\\infty |x+t| Q(x+t)d t<\\infty ,\\\\& \\int _a^\\infty x^2\\int _x^\\infty \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert Q(x+t)d t\\le \\\\& \\qquad \\le \\int _a^\\infty Q(2x)d x \\ \\sup _{x\\ge a} \\int _x^\\infty \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert (1 +(x+t)^2)d t<\\infty ,$ and (REF ), we arrive at (REF ).",
"In summary, we have obtained the following necessary conditions for the scattering data: Theorem 4.8 The scattering data $\\nonumber {\\mathcal {S}} = \\Big \\lbrace & R_+(\\lambda ),\\,T_+(\\lambda ),\\, \\lambda \\in \\sigma _+^{\\mathrm {u,l}}; \\,R_-(\\lambda ),\\,T_-(\\lambda ),\\, \\lambda \\in \\sigma _-^{\\mathrm {u,l}};\\\\& \\lambda _1,,\\lambda _2,\\dots \\in \\mathbb {R}\\setminus (\\sigma _+\\cup \\sigma _-),\\,\\gamma _1^\\pm , \\gamma _2^\\pm ,\\dots \\in \\mathbb {R}_+\\Big \\rbrace $ possess the properties listed in Theorem REF , REF , REF , and REF , and Lemma REF , REF , and REF .",
"The functions $F_\\pm (x,y)$ defined in (REF ), possess the properties listed in Lemma REF ."
],
[
"Acknowledgements",
"I want to thank Ira Egorova and Gerald Teschl for many discussions on this topic."
],
[
"The direct scattering problem",
"Consider the Schrödinger equation $\\left(-\\frac{d^2}{dx^2}+q(x)\\right) y(x)=zy(x), \\quad z\\in $ with a potential $q(x)$ satisfying the following condition $\\pm \\int _0^{\\pm \\infty } (1+x^2)\\vert q(x)-p_\\pm (x)\\vert dx <\\infty .$ Then there exist two solutions, the so-called Jost solutions $\\phi _\\pm (z,x)$ , which are asymptotically close to the background Weyl solutions $\\psi _\\pm (z,x)$ of equation (REF ) as $x\\rightarrow \\pm \\infty $ and they can be represented as $\\phi _\\pm (z,x)=\\psi _\\pm (z,x)\\pm \\int _x^{\\pm \\infty } K_\\pm (x,y)\\psi _\\pm (z,y)dy.$ Here $K_\\pm (x,y)$ are real-valued functions, which are continuously differentiable with respect to both parameters and satisfy the estimate $\\vert K_\\pm (x,y) \\vert \\le C_\\pm (x)Q_\\pm (x+y)=\\pm C_\\pm (x)\\int _{\\frac{x+y}{2}}^{\\pm \\infty }\\vert q(t)-p_\\pm (t)\\vert dt,$ where $C_\\pm (x)$ are continuous, positive, monotonically decreasing functions, and therefore bounded as $x\\rightarrow \\pm \\infty $ .",
"Furthermore, $\\left|\\frac{d K_\\pm (x,y)}{dx}\\right|+\\left|\\frac{d K_\\pm (x,y)}{dy}\\right|\\le C_\\pm (x)\\left(\\left|q_\\pm \\left(\\frac{x+y}{2}\\right)\\right|+Q_\\pm (x+y)\\right)$ and $\\pm \\int _a^{\\pm \\infty }(1+x^2)\\left|\\frac{d}{dx}K_\\pm (x,x)\\right|dx<\\infty , \\quad \\forall a\\in \\mathbb {R}.$ For more information we refer to [12].",
"Moreover, for $\\lambda \\in \\sigma _\\pm ^u\\cup \\sigma _\\pm ^l$ a second pair of solutions of (REF ) is given by $\\overline{\\phi _\\pm (\\lambda ,x)}=\\breve{\\psi }_\\pm (\\lambda ,x)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,y)\\breve{\\psi }_\\pm (\\lambda ,y)dy, \\quad \\lambda \\in \\sigma _\\pm ^u\\cup \\sigma _\\pm ^l.$ Note $\\breve{\\psi }_\\pm (\\lambda ,x)=\\overline{\\psi _\\pm (\\lambda ,x)}$ for $\\lambda \\in \\sigma _\\pm $ .",
"Unlike the Jost solutions $\\phi _\\pm (z,x)$ , these solutions only exist on the upper and lower cuts of the spectrum and cannot be continued to the whole complex plane.",
"Combining (REF ), (REF ), (REF ), and (REF ), one obtains $W(\\phi _\\pm (\\lambda ),\\overline{\\phi _\\pm (\\lambda )})=\\pm g(\\lambda )^{-1}.$ In the next lemma we want to point out, which properties of the background Weyl solutions are also inherited by the Jost solutions.",
"Lemma 3.1 The Jost solutions $\\phi _\\pm (z,x)$ have the following properties: The function $\\phi _\\pm (z,x)$ considered as a function of $z$ , is holomorphic in the domain $(\\sigma _\\pm \\cup M_\\pm )$ , and has simple poles at the points of the set $M_\\pm $ .",
"It is continuous up to the boundary $\\sigma _\\pm ^u \\cup \\sigma _\\pm ^l$ except at the points from $\\hat{M}_\\pm $ .",
"Moreover, we have $\\nonumber \\phi _\\pm (z,x)\\in L^2(\\mathbb {R}_\\pm ), \\quad z\\in \\sigma _\\pm $ For $E\\in \\hat{M}_\\pm $ they satisfy $\\nonumber \\phi _\\pm (z,x)=O\\left(\\frac{1}{\\sqrt{z-E}}\\right), \\quad \\text{ as } z\\rightarrow E\\in \\hat{M}_\\pm ,$ where the $O((z-E)^{-1/2})$ -term depends on $x$ .",
"At the band edges of the spectrum we have the following behavior: $\\lim _{z\\rightarrow E}\\phi _\\pm (z,x)-\\overline{\\phi _\\pm (z,x)}=0\\quad \\text{ for } \\quad E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm ,$ and $\\phi _\\pm (z,x)+\\overline{\\phi _\\pm (z,x)}=O(1) \\quad \\text{ for } \\quad z \\text{ near }E\\in \\hat{M}_\\pm ,$ where the $O(1)$ -term depends on $x$ .",
"Everything follows from the fact that these properties are only dependent on $z$ and therefore the transformation operator does not influence them.",
"Now we want to characterize the spectrum of our operator $L$ , which consists of an (absolutely) continuous part, $\\sigma =\\sigma _+\\cup \\sigma _-$ and an at most countable number of discrete eigenvalues, which are situated in the gaps, $\\sigma _d\\subset \\mathbb {R}\\backslash \\sigma $ .",
"In particular every gap can only contain a finite number of discrete eigenvalues (cf.",
"[15], [16], and [24]) and thus they cannot cluster.",
"For our purposes it will be convenient to write $\\sigma =\\sigma _-^{(1)}\\cup \\sigma _+^{(1)}\\cup \\sigma ^{(2)},$ with $\\sigma ^{(2)}:=\\sigma _-\\cap \\sigma _+, \\quad \\sigma _\\pm ^{(1)}=\\mathop {\\rm clos}(\\sigma _\\pm \\backslash \\sigma ^{(2)}).$ It is well-known that a point $\\lambda \\in \\mathbb {R}\\backslash \\sigma $ corresponds to the discrete spectrum if and only if the two Jost solutions are linearly dependent, which implies that we should investigate $W(z):=W(\\phi _-(z,.",
"), \\phi _+(z,.",
")),$ the Wronskian of the Jost solutions.",
"This is a meromorphic function in the domain $\\sigma $ , with possible poles at the points $M_+\\cup M_- \\cup (\\hat{M}_+\\cap \\hat{M}_-)$ and possible square root singularities at the points $(\\hat{M}_+\\cup \\hat{M}_-)\\backslash (\\hat{M}_+\\cap \\hat{M}_-)$ .",
"For investigating the function $W(z)$ in more detail, we will multiply the possible poles and square root singularities away.",
"Thus we define locally in a small neighborhood $U_j^\\pm $ of the j'th gap $[E_{2j-1}^\\pm ,E_{2j}^\\pm ]$ , where $j=1,2,\\dots $ $\\tilde{\\phi }_{j,\\pm }(z,x)=\\delta _{j,\\pm }(z)\\phi _\\pm (z,x),$ where $\\delta _{j,\\pm }(z)={\\left\\lbrace \\begin{array}{ll}z-\\mu _j^\\pm , & \\text{ if } \\mu _j^\\pm \\in M_\\pm ,\\\\1, & \\text{else}\\end{array}\\right.",
"}$ and $\\hat{\\phi }_{j,\\pm }(z,x)=\\hat{\\delta }_{j,\\pm }(z)\\phi _\\pm (z,x),$ where $\\hat{\\delta }_{j,\\pm }(z)={\\left\\lbrace \\begin{array}{ll}z-\\mu _j^\\pm , & \\text{ if } \\mu _j^\\pm \\in M_\\pm ,\\\\\\sqrt{z-\\mu _j^\\pm }, & \\text{ if } \\mu _j^\\pm \\in \\hat{M}_\\pm ,\\\\1, & \\text{ else}.",
"\\end{array}\\right.",
"}$ Correspondingly, we set $\\tilde{W}(z)=W(\\tilde{\\phi }_-(z,.",
"),\\tilde{\\phi }_+(z,.",
")), \\quad \\hat{W}(z)=W(\\hat{\\phi }_-(z,.",
"),\\hat{\\phi }_+(z,.",
")).$ Here we use the definitions $\\tilde{\\phi }_\\pm (z,x)={\\left\\lbrace \\begin{array}{ll}\\tilde{\\phi }_{j,\\pm }(z,x), & \\text{ for } z\\in U_j^\\pm , j=1,2,\\dots ,\\\\\\phi _\\pm (z,x), & \\text{ else },\\end{array}\\right.",
"}$ $\\hat{\\phi }_\\pm (z,x)={\\left\\lbrace \\begin{array}{ll}\\hat{\\phi }_{j,\\pm }(z,x), & \\text{ for } z\\in U_j^\\pm , j=1,2,\\dots ,\\\\\\phi _\\pm (z,x), & \\text{ else }.",
"\\end{array}\\right.",
"}$ and we will choose $U_j^+=U_m^-$ , if $[E_{2j-1}^+, E_{2j}^+]\\cap [E_{2m-1}^-,E_{2m}^-]\\ne \\emptyset $ .",
"Analogously, one can define $\\delta _\\pm (z)$ and $\\hat{\\delta }_\\pm (z)$ .",
"Note that the function $ \\hat{W}(z)$ is holomorphic in the domain $U_j^\\pm \\cap (\\sigma )$ and continuous up to the boundary.",
"But unlike the functions $W(z)$ and $\\tilde{W}(z)$ it may not take real values on the set $\\mathbb {R}\\backslash \\sigma $ and complex conjugated values on the different sides of the spectrum $\\sigma ^u \\cup \\sigma ^l$ inside the domains $U_j^\\pm $ .",
"That is why we will characterize the spectral properties of our operator $L$ in terms of the function $\\tilde{W}(z)$ which can have poles at the band edges.",
"Since the discrete spectrum of our operator $L$ is at most countable, we can write it as $\\sigma _d=\\bigcup _{n=1}^\\infty \\sigma _n\\subset \\mathbb {R}\\backslash \\sigma ,$ where $\\sigma _n=\\lbrace \\lambda _{n,1}, \\dots ,\\lambda _{n,k(n)}\\rbrace , \\quad n\\in \\mathbb {N}$ and $k(n)$ denotes the number of eigenvalues in the n'th gap of $\\sigma $ .",
"For every eigenvalue $\\lambda _{n,m}$ we can introduce the corresponding norming constants $(\\gamma _{n,m}^\\pm )^{-2}=\\int _{\\mathbb {R}}\\tilde{\\phi }_{\\pm }^2(\\lambda _{n,m},x)dx.$ Now we begin with the study of the properties of the scattering data.",
"Therefore we introduce the scattering relations $T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)=\\overline{\\phi _\\pm (\\lambda ,x)}+R_\\pm (\\lambda )\\phi _\\pm (\\lambda ,x), \\quad \\lambda \\in \\sigma _\\pm ^{u,l},$ where the transmission and reflection coefficients are defined as usual, $T_\\pm (\\lambda ):=\\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\pm (\\lambda ))}{W(\\phi _\\mp (\\lambda ),\\phi _\\pm (\\lambda ))}, \\quad R_\\pm (\\lambda ):=-\\frac{W(\\phi _\\mp (\\lambda ),\\overline{\\phi _\\pm (\\lambda ))}}{W(\\phi _\\mp (\\lambda ),\\phi _\\pm (\\lambda ))}, \\quad \\lambda \\in \\sigma _\\pm ^{u,l}$ Theorem 3.2 For the scattering matrix the following properties are valid: $T_\\pm (\\lambda ^u)=\\overline{T_\\pm (\\lambda ^l)}$ and $R_\\pm (\\lambda ^u)=\\overline{R_\\pm (\\lambda ^l)}$ for $\\lambda \\in \\sigma _\\pm $ .",
"$\\dfrac{T_\\pm (\\lambda )}{\\overline{T_\\pm (\\lambda )}}=R_\\pm (\\lambda )$ for $\\lambda \\in \\sigma _\\pm ^{(1)}$ .",
"$1-\\vert R_\\pm (\\lambda )\\vert ^2=\\dfrac{g_\\pm (\\lambda )}{g_\\mp (\\lambda )}\\vert T_\\pm (\\lambda )\\vert ^2$ for $\\lambda \\in \\sigma ^{(2)}$ .",
"$\\overline{R_\\pm (\\lambda )}T_\\pm (\\lambda )+R_\\mp (\\lambda )\\overline{T_\\pm (\\lambda )}=0$ for $\\lambda \\in \\sigma ^{(2)}$ .",
"(i) and (iv) follow from (REF ), (REF ), (REF ), and Lemma REF For showing (ii) observe that $\\tilde{\\phi }_\\mp (\\lambda ,x)\\in \\mathbb {R}$ as $\\lambda \\in \\mathop {\\rm int}(\\sigma _\\pm ^{(1)})$ , which implies (ii).",
"To show (iii), assume $\\lambda \\in \\mathop {\\rm int}\\sigma ^{(2)}$ , then by (REF ) $\\nonumber \\vert T_\\pm \\vert ^2W(\\phi _\\mp ,\\overline{\\phi _\\mp })=(\\vert R_\\pm \\vert ^2-1)W(\\phi _\\pm ,\\overline{\\phi _\\pm }).$ Thus using (REF ) finishes the proof.",
"Theorem 3.3 The transmission and reflection coefficients have the following asymptotic behavior, as $\\lambda \\rightarrow \\infty $ for $\\lambda \\in \\sigma ^{(2)}$ outside a small $\\varepsilon $ neighborhood of the band edges of $\\sigma ^{(2)}$ : $R_\\pm (\\lambda )&=O(\\vert \\lambda \\vert ^{-1/2}),\\\\T_\\pm (\\lambda )&=1+O(\\vert \\lambda \\vert ^{-1/2}).$ The asymptotics can only be valid for $\\lambda \\in \\sigma ^{(2)}$ outside an $\\varepsilon $ neighborhood of the band edges, because the Jost solutions $\\phi _\\pm $ might have square root singularities there.",
"At first we will investigate $W(\\phi _-(\\lambda ,0),\\phi _+(\\lambda ,0))$ : $\\phi _-(\\lambda ,0)\\phi _+^\\prime (\\lambda ,0)=& \\left(1+\\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy\\right)\\\\ \\nonumber & \\times \\left(m_+(\\lambda )-K_+(0,0)+\\int _0^\\infty K_{+,x}(0,y)\\psi _+(\\lambda ,y)dy\\right).$ Using (cf.",
"()) $\\psi _\\pm ^\\prime (\\lambda ,x)=m_\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,x),$ we can write $\\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy= \\int _{-\\infty }^0 \\frac{K_-(0,y)}{m_-(\\lambda ,y)}\\psi _-^\\prime (\\lambda , y)dy.$ Hence $\\nonumber \\int _{-\\infty }^0 K_-(0,y)\\psi _-(\\lambda ,y)dy=\\frac{K_-(0,0)}{m_-(\\lambda )}+I_1(\\lambda ),$ $I_1(\\lambda )=-\\int _{-\\infty }^0\\left(K_{-,y}(0,y)\\frac{\\psi _-(\\lambda ,y)}{m_-(\\lambda ,y)}-K_-(0,y)\\psi _-(\\lambda ,y)\\frac{m_-^\\prime (\\lambda ,y)}{m_-(\\lambda ,y)^2}\\right)dy.$ Here it should be noticed that $m_\\pm (\\lambda ,y)^{-1}$ has no pole, because (see e.g.",
"[17]) $G_\\pm (\\lambda ,y)N_\\pm (\\lambda ,y)+H_\\pm (\\lambda ,y)^2=Y_\\pm (\\lambda ),$ where $N_\\pm (\\lambda ,y)=-(\\lambda -\\nu _0^\\pm (y))\\prod _{j=1}^\\infty \\frac{\\lambda -\\nu _j^\\pm (y)}{E_{2j-1}^\\pm },$ with $\\nu _0^\\pm (y)\\in (-\\infty ,E_0^\\pm ]$ and $\\nu _j^\\pm (y)\\in [E_{2j-1}^\\pm ,E_{2j}^\\pm ]$ .",
"Thus we obtain $m_\\pm (\\lambda ,y)^{-1}=\\frac{G_\\pm (\\lambda ,y)}{H_\\pm (\\lambda ,y)\\pm Y_\\pm (\\lambda )^{1/2}}=-\\frac{H_\\pm (\\lambda ,y)\\mp Y_\\pm (\\lambda )^{1/2}}{N_\\pm (\\lambda ,y)},$ and therefore $\\frac{K_-(0,0)}{m_-(\\lambda )}=O(\\frac{1}{\\sqrt{\\lambda }})$ .",
"Moreover $I_1(\\lambda )=O\\big (\\frac{1}{\\sqrt{\\lambda }}\\big )$ as the following estimates show: $\\vert I_1(\\lambda )& \\vert \\le \\int _{-\\infty }^0 \\vert K_{-,y}(0,y)\\frac{\\psi _-(\\lambda ,y)}{m_-(\\lambda ,y)}\\vert dy +\\int _{-\\infty }^0\\vert K_-(0,y)\\psi _-(\\lambda ,y)\\frac{m_-^\\prime (\\lambda ,y)}{m_-(\\lambda ,y)^2}\\vert dy \\\\ \\nonumber &\\le \\frac{C}{\\sqrt{\\lambda }}\\int _{-\\infty }^0(\\vert q(y)-p_-(y) \\vert +Q_-(y))dy,$ where we used that $\\vert \\psi _\\pm (\\lambda ,y)\\vert =\\vert \\frac{G _\\pm (\\lambda ,y)}{G_\\pm (\\lambda ,0)}\\vert =O(1)$ and $m_\\pm ^{-1}(\\lambda ,y)=O\\left(\\frac{1}{ \\sqrt{\\lambda }}\\right)$ for all $y$ by the quasi-periodicity, together with (REF ) and $\\psi _\\pm ^{\\prime \\prime }(\\lambda ,x)=m_\\pm (\\lambda ,x)^2\\psi _\\pm (\\lambda ,x)+m_\\pm ^\\prime (\\lambda ,x)\\psi _\\pm (\\lambda ,x).$ Making the same conclusions as before, one obtains $\\nonumber \\int _0^\\infty K_{+,x}(0,y)\\psi _+(\\lambda ,y)dy=O(1).$ In a similar manner one can investigate $\\nonumber \\phi _-^\\prime (\\lambda ,0)\\phi _+(\\lambda ,0)=&\\left(m_-(\\lambda )+K_-(0,0)+\\int _{-\\infty }^0 K_{-,x}(0,y)\\psi _-(\\lambda ,y)dy\\right) \\\\&\\times \\left(1+\\int _0^\\infty K_+(0,y)\\psi _+(\\lambda ,y)dy\\right),$ where $\\int _{-\\infty }^0 K_{-,x}(0,y)\\psi _-(\\lambda ,y)dy=O(1),$ $\\int _0^\\infty K_+(0,y)\\psi _+(\\lambda ,y)dy=-\\frac{K_+(0,0)}{m_+(\\lambda )}+I_2(\\lambda ),$ $I_2(\\lambda )=-\\int _0^\\infty \\left(K_{+,y}(0,y)\\frac{\\psi _+(\\lambda ,y)}{m_+(\\lambda ,y)}-K_+(0,y)\\psi _+(\\lambda ,y)\\frac{m_+^\\prime (\\lambda ,y)}{m_+(\\lambda ,y)^2}\\right)dy,$ and $I_2(\\lambda )=O\\big (\\frac{1}{\\sqrt{\\lambda }}\\big )$ .",
"Thus combining all the informations we obtained so far yields $\\nonumber W(\\phi _-(\\lambda ),\\phi _+(\\lambda ))& = m_+(\\lambda )-m_-(\\lambda )+ K_-(0,0)\\left(\\frac{m_+(\\lambda )-m_-(\\lambda )}{m_-(\\lambda )}\\right)\\\\ \\nonumber &+ K_+(0,0)\\left(\\frac{m_-(\\lambda )-m_+(\\lambda )}{m_+(\\lambda )}\\right)+O(1).$ and therefore, using (REF ), $T_\\pm (\\lambda )=1+O\\left(\\frac{1}{\\sqrt{\\lambda }}\\right).$ Analogously one can investigate the behavior of $W(\\phi _\\mp (\\lambda ),\\overline{\\phi _\\pm (\\lambda )}$ to obtain $R_\\pm (\\lambda )=O\\Big (\\frac{1}{\\sqrt{\\lambda }}\\Big )$ .",
"Theorem 3.4 The functions $T_\\pm (\\lambda )$ can be extended analytically to the domain $(\\sigma \\cup M_\\pm \\cup \\breve{M}_\\pm )$ and satisfy $\\frac{-1}{T_+(z)g_+(z)}=\\frac{-1}{T_-(z)g_-(z)}=:W(z),$ where $W(z)$ possesses the following properties: The function $\\tilde{W}$ is holomorphic in the domain $U_j^\\pm \\cap (\\sigma )$ , with simple zeros at the points $\\lambda _k$ , where $\\left( \\frac{d\\tilde{W}}{dz}(\\lambda _k)\\right)^2=(\\gamma _{n,k}^+ \\gamma _{n,k}^-)^{-2}.$ Besides it satisfies $\\overline{\\tilde{W}(\\lambda ^u)}=\\tilde{W}(\\lambda ^l), \\quad \\lambda \\in U_j^\\pm \\cap \\sigma \\quad \\text{and} \\quad \\tilde{W}(\\lambda )\\in \\mathbb {R}, \\quad \\lambda \\in U_j^\\pm \\cap (\\mathbb {R}\\backslash \\sigma ).$ The function $\\hat{W}(z)$ is continuous on the set $U_j^\\pm \\cap \\sigma $ up to the boundary $\\sigma ^l\\cup \\sigma ^u$ .",
"It can have zeros on the set $\\partial \\sigma \\cup (\\partial \\sigma _+^{(1)} \\cap \\partial \\sigma _-^{(1)}) $ and does not vanish at any other points of $\\sigma $ .",
"If $\\hat{W}(E)=0$ as $E\\in \\partial \\sigma \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)})$ , then $\\hat{W}(z)=\\sqrt{z-E}(C(E)+o(1))$ , $C(E)\\ne 0$ .",
"Except for (REF ) everything follows from the corresponding properties of $\\phi _\\pm (z,x)$ .",
"Therefore assume $\\hat{W}(\\lambda _0)=0$ for some $\\lambda _0\\in \\sigma $ , then $\\tilde{\\phi }_\\pm (\\lambda _0,x)=c_\\pm \\tilde{\\phi }_\\mp (\\lambda _0,x),$ for some constants $c_\\pm $ , which satisfy $c_-c_+=1$ .",
"Moreover, every zero of $\\tilde{W}$ (or $\\hat{W}$ ) outside the continuous spectrum, is a point of the discrete spectrum of $L$ and vice versa.",
"Denote by $\\gamma _\\pm $ the corresponding norming constants defined in (REF ) for some fixed point $\\lambda _0$ of the discrete spectrum.",
"Proceeding as in [21] one obtains $W\\big (\\tilde{\\phi }_\\pm (\\lambda _0,0), \\frac{d}{d\\lambda }\\tilde{\\phi }_\\pm (\\lambda _0,0)\\big )=\\int _0^{\\pm \\infty } \\tilde{\\phi }_\\pm ^2(\\lambda _0,x)dx.$ Thus using (REF ) and (REF ) yields $\\gamma _\\pm ^{-2}& =\\mp c_\\pm ^2\\int _0^{\\mp \\infty }\\tilde{\\phi }_\\mp ^2(\\lambda _0,x)dx \\pm \\int _0^{\\pm \\infty }\\tilde{\\phi }_\\pm ^2(\\lambda _0,x)dx\\\\ \\nonumber & =\\mp c_\\pm ^2 W\\big ( \\tilde{\\phi }_\\mp (\\lambda _0,0),\\frac{d}{d\\lambda }\\tilde{\\phi }_\\mp (\\lambda _0,0)\\big )\\pm W\\big (\\tilde{\\phi }_\\pm (\\lambda _0,0), \\frac{d}{d\\lambda }\\tilde{\\phi }_\\pm (\\lambda _0,0)\\big ) \\\\ \\nonumber & =c_\\pm \\frac{d}{d\\lambda }W(\\tilde{\\phi }_-(\\lambda _0),\\tilde{\\phi }_+(\\lambda _0)).$ Applying now $c_-c_+=1$ , we obtain (REF ).",
"The continuity of $\\hat{W}(z)$ up to the boundary follows immediately from the corresponding properties of $\\hat{\\phi }_\\pm (z,x)$ .",
"Now we will investigate the possible zeros of $\\hat{W}(\\lambda )$ for $\\lambda \\in \\sigma $ .",
"Assume $W(\\lambda _0)=0$ for some $\\lambda _0\\in \\mathop {\\rm int}(\\sigma ^{(2)})$ .",
"Then $\\phi _+(\\lambda _0,x)=c\\phi _-(\\lambda _0,x)$ and $\\overline{\\phi _+(\\lambda _0,x)}=\\overline{c}\\overline{\\phi _-(\\lambda _0,x)}$ .",
"Thus $W(\\phi _+,\\overline{\\phi _+})=\\vert c\\vert ^2 W(\\phi _-, \\overline{\\phi _-})$ and therefore $\\mathop {\\rm sign}g_+(\\lambda _0)=-\\mathop {\\rm sign}g_-(\\lambda _0)$ by (REF ), contradicting (REF ).",
"Next let $\\lambda _0\\in \\mathop {\\rm int}(\\sigma _\\pm ^{(1)})$ and $\\tilde{W}(\\lambda _0)=0$ , then $\\phi _\\pm (\\lambda _0,x)$ and $\\overline{\\phi _\\pm (\\lambda _0,x)}$ are linearly independent and bounded, moreover $\\tilde{\\phi }_\\mp (\\lambda _0,x)\\in \\mathbb {R}$ .",
"Therefore $\\tilde{W}(\\lambda _0)=0$ implies that $\\tilde{\\phi }_\\mp =c_1^\\pm \\phi _\\pm =c_2^\\pm \\overline{\\phi _\\pm }$ and thus $W(\\phi _\\pm ,\\overline{\\phi _\\pm })=0$ , which is impossible by (REF ).",
"Note that in this case $\\lambda _0$ can coincide with a pole $\\mu \\in M_\\mp $ .",
"Since $\\hat{W}(\\lambda )\\ne 0$ for $\\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)})\\cup \\mathop {\\rm int}(\\sigma ^{(1)}_+)\\cup \\mathop {\\rm int}(\\sigma ^{(1)}_-)$ , it is left to investigate the behavior at the band edges of $\\sigma _+$ and $\\sigma _-$ .",
"Therefore introduce the local parameter $\\tau =\\sqrt{z-E}$ in a small neighborhood of each point $E\\in \\partial \\sigma _\\pm $ and define $\\dot{y}(z,x)=\\frac{d}{d\\tau }y(z,x)$ .",
"A simple calculation shows that $\\frac{dz}{d\\tau }(E)=0$ , hence for every solution $y(z,x)$ of (REF ), its derivative $\\dot{y}(E,x)$ is again a solution of (REF ).",
"Therefore, the Wronskian $W(y(E),\\dot{y}(E))$ is independent of $x$ .",
"For each $x\\in \\mathbb {R}$ in a small neighborhood of a fixed point $E\\in \\partial \\sigma _\\pm $ we introduce the function $\\hat{\\psi }_{\\pm ,E}(z,x)={\\left\\lbrace \\begin{array}{ll}\\psi _\\pm (z,x), & E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm , \\\\ \\nonumber \\tau \\psi _\\pm (z,x), & E\\in \\hat{M}_\\pm .\\end{array}\\right.",
"}$ Proceeding as in [1] one obtains $W\\Big (\\hat{\\psi }_{\\pm ,E}(E), \\frac{d}{d\\tau }\\hat{\\psi }_{\\pm ,E}(E)\\Big )=\\pm \\lim _{z\\rightarrow E}\\frac{\\alpha \\tau ^\\alpha }{2g_\\pm (z)},$ where $\\alpha =-1$ if $E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm $ and $\\alpha =1$ if $E\\in \\hat{M}_\\pm $ .",
"Using representation () for $\\psi _\\pm (z,x)$ one can show (cf [12]), $\\nonumber \\psi _\\pm (E,x)=\\left(\\frac{G_\\pm (E,x)}{G_\\pm (E,0)}\\right)^{1/2}\\exp \\left(\\pm \\lim _{z\\rightarrow E}\\int _0^x\\frac{Y_\\pm (z)^{1/2}}{G_\\pm (z,\\tau )}d\\tau \\right), \\quad E\\in \\partial \\sigma $ where $\\nonumber \\exp \\left(\\pm \\lim _{z\\rightarrow E}\\int _0^x\\frac{Y_\\pm (z)^{1/2}}{G_\\pm (z,\\tau )}d\\tau \\right)={\\left\\lbrace \\begin{array}{ll}I^{2s+1}, & \\mu _j\\ne E, \\mu _j(x)=E,\\\\I^{2s+1}, & \\mu _j=E, \\mu _j(x)\\ne E,\\\\I^{2s}, & \\mu _j=E, \\mu _j(x)=E,\\\\I^{2s}, & \\mu _j\\ne E, \\mu _j(x)\\ne E,\\end{array}\\right.",
"}$ for $s\\in \\lbrace 0,1\\rbrace $ .",
"Defining $\\nonumber \\hat{\\phi }_{\\pm ,E}(\\lambda ,x)={\\left\\lbrace \\begin{array}{ll}\\phi _\\pm (\\lambda ,x), & E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm ,\\\\\\tau \\phi _\\pm (\\lambda ,x), & E\\in \\hat{M}_\\pm ,\\end{array}\\right.",
"}$ we can conclude using (REF ) that $\\overline{\\phi _\\pm (E,x)}=\\phi _\\pm (E,x), \\quad \\text{ for } E\\in \\partial \\sigma _\\pm \\backslash \\hat{M}_\\pm .$ Moreover, for $E\\in \\hat{M}_\\pm $ , ${\\left\\lbrace \\begin{array}{ll}\\overline{\\hat{\\phi }_{\\pm ,E}(E,x)}=-\\hat{\\phi }_{\\pm ,E}(E,x), & \\text{ a left band edge from } \\sigma _\\pm ,\\\\\\overline{\\hat{\\phi }_{\\pm ,E}(E,x)}=\\hat{\\phi }_{\\pm ,E}(E,x), & \\text{ a right band edge from } \\sigma _\\pm .\\end{array}\\right.",
"}$ If $\\lambda _0=E\\in \\partial \\sigma ^{(2)}\\cap \\mathop {\\rm int}(\\sigma _\\pm )\\subset \\mathop {\\rm int}(\\sigma _\\pm )$ , then $\\hat{W}(E)=0$ if and only if $W(\\psi _\\pm ,\\hat{\\psi }_{\\mp ,E})(E)=0$ .",
"Therefore, as $\\hat{\\phi }_{\\mp ,E}(E,.",
")$ are either pure real or pure imaginary, $W(\\overline{\\phi _\\pm },\\hat{\\phi }_{\\mp ,E})(E)=0$ , which implies that $\\overline{\\phi _\\pm }(E,x)$ and $\\phi _\\pm (E,x)$ are linearly dependent, a contradiction.",
"Thus the function $\\hat{W}(z)$ can only be zero at points $E$ of the set $\\partial \\sigma \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)})$ .",
"We will now compute the order of the zero.",
"First of all note that the function $\\hat{W}(\\lambda )$ is continuously differentiable with respect to the local parameter $\\tau $ .",
"Since $\\frac{d}{d\\tau }(\\delta _+\\delta _-)(E)=0$ , the function $W(\\hat{\\phi }_{+,E}, \\hat{\\phi }_{-,E})$ has the same order of zero at $E$ as $\\hat{W}(\\lambda )$ .",
"Moreover, if $\\hat{\\delta }_\\pm (E)\\ne 0$ ,then $\\frac{d}{d\\tau }\\hat{\\delta }_\\pm (E)=0$ and if $\\hat{\\delta }_-(E)=\\hat{\\delta }_+(E)=0$ , then $\\frac{d}{d\\tau }(\\tau ^{-2}\\hat{\\delta }_+\\hat{\\delta }_-)(E)=0$ .",
"Hence $\\frac{d}{d\\tau }\\hat{W}(E)=0$ if and only if $\\frac{d}{d\\tau }W(\\hat{\\phi }_{+,E}, \\hat{\\phi }_{-,E})=0$ .",
"Combining now all the informations we obtained so far, we can conclude as follows: if $\\hat{W}(E)=0$ , then $\\hat{\\phi }_{\\pm ,E}(E,.",
")=c_\\pm \\hat{\\phi }_{\\mp ,E}(E,.",
")$ , with $c_-c_+=1$ .",
"Furthermore we can write $\\nonumber \\dot{W}(\\hat{\\phi }_{+,E},\\hat{\\phi }_{-,E})(E)& =W(\\frac{d}{d\\tau }\\hat{\\phi }_{+,E},\\hat{\\phi }_{-,E})(E)-W(\\frac{d}{d\\tau }\\hat{\\phi }_{-,E}, \\hat{\\phi }_{+,E})(E)\\\\ \\nonumber & = c_-W(\\frac{d}{d\\tau }\\hat{\\phi }_{+,E},\\hat{\\phi }_{+,E})(E)-c_+W(\\frac{d}{d\\tau }\\hat{\\phi }_{-,E}, \\hat{\\phi }_{-,E})(E)\\\\ \\nonumber & = c_-W(\\frac{d}{d\\tau }\\hat{\\psi }_{+,E},\\hat{\\psi }_{+,E})(E)-c_+W(\\frac{d}{d\\tau }\\hat{\\psi }_{-,E}, \\hat{\\psi }_{-,E})(E).$ Using (REF ), (REF ), (REF ), and distinguishing several cases as in [1] finishes the proof.",
"Theorem 3.5 The reflection coefficient $R_\\pm (\\lambda )$ satisfies: The reflection coefficient $R_\\pm (\\lambda )$ is a continuous function on the set $\\mathop {\\rm int}(\\sigma _\\pm ^{u,l})$ .",
"If $E\\in \\partial \\sigma _+\\cap \\partial \\sigma _-$ and $\\hat{W}(E)\\ne 0$ , then the function $R_\\pm (\\lambda )$ is also continuous at $E$ .",
"Moreover, $\\nonumber R_\\pm (E)={\\left\\lbrace \\begin{array}{ll}-1 & \\text{ for } E\\notin \\hat{M}_\\pm ,\\\\1 & \\text{ for } E\\in \\hat{M}_\\pm .\\end{array}\\right.",
"}$ At first it should be noted that by Lemma REF the reflection coefficient is bounded, as $\\frac{g_\\pm (\\lambda )}{g_\\mp (\\lambda )}>0$ for $\\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)})$ .",
"Thus, using the corresponding properties of $\\phi _\\pm (z,x)$ , finishes the first part.",
"We proceed as in the proof of [1].",
"By (REF ) the reflection coefficient can be represented in the following form: $R_\\pm (\\lambda )=-\\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\mp (\\lambda ))}{W(\\phi _\\pm (\\lambda ),\\phi _\\mp (\\lambda ))}=\\pm \\frac{W(\\overline{\\phi _\\pm (\\lambda )},\\phi _\\mp (\\lambda ))}{W(\\lambda )},$ and is therefore continuous on both sides of the set $\\mathop {\\rm int}(\\sigma _\\pm )\\backslash (M_\\mp \\cup \\hat{M}_\\mp )$ .",
"Moreover, $\\nonumber \\vert R_\\pm (\\lambda )\\vert =\\left|\\frac{W(\\overline{\\hat{\\phi }_\\pm (\\lambda )},\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )}\\right|,$ where the denominator does not vanish, by assumption and hence $R_\\pm (\\lambda )$ is continuous on both sides of the spectrum in a small neighborhood of the band edges under consideration.",
"Next, let $E\\in \\lbrace E_{2j-1}^\\pm , E_{2j}^\\pm \\rbrace $ with $\\hat{W}(E)\\ne 0$ .",
"Then, if $E\\notin \\hat{M}_\\pm $ , we can write $\\nonumber R_\\pm (\\lambda )=-1\\mp \\frac{\\hat{\\delta }_{j,\\pm }(\\lambda )W(\\phi _\\pm (\\lambda )-\\overline{\\phi _\\pm (\\lambda )},\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )},$ which implies $R_\\pm (\\lambda )\\rightarrow -1$ , since $\\phi _\\pm (\\lambda )-\\overline{\\phi _\\pm (\\lambda )}\\rightarrow 0$ by Lemma REF as $\\lambda \\rightarrow E$ .",
"Thus we proved the first case.",
"If $E\\in \\hat{M}_\\pm $ with $\\hat{W}(E)\\ne 0$ , we use (REF ) in the form $\\nonumber R_\\pm (\\lambda )=1\\pm \\frac{\\hat{\\delta }_{j,\\pm }(\\lambda )W(\\phi _\\pm (\\lambda )+\\overline{\\phi }_\\pm (\\lambda ),\\hat{\\phi }_\\mp (\\lambda ))}{\\hat{W}(\\lambda )},$ which yields $R_\\pm (\\lambda )\\rightarrow 1$ , since $\\hat{\\delta }_{j,\\pm }(\\lambda )\\rightarrow 0$ and $\\phi _\\pm (\\lambda )+\\overline{\\phi _\\pm }(\\lambda )=O(1)$ by Lemma REF as $\\lambda \\rightarrow E$ .",
"This settles the second case."
],
[
"The Gel'fand-Levitan-Marchenko Equation",
"The aim of this section is to derive the Gel'fand-Levitan-Marchenko (GLM) equation, which is also called the inverse scattering problem equation and to obtain some additional properties of the scattering data, as a consequence of the GLM equation.",
"Therefore consider the function $G_\\pm (z,x,y) &= T_\\pm (z)\\phi _\\mp (z,x)\\psi _\\pm (z,y)g_\\pm (z)-\\breve{\\psi }_\\pm (z,x)\\psi _\\pm (z,y)g_\\pm (z)\\\\ \\nonumber & := G^\\prime _\\pm (z,x,y)+G^{\\prime \\prime }_\\pm (z,x,y), \\quad \\pm y >\\pm x,$ where $x$ and $y$ are considered as fixed parameters.",
"As a function of $z$ it is meromorphic in the domain $\\sigma $ with simple poles at the points $\\lambda _k$ of the discrete spectrum.",
"It is continuous up to the boundary $\\sigma ^u\\cup \\sigma ^l$ , except for the points of the set, which consists of the band edges of the background spectra $\\partial \\sigma _+$ and $\\partial \\sigma _-$ , where $G_\\pm (z,x,y)=O((z-E)^{-1/2}) \\quad \\text{ as } \\quad E\\in \\partial \\sigma _+\\cup \\partial \\sigma _-.$ Outside a small neighborhood of the gaps of $\\sigma _+$ and $\\sigma _-$ , the following asymptotics as $z\\rightarrow \\infty $ are valid: $\\nonumber \\phi _\\mp (z,x)&=\\mathrm {e}^{\\mp I\\sqrt{z}x(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1/2})\\right), \\quad g_\\pm (z)=\\frac{-1}{2I\\sqrt{z}}+O(z^{-1}),\\\\ \\nonumber \\breve{\\psi }_\\pm (z,x)& =\\mathrm {e}^{\\mp I\\sqrt{z}x(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1})\\right), \\quad T_\\pm (z)=1+O(z^{-1/2}),\\\\ \\nonumber \\psi _\\pm (z,y) & = \\mathrm {e}^{\\pm I\\sqrt{z}y(1+O(\\frac{1}{z}))}\\left(1+O(z^{-1})\\right),$ and the leading term of $\\phi _\\mp (z,x)$ and $\\breve{\\psi }_\\pm (z,x)$ are equal, thus $G_\\pm (z,x,y)=\\mathrm {e}^{\\pm I\\sqrt{z}(y-x)(1+O(\\frac{1}{z}))}O(z^{-1}), \\quad \\pm y > \\pm x.$ Figure: Contours Γ ε,n \\Gamma _{\\varepsilon ,n}Consider the following sequence of contours $\\Gamma _{\\varepsilon ,n,\\pm }$ , where $\\Gamma _{\\varepsilon ,n,\\pm }$ consists of two parts for every $n\\in \\mathbb {N}$ and $\\varepsilon \\ge 0$ : $C_{\\varepsilon ,n,\\pm }$ consists of a part of a circle which is centered at the origin and has as radii the distance from the origin to the midpoint of the largest band of $[E_{2n}^\\pm ,E_{2n+1}^\\pm ]$ , which lies inside $\\sigma ^{(2)}$ , together with a part wrapping around the corresponding band of $\\sigma $ at a small distance, which is at most $\\varepsilon $ , as indicated by figure 1.",
"Each band of the spectrum $\\sigma $ , which is fully contained in $C_{\\varepsilon ,n,\\pm }$ , is surrounded by a small loop at a small distance from $\\sigma $ not bigger than $\\varepsilon $ .",
"W.l.o.g.",
"we can assume that all the contours are non-intersecting.",
"Using the Cauchy theorem, we obtain $\\nonumber \\frac{1}{2\\pi I}\\oint _{\\Gamma _{\\varepsilon ,n,\\pm }} G_\\pm (z,x,y)dz=\\sum _{\\lambda _k \\in \\mathop {\\rm int}(\\Gamma _{\\varepsilon ,n,\\pm })}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm (z,x,y), \\quad \\varepsilon >0.$ By (REF ) the limit value of $G_\\pm (z,x,y)$ as $\\varepsilon \\rightarrow 0$ is integrable on $\\sigma $ , and the function $G_\\pm ^{\\prime \\prime }(z,x,y)$ has no poles at the points of the discrete spectrum, thus we arrive at $\\frac{1}{2\\pi I}\\oint _{\\Gamma _{0,n,\\pm }}G_\\pm (z,x,y)dz=\\sum _{\\lambda _k \\in \\mathop {\\rm int}(\\Gamma _{0,n,\\pm })}\\mathop {\\rm Res}_{\\lambda _k}G^\\prime _\\pm (z,x,y),\\quad \\pm y > \\pm x.$ Estimate (REF ) allows us now to apply Jordan's lemma, when letting $n\\rightarrow \\infty $ , and we therefore arrive, up to that point only formally, at $\\frac{1}{2\\pi I}\\oint _{\\sigma }G_\\pm (\\lambda ,x,y)d\\lambda =\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y), \\quad \\pm y>\\pm x.$ Next, note that the function $G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)$ does not contribute to the left part of (REF ), since $G_\\pm ^{\\prime \\prime }(\\lambda ^u,x,y)=G_\\pm ^{\\prime \\prime }(\\lambda ^l,x,y)$ for $\\lambda \\in \\sigma _\\mp ^{(1)}$ and, hence $\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)d\\lambda =0$ .",
"In addition, $\\oint _{\\sigma _\\pm }G_\\pm ^{\\prime \\prime }(\\lambda ,x,y)d\\lambda =0$ for $x\\ne y$ by Lemma REF (iv).",
"Therefore we arrive at the following equation, $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\pm }G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y), \\quad \\pm y>\\pm x.$ To make our argument rigorous we have to show that the series of contour integrals along the parts of the spectrum contained in $C_{0,n,\\pm }$ on the left hand side of (REF ) converges as $n\\rightarrow \\infty $ and that the contribution of the integrals along the circles $C_{0,n,\\pm }$ converges against zero as $n\\rightarrow \\infty $ , by applying Jordan's lemma.",
"This will be done next.",
"Using (REF ), (REF ), (REF ), and (REF ), we obtain $\\nonumber \\frac{1}{2\\pi I} & \\oint _{\\sigma _\\pm } G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\oint _{\\sigma _\\pm }T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & = \\oint _{\\sigma _\\pm } \\Big (R_\\pm (\\lambda )\\phi _\\pm (\\lambda ,x)+\\overline{\\phi _\\pm (\\lambda ,x)}\\Big )\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & =\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )+\\oint _{\\sigma _\\pm } \\breve{\\psi }_\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ) \\\\ \\nonumber & \\quad \\pm \\int _x^{\\pm \\infty } dt K_\\pm (x,t)\\Big (\\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,t)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )+\\delta (t-y)\\Big )\\\\ \\nonumber & = F_{r,\\pm }(x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{r,\\pm }(t,y)dt+K_\\pm (x,y),$ where $F_{r,\\pm }(x,y)=\\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ).$ Now properties (ii) and (iii) from Lemma REF imply that $\\vert R_\\pm (\\lambda )\\vert <1 \\quad \\text{for} \\quad \\lambda \\in \\mathop {\\rm int}(\\sigma ^{(2)}), \\quad \\vert R_\\pm (\\lambda )\\vert =1 \\quad \\text{for} \\quad \\lambda \\in \\sigma _\\pm ^{(1)}.$ and by () we can write $\\nonumber F_{r,\\pm } (x,y) & =\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda )\\\\ \\nonumber & = -\\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\frac{(G_\\pm (\\lambda ,x)G_\\pm (\\lambda ,y))^{1/2}}{2Y_\\pm (\\lambda )^{1/2}}\\exp (\\eta _\\pm (\\lambda ,x)+\\eta _\\pm (\\lambda ,y))d\\lambda ,$ with $\\nonumber \\eta _\\pm (\\lambda ,x):=\\pm \\int _0^x \\frac{Y_\\pm (\\lambda )^{1/2}}{G_\\pm (\\lambda ,\\tau )}d\\tau \\in I\\mathbb {R}.$ We are now ready to prove the following lemma.",
"Lemma 4.1 The sequence of functions $F_{r,n,\\pm }(x,y)=\\oint _{\\sigma _\\pm \\cap \\Gamma _{0,n,\\pm }} R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y) d\\rho _\\pm (\\lambda ),$ is uniformly bounded with respect to $x$ and $y$ , that means for all $n\\in \\mathbb {N}$ , $\\vert F_{r,n,\\pm }(x,y)\\vert \\le C$ .",
"Moreover, $F_{r,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{r,\\pm }(x,y) = \\oint _{\\sigma _\\pm } R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ),$ which is again uniformly bounded with respect to $x$ and $y$ .",
"In particular, $F_{r,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"For $\\lambda \\in \\sigma _\\pm $ as $\\lambda \\rightarrow \\infty $ we have the following asymptotic behavior in a small neighborhood $V_{n}^\\pm $ of $E=E_{n}^\\pm $ $\\nonumber \\vert R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )\\vert =O\\Big (\\frac{\\sqrt{E_{2j}^\\pm -E_{2j-1}^\\pm }}{\\sqrt{\\lambda (\\lambda -E)}}\\Big ),$ in a small neighborhood $W_n^\\pm $ of $E=E_n^\\mp $ , if $E\\in \\sigma _\\pm $ $\\nonumber R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=O(\\frac{1}{\\sqrt{\\lambda }}),$ and for $\\lambda \\in \\sigma _\\pm \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_{n}^\\pm \\cup W_{n}^\\pm )$ $R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=\\exp (\\pm I\\sqrt{\\lambda }(\\vert x\\vert +\\vert y\\vert )(1+O(\\frac{1}{\\lambda })))\\Big (\\frac{C}{\\lambda }+O\\Big (\\frac{1}{\\lambda ^{3/2}}\\Big )\\Big ).$ These estimates are good enough to show that $F_{r,\\pm }(x,y)$ exists, if we choose $V_n^\\pm $ and $W_n^\\pm $ in the following way: We choose $V_n^\\pm \\subset \\sigma _\\pm ^{(1)}\\cup \\sigma ^{(2)}$ , if $E_n^\\pm $ is a band edge of $\\sigma _\\pm ^{(1)}$ , such that $V_n^\\pm $ consists of the corresponding band of $\\sigma _\\pm ^{(1)}$ together with the following part of $\\sigma _\\pm ^{(2)}$ with length $E_n^\\pm -E_{n-1}^\\pm $ , if $n$ is even and $E_{n+1}^\\pm -E_n^\\pm $ , if $n$ is odd.",
"If $E_n^+$ is a band edge of $\\sigma ^{(2)}$ , we choose $V_n^\\pm \\subset \\sigma ^{(2)}$ , where the length of $V_n^\\pm $ is equal to the length of the gap pf $\\sigma _\\pm $ next to it.",
"We set $W_n^\\pm \\subset \\sigma ^{(2)}$ with length $3(E_n^\\mp -E_{n-1}^\\mp )$ , if $n$ is even and $3(E_{n+1}^\\mp -E_n^\\mp )$ , if $n$ is odd, centered at the midpoint of the corresponding gap in $\\sigma _\\mp $ .",
"As we are working in the Levitan class and we therefore know that $\\sum _{n=1}^\\infty (E_{2n-1}^\\pm )^{l^\\pm }(E_{2n}^\\pm -E_{2n-1}^\\pm )<\\infty $ for some $l^\\pm >1$ , we obtain that the sequences belonging to $V_n^\\pm $ and $W_n^\\pm $ converge.",
"As far as the behavior along the spectrum away from the band edges of $\\sigma _+$ and $\\sigma _-$ is concerned observe that $\\vert \\exp (\\pm I\\sqrt{\\lambda }(x+y)O(\\frac{1}{\\lambda }))\\vert \\le 1+ (x+y)O(\\frac{1}{\\sqrt{\\lambda }}),\\quad \\lambda \\rightarrow \\infty .$ and therefore $R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )=\\exp (\\pm I\\sqrt{\\lambda }(x+y))\\Big (\\frac{C}{\\lambda }+(1+x+y)O\\Big (\\frac{1}{\\lambda ^{3/2}}\\Big )\\Big ).$ To show the convergence of the series $F_{r,\\pm }(x,y)$ for fixed $x$ and $y$ , we split the integral along the spectrum $\\sigma $ up into three integrals along $\\bigcup _{n\\in \\mathbb {N}} V_n^\\pm $ , $\\bigcup _{n\\in \\mathbb {N}} W_n^\\pm $ , and $\\sigma \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_n^\\pm \\cup W_n^\\pm )$ respectively.",
"As far as the integral along $\\bigcup _{n\\in \\mathbb {N}} V_n^pm$ is concerned observe that the integrand has a square root singularity at the boundary and is therefore integrable along $V_n^\\pm $ for all $n\\in \\mathbb {N}$ .",
"Since we are working within the Levitan class the sum over all $n\\in \\mathbb {N}$ converges.",
"The integrand can be uniformly bounded for all $\\lambda \\in W_n^\\pm $ such that $\\lambda \\ge 1$ .",
"Since there are only finitely many $n\\in \\mathbb {N}$ such that $W_n^\\pm \\subset [0,1]$ , the corresponding series converges by the definition of the Levitan class.",
"Thus it is left to consider the integral along $\\sigma \\backslash \\bigcup _{n\\in \\mathbb {N}}(V_n^\\pm \\cup W_n^\\pm )$ :=I.",
"Direct computation yields $\\nonumber \\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{C}{\\lambda }d\\lambda & =\\pm \\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{2C}{I\\lambda ^{1/2}(x+y)}\\vert _a^b\\\\ \\nonumber & \\pm \\int _a^b\\exp (\\pm I\\sqrt{\\lambda }(x+y))\\frac{C}{I\\lambda ^{3/2}(x+y)}d\\lambda ,$ which is finite since by assumption $\\pm x\\le \\pm y$ .",
"Hence one possibility to see that the corresponding series of integrals converges is to integrate first the function describing the asymptotic behavior along $[E_0^\\pm ,\\infty ]$ and substract from it the series of integrals corresponding to the $[E_0^\\pm ,\\infty ]\\cap I^c$ .",
"Since every interval belonging to the complement belongs to a neighborhood of the gaps of $\\sigma ^{(2)}$ and the integrand can be uniformly bounded, the definition of the Levitan class implies that this series converges.",
"Similarly we conclude $\\nonumber \\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))(x+y)O(\\frac{1}{\\lambda ^{3/2}})d\\lambda & =\\exp (\\pm I\\sqrt{\\lambda }(x+y))O(\\frac{1}{\\lambda })\\vert _a^b\\\\ \\nonumber & +\\int _a^b \\exp (\\pm I\\sqrt{\\lambda }(x+y))O(\\frac{1}{\\lambda ^{2}})d\\lambda $ Note that since we are working within the Levitan class all estimates are independent of $x$ and $y$ .",
"For investigating the other terms, we will need the following lemma, which is taken from [9]: Lemma 4.2 Suppose in an integral equation of the form $f_\\pm (x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)f_\\pm (t,y)dt=g_\\pm (x,y), \\quad \\pm y>\\pm x,$ the kernel $K_\\pm (x,y)$ and the function $g_\\pm (x,y)$ are continuous for $\\pm y>\\pm x$ , $\\nonumber \\vert K_\\pm (x,y) \\vert \\le C_\\pm (x) Q_\\pm (x+y),$ and for $g_\\pm (x,y)$ one of the following estimates hold $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)Q_\\pm (x+y), \\quad \\text{ or }$ $\\vert g_\\pm (x,y) \\vert \\le C_\\pm (x)(1+\\max (0,\\pm x)).$ Furthermore assume that $\\pm \\int _0^{\\pm \\infty } (1+\\vert x \\vert ^2)\\vert q(x)-p_\\pm (x)\\vert dx<\\infty .$ Then (REF ) is uniquely solvable for $f_\\pm (x,y)$ .",
"The solution $f_\\pm (x,y)$ is also continuous in the half-plane $\\pm y >\\pm x$ , and for it the estimate (REF ) respectively (REF ) is reproduced.",
"Moreover, if a sequence $g_{n,\\pm }(x,y)$ satisfies (REF ) or (REF ) uniformly with respect to $n$ and pointwise $g_{n,\\pm }(x,y)\\rightarrow 0$ , for $\\pm y>\\pm x$ , then the same is true for the corresponding sequence of solutions $f_{n,\\pm }(x,y)$ of (REF ).",
"For a proof we refer to [9].",
"Remark 4.3 An immediate consequence of this lemma is the following.",
"If $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ denotes a bounded function, then $\\vert g_\\pm (x,y)\\vert \\le C_\\pm (x)(1+\\max (0,\\pm x))$ and therefore $\\vert f_\\pm (x,y)\\vert \\le C_\\pm (x)(1+\\max (0,\\pm x))$ .",
"Rewriting this integral equation as follows $f_\\pm (x,y)=g_\\pm (x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)f_\\pm (t,y)dy,$ we obtain that the absolute value of the right hand side is smaller than a bounded function $\\tilde{C}_\\pm (x)$ by using (REF ) and (REF ), and hence the same is true for the left hand side.",
"In particular if $C_\\pm (x)$ is a decreasing function the same will be true for $\\tilde{C}_\\pm (x)$ .",
"We will now continue the investigation of our integral equation.",
"Lemma 4.4 The sequence of functions $F_{h,n,\\pm }(x,y)=\\int _{\\sigma _\\pm ^{(1),u}\\cap \\Gamma _{0,n,\\pm }} \\vert T_\\mp (\\lambda )\\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda )$ is uniformly bounded, that means for all $n\\in \\mathbb {N}$ , $\\vert F_{h,n,\\pm }(x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ are monotonically decrasing functions as $x\\rightarrow \\pm \\infty $ .",
"Moerover, $F_{h,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{h,\\pm }(x,y) =\\int _{\\sigma _\\mp ^{(1),u}} \\vert T_\\mp (\\lambda ) \\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y) d\\rho _\\mp (\\lambda ),$ which is again bounded by some monotonically increasing function.",
"In particular, $F_{h,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"On the set $\\sigma _\\mp ^{(1)}$ both the numerator and the denominator of the function $G_\\pm ^\\prime (\\lambda ,x,y)$ have poles (resp.",
"square root singularities) at the points of the set $\\sigma _\\mp ^{(1)}\\cap (M_\\pm \\cup (\\partial \\sigma _+^{(1)}\\cap \\partial \\sigma _-^{(1)}))$ (resp.",
"$\\sigma _\\mp ^{(1)}\\cap (M_\\mp \\backslash (M_\\mp \\cap M_\\pm ))$ , but multiplying them, if necessary away, we can avoid singularities.",
"Hence, w.l.o.g., we can suppose $\\sigma _\\mp ^{(1)}\\cap (M_{r,+}\\cup M_{r,-})=\\emptyset $ .",
"Thus we can write $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda .$ For investigating this integral we will consider, using (REF ), $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\mp (\\lambda ) &\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\mp (\\lambda )d\\lambda \\\\ \\nonumber & = \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}\\phi _\\mp (\\lambda ,x)\\Big (\\overline{\\phi _\\mp (\\lambda ,y)}+R_\\mp (\\lambda )\\phi _\\mp (\\lambda ,y)\\Big )g_\\mp (\\lambda )d\\lambda .$ First of all note that the integrand, because of the representation on the right hand side, can only have square root singularities at the boundary $\\partial \\sigma _\\mp ^{(1)}$ and we therefore have $\\nonumber \\int _{\\sigma _\\mp ^{(1)}\\cap [E_{2n-1}^\\pm ,E_{2n}^\\pm ]}& \\vert \\phi _\\mp (\\lambda ,x) \\Big (\\overline{\\phi _\\mp (\\lambda ,y)}+ R_\\mp (\\lambda )\\phi _\\mp (\\lambda ,y)\\Big )g_\\mp (\\lambda )\\vert d\\lambda \\\\ \\nonumber & \\le 2\\int _{\\sigma _\\mp ^{(1)}\\cap [E_{2n-1}^\\pm , E_{2n}^\\pm ]}\\vert \\phi _\\mp (\\lambda ,x)\\phi _\\mp (\\lambda ,y)g_\\mp (\\lambda )\\vert d\\lambda \\\\ \\nonumber & \\le C_\\pm (y)C_\\pm (x)\\left( \\frac{(E_{2n}^\\pm -E_{2n-1}^\\pm )}{\\sqrt{\\lambda -E_0^\\mp }}+\\frac{\\sqrt{E_{2n}^\\pm -E_{2n-1}^\\pm }}{\\sqrt{\\lambda -E_0^\\mp }}\\right),$ where $E_{2n-1}^\\pm $ and $E_{2n}^\\pm $ denote the edges of the gap of $\\sigma _\\pm $ in which the corresponding part of $\\sigma _\\mp ^{(1)}$ lies and $C_\\pm (x)$ denote monotonically decreasing functions from now on.",
"Therefore as we are working in the Levitan class and by separating $\\sigma _\\mp ^{(1)}$ into the different parts, one obtains that $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (y)C_\\pm (x).$ Thus we can now apply Lemma REF , and hence $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (y)C_\\pm (x)(1+\\max (0,\\pm y)).$ Note that we especially have, because of (REF ), $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\phi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (x)$ Therefore we can conclude that for fixed $x$ and $y$ the left hands side of (REF ) exists and satisfies $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}T_\\pm (\\lambda )\\phi _\\mp (\\lambda ,x)\\psi _\\pm (\\lambda ,y)g_\\pm (\\lambda )d\\lambda \\vert \\le C_\\pm (x),$ and hence $\\vert \\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G^{\\prime }_\\pm (z,x,y)dz\\vert \\le C_\\pm (x).$ We will now rewrite the integrand in a form suitable for our further purposes.",
"Namely, since $\\psi _\\pm (\\lambda ,x)\\in \\mathbb {R}$ as $\\lambda \\in \\sigma _\\mp ^{(1)}$ , we have $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =\\frac{1}{2\\pi I}\\int _{\\sigma _\\mp ^{(1),u}}\\psi _\\pm (\\lambda ,y)\\Big (\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}-\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}\\Big )d\\lambda $ Moreover, (REF ) and Lemma REF (ii) imply $\\nonumber \\overline{\\phi _\\mp (\\lambda ,x)}=T_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)-\\frac{T_\\mp (\\lambda )}{\\overline{T_\\mp (\\lambda )}}.$ Therefore, $\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}-\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}& =\\phi _\\mp (\\lambda ,x)\\Big (\\frac{1}{W(\\lambda )}+\\frac{T_\\mp (\\lambda )}{\\overline{T_\\mp (\\lambda )W(\\lambda )}}\\Big )-\\frac{T_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)}{\\overline{W(\\lambda )}}\\\\ \\nonumber & =\\phi _\\mp (\\lambda ,x)\\frac{2\\mathop {\\rm Re}(T_\\mp ^{-1}(\\lambda )\\overline{W(\\lambda )})T_\\mp (\\lambda )}{\\vert W(\\lambda )\\vert ^2}-\\frac{T_\\mp (\\lambda )\\phi _\\pm (\\lambda )}{\\overline{W(\\lambda )}}.$ But by (REF ) $T_\\mp ^{-1}(\\lambda )\\overline{W(\\lambda )}=\\vert W(\\lambda )\\vert ^2g_\\mp (\\lambda )\\in I\\mathbb {R}, \\quad \\text{ for }\\lambda \\in \\sigma _\\mp ^{(1)},$ and therefore the first summand of (REF ) vanishes.",
"Using now $\\overline{W}=(\\overline{T_\\mp }g_\\mp )^{-1}$ we arrive at $\\frac{\\overline{\\phi _\\mp (\\lambda ,x)}}{\\overline{W(\\lambda )}}-\\frac{\\phi _\\mp (\\lambda ,x)}{W(\\lambda )}=\\vert T_\\mp (\\lambda )\\vert ^2g_\\mp (\\lambda )\\phi _\\pm (\\lambda ,x)$ and hence $\\frac{1}{2\\pi I}\\oint _{\\sigma _\\mp ^{(1)}}G_\\pm ^\\prime (\\lambda ,x,y)d\\lambda =F_{h,\\pm }(x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{h,\\pm }(t.y)dt,$ where $F_{h,\\pm }(x,y)=\\int _{\\sigma _\\pm ^{(1),u}}\\vert T_\\mp (\\lambda )\\vert ^2\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda ),$ and $\\vert F_{h,\\pm }(x,y)\\vert \\le C_\\pm (x)C_\\pm (y)$ by Lemma REF .",
"The partial sums $F_{h,n,\\pm }(x,y)$ can be investigated similarly We will now investigate the r.h.s.",
"of (REF ) and (REF ).",
"Therefore we consider first the question of the existence of the right hand side: To prove the boundedness of the corresponding series on the left hand side, it is left to investigate the series, which correspond to the circles.",
"We will derive the necessary estimates only for the part of the n'th circle $K_{R_{n,\\pm }}$ , where $R_{n,\\pm }$ denotes the radius, in the upper half plane as the part in the lower half plane can be considered similarly.",
"We have $\\nonumber \\vert \\int _{K_{R_{n},\\pm }}G_\\pm (z,x,y)dz\\vert & \\le \\int _0^\\pi C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )\\sin (\\theta /2)}d\\theta \\\\ \\nonumber & \\le \\int _0^{\\pi /2} C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )\\sin (\\eta )} d\\eta \\\\ \\nonumber & \\le \\int _0^{\\pi /2} C \\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu ) 2\\frac{\\eta }{\\pi }}d\\eta \\\\ \\nonumber &\\le C\\frac{1}{\\sqrt{R}(x-y)(1-\\nu )}\\mathrm {e}^{\\pm \\sqrt{R}(x-y)(1-\\nu )2\\frac{\\eta }{\\pi }}\\vert _0^{\\frac{\\pi }{2}},$ where $C$ and $\\nu $ denote some constant, which are dependent on the radius (cf.",
"Lemma REF ).",
"Therefore as already mentioned the part belonging to the circles converges against zero and hence the same is true for the corresponding series, by Jordan's lemma.",
"Thus we obtain that the sequence of partial sums on the right hand side of (REF ) and (REF ) is uniformly bounded and we are therefore ready to prove the following result.",
"Lemma 4.5 The sequence of functions $F_{d,n,\\pm }(x,y)=\\sum _{\\lambda _k\\in \\sigma _d\\cap \\Gamma _{0,n,\\pm }}(\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y)$ is uniformly bounded, that means for all $n\\in \\mathbb {N}$ , $\\vert F_{d,n,\\pm }(x,y)\\vert \\le C_\\pm (x)$ , where $C_\\pm (x)$ are monotonically increasing functions.",
"Moreover, $F_{d,n,\\pm }(x,y)$ converges uniformly as $n\\rightarrow \\infty $ to the function $F_{d,\\pm }(x,y) = \\sum _{\\lambda _k\\in \\sigma _d}(\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y),$ which is again bounded by some monotonically increasing function.",
"In particular, $F_{d,\\pm }(x,y)$ is continuous with respect to $x$ and $y$ .",
"Applying (REF ), (REF ), (REF ), (REF ), and (REF )to the right hand side of (REF ), yields $\\nonumber \\sum _{\\lambda _k\\in \\sigma _d} \\mathop {\\rm Res}_{\\lambda _k}G_\\pm ^\\prime (\\lambda ,x,y)&=-\\sum _{\\lambda _k\\in \\sigma _d}\\mathop {\\rm Res}_{\\lambda _k}\\frac{\\tilde{\\phi }_\\mp (\\lambda ,x)\\tilde{\\psi }_\\pm (\\lambda ,y)}{\\tilde{W}(\\lambda )} \\\\ \\nonumber & = -\\sum _{\\lambda _k\\in \\sigma _d}\\frac{\\tilde{\\phi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y)}{\\tilde{W}^\\prime (\\lambda _k)c_{k,\\pm }}\\\\ \\nonumber & =-\\sum _{\\lambda _k\\in \\sigma _d}(\\gamma _k^\\pm )^2\\tilde{\\phi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y) \\\\ \\nonumber &= -F_{d,\\pm }(x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{d,\\pm }(t,y)dt,$ where $F_{d,\\pm }(x,y):=\\sum _{\\lambda _k\\in \\sigma _d} (\\gamma _k^\\pm )^2\\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y).$ Thus we obtained the following integral equation, $F_{d,\\pm }(x,y)& = -K_\\pm (x,y)-F_{c,\\pm }(x,y)\\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{c,\\pm }(t,y)dt\\\\ \\nonumber & \\mp \\int _x^{\\pm \\infty }K_\\pm (x,t)F_{d,\\pm }(t,y)dt,$ which we can now solve for $F_{d,\\pm }(x,y)$ using again Lemma REF and hence $F_{d,\\pm }(x,y)$ exists and satisfies the given estimates.",
"The corresponding partial sums can be investigated analogously using the considerations from above.",
"Putting everything together, we see that we have obtained the GLM equation.",
"Theorem 4.6 The GLM equation has the form $K_\\pm (x,y)+F_\\pm (x,y)\\pm \\int _x^{\\pm \\infty }K_\\pm (x,t)F_\\pm (t,y)dt=0, \\quad \\pm (y-x)>0,$ where $F_\\pm (x,y)& = \\oint _{\\sigma _\\pm }R_\\pm (\\lambda )\\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\pm (\\lambda ) \\\\ \\nonumber & +\\int _{\\sigma _\\mp ^{(1),u}} \\vert T_\\mp (\\lambda )\\vert ^2 \\psi _\\pm (\\lambda ,x)\\psi _\\pm (\\lambda ,y)d\\rho _\\mp (\\lambda ) \\\\ \\nonumber & +\\sum _{k=1}^\\infty (\\gamma ^\\pm _k)^2 \\tilde{\\psi }_\\pm (\\lambda _k,x)\\tilde{\\psi }_\\pm (\\lambda _k,y).$ Moreover, we have Lemma 4.7 The function $F_\\pm (x,y)$ is continuously differentiable with respect to both variables and there exists a real-valued function $q_\\pm (x)$ , $x\\in \\mathbb {R}$ with $\\pm \\int _a^{\\pm \\infty }(1+x^2)\\vert q_\\pm (x)\\vert dx<\\infty , \\quad \\text{ for all } a\\in \\mathbb {R},$ such that $\\vert F_\\pm (x,y)\\vert \\le \\tilde{C}_\\pm (x)Q_\\pm (x+y),$ $\\left|\\frac{d}{dx}F_\\pm (x,y)\\right|\\le \\tilde{C}_\\pm (x)\\left(\\left|q_\\pm \\left(\\frac{x+y}{2}\\right)\\right|+Q_\\pm (x+y)\\right),$ $\\pm \\int _a^{\\pm \\infty }\\left|\\frac{d}{dx}F_\\pm (x,x)\\right|(1+x^2)dx<\\infty ,$ where $Q_\\pm (x)=\\pm \\int _{\\frac{x}{2}}^{\\pm \\infty }\\vert q_\\pm (t) \\vert dt,$ and $\\tilde{C}_\\pm (x)>0$ is a continuous function, which decreases monotonically as $x\\rightarrow \\pm \\infty $ .",
"Applying once more Lemma REF , one obtains (REF ).",
"Now, for simplicity, we will restrict our considerations to the + case and omit + whenever possible.",
"Set $Q_1(u)=\\int _u^\\infty Q(t)dt$ .",
"Then, using (REF ), the functions $Q(x)$ and $Q_1(x)$ satisfy $\\int _a^\\infty Q_1(t)dt<\\infty , \\quad \\int _a^\\infty Q(t)(1+\\vert t\\vert )dt<\\infty .$ Differentiating (REF ) with respect to $x$ and $y$ yields $\\vert F_x(x,y)\\vert \\le \\vert K_x(x,y)\\vert +\\vert K(x,x)F(x,y)\\vert +\\int _x^\\infty \\vert K_x(x,t)F(t,y)\\vert dt,$ $F_y(x,y)+K_y(x,y)+\\int _x^\\infty K(x,t)F_y(t,y)dt=0.$ We already know that the functions $Q(x)$ , $Q_1(x)$ , $C(x)$ , and $\\tilde{C}(x)$ are monotonically decreasing and positive.",
"Moreover, $\\int _x^\\infty \\Big (\\left|q_+\\Big (\\frac{x+t}{2}\\Big )\\right|+Q(x+t)\\Big )Q(t+y)dt\\le (Q(2x)+Q_1(2x))Q(x+y),$ thus we can estimate $F_x(x,y)$ and $F_y(x,y)$ can be estimates using (REF ) and the method of successive approximation.",
"It is left to prove (REF ).",
"Therefore consider (REF ) for $x=y$ and differentiate it with respect to $x$ : $\\frac{d F(x,x)}{dx}+\\frac{d K(x,x)}{dx}-K(x,x)F(x,x)+\\int _x^\\infty (K_x(x,tF(t,x)+K(x,t)F_y(t,x))dt=0.$ Next (REF ) and (REF ) imply $\\vert K(x,y)F(x,x)\\vert \\le \\tilde{C}(a)C(a)Q^2(2x), \\quad \\text{ for } x>a,$ where $\\int _a^\\infty (1+x^2)Q^2(2x)dx<\\infty $ .",
"Moreover, by (REF ) and (REF ) $\\nonumber \\left|K_x^\\prime (x,t) F(t,x)\\right| + \\left|K(x,t) F_y^\\prime (t,x)\\right|\\le 4\\tilde{C}(a)\\hat{C}(a)\\Bigl \\lbrace \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert Q(x+t) +Q^2(x+t)\\Bigr \\rbrace ,$ together with the estimates $& \\int _a^\\infty d x\\,x^2\\int _x^\\infty Q^2(x+t)d t\\le \\int _a^\\infty |x| Q(2x)d x\\ \\sup _{x\\ge a}\\int _x^\\infty |x+t| Q(x+t)d t<\\infty ,\\\\& \\int _a^\\infty x^2\\int _x^\\infty \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert Q(x+t)d t\\le \\\\& \\qquad \\le \\int _a^\\infty Q(2x)d x \\ \\sup _{x\\ge a} \\int _x^\\infty \\Bigl \\vert q\\Bigl (\\frac{x+t}{2}\\Bigr )\\Bigr \\vert (1 +(x+t)^2)d t<\\infty ,$ and (REF ), we arrive at (REF ).",
"In summary, we have obtained the following necessary conditions for the scattering data: Theorem 4.8 The scattering data $\\nonumber {\\mathcal {S}} = \\Big \\lbrace & R_+(\\lambda ),\\,T_+(\\lambda ),\\, \\lambda \\in \\sigma _+^{\\mathrm {u,l}}; \\,R_-(\\lambda ),\\,T_-(\\lambda ),\\, \\lambda \\in \\sigma _-^{\\mathrm {u,l}};\\\\& \\lambda _1,,\\lambda _2,\\dots \\in \\mathbb {R}\\setminus (\\sigma _+\\cup \\sigma _-),\\,\\gamma _1^\\pm , \\gamma _2^\\pm ,\\dots \\in \\mathbb {R}_+\\Big \\rbrace $ possess the properties listed in Theorem REF , REF , REF , and REF , and Lemma REF , REF , and REF .",
"The functions $F_\\pm (x,y)$ defined in (REF ), possess the properties listed in Lemma REF ."
],
[
"Acknowledgements",
"I want to thank Ira Egorova and Gerald Teschl for many discussions on this topic."
]
] | 1204.0890 |
[
[
"Degreewise n-projective and n-flat model structures on chain complexes"
],
[
"Abstract In the paper \"Cotorsion Pairs in C(R-Mod)\", the authors construct an abelian model structure on the category of chain complexes Ch(R), where the class of cofibrant objects is given by the class of degreewise projective chain complexes.",
"Using a generalization of a well known theorem by I. Kaplansky, we generalize the method used in that paper in order to obtain, for each integer n>1, a new abelian model structure on Ch(R), where the class of cofibrant objects is the class of chain complexes whose terms have projective dimension smaller or equal than n (dw-n-projective complexes), provided the ring R is noetherian.",
"We also present another method, based on the paper \"Covers and Envelopes in Grothendieck Categories: Flat Covers of Complexes with Applications\", to construct this model structure.",
"This method also works to construct an abelian model structure whose cofibrant objects are the dw-n-flat complexes."
],
[
"Introduction",
"A cotorsion pair in an abelian category $\\mathcal {C}$ is a pair $(\\mathcal {A, B})$ , where $\\mathcal {A}$ and $\\mathcal {B}$ are classes of objects of $\\mathcal {C}$ such that they are orthogonal to each other with respect to the ${\\rm Ext}$ functor.",
"A model category is a bicomplete category with three classes of morphisms, called cofibrations, fibrations and weak equivalences, satisfying certain conditions.",
"It turns out to be that these two notions have a deep connection.",
"As far as the author knows, the first person who described this connection was M. Hovey in the paper Cotorsion pairs, model category structures and representation theory, where he proved that any two compatible and complete cotorsion pairs $(\\mathcal {A}, \\mathcal {B} \\cap \\mathcal {E})$ and $(\\mathcal {A} \\cap \\mathcal {E}, \\mathcal {B})$ , in a bicomplete abelian category $\\mathcal {C}$ , give rise to a unique abelian model structure on $\\mathcal {C}$ where $\\mathcal {A}$ is the class of cofibrant objects, $\\mathcal {B}$ is the class of fibrant objects, and $\\mathcal {E}$ is the class of trivial objects.",
"From this point there has been an increasing interest in constructing new model structures, specially on $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ .",
"One of the most influential researchers in this matter has been J. Gillespie, who has provided several results that allows us to induce cotorsion pairs in the category ${\\rm Ch}(\\mathcal {C})$ of chain complexes over an abelian category $\\mathcal {C}$ , from a certain cotorsion pair in $\\mathcal {C}$ .",
"One of those results states that given a cotorsion pair $(\\mathcal {A, B})$ in an abelian category $\\mathcal {C}$ with enough projective and injective objects, there exist two cotorsion pairs in ${\\rm Ch}(\\mathcal {C})$ given by $({\\rm dw}\\widetilde{\\mathcal {A}}, ({\\rm dw}\\widetilde{\\mathcal {A}})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {A}}, ({\\rm ex}\\widetilde{\\mathcal {A}})^\\perp )$ , where ${\\rm dw}\\widetilde{\\mathcal {A}}$ is the class of chain complexes $X$ such that $X_m \\in \\mathcal {A}$ for every $m \\in \\mathbb {Z}$ , and ${\\rm ex}\\widetilde{\\mathcal {A}} = {\\rm dw}\\widetilde{\\mathcal {A}} \\cap \\mathcal {E}$ where $\\mathcal {E}$ is the class of exact complexes.",
"As an example, if $\\mathcal {P}_0$ denotes the class of projective modules in the category $ R{\\rm {\\bf -Mod}} $ of left $R$ -modules, then the cotorsion pair $(\\mathcal {P}_0, R{\\rm {\\bf -Mod}} )$ induces two cotorsion pairs $({\\rm dw}\\widetilde{\\mathcal {P}_0}, ({\\rm dw}\\widetilde{\\mathcal {P}_0})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_0}, ({\\rm ex}\\widetilde{\\mathcal {P}_0})^\\perp )$ .",
"In [4], the authors prove that these pairs are compatible and complete, with the help of a theorem by I. Kaplansky, namely that every projective module can be written as a direct sum of countably generated projective modules.",
"Then, using [10], they get a new abelian model structure on $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ where the class of cofibrant objects is the class ${\\rm dw}\\widetilde{\\mathcal {P}_0}$ , which we shall call the class of degreewise projective complexes.",
"We shall refer to this model structure as the dw-projective model structure.",
"In [2] it is proven that if $\\mathcal {P}_n$ denotes the class of left $R$ -modules with projective dimension at most $n$ , then $(\\mathcal {P}_n, \\mathcal {P}_n^\\perp )$ is a complete and hereditary cotorsion pair.",
"It follows we have two induced cotorsion pairs $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ .",
"Our goal is to prove that these two cotorsion pairs are complete for every $n > 0$ , in order two obtain a new abelian model structure on $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ such that ${\\rm dw}\\widetilde{\\mathcal {P}_n}$ is the class of cofibrant objects.",
"This paper is organized as follows.",
"First, we recall some definitions and introduce the notation we shall use.",
"Then, we shall give a “generalization” of the Kaplansky Theorem in $\\mbox{Mod-}R$ provided that $R$ is left noetherian.",
"Specifically, we shall prove that every module of projective dimension $\\le n$ has a $\\mathcal {P}_n^{\\aleph _0}$ -filtration, where $\\mathcal {P}^{\\aleph _0}_n$ is the set of all modules $M$ for which there exists an exact sequence $ 0 \\longrightarrow P_n \\longrightarrow \\cdots \\longrightarrow P_1 \\longrightarrow P_0 \\longrightarrow M \\longrightarrow 0 $ where $P_k$ is a countably generated projective module, for every $0 \\le k \\le n$ .",
"Using this result, we shall prove that $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ are complete cotorsion pairs.",
"Then, we shall give another method to prove the previous result.",
"The interesting thing of this other method, based on arguments appearing in the proof of [1], is that it can be applied to show that $({\\rm dw}\\widetilde{\\mathcal {F}_n}, ({\\rm dw}\\widetilde{\\mathcal {F}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {F}_n}, ({\\rm ex}\\widetilde{\\mathcal {F}_n})^\\perp )$ are complete cotorsion pairs, where $\\mathcal {F}_n$ denotes the class of left $R$ -modules having flat dimension at most $n$ .",
"At the end of the paper, we shall give some comments concerning the dw-$n$ -projective and dw-$n$ -flat model structures."
],
[
"Preliminaries",
"This section is devoted to recall some notions and to introduce part of the notation we shall use throughout the paper.",
"From now on, we work in the category $ R{\\rm {\\bf -Mod}} $ of left $R$ -modules, and the category $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ of chain complexes over $ R{\\rm {\\bf -Mod}} $ .",
"Given a chain complex $X = (X_m)_{m \\in \\mathbb {Z}}$ with boundary maps $\\partial ^X_m : X_m \\longrightarrow X_{m-1}$ , we shall denote $Z_m(X) := {\\rm Ker}(\\partial ^X_m)$ .",
"A chain complex $X$ is said to be exact if $Z_m(X) = \\partial _{m+1}(X_{m+1})$ , for every $m \\in \\mathbb {Z}$ .",
"A chain complex $Y$ is said to be a subcomplex of $X$ if there exists a monomorphism $i : Y \\longrightarrow X$ .",
"Then we can define the quotient complex $X / Y$ as the complex whose components are given by $(X / Y)_m = X_m / Y_m$ and whose boundary maps $\\partial ^{X / Y}_m : X_m / Y_m \\longrightarrow X_{m-1} / Y_{m-1}$ are given by $ x + Y_m \\mapsto \\partial ^X_m(x) + Y_{m-1}.",
"$ Let $\\mathcal {C}$ be either $ R{\\rm {\\bf -Mod}} $ or $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ .",
"Let $\\mathcal {A}$ and $\\mathcal {B}$ be two classes of objects in $\\mathcal {C}$ .",
"The pair $(\\mathcal {A, B})$ is called a cotorsion pair in $\\mathcal {C}$ if the following conditions are satisfied: (1) $\\mathcal {A} = \\mbox{}^{\\perp }\\mathcal {B} := \\lbrace X \\in {\\rm Ob}(\\mathcal {C}) \\mbox{ / } {\\rm Ext}^1(X, B) = 0 \\mbox{ for every }B \\in \\mathcal {B} \\rbrace $ .",
"(2) $\\mathcal {B} = \\mathcal {A}^{\\perp } := \\lbrace X \\in {\\rm Ob}(\\mathcal {C}) \\mbox{ / } {\\rm Ext}^1(A, X) = 0 \\mbox{ for every }A \\in \\mathcal {A} \\rbrace $ .",
"A cotorsion pair $(\\mathcal {A, B})$ in $\\mathcal {C}$ is said to be complete if: (a) $(\\mathcal {A, B})$ has enough projectives: for every object $X$ there exist objects $A \\in \\mathcal {A}$ and $B \\in \\mathcal {B}$ , and a short exact sequence $ 0 \\longrightarrow B \\longrightarrow A \\longrightarrow X \\longrightarrow 0.",
"$ (b) $(\\mathcal {A, B})$ has enough injectives: for every object $X$ there exist objects $A^{\\prime } \\in \\mathcal {A}$ and $B^{\\prime } \\in \\mathcal {B}$ , and a short exact sequence $ 0 \\longrightarrow X \\longrightarrow B^{\\prime } \\longrightarrow A^{\\prime } \\longrightarrow 0.",
"$ A cotorsion pair $(\\mathcal {A, B})$ is said to be cogenerated by a set $\\mathcal {S} \\subseteq \\mathcal {A}$ if $\\mathcal {B} = \\mathcal {S}^\\perp $ .",
"There is a wide range of complete cotorsion pairs, thanks to the following result, known as the Eklof and Trlifaj Theorem.",
"Theorem 2.1 [5] Every cotorsion pair in $\\mathcal {C}$ cogenerated by a set is complete.",
"Example 2.1 (1) If $\\mathcal {P}_0$ denotes the class of projective modules, then $(\\mathcal {P}_0, R{\\rm {\\bf -Mod}} )$ is a cotorsion pair.",
"Since every projective module is a direct summand of a free module, and $R$ is projective, one can show that $(\\mathcal {P}_0, R{\\rm {\\bf -Mod}} )$ is cogenerated by the set $\\lbrace R \\rbrace $ and hence it is complete.",
"(2) Similarly, if $\\mathcal {I}_0$ denotes the class of injective modules, then $( R{\\rm {\\bf -Mod}} , \\mathcal {I}_0)$ is a cotorsion pair.",
"Using the Baer's Criterion, one can show that $( R{\\rm {\\bf -Mod}} , \\mathcal {I}_0)$ is cogenerated by the set of modules of the form $R/I$ , where $I$ is a left ideal of $R$ .",
"So $( R{\\rm {\\bf -Mod}} , \\mathcal {I}_0)$ is a complete cotorsion pair.",
"(3) A less trivial example of a complete cotorsion pair is given by the flat cotorsion pair $(\\mathcal {F}_0, \\mathcal {F}_0^\\perp )$ , where $\\mathcal {F}_0$ is the class of flat modules.",
"This result was proven by Edgard E. Enochs by using the Eklof and Trlifaj Theorem.",
"Enochs proved that the pair $(\\mathcal {F}_0, \\mathcal {F}_0^\\perp )$ is cogenerated by the set $\\mathcal {S} = \\lbrace S \\in \\mathcal {F}_0 \\mbox{ : } {\\rm Card}(S) \\le \\kappa \\rbrace $ , where $\\kappa $ is an infinite cardinal with $\\kappa \\ge {\\rm Card}(R)$ .",
"(4) The following example is probably the most important cotorsion pair we shall consider in this paper, the pair $(\\mathcal {P}_n, \\mathcal {P}_n^\\perp )$ , where $\\mathcal {P}_n$ is the class of modules which have projective dimension $\\le n$ .",
"Recall that a module $M$ has projective dimension $\\le n$ is there exists an exact sequence $ 0 \\longrightarrow P_n \\longrightarrow P_{n-1} \\longrightarrow \\cdots \\longrightarrow P_1 \\longrightarrow P_0 \\longrightarrow M \\longrightarrow 0, $ such that $P_k$ is a projective module, for every $0 \\le k \\le n$ .",
"Such a sequence is called a projective resolution of $M$ of length $n$.",
"We shall refer to the modules in $\\mathcal {P}_n$ as $n$ -projective modules.",
"In [2], the authors proved that $(\\mathcal {P}_n, \\mathcal {P}_n^\\perp )$ is a cotorsion pair cogenerated by the set of all $n$ -projective modules whose cardinality is less or equal than a given infinite cardinal $\\kappa $ with $\\kappa \\ge {\\rm Card}(R)$ .",
"(5) In a similar way, consider the class $\\mathcal {F}_n$ of modules $M$ such that $M$ has flat dimension at most $n$ , or equivalently, there is an exact sequence $ 0 \\longrightarrow F_n \\longrightarrow \\cdots \\longrightarrow F_1 \\longrightarrow F_0 \\longrightarrow M \\longrightarrow 0 $ where $F_k$ is a flat module, for every $0 \\le k \\le n$ .",
"This sequence is called a flat resolution of length $n$.",
"In [9], it is proven that $(\\mathcal {F}_n, \\mathcal {F}^\\perp _n)$ is a complete cotorsion pair.",
"Here we shall give a easier proof of this fact.",
"Now we recall the notion of a model category.",
"Given a category $\\mathcal {C}$ , a map $f$ in $\\mathcal {C}$ is a retract of a map $g$ in $\\mathcal {C}$ if there is a commutative diagram of the form $ \\begin{tikzpicture}(m) [matrix of math nodes, row sep=2.5em, column sep=3em]{ A & C & A \\\\ B & D & B \\\\ };[-latex](m-1-1) edge (m-1-2) edge node[left] {f} (m-2-1)(m-1-2) edge (m-1-3) edge node[left] {g} (m-2-2)(m-1-3) edge node[right] {f} (m-2-3)(m-2-1) edge (m-2-2)(m-2-2) edge (m-2-3);\\end{tikzpicture} $ where the horizontal composites are identities.",
"Let $f : A \\longrightarrow B$ and $g : C \\longrightarrow D$ be two maps in $\\mathcal {C}$ .",
"We shall say that $f$ has the left lifting property with respect to $g$ (or that $g$ has the right lifting property with respect to $f$ ) if for every pair of maps $u : A \\longrightarrow C$ and $v : B \\longrightarrow D$ with $g \\circ u = v \\circ f$ , there exists a map $d : B \\longrightarrow C$ such that $g \\circ d = v$ and $d \\circ f = u$ .",
"$ \\begin{tikzpicture}(m) [matrix of math nodes, row sep=1em, column sep=3em]{ A & C & & & & A & C \\\\ & & \\mbox{} & & \\mbox{} & \\\\ B & D & & & & B & D \\\\ };[-latex](m-1-1) edge node[left] {f} (m-3-1) edge node[above] {u} (m-1-2)(m-1-6) edge node[left] {f} (m-3-6) edge node[above] {u} (m-1-7)(m-3-1) edge node[below] {v} (m-3-2)(m-1-2) edge node[right] {g} (m-3-2)(m-3-6) edge node[below] {v} (m-3-7)(m-1-7) edge node[right] {g} (m-3-7);[dotted, ->](m-3-6) edge node[above, sloped] {\\exists \\mbox{ } d} (m-1-7);[->, decoration={zigzag,segment length=4,amplitude=.9, post=lineto,post length=2pt},font=\\scriptsize , line join=round](m-2-3) edge[decorate] (m-2-5);\\end{tikzpicture} $ A model category is a bicomplete category $\\mathcal {C}$ equipped with three classes of maps named cofibrations, fibrations and weak equivalences, satisfying the following properties: (1) 3 for 2: If $f$ and $g$ are maps of $\\mathcal {C}$ such that $g\\circ f$ is defined and two of $f$ , $g$ and $g\\circ f$ are weak equivalences, then so is the third.",
"(2) If $f$ and $g$ are maps of $\\mathcal {C}$ such that $f$ is a retract of $g$ and $g$ is a weak equivalence, cofibration, or fibration, then so is $f$ .",
"Define a map to be a ${\\bf trivial \\ cofibration}$ if it is both a weak equivalence and a cofibration.",
"Similarly, define a map to be a trivial fibration if it is both a weak equivalence and a fibration.",
"(3) Trivial cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect to trivial fibrations.",
"(4) Every map $f$ can be factored as $f = \\alpha \\circ \\beta = \\gamma \\circ \\delta $ , where $\\alpha $ (resp.",
"$\\delta $ ) is a cofibration (resp.",
"fibration), and $\\gamma $ (resp.",
"$\\beta $ ) is a trivial cofibration (resp.",
"trivial fibration).",
"An object $X$ in $\\mathcal {C}$ is called cofibrant if the map $0 \\longrightarrow X$ is a cofibration, fibrant if the map $X \\longrightarrow 1$ is a fibration, and trivial if the map $0 \\longrightarrow X$ is a weak equivalence, where 0 and 1 denote the initial and terminal objects of $\\mathcal {C}$ , respectively.",
"Given a bicomplete abelian category $\\mathcal {C}$ , a model structure on it is said to be abelian if the following conditions are satisfied: (a) A map is a cofibration if and only if it is a monomorphism with cofibrant cokernel.",
"(b) A map if a fibration if and only if it is an epimorphism with fibrant kernel."
],
[
"Degreewise $n$ -projective complexes",
"We begin this section with the notion of a filtration.",
"Let $\\mathcal {C}$ be either $ R{\\rm {\\bf -Mod}} $ or $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ .",
"Given an object $X \\in \\mathcal {C}$ , by a filtration of $X$ indexed by an ordinal $\\lambda $ we shall mean a family $(X^\\alpha \\mbox{ : } \\alpha < \\lambda )$ of subobjects of $X$ such that: (1) $X = \\bigcup _{\\alpha < \\lambda } X_\\alpha $ .",
"(3) $X^\\alpha $ is a subobject of $X^{\\alpha ^{\\prime }}$ whenever $\\alpha \\le \\alpha ^{\\prime }$ .",
"(4) $X^\\beta = \\bigcup _{\\alpha < \\beta } X^\\alpha $ for any limit ordinal $\\beta < \\lambda $ .",
"If $\\mathcal {S}$ is some class of objects in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ , we say that a filtration $(X^\\alpha \\mbox{ : } \\alpha < \\lambda )$ of $X$ is a $\\mathcal {S}$ -filtration if for each $\\alpha + 1 < \\lambda $ we have that $X_0$ and $X^{\\alpha + 1} / X^\\alpha $ are isomorphic to an element of $\\mathcal {S}$ .",
"The construction of the model structure given in [4] is based on a theorem by I. Kaplansky, namely: Theorem 3.1 (Kaplansky's Theorem) If $P$ is a projective module then $P$ is a direct sum of countable generated projective modules.",
"So when one thinks of a possible generalization of the dw-projective model structure for $n$ -projective modules, a good question would be if it is possible to generalize the Kaplansky's Theorem for such modules.",
"Let $M \\in \\mathcal {P}_n$ be an $n$ -projective module: $ 0 \\longrightarrow P_n \\longrightarrow \\cdots \\longrightarrow P_1 \\longrightarrow P_0 \\longrightarrow M \\longrightarrow 0.",
"$ By Kaplansky's Theorem we can write $P_k = \\bigoplus _{i \\in I_k} P^i_k$ , where $P^i_k$ is a countably generated projective module, for every $i \\in I_k$ and every $0 \\le k \\le n$ .",
"Then we can rewrite the previous resolution as $ 0 \\longrightarrow \\bigoplus _{i \\in I_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I_0} P^i_0 \\longrightarrow M \\longrightarrow 0.",
"$ From now on we shall write any projective resolution of length $n$ by using such direct sum decompositions.",
"We shall denote by $\\mathcal {P}_n^{\\aleph _0}$ the set of all modules $M$ having a projective resolution as above, where $I_k$ is a countable set for each $0 \\le k \\le n$ .",
"For any class of modules $\\mathcal {A}$ , we denote by ${\\rm dw}\\widetilde{\\mathcal {A}}$ (resp.",
"${\\rm ex}\\widetilde{\\mathcal {A}}$ ) the class of (resp.",
"exact) chain complexes such that each term belongs to $\\mathcal {A}$ .",
"We shall prove that $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ is a cotorsion pair cogenerated by the set ${\\rm dw}\\widetilde{\\mathcal {P}_n^{\\aleph _0}}$ .",
"We shall name ${\\rm dw}\\widetilde{\\mathcal {P}_n}$ the class of dw-$n$ -projective chain complexes.",
"The fact that $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ is a cotorsion pair in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ is a consequence of the following result (which is proven by its author for any abelian category): Proposition 3.1 [8] Let $(\\mathcal {A, B})$ be a cotorsion pair in $ R{\\rm {\\bf -Mod}} $ .",
"Then $({\\rm dw}\\widetilde{\\mathcal {A}}, ({\\rm dw}\\widetilde{\\mathcal {A}})^\\perp )$ is a cotorsion pair in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ .",
"We shall prove that every dw-$n$ -projective complex has a ${\\rm dw}\\widetilde{\\mathcal {P}_n^{\\aleph _0}}$ -filtration.",
"Then the completeness of $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ shall be a consequence of Theorem REF and the following result: Proposition 3.2 Let $(\\mathcal {A, B})$ be a cotorsion pair in $\\mathcal {C} = R{\\rm {\\bf -Mod}} , {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ and let $\\mathcal {S} \\subseteq \\mathcal {A}$ be a set of objects of $\\mathcal {C}$ .",
"If every $A \\in \\mathcal {A}$ has a $\\mathcal {S}$ -filtration, then $(\\mathcal {A, B})$ is cogenerated by $\\mathcal {S}$ .",
"Before proving this, we need the following result known as the Eklof's Lemma.",
"For a proof of this we refer the reader to [9] or [6].",
"Lemma 3.1 (Eklof's Lemma) In $\\mathcal {C} = R{\\rm {\\bf -Mod}} , {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ let $A$ and $B$ be two objects.",
"If $A$ has a $\\mbox{}^\\perp \\lbrace B \\rbrace $ -filtration, then $A \\in \\mbox{}^\\perp \\lbrace B\\rbrace $ .",
"[Proof of Proposition REF :] Consider the cotorsion pair $(\\mbox{}^\\perp (\\mathcal {S}^\\perp ), \\mathcal {S}^\\perp )$ .",
"We shall show that $(\\mbox{}^\\perp ({S}^\\perp ), \\mathcal {S}^\\perp ) = (\\mathcal {A, B})$ .",
"It suffices to show that $\\mathcal {B} = \\mathcal {S}^\\perp $ , since this equality implies $\\mathcal {A} = \\mbox{}^\\perp \\mathcal {B} = \\mbox{}^\\perp (\\mathcal {S}^\\perp )$ .",
"Since $\\mathcal {S} \\subseteq \\mathcal {A}$ , we have $\\mathcal {B} = \\mathcal {A}^\\perp \\subseteq \\mathcal {S}^\\perp $ .",
"Now let $Y \\in \\mathcal {S}^\\perp $ , $A \\in \\mathcal {A}$ and let $(A_\\alpha \\mbox{ : } \\alpha < \\lambda )$ be an $\\mathcal {S}$ -filtration of $A$ .",
"We have ${\\rm Ext}^1(A_0, Y) & = {\\rm Ext}^1(0, Y) = 0, \\\\{\\rm Ext}^1(A_{\\alpha + 1} / A_\\alpha , Y) & = , \\mbox{ whenever $\\alpha + 1 < \\lambda $},$ since $A_0$ and $A_{\\alpha + 1} / A_\\alpha $ are isomorphic to objects in $\\mathcal {C}$ .",
"Then $(A_\\alpha \\mbox{ : } \\alpha < \\lambda )$ is a $\\mbox{}^\\perp \\lbrace Y\\rbrace $ -filtration of $A$ .",
"By the Eklof's Lemma, we have ${\\rm Ext}^1(A, Y) = 0$ , i.e.",
"$Y \\in \\mathcal {A}^{\\perp } = \\mathcal {B}$ since $A$ is any module in $\\mathcal {A}$ .",
"Hence $\\mathcal {S}^\\perp \\subseteq \\mathcal {B}$ .",
"In order to construct ${\\rm dw}\\widetilde{\\mathcal {P}_n^{\\aleph _0}}$ -filtrations of dw-$n$ -projective complexes, we need the following generalization of the Kaplansky's Theorem: Lemma 3.2 (Kaplansky's Theorem fon $n$ -projective modules) Let $R$ be a noetherian ring.",
"Let $M \\in \\mathcal {P}_n$ and let $N$ be a countably generated submodule of $M$ .",
"Then there exists a $\\mathcal {P}^{\\aleph _0}_n$ -filtration of $M$ , say $(M_\\alpha : \\alpha < \\lambda )$ with $\\lambda > 1$ , such that $M_1 \\in \\mathcal {P}^{\\aleph _0}_n$ and $N \\subseteq M_1$ .",
"Let $M \\in \\mathcal {P}_n$ and let $ 0 \\longrightarrow \\bigoplus _{i \\in I_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I_0} P^i_0 \\longrightarrow M \\longrightarrow 0 $ be a projective resolution of $M$ .",
"We shall construct a $\\mathcal {P}^{\\aleph _0}_n$ -filtration $(M_\\alpha \\mbox{ : } \\alpha < \\lambda )$ of $M$ , with $N \\subseteq M_1$ , by using transfinite induction.",
"For $\\alpha = 0$ set $M_0 = 0$ .",
"Now we construct $M_1$ .",
"Let $\\mathcal {G}$ be a countable set of generators of $N$ .",
"Since $f_0$ is surjective, for every $g \\in \\mathcal {G}$ we can choose $y_g \\in \\bigoplus _{i \\in I_0} P^i_0$ such that $g = f_0(y_g)$ .",
"Consider the set $Y = \\lbrace y_g \\mbox{ : } g \\in \\mathcal {G} \\rbrace $ .",
"Since $Y$ is a countable subset of $\\bigoplus _{i \\in I_0} P^i_0$ , we have that $\\left< Y \\right>$ is a countably generated submodule of $P_0$ .",
"Choose a countable subset $I_0^{1, 0} \\subseteq I_0$ such that $\\left< Y \\right> \\subseteq \\bigoplus _{i \\in I^{1, 0}_0} P^i_0$ .",
"Then $f_0\\left( \\left< Y \\right> \\right) \\subseteq N$ .",
"Consider ${\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right)$ .",
"Since $\\bigoplus _{i \\in I_0^{1, 0}} P^i_0$ is countably generated and ${\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right)$ is a submodule of $\\bigoplus _{i \\in I_0^{1, 0}} P^i_0$ , we have that ${\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right)$ is also countably generated, since $R$ is noetherian.",
"Let $\\mathcal {B}$ be a countable set of generators of ${\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right)$ .",
"Let $b \\in \\mathcal {B}$ , then $f(b) = 0$ and by exactness of the above sequence there exists $y_b \\in \\bigoplus _{i \\in I_1} P^i_1$ such that $b = f_1(y_b)$ .",
"Let $Y^{\\prime } = \\lbrace y_b \\mbox{ : } b \\in \\mathcal {B} \\rbrace $ .",
"Note that $Y^{\\prime }$ is a countable subset of $(f_1)^{-1}\\left( {\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right) \\right)$ .",
"Then $\\left< Y^{\\prime } \\right>$ is a countably generated submodule of $\\bigoplus _{i \\in I_1} P^i_1$ .",
"Hence there exists a countable subset $I_1^{1, 0} \\subseteq I_1$ such that $\\bigoplus _{i \\in I_1^{1, 0}} P^i_1 \\supseteq \\left< Y^{\\prime } \\right>$ .",
"Thus $f_1 \\left( \\bigoplus _{i \\in I^{1, 0}_1} P^i_1 \\right) \\supseteq f_1(\\left< Y^{\\prime } \\right>)$ .",
"Now let $z \\in {\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right)$ .",
"Then $z = r_1 b_1 + \\cdots + r_m b_m$ , where each $b_j \\in \\mathcal {B}$ .",
"Since $b_j = f_1(y_{b_j})$ with $y_{b_j} \\in Y^{\\prime }$ , we get $z = f_1(r_1y_{b_1} + \\cdots + r_m y_{b_m}) \\in f_1(\\left< Y^{\\prime } \\right>)$ .",
"Hence, ${\\rm Ker}\\left( \\left.",
"f_0 \\right|_{\\bigoplus _{i \\in I_0^{1, 0}} P^i_0} \\right) \\subseteq f_1(\\left< Y^{\\prime } \\right>) \\subseteq f_1 \\left( \\bigoplus _{i \\in I_1^{1, 0}} P^i_1 \\right)$ .",
"Use the same argument to find a countable subset $I_2^{1, 0} \\subseteq I_2$ such that $f_2\\left( \\bigoplus _{i \\in I_2^{1, 0}} P^i_2 \\right) \\supseteq {\\rm Ker}\\left( f_1|_{\\bigoplus _{i \\in I_1^{1, 0}} P^i_1} \\right)$ .",
"Repeat the same argument until find a countable subset $I_n^{1, 0} \\subseteq I_n$ such that $f_n \\left( \\bigoplus _{i \\in I_n^{1, 0}} P^i_n \\right) \\supseteq {\\rm Ker}\\left( f_{n-1}|_{\\bigoplus _{i \\in I_{n-1}^{1, 0}} P^i_{n-1}} \\right)$ .",
"Now, $f_n\\left( \\bigoplus _{i \\in I_n^{1,0}} P^i_n \\right)$ is a countably generated submodule of $\\bigoplus _{i \\in I_{n-1}} P^i_{n-1}$ .",
"Then choose a countable subset $I_{n-1}^{1,0} \\subseteq I_{n-1}^{1, 1} \\subseteq I_{n-1}$ such that $f_n \\left( \\bigoplus _{i \\in I_n^{1, 0}} P^i_n \\right) \\subseteq \\bigoplus _{i \\in I_{n-1}^{1,1}} P^i_{n-1}$ .",
"Repeat this process until find a countable subset $I^{1,0}_0 \\subseteq I^{1,1}_0 \\subseteq I_0$ satisfying $f_1\\left( \\bigoplus _{i \\in I_1^{1,1}} P^i_1 \\right)$ $\\subseteq \\bigoplus _{i \\in I_0^{1,1}} P^i_0$ .",
"Now choose a countable subset $I_1^{1,1} \\subseteq I_1^{1,2} \\subseteq I_1$ such that $f_1\\left( \\bigoplus _{i \\in I_2^{1, 2}} P^i_1 \\right)$ $\\supseteq {\\rm Ker}\\left( f_0|_{\\bigoplus _{i \\in I_0^{1,1}} P^i_0} \\right)$ .",
"What we have been doing so far is called the zig-zag procedure.",
"Keep repeating this procedure infinitely many times, and set $I^1_k = \\bigcup _{m \\ge 0} I_k^{1, m}$ , for every $0 \\le k \\le n$ .",
"By construction, we get the following exact sequence $ 0 \\longrightarrow \\bigoplus _{i\\in I_n^1} P^i_n \\longrightarrow \\bigoplus _{i \\in I_{n-1}^1} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^1_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^1_0} P^i_0 \\longrightarrow M_1 \\longrightarrow 0 $ where $x \\in M_1 := {\\rm CoKer}\\left( \\bigoplus _{i \\in I^1_1} \\longrightarrow \\bigoplus _{i \\in I_0^1} P^i_0 \\right) \\subseteq M$ and $N \\subseteq M_1$ .",
"We take the quotient of the resolution of $M$ by the resolution of $M^{\\prime }$ , and get the following commutative diagram: $ \\begin{tikzpicture}(m) [matrix of math nodes, row sep=2em, column sep=1.5em]{ & 0 & & 0 & 0 & 0 \\\\ 0 & \\bigoplus _{i \\in I^1_n} P^i_n & \\cdots & \\bigoplus _{i \\in I^1_1} P^i_1 & \\bigoplus _{i \\in I^1_0} P^i_0 & M_1 & 0 \\\\ 0 & \\bigoplus _{i \\in I_n} P^i_n & \\cdots & \\bigoplus _{i \\in I_1} P^i_1 & \\bigoplus _{i \\in I_0} P^i_0 & M & 0 \\\\ 0 & \\bigoplus _{i \\in I_n - I^1_n} P^i_n & \\cdots & \\bigoplus _{i \\in I_1 - I^1_1} P^i_1 & \\bigoplus _{i \\in I_0 - I^1_0} P^i_0 & M / M_1 & 0 \\\\ & 0 & & 0 & 0 & 0 \\\\ };[->](m-1-2) edge (m-2-2) (m-1-4) edge (m-2-4) (m-1-5) edge (m-2-5) (m-1-6) edge (m-2-6)(m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) edge (m-3-2) (m-2-3) edge (m-2-4) (m-2-4) edge (m-2-5) edge (m-3-4) (m-2-5) edge (m-2-6) edge (m-3-5) (m-2-6) edge (m-2-7) edge (m-3-6)(m-3-1) edge (m-3-2) (m-3-2) edge (m-3-3) edge (m-4-2) (m-3-3) edge (m-3-4) (m-3-4) edge (m-3-5) edge (m-4-4) (m-3-5) edge (m-3-6) edge (m-4-5) (m-3-6) edge (m-3-7) edge (m-4-6)(m-4-1) edge (m-4-2) (m-4-2) edge (m-4-3) edge (m-5-2) (m-4-3) edge (m-4-4) (m-4-4) edge (m-4-5) edge (m-5-4) (m-4-5) edge (m-4-6) edge (m-5-5) (m-4-6) edge (m-4-7) edge (m-5-6);\\end{tikzpicture} $ where the third row is an exact sequence since the class of exact complexes is closed under taking cokernels.",
"Then we have a projective resolution of length $n$ for $M / M_1$ .",
"Repeat the same procedure above for $M / M_1$ , by choosing $x^1 + M_1 \\in M/M_1 - \\left\\lbrace 0 + M_1 \\right\\rbrace $ , in order to get an exact sequence $ 0 \\longrightarrow \\bigoplus _{i \\in I^2_n - I^1_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^2_1 - I^1_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^2_0 - I^1_0} P^i_0 \\longrightarrow M_2 / M_1 \\longrightarrow 0, $ for some module $M_1 \\subseteq M_2 \\subseteq M$ , such that $I^2_k - I^1_k$ is countable for every $0 \\le k \\le n$ .",
"Note that $ 0 \\longrightarrow \\bigoplus _{i \\in I^2_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I^2_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^2_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^2_0} P^i_0 \\longrightarrow M_2 \\longrightarrow 0 $ is a projective resolution of $M_2$ , since we have a commutative diagram $ \\begin{tikzpicture}(m) [matrix of math nodes, row sep=2em, column sep=1.5em]{ & 0 & & 0 & 0 & 0 \\\\ 0 & \\bigoplus _{i \\in I^1_n} P^i_n & \\cdots & \\bigoplus _{i \\in I^1_1} P^i_1 & \\bigoplus _{i \\in I^1_0} P^i_0 & M_1 & 0 \\\\ 0 & \\bigoplus _{i \\in I^2_n} P^i_n & \\cdots & \\bigoplus _{i \\in I^2_1} P^i_1 & \\bigoplus _{i \\in I^2_0} P^i_0 & M_2 & 0 \\\\ 0 & \\bigoplus _{i \\in I^2_n - I^1_n} P^i_n & \\cdots & \\bigoplus _{i \\in I^2_1 - I^1_1} P^i_1 & \\bigoplus _{i \\in I^2_0 - I^1_0} P^i_0 & M_2 / M_1 & 0 \\\\ & 0 & & 0 & 0 & 0 \\\\ };[->](m-1-2) edge (m-2-2) (m-1-4) edge (m-2-4) (m-1-5) edge (m-2-5) (m-1-6) edge (m-2-6)(m-2-1) edge (m-2-2) (m-2-2) edge (m-2-3) edge (m-3-2) (m-2-3) edge (m-2-4) (m-2-4) edge (m-2-5) edge (m-3-4) (m-2-5) edge (m-2-6) edge (m-3-5) (m-2-6) edge (m-2-7) edge (m-3-6)(m-3-1) edge (m-3-2) (m-3-2) edge (m-3-3) edge (m-4-2) (m-3-3) edge (m-3-4) (m-3-4) edge (m-3-5) edge (m-4-4) (m-3-5) edge (m-3-6) edge (m-4-5) (m-3-6) edge (m-3-7) edge (m-4-6)(m-4-1) edge (m-4-2) (m-4-2) edge (m-4-3) edge (m-5-2) (m-4-3) edge (m-4-4) (m-4-4) edge (m-4-5) edge (m-5-4) (m-4-5) edge (m-4-6) edge (m-5-5) (m-4-6) edge (m-4-7) edge (m-5-6);\\end{tikzpicture} $ where the first and third rows are exact sequences, and then so is the second since the class of exact complexes is closed under extensions.",
"We have that $M_1$ and $M_2$ are $n$ -projective modules such that $M_1 \\in \\mathcal {P}^{\\aleph _{0}}_n, M_2 / M_1 \\in \\mathcal {P}^{\\aleph _0}_n$ .",
"Now suppose that there is an ordinal $\\beta $ such that: (1) $M_\\alpha $ is an $n$ -projective module, for every $\\alpha < \\beta $ .",
"(2) $M_\\alpha \\subseteq M_{\\alpha ^{\\prime }}$ whenever $\\alpha \\le \\alpha ^{\\prime } < \\beta $ .",
"(3) $M_{\\alpha + 1} / M_\\alpha \\in \\mathcal {P}^{\\aleph _0}_n$ whenever $\\alpha + 1 < \\beta $ .",
"(4) $M_\\gamma = \\bigcup _{\\alpha < \\gamma } M_\\alpha $ for every limit ordinal $\\gamma < \\beta $ .",
"If $\\beta $ is a limit ordinal, then set $M_\\beta = \\bigcup _{\\alpha < \\beta } M_\\alpha $ .",
"Otherwise there exists an ordinal $\\alpha < \\beta $ such that $\\beta = \\alpha + 1$ .",
"In this case, construct $M_{\\alpha + 1} \\in \\mathcal {P}_n$ from $M_\\alpha $ as we constructed $M_2$ from $M_1$ , such that $M_{\\alpha + 1} / M_\\alpha \\in \\mathcal {P}^{\\aleph _0}_n$ .",
"By transfinite induction, we obtain a $\\mathcal {P}_n^{\\aleph _0}$ -filtration $(M_\\alpha \\mbox{ : } \\alpha < \\lambda )$ of $M$ , for some ordinal $\\lambda $ , such that $M_1 \\supseteq N$ and $M_1 \\in \\mathcal {P}^{\\aleph _0}_n$ .",
"From now on, $R$ shall be a noetherian ring.",
"Now we are ready to prove the main result of this section.",
"Theorem 3.2 Every chain complex $X \\in {\\rm dw}\\widetilde{\\mathcal {P}_n}$ has a ${\\rm dw}\\widetilde{\\mathcal {P}_n^{\\aleph _0}}$ -filtration.",
"Let $X \\in {\\rm dw}\\mathcal {P}_n$ and write $ X = \\cdots \\longrightarrow X_{k+1} \\stackrel{\\partial _{k+1}}{\\longrightarrow }X_k \\stackrel{\\partial _k}{\\longrightarrow }X_{k-1} \\longrightarrow \\cdots .",
"$ For each $k$ one has a projective resolution of $X_k$ of length $n$ : $ 0 \\longrightarrow \\bigoplus _{i \\in I_n(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1(k)} P^i_1(k) \\longrightarrow \\bigoplus _{i \\in I_0(k)} P^i_0(k) \\longrightarrow X_k \\longrightarrow 0.",
"$ We shall construct a ${\\rm dw}\\widetilde{\\mathcal {P}_n^{\\aleph _0}}$ -filtration of $X$ by using transfinite induction.",
"For $\\alpha = 0$ set $X^0 = 0$ .",
"For $\\alpha = 1$ , choose $m \\in \\mathbb {Z}$ .",
"Let $S$ be a countably generated submodule of $X_m$ .",
"By the previous lemma, there exists a submodule $\\mathcal {P}^{\\aleph _0}_n \\ni X^1_m \\subseteq X_m$ such that $S \\subseteq X^1_m$ .",
"Note that $X^1_m$ is also countably generated.",
"Then $\\partial _m(X^1_m)$ is a countably generated submodule of $X_{m-1}$ , and so there exists $\\mathcal {P}^{\\aleph _0}_n \\ni X^1_{m-1} \\subseteq X_{m-1}$ such that $\\partial _m(X^1_m) \\subseteq X^1_{m-1}$ .",
"Repeat the same procedure infinitely many times in order to obtain a subcomplex $ X^1 = \\cdots \\longrightarrow X^1_{k+1} \\longrightarrow X^1_k \\longrightarrow X^1_{k-1} \\longrightarrow \\cdots $ of $X$ such that $X^1_k \\in \\mathcal {P}^{\\aleph _0}_n$ for every $k \\in \\mathbb {Z}$ (we are setting $X^1_k = 0$ for every $k > m$ ).",
"Hence $X^1 \\in {\\rm dw}\\widetilde{\\mathcal {P}^{\\aleph _0}_n}$ .",
"Note from the proof of the previous lemma that the quotient $X / X^1$ is in ${\\rm dw}\\widetilde{\\mathcal {P}_n}$ .",
"We have $ X / X^1 = \\cdots \\longrightarrow X_{k+1} / X^1_{k+1} \\longrightarrow X_k / X^1_k \\longrightarrow X_{k-1} / X^1_{k-1} \\longrightarrow \\cdots , $ where for every $k \\le m$ one has the following projective resolutions of length $n$ for $X^1_k$ and $X_k / X^1_{k}$ : $0 & \\longrightarrow \\bigoplus _{i \\in I_n^1(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^1_1(k)} P^i_1(k) \\longrightarrow \\bigoplus _{i \\in I^1_0(k)} P^i_0(k) \\longrightarrow X^1_k \\longrightarrow 0, \\\\0 & \\longrightarrow \\bigoplus _{i \\in I_n(k) - I_n^1(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_0(k) - I^1_0(k)} P^i_0(k) \\longrightarrow X_k / X^1_k \\longrightarrow 0.$ Apply the same procedure above to the complex $X / X^1$ , in order to get a subcomplex $ X^2 / X^1 = \\cdots \\longrightarrow X^2_{k+1} / X^1_{k+1} \\longrightarrow X^2_k / X^1_k \\longrightarrow X^2_{k-1} / X^1_{k-1} \\longrightarrow \\cdots $ of $X / X^1$ , such that for each $k \\in \\mathbb {Z}$ one has the following projective resolution of length $n$ for $X^2_k / X^1_k$ : $ 0 \\longrightarrow \\bigoplus _{i \\in I^2_n - I^1_n} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^2_1 - I^1_1} P^i_1(k) \\longrightarrow \\bigoplus _{i \\in I^2_0 - I^1_0} P^i_0(k) \\longrightarrow X^2_k / X^1_k \\longrightarrow 0, $ where each $I^2_j - I^1_j \\subseteq I_j$ is countable.",
"Now consider the complex $ X^2 = \\cdots \\longrightarrow X^2_{k+1} \\longrightarrow X^2_k \\longrightarrow X^2_{k-1} \\longrightarrow \\cdots .",
"$ As we did in the proof of the previous lemma, we have that $ 0 \\longrightarrow \\bigoplus _{i \\in I^2_n(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^2_1(k)} P^i_1(k) \\longrightarrow \\bigoplus _{i \\in I^2_0(k)} P^i_0(k) \\longrightarrow X^2_k \\longrightarrow 0 $ is an exact sequence.",
"So $X^2_k \\in \\mathcal {P}_n$ for every $k \\in \\mathbb {Z}$ , and hence $X^2 \\in {\\rm dw}\\widetilde{\\mathcal {P}_n}$ , with $X^2 / X^1 \\in {\\rm dw}\\widetilde{\\mathcal {P}^{\\aleph _0}_n}$ .",
"The rest of the proof follows by transfinite induction, as in the end of the proof of the previous lemma."
],
[
"Exact degreewise $n$ -projective complexes",
"Consider the class of exact dw-$n$ -projective complexes ${\\rm ex}\\widetilde{\\mathcal {P}_n} = {\\rm dw}\\widetilde{\\mathcal {P}_n} \\cap \\mathcal {E}$ , where $\\mathcal {E}$ denotes the class of exact complexes.",
"The goal of this section is to prove that $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ is a complete cotorsion pair.",
"This pair is a cotorsion pair by the following result by Gillespie: Proposition 4.1 [8] Let $(\\mathcal {A, B})$ be a cotorsion pair in an abelian category $\\mathcal {C}$ with enough projective and injective objects.",
"If $\\mathcal {B}$ contains a cogenerator of finite injective dimension then $({\\rm ex}\\widetilde{\\mathcal {A}}, ({\\rm ex}\\widetilde{\\mathcal {A}})^\\perp )$ is a cotorsion pair.",
"Recall that a cogenerator in an abelian category $\\mathcal {C}$ is an object $C$ such that for every nonzero object $H$ there exists a nonzero morphism $f : H \\longrightarrow C$ .",
"For example, $\\mbox{Mod-}R$ has a an injective cogenerator given by the abelian group ${\\rm Hom}(R, \\mathbb {Q} / \\mathbb {Z})$ of group homomorphisms and providing it with the scalar multiplication defined by $ f \\cdot r : R \\longrightarrow \\mathbb {Q} / \\mathbb {Z}, \\mbox{ } s \\mapsto f(rs) $ for $f \\in {\\rm Hom}(R, \\mathbb {Q} / \\mathbb {Z})$ and $r \\in R$ (see [3] for details).",
"Since ${\\rm Hom}(R, \\mathbb {Q} / \\mathbb {Z}) \\in \\mathcal {P}_n^\\perp $ , we have that $\\left( {\\rm ex}\\mathcal {P}_n, ({\\rm ex}\\mathcal {P}_n)^\\perp \\right)$ is a cotorsion pair.",
"Given a module $M \\in \\mathcal {P}_n$ , consider a projective resolution of $M$ of length $n$ : $ 0 \\longrightarrow \\bigoplus _{i \\in I_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I_0} P^i_0 \\longrightarrow M \\longrightarrow 0 \\mbox{ \\ ($\\ast $)}.",
"$ We shall say that a projective resolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0} P^i_0 \\longrightarrow N \\longrightarrow 0 \\mbox{ \\ ($\\ast \\ast $)} $ is a nice subresolution of ($\\ast $ ) if $I^{\\prime }_k \\subseteq I_k$ for every $0\\le k \\le n$ and $N \\subseteq M$ .",
"From now on, fix an infinite cardinal $\\kappa $ such that $\\kappa \\ge {\\rm Card}(R)$ .",
"We shall say that a set $S$ is small if ${\\rm Card}(S) \\le \\kappa $ .",
"We shall also say that a chain complex $X = (X_m)_{m \\in \\mathbb {Z}}$ is small if ${\\rm Card}(X) \\le \\kappa $ , where $ {\\rm Card}(X) := \\sum _{m \\in \\mathbb {Z}} {\\rm Card}(X_m).",
"$ So a complex $X$ is small if and only if each term $X_m$ is a small set.",
"Note that if $M$ is an $n$ -projective module with a resolution given by ($\\ast $ ), then it is small if and only if ${\\rm Card}(I_k) \\le \\kappa $ for every $0 \\le k \\le n$ .",
"Let $\\mathcal {P}^{\\le \\kappa }_n$ denote the set of $n$ -projective modules with a small projective resolution.",
"If we consider the resolutions ($\\ast $ ) and ($\\ast \\ast $ ) above, then note that ($\\ast \\ast $ ) is a small and nice subresolution of ($\\ast $ ) if each $I^{\\prime }_k$ is a small subset of $I_k$ .",
"Consider the set $ {\\rm ex}\\widetilde{\\mathcal {P}_n^{\\le \\kappa }} = \\lbrace X \\in {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) \\mbox{ : } X \\mbox{ is exact and $X_m \\in \\mathcal {P}^{\\le \\kappa }_n$ for every $m \\in \\mathbb {Z}$} \\rbrace .",
"$ We shall prove that every exact dw-$n$ -projective complex has a ${\\rm ex}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ -filtration.",
"Lemma 4.1 Let $M \\in \\mathcal {P}_n$ with a projective resolution given by ($\\ast $ ).",
"For every submodule $N \\subseteq M$ with ${\\rm Card}(N) \\le \\kappa $ , there exists a small and nice subresolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0} P^i_0 \\longrightarrow N^{\\prime } \\longrightarrow 0 \\mbox{ \\ ($\\ast \\ast \\ast $)} $ of ($\\ast $ ) such that $N \\subseteq N^{\\prime }$ .",
"Moreover, if $N$ has an small and nice subresolution of $M$ , then ($\\ast \\ast \\ast $ ) can be constructed in such a way that it contains the given resolution of $N$ .",
"Since $f_0$ is surjective, for every $x \\in N$ choose $y_x \\in \\bigoplus _{i \\in I_0} P^i_0$ such that $x = f_0(y_x)$ .",
"Let $Y = \\lbrace y_x \\mbox{ : } x \\in N \\rbrace $ .",
"Note that $\\left< Y \\right>$ is a small submodule of $\\bigoplus _{i \\in I_0} P^i_0$ .",
"So there exists a small subset $I^0_0 \\subseteq I_0$ such that $\\left< Y \\right> \\subseteq \\bigoplus _{i \\in I^0_0} P^i_0$ .",
"We have $f_0\\left( \\bigoplus _{i \\in I^0_0} P^i_0 \\right) \\supseteq N$ .",
"Now consider the submodule ${\\rm Ker}\\left( f_0|_{\\bigoplus _{i \\in I^0_0}} P^i_0 \\right)$ of $f_0\\left( \\bigoplus _{i \\in I^0_0} P^i_0 \\right)$ , which is small since $f_0\\left( \\bigoplus _{i \\in I^0_0} P^i_0 \\right)$ is.",
"Then we can choose a small subset $I^0_1 \\subseteq I_1$ such that $f_1\\left( \\bigoplus _{i \\in I^0_1} P^i_1 \\right) \\supseteq {\\rm Ker}\\left( f_0|_{\\bigoplus _{i \\in I^0_0} P^i_0} \\right)$ .",
"Repeat the same argument until find a small subset $I^0_n \\subseteq I_n$ such that $f_n\\left( \\bigoplus _{i \\in I^0_n} P^i_n \\right) \\supseteq {\\rm Ker}\\left( f_{n-1}|_{\\bigoplus _{i \\in I^0_{n-1}} P^i_{n-1}} \\right)$ .",
"Since $f_n\\left( \\bigoplus _{i \\in I^0_n} P^i_n \\right)$ is a small submodule of $\\bigoplus _{i \\in I_{n-1}} P^i_{n-1}$ , we can choose a small subset $I^0_{n-1} \\subseteq I^1_{n-1} \\subseteq I_{n-1}$ such that $f_n\\left( \\bigoplus _{i \\in I^0_n} P^i_n \\right) \\subseteq \\bigoplus _{i \\in I^1_{n-1}} P^i_{n-1}$ .",
"From this point just use the zig-zag procedure in order to get small subsets $I^{\\prime }_k = \\bigcup _{j \\ge 0} I^j_k \\subseteq I_k$ and an exact sequence $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n} P^i_n \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_{n-1}} P^i_{n-1} \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0} P^i_0 \\longrightarrow N^{\\prime } \\longrightarrow 0 $ where $N^{\\prime } := {\\rm CoKer}\\left( \\bigoplus _{i \\in I^{\\prime }_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0} P^i_0 \\right)$ and $N \\subseteq N^{\\prime } \\subseteq M$ .",
"Now suppose that $N$ has a small and nice subresolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime N}_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^N_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^N_0} P^i_0 \\longrightarrow N \\longrightarrow 0 $ of ($\\ast $ ).",
"Take the quotient of ($\\ast $ ) by this resolution and get $ 0 \\longrightarrow \\bigoplus _{i \\in I_n - I^N_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1 - I^N_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I_0 - I^N_0} P^i_0 \\longrightarrow M / N \\longrightarrow 0.",
"$ Repeat the argument above using this sequence and the small submodule $\\left< z + N \\right>$ , where $z \\notin N$ .",
"Then we get a projective subresolution of the previous one: $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n - I^N_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1 - I^N_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0 - I^N_0} P^i_0 \\longrightarrow N^{\\prime } / N \\longrightarrow 0 $ where each set $I^{\\prime }_k - I^N_k$ is a small set.",
"As we did in the proof of Lemma REF , we have that $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0} P^i_0 \\longrightarrow N^{\\prime } \\longrightarrow 0 $ is small and nice subresolution of ($\\ast $ ), and that contains the resolution of $N$ as a nice subresolution.",
"Lemma 4.2 Let $X \\in {\\rm dw}\\widetilde{\\mathcal {P}_n}$ and let $Y$ be a bounded above subcomplex of $X$ such that ${\\rm Card}(Y_k) \\le \\kappa $ for every $k \\in \\mathbb {Z}$ .",
"Then there exists a (bounded above) subcomplex $Y^{\\prime }$ of $X$ such that $Y \\subseteq Y^{\\prime }$ and $Y^{\\prime } \\in {\\rm dw}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"We are given the following commutative diagram $ \\begin{tikzpicture}(m) [matrix of math nodes, row sep=1.5em, column sep=3em]{ Y =\\mbox{ } \\cdots & 0 & Y_m & Y_{m-1} & \\cdots \\\\ X =\\mbox{ } \\cdots & X_{m+1} & X_m & X_{m-1} & \\cdots \\\\ };[->](m-1-1) edge (m-1-2)(m-1-2) edge (m-1-3) edge (m-2-2)(m-1-3) edge node[above] {\\partial _m} (m-1-4) edge (m-2-3)(m-1-4) edge (m-1-5) edge (m-2-4)(m-2-1) edge (m-2-2) (m-2-2) edge node[above] {\\partial _{m+1}} (m-2-3) (m-2-3) edge node[above] {\\partial _m} (m-2-4) (m-2-4) edge (m-2-5);\\end{tikzpicture} $ Since $X_m$ is an $n$ -projective module, we have a projective resolution $ 0 \\longrightarrow \\bigoplus _{i \\in I_n(m)} P^i_n(m) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1(m)} P^i_1(m) \\longrightarrow \\bigoplus _{i \\in I_0(m)} P^i_0(m) \\longrightarrow X_m \\longrightarrow 0.",
"$ By the previous lemma, there exists a submodule $Y^{\\prime }_m$ of $X_m$ containing $Y_m$ , along with a small and nice subresolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_n(m)} P^i_n(m) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_1(m)} P^i_1(m) \\longrightarrow \\bigoplus _{i \\in I^{\\prime }_0(m)} P^i_0(m) \\longrightarrow Y^{\\prime }_m \\longrightarrow 0.",
"$ Note that ${\\rm Card}(\\partial _m(Y^{\\prime }_m) + Y_{m-1}) \\le \\kappa $ and $Y_{m-1} \\subseteq \\partial _m(Y^{\\prime }_m) + Y_{m-1} \\subseteq X_{m-1}$ .",
"Now choose a submodule $Y^{\\prime }_{m-1} \\subseteq X_{m-1}$ such that $\\partial _m(Y^{\\prime }_m) + Y_{m-1} \\subseteq Y^{\\prime }_{m-1}$ and $Y^{\\prime }_{m-1}$ has a small and nice subresolution of a fixed resolution of $X_{m-1}$ .",
"Repeat this process infinitely many times in order to obtain a complex $ Y^{\\prime } = \\cdots \\longrightarrow 0 \\longrightarrow Y^{\\prime }_m \\longrightarrow Y^{\\prime }_{m-1} \\longrightarrow \\cdots $ such that $Y \\subseteq Y^{\\prime } \\subseteq X$ and $Y^{\\prime } \\in {\\rm dw}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"Theorem 4.1 Every $X \\in {\\rm ex}\\widetilde{\\mathcal {P}_n}$ has a ${\\rm ex}\\widetilde{\\mathcal {P}_n^{\\le \\kappa }}$ -filtration.",
"Let $X \\in {\\rm ex}\\widetilde{\\mathcal {P}_n}$ .",
"We construct a ${\\rm ex}\\widetilde{\\mathcal {P}_n^{\\le \\kappa }}$ -filtration of $X$ using transfinite induction.",
"For $\\alpha = 0$ set $X^0 = 0$ .",
"For the case $\\alpha = 1$ let $m \\in \\mathbb {Z}$ be arbitrary and let $T_1 \\subseteq X_m$ be a small submodule of $X_m$ .",
"Then there exists a small submodule $Y^1_m$ of $X_m$ such that $T_1 \\subseteq Y^1_m$ and that $Y^1_m$ has a small and nice projective subresolution of a given resolution of $X_m$ .",
"Note that $\\partial _m(Y^1_m)$ is a submodule of $X_{m-1}$ with cardinality $\\le \\kappa $ , so there exists a submodule $Y^1_{m-1}$ of $X_{m-1}$ such that $\\partial _m(Y^1_m) \\subseteq Y^1_{m-1}$ and that $Y^1_{m-1}$ has a small and nice projective subresolution of a given resolution of $X_{m-1}$ .",
"Keep repeating this argument infinitely many times.",
"We obtain a complex $ Y^1 = \\cdots \\longrightarrow 0 \\longrightarrow Y^1_m \\longrightarrow Y^1_{m-1} \\longrightarrow \\cdots $ which is a subcomplex of $X$ and $Y^1 \\in {\\rm dw}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"Note that $Y^1$ is not necessarily exact.",
"We shall construct a complex $X^1$ from $Y^1$ such that $X^1 \\subseteq X$ and $X^1 \\in {\\rm ex}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"The rest of this proof uses an argument similar to the one used in [4].",
"Fix any $p \\in \\mathbb {Z}$ .",
"Then ${\\rm Card}(Y^1_p) \\le \\kappa $ and so ${\\rm Card}(Z_p Y^1) \\le \\kappa $ .",
"Since $X$ is exact and ${\\rm Card}(Z_p Y^1) \\le \\kappa $ , there exists a submodule $U \\subseteq X_{p+1}$ with ${\\rm Card}(U) \\le \\kappa $ such that $Z_p Y^1 \\subseteq \\partial _{p+1}(U)$ .",
"Let $C^1$ be a small subcomplex of $X$ such that $U \\subseteq C_{p+1}$ , $C_j = 0$ for every $j > p+1$ , and that each $C_j$ with $j \\le p$ has a small and nice projective subresolution of a given resolution of $X_j$ .",
"Since $Y^1 + C$ is a bounded above subcomplex of $X$ , there exists a small subcomplex $Y^2$ of $X$ such that $Y^1 + C \\subseteq Y^2$ and that each $Y^2_j$ has a small and nice projective subresolution of a given resolution of $X_j$ .",
"Note that $Z_p Y^1 \\subseteq \\partial _{p+1}(Y^2_{p+1})$ .",
"Construct $Y^3$ from $Y^2$ as we just constructed $Y^2$ from $Y^1$ , and so on, making sure to use the same $p \\in \\mathbb {Z}$ at each step.",
"Set $X^1 = \\bigcup _{j = 1}^\\infty Y^j \\subseteq X$ .",
"Note that $X^1$ is exact at $p$ .",
"Repeat this argument to get exactness at any level.",
"So we may assume that $X^1$ is an exact complex.",
"Every $X^1_k$ has a small and nice projective subresolution of the given resolution of $X_k$ .",
"For every $j$ one has a projective subresolution of the form $ 0 \\longrightarrow \\bigoplus _{i \\in I^j_{n}(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^j_1(k)} P^i_1(k) \\longrightarrow \\bigoplus _{i \\in I^j_0(k)} P^i_0(k) \\longrightarrow Y^j_k \\longrightarrow 0, $ where $I^1_l(k) \\subseteq I^2_l(k) \\subseteq \\cdots $ for every $0\\le l \\le n$ , by Lemma REF .",
"If we take the union of all of the previous sequences, then we obtain the following exact sequence: $ 0 \\longrightarrow \\bigoplus _{i \\in \\bigcup _{j\\ge i} I^j_{n}(k)} P^i_n(k) \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in \\bigcup _{j \\ge 1} I^j_0(k)} P^i_0(k) \\longrightarrow \\bigcup _{j \\ge 1} Y^j_k = X^1_j \\longrightarrow 0, $ where $\\bigcup _{j \\ge 1} I^j_l(k) \\subseteq I_l(k)$ for every $0 \\le l \\le n$ .",
"Therefore, $X^1\\in {\\rm ex}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"Now consider the quotient complex $ X / X^1 = \\cdots \\longrightarrow X_{k+1} / X^1_{k+1} \\longrightarrow X_k / X^1_k \\longrightarrow X_{k-1} / X^1_{k-1} \\longrightarrow \\cdots .",
"$ Note that each $X_k / X^1_k$ is $n$ -projective and that $X / X^1$ is exact.",
"We apply the same procedure above to the complex $X / X^1$ in order to get a complex $X^2 / X^1 \\subseteq X / X^1$ such that $X^2 / X^1 \\in {\\rm ex}\\widetilde{\\mathcal {P}_n^{\\le \\kappa }}$ .",
"Note that $X^2$ is an exact complex since the class of exact complexes is closed under extensions, and so $X^2 \\in {\\rm ex}\\widetilde{\\mathcal {P}_n}$ .",
"The rest of the proof follows by using transfinite induction.",
"It follows by Proposition REF and the Eklof and Trlifaj Theorem that $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ is a complete cotorsion pair cogenerated by ${\\rm ex}\\widetilde{\\mathcal {P}_n^{\\le \\kappa }}$ ."
],
[
"The staircase zig-zag argument",
"In this section, we present another method to prove that the induced cotorsion pairs $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ are complete.",
"This method also applies for the cotorsion pairs $({\\rm dw}\\widetilde{\\mathcal {F}_n}, ({\\rm dw}\\widetilde{\\mathcal {F}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {F}_n}, ({\\rm ex}\\widetilde{\\mathcal {F}_n})^\\perp )$ .",
"From now on, let $\\mathcal {A}$ denote either the class of projective modules or the class of flat modules, and let $\\mathcal {A}_n$ denote the class of all modules $M$ having an $\\mathcal {A}$ -resolution of length $n$ : $ 0 \\longrightarrow A_n \\longrightarrow \\cdots \\longrightarrow A_1 \\longrightarrow A_0 \\longrightarrow M \\longrightarrow 0, $ where $A_k \\in \\mathcal {A}$ for every $0 \\le k \\le n$ .",
"Note that $\\mathcal {A}_0 = \\mathcal {A}$ .",
"As we mentioned before, in [9] it is proven that $(\\mathcal {F}_n, \\mathcal {F}^\\perp _n)$ is a complete cotorsion pair.",
"We give a simpler proof of this fact.",
"Lemma 5.1 Let $M \\in \\mathcal {F}_n$ with a flat resolution $ 0 \\longrightarrow F_n \\stackrel{f_n}{\\longrightarrow }F_{n-1} \\longrightarrow \\cdots \\longrightarrow F_1 \\stackrel{f_1}{\\longrightarrow }F_0 \\stackrel{f_0}{\\longrightarrow }M \\longrightarrow 0 \\mbox{ \\ (1)} $ and let $N$ be a small submodule of $M$ .",
"Then there exists a flat subresolution $ 0 \\longrightarrow S^{\\prime }_n \\longrightarrow \\cdots \\longrightarrow S^{\\prime }_1 \\longrightarrow S^{\\prime }_0 \\longrightarrow N^{\\prime } \\longrightarrow 0 $ of (1) such that $S^{\\prime }_k$ is a small and pure submodule of $F_k$ , for every $0 \\le k \\le n$ , and such that $N \\subseteq N^{\\prime }$ .",
"Moreover, if $N$ has a subresolution of (1) $ 0 \\longrightarrow S_n \\longrightarrow \\cdots \\longrightarrow S_1 \\longrightarrow S_0 \\longrightarrow N \\longrightarrow 0 $ where $S_k$ is a small and pure submodule of $F_k$ , for every $0 \\le k \\le n$ , then the above resolution of $N^{\\prime }$ can be constructed in such a way that it contains the resolution of $N$ .",
"First, note that for every flat module $F$ and for every small submodule $N \\subseteq F$ , there exists a small and pure submodule $S \\subseteq F$ such that $N \\subseteq S$ (for a proof of this, see [6]).",
"For every $x \\in N$ there exists $y_x \\in F_0$ such that $x = f_0(y_x)$ .",
"Let $Y$ be the set $\\lbrace y_x \\mbox{ : } x \\in N \\mbox{ and } f_0(y_x) = x \\rbrace $ and consider the submodule $\\left< Y \\right> \\subseteq F_0$ .",
"Since $\\left< Y \\right>$ is small, there exists a small pure submodule $S_0(1) \\subseteq F_0$ such that $\\left< Y \\right> \\subseteq S_0(1)$ .",
"Note that $f_0(S_0(1)) \\supseteq N$ .",
"Now consider ${\\rm Ker}\\left( f_0|_{S_0(1)} \\right)$ and let $A$ be a set of preimages of ${\\rm Ker}\\left( f_0|_{S_0(1)} \\right)$ such that $f_1\\left( \\left< A \\right> \\right) \\supseteq {\\rm Ker}\\left( f_0|_{S_0(1)} \\right)$ .",
"It is easy to see that $\\left< A \\right>$ is a small submodule of $F_1$ , so we can embed it into a small pure submodule $S_1(1) \\subseteq F_1$ .",
"Hence we have $f_1(S_1(1)) \\supseteq {\\rm Ker}\\left( f_0|_{S_0(1)} \\right)$ .",
"Now consider ${\\rm Ker}\\left( f_1|_{S_1(1)} \\right)$ and repeat the same process above in order to find a small pure submodule $S_2(1) \\subseteq F_2$ such that $f_2(S_2(1)) \\supseteq {\\rm Ker}\\left( f_1|_{S_1(1)} \\right)$ .",
"Keep doing this until find a small pure submodule $S_n(1) \\subseteq F_n$ such that $f_n(S_n(1)) \\supseteq {\\rm Ker}\\left( f_{n-1}|_{S_{n-1}(1)} \\right)$ .",
"Now $f_n(S_n(1))$ is a small submodule of $F_{n-1}$ , so there is a small pure submodule $S_{n-1}(2) \\subseteq F_{n-1}$ such that $f_n(S_n(1)) \\subseteq S_{n-1}(2)$ .",
"Repeat this process until find a small pure submodule $S_0(2) \\subseteq F_0$ such that $f_1(S_1(2)) \\subseteq S_0(2)$ .",
"If we now consider ${\\rm Ker}\\left( f_0|_{S_0(2)} \\right) \\subseteq F_0$ , we repeat the same argument above to find a small pure submodule $S_1(3) \\subseteq F_1$ such that $f_1(S_1(3)) \\supseteq {\\rm Ker}\\left( f_0|_{S_0(2)} \\right)$ .",
"Keep repeating this zig-zag procedure infinitely many times and set $S_k = \\bigcup _{i \\ge 1} S_k(i)$ , for every $0 \\le k \\le n$ .",
"Note that each $S_k$ is a pure submodule of $F_k$ .",
"By construction, we get an exact complex $ 0 \\longrightarrow S_n \\longrightarrow S_{n-1} \\longrightarrow \\cdots \\longrightarrow S_1 \\longrightarrow S_0 \\longrightarrow Q \\longrightarrow 0, \\mbox{ \\ \\ \\ (2)} $ where $Q = {\\rm CoKer}(f_1|_{S_1}) \\subseteq M$ .",
"If we take the quotient of (1) by (2), we get an exact complex $ 0 \\longrightarrow F_n / S_n \\longrightarrow F_{n-1} / S_{n-1} \\longrightarrow \\cdots \\longrightarrow F_1 / S_1 \\longrightarrow F_0 / S_0 \\longrightarrow M / Q \\longrightarrow 0.",
"$ Since each $S_k$ is a pure submodule of $F_k$ , we know that $S_k$ and $F_k / S_k$ are flat modules.",
"Therefore, $Q$ is a small $n$ -flat submodule with $N \\subseteq Q$ such that $M / Q$ is also $n$ -flat.",
"The rest of the proof follows as in Lemma REF .",
"Remark 5.1 From Lemma REF and Lemma REF , we have that for every $A \\in \\mathcal {A}_n$ and for every small submodule $0 \\ne N \\subseteq A$ , there exists a small submodule $A^{\\prime } \\subseteq A$ in $\\mathcal {A}$ such that $N \\subseteq A^{\\prime }$ and $A / A^{\\prime } \\in \\mathcal {A}$ .",
"Theorem 5.1 Let $X \\in {\\rm ex}\\widetilde{\\mathcal {A}_n}$ and let $x \\in X$ (i.e.",
"$x \\in X_m$ for some $m \\in \\mathbb {Z}$ ).",
"Then there exists a small complex $Y \\in {\\rm ex}\\widetilde{\\mathcal {A}^{\\le \\kappa }_n}$ such that $x \\in Y$ and $X / Y \\in {\\rm ex}\\widetilde{\\mathcal {A}_n}$ .",
"The following proof is based on an argument given in [1], where the authors prove that $({\\rm dw}\\widetilde{\\mathcal {F}_0}, ({\\rm dw}\\widetilde{\\mathcal {F}_0})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {F}_0}, ({\\rm ex}\\widetilde{\\mathcal {F}_0})^\\perp )$ are complete cotorsion pairs.",
"We shall call this argument the staircase zig-zag.",
"[Proof of Theorem REF ] Assume without loss of generality that $x \\in X_0$ .",
"Consider the submodule $\\left< x \\right> \\subseteq X_0$ .",
"Since $X_0 \\in \\mathcal {A}$ and $\\left< x \\right>$ is small, we can embed $\\left< x \\right>$ into a submodule $\\mathcal {A}^{\\le \\kappa } \\ni Y^1_0 \\subseteq X_0$ such that $X_0 / Y^1_0 \\in \\mathcal {A}$ .",
"Since $X$ is exact, we can construct a small and exact subcomplex $ L^1 \\mbox{ = } \\cdots L^1_2 \\longrightarrow L^1_1 \\longrightarrow Y^1_0 \\longrightarrow \\partial _0(Y^1_0) \\longrightarrow 0 \\longrightarrow \\cdots .",
"$ Since $\\partial _0(Y^1_0)$ is small, there exists a submodule $\\mathcal {A}^{\\le \\kappa } \\ni Y_{-1}^2 \\subseteq X_{-1}$ such that $X_{-1} / Y_{-1}^2 \\in \\mathcal {A}$ .",
"Now construct a small and exact subcomplex $ L^2 \\mbox{ = } \\cdots L^2_2 \\longrightarrow L^2_1 \\longrightarrow L^2_0 \\longrightarrow Y^2_{-1} \\longrightarrow \\partial _{-1}(Y^2_{-1}) \\longrightarrow 0 \\longrightarrow \\cdots .",
"$ Note that it is possible to construct $L^2$ containing $L^1$ .",
"Now embed $L^2_0$ into a submodule $\\mathcal {A}^{\\le \\kappa } \\ni Y^3_0 \\subseteq X_0$ such that $X_0 / Y^3_0 \\in \\mathcal {A}$ .",
"Again, construct a small and exact subcomplex $ L^3 \\mbox{ = } \\cdots \\longrightarrow L^3_2 \\longrightarrow L^3_1 \\longrightarrow Y^3_0 \\longrightarrow Y^2_{-1} + \\partial _0(Y_0^3) \\longrightarrow \\partial _{-1}(Y_{-1}^2) \\longrightarrow 0 \\longrightarrow \\cdots $ containing $L^2$ .",
"Now let $Y^4_1 \\in \\mathcal {A}^{\\le \\kappa }$ be a submodule of $X_1$ containing $L^3_1$ such that $X_1 / Y^4_1 \\in \\mathcal {A}$ , and construct an exact and small complex $ L^4 \\mbox{ = } \\cdots \\longrightarrow L^4_2 \\longrightarrow Y^4_1 \\longrightarrow Y^3_0 + \\partial _1(Y_1^4) \\longrightarrow Y^2_{-1} + \\partial _0(Y_0^3) \\longrightarrow \\partial _{-1}(Y^2_{-1}) \\longrightarrow 0 \\longrightarrow \\cdots $ containing $L^3$ .",
"Embed $Y^3_0 + \\partial _1(Y^4_0)$ into a submodule $\\mathcal {A}^{\\le \\kappa } \\ni Y^5_0 \\subseteq X_0$ such that $X_0 / Y^5_0 \\in \\mathcal {A}$ .",
"Construct an exact and small subcomplex $ L^5 \\mbox{ = } \\cdots \\longrightarrow L^5_2 \\longrightarrow L^5_1 \\longrightarrow Y^5_0 \\longrightarrow Y^2_{-1} + \\partial _0(Y^5_0) \\longrightarrow \\partial _{-1}(Y^2_{-1}) \\longrightarrow 0 \\longrightarrow \\cdots $ containing $L^4$ .",
"In a similar way, construct small and exact complexes $L^6 & = \\cdots \\longrightarrow L^6_1 \\longrightarrow L^6_0 \\longrightarrow Y^6_{-1} \\longrightarrow \\partial _{-1}(Y^6_{-1}) \\longrightarrow 0 \\longrightarrow \\cdots , \\\\L^7 & = \\cdots \\longrightarrow L^7_1 \\longrightarrow L^7_0 \\longrightarrow L^7_{-1} \\longrightarrow Y^7_{-2} \\longrightarrow \\partial _{-2}(Y_{-2}^7) \\longrightarrow 0 \\longrightarrow \\cdots ,$ such that $Y_{-1}^6 \\in \\mathcal {A}$ is a small submodule of $X_{-1}$ containing $Y^6_{-1} + \\partial _0(Y^5_0)$ , and $Y_{-2}^7 \\in \\mathcal {A}$ is a small submodule of $X_{-2}$ containing $\\partial _{-1}(Y_{-1}^6)$ .",
"We have the following commutative diagram of subcomplexes of $X$ : where the $k+1$ -th complex can be constructed in such a way that it contains the $k$ -th complex.",
"Note that the submodules $S^k_i$ appear according to the following pattern: Let $Y = \\bigcup _{n \\ge 1} L^n$ , where $Y_i = \\bigcup _{n \\ge 1} (L^n)_i$ .",
"It is clear that $Y$ is an exact complex.",
"We check that $Y$ is also a ${\\rm dw}\\widetilde{\\mathcal {A}_n}$ .",
"For example, consider $ Y_0 = Y^1_0 \\cup L^2_0 \\cup Y^3_0 \\cup (Y^3_0 + \\partial _1(Y^4_1)) \\cup Y^5_0 \\cup \\cdots = Y^1_0 \\cup Y^3_0 \\cup Y^5_0 \\cup \\cdots .",
"$ It is clear that $Y_0$ is small.",
"At this point, we split the proof in two cases: (1) $\\mathcal {A} = \\mathcal {P}_0$ : Consider a projective resolution of $X_0$ of length $n$ , say $ 0 \\longrightarrow \\bigoplus _{i \\in I_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I_0} P^i_0 \\longrightarrow X_0 \\longrightarrow 0 \\mbox{ \\ (1)}, $ where each direct sum is a direct sum of countably generated projective modules.",
"By Lemma REF , we can construct $Y^1_0$ containing $\\left< x \\right>$ with a subresolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^1_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^1_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^1_0} P^i_0 \\longrightarrow Y^1_0 \\longrightarrow 0 \\mbox{ \\ (2)}, $ where each $I^1_k$ is a small subset of $I_k$ .",
"Note that the quotient of (1) by (2) yields a projective resolution of $X_0 / Y^1_0$ of length $n$ , so $X_0 / Y^1_0 \\in \\mathcal {P}_n$ .",
"Using Lemma REF again, we can construct a subresolution $ 0 \\longrightarrow \\bigoplus _{i \\in I^3_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^3_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^3_0} P^i_0 \\longrightarrow Y^3_0 \\longrightarrow 0 \\mbox{ (3)} $ containing (2) such that $X_0 / Y^3_0 \\in \\mathcal {P}_n$ .",
"We keep applying Lemma REF to get an ascending chain of subresolutions of (1): $0 & \\longrightarrow \\bigoplus _{i \\in I^1_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^1_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^1_0} P^i_0 \\longrightarrow Y^1_0 \\longrightarrow 0 \\\\0 & \\longrightarrow \\bigoplus _{i \\in I^3_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^3_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^3_0} P^i_0 \\longrightarrow Y^3_0 \\longrightarrow 0 \\\\0 & \\longrightarrow \\bigoplus _{i \\in I^5_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in I^5_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in I^5_0} P^i_0 \\longrightarrow Y^5_0 \\longrightarrow 0 \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots $ Now we take the union of this ascending chain and get an exact complex $0 & \\longrightarrow \\bigcup _j \\bigoplus _{i \\in I^j_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigcup _j \\bigoplus _{i \\in I^j_1} P^i_1 \\longrightarrow \\bigcup _j \\bigoplus _{i \\in I^j_0} P^i_0 \\longrightarrow \\bigcup _j Y^j_0 \\longrightarrow 0 \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ = \\\\0 & \\longrightarrow \\bigoplus _{i \\in \\bigcup _j I^j_n} P^i_n \\longrightarrow \\cdots \\longrightarrow \\bigoplus _{i \\in \\bigcup _j I^j_1} P^i_1 \\longrightarrow \\bigoplus _{i \\in \\bigcup _j I^j_0} P^i_0 \\longrightarrow Y_0 \\longrightarrow 0 \\mbox{ \\ (4)}$ Since each $\\bigcup _j I^j_k$ is a small subset of $I_k$ , we have that the previous sequence is a $\\mathcal {P}^{\\le \\kappa }_0$ -subresolution of (1).",
"Note also that the quotient of (1) by (4) yields a projective resolution of $X_0 / Y_0$ of length $n$ .",
"Then $Y_0 \\in \\mathcal {P}^{\\le \\kappa }_n$ .",
"In a similar way, we can show that $Y_m \\in \\mathcal {P}^{\\le \\kappa }_n$ and $X_m / Y_m \\in \\mathcal {P}_n$ , for every $m \\in \\mathbb {Z}$ .",
"Hence, $Y \\in {\\rm ex}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"Note also that the quotient of exact complexes is exact, so we also have $X / Y \\in {\\rm ex}\\widetilde{\\mathcal {P}^{\\le \\kappa }_n}$ .",
"(2) $\\mathcal {A} = \\mathcal {F}_0$ : Consider a flat resolution of $X_0$ of length $n$ , say $ 0 \\longrightarrow F_n \\longrightarrow \\cdots \\longrightarrow F_1 \\longrightarrow F_0 \\longrightarrow X_0 \\longrightarrow 0 \\mbox{ \\ (1')}.",
"$ By Lemma REF , we can construct a subresolution $ 0 \\longrightarrow S^1_n \\longrightarrow \\cdots \\longrightarrow S^1_1 \\longrightarrow S^1_0 \\longrightarrow Y^1_0 \\longrightarrow 0, $ where $\\left< x \\right> \\subseteq Y^1_0$ , and each $S^1_k$ is a small and pure submodule of $F_k$ .",
"As we did in the previous case, applying Lemma REF infinitely many times, we can get an ascending chain of subresolutions $0 & \\longrightarrow S^1_n \\longrightarrow \\cdots \\longrightarrow S^1_1 \\longrightarrow S^1_0 \\longrightarrow Y^1_0 \\longrightarrow 0 \\\\0 & \\longrightarrow S^3_n \\longrightarrow \\cdots \\longrightarrow S^3_1 \\longrightarrow S^3_0 \\longrightarrow Y^3_0 \\longrightarrow 0 \\\\0 & \\longrightarrow S^5_n \\longrightarrow \\cdots \\longrightarrow S^5_1 \\longrightarrow S^5_0 \\longrightarrow Y^5_0 \\longrightarrow 0 \\\\& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\vdots $ Taking the union of these subresolutions yields an exact sequence $ 0 \\longrightarrow \\bigcup _j S^j_n \\longrightarrow \\cdots \\longrightarrow \\bigcup _j S^j_1 \\longrightarrow \\bigcup _j S^j_0 \\longrightarrow Y_0 \\longrightarrow 0 \\mbox{ \\ (2')}, $ where each $\\bigcup _j S^j_k$ is a small and pure submodule of $F_k$ , and so it is flat and the quotient $F_k / \\bigcup _j S^j_k$ is also flat.",
"Then we have that $Y_0 \\in \\mathcal {F}^{\\le \\kappa }_n$ and $X_0 / Y_0 \\in \\mathcal {F}_n$ (take the quotient of (1') by (2')).",
"In a similar way, we can show that $Y_m \\in \\mathcal {F}^{\\le \\kappa }_n$ and $X_m / Y_m \\in \\mathcal {F}_n$ , for every $m \\in \\mathcal {Z}$ .",
"It follows $Y \\in {\\rm ex}\\widetilde{\\mathcal {F}^{\\le \\kappa }_n}$ and $X / Y \\in {\\rm ex}\\widetilde{\\mathcal {F}_n}$ .",
"In both cases, we have constructed a subcomplex $Y \\subseteq X$ containing $x$ such that $Y \\in {\\rm ex}\\widetilde{\\mathcal {A}^{\\le \\kappa }_n}$ and $X / Y \\in {\\rm ex}\\widetilde{\\mathcal {A}_n}$ .",
"Remark 5.2 Note that using the staircase zig-zag method, there is no need to assume that $R$ is right noetherian."
],
[
"dw-$n$ -projective and dw-{{formula:2f35820b-aa3e-4dec-9b87-40f9ab855279}} -flat model structures",
"In this section we obtain two model structures on $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ , one from the two complete cotorsion pairs $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ , and the other one from $({\\rm dw}\\widetilde{\\mathcal {F}_n}, ({\\rm dw}\\widetilde{\\mathcal {F}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {F}_n}, ({\\rm ex}\\widetilde{\\mathcal {F}_n})^\\perp )$ .",
"Recall that a subcategory $\\mathcal {D}$ of an abelian category $\\mathcal {C}$ is said to be thick if the following two conditions are satisfied: (a) $\\mathcal {D}$ is closed under retracts, i.e., given a sequence $ D^{\\prime } \\stackrel{f}{\\longrightarrow }D \\stackrel{g}{\\longrightarrow }D^{\\prime } $ with $g \\circ f = {\\rm id}_{D^{\\prime }}$ and $D \\in \\mathcal {D}$ , then $D^{\\prime } \\in \\mathcal {B}$ .",
"(b) If two out of three of the terms in a short exact sequence $ 0 \\longrightarrow D^{\\prime \\prime } \\longrightarrow D \\longrightarrow D^{\\prime } \\longrightarrow 0 $ are in $\\mathcal {D}$ , then so is the third.",
"For example, the class $\\mathcal {E}$ of exact complexes is thick.",
"The following theorem by Hovey describes how to get an abelian model structure from two complete cotorsion pairs.",
"Theorem 6.1 [10] Let $\\mathcal {C}$ be a bicomplete abelian category with enough projective and injective objects, and let $(\\mathcal {A}, \\mathcal {B} \\cap \\mathcal {E})$ and $(\\mathcal {A} \\cap \\mathcal {E}, \\mathcal {B})$ be two complete cotorsion pairs in $\\mathcal {C}$ such that the class $\\mathcal {E}$ is thick.",
"Then there is a unique abelian model structure on $\\mathcal {C}$ such that $\\mathcal {A}$ is the class of cofibrant objects, $\\mathcal {B}$ is the class of fibrant objects, and $\\mathcal {E}$ is the class of trivial objects.",
"Cotorsion pairs of the form $(\\mathcal {A}, \\mathcal {B} \\cap \\mathcal {E})$ and $(\\mathcal {A} \\cap \\mathcal {E}, \\mathcal {B})$ are called compatible by Gillespie in [8].",
"We shall use the following result from [4] to show that $({\\rm dw}\\widetilde{\\mathcal {P}_n}, ({\\rm dw}\\widetilde{\\mathcal {P}_n})^\\perp )$ and $({\\rm ex}\\widetilde{\\mathcal {P}_n}, ({\\rm ex}\\widetilde{\\mathcal {P}_n})^\\perp )$ are compatible cotorsion pairs.",
"Lemma 6.1 [4] If $(\\mathcal {C}, \\mathcal {D}^{\\prime })$ is a cotorsion pair and $(\\mathcal {U, V})$ is a complete and hereditary cotorsion pair in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ (i.e.",
"$\\mathcal {V}$ contains the injective complexes and it is closed under extensions and under taking cokernels of monomorphisms), and if $\\mathcal {U} \\subseteq \\mathcal {C}$ then when $(\\mathcal {C} \\cap \\mathcal {V})^\\perp = \\mathcal {D}$ , we have $\\mathcal {D}^{\\prime } = \\mathcal {D} \\cap \\mathcal {V}$ .",
"In the previous lemma, put $(\\mathcal {C}, \\mathcal {D}^{\\prime }) = ({\\rm dw}\\widetilde{\\mathcal {A}_n}, ({\\rm dw}\\widetilde{\\mathcal {A}_n})^\\perp )$ and $(\\mathcal {U}, \\mathcal {V}) = (\\mbox{}^\\perp \\mathcal {E}, \\mathcal {E})$ .",
"In [4] it is proven that $(\\mbox{}^\\perp \\mathcal {E}, \\mathcal {E})$ is cogenerated by a set, so it is complete.",
"In [7], it is proven that $\\mbox{}^\\perp \\mathcal {E}$ is the class of dg-projective complexes, where a complex $X$ is dg-projective if $X_m \\in \\mathcal {P}_0$ for every $m \\in \\mathbb {Z}$ and every map $X \\longrightarrow Y$ is nullhomotopic whenever $Y$ is exact.",
"Hence, it is clear that $\\mbox{}^\\perp \\mathcal {E} \\subseteq {\\rm dw}\\widetilde{\\mathcal {A}_n}$ , i.e.",
"$\\mathcal {U} \\subseteq \\mathcal {C}$ .",
"Setting $\\mathcal {D} = (\\mathcal {C} \\cap \\mathcal {V})^\\perp = ({\\rm dw}\\widetilde{\\mathcal {A}_n} \\cap \\mathcal {E})^\\perp $ , we have $({\\rm dw}\\widetilde{\\mathcal {A}_n})^\\perp = ({\\rm dw}\\widetilde{\\mathcal {A}_n} \\cap \\mathcal {E})^\\perp \\cap \\mathcal {E}$ .",
"So we obtain $({\\rm dw}\\widetilde{\\mathcal {A}_n}, ({\\rm dw}\\widetilde{\\mathcal {A}_n})^\\perp ) & = ({\\rm dw}\\widetilde{\\mathcal {A}_n}, ({\\rm dw}\\widetilde{\\mathcal {A}_n} \\cap \\mathcal {E})^\\perp \\cap \\mathcal {E}), \\\\({\\rm ex}\\widetilde{\\mathcal {A}_n}, ({\\rm ex}\\widetilde{\\mathcal {A}_n})^\\perp ) & = ({\\rm dw}\\widetilde{\\mathcal {A}_n} \\cap \\mathcal {E}, ({\\rm dw}\\widetilde{\\mathcal {A}_n} \\cap \\mathcal {E})^\\perp ).",
"\\\\$ From Theorem REF and the previous equalities, we have: Corollary 6.1 There exists a unique abelian model structure in $ {\\rm {\\bf Ch}}(R{\\rm {\\bf -Mod}}) $ such that ${\\rm dw}\\widetilde{\\mathcal {A}_n}$ is the class of cofibrant objects, $({\\rm ex}\\widetilde{\\mathcal {A}_n})^\\perp $ is the class of fibrant objects, and $\\mathcal {E}$ is the class of trivial objects.",
"In the case $\\mathcal {A} = \\mathcal {P}_0$ we name this structure the dw-$n$ -projective model structure, and when $\\mathcal {A} = \\mathcal {F}_0$ we name it the dw-$n$ -flat model structure.",
"Remark 6.1 As far as the author knows, the dw-projective model structure first appeared in [4], while the dw-flat model did in [8]."
]
] | 1204.1358 |
[
[
"Multiple-channel generalization of Lellouch-Luscher formula"
],
[
"Abstract We generalize the Lellouch-Luscher formula, relating weak matrix elements in finite and infinite volumes, to the case of multiple strongly-coupled decay channels into two scalar particles.",
"This is a necessary first step on the way to a lattice QCD calculation of weak decay rates for processes such as D -> pi pi and D -> KK.",
"We also present a field theoretic derivation of the generalization of Luscher's finite volume quantization condition to multiple two-particle channels.",
"We give fully explicit results for the case of two channels, including a form of the generalized Lellouch-Luscher formula expressed in terms of derivatives of the energies of finite volume states with respect to the box size.",
"Our results hold for arbitrary total momentum and for degenerate or non-degenerate particles."
],
[
"Introduction",
"Lattice calculations have made considerable progress toward a first-principles determination of the $K\\rightarrow \\pi \\pi $ weak decay amplitudes [1], [2].",
"The methodology is now in place, results for the $I=2$ final state with a complete error budget are available [2], and complete results for the more challenging $I=0$ final states should become available in the next few years.",
"At that stage we will finally learn whether and in what manner QCD can explain the $\\Delta I=1/2$ rule and the observed CP-violation rate in $K\\rightarrow \\pi \\pi $ decays.",
"Encouraged by this progress, it is natural to consider what information lattice calculations might eventually offer concerning the decays of heavier mesons.",
"For example, the LHCb experiment recently reported evidence for CP-violation in (the difference of) $D^0\\rightarrow \\pi ^+\\pi ^-$ and $D^0 \\rightarrow K^+ K^-$ decays [3].",
"Although the rate is larger than naive expectations from the Standard Model (SM), there is, at present, sufficient uncertainty in the SM prediction for it to be consistent with the LHCb result (see, e.g.",
"Refs.",
"[4], [5], [6], [7], [8], [9]).",
"This raises the obvious question of whether a calculation using lattice methods is feasible.",
"The aim of this paper is to take a first step in developing the methodology for such a calculation.",
"We show how, if one can ignore all but two-particle channels, then a generalization of the work of Lüscher, and of Lellouch and Lüscher, would allow, in principle, a calculation of the required matrix elements from lattice calculations in a finite volume.",
"In practice, however, channels with more than two particles are coupled by the strong interactions to $\\pi \\pi $ and $K\\overline{K}$ , e.g.",
"the four pion channel, and they cannot be ignored at center of mass (CM) energies as high as the $D^0$ -meson mass ($M_{\\!",
"D^0} = 1865 \\mathop {\\rm MeV}\\nolimits $ ).",
"Thus our method would yield only semi-quantitative results for the desired matrix elements.",
"Nevertheless, it is a necessary first step, and work is underway to extend the methodology to channels with multiple particles (see, e.g., Ref. [10]).",
"It is instructive to recall the three essential ingredients needed for the lattice calculation of $K\\rightarrow \\pi \\pi $ amplitudes.",
"First, one needs to know the relation between the energies of two-pion states in a finite box and the infinite-volume scattering amplitude.",
"This was worked out by Lüscher in Refs.",
"[11], [12], [13], [14] (and generalized to a moving frame in Refs.",
"[15], [16], [17]).",
"Second, one needs the relation between the matrix element that one can determine on the lattice, which connects a kaon to a finite-volume two pion state, and the infinite volume matrix element which determines the decay rate.",
"This was provided by Lellouch and Lüscher in Ref.",
"[18] (and generalized to a moving frame in Refs.",
"[16], [17]).",
"Finally, one must calculate the large number of Wick contractions that contribute, including several quark-disconnected contractions requiring special methods and high statistics.",
"In this stage one also extrapolates to physical quark masses.",
"This entire program has been carried out for the $I=2$ final state [2], and a successful pilot calculation has been done for the more challenging $I=0$ case [1].",
"The calculation of $D^0$ decays is considerably more challenging.",
"In particular, the first two of the three aforementioned ingredients need to be generalized to account for the opening of many channels.",
"If we focus on the $I=0$ final state, then strong-interaction rescattering connects two-pion final states to those with four, six, etc.",
"pions, as well as $K\\overline{K}$ and $\\eta \\eta $ states.It is important to note that the fact that the $D^0$ has a very large number of decay channels [19] is not itself a concern, but rather that, having fixed the final-state quantum numbers, in our case to $I=0$ , there are still a large number of states.",
"In a lattice calculation, one can separately consider the decays to states with differing strong-interaction quantum numbers.",
"As already noted, we consider here only the case in which several two-particle channels are open, which for the $D^0$ would mean keeping the $\\pi \\pi $ , $K\\overline{K}$ and $\\eta \\eta $ channels while ignoring those with four or more pions.",
"We make this approximation not because we think that it is a good description of reality at the $D^0$ mass, but rather because it is a necessary first step towards the required formalism.",
"Within this approximation, we provide here the generalization of both the Lüscher quantization condition and the Lellouch-Lüscher (LL) formula.",
"These generalizations are useful also in many other systems.",
"For example, the quantization condition allows the determination of the parameters of the $S$ -matrix in the $I=0$ channel above the two kaon threshold (and thus in the region of the $f_0(980)$ resonance), because the coupling to four or more pions remains weak for such energies.",
"The same should be true in the $I=1$ case, where $K\\overline{K}$ and $\\eta \\pi $ are the dominant channels in the vicinity of the $a_0(980)$ .",
"The multi-channel LL formula can be used to calculate $K \\rightarrow \\pi \\pi $ amplitudes including isospin breaking (so that $\\pi ^{\\pm }$ and $\\pi ^0$ are not degenerate).",
"Generalization to baryon decays are also possible, but this requires dealing with particles with spin, which we do not attempt here.",
"There have been a large number of recent papers studying the generalization of the Lüscher quantization condition to multiple two-body channels [20], [21], [22], [23] and assessing its utility.",
"The work of Ref.",
"[20] uses non-relativistic quantum mechanics, while Ref.",
"[21] is based on a non-relativistic effective field theory.",
"References [22] and [23] are based on relativistic field theory, and give an explicit result [Eq.",
"(3.5) of Ref.",
"[22]] for the case of two s-wave channels in which the total momentum vanishes and in which the contributions from higher partial waves are assumed negligible.",
"We also note that the multiple channel problem has been studied using an alternative approach based on the Bethe-Salpeter wavefunction [24].",
"We provide here, as a step on the way to the generalized LL formula, a derivation of the multiple-channel quantization condition within quantum field theory.",
"We include all allowed mixing between different partial waves.",
"No assumptions about the form of the interactions are needed, aside from the proviso, common to all approaches, that the range of the interaction must be smaller than the box size.",
"Also, our result holds for any value of the total momentum $\\vec{P}$ of the two particle system (i.e.",
"it holds for a moving or a stationary frame), and for either degenerate or non-degenerate particles in each channel.",
"We follow closely the approach of Ref.",
"[16], which presented a generalization of Lüscher's single-channel quantization condition to a moving frame.",
"Indeed, we find that the most general form of the final result, given in Eq.",
"(REF ), is identical in form to that of Ref.",
"[16] (modulo some minor changes in notation).",
"After deriving the general quantization condition in Sec.",
"we restrict our considerations to the simplified situation in which only s-wave scattering is included.",
"We focus on the case with two channels (suggestively labeled $\\pi \\pi $ and $K\\overline{K}$ ), although we also provide the generalization to more than two channels.",
"In the infinite volume theory, the two-channel system is described by a $2 \\times 2$ $S$ -matrix which, due to unitarity and symmetry, is determined by three real parameters [see Eq.",
"(REF ) below].",
"We use a particular parametrization of $S$ to rewrite our quantization condition in a convenient, pure real form [Eq.",
"(REF )].",
"We explain how our result is equivalent to that of Ref.",
"[22] in the case of a stationary frame.",
"As is discussed in Refs.",
"[21], [22], [23], three independent pieces of information are needed to determine the three independent $S$ -matrix parameters at each center of mass energy, $E^*$ .",
"References [22] and [23] discuss in some detail the prospects for using either twisted boundary conditions or uneven box sizes for this purpose.",
"We restrict ourselves here to an alternative approach, also mentioned in Refs.",
"[22] and [23], of using three different choices for the parameters $\\lbrace L,\\vec{P}\\rbrace $ , where $L$ is the box size.",
"(We assume a cubic box and periodic boundary conditions.)",
"The parameters $\\lbrace L,\\vec{P}\\rbrace $ must be tuned such that there is a two-particle state in the spectrum having the desired value of $E^*$ .",
"In this way one obtains three independent conditions, and can solve for the $S$ -matrix parameters at the chosen value of $E^*$ .",
"Turning now to the LL formula, we follow the same approach as used by Lellouch and Lüscher in Ref. [18].",
"Specifically, we add a $D$ -meson to our two-channel system and analyze the effect of an infinitesimal weak perturbation on the quantization condition.",
"This yields a relation between a finite-volume weak matrix element and a linear combination of the desired infinite-volume matrix elements.",
"In Sec.",
"we present a derivation of the relation which follows closely the original LL work.",
"In the final result, Eq.",
"(REF ), the coefficients relating finite and infinite volume matrix elements are given in terms of the $S$ -matrix parameters and their derivatives, evaluated at the decay particle's mass.",
"These can be calculated using the multiple-channel quantization condition, as sketched above.",
"It turns out that three different lattice matrix elements are needed to separately determine the two infinite-volume matrix elements.",
"Note that this is the same as the number needed to determine the $S$ -matrix parameters.",
"For more than two channels this correspondence no longer holds.",
"Since the $S$ -matrix parameters and their derivatives are ultimately determined from the spectral energies, it should be possible to write a form for the generalized LL formula in terms of the spectral energies and their derivatives alone.",
"We derive such a form in Sec. .",
"The result, Eq.",
"(REF ), is probably more useful in practice than Eq.",
"(REF ).",
"The second derivation also brings out an important feature of the generalized LL formula.",
"Finite-volume energy eigenstates in the coupled-channel theory can be written as linear combinations of infinite volume $\\pi \\pi $ and $K\\overline{K}$ states having (in our case) $\\ell =0$ as well as the higher values of $\\ell $ allowed by the cubic symmetry of the box.",
"The LL methodology is (as noted in the original paper) simply a trick to determine the coefficients of the relevant $\\pi \\pi $ and $K\\overline{K}$ states.",
"This point has also been stressed recently by Ref.",
"[25] in a different context.",
"Our second derivation makes clear that, irrespective of the details of the weak Hamiltonian, one always obtains the same linear combination of $\\pi \\pi $ and $K\\overline{K}$ states, and that this feature holds for any number of channels.",
"The remainder of this article is organized as follows.",
"In the following section we give our derivation of the multiple channel quantization condition.",
"In Sec.",
"we restrict to $s$ -wave scattering and derive a useful form of the condition.",
"The multiple channel generalization of the LL formula is then derived in Sec.",
", and the alternative derivation is presented in Sec. .",
"We conclude in Sec. .",
"We include an appendix, in which we discuss the generalization of Watson's theorem to two channels.",
"The generalization of Lüscher's quantization formula to multiple channels for arbitrary $\\vec{P}$ using field-theoretic methods (the work described in our Secs.",
"and ) has also been considered by Briceno and Davoudi [26].",
"Our results are in complete agreement (although we use a different parametrization of the $S$ -matrix).",
"Their paper is being released simultaneously with the present article."
],
[
"Multiple channel extension of quantization condition",
"In this section we derive an extension to multiple two-body channels of the Lüscher quantization condition, which relates the infinite volume scattering amplitudes to finite volume energy levels.",
"We assume throughout a cubic spatial volume with extent $L$ and periodic boundary conditions.",
"The (Minkowski) time direction is taken to be infinite.",
"The total momentum $\\vec{P}= \\frac{2\\pi \\vec{n}_P}{L} \\hspace{30.0pt} ( \\vec{n}_P \\in \\mathbb {Z}^3 )$ is fixed but arbitrary, i.e.",
"the quantization condition we derive holds for a “moving frame” as well as a stationary frame.",
"We first consider the case of only two open channels, describing the extension to an arbitrary number of channels at the end of this section.",
"We take each channel to contain two massive, spinless particles.",
"The particles of channel one are labeled pions and are taken to be identical with mass $m_1=M_\\pi $ .",
"The particles of channel two, called kaons, are taken non-identical, though still degenerate, with mass $m_2=M_K$ .",
"What we have in mind is that the first channel corresponds to the $I=0$ $\\pi \\pi $ state, and the second to the $I=0$ $K \\overline{K}$ state.",
"Including both identical and non-identical pairs allows us to display the factors of $1/2$ that appear in the former case.",
"We consider degenerate particles to simplify the presentation, but describe the generalization to non-degenerate masses at the end of this section.",
"For concreteness, and to match the physical ordering, we take the pion to be lighter than the kaon.",
"For our results to hold, we must assume that the thresholds for three or more particles lie above the two kaon threshold.",
"If we assume a G-parity like symmetry, so that only even numbers of pions can couple to a two-pion state, then the ordering we need is $2 M_\\pi < 2 M_K < E^* < 4 M_\\pi \\,,$ where $E^*$ is the center of mass (CM) energy.",
"The only possible scattering events are then $\\begin{split}1 \\rightarrow 1\\!",
":& \\hspace{10.0pt} \\pi \\ \\pi \\rightarrow \\pi \\ \\pi \\\\1 \\rightarrow 2\\!",
":& \\hspace{10.0pt} \\pi \\ \\pi \\rightarrow K \\ \\overline{K}\\\\2 \\rightarrow 1\\!",
":& \\hspace{10.0pt} K \\ \\overline{K} \\rightarrow \\pi \\ \\pi \\\\2 \\rightarrow 2\\!",
":& \\hspace{10.0pt} K \\ \\overline{K} \\rightarrow K \\ \\overline{K}\\,.\\end{split}$ If $E^*$ drops below $2 M_K$ , only the $\\pi \\pi $ channel is open and the problem reduces to that discussed by Lüscher [11], [12], [13], [14].",
"The inequality $2 M_K< 4 M_\\pi $ does not, of course, hold for physical pions and kaons—the four and six pion thresholds occur below that for two kaons.",
"Nevertheless, the coupling to these higher multiplicity channels is weak at low energies, and our results should still hold approximately as long as we are not too far above the two kaon threshold.",
"Indeed, it may be that, in the $I=0$ case, the $\\eta \\eta $ channel becomes important before that with four or more pions.",
"If so, our formalism would still apply, generalized to three channels as described below.",
"The approximation of ignoring channels with more than two particles will become increasingly poor as the energy increases, and will likely give only a rough guide by the $D$ mass.",
"A qualitative indication of this (ignoring differences in phase space) is that the $f_0(1500)$ has a 50% branching fraction to $4\\pi $ , while the branches to $\\pi \\pi $ , $K\\overline{K}$ and $\\eta \\eta $ are $\\sim 35\\%$ , $9\\%$ and $5\\%$ , respectively [19].",
"The two channel quantization condition is obtained by a straightforward generalization of the single-channel approach of Ref. [16].",
"To make this note somewhat independent of that reference, we reiterate some of the pertinent details.",
"We begin by introducing a two body interpolating field $\\sigma (x)$ (not necessarily local) which couples to both channels.",
"Following Ref.",
"[16] we then define $C_L(P) = \\int _{L;x} e^{i(-\\vec{P} \\cdot \\vec{x} + E x^{0})}\\langle 0\\vert \\sigma (x) \\sigma ^\\dagger (0) \\vert 0 \\rangle $ where $P = (E, \\vec{P})$ is the total four momentum of the two particle system (in the frame where the finite volume condition is applied), and $\\int _{L;x} = \\int _L d^4 x$ is the spacetime integral over finite volume.",
"The relation to the CM energy used above is $E^* = \\sqrt{E^2 - \\vec{P}^{2}} \\,.$ The poles of $C_L$ give the energy spectrum of the finite volume theory, and thus the condition that $C_{L}$ diverge is precisely the quantization condition we are after.",
"Figure: (a) The initial series of ladder diagrams which builds upC L C_L [see Eq. ()].",
"The Bethe-Salpeter kernels iKi Kare connected by fully dressed propagators.",
"The dashed rectangleindicates finite volume momentum sum/integrals.",
"(b) and (c) The serieswhich build up the matrix element AA and the scattering amplitudeiℳi \\mathcal {M}.",
"Note that these series contain only the momentumintegrals appropriate to infinite volume.",
"(d) The resultingseries for the subtracted correlator [see Eqs.",
"() and()].",
"Each dashed vertical line indicates aninsertion of ℱ\\mathcal {F}, which carries the entire volume dependence(neglecting exponentially suppressed dependence).To proceed to a more useful form of the condition, we follow Ref.",
"[16] and write $C_L$ in terms of the Bethe-Salpeter kernel, as illustrated in Fig.",
"REF (a): $C_L(P) = \\int _{L;q} \\sigma _{j,q} \\big [z^2 \\Delta ^2 \\big ]_{jk,q}\\sigma ^\\dagger _{k,q}\\\\+ \\int _{L;q,q^{\\prime }} \\sigma _{j,q} \\big [z^2\\Delta ^2 \\big ]_{jk,q}i K_{kl;q,q^{\\prime }} \\big [z^2 \\Delta ^2 \\big ]_{lm,q^{\\prime }}\\sigma ^\\dagger _{m,q^{\\prime }} + \\cdots \\,.$ The notation here is as follows.",
"Indices $j$ , $k$ , $l$ and $m$ refer to the channel, and take the values 1 or 2.",
"The two particle intermediate states are summed/integrated as is appropriate to finite volume $\\int _{L,q} = \\frac{1}{L^3} \\sum _{\\vec{q}} \\int \\frac{d q^0}{2 \\pi }\\,.$ The summand/integrand includes the product of two fully dressed propagators $\\big [z^2 \\Delta ^2 \\big ]_{ij,q} = \\delta _{ij} \\eta _i\\big [z_i(q) \\Delta _i(q) \\big ] \\big [ z_i(P-q) \\Delta _i(P-q) \\big ] \\,,$ where $z_j(q) \\Delta _j(q) &=\\int d^4 x e^{i q x} \\langle \\phi _j(x) \\phi _j(0) \\rangle \\\\\\Delta _j(q) &= \\frac{i}{q^2 - m_j^2 + i \\epsilon } \\,.$ Here $\\phi _1$ and $\\phi _2$ are interpolating fields for pions and kaons, respectively, chosen such that $z_j=1$ on shell.",
"$\\eta _1=1/2$ and $\\eta _2 = 1$ account for the symmetry factors of the diagrams.",
"$K$ is related to the Bethe-Salpeter kernel $i K_{ij;q,q^{\\prime }} = i BS_{ij}(q, P-q,- q^{\\prime },- P+q^{\\prime })\\,,$ with $BS_{ij}$ the sum of all amputated $j\\rightarrow i$ scattering diagrams which are two-particle-irreducible in the $s$ -channel (with particles of either type).",
"Finally, $\\sigma _{j,q}$ and $\\sigma ^\\dagger _{j,q^{\\prime }}$ describe the coupling of the operators $\\sigma $ and $\\sigma ^\\dagger $ to the two-particle channel $j$ .",
"Their detailed form is not relevant; all we need to know is that they are regular functions of $q$ .",
"We emphasize two important features of Eq.",
"(REF ).",
"First, it does not rely on any choice of interactions between the pions and kaons, such as those predicted by chiral perturbation theory.",
"All the quantities that enter can be written in terms of non-perturbatively defined correlation functions.",
"Second, the kernel $i K$ and the propagator dressing function $z$ have only exponentially suppressed dependence on the volume [12].",
"Thus, if $L$ is large enough that such dependence is negligible (as we assume hereafter), we can take $i K$ and $z$ to have their infinite-volume forms.",
"The dominant power-law volume dependence enters through the momentum sums in the two-particle loops.",
"To extract this dependence, we use the identity derived in Ref.",
"[16], which relates these sums for a moving frame to infinite-volume momentum integrals plus a residue.",
"Before stating the identity we recall the relevant notation.",
"For any four vector $k^\\mu = (k^0,\\vec{k})$ in the moving frame, $k^{\\mu *} = (k^{0*},\\vec{k}^*)$ is the result of a boost to the CM frame.",
"In particular, the total four-momentum $(E,\\vec{P})$ boosts to $(E^*,\\vec{0})$ in the CM frame.",
"We also need the quantities $q_j^*=\\sqrt{(E^*)^2/4-m_j^2} \\,,$ which are the momenta of a pion ($j=1$ ) or kaon ($j=2$ ) in the CM frame.",
"The identity then reads (no sum on $i$ here): $\\int _{L;k} f(k) \\eta _i \\Delta _i(k) \\Delta _i(P-k) g(k) =\\\\\\int _{\\infty ;k} f(k) \\eta _i \\Delta _i(k) \\Delta _i(P-k) g(k)\\\\+ \\int d \\Omega _{q^*} d\\Omega _{q^{*^{\\prime }}} \\ f_i^*(\\hat{q}^*) \\ \\mathcal {F}_{ii}(\\hat{q}^*, \\hat{q}^{^{\\prime }*})g_i^*(\\hat{q}^{^{\\prime }*})\\,,$ with $\\int _{\\infty ;k}= \\int \\frac{d^4k}{(2\\pi )^4} \\,.$ We introduce two functions $f(k)$ and $g(k)$ to correspond to the momentum dependence entering from the left and right of the loop integrals, as well as that from the dressing functions [see Fig.",
"REF (a)].",
"The functions $f$ and $g$ must have ultraviolet behavior that renders the integral/sum convergent.",
"In addition, the branch cuts they contain, corresponding to four or more intermediate particles, must be such that, after the $k^0$ contour integration, they introduce no singularities for real $\\vec{k}$ .",
"This condition holds when $0< E^* < 4 M_\\pi $ .",
"The last line of (REF ) depends on the values of the functions $f$ and $g$ when the two particles are on-shell, and thus only on the direction of the CM momentum, $\\hat{q}^*$ .",
"Specifically, if $q_i^\\mu $ is the moving frame momentum that boosts to the on-shell momentum $(E^*/2, \\vec{q}_i\\,^*)$ , then $f_i^*(\\hat{q}^*) = f(q_i)\\,, \\quad g_i^*(\\hat{q}^*) = g(q_i)\\,.$ Finally the quantity ${\\mathcal {F}}$ , which depends on $q^*$ , $L$ and the particle mass, contains the power-law finite-volume dependence of the loop sum/integral.The result (REF ) is equivalent to Eqs.",
"(41-42) of Ref.",
"[16], although we have done some further manipulations to the last line of (REF ) to bring it into a matrix form.",
"Also, we have included a factor of $\\eta _i$ in ${\\cal F}$ , rather than keeping it explicitly as in Ref. [16].",
"Its form is given below in Eqs.",
"(REF )-(REF ).",
"Note that it is diagonal in channel space, i.e.",
"it cannot change pions into kaons.",
"It can, however, insert angular momentum, due to the breaking of rotation symmetry by the cubic box.",
"The key point of the identity is that the difference between finite and infinite volume integrals depends on on-shell values of the integrand, allowing the finite-volume dependence to be expressed in terms of physical quantities.",
"Applying the identity to each loop integral in Fig.",
"REF (a), one then rearranges the series by grouping terms with the same number of insertions of ${\\mathcal {F}}$ .",
"The volume-independent term with no ${\\mathcal {F}}$ insertions is of no interest, since it does not lead to poles.",
"Thus we drop it and consider the difference $C_{\\rm sub}(P) \\equiv C_L(P) - C_{\\infty }(P) \\,.$ In the remaining diagrams with $\\mathcal {F}$ insertions, all terms to the left of the first ${\\mathcal {F}}$ and to the right of the last are grouped and summed into new endcaps which we label $A_j$ and $A^{\\prime }_j$ [see Fig.",
"REF (b)].",
"These quantities equal certain matrix elements of the interpolating field $\\sigma $ [16] $A_j(\\hat{k}^*) & \\equiv \\langle \\vec{k}^*,-\\vec{k}^*;\\ j;\\ out \\vert \\sigma ^\\dagger (0) \\vert 0 \\rangle _{\\vert \\vec{k}^* \\vert = q^*_j} \\\\A^{\\prime }_j(\\hat{k}^*) & \\equiv \\langle 0 \\vert \\sigma (0)\\vert \\vec{k}^*,-\\vec{k}^*;\\ j;\\ in \\rangle _{\\vert \\vec{k}^* \\vert = q^*_j} \\,.$ In contrast to [16] we include no wavefunction renormalization factors, because our single particle interpolating fields satisfy on-shell renormalization conditions.",
"Having summed up the ends the next step is to do the same for the series which appears between adjacent ${\\mathcal {F}}$ insertions [Fig.",
"REF (c)].",
"As indicated in the figure, this series generates the infinite volume scattering amplitude $i \\mathcal {M}_{ij}$ .",
"We thus deduce an alternative series for $C_{sub}$ built from $A$ , $A^{\\prime }$ and $i \\mathcal {M}$ s, all connected by $\\mathcal {F}$ s [Fig.",
"REF (d)].",
"We stress that the analysis just performed is a straightforward generalization of the single channel analysis of Ref. [16].",
"All that has changed is that ${\\mathcal {F}}$ and ${\\cal M}$ are now $2\\times 2$ matrices in channel space, and $A$ and $A^{\\prime }$ vectors.",
"To proceed, we decompose $A$ , $A^{\\prime }$ , $\\mathcal {M}$ and $\\mathcal {F}$ in spherical harmonics, defining coefficients via $A_j(\\hat{k}^*) & \\equiv \\sqrt{4 \\pi } A_{j;\\ell , m} Y_{\\ell , m}(\\hat{k}^*)\\\\A^{\\prime }_j(\\hat{k}^*) & \\equiv \\sqrt{4 \\pi } A^{\\prime }_{j;\\ell , m}Y^*_{\\ell ,m}(\\hat{k}^*)\\\\\\begin{split}\\mathcal {M}_{ij}(\\hat{k}^*, \\hat{k}^{^{\\prime }*}) & \\equiv \\\\ & \\hspace{-20.0pt} 4\\pi \\mathcal {M}_{ij; \\ell _1,m_1;\\ell _2,m_2} Y_{\\ell _1,m_1}(\\hat{k}^*)Y^*_{\\ell _2,m_2}(\\hat{k}^{^{\\prime }*})\\end{split} \\\\\\begin{split}\\mathcal {F}_{ij}(\\hat{k}^*, \\hat{k}^{^{\\prime }*}) & \\equiv \\\\ & \\hspace{-20.0pt} -\\frac{1}{4 \\pi } F_{ij; \\ell _1,m_1;\\ell _2,m_2} Y_{\\ell _1,m_1}(\\hat{k}^*)Y^*_{\\ell _2,m_2}(\\hat{k}^{^{\\prime }*}) \\,,\\end{split}$ where a sum over all $\\ell $ 's and $m$ 's is implicit.",
"The factors of $4\\pi $ are present so that we match the conventions of Ref. [16].",
"They imply, for example, that for a purely s-wave amplitude, ${\\cal M}$ is the same in the two bases (for the $4\\pi $ cancels with the two spherical harmonics).",
"The kinematical factor $F$ is given in Ref.",
"[16] (aside from the above-noted factor of $\\eta _i$ ) and takes the formAn additional difference from Ref.",
"[16] is the appearance of $\\mathrm {Re\\,}q^*_i$ rather than $q^*$ .",
"This is discussed in the next section.",
"$F_{ij; \\ell _1, m_1; \\ell _2, m_2}\\equiv \\delta _{ij} F_{i; \\ell _1, m_1; \\ell _2, m_2}\\\\= \\delta _{ij} \\eta _i \\bigg [\\frac{\\mathrm {Re\\,}q^*_i}{8 \\pi E^*} \\delta _{\\ell _1 \\ell _2} \\delta _{m_1 m_2}\\\\\\hspace{-10.0pt} - \\frac{i}{2 E^*} \\sum _{\\ell ,m} \\frac{\\sqrt{4\\pi }}{q_i^{*\\,\\ell }} c^P_{\\ell m}(q_i^{*\\,2}) \\int d \\Omega \\;Y^*_{\\ell _1,m_1} Y^*_{\\ell ,m} Y_{\\ell _2,m_2} \\bigg ] \\,.$ Here the volume-dependence enters through the sumsWe are slightly abusing the notation here for the sake of clarity.",
"$c^P_{\\ell m}$ depends not only on $q^{*\\, 2}$ but also on $m_i$ , but we keep the latter dependence implicit.",
"The dependence is made explicit at the end of this section.",
"$c^P_{\\ell m}(q^{*\\, 2}) = \\frac{1}{L^3} \\sum _{\\vec{k}}\\frac{\\omega _k^*}{\\omega _k} \\frac{e^{\\alpha (q^{*\\, 2}-k^{*\\,2})}}{q^{*\\, 2}-k^{*\\, 2}} k^{*\\, \\ell } \\sqrt{4 \\pi } Y_{\\ell , m}(\\hat{k}^*)\\\\- \\delta _{\\ell 0} \\ \\ \\mathcal {P} \\!",
"\\int \\frac{d^3 k^*}{(2\\pi )^3}\\frac{e^{\\alpha (q^{*\\, 2}-k^{*\\, 2})}}{q^{*\\, 2}-k^{*\\, 2}}\\,,$ with $\\omega _k=\\sqrt{\\vec{k}^2+m_i^2}$ being the energy of a particle with momentum $\\vec{k}$ , and $\\omega _k^*$ the energy after boosting to the CM frame.",
"The properties of these sums are discussed in Ref. [16].",
"We are now in a position to write down the final result.",
"The series indicated in Fig.",
"REF (d) gives $C_{\\rm sub}(P) &= - \\sum _{n=0}^\\infty A^{\\prime } F \\left[ - i \\mathcal {M} F \\right]^n A \\,,\\\\&= - A^{\\prime } \\frac{1}{F^{-1} + i {\\cal M}} A\\,.$ Here all indices are left implicit and may be restored in the obvious way.",
"For example, $A^{\\prime } F \\mathcal {M} F A = A^{\\prime }_{i;\\ell _1,m_1} F_{ij;\\ell _1,m_1;\\ell _2,m_2}\\\\ \\mathcal {M}_{jk;\\ell _2,m_2;\\ell _3,m_3} F_{kl;\\ell _3,m_3;\\ell _4,m_4}A_{l,\\ell _4,m_4} \\,.$ As $C_\\infty $ has no poles in the region of $E^*$ that we consider (below $4 M_\\pi $ ), the poles in $C_L$ must match the poles in $C_{\\rm sub}$ .",
"The desired quantization condition is then just that the matrix between $A^{\\prime }$ and $A$ have a divergent eigenvalue.",
"This may be written as $\\det \\left(F^{-1} + i \\mathcal {M}\\right) = 0\\,,$ where we recall that the matrices now act in the product of the two-dimensional channel space and the infinite-dimensional angular-momentum space.",
"More precisely, $F$ is diagonal in channel space but has off-diagonal elements between different angular momentum sectors (as allowed by the symmetries of the cubic box and the momentum $\\vec{P}$ ), while ${\\cal M}$ is diagonal in angular momentum but off-diagonal in channel space.",
"Equation (REF ) is the main result of this section.",
"It has exactly the same form as that for the single channel given in Ref.",
"[16] (aside from the change of notation in which symmetry factors are contained in $F$ rather than kept explicit).",
"The generalization to more than two two-particle channels is now immediate.",
"As long as $E^*$ is kept below the four particle threshold of the lightest particle the arguments above go through in the same manner.",
"One need only extend the values of the channel indices, taking care to include the appropriate symmetry factor $\\eta _j$ for each channel.",
"The final result then has exactly the form of Eq.",
"(REF ).",
"To make the formal expression (REF ) useful in practice one assumes that there is some $\\ell _{\\rm max}$ , above which the partial wave amplitudes are negligible $\\mathcal {M}^{\\ell > \\ell _{\\rm max}}_{ij} = 0 \\,.$ One can then show that, although $F$ couples $\\ell \\le \\ell _{\\rm max}$ to $\\ell > \\ell _{\\rm max}$ , the projection contained in ${\\cal M}$ is sufficient to collapse the required determinant to that in the $\\ell \\le \\ell _{\\rm max}$ subspace.",
"The argument for this result is given for one channel in Ref.",
"[16] and generalizes trivially to the multiple channel case.",
"Thus one finds that Eq.",
"(REF ) still holds, but with $\\mathcal {M}$ and $F$ now understood to be finite dimensional matrices both in channel space and in the partial wave basis, with $\\ell $ running up to $\\ell _{max}$ .",
"To conclude this section we comment briefly on two generalizations of the result.",
"We first consider the case when not just a single $\\sigma $ but rather a set of operators $\\lbrace \\sigma ^a\\rbrace $ is of interest.",
"This is likely to be the case in practice since multiple operators may be needed to find combinations with good overlaps with the finite-volume eigenstates.",
"If there are $n$ such operators, then $C_L$ generalizes to an $n \\times n$ matrix: $C^{ab}_{L}(P) = \\int _{L;x} e^{i(- \\vec{P} \\cdot \\vec{x} + E x^0)} \\langle 0 \\vert \\sigma ^a(x) \\sigma ^{\\dagger b}(0) \\vert 0 \\rangle \\,.$ The generalization of Eq.",
"() is effected by replacing $A^{\\prime }$ with an $n \\times 2$ matrix $\\begin{pmatrix} A^{\\prime }_1 & A^{\\prime }_2 \\end{pmatrix}\\longrightarrow \\begin{pmatrix} A^{^{\\prime }a=1}_1 & A^{^{\\prime }a=1}_2\\\\ A^{^{\\prime }a=2}_1 & A^{^{\\prime }a=2}_2 \\\\ \\vdots & \\vdots \\end{pmatrix}$ and $A$ with a $2 \\times n$ matrix $\\begin{pmatrix} A_1 \\\\ A_2 \\end{pmatrix}\\longrightarrow \\begin{pmatrix} A^{b=1}_1 & A^{b=2}_1 & \\cdots \\\\ A^{b=1}_2 & A^{b=2}_2 & \\cdots \\end{pmatrix} \\,.$ The key point, however, is that the matrix between $A^{\\prime }$ and $A$ is unchanged, so that the quantization condition (REF ) is unaffected.",
"This is as expected, since the operators used to couple to states cannot affect the eigenstates themselves.",
"The second generalization is to the case of non-degenerate particles.",
"The expressions given above remain valid as long as one makes three changes.",
"First, the symmetry factors $\\eta _i$ become unity for all non-degenerate channels.",
"Second, $q_i^*$ in Eqs.",
"(REF ) is replaced by the solution of $E^* = \\sqrt{q_i^{*\\,2} + M_{ia}^2} + \\sqrt{q_i^{*\\,2} + M_{ib}^2}\\,,$ which is the CM momentum when the channel contains particles of masses $M_{ia}$ and $M_{ib}$ .",
"Third, when evaluating $c^P_{\\ell m}$ using Eq.",
"(REF ), one should use one of the masses $M_{ia}$ or $M_{ib}$ when determining $\\omega _k$ , $\\omega _k^*$ and $\\vec{k}^*$ .",
"One can show that both choices lead to the same result.",
"The third change emphasizes that the kinematic functions $c^P_{\\ell m}$ depend not only on $q_i^*$ but also on the particle masses.",
"This can be made explicit by rewriting them in terms of a generalization of the zeta-function introduced in Ref. [15].",
"The result is [27], [28], [29], [26] $c_{\\ell m}^P(q^{*\\,2}) &=-\\frac{\\sqrt{4\\pi }}{\\gamma L^3}\\left(\\frac{2\\pi }{L}\\right)^{\\ell -2}{\\cal Z}_{\\ell m}^P[1;(q^* L/2\\pi )^2]\\,,\\\\{\\cal Z}_{\\ell m}^P[s;x^2]&=\\sum _{\\vec{n}} \\frac{r^\\ell Y_{\\ell m}(\\hat{r})}{(r^2-x^2)^s}\\,,$ where $\\gamma =E/E^*$ , $\\vec{n}$ runs over integer vectors, and $\\vec{r}$ is obtained from $\\vec{n}$ by $r_\\parallel =\\gamma ^{-1}[n_\\parallel - c\\vec{n}_P]$ and $r_\\perp =n_\\perp $ , where parallel and perpendicular are relative to $\\vec{P}$ , and $2 c =(1+(M_{1a}^2-M_{1b}^2)/E^{*\\, 2})$ .",
"The sum in ${\\cal Z}_{\\ell m}$ can be regulated by taking $s > (3+\\ell )/2$ and then analytically continuing to $s=1$ .",
"This shows that mass dependence enters through the differenceThe apparent lack of symmetry under the interchange $M_{ia} \\leftrightarrow M_{ib}$ can be understood as follows.",
"One can show that ${\\cal Z}_{\\ell m}^P \\rightarrow (-)^\\ell {\\cal Z}_{\\ell m}^P$ under this interchange (so that for degenerate masses the zeta-functions for odd $\\ell $ vanish [15]).",
"This sign flip for odd $\\ell $ must hold also for the $c^P_{\\ell m}$ , and it does because the interchange of masses leads to $\\vec{k}^* \\rightarrow -\\vec{k}^*$ at the pole.",
"The sign flip is canceled in the expression for ${\\cal F}$ , Eq.",
"(REF ), since the product $Y_{\\ell _1,m_1}(\\vec{k}^*) Y^*_{\\ell _2,m_2}(\\vec{k}^{^{\\prime }*})$ also changes sign.",
"This is because, when $\\ell $ is odd, the integral over $d\\Omega $ in the definition of $F$ , Eq.",
"(REF ), enforces that $\\ell _1+\\ell _2$ is odd.",
"The overall effect is that the quantization condition is symmetric under mass interchange, as it must be.",
"$M_{ia}^2-M_{ib}^2$ .",
"One can derive (REF ) by generalizing the method used for the degenerate case in Ref.",
"[16]."
],
[
"Multiple-channel quantization condition for s-wave scattering",
"For the remainder of this article we focus on the simplest case, $\\ell _{max}=0$ , in which only s-wave scattering is significant.",
"In this section we determine the explicit form for the finite-volume quantization condition when there are two channels.",
"We also present compact forms for the condition when an arbitrary number of two particle channels are open.",
"With only s-wave scattering, the two channel quantization condition takes the form $[(F^s_1)^{-1} + i \\mathcal {M}^s_{11}][(F^s_2)^{-1}+i \\mathcal {M}^s_{22}]\\\\ - [i \\mathcal {M}^s_{12}][i \\mathcal {M}^s_{21}] = 0 \\,,$ where $F^s_i & = \\eta _i \\left[\\frac{\\mathrm {Re\\,}q^*_i}{8 \\pi E^*} - \\frac{i}{2E^*}c^{P}_i \\right]\\\\c^P_i & \\equiv c^P(q_i^{*\\,2}) \\equiv c^P_{00}(q_i^{*\\,2}) \\,,$ and the superscript on $F$ and ${\\cal M}$ is a reminder that only $\\ell =0$ contributes.",
"To simplify Eq.",
"(REF ), and in particular to re-express it as an equation between real quantities, it is useful to recall first the single-channel analysis.",
"This has the additional benefit of showing how the two-channel result collapses to the known single-channel result in the appropriate kinematic regime, namely $2 M_\\pi < E^* < 2 M_K \\,.$ In this regime $q^*_2$ becomes imaginary, and the second channel contributes negligibly because $c^P$ [Eq.",
"(REF )] becomes exponentially volume-suppressed and $\\mathrm {Re\\,} q^*$ in $F_2$ [Eq.",
"(REF )] vanishes.The appearance of $\\mathrm {Re\\,}q^*$ rather than $q^*$ in $F_i$ can be understood by reviewing the derivation of $F$ in Ref. [16].",
"The term enters as the difference between principal part and $i \\epsilon $ prescriptions.",
"When $q^*$ is imaginary there is no pole and different ways of regulating give the same result.",
"Sending $F_2\\rightarrow 0$ we find that the quantization condition becomes $[\\mathcal {M}_{11}^s]^{-1} = \\eta _1 \\left[ - \\frac{i q_1^*}{8 \\pi E^*} - \\frac{1}{2 E^*} c^{P}(q_1^{*\\,2}) \\right] \\,.$ Note that the pion momentum $q^*_1$ is real for the energy region considered.",
"Naively one might think that Eq.",
"(REF ) gives two conditions, the separate vanishing of the real and imaginary parts.",
"This is not the case, however, because the vanishing of the imaginary part is a volume-independent condition which is guaranteed to hold by the unitarity of the $S$ -matrix.",
"This can be seen by expressing ${\\cal M}$ in terms of the real phase shift $\\delta (q^*)$ , $\\mathcal {M}_{11}^s = \\frac{8 \\pi E^*}{\\eta _1 q_1^*} \\left[\\frac{e^{2i\\delta (q_1^*)}-1}{2i}\\right] = \\left[ \\frac{\\eta _1q_1^*}{8 \\pi E^*}\\left[\\cot \\delta (q_1^*) - i \\right] \\right]^{-1} \\,.$ Here $e^{2i\\delta }$ is the one dimensional unitary $S$ -matrix in the partial wave basis.",
"Given Eq.",
"(REF ), it is manifest that the imaginary part of Eq.",
"(REF ) holds automatically.",
"The real part of (REF ) then gives the moving frame generalization of the Lüscher result in the familiar partial wave form [13], [16], [17] $\\tan \\delta (q_1^*) = - \\tan \\phi ^P(q_1^*) \\,,$ where $\\tan \\phi ^P(q^*) = \\frac{q^*}{4 \\pi } \\left[c^{P}(q^{*\\,2}) \\right]^{-1} \\,.$ We now return to the CM energies for which both channels are open, $2 M_K < E^* < 4 M_\\pi $ , and generalize Eq.",
"(REF ).",
"The first step is to recall the relationship between the scattering amplitude and the $S$ -matrix.",
"Unitarity implies that ${\\cal M}^s-{\\cal M}^{s\\,\\dagger } = i {\\cal M}^{s\\,\\dagger } P^2 {\\cal M}^s\\,,$ where $P^2$ is a diagonal matrix containing the phase-space factors, whose square root is $P = \\frac{1}{\\sqrt{4 \\pi E^*}}\\begin{pmatrix}\\sqrt{\\eta _1 q_1^*} & 0 \\\\ 0 & \\sqrt{\\eta _2 q_2^*} \\\\\\end{pmatrix}\\,.$ We note that, when expressed in terms of $q^*$ , the form of $P$ is still valid if the two particles in the channel are non-degenerate.",
"We also note that the form (REF ) holds for an arbitrary number of two-particle, s-wave channels, with $P$ generalized in the obvious way.",
"The solution to the unitarity relation is $i \\mathcal {M}^s = P^{-1} \\left( S^s - 1 \\right) P^{-1}$ where $S^s$ is a dimensionless, unitary $2\\times 2$ matrix.",
"To proceed, we need to parametrize $S^s$ .",
"First we note that $S^s$ can be taken to be symmetric.",
"This is because of the T-invariance of the strong interactions, together with the fact that angular momentum eigenstates have definite T-parity (in our case, positive).",
"Thus in the $2\\times 2$ case, $S^s$ is determined by three real parameters.",
"We use the “eigenphase convention” of Blatt and Biedenharn [30], $S^s =\\begin{pmatrix}\\operatorname{c}_\\epsilon & - \\!",
"\\operatorname{s}_\\epsilon \\\\ \\operatorname{s}_\\epsilon & \\operatorname{c}_\\epsilon \\end{pmatrix}\\begin{pmatrix}e^{2 i \\delta _{\\alpha }} & 0 \\\\ 0 & e^{2 i \\delta _{\\beta }}\\end{pmatrix}\\begin{pmatrix}\\operatorname{c}_\\epsilon & \\operatorname{s}_\\epsilon \\\\ - \\!",
"\\operatorname{s}_\\epsilon & \\operatorname{c}_\\epsilon \\end{pmatrix} \\,,$ where the notation $\\operatorname{s}_x = \\sin x$ and $\\operatorname{c}_x = \\cos x$ will be used throughout.",
"The three real parameters $\\delta _\\alpha $ , $\\delta _\\beta $ , and $\\epsilon $ generalize the single $\\delta $ which appears in the one channel case.",
"The parameter $\\epsilon $ quantifies the mixing between the mass eigenstates of channels one and two (the pions and kaons) and the $S$ -matrix eigenstates.",
"The phases $\\delta _\\alpha $ and $\\delta _\\beta $ are, for arbitrary $\\epsilon $ , associated with both channels.",
"Of course, in the limit $\\epsilon \\rightarrow 0$ they reduce, respectively, to the phase shifts of pion and kaon elastic scattering.",
"Substituting the form of $S^s$ into Eq.",
"(REF ) and then placing this in Eq.",
"(REF ) and simplifying, we deduceWe emphasize that the physical content of Eq.",
"(REF ), namely that there is a relation between scattering amplitudes and energy levels, does not depend on the parametrization chosen for the matrix $S^s$ .",
"This is clear either from Eq.",
"(REF ) or from Eq.",
"(REF ) below.",
"An advantage of our choice of parametrization is that it shows that Eq.",
"(REF ) only implies one real condition (rather than two), an observation which must hold for any parameterization.",
"We also note that the freedom to independently change the phases of $\\pi \\pi $ and $K\\overline{K}$ states, which leads to $S^s\\rightarrow U^\\dagger S^s U$ , with $U$ a diagonal unitary matrix, does not change the quantization condition, as can be seen most easily from Eq.",
"(REF ) below.",
"$\\left[ \\tan \\delta _\\alpha + \\tan \\phi ^P(q^*_1) \\right] \\left[ \\tan \\delta _\\beta + \\tan \\phi ^P(q^*_2) \\right] \\\\ + \\sin ^2 \\!",
"\\epsilon \\,\\left[ \\tan \\delta _\\alpha - \\tan \\delta _\\beta \\right] \\left[\\tan \\phi ^P(q^*_1) - \\tan \\phi ^P(q^*_2) \\right] \\\\ = 0 \\,.$ This is the main result of this section.",
"One can use it in one of two ways: to predict the spectrum given knowledge of the scattering amplitude from experiment or a model, or to determine the $S$ -matrix parameters from a lattice calculation of the spectrum.",
"In the former case, we note that all quantities appearing in (REF ), i.e.",
"$\\delta _\\alpha $ , $\\delta _\\beta $ , $\\epsilon $ , $q^*_i$ and $\\phi ^P$ , are functions of $E^*$ .",
"One can thus search, at given spatial extent $L$ and total momentum $\\vec{P}$ , for values of $E^*$ which satisfy Eq.",
"(REF ).",
"If the condition holds for a particular $E^*_k$ , then $E_k(L;\\vec{n}_P) = \\sqrt{E^{*2}_k + \\vec{P}^{\\,2}}$ is in the spectrum of the finite-volume moving-frame Hamiltonian.",
"Here we choose to write $E_k$ as a function of $\\vec{n}_P$ rather than $\\vec{P}$ , since, in practice, it is the former quantity which is held fixed as one varies $L$ .",
"The second use of (REF ) is the most relevant for the discussion in subsequent sections.",
"For a given choice of $E^*$ , one finds, through a lattice calculation, three pairs $\\lbrace L,\\vec{n}_P\\rbrace $ for which there is a spectral line $E_k$ such that $E^*_k$ [defined in Eq.",
"(REF )] is equal to $E^*$ .",
"One could use a fixed $\\vec{n}_P$ and consider multiple spectral lines (the simplest choice conceptually), or use three different choices of $\\vec{n}_P$ (probably more practical since one would not need to determine so many excited levels).",
"In either case, one ends up with three versions of Eq.",
"(REF ), all containing the desired quantities $\\delta _\\alpha (E^*)$ , $\\delta _\\beta (E^*)$ and $\\epsilon (E^*)$ , but having different values of the $\\phi ^P(q^*_j)$ .",
"Solving these equations one determines, rather than the angles themselves, the quantities $\\tan \\delta _\\alpha $ , $\\tan \\delta _\\beta $ , and $\\sin ^2 \\epsilon $ at CM energy $E^*$ .",
"For our discussion we therefore restrict $\\delta _{\\alpha ,\\beta } \\in [-\\pi /2,\\pi /2] \\mathrm {\\ \\ and\\ \\ } \\epsilon \\in [0,\\pi /2] \\,.$ Having determined the restricted phases over a range of energies, one can afterward relax the constraints in order to build continuous curves as a function of energy.",
"We direct the reader to Refs.",
"[22], [23] for discussion of other methods for extracting the three scattering parameters.",
"We emphasize that Eq.",
"(REF ) has a very intuitive form.",
"If $\\delta _{\\alpha } = \\delta _{\\beta }$ or $m_1 = m_2$ or $\\epsilon =0$ then the second line vanishes and the result reduces to two copies of the one channel quantization condition [Eq.",
"(REF )].",
"To see that this makes sense, note that for identical phase shifts, the $\\epsilon $ matrix commutes through the phase matrix and we recover two uncoupled channels.",
"Similarly if the masses are degenerate then the eigenstates of the $S$ -matrix will also be mass eigenstates leading to the decoupled form.",
"Finally, the decoupling for $\\epsilon =0$ is an obvious property of the parametrization [Eq.",
"(REF )].",
"An alternative solution to the unitarity relation (REF ) can be given in terms of the $K$ -matrix used in Ref. [22].",
"Specifically, (REF ) is satisfied if $\\left({\\cal M}^{s}\\right)^{-1} = M - i P^2/2\\,,$ with $M$ any real symmetric $2\\times 2$ matrix.",
"If we set $M = \\frac{1}{8\\pi E^*} \\Vert \\sqrt{\\eta }\\Vert \\, K^{-1}\\, \\Vert \\sqrt{\\eta }\\Vert \\,,$ [where double bars denote a diagonal matrix, so that $\\Vert \\eta \\Vert ={\\rm diag}(\\sqrt{\\eta _1},\\sqrt{\\eta _2})$ ], and further set $\\vec{P}=0$ , then one can show that the two-channel quantization condition given above can be manipulated into the form given in Eq.",
"(3.5) of Ref.",
"[22] in terms of the real, symmetric matrix $K$ .",
"We now generalize Eq.",
"(REF ) to $N$ s-wave channels.",
"As noted above, the form of the unitarity relation (REF ) holds for any $N$ , and the same is true for its solution (REF ).",
"In the latter, the $N$ channel $S$ -matrix can be parametrized asThe remainder of this paper is limited to the $s$ -wave, so we drop the superscript ${}^s$ hereafter.",
"$S = R^{-1} \\, \\big \\Vert e^{2i\\delta } \\big \\Vert \\, R \\,,$ with $R$ an $SO(N)$ matrix, and $\\big \\Vert e^{2i\\delta } \\big \\Vert = {\\rm diag} \\big [e^{2i\\delta _\\alpha },e^{2i \\delta _\\beta }, \\cdots \\big ] \\,.$ Together with Eqs.",
"(REF ) and (REF ) one needs the $N \\times N$ generalization of $F$ , which has been discussed in the previous section.",
"From these definitions one can straightforwardly work out the quantization condition for $N$ coupled channels.",
"We conclude this section by describing two additional ways of writing the quantization condition, both of which make the higher channel generalization especially clear.",
"Observe first that, for any number of open channels, $F^{-1} = P^{-1} \\, \\big \\Vert 1-e^{-2i\\phi } \\big \\Vert \\, P^{-1}\\,.$ Combining this with (REF ), it follows that $F^{-1} + i {\\cal M}= P^{-1} \\Big [ S - \\big \\Vert e^{-2i\\phi } \\big \\Vert \\Big ]P^{-1} \\,.$ Since $P^{-1}$ has no singularities in the kinematic regime we consider, the quantization condition can be rewritten as $\\det \\Big [ S - \\big \\Vert e^{-2i\\phi } \\big \\Vert \\Big ] = 0 \\,.$ This form shows that the symmetry factors cancel from the quantization condition in general.",
"Although Eq.",
"(REF ) looks like it will lead to one complex and thus two real conditions, it turns out that it leads only to a single real condition.",
"This follows from the identity $\\big \\Vert 1 + i t_\\phi \\big \\Vert \\times \\Big [ S -\\big \\Vert e^{-2i\\phi } \\big \\Vert \\Big ] \\times \\Big [ R^{-1} \\big \\Vert 1 - it_\\delta \\big \\Vert R \\Big ]\\\\= 2 i \\Big [ R^{-1} \\big \\Vert t_\\delta \\big \\Vert R + \\big \\Vert t_\\phi \\big \\Vert \\Big ] \\,,$ where $t_x = \\tan x$ .",
"It gives a manifestly real rewriting of the quantization condition, $\\det \\Big [ R^{-1} \\big \\Vert t_\\delta \\big \\Vert R + \\big \\Vert t_\\phi \\big \\Vert \\Big ] = 0 \\,.$ This form leads directly to the result (REF ) in the two channel case, and collapses to the single-channel result (REF ) for any channel that decouples from the rest.",
"If any of the channels contain non-degenerate particles, this enters only through the values of the kinematic functions $t_\\phi $ , as discussed in the previous section."
],
[
"Multiple channel extension of the Lellouch-Lüscher formula",
"Having found the two channel quantization condition, we are now in a position to work out the two channel generalization of the LL formula which relates weak matrix elements in finite and infinite volume.",
"The derivation follows the original work by Lellouch and Lüscher, Ref.",
"[18], which was extended to a moving frame by Refs.",
"[16], [17].",
"We begin by introducing a third channel which is decoupled from the original two.",
"This contains a single particle, which we call a $D$ -meson, whose mass satisfies $M_D>2 M_\\pi , 2 M_K\\,.$ We next introduce a weak perturbation to the Hamiltonian density $\\mathcal {H}(x) \\rightarrow \\mathcal {H}(x) + \\lambda \\mathcal {H}_W(x) \\,,$ where $\\lambda $ is a parameter which can be varied freely and can, in particular, be taken arbitrarily small.",
"The perturbation $\\mathcal {H}_W$ is defined to couple channels one and two (pions and kaons) to the third ($D$ -meson) and nothing more.",
"It is convenient to choose it to be invariant under time reversal (T) symmetry.",
"The generalization to perturbations which are not T invariant is described at the end of the section.",
"Consider now the finite volume spectrum, first in the absence of the perturbation.",
"The spectrum of two-particle states with $\\vec{P}=2\\pi \\vec{n}_P/L$ is determined by Eq.",
"(REF ).",
"It is $L$ -dependent and $L$ can therefore be tuned to make one of the levels, call it $k_D$ , degenerate with the energy of a single (moving) $D$ meson $E_{k_D} = \\sqrt{M_D^2 + \\vec{P}^{\\,2}}$ With $L$ fixed in this way, we now turn on the weak interaction.",
"At leading order in degenerate perturbation theory this changes the energies to $E^{(1)} = E^{(0)} \\pm \\lambda V \\vert M_W \\vert $ where $V=L^3$ , $E^{(0)} = E_{k_D}$ , and the finite-volume matrix element is $M_W & = {}_L\\langle k_D \\vert \\mathcal {H}_W(0) \\vert D \\rangle _L.$ The subscripts $L$ on the states indicate that they are normalized to unity, unlike the relativistically normalized infinite volume states.",
"Superscripts ${}^{(1)}$ are used throughout this section to indicate that the quantity includes both the leading order and the $\\mathcal {O}(\\lambda )$ correction, while superscripts ${}^{(0)}$ indicate the unperturbed quantity.",
"The effect of the perturbation may also be written in terms of the CM energy as $E^{*(1)} & = M_D \\pm \\lambda \\Delta E^* \\\\ \\Delta E^* & =\\frac{E^{(0)} V \\vert M_W \\vert }{M_D} \\,.$ Figure: The diagram giving rise tothe amplitude perturbation Δℳ\\Delta \\mathcal {M} [See Eq.",
"()].Of course, in addition to affecting the finite volume energy spectrum, the weak perturbation also changes the infinite volume scattering amplitudes.",
"The leading order effect is generated by the diagram of Fig.",
"REF , which gives $\\mathcal {M}^{(1)} = \\mathcal {M}^{(0)} \\mp \\lambda \\Delta \\mathcal {M}$ with $\\Delta \\mathcal {M}_{j,k} =\\frac{\\langle j \\vert \\mathcal {H}_W(0)\\vert D \\rangle \\langle D \\vert \\mathcal {H}_W(0) \\vert k \\rangle }{2 E^{(0)} V \\vert M_W \\vert } \\,.$ The indices $j$ and $k$ run over the two two-particle channels.",
"This perturbation may equivalently be represented through shifts in $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ $\\delta ^{(1)}_\\alpha (E^*) & = \\delta ^{(0)}_\\alpha (E^*) \\pm \\lambda \\Delta \\delta _\\alpha (E^*) \\\\ \\delta ^{(1)}_\\beta (E^*) & =\\delta ^{(0)}_\\beta (E^*) \\pm \\lambda \\Delta \\delta _\\beta (E^*) \\\\\\epsilon ^{(1)}(E^*) & = \\epsilon ^{(0)}(E^*) \\pm \\lambda \\Delta \\epsilon (E^*) \\,,$ The explicit forms of the perturbed phases are given in Eqs.",
"(REF )-() below.",
"The calculation now proceeds as follows.",
"When the quantities ${\\delta ^{(0)}_\\alpha (E^*)},\\ {\\delta ^{(0)}_\\beta (E^*)},\\ \\mathrm {and}\\ {\\epsilon ^{(0)}(E^*)}$ are placed in the quantization condition [Eq.",
"(REF )], the condition is satisfied by construction at $E^{*(0)}=M_D$ .",
"Alternatively if one places $\\delta ^{(1)}_\\alpha (E^*),\\ \\delta ^{(1)}_\\beta (E^*),\\ \\mathrm {and}\\ \\epsilon ^{(1)}(E^*)$ into the same condition then it must be satisfied when evaluated at the perturbed CM energy $E^{*(1)}$ , but only to linear order in $\\lambda $ .",
"The constant order term in the $\\lambda $ expansion is just the unperturbed condition, and so it is the vanishing of the $\\mathcal {O} (\\lambda )$ term that is of interest.",
"The condition that this term vanish gives the relation between finite and infinite volume weak matrix elements that we are after.",
"The only detail left to discuss, before substituting into the quantization condition and expanding in $\\lambda $ , is the explicit forms of the amplitude corrections to $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ .",
"Before these are found it is useful to determine the constraints on the infinite volume matrix elements which arise from Watson's theorem.",
"As shown in App.",
", time reversal invariance and unitarity constrain the matrix elements to be such that the following two quantities are real: $\\hspace{-5.0pt}v_1 & = e^{-i \\delta _\\alpha } \\left[\\sqrt{q^*_{1}\\eta _1} A_{D\\rightarrow \\pi \\pi } \\operatorname{c}_\\epsilon +\\sqrt{q^*_{2}\\eta _2} A_{D \\rightarrow K K} \\operatorname{s}_\\epsilon \\right]\\,, \\\\\\hspace{-5.0pt} v_2 & = e^{-i \\delta _\\beta } \\left[ - \\sqrt{q^*_{1}\\eta _1}A_{D \\rightarrow \\pi \\pi } \\operatorname{s}_\\epsilon + \\sqrt{q^*_{2}\\eta _2}A_{D \\rightarrow K K} \\operatorname{c}_\\epsilon \\right]\\,.$ Here $A_{D \\rightarrow \\pi \\pi } \\equiv \\langle \\pi \\pi \\vert \\mathcal {H}_W(0) \\vert D \\rangle \\,,$ and similarly for the $K \\overline{K}$ case, normalized so that the decay rates to each channel are $\\Gamma _{D\\rightarrow j} = \\frac{q^*_j \\eta _j}{8 \\pi M_D^2} |A_{D\\rightarrow j}|^2= \\frac{1}{2 M_D} P_{jj}^2 |A_{D\\rightarrow j}|^2\\,.$ This relation holds also if the particles in a channel are non-degenerate (requiring $\\eta =1$ ).",
"All energy-dependent parameters in (REF ) and (), i.e.",
"$\\delta _\\alpha $ , $\\delta _\\beta $ , $\\epsilon $ and $q^*_j$ , are to be evaluated at $E^*=M_D$ .",
"The results (REF ) and () hold when the phases of the states are chosen so that the $S$ -matrix is symmetric (as is possible given T invariance).",
"This does not determine the signs of the two matrix elements, and these signs are unphysical.",
"More precisely, the relative sign ambiguity is the same as the ambiguity in the sign of $\\epsilon $ , so once we have fixed the latter to be positive, the relative sign is physical.",
"The overall sign remains unphysical, and can be chosen, for example to set $v_1 \\ge 0$ , Inverting the relations (REF ) and () yields $A_{D \\rightarrow \\pi \\pi }& = \\frac{1}{\\sqrt{q^{*}_{1}\\eta _1}} \\left[v_1 e^{i \\delta _\\alpha } \\operatorname{c}_\\epsilon - v_2 e^{i \\delta _\\beta }\\operatorname{s}_\\epsilon \\right]\\\\A_{D \\rightarrow K K}& = \\frac{1}{\\sqrt{q^{*}_{2}\\eta _2}} \\left[v_1 e^{i\\delta _\\alpha } \\operatorname{s}_\\epsilon + v_2 e^{i \\delta _\\beta } \\operatorname{c}_\\epsilon \\right]\\,.$ Inserting these in $\\Delta {\\cal M}$ , Eq.",
"(REF ), and using the relation between ${\\cal M}$ and $S$ , Eq.",
"(REF ), and the parametrization of $S$ , Eq.",
"(REF ), we find that perturbations to $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ are real.",
"This is a consistency check on the calculation (or an alternative derivation of the Watson's theorem constraint).",
"Specifically, we find $\\Delta \\delta _{\\alpha } & = - \\mathcal {N} v_1^2 \\\\\\Delta \\delta _{\\beta } & = - \\mathcal {N} v_2^2 \\\\\\Delta \\epsilon & = - \\mathcal {N} \\frac{v_1 v_2}{\\operatorname{c}_\\alpha \\operatorname{c}_\\beta (\\operatorname{t}_\\alpha -\\operatorname{t}_\\beta )}$ where $\\operatorname{t}_\\alpha = \\tan \\delta _\\alpha $ , etc., and $\\mathcal {N} = \\frac{1}{16 \\pi E^{(0)} M_D V \\vert M_W\\vert } \\,.$ We now have all the ingredients to substitute into the quantization condition and determine the LL generalization.",
"We emphasize that, when the expansion in $\\lambda $ is performed, $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ each contribute both from the amplitude corrections of Eqs.",
"(REF )-() and from the shift in the energy, (REF ).",
"The other contributions arise from the energy dependence of $\\phi _i = \\phi ^P(q^*_i)$ .",
"Substituting and simplifying, we find the main result of this section $\\mathcal {C}_{1} v_1^2 + \\mathcal {C}_{2} v_2^2 + \\mathcal {C}_{1 2} v_1 v_2= \\mathcal {C}_{M^2} \\vert M_W \\vert ^2$ where $\\begin{split}\\mathcal {C}_{1} & = \\frac{\\pi }{16} \\frac{\\operatorname{t}_1 + \\operatorname{t}_2 + 2\\operatorname{t}_\\beta + (\\operatorname{t}_2 - \\operatorname{t}_1) (1 - 2 \\operatorname{s}_\\epsilon ^2)}{\\operatorname{c}_\\alpha ^2}\\end{split}\\\\\\begin{split}\\mathcal {C}_{2} & = \\frac{\\pi }{16} \\frac{\\operatorname{t}_1 + \\operatorname{t}_2 + 2\\operatorname{t}_\\alpha + (\\operatorname{t}_1 - \\operatorname{t}_2) (1 - 2 \\operatorname{s}_\\epsilon ^2)}{\\operatorname{c}_\\beta ^2}\\end{split}\\\\\\begin{split}\\mathcal {C}_{12} & = \\frac{\\pi }{4} (\\operatorname{t}_1 - \\operatorname{t}_2)\\frac{\\operatorname{s}_{\\epsilon }\\!",
"\\operatorname{c}_{\\epsilon }}{\\operatorname{c}_\\alpha \\!",
"\\operatorname{c}_\\beta }\\end{split} \\\\\\begin{split}\\mathcal {C}_{M^2} & = \\frac{\\pi ^2 M_D V^2 (E^{(0)})^2}{2}\\bigg [\\frac{\\operatorname{t}^{\\prime }_1}{q_1^*}\\left(\\operatorname{t}_2 + \\operatorname{t}_\\beta + (\\operatorname{t}_\\alpha -\\operatorname{t}_\\beta )\\operatorname{s}_\\epsilon ^2 \\right) \\\\ & \\hspace{50.0pt} + \\frac{\\operatorname{t}^{\\prime }_2}{q_2^*}\\left(\\operatorname{t}_1 + \\operatorname{t}_\\alpha + (\\operatorname{t}_\\beta -\\operatorname{t}_\\alpha ) \\operatorname{s}_\\epsilon ^2 \\right) \\\\ & \\hspace{50.0pt}+ \\frac{4 \\operatorname{t}_\\alpha ^{\\prime }}{M_D}\\left(\\operatorname{t}_2 + \\operatorname{t}_\\beta + (\\operatorname{t}_1 - \\operatorname{t}_2) \\operatorname{s}_\\epsilon ^2 \\right)\\\\ & \\hspace{50.0pt}+ \\frac{4 \\operatorname{t}_\\beta ^{\\prime }}{M_D} \\left(\\operatorname{t}_1 +\\operatorname{t}_\\alpha + (\\operatorname{t}_2 - \\operatorname{t}_1) \\operatorname{s}_\\epsilon ^2 \\right) \\\\ &\\hspace{50.0pt}+ \\frac{4 \\,{\\operatorname{s}^{2}_\\epsilon }^{\\prime }}{M_D} (\\operatorname{t}_1 - \\operatorname{t}_2)(\\operatorname{t}_\\alpha - \\operatorname{t}_\\beta ) \\bigg ] \\,,\\end{split}$ and where we use the additional notation $\\operatorname{t}_i = \\tan \\phi ^P[q^*_i] \\,.$ All quantities are evaluated at the $D$ mass, and we have dropped the superscript ${}^{(0)}$ .",
"The primes on $\\phi _i$ indicate derivatives with respect to $q^*_i$ while those on $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ indicate derivatives with respect to $E^*$ .",
"In each case, these are the natural variables on which the quantities depend.",
"We have checked that this formula reduces to (two copies of) the single-channel LL result if $\\epsilon \\rightarrow 0$ .",
"We now describe how the result (REF ) can be used in practice.",
"A lattice calculation yields the finite-volume matrix element $|M_W|$ , and the aim is to determine the infinite-volume matrix elements $A_{D\\rightarrow \\pi \\pi }$ and $A_{D\\rightarrow KK}$ .",
"Using the generalized quantization condition (REF ) for three different spectral lines (all chosen to have $E^*=M_D$ ) one can determine $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ as described in the previous section.",
"Repeating the procedure at a slightly different energy allows a numerical determination of the required derivatives.",
"One now evaluates $|M_W|$ at the degenerate point on one of the spectral lines.",
"The knowledge of the $S$ -matrix parameters and their derivatives, together with the value of $L$ , allows one to calculate the values of the four $\\mathcal {C}$ 's [Eqs.",
"(REF )-()].",
"Combined with the value of $|M_W|$ , one then finds from Eq.",
"(REF ) a quadratic constraint on $v_1$ and $v_2$ .",
"Repeating the procedure for a second spectral line gives an independent constraint, which allows for the determination of $v_1$ and $v_2$ up to a two-fold ambiguity corresponding to the unknown relative sign.",
"Finally, repeating for a third spectral line resolves the sign ambiguity.",
"With $v_1$ and $v_2$ determined in this way, one can obtain the infinite-volume matrix elements using Eqs.",
"(REF ) and ().",
"Although this procedure is rather elaborate, we note that (for the case of two channels) three spectral lines are needed both for the determination of the parameters of the $S$ -matrix and of the LL factors.",
"We conclude this section by commenting that Eq.",
"(REF ) factors as ${\\rm sgn}(\\mathcal {C}_1)(c_1 v_1 + c_2 v_2)^2 =\\mathcal {C}_{M^2} \\vert M_W \\vert ^2$ where $c_1 = \\sqrt{\\vert \\mathcal {C}_1 \\vert } \\hspace{25.0pt}c_2 = {\\rm sgn}[\\mathcal {C}_1 \\mathcal {C}_{12}] \\sqrt{\\vert \\mathcal {C}_2 \\vert } \\,.$ The only new information encoded in Eqs.",
"(REF ) and (REF ) relative to Eq.",
"(REF ) is that $4 \\mathcal {C}_1 \\mathcal {C}_2 = \\mathcal {C}_{12}^2 \\,,$ which can be shown to hold by applying Eq.",
"(REF ) to Eqs.",
"(REF )-().",
"Although the factorized form (REF ) is simpler, it does not reduce the number of values of $L$ that are needed because there remains a sign ambiguity (from the square root) at each $L$ .",
"What it does make clear, however, is that the generalized LL condition will fail when the signs of $\\mathcal {C}_1$ and $\\mathcal {C}_{M^2}$ are opposite.",
"Presumably this cannot happen for physical values of the phase shifts.",
"We stress that this issue also arises in the original one-channel set-up, where the LL formula only makes sense if $\\frac{d (\\delta + \\phi ^P)}{d q^*} > 0\\,.$ We return to these sign constraints in the next section.",
"The form (REF ) also allows one to write the LL condition as a factored form in terms of the desired matrix elements, $\\vert c_\\pi A_{D \\rightarrow \\pi \\pi } + c_K A_{D \\rightarrow K K}\\vert ^2 = \\vert \\mathcal {C}_{M^2} \\vert \\vert M_W \\vert ^2 \\,,$ where $c_\\pi $ and $c_K$ are complex, and can be determined from the above results.",
"As this equality holds for any T-invariant form of weak perturbation and for any decay particle, it must imply a relation between finite and infinite volume states ${}_L\\langle k_D \\vert \\propto c_\\pi \\langle \\pi \\pi ,{\\rm out} \\vert + c_K \\langle K \\overline{K}, {\\rm out} \\vert + \\dots \\,.$ Here the ellipsis indicates the $\\pi \\pi $ and $K \\overline{K}$ states of higher angular momentum which are needed to satisfy the periodic boundary conditions.",
"Indeed, as noted in the original derivation of Ref.",
"[18], the use of the $D$ -meson is simply a trick to obtain the normalization factors.In the one-channel case, an alternative line of argument has been developed for obtaining the LL relation, based on matching the density of two-particle states in finite and infinite volumes [31].",
"In the present case, we do not see how to use this approach to determine the relative normalization, $c_K/c_\\pi $ , of the two components in the finite volume state.",
"Thus we think that this approach could provide only a consistency check.",
"It follows that Eq.",
"(REF ) must also hold for perturbations which are not T-invariant.",
"The appearance of the linear combination in Eq.",
"(REF ) can be better understood from an alternative derivation of the LL formula, to which we now turn."
],
[
"Alternative derivation of Lellouch-Lüscher formula",
"In this section we present a different derivation of the two channel LL relation which has the following advantages: (a) it does not require determining the shifts $\\Delta \\delta _\\alpha $ , $\\Delta \\delta _\\beta $ and $\\Delta \\epsilon $ , but rather works directly with the change in ${\\cal M}$ ; (b) it gives one directly a condition with the factored form, proportional to the left hand side of Eq.",
"(REF ); (c) it allows one to rewrite the LL condition in a simpler form in which the only inputs required are the derivatives of the energies with respect to $L$ along the three spectral lines.",
"This form is likely to be more practical.",
"We work directly with the condition $\\det (F^{-1} + i {\\cal M})=0$ , and keep results for general number of channels, $N$ , as far as possible.",
"We begin by defining ${\\cal X}&=& F^{-1} + i {\\cal M}\\\\ {\\cal Y}&=& S - \\big \\Vert e^{-2i\\phi } \\big \\Vert \\,.$ and recall from Eq.",
"(REF ) that ${\\cal X}= P^{-1} {\\cal Y}P^{-1} \\,.$ The quantization condition $\\det {\\cal X}=0$ is equivalent to ${\\cal X}$ having an eigenvector with vanishing eigenvalue.",
"We label this eigenvector $\\overrightarrow{e}^X$ .",
"Note also that the symmetry of ${\\cal X}$ implies $(\\overrightarrow{e}^{X})^{Tr}=\\overleftarrow{e}^X$ is a left eigenvector, also with zero eigenvalue.",
"Now we can formulate the LL condition in a relatively compact form.",
"As above, let ${\\cal M}^{(0)}$ be the scattering amplitude at CM energy $E^*=M_D$ .",
"Similarly, let $F^{(0)}$ be the finite-volume factor at this CM energy and for one of the values of box size $L$ for which the quantization condition holds.",
"Then for ${\\cal X}^{(0)} \\equiv (F^{(0)})^{-1}+ i {\\cal M}^{(0)}\\,,$ we have $\\overleftarrow{e}^{X} {\\cal X}^{(0)} \\overrightarrow{e}^{X} = 0 \\,.$ Now, while holding $L$ fixed, we change the energy by $\\pm \\lambda \\Delta E = \\pm \\lambda V |M_W|$ and change ${\\cal M}$ to ${\\cal M}^{(0)}\\mp \\lambda \\Delta {\\cal M}$ , and require that the quantization condition still hold.",
"Thus we have, to linear order in $\\lambda $ , $\\det ({\\cal X}^{(0)} +\\lambda \\Delta {\\cal X}) = 0 \\,,$ where $\\Delta {\\cal X}= \\pm \\Delta E \\frac{\\partial {\\cal X}}{\\partial E}\\Bigg |_L \\mp i \\Delta {\\cal M}\\,.$ It follows that there must be a new eigenvector of the form $\\overrightarrow{e}^{X} + \\lambda \\Delta \\overrightarrow{e}^{X}$ which is annihilated by the perturbed matrix.",
"From the $\\mathcal {O}(\\lambda )$ term in $\\left[ \\overleftarrow{e}^{X} + \\lambda \\Delta \\overleftarrow{e}^{X}\\right] \\left[ {\\cal X}^{(0)} +\\lambda \\Delta {\\cal X}\\right] \\left[\\overrightarrow{e}^{X} + \\lambda \\Delta \\overrightarrow{e}^{X} \\right]=0 \\,,$ we deduce $\\overleftarrow{e}^X \\Delta {\\cal X}\\overrightarrow{e}^X = 0 \\,.$ Using the explicit form of $\\Delta {\\cal X}$ this becomes $\\Delta E\\; \\overleftarrow{e}^X \\frac{\\partial {\\cal X}}{\\partial E}\\Bigg |_L\\overrightarrow{e}^X = \\overleftarrow{e}^X i \\Delta {\\cal M}\\; \\overrightarrow{e}^X \\,,$ where the derivative is evaluated at $E^*=M_D$ .",
"We can slightly simplify this result by expressing the left hand side in terms of ${\\cal Y}$ rather than ${\\cal X}$ , and thus removing factors of $P^{-1}$ .",
"The point is that, when the derivative acts on the $P^{-1}$ factors in ${\\cal X}$ , the contribution to the left hand side vanishes, since one can still act (either to the left or the right) on the zero-eigenvector.",
"Thus we can rewrite the condition in terms of the zero eigenvector of ${\\cal Y}$ , which is $\\overrightarrow{e}^{Y} = P^{-1} \\overrightarrow{e}^X\\,.$ The new form is $\\Delta E\\; \\overleftarrow{e}^Y \\frac{\\partial {\\cal Y}}{\\partial E}\\Bigg |_L\\overrightarrow{e}^Y =\\overleftarrow{e}^X [ i \\Delta {\\cal M}] \\overrightarrow{e}^X \\,.$ We now focus on the $2\\times 2$ case.",
"To proceed, we need the explicit form for $\\overrightarrow{e}^Y$ , which is given, up to an overall normalization factor, by $\\overrightarrow{e}^Y = \\begin{bmatrix} 1 \\\\ z e^{i(\\phi _2-\\phi _1)} \\end{bmatrix}$ where $z$ is the real quantity $z= t_\\epsilon \\frac{\\sin (\\delta _\\beta +\\phi _1)}{\\sin (\\delta _\\beta +\\phi _2)} \\,.$ It is clear from the form of Eq.",
"(REF ) and the relation (REF ) that the normalization of $\\overrightarrow{e}^Y$ is irrelevant and so we have chosen a relatively simple unnormalized form.",
"We evaluate the right hand side of Eq.",
"(REF ) using the form of $\\Delta {\\cal M}$ , Eq.",
"(REF ).",
"It is immediately apparent that the result factorizes, given that $\\Delta {\\cal M}$ is an outer product.",
"This will hold for all $N$ .",
"In the $N=2$ case we have $\\overleftarrow{e}^X [ i \\Delta {\\cal M}] \\overrightarrow{e}^X= i e^{-2i\\phi _1}\\frac{M_\\infty ^2}{8 \\pi M_D E^{(0)} V |M_W|} \\,,$ where $M_\\infty =\\\\e^{i\\phi _1} \\sqrt{q_1^*\\eta _1}\\, A_{D \\rightarrow \\pi \\pi } + z\\, e^{i\\phi _2} \\sqrt{q_2^*\\eta _2}\\, A_{D \\rightarrow K K}\\,.$ Here we have used the assumed T-invariance of ${\\cal H}_W$ .",
"We have pulled out the phase $e^{-2i\\phi _1}$ so that $M_\\infty $ is real.",
"Its reality is not obvious, but can be established using the results derived from Watson's theorem and given in App. .",
"In particular, an algebraic exercise shows that $M_\\infty = \\\\\\sin (\\phi _1-\\phi _2) \\left[ -v_1\\frac{c_\\epsilon }{\\sin (\\delta _\\alpha \\!+\\!\\phi _2)} +v_2\\frac{s_\\epsilon }{\\sin (\\delta _\\beta \\!+\\!\\phi _2)}\\right] \\,,$ and we recall that the quantities $v_1$ and $v_2$ [defined above in Eqs.",
"(REF )-()] are real.",
"The result (REF ) makes clear that, for any choice of ${\\cal H}_W$ , one ends up with the matrix element to a given (complex) linear combination of $\\langle \\pi \\pi |$ and $\\langle K \\overline{K}|$ states, since all the factors ($\\phi _1$ , $\\phi _2$ and $z$ ) are determined by $E^{(0)}$ and $L$ .",
"Indeed, what the LL method has allowed us to do is determine the coefficients of the s-wave $\\langle \\pi \\pi |$ and $\\langle K \\overline{K}|$ components within the finite-volume state.",
"As mentioned above, this decomposition has nothing to do with ${\\cal H}_W$ , and thus we can use the result for any ${\\cal H}_W$ , including one involving T-violation.",
"By comparing the result (REF ) to the general decomposition of the finite-volume state, Eq.",
"(REF ), we can read off the ratio of the coefficients, $\\frac{c_K}{c_\\pi } = e^{i(\\phi _2-\\phi _1)}\\, z\\,\\sqrt{\\frac{q_2^*\\eta _2}{q_1^*\\eta _1}}\\,.$ It is interesting that the relative phase between $c_K$ and $c_\\pi $ is determined by the kinematic phases $\\phi _j$ .",
"Given the form of $\\Delta {\\cal M}$ , and the fact that, in Eq.",
"(REF ), it is sandwiched between $\\overleftarrow{e}^X$ and $\\overrightarrow{e}^X$ , it follows that the zero eigenvector itself gives the relative size of the $\\pi \\pi $ and $K\\overline{K}$ contributions: $\\overrightarrow{e}^X \\propto \\begin{pmatrix} c_\\pi \\\\ c_K \\end{pmatrix}\\,.$ This illustrates in a direct way that the linear combination which appears is completely independent of the form of ${\\mathcal {H}}_W$ , since the eigenvector of ${\\cal X}$ knows nothing about this perturbation.",
"Having discussed the right hand side of Eq.",
"(REF ) in some detail we now turn to the left.",
"Specifically, we show that it is possible to write the left-hand side in terms of the derivative of the spectral energy with respect to $L$ .",
"To motivate this form, we first recall that the LL result of the previous section depends on $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ and their derivatives, evaluated at $E^* = M_D$ .",
"As described in Sec.",
", the three $S$ -matrix parameters may be determined, using Eq.",
"(REF ), by finding three different pairs $\\lbrace L,\\vec{n}_P\\rbrace $ for which there is a spectral line $E_k(L;\\vec{n}_P)$ satisfying $E_k^*(L;\\vec{n}_P)=M_D$ [see Eq.",
"(REF )].",
"Furthermore, by slightly changing the three $L$ values, one can determine $\\delta _\\alpha $ , $\\delta _\\beta $ and $\\epsilon $ at slightly different energies and thus deduce the derivatives at $M_D$ .",
"The point of reiterating these steps is to note that, since the lattice simulation actually gives the energy spectrum as a function of $L$ , it would be preferable if the LL result could be rewritten directly in terms of the properties of the spectrum.",
"In this way the extra step of separately working out the phase shifts and their derivatives would be avoided.",
"This turns out to be possible, as we now show.",
"We use the quantization condition in the form $\\det {\\cal Y}=0$ .",
"To stay on a spectral line $E_k(L;\\vec{n}_P)$ as we vary $E$ away from the moving frame $D$ -meson energy $E^{(0)}$ , we need to vary $L$ in such a way that this condition remains fulfilled.",
"We note that, while $F$ depends on both $E^*$ and $L$ , $S$ depends only on $E^*$ .",
"Thus we use $E^*$ and $L$ as independent variables.",
"Then the condition to stay on a spectral line becomes $0= \\overleftarrow{e}^Y \\left[\\Delta E^* \\frac{\\partial {\\cal Y}}{\\partial E^*}\\Bigg |_L + \\Delta L \\frac{\\partial {\\cal Y}}{\\partial L}\\Bigg |_{E^*}\\right] \\overrightarrow{e}^Y \\,,$ which leads to $\\frac{d E_k^*}{d L}\\Bigg |_{\\rm line} = -\\frac{\\overleftarrow{e}^Y \\frac{\\partial {\\cal Y}}{\\partial L}\\overrightarrow{e}^Y}{\\overleftarrow{e}^Y \\frac{\\partial {\\cal Y}}{\\partial E^*} \\overrightarrow{e}^Y} \\,.$ Here, in the left-hand side, the subscript “line” indicates that the derivative is along a spectral line with fixed $\\vec{n}_P$ .",
"The key features of Eq.",
"(REF ) are that the denominator on the right-hand side is, up to a simple overall factor, equal to the quantity appearing on the left hand side of the Eq.",
"(REF ), while the numerator is a kinematic factor.",
"Specifically, using $\\overleftarrow{e}^Y \\frac{\\partial {\\cal Y}}{\\partial E^*} \\overrightarrow{e}^Y = \\frac{E^*}{E} \\overleftarrow{e}^Y\\frac{\\partial {\\cal Y}}{\\partial E} \\overrightarrow{e}^Y \\,,$ and $\\frac{d E_k}{d L}\\Bigg |_{\\rm line} =\\frac{E_k^*}{E_k}\\frac{d E_k^*}{d L}\\Bigg |_{\\rm line} -\\frac{\\vec{P}^2}{EL}$ (which follows since $E^2=(E^*)^2 + (\\vec{P} L)^2/L^2$ and because $\\vec{P} L$ is fixed along the spectral line), we find $\\overleftarrow{e}^Y \\frac{\\partial {\\cal Y}}{\\partial E} \\overrightarrow{e}^Y =- 2 i e^{-2i\\phi _1}\\frac{\\frac{\\partial \\phi _1}{\\partial L}+ z^2 \\frac{\\partial \\phi _2 }{\\partial L}}{\\frac{d E_k}{d L}\\big |_{\\rm line} + \\frac{\\vec{P}^2}{E L}} \\,.$ Combining this with (REF ) and (REF ) we conclude $\\frac{M_\\infty ^2}{16 \\pi M_D E^{(0)} V^2 |M_W|^2} &=& -\\frac{\\frac{\\partial \\phi _1 }{\\partial L} + z^2 \\frac{\\partial \\phi _2}{\\partial L} }{\\frac{d E_k}{d L}\\big |_{\\rm line} +\\frac{\\vec{P}^2}{E^{(0)} L}} \\,.$ We thus have found an alternative form of the LL relation which is simpler than Eq.",
"(REF ), and also likely to be more practical.",
"The single channel version of (REF ) is instructive.",
"It can be written, using Eq.",
"(REF ), in terms of the decay rate: $\\Gamma _{D\\rightarrow \\pi \\pi } =\\frac{2 E^{(0)} V^2 |M_W|^2}{M_D} \\left[ \\frac{- \\frac{\\partial \\phi }{\\partial L} }{\\frac{d E_k}{d L}\\big |_{\\rm line} + \\frac{\\vec{P}^2}{E^{(0)} L}} \\right]\\,.$ This form holds both for identical and non-identical particles, with the symmetry factor being contained in $\\Gamma $ .",
"It also sheds light on the sign constraints discussed in the previous section.",
"The right-hand side must be positive.",
"Based on numerical studies, we find that $\\partial \\phi /\\partial L$ is always positive, implying that the denominator, which is proportional to $d E_k^*/dL$ , must be negative.",
"The same holds for the two-channel result, Eq.",
"(REF ).",
"In order for the right-hand side to be positive, the denominator must be negative.",
"Since we could do the LL analysis on almost any spectral line, this appears to imply that $d E_k^*/dL < 0$ in general.",
"The only exception is for a state with $E_k^*$ below the two particle threshold.",
"Such a state occurs, for example, as the lowest energy state for $\\vec{P}=0$ if there is an attractive interaction.",
"For such a state one has $d E_k^*/dL=dE_k/dL > 0$ , i.e.",
"of the “wrong” sign.",
"But in this case the LL analysis does not apply, because the particle lies below threshold in infinite volume."
],
[
"Conclusions",
"We have presented two new results in this article.",
"First, a field-theoretic derivation of the generalization of Lüscher's quantization formula to the case of multiple strongly coupled two-particle channels (where the particles are spinless).",
"Second, the generalization to multiple channels of the Lellouch-Lüscher formula relating finite-volume and infinite-volume matrix elements.",
"We also have explained in some detail how, in the case of two channels, one can use these results to determine the infinite-volume decay amplitudes of a particle which is coupled by a weak interaction to the two-body channels.",
"As already noted in the introduction, this is but a step on the way toward our “dream” application, namely the calculation of $D^0\\rightarrow \\pi \\pi $ and $D^0\\rightarrow K\\overline{K}$ amplitudes.",
"To achieve that goal, one will also need to include the channels with four or more pions.",
"These are significant once one approaches the energy $M_D$ .",
"Work in this direction is underway.",
"An example where our formalism should be useful with minimal approximation is the determination of the isospin breaking in $K\\rightarrow \\pi \\pi $ decays.",
"Given the mass splitting between charged and neutral pions, there are really two two-body channels to consider, and in this case the coupling to the four pion channel is very small and can reasonably be neglected."
],
[
"Acknowledgments",
"This work is supported in part by the US DOE grant DE-FG02-96ER40956.",
"We thank Raul Briceno, Zohreh Davoudi, Harvey Meyer and Ulf Meissner for discussions and comments on the manuscript.",
"This work was facilitated in part by the workshop “New Physics from Heavy Quarks in Hadron Colliders,” which was sponsored by the University of Washington and supported by the DOE."
],
[
"Two-channel Watson's theorem ",
"In this appendix we work out the consequences of Watson's theorem for the phases of the matrix elements of interest, $\\langle \\pi \\pi \\vert \\mathcal {H}_W(0) \\vert D \\rangle $ and $\\langle K \\overline{K} \\vert \\mathcal {H}_W(0) \\vert D \\rangle $ .",
"We assume at first that $\\mathcal {H}_W$ is T invariant, and describe the generalization to non-invariant Hamiltonians at the end.",
"We closely follow the textbook presentation given in Ref. [32].",
"We consider the $3\\times 3$ $S$ -matrix with the three states being the hypothetical $D$ meson (at rest) and the s-wave $\\pi \\pi $ and $K\\overline{K}$ states.",
"We assume that we are in the kinematic regime described in the main text, so that the $3\\times 3$ $S$ -matrix is unitary.",
"Although we introduce a weak coupling between the $D$ and the two particle states, so that the $D$ is a resonance, its width is of second-order in the weak interaction and thus can be ignored at the linear order to which we work.",
"Thus it is valid to treat it as an asymptotic state.",
"Watson's theorem follows by breaking the $S$ -matrix into a strong part $S^{(0)}$ and a weak part $S^{W}$ .",
"The strong part is T invariant, and, since we use states which have definite (positive) T-parity, can be taken to be symmetric.",
"This fixes the phases of the $\\pi \\pi $ and $K\\overline{K}$ states, though not their overall signs.",
"Extending the dimensionless, strong-coupling $S$ -matrix of Eq.",
"(REF ) to include the $D$ gives $S^{(0)} =\\begin{pmatrix}1 & 0 \\\\ 0 & S^s\\end{pmatrix} \\,,$ where 1 is the $1\\times 1$ identity and $S^s$ is the $2\\times 2$ s-wave $S$ -matrix given in (REF ).",
"The weak part only contains couplings between the $D$ and the two-particle states, and in $3\\times 3$ notation is $S^{W} =\\begin{pmatrix}0 & S_{D,\\pi \\pi }^{W} & S_{D,K K}^{W} \\\\ S_{\\pi \\pi ,D}^{W} &0 & 0 \\\\ S_{K K,D}^{W} & 0 & 0\\end{pmatrix}\\,.$ The assumed T invariance implies that it, too, is symmetric.",
"The non-zero elements of $S^{W}$ are proportional to the desired matrix elements $S_{j,D}^{W}&= c P_{jj} \\langle j |[-i \\mathcal {H}_W(0)]| D\\rangle \\,,$ where $j=1,2$ runs over the $\\pi \\pi $ and $K\\overline{K}$ channels, $P$ is the square root of the phase space factor defined in Eq.",
"(REF ), and $c$ is a known real constant whose value will not be needed.",
"Unitarity of the complete $S$ -matrix implies that the terms linear in the weak interaction satisfy $i S^{W} = S^{(0)} \\big (i S^{W} \\big )^\\dagger S^{(0)} \\,.$ This implies that $i S_{j,D}^{W} = S^s_{jk} \\big (i S_{D,k}^{W}\\big )^*= S^s_{jk} \\big (i S_{k,D}^{W}\\big )^*\\,,$ where in the last step we have used the symmetry of $S^{W}$ .",
"Using the explicit form for the two-channel $S$ -matrixFor simplicity of presentation, we are here using $\\delta _1=\\delta _\\alpha $ and $\\delta _2=\\delta _\\beta $ .",
"$S^s = R^{-1}\\begin{pmatrix}e^{2i \\delta _{1}} & 0 \\\\ 0 & e^{2i\\delta _{2}}\\end{pmatrix}R\\,,$ with $R =\\begin{pmatrix}\\operatorname{c}_\\epsilon & \\operatorname{s}_\\epsilon \\\\ - \\operatorname{s}_\\epsilon & \\operatorname{c}_\\epsilon \\end{pmatrix} \\,,$ we find $i R_{jk} S_{k,D}^{W} =e^{2i \\delta _{j}} \\big (iR_{jk} S_{k,D}^{W}\\big )^*\\,.$ It follows that the phase of $i R_{jk} S_{k,D}^{W}$ is $e^{i\\delta _j}$ .",
"This is the desired generalization of Watson's theorem to two channels.",
"Thus the quantities $v_j = e^{-i\\delta _j} \\frac{1}{c} \\sqrt{4\\pi E^*}\\, i R_{jk} S_{k,D}^{W}$ are real.",
"Using (REF ) we can rewrite the $v_j$ as in Eqs.",
"(REF ) and ().",
"If the weak interaction is not T invariant, then $S_{j,D}^{W}$ will contain some number of T-violating phases.",
"Since we are working to linear order in the weak interaction, we can break up $\\mathcal {H}_W$ into parts each with a single T-violating phase and treat each separately.",
"Each such part has an overall phase $e^{i\\phi _T}$ , and the symmetry of the S-matrix is replaced by $S_{D,k}^{W}(\\phi _T) = S_{k,D}^{W}(-\\phi _T)\\,.$ However, if we first pull out the overall phase by hand, then the symmetry of $\\Delta S$ is restored, and Watson's theorem applies to the residue."
]
] | 1204.0826 |
[
[
"Transmission resonance spectroscopy in the third minimum of 232Pa"
],
[
"Abstract The fission probability of 232Pa was measured as a function of the excitation energy in order to search for hyperdeformed (HD) transmission resonances using the (d,pf) transfer reaction on a radioactive 231Pa target.",
"The experiment was performed at the Tandem accelerator of the Maier-Leibnitz Laboratory (MLL) at Garching using the 231Pa(d,pf) reaction at a bombarding energy of E=12 MeV and with an energy resolution of dE=5.5 keV.",
"Two groups of transmission resonances have been observed at excitation energies of E=5.7 and 5.9 MeV.",
"The fine structure of the resonance group at E=5.7 MeV could be interpreted as overlapping rotational bands with a rotational parameter characteristic to a HD nuclear shape.",
"The fission barrier parameters of 232Pa have been determined by fitting TALYS 1.2 nuclear reaction code calculations to the overall structure of the fission probability.",
"From the average level spacing of the J=4 states, the excitation energy of the ground state of the 3rd minimum has been deduced to be E(III)=5.05 MeV."
],
[
"Transmission resonance spectroscopy in the third minimum of $^{232}$ Pa L. Csige$^{1,2}$ M. Csatlós$^2$ T. Faestermann$^3$ J. Gulyás$^2$ D. Habs$^{1,4}$ R. Hertenberger$^1$ M. Hunyadi$^2$ A. Krasznahorkay$^2$ H. J. Maier$^1$ P. G. Thirolf$^1$ H.-F. Wirth$^3$ $^1$ Ludwig-Maximilians-Universität München, D-85748 Garching, Germany $^2$ Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI), Post Office Box 51, H-4001 Debrecen, Hungary $^3$ Technische Universität München, D-85748 Garching, Germany $^4$ Max-Planck-Institute for Quantum Optics, D-85748 Garching, Germany The fission probability of $^{232}$ Pa was measured as a function of the excitation energy in order to search for hyperdeformed (HD) transmission resonances using the ($d,pf$ ) transfer reaction on a radioactive $^{231}$ Pa target.",
"The experiment was performed at the Tandem accelerator of the Maier-Leibnitz Laboratory (MLL) at Garching using the $^{231}$ Pa($d,pf$ ) reaction at a bombarding energy of $E_{\\text{d}}$ =12 MeV and with an energy resolution of $\\Delta E$ =5.5 keV.",
"Two groups of transmission resonances have been observed at excitation energies of $E^*$ =5.7 and 5.9 MeV.",
"The fine structure of the resonance group at $E^*$ =5.7 MeV could be interpreted as overlapping rotational bands with a rotational parameter characteristic to a HD nuclear shape ($\\hbar ^2/2\\Theta $ =2.10$\\pm $ 0.15 keV).",
"The fission barrier parameters of $^{232}$ Pa have been determined by fitting TALYS 1.2 nuclear reaction code calculations to the overall structure of the fission probability.",
"From the average level spacing of the $J$ =4 states, the excitation energy of the ground state of the 3rd minimum has been deduced to be $E_{\\text{III}}$ =5.05$^{+0.40}_{-0.10}$ MeV.",
"21.10.Re; 24.30.Gd; 25.85.Ge; 27.90.+b The observation of discrete $\\gamma $ transitions between hyperdeformed (HD) nuclear states represents one of the last frontiers of high-spin physics.",
"Although a large community with 4$\\pi $ $\\gamma $ arrays was searching for HD states in very long experiments, no discrete HD $\\gamma $ transition was found in the mass region of $A\\approx $ 100-130 [1], [2], [3], [4], [5].",
"On the other hand, the existence of low-spin hyperdeformation in the third minimum of the fission barrier is established both experimentally and theoretically in the actinide region [6], [7].",
"Observing transmission resonances as a function of the excitation energy caused by resonant tunneling through excited states in the third minimum of the potential barrier can specify the excitation energies of the HD states.",
"Moreover, the observed states could be ordered into rotational bands and the moments of inertia of these bands can characterize the underlying nuclear shape, proving that these states have indeed a HD configuration.",
"Regarding hyperdeformation, the double-odd nucleus $^{232}$ Pa is of great interest.",
"Even though low-spin hyperdeformation has already proved to be a general feature of uranium [8], [9], [10] and thorium isotopes [11], no HD state has been found in protactinium isotopes so far.",
"In particular, the level scheme of the odd-odd $^{232}$ Pa is completely unknown in the 1st minimum of the potential barrier, only the ground-state properties are known at present (I$_{\\text{gs}}^{\\pi }$ =2$^-$ ) [12].",
"The fine structure of the fission resonances of $^{232}$ Pa has been studied so far only via the ($n,f$ ) reaction [13] with high resolution, but the results showed no convincing evidence on the existence of HD states.",
"A possible reason was the rather limited momentum transfer of the ($n,f$ ) reaction at that low neutron energy ($E_{\\text{n}}\\approx $ 100 keV), which did not allow for the population of rotational bands.",
"In contrast, the ($d,p$ ) reaction can transfer considerable angular momentum, thus full sequences of rotational states with higher spins can be excited.",
"The experimental ($n,f$ ) cross-section was used very recently to deduce the fission barrier parameters of $^{232}$ Pa by performing cross-section calculations with the EMPIRE 2.19 nuclear reaction code [14], in which the optical model for fission was extended to treat double- and triple-humped fission barriers.",
"The fission barrier parameters of $^{232}$ Pa were determined to be $E_{\\text{A}}$ =5.92, $E_{\\text{BI}}$ =6.3 and $E_{\\text{BII}}$ =6.34 MeV [15].",
"This result suggested to expect the appearance of HD resonances in the excitation energy region between $E^*$ =5.9 and $E^*$ =6.3 MeV.",
"In our experiment, the fission probability of $^{232}$ Pa was measured as a function of the excitation energy with high resolution in order to search for HD rotational bands using the $^{231}$ Pa($d,pf$ ) reaction.",
"The experiment was carried out at the Tandem accelerator of the Maier-Leibnitz-Laboratory (MLL) at Garching employing the $^{231}$ Pa($d,pf$ ) reaction with a bombarding energy of $E_{\\text{n}}$ =12 MeV to investigate the fission probability of $^{232}$ Pa in the excitation energy region of $E^*$ =5.5-6.2 MeV.",
"Enriched (99%), 70 $\\mu $ g/cm$^2$ thick radioactive target of $^{231}$ Pa was used on a 20 $\\mu $ g/cm$^2$ thick carbon backing.",
"The ground-state $Q$ -value for the reaction is $Q$ =3.324 MeV, which was calculated using the NNDC $Q$ -value calculator.",
"The excitation energy of the fissioning nucleus was derived from the kinetic energy of the outgoing protons, that was measured by the Garching Q3D magnetic spectrograph [16] set at $\\Theta _{\\text{lab}}$ =139.4$^{\\circ }$ relative to the beam direction.",
"The well-known lines of the $^{208}$ Pb($d,p$ ) reaction were applied to perform the energy calibration of the focal plane detector [17].",
"The experimental energy resolution was deduced to be $\\Delta E$ =5.5 keV (FWHM) in the energy region of our interest.",
"Fission fragments were detected in coincidence with the outgoing protons by two position-sensitive avalanche detectors (PSADs) with a solid angle coverage of 20% of 4$\\pi $ .",
"Figure: The measured fission probability of 232 ^{232}Pa in the excitation energy range of E * E^*=5.5-6.2 MeV.",
"Two resonance groups have been observed around E * E^*=5.75 and 5.9 MeV, respectively, in agreement with the results of a previous (n,fn,f) experiment .",
"Below E * E^*=5.82 MeV a magnified scale of the y-axis was used for better visibility of the resonance structure.The measured high-resolution fission probability spectrum of $^{232}$ Pa is shown in Fig.",
"REF as a function of the excitation energy of the fissioning nucleus in the region of $E^*$ =5.5-6.2 MeV.",
"The random coincidence contribution was subtracted by using the well-defined flight time difference of protons and fission fragments.",
"Two resonance groups can be clearly seen at $E^*$ =5.75 and 5.9 MeV in a fair agreement with the results of a previous ($n,f$ ) experiment [13].",
"Below $E^*$ =5.82 MeV a magnified scale of the y-axis was used for better visibility of the resonance structure.",
"In the ($n,f$ ) experiment, low-energy neutrons ($E_{\\text{n}}$ =120-420 keV) were used to populate the states in the compound nucleus.",
"In this case s-wave neutron capture is the dominant process and the transfer momentum is principally limited to 1$\\hbar $ , thus rotational bands cannot be excited.",
"On the other hand, the fission fragment angular distribution (FFAD) data supported a $K$ =3$^+$ assignment for the resonance at $E_{\\text{n}}$ =156.7 keV.",
"Together with a possible $K$ =3$^-$ assignment for the resonance at $E_{\\text{n}}$ =173.3 keV, which could not be ruled out by the FFAD data, these two resonances could be the bandheads of two close-lying $K$ -bands with opposite parities, a well-known consequence of the octupole deformation in the HD minimum of the fission barrier.",
"However, having no information on the moment of inertia, this result could not be considered as a clear evidence on the existence of a HD minimum as also stated by the authors.",
"Figure: The result of the Markov-chain peak searching algorithm applied to the histogram, which is shown in Fig.",
"(E * E^*=5.7-5.8 MeV).",
"The continuous fission background stemming from the non-resonant tunneling process is subtracted.",
"The positions of the identified resonances are indicated by arrows.Due to the low neutron separation energy of $^{232}$ Pa ($S_{\\text{n}}$ =5.455 MeV), the fission probability is rather small, which resulted in a very limited statistics at deep sub-barrier energies.",
"Therefore, to suppress the statistical fluctuations of the excitation energy spectrum and to identify the resonances unequivocally, we applied a widely used peak-searching method, the so-called Markov-chain algorithm [18] to the data.",
"This method can also be used to subtract the continuous, exponentially rising fission background originating from the non-resonant tunneling process through the fission barrier.",
"In the generated spectrum, a number of sharp resonances could be clearly identified between $E^*$ =5.7 and 5.8 MeV as indicated by arrows in Fig.",
"REF .",
"The limited statistics of the experiment did not allow to extract the spin information from the angular data.",
"However, based on the results of the ($n,f$ ) experiment [13], the first resonance around $E^*$ =5.72 MeV can be identified as the bandhead of a $K$ =3 band with more members in the present experiment owing to the larger transfer momentum of the ($d,p$ ) reaction.",
"The resonance group at $E^*$ =5.9 MeV could not be resolved into individual resonances as a consequence of the large level density (thus strongly overlapping) at this high excitation energy.",
"To allow for an identification of the underlying structure as either resulting from a hyperdeformed (HD) or superdeformed (SD) configuration, the observed resonances have been fitted with overlapping rotational bands assuming both scenarios.",
"Gaussians were used to describe the different band members with a width fixed to the experimental resolution ($\\Delta E$ =5.5 keV).",
"During the fitting procedure, the energy of the bandheads and the intensity of the band members were treated as free parameters and a common rotational parameter was adopted for each band ($\\hbar ^2/2\\Theta $ =2.10$\\pm $ 0.15 keV for the HD scenario).",
"Since the population of the different spins varies only slightly with excitation energy, the intensity ratios of the band members were kept to be constant.",
"The result of the fitting procedure is presented in Fig.",
"REF .",
"The picket fence structure of the three rotational bands is indicated in the figure as well as the quality of the fit ($\\chi ^2/F$ =1.03 with $F$ =99).",
"Given the resonance positions determined by the Markov-chain algorithm, we also tested the assumption of an underlying SD rotational band configuration generating the observed resonance structure.",
"In this scenario (Fig.",
"REF ), our data could be described by four rotational bands with $K$ value assignments of $K$ =3,2,2 and 3, respectively.",
"In this case, the rotational parameter of the bands was $\\hbar ^2/2\\Theta $ =3.3$\\pm $ 0.2 keV, which is characteristic to SD nuclear shapes.",
"The quality of the fit is $\\chi ^2/F$ =1.09 ($F$ =99).",
"However, there are several arguments disfavoring this interpretation.",
"As one can see in Fig.",
"REF , only two resonances could be combined to form the first SD rotational band (K=3), while at expected positions of further members no resonances were observed.",
"Moreover, we have no proof on the existence of a third member of the last $K$ =3 band (expected at $E^*$ =5800 MeV) due to the high level density in the second resonance structure around $E^*$ =5.9 MeV.",
"The FFAD data [13] also disagree with the assignment of $K$ =2 for the second and third rotational band.",
"Furthermore, the level density should be much higher in the deep second minimum ($E_{\\text{II}}$ =1.9 MeV according to Ref.",
"[15]) at this high excitation energy ($E$ =5.7 MeV).",
"Figure: Excitation energy spectrum of 232 ^{232}Pa with statistical errors.",
"The result of the fitting procedure with HD rotational bands is indicated by the solid line.",
"The picket fence structure of the rotational bands together with the KK values of the bands are also shown.",
"The quality of the fit (the reduced χ 2 \\chi ^2 value) is χ 2 /F\\chi ^2/F=1.03.Figure: Excitation energy spectrum of 232 ^{232}Pa with the result of the fitting procedure (solid line) assuming SD rotational bands.",
"The picket fence structure and the KK values of the bands are also indicated.",
"The dashed lines represent the missing members of the bands.",
"The quality of the fit (the reduced χ 2 \\chi ^2 value) is χ 2 /F\\chi ^2/F=1.09.As a conclusion, the observed resonance fine structure can most convincingly be described as a sequence of three overlapping rotational bands with assignments of $K$ =3,4 and 4 for the bandheads at $E^*$ =5717, 5740 and 5745 keV, respectively, and with a rotational parameter ($\\hbar ^2/2\\Theta $ =2.10$\\pm $ 0.15 keV) characteristic for HD nuclear shapes.",
"The depth of the third minimum was determined by comparing the experimentally obtained average level spacings of the $J$ =4 members ($D_{J=4}$ =9 keV) of the HD rotational bands (Fig.",
"REF ) with the calculated ones using the back-shifted Fermi gas (BSFG) description of the level density.",
"The level density of a given nucleus has usually been determined by adjusting the level density parameters to obtain the best description of the low-energy cumulative discrete level schemes as well as the s-wave neutron resonance spacings.",
"However, in the case of $^{232}$ Pa no level scheme is available as already pointed out, so we could not extract the NLD parameters this way.",
"On the other hand, systematic investigations of the NLD showed, that very simple analytic expressions can be used to estimate the NLD parameters involving some basic nuclear quantities like the shell correction energy and the deuteron pairing energy [19].",
"Following the concept of Ref.",
"[20], the back-shifted Fermi gas (BSFG) level density parameters of $^{232}$ Pa were estimated to be $a$ =23.55 MeV$^{-1}$ and $E_1$ =-1.103 MeV.",
"In order to deduce the excitation energy $E_{\\text{III}}$ of the ground state in the third minimum, an excitation energy of $U_{\\text{III}}$ =$E-E_1-E_{\\text{III}}$ was used in the formulas of Ref.",
"[20], where $E_1$ and $E_{\\text{III}}$ stand for the energy backshift of the 1st and the 3rd potential minimum (with respect to the 1st minimum), respectively.",
"Figure: Experimental and calculated average level spacings of the JJ=4 states in the function of the excitation energy for 232 ^{232}Pa.",
"Calculated level spacings are indicated by a solid and a dashed line for representing the values in the first and in the third potential minimum (with E III E_{\\text{III}}=5.05 MeV), respectively.",
"Within the calculation of the BSFG level densities, we used the same parametrization as in Ref. .",
"The error bar of the experimental point (circle) represents the upper and lower limits of the observed JJ=4 level spacings.To match the calculated level spacings to our experimental point (circle in Fig.",
"REF ) the value of $E_{\\text{III}}$ was varied.",
"We obtained the best description with $E_{\\text{III}}$ =5.05 MeV for the excitation energy of the ground state in the third minimum as indicated by solid line in Fig.",
"REF .",
"To estimate the uncertainty of $E_{\\text{III}}$ , we used an upper and a lower limit for the average level spacings.",
"The smallest observed experimental spacing ($D$ =5 keV) was taken as the lower limit, while the upper limit was chosen to be $D$ =24 keV by assuming three equally distributed $J$ =4 states in the excitation energy range of $E^*$ =5.7-5.8 MeV.",
"These limits are indicated in Fig.",
"REF as asymmetric error bars of the experimental point.",
"Our final result is $E_{\\text{III}}$ =5.05$^{+0.40}_{-0.10}$ MeV, which indicates a less deep minimum for $^{232}$ Pa in contrast to our previous results on the even-even uranium isotopes [7], while, however, still being significantly deeper than claimed in a recent theoretical study in this mass range [21].",
"On the other hand, our present result is in a good agreement with the result of Ref.",
"[15], where the third minimum was found to be $E_{\\text{III}}$ =5.4 MeV.",
"To extract the fission barrier parameters of $^{232}$ Pa, we performed cross-section calculations on the $^{231}$ Pa($d,pf$ ) reaction using the TALYS 1.2 nuclear reaction code [22], the only available code that can calculate exclusive fission cross-sections with particle spectra for transfer reactions.",
"In the code, the fission transmission coefficients are calculated following the concept of the Hill-Wheeler formalism, which then enter the Hauser-Feshback statistical model to compete with the particle and photon emission.",
"The fission barrier is parametrized by smoothly joint parabolas, and the barrier parameters, namely the heights ($E_{\\text{A,B1,B2}}$ ) and curvature energies ($\\hbar \\omega _{\\text{A,B1,B2}}$ ) of a triple-humped fission barrier, are given as input parameters.",
"A very important ingredient of the cross-section calculations is the nuclear level density (NLD), both at the equilibrium deformation and at the saddle points.",
"In contrast to the level densities at normal deformation, the saddle level densities generally suffer from a serious lack of experimental information, however, a good approximation can be obtained by introducing additional constants to the ground-state NLD to describe the rotational and vibrational enhancements at large deformations.",
"Figure: Experimental fission probability of 232 ^{232}Pa measured in the present experiment (below E * E^*=6.2 MeV) and in a previous, low-resolution measurement (above E * E^*=6.2 MeV) together with the result of the TALYS 1.2 calculation (continuous line).In Fig.",
"REF the experimental fission probability of $^{232}$ Pa is shown in the excitation energy interval of $E^*=$ 5.2-7.25 MeV together with the result of the TALYS calculation (represented by a solid line) and the obtained fission barrier parameters.",
"The data points of the present experiment ($E^*<$ 6.2 MeV) were extended by a result of a previous, low-resolution ($\\Delta E$ =55 keV) measurement [23] to cover a larger energy range.",
"Class-II (SD) and class-III (HD) states were not introduced into the calculations, so the resonance region could not be reproduced at low excitation energies, however, at this level we aimed at extracting the barrier parameters from the overall structure, the slope and the saturation of the fission probability.",
"Nevertheless, our final parameter set is in good agreement with the results of Ref.",
"[15], [24], where the EMPIRE 2.19 nuclear code was used to calculate the neutron-induced fission cross-section of $^{232}$ Pa and fitted to the experimental cross-section.",
"However, our calculation suggests a slightly lower inner barrier ($E_{\\text{A}}$ =5.1 MeV), taking into account also the relatively large uncertainty of the determination of the inner barrier height.",
"Curvature energies of $\\hbar \\omega _{\\text{A,B1,B2}}$ =1.0 MeV have been used in the calculations.",
"Our fission barrier parameters are consistent with the appearance of class-III resonances between $E^*$ =5.7 and 5.8 MeV and disfavoring the SD interpretation of the resonances.",
"Fig.",
"REF shows the triple-humped fission barrier of $^{232}$ Pa as a result of the present study.",
"The energy region of the observed HD resonances is indicated by two dashed lines.",
"Figure: Triple-humped fission barrier of 232 ^{232}Pa as a result of the present study.",
"The energy region of the observed HD resonances is indicated by horizontal dashed lines (and the hatched area).",
"The curvature energies are ℏω A,B1,B2 \\hbar \\omega _{\\text{A,B1,B2}}=1.0 MeV.Figure REF shows our present results on the inner [Fig.",
"REF (a), as a function of the nuclear charge] and outer [Fig.",
"REF (b), as a function of the fissility parameter] barrier heights of $^{232}$ Pa together with the most recent experimental (empirical) and theoretical fission barrier parameters of even-even actinide nuclei in order to visualize systematic trends.",
"The data points were taken from Ref.",
"[25] (open circles), Ref.",
"[9], [8] (full squares), Ref.",
"[26] (open triangles) and Ref.",
"[27], [28] (open stars).",
"For triple-humped barriers, the average of the two outer barriers ($<E_{\\text{B1}},E_{\\text{B2}}>$ ) is indicated.",
"The data for the inner and outer barrier heights ($E_{\\text{A}}$ and $E_{\\text{B}}$ ) reveal clear trends as a function of the atomic number and fissility parameter, respectively, as illustrated by the two solid lines.",
"Our new data points for $^{232}$ Pa (full triangles) agree reasonably well with these observed tendencies.",
"The dashed line in panel (a) shows the tendency of empirical inner barrier heights determined by using the double-humped fission barrier concept [25], which failed in predicting the most characteristic features of the fission cross-sections of the light actinides and gave rise to the well-known ”Thorium-anomaly” problem, which was resolved by introducing the triple-humped fission barrier concept.",
"Figure: Calculated and experimental (a) inner (E A E_{\\text{A}}) and (b) outer (E B E_{\\text{B}}) barrier heights as a function of the atomic number (ZZ) and fissility parameter (Z 2 /AZ^2/A), respectively, for actinide nuclei in the region of the ”island of fission isomers” (ZZ=88-97).",
"Clear tendencies can be seen for both barriers as illustrated by solid lines.",
"The data points were taken from Ref.",
"(open circles), Ref.",
", (full squares), Ref.",
"(open triangles) and Ref.",
", (open stars).",
"Present results on 232 ^{232}Pa are also shown (full triangles).",
"The dashed line represents the tendency of empirical inner barrier heights, which were determined by using the double-humped fission barrier concept .Summarizing our results, we measured the fission probability of $^{232}$ Pa with high resolution using the $^{231}$ Pa($d,pf$ ) transfer reaction to deduce the fission barrier parameters of $^{232}$ Pa and search for hyperdeformed fission resonances.",
"Sharp transmission resonances have been observed at excitation energies between $E^*$ =5.7-5.8 MeV.",
"These resonance structures could be interpreted most convincingly as overlapping rotational bands with a moment of inertia characteristic of hyperdeformed nuclear shapes ($\\hbar ^2/2\\Theta $ =2.10$\\pm $ 0.15 keV).",
"We found, for the first time, conclusive evidence on hyperdeformed configurations in protactinium isotopes and even more general in an odd-odd nucleus.",
"The fission barrier parameters of $^{232}$ Pa have been deduced by fitting calculated fission probability to the experimental values using the TALYS 1.2 nuclear reaction code.",
"From the average level spacing of the observed resonances the excitation energy of the ground state of the 3rd was determined to be $E_{\\text{III}}$ =5.05$^{+0.40}_{-0.10}$ MeV corresponding to a depth of the third well of 1.25$^{+0.10}_{-0.40}$ MeV.",
"The deduced fission barrier parameters agrees reasonably well with the results of Ref.",
"[15] and support our interpretation of the hyperdeformed fission resonances.",
"The work has been supported by DFG under HA 1101/12-2 and UNG 113/129/0, the DFG Cluster of Excellence ”Origin and Structure of the Universe”, and the Hungarian OTKA Foundation No.",
"K72566."
]
] | 1204.0924 |
[
[
"Universal Constraints on Low-Energy Flavour Models"
],
[
"Abstract It is pointed out that in a general class of flavour models one can identify certain universally present FCNC operators, induced by the exchange of heavy flavour messengers.",
"Their coefficients depend on the rotation angles that connect flavour and fermion mass basis.",
"The lower bounds on the messenger scale are derived using updated experimental constraints on the FCNC operators.",
"The obtained bounds are different for different operators and in addition they depend on the chosen set of rotations.",
"Given the sensitivity expected in the forthcoming experiments, the present analysis suggests interesting room for discovering new physics.",
"As the highlights emerge the leptonic processes, $\\mu\\rightarrow e\\gamma$, $\\mu\\rightarrow eee$ and $\\mu\\rightarrow e$ conversion in nuclei."
],
[
"Introduction",
"One of the reasons to go beyond the Standard Model (SM) is to explain the observed hierarchies in fermion masses and mixing.",
"Any physics beyond the SM is expected to contain new sources of flavour violation and is therefore strongly constrained by experimental data.",
"An interesting hypothesis, known under the name of Minimal Flavour Violation(MFV), is that in the extensions of the SM , as in the SM itself, the maximal fermion flavour symmetry $SU(3)^5$ is broken only by the Yukawa couplings [1].",
"Under that hypothesis, the new sources of flavour violation can be described in terms of higher dimension effective operators, symmetric under $SU(3)^5$ , with Yukawa couplings included as spurion fields.",
"The smallness of some of the Yukawa couplings provides a strong suppression of the FCNC and CP violating effects and therefore MFV is consistent with a new scale as low as a few TeV.",
"However, in most explicit models aiming at explaining the Yukawa coupling pattern constructed so far, the initial flavour symmetry group is much smaller than $SU(3)^5$ .",
"One may then expect potentially more dangerous sources of the FCNC effects and, in consequence, the need for a much higher UV completion scale.",
"It is the purpose of this paper to investigate that question in a large class of flavour models based on horizontal symmetries and to find experimental observables most sensitive to such completions.",
"Similarly as under the MFV hypothesis, once a particular flavour symmetry is assumed, one can construct all effective low-energy operators using a spurion analysis, with the symmetry breaking fields playing the role of spurions [2].",
"Such an analysis has to be repeated for each chosen symmetry group.",
"Furthermore, although in principle all operators allowed by the symmetry arguments are expected to be generated by the (unspecified) UV completions, their coefficients do depend on the UV dynamics and therefore are not under control in such an approach.",
"In this paper we do not take that path.",
"Instead, we point out that in a general class of flavour models one can identify certain patterns of universally present effective FCNC operators.",
"They depend only on the rotation angles transforming the light fermion fields from the flavour basis to the mass eigenstate basis, and the minimal bounds on the scale of UV completions can be estimated for different qualitative pattern of the rotation angles.",
"This is possible because in the flavour basis certain operators are unavoidably present as a consequence of the dynamics supposed to explain the hierarchical Yukawa couplings.",
"For instance, if some flavour diagonal effective operators with non-equal coefficients were always present, our point would be immediately obvious.",
"As we shall see, it is easy to identify the minimal set of operators in the flavour diagonal basis and the above conclusion remains valid.",
"Thus, one can determine the minimal FCNC effects which do not depend on the details of the flavour symmetry.",
"We investigate models based on abelian and non-abelian flavour symmetries.",
"Effective higher dimension operators such as 4-fermion operators or 2-fermion penguin operators necessarily originate from integrating out bosonic degrees of freedom.",
"Here we analyse the flavour violating effects arising from an exchange of heavy scalar degrees of freedom (possibly together with heavy fermions) with the mass scale $M$ that are an integral part of many flavour models.",
"These effects are suppressed by $1/M^2$ .",
"We obtain an absolute lower bound on the scale $M$ of the order of 20 TeV in generic abelian flavour models.",
"In non-abelian models, on the other hand, additional suppression factors in the FCNC effects allow $M$ to be as low as the TeV scale.",
"There are of course also other scalar bosonic degrees of freedom whose integration out generates flavour violating higher dimensional operator.",
"Always present are the SM vector or Higgs bosons.",
"The four-fermion operators generated at tree-level are then suppressed by $1/M^2$ and additional powers of $v^2/M^2$ since they arise from SU(2)$_L$ breaking effects.",
"The case in which these effects are the only new source of flavour-violation has been studied in [3], implying a bound on the scale of the order of roughly 1 TeV.",
"In supersymmetric models flavour violation enters low energy physics through sparticle exchange.",
"Since the suppression scale is only $1/m_{\\rm SUSY}^2$ , these effects can be relevant even if the flavour sector lives at very large scales provided the SUSY breaking scale is even higher [4].",
"Finally, four-fermion operators could arise from an exchange of low-energy flavour gauge bosons as it has been recently studied in [5], [6].",
"In Section 2 we shortly review the UV completion of flavour models in order to show that there must always exist heavy fields that couple to the light fermions and we study the minimal, unavoidable effective flavour violating operators which arise from the exchange of these messenger fields.",
"In section 3 we compare the obtained universal pattern of the minimal FCNC effects with the experimental bounds on flavour-violating operators and obtain constraints on the messenger scale depending on light rotation angles.",
"We then make the additional assumption of a messenger sector compatible with SU(5), which allows us to include other relevant operators in the discussion leading to a variety of correlations between experimental observables.",
"We finally conclude in section 4."
],
[
"Model Setup",
"We consider models with a general flavour symmetry groupFor our analysis it is irrelevant whether the group is discrete, global or gauged.",
"$G_F$ spontaneously broken by the vevs of scalar fields $\\phi _I$ that will be called flavons in the following.",
"The SM Yukawa couplings arise from higher-dimensional $G_F$ invariant operators involving the flavons [7], [8], [9]: $\\mathcal {L}_{yuk} & = y_{ij}^U ~\\overline{q}_{L i} u_{Rj}\\, \\tilde{h} + y_{ij}^D~ \\overline{q}_{L i} d_{Rj} \\, h +{\\rm h.c.} &y_{ij}^{U,\\,D} & \\sim \\prod _{I} \\left( \\frac{\\langle \\phi _I\\rangle }{M} \\right)^{n^{U,\\,D}_{I,ij}},$ where the suppression scale $M \\gtrsim \\langle \\phi _I\\rangle $ is the typical scale of the flavour sector dynamics.",
"The coefficients of the effective operators are assumed to be $\\mathcal {O}\\left( 1 \\right)$ , so that the hierarchy in the Yukawa matrices arise exclusively from the small order parameters $\\epsilon _I \\equiv {\\langle \\phi _I\\rangle }/{M}$ .",
"The transformation properties of the SM fields and the flavons under $G_F$ are properly chosen, so that $\\epsilon _I$ together with their exponents $n^{U,D}_{I,ij}$ reproduce the observed hierarchy of fermion masses of mixing.",
"In order to UV-complete these models one has to “integrate in” new heavy fields at the scale $M$ .",
"These messenger fields are vectorlike and charged under $G_F$ .",
"In order to generate the effective Yukawas of Eq.",
"(REF ), the messengers must couple to SM fermions and flavons and, depending on their nature, they mix either with the SM fermions or with the SM Higgs.",
"In the first case one has to introduce vectorlike fermions with the quantum numbers of the SM fermions (see Fig.",
"REF ).",
"In the second case one introduces scalar fields with the quantum numbers of the SM Higgs field (see Fig.",
"REF ).",
"In the fundamental theory, small fermion masses arise from a small mixing of light and heavy fermions for the first possibility, while they arise from small vevs of the heavy scalars in the second case.",
"We refer to these two possibilities as “Fermion UV completion” (FUVC) and “Higgs UV completion” (HUVC), respectively.",
"The only interactions that are relevant for our discussion are the ones that involve messenger fields and SM fermions.",
"The rest of the Lagrangian is only responsible for generating the mixing between light and heavy states and do not affect the minimal flavour effects we are going to discuss in the next section.",
"The relevant interactions can be readily seen in Figs.",
"REF and REF .",
"In particular, in the case of FUVC we have couplings among (mainly) light and heavy fermions of the schematic form: $\\mathcal {L} & \\supset \\alpha ^Q ~ \\overline{q}_{L i} {Q_R}_\\alpha \\,\\phi _I + \\alpha ^D ~\\overline{D}_{L \\beta } d_{R j} \\,\\phi _J +{\\rm h.c.},$ while in the HUVC we are only interested in interactions involving SM fermions and Higgs messengers: $\\mathcal {L} & \\supset \\lambda ^D_{ij}~ \\overline{q}_{L i} d_{R j} \\, H_\\alpha +{\\rm h.c.}$ Since the hierarchy is supposed to arise from the flavour symmetry breaking alone, we can assume that all dimensionless couplings in the fundamental Lagrangian are $\\mathcal {O}\\left( 1 \\right)$ .",
"A more detailed discussion of the structure of the messenger sector is presented in [4].",
"Figure: Schematic diagram for the Fermion UVC.Figure: Schematic diagram for the Higgs UVC."
],
[
"Model-universal FCNC effective operators",
"We now want to derive the effective flavour-violating operators that arise from messenger exchange independently of the details of the particular flavour model.",
"For this we consider flavour-conserving operators, which then induce FCNC effects in the mass basis that only depend on light rotation angles.",
"Let us first assume the presence of a coupling in the messenger Lagrangian of the form ${\\cal L} \\supset \\alpha \\, \\overline{f}_{Li} X_R Y,$ where $\\alpha \\sim \\mathcal {O}\\left( 1 \\right)$ , $f_{Li}$ is a (mainly) light fermion and $X_R$ and $Y$ are a fermion and a scalar of which at least one is a heavy messenger, cf. Eqns.",
"(REF , REF ).",
"From the box diagram with $X,Y$ propagating in the loop (see Fig.",
"REF a) we get the effective operator ${\\cal L}_{eff} \\supset \\frac{|\\alpha |^4}{16 \\pi ^2 M^2} (\\overline{f}_{Li} \\gamma ^\\mu f_{Li})^2,$ where $M$ is the heaviest mass in the loop and we neglected factors of $\\mathcal {O}\\left( 1 \\right)$ .",
"We can use the same coupling also to write down a penguin diagram with a mass insertion in the external fermion line (see Fig.",
"REF b).",
"This generates the dipole operator ${\\cal L}_{eff} \\supset \\frac{|\\alpha |^2}{16 \\pi ^2 M^2} m_i \\, \\overline{f}_{Li} \\sigma ^{\\mu \\nu } f_{Ri} F_{\\mu \\nu },$ where $m_i$ is the light fermion mass and we have again estimated the coefficient very roughly.",
"The couplings $ {\\cal L} \\sim \\alpha \\, \\overline{f}_{Li} X_R Y$ are indeed present in the messenger sector of every flavour model.",
"This is clear at least for $i=1,2$ , since the light generations have to couple to some messengers in order to obtain a full rank mass matrix.",
"The same argument holds for $i=3$ in the downExcept in models where the bottom mass arises at the renormalisable level, like in 2HDM with large $\\tan {\\beta }$ .",
"and charged lepton sector, but in general not in the up sector, since the top quark can get massive without coupling to the messenger sector.",
"Finally there are also certain tree-level operators which unavoidably arise in Higgs UV completions.",
"Consider for example the couplings of the heavy neutral Higgs fields to the down quarks ${\\cal L} \\supset \\lambda _{ij} \\overline{d}_{Li} d_{Rj} H.$ In order to get a full rank down mass matrix, at least one of the four couplings $\\lambda _{11}, \\lambda _{12}, \\lambda _{21},\\lambda _{22}$ has to be non-zero.",
"By integrating out $H$ at tree-level (see Fig.",
"REF c) we therefore obtain at least one of the four operators ${\\cal L}_{eff} \\supset \\frac{|\\lambda _{ij}|^2}{M^2} (\\overline{d}_{Li} d_{Rj}) (\\overline{d}_{Rj} d_{Li}) \\qquad (i,j=1,2)$ Again the same reasoning goes through in the up 1-2 sector, and in every charged lepton and down sector (e.g.",
"at least one of the four operators in Eq.",
"(REF ) must exist also for $i,j=1,3$ and $i,j = 2,3$ ), but in most generality not in sectors involving the top quark.",
"In principle there are tree-level operators also in fermion UV completions arising from flavon exchange.",
"As can be seen from Eq.",
"(REF ), these operators are suppressed by $m_i m_j / M_{\\phi }^2 \\langle \\phi \\rangle ^2$ , where $m_i$ are the light fermion masses, $M_\\phi $ is the flavon mass and $\\langle \\phi \\rangle $ its vev.",
"For flavon mass and vev not far from the messenger scale this contribution scales with the fourth power of the inverse messenger scale and gives only very mild bounds on the messenger scale [8].",
"We therefore neglect these contributions in the following.",
"Having collected the unavoidable flavour-conserving operators in Eqns.",
"(REF )–(REF ), we now go to the mass basis using approximate transformations of this kind: $d_{Li} \\rightarrow d_{Li} + \\sum _{j \\ne i} \\theta ^{DL}_{ij} d_{Lj}\\,,$ in order to obtain flavour-violating operators.",
"In specific models there are in general other contributions to such operators that could be even larger.",
"However, the contribution discussed above does not depend on the details of the flavour symmetry and its breaking pattern and therefore allows to estimate minimal predictions for the operator coefficients.",
"Notice that in abelian models there is no reason for cancellations among different contributions to a given FCNC effective operator generated by Eq.",
"(REF ), since the operators in Eqns.",
"(REF )–(REF ) arise from integrating out flavour messengers that by construction have different $\\mathcal {O}\\left( 1 \\right)$ couplings to light fermions, i.e.",
"the breaking of flavour universality is O(1).",
"In non-abelian flavour models those couplings are universal (controlled by the symmetry) and the breaking of flavour universality is suppressed by small order parameters.",
"This is because in those models the operator of Eq.",
"(REF ) in the flavour basis is given by (restricting to the 1-2 sector) ${\\cal L}_{eff} \\sim |\\alpha |^4 \\left( \\overline{f}_{L1} \\gamma ^\\mu f_{L1} + \\overline{f}_{L2} \\gamma ^\\mu f_{L2} \\right)^2,$ which clearly remains in this form after rotating to the mass basis.",
"Flavor transitions are only generated upon including a universality breaking term ${\\cal L}_{eff} \\sim |\\alpha |^4 \\left( \\overline{f}_{L1} \\gamma ^\\mu f_{L1} + \\overline{f}_{L2} \\gamma ^\\mu f_{L2} + \\Delta _{12} \\overline{f}_{L2} \\gamma ^\\mu f_{L2} \\right)^2.$ Each flavor transition in the operator in the mass basis is then suppressed by $\\Delta _{12}$ that depends on the flavon vevs that are responsible for universality breaking.",
"Therefore in non-abelian models there is an additional suppression of the above effect that depends on the particular flavour transition.",
"In principle this suppression can be as large as in MFV, but in models with a single non-abelian factor, like in most explicit models constructed so far, the additional suppression factor is rather mild compared to MFV.",
"To estimate this factor we consider the case of a SU(3)$_F$ model [10] with all quarks transforming as a ${\\bf 3}$ and flavons as ${\\bf \\overline{3}}$ .In models with both singlets and triplets there is no additional suppression of flavour transitions involving singlets, but possibly larger suppression for transitions involving only triplets.",
"The flavons get hierarchical vevs that induce the quark masses.",
"The flavon vev responsible for universality breaking in the $i$ -$j$ sector is therefore roughly given by the square root of the Yukawa coupling $\\sqrt{y_{jj}}$ .",
"The situation is similar in the case of some U(2)$_F$ flavour models [11], where transitions in 1-3 and 2-3 sector are always unsuppressed.",
"Taking into account the general case of two Higgs doublet models (2HDM), one finds the following suppression factors for each flavour transitions (two flavon insertions): Table: Additional suppression factors of flavour transitions in simple non-abelian models.where $ \\tan {\\beta } = v_u/v_d$ and $\\epsilon $ is of the order of the Cabibbo angle.",
"This table implies for example that the coefficients of the four-fermion operators relevant for $K-\\overline{K}$ mixing would get an additional suppression of $\\epsilon ^{10} \\tan {\\beta }^2$ (we need two flavour transitions) with respect to the abelian case.",
"The model-independent bounds on effective operator coefficients in abelian flavour models can be therefore easily extended to simple non-abelian groups.",
"Moreover, the abelian case is relevant for non-abelian models with some SM fermions transforming as singlets of the non-abelian flavour group.",
"We can now use the procedure outlined above in order to obtain estimates for the minimal coefficients of FCNC operators induced by the messenger sector of generic flavour models.",
"The most interesting of these operators are those related to $\\Delta F=2$ processes and the LFV decays $\\mu \\rightarrow eee$ , $\\mu \\rightarrow e \\gamma $ .",
"In Table REF we show the estimates of the operator coefficients separately for Higgs and fermion UVC for abelian models and add for comparison the MFV prediction.",
"A few comments regarding the estimation of the Wilson coefficients are in order: The expressions in Table REF are valid for abelian flavour models, but can be generalised to simple non-abelian groups using the suppression factors provided in Table REF .",
"The expressions involving b-quarks is valid in general only if $m_b$ does not arise at the renormalisable level.",
"The estimate on the tree-level contribution to $\\mu \\rightarrow e e e$ in Higgs UVCs is rather conservative, since it accounts for the possibility that $\\lambda ^E_{11}, \\lambda ^E_{12}, \\lambda ^E_{21}$ all vanish.",
"Still this minimal effect can be sizable if the rotations are large.",
"In summary one can obtain minimal predictions of the coefficients of certain flavour-violating effective operators, which do not depend on the details of the flavour model but only on the light fermion mass matrix.",
"These estimates are derived from the coefficients of flavour-diagonal operators (which can be easily obtained in generic flavour models) and the corresponding rotation angles.",
"In non-abelian models one has to take into account the additional suppression discussed above.",
"Table: Relevant operators and their minimal Wilson coefficients in units of 1/M 2 1/M^2 for HUVC, FUVC and MFV.",
"Here L≃1/16π 2 L\\simeq 1/16 \\pi ^2, X,Y=L,RX, Y = L,R with Y≠XY \\ne X."
],
[
"Phenomenological Implications",
"The predictions for the minimal flavour-violating effects obtained in the previous section are to be compared with experimental bounds for those operators.",
"In Table REF we list the present bounds on the Wilson coefficients of the relevant operators for a suppression scale of 1 TeV.For completeness, we include the bounds to left-right vector operators like $(\\overline{s}_L\\gamma ^\\mu d_L) (\\overline{s}_R\\gamma ^\\mu d_R)$ , even though we do not use them in our analysis, as in HUVC they give a negligible constraint compared to the scalar operators like $(\\overline{s}_Ld_R) (\\overline{s}_Rd_L)$ , while in FUVC they do not arise in a model-independent way.",
"Table: Relevant processes and corresponding operators with bounds on Wilson coefficients.",
"Values in [ ] are for expected future experimental bounds.",
"X,Y=L,RX, Y = L,R with Y≠XY \\ne X.The bounds on the Wilson coefficients of the $\\Delta F=2$ operators have been obtained as in [12], [13], taking into account the QCD running of the operators (from 1 TeV) and imposing as a condition for the meson mass splittings $(\\Delta m)^{\\rm NP}\\le (\\Delta m)^{\\rm exp}$ , where $(\\Delta m)^{\\rm NP}$ represent the flavour messenger contributions and $(\\Delta m)^{\\rm exp}$ the experimental measured values reported in [14].The bounds in the $D-\\overline{D}$ system are in good agreement with the values reported in the literature [15], once different conventions in the definition of the operators are taken into account.",
"For the CPV observables, we imposed $\\epsilon _K^{NP}\\le 0.6\\times \\epsilon _K^{\\rm exp}$ in the $K-\\overline{K}$ system [12], [13], and in the $B_{d,s}$ and $D$ sectors we required that the total prediction (including the SM contribution with the corresponding uncertainty) is within the 2$\\sigma $ experimental ranges $\\begin{tabular}{cc} 0.62 \\le S_{\\psi K_S} \\le 0.72 \\cite {PDG} & - 0.23 \\le S_{\\psi \\phi } \\le 0.53 \\cite {LHCbpsiphi} \\\\& \\\\ 0.5 \\le |q/p|_D \\le 1.2 \\cite {PDG} & - 4 \\times 10^{-3} \\le A_\\Gamma \\le 7 \\times 10^{-3} \\cite {PDG}.\\end{tabular}$ The bounds on the imaginary parts of the Wilson coefficients have been taken to be the largest possible values consistent with the above ranges combined with the constraints on $\\Delta m$ .",
"The future bounds on the Wilson coefficients have been obtained using the expected LHCb sensitivities on CPV observables in the $D$ and $B_s$ sectors given in [17].",
"In particular, we consider a future sensitivity on $|q/p|_D$ at the level of $10^{-3}$ , of the order of the naive SM prediction, even though we cannot exclude that the SM contribution is actually much larger, due to large long distance uncertainties.",
"Of course, the future bounds reported in the table are valid under the hypothesis that no CPV is observed with an experimental sensitivity at the level mentioned above.",
"Note that, in this case, the future bounds in the $D-\\overline{D}$ system will be almost as strong as in the $K$ sector.",
"The bounds for the LFV processes have been computed taking into account the recent 90% CL limit ${\\rm BR}(\\mu \\rightarrow e\\gamma )< 2.4\\times 10^{-12}$ obtained by the MEG experiment [18].",
"The future bounds correspond to the expected sensitivities for BR($\\mu \\rightarrow e \\gamma $ ) $\\sim 10^{-13}$ [18] and BR($\\mu \\rightarrow e e e$ ) $\\sim 10^{-16}$ [19].",
"Using the information in Tables REF and REF we can easily estimate the bounds on the messenger scale separately for each flavour transition as a function of the rotation angles.We neglect running effects above 1 TeV.",
"Of course in the SM only the difference of left-handed rotations in the up and down sector are observable.",
"In particular right-handed rotations are not constrained, though in many flavour models they are roughly of the same order as the left-handed ones.",
"Left-handed rotations have to be smaller or equal than the corresponding CKM entries in the absence of cancellations between up and down sector, which is what we assume in the following.",
"Note that also the left-handed rotations in each sector can be complex, since the field redefinitions are already used to absorb the 5 phases of the CKM matrix given by the difference between up and down sector left-handed rotations.",
"For a given quark flavour transition we consider six different cases for the rotation angles: the left-handed rotation is either zero or given by the CKM value for this transition, and the right-handed rotation is either maximalMaximal rotation means $1/\\sqrt{2}$ , but we will simply write “1” in the tables.",
"To calculate the bound we take into account all additional factors of $\\sqrt{2}$ that arise in this case., zero, or equal to the left-handed rotation.",
"Since there is no effect when both rotations vanish and the predictions are symmetric in the exchange $L \\leftrightarrow R$ , only four combinations are relevant.",
"We also distinguish the general case of complex rotation angles from the specific scenario with vanishing phases, in which case only the bounds from CP conserving $\\Delta F =2$ observables apply.",
"Finally we distinguish between fermionic and Higgs messengers, the only difference being that in the Higgs UV completion scalar-scalar operators with 1-2 transitions arise at tree-level while they are negligible in the fermionic messenger case.",
"The results are shown in Tables REF and REF .",
"These numbers give a rough impression of the importance of the minimal effect that we are discussing (up to unknown $\\mathcal {O}\\left( 1 \\right)$ coefficients).",
"Indeed for large and complex right-handed rotations in the down 1-2 sector, this effect is sensitive to Higgs messenger scales up to $\\mathcal {O}\\left( 10^5 \\right)$ TeV, which is roughly the magnitude of the new physics scale one obtains without any suppression [12] (up to a factor of 1/$\\sqrt{\\epsilon }$ ).",
"Fermionic messengers give weaker constraints, because of the absence of tree-level effects, but nevertheless are sizable in many cases and can test messenger scales up to $\\mathcal {O}\\left( 10^3 \\right)$ TeV.",
"Table: Constraints from K-K ¯K - \\overline{K} (up) and D-D ¯D - \\overline{D} (down) mixing on the messenger scale in TeV for Higgs and fermion UV completions with real and complex (*) rotations angles.",
"Values in [ ] correspond to the expected future experimental sensitivitiesThe most interesting aspect of these tables regards the 1-2 sector.",
"Since the left-handed rotation must be of the order of the Cabibbo angle ($\\approx \\epsilon $ ) either in the up or in the down sector or both, the messenger scale must be larger than the smallest entry in Table REF .",
"We therefore obtain an overall minimal bound on the messenger scale given by 19 TeV for the case that the rotation angle is real and comes from the down sector.",
"Since in non-abelian models there are additional suppression factors (cf.",
"Table REF ), the minimal effects alone do not exclude the possibility that the messenger fields of such models could be as light as a TeV and therefore in the reach of LHC.",
"Table: Constraints from B d -B ¯ d B_d - \\overline{B}_d (up) and B s -B ¯ s B_s - \\overline{B}_s (down) mixing on the messenger scale in TeV for Higgs and fermion UV completions with real and complex (*) rotations angles.",
"Values in [ ] correspond to the expected future experimental sensitivities.Despite the fact that the constraints in the 1-3 and 2-3 sector are much weaker, they are still relevant since minimal effects in the 1-2 sector do not imply small effects in other sectors where rotations can be large, giving bounds on the messenger scale comparable or even larger than 19 TeV (cf.",
"Table REF ).",
"In particular this is true in non-abelian models where transitions in the 1-2 sector are much stronger suppressed compared to sectors involving the 3rd family, whereas, in abelian models, it is clear from comparing Tables REF and REF that large effects in the 2-3 sector are only possible if the phases in the 1-2 sector are sufficiently suppressed.",
"The possibility of large CP violation in $B_s$ mixing is especially interesting, since the experimental sensitivity will be improved by LHCb in the near future.",
"For lepton flavour transitions we consider six possibilities for the rotation angles: the right-handed rotation can be maximal, zero or $\\epsilon $ , as it can be the case in SU(5), and the left-handed rotation is either maximal, zero, or equal to the right-handed rotation.",
"Since there is no effect when both rotation vanish and the predictions are symmetric in the exchange $L \\leftrightarrow R$ , only five combinations are relevant.",
"CP violating effects do not play a role here, but we still distinguish between fermionic and Higgs messengers, again the only difference being that in the Higgs UVC certain operators can arise at tree-level.",
"Table: Constraints from μ→eγ\\mu \\rightarrow e \\gamma and μ→eee\\mu \\rightarrow eee on the messenger scale in TeV for Higgs and fermion UV completions.",
"Values in [ ] are for expected future experimental bounds on these processes.The results are shown in Table REF .",
"For large rotation angles the minimal LFV effect is sensitive to messenger scales of the order of 50 TeV with present exclusion limits and up to hundreds of TeV with future limits.",
"The most interesting aspect is that the branching ratio ${\\rm BR}(\\mu \\rightarrow eee )$ can be substantially enhanced with respect to ${\\rm BR}(\\mu \\rightarrow e \\gamma )$ , provided that the corresponding tree-level operator is sizable.",
"In general, there is a lower bound on the ratio ${\\rm BR}(\\mu \\rightarrow eee )$ /${\\rm BR}(\\mu \\rightarrow e \\gamma )$ from the dipole transition $\\mu \\rightarrow e \\gamma ^*$ approximately given by $6 \\times 10^{-3}$ [20].",
"This bound is saturated in many SM extensions like SUSY, and is indeed close to the ratio one obtains here when both operators arise at loop-level and have the same angle dependence, as it is the case for fermionic messengers.",
"In the case of Higgs UVC instead $\\mu \\rightarrow eee$ can arise through a tree-level exchange of a heavy Higgs messenger.",
"The minimal effect is suppressed by the product of three rotations, but can win over the loop induced one when rotations are large.",
"In this case ${\\rm BR}(\\mu \\rightarrow eee )$ /${\\rm BR}(\\mu \\rightarrow e \\gamma )$ can be as large as 19 if the rotation angles are maximal.",
"This can be read from Table REF using the formula $\\frac{{\\rm BR}(\\mu \\rightarrow eee )}{{\\rm BR}(\\mu \\rightarrow e \\gamma )} = \\left( \\frac{M_{\\mu \\rightarrow eee}}{M_{\\mu \\rightarrow e \\gamma }} \\right)^4 \\, \\frac{{\\rm BR^{exp}}(\\mu \\rightarrow eee )}{{\\rm BR^{exp}}(\\mu \\rightarrow e \\gamma )},$ where BR$^{\\rm exp}(\\mu \\rightarrow X)$ denotes the present experimental limit and $M_{\\mu \\rightarrow X}$ the corresponding bound on the mass scale of the effective operator shown in the table.",
"There are good prospects to test this ratio in the future.",
"In fact, if $\\mu \\rightarrow e \\gamma $ is observed at the MEG experiment with ${\\rm BR}(\\mu \\rightarrow e \\gamma )\\gtrsim 10^{-13}$ , $\\mu \\rightarrow eee$ generated at tree-level (HUVC) can still have a rate close to the present experimental limit (${\\rm BR}(\\mu \\rightarrow eee )< 10^{-12}$ ), whereas in the case when $\\mu \\rightarrow eee$ arises at loop level, the rate is expected to be ${\\rm BR}(\\mu \\rightarrow eee ) \\gtrsim 8 \\times 10^{-16}$ (irrespectively of the mixing angles), which is still in the reach of the future experimental sensitivity.",
"As an illustration of the magnitude of the effects we estimate the branching ratios for a reference scale of 19 TeV (as needed to satisfy the quark sector constraints): ${\\rm BR}(\\mu \\rightarrow e \\gamma ) & \\simeq 5.3 \\times 10^{-12} \\left( \\frac{19 ~{\\rm TeV}}{M} \\right)^4 \\left( \\frac{{\\rm max}(\\theta ^{EL}_{12}, \\theta ^{ER}_{12})}{\\epsilon } \\right)^2 \\\\\\nonumber \\\\{\\rm BR}(\\mu \\rightarrow eee) & \\simeq 2.9 \\times 10^{-13} \\left( \\frac{19 ~{\\rm TeV}}{M} \\right)^4 \\left( \\frac{{\\rm max}(\\theta ^{EL}_{12}, \\theta ^{ER}_{12})}{\\epsilon } \\right)^2 \\left( \\frac{\\theta ^{EL}_{12} }{\\epsilon } \\right)^2 \\left( \\frac{\\theta ^{ER}_{12} }{\\epsilon } \\right)^2,$ where we assumed HUVC for the $\\mu \\rightarrow eee$ rate.",
"Notice that the bounds from the quark sector still allow for rates of LFV observables in the reach of running or future experiments.",
"Let us summarise the main points of the above discussion (valid up to $\\mathcal {O}\\left( 1 \\right)$ coefficients): The messenger scale in abelian models has to be larger than 19 TeV.",
"This minimal bound does not prevent large effects in $B_q - \\overline{B}_q$ mixing and LFV decays, because the rotations in the corresponding sectors could be large.",
"In non-abelian models the minimal effects do not exclude messengers at the TeV scale and thus in the reach of LHC.",
"${\\rm BR}(\\mu \\rightarrow eee )$ /${\\rm BR}(\\mu \\rightarrow e \\gamma )$ can be as large as $\\mathcal {O}\\left( 10 \\right)$ for large leptonic rotations."
],
[
"Predictions in SU(5)",
"The number of completely model-independent operators that are subject to sizable constraints is restricted to $\\Delta F=2 $ and dipole operators involving 1-2 flavour transitions.",
"Only with additional assumptions one can make further statements on e.g.",
"two-quark-two-lepton ($2q2\\ell $ ) operators.",
"A particular well-motivated and predictive assumption is that the flavour sector is compatible with an (approximate) SU(5) GUT structure, which connects lepton and quark operators and correlates the charged lepton and down quark mass matrix.",
"In particular this implies: (i) the existence of heavy states that couple both to quarks and leptons, so that diagrams as in Fig.",
"REF unambiguously induce ($2q2\\ell $ ) operators; (ii) $M_d \\approx M_e^T$ and therefore $\\theta ^{DR}_{ij} \\approx \\theta ^{EL}_{ij}$ and $\\theta ^{DL}_{ij} \\approx \\theta ^{ER}_{ij}$ .Since we are doing order of magnitude estimates, we neglect the high-energy corrections to this relation, such as the Georgi-Jarlskog factor [21], necessary to correctly account for the low-energy mass ratios of the first two generations leptons and down quarks.",
"We list the interesting operators along with the bounds on their Wilson coefficients in Table REF .",
"These bounds have been obtained using the formulae in [22].",
"For the operators contributing to $B_{d,s} \\rightarrow \\mu ^+ \\mu ^-$ decays, we have used the new LHCb 95% CL limits [23]: ${\\rm BR}(B_{d} \\rightarrow \\mu ^+ \\mu ^-)< 1.0 \\times 10^{-9} \\qquad {\\rm BR}(B_{s} \\rightarrow \\mu ^+ \\mu ^-)< 4.5 \\times 10^{-9}.$ As future bounds on these processes we have taken the values corresponding to the SM predictions, while for $\\mu \\rightarrow e$ conversion in nuclei we have considered the future sensitivity ${\\rm CR}(\\mu \\rightarrow e ~ {\\rm in~ Ti}) \\sim 5\\times 10^{-17}$ [24].",
"Again we compare the bounds with the predictions based on the minimal effects from messenger exchange.",
"In Table REF we show the estimates of the operator coefficients separately for Higgs and fermion UVC and add for comparison the MFV prediction.",
"Table: Relevant operators and their minimal Wilson coefficients in units of 1/M 2 1/M^2 for HUVC, FUVC and MFV.",
"Here L≃1/16π 2 L\\simeq 1/16 \\pi ^2, X,Y=L,R;Y≠XX, Y = L,R; Y \\ne X.For each flavour transition we then use this table to calculate the bounds on the messenger scale for given rotation angles and look for correlations.",
"Now only four combinations of rotation angles are relevant and again we distinguish between fermionic and Higgs messengers, the only difference being that in the Higgs UVC some operators can arise at tree-level.",
"Table: 1-2 sector constraints on the messenger scale in TeV for fermion (up) and Higgs (down) UVCs assuming SU(5).Values in [ ] give the expected future bounds.The results for the 1-2 sector are shown in Table REF .",
"Note that the processes $K_L \\rightarrow \\mu ^+ \\mu ^-$ and $K_L \\rightarrow e^+ e^-$ do not appear in these tables, since the new physics contribution to these processes is negligible if the constraints on the messenger scale from correlated processes are fulfilled.",
"In the case of fermion UVC (Table REF , up) the strongest bound on the scale comes from $K - \\overline{K}$ mixing except for small real angles, in which case $\\mu \\rightarrow e \\gamma $ can be more constraining than $\\Delta m_K$ .",
"The most interesting aspect is that the ratio ${\\rm CR}(\\mu \\rightarrow e ~ {\\rm in ~Ti})/{\\rm BR}(\\mu \\rightarrow e \\gamma )$ is always $\\gtrsim \\mathcal {O}\\left( 0.1 \\right)$ , much larger than the typical SUSY prediction $\\sim \\alpha _{em}$ .",
"Again this follows from the bounds in Table REF using $\\frac{{\\rm CR}(\\mu \\rightarrow e~ {\\rm in ~Ti})}{{\\rm BR}(\\mu \\rightarrow e \\gamma )} = \\left( \\frac{M_{\\mu \\rightarrow e~ {\\rm in ~Ti}}}{M_{\\mu \\rightarrow e \\gamma }} \\right)^4 \\, \\frac{{\\rm CR^{exp}}(\\mu \\rightarrow e~ {\\rm in ~Ti} )}{{\\rm BR^{exp}}(\\mu \\rightarrow e \\gamma )},$ where ${\\rm CR^{exp}(\\mu \\rightarrow e~ {\\rm in ~Ti} )} = 4.3 \\times 10^{-12}$ and the notation is as in Eq.",
"(REF ).",
"The correlations in the $\\mu -e$ sector are such that, if MEG finds evidence for $\\mu \\rightarrow e\\gamma $ with ${\\rm BR}(\\mu \\rightarrow e\\gamma )\\gtrsim 10^{-13}$ , then we have using again Eqns.",
"(REF , REF ) ${\\rm BR}(\\mu \\rightarrow eee) & \\gtrsim 8 \\times 10^{-16} & {\\rm CR}(\\mu \\rightarrow e ~ {\\rm in~ Ti}) & \\gtrsim 10^{-15},$ i.e.",
"other LFV processes must be observed at future experiments.",
"If, on the other hand, MEG does not observe $\\mu \\rightarrow e\\gamma $ (setting a bound on the scale up to $M\\gtrsim 75$ TeV), there is still the possibility to discover $\\mu \\rightarrow eee$ and $\\mu \\rightarrow e$ conversion in Nuclei (since future experiments will test larger scales, up to 120 TeV and 290 TeV respectively).",
"Notice that the future sensitivity for $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow eee$ and $\\mu \\rightarrow e ~ {\\rm in~ Ti}$ is always beyond the bound from $K - \\overline{K}$ mixing, provided that the CPV phases are sufficiently suppressed.",
"In the case of Higgs messengers (Table REF , down), the strongest bound is set either by $K - \\overline{K}$ mixing observables or $K_L \\rightarrow \\mu ^{\\pm } e^{\\mp }$ , that can now arise at tree-level.",
"This latter process is the most constraining (even in the CPV case) if at least one of the angles is very small, so that the tree-level contribution to $K - \\overline{K}$ vanishes.",
"The strong bounds from the Kaon sector imply that BR($\\mu \\rightarrow e\\gamma $ ) is always suppressed below the $10^{-16}$ level, i.e.",
"far beyond the reach of MEG, and $\\mu \\rightarrow eee$ cannot be observed neither in this scenario.",
"The only possible deviations from the SM can be then observed in the Kaon system and for $\\mu \\rightarrow e$ conversion in nuclei, for which can be as large as ${\\rm CR}(\\mu \\rightarrow e ~ {\\rm in~ Ti}) \\simeq 10^{-17}\\div 10^{-18}$ , provided that CP violating phases are sufficiently suppressed.",
"Let us briefly comment on the 1-3 and 2-3 sectors.",
"Besides the $B_d$ and $B_s$ mixing observables (respectively $\\Delta m_{B_d}$ , $S_{\\psi K_s}$ and $\\Delta m_{B_s}$ , $S_{\\psi \\phi }$ ) discussed in Table REF , we consider $B_{d,s}\\rightarrow \\ell ^+\\ell ^-$ .The LFV $\\tau $ decays and $b\\rightarrow s\\gamma $ give negligible bounds on the messenger scale compared to other observables.",
"For the fermion UVC, all processes in these sectors give a very weak constraint on the messenger scale (below the 1 TeV level), much below the minimal bound of 19 TeV, unless the RH rotations are $\\mathcal {O}(1)$ .",
"In this case, deviations from the SM are only possible for $\\Delta F=2$ observables, since $\\Delta m_{B_d}$ and $S_{\\psi K_s}$ give at present a bound on the scale of about 25 TeV and the future LHCb sensitivity to $S_{\\psi \\phi }$ can constrain scales up to 35 TeV.",
"The same results qualitatively hold in the case of Higgs UVC, but for $B_{d,s}\\rightarrow \\ell ^+\\ell ^-$ , that can now arise at tree-level.",
"Larger rates than in the SM are then possible and, in particular, the present experimental bounds for $B_{d,s}\\rightarrow \\mu ^+\\mu ^-$ can be easily saturated.",
"Interestingly, the ratio BR($B_{d}\\rightarrow \\mu ^+\\mu ^-$ )/BR($B_{s}\\rightarrow \\mu ^+\\mu ^-$ ) can be even $\\mathcal {O}(1)$ , contrary to the SM (and MFV) prediction.",
"Let us summarise the phenomenological consequences (valid up to $\\mathcal {O}\\left( 1 \\right)$ factors) separately for Higgs and fermion UVCs.",
"FUVC With the present experimental bounds the ratio ${\\rm CR}(\\mu \\rightarrow e ~ {\\rm in ~Ti})/{\\rm BR}(\\mu \\rightarrow e \\gamma )$ is $\\gtrsim \\mathcal {O}\\left( 0.1 \\right)$ .",
"Evidence for $\\mu \\rightarrow e \\gamma $ at MEG would imply $\\mu \\rightarrow eee$ and $\\mu -e$ conversion in the reach of future experiments.",
"Deviation from the SM in 1-3 and 2-3 transitions can only occur in $\\Delta F =2 $ observables.",
"HUVC Kaon sector bounds prevent observation of LFV processes except $\\mu - e$ conversion.",
"${\\rm BR}(B_{d} \\rightarrow \\mu ^+ \\mu ^-)$ and ${\\rm BR}(B_{s} \\rightarrow \\mu ^+ \\mu ^-)$ can saturate present bounds and be comparable to each other, contrary to MFV."
],
[
"Conclusions",
"We have discussed model-independent minimal flavour-violating effects induced by messenger sectors in models of fermion masses and mixing based on horizontal symmetries.",
"In the flavour basis, integrating out the messengers induces higher-dimensional flavour-violating operators, whose coefficients depend on the specific flavour symmetry and breaking pattern, and flavour conserving operators.",
"After rotating to the fermion mass basis, the latter also contribute to the flavour-violating operators, providing a minimal contribution to flavour changing neutral currents.",
"This contribution is model independent in the case of abelian flavour symmetries (and to large extent for the non-abelian case too).",
"The coefficients of these operators only depend on light rotation angles (i.e.",
"on the structure of the Yukawa matrices).",
"In non-abelian models these coefficients are further suppressed by the breaking of flavour universality that is related to small flavon vevs.",
"For those minimal, universal contributions to the FCNC and CPV effects we have derived the bounds on the mass scale of messengers.",
"They are valid (up to $\\mathcal {O}\\left( 1 \\right)$ coefficients) for any abelian model and can be easily applied to a large class of non-abelian models taking into account additional suppression factors.",
"Moreover, the abelian case is relevant for non-abelian models with some SM fermions transforming as singlets of the non-abelian flavour group.",
"The obtained lower bounds on the messenger scale are different for different operators and in addition they depend on the chosen set of rotations.",
"Given the sensitivity expected in the forthcoming experiments, that leaves interesting room for discovering new physics and for testing fermion mass models.",
"As the highlights of our analysis emerge the leptonic processes, $\\mu \\rightarrow e \\gamma $ , $\\mu \\rightarrow eee$ and $\\mu \\rightarrow e$ conversion in nuclei.",
"In more detail, we find that the interplay of $K-\\overline{K}$ and $D-\\overline{D}$ mixing implies that the messenger scale in abelian models has to be larger than about 20 TeV.",
"Though quite strong, this minimal bound does not prevent large effects in $B_q - \\overline{B}_q$ mixing and in LFV decays.",
"In particular, the ratio ${\\rm BR}(\\mu \\rightarrow eee )$ /${\\rm BR}(\\mu \\rightarrow e \\gamma )$ can be as large as $\\mathcal {O}\\left( 10 \\right)$ for large leptonic rotations, contrary to SUSY scenarios.",
"More generally, the BRs for both processes can be large enough to be within the reach of the future experiments.",
"In non-abelian models the additional suppression factor on 1-2 flavour transitions (at least $\\sim \\epsilon ^2$ ) allows messengers at the TeV scale, and therefore possibly in the reach of LHC.",
"Assuming approximately SU(5)-symmetric Yukawas, we can include in the analysis relevant $2q2\\ell $ operators and provide correlations between quark and lepton sector.",
"We find that for the UV completion with heavy fermions the ratio ${\\rm CR}(\\mu \\rightarrow e ~ {\\rm in ~Ti})/{\\rm B R}(\\mu \\rightarrow e \\gamma )$ is $\\gtrsim \\mathcal {O}\\left( 0.1 \\right)$ .",
"An evidence for $\\mu \\rightarrow e \\gamma $ at MEG would imply $\\mu \\rightarrow eee$ and $\\mu -e$ conversion in the reach of future experiments.",
"Moreover, deviation from the SM in 1-3 and 2-3 transitions can only occur in $\\Delta F =2 $ observables.",
"For the UV completion with heavy Higgses, on the other hand, the bounds from the Kaon sector prevent the observation of LFV processes at running/future experiments, with the possible exception of $\\mu \\rightarrow e$ conversion in Nuclei.",
"Also, in this case, the new contributions to $B_{d,s} \\rightarrow \\mu ^+ \\mu ^-$ can saturate the present experimental limits and give the two processes at comparable rates, contrary to the SM and MFV scenarios."
],
[
"Acknowledgements",
"We would like to thank S. Davidson, C. Hagedorn, G. Isidori and P. Paradisi for useful discussions and communications.",
"The Feynman diagrams have been drawn using JaxoDraw [25].",
"We thank the Theory Division of CERN for hospitality during several stages of this work.",
"L.C.",
"and R.Z.",
"are grateful to the Institute of Theoretical Physics of the University of Warsaw for kind hospitality and financial support during their stays in Warsaw.",
"S.P.",
"and R.Z.",
"acknowledge support of the TUM-IAS funded by the German Excellence Initiative.",
"This work has been partially supported by the contract PITN-GA-2009-237920 UNILHC and by the National Science Centre in Poland under research grant DEC-2011/01/M/ST2/02466."
]
] | 1204.1275 |
[
[
"Variable $G$ Correction for Dark Energy Model in Higher Dimensional\n Cosmology"
],
[
"Abstract In this work, we have considered $N (=4+d)$-dimensional Einstein field equations in which 4-dimensional space-time which is described by a FRW metric and that of the extra $d$-dimensions by an Euclidean metric.",
"We have calculated the corrections to statefinder parameters due to variable gravitational constant $G$ in higher dimensional Cosmology.",
"We have considered two special cases whether dark energy and dark matter interact or not.",
"In a universe where gravitational constant is dynamic, the variable $G$-correction to statefinder parameters is inevitable.",
"The statefinder parameters are also obtained for generalized Chaplygin gas in the effect of the variation of $G$ correction."
],
[
"Introduction",
"Cosmological observations obtained by various cosmic explorations of supernova of type Ia [1], CMB analysis of WMAP data [2], extragalactic explorer SDSS [3] and X-ray [4] convincingly indicate that the observable universe is experiencing an accelerated expansion.",
"Although the simplest and natural solution to explain this cosmic behavior is the consideration of a cosmological constant [5], however it leads to two relevant problems (namely the “fine-tuning” and the “coincidence” one).",
"Recently new dynamical nature of dark energy are considered in the literature, at least in an effective level, originating from various fields, including a canonical scalar field (quintessence) [6], a phantom field, that is a scalar field with a negative sign of the kinetic term [7], or the combination of quintessence and phantom in a unified model named quintom [8].",
"There are some numerous indications that $G$ can be varying and that there is an upper limit to that variation, with respect to time or with the expansion of the universe [9].",
"In this connection the most significant evidences come from the observations of Hulse-Taylor binary pulsar [11], [10], helio-seismological data [12], Type Ia supernova observations [1] and astereoseismological data coming from the pulsating white dwarf star G117-B15A [13]; all the above evidences combined lead to $|\\dot{G}/G| \\le 4.10 \\times 10^{-11} yr^{-1}$ , for $z\\lesssim 3.5$ , thereby suggesting a mild variation on cosmic level [14].",
"On a more theoretical level, varying gravitational constant has some benefits too, for instance it can help alleviating the dark matter problem [15], the cosmic coincidence problem [16] and the discrepancies in Hubble parameter value [17].",
"In literature, a variable gravitational constant has been accommodated in gravity theories including the Kaluza-Klein [18], Brans-Dicke framework [19] and scalar-tensor theories [20].",
"From the perspective of new gravitational theories including string theory and braneworld models, (see [21] and references therein), there are a lot of speculations that there could be extra dimensions of space (6 extra dimensions of space in the string theories) besides there are no convincing evidences for their existence.",
"The size of these dimensions (whether small as Planck scale or infinitely long like usual dimensions) is still open to debate.",
"In literature, various theoretical models with extra dimensions have been constructed to account dark energy [22].",
"Previously some works on variable $G$ correction have been investigated [23] to find the statefinder parameters for several dark energy models.",
"The main motivation of this work is to investigate the role of statefinder parameters [24] (which can be written in terms of some observable parameters) in higher dimensional cosmology assuming a varying gravitational constant $G$ for interacting, non-interacting and generalized Chaplygin gas models."
],
[
"Basic Equations and Solutions",
"We consider homogeneous and anisotropic $N$ -dimensional space-time model described by the line element [25], [26] $ds^{2}=ds^{2}_{FRW}+\\sum _{i=1}^{d}b^{2}(t)dx_{i}^{2},$ where $d$ is the number of extra dimensions $(d=N-4)$ and $ds^{2}_{FRW}$ represents the line element of the FRW metric in four dimensions is given by $ds^{2}_{F RW}=-dt^{2}+a^{2}(t)\\left[\\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2})\\right],$ where $a(t)$ and $b(t)$ are the functions of $t$ alone represents the scale factors of 4-dimensional space time and extra dimensions respectively.",
"Here $k ~(=0, ~\\pm 1)$ is the curvature index of the corresponding 3-space, so that the above model of the Universe is described as flat, closed and open respectively.",
"The Einstein's field equations for the above non-vacuum higher dimensional space-time symmetry are $3\\left(\\frac{\\dot{a}^{2}+k}{a^{2}}\\right)=\\frac{\\ddot{D}}{D}-\\frac{d^{2}}{8}\\frac{\\dot{b}^{2}}{b^{2}}+\\frac{d}{8} \\frac{\\dot{b}^{2}}{b^{2}}+8\\pi G \\rho $ $2\\frac{\\ddot{a}}{a}+\\frac{\\dot{a}^{2}+k}{a^{2}}=\\frac{\\dot{a}}{a}\\frac{\\dot{D}}{D}+\\frac{d^{2}}{8}\\frac{\\dot{b}^{2}}{b^{2}}-\\frac{d}{8} \\frac{\\dot{b}^{2}}{b^{2}}-8\\pi G p$ and $\\frac{\\ddot{b}}{b}+3\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}=-\\frac{\\dot{D}}{D}\\frac{\\dot{b}}{b}+\\frac{\\dot{b}^{2}}{b^{2}}-\\frac{8\\pi G p}{2}$ where $\\rho $ and $p$ are energy density and isotropic pressure of the fluid filled in the universe respectively.",
"We choose, $D^{2}=b^{d}(t)$ , so we have $\\frac{\\dot{D}}{D}=\\frac{d}{2}\\frac{\\dot{b}}{b}$ and $\\frac{\\ddot{D}}{D}=\\frac{d}{2}\\frac{\\ddot{b}}{b}+\\frac{d^{2}-2d}{4}\\frac{\\dot{b}^{2}}{b^{2}}$ .",
"Hence the equations (1), (2) and (3) become $3\\left(\\frac{\\dot{a}^{2}+k}{a^{2}}\\right)=\\frac{d}{2}\\frac{\\ddot{b}}{b}+\\frac{d^{2}-2d }{4}\\frac{\\dot{b}^{2}}{b^{2}}-\\frac{d^{2}}{8}\\frac{\\dot{b}^{2}}{b^{2}}+\\frac{d}{8} \\frac{\\dot{b}^{2}}{b^{2}}+8\\pi G \\rho ,$ $2\\frac{\\ddot{a}}{a}+\\frac{\\dot{a}^{2}+k}{a^{2}}=\\frac{d}{2}\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}+\\frac{d^{2}}{8}\\frac{\\dot{b}^{2}}{b^{2}}-\\frac{d}{8} \\frac{\\dot{b}^{2}}{b^{2}}-8\\pi G p,$ and $\\frac{\\ddot{b}}{b}+3\\frac{\\dot{a}}{a}\\frac{\\dot{b}}{b}=-\\frac{\\dot{d}}{2}\\frac{\\dot{b^{2}}}{b^{2}}+\\frac{\\dot{b}^{2}}{b^{2}}-\\frac{8\\pi G p}{2}.$ Defining $H_{1}=\\frac{\\dot{a}}{a}$ , $H_{2}=\\frac{\\dot{b}}{b}$ we have from (6) to (8), $3 H_{1}^{2}+3 \\frac{k}{a^{2}}-\\frac{d}{2}\\dot{H}_{2}+\\left(-\\frac{d}{8}-\\frac{d^{2}}{8}\\right)H_{2}^{2}=8 \\pi G \\rho ,$ $3 H_{1}^{2}+ \\frac{k}{a^{2}}+2\\dot{H}_{1}-\\frac{d}{2}H_{1}H_{2}+\\left(\\frac{d}{8}-\\frac{d^{2}}{8}\\right)H_{2}^{2}=-8 \\pi G p,$ $\\dot{H}_{2}+3 H_{1} H_{2}-\\frac{d}{2}H_{2}^{2}=-\\frac{8 \\pi Gp}{2}.$ Now consider the universe is filled with the dark matter (with negligible pressure) and dark energy.",
"Assuming $p=\\omega \\rho _{x}$ , $\\rho = \\rho _{m}+\\rho _{x}$ , where $\\rho _{m}$ and $\\rho _{x}$ are the energy densities of dark matter and dark energy respectively, $\\omega $ is the equation of state parameter for dark energy.",
"Note that $\\omega $ is a dynamical time dependent parameter and will be useful in later calculations.",
"Now eliminating $\\dot{H}_{1}$ , $\\dot{H}_{2}$ from (9), (10) and (11) we have, $24 k= a^{2}\\left(-24 H_{1}^{2}-12 d H_{1}H_{2}-(d-1)d H_{2}^{2}+16\\pi G (4 \\rho _{m}+(4-d \\omega )\\rho _{x})\\right)$ This equation can be written as $(d+3)^{2}H^{2}+3H_{1}^{2}-\\frac{d(d-1)}{2}H_{2}^{2}+\\frac{12k}{a^{2}}=32 \\pi G \\rho _{m}+(4-d \\omega )8 \\pi G \\rho _{x}$ where $H$ is the Hubble parameter defined by $H=\\frac{1}{d+3}\\left(3 H_{1}+d H_{2}\\right)$ .",
"This can be written as $\\Omega +\\frac{3}{(d+3)^{2}}\\Omega _{1}-\\frac{d(d-1)}{2}\\Omega _{2}+\\frac{12}{(d+3)^{2}}\\Omega _{k}=\\frac{4}{d+3}\\Omega _{m}+\\frac{4-d\\omega }{d+3}\\Omega _{x}$ where $\\Omega _{1}=\\frac{H_{1}^{2}}{H^{2}}$ , $\\Omega _{2}=\\frac{H_{2}^{2}}{H^{2}}$ are dimensionless parameters and $\\Omega _{m}=\\frac{8 \\pi G \\rho _{m}}{(d+3)H^{2}}$ , $\\Omega _{x}=\\frac{8 \\pi G \\rho _{x}}{(d+3)H^{2}}$ are fractional density parameters, $\\Omega _{k}=\\frac{k}{a^{2}H^{2}}$ is another dimensionless parameter, represents the contribution in the energy density from the spatial curvature and $\\Omega $ is the total density parameter.",
"Now solving (9), (10) and (11) we have the solutions of $\\dot{H}_{1}$ and $\\dot{H}_{2}$ as $\\dot{H}_{1}=\\frac{1}{24}\\left(-48H_{1}^{2}+d(d-1)H_{2}^{2}-\\frac{24 k}{a^{2}}+8 \\pi G(4\\rho _{m}-(-4+(d+12)\\omega )\\rho _{x})\\right)$ and $\\dot{H}_{2}=\\frac{2}{d}\\left( 3H_{1}^{2}-\\frac{1}{8}d(d+1)H_{2}^{2}+\\frac{3 k}{a^{2}}-8 \\pi G(\\rho _{m}+\\rho _{x}) \\right)$ The deceleration parameter $q=-1-\\frac{\\dot{H}}{H^{2}}$ is given by in terms of dimensionless parameters $q=-1-\\frac{3}{d+3}\\Omega _{k}+\\frac{d}{8}\\Omega _{2}+\\frac{3}{2}\\Omega _{m}+\\frac{12+12\\omega +d\\omega }{8}\\Omega _{x}$ and the derivative of deceleration parameter is obtained as $\\dot{q}=\\frac{3}{d+3}H \\Omega _{k}\\left(2\\sqrt{\\Omega _{1}}-2(q+1)\\right)+\\frac{d}{8}\\dot{\\Omega }_{2}+\\frac{3}{2}\\dot{\\Omega }_{m}+\\frac{12+12\\omega +d \\omega }{8}\\dot{\\Omega }_{x}+\\frac{d+12}{8}\\dot{\\omega }\\Omega _{x}$ Also from (14) we obtain the expression of the total density parameter in the form $\\Omega =\\frac{4}{d+3}\\frac{\\rho }{\\rho _{cr}}-\\frac{d\\omega }{d+3}\\Omega _{x}-\\frac{12}{(d+3)^{2}}\\Omega _{k}-\\frac{3}{(d+3)^{2}}\\Omega _{1}+\\frac{d(d-1)}{2}\\Omega _{2}$ Now define the critical density, $\\rho _{cr}=\\frac{3 H^{2}}{8 \\pi G(t)}~~~~~~ \\text{which gives afterdifferentiation}~~ \\dot{\\rho }_{cr}=\\rho _{cr}\\left(2\\frac{\\dot{H}}{H}-\\frac{\\dot{G}}{G}\\right)$ which implies $\\dot{\\rho }_{cr}=-H \\rho _{cr}(2(1+q)+\\triangle G)$ where, $\\triangle G\\equiv \\frac{G^{\\prime }}{G}, \\dot{G}=H G^{\\prime }$ (prime denotes differentiation with respect to $x\\equiv \\ln a$ ).",
"The benefit of the previous rule $\\dot{G}=H G^{\\prime }$ relates the variations in $G$ with respect to time $\\dot{G}$ and the expansion of the universe $G^{\\prime }$ .",
"Differentiating (19) we have $\\dot{\\Omega }=\\frac{4}{d+3}\\frac{\\dot{\\rho }}{\\rho _{cr}}+\\frac{4H(2(1+q)+\\triangle G)}{d+3}\\frac{\\rho }{\\rho _{cr}}-\\frac{24H}{(d+3)^{2}}\\Omega _{k}(\\sqrt{\\Omega _{1}}-(q+1))-\\frac{3}{(d+3)^{2}}\\dot{\\Omega _{1}}+\\frac{d(d-1)}{2}\\dot{\\Omega }_{2}-\\frac{d}{d+3}\\omega \\dot{\\Omega }_{x}-\\frac{d}{d+3}\\dot{\\omega }\\Omega _{x}$ where $\\dot{\\Omega _{1}}$ and $\\dot{\\Omega }_{2}$ are given by $\\dot{\\Omega _{1}}=H \\left[ \\Omega _{1}^{1/2}\\Omega _{k}-3\\Omega _{1}^{2}-\\frac{3d}{2(d+3)}\\Omega _{1}^{3/2}\\Omega _{2}^{1/2}+\\frac{d(d-1)}{8}\\Omega _{1}^{1/2}\\Omega _{2}-(d+3)\\omega \\Omega _{1}^{1/2}\\Omega _{x}-\\frac{9}{d+3}\\Omega _{1}\\Omega _{k}+\\frac{d}{2}\\Omega _{1}\\Omega _{2}^{1/2}\\right.$ $\\left.+\\frac{d(d+7)}{8(d+3)}\\Omega _{1} \\Omega _{2}+4\\Omega _{1}\\Omega _{m}+(4+3\\omega )\\Omega _{1} \\Omega _{x} \\right]$ $\\dot{\\Omega }_{2}=H\\left[\\frac{12}{d}\\Omega _{1}^{3/2}-\\frac{3}{d+3}\\Omega _{1}^{3/2}\\Omega _{2}^{1/2}-\\frac{d+1}{2}\\Omega _{1}^{1/2}\\Omega _{2}-\\frac{3d}{2(d+3)}\\Omega _{1}^{3/2}\\Omega _{2}+\\frac{d(d+7)}{8(d+3)}\\Omega _{1}^{1/2}\\Omega _{2}^{3/2}+\\frac{12}{d}\\Omega _{1}^{1/2}\\Omega _{k}\\right.$ $\\left.-\\frac{9}{d+3}\\Omega _{1}^{1/2}\\Omega _{2}^{1/2}\\Omega _{k}-\\frac{4(d+3)}{d}\\Omega _{1}^{1/2}(\\Omega _{m}+\\Omega _{x})+\\Omega _{1}^{1/2}\\Omega _{2}^{1/2}\\left(4\\Omega _{m}+(4+3\\omega )\\Omega _{x}\\right)\\right]$ The trajectories in the {$r,s$ } plane corresponding to different cosmological models depict qualitatively different behaviour.",
"The statefinder diagnostic along with future SNAP observations may perhaps be used to discriminate between different dark energy models.",
"The above statefinder diagnostic pair for cosmology are constructed from the scale factor $a$ .",
"The statefinder parameters are given by [24] $r=\\frac{\\dddot{a}}{aH^{2}}~,~~s=\\frac{r-1}{3(q-1/2)}$ Now we obtain the expressions for $r$ and $s$ as follows $r=\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}+(12+(d+12)\\omega )\\Omega _{x}\\left(-\\frac{3}{8}+\\frac{3}{4}\\Omega _{m}-\\frac{3}{4(d+3)}\\Omega _{k}+\\frac{d}{16}\\Omega _{2}-\\frac{1}{8}\\frac{\\dot{\\Omega }_{x}}{H}\\right)-\\frac{3}{d+3}\\Omega _{k}(3\\Omega _{m}-3+2\\Omega _{1}^{1/2})$ $-\\frac{3d}{4(d+3)}\\Omega _{2}\\Omega _{k}+\\frac{3d}{8}\\Omega _{2}(2\\Omega _{m}-1)+4\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}-\\frac{9}{2}\\Omega _{m}+\\frac{1}{32}[(12+(d+12)\\omega )^{2}\\Omega _{x}^{2}]-\\frac{(d+12)}{8}\\frac{\\dot{\\omega }}{H}\\Omega _{x}+1$ $s=\\frac{8(d+3)}{3[d(d+3)\\Omega _{2}-24\\Omega _{k}+(d+3)\\left(-12+12\\Omega _{m}+(12+(d+12)\\omega )\\Omega _{x}\\right)]}\\left[\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\right.$ $+(12+(d+12)\\omega )\\Omega _{x}\\left(-\\frac{3}{8}+\\frac{3}{4}\\Omega _{m}-\\frac{3}{4(d+3)}\\Omega _{k}+\\frac{d}{16}\\Omega _{2}-\\frac{1}{8}\\frac{\\dot{\\Omega }_{x}}{H}\\right)-\\frac{3}{d+3}\\Omega _{k}(3\\Omega _{m}-3+2\\Omega _{1}^{1/2})~~~~~~~~~~~~~~~~~~~~$ $\\left.-\\frac{3d}{4(d+3)}\\Omega _{2}\\Omega _{k}+\\frac{3d}{8}\\Omega _{2}(2\\Omega _{m}-1)+4\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}-\\frac{9}{2}m+\\frac{1}{32}\\left((12+(d+12)\\omega )^{2}\\Omega _{x}^{2}\\right)-\\frac{(d+12)}{8}\\frac{\\dot{\\omega }}{H}\\Omega _{x}\\right]$ This is the expressions for $\\lbrace r,s\\rbrace $ parameters in terms of fractional densities of dark energy model in higher dimensional cosmology for closed (or open) universe where the derivative of the density parameters i.e., $\\dot{\\Omega }_{1}$ and $\\dot{\\Omega }_{2}$ are given in equation (23) and (24).",
"Now in the following subsections, we shall analyze the statefinder parameters for the non-interacting and interacting dark energy models."
],
[
"Non-interacting Dark Energy Model",
"In this subsection we study the model of non-interacting case where the dark energy and dark matter do not interact with each other.",
"We assume that dark matter and dark energy are separately conserved.",
"So the continuity equation for cold dark matter is $\\dot{\\rho }_{m}+(d+3) H \\rho _{m}=0$ and for dark energy is $\\dot{\\rho }_{x}+(d+3) H (1+\\omega )\\rho _{x}=0$ .",
"So solving (22) for two different cases we have the expressions of $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ as: $\\dot{\\Omega }_{m}=\\frac{1}{2(d+3)(d+3+d\\omega )}\\left[-6\\dot{\\Omega _{1}}+d(d-1)(d+3)^{2}\\dot{\\Omega }_{2}+2\\left(24H(q+1-\\Omega _{1}^{1/2})\\Omega _{k}+4H((d+3)(2q-1-d+\\triangle G)\\right.\\right.$ $\\left.\\left.+d(2q+\\triangle G+2)\\omega +d(d+3)\\omega ^{2})\\Omega _{m}-d(4H(2q+2+\\triangle G)\\omega +(d+3)\\dot{\\omega })\\Omega _{x}\\right)\\right]$ $\\dot{\\Omega }_{x}=\\frac{1}{2(d+3)(d+3+d\\omega )}\\left[-6\\dot{\\Omega _{1}}+d(d-1)(d+3)^{2}\\dot{\\Omega }_{2}+2\\left(24H(q+1-\\Omega _{1}^{1/2})\\Omega _{k}-(d+3)(4(d+3)H(\\omega +1)\\Omega _{m}\\right.\\right.$ $\\left.",
"\\left.+(-4H(2q+2+\\triangle G )+d\\dot{\\omega })\\Omega _{x})\\right)\\right]$ In the equations (25) and (26), we have calculated the general expressions of the statefinder parameters $\\lbrace r, s\\rbrace $ .",
"In this non-interacting dark energy model, the above parameters are also same where the $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ are given by the equations (27) and (28)."
],
[
"Interacting Dark energy Model",
"In this subsection we study the model of interacting case where the dark energy and dark matter are interact with each other.",
"These models describe an energy flow between the components i.e.",
"they not separately conserved.",
"According to recent observational data of Supernovae and CMB the present evolution of the Universe permit the energy transfer decay rate proportional to present value of the Hubble parameter.",
"Many authors have widely studied this interacting model.",
"In Pavon and Zimdahl [27] state that the unknown nature of dark energy and dark matter make no contradiction about their mutual interaction.",
"In Zhang and Olivers et al [28] showed that the theoretical interacting model are consistent with the type Ia supernova and CMB observational data.",
"Here we assume that the dark energy and dark matter are interacting with each other, so the continuity equations of dark matter and dark energy become $\\dot{\\rho }_{m}+(d+3) H \\rho _{m}=Q$ and $\\dot{\\rho }_{x}+(d+3) H (1+\\omega )\\rho _{x}=-Q$ where $Q$ is is the interacting term which is a arbitrary function.",
"This interacting term determine the direction of the energy flow both sides of the dark matter and dark energy.",
"In general this term can be choose as a function of different cosmological parameters like Hubble parameter and dark energy or dark matter density.",
"in this work we choose $Q=(d+3) \\delta H\\rho _{x}$ where $\\delta $ is a couple constant.",
"The positive $\\delta $ represents the energy transfer from dark energy to dark matter.",
"If $\\delta =0$ the above model transfer to non-interacting case.",
"Here negative $\\delta $ is not considered as it can violate the thermodynamical laws of the universe.",
"So the $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ are given by the equations $\\dot{\\Omega }_{m}=\\frac{1}{2(d+3)(d+3+d\\omega )}\\left[-6\\dot{\\Omega _{1}}+d(d-1)(d+3)^{2}\\dot{\\Omega }_{2}+2\\left(24H(q+1-\\Omega _{1}^{1/2})\\Omega _{k}+4H((d+3)( 2q-1-d+\\triangle G)\\right.\\right.$ $\\left.\\left.+d(2+\\triangle G+2q)\\omega +d(d+3)\\omega ^{2})\\Omega _{m}+(4H(-d(2q+2+\\triangle G)\\omega +(d+3)\\delta (d+3+2 d\\omega ))-d(d+3)\\dot{\\omega })\\Omega _{x}\\right)\\right]$ $\\dot{\\Omega }_{x}=\\frac{1}{2(d+3)( d+3+d\\omega )}\\left[-6\\dot{\\Omega _{1}}+d(d-1)(d+3)^{2}\\dot{\\Omega }_{2}+2\\left(24H(q+1-\\Omega _{1}^{1/2})\\Omega _{k}\\right.\\right.~~~~~~~~~~~~~~~~~~~~~~~~~~$ $\\left.\\left.-(d+3)(4H(d+3)(1+\\omega )\\Omega _{m}+(-4H(2q+2+\\triangle G)+4(d+3) H \\delta +d \\dot{\\omega })\\Omega _{x})\\right)\\right]$ In the equations (25) and (26), we have calculated the general expressions of the statefinder parameters $\\lbrace r, s\\rbrace $ .",
"In this interacting dark energy model, the above parameters are also same where the $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ are given by the equations (31) and (32)."
],
[
"Generalized Chaplygin gas",
"It is well known to everyone that Chaplygin gas provides a different way of evolution of the universe and having behaviour at early time as presureless dust and as cosmological constant at very late times, an advantage of generalized Chaplygin gas (GCG), that is it unifies dark energy and dark matter into a single equation of state.",
"This model can be obtained from generalized version of the Born-Infeld action.",
"The equation of state for generalized Chaplygin gas is [29] $p_{x}=-\\frac{A}{\\rho _{x}^{\\alpha }}$ where $0<\\alpha <1$ and $A>0$ are constants.",
"Inserting the above equation of state (33) of the GCG into the non-interacting energy conservation equation we have $\\rho _{x}=\\left[A+\\frac{B}{(a^{3}b^{d})^{(\\alpha +1)}}\\right]^{\\frac{1}{\\alpha +1}}$ where $B$ is an integrating constant.",
"$\\omega =-A \\left(A +\\frac{B}{(a^{3}b^{d})^{(\\alpha +1)}}\\right)^{-1}$ Differentiating (35) we have $\\frac{\\dot{\\omega }}{H}=-(d+3) A B (1 + \\alpha ) \\frac{1}{(a^{3}b^{d})^{(\\alpha +1)}}\\left(A + \\frac{B}{(a^{3}b^{d})^{(\\alpha +1)}}\\right)^{-2}$ Now putting (36) in (25) and (26), we have $r=\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}-\\frac{9}{2}\\Omega _{m}+\\frac{9}{2}\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}+\\frac{9}{d+3}\\Omega _{k}-\\frac{6}{d+3}\\Omega _{1}^{1/2}\\Omega _{k}-\\frac{3d}{d+3}\\Omega _{2}\\Omega _{k}-\\frac{3d}{8}\\Omega _{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $-\\frac{36B^{2}-3a^{6}A^{2}b^{2d}(a^{3}b^{d})^ {2\\alpha }d-AB(a^{3}b^{d})^{\\alpha +1}(d(d+18)+(d+3)(d+12)\\alpha )}{8(A(a^{3}b^{d})^{\\alpha +1}+B)^{2}}\\Omega _{x}+\\frac{1}{32}\\left(-12+\\frac{A(d+12)}{A+(a^{3}b^{d})^{-\\alpha -1}B}\\right)^{2}\\Omega _{x}^{2}$ $+\\left(12-\\frac{A(d+12)}{A+(a^{3}b^{d})^{-\\alpha -1}B}\\right)\\left(\\frac{3}{4}\\Omega _{m}\\Omega _{x}-\\frac{\\dot{\\Omega }_{x}}{8H}\\right)+\\left(\\frac{B(d+12)\\Omega _{x}}{A(a^{3}b^{d})^{\\alpha +1}+B}+12\\Omega _{m}-d\\Omega _{x}\\right)\\left(\\frac{d}{16}\\Omega _{2}-\\frac{3}{4(d+3)}\\Omega _{k}\\right)+1$ $s=\\frac{1}{-\\frac{9}{2}+\\frac{3d\\Omega _{2}}{8}-\\frac{9\\Omega _{k}}{d+3}+\\frac{9\\Omega _{m}}{2}-\\frac{3d\\Omega _{x}}{8}+\\frac{3}{8}\\frac{B(d+12)\\Omega _{x}}{A(a^{3}b^{d})^{\\alpha +1}+B}}\\left[\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}-\\frac{9}{2}\\Omega _{m}+4\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}+\\frac{9}{d+3}\\Omega _{k}-\\frac{6}{d+3}\\Omega _{1}^{1/2}\\Omega _{k}-\\frac{3d}{8}\\Omega _{2} \\right.$ $-\\frac{36B^{2}-3a^{6}A^{2}b^{2d}(a^{3}b^{d})^ {2\\alpha }d-AB(a^{3}b^{d})^{\\alpha +1}(d(d+18)+(d+3)(d+12)\\alpha )}{8(A(a^{3}b^{d})^{\\alpha +1}+B)^{2}}\\Omega _{x}+\\frac{1}{32}\\left(-12+\\frac{A(d+12)}{A+(a^{3}b^{d})^{-\\alpha -1}B}\\right)^{2}\\Omega _{x}^{2}$ $\\left.-\\frac{3d}{d+3}\\Omega _{2}+\\left(12-\\frac{A(d+12)}{A+(a^{3}b^{d})^{-\\alpha -1}B}\\right)\\left(\\frac{3}{3}\\Omega _{m}\\Omega _{x}-\\frac{\\dot{\\Omega }_{x}}{8H}\\right)+\\left(\\frac{B(d+12)\\Omega _{x}}{A(a^{3}b^{d})^{\\alpha +1}+B}+12\\Omega _{m}-d\\Omega _{x}\\right)\\left(\\frac{d}{16}\\Omega _{2}-\\frac{3}{4(d+3)}\\Omega _{k}\\right)\\right]$ These are the expressions for $\\lbrace r,s\\rbrace $ parameters in terms of fractional densities for non-interacting case of generalized Chaplygin gas model in higher dimensional Cosmology, where $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ are given by the equations (27) and (28).",
"Again inserting the equation of state (33) of the GCG into the interacting energy conservation equation (30) we have $\\rho _{x}=\\left[\\frac{A}{\\delta +1}+\\frac{B}{(\\delta +1)(a^{3}b^{d})^{(\\delta +1)(\\alpha +1)}}\\right]^{\\frac{1}{\\alpha +1}}$ and $\\omega =-A \\left(\\frac{A}{\\delta +1} +\\frac{B}{(\\delta +1)(a^{3}b^{d})^{(\\delta +1)(\\alpha +1)}}\\right)^{-1}$ Differentiating (40) we have $\\frac{\\dot{\\omega }}{H}=-(d+3) A B (1 + \\alpha ) \\frac{1}{(a^{3}b^{d})^{(\\delta +1)(\\alpha +1)}}\\left(\\frac{A}{\\delta +1} +\\frac{B}{(\\delta +1)(a^{3}b^{d})^{(\\delta +1)(\\alpha +1)}}\\right)^{-2}$ Now putting (41) in (25) and (26), we have $r=\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}-\\frac{9}{2}\\Omega _{m}+\\frac{9}{2}\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}+\\frac{9}{d+3}\\Omega _{k}-\\frac{6}{d+3}\\Omega _{1}^{1/2}\\Omega _{k}-\\frac{3d}{d+3}\\Omega _{2}\\Omega _{k}-\\frac{3d}{8}\\Omega _{2}-\\frac{9}{d+3}\\Omega _{k}\\Omega _{m}+\\frac{3d}{4}\\Omega _{2}\\Omega _{m}$ $+\\frac{AB(a^{3}b^{d})^{(1+\\alpha )(1+\\delta )}(d+3)(d+12)(1+\\alpha )(1+\\delta )^{2}}{8(A(a^{3}b^{d})^{(1+\\alpha )(1+\\delta )}+B)^{2}}+\\left(\\frac{d}{16}\\Omega _{2}+\\frac{3}{4}\\Omega _{m}-\\frac{3}{4(d+3)}\\Omega _{k}-\\frac{3}{8}\\right)$ $\\times \\left(-12+\\frac{A(d+12)(1+\\delta )}{A+B(a^{3}b^{d})^{-(1+\\alpha )(1+\\delta )}}\\right)-\\left(-12+\\frac{A(d+12)(1+\\delta )}{A+B(a^{3}b^{d})^{-(1+\\alpha )(1+\\delta )}}\\right)\\frac{\\dot{\\Omega }_{x}}{8H}+1$ $s=\\frac{1}{3d\\Omega _{2}-\\frac{72}{d+3}\\Omega _{k}+36\\Omega _{m}-36+3\\left(-12+\\frac{A(d+12)(1+\\delta )}{A+B(a^{3}b^{d})^{-(1+\\alpha )(1+\\delta )}}\\right)\\Omega _{x}}\\left[\\frac{d^{2}}{32}\\Omega _{2}^{2}-\\frac{d}{8}\\frac{\\dot{\\Omega }_{2}}{H}-\\frac{9}{2}\\Omega _{m}+\\frac{9}{2}\\Omega _{m}^{2}-\\frac{3}{2}\\frac{\\dot{\\Omega }_{m}}{H}+\\frac{9}{d+3}\\Omega _{k} \\right.$ $-\\frac{6}{d+3}\\Omega _{1}^{1/2}\\Omega _{k}-\\frac{3d}{d+3}\\Omega _{2}\\Omega _{k}-\\frac{3d}{8}\\Omega _{2}-\\frac{9}{d+3}\\Omega _{k}\\Omega _{m}+\\frac{3d}{4}\\Omega _{2}\\Omega _{m}+\\frac{AB(a^{3}b^{d})^{(1+\\alpha )(1+\\delta )}(d+3)(d+12)(1+\\alpha )(1+\\delta )^{2}}{8(A(a^{3}b^{d})^{(1+\\alpha )(1+\\delta )}+B)^{2}}$ $\\left.+\\left(\\frac{d}{16}\\Omega _{2}+\\frac{3}{4}\\Omega _{m}-\\frac{3}{4(d+3)}\\Omega _{k}-\\frac{3}{8}\\right)\\times \\left(-12+\\frac{A(d+12)(1+\\delta )}{A+B(a^{3}b^{d})^{-(1+\\alpha )(1+\\delta )}}\\right)-\\left(-12+\\frac{A(d+12)(1+\\delta )}{A+B(a^{3}b^{d})^{-(1+\\alpha )(1+\\delta )}}\\right)\\frac{\\dot{\\Omega }_{x}}{8H}\\right]$ These are the expressions for $\\lbrace r,s\\rbrace $ parameters in terms of fractional densities for interacting case of generalized Chaplygin gas model in higher dimensional Cosmology, where $\\dot{\\Omega }_{m}$ and $\\dot{\\Omega }_{x}$ are given by the equations (31) and (32)."
],
[
"Conclusions",
"In this work, we have considered $N~(=4+d)$ -dimensional Einstein field equations in which 4-dimensional space-time is described by a FRW metric and that of the extra $d$ -dimensions by an Euclidean metric.",
"We have calculated the corrections to statefinder parameters $\\lbrace r,s\\rbrace $ and deceleration parameter $q$ due to variable gravitational constant $G$ in higher dimensional Cosmology.",
"These corrections are relevant because several astronomical observations provide constraints on the variability of $G$ .",
"We have first assumed that the dark energy do not interact with dark matter.",
"Next we have considered the dark energy and dark matter are not separately conserved i.e., they interact with each other with a particular interacting term in the form $Q=(d+3)\\delta H \\rho _{x}$ where $\\delta $ is a couple constant.",
"In both the cases, the statefinder parameters have been found in terms of the dimensionless density parameters as well as EoS parameter $\\omega $ and the Hubble parameter.",
"An important thing to note is that these are the $G$ -corrected statefinder parameters and they remain geometrical parameters as previous.",
"Because, the parameter $\\triangle G$ is a pure number and is independent of the geometry.",
"Finally we have analyzed the above statefinder parameters in terms of some observable parameters for the non-interacting and interacting cases when the universe is filled with generalized Chaplygin gas.",
"These dynamical statefinder parameters may generate different stages of the anisotropic universe in higher dimensional Cosmology if the observable parameters are known for interacting and non-interacting models."
]
] | 1204.0925 |
[
[
"The stellar metallicity distribution of disc galaxies and bulges in\n cosmological simulations"
],
[
"Abstract By means of high-resolution cosmological hydrodynamical simulations of Milky Way-like disc galaxies, we conduct an analysis of the associated stellar metallicity distribution functions (MDFs).",
"After undertaking a kinematic decomposition of each simulation into spheroid and disc sub-components, we compare the predicted MDFs to those observed in the solar neighbourhood and the Galactic bulge.",
"The effects of the star formation density threshold are visible in the star formation histories, which show a modulation in their behaviour driven by the threshold.",
"The derived MDFs show median metallicities lower by 0.2-0.3 dex than the MDF observed locally in the disc and in the Galactic bulge.",
"Possible reasons for this apparent discrepancy include the use of low stellar yields and/or centrally-concentrated star formation.",
"The dispersions are larger than the one of the observed MDF; this could be due to simulated discs being kinematically hotter relative to the Milky Way.",
"The fraction of low metallicity stars is largely overestimated, visible from the more negatively skewed MDF with respect to the observational sample.",
"For our fiducial Milky Way analog, we study the metallicity distribution of the stars born \"in situ\" relative to those formed via accretion (from disrupted satellites), and demonstrate that this low-metallicity tail to the MDF is populated primarily by accreted stars.",
"Enhanced supernova and stellar radiation energy feedback to the surrounding interstellar media of these pre-disrupted satellites is suggested as an important regulator of the MDF skewness."
],
[
"Introduction",
"Galaxy formation and evolution is a distinctly multi-disciplinary field, connecting fundamental cosmology and structure formation, through stellar astrophysics, nucleosynthesis, and therefore atomic physics.",
"These extremes manifest themselves in the appearance and characteristics of the stellar populations which comprise the galaxies we observe empirically.",
"Because of the obvious, deep-rooted, interest in understanding the origin of our own Milky Way, there are intense efforts underway to understand the underlying physics driving disc galaxy formation within the concordant $\\Lambda $ CDM cosmology.",
"From an empirical perspective, the Milky Way clearly possesses the greatest wealth of observational constraints to any formation scenario, from accurate 6d phase-space coordinates (positions and velocities), chemical abundances, and ages, for massive numbers of halo, disc, and bulge stars (e.g., Freeman & Bland-Hawthorn 2002).",
"Galactic chemical evolution models developed in a cosmological framework are particularly fruitful tools to derive crucial information regarding the star formation histories (SFH) of galaxies, on the ages of the stellar populations, and on the gas accretion and outflow histories.",
"A key observable for constraining any such model is the metallicity distribution function (MDF) of the stars of the various sub-components of a galaxy.",
"The MDF bears information concerning the star formation history of our Galaxy (e.g.",
"Tinsley 1980; Matteucci & Brocato 1990; Pagel & Tautvaisiene 1995; Caimmi 1997; Haywood 2006), which is directly linked to the merging history of its progenitors, i.e.",
"the “building blocks” of the various components.",
"A detailed study of the impact of the accretion history on the shape of the MDF can be found in Font et al.",
"(2006), where it was shown that the earlier the major accretion epoch of satellites of the central galaxy, the more the MDF peak is shifted toward lower metallicities.",
"These works show how the study of the MDF is important to gain information on the evolution of both the dark matter content of these systems and of the baryonic matter, governed by various physical processes which lead ultimately to self-regulated star formation.",
"The metal content of galaxies grows with time (modulo dilution effects from metal-poor infall and metal-rich outflows), hence the disc MDF allows us to track the enrichment history of the Milky Way empirically and the enrichment history of simulated Milky Way-like analogs disc through model `deconstruction'.",
"The extreme, metal-poor tail of the MDF provides crucial information on the earliest enrichment phases of the disc, while the most metal-rich stars bear the imprint of the latest galactic evolutionary stages.",
"In what follows, we study the MDFs of the discs and bulges associated with a family of high-resolution cosmological hydrodynamical simulations.",
"These tools represent an ideal instrument to follow the dynamical and chemical evolution of the Galactic stellar populations from first principles, with a detailed knowledge at any timestep of the spatial distibution of gaseous and stellar matter.",
"This facilitates comparison with observational data, in particular local solar neighbourhood stars, since in the simulations, the position of each stellar particle is known and it is hence straightforward to select physical regions whose properties can be associated to those observed in the solar neighbourhood.",
"The specific simulations employed in our analysis are described in §2, the results presented in §3, and our conclusions drawn in §4."
],
[
"Model Description",
"The six simulations employed here are drawn from the MUGS sample (Stinson et al.",
"2010).",
"From these, we derive the MDFs associated with their analogous `solar neighbourhoods' and `bulges', and contrast them with those measured in the Milky Way.",
"Below, we provide an overview of the six simulations, along with the kinematic decomposition employed to separate disc stars from spheroid stars.",
"A more extensive background to the simulations is provided by Stinson et al.",
"(2010), while the radial and vertical metallicity gradients are explored by Pilkington et al.",
"(2012a)."
],
[
"The MUGS Simulations",
"The MUGS simulations were run using the gravitational N-body $+$ SPH code Gasoline (Wadsley et al.",
"2004).",
"Here, we provide a brief overview of the star formation and feedback recipes employed, as they impact most directly on the chemical abundances associated with the stellar populations; full details of the simulation are given in Stinson et al.",
"(2010).",
"The basic star formation and supernova feedback follows the `blastwave scenario' of Stinson et al.",
"(2006); stars can form from SPH gas particles which meet specific density ($>$ 1 cm$^{-3}$ ), temperature ($<$ 15000 K) and convergent flow criteria.",
"When these are met, stars can form with a star formation rate $dM_{\\star }/dt$ given by $\\frac{dM_{\\star }}{dt}=c^{\\star }\\frac{M_{gas}}{t_{dyn}},$ where c$^{\\star }=0.015$ represents the star formation efficiency.",
"This quantity is tuned in order to match the Kennicutt law for star formation in local disc galaxies (Kennicutt 1998).",
"The quantity $M_{gas}$ is the mass of the star-forming gas particle, while $ t_{dyn}$ is its dynamical timescale.",
"Within each `star' particle, `individual' star masses are distributed according to a Kroupa, Tout & Gilmore (1993) initial mass function (IMF), with lower and upper mass limits of 0.1 M$_\\odot $ and 100 M$_\\odot $ , respectively.",
"Stars with masses between 8 M$_\\odot $ and 40 M$_\\odot $ are assumed to explode as Type II supernovae (SNeII).",
"Each supernova is assumed to possess an energy of 10$^{51}$ erg, and we assume 40% of this energy is made available in the form of thermal energy to the surrounding interstellar medium (ISM).",
"The heavy elements restored to the ISM by SNeII in this version of Gasoline are O and Fe.",
"Analytical power-law fits in mass were made using the yields of Woosley & Weaver (1995), convolved with the aforementioned Kroupa et al.",
"(1993) IMF, to derive the mass fraction of metals ejected by SNeII for each stellar particle.",
"These elements are returned on the timescale of the lifetimes of the individual stars comprising the IMF, after Raiteri et al.",
"(1996).",
"Type Ia supernovae (SNeIa) are included within Gasoline, patterned again after the Raiteri et al.",
"implementation of the Greggio & Renzini (1983) single-degenerate progenitor model.",
"Each SNeIa is assumed to return 0.63 M$_{\\odot }$ of iron and 0.13 M$_{\\odot }$ of oxygen to the ISM.",
"This is an important feature of the code, which differentiates it from other previous attempts to model chemical abundances in simulations.",
"In the past,cosmological codes tracked the total gas metallicity (Z), under the assumption of the instantaneous recycling approximation, i.e.",
"neglecting the time delay between star formation and the energetic and chemical feedback from stellar winds and SNe (e. g., Sanchez-Blazquez et al.",
"2009).",
"Such codes are not suited to study elements such as Fe, mainly produced by type Ia SNe on timescales varing from 0.03 Gyr up to several Gyr, but which are crucial since they are primary metallicity tracers, in particular in observational studies of the stellar MDF.",
"In this paper, we take into account finite time delays for the main channels for Fe production, i.e.",
"type Ia and type II SNe, hence our study should be regarded as a significant step forward with respect to previous studies of chemical abundances in cosmological simulations.",
"Other recent chemical evolution studies in fully cosmological disc simulations relaxing the instantaneous recycling approximation include Rahimi et al.",
"(2010) and Few et al.",
"(2012).",
"The contribution of single low and intermediate mass asymptotic giant branch stars is not included in these runs, but for the analysis of oxygen and iron, this is negligible with respect to the contributions of SNeII and SNeIa.",
"The total metallicity in this version of Gasoline is tracked by assuming Z$\\equiv $ O+Fe.By assuming Z$\\equiv $ O+Fe, we underestimate the global metal production rate by roughly a factor of two, which leads to a parallel underestimate in the gas cooling rate, and hence star formation rate (Pilkington et al.",
"2012a).",
"Given the strong non-linearity of the dependence of the feedback and cooling processes on the global metallicity, the only way to quantify how this alters our results would be to re-run all the simulations with the correct chemical evolution prescriptions; to this purpose, our next generation of runs with Gasoline will employ a more complete chemical `network', ensuring that $\\sim $ 90% of the global metallicity `Z' will be tracked element-by-element (see Pilkington et al.",
"2012b).",
"For these runs, only the solar metallicity yields were employed, and long-lived SNeIa progenitors (i.e.",
"those in binary systems with companions having mass $m$$<$ 1.5 M$_\\odot $ ) were neglected.",
"For a Kroupa et al.",
"(1993) IMF, in a simple stellar population the amount of Fe produced by the progenitors with mass $m$$<$ 1.5 M$_\\odot $ is only $\\sim $ 20% of the amount produced by all progenitors, i.e.",
"with masses ranging from $0.8 M_{\\odot }$ to $8 M_{\\odot }$ .",
"While not important for our MDF work here, the systematic neglect of long-lived SNeIa progenitors could have a potential impact on the [O/Fe]-[Fe/H] relationship (Pilkington et al., in preparation).",
"For our analysis, we selected six galaxies with the most prominent discs, following the same criteria as described in Pilkington et al.",
"(2012a)The selected galaxies are those for which there was unequivocal identification of the disc (from angular momentum arguments constructed from the gas and young star distributions, see Stinson et al.",
"2010).",
"In this way, we are able to eliminate extreme values of bulge-to-total, but formally, we only included those disks for which alignment based upon the gas/young stars was obvious.. Each of the six MUGS simulations analysed here were run in a 50$h^{-1}$ Mpc cosmological box with `volume renormalization' to ensure higher space and force resolution in the region centred on the central galaxy (Klypin et al.",
"2001).",
"For each simulation, the $z$ =0 output was examined to find sufficiently isolated halos within the mass range 5$\\times $ 10$^{11}$ M$_{\\odot }$ and 2$\\times $ 10$^{12}$ M$_{\\odot }$ .",
"Sixteen halos within this range were selected at random and re-run at higher resolution (9 of which are described by Stinson et al.",
"2010, with a further 7 now having been realised subsequent to the publication of the first sample).",
"Six galaxies were chosen from the MUGS suite': g1536, g24334, g28547, g422, g8893, and g15784.",
"The system g15784 is our adopted fiducial Milky Way analog, owing to its total mass and its baryonic mass in the disc, both similar to the values calculated with up-to-date dynamical models of our Galaxy (see McMillan 2011).",
"In Table REF , we list the key properties for each of the simulations employed here.",
"Following the same notation as the original MUGS simulations, galaxies are identified using the group number from the original friends-of-friends galaxy catalogue.",
"The first column contains the galaxy name; the second, third, and fourth columns are, for each galaxy, the total (baryonic and non-baryonic) mass, the gas mass, and the stellar mass inside the virial radius $R_{vir}$ , respectively.",
"The fifth, sixth, and seventh columns are the corresponding number of dark matter particles, gas particles, and stellar particles, respectively.",
"The eigth and ninth columns are the total disc mass and the total bulge mass assigned to each sub-component, after application of the kinematic decomposition and spatial cuts described in Sections 2.2 and 2.3.",
"The final two columns are the half mass radius for the spheroid component ($R_{eff}$ ) and the disc scalelength ($R_d$ ), calculated by means of an exponential profile fit to the disc component."
],
[
"Kinematic Decomposition",
"To isolate the stellar components associated with the simulated bulge and disc for each simulation, we performed kinematic decompositions, after Abadi et al.",
"(2003).",
"We first centre and align the angular momentum vectors of the baryons with the $z$ -axis of the volume, and remove any systemic velocity associated with the simulated galaxy, for ease of subsequent decomposition.",
"We next compute the Lindblad Diagram of all stellar particles in the inner region of the halo ($r$$<$ 30 kpc), i.e.",
"the $z$ -component of the specific angular momentum as a function of the specific binding energy.",
"An example is shown in Fig REF (top left panel) for the fiducial simulation g15784.",
"The distribution of the orbital circularity $\\epsilon _J$ is then constructed, where $\\epsilon _J = J_z /J_{circ}(E)$ where $J_z$ is the $z$ -component of the specific angular momentum and $J_{circ}(E)$ the angular momentum of a circular orbit at a given specific binding energy.",
"The $J_{circ}(E)$ curve (shown as a white line in the top-left panel of Fig.",
"REF ) indicates the location of circular orbits corotating with the disc.",
"The spheroid component (comprised of both bulge and inner halo stars) is defined a priori to be a symmetric distribution centered on $J_z/J_{circ}(E)=0$ .",
"This means that, by construction, the bulge/inner halo components are assumed to be non-rotating.",
"Since we are here interested in distinguishing the bulge from the disc component, we assume that the disc is made by all the stars except those belonging to the bulge.",
"Star particles with negative circularity are assigned to the spheroid component; those with positive circularity are randomly assigned to either the spheroid or disc components, weighted by the likelihood imposed by the relative numbers of both components at a given positive $J_z/J_{circ}(E)$ .",
"To prevent stars of nearby satellites from being included in our analysis, we perform a further spatial cut as described in REF .",
"This method, while somewhat arbitrary in its definition of the components, does allow one to decompose in an objective, 'hands-off' manner, the stellar component into spheroidal and disc component, as shown graphically in the bottom panels of Fig.",
"REF .",
"Figure: Kinematic decomposition of g15784.",
"Top left panel: thezz-component of the specific angular momentum shown as a functionof specific binding energy for all stars within 30 kpc of thecentre of the galaxy.",
"The white curve shows the J circ (E)J_{circ}(E)location.",
"The colours scale with the density of particles in eachpixel, with redder colours corresponding to higher densities.",
"Top rightpanel: distribution of the orbital circularity of the stellarcomponent (black line) and the decomposition into spheroid (orangeline) and disc (blue line) components.",
"Bottom panels: images ofthe spheroid (left) and disc (right) components.",
"The colours scalewith the density of particles in each pixel, with redder colourscorresponding to higher densities.",
"The sub-structure/peak near J z /J circ (E)∼+0.1J_{z}/J_{circ}(E) \\sim +0.1 isassociated with the satellite seen at (x,z)∼(-10,+10)(x,z) \\sim (-10,+10) in the lower-right panel."
],
[
"Spatial Cuts",
"In addition to the kinematic decomposition of Sect.REF , for each galaxy we also applied a spatial cut to both the spheroid and disc components.",
"Spheroid stars within $R_{b,cut}$ =1$-$ 3 kpc are assigned to the `bulge', a radial `cut' which qualitatively corresponds to the spatial extent of the simulated bulge, although we have confirmed that the exact value selected within this range does not impact on our conclusions.",
"Further, for each simulation, we assign stars to the disc should they lie within 4 kpc of the mid-plane, beyond the aforementioned bulge-disc radial cut, i.e.",
"$r^{\\prime } = \\sqrt{X^2+Y^2}>R_{b,cut}$ .",
"For one simulation (g24334), an additional spatial cut was applied, in order to remove the presence of a dwarf satellite which, at redshift $z$ =0, is passing through the disk at $r^\\prime $$\\sim $ 5 kpc.",
"The disc for g24334 was, instead, defined by $R_{b,cut}$$<$$r^\\prime $$<$ 4 kpc.",
"Table: Main features of the simulated galaxies from the MUGS sample.",
"All masses are expressed in units of 10 10 M ⊙ 10^{10} M_{\\odot } and all radii are in kpc.Figure: Star formation histories (SFHs) of the MUGS galaxies consideredin this work.",
"In each panel, the black solid lines are SFHs computed for all the stars insideR vir R_{vir}.",
"The blue dashed lines and the red dotted lines arethe SFHs for stars belonging to the disc and the bulge, respectively,after performing a kinematic cut as described in Sect.",
".Figure: Star formation histories (SFHs) of the MUGS galaxies consideredin this work.",
"In each panel, the black solid lines areas in Fig.",
",whereas the blue dashed lines and the red dotted lines arethe SFHs for stars belonging to the disc and the bulge, respectively,after performing a pure spatial cut as described in Sect.",
".Figure: Star formation histories (SFHs) of the MUGS discs in differentspatial regions.The red dashed lines, blue solid lines, and green dash-dottedlines represent the SFHs computed considering the star particles whichat the present time are located at radii rr<<2 R d R_{d}, between2 R d R_{d} and 3 R d R_{d}, and radii rr>>3 R d R_{d}, respectively.To trace the SFH of the innermost disc regions, disc particles within typicalbulge radii (i.e.",
"at rr<<1-3 kpc) have been excluded.The thick solid red lines are pure spatially cut bulge SFHs, i.e.",
"they represent SFHscalculated considering all the star particles in the innermost “bulge” regions, regardless of theirkinematics.Figure: “Solar neighbourhood” star formation histories of the MUGS galaxies and effect ofbulge stars with positive circularities.",
"The solid lines are the SFHs calculated after a kinematiccut as described in Sect.",
"plus a spatial cut as in Sect.",
".The dashed lines are the result of a spatial cut and the exclusion of all the star particleswith J z /J circ (E)<0.8J_{z}/J_{circ}(E) < 0.8."
],
[
"Star Formation Histories",
"In Fig.",
"REF , the star formation histories (SFHs) of the six MUGS galaxies are shown.",
"In each panel, we show the SFH computed considering all the star particles within the virial radius (solid lines), that of the disc (dashed lines), and the bulge (dotted lines).",
"The disc and bulge SFHs reported in Fig.",
"REF have been derived after performing the kinematic decompostion described in Sect.",
"REF .",
"All the SFHs show similar behaviour, i.e.",
"an $\\sim $ 20$-$ 30 M$_\\odot $ /yr peak during the first few Gyrs, followed by an exponential and fairly continuous decline at later times with a timescale ranging from $\\sim $ 4$-$ 7 Gyr.",
"In two cases (g24334 and g422), the early evolutionary phases are dominated by more intense centrally-concentrated star formation.",
"In the other systems, throughout their whole history (in general), the discs show SFRs higher than those of their associated bulges.",
"Moreover, the disc SFHs show higher present-day SFRs than the bulges by factors of a few to ten.",
"In a few cases, such as g1536, the bulge SFHs are somewhat higher than those encountered in nature, at least over the last $\\sim $ 5 Gyrs of the simulation.",
"In general, the present-day bulge SFRs are higher than those seen in the sample of Fisher et al (2009) by factors of a few, except for g28547, which sits at the median of the aforementioned sample.",
"That said, it is important to remind the reader that in this suite of simulations, the only form for star formation `quenching' present is that associated with feedback from SNe; in general, this is not sufficient for producing passive spheroids in cosmological simulations (e.g.",
"Kawata & Gibson 2005) or semi-analytic models (e.g.",
"Calura & Menci 2009).",
"In Fig.",
"REF , we show the SFHs for discs and bulges obtained solely via the spatial cut criteria described in Sect.",
"REF .",
"All the MUGS galaxies suffer an excessive central concentration of mass and of star formation, as already reported by Stinson et al.",
"(2010), who performed a careful analysis of the rotation curves of various systems, and of Pilkington et al.",
"(2012a), who studied the radial metallicity gradients.",
"As Fig.",
"REF shows, if we assume that all the star particles included within the innermost 1-3 kpc belong to the bulges, we end up with unnaturally high present-day values for the bulge SFHs, which reflect this well-known issue of an excess of mass in the centre of the simulated galaxies.",
"Several works have already shown that the problem regarding the central concentration of mass may be partially alleviated by increasing the simulation resolution (e.g.",
"Pilkington et al.",
"2011; Brook et al.",
"2012).",
"Further tests are needed to understand to what extent resolution may help in ameliorating this problem.",
"As reported by Stinson et al.",
"(2010), in general MUGS bulges are bluer than real bulges, and that quenching star formation sooner would produce bulges that better match the red sequence.",
"In fact, MUGS galaxies do not include AGN feedback, which might significantly help driving star-forming gas out of the central regions of galaxies and limiting the mass of bulges, as already shown in semi analytic models and even on mass scales comparable to the one of the MW bulge (e.g., Calura & Menci 2011).",
"In Fig.",
"REF , we show the SFHs for different spatial regions within each of the galaxies in our sample.",
"For each stellar particle, we have calculated its present-day distance from the centre of the disc and divided the disc into three different regions: (1) the innermost one, in which all the particles lie within the inner 2 $R_{d}$ , excluding those in disc regions within radii typical of bulges (1-3 kpc); (2) an annulus encompassing the particles with distances 2 $R_{d}$$<$$r$$<$ 3 $R_{d}$ , and (3) the disc outskirts, including all the particles with distances beyond 3 $R_{d}$ .",
"In each panel of Fig.",
"REF , the thick solid red lines represent pure spatially-cut bulge SFHs, i.e.",
"SFHs calculated considering all the star particles in the innermost “bulge” regions, regardless of their kinematics.",
"It is important to note that in Fig.",
"REF we are showing SFHs calculated in different spatial regions considering the present-day position of each stellar particle, and not the position where the stellar particles were formed.",
"The bulge SFHs dominate the overall star formation budget througout most of the cosmic time.",
"After having excluded the star particles in the bulge regions, the SFHs of the innermost disc regions are in general comparable to those at larger distances.",
"At late times, the outermost parts tend to show `oscillating' SFHs at late times ($>$ 5 Gyrs).",
"This effect is due to the adoption of a star formation density threshold ($>$ 1 cm$^{-3}$ ); the outermost regions of the discs are characterised by lower densities, hence more likely to present such a modulating star formation behaviour, once the gas density is comparable to that of the threshold value.",
"This phenomenon can also be seen in non-cosmological chemical evolution models for disc galaxies (e.g.",
"Chiappini et al.",
"2001; Cescutti et al.",
"2007) which adopt a density threshold for star formation.",
"We refer the reader to a more targeted analysis of the temporal evolution of the radial star formation rate profiles within simulated cosmological discs by Pilkington et al.",
"(2012a).",
"It is also important to recall that radial migration is likely to occur within these simulations.",
"Sanchez-Blazquez et al.",
"(2009) presented the analysis of a cosmological disc simulation, comparable in mass and kinematic heating profile (e.g.",
"House et al.",
"2011) to those analysed here, and found that the mean radial distance traversed by the disc star particles was $\\sim $ 1.7 kpc.",
"The effects of stellar migration are not taken into account in Fig.",
"REF .",
"To assess the effect of bulge stars with positive circularities, in Fig.",
"REF we show the “solar neigbourhood” SFHs computed as in Fig.",
"REF , compared to the SFHs computed considering the star particles with 2 $R_{d}$$<$$r$$<$ 3 $R_{d}$ and having excluded the ones with $J_z/J_{circ}(E)<0.8$ .",
"As visible in Fig.",
"1, at this $J_z/J_{circ}(E)$ value the bulge circularity distribution is very low, whereas the one for the disc is close to the peak value.",
"This ensures the removal of a substantial fraction of particles with bulge kinematics, while at the same time, retaining the majority of disc particles.",
"The difference bewteen the two SFHs is in most of the cases more visible at early (i.e.",
"$<$ 8 Gyr) times, when the bulge SFH was particularly intense, as seen in Fig.",
"REF .",
"Overall, the exclusion of star particles with $J_z/J_{circ}(E)<0.8$ does not seem to affect the global shape of the SFHs.",
"In the current analysis, the use of a spatial or kinematical definition of the solar neighbourhood region does not substantially affect our results on the metallicity distribution.",
"There is a low contamination from low angular momentum, high metallicity bulge star particles at large distances from the centre.",
"The effects of bulge stars with positive circularities on the solar neighbourhod MDF will be studied in detail later in Sect.",
"REF ."
],
[
"Metallicity Distribution Functions in the Discs",
"In Fig.",
"REF , we show the MDFs of our six simulated galaxies.",
"In each panel, we show the MDF calculated using all the stellar particles (i) located within the virial radius $R_{vir}$ , (ii) in the disc after the kinematic decomposition described in Sect.",
"REF and (iii) after a kinematic descomposition plus a spatial cut as described in Sect.",
"REF .",
"The MDFs for the particles included within $R_{vir}$ show several peaks, each corresponding to stellar populations associated with various kinematic components.",
"A representative case is that of our fiducial simulation g15784, whose MDF shows a high-metallicity peak at [Fe/H]$\\sim $ 0.2 and a broader, more significant, peak at lower metallicity (near [Fe/H]$\\sim $$-$ 0.2).",
"The disc MDFs are in most cases very similar to the ones calculated using all the star particles within $R_{vir}$ .",
"Performing the spatial cut of §REF , we can see that the high-metallicity peak of the MDF has been removed (g1536 and g15784) or substantially reduced (g28547).",
"This is because in any galaxy, the highest metallicity stellar particles tend to reside near the centre, similar indeed to what is encountered in nature, including our own Milky Way, where the metal-rich stars are found preferentially in the bulge and inner disc.",
"In some cases (g28547 and g8893), the multi-peaked structure of the MDF is still present (or even exacerbated) after performing the spatial cut.",
"This is due to their particular SFHs, which tend to show several late-time star formation episodes.",
"A similar behaviour was found for dwarf galaxies within a semi-analytic galaxy formation model (Calura & Menci 2009), where multi-peak SFHs tend to be associated with complex multi-component stellar metallicity distributions.",
"Finally, we stress that a comprehensive analysis of the origin of the metallicity gradients in the MUGS discs, including a few cases described here, can be found in Pilkington et al.",
"(2012a).",
"In general, MUGS galaxies can account for the slope of the metallicity gradient observed today in young stars in the Milky Way and in HII regions in local discs.",
"The analysis of Pilkington et al.",
"(2012a) showed that the metallicity zero-point of the MUGS galaxies is offset by 0.2-0.3 dex from those in nature, but this does not impact on the determination of the gradients therein."
],
[
"Metallicity Distribution Functions in the Solar Neighbourhood",
"In Fig.",
"REF , we show the MDFs calculated for each simulation using star particles situated within a circular annulus of 2$R_{d}$$<$$r$$<$ 3$R_{d}$ , compared with the observational MDF in the solar neighbouhood.",
"The observational MDF adopted here is calculated from the Geneva Copenhagen survey (GCS) sample of solar-neighbourhood stars (Nordström et al.",
"2004; Holmberg, Nordström & Andersen 2007).",
"Nordström et al.",
"(2004) obtained Strömgren photometry and radial velocities for a magnitude-limited sample of $\\sim $ 17000 F and G dwarfs.",
"From the photometry, they estimated metallicities and ages.",
"There has been some debate about the calibration of the metallicities and ages (Haywood 2006; Holmberg et al.",
"2007; Haywood 2008, Casagrande et al.",
"2011).",
"Recently, the re-calibrated data from Holmberg et al.",
"(2009) became available, and it is with the MDF drawn from these data that we compare our model predictions.",
"Following Holmberg et al.",
"(2009), we define a 'clean' sub-sample by removing (i) binary stars, (ii) stars for which the uncertainty in age is $>$ 25%, (iii) stars for which the uncertainty in trigonometric parallax is $>13$ %, and (iv) stars for which a 'null' entry was provided for any of the parallax, age, metallicity, or their associated uncertainties.",
"This 'clean' sub-sample consists of $\\sim 4000$ stars.",
"The region including the star particles with distances $r$ in the range 2$R_{d}$$<$$r$$<$ 3$R_{d}$ are analogous `solar neighbourhoods' for the MUGS simulations; in the following, we will employ the MDF calculated for the star particles included in these region for our comparison with extant data of the Milky Way's solar neighbourhood.",
"The solid lines in Fig.",
"REF are the MDFs calculated after a kinematic cut as described in Sect.",
"REF plus a spatial cut as explained in Sect.",
"REF .",
"The dashed lines are MDFs after applying a spatial cut and by excluding all the star particles with $J_{z}/J_{circ}(E) < 0.8$ .",
"A comparison of the solid and dashed lines helps in understanding the role of bulge stars with positive circularities in the 'solar neighbourhood' MDF: in all the panels, the MDFs computed in these two different ways are very similar, hence the contribution from bulge star particles at radial distances $> 2 R_{eff}$ is expected to be low.",
"Also the MDFs computed by means of a solely spatial cut (dotted lines in Fig.",
"REF ) are very similar to the ones computed with a spatial plus kinematic cut.",
"This shows that our results concerning the “solar neighbourhood” MDF are not overly sensitive to a kinematic selection of disc stars.",
"This is an encouraging aspect, since for the observational sample we are comparing our results with, it is impossible to perform any such distinction between bulge and disc stars.",
"The fact that the results are stable against the definition of “solar neighbrouhood” indicates that our results should be considered robust.",
"Concerning the comparison between model results and the observed MDF (solid histograms in Fig.",
"REF ), the first striking difference regards the position of the peak metallicities, with the model MDFs peaking at [Fe/H] values $\\sim 0.2 -0.3$ dex lower than the one of the GCS sample.",
"This is an indication that the average stellar metallicity in the simulations is lower with respect to the one observed in the solar neighbourhood.",
"Another important aspect emerging from this comparison regards the relative fraction of low metallicity stars, which in the simulated galaxies is substantially higher, and which will be quantified and discussed in more detail in Sect.",
"REF ."
],
[
"Statistical analysis of the MDF in simulations and in the solar neighbourhood",
"A comparison of the main statistical features of the predicted MDFs and the empirical MDF of the Milky Way's solar neighbourhood is provided in Tab.",
"REF .",
"The characteristics noted there are patterned closely upon those performed by Kirby et al.",
"(2011) in their analysis of the MDFs of Local Group dwarf spheroidals and by Pilkington et al (2012b) in their analysis of the MDFs of simulated dwarf disc galaxies.",
"In the second and third columns, we list the mean and the median of the MDF for a given simulation, whose name is reported in the first column.",
"In this table, we use 5-$\\sigma $ clipping of outliers from the distribution before deriving MDF shape characteristics.",
"The dispersion $\\sigma $ is reported in the fourth column, whereas the interquartile range (IQR), the interdecile (IDR), intercentile range (ICR), inter tenth percent (ITR), the skewness, and the kurtosis are reported in the fifth, sixth, seventh, eigth, ninth amd tenth columns, respectively.",
"The dispersion $\\sigma $ , the IQR, IDR, ICR and ITR are different measures of the width of the distribution, while the skewness is a measure of the symmetry, with positive and negative values indicating an MDF skewed to the right (towards positive metallicities) and to the left (towards negative metallicities), respectively.",
"The kurtosis (or peakedness) indicates the degree to which the MDF is peaked with respect to a normal distribution: high kurtosis values ($\\gg 1$ ) signify a distribution with extended tails, while lower values signify light tails.",
"As already noted, the model MDFs show median metallicities that are systematically lower than the empirical MDF, with offsets ranging from $-$ 0.14 dex (considering the median as representative of the peak position) to $-$ 0.45 dex.",
"In the fiducial simulation (g15784), the offset between the peaks is $-$ 0.32 dex.",
"The discrepancy is lower for g24334, for which we are only considering the innermost regions since its disk scalelength is only 1 kpc.",
"It is therefore not surprising that the mean and median stellar metallicities for this system are larger with respect to the others, due to the presence of a significant metallicity gradient (Pilkington et al.",
"2012a), a point to which we return below."
],
[
"Possible reasons for the discrepancies between simulations and observations",
"The apparent discrepancy between the MDF peaks of the simulated 'solar neighbourhood' and that observed in the Milky Way can be ascribed to several reasons.",
"First, the chemical evolution prescriptions incorporated within Gasoline (Raiteri et al.",
"1996) predict the cumulative Fe mass produced after 10 Gyr by a 1 $M_{\\odot }$ simple stellar population, employing a Kroupa et al.",
"(1993) IMF is 0.0007 M$_{\\odot }$ , a factor of two lower than that reported by Portinari et al.",
"(2004), in their study of the impact of the IMF on various local chemical evolution observables.",
"This discrepancy is due in part to a different SNeIa frequency (7% in the case of Portinari et al.",
"2004 versus 5% adopted within Gasoline), and to a lower Fe yield from SNeII (in Gasoline, SNeII form a $1M_{\\odot }$ SSP produces 0.00022 M$_{\\odot }$ of Fe, versus 0.00048 M$_{\\odot }$ according to Portinari et al.",
"2004) and to a slightly lower mass of Fe produced by a single SNIa (0.63 M$_{\\odot }$ versus 0.7 M$_{\\odot }$ used by Portinari et al.",
"2004).",
"Therefore, the nucleosynthesis prescriptions adopted within Gasoline tend to produce less Fe with respect to other chemical evolution models designed to reproduce local constraints, and this is certainly one of the reasons for the lower metallicity peaks of the simulated MDFs.",
"Another reason is linked to the formalism used here to model SNeIa: here, the contribution of SNeIa progenitors with mass $m$$<$ 1.5 M$_\\odot $ was neglected, and this leads to the underestimation of the Fe mass in stellar particles older than 5 Gyr.",
"In fact, by integrating the type Ia SN rate of Greggio & Renzini (1983) from $T_{0}$ =0 Gyr to $T_{1}$ =5 Gyr, corresponding to the lifetime of a 1.5 M$_\\odot $ star, and comparing this number to the integral of the rate over one Hubble time, one can show that the total Fe production from stars down to the present turnoff mass (0.8 M$_{\\odot }$ ) is underestimated by $\\sim $ 0.1 dex.",
"Other effects which could cause a loss of metals and a consequent low stellar metallicity in the disc would be metal ejection during mergers occurring at early times.",
"In this case, we should be finding mean metallicities higher in the gas (including both cold and warm components) with respect to the stars.",
"However, a preliminary estimate of gas and stellar metallicity gradients in the simulated discs do not seem to support this scenario: in the g15784 simulation, the mass-weighted average metallicity calculated for the gas particles belonging to the most massive galaxy (i.e.",
"the g15784 disc) is $<Z>_{g}=0.0005$ , whereas the analogous stellar mean metallicity is $<Z>_{*}=0.007$ .",
"Finally, as indicated by a parallel study of the evolution of the metallicity gradients in the MUGS galaxies (Pilkington et al.",
"2012a), the star formation threshold may contribute to a more centrally concentrated star formation history, in particular during the early stages.",
"If star formation in the disk is underestimated in the models at early times, this may lead to steep abundance gradients and low metallicity star formation in the early disc, resulting in a lower present-day metallicity in this region.",
"The model MDFs appear broader than the observed one, as indicated by the $\\sigma $ , IDR-ITR values.",
"This could be related to radial migration of stellar particles, which can broaden the MDF (Schoenrich & Binney 2009) and whose effect could be enhanced by the fact that these simulations (like most cosmological disc simulations) are substantially hotter (kinematically speaking) relative to the Milky Way (e.g.",
"House et al.",
"2011).",
"Metal circulation in the disc and outwards could play some role as well in broadening the MDF.",
"The skewness values vary from disc to disc, however, in most cases, the simulated MDFs tend to show more negative skewness with respect to the observed MDF of the solar neighbourhood, which relates mostly to their over-populated low-metallicity tails.",
"Finally, the kurtosis values are higher than the one derived for the GCS sample, again due to heavy low metallicity tails.",
"It is worth noting that the use of a 5-$\\sigma $ clipping limits the effects of the presence of extreme low-metallicity tails which, without taking into account any clipping, would give rise to even higher kurtosis values.",
"Later in Section REF , we will see how the age-metallicity relation may be regarded as a useful diagnostic to understand in better detail the implications of our MDF study and the reasons for the discrepancies between the theoretical and obserevd MDFs.",
"Figure: Metallicity distribution functions for the six MUGS discs.Thick solid lines: MDF calculated considering all the stellar particleswithin the virial radius.",
"Thin dotted lines: MDFs for the particlesbelonging to the discs after the kinematic decomposition described in§.",
"Thin dashed lines: MDF calculated for particles in thedisc and after the additional spatial cut described in §,i.e.",
"considering the star particles in the disc within 4 kpc of themid-plane and at galactocentric radii R̾R b ̾R_b.Figure: Metallicity distribution functions for the 'solar neighbourhood'of each of the six MUGS discs, compared to the MDFobserved in the solar neighbourhood as forthe 'clean' sub-sample drawnfrom theGeneva-Copenhagen Survey (Holmberg et al.",
"2009).The solid histrograms represent the observed MDF.The solid lines are MDFs calculated after a kinematiccut as described in Sect.",
"plus a spatial cut (i.e.considering an annulusencompassing all the particles with distances rr from the centre in the range2 R d R_{d}<<rr<<3 R d R_{d})as in Sect.",
".The Dashed lines are MDFs after a spatial cut and the exclusion of all the star particleswith J z /J circ (E)<0.8J_{z}/J_{circ}(E) < 0.8.Dotted lines: MDFs obtained with a solely spatial cut, i.e.",
"considering the star particles in the annulus withoutany other kinematical selection.Figure: Metallicity distribution functions for the bulges of each ofthe six MUGS galaxies.",
"Solid lines: MDFs calculated by using all theparticles belonging to the spheroid, after application of the kinematicdecomposition described in §.Dashed lines: spheroid MDFrestricted to the innermost regions of the bulge, i.e.",
"computed byperforming a spatial cut on the bulge component.In each panel, the solid histogram is the observational bulge MDF from Zoccali et al.",
"(2008).Figure: Cumulative metallicity distribution functions for the 'solarneighbourhood' of each of the six MUGS galaxies and forthe CGS 'clean' sub-sample, each onenormalized at the integral of the corresponding differential MDF.Figure: Cumulative metallicity distribution functions for the bulges ofeach of the six MUGS galaxies and for the Z08 sample.Each cumulative MDF is normalized to the integral of the corresponding differential MDF.Figure: Age-metallicity relations for the “solar neighbourhood” (upper panel)and of the “bulge” (middle panel) of the simulated galaxy g15784,and of the sub-sample of Holmberg et al.",
"(2009) of solar neighbourhood starsdescribed in Sec. .",
"The colours of each simulated or observed starscales with its age (colour bar on the right)."
],
[
"Possible selection effects in the observed sample",
"In principle, two possible kinds of bias may affect the comparison between data and simulations.",
"One is connected with the observational uncertainties and the sample selection, the other involves the \"representativeness\" of the local sample.",
"Concerning the former, the 'clean' sample of Holmberg et al.",
"(2009) is complete down to $M_{V} \\sim 4.5$ up to about 40-50 pc, thus no significant fraction of F-G stars is expected to be missed.",
"In principle, the choice of using F-G stars (which are long-lived enough to trace the disk star formation over the entire Hubble time) may imply a slightly different range of masses for populations of different metallicities (a metal poor star is hotter than a metal rich star, so for a fixed spectral type it will be slightly less massive), but this effect is likely to be very small and it is unlikely to substantially affect our analysis.",
"The other potential issue stems from the definition of \"solar neighborhood\" itself.",
"Dynamical diffusion of orbitsStellar velocities are randomized through chance encounters with interstellar clouds, gaining energy and increasing the velocity dispersion (e.g.",
"Wielen 1977).",
"allows stars to drift from their birthplaces over time scales of several Gyr and, as a consequence, may deplete the local (i.e.",
"regarding the volume within $\\sim 100$ pc) star formation rate at early epochs (see e.g.",
"Schröder & Pagel 2003).",
"Indeed, different thin disk populations are known to have different scale heights, with their height increasing with age.",
"Since oldest stars are likely the most metal poor, this could imply an observed local metallicity distribution biased against lowest metallicity stars.",
"It is not possible to assess quantitatively the role of each of the effects described in this section.",
"Important constraints may come from the Gaia mission, which will soon take a complete census of stars down to ${M}_{V} = 4.5$ with parallaxes measured with an accuracy better than 10% up to 2–3 kpc, thus extending the solar neighborhood sample to a realistic disk/thick disk sample."
],
[
"Metallicity Distribution Functions in the Bulge",
"The bulge MDFs for the six MUGS galaxies are shown in Fig REF , together with observational data from Zoccali et al.",
"(2008, Z08 hereinafter).",
"The latter are derived from a survey of 800 K-giants in the Galactic bulge, observed at a resolution $R$$\\sim $ 20000.",
"In each panel, we show two MDFs derived from the simulations: one based upon the use of all star particles belonging to the bulge, after application of the kinematic decomposition described in §REF (solid lines), and a second restricted to the innermost regions of the bulge, i.e.",
"computed by performing a spatial cut on the bulge component (dashed lines).",
"As described in §REF , the stellar particles belonging to the bulge region are those at a distance $r = \\sqrt{X^2+Y^2+Z^2}$ $<$ $R_{b,cut}$ (where, 1$<$$R_{b,cut}$$<$ 3 kpc).",
"The main features of the observed MDF for the Galactic bulge and those derived from the simulations are summarised in Table REF .",
"The entries reported in the columns of Table REF are the same as those of Tab.",
"REF , shown in the same order.",
"Both the empirical and simulation-based MDFs show negative skewness, and are more asymmetric than the corresponding MDFs for the solar neighbourhood.",
"The empirical MDF of Zoccali et al.",
"(2008) shows a peak centered nearly at solar [Fe/H] and broader than the one of the solar neighbourhood.",
"The bulge MDF also possesses a negative skewness as the solar neighbourhood.",
"The higher kurtosis of the bulge MDF indicates a more peaked metallicity distribution with respect to that of the solar neighbourhood.",
"In most of the cases, the peak of the model MDF, represented by the mean and median values reported in Tab.",
"REF , is offset with respect to the empirical one by $-$ 0.2 - $-$ 0.3 dex; such an offset is perhaps not surprising, considering the implementation of the stellar yields, as discussed in §REF .",
"The discrepancy is lower only for g24334, a simulation for which the stars accreted from disrupted satellites dominate over the ones formed in its main disc and one whose radial abundance gradient is both steep and shows little temporal evolution, while the others (e.g.",
"g15784 and g422) show gradients which flatten with time (Pilkington et al.",
"2012a).",
"Only g24334 shows the same steep radial gradient today as it showed at redshift $z$$\\sim $ 2.5, leading in part to higher overall metallicity in its innermost regions, which are the ones we are considering as “solar neighbourhood” owing to its small disc size.",
"In all these senses, g24334 differs from the other five MUGS simulations, which each show similar mean and median [Fe/H] values.",
"The dispersions of the simulated bulge MDFs are, in general, in better agreement with the observations than the case of the solar neighbourhood MDFs.",
"A good agreement between model predictions and observations is visible also for the skewness values.",
"The kurtosis values are still higher than the one of the observed bulge MDF, however the agreement is better than in the solar neighbourhood.",
"Two recent studies of red giants in Baade's Window (Hill et al.",
"2011) and dwarf/subgiants in the Galactic bulge (Bensby et al.",
"2011) suggest that the MDF in the inner region has multiple peaks (or is at least double-peaked), with a low-metallicity peak occuring near [Fe/H]$\\sim $$-$ 0.6 (Bensby et al.)",
"or [Fe/H]$\\sim $$-$ 0.3 (Hill et al.",
"), and a higher-metallicity peak centered at [Fe/H]$\\sim $$+$ 0.3.",
"Hill et al.",
"suggest that the low-metallicity stellar component shows a large dispersion, while the high-metallicity component appears narrow.",
"The above results are in qualitative agreement with those shown in Fig REF , in particular as far as the Milky-Way analogue g15784 is concerned.",
"A low-metallicity component peaked at [Fe/H]$\\sim $$-$ 0.3 showing significant dispersion is visible in the top-right panel of Fig REF , with a narrower component centered near [Fe/H]$\\sim $$+$ 0.15.",
"The relative amplitude of the two peaks in g15784 is close to unity, while the aforementioned studies of Hill et al.",
"(2011) and Bensby et al.",
"(2011) suggest that the lower-metallicity peak appears considerably weaker than the one at high-metallicity.",
"Both studies converge to a picture of a longer formation time-scale for the metal-rich component.",
"Such a picture is consistent with the age-metallicity relation predicted for the bulge of g15784, (see Sect.",
"REF ), and with the star formation histories shown in Fig 1.",
"From Fig 1, it is clear that the peak of the star formation occurs at early times.",
"Additionally, while residual star formation is present at relatively recent times, this does not contribute to the bulk of the stellar mass, i.e.",
"most of the stars in the simulated bulges are old and have low metallicity.",
"Another aspect emerging from the studies of Hill et al.",
"(2011) and Bensby et al.",
"(2011) is the uncertainty in the position/metallicity of the metal-poor stellar component, likely due to the very different sample selection criteria between the two studies.",
"For the moment, we do not attempt to perform any more detailed comparison with these results since they are very recent and awaiting further confirmation."
],
[
"The Cumulative Metallicity Distribution",
"A diagnostic often used to investigate in better detail the low metallicity tail of the MDF in chemical evolution models is the cumulative metallicity distribution function.",
"The cumulative MDF, calculated at a given metallicity [Fe/H], represents the number of stars with metallicity lower than [Fe/H].",
"The cumulative MDF reflects essentially the same information as the differential MDF (Caimmi 1997), but it is less sensitive to small number statistics and better tracks the behaviour at low metallicties for both low-metallicity local galaxies (Helmi et al.",
"2006) and the solar neighbourhood.",
"In Fig.",
"REF , we show the cumulative MDF as observed in the solar neighbourhood and as predicted for the six MUGS galaxies.",
"In Fig.",
"REF , each curve is normalized to the integral of the corresponding MDF.",
"From the upper of Fig REF , one can see that in the Milky Way's solar neighbourhood, from the sample of Holmberg et al.",
"(2009) and with the cuts performed in Sect.",
"REF , $\\sim $ 10% of the stars have a metallicity [Fe/H]$<$$-$ 0.5.",
"This is in stark contrast with the simulation results which, without taking into account colour and magnitude selection effects, below [Fe/H]=-0.5 show fractions greater than $\\sim $ 30%.",
"Mass fractions relative to the total MDF for the stars associated with the four metallicity values are tabulated in Tab.",
"REF , where the excess of low metallicity stars is emphasised: at any metallicity value from [Fe/H]=-2 to [Fe/H]=$-0.13$ , the predicted mass fraction is considerably higher than the values obtained by integrating the observational MDF.",
"This discrepancy is certainly related to the metallicity offset visible in the peaks of differential MDF discussed in Sect.",
"REF .",
"However, the excess of very low metallicity stars is to be ascribed to other reasons.",
"Several solutions to this problem could be plausible: one of them is the adoption of a modified IMF, known to alleviate the excess of metal-poor stars in local dwarf galaxies - (see e.g., Calura & Menci 2009).",
"However, since this would cause a larger relative number of high mass stars in a stellar population, this modified IMF would have a strong impact on the abundance ratios, such as the [O/Fe] ratio, and even on the SN feedback.",
"In the solar neighbourhood, an IMF similar to that of Kroupa et al.",
"(1993), as adopted here, reproduces a large set of observational constraints, including the abundance ratios (Calura et al.",
"2010).",
"Further, since a truncated or even slightly top-heavy IMF is known to produce a strong $\\alpha $ enhancement in the abundance ratios (Calura & Menci 2009), this does not seem to represent a proper solution to our problem.",
"Investigations of the abundance ratios in the solar neighbourhood of the MUGS discs will be a useful to test this hypothesis as a solution to the problem related to the excess of metal-poor stars.",
"Such an analysis is currently underway, but deferred to a forthcoming paper.",
"An alternate explanation to the dearth of low metallicity stars in the solar neighbourhood is to invoke a prompt initial enrichment scenario, with a population of objects such as zero metallicity (Pop III) stars (Matteucci & Calura 2005; Ohkubo et al.",
"2006; Greif et al.",
"2007), releasing a sufficient amount of heavy elements to prevent the formation of very low-metallicity stars in galactic discs.",
"Testing the impact of such objects in the chemical enrichment of discs in simulations would be possible by including the yields of very massive stars as provided by, e.g.",
"Ohkubo et al.",
"(2006).",
"This is beyond the scope of the present paper, but under consideration for a future generation of chemo-dynamical simulations.",
"In Fig.",
"REF , we show the cumulative predicted bulge MDF for our MUGS galaxies and that observed in the Galactic bulge by Zoccali et al.",
"(2008), with each cumulative MDF normalized to the integral of the corresponding differential MDF.",
"The mass fractions relative to the total MDF for the stars associated with the various metallicity values are tabulated in Tab.",
"REF .",
"Also in the bulges the simulated galaxies overestimate the number of low metallicity stars at most metallicities.",
"However, aside from the overabundance of low-metallicity stars, the rest of the form of the model MDF is quite consistent with the observational MDF of Z08.",
"Again, it will be interesting to see in the future if the predicted overabundance of low metallicity stars is due to sampling effects in the observational MDF and if this could be alleviated with different nucleosynthesis prescriptions for zero metallicity stars."
],
[
"The Age-Metallicity Relation",
"In Fig.",
"REF , we are showing the age-metallicity relation (AMR) in the “solar neighbourhood” of our fiducial Milky Way analogue g15784 (upper panel), of its bulge (middle panel) and of our GCS 'clean' sub-sample (lower panel).",
"From this figure, it is possible to appreciate how the slope of the AMR is reflecting the MDF problems described in the previous sections, in particular the over-populated low metallicity tails visible in the theoretical MDFs and their negative skewness (see also the parallel study of the MDF in dwarf spiral galaxies of Pilkington et al.",
"2012b).",
"In Fig.",
"REF , the metallicities of the stars in the Milky Way’s solar neighbourhood (GCS) are higher than those of the g15784 “solar neigbourhood”.",
"This should not be surprising, in the light of the discussion in Sect.",
"REF , in particular regarding the MDF peak metallicity of our simulations, lower than the one of the GCS.",
"A striking feature of Fig.",
"REF is that the AMR of the GCS solar neighbourhood sub-sample is essentially flat.",
"The AMR predicted for the g15784 solar neighbouhood is predominantly flat, but significant deviations are clearly visible at ages $> 10$ Gyr, where the g15784 simulation shows essentially a correlated AMR.",
"In shape, this relation is similar to those predicted by classical galactic chemical evolution models (e.g.",
"Fenner & Gibson 2003, Haywood 2006; Spitoni et al.",
"2009).",
"These hints for a correlated AMR are to be associated to the excessive negative skewed MDFs of the simulations: the excessively negative skewness is mostly due to low metallicity star particles with ages $>10$ Gyr, older than the stars which settle on the plateau of AMR relation.",
"Furthermore, the dearth of stars older than 13 Gyr is striking as well, and is at variance with the results of non-cosmological homogeneous galactic chemical evolution models (Matteucci et al.",
"2009), which in general are successful in reproducing the local MDF.",
"Perhaps more hints on this aspect could come from studies with inhomogenous chemical evolution models (Oey 2000; Cescutti 2008) , which allow to explore the parameter space nearly as fast as homogeneous models, but more suited to explore the causes of asymmetric or multi-peaked MDFs or the dispersion and slope of the AMR.",
"Also the bulge of g15784 sees a flat age-metallicity relation for stars born with [Fe/H]$\\sim $$+$ 0.15.",
"The formation of these stars, which would populate the high-metallicity peak in the bulge MDF, extends over a period of $\\sim $ 10 Gyrs or more.",
"Most of the stars older than $\\sim $ 10 Gyrs have clearly formed with a lower metallicity and present a correlated AMR similar to the solar neighbourhood of g15784 .",
"A more-thorough study of the AMR of MUGS simulations in both bulges and discs will be presented in Bailin et al., in prep."
],
[
"MDFs of In-situ Stars vs Accreted Stars",
"Important clues as to the origin of various stellar populations formed at different metallicities may come from the study of the separate MDFs of the stars born in situ as opposed to those accreted (Sales et al.",
"2009; Roskar et al.",
"2008; Sanchez-Blazquez et al.",
"2009; Rahimi et al.",
"2010).",
"We define in situ stars as being those born within a cylinder with a radius linearly increasing with cosmic time, constrained to a present-day current value of $\\sim $ 30 kpc, with a time-independent height of $\\pm $ 2 kpc (House et al.",
"2011).",
"The choice of these values for radius and height were suggested after visual inspection of the appearence of the galactic disc at various cosmic times and redshifts.",
"The choice for the height does not have any impact on our results (for values on the order of a few kpc); the value chosen is conservative and satisfies our desire to exclude the effects of satellites.",
"In other words, in situ stars are born `locally', at small distances from the main progenitor, while accreted stars form within satellites and are accreted later.",
"Such a cylindrical volume encompasses both bulge and disc stars; no kinematical distinction is performed in this case, hence our results are valid for the whole galaxy including all its kinematical components.",
"In this section, we will only show the results regarding our Milky Way fiducial (g15784) - partly because it does represent the best analogue to our own Galaxy (see also Brook et al.",
"2011; 2012), and partly because its output temporal cadence was the highest, ensuring the greatest wealth of data with which to work.",
"We did examine all relevant metrics which could be derived with the more limited information available to us at the time of analysis and find that none of our results are tied to this one system.",
"This is consistent with the results of Pilkington et al.",
"(2012a), regarding the similarity of the metallicity gradients within the MUGS galaxies.Modulo the aforementioned discrepant (but understood) g24334.",
"The SFHs of the stellar particles classified as in situ and accreted are shown for the MUGS galaxy g15784 in Fig.",
"REF .",
"The accreted stars form mostly at early times ($t_{\\rm form}$$<$ 2 Gyrs, where the maximum of the `accreted' SFH lies).",
"Another peak within the `accreted' SFH occurs more recently (12$̶t_{\\rm form}̶$ 13 Gyr).",
"The accreted stars contribute roughly 1/3 of the present-day stellar mass of the system; the majority of the stars are born in situ via a merger event and/or local star formation episode.",
"In Fig.",
"REF , we show the total MDF (solid lines in both panels, normalised to unity in all cases) for g15784, alongside the MDFs computed considering only the stars born in situ (dotted lines) and those accreted (dashed lines).",
"We show here the MDFs based upon the total global metallicity Z, rather than just Fe, as the primary MDF metrics in which we are interested are insensitive to this choice.",
"The in situ MDF does not differ significantly from that of the total : both functions show two peaks (at [Z/H]=$-$ 0.2 and [Z/H]=$+$ 0.1).",
"While we have not performed any kinematic decomposition here, it is quite obvious from the results discussed previously that the high metallicity peak is populated by the stars of the central region (i.e.",
"bulge), whereas the lower-metallicity peak is generated by disc stars.",
"The accreted stellar populations result in an MDF centered at [Z/H]$\\sim $$-$ 0.1; furthermore, the MDF for accreted stars does not show any peak at higher metallicities present in the in situ MDF, and the low metallicity tail is more pronounced than that seen within the in situ population, indicating a larger relative fraction of low metallicity stars.",
"It is not surprising that the accreted stars have lower metallicity, as they come from disrupted satellites which have lower mass than the central galaxy g15784, hence they should have lower metallicity stars, consistent with what the well-known mass-metallicity relation suggests in local and high-redshift galaxies (Maiolino et al.",
"2008; Calura et al.",
"2009).",
"In the upper panel of Fig.",
"REF , we show the age distribution function (ADF) of the stars born in situ and those accreted.",
"The ADF shows that the accreted stars are mainly older than 10 Gyrs, whereas the stars formed in situ show a broad range of ages.",
"It is interesting to examine the cumulative MDF calculated for the stars born in situ and for those accreted (Fig.",
"REF ).",
"The low metallicity tail (i.e.",
"stars with [Z/H]$<$$-$ 4) is populated almost exclusively by accreted stars: this means that for g15784, the lowest metallicity stars are of extragalactic origin.",
"We have investigated the kinetic energy of the stellar particles belonging to the two populations, but no clear distinction was found.",
"This is consistent with that found by Rahimi et al.",
"(2010), i.e.",
"that an early accretion event is unlikely to leave any strong signature in any obvious physical property of the present-time stellar populations.",
"Table: Main features of the MDF observed in the Milky Way's solarneighbourhood based upon theHolmberg et al.",
"(2009) GCS empirical dataset and those of the simulated “solar neighbourhood”MDFs for the six MUGS galaxies.Table: Main features of the MDF observed in the Milky Way's bulgeand those of the simlated “bulge” MDFs for the six MUGS galaxies.Table: Cumulative MDF calculated at various metallicity values for the observed solar neighbourhood and for the solar neighbourhood analog sampleof our six MUGS galaxies.Table: Cumulative MDF calculated at various metallicity values for the observed Bulge and for the “bulge” of our six MUGS galaxies."
],
[
"Conclusions",
"We have analysed the MDFs constructed from a suite of six high-resolution hydrodynamical disc galaxy simulations.",
"Both kinematic decompositions and spatial cuts were performed on each, in order to isolate samples of analogous `solar neighbourhood' and `bulge' samples, for comparison with corresponding datasets from the Milky Way.",
"Our main conclusions can be summarised as follows.",
"In general, after having performed a kinematical decomposition of discs and bulges, in most of the cases the star formation histories of the discs dominate over those of the bulges.",
"On the other hand, if we define discs and bulges on a solely spatial basis, bulges have unnaturally high present-day SFR values , which reflect the well-known issue of an excessive central concentration of mass in the simulated galaxies.",
"Increasing resolution may help alleviate this problem, however it is not yet clear to what extent.",
"At the present time, an excess of star formation is visible in the simulated bulges.",
"To limit this phenomenon, the next generation of simulations will have to include mechanisms of star formation quenching such as AGN feedback, which, in semi-analytic models of galaxy formation, have turned out to be efficient in decreasing star formation timescales in spheroids.",
"The `oscillating' behaviour of star formation in the outermost parts is due to the adopted star formation density threshold which acts to control the ability of the low-density regions in the outskirts to undergo substantial and sustained star formation.",
"The MDFs derived using all the stellar particles situated within the virial radius possess a number of `peaks', each associated with stellar populations belonging to various kinematic components.",
"Applying a spatial cut, in particular removing central stars within $R_{b,cut}$ =1$-$ 3 kpc from the centre, has a significant effect on the MDF in that the highest metallicity peak is generally removed.",
"As in nature, these highest metallicity stellar particles tend to reside in the central region, on average.",
"A region analogous to the solar neighbourhood was defined in each system, by considering all the star particles contained within an annulus 2$R_{d}$$<$$r$$<$ 3$R_{d}$ .",
"The MDF of these stars show median metallicities lower by 0.2$-$ 0.3 dex than that of the Milky Way's solar neighbourhood.",
"This can be traced to several reasons, including the lower stellar yields implemented within Gasoline.",
"The predicted distributions are broader than the one observed in the solar neighbourhood, consistent with other studies of the MDF with cosmological simulations (Tissera et al.",
"2012).",
"In our simulations, the overly broad MDFs are related to discs kinematically hotter relative to the Milky Way (House et al.",
"2011).",
"The derived MDFs possess, on average, more negative skewness and higher kurtosis than those encountered in nature; such a result is traced to the more highly populated low-metallicity `tails' in the simulations' MDFs.",
"The MDFs derived for stars in the bulges are in reasonable agreement with that observed in the Milky Way.",
"While the median metallicities are somewhat lower, the inferred MDFs' dispersions and skewness are consistent with the Milky Way.",
"The kurtosis values are higher than the one of the observed bulge MDF, however the agreement is better than in the solar neighbourhood.",
"The prevalence of the low-metallicity tails are emphasised by examining the cumulative MDFs.",
"In the solar neighbourhood, the predicted relative number of stars with [Fe/H]$<$$-$ 3 with respect to the number of stars with [Fe/H]$<$$-$ 2 is of the order 10%, whereas in the Milky Way this ratio is effectively zero.",
"Solutions to this problem include prompt initial enrichment by a population of short-lived zero metallicity stars or perhaps an alternate treatment of metal diffusion (Pilkington et al.",
"2012b).",
"For the fiducial simulation (g15784), we studied the star formation history and the MDF of the stellar populations born in situ and of the those accreted subsequent to satellite disruption.",
"An early accretion episode generated a population of stars older than $\\sim $ 10 Gyrs, whereas the stars formed in situ show a broad range of ages.",
"The low-metallicity tail of the MDF is populated mostly by accreted stars; this means that for g15784, the majority of the lowest metallicity stars are of extragalactic origin.",
"In the future, we plan to extend our study on the metallicity distribution function in MUGS galaxies to various kinematical components, including the halos, and to examine its variation with respect to position in the galaxy.",
"A recent extensive work which will be useful to test our simulations is the one of Schlesinger et al.",
"(2012), where by means of the SEGUE sample of G and K dwarf stars, the variations of the MDF in the Galaxy with radius and height has been investigated.",
"A comparison of the results from various suites of disc galaxy simulations, such as Pilkington et al.",
"(2012a), will be also useful to better understand the effects of the sub-grid physics on the MDFs.",
"Figure: In situ (dotted line) vs accreted (dashed line)star formation history as a function of cosmic time for the MUGS galaxy g15784.The solid linecorresponds to the the total (in situ + accreted) starformation history.Figure: In situ (dotted line) vs accreted (dashed line)metallicity distribution functions (lower panel) and age distributionfunctions (upper panel) for the MUGS galaxy g15784.",
"The solid lines correspond to the total (insitu + accreted) distribution functions.Figure: In situ (dotted line) vs accreted (dashed line)cumulative metallicity distribution functions for the MUGS galaxy g15784.",
"The two curves arenormalised to unity at [Z/H]=++0.5."
],
[
"Acknowledgments",
"FC wish to thank Simona Bellavista for some useful suggestions and M. Bellazzini for interesting discussions.",
"BKG, CBB and LM-D acknowledge the support of the UK's Science & Technology Facilities Council (ST/F002432/1, ST/G003025/1).",
"BKG, KP, and CGF acknowledge the generous visitor support provided by Saint Mary's University and Monash University.",
"This work was made possible by the University of Central Lancashire's High Performance Computing Facility, the UK's National Cosmology Supercomputer (COSMOS), NASA's Advanced Supercomputing Division, TeraGrid, the Arctic Region Supercomputing Center, the University of Washington and the Shared Hierarchical Academic Research Computing Network (SHARCNET).",
"This paper makes use of simulations performed as part of the SHARCNET Dedicated Resource project: 'MUGS: The McMaster Unbiased Galaxy Simulations Project' (DR316, DR401 and DR437).",
"We thank the DEISA consortium, co-funded through EU FP6 project RI-031513 and the FP7 project RI-222919, for support within the DEISA Extreme Computing Imitative."
]
] | 1204.1051 |
[
[
"Constraints on Galileon-induced precessions from solar system orbital\n motions"
],
[
"Abstract We use latest data from solar system planetary orbital motions to put constraints on some Galileon-induced precessional effects.",
"Due to the Vainshtein mechanism, the Galileon-type spherically symmetric field of a monopole induces a small, screened correction proprtional to \\sqrt{r} to its usual r^-1 Newtonian potential which causes a secular precession of the pericenter of a test particle.",
"In the case of our solar system, latest data from Mars allow to constrain the magnitude of such an interaction down to \\alpha <= 0.3 level.",
"Another Galileon-type effect which might impact solar system dynamics is due to an unscreened constant gradient induced by the peculiar motion of the Galaxy.",
"The magnitude of such an effect, depending on the different gravitational binding energies of the Sun and the planets, is \\xi <= 0.004 from the latest bounds on the supplementary perihelion precession of Saturn."
],
[
"Introduction",
"Galileon theorySee, e.g., [1] for a recent review.",
"[2] , originally formulated in 4D flat spacetime and later generalized also to curved backgrounds [3], yields a modified gravity model relying upon anAt least, in its original formulation.",
"Multi-Galileon scenarios, introduced for the first time in [4], appeared to cope with certain potential inconsistencies of the single-Galileon model.",
"additional light scalar field $\\pi $ , the Galileon, endowed with derivatve self-interactionsThey can help in preserving the observed agreement of the General Theory of Relativity with solar system phenomenology by partly screening the effects of the extra-scalar field at relatively short distances.",
"See below and [5].",
"; for a recent overview of this$-$ and of many others$-$ modified model of gravity, see, e.g., [5].",
"The name comes from the fact that the Lagrangian for $\\pi $ is left unchanged by a transformation of it which generalizes the usual Galilean invariance.",
"The Galileon scenario aims to explainSee [6], [7] for an earlier formulation in a different context.",
"the observed acceleration of the Universe [8], [9], [10], [11], [12], [13] without resorting to Dark Energy by modifying the known laws of gravitational interaction at large distances.",
"At the same time, it avoids to compromise their agreement with observations at solar system scales through a screening based on an implementation of the Vainshtein mechanism [14].",
"The Vainshtein mechanism was first shown to work in all its generality [15], and then in the decoupling limit as well [16].",
"Inspired to the multidimensional braneworld model by Dvali, Gabadadze and Porrati (DGP) [17], [18], [19], [20], the Galileon theory, which has been shown the emerge explicitly from higher-dimensional braneworld scenarios and massive gravity [21], [22], [23], is able to cure certain drawbacks of the latter like the appearance of ghosts in the self-accelerated branch [24].",
"Nonetheless, the single-Galileon scenario is not entirely consistent because of certain issues appearing both at classical and quantum scales, along with unwanted superluminal features [5].",
"A bi-Galileon theory [25] seems able to successfully face such problems.",
"In this paper, we want to effectively put quantitative constraints on some consequences of the Galileon scenario with orbital motions in our solar system.",
"For other constraints on Galileon-type interactions inferred from cosmological observations of different phenomena, see, e.g., [26], [27], [28].",
"As an exciting perspective, local observations can fruitfully shed light on the Dark Energy problem and nature of gravity.",
"Indeed, as we will show in the next Sections, Galileon induces various kinds of local, non-negligible orbital effects, given the present-day level of accuracy in constraining non-standard planetary orbital precessions from solar system observations [29].",
"In particular, in Section we look at the small orbital precessions induced by a single Galileon-type field of a central monopole.",
"In Section we inspect the effects caused by the unscreened Galileon constant gradient due to the large scale-induced peculiar motion of the Galaxy.",
"As a result, a small differential Sun-planets extra-acceleration should occur in view of their different gravitational self-energies.",
"For other proposed tests involving different astronomical systems like compact objects in the Milky Way crossing the Galactic plane, and stars and the central black hole in M87, see [30].",
"As a general cautionary remark in using certain scenarios less directly accessible and known with respect to our solar system, we want to note that they may typically suffer from large systematic effects whose accurate knowledge is often lacking, contrary to Sun's planetary arena.",
"Moreover, in some cases they are also quite model-dependent in the sense that they heavily rely upon theoretical assumptions which are still speculative since they have not yet been tested independently with a variety of different phenomena, or have not yet been directly tested at all.",
"It should also be remarked that some astrophysical phenomena like the behavior of black holes in extragalctic scenarios may crucially depend on the composition, formation and dynamical history of the systems considered.",
"This is why we adopt well known and largely tested orbital motions of some natural bodies in our solar system.",
"Section summarize our findings."
],
[
"The orbital precessions due to a single Galileon-type field of a central monopole",
"As a result of the Vainshtein mechanism, one of the forms in which the Galileon extra-potential for a localized source of mass $M$ can appear is [2], [31], [5] $U_{\\rm Gal} = \\alpha H_0\\sqrt{GM r},$ where $G$ is the Newtonian constant of gravitation and [32] $H_0=73.8\\ {\\rm km\\ s^{-1}\\ Mpc^{-1}}= 2.4\\times 10^{-18}\\ {\\rm s^{-1}}$ is the current value of the Hubble parameter.",
"We emphasize that, in principle, a free parameter-a mass scale-which does not necessarily coincide with $H_0$ can be present in eq.",
"(REF ).",
"Thus, the final results of our analysis should depend on such a free parameter and the coupling constant $\\alpha $ , which is expected to be of the order of unity if the aforementioned free mass parameter does not exactly coincide with $H_0$.",
"eq.",
"(REF ) comes from one of the non-linear terms in the spherically symmetric, static equation of motion for $\\pi $ near an object of mass $M$ [2], [31], [5]; such non-linearities become dominant just at relatively short distances.",
"Notably, it is the same non-linear interaction which allows for both the Vainshtein screening and for the self-acceleration of the Universe [2].",
"As far as the solar system is concerned, the Galileon becomes dominant with respect to the usual Newtonian potential $U_{\\rm N}$ at distances of the order of the Vainshtein radius for the Sun which amounts to about 240 pc.",
"Instead, throughout the planetary region it is $\\left|{U_{\\rm Gal}\\over U_{\\rm N}}\\right|=\\left|\\alpha \\right| H_0\\sqrt{{r^3\\over GM}} \\sim 10^{-12}-10^{-9};$ the screening of the Galileon is, thus, quite effective when the potentials are compared.",
"Nonetheless, it turns out that eq.",
"(REF ) affects planetary orbital motions with secular precessional effects whose expected magnitude may not be hopelessly negligible, being, thus, possible to effectively constrain it with present-day observations.",
"Moreover, the expected improvements in the knowledge of the orbital motions of some of the planets of the solar system currently orbited by accurately tracked spacecrafts like MESSENGER (Mercury) and Cassini (Saturn) will further strengthen the bounds on the Galileon-type interaction.",
"In view of the screening of eq.",
"(REF ), the Galileon-induced planetary orbital effects can be worked out with standard perturbative techniques [33].",
"The average of eq.",
"(REF ) over a full orbital period $P_{\\rm b}$ of a test particle, obtained by using the true anomaly $f$ as fast variable of integration, is $\\left\\langle U_{\\rm Gal}\\right\\rangle \\nonumber & = -{\\alpha H_0\\sqrt{GM a}\\over 3\\pi }\\left\\lbrace \\sqrt{1-e}\\left[-4E\\left({2e\\over -1+e}\\right) +\\left(1 + e\\right)K\\left({2e\\over -1+e}\\right)\\right] + \\right.",
"\\\\ \\nonumber \\\\& + \\left.",
"\\sqrt{1 + e}\\left[-4E\\left({2e\\over 1+e}\\right) +\\left(1-e\\right)K\\left({2e\\over 1 + e}\\right)\\right]\\right\\rbrace ,$ whereHere $\\pi $ is not the Galileon scalar field, being, instead, the usual ratio of circumference to diameter in Euclidean geometry.",
"$E$ is the complete elliptic integral and $K$ is the complete elliptic integral of the first kind; $a$ and $e$ are the semimajor axis and the eccentricity of the test particle's orbit.",
"By using eq.",
"(REF ) in the Lagrange planetary equations [33] for the variation of the osculating Keplerian orbital elements of a test particle, it follows $\\left\\langle \\frac{{{d}}{a}}{{{d}}{t}}\\right\\rangle & = 0, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{e}}{{{d}}{t}}\\right\\rangle & = 0, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{I}}{{{d}}{t}}\\right\\rangle & = 0, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\mathit {\\Omega }}}{{{d}}{t}}\\right\\rangle & = 0, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\varpi }}{{{d}}{t}}\\right\\rangle \\nonumber & = {\\alpha H_0\\over 2\\pi e^2}\\left\\lbrace \\left(-1 + e\\right)\\sqrt{1 + e}\\ E\\left({2e\\over e - 1}\\right) -\\left(1+e\\right)\\sqrt{1-e}\\ E\\left({2e\\over 1 + e}\\right) +\\right.\\\\ \\nonumber \\\\& + \\left.",
"\\left(1 - e^2\\right)\\left[\\sqrt{1+e}\\ K\\left({2e\\over -1+e}\\right) + \\sqrt{1 - e}\\ K\\left({2e\\over 1 + e}\\right)\\right] \\right\\rbrace , \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\mathcal {M}}}{{{d}}{t}}\\right\\rangle \\nonumber & = {\\alpha H_0\\over 2\\pi e^2\\sqrt{1-e^2}}\\left\\lbrace \\left(1+{5\\over 3}e^2\\right)\\left[\\left(1 - e\\right)\\sqrt{1+e}\\ E\\left({2e\\over -1+e}\\right)+\\left(1+e\\right)\\sqrt{1-e}\\ E\\left({2e\\over 1+e}\\right)\\right] -\\right.\\\\ \\nonumber \\\\& - \\left.",
"\\left(1 -{4\\over 3}e^2 + {e^4\\over 3} \\right)\\left[\\sqrt{1+e}\\ K\\ \\left({2e\\over -1+e}\\right) +\\sqrt{1-e}\\ K\\left({2e\\over 1+e}\\right)\\right] \\right\\rbrace ,$ where $I$ is the inclination of the orbital plane to the reference $\\lbrace x,y\\rbrace $ plane adopted, $\\mathit {\\Omega }$ is the longitude of the ascending nodeIt is an angle in the reference $\\lbrace x,y\\rbrace $ plane counted from the reference $x$ direction to the line of the nodes, which is the intersection of the orbital plane to the $\\lbrace x,y\\rbrace $ plane itself., $\\varpi $ is the longitude of pericenterIt is a “dogleg” angle since it is defined as $\\varpi \\doteq \\mathit {\\Omega }+\\omega ,$ where $\\omega $ is the argument of pericenter lying in the orbital plane and counted from the line of the nodes to the point of closest approach., and $\\mathcal {M}$ is the mean anomalyIt is defined as $\\mathcal {M}\\doteq n_{\\rm b}\\left(t-t_0\\right),$ where $n_{\\rm b}\\doteq \\sqrt{GM/a^3}$ is the Keplerian mean motion $t_0$ is the time of passage at pericenter.. We remark that eq.",
"(REF )-eq.",
"() are exact in the sense that no a-priori simplifying assumptions on the orbital geometry of the test particle were assumed.",
"Notice also that eq.",
"(REF )-eq.",
"() depend on the orbital configuration of the test particle through the eccentricity $e$ ; they are independent of the semimajor axis $a$ .",
"To order $\\mathcal {O}(e^2)$ , eq.",
"()-eq.",
"() reduce to $\\left\\langle \\frac{{{d}}{\\varpi }}{{{d}}{t}}\\right\\rangle & = -{3\\alpha H_0\\over 8}\\left(1 - {13\\over 32}e^2\\right), \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\mathcal {M}}}{{{d}}{t}}\\right\\rangle & ={11\\alpha H_0\\over 8}\\left(1 - {39\\over 352}e^2\\right).$ We notice that the spherical symmetry of eq.",
"(REF ) would immediately allow to infer that no secular effects at all can occur for $a,e,I,\\mathit {\\Omega }$ .",
"Indeed, the Lagrange rate equation for $a$ [33] contains the partial derivative of the averaged perturbing potential $\\left\\langle U_{\\rm pert}\\right\\rangle $ with respect to $\\mathcal {M}$ , which is proportional to $t$ .",
"The Lagrange equations for $e,I,\\mathit {\\Omega }$ [33] are formed with the partial derivatives of $\\left\\langle U_{\\rm pert}\\right\\rangle $ with respect to $\\mathit {\\Omega },\\omega , I$ (and $\\mathcal {M}$ as well in the Lagrange equation for $e$ ), which are absent in all spherically symmetric perturbing potentials.",
"Table: Supplementary precessions Δϖ ˙,ΔΩ ˙\\Delta \\dot{\\varpi }, \\Delta \\dot{\\mathit {\\Omega }} ofperihelia and nodes of some planets of our solar system estimated by Fienga et al.",
"with the INPOP10a ephemerides.",
"Data from Messenger and Cassini were used.",
"All standard Newtonian/Einsteinian dynamics was modeled , with the exception of the solar Lense-Thirring effect.",
"However, it is relevant only for Mercury, given the uncertainties released.",
"The reference {x,y}\\lbrace x,y\\rbrace plane is the mean Earth's equator at J2000.02000.0.",
"mas cty -1 ^{-1} stands for milliarcseconds per century.From latest Mars data, summarized in Table REF , it follows $\\alpha = 0.07\\pm 0.3$"
],
[
"Differential orbital precessions induced by unscreened large scale structure Galileon",
"According to Hui and Nicolis [30], the Galileon symmetry allows for the existence of an additional constant gradient in any solution of the Galileon equation.",
"In the case of a background external field, such an extraIn [30], it is $\\pi =\\alpha \\varphi $ .",
"$\\nabla \\varphi _{\\rm ext}$ would impart an extra-acceleration to a body of mass $M$ $A=-\\alpha \\left({Q\\over M}\\right)\\nabla \\varphi _{\\rm ext},$ where $Q$ is a conserved scalar charge which coincides with $M$ if the self-gravitational binding energy is negligible; for a compact object like a neutron star $Q/M<1$ , while $Q/M=0$ for a black hole.",
"If, for a given object, such a constant gradient actually exists or not depends on the boundary conditions of the specific scenario considered.",
"In the case of a galaxy, they are yielded by the surrounding large scale structure.",
"Hui and Nicolis [30] pointed out that it would be able to induce a long-wavelength scalar field which, on the scale of a galaxy, would resemble just a constant gradient.",
"It would be able to penetrate the Vainshtein zone of the galaxy, acting unsuppressed on it and on all its constituents which, thus, would experience a differential fall according to their $Q/M$ ratios.",
"In principle, also a Sun-planet pair should experience such a small differential tug because, although non-relativistic, our star has a gravitational binding self-energy much larger than the planets.",
"Thus, it is possible to analyze perturbatively its effects on the orbital motion in detail.",
"It can be anticipated that, in view of the anisotrpic character of the interaction mediated by $\\nabla \\varphi _{\\rm ext}$ , the resulting orbital perturbations should likely not be limited to just the perihelion and the mean anomaly.",
"Indeed, the averaged perturbed potential of the relative Sun-planet motion due to the Galileon-induced large scale structure effect is $\\left\\langle U_{\\rm lss}\\right\\rangle \\nonumber & =\\left\\langle \\xi \\nabla \\varphi _{\\rm ext}\\cdot r\\right\\rangle = -{3\\over 2}\\xi ae A \\left\\lbrace \\cos \\omega \\left(\\hat{k}_x\\cos \\mathit {\\Omega }+ \\hat{k}_y\\sin \\mathit {\\Omega }\\right) + \\right.",
"\\\\ \\nonumber \\\\& + \\left.\\sin \\omega \\left[\\hat{k}_z\\sin I+\\cos I\\left(\\hat{k}_y\\cos \\mathit {\\Omega }-\\hat{k}_x\\sin \\mathit {\\Omega }\\right)\\right]\\right\\rbrace ,$ where $\\xi $ accounts for the Sun-planet difference in their $Q/M$ , and $A &\\doteq \\left|\\nabla \\varphi _{\\rm ext}\\right|, \\\\ \\nonumber \\\\\\hat{k}&\\doteq {\\nabla \\varphi _{\\rm ext}\\over \\left|\\nabla \\varphi _{\\rm ext}\\right|}.$ As expected, the Lagrange planetary equations, applied to eq.",
"(REF ), yield the following non-zero secular rates of changes $\\left\\langle \\frac{{{d}}{a}}{{{d}}{t}}\\right\\rangle & = 0, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{e}}{{{d}}{t}}\\right\\rangle \\nonumber & = -{3\\xi A\\sqrt{1-e^2}\\over 2n_{\\rm b}a}\\left[\\hat{k}_z\\sin I\\cos \\omega + \\cos I\\cos \\omega \\left(\\hat{k}_y\\cos \\mathit {\\Omega }-\\hat{k}_x\\sin \\mathit {\\Omega }\\right) -\\right.\\\\ \\nonumber \\\\& - \\left.\\sin \\omega \\left(\\hat{k}_x\\cos \\mathit {\\Omega }+\\hat{k}_y\\sin \\mathit {\\Omega }\\right) \\right], \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{I}}{{{d}}{t}}\\right\\rangle & = {3\\xi A e\\cos \\omega \\left[\\hat{k}_z\\cos I+ \\sin I\\left(\\hat{k}_x\\sin \\mathit {\\Omega }-\\hat{k}_y\\cos \\mathit {\\Omega }\\right) \\right]\\over 2n_{\\rm b}a}, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\mathit {\\Omega }}}{{{d}}{t}}\\right\\rangle & = {3\\xi A e\\csc I\\sin \\omega \\left[\\hat{k}_z\\cos I+ \\sin I\\left(\\hat{k}_x\\sin \\mathit {\\Omega }-\\hat{k}_y\\cos \\mathit {\\Omega }\\right)\\right]\\over 2 n_{\\rm b}a\\sqrt{1-e^2}}, \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\varpi }}{{{d}}{t}}\\right\\rangle \\nonumber & = {3\\xi A\\over 2en_{\\rm b}a\\sqrt{1-e^2}}\\left\\lbrace \\left(1-e^2\\right)\\cos \\omega \\left(\\hat{k}_x\\cos \\mathit {\\Omega }+ \\hat{k}_y\\sin \\mathit {\\Omega }\\right) +\\right.\\\\ \\nonumber \\\\& + \\left.",
"\\sin \\omega \\left[\\left(\\cos I-e^2\\right)\\left(\\hat{k}_y\\cos \\mathit {\\Omega }-\\hat{k}_x\\sin \\mathit {\\Omega }\\right) + \\hat{k}_z\\left(1 - e^2 +\\cos I\\right)\\tan \\left({I\\over 2}\\right) \\right] \\right\\rbrace , \\\\ \\nonumber \\\\\\left\\langle \\frac{{{d}}{\\mathcal {M}}}{{{d}}{t}}\\right\\rangle \\nonumber & = -{3\\xi A\\left(1 + e^2\\right)\\over 2en_{\\rm b}a}\\left\\lbrace \\cos \\omega \\left(\\hat{k}_x\\cos \\mathit {\\Omega }+\\hat{k}_y\\sin \\mathit {\\Omega }\\right) +\\right.\\\\ \\nonumber \\\\& + \\left.\\sin \\omega \\left[\\hat{k}_z\\sin I+\\cos I\\left(\\hat{k}_y\\cos \\mathit {\\Omega }-\\hat{k}_x\\sin \\mathit {\\Omega }\\right)\\right] \\right\\rbrace .$ Also the long-term rates of change of eq.",
"(REF )-eq.",
"() are exact in the sense that no a-priori simplifying assumptions on the orbital configuration of the test particle were assumed.",
"They are proportional to $\\sqrt{a}$ , so that they are larger for the outer planets.",
"For a more realistic contact with actual observations, which are always referred to a specific reference frame, we also did not a-priori align the unit vector $\\hat{k}$ of the external gradient to any specific direction.",
"It is important to notice that our results eq.",
"(REF )-eq.",
"() are quite general, so that they do not necessarily refer to a Sun-planet pair, being valid for other systems including a compact object as well.",
"The peculiar motion of the Milky Way [34] can be assumed coincident with that of the Local Group with respect to the Cosmic Microwave Background (CMB) rest frame.",
"It occurs at [35] $v_{\\rm LG} = 627\\ {\\rm km\\ s^{-1}}$ towards ${\\rm RA} & = 11.11\\ {\\rm hr}, \\\\ \\nonumber \\\\{\\rm DEC} & = -27.33\\ {\\rm deg},$ corresponding to $\\hat{k}_x& = 0.8717,\\\\ \\nonumber \\\\\\hat{k}_y& = 0.1711,\\\\ \\nonumber \\\\\\hat{k}_z& = -0.459.$ The magnitude of the constant gradient can be approximated to the product of the speed of the peculiar motion times the Hubble parameter [30].",
"In the case of the Milky Way, from eq.",
"(REF ) we have $A\\sim v_{\\rm LG} H_0 = 1.5\\times 10^{-12}\\ {\\rm m\\ s^{-2}}.$ From eq.",
"() and the uncertainty for the supplementary perihelion precession of Saturn in Table REF it turns out $\\left|\\xi \\right| \\le 0.004.$"
],
[
"Summary and conclusions",
"As far as the Galileon-type spherically symmetric field of a central monopole is concerned, we looked at the term $\\propto H_0\\sqrt{r}$ arising from the Vainshtein mechanism.",
"For the Sun and its planets, it is screened to a high level with respect to the usual $r^{-1}$ Newtonian potential; nonetheless, it induces non-vanishing secular precessions of the longitude of perihelion and the mean anomaly which depend only on the planetary eccentricities.",
"Interestingly, their expected magnitudes are rather close to the present-day level of accuracy in constraining the planetary supplementary precessions, of the order of $\\lesssim 1$ milliarcseconds per century.",
"Thus, we were able to effectively constrain the strength parameter $\\alpha $ of the residual Galileon interaction down to $\\left|\\alpha \\right| \\le 0.3$ from Mars data.",
"Then, we looked at the unscreened Galileon-type local orbital effects caused by the Galactic peculiar motion due to large scale structures.",
"They are proportional to the difference of the gravitational self-energies between the Sun and its planets.",
"Because of the anisotropy of such an interaction, all the osculating Keplerian orbital elements of a test particle undergo non-zero long-term variations, apart from the semimajor axis.",
"By using the supplementary perihelion precession of Saturn, we were able to infer $\\left|\\xi \\right|\\le 0.004$ for the strength parameter of such an interaction.",
"Further gathering and processing data from ongoing and forthcoming spacecraft-based missions orbiting some planets of the solar system like Mercury (MESSENGER, BepiColombo) and Saturn (Cassini) will allow to enhance such thrilling opportunity of shedding light on crucial cosmological issues from local observations."
],
[
"Acknowledgments",
"I thank S. Deser and C. de Rham for having pointed out to me relevant references.",
"I am also grateful to an anonymous referee for competent and useful remarks."
]
] | 1204.0745 |
[
[
"A non-type (D) linear isometry"
],
[
"Abstract Previous constructions of non-type (D) maximal monotone operators were based on the non-type (D) operators introduced by Gossez, and the construction of such operators or the proof that they were non-type (D) were not straightforward.",
"The aim of this paper is to present a very simple non-type (D) linear isometry."
],
[
"Introduction",
"Maximal Monotone operators were defined and used in the early sixties as a theoretical framework for the study of electrical networks, and, later on, for the study of non-linear partial differential equations.",
"The first works on monotone operators were due to Zarantonello [14], Minty [11], Kato [10], Browder [4], Rockafellar [12], Brézis [3], among others.",
"Since then, monotone operators were object of intense study.",
"See [2] for a survey on the subject.",
"Fitzpatrick proved that any maximal monotone operator can be represented by convex functions by providing the explicit formula for one of these functions.",
"Therefore, is quite natural to ask the inverse question: under which conditions a convex function represents a maximal monotone operator.",
"Characterizations of the convex functions which represent such operators in reflexive Banach spaces where presented in [6].",
"In non-reflexive Banach spaces, a characterization of convex functions which represents a sub-class of maximal monotone operators (those of type (D)) where presented in [13].",
"In the non-reflexive space $\\ell ^1$ , non-type (D) maximal monotone operators where presented by Gossez in [8], [9].",
"Since then, examples of non-type (D) (maximal monotone) operators in $c_0$ where presented in [5], and such examples in James spaces where presented in [1].",
"So far, the examples of non-type (D) maximal monotone operators were constructed using always Gossez's original examples and ideas in their construction.",
"The technicalities presented in the construction of previous non-type (D) operator may lead one to believe that operators of this type are inherently complex and/or pathological.",
"Our aim is to present very simple linear isometry which happens to be a non-type (D) maximal monotone operator."
],
[
"Notation and basics definitions",
"Let $A$ and $B$ be arbitrary sets.",
"A point-to-set operator $T:A\\rightrightarrows B$ of $A$ in $B$ , is a triple $(A,B,G)$ where $G\\subset A\\times B$ .",
"The set $G$ , called the graph of $T$ , is denoted as $\\operatorname{gra}(T)=G$ and, for $a\\in A$ $T(a)=\\lbrace b\\;|\\; (a,b)\\in \\operatorname{gra}(T)\\rbrace $ Hence, $T:A\\rightrightarrows B$ may also be regarded as a map of $A$ in to $\\wp (B)$ , the power set of $B$ , and be denoted as $T:A\\rightarrow \\wp (B)$ .",
"From now on we identify a map $F:A\\rightarrow B$ with the operator $F:A\\rightrightarrows B$ with the same graph.",
"Let $X$ be a real Banach space, with topological dual $X^*$ .",
"We will use the notation $\\langle {x},{x^*}\\rangle =\\langle {x^*},{x}\\rangle =x^*(x),\\qquad \\forall x\\in X,x^*\\in X^*.$ The weak-$*$ topology of $X^*$ is the smallest topology (in $X^*$ ), in which the maps $X^*\\ni x^*\\mapsto \\langle {x},{x^*}\\rangle $ are continuous for each $x\\in X$ .",
"The bidual of $X$ is $X^{**}=(X^*)^*$ and the canonical injection of $X$ in to $X^{**}$ is $J:X\\rightarrow X^{**}, \\qquad \\langle {x^*},{J(x)}\\rangle =\\langle {x^*},{x}\\rangle \\;\\forall x^*\\in X^*.$ Note that this map is a linear isometry.",
"From now on, we identify $X$ with its image under the canonical injection of $X$ into $X^{**}$ .",
"The space $X$ is non-reflexive if $J$ is not onto, which under the above convention means $X\\subsetneq X^{**}$ .",
"A point-to-set operator $T:X\\rightrightarrows X^*$ (respectively, $T:X^*\\rightrightarrows X$ ) is monotone if for any $(x,x^*),(y,y^*)\\in \\operatorname{gra}(T)$ (respectively, for any $(x^*,x),(y^*,y)\\in \\operatorname{gra}(T)$ ) $\\langle {x-y},{x^*-y^*}\\rangle \\ge 0$ and is maximal monotone if it is monotone and its graph is maximal, with respect to the partial order of inclusion, in the family of graphs of maximal monotone operators from $X$ to $X^*$ (respectively, from $X^*$ to $X$ ) Define, for $T:X\\rightrightarrows X^*$ the operator $\\overline{T}:X^{**}\\rightrightarrows X^*$ which graph is given by the limits, in the weak-$*\\times $ strong topology of $X^{**}\\times X^*$ , of bounded nets of elements in the graph of $T$ .",
"A maximal monotone operator $T:X\\rightrightarrows X^*$ is of type (D) [7], if every point $(x^{**},x^*)\\in X^{**}\\times X^*$ such that $\\langle {x^*-y^*},{x^{**}-y}\\rangle \\ge 0,\\quad \\forall \\,(y,y^*)\\in \\operatorname{gra}(T),$ is contained in the graph of $\\overline{T}$ .",
"This is equivalent to the fact of $\\overline{T}:X^{**}\\rightrightarrows X^*$ being maximal monotone."
],
[
"A self-canceling non-type (D) maximal monotone linear isometry",
"The next theorem is our main result.",
"Theorem 3.1 Let $X$ be a non-reflexive real Banach space.",
"The operator $T:X\\times X^*\\rightarrow X^{*}\\times X^{**}, \\qquad T(x,x^*)=(-x^*,x)$ is a non-type (D) maximal monotone linear isometry with infinitely many maximal monotone extensions to $X^{**}\\times X^{***}\\rightrightarrows X^*\\times X^{**}$ .",
"It follows trivially from its definition that $T$ is a linear monotone isometry.",
"Since $T$ is a continuous monotone map, it is maximal monotone.",
"Take $(p,p^*)\\in \\operatorname{gra}\\overline{T}$ .",
"This means that there exists a bounded net of elements in the graph of $T$ $\\big \\lbrace \\left((x_i,x^*_i),(y^*_i,y^{**}_i)\\right)\\big \\rbrace _{i\\in I}$ which converges in the weak-$*\\times $ strong topology to $(p,p^*)$ .",
"Using the definition of $T$ we have $x^*_i=-y^*_i,\\qquad y^{**}_i=x_i.$ Since $\\lbrace (y^*_i,y^{**}_i) \\rbrace $ converges in the norm topology, $\\lbrace (x_i,x^*_i)\\rbrace $ also converges in the strong topology and its limits belong to $X\\times X^*$ .",
"Therefore for some $x\\in X$ , $x^*\\in X^*$ , $ p=(x,x^*)\\in X\\times X^*,\\qquad p^*=(-x^*,x)$ Using again the definition of $T$ we have $(p,p^*)\\in \\operatorname{gra}(T)$ .",
"Altogether we proved that $\\operatorname{gra}(\\overline{T})=\\operatorname{gra}(T).$ Now we will prove that $\\overline{T}:X^{***}\\times X^{**}\\rightrightarrows X^*\\times X^{**}$ has infinitely many maximal monotone extensions.",
"Since $X$ is non-reflexive and a closed subspace of $X^{**}$ , there exist $x_0^{**}\\in X^{**}\\setminus X$ and $w_0\\in X^{***}$ such that $\\langle {x^{**}_0},{w_0}\\rangle =1,\\qquad w_0(x)=0,\\;\\forall x\\in X$ Define, for each $t\\in (0,\\infty )$ , $p_t=(t x^{**}_0,(1/t)w_0),\\qquad q_t=(0,tx^{**}_0) .$ Take $(x,x^*)\\in X\\times X^*$ .",
"Direct calculation yields: $\\langle {(x,x^*)-p_t},{(-x^*,x)-q_t}\\rangle &=\\langle {(x-tx_0^{**},x^*-(1/t)w_0)},{(-x^*,x-tx_0^{**})}\\rangle \\\\&=\\langle {x-tx_0^{**}},{-x^*}\\rangle +\\langle {x^*-(1/t)w_0},{x-tx_0^{**}}\\rangle =1$ where the last equality follows from (REF ).",
"Therefore, the operator $A_t:X^{**}\\times X^{***}\\rightrightarrows X^{*}\\times X^{**}$ with graph $\\operatorname{gra}(A_t)=\\operatorname{gra}{T}\\cup \\lbrace (p_n,q_t)\\rbrace .$ is monotone.",
"Hence, for each $t>0$ , there exists a maximal monotone extension of $A_t$ , say $B_t$ .",
"If $t,s>0$ , $t\\ne s$ , then $\\langle {p_t-p_s},{q_t-q_s}\\rangle &= \\left\\langle {\\bigg ((t-s)x^{**}_0,(1/t-1/s)w_0\\bigg )},{\\bigg (0,(t-s)x_0^{**}\\bigg )}\\right\\rangle \\\\&=(t-s)(1/t-1/s) =-\\frac{(t-s)^2}{ts}<0$ which prove that $(p_t,q_t)\\in B_t\\setminus B_s$ and, in particular $B_t\\ne B_s$ .",
"Observe that if $X$ is a James's space, then $X\\times X^*$ is a non-reflexive Banach spaces which is isometric to its dual.",
"Indeed, if $A:X\\rightarrow X^{**}$ is an isometry, then $\\mathbb {A}:X\\times X^*\\rightarrow X^*\\times X^{**},\\qquad \\mathbb {A}(x,x^*)=(x^*, A(x))$ is such an isometry."
]
] | 1204.1090 |
[
[
"Fixed-b Subsampling and Block Bootstrap: Improved Confidence Sets Based\n on P-value Calibration"
],
[
"Abstract Subsampling and block-based bootstrap methods have been used in a wide range of inference problems for time series.",
"To accommodate the dependence, these resampling methods involve a bandwidth parameter, such as subsampling window width and block size in the block-based bootstrap.",
"In empirical work, using different bandwidth parameters could lead to different inference results, but the traditional first order asymptotic theory does not capture the choice of the bandwidth.",
"In this article, we propose to adopt the fixed-b approach, as advocated by Kiefer and Vogelsang (2005) in the heteroscedasticity-autocorrelation robust testing context, to account for the influence of the bandwidth on the inference.",
"Under the fixed-b asymptotic framework, we derive the asymptotic null distribution of the p-values for subsampling and the moving block bootstrap, and further propose a calibration of the traditional small-b based confidence intervals (regions, bands) and tests.",
"Our treatment is fairly general as it includes both finite dimensional parameters and infinite dimensional parameters, such as marginal distribution function and normalized spectral distribution function.",
"Simulation results show that the fixed-b approach is more accurate than the traditional small-b approach in terms of approximating the finite sample distribution, and that the calibrated confidence sets tend to have smaller coverage errors than the uncalibrated counterparts."
],
[
"Introduction",
"Subsampling and block-based bootstrap methods have been widely used in inference problems for time series; see Politis et al.",
"(1999a) and Lahiri (2003) for book-length treatments of these important resampling methods.",
"To accommodate the unknown time series dependence nonparametrically, these methods introduce a bandwidth parameter $l_n$ , such as the block size in the block-based bootstrap and the subsampling window width in subsampling.",
"The bandwidth $l_n$ plays an important role in the finite sample performance of subsampling or block bootstrap based inference.",
"Intuitively, if the bandwidth (or block size) is too small, it may not capture the dependence in a time series sufficiently, whereas if it is too large, the number of blocks for subsampling/resampling is too small to lead to a good approximation of finite sample distribution.",
"Statistically speaking, the bandwidth $l_n$ is a smoothing parameter as it usually leads to a bias-variance tradeoff in variance estimation or size-power tradeoff in testing on the basis of subsampling and block bootstrap.",
"In the traditional asymptotic theory, $l_n$ goes to infinity as sample size $n$ goes to infinity and the fraction $b=l_n/n$ goes to zero, which is a necessary condition for the general consistency of subsampling and block-based bootstrap methods without additional assumptions.",
"Therefore, the role of $l_n$ (or $b$ ) does not show up in the conventional first order asymptotics, although in practice the choice of $l_n$ does affect the subsampling/block bootstrap distribution estimator and related operating characteristics.",
"In this paper, we aim to offer a new perspective on the use of these smoothing parameter dependent resampling methods based on the so-called fixed-$b$ approach, which was first proposed by Kiefer and Vogelsang (2005) in the context of heteroscedasticity-autocorrelation robust (HAR) testing.",
"It was found that the asymptotic distribution obtained under the fixed-$b$ framework (i.e.",
"$b\\in (0,1]$ is held fixed in the asymptotics) provides a better approximation to the sampling distribution of the studentized test statistic than its counterpart obtained under the small-$b$ framework (i.e., $b\\rightarrow 0$ as $n\\rightarrow \\infty $ ).",
"See Jansson (2004) and Sun et al.",
"(2008) for rigorous theoretical justifications.",
"The fixed-$b$ approach has the advantage of accounting for the effect of the bandwidth, as different bandwidth parameters correspond to different limiting (null) distributions.",
"The literature on inference using the fixed-$b$ approach and its variants has been growing steadily; see Hashimzade and Vogelsang (2008), Sun et al.",
"(2008), Shao (2010a), Goncalves and Vogelsang (2011), and Sayginsoy and Vogelsang (2011) among others for recent contributions.",
"In this paper, we adopt Kiefer and Vogelsang's fixed-$b$ approach and investigate its possible gain in the context of subsampling [Politis and Romano (1994)] and the moving block bootstrap [Künsch (1989) and Liu and Singh (1992)].",
"The extension to other bandwidth-dependent bootstrap methods, such as the tapered block bootstrap [Paparoditis and Politis (2001, 2002), Shao (2010b)] and the dependent wild bootstrap [Shao (2010c)] are possible but are not pursued here.",
"Under the fixed-$b$ asymptotics, Lahiri (2001) showed that the subsampling and the moving block bootstrap approximations are no longer consistent in the case of sample mean, which seems to suggest that a direct application of the fixed-$b$ approach is fruitless.",
"A novel feature of our extension is that we study the limiting null distribution of the p-value, which is $U(0,1)$ (i.e., uniform distribution on $[0,1]$ ) under the small-$b$ asymptotics, but is dependent upon $b$ and differs from $U(0,1)$ under the fixed-$b$ asymptotics.",
"For a scalar parameter, we calibrate the nominal coverage level on the basis of the pivotal limiting null distribution of the p-value under the fixed-$b$ framework, and modify the small-$b$ based confidence interval by inverting the corresponding test.",
"Thus the impact of the bandwidth parameter $l_n$ on the subsampling/block bootstrap distribution approximation is captured to the first order using a p-value based adjustment.",
"Simulation studies are conducted to demonstrate that the fixed-$b$ approach delivers confidence intervals of better coverage in most situations and that the fixed-$b$ based intervals are slightly wider than the small-$b$ counterparts, consistent with early findings associated with the fixed-b approach; see e.g.",
"Kiefer and Vogelsang (2005).",
"So far the use of the fixed-$b$ approach has been restricted to the inference of a finite dimensional parameter.",
"Since the subsampling and moving block bootstrap have also been used in the inference of infinite dimensional parameters, such as marginal distribution function and (normalized) spectral distribution function of a stationary time series, we explore an extension of the fixed-$b$ idea to construct confidence bands for these infinite dimensional parameters.",
"Unlike the case of a scalar parameter, the limiting null distribution of the subsampling-based p-value is not pivotal under the fixed-$b$ asymptotics and it depends on the unknown dependence structure of the underlying process.",
"To alleviate the problem, we apply the subsampling method to approximate the sampling distribution of the p-value so inference becomes feasible.",
"This double subsampling approach is also used in constructing the confidence region for a vector parameter.",
"The remainder of the article is organized as follows.",
"Section introduces an extension of the fixed-$b$ approach to subsampling and the moving block bootstrap in the mean case.",
"Section lays out a general framework and describes the fixed-$b$ based confidence interval (region) for a finite dimensional parameter and some implementational issues.",
"Section presents an extension of the fixed-$b$ approach to confidence band construction for the marginal distribution function and normalized spectral distribution function.",
"Simulation results are reported in Section .",
"Section concludes and Section contains some technical details."
],
[
"Inference for the mean",
"To help the reader understand the essence of the fixed-$b$ approach, we shall focus on the simple problem: inference for the mean of a stationary time series.",
"Suppose we want to test $H_0: \\mu =\\mu _0$ versus $H_1:\\mu \\ne \\mu _0$ based on the observations $\\lbrace X_t\\rbrace _{t=1}^{n}$ from a univariate stationary time series with $\\mathbb {E}(X_t)=\\mu $ .",
"Under suitable moment and weak dependence conditions, we have $\\sqrt{n}(\\bar{X}_n-\\mu )\\rightarrow _{D} N(0,\\sigma ^2)$ where $\\sigma ^2=\\sum _{k\\in \\mathbb {Z}}\\gamma (k)$ is the long run variance with $\\gamma (k)={\\mbox{cov}}(X_0,X_k)$ and “$\\rightarrow _{D}$ ” denotes convergence in distribution.",
"The scale parameter $\\sigma ^2$ can be consistently estimated by the so-called lag window estimator $\\hat{\\sigma }_n^2=\\sum _{j=1-n}^{n-1}K(j/l)\\hat{\\gamma }_n(j)$ , where $l=l_n$ is a bandwidth parameter, $K(\\cdot )$ is a kernel function and $\\hat{\\gamma }_n(j)=n^{-1}\\sum _{k=|j|+1}^{n}(X_k-\\bar{X}_n)(X_{k-|j|}-\\bar{X}_n)$ is the sample autocovariance at lag $j$ .",
"A natural test statistic is $G_n=n(\\bar{X}_n-\\mu _0)^2/\\hat{\\sigma }_n^2$ .",
"To ensure the consistency of $\\hat{\\sigma }_n^2$ as an estimator of $\\sigma ^2$ , the bandwidth parameter $l=bn$ , where $b\\in (0,1]$ , typically satisfies $1/l+l/n=o(1)$ (i.e., $b + n^{-1}/b =o(1)$ ) as $n\\rightarrow \\infty $ .",
"This is the so-called small-$b$ asymptotics, under which the limiting null distribution of $G_n$ is the distribution of $\\chi _1^2$ .",
"Under the fixed-$b$ asymptotics, the ratio of bandwidth to sample size $b$ is held fixed and $G_n$ converges in distribution (under the null) to $U(b)$ , whose detailed form can be found in Kiefer and Vogelsang (2005).",
"The distribution of $U(b)$ depends on the kernel $K$ and $b$ , so different choices of the kernel and bandwidth lead to different limiting null distributions.",
"From both empirical and theoretical perspectives, the fixed-$b$ approach has been shown to provide a more accurate approximation to the finite sample distribution of $G_n$ than the small-$b$ counterpart under the null, so it corresponds to better size in hypothesis testing.",
"Owing to the duality between confidence interval construction and hypothesis testing, the interval delivered by the fixed-$b$ approach tends to have an empirical coverage closer to the nominal one.",
"In the next two subsections, we describe an extension of the fixed-$b$ approach to subsampling and the moving block bootstrap for the mean inference problem.",
"A further extension to the inference of a finite dimensional parameter is made in Section .",
"Throughout, we use $\\lfloor a\\rfloor $ to denote the integer part of $a\\in \\mathbb {R}$ and $\\lceil a\\rceil $ to denote the smallest integer larger than or equal to $a$ .",
"The symbol $N(\\mu ,\\Sigma )$ denotes the normal distribution with mean $\\mu $ and covariance matrix $\\Sigma $ ."
],
[
"Subsampling",
"For the inference of the mean, the subsampling method approximates the sampling distribution of $\\sqrt{n}(\\bar{X}_n-\\mu )$ with the empirical distribution generated by its subsample counterpart $\\sqrt{l}(\\bar{X}_{j,j+l-1}-\\bar{X}_n)$ , where $\\bar{X}_{j,j+l-1}=l^{-1}\\sum _{i=j}^{j+l-1}X_i$ , $j=1,\\cdots ,N=n-l+1$ .",
"Let $L_{n,l}(x)=N^{-1}\\sum _{j=1}^{N}{\\bf 1}\\lbrace \\sqrt{l}(\\bar{X}_{j,j+l-1}-\\bar{X}_n)\\le x\\rbrace $ be the subsampling approximation, where ${\\bf 1}(A)$ denotes the indicator function of the set $A$ .",
"For a given $\\alpha \\in (0,1)$ (say, $\\alpha =0.05$ or $0.1$ ), we define the subsampling-based critical values as $c_{n,l}(1-\\alpha )=\\inf \\lbrace x: L_{n,l}(x)\\ge 1-\\alpha \\rbrace $ .",
"Then a $100(1-\\alpha )\\%$ (one-sided) confidence interval is $[\\bar{X}_n-n^{-1/2}c_{n,l}(1-\\alpha ),\\infty )$ and the $100(1-\\alpha )\\%$ (two-sided) equal-tailed confidence interval is $[\\bar{X}_n-n^{-1/2}c_{n,l}(1-\\alpha /2),\\bar{X}_n-n^{-1/2}c_{n,l}(\\alpha /2)]$ .",
"In the testing context, if the alternative hypothesis is $H_1: \\mu >\\mu _0$ , then we reject the null hypothesis at the significance level $\\alpha $ if $\\mu _0\\notin [\\bar{X}_n-n^{-1/2}c_{n,l}(1-\\alpha ),\\infty )$ ; if the alternative hypothesis is $H_1:\\mu <\\mu _0$ , then the null is rejected provided that $\\mu _0\\notin (-\\infty ,\\bar{X}_n-n^{-1/2}c_{n,l}(\\alpha )]$ , which is also a one-sided confidence interval with nominal level $(1-\\alpha )$ ; If the alternative hypothesis is $H_1:\\mu \\ne \\mu _0$ , then the null hypothesis is rejected when $\\mu _0\\notin [\\bar{X}_n-n^{-1/2}c_{n,l}(1-\\alpha /2),\\bar{X}_n-n^{-1/2}c_{n,l}(\\alpha /2)]$ .",
"The above inference is based on the traditional small-$b$ based asymptotic theory, under which $\\sup _{x\\in \\mathbb {R}} | L_{n,l}(x)-\\Phi (x/\\sigma )|= o_p(1)$ , where $\\Phi $ is the distribution function for the standard normal distribution, and $\\sup _{x\\in \\mathbb {R}} |L_{n,l}(x)-P\\lbrace \\sqrt{n}(\\bar{X}_n-\\mu )\\le x\\rbrace |=o_p(1)$ , i.e., the subsampling method provides a consistent approximation to the sampling distribution of $\\sqrt{n}(\\bar{X}_n-\\mu )$ and its limiting distribution.",
"Note that we are using the data-centered subsampling distribution for testing as recommended by Berg et al.",
"(2010).",
"Under the fixed-$b$ framework, $L_{n,l}$ does not converge to $N(0,\\sigma ^2)$ in distribution.",
"Instead, Lahiri (2001) showed that the limit of $L_{n,l}(x)$ is $ (1-b)^{-1}\\int _0^{1-b}{\\bf 1}[\\lbrace W(b+t)-W(t)-bW(1)\\rbrace \\sigma /\\sqrt{b}\\in (-\\infty ,x]] dt,$ where $W(t)$ is a standard Brownian motion.",
"Since $L_{n,l}$ converges to a random measure, the subsampling-based inference is asymptotically invalid under the fixed-$b$ framework.",
"To alleviate the problem, we shall modify the traditional subsampling-based inference procedure by considering the subsampling-based p-value and its limiting null distribution.",
"For the one-sided alternative hypothesis $H_1:\\mu >\\mu _0$ , we define the p-value as $pval_{n,l}^{\\mbox{SUB}}:=\\frac{1}{N}\\sum _{j=1}^{N} {\\bf 1}\\lbrace \\sqrt{n}(\\bar{X}_n-\\mu _0)\\le \\sqrt{l}(\\bar{X}_{j,j+l-1}-\\bar{X}_n)\\rbrace .$ Under the small-$b$ asymptotics, it can be shown that $pval_{n,l}^{\\mbox{SUB}}$ converges to $U[0,1]$ in distribution under the null, whereas under the fixed-b asymptotics, its limiting null distribution is the distribution of $G(b)$ , where $G(b)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}[W(1)\\le \\lbrace W(b+t)-W(t)-bW(1)\\rbrace /\\sqrt{b}] dt.$ Note that the nuisance parameter $\\sigma $ is canceled out in $G(b)$ , which is pivotal for a given $b$ .",
"Let $G_{\\alpha }(b)$ denote the $100\\alpha \\%$ quantile of the distribution $G(b)$ .",
"Then at the significance level $\\alpha $ , we reject the null and favor the alternative $H_1:\\mu >\\mu _0$ , if the (realized) p-value is smaller than $G_{\\alpha }(b)$ .",
"Correspondingly, a one-sided confidence interval under the fixed-$b$ asymptotics can be obtained by inverting the test, i.e., $\\lbrace \\mu : \\frac{1}{N}\\sum _{j=1}^{N} {\\bf 1}\\lbrace \\sqrt{n}(\\bar{X}_n-\\mu )\\le \\sqrt{l}(\\bar{X}_{j,j+l-1}-\\bar{X}_n)\\rbrace \\ge G_{\\alpha }(b)\\rbrace $ , which is $\\lbrace \\mu : \\sqrt{n}(\\bar{X}_n-\\mu )\\le c_{n,l}(1-G_{\\alpha }(b))\\rbrace =[\\bar{X}_n-n^{-1/2}c_{n,l}(1-G_{\\alpha }(b)),\\infty ).$ Compared to the conventional subsampling-based confidence interval, the difference lies in the replacement of $\\alpha $ by $G_{\\alpha }(b)$ in $c_{n,l}$ .",
"Note that $\\alpha $ is the $100\\alpha \\%$ quantile of $U(0,1)$ , which is the limiting null distribution of the p-value under the small-$b$ asymptotics.",
"In a similar manner, we can obtain the $100(1-\\alpha )\\%$ two sided equal-tailed confidence interval for $\\mu $ as $[\\bar{X}_n-n^{-1/2}c_{n,l}(1-G_{\\alpha /2}(b)),\\bar{X}_n-n^{-1/2}c_{n,l}(G_{\\alpha /2}(b))]$ and another one-sided confidence interval $(-\\infty ,\\bar{X}_n-n^{-1/2}c_{n,l}(G_{\\alpha }(b))]$ under the fixed-$b$ asymptotics.",
"One can view the fixed-$b$ based inference as a way of calibrating the small-$b$ counterpart with the level $\\alpha $ adjusted by $G_{\\alpha }(b)$ , so the effect of $b$ on the inference is taken into account.",
"If one wants to construct a symmetric two sided confidence interval for $\\mu $ , then one can approximate the sampling distribution of $\\sqrt{n}|\\bar{X}_n-\\mu |$ by $\\widetilde{L}_{n,l}(x)=N^{-1}\\sum _{j=1}^{N}{\\bf 1}(\\sqrt{l}|\\bar{X}_{j,j+l-1}-\\bar{X}_n|\\le x)$ .",
"Letting $\\widetilde{c}_{n,l}(1-\\alpha )=\\inf \\lbrace x: \\widetilde{L}_{n,l}(x)\\ge 1-\\alpha \\rbrace $ , then the $100(1-\\alpha )\\%$ symmetric confidence interval for $\\mu $ is $[\\bar{X}_n-n^{-1/2}\\widetilde{c}_{n,l}(1-\\alpha ),\\bar{X}_n+n^{-1/2}\\widetilde{c}_{n,l}(1-\\alpha )]$ under the small-$b$ asymptotic theory.",
"The p-value is defined as $\\widetilde{pval}_{n,l}^{SUB}=\\frac{1}{N}\\sum _{j=1}^{N} {\\bf 1}\\lbrace \\sqrt{n}|\\bar{X}_n-\\mu _0|\\le \\sqrt{l}|\\bar{X}_{j,j+l-1}-\\bar{X}_n|\\rbrace .$ Under the fixed-$b$ asymptotics, the limiting null distribution of $\\widetilde{pval}_{n,l}^{SUB}$ is the distribution of $\\widetilde{G}(b)$ , where $\\widetilde{G}(b)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\lbrace |W(1)|\\le |W(b+t)-W(t)-bW(1)|/\\sqrt{b}\\rbrace dt.$ Let $\\widetilde{G}_{\\alpha }(b)$ denote the $100\\alpha \\%$ quantile of the distribution $\\widetilde{G}(b)$ .",
"Then the fixed-$b$ based $100(1-\\alpha )\\%$ symmetric confidence interval is $\\lbrace \\mu : N^{-1}\\sum _{j=1}^{N}{\\bf 1}(\\sqrt{n}|\\bar{X}_n-\\mu |\\le \\sqrt{l}|\\bar{X}_{j,j+l-1}-\\bar{X}_n|) \\ge \\widetilde{G}_{\\alpha }(b)\\rbrace $ , i.e., $[\\bar{X}_n-n^{-1/2}\\widetilde{c}_{n,l}(1-\\widetilde{G}_{\\alpha }(b)),\\bar{X}_n+n^{-1/2}\\widetilde{c}_{n,l}(1-\\widetilde{G}_{\\alpha }(b))].$"
],
[
"Moving block bootstrap",
"In this subsection, we shall consider the approximation of the sampling distribution of $\\sqrt{n}(\\bar{X}_n-\\mu )$ by the moving block bootstrap (MBB).",
"Denote the MBB sample by $\\lbrace X_1^*(l),\\cdots ,X_n^*(l)\\rbrace $ with the dependence on the block size $l$ being explicit.",
"Then we approximate the sampling distribution of $\\sqrt{n}(\\bar{X}_n-\\mu )$ by the conditional distribution of $\\sqrt{n}\\lbrace \\bar{X}_n^*(l)-\\bar{X}_n\\rbrace $ or $\\sqrt{n}[\\bar{X}_n^*(l)-\\mathbb {E}^*\\lbrace \\bar{X}_n^*(l)\\rbrace ]$ given the data, where $\\bar{X}_n^*(l)=n^{-1}\\sum _{t=1}^{n}X_t^*(l)$ is the sample mean for the bootstrap sample, $\\mathbb {E}^*$ and ${\\mbox{var}}^*$ are used to denote the conditional expectation and variance, respectively.",
"To avoid the issue of centering, we could use the circular bootstrap [Politis and Romano (1992)], which is asymptotically equivalent to the moving block bootstrap [Lahiri (2003)].",
"For simplicity, we shall focus on the bootstrap approximation based on $\\sqrt{n}\\lbrace \\bar{X}_n^*(l)-\\bar{X}_n\\rbrace $ .",
"The same idea can be applied to the other bootstrap approximation.",
"We define the MBB-based p-value as $pval_{n,l}^{\\mbox{MBB}}:=E^*[{\\bf 1}\\lbrace \\sqrt{n}(\\bar{X}_n-\\mu _0)\\le \\sqrt{n}(\\bar{X}_{n}^*(l)-\\bar{X}_n)\\rbrace ],$ which corresponds to the alternative $H_1:\\mu >\\mu _0$ .",
"Under the small-$b$ asymptotics, the p-value $pval_{n,l}^{\\mbox{MBB}}$ is expected to converge to $U(0,1)$ in distribution under the null, although we are unaware of a formal proof.",
"Under the fixed-$b$ asymptotics, we assume $R_b=n/l=1/b$ (i.e.",
"reciprocal of $b$ ) to be an integer for the ease of our discussion.",
"Then $\\bar{X}_{n}^*(l)=n^{-1}\\sum _{j=1}^{R_b l}X_j^*(l)=n^{-1}\\sum _{j=1}^{R_b} v_j^*$ , where, conditional on the data, $\\lbrace v_j^*\\rbrace _{j=1}^{R_b}$ are iid (independent and identically distributed) with a discrete uniform distribution, $P(v_1^*=\\sum _{t=j}^{j+l-1}X_t)=1/N$ , $j=1,\\cdots ,N$ .",
"Hence the above p-value is equal to $\\frac{1}{N^{R_b}} \\sum _{j_1,j_2,\\cdots ,j_{R_b}=1}^{N} {\\bf 1}\\left\\lbrace n^{-1/2}\\sum _{h=1}^{R_b}\\sum _{s=j_h}^{j_h+l-1} (X_s-\\mu _0)-n^{1/2}(\\bar{X}_n-\\mu _0) \\ge n^{1/2} (\\bar{X}_n-\\mu _0) \\right\\rbrace .",
"$ Under the fixed-$b$ asymptotics and the null, it converges in distribution to $H(b):=(1-b)^{-R_b} \\int _0^{1-b}\\cdots \\int _{0}^{1-b} {\\bf 1}\\left[\\sum _{h=1}^{R_b} \\lbrace W(t_h+b)-W(t_h)\\rbrace \\ge 2 W(1)\\right] dt_1\\cdots dt_{R_b}$ Let $H_{\\alpha }(b)$ denote the $100\\alpha \\%$ quantile of $H(b)$ .",
"In practice, we usually further approximate the distribution of $\\sqrt{n}\\lbrace \\bar{X}_n^*(l)-\\bar{X}_n\\rbrace $ by taking a finite number of bootstrap samples, say, $\\lbrace X_t^{*,j}(l)\\rbrace _{t=1}^{n}$ , $j=1,\\cdots ,B$ .",
"We approximate the sampling distribution of $\\sqrt{n}(\\bar{X}_n-\\mu )$ by $M_{n,l,B}^*(x)=\\frac{1}{B}\\sum _{j=1}^{B} {\\bf 1}[\\sqrt{n}\\lbrace \\bar{X}_n^{*,j}(l)-\\bar{X}_n\\rbrace \\le x]$ , where $\\bar{X}_n^{*,j}(l)=n^{-1}\\sum _{t=1}^{n}X_t^{*,j}(l)$ .",
"Let $c_{n,l,B}^*(1-\\alpha )=\\inf \\lbrace x:M_{n,l,B}^*(x)\\ge 1-\\alpha \\rbrace $ .",
"The corresponding fixed-$b$ based two sided equal tailed confidence interval for $\\mu $ is then $[\\bar{X}_n-n^{-1/2}c_{n,l,B}^*(1-H_{\\alpha /2}(b)),\\bar{X}_n-n^{-1/2}c_{n,l,B}^*(H_{\\alpha /2}(b))]$ and the one-sided confidence intervals can be formed analogous to those developed for the subsampling method.",
"The details are omitted.",
"The above discussion is based on the assumption that $R_b=1/b$ is an integer.",
"When $R_b$ is not an integer, we use a fraction of the last resampled block to make the bootstrap sample size equal to original sample size.",
"Then the p-value is $&&\\frac{1}{N^{\\lfloor R_b\\rfloor }} \\frac{1}{l\\lfloor R_b\\rfloor +1} \\sum _{j_1,j_2,\\cdots ,j_{\\lfloor R_b\\rfloor }=1}^{N} \\sum _{j_{\\lfloor R_b\\rfloor +1}=1}^{l\\lfloor R_b\\rfloor +1} {\\bf 1}\\left\\lbrace n^{-1/2}\\left(\\sum _{h=1}^{\\lfloor R_b\\rfloor }\\sum _{s=j_h}^{j_h+l-1} (X_s-\\bar{X}_n) \\right.\\right.\\\\&&\\left.\\left.+\\sum _{s=j_{\\lfloor R_b\\rfloor +1}}^{j_{\\lfloor R_b\\rfloor +1}+n-l\\lfloor R_b\\rfloor -1}(X_s-\\bar{X}_n)\\right) \\ge n^{1/2} (\\bar{X}_n-\\mu _0) \\right\\rbrace .$ and its limiting null distribution can be derived similarly.",
"Below we shall focus our discussion on the case $1/b$ is an integer for simplicity.",
"In a similar fashion, if we want to construct an MBB-based symmetric confidence interval for $\\mu $ , we consider the approximation of the sampling distribution of $\\sqrt{n}|\\bar{X}_n-\\mu |$ by the conditional distribution of $\\sqrt{n}|\\bar{X}_n^*(l)-\\bar{X}_n|$ given the data.",
"The corresponding p-value is $&&\\widetilde{pval}_{n,l}^{\\mbox{MBB}}:=E^*\\lbrace {\\bf 1}(\\sqrt{n}|\\bar{X}_n-\\mu _0|\\le \\sqrt{n}|\\bar{X}_{n}^*(l)-\\bar{X}_n|)\\rbrace \\\\&&\\hspace{14.22636pt}=\\frac{1}{N^{R_b}}\\sum _{j_1,j_2,\\cdots ,j_{R_b}=1}^{N} {\\bf 1}\\left( \\left|n^{-1/2} \\sum _{h=1}^{R_b}\\sum _{s=j_h}^{j_h+l-1} (X_s-\\mu _0)- \\sqrt{n} (\\bar{X}_n-\\mu _0)\\right|\\ge \\sqrt{n}|\\bar{X}_n-\\mu _0| \\right)$ and it converges in distribution to $\\widetilde{H}(b)=(1-b)^{-R_b} \\int _0^{1-b}\\cdots \\int _{0}^{1-b} {\\bf 1}\\left(\\left|\\sum _{h=1}^{R_b} \\lbrace W(t_h+b)-W(t_h)\\rbrace -W(1)\\right| \\ge |W(1)| \\right) dt_1\\cdots dt_{R_b}$ under the null.",
"Define $\\widetilde{M}_{n,l,B}^*(x)=\\frac{1}{B}\\sum _{j=1}^{B} {\\bf 1}\\lbrace \\sqrt{n}|\\bar{X}_n^{*,j}(l)-\\bar{X}_n|\\le x\\rbrace $ and $\\widetilde{c}_{n,l,B}^*(1-\\alpha )=\\inf \\lbrace x:\\widetilde{M}_{n,l,B}^*(x)\\ge 1-\\alpha \\rbrace $ .",
"Then the fixed-$b$ based $100(1-\\alpha )\\%$ symmetric confidence interval for $\\mu $ is $[\\bar{X}_n-n^{-1/2}\\widetilde{c}_{n,l,B}^{*}(1-\\widetilde{H}_{\\alpha }(b)),\\bar{X}_n+n^{-1/2}\\widetilde{c}_{n,l,B}^*(1-\\widetilde{H}_{\\alpha }(b))]$ ."
],
[
"Finite dimensional parameter",
"We first introduce some notation.",
"Let $D[0,1]$ be the space of functions on $[0,1]$ which are right continuous and have left limits, endowed with the Skorokhod topology (Billingsley 1968).",
"Denote by “$\\Rightarrow $ \" weak convergence in $D[0,1]$ or more generally in the $\\mathbb {R}^k$ -valued function space $D^k[0,1]$ , where $k\\in \\mathbb {N}$ .",
"Later in Section , we also use “$\\Rightarrow $ \" to denote the weak convergence in $D[0,\\pi ]$ , $D([0,1]\\times [0,\\pi ])$ and $D([-\\infty ,\\infty ]\\times [0,1])$ ."
],
[
"Subsampling",
"Following Politis et al.",
"(1999a), we assume that the parameter of interest is $\\theta (P)\\in \\mathbb {R}^k$ , where $P$ is the joint probability law that governs the $p$ -dimensional stationary sequence $\\lbrace X_t\\rbrace _{t\\in \\mathbb {Z}}$ .",
"Let $\\hat{\\theta }_n=\\hat{\\theta }_n(X_1,\\cdots ,X_n)$ be an estimator of $\\theta =\\theta (P)$ based on the observations $(X_1,\\cdots ,X_n)$ .",
"Further we define the subsampling estimator of $\\theta (P)$ by $\\hat{\\theta }_{j,j+l-1}=\\hat{\\theta }_l(X_j,\\cdots ,X_{j+l-1})$ on the basis of the subsample $(X_j,\\cdots ,X_{j+l-1})$ , $j=1,\\cdots ,N$ .",
"Let $\\Vert \\cdot \\Vert $ be a norm in $\\mathbb {R}^k$ .",
"The subsampling-based distribution estimator of $\\Vert \\sqrt{n}\\lbrace \\hat{\\theta }_n-\\theta (P)\\rbrace \\Vert $ is denoted as $\\widetilde{L}_{n,l}(x)=N^{-1}\\sum _{j=1}^{N}{\\bf 1}(\\Vert \\sqrt{l}(\\hat{\\theta }_{j,j+l-1} -\\hat{\\theta }_n)\\Vert \\le x)$ .",
"In the testing context (say $H_0:\\theta =\\theta _0$ versus $H_1:\\theta \\ne \\theta _0$ ), we define the subsampling based p-value as $\\widetilde{pval}_{n,l}^{SUB}=N^{-1}\\sum _{j=1}^{N}{\\bf 1}(\\Vert \\sqrt{n}(\\hat{\\theta }_n-\\theta )\\Vert \\le \\Vert \\sqrt{l}(\\hat{\\theta }_{j,j+l-1}-\\hat{\\theta }_n)\\Vert ),$ where we do not distinguish $\\theta $ and $\\theta _0$ for notational convenience because they are the same under the null.",
"To obtain the limiting null distribution of the p-value under the fixed-$b$ framework, we further assume $\\theta (P)=T(F)$ , where $F$ is the marginal distribution of $X_1\\in \\mathbb {R}^p$ , and $T$ is a functional that takes value in $\\mathbb {R}^k$ .",
"Then a natural estimator of $T(F)$ is $\\hat{\\theta }_n=T(\\rho _{1,n})$ , where $\\rho _{1,n}=n^{-1}\\sum _{t=1}^{n}\\delta _{X_t}$ is the empirical distribution.",
"Here $\\delta _x$ stands for the point mass at $x$ .",
"Similarly, $\\hat{\\theta }_{j,j+l-1}=T(\\rho _{j,j+l-1})$ , where $\\rho _{j,j+l-1}=l^{-1}\\sum _{h=j}^{j+l-1}\\delta _{X_h}$ .",
"Assume that there is an expansion of $T(\\rho _{1,n})$ in the neighborhood of $F$ , i.e., $T(\\rho _{1,n})=T(F)+n^{-1}\\sum _{t=1}^{n}IF(X_t; F)+R_{1,n},$ where $IF(X_t;F)$ is the influence function of $T$ (Hampel, Ronchetti, Rousseeuw and Stahel, 1986) defined by $IF(x;F)=\\lim _{\\epsilon \\downarrow 0}\\frac{T((1-\\epsilon )F+\\epsilon \\delta _x)-T(F)}{\\epsilon }$ and $R_{1,n}$ is the remainder term.",
"Similarly, $T(\\rho _{j,j+l-1})=T(F)+l^{-1}\\sum _{h=j}^{j+l-1}IF(X_h;F)+R_{j,j+l-1}$ .",
"Below are the two key assumptions we need.",
"(A.1) Assume that $\\mathbb {E}\\lbrace IF(X_j;F)\\rbrace = 0$ and $ n^{-1/2}\\sum _{j=1}^{\\lfloor nr\\rfloor } IF(X_j;F)\\Rightarrow \\Sigma (P)^{1/2} W_k(r)$ , where $\\Sigma (P)$ is a positive definite matrix and $W_k(\\cdot )$ denotes the $k$ -th dimensional vector of independent Brownian motions.",
"(A.2) Assume that $\\sqrt{n}\\Vert R_{1,n}\\Vert =o_p(1)$ and $\\sqrt{l}\\sup _{j=1,\\cdots ,N}\\Vert R_{j,j+l-1}\\Vert =o_p(1)$ .",
"Note that (A.1) is Assumption 1 in Shao (2010a) and its verification has been discussed in Remark 1 therein.",
"The assumption (A.2) is to ensure the negligibility of remainder terms.",
"In the sample mean case, $IF(X_t;F)=(X_t-\\mu )$ and the remainder terms vanish, so (A.2) is automatically satisfied and (A.1) reduces to a functional central limit theorem for the partial sum process of $X_t$ .",
"Theorem 1 Suppose the assumptions (A.1) and (A.2) hold and $b\\in (0,1]$ is held fixed as $n\\rightarrow \\infty $ .",
"The limiting null distribution of $\\widetilde{pval}_{n,l}^{SUB}$ is the distribution of $\\widetilde{G}(b;k)$ , where $\\widetilde{G}(b;k)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}[\\Vert \\Sigma (P)^{1/2} W_k(1)\\Vert \\le \\Vert \\Sigma (P)^{1/2}\\lbrace W_k(b+t)-W_k(t)-bW_k(1)\\rbrace \\Vert /\\sqrt{b}]dt.$ In the special case $k=1$ , $\\widetilde{G}(b;1)=\\widetilde{G}(b)$ .",
"Thus for a scalar parameter, the limiting null distribution of the p-value is pivotal for a given $b$ and the $100(1-\\alpha )\\%$ symmetric confidence interval for $\\theta $ is $ [\\hat{\\theta }_n-n^{-1/2}\\widetilde{c}_{n,l}(1-\\widetilde{G}_{\\alpha }(b)),\\hat{\\theta }_n+n^{-1/2}\\widetilde{c}_{n,l}(1-\\widetilde{G}_{\\alpha }(b))],$ which reduces to (REF ) in the mean case.",
"To conduct the inference for the case $k=1$ , we need to know $G_{\\alpha }(b)$ , $\\widetilde{G}_{\\alpha }(b)$ , $H_{\\alpha }(b)$ and $\\widetilde{H}_{\\alpha }(b)$ .",
"Following the practice of Kiefer and Vogelsang (2005), we first generate the simulated values for $\\alpha =0.05,0.1$ and $b=0.01,0.02,\\cdots ,0.2$ , then fit the quadratic equation $cv(b)=a_0+a_1b + a_2b^2$ to the simulated values by ordinary least squares.",
"The intercept $a_0$ was set to be equal to $\\alpha $ , so that $cv(0)=\\alpha $ .",
"Table REF reports the estimated coefficients and $R^2$ from the regressions (ranging from 0.9584 to 0.9997), which suggests quite satisfactory fits.",
"For $H_{\\alpha }(b)$ and $\\widetilde{H}_{\\alpha }(b)$ , fitting higher order polynomials does not lead to substantial higher $R^2$ .",
"To simulate $G_{\\alpha }(b)$ and $\\widetilde{G}_{\\alpha }(b)$ for a given $\\alpha $ and $b\\in (0,0.2)$ , we generate 5000 iid $N(0,1)$ random variables, and use its normalized partial sum to approximate the standard Brownian motion.",
"For $H_{\\alpha }(b)$ and $\\widetilde{H}_{\\alpha }(b)$ , we approximate $\\mathbb {E}^*$ in the definition of p-value by performing bootstrap 50000 times.",
"50000 monte carlo replications were used for all the cases.",
"For small $\\alpha $ (say $\\alpha =0.01$ ) and relatively large $b$ , say $b=0.15, \\cdots , 0.2$ , our simulated critical values are mostly zero, so we are unable to provide a fitted quadratic equation when $\\alpha $ is very small.",
"Nevertheless, if the goal is to construct a $90\\%$ or $95\\%$ symmetric confidence interval, or a $90\\%$ equal-tailed confidence interval, or a one-sided confidence interval of nominal coverage $90\\%$ or $95\\%$ , Table REF is useful when $b\\in (0,0.2]$ .",
"Please insert Table REF about here!",
"For a vector parameter (i.e.",
"$k\\ge 2$ ), the limiting null distribution of the p-value depends on the unknown long run variance matrix, so is not pivotal in general.",
"One way out is to approximate the limiting null distribution by subsampling.",
"Denote by $n^{\\prime }$ the subsampling width at the second stage.",
"Let $l^{\\prime }=\\lceil n^{\\prime }b\\rceil $ and $N^{\\prime }=n^{\\prime }-l^{\\prime }+1$ .",
"For each subsample $\\lbrace X_t,\\cdots ,X_{t+n^{\\prime }-1}\\rbrace $ , we define the subsampling counterpart of $\\widetilde{pval}_{n,l}^{SUB}$ as $q_{n^{\\prime },t}=(N^{\\prime })^{-1} \\sum _{j=t}^{t+N^{\\prime }-1} {\\bf 1}\\left\\lbrace \\Vert \\sqrt{l^{\\prime }}(\\hat{\\theta }_{j,j+l^{\\prime }-1}-\\hat{\\theta }_{t,t+n^{\\prime }-1})\\Vert \\ge \\Vert \\sqrt{n^{\\prime }}(\\hat{\\theta }_{t,t+n^{\\prime }-1}-\\hat{\\theta }_n)\\Vert \\right\\rbrace $ for $t=1,\\cdots ,n-n^{\\prime }+1$ .",
"Denote the empirical distribution function of $\\lbrace q_{n^{\\prime },t}\\rbrace _{t=1}^{n-n^{\\prime }+1}$ by $Q_{n,n^{\\prime }}(x)=(n-n^{\\prime }+1)^{-1}\\sum _{j=1}^{n-n^{\\prime }+1}{\\bf 1}(q_{n^{\\prime },j}\\le x)$ , which can be used to approximate the sampling distribution or the limiting null distribution of $\\widetilde{pval}_{n,l}^{SUB}$ .",
"Let $c_{n,n^{\\prime },l}(1-\\alpha )=\\inf \\lbrace x: Q_{n,n^{\\prime }}(x)\\ge 1-\\alpha \\rbrace $ .",
"Then the calibrated $100(1-\\alpha )\\%$ subsampling-based confidence region for $\\theta $ is $\\lbrace \\theta \\in \\mathbb {R}^k: ~\\widetilde{pva}_{n,l}^{SUB}~\\mbox{in}~(\\ref {eq:pvaluesub})~\\ge c_{n,n^{\\prime },l}(\\alpha )\\rbrace ,$ whereas the traditional subsampling-based confidence region is $\\lbrace \\theta \\in \\mathbb {R}^k: ~\\widetilde{pva}_{n,l}^{SUB}~\\mbox{in}~(\\ref {eq:pvaluesub})~\\ge \\alpha \\rbrace $ .",
"Theorem 2 Assume that $1/n^{\\prime }+n^{\\prime }/n=o(1)$ and $b\\in (0,1]$ is fixed.",
"Suppose that the process $X_t$ is $\\alpha $ -mixing and $\\widetilde{G}(b;k)$ is a continuous random variable.",
"Then we have $\\sup _{x\\in \\mathbb {R}} |Q_{n,n^{\\prime }}(x)-P(\\widetilde{G}(b;k) \\le x)|=o_p(1).$ Consequently, the asymptotic coverage probability of confidence region in (REF ) is $(1-\\alpha )$ .",
"Remark 1 As we have done subsampling twice, this procedure is naturally called double subsampling in the spirit of double bootstrap.",
"The use of subsampling at the second stage is mainly to approximate the sampling distribution or the limiting null distribution of the p-value, which is unknown under the fixed-$b$ asymptotic framework.",
"Of course, the approximation error depends on the subsampling window size $n^{\\prime }$ at the second stage.",
"If we view $n^{\\prime }/n$ as a fixed constant in the above asymptotics, then the asymptotic coverage of the calibrated confidence region is still different from the nominal level.",
"One can perform further calibration by subsampling, which leads to iterative subsampling, similar to iterative bootstrap in Beran (1987, 1988).",
"In practice, however, the selection of the subsampling window size at each stage usually involves quite expensive computation, and the (finite sample) improvement in coverage errors is not guaranteed by doing subsampling iteratively.",
"As pointed out by a referee, a possible alternative approach is to simulate the asymptotic null distribution of the p-value, i.e.",
"the distribution of $\\widetilde{G}(b;k)$ after plugging in a consistent estimator of long run variance matrix.",
"Note that in general consistent estimation of long run variance matrix also involves the bandwidth selection; see e.g.",
"Politis (2011).",
"Since the above-mentioned double subsampling approach is also applicable to the infinitely dimensional case [see Section ], we shall not pursue this alternative approach."
],
[
"Moving block bootstrap",
"For the moving block bootstrap, we approximate the sampling distribution of $\\Vert \\sqrt{n}\\lbrace \\hat{\\theta }_n-\\theta \\rbrace \\Vert $ by the conditional distribution of $\\sqrt{n}(\\hat{\\theta }_n^*-\\hat{\\theta }_n)$ , where $\\hat{\\theta }_n^*=\\hat{\\theta }_n\\lbrace X_1^*(l),\\cdots ,X_n^*(l)\\rbrace $ .",
"Define the p-value as $\\widetilde{pval}_{n,l}^{MBB}:= \\mathbb {E}^*\\lbrace {\\bf 1}(\\Vert \\sqrt{n}(\\hat{\\theta }_n-\\theta _0)\\Vert \\le \\Vert \\sqrt{n}(\\hat{\\theta }_n^*-\\hat{\\theta }_n)\\Vert )\\rbrace $ .",
"It can be expected that under certain regularity conditions, the limiting null distribution of $\\widetilde{pval}_{n,l}^{MBB}$ is the distribution of $&&\\widetilde{H}(b;k)=\\frac{1}{(1-b)^{R_b}} \\int _0^{1-b}\\cdots \\int _{0}^{1-b} {\\bf 1}\\left[\\left\\Vert \\Sigma (P)^{1/2}\\left[\\sum _{h=1}^{R_b} \\lbrace W_k(t_h+b)-W_k(t_h)\\rbrace -W_k(1)\\right]\\right\\Vert \\right.\\\\&& \\hspace{85.35826pt} \\left.",
"\\ge \\Vert \\Sigma (P)^{1/2} W_k(1)\\Vert \\right] dt_1\\cdots dt_{R_b},$ which coincides with $\\widetilde{H}(b)$ when $k=1$ .",
"When $k\\ge 2$ , the p-value is not asymptotically pivotal under the fixed-$b$ asymptotics, and its sampling distribution can be approximated by subsampling or the moving block bootstrap.",
"Since the idea is similar to the double subsampling procedure described above, we omit the details.",
"We mention in passing that Lee and Lai (2009) have recently studied the benefit of performing double block bootstrap for the smooth function model.",
"The p-value based calibration is closely related to the prepivoting method proposed by Beran (1987, 1988).",
"The p-value of a statistic is itself a statistic that has a pivotal limiting distribution or tends to be more pivotal than the original (unstudentized) statistic.",
"In Beran (1987), the limiting null distribution of the p-value was assumed to be $U(0,1)$ , and he focused on the refinement of the approximation error of sampling distribution of the p-value to $U(0,1)$ by prepivoting and iterative bootstrap.",
"His treatment is quite general but is mainly focused on the iid setting.",
"By contrast, we deal with time series with independent data as a special case and the limiting null distribution of the p-value (under the fixed-$b$ asymptotics) is not $U(0,1)$ .",
"In addition, our calibration can be applied to the inference of infinite dimensional parameters [see Section ], which is not covered by Beran (1987, 1988).",
"Another related calibration method in the bootstrap literature was proposed by Loh (1987, 1991), who calibrated confidence coefficients using a consistent estimate of actual coverage probability.",
"For a given confidence interval, its estimated coverage probability is used to alter the nominal level of the interval, and it is shown that the calibrated interval is asymptotically robust under iid assumptions and some regularity conditions.",
"Similar to Beran's work, Loh's discussion is limited to the iid setting and his calibration method seems only applicable to the inference of finite dimensional parameters.",
"For a comprehensive account of bootstrap iteration and calibration, see Hall (1992).",
"For a finite dimensional parameter, another way of making the statistic more pivotal is to do studentization using a consistent estimate of asymptotic variance of the original statistic.",
"For dependent data, this typically involves the estimation of long run variance using the lag window type estimate.",
"Although theoretically possible, consistent estimation can be difficult to carry out in practice for some statistics.",
"For example, if $k=p=1$ , $\\theta =\\mbox{median}(F)$ and $\\hat{\\theta }_n=\\mbox{median}(X_1,\\cdots ,X_n)$ , then $\\Sigma (P)=\\lbrace 4 g^2(\\theta )\\rbrace ^{-1}\\sum _{k=-\\infty }^{\\infty }{\\mbox{cov}}\\lbrace 1-2{\\bf 1}(X_0\\le \\theta ),1-2{\\bf 1}(X_k\\le \\theta )\\rbrace $ with $g(\\cdot )$ being the density function of $X_1$ .",
"Consistent estimation of $\\Sigma (P)$ involves kernel density estimation for $g(\\theta )$ and long run variance estimation for the transformed series $1-2{\\bf 1}(X_t\\le \\theta )$ , both of which involve the choice of a bandwidth parameter.",
"By contrast, subsampling and the moving block bootstrap can be used to provide consistent variance estimate, which lead to a studentized statistic, or a p-value, which is more pivotal than the original unstudentized statistic.",
"Both methods are relatively easier to implement, although they also require the user to choose the subsampling window width or block size.",
"The self-normalized approach of Shao (2010a), which uses recursive subsample estimates in its studentization, would be another good candidate when a direct consistent long run variance estimation is difficult, although there is an efficiency loss under certain loss functions."
],
[
"Infinite dimensional parameter",
"In previous sections, our discussion focuses on the inference of a finite dimensional parameter, for which a $\\sqrt{n}$ -consistent estimator exists and the asymptotic normality holds.",
"In general, the use of subsampling and block bootstrap methods are not limited to the inference for finite dimensional parameters.",
"In the time series setting, they have been used to provide an approximation of the nonpivotal limiting distribution when the parameter of interest is of infinite dimension, such as marginal distribution function and spectral distribution function of a stationary time series.",
"In what follows, we use $\\Vert F-G\\Vert _{\\infty }$ to denote $\\sup _{x\\in D}|F(x)-G(x)|$ , where $D=[-\\infty ,\\infty ]$ in Section REF and $D=[0,\\pi ]$ in Section REF ."
],
[
"Marginal distribution function",
"Consider a stationary sequence $\\lbrace X_k, k\\in \\mathbb {Z}\\rbrace $ and let $m(s)=P(X_0\\le s)$ be its marginal distribution.",
"Given the observations $\\lbrace X_t\\rbrace _{t=1}^{n}$ , the empirical process is defined as $m_n(s)=n^{-1}\\sum _{k=1}^{n}{\\bf 1}(X_k\\le s)$ .",
"More generally, we define the standardized recursive process $K_n(s,\\lfloor nt\\rfloor )=n^{-1/2}\\sum _{k=1}^{\\lfloor nt\\rfloor }\\lbrace {\\bf 1}(X_k\\le s)-m(s)\\rbrace , ~t\\in [0,1].$ Under certain regularity conditions [see Berkes et al.",
"(2009)], we have that $K_n(s,\\lfloor nt\\rfloor ) \\Rightarrow K(s,t).$ Here $K(s,t)$ , $(s,t)\\in [-\\infty ,\\infty ] \\times [0,1]$ is a two-parameter mean zero Gaussian process with ${\\mbox{cov}}\\lbrace K(s,t),K(s^{\\prime },t^{\\prime })\\rbrace = (t\\wedge t^{\\prime })\\Gamma (s,s^{\\prime }),$ where $\\Gamma (s,s^{\\prime })=\\sum _{k=-\\infty }^{\\infty }{\\mbox{cov}}\\lbrace {\\bf 1}(X_0\\le s),{\\bf 1}(X_k\\le s^{\\prime })\\rbrace $ .",
"To construct a confidence band for $m(\\cdot )$ , we note that by the continuous mapping theorem, (REF ) implies that $\\sqrt{n} \\Vert m_n-m\\Vert _{\\infty } \\rightarrow _D \\sup _{s\\in \\mathbb {R}} |K(s,1)|.$ Since $K(s,1)$ is a Gaussian process with mean zero and unknown covariance ${\\mbox{cov}}\\lbrace K(s,1),K(s^{\\prime },1)\\rbrace =\\Gamma (s,s^{\\prime })$ , direct inference of $m(\\cdot )$ is difficult.",
"To circumvent the difficulty, both the moving block bootstrap and subsampling have been proposed to approximate the limiting distribution $\\sup _{s\\in \\mathbb {R}} |K(s,1)|$ consistently; see Bühlmann (1994), Naik-Nimbalkar and Rajarshi (1994), and Politis et al.",
"(1999b).",
"Below we shall focus our discussion on the subsampling method, and a similar argument applies to the moving block bootstrap approach in view of the argument in Section REF .",
"Let $g_n(t,s)=l^{1/2}\\lbrace m_{t,t+l-1}(s)-m_n(s)\\rbrace $ , $t=1,\\cdots , N=n-l+1$ be the subsampling counterpart of $n^{1/2} \\lbrace m_n(s)-m(s)\\rbrace $ , where $m_{t,t+l-1}(s)=l^{-1}\\sum _{h=t}^{t+l-1} {\\bf 1}(X_h\\le s)$ .",
"Assuming $l/n+1/l=o(1)$ and other regularity conditions, Politis et al.",
"(1999b) showed that the subsampling approximation based on $\\lbrace g_n(t,s)\\rbrace _{t=1}^{N}$ is consistent in certain function space.",
"This implies that the sampling distribution of $\\sqrt{n}\\Vert m_n-m\\Vert _{\\infty }$ (or the distribution of $\\sup _{s\\in \\mathbb {R}} |K(s,1)|$ ) can be consistently approximated by the empirical distribution of $\\lbrace \\sqrt{l}\\Vert m_{t,t+l-1}-m_n\\Vert _{\\infty } \\rbrace _{t=1}^{N}$ .",
"The above result is obtained under the small-$b$ asymptotics.",
"To introduce our calibration method, we again start with the p-value and study its limiting null distribution under the fixed-$b$ asymptotics.",
"For notational simplicity, we do not distinguish the true marginal distribution function $m(x)$ and the hypothesized function $m_0(x)$ , because they are identical under the null hypothesis.",
"Define the p-value $pval_{n,l}^{\\mbox{E}}=N^{-1}\\sum _{j=1}^{N} {\\bf 1}\\left\\lbrace l^{1/2}\\Vert m_{j,j+l-1}-m_n\\Vert _{\\infty } \\ge \\sqrt{n}\\Vert m_n-m\\Vert _{\\infty } \\right\\rbrace .$ Let $b=l/n$ .",
"Under the fixed-$b$ asymptotics, the limiting null distribution of the p-value is the distribution of $J(b)$ , where $J(b):=(1-b)^{-1} \\int _0^{1-b} {\\bf 1} \\left\\lbrace \\sup _{s\\in \\mathbb {R}} |K(s,r+b) - K(s,r) - bK(s,1)|/\\sqrt{b} \\ge \\sup _{s\\in \\mathbb {R}} |K(s,1)|\\right\\rbrace dr.$ Note that the distribution of $J(b)$ is not pivotal for a given $b$ , because it depends upon the Gaussian process $K(s,t)$ , whose covariance structure is tied to the unknown dependence structure of $X_t$ .",
"So subsampling at the first stage is insufficient under the fixed-$b$ asymptotic framework.",
"It is worth noting that in the iid setting, the quantity $\\sqrt{n} \\Vert m_n-m\\Vert _{\\infty }$ is pivotal provided that $m$ is continuous, so the inferential difficulty is mainly caused by the presence of unknown weak dependence.",
"To make the inference feasible, we propose to approximate the sampling distribution of the p-value or its limiting null distribution by subsampling; see Section REF .",
"Let $n^{\\prime }$ be the subsampling window size at the second stage, $l^{\\prime }=\\lceil n^{\\prime }b\\rceil $ and $N^{\\prime }=n^{\\prime }-l^{\\prime }+1$ .",
"For each subsample $\\lbrace X_{t},\\cdots ,X_{t+n^{\\prime }-1}\\rbrace $ , the subsampling counterpart of $pval_{n,l}^{\\mbox{E}}$ is defined as $h_{n^{\\prime },t}=(N^{\\prime })^{-1}\\sum _{j=t}^{t+N^{\\prime }-1} {\\bf 1}\\left\\lbrace \\sqrt{l^{\\prime }}\\Vert m_{j,j+l^{\\prime }-1}-m_{t,t+n^{\\prime }-1}\\Vert _{\\infty } \\ge \\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m_n\\Vert _{\\infty } \\right\\rbrace $ for $t=1,\\cdots ,n-n^{\\prime }+1$ .",
"Then we can approximate the sampling distribution of $pval_{n,l}^{\\mbox{E}}$ or its limit null distribution $J(b)$ by the empirical distribution associated with $\\lbrace h_{n^{\\prime },t}\\rbrace _{t=1}^{n-n^{\\prime }+1}$ , denoted as $J_{n,n^{\\prime }}(x)=(n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1}{\\bf 1}(h_{n^{\\prime },t}\\le x)$ .",
"For a given $\\alpha \\in (0,1)$ , the $100(1-\\alpha )\\%$ traditional subsampling-based confidence band for $m(\\cdot )$ is $\\lbrace m:~m~\\mbox{is a distribution function and}~pval_{n,l}^{\\mbox{E}}~\\mbox{in} ~(\\ref {eq:pvalueband}) \\ge \\alpha \\rbrace ,$ and the calibrated confidence band is $\\lbrace m:~m~\\mbox{is a distribution function and}~pval_{n,l}^{\\mbox{E}}~\\mbox{in} ~(\\ref {eq:pvalueband}) \\ge \\bar{c}_{n,n^{\\prime },l}(\\alpha )\\rbrace ,$ where $\\bar{c}_{n,n^{\\prime },l}(1-\\alpha )=\\inf \\lbrace x: J_{n,n^{\\prime }}(x) \\ge 1-\\alpha \\rbrace $ .",
"The following theorem states the consistency of subsampling at the second stage, which implies that the coverage for the calibrated confidence band is asymptotically correct.",
"Let $\\widetilde{V}_{b}(r,\\epsilon ):=P\\left\\lbrace \\left|\\sup _{s\\in \\mathbb {R}} |K(s,r+b) - K(s,r) - bK(s,1)|/\\sqrt{b} - \\sup _{s\\in \\mathbb {R}} |K(s,1)|\\right| = \\epsilon \\right\\rbrace .$ Theorem 3 Assume that $1/n^{\\prime }+n^{\\prime }/n=o(1)$ , (REF ) and $b\\in (0,1]$ is fixed.",
"(a) The limiting null distribution of the p-value in (REF ) is the distribution of $J(b)$ provided that $\\widetilde{V}_b(r,0)=0$ for every $r\\in [0,1-b]$ .",
"(b) Suppose that the process $X_t$ is $\\alpha $ -mixing, $J(b)$ is a continuous random variable and $\\widetilde{V}_b(r,\\epsilon )=0$ for every $r\\in [0,1-b]$ and $\\epsilon \\ge 0$ .",
"Then we have $\\sup _{x\\in \\mathbb {R}} |J_{n,n^{\\prime }}(x)-P(J(b)\\le x)|=o_p(1).$ Consequently, the asymptotic coverage probability of confidence band in (REF ) is $(1-\\alpha )$ .",
"The conditions on $J(b)$ and $\\widetilde{V}_b(r,\\epsilon )$ are technical ones that are not easy to verify.",
"The verification seems related to the regularity of the distribution of the maximum of Gaussian processes; see Diebolt and Posse (1996), Azaïs and Wschebor (2001) and references therein.",
"We conjecture that they hold for a large class of Gaussian processes.",
"Note that our calibration is based on the subsampling based approximation to sampling distribution of the p-value, which is obtained by doing the subsampling in the first stage.",
"As the p-value is a prepivoted statistic, we are effectively combining the prepivoting idea with subsampling in the infinite dimensional parameter case, for which the usual studentizing technique in the finite dimensional parameter case does not seem to apply.",
"The idea of prepivoting (using the p-value) in the infinite dimensional parameter case seems new and quite general.",
"We can also use the moving block bootstrap in the first stage to obtain a p-value or in the second stage to approximate the sampling distribution of the p-value.",
"But the implementation of the moving block bootstrap in this setting seems very computationally demanding, especially when the block size is chosen through some data driven algorithms.",
"For this reason, we shall focus on the subsampling method in simulation studies for the infinite dimensional case."
],
[
"Spectral distribution function",
"Another infinite dimensional parameter of interest in time series analysis is the spectral distribution function $F(\\lambda )=\\int _0^{\\lambda } f(w)dw$ , $\\lambda \\in [0,\\pi ]$ , where $f(\\cdot )$ is the spectral density function of $\\lbrace X_t\\rbrace $ .",
"Let $I_n(w)=(2\\pi n)^{-1}\\left|\\sum _{t=1}^{n} (X_t-\\bar{X}_n) e^{itw}\\right|^2$ be the periodogram.",
"A commonly used estimator for $F(\\lambda )$ is $F_n(\\lambda )=\\int _0^{\\lambda }I_n(w)dw$ or its discretized version $F_n(\\lambda )=(2\\pi )/n \\sum _{0<2\\pi s/n\\le \\lambda } I_n(2\\pi s/n)$ .",
"It has been shown that the two versions are asymptotically equivalent [Dahlhaus (1985a)] and we shall use the discrete version for the computational convenience.",
"Under certain regularity conditions, we have that $\\sqrt{n}\\lbrace F_n(\\lambda )-F(\\lambda )\\rbrace \\Rightarrow G(\\lambda )$ , where $G(\\lambda )$ is a mean zero Gaussian process with covariance $C(\\lambda ,\\lambda ^{\\prime })={\\mbox{cov}}\\lbrace G(\\lambda ),G(\\lambda ^{\\prime })\\rbrace =2\\pi \\int _0^{\\lambda \\wedge \\lambda ^{\\prime }} f^2(w)dw+ 2\\pi \\int _0^{\\lambda }\\int _0^{\\lambda ^{\\prime }} f_4(w_1,-w_1,-w_2)dw_1 dw_2.$ Here $f_4(\\cdot ,\\cdot ,\\cdot )$ is the fourth order cumulant spectrum.",
"For various sets of conditions for this weak convergence to hold, see Brillinger (1975), Dahlhaus (1985b) and Anderson (1993).",
"Applying the continuous mapping theorem, we get $\\sqrt{n}\\Vert F_n-F\\Vert _{\\infty }\\rightarrow _{D} \\sup _{\\lambda \\in [0,\\pi ]} |G(\\lambda )|.$ Since the covariance of $G(\\lambda )$ depends on unknown second order and fourth order spectrum, the distribution of $\\sup _{\\lambda \\in [0,\\pi ]}|G(\\lambda )|$ is unknown and is usually difficult to estimate directly, which renders the confidence band construction for $F$ a hard task.",
"To alleviate the problem, Politis et al.",
"(1999b) proposed to apply the subsampling method to approximate the limiting distribution in (REF ) and they proved the consistency.",
"See Politis et al.",
"(1993) for some related numerical work.",
"Often in practice, the main interest is on the pattern of dependence described in terms of autocorrelations, then the normalized spectral distribution function $\\widetilde{F}(\\lambda )=F(\\lambda )/F(\\pi )$ , $\\lambda \\in [0,\\pi ]$ , which does not depend on the marginal variance of $X_t$ , is of more practical relevance.",
"Politis et al.",
"(1999b) mentioned that the subsampling method is still consistent in the approximation of the sampling distribution or the limiting distribution of $\\sqrt{n}\\Vert \\widetilde{F}_n-\\widetilde{F}\\Vert _{\\infty }$ , where $\\widetilde{F}_n(\\lambda )=F_n(\\lambda )/F_n(\\pi )$ .",
"To introduce our calibration method, we need to define the estimate of $\\widetilde{F}(\\lambda )$ based on the subsample $(X_t,\\cdots ,X_{t^{\\prime }})$ for $1\\le t<t^{\\prime }\\le n$ .",
"In particular, we define the periodogram on the basis of the subsample $\\lbrace X_t,\\cdots ,X_{t^{\\prime }}\\rbrace $ as $I_{t,t^{\\prime }}(w)=\\lbrace 2\\pi (t^{\\prime }-t+1)\\rbrace ^{-1}|\\sum _{j=t}^{t^{\\prime }}(X_j-\\bar{X}_{t,t^{\\prime }})\\exp (ijw)|^2$ , where $\\bar{X}_{t,t^{\\prime }}=(t^{\\prime }-t+1)^{-1}\\sum _{j=t}^{t^{\\prime }}X_j$ , $F_{t,t^{\\prime }}(\\lambda )=\\int _0^{\\lambda }I_{t,t^{\\prime }}(w) dw$ , and $\\widetilde{F}_{t,t^{\\prime }}(\\lambda )=F_{t,t^{\\prime }}(\\lambda )/F_{t,t^{\\prime }}(\\pi )$ .",
"The subsampling method approximates the sampling distribution of $\\sqrt{n} \\Vert \\widetilde{F}_n-\\widetilde{F}\\Vert _{\\infty }$ by the empirical distribution generated from $ \\sqrt{l}\\Vert \\widetilde{F}_{t,t+l-1}-\\widetilde{F}_{1,n}\\Vert _{\\infty }$ , $t=1,\\cdots ,N$ and the corresponding p-value is $pval_{n,l}^{\\mbox{S}}=N^{-1}\\sum _{t=1}^{N} {\\bf 1}\\left\\lbrace \\sqrt{l}\\Vert \\widetilde{F}_{t,t+l-1}-\\widetilde{F}_{1,n}\\Vert _{\\infty } \\ge \\sqrt{n}\\Vert \\widetilde{F}_n-\\widetilde{F}\\Vert _{\\infty }\\right\\rbrace .$ Under the small-b asymptotics, the limiting null distribution of $pval_{n,l}^{\\mbox{S}}$ is $U(0,1)$ , but under the fixed-$b$ asymptotics, the limiting null distribution is expected to depend on $b$ and the intricate second and fourth order dependence structure of $X_t$ .",
"The derivation of the limiting distribution of the p-value relies on the functional central limit theorem for $\\sqrt{n}\\lbrace F_{1,\\lfloor nr\\rfloor }(\\lambda )-F(\\lambda )\\rbrace $ , $(r,\\lambda )\\in [0,1]\\times [0,\\pi ]$ , which seems unavailable in the literature.",
"In view of Theorem 1 in Shao (2009), Theorem 2 in Shao (2010a), and Theorem 3.3 in Dahlhaus (1985b), we conjecture that $\\sqrt{n}\\lbrace F_{1,\\lfloor nr\\rfloor }(\\lambda )-F(\\lambda )\\rbrace \\Rightarrow H(r,\\lambda ),$ where $H(r,\\lambda )$ is a mean zero Gaussian process with covariance ${\\mbox{cov}}\\lbrace H(r,\\lambda ),H(r^{\\prime },\\lambda ^{\\prime })\\rbrace =(r\\wedge r^{\\prime }) C(\\lambda ,\\lambda ^{\\prime })$ .",
"Let $\\widetilde{H}(r,\\lambda )=\\lbrace H(r,\\lambda ) F(\\pi ) -F(\\lambda ) H(r,\\pi )\\rbrace /F^2(\\pi )$ .",
"Then (REF ) implies that $\\sqrt{n}\\lbrace \\widetilde{F}_{1,\\lfloor nr\\rfloor }(\\lambda )-\\widetilde{F}(\\lambda )\\rbrace \\Rightarrow \\widetilde{H}(r,\\lambda )$ by the continuous mapping theorem and that the limiting null distribution of the p-value is the distribution of $(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left(\\sup _{\\lambda \\in [0,\\pi ]} |\\widetilde{H}(r+b,\\lambda )-\\widetilde{H}(r,\\lambda )-b\\widetilde{H}(1,\\lambda )|/\\sqrt{b}\\ge \\sup _{\\lambda \\in [0,\\pi ]} |\\widetilde{H}(1,\\lambda )|\\right) dr,$ which is not pivotal.",
"Following the calibration idea described in Section REF , we apply the subsampling method to approximate the sampling distribution of the p-value.",
"For a given subsampling block size $n^{\\prime }$ at the second stage, let $l^{\\prime }=\\max (\\lceil n^{\\prime } b\\rceil ,2)$ since a minimum sample size of 2 is needed to estimate the spectral distribution function.",
"Let $N^{\\prime }=n^{\\prime }-l^{\\prime }+1$ and $\\widetilde{h}_{n^{\\prime },t}=(N^{\\prime })^{-1}\\sum _{j=t}^{t+N^{\\prime }-1} {\\bf 1}\\left\\lbrace \\sqrt{l^{\\prime }} \\Vert \\widetilde{F}_{j,j+l^{\\prime }-1}-\\widetilde{F}_{t,t+n^{\\prime }-1}\\Vert _{\\infty } \\ge \\sqrt{n^{\\prime }}\\Vert \\widetilde{F}_{t,t+n^{\\prime }-1}-\\widetilde{F}_n\\Vert _{\\infty } \\right\\rbrace $ for $t=1,\\cdots ,n-n^{\\prime }+1$ .",
"The traditional subsampling-based $100(1-\\alpha )\\%$ confidence band for $\\widetilde{F}(\\cdot )$ is $\\lbrace \\widetilde{F}: \\widetilde{F}~\\mbox{is a normalized spectral distribution function and}~pval_{n,l}^{\\mbox{S}}~\\mbox{in~(\\ref {eq:pvalue2})}\\ge \\alpha \\rbrace .$ In contrast, the calibrated confidence band is $\\lbrace \\widetilde{F}: \\widetilde{F}~\\mbox{is a normalized spectral distribution function and}~pval_{n,l}^{\\mbox{S}}~\\mbox{in~(\\ref {eq:pvalue2})}\\ge \\tilde{c}_{n,n^{\\prime },l}(\\alpha )\\rbrace ,$ where $\\tilde{c}_{n,n^{\\prime },l}(1-\\alpha )=\\inf \\lbrace x: (n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1} {\\bf 1}(\\widetilde{h}_{n^{\\prime },t}\\le x)\\ge 1-\\alpha \\rbrace $ .",
"If (REF ) is true, then the confidence band in (REF ) is expected to have $100(1-\\alpha )\\%$ coverage asymptotically under appropriate mixing and moment conditions and the assumptions that $b\\in (0,1]$ is held fixed and $1/n^{\\prime }+n^{\\prime }/n\\rightarrow 0$ as $n\\rightarrow \\infty $ ."
],
[
"Simulation results",
"In this section, we conduct simulation studies to evaluate the accuracy of the asymptotic approximations provided by both small-$b$ and fixed-$b$ approaches to the finite sample distribution.",
"Specifically, we examine the empirical coverage probabilities and the volumes of confidence sets to see if the fixed-$b$ approach corresponds to smaller coverage errors."
],
[
"Finite sample performance of confidence intervals",
"In this subsection, we consider a univariate stationary time series model with various types of dependence structure.",
"To be specific, we let $X_t=\\mu +u_t,~~u_t=\\rho u_{t-1}+\\epsilon _t+\\theta \\epsilon _{t-1},~\\epsilon _t\\sim iid~ N(0,1).$ We consider (i) AR(1)-$N(0,1)$ error: $(\\rho ,\\theta )=(0,0)$ , $(0.5,0)$ and $(0.8,0)$ ; (ii) MA(1)-$N(0,1)$ error: $(\\rho ,\\theta )=(0,-0.5)$ ; and their corresponding AR(1)-EXP(1) and MA(1)-EXP(1) models, where $\\epsilon _t\\sim iid~\\mbox{EXP}(1)-1$ has mean zero, unit variance but with an asymmetric distribution.",
"Following the suggestion of a referee, we also include two nonlinear time series models: Nonlinear 1, $X_t=0.6\\sin (X_{t-1})+\\epsilon _t$ , where $\\epsilon _t\\sim iid~N(0,1)$ .",
"This model was used in the simulation work of Paparoditis and Politis (2001) and Shao (2010c); Nonlinear 2 (threshold autoregressive model of order 1), $X_t=0.3X_{t-1}{\\bf 1}(X_{t-1}>0)+0.8X_{t-1}{\\bf 1}(X_{t-1}\\le 0)+\\epsilon _t$ , where $\\epsilon _t\\sim iid~N(0,1)$ .",
"Sample size $n=100$ and the number of bootstrap replications is 5000.",
"The bandwidth parameter $l$ varies from 3 to 16, i.e.",
"$b=0.03,0.04,\\cdots ,0.16$ .",
"We examine the empirical coverages and average widths of symmetric confidence intervals for $\\mu =\\mathbb {E}(X_1)$ and the $25\\%$ trimmed mean based on 10000 replications.",
"Nominal coverage is set to be $95\\%$ .",
"For the models with normally distributed errors, the results for the mean case are depicted in Figure REF , in which the left panel shows the empirical coverages and the right panel shows the corresponding ratios of average interval widths (fixed-$b$ over small-$b$ ).",
"The symbols “SS\" and “BB\" stand for subsampling and the moving block bootstrap, respectively.",
"For both subsampling and the moving block bootstrap, the undercoverage occurs and it gets more severe as the dependence strengthens.",
"The empirical coverages for the fixed-$b$ approach are closer to the nominal level than those for the small-$b$ approach, with the difference between two empirical coverages increasing as $b$ gets large.",
"On the other hand, the fixed-$b$ based interval is slightly wider than its small-$b$ counterpart, with the ratio of widths increasing with respect to $b$ in general.",
"These findings are consistent with the intuition that the larger $b$ is, the more accurate the fixed-$b$ based approximation provides relative to its small-$b$ counterpart.",
"The intervals constructed based on the moving block bootstrap have noticeably better coverage than the ones based on subsampling, especially for large $b$ .",
"For the MA(1) model with $\\theta =-0.5$ , there is an overcoverage for the fixed-$b$ based interval, which is usually slightly more conservative than the small-$b$ counterpart.",
"The overcoverage in the case of negatively correlated time series corresponds to the underrejection for Kiefer and Vogelsang's (2005) studentized statistic when using normal approximation (i.e.",
"small-$b$ approach) and $b$ is small (see Figure 1 therein), so our results are in a sense consistent with those in Kiefer and Vogelsang (2005).",
"Practically speaking, the overcoverage is less harmful to the practitioner than the undercoverage, so are less concerned in practice.",
"The results for the models with exponentially distributed errors as presented in Figure REF are very similar to the ones for normally distributed errors, suggesting that the asymmetric shape of exponentially distributed errors has little impact on the finite sample performance.",
"Figures REF and REF present the results for the trimmed mean case, which are fairly similar to the results in the mean case.",
"Additionally, the results for the nonlinear models in Figure REF resemble those for AR(1)-N(0,1) models with $\\rho =0.5$ in both the mean and the trimmed mean case, indicating that nonlinearity does not affect our results much.",
"Due to the duality of confidence interval construction and hypothesis testing, we would expect that the fixed-b approach leads to better size (i.e.",
"size closer to the nominal one) in all the models except MA(1) with $\\theta =-0.5$ , at the sacrifice of (raw) power.",
"The power loss is expected to be moderate because the ratio of fixed-$b$ based interval width over the small-$b$ based interval width is quite close to 1.",
"Overall, the simulation results demonstrate that the fixed-$b$ approach delivers more accurate inference for both subsampling and the moving block bootstrap in most situations owing to its more accurate approximation to the finite sample distribution.",
"Of course, we only show the improved accuracy of the fixed-$b$ approximation for a specific $\\alpha =0.05$ , which is also what Kiefer and Vogelsang did.",
"We also tried $\\alpha =10\\%$ and qualitatively similar results are obtained.",
"It would be interesting to provide some theoretical justifications on the order of the error rejection probability.",
"For the subsampling method, this boils down to the order of $\\sup _{\\alpha \\in [0,1]} |P(\\widetilde{pval}_{n,l}^{SUB}\\le \\alpha )-P(\\widetilde{G}(l/n)\\le \\alpha )|$ under the fixed-$b$ framework.",
"Note that under the small-$b$ framework, the error is $\\sup _{\\alpha \\in [0,1]} |P(\\widetilde{pval}_{n,l}^{SUB}\\le \\alpha )-\\alpha |,$ which is expected to be larger.",
"A formal theoretical proof seems quite challenging and is left for future research."
],
[
"Finite sample performance of confidence regions and confidence bands",
"In this subsection, we examine the coverage probabilities of confidence regions for the vector parameter of mean and median, and confidence bands for the marginal distribution function $m(\\cdot )$ and the normalized spectral distribution function $\\widetilde{F}(\\cdot )$ .",
"Let $\\lbrace X_t\\rbrace _{t=1}^{n}$ be generated from the AR$(1)$ model: $X_t=\\rho X_{t-1} + e_t$ , where $\\rho =-0.6, 0, 0.5, 0.8$ , $e_t\\sim iid~ N(0,1)$ or $\\mbox{EXP}(1)-1$ .",
"Sample size $n=200$ and number of replications is 1000.",
"We use the Euclidean norm in the confidence region construction.",
"For both confidence regions and confidence bands, we compared the following three schemes: (1) traditional subsampling-based confidence region (band); (2) calibrated subsampling-based confidence region (band) with a fixed $n^{\\prime }$ , where $n^{\\prime }=15$ for confidence region construction and $n^{\\prime }=30$ for confidence band construction; (3) calibrated subsampling-based confidence region (band) with $n^{\\prime }$ chosen in a data driven fashion.",
"Here we employ a variant of a block size selection procedure proposed in Bickel and Sakov (2008) for the $m$ out of $n$ bootstrap (also see Götze and Ra$\\breve{c}$ kauskas (2001)), which is closely related to the subsampling method.",
"The use of Bickel and Sakov's automatic bandwidth selection in the subsampling context has been explored in Jach et al.",
"(2011) recently.",
"The procedure consists of the following steps (in the confidence band case): For a predetermined interval $[K_1,K_2]$ and $g\\in (0,1)$ , we consider a sequence of $n_j$ 's of the form $n_j=\\lfloor g^{j-1} K_2\\rfloor , ~\\mbox{for}~ j=1,2,\\cdots , \\lfloor \\log (K_2/K_1)/\\lbrace -\\log (g)\\rbrace \\rfloor $ .",
"For each $n_j$ , find $J_{n,n_j}$ , where $J_{n,n_j}$ is the subsampling-based distribution estimator for the sampling distribution of the p-value.",
"Set $j_0 = \\mbox{argmin}_{j=1,\\cdots , \\lfloor \\log (K_2/K_1)/\\lbrace -\\log (g)\\rbrace \\rfloor }~\\sup _{x\\in \\mathbb {R}} |J_{n,n_j}(x) - J_{n,n_{j+1}}(x)|$ .",
"Then the optimal block size is $g^{j_0}K_2$ .",
"If the difference is minimized for a few values of $j$ , then pick the largest among them.",
"In our simulation experiment, we set $(K_1,K_2,g)=(5,40,0.75)$ for confidence region construction and $(K_1,K_2,g)=(10,60,0.75)$ for confidence band construction, which corresponds to a sequence of block lengths as $(40, 30, 22, 16, 12, 9, 7, 5)$ and $(60, 45, 33, 25,18, 14, 10)$ , respectively.",
"Figures REF and REF depict the empirical coverages and the ratios of the radii of the confidence regions over that delivered by the uncalibrated traditional subsampling-based region for the vector parameter and for the models with normally distributed errors and exponentially distributed errors, respectively.",
"The symbols “Traditional\", “Calibrated (fixed)\" and “Calibrated (data-driven)\" correspond to the schemes (1)-(3) described above.",
"As the findings for the normally distributed case and the exponentially distributed case are very close, we shall only describe the results for the normally distributed case.",
"When $\\rho =0, 0.5,0.8$ , there is an undercoverage associated with the traditional subsampling-based approach and the coverage errors increase with respect to the magnitude of dependence.",
"The improvement in coverage offered by the calibration is apparent in these cases and it holds uniformly over the range of $b$ s under examination.",
"On the other hand, the corresponding radius of the calibrated region is slightly larger than that of the uncalibrated counterpart.",
"In the case $\\rho =-0.6$ , the calibrated region performs worse compared to the traditional counterpart when $b$ is small, but still offer some improvement when $b$ is large.",
"It is not fully clear why this phenomenon occurs.",
"Nevertheless, it suggests that caution has to be exercised in the use of fixed-$b$ based calibration when the autocorrelations of the series have alternating signs.",
"Figures REF -REF have the same format as Figure REF and their right panels show the ratios of the mean band widths over that delivered by the uncalibrated traditional subsampling-based band.",
"For the marginal distribution function, there is an apparent undercoverage for the traditional subsampling-based confidence band in all cases with coverage errors increasing with respect to the magnitude of dependence (compare the plots for $\\rho =0, 0.5, 0.8$ ), especially at small $b$ s. When $\\rho =0,0.5,0.8$ and for almost all $b=0.01,\\cdots ,0.2$ , the coverages delivered by the calibrated bands based on fixed or data driven subsampling width are closer to the nominal level than the traditional counterpart.",
"When $\\rho =-0.6$ , the calibrated bands based on the fixed or data-dependent bandwidths improve the coverage when $b\\ge 0.04$ , but fails to do so when $b=0.01, 0.02, 0.03$ , suggesting that potential improvements can be made about the selection of $n^{\\prime }$ .",
"In all cases, the calibrated bands are slightly wider than the uncalibrated counterpart, but the ratios are quite close to 1.",
"The “better coverage but wider band\" phenomenon is in accordance with the “better coverage but wider interval\" finding in the scalar parameter case.",
"The two calibrated bands perform similarly in most situations, and their performance is strikingly close when $\\rho =0.8$ .",
"As seen from Figure REF , which plots the empirical coverages and ratios of mean band widths with respect to $b=0.04,\\cdots ,0.3$ for the normalized spectral distribution, the improvement of the calibration in terms of coverage error is quite substantial when $\\rho =-0.6,0.5,0.8$ .",
"In the case $\\rho =0$ , the calibrated bands are conservative when $b$ is relatively small, but again provides some improvement when $b$ is close to $0.3$ .",
"Overall the results for the normalized spectral distribution function are qualitatively similar to those for the marginal distribution function.",
"Based on the simulation results for confidence intervals reported in Section REF and for confidence regions and bands reported in this subsection, it appears that the calibration works very effectively when the series is positively dependent."
],
[
"Conclusion",
"Subsampling and block-based bootstrap methods have been shown to be widely applicable to many inference problems in time series analysis.",
"The fixed-$b$ asymptotics developed here explicitly captures the choice of bandwidth parameter in subsampling and the moving block bootstrap, and the resulting first order approximation is expected to be more accurate than that provided by the small-$b$ asymptotics.",
"As demonstrated in Section , the fixed-$b$ based calibrated confidence intervals (regions, bands) provide an unambiguous improvement over the uncalibrated counterparts in terms of coverage errors in most cases considered.",
"Our calibration method is developed by estimating the sampling distribution of the p-value, which relates to the prepivoting method by Beran (1987, 1988) and the confidence coefficient calibration method by Loh (1987).",
"However, our proposal differs from theirs in two important respects: (i) the limiting null distribution of the p-value is not (necessarily) $U(0,1)$ , which is the case for Beran (1987, 1988).",
"In our setting, a pivotal limiting distribution exists in the scalar parameter case, but not in the case of vector parameter and infinite dimensional parameter, for which the subsampling method is used to provide a good approximation; (ii) Their discussions are limited to the iid setting and inference for finite dimensional parameters.",
"In contrast, our treatment goes substantially beyond their developments by allowing for time series data and the inference of infinite dimensional parameters.",
"Coupled with the recently developed fixed-$b$ approach [Kiefer and Vogelsang (2005)] in econometrics literature, we provide a general recipe for the calibration of the traditional resampling-based inference procedures when smoothing parameters, such as window width in subsampling and block size in the moving block bootstrap are used to accommodate the dependence.",
"To conclude the paper, we provide a discussion of open problems and possible extensions.",
"(1) Our method can be used as a calibration tool for a properly chosen smoothing parameter and it is practically important to choose the smoothing parameter in a sensible way.",
"The choice of subsampling width and block size for the block-based bootstrap has been discussed in Chapter 9 of Politis et al.",
"(1999a) and Chapter 7 of Lahiri (2003).",
"It seems natural to ask if it is meaningful to consider the optimal smoothing parameter selection from a fixed-$b$ based viewpoint, as opposed to the small-$b$ based approach [see e.g.",
"Bühlmann and Künsch (1999) and Politis and White (2004)].",
"A high order expansion of the sampling distribution of the p-value under the null and alternative seems needed to tackle this issue.",
"(2) The development in this article is confined to time series, although subsampling and block based bootstrap methods have been extended to spatial settings [see Chapter 5 of Politis et al.",
"(1999a) and Chapter 12 of Lahiri (2003) and references therein].",
"An extension of the fixed-$b$ based calibration idea to spatial settings is expected to be possible but seems nontrivial for irregularly spaced spatial data.",
"(3) In addition, we impose the weak dependence throughout so the asymptotic normality or functional central limit theorem with $\\sqrt{n}$ convergence rate hold.",
"When the time series is long-range dependent, the subsampling method has been proved to be consistent in some situations [see Hall et al.",
"(1998), Nordman and Lahiri (2005)].",
"It would be interesting to extend the fixed-$b$ approach to calibrate the subsampling based inference in these settings.",
"(4) A close relative of the block-based bootstraps is the so-called sieve bootstrap [Bühlmann (1997)], which also involves a bandwidth parameter (i.e., the order of the approximating autoregressive model).",
"It is natural to ask whether it is possible to extend the fixed-$b$ approach to calibrate the sieve bootstrap based confidence sets.",
"We leave these possible extensions for future work."
],
[
"Appendix",
"Proof of Theorem REF : For the convenience of notation, let $Y_h=IF(X_h;F)$ and $\\Delta =\\Sigma (P)^{1/2}$ .",
"Further let $T_{n,j}={\\bf 1}(\\Vert \\sqrt{n}(\\hat{\\theta }_n-\\theta _0)\\Vert \\le \\Vert \\sqrt{l}(\\hat{\\theta }_{j,j+l-1}-\\hat{\\theta }_n)\\Vert )$ and $\\widetilde{T}_{n,j}={\\bf 1}[\\Vert n^{-1/2}\\sum _{j=1}^{n} Y_j \\Vert \\le \\Vert l^{-1/2} \\lbrace \\sum _{h=j}^{j+l-1} Y_h - (l/n)\\sum _{j=1}^{n} Y_j\\rbrace \\Vert ]$ .",
"Then $\\widetilde{pval}_{n,l}^{SUB}=N^{-1}\\sum _{j=1}^{N} T_{n,j}$ .",
"Let $D_n(\\epsilon )=\\lbrace \\Vert \\sqrt{n}R_{1,n}\\Vert <\\epsilon , \\sup _{j=1,\\cdots ,N} \\Vert \\sqrt{l}R_{j,j+l-1}\\Vert <\\epsilon \\rbrace $ for any $\\epsilon >0$ .",
"Then $P\\lbrace D_n(\\epsilon )\\rbrace \\rightarrow 1$ as $n\\rightarrow \\infty $ .",
"On $D_n(\\epsilon )$ , we have that $|T_{n,j}-\\widetilde{T}_{n,j}|\\le {\\bf 1}\\left[\\left|\\left\\Vert n^{-1/2}\\sum _{j=1}^{n} Y_j \\right\\Vert - \\left\\Vert l^{-1/2} \\left\\lbrace \\sum _{h=j}^{j+l-1} Y_h - (l/n)\\sum _{j=1}^{n} Y_j\\right\\rbrace \\right\\Vert \\right|\\le 2\\epsilon \\right].$ So the expression $N^{-1}\\sum _{j=1}^{N} |T_{n,j}-\\widetilde{T}_{n,j}|$ is bounded by $N^{-1}\\sum _{j=1}^{N}{\\bf 1}\\left[\\left|\\left\\Vert n^{-1/2}\\sum _{j=1}^{n} Y_j \\right\\Vert - \\left\\Vert l^{-1/2} \\left\\lbrace \\sum _{h=j}^{j+l-1} Y_h - (l/n)\\sum _{j=1}^{n} Y_j\\right\\rbrace \\right\\Vert \\right|\\le 2\\epsilon \\right],$ which, by the continuous mapping theorem, converges in distribution to $I(b,\\epsilon )$ , where $I(b,\\epsilon ):=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left[ \\left|\\Vert \\Delta W_k(1)\\Vert - \\left\\Vert \\Delta \\lbrace W_k(b+t)-W_k(t)-bW_k(1)\\rbrace /\\sqrt{b}\\right\\Vert \\right|\\le 2\\epsilon \\right]dt.$ It is not hard to see that for each $t\\in [0,1-b]$ , the integrand in $I(b,\\epsilon )\\downarrow 0$ almost surely as $\\epsilon \\downarrow 0$ , which implies that $\\lim _{\\epsilon \\downarrow 0} I(b,\\epsilon ) = 0$ almost surely by the Lebesgue dominated convergence theorem.",
"Since $N^{-1}\\sum _{j=1}^{N}\\widetilde{T}_{n,j}\\rightarrow _{D} \\widetilde{G}(b;k) $ by the continuous mapping theorem, the conclusion follows by letting $\\epsilon \\downarrow 0$ and $n\\rightarrow \\infty $ .",
"We provide a justification for the use of the continuous mapping theorem above.",
"For any $x\\in D^k[0,1]$ , define the functional $f_1(x)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left[ \\left|\\Vert x(1)\\Vert - \\left\\Vert \\lbrace x(b+t)-x(t)-b x(1)\\rbrace /\\sqrt{b}\\right\\Vert \\right|\\le 2\\epsilon \\right]dt.$ and $f_2(x)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left[ \\Vert x(1)\\Vert \\le \\left\\Vert \\lbrace x(b+t)-x(t)-b x(1)\\rbrace /\\sqrt{b}\\right\\Vert \\right]dt.$ To use the continuous mapping theorem, we need to show that both $f_1$ and $f_2$ are $\\Delta W_k(\\cdot )-$ continuous almost surely.",
"We shall focus on $f_1$ and the same argument applies to $f_2$ .",
"Define $D_{f_1}=\\lbrace x: f_1~\\mbox{ is not continuous at}~ x\\rbrace $ .",
"Then $D_{f_1}\\subset \\widetilde{D}_{f_1}=\\lbrace x: \\lambda \\lbrace t\\in [0,1-b]: \\Vert x(1)\\Vert - \\left\\Vert \\lbrace x(b+t)-x(t)-b x(1)\\rbrace /\\sqrt{b}\\right\\Vert = \\pm 2\\epsilon \\rbrace >0\\rbrace ,$ where $\\lambda $ stands for Lebesgue measure.",
"It is enough to show $P(\\Delta W_k(\\cdot )\\in \\widetilde{D}_{f_1})=0$ .",
"To this end, we note that $&&\\mathbb {E}\\int _0^{1-b} {\\bf 1}(\\Vert \\Delta W_k(1)\\Vert - \\left\\Vert \\lbrace \\Delta W_k(b+t)-\\Delta W_k(t)-b \\Delta W_k(1)\\rbrace /\\sqrt{b}\\right\\Vert = \\pm 2\\epsilon )dt\\\\&&\\hspace{28.45274pt}=\\int _0^{1-b} P(\\Vert \\Delta W_k(1)\\Vert - \\left\\Vert \\lbrace \\Delta W_k(b+t)-\\Delta W_k(t)-b \\Delta W_k(1)\\rbrace /\\sqrt{b}\\right\\Vert = \\pm 2\\epsilon )dt=0$ where we have used the fact that for each $t\\in [0,1-b]$ , $P(\\Vert \\Delta W_k(1)\\Vert - \\left\\Vert \\lbrace \\Delta W_k(b+t)-\\Delta W_k(t)-b \\Delta W_k(1)\\rbrace /\\sqrt{b}\\right\\Vert = \\pm 2\\epsilon )=0$ The fact (REF ) can be easily shown by noticing that the joint distribution of $(\\Delta W_k(1),\\lbrace \\Delta W_k(b+t)-\\Delta W_k(t)-b \\Delta W_k(1)\\rbrace /\\sqrt{b})$ is multivariate normal with a positive definite covariance matrix.",
"So $P(\\Delta W_k(\\cdot )\\in \\widetilde{D}_{f_1})=0$ holds and the use of the continuous mapping theorem is justified.",
"The proof is thus complete.",
"$\\diamondsuit $ Proof of Theorem REF : The proof is similar to that of Theorem REF , so we omit the details.",
"$\\diamondsuit $ Proof of Theorem REF : (a) The proof follows from the use of the continuous mapping theorem.",
"Here the mapping $f: D([-\\infty ,\\infty ]\\times [0,1])\\rightarrow \\mathbb {R}$ is defined as $f(x)=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left\\lbrace \\sup _{s\\in \\mathbb {R}} |x(s,r+b) - x(s,r) - bx(s,1)|/\\sqrt{b} \\ge \\sup _{s\\in \\mathbb {R}} |x(s,1)|\\right\\rbrace dr.$ Following the argument in the proof of Theorem REF , we can show that if $\\widetilde{V}_{b}(r,0)=0$ for every $r\\in [0,1-b]$ , then the mapping $f$ is $K$ -continuous, i.e., the probability that the Gaussian process $K(\\cdot ,\\cdot )$ falls into the discontinuity set of $f$ is zero.",
"This completes the proof.",
"(b) In view of the continuity assumption of $J(b)$ and the monotonicity of $J_{n,n^{\\prime }}(x)$ , it suffices to show $J_{n,n^{\\prime }}(x)=P(J(b)\\le x)+o_p(1)$ for each $x\\in \\mathbb {R}$ .",
"Let $\\widehat{h}_{n^{\\prime },t}=(N^{\\prime })^{-1}\\sum _{j=t}^{t+N^{\\prime }-1} {\\bf 1}\\left\\lbrace \\sqrt{l^{\\prime }} \\Vert m_{j,j+l^{\\prime }-1}-m_{t,t+n^{\\prime }-1}\\Vert _{\\infty } \\ge \\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m\\Vert _{\\infty } \\right\\rbrace $ for $t=1,\\cdots ,n-n^{\\prime }+1$ and $\\widehat{J}_{n,n^{\\prime }}(x)=(n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1} {\\bf 1}(\\widehat{h}_{n^{\\prime },t}\\le x)$ .",
"Note that $&&\\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m\\Vert _{\\infty } - \\sqrt{n^{\\prime }} \\Vert m_{n}-m\\Vert _{\\infty } \\le \\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m_n\\Vert _{\\infty }\\\\&&\\hspace{56.9055pt} \\le \\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m\\Vert _{\\infty } + \\sqrt{n^{\\prime }}\\Vert m_{n}-m\\Vert _{\\infty }.$ For any $\\epsilon >0$ , let $E_n(\\epsilon )=\\lbrace \\sqrt{n^{\\prime }} \\Vert m_{n}-m\\Vert _{\\infty } \\le \\epsilon \\rbrace $ and $V_{b}(r,\\epsilon ):=P\\left\\lbrace \\left|\\sup _{s\\in \\mathbb {R}} |K(s,r+b) - K(s,r) - bK(s,1)|/\\sqrt{b} - \\sup _{s\\in \\mathbb {R}} |K(s,1)|\\right|\\le \\epsilon \\right\\rbrace .$ Then $P\\lbrace E_n(\\epsilon )\\rbrace \\rightarrow 1$ as $n\\rightarrow \\infty $ .",
"On $E_n(\\epsilon )$ , we have that for each $t=1,\\cdots , n-n^{\\prime }+1$ , $|h_{n^{\\prime },t}-\\widehat{h}_{n^{\\prime },t}| \\le W_n(t;\\epsilon )$ , where $W_n(t;\\epsilon ) := (N^{\\prime })^{-1}\\sum _{j=t}^{t+N^{\\prime }-1} {\\bf 1}\\left\\lbrace \\left|\\sqrt{l^{\\prime }} \\Vert m_{j,j+l^{\\prime }-1}-m_{t,t+n^{\\prime }-1}\\Vert _{\\infty } - \\sqrt{n^{\\prime }}\\Vert m_{t,t+n^{\\prime }-1}-m\\Vert _{\\infty } \\right| \\le \\epsilon \\right\\rbrace .$ By stationarity, we have that $\\mathbb {E}|h_{n^{\\prime },t}-\\widehat{h}_{n^{\\prime },t}| {\\bf 1}\\lbrace E_n(\\epsilon )\\rbrace &\\le & (N^{\\prime })^{-1}\\sum _{j=1}^{N^{\\prime }} P\\left\\lbrace \\left| \\sqrt{l^{\\prime }} \\Vert m_{j,j+l^{\\prime }-1}-m_{1,n^{\\prime }}\\Vert _{\\infty }-\\sqrt{n^{\\prime }}\\Vert m_{1,n^{\\prime }}-m\\Vert _{\\infty } \\right| \\le \\epsilon \\right\\rbrace \\\\&\\rightarrow & L(b,\\epsilon ):=(1-b)^{-1}\\int _0^{1-b} V_b(r,\\epsilon ) dr$ The above convergence follows from Theorem 3 of Ferguson (1996) and the fact that $W_n(1;\\epsilon )\\rightarrow _{D} J(b,\\epsilon )$ , where $ J(b,\\epsilon ):=(1-b)^{-1}\\int _0^{1-b} {\\bf 1}\\left\\lbrace \\left|\\sup _{s\\in \\mathbb {R}} |K(s,r+b) - K(s,r) - bK(s,1)|/\\sqrt{b} - \\sup _{s\\in \\mathbb {R}} |K(s,1)|\\right|\\le \\epsilon \\right\\rbrace dr $ Again the continuous mapping theorem is invoked to derive the weak convergence of $W_n(1,\\epsilon )$ and following the argument in the proof of Theorem REF , its use can be justified under the assumption that $\\widetilde{V}_b(r,\\epsilon )=0$ for each $r\\in [0,1-b]$ and $\\epsilon \\ge 0$ .",
"Next it is not hard to see that $\\lim _{\\epsilon \\downarrow 0} L(b,\\epsilon )=0$ since $V_b(r,\\epsilon )\\downarrow V_b(r,0)=0$ as $\\epsilon \\downarrow 0$ for every $r\\in [0,1-b]$ .",
"Thus $\\sup _{t=1,\\cdots ,n-n^{\\prime }+1}\\mathbb {E}|h_{n^{\\prime },t}-\\widehat{h}_{n^{\\prime },t}|\\le \\mathbb {E}|h_{n^{\\prime },1}-\\widehat{h}_{n^{\\prime },1}|{\\bf 1}\\lbrace E_n(\\epsilon )\\rbrace +2P(E_n(\\epsilon )^c)\\le \\epsilon $ for large enough $n$ .",
"Furthermore, $&&\\widehat{J}_{n,n^{\\prime }}(x-\\sqrt{\\epsilon })-(n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1} {\\bf 1} \\lbrace |\\widehat{h}_{n^{\\prime },t}-h_{n^{\\prime },t}|\\ge \\sqrt{\\epsilon }\\rbrace \\le J_{n,n^{\\prime }}(x)\\\\&&\\hspace{56.9055pt}\\le \\widehat{J}_{n,n^{\\prime }}(x+\\sqrt{\\epsilon })+(n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1} {\\bf 1} \\lbrace |\\widehat{h}_{n^{\\prime },t}-h_{n^{\\prime },t}|\\ge \\sqrt{\\epsilon }\\rbrace .$ By the Markov inequality, $(n-n^{\\prime }+1)^{-1}\\sum _{t=1}^{n-n^{\\prime }+1} P\\lbrace |\\widehat{h}_{n^{\\prime },t}-h_{n^{\\prime },t}|\\ge \\sqrt{\\epsilon }\\rbrace \\le \\sqrt{\\epsilon }$ .",
"Using the same argument in the proof of Theorem 3.2.1. of Politis et al.",
"(1999a), we can show that $\\widehat{J}_{n,n^{\\prime }}(x)-P\\lbrace J(b)\\le x\\rbrace =o_p(1)$ , which follows from the stationarity and strong mixing properties of $X_t$ and the boundness of $\\lbrace \\widehat{h}_{n^{\\prime },t}\\rbrace _{t=1}^{n-n^{\\prime }+1}$ .",
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]
] | 1204.1035 |
[
[
"Commuting Pauli Hamiltonians as maps between free modules"
],
[
"Abstract We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices.",
"Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules over the translation-group algebra, so homological methods are applicable.",
"In any dimension every point-like charge appears as a vertex of a fractal operator, and can be isolated with energy barrier at most logarithmic in the separation distance.",
"For a topologically ordered system in three dimensions, there must exist a point-like nontrivial charge.",
"If the ground-state degeneracy is upper bounded by a constant independent of the system size, then the topological charges in three dimensions always appear at the end points of string operators.",
"A connection between the ground state degeneracy and the number of points on an algebraic set is discussed.",
"Tools to handle local Clifford unitary transformations are given."
],
[
"Pauli group as a vector space",
"The Pauli matrices $\\sigma _x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\quad \\sigma _y = \\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}, \\quad \\sigma _z = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}$ satisfy $\\sigma _a \\sigma _b = i \\varepsilon _{abc} \\sigma _c, \\quad \\lbrace \\sigma _a , \\sigma _b \\rbrace = 2 \\delta _{ab}.$ Thus, the Pauli matrices together with scalars $\\pm 1, \\pm i$ form a group under multiplication.",
"Given a system of qubits, the set of all possible tensor products of the Pauli matrices form a group, where the group operation is the multiplication of operators.",
"If the system is infinite, physically meaningful operators are those of finite support, i.e., acting on all but finitely many qubits by the identity.",
"We shall only consider this Pauli group of finite support, and call it simply the Pauli group.",
"An element of the Pauli group is called a Pauli operator.",
"Since any two elements of the Pauli group either commute or anti-commute, ignoring the phase factor altogether, one obtains an abelian group.",
"Moreover, since any element $O$ of the Pauli group satisfies $O^2 = \\pm I$ , An action of $\\mathbb {Z}/2\\mathbb {Z}$ on Pauli group modulo phase factors $P / \\lbrace \\pm 1, \\pm i\\rbrace $ is well-defined, by the rule $ n \\cdot O = O^n$ where $n \\in \\mathbb {Z}/2\\mathbb {Z}$ .",
"For $\\mathbb {F}_2 = \\mathbb {Z}/2\\mathbb {Z}$ being a field, $P / \\lbrace \\pm 1, \\pm i\\rbrace $ becomes a vector space over $\\mathbb {F}_2$ .",
"The group of single qubit Pauli operators up to phase factors is identified with the two dimensional $\\mathbb {F}_2$ -vector space.",
"If $\\Lambda $ is the index set of all qubits in the system, the whole Pauli group up to phase factors is the direct sum $\\bigoplus _{i \\in \\Lambda } V_i$ where $V_i$ is the vector space of the Pauli operators for the qubit at $i$ .",
"Explicitly, $I = (00), \\sigma _x = (10), \\sigma _z = (01), \\sigma _y = (11)$ .",
"A multi-qubit Pauli operator is written as a finite product of the single qubit Pauli operators, and hence is written as a binary string in which all but finitely many entries are zero.",
"A pair of entries of the binary string describes a single qubit component in the tensor product expression.",
"The multiplication of two Pauli operators corresponds to entry-wise addition of the two binary strings modulo 2.",
"The commutation relation may seem at first lost, but one can recover it by introducing a symplectic form [10].",
"Let $\\lambda _1 = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}$ be a symplectic form on the vector space $(\\mathbb {F}_2)^2$ of a single qubit Pauli operators.",
"The minus sign is not necessary for qubits, but is for qudits of prime dimensions One can easily check that the commutation relation of two Pauli matrices $O_1, O_2$ is precisely the value of this symplectic form evaluated on the pair of vectors representing $O_1$ and $O_2$ .",
"Two multi-qubit Pauli operator (anti-)commutes if and only if there are (odd)even number of pairs of the anti-commuting single qubit Pauli operators in their tensor product expression.",
"So, the two Pauli operator (anti-)commutes precisely when the value of the direct sum of symplectic form $\\bigoplus _{q \\in \\Lambda } \\lambda _1$ is (non-)zero.",
"($\\Lambda $ could be infinite but the form is well-defined since any vector representing a Pauli operator is of finite support.)",
"We shall call the value of the symplectic form the commutation value."
],
[
"Pauli space on a group",
"Let $\\Lambda $ be the index set of all qubits, and suppose now that $\\Lambda $ itself is an abelian group.",
"There is a natural action of $\\Lambda $ on the Pauli group modulo phase factors induced from the group action of $\\Lambda $ on itself by multiplication.",
"For example, if $\\Lambda = \\mathbb {Z}$ , the action of $\\Lambda $ is the translation on the one dimensional chain of qubits.",
"If $R=\\mathbb {F}_2[\\Lambda ]$ is the group algebra with multiplicative identity denoted by 1, the Pauli group modulo phase factors acquires a structure of an $R$ -module.",
"We shall call it the Pauli module.",
"The Pauli module is free and has rank 2.",
"Let $r \\mapsto \\bar{r}$ be the antipode map of $R$ , i.e., the $\\mathbb {F}_2$ -linear map into itself such that each group element is mapped to its inverse.",
"Since $\\Lambda $ is abelian, the antipode map is an algebra-automorphism.",
"Let the coefficient of $a \\in R$ at $g \\in \\Lambda $ be denoted by $a_g$ .",
"Hence, $a = \\sum _{g \\in \\Lambda } a_g g$ for any $a \\in R$ .",
"One may write $a_g = (a \\bar{g})_1$ .",
"Define $\\mathop {\\mathrm {tr}}(a) = a_1$ for any $a \\in R$ .",
"Proposition 1.1 [10] Let $(a,b), (c,d) \\in R^2$ be two vectors representing Pauli operators $O_1, O_2$ up to phase factors: $O_1 &= \\left( \\bigotimes _{g \\in \\Lambda } (\\sigma _x^{(g)})^{a_g} \\right)\\left( \\bigotimes _{g \\in \\Lambda } (\\sigma _z^{(g)})^{b_g} \\right) ,\\\\O_2 &= \\left( \\bigotimes _{g \\in \\Lambda } (\\sigma _x^{(g)})^{c_g} \\right)\\left( \\bigotimes _{g \\in \\Lambda } (\\sigma _z^{(g)})^{d_g} \\right)$ where $\\sigma ^{(g)}$ denotes the single qubit Pauli operator at $g \\in \\Lambda $ .",
"Then, $O_1$ and $O_2$ commute if and only if $\\mathop {\\mathrm {tr}}\\left(\\begin{pmatrix} \\bar{a} & \\bar{b} \\end{pmatrix}\\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}\\begin{pmatrix} c \\\\ d \\end{pmatrix}\\right) = 0 .$ The commutation value of $(\\sigma _x^{(g)})^{n} (\\sigma _z^{(g)})^{m}$ and $(\\sigma _x^{(g)})^{n^{\\prime }} (\\sigma _z^{(g)})^{m^{\\prime }}$ is $nm^{\\prime } - mn^{\\prime } \\in \\mathbb {F}_2$ .",
"Viewed as pairs of group algebra elements, $(\\sigma _x^{(g)})^{n} (\\sigma _z^{(g)})^{m}$ and $(\\sigma _x^{(g)})^{n^{\\prime }} (\\sigma _z^{(g)})^{m^{\\prime }}$ are $(n g, m g)$ and $(n^{\\prime } g, m^{\\prime } g)$ , respectively.",
"We see that $nm^{\\prime } - mn^{\\prime } = \\mathop {\\mathrm {tr}}\\left(\\begin{pmatrix} n g^{-1} & m g^{-1} \\end{pmatrix}\\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}\\begin{pmatrix} n^{\\prime } g \\\\ m^{\\prime } g \\end{pmatrix}\\right) .$ Since any Pauli operator is a finite product of these, the result follows by linearity.",
"We wish to characterize a $\\mathbb {F}_2$ -subspace $S$ of the Pauli module invariant under the action of $\\Lambda $ , i.e., a submodule, on which the commutation value is always zero.",
"As we will see in the next subsection, this particular subspace yields a local Hamiltonian whose energy spectrum is exactly solvable, which is the main object of this paper.",
"Let $(a,b)$ be an element of $S \\subseteq R^2 = (\\mathbb {F}_2[\\Lambda ])^2$ .",
"For any $r \\in R$ , $(ra, rb)$ must be a member of $S$ .",
"Demanding that the symplectic form on $S$ vanish, by Proposition REF we have $\\mathop {\\mathrm {tr}}(ra \\bar{b} - rb \\bar{a}) = 0 .$ Since $r$ was arbitrary, we must have $a \\bar{b} - b \\bar{a} = 0$ .A symmetric bilinear form $ \\langle r, s \\rangle = \\mathop {\\mathrm {tr}}(r \\bar{s}) $ on $R$ is non-degenerate.",
"Let us denote $\\begin{pmatrix}\\bar{a} & \\bar{b} \\end{pmatrix}$ as $\\begin{pmatrix} a \\\\ b \\end{pmatrix}^\\dagger $ , and write any element of $R^2$ as a $2 \\times 1$ matrix.",
"We conclude that $S$ is a submodule of $R^2$ over $R$ generated by $s_1, \\ldots , s_t$ such that any commutation value always vanishes, if and only if $s_i^\\dagger \\lambda _1 s_j = 0$ for all $i,j = 1,\\ldots ,t$ .",
"The requirement that $\\Lambda $ be a group might be too restrictive.",
"One may have a coarse group structure on $\\Lambda $ , the index set of all qubits.",
"We consider the case that the index set is a product of a finite set and a group.",
"By abuse of notation, we still write $\\Lambda $ to denote the group part, and insist that to each group element are associated $q$ qubits ($q \\ge 1$ ).",
"Thus obtained Pauli module should now be identified with $R^{2q}$ , where $R = \\mathbb {F}_2[\\Lambda ]$ is the group algebra that encodes the notion of translation.",
"We write an element $v$ of $R^{2q}$ by a $2q \\times 1$ matrix, and denote by $v^\\dagger $ the transpose matrix of $v$ whose each entry is applied by the antipode map.",
"We always order the entries of $v$ such that the upper $q$ entries describes the $\\sigma _x$ -part and the lower the $\\sigma _z$ -part.",
"Since the commutation value on $R^{2q}$ is the sum of commutation values on $R^2$ , we have the following: If $S$ is a submodule of $R^{2q}$ over $R$ generated by $s_1, \\ldots , s_t$ , the commutation value always vanishes on $S$ , if and only if for all $i,j = 1, \\ldots , t$ $s_i^\\dagger \\lambda _q s_j = 0$ where $\\lambda _q = \\begin{pmatrix} 0 & \\mathrm {id}_q \\\\ -\\mathrm {id}_q & 0 \\end{pmatrix}$ is a $2q \\times 2q$ matrix.",
"Let us summarize our discussion so far.",
"Proposition 1.2 On a set of qubits $\\Lambda \\times \\lbrace 1,\\ldots ,q\\rbrace $ where $\\Lambda $ is an abelian group, the group of all Pauli operators of finite support up to phase factors, form a free module $P=R^{2q}$ over the group algebra $R = \\mathbb {F}_2[\\Lambda ]$ .",
"The commutation value $\\langle a, b \\rangle = \\mathop {\\mathrm {tr}}( a^\\dagger \\lambda _q b )$ for $a,b \\in P$ is zero if and only if the Pauli operators corresponding to $a$ and $b$ commute.",
"If $\\sigma $ is a $2q \\times t$ matrix whose columns generate a submodule $S \\subseteq P$ , then the commutation value on $S$ always vanishes if and only if $\\sigma ^\\dagger \\lambda _q \\sigma = 0 .$ Proposition REF [10] is a special case of Proposition REF when $\\Lambda $ is a trivial group."
],
[
"Local Hamiltonians on groups",
"Recall that we place $q$ qubits on each site of $\\Lambda $ .",
"The total system of the qubits is $\\Lambda \\times \\lbrace 1,\\ldots ,q\\rbrace $ .",
"Definition 1 Let $H = -\\sum _{g \\in \\Lambda } h_{1,g} + \\cdots + h_{t,g}$ be a local Hamiltonian consisted of Pauli operators that is (i)commuting, (ii) translation-invariant up to signs, and (iii) frustration-free.",
"We call $H$ a code Hamiltonian (also known as stabilizer Hamiltonian).",
"The stabilizer module of $H$ is the submodule of the Pauli module $P$ generated by the images of $h_1,\\ldots ,h_t$ in $P$ .",
"The number of interaction types is $t$ .",
"The energy spectrum of the code Hamiltonian is trivial; it is discrete and equally spaced.",
"Example 1 One dimensional Ising model is the Hamiltonian $H = - \\sum _{i \\in \\mathbb {Z}} \\sigma _z^{(i)} \\otimes \\sigma _z^{(i+1)} .$ The lattice is the additive group $\\mathbb {Z}$ , and the group algebra is $R=\\mathbb {F}_2[x,\\bar{x}]$ .",
"The Pauli module is $R^2$ and the stabilizer module $S$ is generated by $\\begin{pmatrix}0 \\\\1+x\\end{pmatrix} .$ One can view this as the matrix $\\sigma $ of Proposition REF .",
"$H$ is commuting; $\\sigma ^\\dagger \\lambda _1 \\sigma = 0$ .",
"$\\Diamond $"
],
[
"Excitations",
"For a code Hamiltonian $H$ , an excited state is described by the terms in the Hamiltonian that have eigenvalues $-1$ .",
"Each of the flipped terms is interpreted as an excitation.",
"Although the actual set of all possible configurations of excitations that are obtained by applying some operator to a ground state, may be quite restricted, it shall be convenient to think of a larger set.",
"Let $E$ be the set of all configurations of finite number of excitations without asking physical relevance.",
"Since an excitation is by definition a flipped term in $H$ , the set $E$ is equal to the collection of all finite sets consisted of the terms in $H$ .",
"If Pauli operators $U_1, U_2$ acting on a ground state creates excitations $e_1, e_2 \\in E$ , their product $U_1 U_2$ creates excitations $(e_1 \\cup e_2) \\setminus (e_1 \\cap e_2)$ .",
"Here, we had to remove the intersection because each excitation is its own annihilator; any term in the $H$ squares to the identity.",
"Exploiting this fact, we make $E$ into a vector space over $\\mathbb {F}_2$ .",
"Namely, we take formal linear combinations of terms in $H$ with the coefficient $1 \\in \\mathbb {F}_2$ when the terms has $-1$ eigenvalue, and the coefficient $0 \\in \\mathbb {F}_2$ when the term has $+1$ eigenvalue.",
"The symmetric difference is now expressed as the sum of two vectors $e_1 + e_2$ over $\\mathbb {F}_2$ .",
"In view of Pauli group as a vector space, $U_1 U_2$ is the sum of the two vectors $v_1 + v_2$ that respectively represents $U_1, U_2$ .",
"Therefore, the association $U_i \\mapsto e_i$ induces a linear map from the Pauli space to the space of virtual excitations $E$ .",
"The set of all excited states obeys the translation-invariance as the code Hamiltonian $H$ does.",
"So, $E$ is a module over the group algebra $R=\\mathbb {F}_2[\\Lambda ]$ .",
"The association $U_i \\mapsto e_i$ clearly respects this translation structure.",
"Our discussion is summarized by saying that the excitations are described by an $R$ -linear map $\\epsilon : P \\rightarrow E$ from the Pauli module $P$ to the module of virtual excitations $E$.",
"As the excitation module is the collection of all finite sets of the terms in $H$ , we can speak of the module of generator labels $G$ , which is equal to $E$ as an $R$ -module.",
"$G$ is a free module of rank $t$ if there are $t$ types of interaction.",
"The matrix $\\sigma $ introduced in Section REF can be viewed as $\\sigma : G \\rightarrow P$ from the module of generator labels to the Pauli module.",
"Proposition 1.3 If $\\sigma $ is the generating map for the stabilizer module of a code Hamiltonian, then $\\epsilon = \\sigma ^\\dagger \\lambda _q .$ The matrix $\\epsilon $ can be viewed as a generalization of the parity check matrix of the standard theory of classical or quantum error correcting codes [17], [20], [21], [22], when a translation structure is given.",
"This is a simple corollary of Proposition REF .",
"Let $h_{i,g}$ be the terms in the Hamiltonian where $i = 1,\\ldots , t$ , and $ g \\in \\Lambda $ .",
"In the Pauli module, they are expressed as $g h_i$ where $h_i$ is the $i$ -th column of $\\sigma $ .",
"For any $u \\in P$ , let $\\epsilon (u)_i$ be the $i$ -th component of $\\epsilon (u)$ .",
"By definition, $\\epsilon ( u )_i= \\sum _{g \\in \\Lambda } g~ \\mathop {\\mathrm {tr}}\\left( (g h_i)^\\dagger \\lambda _q u \\right)= \\sum _{g \\in \\Lambda } g~ \\mathop {\\mathrm {tr}}\\left( \\bar{g} h_i^\\dagger \\lambda _q u \\right)= h_i^\\dagger \\lambda _q u$ Thus, $h_i^\\dagger \\lambda _q$ is the $i$ -th row of $\\epsilon $ .",
"Remark 1 The commutativity condition in Proposition REF of the code Hamiltonian is recast into the condition that $G \\xrightarrow{} P \\xrightarrow{} E$ be a complex, i.e., $\\epsilon \\circ \\sigma = 0$ .",
"Equivalently, $\\mathop {\\mathrm {im}}\\sigma \\subseteq (\\mathop {\\mathrm {im}}\\sigma )^\\perp = \\ker \\epsilon $ where $\\perp $ is with respect to the symplectic form."
],
[
"Equivalent Hamiltonians",
"The stabilizer module entirely determines the physical phase of the code Hamiltonian in the following sense.",
"Proposition 2.1 Let $H$ and $H^{\\prime }$ be code Hamiltonians on a system of qubits, and suppose their stabilizer modules are the same.",
"Then, there exists a unitary $U = \\bigotimes _{g \\in \\Lambda } U_{g}$ mapping the ground space of $H$ onto that of $H^{\\prime }$ .",
"Moreover, there exist a continuous one-parameter family of gapped Hamiltonians connecting $U H U^\\dagger $ and $H^{\\prime }$ .",
"Let $\\lbrace p_\\alpha \\rbrace $ be a maximal set of $\\mathbb {F}_2$ -linearly independent Pauli operators of finite support that generates the common stabilizer module $S$ .",
"$\\lbrace p_\\alpha \\rbrace $ is not necessarily translation-invariant.",
"Any ground state $\\left| {\\psi } \\right\\rangle $ of $H$ is a common eigenspace of $\\lbrace p_\\alpha \\rbrace $ with eigenvalues $p_\\alpha \\left| {\\psi } \\right\\rangle = e_\\alpha \\left| {\\psi } \\right\\rangle $ , $e_\\alpha = \\pm 1$ .",
"Similarly, the ground space of $H^{\\prime }$ gives the eigenvalues $e^{\\prime }_\\alpha = \\pm 1$ for each $p_\\alpha $ .",
"The abelian group generated by $\\lbrace p_\\alpha \\rbrace $ is precisely the vector space $S$ , and the assignment $p_\\alpha \\mapsto e_\\alpha $ defines a dual vector on $S$ .",
"If $U$ is a Pauli operator of possibly infinite support, then $ p_\\alpha U \\left| {\\psi } \\right\\rangle = e^{\\prime \\prime }_\\alpha e_\\alpha U \\left| {\\psi } \\right\\rangle $ for some $e^{\\prime \\prime }_\\alpha = \\pm 1$ , where $e^{\\prime \\prime }_\\alpha $ is determined by the commutation relation between $U$ and $p_\\alpha $ .",
"Thus, the first statement follows if we can find $U$ such that the commutation value between $U$ and $p_\\alpha $ is precisely $e^{\\prime \\prime }_\\alpha $ .",
"This is always possible since the dual space of the vector space $P$ is isomorphic to the direct product $\\prod _{\\Lambda \\times \\lbrace 1,\\ldots , q\\rbrace } \\mathbb {F}_2^2$ , which is vector-space-isomorphic to the Pauli group of arbitrary support up to phase factors.If $V$ is a finite dimensional vector space over some field,the dual vector space of $\\bigoplus _I V$ is isomorphic to $\\prod _{I} V$ where $I$ is an arbitrary index set.",
"Now, $U H U^\\dagger $ and $H^{\\prime }$ have the same eigenspaces, and in particular, the same ground space.",
"Consider a continuous family of Hamiltonians $H(u,u^{\\prime }) = u U H U^\\dagger + u^{\\prime } H^{\\prime }$ where $u,u^{\\prime } \\in \\mathbb {R}$ .",
"It is clear that $H = H(1,0) \\rightarrow H(1,1) \\rightarrow H(0,1) = H^{\\prime }$ is a desired path.",
"The criterion of Proposition REF to classify the physical phases is too narrow.",
"Physically meaningful universal properties should be invariant under simple and local changes of the system.",
"More concretely, Definition 2 Two code Hamiltonians $H$ and $H^{\\prime }$ are equivalent if their stabilizer modules become the same under a finite composition of symplectic transformations, coarse-graining, and tensoring ancillas.",
"We shall define the symplectic transformations, the coarse-graining, and the tensoring ancillas shortly."
],
[
"Symplectic transformations",
"Definition 3 A symplectic transformation $T$ is an automorphism of the Pauli module induced by a unitary operator on the system of qubits such that $T^\\dagger \\lambda _q T = \\lambda _q$ where $\\dagger $ is the transposition followed by the entry-wise antipode map.",
"When the translation group is trivial these transformations are given by so-called Clifford operators.",
"See [11].",
"Only the unitary operator on the physical Hilbert space that respects the translation can induce a symplectic transformation.",
"By definition, a symplectic transformation maps each local Pauli operator to a local Pauli operator, and preserves the commutation value for any pair of Pauli operators.",
"Proposition 2.2 Any two unitary operators $U_1, U_2$ that induce the same symplectic transformation differ by a Pauli operator (of possibly infinite support).",
"If the translation group is trivial, the proposition reduces to Theorem 15.6 of [11] The symplectic transformation induced by $U = U_1^\\dagger U_2$ is the identity.",
"Hence, $U$ maps each single qubit Pauli operator $\\sigma _{x,z}^{(g,i)}$ to $\\pm \\sigma _{x,z}^{(g,i)}$ .",
"By the argument as in the proof of Proposition REF , there exists a Pauli operator $O$ of possibly infinite support that acts the same as $U$ on the system of qubits.",
"Since Pauli operators form a basis of the operator algebra of qubits, we have $O=U$ .",
"The effect of a symplectic transformation on the generating map $\\sigma $ is a matrix multiplication on the left.",
"$\\sigma \\rightarrow U \\sigma $ For example, the following is induced by uniform Hadamard, controlled-Phase, and controlled-NOT gates.",
"For notational clarity, define $E_{i,j}(a)\\ (i \\ne j)$ as the row-addition elementary $2q \\times 2q$ matrix $\\left[ E_{i,j}(a) \\right]_{\\mu \\nu } = \\delta _{\\mu \\nu } + \\delta _{\\mu i} \\delta _{\\nu j} a$ where $\\delta _{\\mu \\nu }$ is the Kronecker delta and $a \\in R = \\mathbb {F}_2[\\Lambda ]$ .",
"Recall that we order the components of $P$ such that the first half components are for $\\sigma _x$ -part, and the second half components are for $\\sigma _z$ -part.",
"Definition 4 The following are elementary symplectic transformations: (Hadamard) $E_{i,i+q}(-1) E_{i+q,i}(1) E_{i,i+q}(-1)$ where $1 \\le i \\le q$ , (controlled-Phase) $E_{i+q,i}(f)$ where $f = \\bar{f}$ and $1 \\le i \\le q$ , (controlled-NOT) $E_{i,j}(a) E_{j+q,i+q}(-\\bar{a})$ where $1 \\le i \\ne j \\le q$ .",
"For the case of a trivial translation group, these transformations explicitly appear in [10] and [11].",
"Recall that the Hadamard gate is a unitary transformation on a qubit given by $U_H = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}1 & 1 \\\\1 & -1\\end{pmatrix}$ with respect to basis $\\lbrace \\left| {0} \\right\\rangle , \\left| {1} \\right\\rangle \\rbrace $ .",
"At operator level, $U_H X U_H^\\dagger = Z, \\quad U_H Z U_H^\\dagger = X$ where $X$ and $Z$ are the Pauli matrices $\\sigma _x$ and $\\sigma _z$ , respectively.",
"Thus, the application of Hadamard gate on every $i$ -th qubit of each site of $\\Lambda $ swaps the corresponding $X$ and $Z$ components of $P$ .",
"The controlled phase gate is a two-qubit unitary operator whose matrix is $U_P =\\begin{pmatrix}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 1 & 0 \\\\0 & 0 & 0 & -1\\end{pmatrix}$ with respect to basis $\\lbrace \\left| {00} \\right\\rangle , \\left| {01} \\right\\rangle , \\left| {10} \\right\\rangle , \\left| {11} \\right\\rangle \\rbrace $ .",
"At operator level, $U_P (X \\otimes I ) U_P^\\dagger = X \\otimes Z, & &U_P (Z \\otimes I ) U_P^\\dagger = Z \\otimes I, \\\\U_P (I \\otimes X ) U_P^\\dagger = Z \\otimes X, & &U_P (I \\otimes Z ) U_P^\\dagger = I \\otimes Z.$ Note that since $U_P$ is diagonal, any two $U_P$ on different pairs of qubits commute.",
"Let $(g,i)$ denote the $i$ -th qubit at $g \\in \\Lambda $ .",
"The uniform application $U^{(i)}_g = \\prod _{h \\in \\Lambda } U_P( (h,i), (h+g,i) )$ of $U_P$ throughout the lattice $\\Lambda $ such that each $U_P( (h,i), (h+g,i) )$ acts on the pair of qubits $(h,i)$ and $(h+g,i)$ is well-defined.",
"From the operator level calculation of $U_P$ , we see that $U^{(i)}_g$ induces $P \\ni (\\ldots ,x_i,\\ldots , z_i, \\ldots ) \\mapsto ( \\ldots , x_i,\\ldots , z_i + (g+ \\bar{g})x_i, \\ldots ) \\in P$ on the Pauli module, which is represented as $E_{i+q,i}(g+\\bar{g})$ .",
"The composition $U^{(i)}_{g_1} U^{(i)}_{g_2} \\cdots U^{(i)}_{g_n}$ of finitely many controlled-Phase gates $U^{(i)}_g$ with different $g$ is represented as $E_{i+q,i}(f)$ where $f = \\bar{f} = \\sum _{k=1}^{n} g_k + \\bar{g}_k$ .",
"The single qubit phase gate $\\begin{pmatrix}1 & 0 \\\\0 & i\\end{pmatrix}$ maps $X \\leftrightarrow Y$ and $Z \\mapsto Z$ .",
"On the Pauli module $P$ , it is $P \\ni (\\ldots ,x_i,\\ldots , z_i, \\ldots )^T \\mapsto ( \\ldots , x_i,\\ldots , z_i + x_i, \\ldots )^T \\in P .$ which is $E_{i+q,i}(1)$ .",
"Note that any $f \\in R$ such that $f = \\bar{f}$ is always of form $f = \\sum g_k + \\bar{g}_k$ or $f = 1 + \\sum g_k + \\bar{g}_k$ where $g_k$ are monomials.",
"Thus, the Phase gate and the controlled-Phase gate induce transformations $E_{i+q,i}(f)$ where $f = \\bar{f}$ .",
"The controlled-NOT gate is a two-qubit unitary operator whose matrix is $U_N =\\begin{pmatrix}1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 \\\\0 & 0 & 0 & 1 \\\\0 & 0 & 1 & 0\\end{pmatrix}$ with respect to basis $\\lbrace \\left| {00} \\right\\rangle , \\left| {01} \\right\\rangle , \\left| {10} \\right\\rangle , \\left| {11} \\right\\rangle \\rbrace $ .",
"That is, it flips the target qubit conditioned on the control qubit.",
"At operator level, $U_N (X \\otimes I ) U_N^\\dagger = X \\otimes X, & &U_N (Z \\otimes I ) U_N^\\dagger = Z \\otimes I, \\\\U_N (I \\otimes X ) U_N^\\dagger = I \\otimes X, & &U_N (I \\otimes Z ) U_N^\\dagger = Z \\otimes Z.$ If $i < j$ , the uniform application $U^{(i,j)}_g = \\bigotimes _{h \\in \\Lambda } U_P( (h,i), (h+g,j) )$ such that each $U_N( (h,i), (h+g,j) )$ acts on the pair of qubits $(h,i)$ and $(h+g,j)$ with one at $(h,i)$ being the control induces $P \\ni & (\\ldots ,x_i,\\ldots ,x_j,\\ldots , z_i, \\ldots , z_j, \\ldots )^T \\\\& \\mapsto ( \\ldots , x_i, \\ldots , x_j + g x_i, \\ldots , z_i + \\bar{g} z_j, \\ldots , z_j, \\ldots )^T \\in P .$ Thus, any finite composition of controlled-NOT gates with various $g$ is of form $E_{i,j}(a) E_{j+q,i+q}(\\bar{a})$ .",
"It might be useful to note that the controlled-NOT and the Hadamard combined, induces a symplectic transformation (controlled-NOT-Hadamard) $E_{i+q,j}(a) E_{j+q,i}(\\bar{a})$ where $a \\in R$ and $ 1 \\le i \\ne j \\le q$ .",
"Remark that an arbitrary row operation on the upper $q$ components can be compensated by a suitable row operation on the lower $q$ components so as to be a symplectic transformation."
],
[
"Coarse-graining",
"Not all unitary operators conform with the lattice translation.",
"In Example REF the lattice translation has period 1.",
"Then, for example, the Hadamard gate on every second qubit does not respect this translation structure; it only respects a coarse version of the original translation.",
"We need to shrink the translation group to treat such unitary operators.",
"Let $\\Lambda $ be the original translation group of the lattice with $q$ qubits per site, and $\\Lambda ^{\\prime }$ be its subgroup of finite index: $|\\Lambda /\\Lambda ^{\\prime }| = c < \\infty $ .",
"The total set of qubits $\\Lambda \\times \\lbrace 1,\\ldots ,q\\rbrace $ is set-theoretically the same as $\\Lambda ^{\\prime } \\times \\lbrace 1, \\ldots , c \\rbrace \\times \\lbrace 1,\\ldots ,q\\rbrace = \\Lambda ^{\\prime } \\times \\lbrace 1, \\ldots , cq\\rbrace $ .",
"We take $\\Lambda ^{\\prime }$ as our new translation group under coarse-graining.",
"The Pauli group modulo phase factors remains the same as a $\\mathbb {F}_2$ -vector space for it depends only on the total index set of qubits.",
"We shall say that the system is coarse-grained by $R^{\\prime }=\\mathbb {F}_2[\\Lambda ^{\\prime }]$ if we restrict the scalar ring $R$ to $R^{\\prime }$ for all modules pertaining to the system.",
"For example, suppose $\\Lambda = \\mathbb {Z}^2$ , so the original base ring is $R = \\mathbb {F}_2[x,y,\\bar{x},\\bar{y}]$ .",
"If we coarse-grain by $R^{\\prime } = \\mathbb {F}_2[x^{\\prime },y^{\\prime }, \\bar{x}^{\\prime }, \\bar{y}^{\\prime }]$ where $x^{\\prime } = x^2, y^{\\prime } = y^2$ , we are taking the sites $1,x,y,xy$ of the original lattice as a single new site."
],
[
"Tensoring ancillas",
"We have considered possible transformations on the stabilizer modules of code Hamiltonians, and kept the underlying index set of qubits invariant.",
"It is quite natural to allow tensoring ancilla qubits in trivial states.",
"In terms of the stabilizer module $S \\subseteq P=R^{2q}$ , it amounts to embed $S$ into the larger module $R^{2q^{\\prime }}$ where $q^{\\prime } > q$ .",
"Concretely, let $\\sigma = \\begin{pmatrix} \\sigma _X \\\\ \\sigma _Z \\end{pmatrix}$ be the generating matrix of $S$ as in Proposition REF .",
"By tensoring ancilla, we embed $S$ as $\\begin{pmatrix} \\sigma _X \\\\ \\sigma _Z \\end{pmatrix}\\rightarrow \\begin{pmatrix}\\sigma _X & 0 \\\\0 & 0 \\\\\\sigma _Z & 0 \\\\0 & 1\\end{pmatrix} .$ This amounts to taking the direct sum of the original complex $G \\xrightarrow{} P \\xrightarrow{} E$ and the trivial complex $0\\rightarrow R\\xrightarrow{}R^2\\xrightarrow{}R\\rightarrow 0$ to form $G \\oplus R \\xrightarrow{} P \\oplus R^2 \\xrightarrow{} E \\oplus R .$"
],
[
"Topological order",
"From now on we assume that $\\Lambda $ is isomorphic to $\\mathbb {Z}^D$ as an additive group.",
"$D$ shall be called the spatial dimension of $\\Lambda $ .",
"Definition 5 Let $\\sigma : G \\rightarrow P$ be the generating map for the stabilizer module of a code Hamiltonian $H$ .",
"We say $H$ is exact if $(\\mathop {\\mathrm {im}}\\sigma )^\\perp = \\mathop {\\mathrm {im}}\\sigma $ , or equivalently $G \\xrightarrow{} P \\xrightarrow{} E$ is exact, i.e., $\\ker \\epsilon = \\mathop {\\mathrm {im}}\\sigma $ .",
"It follows that the exactness condition is a property of the equivalence class of code Hamiltonians in the sense of Definition REF .",
"By imposing periodic boundary conditions, a translation-invariant Hamiltonian yields a family of Hamiltonians $\\lbrace H(L) \\rbrace $ defined on a finite system consisted of $L^D$ sites.",
"One might be concerned that some $H(L)$ would be frustrated.",
"We intentionally exclude such a situation.",
"The frustration might indeed occur, but it can easily be resolved by choosing the signs of terms in the Hamiltonian.",
"In this way, one might loose the translation-invariance in a strict sense.",
"However, we retain the physical phase regardless of the sign choice because different sign choices are related by a Pauli operator acting on the whole system which is a product unitary operator.",
"Hence, the entanglement property of the ground state and the all properties of excitations do not change.",
"Definition 6 Let $H(L)$ be Hamiltonians on a finite system of linear size $L$ in $D$ dimensional physical space, and $\\Pi _L$ be the corresponding ground space projector.",
"$H(L)$ is called topologically ordered if for any $O$ supported inside a hypercube of size $(L/2)^D$ one has $\\Pi _L O \\Pi _L \\propto \\Pi _L .$ This means that no local operator is capable of distinguishing different ground states.",
"This condition is trivially satisfied if $H(L)$ has a unique ground state.",
"A technical condition that is used in the proof of the stability of topological order against small perturbations is the following `local topological order' condition [24], [25], [26].",
"We say a diamond region $A(r)$ of radius $r$ at $o \\in \\mathbb {Z}^D$ for the set $A(r)_o = \\left\\lbrace (i_1,\\ldots ,i_D) + o \\in \\mathbb {Z}^D ~| \\sum _\\mu |i_\\mu | \\le r \\right\\rbrace .$ Definition 7 Let $H(L)$ be code Hamiltonians on a finite system of linear size $L$ in $D$ dimensional physical space.",
"For any diamond region $A=A(r)$ of radius $r$ , let $\\Pi _A$ be the projector onto the common eigenspace of the most negative eigenvalues of terms in the Hamiltonian $H(L)$ that are supported in $A$ .",
"For $b > 0$ , denote by $A^b$ the distance $b$ neighborhood of $A$ .",
"$H(L)$ is called locally topologically ordered if there exists a constant $b > 0$ such that for any operator $O$ supported on a diamond region $A$ of radius $r < L/2$ one has $\\Pi _{A^b} O \\Pi _{A^b} \\propto \\Pi _{A^b} .$ Since any operator is a $\\mathbb {C}$ -linear combination of Pauli operators, if Eq.",
"(REF ),(REF ) are satisfied for Pauli operators, then the (local) topological order condition follows.",
"If a Pauli operator $O$ is anti-commuting with a term in a code Hamiltonian $H(L)$ , The left-hand side of Eq.",
"(REF ),(REF ) are identically zero.",
"In this case, there is nothing to be checked.",
"If $O$ acting on $A$ is commuting with every term in $H(L)$ supported inside $A^b$ , Eq.",
"(REF ) demands that it act as identity on the ground space, i.e., $O$ must be a product of terms in $H(L)$ up to $\\pm i,\\pm 1$ .",
"Eq.",
"(REF ) further demands that $O$ must be a product of terms in $H(L)$ supported inside $A^b$ up to $\\pm i,\\pm 1$ .",
"Lemma 3.1 A code Hamiltonian $H$ is exact if and only if $H(L)$ is locally topologically ordered for all sufficiently large $L$ .",
"In order to see this, it will be important to use Laurent polynomials to express elements of the group algebra $R = \\mathbb {F}_2[\\mathbb {Z}^D] \\cong \\mathbb {F}_2[x_1,x_1^{-1},\\ldots ,x_D,x_D^{-1}]$ .",
"See also [18].",
"For example, $x y^2 z^2 + x y^{-1} \\quad \\Longleftrightarrow \\quad 1(1,2,2)+1(1,-1,0) .$ The sum of the absolute values of exponents of a monomial will be referred to as absolute degree.",
"The absolute degree of a Laurent polynomial is defined to be the maximum absolute degree of its terms.",
"The degree measures the distance or size in the lattice.",
"The Laurent polynomial viewpoint enables us to apply Gröbner basis techniques.",
"The long division algorithm for polynomials in one variable yields an effective and efficient test whether a given polynomial is divisible by another.",
"When two or more but finitely many variables are involved, a more general question is how to test whether a given polynomial is a member of an ideal.",
"For instance, $f=xy-1$ is a member of an ideal $J = (x-1,y-1)$ because $xy -1 = y(x-1) + (y-1)$ .",
"But, $g=xy$ is not a member of $J$ because $g = y(x-1) + (y-1) + 1$ and the `remainder' 1 cannot be removed.",
"Here, the first term is obtained by looking at the initial term $xy$ of $f$ and comparing with the initial terms $x$ and $y$ of the generators of $J$ .",
"While one tries to eliminate the initial term of $f$ and to eventually reach zero, if one cannot reach zero as for $g$ , then the membership question is answered negatively.",
"Systematically, an well-ordering on the monomials, i.e., a term order, is defined such that the order is preserved by multiplications.",
"And a set of generators $\\lbrace g_i \\rbrace $ for the ideal is given with a special property that any element in the ideal has an initial term (leading term) divisible by an initial term of some $g_i$ .",
"A Gröbner basis is precisely such a generating set.",
"This notion generalizes to free modules over polynomial ring by refining the term order with the basis of the modules.",
"An example is as follows.",
"Let $\\sigma _1 =\\begin{pmatrix}\\mathbf {x^2} - y \\\\x^2 + 1\\end{pmatrix} \\quad \\sigma _2 =\\begin{pmatrix}1 \\\\\\mathbf {y}\\end{pmatrix}$ generate a submodule $M$ of $S^2$ where $S = \\mathbb {F}[x,y]$ is a polynomial ring.",
"They form a Gröbner basis, and the initial terms are marked as bold.",
"A member of $S^2$ $\\begin{pmatrix}x^2 + x^2 y - y^2 \\\\y+2 x^2 y\\end{pmatrix}$ is in $M$ because the following “division” results in zero.",
"$\\begin{pmatrix}x^2+\\mathbf {x^2 y}-y^2 \\\\y+2 x^2 y\\end{pmatrix}\\xrightarrow{}\\begin{pmatrix}x^2 \\\\\\mathbf {x^2 y}\\end{pmatrix}\\xrightarrow{} 0$ A comprehensive material can be found in [27].",
"The situation for Laurent polynomial ring is less discussed, but is not too different.",
"A direct treatment is due to Pauer and Unterkircher [28].",
"One introduces a well-order on monomials, that is preserved by multiplications with respect to a so-called cone decomposition.",
"An ideal $J$ over a Laurent polynomial ring can be thought of as a collection of configurations of coefficient scalars written on the sites of the integral lattice $\\mathbb {Z}^D$ .",
"If we take a cone, say, $C = \\lbrace (i_1,i_2,i_3) \\in \\mathbb {Z}^3 | i_1 \\le 0, i_2 \\ge 0, i_3 \\ge 0 \\rbrace ,$ then $J_C = J \\cap \\mathbb {F}[C]$ looks very similar to an ideal $I$ over a polynomial ring $\\mathbb {F}[x,y,z]$ .",
"Concretely, $I$ can be obtained by applying $x^{-1} \\mapsto x, y \\mapsto y, z \\mapsto z$ to $J_C$ .",
"The initial terms of $J_C$ should be treated similarly as those in $I$ .",
"This is where the cone decomposition plays a role.",
"The lattice $\\mathbb {Z}^D$ decomposes into $2^D$ cones, and the initial terms of $J$ is considered in each of the cones.",
"Correspondingly, a Gröbner basis is defined to generate the initial terms of a given module in each of the cones.",
"An intuitive picture for the division algorithm is to consider the support of a Laurent polynomial as a finite subset of $\\mathbb {Z}^D$ around the origin (the least element of $\\mathbb {Z}^D$ ), and to eliminate outmost points so as to finally reach the origin.",
"If $m$ is a column matrix of Laurent polynomials, each step in the division algorithm by a Gröbener basis $\\lbrace g\\rbrace $ replaces $m$ with $m^{\\prime } = m - c g$ , where $c$ is a monomial, such that the initial term of $m^{\\prime }$ is strictly smaller than that of $m$ .",
"Note that the absolute degree of $c$ does not exceed that of $m$ .Strictly speaking, one can introduce a term order such that this is true.",
"We have to show that if $v \\in \\ker \\epsilon = \\mathop {\\mathrm {im}}\\sigma $ is supported in the diamond of radius $r$ centered at the origin, then $v$ can be expressed as a linear combination $v = \\sum _i c_i \\sigma _i$ of the columns $\\sigma _i$ of $\\sigma $ such that the coefficients $c_i \\in R$ have absolute degree not exceeding $w+r$ .",
"for some fixed $w$ .",
"A Gröbner basis [28] is computed solely from the matrix $\\sigma $ , and the division algorithm yields desired $c_i$ .This part can be adapted to an error correcting procedure or a decoder.",
"The bottleneck of the universal decoder presented in [9] is the routine that tests whether a given cluster of excitations can be created by a Pauli operator supported in the box that envelops the cluster.",
"The Gröbner basis for $\\mathop {\\mathrm {im}}\\epsilon $ in the degree monomial order provides a fast algorithm for it: The division algorithm yields zero remainder with respect to the Gröbner basis, if and only if the given cluster is in $\\mathop {\\mathrm {im}}\\epsilon $ .",
"Note also that this argument proves that the topological order condition as defined in [9] is always satisfied if the code Hamiltonian is exact.",
"Conversely, suppose $v \\in \\ker \\epsilon $ .",
"We have to show $v \\in \\mathop {\\mathrm {im}}\\sigma $ .",
"Choose so large $L$ that the Pauli operator $O$ representing $v$ is contained in a pyramid region far from the boundary.",
"The local topological order condition implies that $O$ is a product of terms near the pyramid region.",
"Since this product expression is independent of the boundary, we see $v \\in \\mathop {\\mathrm {im}}\\sigma $ .",
"The Buchsbaum-Eisenbud theorem [29] below characterizes an exact sequence from the properties of connecting maps.",
"(See also [27],[30].)",
"A few notions should be recalled.",
"Let $\\mathbf {M}$ be a matrix, not necessarily square, over a ring.",
"A minor is the determinant of a square submatrix of $\\mathbf {M}$ .",
"$k$ -th determinantal ideal $I_k(\\mathbf {M})$ is the ideal generated by all $k \\times k$ minors of $\\mathbf {M}$ .",
"It is not hard to see that the determinantal ideal is invariant under any invertible matrix multiplication on either side.",
"The rank of $\\mathbf {M}$ is the largest $k$ such that $k$ -th determinantal ideal is nonzero.",
"Thus, the rank of a matrix over an arbitrary ring is defined, although the dimension of the image in general is not defined or is infinite.",
"The 0-th determinantal ideal is taken to be the unit ideal by convention.",
"For a map $\\phi $ between free modules, we write $I(\\phi )$ to denote the $k$ -th determinantal ideal of the matrix of $\\phi $ where $k$ is the rank of that matrix.",
"Fitting Lemma [27] states that determinantal ideals only depend on $\\mathop {\\mathrm {coker}}\\phi $ .",
"The (Krull) dimension of a ring is the supremum of lengths of chains of prime ideals.",
"Here, the length of a chain of prime ideals $\\mathfrak {p}_0 \\subsetneq \\mathfrak {p}_1 \\subsetneq \\cdots \\subsetneq \\mathfrak {p}_n$ is defined to be $n$ .",
"Most importantly, the dimension of $\\mathbb {F}[x_1,\\ldots ,x_n]$ is $n$ where $\\mathbb {F}$ is a field, as $(0) \\subset (x_1) \\subset (x_1,x_2) \\subset \\cdots \\subset (x_1,\\ldots ,x_n) .$ Dimensions are in general very subtle, but intuitively, it counts the number of independent `variables.'",
"Geometrically, a ring is a function space of a geometric space, and the independent variables define a coordinate system on it.",
"So the Krull dimension correctly captures the intuitive dimension.",
"For instance, $y-x^2=0$ defines a parabola in a plane, and the functions that vanish on the parabola form an ideal $(y-x^2) \\subset \\mathbb {F}[x,y]$ .",
"Thus, the function space is identified with $\\mathbb {F}[x,y]/(y-x^2) \\cong \\mathbb {F}[x]$ , whose Krull dimension is, as expected, 1.",
"Facts we need are quite simple: In a zero-dimensional ring, every prime ideal is maximal.",
"$\\dim R = \\dim \\mathbb {F}_2[x_1^{\\pm 1},\\ldots ,x_D^{\\pm 1}] = D$ When $I$ is an ideal of $R$ , $\\dim R/I + \\mathop {\\mathrm {codim}}I = D$ .The codimension or height of a prime ideal $\\mathfrak {p}$ is the supremum of the lengths of chains of prime ideals contained in $\\mathfrak {p}$ .",
"That is, the codimension of $\\mathfrak {p}$ is the Krull dimension of the local ring $R_\\mathfrak {p}$ .",
"The codimension of an arbitrary ideal $I$ is the minimum of codimensions of primes that contain $I$ .",
"If $S$ is an affine domain, i.e., a homomorphic image of a polynomial ring over a field with finitely many variables such that $S$ has no zero-divisors, it holds that $\\mathop {\\mathrm {codim}}I + \\dim R/I = \\dim S$ [27].",
"We shall be dealing with three different kinds of `dimensions': The first one is the spatial dimension $D$ , which has an obvious physical meaning.",
"The second one is the Krull dimension of a ring, just introduced.",
"The Krull dimension is upper bounded by the spatial dimension in any case.",
"The last one is the dimension of some module as a vector space.",
"Recall that all of our base ring contains a field – $\\mathbb {F}_2$ for qubits.",
"The vector space dimension arises naturally when we actually count the number of orthogonal ground states.",
"The dimension as a vector space will always be denoted with a subscript like $\\dim _{\\mathbb {F}_2}$ .",
"Proposition 3.2 [29]The original result is stronger than what is presented here.",
"It is stated with the depths of the determinantal ideals.",
"If a complex of free modules over a ring $0 \\rightarrow F_n \\xrightarrow{} F_{n-1} \\rightarrow \\cdots \\rightarrow F_1 \\xrightarrow{} F_0$ is exact, then $\\mathop {\\mathrm {rank}}F_k = \\mathop {\\mathrm {rank}}\\phi _k + \\mathop {\\mathrm {rank}}\\phi _{k+1}$ for $k=1,\\ldots ,n-1$ $\\mathop {\\mathrm {rank}}F_n = \\mathop {\\mathrm {rank}}\\phi _n$ .",
"$I(\\phi _k)=(1)$ or else $\\mathop {\\mathrm {codim}}I(\\phi _{k}) \\ge k$ for $k=1,\\ldots ,n$ .",
"Remark 2 For an exact code Hamiltonian, we have a exact sequence $G \\xrightarrow{} P \\xrightarrow{} E$ .",
"As we will see in Lemma REF , $\\mathop {\\mathrm {coker}}\\sigma $ has a finite free resolution, and we may apply the Proposition REF .",
"Since $\\overline{ I_k(\\sigma ) } = I_k(\\epsilon )$ for any $k \\ge 0$ , we have $2q = \\mathop {\\mathrm {rank}}P = \\mathop {\\mathrm {rank}}\\sigma + \\mathop {\\mathrm {rank}}\\epsilon = 2~ \\mathop {\\mathrm {rank}}\\sigma .$ The size $2q \\times t$ of the matrix $\\sigma $ satisfies $t \\ge q$ .",
"If $I_q(\\sigma ) \\ne R$ , then $\\mathop {\\mathrm {codim}}I_q(\\sigma ) \\ge 2$ ."
],
[
"Ground state degeneracy",
"Let $H(L)$ be the Hamiltonians on finite systems obtained by imposing periodic boundary conditions as in Section .",
"A symmetry operator of $H(L)$ is a $\\mathbb {C}$ -linear combination of Pauli operator that commutes with $H(L)$ .",
"In order for a Pauli symmetry operator to have a nontrivial action on the ground space, it must not be a product of terms in $H(L)$ .",
"In addition, since $H(L)$ is a sum of Pauli operators, a symmetry Pauli operator must commute with each term in $H(L)$ .",
"Hence, a symmetry Pauli operator $O$ with nontrivial action on the ground space must have image $v$ in the Pauli module such that $v(O) \\in \\ker \\epsilon _L \\setminus \\mathop {\\mathrm {im}}\\sigma _L$ where $G / \\mathfrak {b}_L G \\xrightarrow{} P / \\mathfrak {b}_L P \\xrightarrow{} E / \\mathfrak {b}_L E$ and $\\mathfrak {b}_L = (x^L_1 -1,\\ldots , x^L_D -1) \\subseteq R,$ which effectively imposes the periodic boundary conditions.",
"Since each term in $H(L)$ acts as an identity on the ground space, if $O^{\\prime }$ is a term in $H(L)$ , the symmetry operator $O$ and the product $OO^{\\prime }$ has the same action on the ground space.",
"$OO^{\\prime }$ is expressed in the Pauli module as $v(O) + v^{\\prime }(O^{\\prime })$ for some $v^{\\prime } \\in \\mathop {\\mathrm {im}}\\sigma _L$ .",
"Therefore, the set of Pauli operators of distinct actions on the ground space is in one-to-one correspondence with the factor module $K(L) = \\ker \\epsilon _L ~/~ \\mathop {\\mathrm {im}}\\sigma _L .$ The vector space dimension $\\dim _{\\mathbb {F}_2} K(L)$ is precisely the number of independent Pauli operators that have nontrivial action on the ground space.",
"Since $\\ker \\epsilon _L = (\\mathop {\\mathrm {im}}\\sigma _L)^\\perp $ by definition of $\\epsilon $ , and $\\mathop {\\mathrm {im}}\\sigma _L$ as an $\\mathbb {F}_2$ -vector space is a null space of the symplectic vector space $P/\\mathfrak {b}_L P$ , it follows that $\\ker \\epsilon _L = \\mathop {\\mathrm {im}}\\sigma _L \\oplus W$ for some hyperbolic subspace $W$ .",
"The quotient space $K(L) \\cong W$ is thus hyperbolic and has even vector space dimension $2k$ .",
"Choosing a symplectic basis for $K(L)$ , it is clear that $K(L)$ represents the tensor product of $k$ qubit-algebras.",
"Therefore, the ground space degeneracy is exactly $2^k$ [22], [10].",
"In the theory of quantum error correcting codes, $k$ is called the number of logical qubits, and the elements of $K(L)$ are called the logical operators.",
"In this section, $k$ will always denote $\\frac{1}{2} \\dim _{\\mathbb {F}_2} K$ .",
"Definition 8 The associated ideal for a code Hamiltonian is the $q$ -th determinantal ideal $I_q(\\sigma ) \\subseteq R$ of the generating map $\\sigma $ .",
"Here, $q$ is the number of qubits per site.",
"The characteristic dimension is the Krull dimension $\\dim R / I_q(\\sigma )$ .",
"The associated ideals appears in Buchsbaum-Eisenbud theorem (Proposition REF ), which says that the homology $K(L)$ is intimately related to the associated ideal.",
"Imposing boundary conditions such as $x^L=1$ amounts to treating $x$ not as variables any more, but as a `solution' of the equation $x^L-1=0$ .",
"In order for $K(L)$ to be nonzero, the `solution' $x$ should make the associated ideal to vanish.",
"Hence, by investigating the solutions of $I_q(\\sigma )$ one can learn about the relation between the degeneracy and the boundary conditions.",
"Roughly, a large number of solutions of $I_q(\\sigma )$ compatible with the boundary conditions means a large degeneracy.",
"As $d = \\dim R/I_q(\\sigma )$ is the geometric dimension of the algebraic set defined by $I_q(\\sigma )$ , a larger $d$ means a larger number of solutions.",
"Hence, the characteristic dimension $d$ controls the growth of the degeneracy as a function of the system size.",
"For example, consider a chain complex over $R = \\mathbb {F}[x^{\\pm 1},y^{\\pm 1}]$ .",
"$0 \\rightarrow R^1\\xrightarrow{}R^2\\xrightarrow{}R^1$ It is exact at $R^2$ .",
"The smallest nonzero determinantal ideal $I$ for either $\\partial _1$ or $\\partial _2$ is $I=(x-1,y-1)$ .",
"If we impose `boundary conditions' such that $x=1$ and $y=1$ , then $I$ becomes zero, and according to Buchsbaum-Eisenbud theorem, the homology $K$ at $R^2$ should be nontrivial.",
"Since the solution of $I$ consists of a single point $(1,1)$ on a 2-plane, it is conceivable that `boundary conditions' of form $\\mathfrak {b}_L$ would always give $K(L)$ of a constant $\\mathbb {F}$ -dimension, which is true in this case.",
"If we insist that the complex is over $R^{\\prime } = \\mathbb {F}[x^{\\pm 1},y^{\\pm 1},z^{\\pm 1}]$ , then the zero set of $I$ is a line $(1,1,z)$ in 3-space; there are many `solutions.'",
"In this case, $K^{R^{\\prime }}(L)$ has $\\mathbb {F}$ -dimension $2L$ .",
"An obvious example where the homology $K$ is always zero regardless of the boundary conditions is this: $0 \\rightarrow R^1\\xrightarrow{}R^2\\xrightarrow{}R^1$ Here, the determinantal ideal is $(1)=R$ , and thus has no solution.",
"The intuition from these examples are made rigorous below."
],
[
"Condition for degenerate Hamiltonians",
"A routine yet very important tool is localization.",
"The origin of all difficulties in dealing with general rings is that nonzero elements do not always have multiplicative inverse; one cannot easily solve linear equations.",
"The localization is a powerful technique to get around this problem.",
"As we build rational numbers from integers by declaring that nonzero numbers have multiplicative inverse, the localization enlarges a given ring and formally allows certain elements to be invertible.",
"It is necessary and sometimes desirable not to invert all nonzero elements, in order for the localization to be useful.",
"For a consistent definition, we need a multiplicatively closed subset $S$ containing 1, but not containing 0, of a ring $R$ and declare that the elements of $S$ is invertible.",
"The new ring is written as $S^{-1}R$ , in which a usual formula $\\frac{r_1}{s_1} + \\frac{r_2}{s_2} = \\frac{r_1 s_2 + r_2 s_1}{s_1 s_2}$ holds.",
"The original ring naturally maps into $S^{-1}R$ as $\\phi : r \\mapsto \\frac{r}{1}$ .",
"The localization means that one views all data as defined over $S^{-1}R$ via the natural map $\\phi $ .It is a functor from the category of $R$ -modules to that of $S^{-1}R$ -modules.",
"A localized ring, by definition, has more invertible elements, and hence has less nontrivial ideals.",
"In fact, our Laurent polynomial ring is a localized ring of the polynomial ring by inverting monomials, e.g., $\\lbrace x^i y^j | i,j \\ge 0 \\rbrace $ .",
"Nontrivial ideals such as $(x)$ or $(x,y)$ in the polynomial ring become the unit ideal $(1)$ in the Laurent polynomial ring.",
"Further localizations in this paper are with respect to prime ideals.",
"In this case, we say the ring is localized at a prime ideal $\\mathfrak {p}$.",
"A prime ideal $\\mathfrak {p}$ has a defining property that $a b \\notin \\mathfrak {p}$ whenever $a \\notin \\mathfrak {p}$ and $b \\notin \\mathfrak {p}$ .",
"Thus, the set-theoretic complement of $\\mathfrak {p}$ is a multiplicatively closed set containing 1.",
"In $(R\\setminus \\mathfrak {p})^{-1}R$ , denoted by $R_\\mathfrak {p}$ , any element outside $\\mathfrak {p}$ is invertible, and therefore $\\mathfrak {p}$ becomes a unique maximal ideal of $R_\\mathfrak {p}$ .",
"Moreover, the localization sometimes simplifies the generators of an ideal.",
"For instance, if $R=\\mathbb {F}[x,x^{-1}]$ and $\\mathfrak {p}= (x-1)$ , the ideal $((x -1)(x^5-x+1)) \\subseteq R$ localizes to $(x-1)_\\mathfrak {p}\\subseteq R_\\mathfrak {p}$ since $x^5-x+1$ is an invertible element of $R_\\mathfrak {p}$ .",
"An important fact about the localization is that a module is zero if and only if its localization at every prime ideal is zero.",
"Further, the localization preserves exact sequences.",
"So we can analyze a complex by localizing at various prime ideals.",
"For a thorough treatment about localizations, see Chapter 3 of [31].",
"The term `localization' is from geometric considerations where a ring is viewed as a function space on a geometric space.",
"Lemma 4.1 Let $I$ be the associated ideal of an exact code Hamiltonian, and $\\mathfrak {m}$ be a prime ideal of $R$ .",
"Then, $I \\lnot \\subseteq \\mathfrak {m}$ implies that the localized homology $K(L)_\\mathfrak {m}= \\ker (\\epsilon _L)_\\mathfrak {m}~/~ \\mathop {\\mathrm {im}}(\\sigma _L)_\\mathfrak {m}$ is zero for all $L \\ge 1$ .",
"It is a simple variant of a well-known fact that a module over a local ring is free if its first non-vanishing Fitting ideal is the unit ideal [30].",
"Recall that the localization and the factoring commute.",
"By assumption, $(I_q(\\epsilon ))_\\mathfrak {m}= \\overline{ (I_q(\\sigma ))_\\mathfrak {m}} = (1) = R_\\mathfrak {m}=: S$ .",
"Recall that the local ring $S$ has the unique maximal ideal $\\mathfrak {m}$ , and any element outside the maximal ideal is a unit.",
"If every entry of $\\epsilon $ is in $\\mathfrak {m}$ , then $I_q(\\epsilon ) \\subseteq \\mathfrak {m}\\ne S$ .",
"Therefore, there is a unit entry, and by column and row operations, $\\epsilon $ is brought to $\\epsilon \\cong \\begin{pmatrix}1 & 0 \\\\0 & \\epsilon ^{\\prime }\\end{pmatrix}$ where $\\epsilon ^{\\prime }$ is a submatrix.",
"It is clear that $I_{q-1}(\\epsilon ^{\\prime }) \\subseteq I_q(\\epsilon )$ since any $q-1 \\times q-1$ submatrix of $\\epsilon ^{\\prime }$ can be thought of as a $q \\times q$ submatrix of $\\epsilon $ where the first column and first row have the unique nonzero entry 1 at $(1,1)$ .",
"It is also clear that $I_{q-1}(\\epsilon ^{\\prime }) \\supseteq I_q(\\epsilon )$ since any $q \\times q$ submatrix of $\\epsilon $ contains either zero row or column, or the $(1,1)$ entry 1 of $\\epsilon $ .",
"Hence, $I_{q-1}(\\epsilon ^{\\prime }) = (1)$ , and we can keep extracting unit elements into the diagonal by row and column operations [30].",
"After $q$ steps, $t \\times 2q$ matrix $\\epsilon $ becomes precisely $\\epsilon \\cong \\begin{pmatrix}\\mathrm {id}_q & 0 \\\\0 & 0\\end{pmatrix}$ where $\\mathrm {id}_q$ is the $q \\times q$ identity matrix.",
"Since localization preserves the exact sequence $G \\rightarrow P \\rightarrow E$ , $\\sigma $ maps to the lower $q$ components of $P$ with respect to the basis where $\\epsilon $ is in the above form.",
"Since $I_q(\\sigma ) = (1)$ , we must have (after basis change) $\\sigma \\cong \\begin{pmatrix}0 & 0 \\\\\\mathrm {id}_q & 0\\end{pmatrix}.$ Therefore, even after factoring by the proper ideal $\\mathfrak {b}_L$ , the homology $K(L) = \\ker \\epsilon _L ~/~ \\mathop {\\mathrm {im}}\\sigma _L$ is still zero.",
"Corollary 4.2 The associated ideal of an exact code Hamiltonian is the unit ideal, i.e., $I_q(\\sigma ) = R$ , if and only if $K(L) = \\ker \\epsilon _L ~/~ \\mathop {\\mathrm {im}}\\sigma _L = 0$ for all $L \\ge 1$ .",
"If $I(\\sigma ) = R$ , $I(\\sigma )$ is not contained in any prime ideal $\\mathfrak {m}$ .",
"The above lemma says $K(L)_\\mathfrak {m}= 0$ .",
"Since a module is zero if and only if its localization at every prime ideal is zero, $K(L) = 0$ for all $L \\ge 1$ .",
"For the converse, observe that if $\\mathbb {F}$ is any extension field of $\\mathbb {F}_2$ , for any $\\mathbb {F}_2$ -vector space $W$ , we have $\\dim _\\mathbb {F}\\mathbb {F}\\otimes _{\\mathbb {F}_2} W = \\dim _{\\mathbb {F}_2} W$ .",
"We replace the ground field $\\mathbb {F}_2$ with its algebraic closure $\\mathbb {F}^a$ to test whether $K(L) \\ne 0$ .",
"If $I_q(\\sigma )$ is not the unit ideal, then it is contained in a maximal ideal $\\mathfrak {m}\\subsetneq R$ .",
"By Nullstellensatz, $\\mathfrak {m}= (x_1 - a_1,\\ldots ,x_D - a_D)$ for some $a_i \\in \\mathbb {F}^a$ .",
"Since in $R$ any monomial is a unit, we have $a_i \\ne 0$ .",
"Therefore, there exists $L \\ge 1$ such that $a_i^L = 1$ and $2 \\nmid L$ .",
"The equation $x^L-1=0$ has no multiple root.",
"We claim that $K(L) \\ne 0$ .",
"It is enough to verify this for the localization at $\\mathfrak {m}$ .",
"Since anything outside $\\mathfrak {m}$ is a unit in $R_\\mathfrak {m}$ and each $x_i^L-1$ contains exactly one $x_i - a_i$ factor, we see $(\\mathfrak {b}_L)_\\mathfrak {m}= \\mathfrak {m}_\\mathfrak {m}$ .",
"Therefore, $(\\epsilon _L)_\\mathfrak {m}= \\epsilon _\\mathfrak {m}/ (\\mathfrak {b}_L)_\\mathfrak {m}$ and $(\\sigma _L)_\\mathfrak {m}= \\sigma _\\mathfrak {m}/ (\\mathfrak {b}_L)_\\mathfrak {m}$ is a matrix over the field $R/\\mathfrak {m}= \\mathbb {F}^a$ .",
"Since $I_q(\\sigma ) \\subseteq \\mathfrak {m}$ , we have $I_q(\\sigma _L)_\\mathfrak {m}= 0$ .",
"That is, $\\mathop {\\mathrm {rank}}_{\\mathbb {F}^a} (\\sigma _L)_\\mathfrak {m}< q$ .",
"It is clear that $\\dim _{\\mathbb {F}^a} K(L)_\\mathfrak {m}= \\dim _{\\mathbb {F}^a} \\ker (\\epsilon _L)_\\mathfrak {m}/ \\mathop {\\mathrm {im}}(\\sigma _L)_\\mathfrak {m}\\ge 1$ .",
"This corollary says that in order to have a degenerate Hamiltonian $H(L)$ , one must have a proper associated ideal.",
"We shall simply speak of a degenerate code Hamiltonian if its associated ideal is proper."
],
[
"Counting points in an algebraic set",
"It is important that the factor ring $R/\\mathfrak {b}_L = \\mathbb {F}_2 [x_1, \\ldots , x_D]~/~(x^L_1 -1,\\ldots ,x^L_D -1)$ is finite dimensional as a vector space over $\\mathbb {F}_2$ , and hence is Artinian.",
"In fact, $\\dim _{\\mathbb {F}_2} R/\\mathfrak {b}_L = L^D$ .",
"This ring appears also in [18].",
"Due to the following structure theorem of Artinian rings, $K(L)$ can be explicitly analyzed by the localizations.",
"Proposition 4.3 [31][27] Let $S$ be an Artinian ring.",
"(For example, $S$ is a homomorphic image of a polynomial ring over finitely many variables with coefficients in a field $\\mathbb {F}$ , and is finite dimensional as a vector space over $\\mathbb {F}$ .)",
"Then, there are only finitely many maximal ideals of $S$ , and $S \\cong \\bigoplus _\\mathfrak {m}S_\\mathfrak {m}$ where the sum is over all maximal ideals $\\mathfrak {m}$ of $S$ and $S_\\mathfrak {m}$ is the localization of $S$ at $\\mathfrak {m}$ .",
"The following calculation tool is sometimes useful.",
"Recall that a group algebra is equipped with a non-degenerate scalar product $\\langle v,w \\rangle = \\mathop {\\mathrm {tr}}(v \\bar{w})$ .",
"This scalar product naturally extends to a direct sum of group algebras.",
"Lemma 4.4 Let $\\mathbb {F}$ be a field, and $S = \\mathbb {F}[\\Lambda ]$ be the group algebra of a finite abelian group $\\Lambda $ .",
"If $N$ is a submodule of $S^n$ , then the dual vector space $N^*$ is vector-space isomorphic to $S^n / N^\\perp $ , where $\\perp $ is with respect to the scalar product $\\langle \\cdot , \\cdot \\rangle $ .",
"Consider $\\phi : S^n \\ni x \\mapsto \\langle \\cdot , x \\rangle \\in N^*$ .",
"The map $\\phi $ is surjective since the scalar product is non-degenerate and $S^n$ is a finite dimensional vector space.",
"The kernel of $\\phi $ is precisely $N^\\perp $ .",
"Corollary 4.5 Put $2k = \\dim _{\\mathbb {F}_2} K(L)$ .",
"Then, $k = qL^D - \\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {im}}\\sigma _L = \\dim _{\\mathbb {F}_2} \\ker \\epsilon _L - qL^D.$ Further, if $q=t$ , then $k = \\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {coker}}\\epsilon _L .$ The first formula is a rephrasing of the fact that the number of encoded qubits is the total number of qubits minus the number of independent stabilizer generators [22], [10].",
"Put $S = R/\\mathfrak {b}_L$ .",
"If $v_1,\\ldots , v_t$ denote the columns of $\\sigma _L$ , we have $\\ker \\sigma _L^\\dagger = \\lambda _q \\ker \\epsilon _L= \\bigcap _i v_i^\\perp = \\left( \\sum _i S v_i \\right)^\\perp = \\left( \\mathop {\\mathrm {im}}\\sigma _L \\right)^\\perp .$ Hence, $\\dim _{\\mathbb {F}_2} \\ker \\epsilon _L = \\dim _{\\mathbb {F}_2} S^{2q} - \\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {im}}\\sigma _L.$ Since $\\dim _{\\mathbb {F}_2} S = L^D$ and $K(L) = \\ker \\epsilon _L / \\mathop {\\mathrm {im}}\\sigma _L$ , the first claim follows.",
"Since $\\mathop {\\mathrm {im}}\\sigma _L \\cong S^t / \\ker \\sigma _L$ , if $t=q$ , we have $k = \\dim _{\\mathbb {F}_2} \\ker \\sigma _L$ by the first claim.",
"From Eq.",
"(REF ), we conclude that $k = \\dim _{\\mathbb {F}_2} S^t / \\mathop {\\mathrm {im}}\\sigma _L^\\dagger = \\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {coker}}\\epsilon _L$ .",
"We will apply these formulas in Section .",
"The characteristic dimension is related to the rate at which the degeneracy increases as the system size increases in the following sense.",
"Recall that $2k = \\dim _{\\mathbb {F}_2} K(L)$ and the ground state degeneracy is $2^k$ .",
"Lemma 4.6 Suppose $2 \\nmid L$ .",
"Let $\\mathbb {F}^a$ be the algebraic closure of $\\mathbb {F}_2$ .",
"If $N$ is the number of maximal ideals in $\\mathbb {F}^a \\otimes _{\\mathbb {F}_2} R$ that contains $\\mathfrak {b}_L + I_q(\\sigma )$ , then $N \\le \\dim _{\\mathbb {F}_2} K(L) \\le 2q N.$ We replace the ground field $\\mathbb {F}_2$ with $\\mathbb {F}^a$ .",
"Any maximal ideal of an Artinian ring $\\mathbb {F}^a[x_i^{\\pm 1}]/\\mathfrak {b}_L$ is of form $\\mathfrak {m}= (x_1 - a_1, \\ldots , x_D - a_D)$ where $a_i^L = 1$ by Nullstellensatz.",
"Since $2 \\nmid L$ , we see that $(\\mathfrak {b}_L)_\\mathfrak {m}= \\mathfrak {m}_\\mathfrak {m}$ and that $(R/\\mathfrak {b}_L)_\\mathfrak {m}\\cong \\mathbb {F}^a$ is the ground field.",
"(See the proof of Corollary REF .)",
"Now, $I_q(\\sigma ) + \\mathfrak {b}_L \\subseteq \\mathfrak {m}$ iff $I_q(\\sigma )_\\mathfrak {m}+ (\\mathfrak {b}_L)_\\mathfrak {m}\\subseteq \\mathfrak {m}_\\mathfrak {m}= (\\mathfrak {b}_L)_\\mathfrak {m}$ iff $I_q(\\sigma )$ becomes zero over $R_\\mathfrak {m}/ (\\mathfrak {b}_L)_\\mathfrak {m}\\cong \\mathbb {F}^a$ iff $1 \\le \\dim _{\\mathbb {F}^a} K(L)_\\mathfrak {m}\\le 2q$ .",
"Since by Proposition REF , $\\dim _{\\mathbb {F}^a} K(L)$ is a finite direct sum of localized ones, we are done.",
"Lemma 4.7 Let $I$ be an ideal such that $\\dim R/I = d$ .",
"We have $\\dim _{\\mathbb {F}_2} R / (I+\\mathfrak {b}_L) \\le cL^d$ for all $L \\ge 1$ and some constant $c$ independent of $L$ .",
"We replace the ground field with its algebraic closure $\\mathbb {F}^a$ .",
"Write $\\tilde{x}_i$ for the image of $x_i$ in $R/I$ .",
"By Noether normalization theorem [27], there exist $y_1,\\ldots , y_d \\in R/I$ such that $R/I$ is a finitely generated module over $\\mathbb {F}^a[y_1,\\ldots ,y_d]$ .",
"Moreover, one can choose $y_i = \\sum _{j=1}^D M_{ij} \\tilde{x}_j$ for some rank $d$ matrix $M$ whose entries are in $\\mathbb {F}^a$ .",
"Making $M$ into the reduced row echelon form, we may assume $y_i = \\tilde{x}_i + \\sum _{j>d} a_{ij} \\tilde{x}_j$ for each $1 \\le i \\le d$ .",
"Let $S=\\mathbb {F}^a[z_1,\\ldots ,z_D]$ be a polynomial ring in $D$ variables.",
"Let $\\phi : S \\rightarrow R/( I + \\mathfrak {b}_L )$ be the ring homomorphism such that $z_i \\mapsto y_i$ for $1 \\le i \\le d$ and $z_j \\mapsto \\tilde{x}_j$ for $ d < j \\le D$ .",
"By the choice of $y_i$ , $\\phi $ is clearly surjective.",
"Consider the ideal $J$ of $S$ generated by the initial terms of $\\ker \\phi $ with respect to the lexicographical monomial order in which $z_1 \\prec \\cdots \\prec z_D$ .",
"Since $\\tilde{x}_j$ is integral over $\\mathbb {F}[y_1,\\ldots ,y_d]$ , the monomial ideal $J$ contains $z_j^{n_j}$ for some positive $n_j$ for $d < j \\le D$ .",
"Here, $n_j$ is independent of $L$ .",
"Since $z_i^L \\in J$ for $1 \\le i \\le d$ , we conclude that $\\dim _{\\mathbb {F}^a} R/(I+\\mathfrak {b}_L) = \\dim _{\\mathbb {F}^a} S/J \\le L^d \\cdot n_{d+1} n_{d+2} \\cdots n_{D}$ by Macaulay theorem [27].",
"Corollary 4.8 If $2 \\nmid L$ , and $d = \\dim R/I_q(\\sigma )$ is the characteristic dimension of a code Hamiltonian, then $\\dim _{\\mathbb {F}_2} K(L) \\le c L^d$ for some constant $c$ independent of $L$ .",
"If $J = \\mathfrak {b}_L + I(\\sigma )$ , $N$ in Lemma REF is equal to $\\dim _{\\mathbb {F}^a} \\mathbb {F}^a \\otimes R /\\mathop {\\mathrm {rad}} J$ .",
"This is at most $\\dim _{\\mathbb {F}^a} \\mathbb {F}^a \\otimes R / J = \\dim _{\\mathbb {F}_2} R/J$ .",
"Lemma 4.9 Let $d$ be the characteristic dimension.",
"There exists an infinite set of integers $\\lbrace L_i \\rbrace $ such that $\\dim _{\\mathbb {F}_2} K(L_i) \\ge {L_i}^d /2$ We replace the ground field with its algebraic closure $\\mathbb {F}^a$ .",
"Let $\\mathfrak {p}^{\\prime } \\supseteq I(\\sigma )$ be a prime of $R$ of codimension $D-d$ .",
"Let $\\mathfrak {p}$ be the contraction (pull-back) of $\\mathfrak {p}^{\\prime }$ in the polynomial ring $S = \\mathbb {F}^a[x_1,\\ldots ,x_D]$ .",
"Since the set of all primes of $R$ is in one-to-one correspondence with the set of primes in $S$ that does not include monomials, it follows that $\\mathfrak {p}$ has codimension $D-d$ and does not contain any monomials.",
"Let $V$ denote the affine variety defined by $\\mathfrak {p}=(g_1,\\ldots ,g_n)$ .",
"Since $\\mathfrak {p}$ contains no monomials, $V$ is not contained in any hyperplanes $x_i = 0$ ($i = 1,\\ldots , D$ ).",
"Let $A_1$ be a finite subfield of $\\mathbb {F}^a$ that contains all the coefficients of $g_i$ , so $V$ can be defined over $A_1$ .",
"Let $A_n \\subseteq \\mathbb {F}^a$ be the finite extension fields of $A_1$ of extension degree $n$ .",
"Put $L_n = |A_n|-1$ .",
"For any subfield $A$ of $\\mathbb {F}^a$ , let us say a point of $V$ is rational over $A$ if its coordinates are in $A$ .",
"The number $N^{\\prime }(L_n)$ of points $(a_i) \\in V$ satisfying $a_i^{L_n} = 1$ is precisely the number of the rational points of $V$ over $A_n$ that are not contained in the hyperplanes $x_i=0$ .",
"Since $I(\\sigma ) \\subseteq \\mathfrak {p}^{\\prime }$ , the number $N$ in Lemma REF is at least $N^{\\prime }(L_n)$ .",
"It remains to show $N^{\\prime }(L_n) \\ge L_n^d /2$ for all sufficiently large $n$ .",
"This follows from the result by Lang and Weil [32], which states that the number of points of a projective variety of dimension $d$ that are rational over a finite field of $m$ elements is $m^d + O\\left(m^{d-\\frac{1}{2}} \\right)$ asymptotically in $m$ .",
"Since Lang-Weil theorem is for projective variety and we are with an affine variety $V$ , we need to subtract the number of points in the hyperplanes $x_i = 0$ ($i = 0,1,\\ldots ,D$ ) from the Zariski closure of $V$ .",
"The subvarieties in the hyperplanes, being closed, have strictly smaller dimensions, and we are done."
],
[
"Fractal operators and topological charges",
"This section is to provide a characterization of topological charges, and their dynamical properties.",
"Before we turn to a general characterization and define fractal operators, let us review familiar examples.",
"Note that for two dimensions the base ring is $R = \\mathbb {F}_2[x,\\bar{x}, y, \\bar{y}]$ .",
"Example 2 (Toric Code) Although the original two-dimensional toric code has qubits on edges [1], we put two qubits per site of the square lattice to fit it into our setting.",
"Concretely, the first qubit to each site represents the one on its east edge, and the second qubit the one on its north edge.",
"With this convention, the Hamiltonian is the negative sum of the following two types of interactions: $@!0{XI @{-}[r] & XX @{-}[d] \\\\II @{-}[u] & IX @{-}[l]} \\quad @!0{ZI @{-}[r] & II @{-}[d] \\\\ZZ @{-}[u] & IZ @{-}[l]} \\quad \\quad @!0{y @{-}[r] & xy @{-}[d] \\\\1 @{-}[u] & x @{-}[l]}$ where we used $X,Z$ to abbreviate $\\sigma _x,\\sigma _z$ , and omitted the tensor product symbol.",
"Here, the third square specifies the coordinate system of the square lattice.",
"Since there are $q=2$ qubits per site, the Pauli module is of rank 4.",
"The corresponding generating map $\\sigma : R^2 \\rightarrow R^4$ is given by the matrix $\\sigma _{\\text{2D-toric}} =\\begin{pmatrix}y+xy & 0 \\\\x+xy & 0 \\\\\\hline 0 & 1 + y \\\\0 & 1 + x\\end{pmatrix}\\cong \\begin{pmatrix}1 + \\bar{x} & 0 \\\\1 + \\bar{y} & 0 \\\\\\hline 0 & 1 + y \\\\0 & 1 + x\\end{pmatrix} .$ Here, the each column expresses each type of interaction.",
"It is clear that $\\epsilon _{\\text{2D-toric}} = \\sigma ^\\dagger \\lambda _2 =\\begin{pmatrix}0 & 0 & 1 + x & 1 + y \\\\1+ \\bar{y} & 1+ \\bar{x} & 0 & 0\\end{pmatrix}$ and $\\ker \\epsilon = \\mathop {\\mathrm {im}}\\sigma $ ; the two dimensional toric code satisfies our exactness condition.",
"The associated ideal is $I(\\sigma ) = ( (1+x)^2, (1+x)(1+y), (1+y)^2 )$ .",
"The characteristic dimension is $\\dim R / I(\\sigma ) = 0$ .",
"Note also that $\\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon = (x-1,y-1)$ .",
"The electric and magnetic charge are represented by $\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}, \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\in E \\setminus \\mathop {\\mathrm {im}}\\epsilon $ , respectively.",
"The connection with cellular homology should be mentioned.",
"$\\sigma $ can be viewed as the boundary map from the free module of all 2-cells with $\\mathbb {Z}_2$ coefficients of the cell structure of 2-torus induced from the tessellation by the square lattice.",
"Then, $\\epsilon $ is interpreted as the boundary map from the free module of all 1-cells to that of all 0-cells.",
"$\\sigma $ or $\\epsilon $ is actually the direct sum of two boundary maps.",
"Indeed, the space $K(L) = \\ker \\epsilon _L / \\mathop {\\mathrm {im}}\\sigma _L$ of operators acting on the ground space (logical operators) has four generators $l_y(X) = \\begin{pmatrix} 1+y+\\cdots +y^{L-1} \\\\ 0 \\\\ 0 \\\\ 0 \\end{pmatrix}, & &l_x(X) = \\begin{pmatrix} 0 \\\\ 1+ x+ \\cdots + x^{L-1} \\\\ 0 \\\\ 0 \\end{pmatrix}, \\\\l_x(Z) = \\begin{pmatrix} 0 \\\\ 0 \\\\ 1+x+\\cdots +x^{L-1} \\\\ 0 \\end{pmatrix}, & &l_y(Z) = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1+y+\\cdots +y^{L-1} \\end{pmatrix},$ which correspond to the usual nontrivial first homology classes of 2-torus.",
"The description by the cellular homology might be advantageous for the toric code over our description with pure Laurent polynomials; in this way, it is clear that the toric code can be defined on an arbitrary tessellation of compact orientable surfaces.",
"However, it is unclear whether this cellular homology description is possible after all for other topologically ordered code Hamiltonians.",
"$\\Diamond $ Example 3 (2D Ising model on square lattice) The Ising model has nearest neighbor interactions that are horizontal and vertical.",
"In our formalism, they are represented as $1+x$ and $1+y$ .",
"Thus, $\\sigma _\\text{2D Ising} =\\begin{pmatrix}0 & 0 \\\\1+x & 1+y\\end{pmatrix} .$ As it is not topologically ordered, the complex $G \\rightarrow P \\rightarrow E$ is not exact.",
"Moreover, $\\sigma $ is not injective.",
"$\\sigma _\\text{2D Ising;1} =\\begin{pmatrix}1+y \\\\1+x\\end{pmatrix}$ generates the kernel of $\\sigma $ .",
"That is, the complex $0 \\rightarrow G_1 \\xrightarrow{} G \\xrightarrow{} P$ is exact.",
"$\\Diamond $ In both examples, there exist isolated excitations.",
"In the toric code, the isolated excitation can be (topologically) nontrivial since the electric charge is not in $\\mathop {\\mathrm {im}}\\epsilon $ .",
"On the contrary, in 2D Ising model, any isolated excitation is actually created by an operator of finite support because any excitation created by some Pauli operator appears as several connected loops.",
"This difference motivates the following definition for charges.",
"Let $\\tilde{R}$ be the set of all $\\mathbb {F}_2$ -valued functions on the translation group $\\Lambda $ , not necessarily finitely supported.",
"For instance, if $\\Lambda = \\mathbb {Z}$ , $\\tilde{f} = \\cdots + x^{-4} + x^{-2} + 1 + x^2 + x^4 + \\cdots \\in \\tilde{R}$ represents a function whose value is 1 at even lattice points, and 0 at odd points.",
"Note that $\\tilde{R}$ is a $R$ -module, since the multiplication is a convolution between an arbitrary function and a finitely supported function.",
"For example, $(1+x) \\cdot \\tilde{f} &= \\cdots + x^{-2} + x^{-1} + 1 + x + x^2 + \\cdots , \\\\(1+x)^2 \\cdot \\tilde{f} &= 0 .$ Let $\\tilde{P} = \\tilde{R}^{2q}$ be the module of Pauli operators of possibly infinite support.",
"Similarly, let $\\tilde{E}$ be the module of virtual excitations of possibly infinitely many terms.",
"Formally, $\\tilde{P}$ is the module of all $2q$ -tuples of functions on the translation-group, and $\\tilde{E}$ is that of all $t$ -tuples.",
"Clearly, $P \\subseteq \\tilde{P}$ and $E \\subseteq \\tilde{E}$ .",
"The containment is strict if and only if the translation-group is infinite.",
"Since the matrix $\\epsilon $ consists of Laurent polynomials with finitely many terms, $\\epsilon :P \\rightarrow E$ extends to a map from $\\tilde{P}$ to $\\tilde{E}$ .",
"Definition 9 A (topological) charge $e = \\epsilon ( \\tilde{p} ) \\in E$ is an excitation of finite energy (an element of the virtual excitation module) created by a Pauli operator $\\tilde{p} \\in \\tilde{P}$ of possibly infinite support.",
"A charge $e$ is called trivial if $e \\in \\epsilon (P)$ .",
"By definition, the set of all charges modulo trivial ones is in one-to-one correspondence with the superselection sectors.",
"According to the definition, any charge of 2D Ising model is trivial.",
"A nontrivial charge may appear due to the following fractal generators.",
"Definition 10 We call zero-divisors on $\\mathop {\\mathrm {coker}}\\epsilon $ as fractal generators.",
"In other words, an element $f \\in R \\setminus \\lbrace 0\\rbrace $ is a fractal generator if there exists $v \\in E \\setminus \\mathop {\\mathrm {im}}\\epsilon $ such that $f v \\in \\mathop {\\mathrm {im}}\\epsilon $ .",
"There is a natural reason the fractal generator deserves its name.",
"Consider a code Hamiltonian with a single type of interaction: $t=1$ .",
"So each configuration of excitations is described by one Laurent polynomial.",
"For example, in two dimensions, $f = 1 + x + y = \\epsilon (p)$ represents three excitations, one at the origin of the lattice and the others at $(1,0)$ and $(0,1)$ created by a Pauli operator represented by $p$ .",
"(This example is adopted from [33].)",
"In order to avoid repeating phrase, let us call each element of the Pauli module a Pauli operator, and instead of using multiplicative notation we use module operation $+$ to mean the product of the corresponding Pauli operators.",
"Consider the Pauli operator $fp = p + x p + y p \\in P$ .",
"It describes the Pauli operator $p$ at the origin multiplied by the translations of $p$ at $(1,0)$ and at $(0,1)$ .",
"So $fp$ consists of three copies of $p$ .",
"This Pauli operator maps the ground state to the excited state $f^2 = 1 + x^2 + y^2$ .",
"The number of excitations is still three, but the excitations at $(1,0), (0,1)$ have been replaced by those at $(2,0),(0,2)$ .",
"Similarly, the Pauli operator $f^{2+1} p = f^2 (fp)$ consists of three copies of $fp$ , or $3^2$ copies of $p$ .",
"The excited state created by $f^3 p$ is $f^4 = (f^2)^2 = 1 + x^{2^2} + y^{2^2}$ .",
"Still it has three excitations, but they are further apart.",
"The Pauli operator $f^{2^n -1} p$ consists of $3^n$ copies of $p$ in a self-similar way, and the excited state caused by $f^{2^n -1} p$ consists of a constant number of excitations.",
"More generally, if there are $t > 1$ types of terms in the Hamiltonian, the excitations are described by a $t \\times 1$ matrix.",
"If it happens to be of form $f v$ for some $f \\in R$ consisted of two or more terms, there is a family of Pauli operators $f^{2^n -1} p$ with self-similar support such that it only creates a bounded number of excitations.",
"An obvious but uninteresting way to have such a situation is to put $f v = \\epsilon (f p^{\\prime })$ for a Pauli operator $p^{\\prime }$ where $v = \\epsilon ( p^{\\prime } )$ .",
"Our definition avoids this triviality by requiring $v \\notin \\mathop {\\mathrm {im}}\\epsilon $ .",
"The reader may wish to compare the fractals with finite cellular automata [12].",
"Proposition 5.1 [34] Suppose $\\mathop {\\mathrm {coker}}\\epsilon \\ne 0$ .",
"Then, the following are equivalent: There does not exist a fractal generator.",
"$\\mathop {\\mathrm {coker}}\\epsilon $ is torsion-free.",
"There exists a free $R$ -module $E^{\\prime }$ of finite rank such that $P \\xrightarrow{} E \\rightarrow E^{\\prime }$ is exact.",
"The first two are equivalent by definition.",
"The sequence above is exact if and only if $ 0 \\rightarrow \\mathop {\\mathrm {coker}}{\\epsilon } \\rightarrow E^{\\prime } $ is exact.",
"Since $\\mathop {\\mathrm {coker}}\\epsilon $ has a finite free resolution, the second is equivalent to the third.",
"The following theorem states that the fractal operators produces all nontrivial charges.",
"Theorem 1 Suppose $\\Lambda = \\mathbb {Z}^D$ is the translation-group of the underlying lattice.",
"The set of all charges modulo trivial ones is in one-to-one correspondence with the torsion submodule of $\\mathop {\\mathrm {coker}}\\epsilon $ .",
"To illustrate the idea of the proof, consider a (classical) excitation mapIt is classical because it is not derived from an interesting quantum commuting Pauli Hamiltonian.",
"For a classical Hamiltonian where all terms are tensor products of $\\sigma _z$ , there is no need to keep a $t \\times 2q$ matrix $\\epsilon $ since the right half $\\epsilon $ is zero.",
"Just the left half suffices, which can be arbitrary since the commutativity equation $\\epsilon \\lambda \\epsilon ^\\dagger =0$ is automatic.",
"Nevertheless, the excitations and fractal operators are relevant.",
"Our proof of the theorem is not contingent on the commutativity equation.",
"$\\phi =\\begin{pmatrix}1+x+y & 0 \\\\0 & 1+x \\\\0 & 1+y \\\\\\end{pmatrix}: R^2 \\rightarrow R^3 .$ A nonzero element $f=1+x+y \\in R$ is a fractal generator since $\\begin{pmatrix} 1 & 0 & 0 \\end{pmatrix}^T \\notin \\mathop {\\mathrm {im}}\\phi $ and $(1+x+y)\\begin{pmatrix} 1 & 0 & 0 \\end{pmatrix}^T \\in \\mathop {\\mathrm {im}}\\phi $ ; $f$ is a zero-divisor on a torsion element $\\begin{pmatrix} 1 & 0 & 0 \\end{pmatrix}^T \\in \\mathop {\\mathrm {coker}}\\epsilon $ .",
"It is indeed a charge since $\\phi (\\tilde{f} \\begin{pmatrix} 1 & 0 \\end{pmatrix}^T) = \\begin{pmatrix} 1 & 0 & 0 \\end{pmatrix}^T$ where $\\tilde{f} = \\lim _{n \\rightarrow \\infty } f^{2^n-1} \\in \\mathbb {F}[[x,y]]$ is a formal power series, which can be viewed as an element of $\\tilde{R}$ .",
"The limit is well-defined since $f^{2^{n+1}-1} - f^{2^n -1}$ only contains terms of degree $2^n$ or higher.",
"That is to say, only higer order `corrections' are added and lower order terms are not affected.",
"Of course, there is no natural notion of smallness in the ring $\\mathbb {F}[x,y]$ .",
"But one can formally call the members of the ideal power $(x,y)^n \\subseteq \\mathbb {F}[x,y]$ small.",
"It is legitimate to introduce a topology in $R$ defined by the ever shrinking ideal powers $(x,y)^n$ .",
"They play a role analogous to the ball of radius $1/n$ in a metric topological space.",
"The completion of $\\mathbb {F}[x,y]$ where every Cauchy sequence with respect to this topology is promoted to a convergent sequence, is nothing but the formal power series ring $\\mathbb {F}[[x,y]]$ .",
"For a detailed treatment, see Chapter 10 of [31].",
"The completion and the limit only make sense in the polynomial ring $\\mathbb {F}[x,y]$ .",
"The reason $\\tilde{f}$ is well-defined is that $f \\in \\mathbb {F}[x^{\\pm 1},y^{\\pm 1}]$ is accidentally expressed as a usual polynomial with lowest order term 1.",
"In the proof below we show that every fractal generator can be expressed in this way.",
"Hence, a torsion element of $\\mathop {\\mathrm {coker}}\\epsilon $ is really a charge.",
"For a module $M$ , let $T(M)$ denote the torsion submodule of $M$ : $T(M) = \\lbrace m \\in M \\ |\\ \\exists \\, r \\in R \\setminus \\lbrace 0\\rbrace \\text{ such that } rm = 0 \\rbrace $ Suppose first that $T(\\mathop {\\mathrm {coker}}\\epsilon ) = 0$ .",
"We claim that in this case there is no nontrivial charge.",
"Let $e = \\epsilon ( \\tilde{p} ) \\in E$ be a charge, where $\\tilde{p} \\in \\tilde{P}$ .",
"By Proposition REF we have an exact sequence of finitely generated free modules $ P \\xrightarrow{} E \\xrightarrow{} E_1$ .",
"Since the matrix $\\epsilon _1$ is over $R$ , the complex extends to a complex of modules of tuples of functions on the translation-group.",
"$\\tilde{P} \\xrightarrow{} \\tilde{E} \\xrightarrow{} \\tilde{E}_1$ (This extended sequence may not be exact.)",
"Then, $\\epsilon _1(e) = \\epsilon _1( \\epsilon ( \\tilde{p} ) ) = 0$ since $\\epsilon _1 \\circ \\epsilon = 0$ identically.",
"But, $e \\in E$ , and therefore, $e \\in \\ker \\epsilon _1 \\cap E = \\epsilon (P)$ .",
"It means that $e$ is a trivial charge, i.e., $e$ maps to zero in $\\mathop {\\mathrm {coker}}\\epsilon $ , and proves the claim.One may wish to consider $\\epsilon $ to consist of the second column of $\\phi $ above.",
"Then $\\epsilon _1 = \\begin{pmatrix} 1+y & -1-x \\end{pmatrix}$ .",
"Now, allow $\\mathop {\\mathrm {coker}}(P \\xrightarrow{} E)$ to contain torsion elements.",
"$Q = (\\mathop {\\mathrm {coker}}\\epsilon ) / T(\\mathop {\\mathrm {coker}}\\epsilon )$ is torsion-free, and is finitely presented as $Q = \\mathop {\\mathrm {coker}}( \\epsilon ^{\\prime } : P^{\\prime } \\rightarrow E )$ where $P^{\\prime }$ is a finitely generated free module.",
"In fact, we may choose $\\epsilon ^{\\prime }$ by adding more columns representing the generators of the torsion submodule of $\\mathop {\\mathrm {coker}}\\epsilon $ to the matrix $\\epsilon $ .",
"$\\epsilon = \\begin{pmatrix} \\# & \\# \\\\ \\# & \\# \\end{pmatrix} \\quad \\quad \\epsilon ^{\\prime }= \\begin{pmatrix} \\# & \\# & * & * \\\\ \\# & \\# & * & * \\end{pmatrix}$ Then, $P$ can be regarded as a direct summand of $P^{\\prime }$ .If we take $\\epsilon = \\phi $ above, then $\\epsilon ^{\\prime } = \\begin{pmatrix}1+x+y & 0 & 1 \\\\0 & 1+x & 0 \\\\0 & 1+y & 0 \\\\\\end{pmatrix}.$ Note that $P^{\\prime } = P \\oplus R$ .",
"Let $e = \\epsilon ( \\tilde{p} ) \\in E$ be any charge.",
"Since the matrix $\\epsilon ^{\\prime }$ contains $\\epsilon $ as submatrix, we may write $e = \\epsilon ^{\\prime }(\\tilde{p}) \\in E$ .",
"Since $T(\\mathop {\\mathrm {coker}}\\epsilon ^{\\prime })=0$ , we see by the first part of the proof that $e = \\epsilon ^{\\prime }( p^{\\prime } )$ for some $p^{\\prime } \\in P^{\\prime }$ .",
"Then, $e$ maps to zero in $Q$ , and it follows that $e$ maps into $T(\\mathop {\\mathrm {coker}}\\epsilon )$ in $\\mathop {\\mathrm {coker}}\\epsilon $ .",
"In other words, the equivalence class of $e$ modulo trivial charges is a torsion element of $\\mathop {\\mathrm {coker}}\\epsilon $ .",
"Conversely, we have to prove that for every element $e \\in E$ such that $fe = \\epsilon (p)$ for some $f \\in R \\setminus \\lbrace 0\\rbrace $ and $p \\in P$ , there exists $\\tilde{p} \\in \\tilde{P}$ such that $ e = \\epsilon ( \\tilde{p} )$ .",
"Here, $\\tilde{P}$ is the module of all $2q$ -tuples of $\\mathbb {F}_2$ -valued functions on the translation-group.",
"Consider the lexicographic total order on $\\mathbb {Z}^D$ in which $x_1 \\succ x_2 \\succ \\cdots \\succ x_D$ .",
"It induces a total order on the monomials of $R$ .",
"Choose the least term $f_0$ of $f$ .",
"By multiplying $f_0^{-1}$ , we may assume $f_0 = 1$ .If $D=1$ , $f$ would be a polynomial of nonnegative exponents with the lowest order term being 1.",
"If $D=2$ and $f = y + y^2 + x$ , then the least term is $y$ .",
"After multiplying $f_0^{-1}$ , it becomes $1+y+xy^{-1}$ .",
"We claim that the sequence $f,\\ \\ f^2 f,\\ \\ f^4 f^2 f,\\ \\ \\ldots ,\\ \\ f^{2^n}f^{2^{n-1}} \\cdots f^2 f,\\ \\ \\ldots $ converges to $\\tilde{f} \\in \\tilde{R}$ , where $\\tilde{R}$ is the set of all $\\mathbb {F}_2$ -valued functions on $\\Lambda $ .",
"Given the claim, since $f^{2^n} e = e + (f-1)^{2^n}e = \\epsilon (f^{2^{n-1}} \\cdots f^2 f p)$ where $p \\in P$ , we conclude that $e = \\epsilon ( \\tilde{f} p )$ is a charge.",
"If $f$ is of nonnegative exponents, and hence $f \\in S=\\mathbb {F}_2[x_1,\\ldots ,x_D]$ , then the claim is clearly true.",
"Indeed, the positive degree terms of $f^{2^n} = 1 + (f-1)^{2^n}$ are in the ideal power $(x_1,\\ldots ,x_D)^{2^p} \\subset S$ .",
"Therefore, the sequence Eq.",
"(REF ) converges in the formal power series ring $\\mathbb {F}_2[[x_1,\\ldots ,x_D]]$ , which can be regarded as a subset of $\\tilde{R}$ .",
"If $f$ is not of nonnegative exponents, one can introduce the following change of basis of the lattice $\\mathbb {Z}^D$ such that $f$ becomes of nonnegative exponents.",
"In other words, the sequence Eq.",
"(REF ) is in fact contained in a ring that is isomorphic to the formal power series ring, where the convergence is clear.",
"For any nonnegative integers $m_1,\\ldots ,m_{D-1}$ , define a linear transformation $\\zeta _m =\\zeta _{(m_1,m_2,\\ldots ,m_{D-1})} :\\begin{pmatrix}a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_D\\end{pmatrix}\\mapsto \\begin{pmatrix}a^{\\prime }_1 \\\\ a^{\\prime }_2 \\\\ \\vdots \\\\ a^{\\prime }_D\\end{pmatrix}=\\begin{pmatrix}1 & 0 & 0 & \\cdots & 0 \\\\m_1 & 1 & 0 & & 0 \\\\m_1 & m_2 & 1 & & 0 \\\\\\vdots & & & \\ddots & \\vdots \\\\m_1 & m_2 & \\cdots & m_{D-1}& 1 \\\\\\end{pmatrix}\\begin{pmatrix}a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_D\\end{pmatrix}\\text{ on } \\mathbb {Z}^D .$ $\\zeta _m$ induces the map $x_1^{a_1} \\cdots x_D^{a_D} \\mapsto x_1^{a^{\\prime }_1} \\cdots x_D^{a^{\\prime }_D}$ on $R$ .",
"Let $u=x_1^{a_1} \\cdots x_D^{a_D}$ be an arbitrary term of $f$ other than 1, so $u \\succ 1$ .",
"For the smallest $i \\in \\lbrace 1,\\ldots ,D\\rbrace $ such that $a_i \\ne 0$ , one has $a_i > 0$ due to the lexicographic order.",
"Hence, if we choose $m_i$ large enough and set $m_j=0\\,(j\\ne i)$ , then $\\zeta _m (u)$ has nonnegative exponents.",
"Since any $\\zeta _{m}$ maps a nonnegative exponent term to a nonnegative exponent term, and there are only finitely many terms in $f$ , it follows that there is a finite composition $\\zeta $ of $\\zeta _m$ 's which maps $f$ to a polynomial of nonnegative exponents.For our previous example $f = 1+y+xy^{-1}$ , one takes $\\zeta : x^i y^j \\mapsto x^i y^{i+j}$ , so $\\zeta (f) = 1 + y + x$ .",
"Since a nontrivial charge $v$ has finite size anyway (the maximum exponent minus the minimum exponent of the Laurent polynomials in the $t \\times 1$ matrix $v$ ), we can say that the charge $v$ is point-like.",
"Moreover, we shall have a description how the point-like charge can be separated from the other by a local process.",
"By the local process we mean a sequence of Pauli operators $[[o_1, \\ldots , o_n]]$ such that $o_{i+1} - o_i$ is a monomial.",
"The number of excitations, i.e., energy, at an instant $i$ will be the number of terms in $\\epsilon (o_i)$ .",
"Theorem 2 [33] If there is a fractal generator of a code Hamiltonian, then for all sufficiently large $r$ , there is a local process starting from the identity by which a point-like charge is separated from the other excitations by distance at least $2^r$ .",
"One can choose the local process in such a way that at any intermediate step there are at most $c r$ excitations for some constant $c$ independent of $r$ .",
"For notational simplicity, we denote the local process $[[o_1,\\ldots ,o_n]]$ by $s = [o_1,~ o_2-o_1,~ o_3 - o_2, \\ldots , o_n - o_{n-1}].$ It is a recipe to construct $o_n$ , consisted of single qubit operators.",
"$o_n$ can be expressed as “$o_n = \\int s$ ”, the sum of all elements in the recipe.",
"Let $f$ be a fractal generator, and put $fv = \\epsilon (p)$ where $v \\notin \\mathop {\\mathrm {im}}\\epsilon $ .",
"We already know $v$ is a point-like nontrivial charge.",
"Write $p = \\sum _{i=1}^n p_i, \\quad f = \\sum _{i=1}^l f_i$ where each of $p_i$ and $f_i$ is a monomial.",
"Let $s_0 = [ 0, p_1, p_2, \\ldots , p_{n}]$ be a recipe for constructing $p$ ; $\\int s_0 = p$ .",
"Given $s_i$ , define inductively $s_{i+1} = (f_1^{2^i} \\cdot s_i) \\circ (f_2^{2^i} \\cdot s_i) \\circ \\cdots \\circ (f_l^{2^i} \\cdot s_i)$ where $\\circ $ denotes the concatenation and $f_i \\cdot [u_1,\\ldots ,u_{n^{\\prime }}] = [f_i u_1, \\ldots , f_i u_{n^{\\prime }}]$ .",
"It is clear that $s_{i+1}$ constructs the Pauli operator $\\int s_{r}= f^{2^{r-1}} \\int s_{r-1} = f^{2^{r-1}} f^{2^{r-2}} \\int s_{r-2}= f^{2^{r-1} + 2^{r-2} + \\cdots + 1} \\int s_0 = f^{2^r-1} p$ whose image under $\\epsilon $ is $f^{2^r}v$ .",
"Thus, if $r$ is large enough so that $2^r$ is greater than the size of $v$ , the configuration of excitations is precisely $l$ copies of $v$ .",
"The distance between $v$ 's is at least $2^r$ minus twice the size of $v$ .",
"Therefore, there is a constant $e > 0$ such that for any $r \\ge 0$ the energy of $f^{2^r}v \\in E$ is $\\le e$ .",
"Let $\\Delta (r)$ be the maximum energy during the process $s_r$ .",
"We prove by induction on $r$ that $\\Delta (r) \\le el (r+1).$ When $r = 0$ , it is trivial.",
"In $s_{r+1}$ , the energy is $\\le \\Delta (r)$ until $f_1^{2^r} s_r$ is finished.",
"At the end of $f_1^{2^r} s_r$ , the energy is $\\le e $ .",
"During the subsequent $f_2^{2^r} s_r$ , the energy is $\\le \\Delta (r) + e$ , and at the end of $(f_1^{2^r} s_r) \\circ (f_2^{2^r} s_r)$ , the energy is $\\le 2 e$ .",
"During the subsequent $f_j^{2^r} s_r$ , the energy is $\\le \\Delta (r) + je$ .",
"Therefore, $\\Delta (r+1) \\le \\Delta (r) + el \\le el(r+2)$ by the induction hypothesis.",
"Fractal operators appear in Newman-Moore model [33] where classical spin glass is discussed.",
"Their model has generating matrix $\\sigma = \\begin{pmatrix} 0 & 1+x+y \\end{pmatrix}^T$ .",
"The theorem is a simple generalization of Newman and Moore's construction.",
"Another explicit example of fractal operators in a quantum model can be found in [8].",
"Note that the notion of fractal generators includes that of `string operators'.",
"In fact, a fractal generator that contains exactly two terms gives a family of nontrivial string segments of unbounded length, as defined in [7].",
"Below, we point out a couple of sufficient conditions for nontrivial charges, or equivalently, fractal generators to exist.",
"Proposition 5.2 For code Hamiltonians, the existence of a fractal generator is a property of an equivalence class of Hamiltonians in the sense of Definition REF .",
"Suppose $\\mathop {\\mathrm {im}}\\sigma = \\mathop {\\mathrm {im}}\\sigma ^{\\prime }$ .",
"Each column of $\\sigma ^{\\prime }$ is a $R$ -linear combination of those of $\\sigma $ , and vice versa.",
"Thus, there is a matrix $B$ and $B^{\\prime }$ such that $\\epsilon ^{\\prime } = B\\epsilon $ and $\\epsilon = B^{\\prime } \\epsilon ^{\\prime }$ .",
"$BB^{\\prime }$ and $B^{\\prime }B$ are identity on $\\mathop {\\mathrm {im}}\\epsilon ^{\\prime }$ and $\\mathop {\\mathrm {im}}\\epsilon $ respectively.",
"In particular, $B^{\\prime }$ and $B$ are injective on $\\mathop {\\mathrm {im}}\\epsilon ^{\\prime }$ and $\\mathop {\\mathrm {im}}\\epsilon $ respectively.",
"Suppose $f$ is a fractal generator for $\\epsilon $ , i.e., $fv = \\epsilon p \\ne 0$ .",
"Then, $0 \\ne Bfv = f B v = B\\epsilon (p) = \\epsilon ^{\\prime }(p)$ .",
"If $Bv \\in \\mathop {\\mathrm {im}}\\epsilon ^{\\prime }$ , then $v = B^{\\prime }Bv \\in \\mathop {\\mathrm {im}}\\epsilon $ , a contradiction.",
"Therefore, $f$ is also a fractal generator for $\\epsilon ^{\\prime }$ .",
"By symmetry, a fractal generator for $\\epsilon ^{\\prime }$ is a fractal generator for $\\epsilon $ , too.",
"Suppose $R^{\\prime } \\subseteq R$ is a coarse-grained base ring.",
"If $\\mathop {\\mathrm {coker}}\\epsilon $ is torsion-free as an $R$ -module, then so it is as an $R^{\\prime }$ -module.",
"If $f \\in R$ is a fractal generator, the determinant of $f$ as a matrix over $R^{\\prime }$ is a fractal generator.",
"A symplectic transformation or tensoring ancillas does not change $\\mathop {\\mathrm {coker}}\\epsilon $ .",
"Proposition 5.3 For any ring $S$ and $t \\ge 1$ , if $0 \\rightarrow S^t \\rightarrow S^{2t} \\xrightarrow{} S^t$ is exact and $I(\\phi ) \\ne S$ , then $\\mathop {\\mathrm {coker}}\\phi $ is not torsion-free.",
"In particular, for a degenerate exact code Hamiltonian, if $\\sigma $ is injective, then there exists a fractal generator.",
"By Proposition REF , $\\mathop {\\mathrm {rank}}\\phi = t$ .",
"Since $0 \\subsetneq I_t(\\phi ) \\subsetneq S$ is the initial Fitting ideal, we have $0 \\ne \\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\phi \\ne S$ .",
"That is, $\\mathop {\\mathrm {coker}}\\phi $ is not torsion-free.",
"For the second statement, set $S=R$ .",
"If $\\sigma $ is injective, we have an exact sequence $0 \\rightarrow G \\xrightarrow{} P \\xrightarrow{} E .$ By Remark REF , $t = \\mathop {\\mathrm {rank}}G = \\mathop {\\mathrm {rank}}\\sigma = \\mathop {\\mathrm {rank}}\\epsilon = q$ .",
"Proposition 5.4 Suppose the characteristic dimension is $D-2$ for a degenerate exact code Hamiltonian.",
"Then, there exists a fractal generator.",
"Suppose on the contrary there are no fractal generators.",
"Then, by Proposition REF , $G \\xrightarrow{} P \\xrightarrow{} E \\rightarrow E^{\\prime }$ is exact for some finitely generated free module $E^{\\prime }$ .",
"Since $\\mathop {\\mathrm {coker}}\\epsilon $ has finite free resolution by Lemma REF , Proposition REF implies $\\mathop {\\mathrm {codim}}I(\\sigma ) \\ge 3$ unless $I(\\sigma ) = R$ .",
"But, $\\mathop {\\mathrm {codim}}I(\\sigma ) = 2$ and $I(\\sigma ) \\ne R$ by Corollary REF .",
"This is a contradiction."
],
[
"One dimension",
"The group algebra $R = \\mathbb {F}_2[x,\\bar{x}]$ for the one dimensional lattice $\\mathbb {Z}$ is a Euclidean domain where the degree of a polynomial is defined to be the maximum exponent minus the minimum exponent.",
"(In particular, any monomial has degree 0.)",
"Given two polynomials $f,g$ in $R$ , one can find their $\\gcd $ by the Euclid's algorithm.",
"It can be viewed as a column operation on the $1 \\times 2$ matrix $\\begin{pmatrix} f & g \\end{pmatrix}$ .",
"Similarly, one can find $\\gcd $ of $n$ polynomials by column operations on $1 \\times n$ matrix $\\begin{pmatrix}f_1 & f_2 & \\cdots & f_n\\end{pmatrix} .$ The resulting matrix after the Euclid's algorithm will be $\\begin{pmatrix}\\gcd (f_1,\\ldots ,f_n) & 0 & \\cdots & 0\\end{pmatrix} .$ Given a matrix $\\mathbf {M}$ of univariate polynomials, we can apply Euclid's algorithm to the first row and first column by elementary row and column operations in such a way that the degree of $(1,1)$ -entry $\\mathbf {M}_{11}$ decreases unless all other entries in the first row and column are divisible by $\\mathbf {M}_{11}$ .",
"Since the degree cannot decrease forever, this process must end with all entries in the first row and column being zero except $\\mathbf {M}_{11}$ .",
"By induction on the number of rows or columns, we conclude that $\\mathbf {M}$ can be transformed to a diagonal matrix by the elementary row and column operations.",
"This is known as the Smith's algorithm.",
"The following is a consequence of the finiteness of the ground field.",
"Lemma 6.1 Let $\\mathbb {F}$ be a finite field and $S = \\mathbb {F}[x]$ be a polynomial ring.",
"Let $\\phi : S \\xrightarrow{} S$ be a $1 \\times 1$ matrix such that $f(0) \\ne 0$ .",
"$\\phi $ can be viewed as an $n \\times n$ matrix acting on the free $S^{\\prime }$ -module $S$ where $S^{\\prime } = \\mathbb {F}[x^{\\prime }]$ and $x^{\\prime } =x^n$ .",
"Then, for some $n \\ge 1$ , the matrix $\\phi $ is transformed by elementary row and column operations into a diagonal matrix with entries 1 or $x^{\\prime }-1$ .",
"The number of $x^{\\prime }-1$ entries in the transformed $\\phi $ is equal to the degree of $f$ .",
"The splitting field $\\tilde{\\mathbb {F}}$ of $f(x)$ is a finite extension of $\\mathbb {F}$ .",
"Since $\\tilde{\\mathbb {F}}$ is finite, every root of $f(x)$ is a root of $x^{n^{\\prime }} -1$ for some $n^{\\prime } \\ge 1$ .",
"Choose an integer $p \\ge 1$ such that $2^p$ is greater than any multiplicity of the roots of $f(x)$ .",
"Then, clearly $f(x)$ divides $(x^{n^{\\prime }}-1)^{2^p} = x^{2^p n^{\\prime }} - 1$ .",
"Let $n$ be the smallest positive integer such that $f(x)$ divides $x^n-1$ .This part is well-known, at least in the linear cyclic coding theory [17].",
"Consider the coarse-graining by $S^{\\prime } = \\mathbb {F}[x^{\\prime }]$ where $x^{\\prime }=x^n$ .",
"$S$ is a free $S^{\\prime }$ -module of rank $n$ , and $(f)$ is now an endomorphism of the module $S$ represented as an $n \\times n$ matrix.",
"Since $f(x)g(x) = x^n -1$ for some $g(x) \\in \\mathbb {F}[x]$ , we have $A B = (x^{\\prime }-1) \\mathrm {id}_n$ where $x^{\\prime } = x^n$ , and $A, B$ are the matrix representation of $f(x)$ and $g(x)$ respectively as endomorphisms.",
"$A$ and $B$ have polynomial entries in variable $x^{\\prime }$ .",
"The determinants of $A,B$ are nonzero for their product is $(x^{\\prime }-1)^n \\ne 0$ .",
"Let $E_1$ and $E_2$ be the products of elementary matrices such that $A^{\\prime } = E_1 A E_2$ is diagonal.",
"Such matrices exist by the Smith's algorithm.",
"Put $B^{\\prime } = E_2^{-1} B E_1^{-1}$ .",
"Then, $A^{\\prime } B^{\\prime } = E_1 A E_2 E_2^{-1} B E_1^{-1} = E_1 A B E_1^{-1} = (x^{\\prime }-1)\\mathrm {id}_n .$ Since $A^{\\prime }$ and $I_n$ are diagonal of non-vanishing entries, $B^{\\prime }$ must be diagonal, too.",
"It follows that the diagonal entries of $A^{\\prime }$ divides $(x^{\\prime }-1)$ ; that is, they are 1 or $x^{\\prime }-1$ .",
"The number of entries $x^{\\prime }-1$ can be counted by considering $S/(f(x))$ as an $\\mathbb {F}$ -vector space.",
"It is clear that $\\dim _{\\mathbb {F}} S/(f(x)) = \\deg f(x)$ .",
"$S/(f(x)) = \\mathop {\\mathrm {coker}}\\phi $ viewed as a $S^{\\prime }$ -module is isomorphic to $S^{\\prime n} / \\mathop {\\mathrm {im}}A^{\\prime }$ , the vector space dimension of which is precisely the number of $x^{\\prime }-1$ entries in $A^{\\prime }$ .",
"For example, consider $f(x) = x^2 + x + 1 \\in S = \\mathbb {F}_2[x]$ .",
"It is the primitive polynomial of the field $\\mathbb {F}_4$ of four elements over $\\mathbb {F}_2$ .",
"Any element in $\\mathbb {F}_4$ is a solution of $x^4-x=0$ .",
"Since $f(0) = 1$ , we see that $n=3$ is the smallest integer such that $f(x)$ divides $x^n-1$ .",
"As a module over $S^{\\prime } = \\mathbb {F}_2[x^3]$ , the original ring $S$ is free with (ordered) basis $\\lbrace 1, x, x^2\\rbrace $ .",
"The multiplication by $x$ on $S$ viewed as an endomorphism has a matrix representation $x =\\begin{pmatrix}0 & 0 & x^3 \\\\1 & 0 & 0 \\\\0 & 1 & 0\\end{pmatrix} .$ Thus, $f(x)$ as an endomorphism of $S^{\\prime }$ -module $S$ has a matrix representation as follows.",
"$f(x) =\\begin{pmatrix}1 & x^3 & x^3 \\\\1 & 1 & x^3 \\\\1 & 1 & 1\\end{pmatrix}\\cong \\begin{pmatrix}1 & 0 & 0 \\\\0 & x^3 +1 & 0 \\\\0 & 0 & x^3 +1\\end{pmatrix}$ Here, the second matrix is obtained by row and column operations.",
"There are 2 diagonal entries $x^3+1$ as $f(x)$ is of degree 2.",
"Theorem 3 If $\\Lambda = \\mathbb {Z}$ , any system governed by a code Hamiltonian is equivalent to finitely many copies of Ising models, plus some non-interacting qubits.",
"In particular, the topological order condition is never satisfied.",
"Yoshida [13] arrived at a similar conclusion assuming that the ground space degeneracy when the Hamiltonian is defined on a ring should be independent of the length of the ring.",
"If translation group is trivial, the proof below reduces to a well-known fact that the Clifford group is generated by controlled-NOT, Hadamard, and Phase gates [11].",
"The proof in fact implies that the group of all symplectic transformations in one-dimension is generated by elementary symplectic transformations of Section REF .",
"We will make use of the elementary symplectic transformations and coarse-graining to deform $\\sigma $ to a familiar form.",
"Recall that for any elementary row-addition $E$ on the upper block of $\\sigma $ there is a unique symplectic transformation that restricts to $E$ .",
"Applying Smith's algorithm to the first row and the first column of $2q \\times t$ matrix $\\sigma $ , one gets $\\begin{pmatrix}f_1 & 0 \\\\0 & A \\\\\\hline g_1 & g_2 \\\\\\vdots & B\\end{pmatrix}$ by elementary symplectic transformations.",
"Let $1 \\le i < j \\le q$ be integers.",
"If some $(1,q+j)$ -entry is not divisible by $f_1$ , apply Hadamard on $j$ -th qubit to bring $(q+j)$ -th row to the upper block, and then run Euclid's algorithm again to reduce the degree of $(1,1)$ -entry.",
"The degree is a positive integer, so this process must end after a finite number of iteration.",
"Now every $(q+j,1)$ -entry is divisible by $f_1$ and hence can be made to be 0 by the controlled-NOT-Hadamard: $\\begin{pmatrix}f_1 & 0 \\\\0 & A \\\\\\hline g_1 & g_2 \\\\0 & B\\end{pmatrix} .$ Further we may assume $\\deg f_1 \\le \\deg g_1$ .",
"Since $\\sigma ^\\dagger \\lambda _q \\sigma = 0$ , we have a commutativity condition $\\bar{f}_1 g_1 - \\bar{g}_1 f_1 = 0 .$ Write $f_1 = \\alpha x^a + \\cdots + \\beta x^b $ and $ g_1 = \\gamma x^c + \\cdots + \\delta x^d $ where $ a \\le b$ and $c \\le d$ and $\\alpha ,\\beta ,\\gamma ,\\delta \\ne 0$ .",
"Then, $\\bar{f}_1 g_1 = \\beta \\gamma x^{c-b} + \\cdots + \\alpha \\delta x^{d-a}$ .",
"Since $f_1 \\bar{g}_1 = \\bar{f}_1 g_1$ , it must hold that $-(c-b)=d-a$ and $\\alpha \\delta = \\beta \\gamma $ .",
"Since $\\deg f_1 \\le \\deg g_1$ , we have $d-b = -(c-a) \\ge 0$ .",
"The controlled-Phase $E_{1+q,1}( -(x^{d-b} + x^{c-a})\\delta / \\beta )$ will decrease the degree of $g_1$ by two, which eventually becomes smaller than $\\deg f_1$ .",
"One may then apply Hadamard to swap $f_1$ and $g_1$ .",
"Since the degree of $(1,1)$ -entry cannot decrease forever, the process must end with $g_1 = 0$ .",
"The commutativity condition between $i$ -th($i>1$ ) column and the first is $ f_1 \\bar{g}_i = 0 $ .",
"Since $f_1 \\ne 0$ , we get $g_i = 0$ : $\\begin{pmatrix}f_1 & 0 \\\\0 & A \\\\\\hline 0 & 0 \\\\0 & B\\end{pmatrix} .$ Continuing, we transform $\\sigma $ into a diagonal matrix.",
"(We have shown that $\\sigma $ can be transformed via elementary symplectic transformations to the Smith normal form.)",
"Now the Hamiltonian is a sum of non-interacting purely classical spin chains plus some non-interacting qubits ($f_i = 0$ ).",
"It remains to classify classical spin chains whose stabilizer module is generated by $\\begin{pmatrix}f\\end{pmatrix}$ where we omitted the lower half block.",
"We can always choose $f=f(x)$ such that $f(x)$ has only non-negative exponents and $f(0)\\ne 0$ since $x$ is a unit in $R$ .",
"Lemma REF says that $(f)$ becomes a diagonal matrix of entries 1 or $x^{\\prime }-1$ after a suitable coarse-graining followed by a symplectic transformation and column operations.",
"1 describes the ancilla qubits, and $x^{\\prime }-1 = x^{\\prime }+1$ does the Ising model."
],
[
"Two dimensions",
"In the following two sections we will be mainly interested in exact code Hamiltonians.",
"If $D=2$ , the lattice is $\\Lambda = \\mathbb {Z}^2$ , and our base ring is $R = \\mathbb {F}_2 [x,\\bar{x}, y, \\bar{y}]$ .",
"The following asserts that the local relations — a few terms in the Hamiltonian that multiply to identity in a nontrivial way as in 2D Ising model, or the kernel of $\\sigma $ — among the terms in a code Hamiltonian, can be completely removed for exact Hamiltonians in two dimensions [14].",
"We prove a more general version.",
"Lemma 7.1 If $G \\xrightarrow{} P \\xrightarrow{} E$ is exact over $R = F_2[ x_1, \\bar{x}_1, \\ldots , x_D, \\bar{x}_D ]$ , There exists $\\sigma ^{\\prime } : G^{\\prime } \\rightarrow P$ such that $\\mathop {\\mathrm {im}}\\sigma ^{\\prime } = \\mathop {\\mathrm {im}}\\sigma $ and $0 \\rightarrow G_{D-2} \\rightarrow \\cdots \\rightarrow G_1 \\rightarrow G^{\\prime } \\xrightarrow{} P \\xrightarrow{} E$ is an exact sequence of free $R$ -modules.",
"If $D=2$ , one can choose $\\sigma ^{\\prime }$ to be injective.",
"The lemma is almost the same as the Hilbert syzygy theorem [27] applied to $\\mathop {\\mathrm {coker}}\\epsilon $ , which states that any finitely generated module over a polynomial ring with $n$ variables has a finite free resolution of length $\\le n$ , by finitely generated free modules.",
"A difference is that our two maps on the far right in the resolution has to be related as $\\epsilon = \\sigma ^\\dagger \\lambda $ .",
"To this end, we make use of a constructive version of Hilbert syzygy theorem via Gröbner basis.",
"Proposition 7.2 [27] Let $\\lbrace g_1, \\ldots , g_n \\rbrace $ be a Gröbner basis of a submodule of a free module $M_0$ over a polynomial ring.",
"Then, the S-polynomials $\\tau _{ij}$ of $\\lbrace g_i \\rbrace $ in the free module $M_1 = \\bigoplus _{i=1}^n S e_i$ generate the syzygies for $\\lbrace g_i\\rbrace $ .",
"If the variable $x_1,\\ldots , x_s$ are absent from the initial terms of $g_i$ , one can define a monomial order on $M_1$ such that $x_1,\\ldots ,x_{s+1}$ is absent from the initial terms of $\\tau _{ij}$ .",
"If all variables are absent from the initial terms of $g_i$ , then $M_0/(g_1,\\ldots ,g_n)$ is free.",
"Without loss of generality assume that the $t \\times 2q$ matrix $\\epsilon $ have entries with nonnegative exponents, so $\\epsilon $ has entries in $S=\\mathbb {F}_2[x_1,\\ldots ,x_D]$ .",
"Below, every module is over the polynomial ring $S$ unless otherwise noted.",
"Let $E_+$ be the free $S$ -module of rank equal to $\\mathrm {rank}_R~ E$ .",
"If $g_1,\\cdots , g_{2q}$ are the columns of $\\epsilon $ , apply Buchberger's algorithm to obtain a Gröbner basis $g_1,\\cdots , g_{2q}, \\ldots , g_n$ of $\\mathop {\\mathrm {im}}\\epsilon $ .",
"Let $\\epsilon ^{\\prime }$ be the matrix whose columns are $g_1,\\ldots , g_n$ .",
"We regard $\\epsilon ^{\\prime }$ as a map $M_0 \\rightarrow E_+$ .",
"By Proposition REF , the initial terms of the syzygy generators (S-polynomials) $\\tau _{ij}$ for $\\lbrace g_i \\rbrace $ lacks the variable $x_1$ .",
"Writing each $\\tau _{ij}$ in a column of a matrix $\\tau _1$ , we have a map $\\tau _1 : M_1 \\rightarrow M_0$ .",
"By induction on $D$ , we have an exact sequence $M_{D} \\xrightarrow{} M_{D-1} \\xrightarrow{}\\cdots \\xrightarrow{} M_0 \\xrightarrow{} E_+$ of free $S$ -modules, where the initial terms of columns of $\\tau _D$ lack all the variables.",
"By Proposition REF again, $M^{\\prime }_{D-1} = M_{D-1} / \\mathop {\\mathrm {im}}\\tau _{\\tau _D}$ is free.",
"Since $\\ker \\tau _{D-1} = \\mathop {\\mathrm {im}}\\tau _D$ , we have $0 \\rightarrow M^{\\prime }_{D-1} \\xrightarrow{}\\cdots \\xrightarrow{} M_0 \\xrightarrow{} E_+$ Since $g_{2q+1},\\ldots ,g_n$ are $S$ -linear combinations of $g_1,\\ldots , g_{2q}$ , there is a basis change of $M_0$ so that the matrix representation of $\\epsilon ^{\\prime }$ becomes $\\epsilon ^{\\prime } \\cong \\begin{pmatrix}\\epsilon & 0\\end{pmatrix} .$ With respect to this basis of $M_0$ , the matrix of $\\tau _1$ is $\\tau _1 \\cong \\begin{pmatrix}\\tau _{1u} \\\\\\tau _{1d}\\end{pmatrix}$ where $\\tau _{1u}$ is the upper $2q \\times t^{\\prime }$ submatrix.",
"Since $\\ker \\epsilon ^{\\prime } = \\mathop {\\mathrm {im}}\\tau _1$ , The first row $r$ of $\\tau _{1d}$ should generate $1 \\in S$ .",
"(This property is called unimodularity.)",
"Quillen-Suslin theorem [35] states that there exists a basis change of $M_1$ such that $r$ becomes $\\begin{pmatrix}1 & 0 & \\cdots & 0 \\end{pmatrix}$ .",
"Then, by some basis change of $M_0$ , one can make $\\epsilon ^{\\prime } \\cong \\begin{pmatrix}\\epsilon & 0\\end{pmatrix} ,\\quad \\tau _{1d} \\cong \\begin{pmatrix}1 & 0 \\\\0 & \\tau ^{\\prime }_{1d}\\end{pmatrix} .$ where $\\tau ^{\\prime }_{1d}$ is a submatrix.",
"By induction on the number of rows in $\\tau _{1d}$ , we deduce that the matrix of $\\tau _1$ can be brought to $\\epsilon ^{\\prime } \\cong \\begin{pmatrix}\\epsilon & 0\\end{pmatrix} ,\\quad \\tau _1 \\cong \\begin{pmatrix}\\sigma ^{\\prime \\prime } & \\sigma ^{\\prime } \\\\I & 0\\end{pmatrix}$ Note that $\\epsilon \\sigma ^{\\prime \\prime } = 0$ and $\\epsilon \\sigma ^{\\prime } = 0$ .",
"The basis change of $M_0$ by $\\begin{pmatrix} I & -\\sigma ^{\\prime \\prime } \\\\ 0 & I \\end{pmatrix}$ gives $\\epsilon ^{\\prime } \\cong \\begin{pmatrix}\\epsilon & 0\\end{pmatrix} ,\\quad \\tau _1 \\cong \\begin{pmatrix}0 & \\sigma ^{\\prime } \\\\I & 0\\end{pmatrix} .$ The kernel of $\\begin{pmatrix} \\sigma ^{\\prime } \\\\ 0 \\end{pmatrix}$ determines $\\ker \\tau _1 = \\mathop {\\mathrm {im}}\\tau _2$ .",
"Let $M^{\\prime }_1$ denote the projection of $M_1$ such that the sequence $0 \\rightarrow M^{\\prime }_{D-1} \\xrightarrow{}\\cdots \\rightarrow M_2 \\rightarrow M^{\\prime }_1 \\xrightarrow{} M^{\\prime }_0 \\xrightarrow{} E_+$ of free $S$ -modules is exact.",
"Taking the ring of fractions with respect to the multiplicatively closed set $U = \\lbrace x_1^{i_1} \\cdots x_D^{i_D} | i_1,\\ldots ,i_D \\ge 0\\rbrace ,$ we finally obtain the desired exact sequence over $U^{-1}S = R$ with $P = U^{-1}M^{\\prime }_0$ and $E = U^{-1}E_+$ .",
"Since $\\mathop {\\mathrm {im}}\\sigma = \\ker \\epsilon $ , we have $\\mathop {\\mathrm {im}}\\sigma ^{\\prime } = \\mathop {\\mathrm {im}}\\sigma $ .",
"Lemma 7.3 Let $R$ be a Laurent polynomial ring in $D$ variables over a finite field $\\mathbb {F}$ , and $N$ be a module over $R$ .",
"Suppose $J = \\mathop {\\mathrm {ann}}_R N$ is a proper ideal such that $\\dim R/J = 0$ .",
"Then, there exists an integer $L \\ge 1$ such that $\\mathop {\\mathrm {ann}}_{R^{\\prime }} N = (x_1^L -1, \\ldots , x_D^L -1) \\subseteq R^{\\prime }$ where $R^{\\prime } = \\mathbb {F}[x_1^{\\pm L},\\ldots ,x_D^{\\pm L}]$ is a subring of $R$ .",
"This is a variant of Lemma REF .",
"The annihilator $J = \\mathop {\\mathrm {ann}}_R N$ is the set of all elements $r \\in R$ such that $r n = 0$ for any $n \\in N$ .",
"It is an ideal; if $r_1, r_2 \\in \\mathop {\\mathrm {ann}}_R N$ , then $r_1+r_2$ is an annihilator since $(r_1 + r_2)n = r_1 n + r_2 n = 0$ , and $a r_1 \\in \\mathop {\\mathrm {ann}}_R N$ for any $a \\in R$ since $(a r_1) n = a(r_1 n)=0$ .",
"If $R^{\\prime } \\subseteq R$ is a subring and $N$ is an $R$ -module, $N$ is an $R^{\\prime }$ -module naturally.",
"Clearly, $J^{\\prime } = \\mathop {\\mathrm {ann}}_{R^{\\prime }} N$ is by definition equal to $(\\mathop {\\mathrm {ann}}_{R} N) \\cap R^{\\prime }$ .",
"Note that $J^{\\prime }$ is the kernel of the composite map $R^{\\prime } \\hookrightarrow R \\rightarrow R/J$ .",
"Hence, we have an algebra homomorphism $\\varphi ^{\\prime } : R^{\\prime }/J^{\\prime } \\rightarrow R/J$ .",
"Although $R^{\\prime }$ is a subring, it is isomorphic to $R$ via the correspondence $x_i^L \\leftrightarrow x_i$ .",
"Therefore, we may view $\\varphi ^{\\prime }$ as a map $\\varphi : R/I \\rightarrow R/J$ for some ideal $I \\subseteq R$ .",
"It is a homomorphism such that $\\varphi (x_i) = x_i^L$ .",
"Considering the algebras as the set of all functions on the algebraic sets $V(I)$ and $V(J)$ defined by $I$ and $J$ , respectively, we obtain a map $\\hat{\\varphi }: V(J) \\rightarrow V(I)$ .",
"Intuitively, $\\hat{\\varphi }$ maps each point $(a_1,\\ldots ,a_D) \\in \\mathbb {F}^D$ to $(a_1^L,\\ldots ,a_D^L) \\in \\mathbb {F}^D$ .",
"In a finite field, any nonzero element is a root of unity.",
"Since $\\dim R/J = 0$ , which means that $V(J)$ is a finite set, we can find a certain $L$ so $V(I)$ would consist of a single point.",
"A formal proof is as follows.",
"Since $R$ is a finitely generated algebra over a field, for any maximal ideal $\\mathfrak {m}$ of $R$ , the field $R/\\mathfrak {m}$ is a finite extension of $\\mathbb {F}$ (Nullstellensatz [27]).",
"Hence, $R/\\mathfrak {m}$ is a finite field.",
"Since $x_i$ is a unit in $R$ , the image $a_i \\in R/\\mathfrak {m}$ of $x_i$ is nonzero.",
"$a_i$ being an element of finite field, a power of $a_i$ is 1.",
"Therefore, there is a positive integer $n$ such that $\\mathfrak {b}_n = (x_1^n-1,\\ldots ,x_D^n-1) \\subseteq \\mathfrak {m}$ .",
"Since $x^n-1$ divides $x^{nn^{\\prime }}-1$ , we see that there exists $n \\ge 1$ such that $\\mathfrak {b}_n \\subseteq \\mathfrak {m}_1 \\cap \\mathfrak {m}_2$ for any two maximal ideals $\\mathfrak {m}_1, \\mathfrak {m}_2$ .",
"One extends this by induction to any finite number of maximal ideals.",
"Since $\\dim R/J = 0$ , any prime ideal of $R/J$ is maximal and the Artinian ring $R/J$ has only finitely many maximal ideals.",
"$\\mathrm {rad}~ J$ is then the intersection of the contractions (pull-backs) of these finitely many maximal ideals.",
"Therefore, there is $n \\ge 1$ such that $\\mathfrak {b}_n \\subseteq \\mathrm {rad}~ J .$ Since $R$ is Noetherian, $(\\mathrm {rad}~J)^{p^r} \\subseteq J$ for some $r \\ge 0$ where $p$ is the characteristic of $\\mathbb {F}$ .",
"Hence, we have $\\mathfrak {b}_{np^r} \\subseteq \\mathfrak {b}_n^{p^r} \\subseteq (\\mathrm {rad}~J)^{p^r} \\subseteq J.$ Let $L = np^r$ .",
"If $R^{\\prime } = \\mathbb {F}[x_1^L,\\bar{x}_1^L,\\ldots ,x_D^L,\\bar{x}_D^L]$ , $\\mathop {\\mathrm {ann}}_{R^{\\prime }} N$ is nothing but $J \\cap R^{\\prime }$ .",
"We have just shown $\\mathfrak {b}_L \\cap R^{\\prime } \\subseteq J \\cap R^{\\prime }$ .",
"Since $J$ is a proper ideal, we have $1 \\notin J \\cap R^{\\prime }$ .",
"Thus, $\\mathfrak {b}_L \\cap R^{\\prime }= J \\cap R^{\\prime }$ since $\\mathfrak {b}_L \\cap R^{\\prime }$ is maximal in $R^{\\prime }$ .",
"Theorem 4 For any two dimensional degenerate exact code Hamiltonian, there exists an equivalent Hamiltonian such that $\\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon = (x-1,y-1).$ Thus, $\\mathop {\\mathrm {coker}}\\epsilon $ is a torsion module.",
"The content of Theorem REF is presented in [14].",
"We will comment on it after the proof.",
"By Lemma REF , we can find an equivalent Hamiltonian such that the generating map $\\sigma $ for its stabilizer module is injective: $0 \\rightarrow G \\xrightarrow{} P .$ Let $t$ be the rank of $G$ .",
"The exactness condition says $0 \\rightarrow G \\xrightarrow{} P \\xrightarrow{} E$ is exact where $\\epsilon = \\sigma ^\\dagger \\lambda _q$ and $E$ has rank $t$ .",
"Applying Proposition REF , since $\\overline{ I(\\sigma ) } = I(\\epsilon )$ and hence in particular $\\mathop {\\mathrm {codim}}I(\\sigma ) = \\mathop {\\mathrm {codim}}I(\\epsilon )$ , we have that $q=t$ and $\\mathop {\\mathrm {codim}}I(\\epsilon ) \\ge 2$ if $I(\\epsilon ) \\ne R$ .",
"But, $I(\\epsilon ) \\ne R$ by Corollary REF .",
"Since $q=t$ , $I(\\epsilon )$ is equal to the initial Fitting ideal, and therefore has the same radical as the annihilator of $\\mathop {\\mathrm {coker}}\\epsilon = E/ \\mathop {\\mathrm {im}}\\epsilon $ .",
"(See [27] or [35].)",
"In particular, $\\dim R / (\\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon ) = 0$ .",
"Apply Lemma REF to conclude the proof.",
"An interpretation of the theorem is the following.",
"For systems of qubits, Theorem REF says that $x+1$ and $y+1$ are in $\\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon $ .",
"In other words, any element $v$ of $E$ is a charge, and a pair of $v$ 's of distance 1 apart can be created by a local operator.",
"Equivalently, $v$ can be translated by distance 1 by the local operator.",
"Since translation by distance 1 generates all translations of the lattice, we see that any excitation can be moved through the system by some sequence of local operators.",
"This is exactly what happens in the 2D toric code: Any excited state is described by a configuration of magnetic and electric charge, which can be moved to a different position by a string operator.",
"Moreover, since $(x-1,y-1) = \\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon $ , the action of $x,y \\in R$ on $\\mathop {\\mathrm {coker}}\\epsilon $ is the same as the identity action.",
"Therefore, the $R$ -module $\\mathop {\\mathrm {coker}}\\epsilon $ is completely determined up to isomorphism by its dimension $k$ as an $\\mathbb {F}_2$ -vector space.",
"In particular, $\\mathop {\\mathrm {coker}}\\epsilon $ is a finite set, which means there are finitely many charges.",
"The module $K(L)$ of Pauli operators acting on the ground space (logical operators), can be viewed as $K(L) = {\\mathrm {Tor}}_1(\\mathop {\\mathrm {coker}}\\epsilon , R/\\mathfrak {b}_L)$ .",
"Thus, $K(L)$ is determined by $k$ up to $R$ -module isomorphisms.",
"This implies that the translations of a logical operator are all equivalent.",
"It is not too obvious at this moment whether the symplectic structure, or the commutation relations among the logical operators, of $K(L)$ is also completely determined.",
"Yoshida [13] argued a similar result assuming that the ground state degeneracy should be independent of system size.",
"Bombin [14] later claimed without the constant degeneracy assumption that one can choose locally independent stabilizer generators in a `translationally invariant way' in two dimensions, for which Lemma REF is a generalization, and that there are finitely many topological charges, which is immediate from Theorem REF since $\\mathop {\\mathrm {coker}}\\epsilon $ is a finite set.",
"The claim is further strengthened assuming extra conditions by Bombin et al.",
"[15], which can be summarized by saying that $\\sigma $ is a finite direct sum of $\\sigma _\\text{2D-toric}$ in Example REF .",
"Remark 3 Although the strings are capable of moving charges on the lattice, it could be very long compared to the interaction range.",
"Consider $\\epsilon _p =\\begin{pmatrix}p(x) & p(y) & 0 & 0 \\\\0 & 0 & p(\\bar{y}) & -p(\\bar{x})\\end{pmatrix}$ where $p$ is any polynomial.",
"It defines an exact code Hamiltonian.",
"For instance, the choice $p(t) = t-1$ reproduces the 2D toric code of Example REF .",
"Now let $p(t)$ be a primitive polynomial of the extension field $\\mathbb {F}_{2^w}$ over $\\mathbb {F}_2$ .",
"$p(t)$ has coefficients in the base field $\\mathbb {F}_2$ and factorizes in $\\mathbb {F}_{2^w}$ as $p(t) = (t-\\theta )(t-\\theta ^2)(t-\\theta ^{2^2})\\cdots (t-\\theta ^{2^{w-1}})$ .",
"(See [35].)",
"The multiplicative order of $\\theta $ is $N = 2^w -1$ .",
"The degree $w$ of $p(t)$ may be called the interaction range.",
"If the charge $e = \\begin{pmatrix}1 & 0 \\end{pmatrix}^T$ at $(0,0)\\in \\mathbb {Z}^2$ is transported to $(a,b) \\in \\mathbb {Z}^2 \\setminus \\lbrace (0,0) \\rbrace $ by some finitely supported operator, we have $(x^a y^b -1)e \\in \\mathop {\\mathrm {im}}\\epsilon $ .",
"That is, $x^a y^b -1 \\in (p(x),p(y))$ .",
"Substituting $x \\mapsto \\theta $ and $y \\mapsto \\theta ^{2^m}$ , we see that $\\theta ^{a+2^m b} = 1$ or $a+2^m b \\equiv 0 \\pmod {N}$ for any $m \\in \\mathbb {Z}$ .",
"In other words, $a \\equiv -b \\equiv -2b \\pmod {N}$ .",
"It follows that $|a| + |b| \\ge \\frac{N}{2-1}$ .",
"Therefore, the length of the string segment transporting a charge is exponential in the interaction range $w$ ."
],
[
"Three dimensions",
"In the previous section, we derived a consequence of the exactness of code Hamiltonians.",
"The two-dimensional Hamiltonian was special so we were able to characterize the behavior of the charges more or less completely.",
"Here, we prove a weaker property of three dimensions that there must exist a nontrivial charge for any exact code Hamiltonian.",
"It follows from Theorems REF ,REF that such a charge can spread through the system by surmounting the logarithmic energy barrier.",
"Lemma 8.1 Suppose $D=3$ , $0 \\rightarrow G_1 \\xrightarrow{} G \\xrightarrow{} P \\xrightarrow{} E$ is exact, and $I(\\sigma ) \\subseteq \\mathfrak {m}= (x-1,y-1,z-1)$ .",
"Then, $\\mathop {\\mathrm {coker}}\\epsilon $ is not torsion-free.",
"Suppose on the contrary $\\mathop {\\mathrm {coker}}\\epsilon $ is torsion-free.",
"We have an exact sequence $0 \\rightarrow G_1 \\xrightarrow{} G \\xrightarrow{} P \\xrightarrow{} E \\rightarrow E^{\\prime }.$ If $G_1 = 0$ , Proposition REF implies the conclusion.",
"So we assume $G_1 \\ne 0$ , and therefore we have $I(\\sigma _1) = R$ by Proposition REF .",
"Let us localize the sequence at $\\mathfrak {m}$ , so $I(\\sigma _1)_\\mathfrak {m}= R_\\mathfrak {m}$ .",
"Since $\\mathop {\\mathrm {rank}}(G_1)_\\mathfrak {m}= \\mathop {\\mathrm {rank}}(\\sigma _1)_\\mathfrak {m}$ , the matrix of $(\\sigma _1)_\\mathfrak {m}$ becomes $(\\sigma _1)_\\mathfrak {m}=\\begin{pmatrix}0 \\\\I\\end{pmatrix}$ for some basis of $(G_1)_\\mathfrak {m}$ and $G_\\mathfrak {m}$ .",
"See the proof of Lemma REF .",
"In other words, there is an invertible matrix $B \\in \\mathrm {GL}_{ t \\times t}( R_\\mathfrak {m})$ such that $\\sigma _\\mathfrak {m}B =\\begin{pmatrix}\\tilde{\\sigma }& 0\\end{pmatrix}$ where $\\tilde{\\sigma }$ is the $2q \\times t^{\\prime }$ submatrix.",
"Note that the antipode map is a well-defined automorphism of $R_\\mathfrak {m}$ since $\\overline{\\mathfrak {m}}= \\mathfrak {m}$ .",
"Since $\\epsilon = \\sigma ^\\dagger \\lambda _q$ , we have $B^\\dagger \\epsilon _\\mathfrak {m}=\\begin{pmatrix}\\tilde{\\sigma }^\\dagger \\\\0\\end{pmatrix} \\lambda _q=\\begin{pmatrix}\\tilde{\\sigma }^\\dagger \\lambda _q \\\\0\\end{pmatrix} .$ Therefore, we get a new exact sequence $0 \\rightarrow G^{\\prime } \\xrightarrow{} P_\\mathfrak {m}\\xrightarrow{} R_\\mathfrak {m}^{t^{\\prime }}$ where $G^{\\prime } = G_\\mathfrak {m}/ \\mathop {\\mathrm {im}}(\\sigma _1)_\\mathfrak {m}$ is a free $R_\\mathfrak {m}$ -module and $t^{\\prime } = \\mathop {\\mathrm {rank}}G^{\\prime }$ .",
"It is clear that $\\mathop {\\mathrm {rank}}\\tilde{\\epsilon }= \\mathop {\\mathrm {rank}}\\tilde{\\sigma }$ .",
"Setting $S = R_\\mathfrak {m}$ in Proposition REF implies that $\\mathop {\\mathrm {coker}}\\tilde{\\epsilon }$ is not torsion-free.",
"But, since we are assuming $\\mathop {\\mathrm {coker}}\\epsilon _m$ is torsion-free, $\\mathop {\\mathrm {coker}}\\tilde{\\sigma }^\\dagger $ is also torsion-free by Eq.",
"(REF ).",
"This is a contradiction.",
"Theorem 5 For any three-dimensional, degenerate and exact code Hamiltonian, there exists a fractal generator.",
"By Lemma REF , there exists an equivalent Hamiltonian such that $0 \\rightarrow G_1 \\xrightarrow{} G \\xrightarrow{} P \\xrightarrow{} E$ is exact.",
"The existence of a fractal generator is a property of the equivalence class by Proposition REF .",
"If we can find a coarse-graining such that $I(\\sigma ^{\\prime }) \\subseteq (x^{\\prime }-1,y^{\\prime }-1,z^{\\prime }-1)$ , then Lemma REF shall imply the conclusion.",
"Recall that $\\epsilon _L$ and $\\sigma _L$ denote the induced maps by factoring out $\\mathfrak {b}_L = (x^L-1,y^L-1,z^L-1)$ .",
"See Sec. .",
"There exists $L$ such that $K(L) = \\ker \\epsilon _L / \\mathop {\\mathrm {im}}\\sigma _L \\ne 0$ by Corollary REF .",
"Consider the coarse-grain by $x^{\\prime }=x^L,~ y^{\\prime }=y^L,~ z^{\\prime }=z^L$ .",
"Let $R^{\\prime } = F_2[x^{\\prime \\pm 1}, y^{\\prime \\pm 1}, z^{\\prime \\pm 1}]$ denote the coarse-grained base ring.",
"If $K^{\\prime }(L^{\\prime })$ denotes $\\ker \\epsilon ^{\\prime }_{L^{\\prime }} / \\mathop {\\mathrm {im}}\\sigma ^{\\prime }_{L^{\\prime }}$ as $R^{\\prime }$ -module, we see that $K^{\\prime }(1) = K(L)$ as $\\mathbb {F}_2$ -vector space.",
"In particular, $K^{\\prime }(1) \\ne 0$ .",
"Put $\\mathfrak {m}= (x^{\\prime }-1,y^{\\prime }-1,z^{\\prime }-1) = \\mathfrak {b}^{\\prime }_1 \\subseteq R^{\\prime }$ .",
"Then, $K^{\\prime }(1)_{\\mathfrak {m}} = K^{\\prime }(1) \\ne 0$ .",
"By Lemma REF , we have $I(\\sigma ^{\\prime }) \\subseteq \\mathfrak {m}$ .",
"Yoshida argued that when the ground state degeneracy is constant independent of system size there exists a string operator [16].",
"To prove it, we need an algebraic fact.",
"Proposition 8.2 Let $M$ be a finitely presented $R$ -module, and $T$ be its torsion submodule.",
"Let $I$ be the first non-vanishing Fitting ideal of $M$ .",
"Then, $\\mathop {\\mathrm {rad}}I \\subseteq \\mathop {\\mathrm {rad}}\\mathop {\\mathrm {ann}}T .$ Let $\\mathfrak {p}$ be any prime ideal of $R$ such that $I \\lnot \\subseteq \\mathfrak {p}$ .",
"By the calculation of the proof of Lemma REF , $M_\\mathfrak {p}$ is a free $R_\\mathfrak {p}$ -module, and hence is torsion-free.",
"Since $T$ is embedded in $M$ , it follows that $T_\\mathfrak {p}= 0$ , or equivalently, $\\mathop {\\mathrm {ann}}T \\lnot \\subseteq \\mathfrak {p}$ .",
"Since the radical of an ideal is the intersection of all primes containing it [31], the claim is proved.",
"Corollary 8.3 Let $T$ be the set of all point-like charges modulo locally created ones of a degenerate and exact code Hamiltonian in three dimensions of characteristic dimension zero.",
"Then, one can coarse-grain the lattice such that $\\mathop {\\mathrm {ann}}T = (x-1,y-1,z-1) .$ The corollary says that any point-like charge is attached to strings and is able to move freely through the lattice.",
"The condition is implied by Lemma REF if the ground state degeneracy is constant independent of the system size when defined on a periodic lattice.",
"By Theorem REF , $T$ is the torsion submodule of $\\mathop {\\mathrm {coker}}\\epsilon $ .",
"By Theorem REF , $T$ is nonzero.",
"Setting $M = \\mathop {\\mathrm {coker}}\\epsilon $ in Proposition REF , the associated ideal $I_q(\\epsilon )$ is the first non-vanishing Fitting ideal of $M$ .",
"Since $\\dim R / I_q(\\epsilon ) = 0$ by assumption, we have $\\dim R / \\mathop {\\mathrm {ann}}T = 0$ .",
"Lemma REF implies the claim."
],
[
"More examples",
"Example 4 (Toric codes in higher dimensions) Any higher dimensional toric code can be treated similarly as for two dimensional case.",
"In three dimensions one associates each site with $q=3$ qubits.",
"It is easily checked that $\\sigma _\\text{3D-toric} =\\begin{pmatrix}1 + \\bar{x} & 0 & 0 & 0 \\\\1 + \\bar{y} & 0 & 0 & 0 \\\\1 + \\bar{z} & 0 & 0 & 0 \\\\\\hline 0 & 0 & 1+z & 1+y \\\\0 & 1+z & 0 & 1+x \\\\0 & 1+y & 1+x & 0 \\\\\\end{pmatrix} .$ Both two- and three-dimensional toric codes have the property that $\\mathop {\\mathrm {coker}}\\epsilon $ is not torsion-free.",
"However, in two dimensions any element of $E$ is a physical charge, whereas in three dimensions $E$ contains physically irrelevant elements.",
"Note that in both cases, $1+x$ and $1+y$ are fractal generators.",
"Being consisted of two terms, they generate the `string operators.'",
"The 4D toric code [4] has $\\sigma _x$ -type interaction and $\\sigma _z$ -type interaction.",
"Originally the qubits are placed on every plaquette of 4D hypercubic lattice; instead we place $q=6$ qubits on each site.",
"The generating map $\\sigma $ for the stabilizer module is written as a $12 \\times 8$ -matrix ($t=8$ ) $\\sigma _\\text{4D-toric} =\\begin{pmatrix}\\sigma _X & 0 \\\\0 & \\sigma _Z\\end{pmatrix}$ where $\\sigma _X &=\\begin{pmatrix}1+y & 1+x & 0 & 0 \\\\1+w & 0 & 0 & 1+x \\\\1+z & 0 & 1+x & 0 \\\\0 & 1+z & 1+y & 0 \\\\0 & 1+w & 0 & 1+y \\\\0 & 0 & 1+w & 1+z\\end{pmatrix}, \\\\\\bar{\\sigma }_Z &=\\begin{pmatrix}0 & 0 & 1+w & 1+z \\\\0 & 1+z & 1+y & 0 \\\\0 & 1+w & 0 & 1+y \\\\1+w & 0 & 0 & 1+x \\\\1+z & 0 & 1+x & 0 \\\\1+y & 1+x & 0 & 0\\end{pmatrix}.$ Note the bar on $\\sigma _Z$ .",
"Theorem REF does not prevent the absence of a fractal generator in four or higher dimensions.",
"Indeed, this 4D toric code lacks any fractal generator.",
"To see this, it is enough to consider $\\sigma _Z$ since $\\overline{\\mathop {\\mathrm {coker}}\\sigma _X^\\dagger } \\cong \\mathop {\\mathrm {coker}}\\sigma _Z^\\dagger $ as $R_4$ -modules, where $R_4 = \\mathbb {F}_2[x^{\\pm 1},y^{\\pm 1},z^{\\pm 1},w^{\\pm 1}]$ .",
"If $\\epsilon _1 =\\begin{pmatrix}1+x & 1+y & 1+z & 1+w\\end{pmatrix} : R_4^4 \\rightarrow R_4,$ then $R_4^6 \\xrightarrow{} R_4^4 \\xrightarrow{} R_4$ is exact.",
"(A direct way to check it is to compute S-polynomials [27] of the entries of $\\epsilon _1$ , and to verify that they all are in the rows of $\\sigma _Z$ .)",
"Hence, $\\mathop {\\mathrm {coker}}\\sigma _Z^\\dagger $ is torsion-free by Proposition REF .",
"For the toric codes in any dimensions, $\\sigma $ has nonzero entries of form $x_i-1$ .",
"The radical of the associated ideal $I(\\sigma )$ is equal to $\\mathfrak {m}= (x_1-1,\\ldots , x_D-1)$ .",
"So $\\mathfrak {m}$ is the only maximal ideal of $R$ that contains $I(\\sigma )$ .",
"The characteristic dimension is zero.",
"If $2 \\nmid L$ , since $(\\mathfrak {b}_L)_{\\mathfrak {m}} = \\mathfrak {m}_{\\mathfrak {m}}$ , $(\\sigma _L)_\\mathfrak {m}$ is a zero matrix.",
"Any other localization of $\\sigma _L$ does not contribute to $\\dim _{\\mathbb {F}_2} K(L)$ by Lemma REF .",
"Therefore, if $2 \\nmid L$ , $K(L)$ has constant vector space dimension independent of $L$ .",
"There is a more direct way to compute the $R$ -module $K(L)$ .",
"For the three-dimensional case, consider a free resolution of $R_3/\\mathfrak {m}$ , where $R_3 = \\mathbb {F}_2[x^{\\pm 1},y^{\\pm 1},z^{\\pm 1}]$ , as $0 \\rightarrow R_3^1\\xrightarrow{}R_3^3\\xrightarrow{}R_3^3\\xrightarrow{}R_3^1\\rightarrow R_3/\\mathfrak {m}\\rightarrow 0$ where $a=1+x$ , $b=1+y$ , and $c=1+z$ .",
"We see that $\\sigma _\\text{3D-toric} = \\bar{\\partial }_3 \\oplus \\partial _2 ,\\quad \\text{and} \\quad \\epsilon _\\text{3D-toric} = \\bar{\\partial }_2 \\oplus \\partial _1 .$ Therefore, $K(L)_\\text{3D-toric} \\cong {\\mathrm {Tor}}_1(\\mathop {\\mathrm {coker}}\\epsilon _\\text{3D-toric},R_3/\\mathfrak {b}_L)&\\cong {\\mathrm {Tor}}_2( \\overline{ R_3/\\mathfrak {m}}, R_3/\\mathfrak {b}_L ) \\oplus {\\mathrm {Tor}}_1( R_3/\\mathfrak {m}, R_3/\\mathfrak {b}_L ).$ Using ${\\mathrm {Tor}}(M,N) \\cong {\\mathrm {Tor}}(N,M)$ and the fact that a resolution of $R_3/\\mathfrak {b}_L$ is Eq.",
"(REF ) with $a,b,c$ replaced by $x^L-1,y^L-1,z^L-1$ , respectively, we have ${\\mathrm {Tor}}_i(R_3/\\mathfrak {m},R_3/\\mathfrak {b}_L) \\cong {\\mathrm {Tor}}_i(R_3/\\mathfrak {m},R_3/\\mathfrak {m}) \\cong (\\mathbb {F}_2)^{_3 C _i}$ for each $0 \\le i \\le 3$ .",
"Therefore, $K(L)_\\text{3D-toric} \\cong (\\mathbb {F}_2)^{_3 C _2} \\oplus (\\mathbb {F}_2)^{_3 C _1} \\cong (\\mathbb {F}_2)^6$ .",
"The four-dimensional case is similar: $K(L)_\\text{4D-toric} \\cong {\\mathrm {Tor}}_2 (R_4/\\mathfrak {m}, R_4/\\mathfrak {b}_L) \\oplus {\\mathrm {Tor}}_2 (\\overline{R_4/\\mathfrak {m}}, R_4/\\mathfrak {b}_L) \\cong \\left( (\\mathbb {F}_2)^{_4 C _2} \\right)^2 .$ The calculation here is closely related to the cellular homology interpretation of toric codes.",
"$\\Diamond $ Example 5 (Wen plaquette [36]) This model consists of a single type of interaction ($t=q=1$ ) $@!0{X @{-}[r] & Y @{-}[d] \\\\Y @{-}[u] & X @{-}[l]} \\quad \\quad \\sigma _\\text{Wen} =\\begin{pmatrix}1 + x + y + xy \\\\\\hline 1 + xy\\end{pmatrix}$ where $X,Y$ are abbreviations of $\\sigma _x,\\sigma _y$ .",
"It is known to be equivalent to the 2D toric code.",
"Take the coarse-graining given by $R^{\\prime } = \\mathbb {F}_2[x^{\\prime },y^{\\prime },\\bar{x}^{\\prime }, \\bar{y}^{\\prime }]$ where $x^{\\prime } = x \\bar{y}, \\quad \\quad y^{\\prime } = y^2 .$ (The coarse-graining considered in this example is intended to demonstrate a non-square blocking of the old lattice to obtain a `tilted' new lattice, and is by no means special.)",
"As an $R^{\\prime }$ -module, $R$ is free with basis $\\lbrace 1, y \\rbrace $ .",
"With the identification $R = (R^{\\prime } \\cdot 1) \\oplus (R^{\\prime } \\cdot y)$ , we have $x \\cdot 1 = x^{\\prime } \\cdot y$ , $x \\cdot y = x^{\\prime }y^{\\prime } \\cdot 1$ , and $y \\cdot 1 = 1 \\cdot y$ , $y \\cdot y = y^{\\prime } \\cdot 1$ .",
"Hence, $x$ and $y$ act on $R^{\\prime }$ -modules as the matrix-multiplications on the left: $x \\mapsto \\begin{pmatrix}0 & x^{\\prime }y^{\\prime }\\\\x^{\\prime } & 0\\end{pmatrix},\\quad y \\mapsto \\begin{pmatrix}0 & y^{\\prime }\\\\1 & 0\\end{pmatrix}.$ Identifying $R^n = [ (R^{\\prime } \\cdot 1) \\oplus (R^{\\prime } \\cdot y) ] \\oplus \\cdots \\oplus [(R^{\\prime } \\cdot 1) \\oplus (R^{\\prime } \\cdot y)] ,$ our new $\\sigma $ on the coarse-grained lattice becomes $\\sigma ^{\\prime } =\\begin{pmatrix}1+x^{\\prime }y^{\\prime } & y^{\\prime }+x^{\\prime }y^{\\prime }\\\\1+x^{\\prime } & 1+x^{\\prime }y^{\\prime }\\\\\\hline 1+x^{\\prime }y^{\\prime } & 0 \\\\0 & 1+x^{\\prime }y^{\\prime }\\end{pmatrix}.$ By a sequence of elementary symplectic transformations, we have $\\sigma ^{\\prime }&\\xrightarrow[E_{1,3}(1)]{E_{2,4}(1)}\\begin{pmatrix}0 & y^{\\prime }+x^{\\prime }y^{\\prime } \\\\1+x^{\\prime } & 0 \\\\1+x^{\\prime }y^{\\prime } & 0 \\\\0 & 1+x^{\\prime }y^{\\prime }\\end{pmatrix}\\xrightarrow[E_{3,2}(y^{\\prime })]{E_{4,1}(\\bar{y}^{\\prime })}\\begin{pmatrix}0 & y^{\\prime }+x^{\\prime }y^{\\prime } \\\\1+x^{\\prime } & 0 \\\\1+y^{\\prime } & 0 \\\\:0 & x^{\\prime }y^{\\prime }+x^{\\prime }\\end{pmatrix}\\\\& \\xrightarrow[\\times \\bar{x}^{\\prime } \\bar{y}^{\\prime }]{\\text{col.2}}\\begin{pmatrix}0 & 1 + \\bar{x}^{\\prime } \\\\1+x^{\\prime } & 0 \\\\1+y^{\\prime } & 0 \\\\0 & 1+\\bar{y}^{\\prime }\\end{pmatrix}\\xrightarrow{}\\begin{pmatrix}1+y^{\\prime } & 0 \\\\1+x^{\\prime } & 0 \\\\0 & 1+ \\bar{x}^{\\prime }\\\\0 & 1 + \\bar{y}^{\\prime }\\\\\\end{pmatrix},$ which is exactly the 2D toric code.",
"$\\Diamond $ Example 6 (Chamon model [37], [38]) This three-dimensional model consists of single type of term in the Hamiltonian.",
"The generating map is $\\sigma _\\text{Chamon} =\\begin{pmatrix}x+\\bar{x} + y + \\bar{y} \\\\\\hline z+\\bar{z} + y + \\bar{y}\\end{pmatrix}.$ Since $\\sigma ^\\dagger \\lambda _1\\begin{pmatrix}0 \\\\1\\end{pmatrix}=(1+ x \\bar{y})\\begin{pmatrix}0 \\\\\\bar{x} + y\\end{pmatrix},$ $1+x \\bar{y}$ is a fractal generator.",
"Consisted of two terms, it generates a string operator.",
"The degeneracy can be calculated using Corollary REF .",
"Assume all the three linear dimensions of the system are even.",
"Put $S = R / (x + \\bar{x} + y + \\bar{y}, z + \\bar{z} + y + \\bar{y}, x^{2l} -1, y^{2m}-1, z^{2n} -1 ).$ Then, the $\\log _2$ of the degeneracy is $k = \\dim _{\\mathbb {F}_2} S$ .",
"In $S$ , we have $x + \\bar{x} = y + \\bar{y} = z + \\bar{z}$ .",
"Since $S$ has characteristic 2, it holds that $w^{p+1} + w^{-p-1} = (w + w^{-1})(w^p + w^{p-2} + \\cdots + w^{-p})$ for $p \\ge 1$ and $w = x,y,z$ .",
"By induction on $p$ , we see that $w^{p} + w^{-p}$ is a polynomial in $w + w^{-1}$ .",
"Therefore, $x^p + \\bar{x} ^p = y^p + \\bar{y}^p = z^p + \\bar{z}^p$ for all $p \\ge 1$ in $S$ .",
"Put $g = \\gcd (l,m,n)$ .",
"Since $x^l + x^{-l} = y^m + y^{-m} = z^n + z^{-n} = 0$ in $S$ , we have $x^g + x^{-g} = y^g + y^{-g} = z^g + z^{-g} = 0$ .",
"Applying Buchberger's criterion with respect to the lexicographic order in which $x \\prec y \\prec z$ , we see that $S = \\mathbb {F}_2[x,y,z] / (z^2 + zx^{2l-1}+ zx+1, y^2 + yx^{2l-1}+yx+1, x^{2g}+1)$ is expressed with a Gröbner basis.",
"Therefore, $k = \\dim _{\\mathbb {F}_2} S = 8 \\gcd (l,m,n).$ $\\Diamond $ Example 7 (Cubic Code) The Hamiltonian of code 1 in [7] is the translation-invariant negative sum of the following two types of interaction terms: $@!0{& IZ @{-}[ld]@{.",
"}[dd] @{-}[rr] & & ZI @{-}[ld] \\\\ZI @{-}[rr] @{-}[dd] & & ZZ @{-}[dd] & \\\\& II @{.",
"}[ld] & & IZ @{-}[uu] @{.",
"}[ll] \\\\IZ @{-}[rr] & & ZI @{-}[ru] }\\quad @!0{& IX @{-}[ld]@{.",
"}[dd] @{-}[rr] & & XI @{-}[ld] \\\\XI @{-}[rr] @{-}[dd] & & II @{-}[dd] & \\\\& XX @{.",
"}[ld] & & IX @{-}[uu] @{.",
"}[ll] \\\\IX @{-}[rr] & & XI @{-}[ru] }\\quad \\quad @!0{& z @{-}[ld]@{.",
"}[dd] @{-}[rr] & & yz @{-}[ld] \\\\xz @{-}[rr] @{-}[dd] & & xyz @{-}[dd] & \\\\& 1 @{.",
"}[ld] & & y @{-}[uu] @{.",
"}[ll] \\\\x @{-}[rr] & & xy @{-}[ru] }$ Here, the third cube specifies the coordinate system of the simple cubic lattice.",
"The corresponding generating map for the stabilizer module is $\\sigma _\\text{cubic-code} =\\begin{pmatrix}1 + xy + yz + zx & 0 \\\\1 + x + y + z & 0 \\\\0 & 1 + \\bar{x} + \\bar{y} + \\bar{z} \\\\0 & 1 + \\bar{x} \\bar{y} + \\bar{y} \\bar{z} + \\bar{z} \\bar{x}\\end{pmatrix}$ The associated ideal is contained in a prime ideal of codimension 2: $I(\\sigma ) \\subseteq ( 1+x+y+z, 1+xy+yz+zx ) = \\mathfrak {p}.$ Since $\\mathop {\\mathrm {codim}}I(\\sigma ) \\ge 2$ , the characteristic dimension is 1.",
"Since $\\mathop {\\mathrm {coker}}\\epsilon _\\text{cubic-code} = R / \\mathfrak {p}\\oplus R / \\bar{\\mathfrak {p}}$ , any nonzero element of $\\mathfrak {p}$ is a fractal generator.",
"Let us explicitly calculate the ground state degeneracy when the Hamiltonian is defined on $L \\times L \\times L$ cubic lattice with periodic boundary conditions.",
"By Corollary REF , $k = \\dim _{\\mathbb {F}_2} R / (\\mathfrak {p}+ \\mathfrak {b}_L) \\oplus R / (\\bar{\\mathfrak {p}}+ \\mathfrak {b}_L) = 2 \\dim _{\\mathbb {F}_2} R / (\\mathfrak {p}+ \\mathfrak {b}_L).$ So the calculation of ground state degeneracy comes down to the calculation of $d = \\dim _{\\mathbb {F}_2} T^{\\prime } / \\mathfrak {p}$ where $T^{\\prime } = \\mathbb {F}_2[x,y,z]/(x^{n_1}-1, y^{n_2}-1, z^{n_3}-1 )$ .",
"We may extend the scalar field to any extension field without changing $d$ .",
"Let $\\mathbb {F}$ be the algebraic closure of $\\mathbb {F}_2$ and let $T=\\mathbb {F}[x,y,z]/(x^{n_1}-1, y^{n_2}-1, z^{n_3}-1 )$ be an Artinian ring.",
"By Proposition REF , it suffices to calculate for each maximal ideal $\\mathfrak {m}$ of $T$ the vector space dimension $d_\\mathfrak {m}= \\dim _{\\mathbb {F}} (T / \\mathfrak {p})_\\mathfrak {m}$ of the localized rings, and sum them up.",
"Suppose $n_1,n_2,n_3 > 1$ .",
"By Nullstellensatz, any maximal ideal of $T$ is of form $\\mathfrak {m}=(x-x_0,y-y_0,z-z_0)$ where $x_0^{n_1}=y_0^{n_2}=z_0^{n_3}=1$ .",
"(If $n_1 = n_2 = n_3 = 1$ , then $T$ becomes a field, and there is no maximal ideal other than zero.)",
"Put $n_i = 2^{l_i}n_i^{\\prime }$ where $n_i^{\\prime }$ is not divisible by 2.",
"Since the polynomial $x^{n_1} -1$ contains the factor $x-x_0$ with multiplicity $2^{l_1}$ , it follows that $T_\\mathfrak {m}= \\mathbb {F}[x,y,z]_\\mathfrak {m}/ ( x^{2^{l_1}}+a^{\\prime },~ y^{2^{l_2}}+b^{\\prime },~ z^{2^{l_3}}+c^{\\prime })$ where $a^{\\prime } = x_0^{2^{l_1}}, b^{\\prime } = y_0^{2^{l_2}}, c^{\\prime } = z_0^{2^{l_3}}$ .",
"Hence, $(T/\\mathfrak {p})_\\mathfrak {m}\\cong \\mathbb {F}[x,y,z] / I^{\\prime }$ where $I^{\\prime } = (x+y+z+1, xy+xz+yz+1 ,~ x^{2^{l_1}}+a^{\\prime },~ y^{2^{l_2}}+b^{\\prime },~ z^{2^{l_3}}+c^{\\prime }).$ If $I^{\\prime } = \\mathbb {F}[x,y,z]$ , then $d_\\mathfrak {m}= 0$ .",
"Without loss of generality, we assume that $l_1 \\le l_2 \\le l_3$ .",
"By powering the first two generators of $I^{\\prime }$ , we see that $(x_0,y_0,z_0)$ must be a solution of them in order for $I^{\\prime }$ not to be a unit ideal.",
"Eliminating $z$ and shifting $x \\rightarrow x+1$ , $y \\rightarrow y+1$ , our objective is to calculate the Gröbner basis for the proper ideal $I = (x^2+xy+y^2, x^{2^{l_1}}+a,~ y^{2^{l_2}}+b )$ where $a=a^{\\prime }+1$ and $b = b^{\\prime }+1$ .",
"So $d_\\mathfrak {m}= \\dim _{\\mathbb {F}} \\mathbb {F}[x,y]/I.$ One can easily deduce by induction that $y^{2^m} + x^{2^m-1}(m x +y) \\in I$ for any integer $m \\ge 0$ .",
"And $b = \\omega a^{2^{l_2 - l_1}}$ for a primitive third root of unity $\\omega $ .",
"So we arrive at $I = ( y^2 + yx + x^2,~ yx^{2^{l_2} -1 } + b ( 1 + l_2 \\omega ^2 ),~ x^{2^{l_1}} + a )$ We apply the Buchberger criterion.",
"If $a \\ne 0$ , i.e., $x_0 \\ne 1$ , then $b \\ne 0$ and $I = (x+(\\omega ^2 + l_2)y, x^{2^{l_1}} + a)$ , so $d_\\mathfrak {m}= 2^{l_1}$ If $a=b=0$ , then $I = (y^2 + yx + x^2, yx^{2^{l_2}-1}, x^{2^{l_1}} )$ .",
"The three generators form Gröbner basis if $l_2 = l_1$ .",
"Thus, in this case, $d_\\mathfrak {m}= 2^{l_1+1}-1$ .",
"If $ l_2 > l_1$ , then $d_\\mathfrak {m}= 2^{l_1+1}$ .",
"To summarize, except for the special point $(1,1,1)\\in \\mathbb {F}^3$ of the affine space, each point in the algebraic set $V = \\left\\lbrace (x,y,z) \\in \\mathbb {F}^3 ~|~\\begin{matrix}x+y+z+1 = xy+xz+yz+1 = 0 \\\\x^{n_1^{\\prime }}-1 = y^{n_2^{\\prime }}-1 = z^{n_3^{\\prime }}-1 = 0\\end{matrix}\\right\\rbrace $ contribute $2^{l_1}$ to $d$ .",
"The contribution of $(1,1,1)$ is either $2^{l_1 +1}$ or $2^{l_1 +1}-1$ .",
"The latter occurs if and only if $l_1$ and $l_2$ , the two smallest numbers of factors of 2 in $n_1,n_2,n_3$ , are equal.",
"Let $d_0 = \\#V$ be the number of points in $V$ .",
"The desired answer is $d = 2^{l_1} (d_0 -1) +{\\left\\lbrace \\begin{array}{ll}2^{l_1 +1} -1 & \\text{if $l_1 = l_2$ } \\\\2^{l_1 +1} & \\text{otherwise}\\end{array}\\right.",
"}$ where $l_1 \\le l_2 \\le l_3$ are the number of factors of 2 in $n_i$ .",
"The algebraic set defined by $(x+y+z+1,~xy+xz+yz+1)$ is the union of two isomorphic lines intersecting only at $x=y=z=1$ , one of which is parametrized by $x \\in \\mathbb {F}$ as $(1+x, 1+\\omega x, 1+\\omega ^2 x) \\in \\mathbb {F}^3,$ and another is parametrized as $(1+x,1+\\omega ^2 x, 1+\\omega x) \\in \\mathbb {F}^3.$ where $\\omega $ is a primitive third root of unity.",
"Therefore, the purely geometric number $d_0 = 2 d_1 -1$ can be calculated by $d_1 = \\deg _x \\gcd \\left( (1+x)^{n_1^{\\prime }}+1, (1+\\omega x)^{n_2^{\\prime }}+1, (1+\\omega ^2 x)^{n_3^{\\prime }}+1 \\right) .$ Using $(\\alpha +\\beta )^{2^p} = \\alpha ^{2^p} + \\beta ^{2^p}$ and $\\omega ^2 + \\omega + 1 = 0$ , one can easily compute some special cases as summarized in the following corollary.",
"$\\Diamond $ Corollary 9.1 Let $2^k$ be the ground state degeneracy of the cubic code on the cubic lattice of size $L^3$ with periodic boundary conditions.",
"($k$ is the number of encoded qubits.)",
"Then $\\frac{k+2}{4}&= \\deg _x \\gcd \\left( (1+x)^{L}+1,~ (1+\\omega x)^{L}+1,~ (1+\\omega ^2 x)^{L}+1 \\right)_{\\mathbb {F}_4} \\\\&= {\\left\\lbrace \\begin{array}{ll}1 & \\text{if $L = 2^p+1$}, \\\\L & \\text{if $L =2^p$}, \\\\L-2 & \\text{if $L = 4^p -1$}, \\\\1 & \\text{if $L = 2^{2p+1} -1$}.\\end{array}\\right.",
"}$ where $\\omega ^2 + \\omega + 1 = 0$ and $p \\ge 1$ is any integer.",
"Example 8 (Levin-Wen fermion model [39]) The 3-dimensional model is originally defined in terms of hermitian bosonic operators $\\lbrace \\gamma ^{ab} \\rbrace _{a,b=1,\\ldots ,6}$ , squaring to identity if nonzero, such that $\\gamma ^{ab}=-\\gamma ^{ba}$ , $[\\gamma ^{ab},\\gamma ^{cd}]=0$ if $a,b,c,d$ are distinct, and $\\gamma ^{ab}\\gamma ^{bc}=i\\gamma ^{ac}$ if $a \\ne c$ .",
"An irreducible representation is given by Pauli matrices acting on $\\mathbb {C}^2 \\otimes \\mathbb {C}^2$ , and their commuting Hamiltonian fits nicely into our formalism.",
"The model was proposed to demonstrate that the point-like excitations may actually be fermions.",
"$\\sigma _\\text{Levin-Wen} &=\\begin{pmatrix}1+z & 1+z & x+y \\\\y+y z & x+x z & x+y \\\\y+z & 1+x & 1+x \\\\y+z & z+x z & y+x y\\end{pmatrix} \\\\\\epsilon _\\text{Levin-Wen} &=\\begin{pmatrix}y+z & y+z & y+y z & 1+z \\\\z+x z & 1+x & x+x z & 1+z \\\\y+x y & 1+x & x+y & x+y\\end{pmatrix}$ Here we multiplied the rows of $\\epsilon _\\text{Levin-Wen}$ by suitable monomials to avoid negative exponents.",
"One readily verifies that $\\ker \\epsilon _\\text{Levin-Wen} = \\mathop {\\mathrm {im}}\\sigma _\\text{Levin-Wen}$ .",
"The model is symmetric under the spatial rotation by $\\pi /3$ about $(1,1,1)$ axis.",
"Indeed, if one changes the variables as $x \\mapsto y \\mapsto z \\mapsto x$ and apply a symplectic transformation $\\omega =\\begin{pmatrix}1 & 0 & 1 & 0 \\\\0 & 1 & 0 & 1 \\\\1 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0\\end{pmatrix} :{\\left\\lbrace \\begin{array}{ll}XI \\mapsto YI \\\\IX \\mapsto IY \\\\ZI \\mapsto XI \\\\IZ \\mapsto IX\\end{array}\\right.}",
",$ then $\\sigma _\\text{Levin-Wen}$ remains the same up to permutations of columns.",
"The torsion submodule $T$ of $C = \\mathop {\\mathrm {coker}}\\epsilon _\\text{Levin-Wen}$ , which describes the point-like charges according to Theorem REF , is $T = R \\cdot \\begin{pmatrix} 1+y \\\\ 1+x \\\\ 0 \\end{pmatrix} .$ In order to see this, first shift the variables $a = x+1, b=y+1, c=z+1$ .",
"Then, $\\epsilon _\\text{Levin-Wen}$ becomes $\\epsilon _\\text{Levin-Wen} =\\begin{pmatrix}b+c & b+c & c+b c & c \\\\a+a c & a & c+a c & c \\\\a+a b & a & a+b & a+b\\end{pmatrix}=: \\phi $ We will verify that $N = C / T$ is torsion-free.",
"A presentation of $N = \\mathop {\\mathrm {coker}}\\phi ^{\\prime }$ is obtained by joining the generator of $T$ to the matrix $\\phi $ .",
"$\\phi ^{\\prime } =\\begin{pmatrix}b+c & b+c & c+b c & c & b \\\\a+a c & a & c+a c & c & a \\\\a+a b & a & a+b & a+b & 0\\end{pmatrix}$ Column operations of $\\phi ^{\\prime }$ give $\\phi ^{\\prime } \\cong \\begin{pmatrix}0 & c & b & 0 & 0 \\\\c & 0 & a & 0 & 0 \\\\b & a & 0 & 0 & 0\\end{pmatrix} =\\begin{pmatrix}\\partial _2 & 0 & 0\\end{pmatrix}$ where $\\partial _2$ is from Eq.",
"(REF ).",
"Therefore, $\\phi ^{\\prime }$ generates the kernel of $\\partial _1$ , and by Proposition REF , $N = \\mathop {\\mathrm {coker}}\\phi ^{\\prime }= \\mathop {\\mathrm {coker}}\\partial _2$ is torsion-free.",
"The torsion submodule $T$ of $C = \\mathop {\\mathrm {coker}}\\phi $ is annihilated by $a$ , $b$ , or $c$ (See Corollary REF ): $a \\begin{pmatrix} b \\\\ a \\\\ 0 \\end{pmatrix} =\\phi \\begin{pmatrix} 1 \\\\ 1+a \\\\ 0 \\\\ a \\end{pmatrix}, \\quad b \\begin{pmatrix} b \\\\ a \\\\ 0 \\end{pmatrix} =\\phi \\begin{pmatrix} 1 \\\\ 1+b \\\\ 1 \\\\ 1 \\end{pmatrix}, \\quad c \\begin{pmatrix} b \\\\ a \\\\ 0 \\end{pmatrix} =\\phi \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 1 \\end{pmatrix}.$ Therefore, $T$ is isomorphic to $\\mathop {\\mathrm {coker}}\\partial _1 \\cong \\mathbb {F}_2$ of Eq.",
"(REF ).",
"The arguments $h_x,h_y,h_z$ of $\\phi $ can be thought of as hopping operators for the charge.",
"According to [39], one can check that the charge is actually a fermion from the commutation values among, for example, $h_x,h_y,\\bar{y} h_y$ .",
"Consider a short exact sequence $0 \\rightarrow T \\rightarrow C \\rightarrow N \\rightarrow 0 .$ The corresponding sequence for 3D toric code splits, i.e., $C \\cong T \\oplus N$ , while this does not.",
"It implies that this model is not equivalent to the 3D toric code.",
"Now we can compute the ground state degeneracy, or $\\dim _{\\mathbb {F}_2} K(L)$ .",
"Tensoring the boundary condition $B = R/\\mathfrak {b}_L = R/(x^L-1,y^L-1,z^L-1)$ to the short exact sequence, we have a long exact sequence $\\cdots \\rightarrow {\\mathrm {Tor}}_1(T,B) \\xrightarrow{} {\\mathrm {Tor}}_1( C, B) \\xrightarrow{} {\\mathrm {Tor}}_1(N,B)\\rightarrow T \\otimes B \\rightarrow C \\otimes B \\rightarrow N \\otimes B \\rightarrow 0 .$ Hence, $K(L) \\cong {\\mathrm {Tor}}_1( C,B)$ has vector space dimension $\\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {im}}\\delta + \\dim _{\\mathbb {F}_2} \\ker \\delta $ .",
"Since the sequence is exact, $\\dim _{\\mathbb {F}_2} \\ker \\delta = \\dim _{\\mathbb {F}_2} \\mathop {\\mathrm {im}}\\delta ^{\\prime }$ .",
"As we have seen in Example REF , ${\\mathrm {Tor}}_1(T,B) & \\cong {\\mathrm {Tor}}_1(R/\\mathfrak {m},B) \\cong (\\mathbb {F}_2)^3 , \\quad \\text{and} \\\\{\\mathrm {Tor}}_1(N,B) & \\cong {\\mathrm {Tor}}_2(R/\\mathfrak {m},B) \\cong (\\mathbb {F}_2)^3 .$ It follows that $\\dim _{\\mathbb {F}_2} K(L) \\le \\dim _{\\mathbb {F}_2} {\\mathrm {Tor}}_1(N,B) + \\dim _{\\mathbb {F}_2} {\\mathrm {Tor}}_1(T,B) = 6$ .",
"It is routine to verify that $\\mathfrak {b}_4 \\subseteq I_2(\\phi ) \\subseteq \\mathfrak {m}:= (x+1,y+1,z+1)$ .",
"Recall the decomposition $K(L) = \\bigoplus _\\mathfrak {p}K(L)_\\mathfrak {p}$ where $\\mathfrak {p}$ runs over all maximal ideals of $R/\\mathfrak {b}_L$ .",
"Due to Lemma REF , this decomposition consists of only one summand $K(L)_\\mathfrak {m}$ .",
"When $L$ is odd, since $(\\mathfrak {b}_L)_\\mathfrak {m}= \\mathfrak {m}_\\mathfrak {m}$ , we know $K(L)_\\mathfrak {m}= K(1)_\\mathfrak {m}$ .",
"Since $\\phi \\mapsto 0$ under $a=b=c=0$ , we see $\\dim _{\\mathbb {F}_2} K(1) = 4$ .",
"The logical operators in this case are $\\begin{pmatrix} 0 \\\\ 0 \\\\ \\widehat{z} \\\\ \\widehat{z} \\end{pmatrix}\\smile \\widehat{x} \\cdot \\widehat{y \\bar{z}}\\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix}\\quad ; \\quad \\begin{pmatrix} \\widehat{x} \\\\ \\widehat{x} \\\\ 0 \\\\ 0 \\end{pmatrix}\\smile \\widehat{z} \\cdot \\widehat{x y}\\begin{pmatrix} 1 \\\\ 1 \\\\ 1 \\\\ 0 \\end{pmatrix}$ where $\\widehat{\\mu }= \\sum _{n=0}^{L-1} \\mu ^n$ so $\\mu \\cdot \\widehat{\\mu }= \\widehat{\\mu }$ , and symplectic pairs are tied.",
"The left elements are string-like, and the right surface-like.",
"When $L$ is even, the following are $\\mathbb {F}_2$ -independent elements of $K(L)$ .",
"As there are 6 in total, the largest possible number, we conclude that $K(L)$ is 6-dimensional, i.e., the number of encoded qubits is 3 when linear dimensions are even.",
"$\\begin{pmatrix} 0 \\\\ 0 \\\\ \\widehat{z} \\\\ \\widehat{z} \\end{pmatrix}\\smile \\widehat{x} ^{\\prime } \\widehat{y} ^{\\prime }\\begin{pmatrix} 1+y \\\\ x+xy \\\\ 0 \\\\ 1+x+y+xy \\end{pmatrix};\\begin{pmatrix} \\widehat{x} \\\\ \\widehat{x} \\\\ 0 \\\\ 0 \\end{pmatrix}\\smile \\widehat{y} ^{\\prime } \\widehat{z} ^{\\prime }\\begin{pmatrix} 1+z \\\\ 1+z \\\\ 1+z \\\\ y+yz \\end{pmatrix};\\begin{pmatrix} \\widehat{y} \\\\ \\widehat{y} \\\\ \\widehat{y} \\\\ \\widehat{y} \\end{pmatrix}\\smile \\widehat{z} ^{\\prime } \\widehat{x} ^{\\prime }\\begin{pmatrix} 0 \\\\ 1+x+z+xz \\\\ 1+x \\\\ 1+x \\end{pmatrix}$ where $\\widehat{\\mu }^{\\prime } = \\sum _{i=0}^{L/2-1} \\mu ^{2i}$ so $(1+\\mu )\\widehat{\\mu }^{\\prime } = \\widehat{\\mu }$ .",
"The pairs are transformed cyclically by $x \\mapsto y \\mapsto z \\mapsto x$ together with the symplectic transformation $\\omega $ of Eq.",
"(REF ).",
"$\\Diamond $"
],
[
"Discussion",
"There are many natural questions left unanswered.",
"Perhaps, it would be the most interesting to answer how much the associated ideal $I(\\sigma )$ determines about the Hamiltonian.",
"Note that the very algebraic set defined by the associated ideal is not invariant under coarse-graining.",
"For instance, in the characteristic dimension zero case, the algebraic set can be a several points in the affine space, but becomes a single point under a suitable coarse-graining.",
"It is reasonable to conceive that the algebraic set is mapped by the affine map $(a_i) \\mapsto (a_i^n)$ under the coarse-graining by $x_i^{\\prime } = x_i^n$ .",
"This is true if $t=q$ , so the $q$ -th determinantal ideal of $\\epsilon $ , being the initial Fitting ideal, has the same radical as $\\mathop {\\mathrm {ann}}\\mathop {\\mathrm {coker}}\\epsilon $ .",
"In fact, we have implicitly used this idea in the proofs of Lemma REF , REF , and Theorem REF .",
"The case $t > q$ is not explicitly handled here.",
"Also, it is interesting on its own to prove or disprove that the elementary symplectic transformations generate the whole symplectic transformation group."
]
] | 1204.1063 |
[
[
"1-- and 0++ heavy four-quark and molecule states in QCD"
],
[
"Abstract We estimate the masses of the 1^{--} heavy four-quark and molecule states by combining exponential Laplace (LSR) and finite energy (FESR) sum rules known perturbatively to lowest order (LO) in alpha_s but including non-perturbative terms up to the complete dimension-six condensate contributions.",
"This approach allows to fix more precisely the value of the QCD continuum threshold (often taken ad hoc) at which the optimal result is extracted.",
"We use double ratio of sum rules (DRSR) for determining the SU(3) breakings terms.",
"We also study the effects of the heavy quark mass definitions on these LO results.",
"The SU(3) mass-splittings of about (50 - 110) MeV and the ones of about (250 - 300) MeV between the lowest ground states and their 1st radial excitations are (almost) heavy-flavour independent.",
"The mass predictions summarized in Table 4 are compared with the ones in the literature (when available) and with the three Y_c(4260,~4360,~4660) and Y_b(10890) 1^{--} experimental candidates.",
"We conclude (to this order approximation) that the lowest observed state cannot be a pure 1^{--} four-quark nor a pure molecule but may result from their mixings.",
"We extend the above analyzes to the 0^{++} four-quark and molecule states which are about (0.5-1) GeV heavier than the corresponding 1^{--} states, while the splittings between the 0^{++} lowest ground state and the 1st radial excitation is about (300-500) MeV.",
"We complete the analysis by estimating the decay constants of the 1^{--} and 0^{++} four-quark states which are tiny and which exhibit a 1/M_Q behaviour.",
"Our predictions can be further tested using some alternative non-perturbative approaches or/and at LHCb and some other hadron factories."
],
[
"Introduction and a short review on the $1^{++}$ channel",
"A large amount of exotic hadrons which differ from the “standard\" $\\bar{c}c$ chamonium and $\\bar{b}b$ bottomium radial excitation states have been recently discovered in $B$ -factories through $J/\\psi \\pi ^+\\pi ^-$ and $\\Upsilon \\pi ^+\\pi ^-$ processes and have stimulated different theoretical interpretations.",
"Most of them have been assigned as four-quarks and/or molecule states [3].",
"In previous papers [1], [2], some of us have studied, using exponential QCD spectral sum rules (QSSR) [4] For reviews, see e.g.",
"[5], [6].",
"and the double ratio of sum rules (DRSR) [7] For some other successful applications, see [8], [9], [10]., the nature of the $X(3872)$ $1^{++}$ states found by Belle [11] and confirmed by Babar [12], CDF [13] and D0 [14].",
"If it is a $(cq)(\\overline{cq})$ four-quark or $D-D^*$ molecule state, one finds for $m_c=1.23$ GeV [1] The two configurations give almost a degenerate mass-value [2].",
": ${X_c}=(3925\\pm 127)~{\\rm MeV}~,$ corresponding to a $t_c$ -value common solution of the exponential Laplace (LSR) and Finite Energy (FESR) sum rules: $\\sqrt{t_c}=(4.15\\pm 0.03)~{\\rm GeV}~,$ while in the $b$ -meson channel, using $m_b=4.26$ GeV, one finds [1]: $X_b=(10144\\pm 104)~\\rm {MeV}~~{\\rm with}~~\\sqrt{t_c}=(10.4\\pm 0.02)~{\\rm GeV},$ where a similar result has been found in [15] using another choice of interpolating current.",
"However, in the case of the $X_c(3872)$ , the previous two configurations are not favoured by its narrow hadronic width ($\\le $ 2.3 MeV), which has lead some of us to propose that it could be, instead, a $\\lambda -J/\\psi $ -type molecule [2] described by the current: $J_\\mu ^\\lambda =\\left({g\\over \\Lambda }\\right)^2_{\\rm eff}(\\bar{c}\\lambda ^a\\gamma ^\\mu c)(\\bar{q}\\lambda _a\\gamma _5q)~,$ where $\\lambda _a$ is the colour matrix , while $g$ and $\\Lambda $ are coupling and scale associated to an effective Van Der Vaals force.",
"In this case, the narrow width of the $X_c$ is mainly due to the extra-gluon exchange which gives a suppression of the order $\\alpha _s^2$ compared to the two former configurations, if one evaluates this width using vertex sum rules.",
"The corresponding mass is slightly lower than the one in Eq.",
"(REF ) [2]: $r\\equiv {X_c^{\\lambda }\\over X_c^{mol}}=0.96\\pm 0.03~~ \\Longrightarrow X_c^{\\lambda }=(3768\\pm 127)~{\\rm MeV}~.$ which (within the errors) also agree with the data.",
"By assuming that the mass of the radial excitation $X^{\\prime }_Q\\approx \\sqrt{t_c}$ , one can also deduce the mass-splitting: $X^{\\prime }_c-X_c\\simeq 225~{\\rm MeV}\\approx X^{\\prime }_b-X_b\\simeq 256~{\\rm MeV}~,$ which is much lower than the ones of ordinary charmonium and bottomium states: ${\\psi }(2S)-{\\psi }(1S)\\simeq 590\\approx {\\Upsilon }(2S)-{\\Upsilon }(1S)\\simeq 560~{\\rm MeV},$ and suggests a completely different dynamics for these exotic states.",
"Comparing the previous results with the observed $Z_b(10610)$ and $Z_b(10650)$ states whose quantum numbers have been assigned to be $1^{++}$ , one can conclude that these observed states are heavier than the 1st radial excitation of the $X_b(10.14)$ expected from QSSR to lowest order in $\\alpha _s$ [1]."
],
[
"QCD Analysis of the $1^{--}$ and {{formula:b84a18f0-3ace-4d24-a583-8be1ea656b04}} channels",
"In the following, we extend the previous analysis to the case of the $1^{--}$ and $0^{++}$ channels and improve some existing analysis from QCD (spectral) sum rules in the $1^{--}$ channel [18], [19].",
"The results will be compared with the experimental $1^{--}$ candidate states: $Y(4260),~~~~~~~ Y(4360), ~~~~~~~ Y(4660)~,~~~~~~~ Y_b(10890)$ seen by Babar [16] and Belle [17], [23] and which decay into $J/\\psi \\pi ^+\\pi ^-$ and $\\Upsilon \\pi ^+\\pi ^-$ around the $\\Upsilon (5S)$ mass.",
"These states cannot be identified with standard $\\bar{c}c$ charmonium and $\\bar{b}b$ bottomium radial excitations and have been assigned in the literature to be four-quark or molecule states or some threshold effects."
],
[
"$\\bullet ~$ QCD input parameters",
"The QCD parameters which shall appear in the following analysis will be the charm and bottom quark masses $m_{c,b}$ , the light quark masses $m_{d,s}$ , the light quark condensates $\\langle \\bar{q}q\\rangle $ and $\\langle \\bar{s}s\\rangle $ , the gluon condensates $ \\langle g^2G^2\\rangle \\equiv \\langle g^2G^a_{\\mu \\nu }G_a^{\\mu \\nu }\\rangle $ and $\\langle g^3G^3\\rangle \\equiv \\langle g^3f_{abc}G^a_{\\mu \\nu }G^b_{\\nu \\rho }G^c_{\\rho \\mu }\\rangle $ , the mixed condensate $\\langle \\bar{q}g\\sigma Gq\\rangle \\equiv {\\langle \\bar{q}g\\sigma ^{\\mu \\nu } (\\lambda _a/2) G^a_{\\mu \\nu }q\\rangle }$ and the four-quark condensate $\\rho \\langle \\bar{q}q\\rangle ^2$ , where $\\rho $ indicates the violation of the four-quark vacuum saturation.",
"Their values are given in Table REF and we shall work with the running light quark parameters: ${\\bar{m}}_s(\\tau )&=&{{\\hat{m}}_s \\over \\left(-\\log { \\sqrt{\\tau }\\Lambda }\\right)^{-2/{\\beta _1}}}\\nonumber \\\\{\\langle \\bar{q}q\\rangle }(\\tau )&=&-\\hat{\\mu }_q^3 \\left(-\\log { \\sqrt{\\tau }\\Lambda }\\right)^{-2/{\\beta _1}}\\nonumber \\\\{\\langle \\bar{q}g\\sigma Gq\\rangle }(\\tau )&=&-M_0^2{\\hat{\\mu }_q^3} \\left(-\\log { \\sqrt{\\tau }\\Lambda }\\right)^{-1/{3\\beta _1}}~,$ where $\\beta _1=-(1/2)(11-2n/3)$ is the first coefficient of the $\\beta $ function for $n$ flavours; $\\hat{m}_s$ and $\\hat{\\mu }_q$ are renormalization group invariant light quark mass and condensate [25], [26].",
"Table: QCD input parameters.",
"For the heavy quark masses, we usethe range spannedby the running MS ¯\\overline{MS} mass m ¯ Q (M Q )\\overline{m}_Q(M_Q) and the on-shell massfrom QCD (spectral) sum rules compiled in pages602 and 603 of the book in and recently obtained in Ref.",
".",
"The values of Λ\\Lambda and μ ^ q \\hat{\\mu }_q havebeen obtained from α s (M τ )=0.325(8)\\alpha _s(M_\\tau )=0.325(8) and from the runningmasses: (m ¯ u +m ¯ d )(2)=7.9(3)(\\overline{m}_u+\\overline{m}_d)(2)=7.9(3) MeV .",
"Theoriginal errors have been multiplied by 2 for a conservative estimate of theerrors.We assume that the $Y$ state is described either by the lowest dimension (without derivative terms) four-quark and molecule $\\bar{D}_sD^*_s$ vector currents $J_\\mu $ given in Tables REF and REF .",
"Unlike the case of baryons where both positive and parity states can couple to the same operator [24], the situation is simpler here as the vector and axial-vector currents have a well-defined quantum numbers to which are associated the $1^{--}$ (resp.",
"$1^{++}$ ) states for the transverse part and the $0^{++}$ (resp.",
"$0^{--}$ ) states for the longitudinal part.",
"In the case of four-quark currents, we can have two-types of lowest derivative vector operators which can mix through the mixing parameter $b$ The $1^{++}$ four-quark state described by the axial-vector current has been analyzed in [1], [2].. Another possible mixing can occur through the renormalization of operators [26], [27] though this type of mixing will only induce an overall effect due to the anomalous dimension which will be relevant at higher order in $\\alpha _s$ but will disappear in the ratio of sum rules used in this paper.",
"For the molecule current, we choose the product of local bilinear current which has the quantum number of the corresponding meson state.",
"In this sense, we have only an unique interpolating current.",
"Observed states can be a mixing of different states associated to each choice of operators and their selection can only be done through the analysis of their decays [2] but this is beyond the scope of this paper.",
"The two-point functions of the $Y_Q~(Q\\equiv c,b)$ (assumed to be a $1^{--}$ vector meson) is defined as: $\\Pi ^{\\mu \\nu }(q)&\\equiv &i\\int d^4x ~e^{iq.x}\\langle 0|T[j^\\mu (x){j^\\nu }^\\dagger (0)]|0\\rangle \\nonumber \\\\&=&-\\Pi ^{(1)}(q^2)(g^{\\mu \\nu }-{q^\\mu q^\\nu \\over q^2})+\\Pi ^{(0)}(q^2){q^\\mu q^\\nu \\over q^2}~,$ where $J^\\mu $ are the interpolating vector currents given Tables REF and REF .",
"We assume that the $Y$ state is described either by the lowest dimension (without derivative terms) four-quark and molecule $\\bar{D}_sD^*_s$ currents given in Tables REF and REF .",
"The two invariants, $\\Pi ^{(1)}$ and $\\Pi ^{(0)}$ , appearing in Eq.",
"(REF ) are independent and have respectively the quantum numbers of the spin 1 and 0 mesons.",
"We can extract $\\Pi _Q^{(1)}$ and $\\Pi ^{(0)}(q^2)$ or the corresponding spectral functions from the complete expression of $\\Pi ^{\\mu \\nu }_Q(q)$ by applying respectively to it the projectors: ${\\cal P}^{(1)}_{\\mu \\nu }= - \\frac{1}{3} \\left( g^{\\mu \\nu } - \\frac{q^\\mu q^\\nu }{q^2} \\right)~~~~~{\\rm and}~~~~~{\\cal P}^{(0)}_{\\mu \\nu }= {q_\\mu q_\\nu \\over q^2}~.$ Due to its analyticity, the correlation function, $\\Pi ^{(1,0)}(q^2)$ in Eq.",
"(REF ), obeys the dispersion relation: $\\Pi ^{(1,0)}(q^2)={1\\over \\pi }\\int _{4m_c^2}^\\infty ds {{\\rm Im}\\:\\Pi ^{(1,0)}(s)\\over s-q^2-i\\epsilon }+\\cdots \\;,$ where $\\mbox{Im}\\:\\Pi ^{(1,0)}(s)$ are the spectral functions.",
"The QCD expressions of these spectral functions are given in Tables REF and REF .",
"$1/q^2$ terms discussed in [44], [45],which are dual to higher order terms of the QCD series will not be included here as we work to leading order.",
"Table: QCD expression of the Four-Quark Spectral Functions to lowest order in α s \\alpha _s and up to dimension-six condensates: Q≡c,bQ\\equiv c,b is the heavy quark field.Table: QCD expression of the Molecule Spectral Functions to lowest order in α s \\alpha _s and up to dimension-six condensates: Q≡c,bQ\\equiv c,b is the heavy quark field, while g ' g^{\\prime } and Λ ' \\Lambda ^{\\prime } are coupling and scale associated to an effective Van Der Vaals force."
],
[
"$1^{--}$ four-quark state mass {{formula:4b452846-2b5f-481b-b61e-02ccade825dc}} from QSSR",
"In the following, we shall estimate the mass of the $1^{--}$ four-quark state $(\\overline{Qq}) (Qq)$ ($Q\\equiv c,~b$ and $q\\equiv u,~d$ quarks), hereafter denoted by $Y_{Qd}$ .",
"In so doing, we shall use the ratios of the Laplace (exponential) sum rule: ${\\cal R}^{LSR}_{Qd}(\\tau )\\equiv -{d\\over d\\tau }{\\rm Log}{\\int _{t_<}^{t_c} dt e^{-t\\tau }{1\\over \\pi }{\\rm Im}\\Pi ^{(1)}(t)}~,$ and of FESR: ${\\cal R}_{Qd}^{FESR}\\equiv {\\int _{t_<}^{t_c}dt ~t^n{1\\over \\pi }{\\rm Im}\\Pi ^{(1)}(t)\\over \\int _{t<}^{t_c} dt ~t^{n-1} {1\\over \\pi }{\\rm Im}\\Pi ^{(1)}(t)}~~~~:~~ n=1~,$ where $t_<$ is the hadronic (quark) threshold.",
"Within the usual duality ansatz “one resonance\" + $\\theta (t-t_c)\\times QCD~continuum$ parametrization of the spectral function, the previous ratios of sum rules give: ${\\cal R}^{LSR}_{Qd}(\\tau )\\simeq M_{Y_{Qd}}^2 \\simeq {\\cal R}_{Qd}^{FESR}~.$ For a discussion more closed to the existing literature which we shall test the reliability in the following, we start to work with the current corresponding to $b=0$ .",
"We shall discuss the more general choice of current when $b$ is a free parameter at the end of this section."
],
[
"$\\bullet ~$ The {{formula:b5dc6b76-5d40-4693-9863-32e009a8ce00}} mass from LSR and FESR for the case {{formula:c9ec37a5-f010-4da7-ac0c-4e2f59ff257d}} =0",
"Using the QCD inputs in Table REF , we show the $\\tau $ -behaviour of $M_{Y_{cd}}$ from ${\\cal R}^{LSR}_{cd}$ in Fig.",
"REF a for $m_c=1.26$ GeV and for different values of $t_c$ .",
"One can notice from Fig.",
"REF a that the $\\tau $ -stability is obtained from $\\sqrt{t_c}\\ge 5.1$ GeV, while the $t_c$ -stability is reached for $\\sqrt{t_c}=7$ GeV.",
"The most conservative prediction from the LSR is obtained in this range of $t_c$ -values for $m_c=1.26$ GeV and gives in units of GeV: $4.79 \\le M_{Y_{cd}}\\le 5.73~~{\\rm for}~~5.02 \\le \\sqrt{t_c}\\le 7~{\\rm and}~ m_c=1.26,\\nonumber \\\\5.29 \\le M_{Y_{cd}}\\le 6.11~~{\\rm for}~~5.5 \\le \\sqrt{t_c}\\le 7~{\\rm and}~m_c=1.47.$ We compare in Fig.",
"REF b), the $t_c$ -behaviour of the LSR results obtained at the $\\tau $ -stability points with the ones from $ {\\cal R}^{FESR}_{cd}$ for the charm quark mass $m_c$ =1.23 GeV (running) and 1.47 GeV (on-shell).",
"One can deduce the common solution in units of GeV: $M_{Y_{cd}}&=& 4.814~~~~{\\rm for}~~~~ \\sqrt{t_c}=5.04(5)~~ {\\rm and}~~m_c=1.26,\\nonumber \\\\&=& 5.409~~~~{\\rm for}~~~~ \\sqrt{t_c}=5.6~~ {\\rm and}~~m_c=1.47~.$ In order to fix the values of $M_{Y_{cd}}$ obtained at this lowest order PT calculations, we can also refer to the predictions of the $J/\\psi $ mass using the LSR at the same lowest order PT calculations and including the condensate contributions up to dimension-six.",
"We observe that the on-shell $c$ -quark mass value tends to overestimate $M_{J/\\psi }$ [2], [28].",
"The same feature happens for the evaluation of the $X(1^{++})$ four-quark state mass [1].",
"Though this observation may not be rigorous as the strength of the radiative corrections is channel dependent, we are tempted to take as a final result in this paper the prediction obtained by using the running mass $\\overline{m}_c(m_c)=1262(17)$ MeV within which it is known, from different examples in the literature, that the PT series converge faster [28] We plan to check this conjecture in a future publication when PT radiative corrections are included..",
"Including different sources of errors, we deduce in MeV We consider this result as an improvement (smaller error) of the one e.g.",
"in [18], [19] where only exponential sum rules have been used.",
"However, the present error and the existing ones in the literature may have been underestimated due to the non-inclusion of the unknown PT radiative corrections and some eventual systematics of the approach.",
": $M_{Y_{cd}}&=&4814(50)_{t_c}(14)_{m_c}(2)_\\Lambda (17)_{\\bar{u}u}(2)_{G^2}(4)_{M^2_0}(13)_{G^3}(6)_\\rho \\nonumber \\\\&=&4814(57)~.$ Using the fact that the 1st FESR moment gives a correlation between the mass of the lowest ground state and the onset of continuum threshold $t_c$ , where its value coincide approximately with the value of the 1st radial excitation mass (see e.g.",
"ref.",
"[39] and some other examples in [5]), we shall approximately identify its value with the one of the radial excitation.",
"In order to take into account the systematics of the approach and some eventual small local duality violation advocated by [46] which can only be detectable in a high-precision analysis like the extraction of $\\alpha _s$ from $\\tau $ -decay [29], [47], we have allowed $t_c$ to move around this intersection point.",
"Assuming that the mass of the radial excitation is approximately $\\sqrt{t_c}$ , one can deduce the mass-splitting: $M^{\\prime }_{Y_{cd}}-M_{Y_{cd}}\\approx 226~{\\rm MeV}~,$ which is similar to the one obtained for the $X(1^{++})$ four-quark state [1].",
"This splitting is much lower than the one intuitively used in the current literature: $M_{\\psi }(2S)-M_{\\psi }(1S)\\simeq 590~{\\rm MeV}~,$ for fixing the arbitrary value of $t_c$ entering in different Borel (exponential) sum rules of the four-quark and molecule states.",
"This difference may signal some new dynamics for the exotic states compared with the usual $\\bar{c}c$ charmonium states and need to be tested from some other approaches such as potential models, heavy quark symmetry, AdS/QCD and lattice calculations.",
"Figure: a) τ\\tau -behaviour of M Y bd (1 -- )M_{Y_{bd}}(1^{--}) from ℛ bd LSR {\\cal R}^{LSR}_{bd} for the current mixing parameter b=0b=0, for different values of t c t_cand for m b =4.17m_b=4.17 GeV.",
"b) t c t_c-behaviour of the LSR results obtained at the τ\\tau -stability points and comparison with the ones from ℛ bd FESR {\\cal R}^{FESR}_{bd} for m b =4.17m_b=4.17 and 4.70 GeV.Using similar analysis for the $b$ -quark, we show the $\\tau $ -behaviour of ${\\cal R}^{LSR}_{bd}(\\tau )$ in Fig.",
"REF a for $m_b=4.17$ GeV and for different values of $t_c$ .",
"In Fig.",
"REF b, the same analysis is shown for $m_b=4.70$ GeV.",
"The most conservative result from the LSR is (in units of GeV) is: $11.0 \\le M_{Y_{bd}}\\le 12.4~{\\rm for}~11.2 \\le \\sqrt{t_c}\\le 14.5~{\\rm and}~ m_b=4.17,\\nonumber \\\\12.1 \\le M_{Y_{bd}}\\le 13.4~{\\rm for}~12.",
"2\\le \\sqrt{t_c}\\le 15.5~{\\rm and}~ m_b=4.70,\\nonumber $ where the lower (resp.",
"higher) values of $t_c$ correspond to the beginning of $\\tau $ (resp.",
"$t_c$ )-stability.",
"We compare in Fig.",
"REF b), the $t_c$ -behaviour of the LSR results obtained at the $\\tau $ -stability points with the ones from $ {\\cal R}^{FESR}_{bd}$ for the $b$ quark mass $m_b$ =4.17 GeV (running) and 4.70 GeV (on-shell).",
"One can deduce the common solution in units of GeV: $M_{Y_{bd}}&=& 11.26~~~~{\\rm for}~~~~ \\sqrt{t_c}=11.57(7)~ {\\rm and}~m_b=4.17\\nonumber \\\\&=& 12.09~~~~{\\rm for}~~~~ \\sqrt{t_c}=12.2~ {\\rm and}~m_b=4.70.$ One can notice, like in the case of the charm quark that the value of the on-shell quark mass tends to give a higher value of $M_{Y_{bd}}$ within this lowest order PT calculations.",
"Considering, like in the case of charm, as a final estimate the one from the running $b$ -quark mass $\\overline{m}_b(m_b)=4177(11)$ MeV [28], we deduce in MeV: $M_{Y_{bd}}&=&11247(45)_{t_c}(8)_{m_b}(2)_\\Lambda (15)_{\\bar{u}u}(1)_{G^2}(1)_{M^2_0}(1)_{G^3}(5)_\\rho \\nonumber \\\\&=&11256(49)~.$ From the previous result, one can deduce the approximate value of the mass-splitting between the 1st radial excitation and the lowest mass ground state: $M^{\\prime }_{Y_{bd}}-M_{Y_{bd}}\\approx M^{\\prime }_{Y_{cd}}-M_{Y_{cd}}\\approx 250~{\\rm MeV}~,$ which are (almost) heavy-flavour independent and also smaller than the one of the bottomium splitting: $M_{\\Upsilon }(2S)-M_{\\Upsilon }(1S)\\simeq 560~{\\rm MeV}~.$ In the following, we shall let the current mixing parameter $b$ defined in Table REF free and study its effect on the results obtained in Eqs.",
"(REF ) and (REF ).",
"In so doing, we fix the values of $\\tau $ around the $\\tau $ -stability point and $t_c$ around the intersection point of the LSR and FESR.",
"The results of the analysis are shown in Fig.",
"REF .",
"We notice that the results are optimal at the value $b=0$ which a posteriori justifies the results obtained previously for $b=0$ .",
"Figure: a) bb-behaviour of M Y cd M_{Y_{cd}} for given values of τ\\tau and t c t_cand for m c =1.26m_c=1.26 GeV; b) the same as a) but for M Y bd M_{Y_{bd}}and for m b =4.17m_b=4.17 GeV.For completing the analysis of the effect of $b$ , we also study the decay constant $f_{Y_Qd}$ defined as: $\\langle 0\\vert j^\\mu _{4q}\\vert Y_{Qd}\\rangle =f_{Y_Qd}M^4_{Y_{Qd}}\\epsilon ^\\mu ~.$ We show the analysis in Fig.",
"REF giving $M_{Y_{Qd}}$ and the corresponding $t_c$ obtained above.",
"One can deduce the optimal values at $b=0$ : $f_{Y_{cd}} \\simeq 0.08~{\\rm MeV}~~~~{\\rm and} ~~~~~f_{Y_{bd}} \\simeq 0.03~{\\rm MeV}~,$ which are much smaller than $f_\\pi =132$ MeV, $f_\\rho \\simeq 215$ MeV and $f_D\\simeq f_B$ =203 MeV [48].",
"On can also note that the decay constant decreases like $1/M_Q$ which can be tested in HQET or/and lattice QCD.",
"Figure: a) τ\\tau -behaviour of f Y cd f_{Y_{cd}} for given values of b=0b=0 and t c t_cand for m c =1.26m_c=1.26 GeV; b) the same as a) but for f Y bd f_{Y_{bd}}and for m b =4.17m_b=4.17 GeV;c) bb-behaviour of f Y cd f_{Y_{cd}} for given values of τ\\tau at the stability and t c t_cd) the same as c) but for f Y bd f_{Y_{bd}}We study the ratio $M_{Y_{Qs}}/M_{Y_{Qd}}$ using double ratio of LSR (DRSR): $r^Q_{sd}\\equiv {\\sqrt{{\\cal R}_{Qs}^{LSR}}\\over \\sqrt{{\\cal R}_{Qd}^{LSR}}}~~~~{\\rm where}~~~~Q\\equiv c,~b~.$ We show the $\\tau $ -behaviour of $r^c_{sd}$ and $r^b_{sd}$ respectively in Figs.",
"REF a and REF b for $m_c=1.26$ GeV and $m_b=4.17$ GeV for different values of $t_c$ .",
"We show, in Fig.",
"REF c and Fig.",
"REF d, the $t_c$ -behaviour of the stabilities or inflexion points for two different values (running and on-shell) of the quark masses.",
"One can see in these figures that the DRSR is very stable versus the $t_c$ variations in the case of the running heavy quark masses.",
"We deduce the corresponding DRSR: $r^c_{sd}&=&1.018(1)_{m_c}(5)_{m_s} (2)_{\\kappa }(2)_{\\bar{u}u}(1)_\\rho ~,\\nonumber \\\\r^b_{sd}&=&1.007(0.5)_{m_b}(2)_{m_s} (0.5)_{\\kappa }(1)_{\\bar{u}u}(0.3)_\\rho ~,$ respectively for $\\sqrt{t_c}=5.1$ and 11.6 GeV.",
"Figure: a) τ\\tau -behaviour of r sd c r^c_{sd} for the current mixing parameter b=0b=0, for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of r sd b r^b_{sd} for different values of t c t_cand for m b =4.17m_b=4.17 GeV;c) t c t_c-behaviour of the inflexion points (or minimas) of r sd c r^c_{sd} from Fig a; d) The same for the bb quark using r sd b r^b_{sd} from Fig b.Using the results for $Y_{Qd}$ in Eqs.",
"(REF ) and (REF ) and the values of the $SU(3)$ breaking ratio in Eq.",
"(REF ), we can deduce the mass of the $Y_{Qs}$ state in MeV: $M_{Y_{cs}}=4900(67)~,~~~~~~~~~~~M_{Y_{bs}}=11334(55)~,$ leading to the $SU(3)$ mass-splitting: $\\Delta M^{Y_c}_{sd}\\approx 87~{\\rm MeV}\\approx \\Delta M^{Y_b}_{sd}\\approx 78~{\\rm MeV}~,$ which is also (almost) heavy-flavour independent.",
"Like in the previous case, we use LSR and FESR for studying the masses of the $\\bar{D}^*_dD_d$ and $\\bar{B}^*_{d}B_{d}$ and DRSR for studying the $SU(3)$ breaking ratios: $r^D_{sd}\\equiv {M_{D^*_sD_s}\\over M_{D^*_dD_d}}~,~~~~~~~~~~~~~~~~~~~r^B_{sd}\\equiv {M_{B^*_sB_s}\\over M_{B^*_dB_d}}~.$ We show their $\\tau $ -behaviour for different values of $t_c$ and for $m_c=1.26$ GeV and $m_b=4.17$ GeV respectively in Figs.",
"REF a,b and REF a,b.",
"The $t_c$ -behaviour of the $\\tau $ -minimas is shown in Fig.",
"REF c,d for the masses and in Fig.",
"REF c,d for the $SU(3)$ breaking ratios.",
"Using the sets ($m_c=1.26$ GeV, $\\sqrt{t_c}=5.58~{\\rm GeV}$ ) and ($m_b=4.17~{\\rm GeV}$ , $\\sqrt{t_c}=11.64(3)~{\\rm GeV}$ ) common solutions of LSR and FESR, one can deduce in MeV: $M_{{D^*_dD_d}}&=&5268(14)_{m_c}(3)_\\Lambda (19)_{\\bar{u}u}(0)_{G^2}(0)_{M^2_0}(2)_{G^3}(5)_\\rho ,\\nonumber \\\\&=&5268(24)~,\\nonumber \\\\M_{{B^*_dB_d}}&=&11302 (20)_{t_c}(9)_{m_b}(2)_\\Lambda (19)_{\\bar{u}u}(0)_{G^2}(0)_{M^2_0}(1)_{G^3}(5)_\\rho \\nonumber \\\\&=&11302 (30)~,\\nonumber \\\\r^D_{sd}&=&1.018(1)_{m_c}(4){m_s}(0.8)_{\\kappa }(0.5)_{\\bar{u}u}(0.2)_{\\rho }(0.1)_{G^3}~,\\nonumber \\\\r^B_{sd}&=&1.006(1)_{m_b}(2){m_s}(1)_{\\kappa }(0.5)_{\\bar{u}u}(0.2)_{\\rho }(0.1)_{G^3}~.$ Using the previous results in Eq.",
"(REF ), one obtains in MeV : $M_{{D^*_sD_s}}=5363(33)~,~~~~~ M_{{B^*_sB_s}}=11370(40)~,$ corresponding to a $SU(3)$ mass-splitting: $\\Delta M^{DD^*}_{sd}\\simeq 95 ~{\\rm MeV} \\approx \\Delta M^{BB^*}_{sd}\\simeq 68~{\\rm MeV}~.$ These results for $M_{DD^*}$ are in the upper part of the range given in [18] due both to the smaller values of $m_c=1.23$ GeV and $\\sqrt{t_c}$ =5.5 GeV used in that paper.",
"Though the $DD^*$ molecule mass is above the $DD^*$ threshold which is similar to the e.g.",
"the case of the $\\pi \\pi $ continuum and $\\rho $ -meson resonance in $e^+e^-$ to the I=1 hadrons channel, one expects that at the $\\tau $ -stability point or inside the sum rule window, where the QCD continuum contribution is minimum while the OPE is still convergent, the lowest ground state dominates the sum rule.",
"Figure: a) τ\\tau -behaviour of M J/ψS 2 M_{J/\\psi S_2} for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of M ΥS 2 M_{\\Upsilon S_2} for different values of t c t_c and for m b =4.17m_b=4.17 GeV;c) t c t_c-behaviour of the extremas in τ\\tau of M J/ψS 2 M_{J/\\psi S_2} for m c =1.26-1.47m_c=1.26-1.47 GeV; d) the same as c) but for M ΥS 2 M_{\\Upsilon S_2} for m b =4.17-4.70m_b=4.17-4.70 GeV.Figure: a) τ\\tau -behaviour of r sd ψ r^\\psi _{sd} for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of r sd Υ r^\\Upsilon _{sd} for different values of t c t_c and for m b =4.17m_b=4.17 GeV;c) t c t_c-behaviour of the extremas in τ\\tau of r sd ψ r^\\psi _{sd} for m c =1.26-1.47m_c=1.26-1.47 GeV; d) the same as c) but for r sd Υ r^\\Upsilon _{sd} for m b =4.17-4.70m_b=4.17-4.70 GeV.Combining LSR and FESR, we consider the mass of the $J/\\psi S_2$ and $\\Upsilon S_2$ molecules in a colour singlet combination, where $S_2\\equiv \\bar{u}u+\\bar{d}d$ is a scalar meson The low-mass $\\pi ^+\\pi ^-$ invariant mass due to the $\\sigma $ meson is expected to result mainly from its gluon rather than from its quark component [49], [50] such that an eventual quark-gluon hybrid meson nature of the $Y_c$ is also possible..",
"In so doing, we work with the LO QCD expression obtained in [19].",
"We show the results versus the LSR variable $\\tau $ in Fig.",
"REF a,b.",
"The $t_c$ -behaviour of different $\\tau $ -extremas is given in Figs.",
"REF c,d from which we can deduce for the running quark masses for $\\sqrt{t_c}= 5.30(2)$ and 10.23(3) GeV in units of MeV: $M_{J/\\psi S_2}&=&5002(20)_{t_c}(8)_{m_c}(2)_\\Lambda (19)_{\\bar{u}u}(9)_{G^2}(0)_{M^2_0}(0)_{G^3}(6)_\\rho \\nonumber \\\\&=&5002(31)~,\\nonumber \\\\M_{\\Upsilon S_2}&=&10015(20)_{t_c}(9)_{m_b}(2)_\\Lambda (16)_{\\bar{u}u}(17)_{G^2}(0)_{M^2_0}(0)_{G^3}(5)_\\rho \\nonumber \\\\&=&10015(33)~.$ The splitting (in units of MeV) with the first radial excitation approximately given by $\\sqrt{t_c}$ is: $M^{\\prime }_{J/\\psi S_2}-M_{J/\\psi S_2}\\approx 298~,~~~~~~ M^{\\prime }_{\\Upsilon S_2}-M_{\\Upsilon S_2}\\approx 213~.$ In the same way, we show in Figs.",
"REF the $\\tau $ and $t_c$ behaviours of the $SU(3)$ breaking ratios, from which, we can deduce: $r^\\psi _{sd}&\\equiv & {M_{J/\\psi S_3}\\over M_{J/\\psi S_2}}=1.022(0.2)_{m_c}(5)_{m_s}(2)_{\\kappa }~, \\nonumber \\\\r^\\Upsilon _{sd}&\\equiv & {M_{\\Upsilon S_3}\\over M_{\\Upsilon S_2}}=1.011(1)_{m_b}(2)_{m_s}(0.2)_{\\kappa }~,$ where $S_3\\equiv \\bar{s}s$ is a scalar meson.",
"Then, we obtain in MeV: $M_{J/\\psi S_3}=5112(41)~,~~~~~~~~~M_{\\Upsilon S_3}=10125(40)~,$ corresponding to the $SU(3)$ mass-splittings: $\\Delta M^{J/\\psi }_{sd}\\simeq \\Delta M^{\\Upsilon }_{sd}\\approx 110~{\\rm MeV}~.$ The mass-splittings in Eq.",
"(REF ) are comparable with the ones obtained previously.",
"Doing the same exercise for the octet current, we deduce the results in Table REF where the molecule associated to the octet current is 100 (resp.",
"250) MeV above the one of the singlet current for $J/\\psi $ (resp $\\Upsilon $ ) contrary to the $1^{++}$ case discussed in [2].",
"The ratio of $SU(3)$ breakings are respectively 1.022(5) and 1.010(2) in the $c$ and $b$ channels which are comparable with the ones in Eq.",
"REF .",
"When comparing our results with the ones in Ref.",
"[19], we notice that the low central value of $M_{J/\\psi S_2}$ obtained there (which we reproduce) corresponds to a smaller value of $m_c=1.23$ GeV and mainly to a low value of $\\sqrt{t_c}=5.1$ GeV which does not coïncide with the common solution $\\sqrt{t_c}=5.3$ GeV from LSR and FESR.",
"On the opposite, the large value of $M_{\\Upsilon S_2}=10.74$ (resp.",
"11.09) GeV obtained there corresponds to a too high value $\\sqrt{t_c}=11.3$ (resp.",
"11.7) GeV compared with the LSR and FESR solution $\\sqrt{t_c}=10.23$ (resp.",
"10.48) GeV for the singlet (resp.",
"octet) current.",
"Figure: a) τ\\tau -behaviour of Y cd 0 {Y^0_{cd}} for the current mixing parameter b=0b=0, for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of Y bd 0 {Y^0_{bd}} for different values of t c t_c and for m b =4.17m_b=4.17 GeV; c) t c t_c-behaviour of the extremas in τ\\tau of Y cd 0 {Y^0_{cd}} for m c =1.26-1.47m_c=1.26-1.47 GeV; d) the same as c) but for Y bd 0 {Y^0_{bd}} for m b =4.17-4.70m_b=4.17-4.70 GeV.Figure: a) bb-behaviour of M Y cd 0 M_{Y^0_{cd}} for given values of τ\\tau and t c t_cand for m c =1.26m_c=1.26 GeV; b) the same as a) but for M Y bd 0 M_{Y^0_{bd}}and for m b =4.17m_b=4.17 GeV.Figure: a) τ\\tau -behaviour of f Y cd 0 f_{Y^0_{cd}} for given values of b=0b=0 and t c t_cand for m c =1.26m_c=1.26 GeV; b) the same as a) but for f Y bd 0 f_{Y^0_{bd}}and for m b =4.17m_b=4.17 GeV;c) bb-behaviour of f Y cd 0 f_{Y^0_{cd}} at the τ\\tau -stability and for a given value of t c t_cd) the same as c) but for f Y bd 0 f_{Y^0_{bd}} Figure: a) τ\\tau -behaviour of r sd 0c r^{0c}_{sd} for the current mixing parameter b=0b=0, for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of r sd 0b r^{0b}_{sd} for different values of t c t_c and for m b =4.17m_b=4.17 GeV;c) t c t_c-behaviour of the extremas in τ\\tau of r sd 0c r^{0c}_{sd} for m c =1.26-1.47m_c=1.26-1.47 GeV; d) the same as c) but for r sd 0b r^{0b}_{sd} for m b =4.17-4.70m_b=4.17-4.70 GeV."
],
[
"$0^{++}$ four-quark and molecule masses from QSSR",
"In the following, we extend the previous analysis to the case of the $0^{++}$ mesons.",
"Figure: a) τ\\tau -behaviour of M D d D d M_{D_dD_d} for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of M B d B d M_{B_dB_d} for different values of t c t_c and for m b =4.17m_b=4.17 GeV;c) t c t_c-behaviour of the extremas in τ\\tau of M D d D d M_{D_dD_d} and for m c =1.26-1.47m_c=1.26-1.47 GeV; d) the same as c) but for M B d B d M_{B_dB_d} and for m b =4.17-4.70m_b=4.17-4.70 GeV.Figure: a) τ\\tau -behaviour of r sd D r^D_{sd} for different values of t c t_cand for m c =1.26m_c=1.26 GeV; b) τ\\tau -behaviour of r sd B r^B_{sd} for different values of t c t_cand for m b =4.17m_b=4.17 GeV; c) t c t_c-behaviour of the inflexion points (or minimas) of r sd D r^D_{sd} from Fig a; d)the same for the bb quark using r sd B r^B_{sd} from Fig b.Table: Masses of the four-quark and molecule states from the present analysis combiningLaplace (LSR) and Finite Energy (FESR).",
"We have used double ratios (DRSR) of sum rulesfor extracting the SU(3)SU(3) mass-splittings.",
"The results correspond to the value of the running heavy quark masses but the SU(3)SU(3) mass-splittings are less affected by such definitions.",
"As already mentioned in the text for simplifying notations, DD and BB denote the scalar D 0 * D^*_0 and B 0 * B^*_0 mesons.",
"The errors do not take into account the unknown ones from PT corrections."
],
[
"$\\bullet ~${{formula:90f1b0bf-17fb-4194-93de-f76d02fcbbb7}} mass and decay constant from LSR and FESR",
"We do the analysis of the $Y_{cd}^0$ and $Y_{bd}^0$ masses using LSR and FESR.",
"We show the results in Figs REF for the current mixing parameter $b=0$ from which we deduce in MeV, for the running quark masses, and respectively for $\\sqrt{t_c}=6.5$ and 13.0 GeV where LSR and FESR match: $M_{Y^0_{cd}}&=& 6125(16)_{m_c}(7)_\\Lambda (44)_{\\bar{u}u}(12)_{G^2}(14)_{\\rho }\\nonumber \\\\&=&6125(51)~{\\rm MeV}~,\\nonumber \\\\M_{Y^0_{bd}}&=& 12542(22)_{t_c}(13)_{m_b}(1)_\\Lambda (7)_{\\bar{u}u}(34)_{G^2}(2)_{\\rho }\\nonumber \\\\&=&12542(43)~{\\rm MeV}~.$ One can notice that the splittings between the lowest ground state and the 1st radial excitation approximately given by $\\sqrt{t_c}$ is in MeV: $M^{\\prime }_{Y^0_{cd}}-M_{Y^0_{cd}}\\approx 375~,~~~~~M^{\\prime }_{Y^0_{bd}}-M_{Y^0_{bd}}\\simeq 464~,$ which is larger than the ones of the $1^{--}$ states, comparable with the ones of the $J/\\psi $ and $\\Upsilon $ , and are (almost) heavy-flavour independent.",
"We show in Fig REF the effect of the choice of $b$ operator mixing parameter on the mass predictions, indicating an optimal value at $b=0$ .",
"For completeness, we show in Fig.",
"REF the $\\tau $ and $b$ behaviours of the decay constants from which we deduce: $f_{Y^0_{cd}} \\simeq 0.12~{\\rm MeV}~~~~{\\rm and} ~~~~~f_{Y^0_{bd}} \\simeq 0.03~{\\rm MeV}~,$ which are comparable with the ones of the spin 1 case in Eq.",
"(REF ).",
"We show in Figs.",
"REF the $\\tau $ and $t_c$ behaviours of the $SU(3)$ breaking ratios for the current mixing parameter $b=0$ : $r^{0Q}_{sd}\\equiv {Y^0_{Qs}\\over Y^0_{Qd}}~:~~~~Q\\equiv c,b~,$ from which we deduce: $r^{0c}_{sd}&=&1.011(2)_{m_c}(3.8){m_s}(1.4)_{\\kappa }(1)_{\\bar{u}u}(0.7)_{\\rho }~,\\nonumber \\\\r^{0b}_{sd}&=&1.004(1)_{m_c}(1.7){m_s}(0.3)_{\\kappa }~,$ leading (in units of MeV) to: $M_{Y^0_{cs}}= 6192(59)~,~~~~~~~~~M_{Y^0_{bs}}= 12592(50)~,$ and the $SU(3)$ mass-splittings: $\\Delta M_{sd}^{Y^0_c}\\simeq 67\\approx \\Delta M_{sd}^{Y^0_b}\\simeq 50~{\\rm MeV}~.$ We show the $\\tau $ and $t_c$ behaviours of the masses $M_{D_{d}D_{d}}$ and $M_{B_{d}B_{d}}$ in Figs REF .",
"Like in previous sections, we consider as a final result (in units of MeV) the one corresponding to the running masses for $\\sqrt{t_c}=6.25(3)$ and 12.02 GeV: $M_{D_{d}D_{d}}&=&5955(24)_{t_c}(14)_{m_c}(5)_\\Lambda (36)_{\\bar{u}u}(4)_{G^2}(4)_{G^3}(12)_{\\rho }\\nonumber \\\\&=&5955(48)~,\\nonumber \\\\M_{B_{d}B_{d}}&=& 11750(12)_{m_b}(4)_\\Lambda (35)_{\\bar{u}u}(7)_{G^2}(3)_{G^3}(12)_{\\rho }\\nonumber \\\\&=&11750(40)$ One can notice that the splittings between the lowest ground state and the 1st radial excitation approximately given by $\\sqrt{t_c}$ is in MeV: $M^{\\prime }_{D_{d}D_{d}}-M_{D_{d}D_{d}}\\approx 290~,~~~~~~M^{\\prime }_{B_{d}B_{d}}-M_{B_{d}B_{d}}\\approx 270~,$ which, like in the case of the $1^{--}$ states are smaller than the ones of the $J/\\psi $ and $\\Upsilon $ , and almost heavy-flavour independent.",
"We show in Fig.",
"REF the $\\tau $ - behaviour of the $SU(3)$ mass ratios for different values of $t_c$ and the $t_c$ behaviour of their $\\tau $ -extremas.",
"Therefore, we deduce: $r^{0D}_{sd}&\\equiv &{M_{D_sD_s}\\over M_{D_dD_d}}=1.015(1)_{m_c}(4){m_s}(2)_{\\kappa }(1)_{\\bar{u}u}(0.5)_{\\rho }~,\\nonumber \\\\r^{0B}_{sd}&\\equiv & {M_{B_sB_s}\\over M_{B_dB_d}}=1.008(1)_{m_c}(4){m_s}(2)_{\\kappa }(1)_{\\bar{u}u}(0.5)_{\\rho }~.$ Using the previous values of $M_{D_dD_d}$ and $M_{B_dB_d}$ , we deduce in MeV: $M_{D_sD_s}=6044(56)~,~~~~~~~~M_{B_sB_s}=11844(50)~,$ which corresponds to a $SU(3)$ splitting: $\\Delta M^{DD}_{sd}\\approx 89~{\\rm MeV} \\approx \\Delta M^{BB}_{sd}\\approx 94~{\\rm MeV}~.$"
],
[
"Summary and conclusions",
"We have studied the spectra of the $1^{--}$ and $0^{++}$ four-quarks and molecules states by combining Laplace (LSR) and finite energy (FESR) sum rules.",
"The $SU(3)$ mass-splittings have been obtained using double ratios of sum rules (DRSR).",
"We consider the present results as improvement of the existing ones in the literature extracted only from LSR where the criterion for fixing the value of the continuum thresholds are often ad hoc or based on the ones of the standard charmonium/bottomium systems mass-splittings which are not confirmed by the present analysis.",
"Our results are summarized in Table REF .",
"We find that : $\\bullet ~$ The three $Y_c(4260,~4360,~4660) $ $1^{--}$ experimental candidates are too low for being pure four-quark or/and molecule $\\bar{D}D^*$ and $J/\\psi S_2$ states but can result from their mixings.",
"The $Y_b(10890)$ is lower than the predicted values of the four-quark and $\\bar{B}B^*$ molecule masses but heavier than the predicted $\\Upsilon S_2$ and $\\Upsilon S_3$ molecule states.",
"Our results may indicate that some other natures (hybrids, threshold effects,...) of these states are not excluded.",
"On can notice that our predictions for the masses are above the corresponding meson-meson thresholds indicating that these exotic states can be weakly bounded.",
"$\\bullet ~$ For the $1^{--}$ , there is a regularity of about (250-300) MeV for the value of the mass-splittings between the lowest ground state and the 1st radial excitation roughly approximated by the value of the continuum threshold $\\sqrt{t_c}$ at which the LSR and FESR match.",
"These mass-splittings are (almost) flavour-independent and are much smaller than the ones of 500 MeV of ordinary charmonium and bottomium states and do not support some ad hoc choice used in the literature for fixing the $t_c$ -values when extracting the optimal results from the LSR.",
"$\\bullet ~$ There is also a regularity of about 50–90 MeV for the $SU(3)$ mass-splittings of the different states which are also (almost) flavour-independent.",
"$\\bullet ~$ The spin 0 states are much more heavier ($\\ge $ 400 MeV) than the spin 1 states, like in the case of hybrid states [5].",
"$\\bullet ~$ The decay constants of the $1^{--}$ and $0^{++}$ four-quark states obtained in Eqs (REF )and (REF ) are much smaller than $f_\\pi $ , $f_\\rho $ and $f_{D,B}$ .",
"Unlike $f_B$ expected to behave as $1/\\sqrt{M_Q}$ , the four-quark states decay constants exhibit a $1/M_Q$ behaviour which can be tested using HQET or/and lattice QCD.",
"It is likely that some other non-perturbative approaches such as potential models, HQET, AdS/QCD and lattice calculations check the previous new features and values on mass-splittings, mass and decay constants derived in this paper.",
"We also expect that present and future experiments (LHCb, Belle, Babar,...) can test our predictions."
],
[
"Acknowledgment",
"This work has been partly supported by the CNRS-IN2P3 within the project Non-Perturbative QCD and Hadron Physics, by the CNRS-FAPESP program and by CNPq-Brazil.",
"S.N.",
"has been partly supported by the Abdus Salam ICTP-Trieste (Italy) as an ICTP consultant for Madagascar.",
"We thank the referee for his comments which lead to the improvements of the original manuscript."
]
] | 1204.1236 |
[
[
"Opinion formation in time-varying social networks: The case of the\n naming game"
],
[
"Abstract We study the dynamics of the naming game as an opinion formation model on time-varying social networks.",
"This agent-based model captures the essential features of the agreement dynamics by means of a memory-based negotiation process.",
"Our study focuses on the impact of time-varying properties of the social network of the agents on the naming game dynamics.",
"In particular, we perform a computational exploration of this model using simulations on top of real networks.",
"We investigate the outcomes of the dynamics on two different types of time-varying data - (i) the networks vary on a day-to-day basis and (ii) the networks vary within very short intervals of time (20 seconds).",
"In the first case, we find that networks with strong community structure hinder the system from reaching global agreement; the evolution of the naming game in these networks maintains clusters of coexisting opinions indefinitely leading to metastability.",
"In the second case, we investigate the evolution of the naming game in perfect synchronization with the time evolution of the underlying social network shedding new light on the traditional emergent properties of the game that differ largely from what has been reported in the existing literature."
],
[
"Introduction",
"Social networks are inherently dynamic.",
"Social interactions and human activities are intermittent, the neighborhood of individuals moving over a geographic space evolves over time, links appear and disappear in the World-Wide-Web.",
"The essence of social network lies in its time-varying nature.",
"Links may exist for a certain time period and may be recurrent.",
"In summary, as time progresses, the societal structure keeps changing.",
"Similarly, with the evolution of time, social conventions, shared cultural and linguistic patterns reshape themselves.",
"Opinions spread, some gets trapped into communities, some crosses the barrier of local groups/communities and become accepted globally among different communities and some die competing with others.",
"Most of these social phenomena can be modeled and analyzed in a time-varying framework.",
"Almost all previous work is limited to the analysis of the Naming Game dynamics on static networks [5], [3], [18], [10], [16], [7], [11], [9], [27].",
"Therefore, in this paper, we focus on the competing opinion formation over time-varying real-world social networks.",
"One way of viewing at time-varying networks is as a series of static graphs accumulated over a fixed time interval; however this kind of networks do not always perfectly capture the temporal ordering of the links appearing in the system which may sometimes lead to over/under-estimation of network topologies.",
"Thus, we plan to investigate the opinion formation process on both the accumulated static graphs as well as on its detailed time-resolved counterpart.",
"In this paper, we focus on the basic Naming Game model (NG) [5] to study how opinions spread with time and how societies move towards consensus in the adoption of a single opinion through negotiation or agree upon multiple opinions due to non-uniform interaction pattern among different communities.",
"The evolution of the system in this model takes place through the usual local pairwise interactions among artificial agents that necessarily captures the generic and essential features of an agreement process.",
"This model was expressly conceived to explore the role of self-organization in the evolution of languages [24], [25] and has acquired a paradigmatic role in semiotic dynamics that studies evolution of languages through invention of new words, grammatical constructions and more specifically, through adoption of new meaning for different words.",
"NG finds wide applications in various fields ranging from artificial sensor network as a leader election model [2] to the social media as an opinion formation model.",
"The minimal Naming Game (NG) consists of a population of $N$ agents observing a single object in the environment (may be a discussion on a particular topic) and opining for that by means of communication to one another through pairwise interactions, in order to reach a global agreement.",
"The agents have at their disposal an internal inventory, in which they can store an unlimited number of different words or opinions.",
"At the beginning, all the individuals have empty inventories.",
"At each time step, the dynamics consists of a pairwise interaction between randomly chosen individuals.",
"The chosen individuals can take part in the interaction as a “speaker” or as a “hearer.” The speaker voices to the hearer a possible opinion for the object under consideration; if the speaker does not have one, i.e., ͑his inventory is empty͒, he invents an opinion͔.",
"In case where he already has many opinions ͑stored in his inventory͒, he chooses one of them randomly.",
"The hearer's move is deterministic: if she possesses the opinion pronounced by the speaker, the interaction is a “success”, and in this case both speaker and hearer retain that opinion as the right one, removing all other competing opinions/words in their inventories; otherwise, the new opinion is included in the inventory of the hearer, without any cancellation of opinions in which case the interaction is termed as a “failure” (see fig REF ).",
"The game is played on a fully connected network, i.e., each agent can, in principle, communicate with all the other agents, and makes two basic assumptions.",
"One assumes that the number of possible opinions is so huge that the probability of a opinion being reinvented is practically negligible (this means that similar opinions is not taken into account here, although the extension is trivially possible).",
"As a consequence, one can reduce, without loss of generality, the environment to be consisting of only one single object/topic of discussion.",
"Figure: (Color online) Agent's interaction rules in basic NG.",
"Suppose there is a topic on which a discussion is going on, say “Who is the greatest tennis player?”.",
"(Top) The speaker chosen at random, opines for “Federer” (also chosen randomly from his inventory of opinions).",
"Now, the hearer (again chosen at random) does not have this opinion in her inventory, and therefore she adds the opinion “Federer” in her inventory and the interaction is a failure.",
"(Bottom) The speaker opines for “McEnroe” and in this case the opinion is present in the hearer's inventory.",
"So, they delete all other opinions except “McEnroe”.",
"The interaction this time is “success”.Although the system reaches a global consensus through the invention and decay of opinions, it is interesting to note the important differences from other opinion formation models.",
"In Axelrod’s model [1], each agent is endowed with a vector of opinions, and can interact with other agents only if their opinions are already close enough; in Sznajd’s model [26] and in the Voter model [15], the opinion can take only two discrete values, and an agent takes deterministically the opinion of one of its neighbors.",
"Further in [12], the opinion is modeled as a unique variable and the evolution of two interacting agents is deterministic.",
"In the Naming Game model on the other hand, each agent can potentially have an unlimited number of possible discrete states (or opinions) at the same time, accumulating in its memory different possible opinions; the agents are able to “wait” before reaching a decision.",
"Moreover, each dynamical step can be seen as a negotiation between a speaker and a hearer, with a certain degree of stochasticity.",
"In this paper, we consider the NG dynamics on two different types of time-varying data; one varying across days while another varying over very short intervals of time (20 seconds).",
"In the first case, we observe that networks with strong community structures delay the convergence due to co-existence of competing and long-lasting clusters of opinions.",
"In the second case, the games are played in perfect synchronization with the time-evolution of the network.",
"In this case, we observe that the global observables are markedly different from the case where the games are played on the static (and composite) version of the same network as well as from the traditional results reported in the literature.",
"The rest of the paper is organized as follows.",
"Section 2 is devoted for the discussion of the state of the art.",
"In Section 3, we describe the datasets on which we investigate the Naming Game dynamics in a time-varying social scenario.",
"Section 4 provides the elaborate model description.",
"In Section 5, we present the results and provide explanations for our findings.",
"Finally, conclusions are drawn in section 6."
],
[
"Related work",
"Most previous studies of the NG model in semiotic/opinion dynamics have focused on populations of agents in which all pairwise interactions are allowed, i.e., the agents are placed on the vertices of a fully connected graph.",
"In statistical mechanics, this topological structure is commonly referred to as “mean-field” topology [5], [6].",
"Apart from mean-field case, the model has also been studied on regular lattices [3], [18]; small world networks [18], [10], [16], [7]; random geometric graphs [18], [17], [14]; and static [11], [9], [27], dynamic [21], and empirical [19] complex networks.",
"Lu et.",
"al.",
"[19] have studied the Naming Game dynamics on a high-school friendship network and have shown that the presence of community structures affect the behavior of the dynamics through the formation of long-living late-stage meta-stable clusters of opinions.",
"Therefore, they propose injection of committed agents (agents who never change their opinion) into the population for fast agreement of the dynamics.",
"Nardini et.",
"al.",
"[21] studied the dynamics of NG on adaptive networks where the connections can be rewired with the evolution of the game.",
"All these prior works have studied the NG dynamics on essentially static networks.",
"Therefore, our study reported here is unique and different from the literature since we consider the evolution of the NG dynamics over time-varying social structure."
],
[
"Datasets",
"For the purpose of the investigation of the NG dynamics on time-varying networks, we consider two specific real-world face-to-face contact datasets and present our results on each of them.",
"Both the datasets are obtained from http://www.sociopatterns.org/datasets/.",
"The first dataset we consider is the face-to-face interaction data of visitors of the Science Gallery in Dublin, Ireland during the spring of 2009 at the event of art-science exhibition “INFECTIOUS: STAY AWAY”.",
"The dataset contains the cumulative daily networks of the visitors for sixty-nine days [13].",
"The nodes represent visitors of the Science Gallery while the edges represent close-range face-to-face proximity (measured using RFID devices carried by each visitor) between the concerned persons.",
"The weights associated with the edges are the number of 20 seconds intervals during which close-range face-to-face proximity could be detected.",
"Thus, these daily networks can be thought of as sixty-nine snapshots of a time-varying societal structure with a periodicity of 24 hours.",
"We will refer to this as the SG dataset.",
"We also consider the time-resolved datasets which are the dynamic counterparts of the daily cumulated contact networks of the SG dataset.",
"From the 69 daily instances, we consider time-resolved contact pattern of four instances day 9, 20, 22 and 26, which can be considered as the representatives of all the instances.",
"These time-resolved data are refered to as SGD dataset.",
"The last dataset we consider is the face-to-face interaction data of the conference attendees of the ACM Hypertext 2009 conference held in Institute for Scientific Interchange Foundation in Turin, Italy, from June 29th to July 1st, 2009, where the SocioPatterns project deployed the Live Social Semantics application.",
"The dataset contains the dynamical network of face-to-face proximity of 115 conference attendees over about 2.5 days.",
"In future reference, we will refer to this as the HT dataset."
],
[
"The model description",
"The basic NG Model can be summarized as follows.",
"At each time step ($t$ = 1, 2, ..) two agents are randomly selected to interact: one of them plays the role of speaker, the other one that of hearer.",
"The interactions obey the following rules The speaker voices an opinion from its list of opinions to the hearer.",
"(If the speaker has more than one opinion on his list, he randomly chooses one; if he has none, he invents one randomly.)",
"If the hearer has this opinion, the communication is termed “successful”, and both players delete all other opinions, i.e., collapse their list of opinions to this one opinion.",
"Therefore, they meet a local agreement.",
"If the hearer does not have the opinion transmitted by the speaker (termed “unsuccessful” communication), she adds the opinion to her list of opinions without any deletion.",
"Note that in this model any agent is free to interact with any other agent, i.e., the underlying social structure is assumed to be fully connected.",
"For the purpose of our analysis however, we assume that the agents are embedded on realistic social networks (i.e., SG and HT) that are continuously varying over time.",
"In this case, although the basic rules of the game remain exactly same, the only issue is to devise a strategy for the speaker-hearer selection.",
"We consider two variants of this selection, the first one being suitable for the SG dataset and the second one for the SGD and HT dataset.",
"Strategy I: Here we randomly select a speaker and preferentially choose a hearer among its neighbors.",
"Our intention here is to simulate an important criterion: we talk most preferably to those with whom we had already met before.",
"This is implemented as follows: The speaker $i$ is selected randomly.",
"The hearer $j$ is selected using the preferential rule, with the probability $p_{ij} = \\frac{w_{ij}}{\\sum _{j=1}^{k}w_{ij}}$ where $w_{ij}$ can be thought of as the total number of contact events between the pair $i$ and $j$ while $k$ is the degree of agent $i$ (i.e, the number of other agents that $i$ is connected to at a particular instant of time).",
"Strategy II: This variant is quite straight-forward.",
"We choose a random speaker and a random hearer among its neighbors to impart the equal importance of each pair of connections.",
"The main quantities of interest which describe the emergent properties of the system are the total number $N_w(t)$ of words/opinions in the system at the time $t$ (i.e., the total size of the memory); the number of different words/opinions $N_d(t)$ in the system at the time $t$ ; the average success rate $S(t)$ , i.e., the probability, computed averaging over many simulation runs, that the chosen agent gets involved in a successful interaction at a given time $t$ .",
"From a global perspective, the quantities which are of interest are the time to reach the global consensus ($t_{conv}$ ), the maximum memory required by the agents during the process ($N_w^{max}$ ) and the time required to reach the memory peak ($t_{max}$ )."
],
[
"Results and discussions",
"In this section, we present the results of the analysis of the NG dynamics on the SG, SGD and the HT datasets."
],
[
"Analysis on day-wise SG dataset",
"We have studied the opinion formation process on the sixty-nine days of close interactions among the visitors for the SG dataset.",
"The primary focus of this study is to find the behavior of the global quantities of the NG dynamics with the evolution of the topology over time.",
"We play the NG on each of these daily networks of sixty-nine days following Strategy I.",
"The networks are not always fully connected.",
"In case of disconnected components, $N_d$ should never converge to 1 and consequently, the emergence of multi-opinion state is observed.",
"Therefore, in this case we redefine $t_{conv}$ as the time to reach the following state: $N_{w} = N $ and $N_d = c$ where $c$ is the number of disconnected components.",
"The natural question that arises is how the opinion dynamics gets affected as the underlying network structure varies over the days.",
"It is interesting to note that the memory peak $N_w^{max}$ and the time to reach the memory peak $t_{max}$ have a strong correlation with the system size $N$ .",
"In fact, they have a linear scaling with the population size $N$ (See fig REF ).",
"This relationship is in agreement with what has been observed in the literature for small-world, scale-free and random networks [11], [7] where both $N_w^{max}$ and $t_{max}$ scales as $O(N)$ .",
"Figure: (Color online) Scaling relation of N w max N_w^{max} and t max t_{max} with population size NN.",
"(a) Temporal behavior of N w max N_w^{max} and the population sizeNN for each of the 69 instances.",
"Data are smoothed by taking 7 point running average.",
"(b) Scaling of N w max N_w^{max} with NN which is linearly fitted.",
"The inset shows the scalingof N w max N_w^{max} in the thresholded networks where the threshold is set to 1, 2 and 5 respectively.",
"All these curves are linearly fitted.",
"(c) Variation of t max t_{max} and population size NN with time.",
"Data are smoothed by taking 7 point running average.",
"(d) Scaling of t max t_{max} with NN which is linearly fitted.",
"The inset shows the scaling of t max t_{max} in the thresholded networks where the threshold is set to 1, 2 and 5 respectively.",
"All these curves are linearly fitted.Nevertheless, these cumulative daily networks do not resemble any of the well-known topologies which will be clear when we dig into the results of $t_{conv}$ .",
"We observe that the behavior of $t_{conv}$ (see fig REF (a), (b)) is not in lines of the existing literature where it is usually noted that $t_{conv} \\sim N^{1.4}$ .",
"Therefore the natural question that needs to be addressed is that what is (are) the property (s) of the underlying network that leads to such a non-conforming behavior of $t_{conv}$ .",
"Figure: (Color online) (a)Temporal behavior of t conv t_{conv} and the population size NN.",
"Data are smoothed by taking 7 point running average.",
"(b) Scatter plot of t conv t_{conv} and NN which could not be fitted with y=x 1.4 y = x^{1.4} (R 2 ≈0.03R^2 \\approx 0.03).",
"(c), (d) and (e)Correlation of t conv t_{conv} with variance of community sizes Var(s)Var (s) detected by various community detection algorithms.",
"The curves are smoothed by taking 20 point running averages.In fact, the answer to this question lies in the common behavior of the real-world social networks.",
"These networks typically consist of a number of communities; nodes within communities are more densely connected, while links bridging communities are sparse.",
"The effect of the community structure plays a dominant role with the emergence of long-lasting multi-opinion states at the late stage of the dynamics which has also been observed in [11] and [19].",
"In fact, each community reaches internal consensus fast but the weak connections between communities are not sufficient for opinions to propagate from one community to the other leading to long multi-opinion states which are also known as “metastable states” in the domain of statistical physics.",
"Formally, a metastable state is a state of the dynamics where global shifts are always possible but progressively more unlikely and the response properties depend on the age of the system [20].",
"Community structures are essentially authentic signatures of metastability which inhibits the dynamics leading to very slow convergence.",
"Presence of community structures slows down the dynamics, however, what renders the system even slower is the presence of different-sized communities.",
"The reason for this is quite straight-forward: the agents that are part of a larger size community have a higher probability of being chosen for a game than those belonging to a smaller size community.",
"This is a reminiscent of the fact that the agents are chosen randomly which automatically increases the chances of landing in a larger size community simply because a larger bulk of the population is confined within this community.",
"Therefore, even when consensus is reached very fast in a large community, the system keeps on choosing agents from this community itself mostly resulting in “success with no outcome”.",
"Further, since the inter-community links are weak, and agents from smaller commuinities are hardly chosen the overall state of the system hardly changes thereby always keeping the agents away from the global consensus.",
"This is reflected through fig REF (c), (d) and (e) where we report the correlation of $t_{conv}$ with the variance of the community sizes.",
"The basic idea is as follows: if a network gets decomposed into m communities each of size $s_1, s_2, .",
".",
"., s_m$ then we calculate the statistical variance of this size distribution and plot it against $t_{conv}$ .",
"For the purpose of community analysis, we use three standard algorithms - Newman and Girvan (NGR) [23], Newman, Clauset and Moore (NCM) [8] and community detection by eigen vector (EV) [22] and in each case we observe that $t_{conv}$ has a strong positive correlation with the variance of the community sizes (see fig REF (c), (d) and (e)).",
"As we have suggested earlier, we consider two variants of pair selection, the weighted and the unweighted one.",
"In this subsection, we attempt to study the effect of edge-weights on the dynamics.",
"The edge weights play significant role in pair-selection and so there is a possibility that this affects the dynamics.",
"However, what we observe here is in the contrary.",
"The global quantities of interest in case where all the neighboring agents are given equal preference remain roughly equivalent to the case where the weights are considered (see fig REF (a), (b) and (c)).",
"The reason behind this is the skewed distribution of edge weights.",
"We find that above 60 % edges on average are low-weight edges which somehow drives the dynamics of the preferential model towards the behavior close to the dynamics of the unweighted NG dynamics (see table REF ).",
"In addition, we also observe a strong correlation between the average degree $\\langle k \\rangle $ and the average strength $\\langle s \\rangle $ (see fig REF (d)).",
"The weighted clustering coefficient $C^w$ is also close to the topological clustering coefficient $C$ (see fig REF (e)).",
"Further, the weighted average nearest neighbor degree $\\langle k_{nn}^w \\rangle $ and the unweighted average nearest neighbor degree $\\langle k_{nn} \\rangle $ are perfectly correlated (see fig REF (f)).",
"Figure: (Color online) Effect of edge weights on the dynamics.",
"(a), (b) and (c) Temporal evolution of N w max N_w^{max}, t max t_{max} and t conv t_{conv} for the weighted and unweighted NG respectively smoothed over a time sliding window of size 7 .",
"(d) Variation of 〈k〉\\langle k \\rangle with 〈s〉\\langle s \\rangle .",
"(e) Variation of C w C^w with CC.",
"(f) Variation of 〈k nn w 〉\\langle k_{nn}^w \\rangle with 〈k nn 〉\\langle k_{nn} \\rangle .Table: Distribution of edge weights averaged over all 69 instances."
],
[
"Examples of individual instances",
"In this subsection, we dig deeper into the individual snapshots to have a more clear understanding of the ongoing dynamics.",
"From the sixty-nine instances, we present four representative cases that roughly capture all the different characteristics found across the instances.",
"Two among these, consist of disconnected components while the other two are single connected components.",
"Further, two of them (one connected and the other disconnected) show fast convergence while another two (again one connected and the other disconnected) show slow convergence triggered by the presence of community structures leading to metastability.",
"Here we propose two metrics to capture the two distinct behaviors of the convergence time.",
"The first one is the average unique words per community which is denoted by $U(t)$ and defined as follows: $U(t) = \\frac{\\sum _{i=1}^{C}|A_i|}{C}$ where $C$ is the number of communities and $A_i$ is the list of unique words in community $i$ .",
"The second metric we propose is the average overlap of unique words across communities which is denoted by $O_c(t)$ and defined as follows: $O_c(t) =\\frac{2}{C(C-1)} \\sum _{i>j} \\frac{2(|A_i \\bigcap A_j|)}{\\sqrt{2(|A_i|^2+|A_j|^2)}}$ Figure: (Color online) (a) and (b) Comparison of the evolution of the total number of words N w (t)N_w(t) and number of different words N d (t)N_d(t) with time on four representative networks.",
"(c)Average number of unique words per community U(t)U(t) evolving over time.",
"(d) Temporal evolution of average overlap of words across communities O c (t)O_c(t).",
"Each point in the above curves represents the averagevalue obtained over 100 simulation runs.We consider the daily networks of 9th, 20th, 22nd and 26th day.",
"The 9th day and 22nd day network structures consist of a single connected component with 200 and 240 nodes respectively while the 20th and 26th daily network consist of multiple disconnected components with 96 and 156 nodes respectively.",
"The evolution of $N_w(t)$ shows a steady growth signifying inventions of new opinions coupled with a series of failure interactions until the maxima is reached (see fig REF (a)).",
"From this point onward, the reorganization phase commences and the players encounter mostly successful interaction resulting in the drop of $N_w(t)$ (fig REF (a)).",
"While for the 20th (disconnected network) and 22nd (connected network) day consensus is reached fast, for the 9th (connected network) and the 26th (disconnected network) day the system gets arrested in a long plateau indicating the presence of metastability and strong community structures.",
"The growth of $N_d(t)$ also signifies similar pattern, steady rise followed by steady fall and a plateau (signifying a strong community structure) in case of the 9th and 26th day (see fig REF (b)).",
"To explore the flat plateau region further we report $U(t)$ and $O_c(t)$ in fig REF (c) and REF (d) respectively.",
"It is interesting to note that both $U(t)$ and $O_c(t)$ show a plateau in case of the 9th and the 26th day which is a signature of the fact that the games played in the plateau region predominantly produces success with no deletion of opinions leading to the emergence of metastability.",
"In this section, we consider the datasets containing dynamic face-to-face interactions.",
"We play the Naming Game on these time-varying networks in complete synchronization with the real time, i.e., a single game is played on a single time-resolved snapshot of the same network.",
"Thus, at each time step $t$ = 1, 2 .",
".",
".",
", the game is played among those agents that are alive at that particular instant of time in the network.",
"We consider the Strategy II where at each time step, we choose a random speaker and a random hearer among its neighbors.",
"Figure: (Color online) (a) and (b) The temporal evolution of N w (t)N_w(t) and N d (t)N_d(t) on time-varying day 9 SGD dataset.",
"The data are averaged over 100 simulation runs.",
"(c) The behavior of the number of inventions of opinions N i (t)N_i(t) over time.",
"(d) Comparison of ΔN w (t)\\Delta N_w(t) with success rate S(t)S(t).",
"(e) Comparison of temporal evolution of ΔN w (t)\\Delta N_w(t) and number of new connections smoothed by taking 20 point running average.",
"(f) Comparison of ΔN w (t)\\Delta N_w(t) with the variance of community sizes (found by NGR, NCM and EV algorithm) evolving over time (the curves are suitably scaled by some constant for the purpose of better visualization).",
"The data are smoothed by taking 20 point running average."
],
[
"Results from SGD dataset",
"In this section, we consider time-resolved dataset of four representatives from the SG dataset.",
"The networks on 9th and 22nd day consist of a single components; however, the 20th and 26th day networks show existence of multiple disconnected components.",
"We analyze each of these time-evolving networks and report the behavior of the global quantities as well as different network properties infuencing the game dynamics.",
"The time evolution of $N_w(t)$ and $N_d(t)$ on the time-varying graph of day 9 (see fig REF (a) and (b)) show a drastically different behavior from the case where these quantities are measured on the static (and composite) counterpart (see fig REF (a) and (b)).",
"The temporal graph shows a slow growth regime followed by a sharp transition, whereas the static counterpart shows steady growth regime followed by a steady fall and finally a long-lasting metastable state (see fig REF (a)).",
"This difference in behavior is due to the fact that in the time-varying case inventions of opinions prevail throughout the dynamics (see fig REF (c)) which prevents the disposal of opinions from the system and hence the memory sizes do not decrease.",
"Further, in fig REF (d) we show how the absolute change in $N_w$ is driven by the success rate; $\\Delta N_w$ increases with a decrease in $S(t)$ while it decreases with an increase in $S(t)$ .",
"Fig REF (e) shows the direct correspondence of the $\\Delta N_w$ with the new connections.",
"Another interesting property which has an impact on the dynamics is the community size.",
"Indeed the variance of the community sizes relates in a similar way to $\\Delta N_w$ (see fig REF (f)).",
"Therefore, the continuous inventions, the influx of new connections (causing more failures) and the fact that the opinions get trapped within local neighborhoods together contribute to the steeply rising memory size over the time evolution of the dynamics.",
"Figure: (Color online) (a) and (b) The temporal evolution of N w (t)N_w(t) and N d (t)N_d(t) on time-varying day 20 SGD dataset.",
"The data are averaged over 100 simulation runs.",
"(c) The behavior of the number of inventions of opinions N i (t)N_i(t) over time.",
"(d) Comparison of ΔN w (t)\\Delta N_w(t) with success rate S(t)S(t).",
"(e) Comparison of temporal evolution of ΔN w (t)\\Delta N_w(t) and number of new connections smoothed by taking 20 point running average.",
"(f) Comparison of ΔN w (t)\\Delta N_w(t) with the variance of community sizes (found by NGR, NCM and EV algorithm) evolving over time (the curves are suitably scaled by some constant for the purpose of better visualization).",
"The data are smoothed by taking 20 point running average.Figure: (Color online) (a) and (b) The temporal evolution of N w (t)N_w(t) and N d (t)N_d(t) on time-varying day 22 SGD dataset.",
"The data are averaged over 100 simulation runs.",
"(c) The behavior of the number of inventions of opinions N i (t)N_i(t) over time.",
"(d) Comparison of ΔN w (t)\\Delta N_w(t) with success rate S(t)S(t).",
"(e) Comparison of temporal evolution of ΔN w (t)\\Delta N_w(t) and number of new connections smoothed by taking 20 point running average.",
"(f) Comparison of ΔN w (t)\\Delta N_w(t) with the variance of community sizes (found by NGR, NCM and EV algorithm) evolving over time (the curves are suitably scaled by some constant for the purpose of better visualization).",
"The data are smoothed by taking 20 point running average.Figure: (Color online) (a) and (b) The temporal evolution of N w (t)N_w(t) and N d (t)N_d(t) on time-varying day 26 SGD dataset.",
"The data are averaged over 100 simulation runs.",
"(c) The behavior of the number of inventions of opinions N i (t)N_i(t) over time.",
"(d) Comparison of ΔN w (t)\\Delta N_w(t) with success rate S(t)S(t).",
"(e) Comparison of temporal evolution of ΔN w (t)\\Delta N_w(t) and number of new connections smoothed by taking 20 point running average.",
"(f) Comparison of ΔN w (t)\\Delta N_w(t) with the variance of community sizes (found by NGR, NCM and EV algorithm) evolving over time (the curves are suitably scaled by some constant for the purpose of better visualization).",
"The data are smoothed by taking 20 point running average.The time-varying networks of 20th, 22nd and 26th day also behave more or less in the same way as for day 9 (see fig REF , fig REF , fig REF ).",
"All these time-varying networks typically show a similar slow growth stage followed by a steep rise which is in contrast to their static counterparts (see fig REF ) where we found mostly 3 distinct phase of the dynamics - a growth stage, followed by a sharp fall due to series of successful interactions and a meta-stable state due to presence of community structures.",
"This discrepancy in the behavior of $N_w(t)$ and $N_d(t)$ curves is due to the fact that the time-varying networks witness a continuous influx of new agents into the system with inventions happening throughout the evolution and the old agents not playing enough games with the new ones to negotiate and agree upon an opinion.",
"In fig REF , we show how the frequency of interaction between a pair of individuals predicts the similarity of the opinions among individuals over different instants of time.",
"We measure the similarity of opinions between a pair of individuals by Jaccard Coefficient (JC) of their inventories.",
"It is formally defined as the size of the intersection divided by the size of the union of the inventories i.e., $JC(A_i,A_j) = \\frac{|A_i\\cap A_j|}{|A_i\\cup A_j|}$ where $A_i$ is $i^{th}$ agent's inventory.",
"From all the graphs, it is evident that there is a trend of having higher similarity in opinions with the higher edge-weight where edge-weight reflects the frequency of interactions between a pair till that particular instant of time.",
"Thus, with frequent meetings, individuals tend to share similar opinion.",
"This usually also happens in real-life scenarios where more we meet more similar-opinionated people we become.",
"Figure: Comparison of mean similarity with edge weights.",
"The graphs on the first row show the similarity results on SGD 9th day for four different instances of time at t=T/4t = T/4, t=T/2t = T/2, t=3T/4t= 3T/4 and t=Tt = T where TT is the total time.",
"Similarly, rows 2, 3 and 4 show similarity results for SGD 20th, 22nd and 26th day respectively at four different time instances -t=T/4t = T/4, t=T/2t = T/2, t=3T/4t= 3T/4 and t=Tt = T. The curves are smoothed by taking 10 point running average.Figure: (Color online) Comparison of the global quantities in the real and the simulated networks.",
"The first row corresponds to the temporal evolution of N w N_w and N d N_d for SGD 9th day network.",
"The second, third and fourth row respectively correspond to the temporal evolution of N w N_w and N d N_d for SGD 20th, 22nd and 26th day network.",
"The datapoints on the curve are averaged over 100 simulation runs for each of 100 network realizations (in case of the simulated network).Results from the control experiments: For the purpose of control experiment, we create simulated versions for each of the four time-varying networks by constructing random edges of same number as in the real network in each 20s time interval.",
"We play the naming game on these simulated networks.",
"These types of networks resemble stochastic networks where edges randomly appear or disappear in each epoch.",
"The behavior of the $N_w(t)$ and $N_d(t)$ (see fig REF ) are not in the lines of what we observe in the real counterparts.",
"In all the simulated networks, the $N_w(t)$ and $N_d(t)$ behave similarly as in case of static Erdős-Rényi graphs [11]."
],
[
"Results from the HT dataset",
"In this section, we consider the second dataset containing dynamic face-to-face interactions among 113 conference attendees.",
"We first study the global behavior of the system through the temporal evolution of three main quantities: the total number $N_w(t)$ ͒ of opinions in the system, the number of different opinions $N_d(t)$ ͒, and the rate of success $S(t)$ .",
"Figure: (Color online) (a) and (b) The temporal evolution of N w (t)N_w(t) and N d (t)N_d(t) on time-varying conference network respectively.",
"The insets show the evolution of N w (t)N_w(t) and N d (t)N_d(t) on their static counterpart.",
"The data are averaged over 1000 simulation runs.",
"(c) The behavior of the number of inventions of opinions N i (t)N_i(t) over time.",
"(d) Comparison of ΔN w (t)\\Delta N_w(t) with success rate S(t)S(t).The curve corresponding to $N_w(t)$ shows an initial slow growth followed by a sharp transition and finally reaching a steady growth regime (see fig REF (a)).",
"Note that this result is markedly in contrast to what would have been observed if the games were played on the composite network constructed at the end of the conference (see fig REF (a) inset).",
"In fact, this result is in contrast to most of the other results that have been reported in the literature so far indicating that the time-varying nature of the underlying societal structure with new connections being formed, old connections being dropped and agents entering, leaving and re-entering the system has a strong impact on the emergent pattern of opinion formation.",
"Similar trends are also observed for $N_d(t)$ - initially a slow growth followed by a sharp transition reaching a peak and finally a drop, however, no way close to 1 (see fig REF (b)).",
"The inset in fig REF (b) shows the evolution of $N_d$ if the games were played on the composite network finally obtained.",
"Initially, as time proceeds, new individuals join the network that increases the number of inventions of new opinions (see fig REF (c)) thus causing a rise in both $N_w(t)$ and $N_d(t)$ .",
"However, later on new inventions stop (fig REF (c)) as the players joining late are less compared to the number that have already joined and are therefore rarely chosen as speakers thus inhibiting new inventions.",
"Hence, $N_d(t)$ is found to drop in the later stage of the dynamics although pointing to a clear existence of multiple opinions.",
"In contrast, $N_w(t)$ doesn't drop because although new opinions are not formed, old opinions trapped in different groups do not get disposed off the system.",
"Further, in fig REF (d) we show how the absolute change in $N_w$ is driven by the rate of success of agents; $\\Delta N_w$ increases with a decrease in $S(t)$ while it decreases with an increase in $S(t)$ .",
"Finally, an important analysis that is required to complete the picture centers around the precise reason for the steady growth in $N_w$ in the final regime of the dynamics.",
"We attempt to provide a plausible explanation for this through a series of results reported in fig REF .",
"Figure: (Color online) (a) Evolution of inventory sizes nn (n=0,1...)(n = 0, 1 .",
".",
".",
").",
"f n (t)f_n(t) is the fraction of agents whose inventory size is nn at time tt.",
"(b) Comparison of temporal evolution of ΔN w (t)\\Delta N_w(t) and number of new connections smoothed by 20 point running average.",
"(c) Comparison of ΔN w (t)\\Delta N_w(t) with the variance of community sizes (found by NGR, NCM and EV algorithm) evolving over time (the curves are suitably scaled by some constant for the purpose of better visualization).",
"The data are smoothed by taking 20-point running average.In fig REF (a), we present the fraction of agents having 0, 1, 2 and more opinions in their inventories.",
"Clearly, with the evolution of system, the fraction of agents with inventory size 0 diminishes; fraction of agents with size 1 increase steadily while that with size 2 is roughly stable; even larger size inventories appear only rarely in the course of the evolution.",
"In addition, we observe that $\\Delta N_w$ has a direct correspondence with the number of new connections acquired by the network at each timestep (see fig REF (b)).",
"These new connections trigger an increase in failure events, thereby increasing $N_w$ ; at the same time success events cannot reduce $N_w$ since in most cases the inventory sizes of the agents are already very low ($\\sim 1$ ) and most of these success events are actually again “success with no outcome”.",
"This last observation indicates that there should be an inherent community structure in this time-varying network and this is made apparent through fig REF (c) where we report the variance of the the size of the communities (using three different algorithms as in the previous cases) and show that this is highly correlated to $\\Delta N_w$ .",
"In summary, the presence of community structure coupled with a continuous influx of new connections (leading to late-stage failures in the system) together lead to the steady growth of $N_w$ in its final regime of evolution.",
"In fig REF , we present mean similarity of opinions among individuals with edge weights in different time instances.",
"In all the instances, there is a positive correlation of having similar opinions with frequency of interactions i.e., higher the frequency of interactions (edge-weight), higher is the similarity in opinion.",
"Figure: Comparison of mean similarity with edge weights on HT dataset.",
"The mean similarity of opinions with edge weights are shown at different instances of time (a) at t=T/4t = T/4, (b) t=T/2t = T/2, (c) t=3T/4t= 3T/4 and (d) t=Tt = T where TT is the total time.The curves are smoothed by taking 40 point running average.Results from control experiments : For this HT dataset also, we create simulated networks where at each 20s time interval, we construct $m$ number of random edges with $m$ being the count of edges that appeared on that time interval in the real network.",
"We observe the two most important observables $N_w(t)$ and $N_d(t)$ by playing naming game on these simulated networks.",
"Both these quantities show a different behavior from its real counterpart.",
"The $N_w(t)$ and $N_d(t)$ in the simulated networks show distinct two regions - a steady growth and then a fall whereas the $N_w(t)$ curve in the real network show a slow growth zone followed by a sharp transition and finally a zone of steady growth.",
"The simulated networks tend to behave as standard Erdős-Rényi graphs.",
"Figure: (Color online) Comparison of the global quantities in the real and the simulated networks.",
"Temporal evolution of (a) N w N_w, (b) N d N_d for HT real and simulated dataset.",
"The datapoints on the curve are averaged over 100 simulation runs for each of 100 network realizations (in case of simulated network)."
],
[
"Conclusions and future work",
"In this paper, we studied the Naming Game as a model of opinion formation on the time-varying social networks.",
"Some of our key observations are: (a) While considering composite snapshots accumulated over a certain period (e.g., 69 instances of SG dataset with each instance being an accumulation of snapshots ) both the maximum memory and the time to reach this memory peak scale as population size ($N$ ); however, the time to reach the consensus strongly depends on the presence of community structure (rather than a straight-forward $N^{1.4}$ scaling); (b) While considering the time-evolution of the network in perfect synchronization with the steps of the game (e.g., SGD and HT dataset) we observe that the emergent behavior of the most important observables (i.e., $N_w(t)$ and $N_d(t)$ ) have a nature that is markedly in contrast to what has been reported so far in the literature thus indicating the strong influence of the underlying societal structure on the dynamics of opinion formation.",
"While in case of SGD, we observe that new inventions along with a continuous influx of new agents keeps both $N_w(t)$ and $N_d(t)$ sharply growing, in case of HT, successful interactions among older agents cause inventions to stop (hence a fall in $N_d(t)$ ) although late-stage failures continue to exist due to influx of new agents thus contributing to a steady growth of $N_w(t)$ in the final phase of the dynamics.",
"The fall of $N_d(t)$ curve is not observed in case of SGD possibly because in this case the games are played over a shorter span of time (1 day) in comparison to HT where the games are played over 2.5 days so that enough successful interactions could be realized.",
"There are quite a few interesting directions that can be explored in the future.",
"One such direction could be to incorporate the dominance index of the agents into the model.",
"Not all actors in a society are equally dominant; while some of the actors are more opinionated and dominant the others might be more passive.",
"This characteristic property can be incorporated into the model by ranking those agents that are more successful in their past interactions as more dominant.",
"In this setting, it would be interesting to investigate the scaling relations most naturally under the constraints that the dominant agents are allowed to speak more.",
"Another direction could be to investigate the effect of the flexibility of the agents in adapting to new opinions (traditionally modeled by a system parameter $\\beta $ that encodes the probability with which the agents update their inventories in case of successful interactions [4]) when they are embedded on time-varying networks.",
"Finally, a thorough analytical estimate of the important dynamical quantities reported only through empirical evidence here is needed to have a “clear-cut” understanding of the emergent behavior of the system."
]
] | 1204.1160 |
[
[
"A cohomological classification of vector bundles on smooth affine\n threefolds"
],
[
"Abstract We give a cohomological classification of vector bundles of rank $2$ on a smooth affine threefold over an algebraically closed field having characteristic unequal to $2$.",
"As a consequence we deduce that cancellation holds for rank $2$ vector bundles on such varieties.",
"The proofs of these results involve three main ingredients.",
"First, we give a description of the first non-stable ${\\mathbb A}^1$-homotopy sheaf of the symplectic group.",
"Second, these computations can be used in concert with F. Morel's ${\\mathbb A}^1$-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in ${\\mathbb A}^1$-homotopy theory) to reduce the classification results to cohomology vanishing statements.",
"Third, we prove the required vanishing statements."
],
[
"Introduction",
"This paper, together with its companions [2], [1], studies questions in the theory of projective modules using the Morel-Voevodsky ${\\mathbb {A}}^1$ -homotopy theory [33].",
"The jumping off point of this approach is F. Morel's ${\\mathbb {A}}^1$ -homotopy classification of vector bundles over smooth affine schemes [31], which can be viewed as an algebro-geometric analog of Steenrod's homotopy classification of topological vector bundles [43].",
"The basic idea is that, by applying the machinery of obstruction theory in the context of ${\\mathbb {A}}^1$ -homotopy theory, one may mimic results about classification of topological vector bundles (see, e.g., [13]).",
"Write ${\\mathscr V}_n(X)$ for the set of isomorphism classes of rank $n$ vector bundles on $X$ .",
"Theorem 1 (See Theorem REF and Corollary REF ) Suppose $X$ is a smooth affine 3-fold over an algebraically closed field $k$ having characteristic unequal to 2.",
"The map assigning to a rank 2 vector bundle $\\mathcal {E}$ on $X$ the pair $(c_1({\\mathcal {E}}),c_2(\\mathcal {E}))$ of Chern classes gives a pointed bijection $\\mathcal {V}_2(X) {\\stackrel{\\sim }{\\;\\longrightarrow \\;}}Pic(X) \\times CH^2(X).$ For context, recall that N. Mohan Kumar and M.P.",
"Murthy proved [29] the existence of a rank 2 vector bundle on a smooth affine threefold over an algebraically closed field having given Chern classes, i.e., the map in the statement was known to be surjective.",
"Mohan Kumar and Murthy also proved that the map $(c_1,c_2,c_3): \\mathcal {V}_3(X) \\rightarrow \\prod _{i=1}^3 CH^i(X)$ , analogous to the one studied above, is bijective.",
"Remark 2 Let $X$ be a smooth affine 3-fold over an algebraically closed field.",
"Theorem REF implies that cancellation holds for vector bundles of rank 2 on $X$ , i.e., if $\\mathcal {E}_1,\\mathcal {E}_2 \\in \\mathcal {V}_2(X)$ satisfy $\\mathcal {E}_1 \\oplus \\mathcal {E} \\cong \\mathcal {E}_2 \\oplus \\mathcal {E}$ for some vector bundle $\\mathcal {E}$ , then $\\mathcal {E}_1 \\cong \\mathcal {E}_2$ .",
"A weaker version of this cancellation statement, with $\\mathcal {E}_1$ free, was established by the second author in [15].",
"Note also that, for vector bundles of rank $\\ne 2$ , the cancellation statement above is classical: it follows from results of Suslin (for rank 3 bundles) [44] and Bass-Schanuel [11] (for bundles of rank $> 3$ ).",
"As mentioned before, we use techniques of obstruction theory to describe the set of ${\\mathbb {A}}^1$ -homotopy classes of maps to an appropriate Grassmannian.",
"There is a version of the Postnikov tower in ${\\mathbb {A}}^1$ -homotopy theory, and the main impediments to applying this method to describe homotopy classes arise from our limited knowledge of ${\\mathbb {A}}^1$ -homotopy sheaves.",
"Classical homotopy groups are notoriously difficult to compute, and ${\\mathbb {A}}^1$ -homotopy groups are no different in this respect.",
"For the result above, we will need information about ${\\mathbb {A}}^1$ -homotopy groups of $BGL_2$ .",
"By means of the exceptional isomorphism $SL_2 \\cong Sp_2$ , the computation can be reduced to that of describing the first “non-stable\" ${\\mathbb {A}}^1$ -homotopy sheaf of the symplectic group.",
"Theorem 3 (See Theorem REF and Lemma REF ) If $k$ is a perfect field having characteristic unequal to 2, there is a canonical short exact sequence of strictly ${\\mathbb {A}}^1$ -invariant sheaves $0 \\longrightarrow {\\mathbf {T}}_4^{\\prime } \\longrightarrow \\pi _2^{{\\mathbb {A}}^1}(SL_2) \\longrightarrow {{\\mathbf {K}}}^{Sp}_3 \\longrightarrow 0.$ Here, ${{\\mathbf {K}}}^{Sp}_3$ is the sheafification of the third symplectic K-theory group for the Nisnevich topology.",
"The sheaf ${\\mathbf {T}}_4^{\\prime }$ sits in an exact sequence of the form $\\mathbf {I}^5 \\longrightarrow \\mathbf {T}^{\\prime }_4 \\longrightarrow \\mathbf {S}^{\\prime }_4 \\longrightarrow 0$ where ${\\mathbf {I}}^5$ is the unramified sheaf associated with the 5th power of the fundamental ideal in the Witt ring, and there is an epimorphism ${{\\mathbf {K}}}^M_4/12 \\rightarrow {\\mathbf {S}}_4^{\\prime }$ that becomes an isomorphism after 2-fold contraction.",
"Remark 4 Wendt [50] provides a rather general description of sections of ${\\mathbb {A}}^1$ -homotopy sheaves of $SL_n$ (and, more generally, Chevalley groups) as “unstable Karoubi-Villamayor K-theory.\"",
"Our approach complements this approach by a description that is amenable to cohomological computation.",
"The proof of Theorem REF relies on the theory of ${\\mathbb {A}}^1$ -fiber sequences attached to $Sp_{2n}$ -bundles developed by Morel [31] and Wendt [51], which is reviewed in Section and complemented in Section .",
"We use some results from the theory of Grothendieck-Witt groups, which is reviewed in Section .",
"Given Theorem REF , the general techniques of obstruction theory, as discussed in Section , reduce the proof of Theorem REF to identifications of and vanishing statements for certain cohomology groups with coefficients in ${\\mathbb {A}}^1$ -homotopy sheaves; these vanishing statements, together with some complements, occupy Sections and .",
"In particular, we establish a new vanishing theorem for certain $\\mathbf {I}^j$ -cohomology groups on smooth affine schemes and a precise description of $H^3_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^{Sp}_3)$ (and its twisted versions) for any smooth scheme $X$ using Chow groups mod 2 and Steenrod operations (see Theorem REF ).",
"Finally, Section discusses compatibility of our computations with complex realization."
],
[
"Acknowledgements",
"This project was initially begun when both authors were attending the “Summer school on rigidity and the conjecture of Friedlander and Milnor\" held in August 2011 at the University of Regensburg.",
"A first version of this paper was put on the ArXiv in May 2012, but the current version differs rather substantially from the initial version both in scope and content; this version owes a debt to many people.",
"First and foremost, the authors would like to thank Fabien Morel for useful discussion of material in and around that of [31]; the results contained therein played a formative role in the ideas of this paper.",
"We have had opportunity to discuss these ideas and results with many people including Matthias Wendt, Chuck Weibel, V. Srinivas, Ben Williams, and Christian Haesemeyer, each of whom contributed useful comments and questions and helped clarify our thoughts.",
"The first author would like to thank Sasha Merkurjev for explanations about the motivic spectral sequence, the product of which, however, no longer appears in this work.",
"The second author would like to thank Marco Schlichting for kindly providing the proof of Lemma REF .",
"Finally, the authors thank the referees of previous versions of this paper for a number of comments that helped to streamline and clarify the discussion, and for weeding out some silly mistakes."
],
[
"${\\mathbb {A}}^1$ -fiber sequences and the stable range for {{formula:3ea1e570-c898-40c4-afa8-10cbf85e9950}}",
"In this section, we review some preliminaries from ${\\mathbb {A}}^1$ -homotopy theory, especially some results from the theory of ${\\mathbb {A}}^1$ -fiber sequences, due to Morel and Wendt, and Morel's classification theorem for vector bundles over smooth affine schemes.",
"We then recall some stabilization results for ${\\mathbb {A}}^1$ -homotopy sheaves of linear and symplectic groups; these results are also due to Morel and Wendt.",
"The ultimate goal of this section is to define the stable range for $Sp_{2n}$ , and understand the ${\\mathbb {A}}^1$ -homotopy sheaves of $Sp_{2n}$ in this range."
],
[
"Preliminaries from ${\\mathbb {A}}^1$ -homotopy theory",
"Assume $k$ is a field.",
"Write $\\mathscr {Sm}_k$ for the category of schemes that are smooth, separated and have finite type over $\\operatorname{Spec}k$ .",
"Set $\\mathscr {Spc}_k := \\Delta ^{\\circ }{\\mathscr {Shv}}_{\\operatorname{Nis}}(\\mathscr {Sm}_k)$ (resp.",
"$\\mathscr {Spc}_{k,\\bullet }$ ) for the category of (pointed) simplicial sheaves on the site of smooth schemes equipped with the Nisnevich topology; objects of this category will be referred to as (pointed) $k$ -spaces, or simply as (pointed) spaces if $k$ is clear from the context.",
"Write $\\mathscr {H}_s^{\\operatorname{Nis}}(k)$ (resp $\\mathscr {H}_{s,\\bullet }^{\\operatorname{Nis}}(k)$ ) for the (pointed) Nisnevich simplicial homotopy category: this category can be obtained as the homotopy category of, e.g., the injective local model structure on $\\mathscr {Spc}_k$ (see, e.g., [33] for details).",
"Write $\\mathscr {H}({k})$ (resp.",
"$\\mathscr {H}_{\\bullet }({k})$ ) for the associated ${\\mathbb {A}}^1$ -homotopy category, which is constructed as a Bousfield localization of $\\mathscr {H}_s^{\\operatorname{Nis}}(k)$ (resp.",
"$\\mathscr {H}_{s,\\bullet }^{\\operatorname{Nis}}(k)$ ).",
"Given two (pointed) spaces $\\mathscr {X}$ and $\\mathscr {Y}$ , we set $[\\mathscr {X},\\mathscr {Y}]_{s} := \\operatorname{Hom}_{\\mathscr {H}_s^{\\operatorname{Nis}}(k)}(\\mathscr {X},\\mathscr {Y})$ and $[\\mathscr {X},\\mathscr {Y}]_{{\\mathbb {A}}^1} := \\operatorname{Hom}_{\\mathscr {H}({k})}(\\mathscr {X},\\mathscr {Y})$ ; morphisms in pointed homotopy categories will be denoted similarly with base-points explicitly written if it is not clear from context.",
"We write $S^i_s$ for the constant sheaf on $\\mathscr {Sm}_k$ associated with the simplicial $i$ -sphere, and ${{\\mathbf {G}}_{m}}$ will always be pointed by 1.",
"If $\\mathscr {X}$ is any space, the sheaf of ${\\mathbb {A}}^1$ -connected components $\\pi _0^{{\\mathbb {A}}^1}({\\mathscr X})$ is the Nisnevich sheaf associated with the presheaf $U \\mapsto [U,\\mathscr {X}]_{{\\mathbb {A}}^1}$ .",
"More generally, the ${\\mathbb {A}}^1$ -homotopy sheaves of a pointed space $(\\mathscr {X},x)$ , denoted $\\pi _i^{{\\mathbb {A}}^1}(\\mathscr {X},x)$ are defined as the Nisnevich sheaves associated with the presheaves $U \\mapsto [S^i_s \\wedge U_+,(\\mathscr {X},x)]_{{\\mathbb {A}}^1}$ .",
"We also write $\\pi _{i,j}^{{\\mathbb {A}}^1}({\\mathscr X},x)$ for the Nisnevich sheafification of the presheaf $U \\mapsto [S^i_s \\wedge {{\\mathbf {G}}_{m}}^{\\wedge j} \\wedge U_+,({\\mathscr X},x)]_{{\\mathbb {A}}^1}$ .",
"For $i = j = 0$ , the morphism of sheaves $\\pi _{0,0}^{{\\mathbb {A}}^1}(\\mathscr {X},x) \\rightarrow \\pi _0^{{\\mathbb {A}}^1}({\\mathscr X})$ obtained by “forgetting the base-point\" is an isomorphism.",
"If $\\mathscr {X}$ is an ${\\mathbb {A}}^1$ -connected space or if the base-point is clear from context, to ease the notational burden, we will sometimes suppress the base-point.",
"A presheaf of sets ${\\mathcal {F}}$ on $\\mathscr {Sm}_k$ is called ${\\mathbb {A}}^1$ -invariant if for any smooth $k$ -scheme $U$ the morphism ${\\mathcal {F}}(U) \\rightarrow {\\mathcal {F}}(U \\times {\\mathbb {A}}^1)$ induced by pullback along the projection $U \\times {\\mathbb {A}}^1\\rightarrow U$ is a bijection.",
"A Nisnevich sheaf of groups $\\mathcal {G}$ is called strongly ${\\mathbb {A}}^1$ -invariant if the cohomology presheaves $H^i_{\\operatorname{Nis}}(\\cdot ,\\mathcal {G})$ are ${\\mathbb {A}}^1$ -invariant for $i = 0,1$ .",
"A Nisnevich sheaf of abelian groups ${\\mathbf {A}}$ is called strictly ${\\mathbb {A}}^1$ -invariant if the cohomology presheaves $H^i_{\\operatorname{Nis}}(\\cdot ,{\\mathbf {A}})$ are ${\\mathbb {A}}^1$ -invariant for every $i \\ge 0$ .",
"We will repeatedly use the fact that strictly ${\\mathbb {A}}^1$ -invariant sheaves are unramified in the sense of [31].",
"One consequence of this is, e.g.",
"by [31], that if $f:\\mathbf {A}\\rightarrow \\mathbf {A}^\\prime $ is a morphism of strictly invariant sheaves then $f$ is an isomorphism (or epimorphism or monomorphism) if and only if the induced map on sections over (finitely generated) extensions of the base field are isomorphisms.",
"Suppose $\\mathcal {G}$ is a strongly ${\\mathbb {A}}^1$ -invariant sheaf of groups.",
"For any smooth $k$ -scheme $U$ , the unit of ${{\\mathbf {G}}_{m}}$ defines a morphism $U \\rightarrow {{\\mathbf {G}}_{m}}\\times U$ .",
"Recall that $\\mathcal {G}_{-1}$ is the sheaf $\\mathcal {G}_{-1}(U) = \\ker (\\mathcal {G}({{\\mathbf {G}}_{m}}\\times U) \\rightarrow \\mathcal {G}(U))$ Iterating this construction one defines $\\mathcal {G}_{-i}$ .",
"If we restrict $()_{-1}$ to the category of strictly ${\\mathbb {A}}^1$ -invariant sheaves of groups, it is an exact functor (see, e.g., [31] or, more precisely, its proof).",
"Associated with any strictly ${\\mathbb {A}}^1$ -invariant sheaf $\\mathbf {A}$ is a Rost-Schmid complex [31], which coincides with the usual Gersten complex [31] and has terms described using contractions of $\\mathbf {A}$ [31].",
"This complex provides a flasque resolution of $\\mathbf {A}$ by [31], and its cohomology groups compute Nisnevich (or Zariski) cohomology groups of $\\mathbf {A}$ [31].",
"Write ${{\\mathbf {K}}}^{MW}_n$ for the $n$ -th unramified Milnor-Witt K-theory sheaf described in [31].",
"As explained in [31], any strictly ${\\mathbb {A}}^1$ -invariant sheaf of the form ${\\mathbf {A}}_{-1}$ admits a canonical action of ${{\\mathbf {G}}_{m}}$ that factors through an action of ${{\\mathbf {K}}}^{MW}_0$ [31].",
"Moreover, the ${{\\mathbf {G}}_{m}}$ -action on ${{\\mathbf {K}}}^{MW}_n$ that arises in this fashion is precisely the one coming from the product map ${{\\mathbf {K}}}^{MW}_0 \\times {{\\mathbf {K}}}^{MW}_n \\rightarrow {{\\mathbf {K}}}^{MW}_n$ .",
"If $X$ is a smooth scheme, ${\\mathcal {L}}$ is a line bundle on $X$ , and ${\\mathbf {A}}_{-1}$ is a strictly ${\\mathbb {A}}^1$ -invariant sheaf, the ${{\\mathbf {G}}_{m}}$ -action just mentioned allows one to “twist\" the sheaf ${\\mathbf {A}}_{-1}$ by ${\\mathcal {L}}$ as follows.",
"Write ${\\mathcal {L}}^{\\circ }$ for the ${{\\mathbf {G}}_{m}}$ -torsor corresponding to the complement of the zero section in ${\\mathcal {L}}$ and define ${\\mathbf {A}}_{-1}({\\mathcal {L}})$ as the sheaf on the small Nisnevich site of $X$ associated with the presheaf assigning to an étale morphism $u: U \\rightarrow X$ the abelian group ${\\mathbf {A}}_{-1}(U) \\otimes _{{\\mathbb {Z}}[{\\mathcal {O}}(U)^{\\times }]} {\\mathbb {Z}}[(u^*{\\mathcal {L}})^{\\circ }(U)]$ .",
"In this case, the twisted Rost-Schmid complex $C_*^{RS}(X,{\\mathcal {L}};{\\mathbf {A}}_{-1})$ defined in [31] provides a flasque resolution of ${\\mathbf {A}}_{-1}({\\mathcal {L}})$ .",
"In the special case where ${\\mathbf {A}} = \\mathbf {I}^{j}$ the unramified sheaf associated with the fundamental ideal in the Witt ring (see [17]), the description of the ${{\\mathbf {K}}}^{MW}_0$ -action on ${\\mathbf {A}}_{-1}=\\mathbf {I}^{j-1}$ from [31] shows that $\\mathbf {I}^j({\\mathcal {L}})$ and the associated twisted Rost-Schmid complex coincide with the twisted sheaf denoted $I^j_{{\\mathcal {L}}}$ and the twisted Gersten-Witt resolution from [17].",
"Likewise, the twisted Rost-Schmid complex of ${{\\mathbf {K}}}^{MW}_n({\\mathcal {L}})$ coincides with the fiber product of [14].",
"These “twisted\" constructions will only reappear in Sections , and .",
"There is a general definition of a fiber sequence in a model category [25].",
"This definition generalizes the usual examples coming from Serre fibrations in topology.",
"We refer the reader to, e.g., [4] for a concise development of these ideas in the context of the ${\\mathbb {A}}^1$ -homotopy category.",
"Just like in classical topology, ${\\mathbb {A}}^1$ -fiber sequences give rise to long exact sequences in ${\\mathbb {A}}^1$ -homotopy sheaves (see, e.g., [4]).",
"The main result about fiber sequences we will use is summarized in the following statement, which is quoted from [51]; in any situation in this paper where a sequence of spaces is asserted to be an ${\\mathbb {A}}^1$ -fiber sequence (and for which no auxiliary reference is given), the sequence has this property because of the following result.",
"Theorem 2.1 (Morel, Moser, Wendt) Assume $F$ is a field, and $(X,x)$ is a pointed smooth $F$ -scheme.",
"If $P \\rightarrow X$ is a $G$ -torsor for $SL_n, GL_n$ or $Sp_{2n}$ , then there is an ${\\mathbb {A}}^1$ -fiber sequence of the form $G \\longrightarrow P \\longrightarrow X.$ If, moreover, $Y$ is a pointed smooth quasi-projective $F$ -scheme equipped with a left action of $G$ , then the associated fiber space, i.e.",
"the quotient $P \\times ^G Y$ , exists as a smooth scheme, and there is an ${\\mathbb {A}}^1$ -fiber sequence of the form $Y \\longrightarrow P \\times ^G Y \\longrightarrow X.$ [Comments on the proof.]",
"As regards attribution: Morel proved the above result for $SL_n$ or $GL_n$ ($n = 1$ or $n \\ge 3$ ) in [31], and Wendt extended his result to treat a rather general class of reductive groups; the case where $G = SL_2$ requires the results of Moser [32].",
"In [51], the existence of fiber sequences such as the above is stated under the apparently additional hypothesis that $F$ be infinite.",
"However, the assumption that $F$ is infinite is used by way of Theorem 3.1 of ibid to guarantee extensibility of $G$ -torsors.",
"In the case of special groups, such as above, this kind of extensibility result is known (e.g., by results of Lindel for vector bundles) without the assumption that the base-field is infinite (see [51]).",
"Finally, assuming $Y$ is quasi-projective, the quotient $P \\times ^G Y$ exists as a smooth scheme.",
"Since $G$ is special (in the sense of Grothendieck-Serre), all $G$ -torsors are Zariski (and hence Nisnevich) locally trivial.",
"Combining these two observations, the notation $P \\times ^G Y$ is unambiguous, i.e., the Nisnevich sheaf quotient coincides with the scheme quotient.",
"We now apply the results on fiber sequences above to the ${\\mathbb {A}}^1$ -fiber sequences $\\begin{split}SL_{n} &\\longrightarrow SL_{n+1} \\longrightarrow SL_{n+1}/SL_{n} \\text{ and }\\\\Sp_{2n} &\\longrightarrow Sp_{2n+2} \\longrightarrow Sp_{2n+2}/Sp_{2n}.\\end{split}$ Each of these fiber sequences gives rise to a long exact sequence in ${\\mathbb {A}}^1$ -homotopy sheaves.",
"The homogeneous spaces that appear in these fiber sequences are “highly ${\\mathbb {A}}^1$ -connected.\"",
"More precisely, we have the following result, whose proof we leave to the reader.",
"Proposition 2.2 The “projection onto the first column\" morphism $SL_{n+1} \\rightarrow {\\mathbb {A}}^{n+1} \\setminus 0$ (resp.",
"$Sp_{2n+2} \\rightarrow {\\mathbb {A}}^{2n+2} \\setminus 0$ ) factors through a morphism $SL_{n+1}/SL_{n} \\rightarrow {\\mathbb {A}}^{n+1} \\setminus 0$ (resp.",
"$Sp_{2n+2}/Sp_{2n} \\rightarrow {\\mathbb {A}}^{2n+2} \\setminus 0$ ) that is Zariski locally trivial with affine space fibers; in particular, this morphism is an ${\\mathbb {A}}^1$ -weak equivalence.",
"Theorem 2.3 ([31]) For any integer $n \\ge 1$ , the space ${\\mathbb {A}}^{n+1} \\setminus 0$ is $(n-1)$ -${\\mathbb {A}}^1$ -connected, and there is a canonical isomorphism $\\pi _{n}^{{\\mathbb {A}}^1}({\\mathbb {A}}^{n+1} \\setminus 0) \\cong {{\\mathbf {K}}}^{\\mathrm {MW}}_{n+1}$ .",
"Combining Theorem REF and Proposition REF , one immediately deduces the following result.",
"Corollary 2.4 (Morel, Wendt) The morphisms $\\pi _i^{{\\mathbb {A}}^1}(SL_{n}) \\rightarrow \\pi _i^{{\\mathbb {A}}^1}(SL_{n+1})$ are epimorphisms for $i \\le n-1$ and isomorphisms for $i \\le n-2$ .",
"The morphisms $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n}) \\rightarrow \\pi _i^{{\\mathbb {A}}^1}(Sp_{2n+2})$ are epimorphisms for $i \\le 2n$ and isomorphisms for $i \\le 2n-1$ .",
"In particular, observe that the sheaves $\\pi _i^{{\\mathbb {A}}^1}(SL_n)$ coincide with the stable groups $\\pi _i^{{\\mathbb {A}}^1}(SL_{\\infty })$ for $i \\le n-2$ , and the sheaves $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n})$ coincide with $\\pi _i^{{\\mathbb {A}}^1}(Sp_{\\infty })$ for $i \\le 2n-1$ .",
"These observations allow us to define the stable range.",
"Definition 2.5 The sheaves $\\pi _i^{{\\mathbb {A}}^1}(SL_n)$ for $i \\le n-2$ and the sheaves $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n})$ for $i \\le 2n-1$ will be said to be in the stable range.",
"Using the stabilization results above, it is possible to deduce a stable range for ${\\mathbb {A}}^1$ -homotopy sheaves of the symplectic group; the corresponding statements for the special linear group were recalled in [2].",
"Given a sheaf of groups $G$ , we write $BG$ for its simplicial classifying space in the sense of [33].",
"Theorem 2.6 For any integers $i,n\\in \\mathbb {N}$ there are canonical isomorphisms $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n}) \\cong \\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_{2n}).$ If $0 \\le i \\le 2n-1$ and, furthermore, the base field $k$ is assumed to have characteristic unequal to 2, there are canonical isomorphisms of the form $\\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_{2n}) \\cong \\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_{\\infty }) \\cong {{\\mathbf {K}}}^{Sp}_{i+1}.$ Since $Sp_{2n}$ is ${\\mathbb {A}}^1$ -connected for any $n\\in \\mathbb {N}$ , we deduce from [31] that $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n})=\\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_{2n})$ for any integer $i$ .",
"By Corollary REF , there are isomorphisms $\\pi _i^{{\\mathbb {A}}^1}(Sp_{2n})=\\pi _i^{{\\mathbb {A}}^1}(Sp_\\infty )$ if $0 \\le i \\le 2n-1$ .",
"Similarly, there are isomorphisms $\\pi _i^{{\\mathbb {A}}^1}(Sp_\\infty )=\\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_\\infty )$ and thus $\\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_{2n})\\cong \\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_\\infty )$ for $0 \\le i \\le 2n-1$ .",
"If $\\mathrm {char}(k)\\ne 2$ , the space ${\\mathbb {Z}}\\times BSp_{\\infty }$ represents symplectic $K$ -theory in $\\mathscr {H}({k})$ (see for instance [35]) and it follows that $\\pi _{i+1}^{{\\mathbb {A}}^1}(BSp_\\infty )={{\\mathbf {K}}}^{Sp}_{i+1}$ ."
],
[
"The first non-stable ${\\mathbb {A}}^1$ -homotopy sheaf of {{formula:f52ff8fc-5ef6-441a-ad98-0acf868f8c76}}",
"The main goal of this section is to describe the first non-stable ${\\mathbb {A}}^1$ -homotopy sheaf for $Sp_{2n}$ and, as a consequence the “next\" non-stable ${\\mathbb {A}}^1$ -homotopy sheaf of $GL_2$ ; this is achieved in Theorem REF .",
"By the results of Section , the first non-stable sheaf is $\\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n})$ , and this sheaf is an extension of a symplectic K-theory sheaf (the stable part) by something “non-stable\".",
"Our goal here is to describe the non-stable part; we will see that it is an extension of a torsion sheaf by a quotient of a sheafification of a power of the fundamental ideal in the Witt ring.",
"To begin, observe that the long exact sequence in ${\\mathbb {A}}^1$ -homotopy sheaves associated with the ${\\mathbb {A}}^1$ -fiber sequence arising from the $Sp_{2n}$ -torsor $Sp_{2n+2} \\rightarrow Sp_{2n+2}/Sp_{2n}$ yields an exact sequence of the form $\\pi _{2n+1}^{{\\mathbb {A}}^1}(Sp_{2n+2}) \\stackrel{\\varphi _{2n+2}}{\\longrightarrow } \\pi _{2n+1}^{{\\mathbb {A}}^1}(Sp_{2n+2}/Sp_{2n}) \\longrightarrow \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\longrightarrow \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n+2}) \\longrightarrow 0.$ The sheaves involving $Sp_{2n+2}$ are already in the stable range by Theorem REF , and since there is an ${\\mathbb {A}}^1$ -weak equivalence $Sp_{2n+2}/Sp_{2n} \\longrightarrow {\\mathbb {A}}^{2n+2} \\setminus 0$ , we obtain an exact sequence of the form ${{\\mathbf {K}}}^{Sp}_{2n+2} \\stackrel{\\varphi _{2n+2}}{\\longrightarrow } {{\\mathbf {K}}}^{MW}_{2n+2} \\longrightarrow \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\longrightarrow {{\\mathbf {K}}}^{Sp}_{2n+1} \\longrightarrow 0.$ If we set ${\\mathbf {T}}^{\\prime }_{2n+2} := \\operatorname{coker}({{\\mathbf {K}}}^{Sp}_{2n+2} \\stackrel{\\varphi _{2n+2}}{\\longrightarrow } {{\\mathbf {K}}}^{MW}_{2n+2}),$ then our goal is to describe ${\\mathbf {T}}^{\\prime }_{2n+2}$ explicitly.",
"To this end, recall from [2] that there is an exact sequence of sheaves of the form ${0[r] & \\mathbf {I}^{2n+3}[r] & {{\\mathbf {K}}}_{2n+2}^{MW}[r] & {{\\mathbf {K}}}_{2n+2}^M[r] & 0}$ where $\\mathbf {I}^{2n+3}$ is the unramified sheaf associated with the $(2n+3)$ th power of the fundamental ideal in the Witt ring (again, see [17]).",
"Composing $\\varphi _{2n+2}$ with the epimorphism ${{\\mathbf {K}}}^{MW}_{2n+2}\\rightarrow {{\\mathbf {K}}}^M_{2n+2}$ , we get a morphism $\\varphi ^\\prime _{2n+2}:{{\\mathbf {K}}}^{Sp}_{2n+2}\\longrightarrow {{\\mathbf {K}}}^M_{2n+2}$ and we set ${\\mathbf {S}}^{\\prime }_{2n+2} := \\operatorname{coker}({{\\mathbf {K}}}^{Sp}_{2n+2} \\stackrel{\\varphi ^\\prime _{2n+2}}{\\longrightarrow } {{\\mathbf {K}}}^{M}_{2n+2}).$ The proof of the following lemma is straightforward.",
"Lemma 3.1 There is an exact sequence of sheaves of the form $\\mathbf {I}^{2n+3}\\longrightarrow {\\mathbf {T}}^{\\prime }_{2n+2}\\longrightarrow {\\mathbf {S}}^{\\prime }_{2n+2}\\longrightarrow 0.$ We will see in Section that the morphism $\\mathbf {I}^{2n+3}\\longrightarrow {\\mathbf {T}}^{\\prime }_{2n+2}$ is, in general, non-trivial.",
"For now, we turn to the task of describing ${\\mathbf {S}}^{\\prime }_{2n+2}$ .",
"The standard inclusion $Sp_{2n+2}\\rightarrow SL_{2n+2}$ induces a commutative diagram ${Sp_{2n+2}[r][d] & Sp_{2n+2}/Sp_{2n}[d] \\\\SL_{2n+2}[r] & SL_{2n+2}/SL_{2n+1}}$ and it is straightforward to check that the right-hand vertical map is an isomorphism.",
"Moreover, both $Sp_{2n+2}/Sp_{2n}$ and $SL_{2n+2}/SL_{2n+1}$ are ${\\mathbb {A}}^1$ -weakly equivalent to $\\mathbb {A}^{2n+2}\\setminus 0$ by Proposition REF , and the commutative diagram of sheaves obtained by applying $\\pi _{2n+1}^{{\\mathbb {A}}^1}(\\_)$ to the above diagram takes the form: ${{{\\mathbf {K}}}_{2n+2}^{Sp}[r]^-{\\varphi _{2n+2}}[d] & {{\\mathbf {K}}}_{2n+2}^{MW}@{=}[d] \\\\\\pi _{2n+1}^{{\\mathbb {A}}^1}(SL_{2n+2})[r] & {{\\mathbf {K}}}_{2n+2}^{MW}.",
"}$ By [2], the map $\\pi _{2n+1}^{{\\mathbb {A}}^1}(SL_{2n+2})\\rightarrow {{\\mathbf {K}}}_{2n+2}^{MW}$ fits in a commutative diagram ${\\pi _{2n+1}^{{\\mathbb {A}}^1}(SL_{2n+2})[r][d]& {{\\mathbf {K}}}_{2n+2}^{MW}[d] \\\\{{\\mathbf {K}}}_{2n+2}^Q[r]_-{\\psi _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^M}$ where the left vertical arrow is the stabilization map (i.e., the morphism on ${\\mathbb {A}}^1$ -homotopy sheaves induced by $SL_{2n+2}\\rightarrow SL_{\\infty }$ ), the right vertical arrow is the canonical epimorphism and $\\psi _{2n+2}$ is the sheafified version of a homomorphism originally defined by Suslin in [45].",
"Combining the two diagrams above, we obtain a diagram of the form ${{{\\mathbf {K}}}_{2n+2}^{Sp}[r]^-{\\varphi _{2n+2}}[d]_-{f_{2n+2,2}} [rd]^-{\\varphi ^\\prime _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^{MW}[d] \\\\{{\\mathbf {K}}}_{2n+2}^Q[r]_-{\\psi _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^M},$ where $f_{2n+2,2}$ is the forgetful homomorphism (which is itself induced by the natural map $BSp_{\\infty }\\rightarrow BSL_{\\infty }$ ; see Section for more information).",
"The image of $\\psi _{2n+2}$ is hard to control in general; however, the natural homomorphism from Milnor $K$ -theory to Quillen $K$ -theory can be sheafified to obtain a morphism of sheaves $\\mu _{2n+2}:{{\\mathbf {K}}}_{2n+2}^M\\rightarrow {{\\mathbf {K}}}_{2n+2}^Q$ and the composite ${{{\\mathbf {K}}}_{2n+2}^M[r]^-{\\mu _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^Q[r]^-{\\psi _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^M}$ is multiplication by $(2n+1)!$ ([2]).",
"Thus $(2n+1)!",
"{{\\mathbf {K}}}_{2n+2}^M\\subset \\mathrm {Im}(\\psi _{2n+2})$ .",
"As in [2], we write $\\mathbf {S}_{2n+2}$ for the cokernel of $\\psi _{2n+2}$ .",
"Proposition 3.2 There is a commutative diagram of epimorphisms of sheaves of the form ${{{\\mathbf {K}}}^M_{2n+2}/2(2n+1)!",
"[r][d] & \\mathbf {S}^\\prime _{2n+2}[d] \\\\{{\\mathbf {K}}}^M_{2n+2}/(2n+1)!",
"[r] & \\mathbf {S}_{2n+2}.",
"}$ The maps $GL_n\\rightarrow Sp_{2n}$ defined by $M\\mapsto \\begin{pmatrix} M & 0 \\\\ 0 & (M^{-1})^t\\end{pmatrix}$ induce a map $BGL_{\\infty }\\rightarrow BSp_{\\infty }$ and thus, by taking ${\\mathbb {A}}^1$ -homotopy sheaves, a morphism of sheaves $H_{2n+2,2}:{{\\mathbf {K}}}^Q_{2n+2}\\rightarrow {{\\mathbf {K}}}_{2n+2}^{Sp}$ .",
"We claim the following diagram commutes: $@C=3em{{{\\mathbf {K}}}^M_{2n+2}[r]^-{\\mu _{2n+2}}@{=}[d] & {{\\mathbf {K}}}^Q_{2n+2}[r]^-{H_{2n+2,2}}@{=}[d] & {{\\mathbf {K}}}_{2n+2}^{Sp}[r]^-{\\varphi ^\\prime _{2n+2}}[d]^-{f_{2n+2,2}} & {{\\mathbf {K}}}^M_{2n+2}@{=}[d] \\\\{{\\mathbf {K}}}^M_{2n+2}[r]_-{\\mu _{2n+2}} & {{\\mathbf {K}}}^Q_{2n+2}[r]_-2 & {{\\mathbf {K}}}_{2n+2}^{Q}[r]_-{\\psi _{2n+2}} & {{\\mathbf {K}}}^M_{2n+2}.",
"}$ Indeed, the middle square commutes by Lemma REF , while the right-hand square commutes by the discussion preceding the statement of the proposition.",
"The composite of the bottom maps is equal to multiplication by $2(2n+1)!$ , and we obtain the required epimorphism of sheaves ${{\\mathbf {K}}}^M_{2n+2}/2(2n+1)!\\rightarrow \\mathbf {S}^\\prime _{2n+2}$ .",
"The diagram ${{{\\mathbf {K}}}^M_{2n+2}/2(2n+1)!",
"[r][d] & \\mathbf {S}^\\prime _{2n+2}[d] \\\\{{\\mathbf {K}}}^M_{2n+2}/(2n+1)!",
"[r] & \\mathbf {S}_{2n+2}}$ commutes by construction, and the result follows from Lemma REF .",
"For convenient reference, we summarize the above results in the following statement.",
"Theorem 3.3 There are exact sequences of the form $\\begin{split}0 \\longrightarrow {\\mathbf {T}}^{\\prime }_{2n+2} \\longrightarrow & \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\longrightarrow {{\\mathbf {K}}}^{Sp}_{2n+1} \\longrightarrow 0, \\text{ and } \\\\0 \\longrightarrow \\mathbf {D}_{2n+3} \\longrightarrow & \\mathbf {T}^{\\prime }_{2n+2} \\longrightarrow \\mathbf {S}^{\\prime }_{2n+2} \\longrightarrow 0,\\end{split}$ where ${\\mathbf {T}}^{\\prime }_{2n+2}$ and ${\\mathbf {S}}^{\\prime }_{2n+2}$ are defined above, and ${\\mathbf {S}}^{\\prime }_{2n+2}$ is a quotient of ${{\\mathbf {K}}}^M_{2n+2}/(2(2n+1)!",
")$ , and $\\mathbf {D}_{2n+3}$ is a quotient of $\\mathbf {I}^{2n+3}$ .",
"Remark 3.4 In Section , we will complement the result above by establishing Lemmas REF and REF .",
"The second result shows that, for $n$ odd, after repeated contraction the epimorphisms ${{\\mathbf {K}}}^M_{2n+2}/(2(2n+1)!)",
"\\rightarrow \\mathbf {S}^{\\prime }_{2n+2}$ are isomorphisms, while the first result establishes a corresponding result for $n$ even (the statement is a just a bit more complicated).",
"We will also show in Lemma REF that $\\mathbf {D}_{2n+3}$ cannot be the zero sheaf for $n$ odd."
],
[
"Grothendieck-Witt groups",
"In this section, we begin by recalling some basic facts about Grothendieck-Witt groups, which are a Waldhausen-style version of hermitian $K$ -theory.",
"The general reference for the material of this section is the work of M. Schlichting [38], [39].",
"The main results are Lemma REF , which was used in the proof of Proposition REF , and Theorem REF .",
"We also spend some time discussing the natural action of ${{\\mathbf {G}}_{m}}$ on contractions of Grothendieck-Witt sheaves, which provides a technical ingredient in the proof of Theorem REF ."
],
[
"Definitions",
"Let $X$ be a smooth scheme with $2\\in {\\mathcal {O}}_X(X)^\\times $ (we keep these assumptions throughout the section, though it is not necessary for some of the arguments).",
"For every such $X$ , and any line bundle ${\\mathcal {L}}$ on $X$ , one has an exact category with weak equivalences and (strong) duality $(\\mathcal {C}^b(X),qis,\\sharp _{{\\mathcal {L}}},\\varpi _{{\\mathcal {L}}})$ in the sense of [39].",
"Here, $\\mathcal {C}^b(X)$ is the category of bounded complexes of locally free coherent sheaves on $X$ , weak equivalences are given by quasi-isomorphisms of complexes, $\\sharp _{{\\mathcal {L}}}$ is the duality functor induced by the functor $\\operatorname{Hom}_{{\\mathcal {O}}_X}(\\_,{\\mathcal {L}})$ , and the natural transformation $\\varpi _{{\\mathcal {L}}}:1\\rightarrow \\sharp _{{\\mathcal {L}}}\\sharp _{{\\mathcal {L}}}$ is induced by the canonical identification of a locally free sheaf with its double dual.",
"The (left) translation (or shift) functor $T:\\mathcal {C}^b(R)\\rightarrow \\mathcal {C}^b(R)$ yields new dualities $\\sharp ^n_{{\\mathcal {L}}}:=T^n\\circ \\sharp _{{\\mathcal {L}}}$ and canonical isomorphisms $\\varpi ^n_{{\\mathcal {L}}}:=(-1)^{n(n+1)/2}\\varpi _{{\\mathcal {L}}}$ .",
"Associated with an exact category with weak equivalences and (strong) duality is a Grothendieck-Witt space and higher Grothendieck-Witt groups, obtained as homotopy groups of the Grothendieck-Witt space [39].",
"We write $\\mathcal {GW}(\\mathcal {C}^b(X),qis,\\sharp ^j_{{\\mathcal {L}}},\\varpi ^j_{{\\mathcal {L}}})$ for the Grothendieck-Witt space of the example described in the previous paragraph.",
"Definition 4.1 For $i\\ge 0$ , we denote by $GW^j_i(X,{{\\mathcal {L}}})$ the group $\\pi _i\\mathcal {GW}(\\mathcal {C}^b(X),qis,\\sharp ^j_{{\\mathcal {L}}},\\varpi ^j_{{\\mathcal {L}}})$ .",
"If ${{\\mathcal {L}}}={\\mathcal {O}}_X$ , we write $GW^j_i(X)$ for $GW^j_i(X,{\\mathcal {O}}_X)$ .",
"Later, we will also use “negative Grothendieck-Witt groups:\" Schlichting constructs a spectrum $\\mathbb {G}W(\\mathcal {C}^b(X),qis,\\sharp ^j_{{\\mathcal {L}}},\\varpi ^j_{{\\mathcal {L}}})$ [39], and the negative Grothendieck-Witt groups are defined as $GW_{-i}^j(X,{{\\mathcal {L}}}):=\\pi _{-i}\\mathbb {G}W(\\mathcal {C}^b(X),qis,\\sharp ^j_{{\\mathcal {L}}},\\varpi ^j_{{\\mathcal {L}}})$ for $i>0$ .",
"For any $j\\in {\\mathbb {Z}}$ , the group $GW_0^j(X,{{\\mathcal {L}}})$ coincides with the Grothendieck-Witt group defined by Balmer-Walter of the triangulated category $D^b(X)$ of bounded complexes of coherent locally free ${\\mathcal {O}}_X$ -modules endowed with the corresponding duality ([40], [48]), and negative Grothendieck-Witt groups coincide with triangular Witt groups as defined by P. Balmer (see, e.g., [5]) under the formula $GW_{-i}^j(X,{{\\mathcal {L}}})=W^{i+j}(X,{{\\mathcal {L}}})$ ; these identifications will be used when we study “contractions\" in Proposition REF .",
"The Grothendieck-Witt groups defined above coincide with hermitian $K$ -theory as defined by M. Karoubi [27], [28] in the case of affine schemes, at least when 2 is invertible (see [38], see also [23]).",
"In particular, given a smooth $k$ -algebra $R$ we have by [40] the identifications $\\begin{split}GW_i^0(R) &= K_iO(R) \\\\GW_i^2(R) &= K_iSp(R).\\end{split}$ There are identifications $GW_i^1(R)={}_{-1}U_i(R)$ and $GW_i^3(R)=U_i(R)$ , where the groups $U_i(R)$ and ${}_{-1}U_i(R)$ are Karoubi's $U$ -groups, and $GW_i^n$ is 4-periodic in $n$ .",
"One can compare Quillen $K$ -theory with higher Grothendieck-Witt groups using the hyperbolic morphisms $H_{i,j}:K_i(X)\\rightarrow GW_i^j(X,{{\\mathcal {L}}})$ and forgetful morphisms $f_{i,j}:GW_i^j(X,{{\\mathcal {L}}})\\rightarrow K_i(X)$ defined for any $i,j\\in \\mathbb {N}$ and any line bundle ${{\\mathcal {L}}}$ over $X$ (note the indexing!).",
"The hyperbolic and forgetful morphisms are connected by means of the Karoubi periodicity exact sequences ([40]) $@C=1.71em{\\ldots [r] & K_i(X)[r]^-{H_{i,j}} & GW_i^j(X,{{\\mathcal {L}}})[r]^-{\\eta } & GW_{i-1}^{j-1}(X,{{\\mathcal {L}}})[r]^-{f_{i-1,j-1}} & K_{i-1}(X)[r]^-{H_{i-1,j}} & GW_{i-1}^j(X,{{\\mathcal {L}}})[r] & \\ldots },$ where $\\eta $ is multiplication by a distinguished element in $GW_{-1}^{-1}(k)$ .",
"The composition $f_{i,j}\\circ H_{i,j}$ is, in general, difficult to understand, but the situation is slightly better when we take $X$ to be (the spectrum of) a field.",
"For any field $F$ , the identification $\\mu _1:K^M_1(F)\\rightarrow K^Q_1(F)$ , induces via cup-product a (functorial in $F$ ) homomorphism $\\mu _i:K^M_i(F)\\rightarrow K^Q_i(F)$ .",
"Lemma 4.2 For any field $F$ having characteristic unequal 2, and for any integers $i,j \\ge 0$ , the following diagram commutes: $@C=6em{K^Q_i(F)[r]^-{f_{i,j}\\circ H_{i,j}} & K^Q_i(F) \\\\K^M_i(F)[r]_-{(1+(-1)^{i+j})Id}[u]^-{\\mu _i} & K^M_i(F).",
"[u]_-{\\mu _i}}$ Let $(\\mathcal {E},\\omega ,\\sharp ,\\eta )$ be an exact category with weak-equivalences and duality in the sense of [39].",
"With any exact category with weak-equivalences, one can associate the hyperbolic category $(\\mathcal {HE},\\omega )$ [39].",
"Its objects are pairs $(X,Y)$ of objects of $\\mathcal {C}$ , a morphism $(X,Y)\\rightarrow (X^\\prime ,Y^\\prime )$ is a pair $(a,b)$ of morphisms of $\\mathcal {C}$ with $a:X\\rightarrow X^\\prime $ and $b:Y^\\prime \\rightarrow Y$ .",
"Such a morphism is a weak-equivalence if $a$ and $b$ are.",
"The switch $(X,Y)\\mapsto (Y,X)$ yields a duality $^\\star $ on $\\mathcal {HE}$ and there is an obvious identification $id:1\\rightarrow ^{\\star \\star }$ .",
"Thus $(\\mathcal {HE},\\omega ,^\\star ,id)$ is an exact category with weak-equivalences and duality.",
"The Grothendieck-Witt space $\\mathcal {GW}(\\mathcal {HE},\\omega ,^\\star ,id)$ is naturally homotopic to the $K$ -theory space $\\mathcal {K}(\\mathcal {E},\\omega )$ [39].",
"In this context, the forgetful functor $f$ reads as $f(X)=(X,X^\\sharp )$ for any $X$ in $\\mathcal {E}$ .",
"On the other hand the hyperbolic functor $H:\\mathcal {HE}\\rightarrow \\mathcal {E}$ is defined by $H(X,Y)=X\\oplus Y^\\sharp $ [38].",
"The composition $fH:(\\mathcal {HE},\\omega ,^\\star ,id)\\rightarrow (\\mathcal {HE},\\omega ,^\\star ,id)$ is then given by $fH(X,Y)=(X\\oplus Y^\\sharp , X^\\sharp \\oplus Y^{\\sharp \\sharp })$ .",
"In particular, this composition is the same for $(\\mathcal {E},\\omega ,\\sharp ,\\eta )$ and $(\\mathcal {E},\\omega ,\\sharp ,-\\eta )$ .",
"We now specialize to the case of $(\\mathcal {C}^b(F),qis,\\sharp ^j,\\varpi ^j)$ .",
"By the discussion of the previous paragraph, the map $fH$ on $K$ -theory sends an object $X$ of $\\mathcal {C}^b(F)$ to $X \\oplus X^{\\sharp }[j]$ .",
"By the additivity theorem (see, e.g., [49]) the induced map $K(F) \\rightarrow K(F)$ is thus $Id + (-1)^j \\tau $ , where $\\tau $ is induced by the involution of $GL(F)$ defined by $G\\mapsto (G^t)^{-1}$ .",
"Now, $\\tau $ acts as the identity on $K_0(F)$ and is easily seen to be multiplication by $-1$ on $K_1(F)$ .",
"J. Hornbostel [24] showed that there are spaces $\\mathscr {GW}^j$ in $\\mathscr {H}_{\\bullet }({k})$ for any $j\\in \\mathbb {Z}$ such that for any smooth scheme $X$ we have $GW_i^j(X)=[S^i_s\\wedge X_+,\\mathscr {GW}^j]_{{\\mathbb {A}}^1}$ (see also [35]).",
"Definition 4.3 For $i\\in \\mathbb {N}$ , we set $\\begin{split}\\mathbf {GW}^j_i &:= \\pi _i^{{\\mathbb {A}}^1}(\\mathscr {GW}^j), \\text{ and } \\\\\\mathbf {GW}^j_{-i} &:= \\pi ^{{\\mathbb {A}}^1}_{0,i}(\\mathscr {GW}^{i+j}).\\end{split}$ Since $GW_i^2(X)=K_iSp(R)$ for any ring $R$ , we have $\\mathbf {GW}_i^2={{\\mathbf {K}}}_i^{Sp}$ for any $i\\in \\mathbb {N}$ .",
"Moreover, it follows from [35] that the sheaves $\\mathbf {GW}^j_{-i}$ are the Nisnevich sheaves associated with the presheaves $U\\mapsto W^{i+j}(U)$ .",
"Witt and Grothendieck-Witt groups are also representable in the stable ${\\mathbb {A}}^1$ -homotopy category of ${\\mathbb {P}}^1$ -spectra ([24]), and therefore the sheaves $\\mathbf {GW}_i^j$ are strictly ${\\mathbb {A}}^1$ -invariant for any $i,j\\in \\mathbb {Z}$ .",
"Using [31] (as well as [35]), we deduce the following result on contractions of Grothendieck-Witt sheaves.",
"Proposition 4.4 For any $i,j\\in \\mathbb {Z}$ , we have $(\\mathbf {GW}^j_{i})_{-1}=\\mathbf {GW}^{j-1}_{i-1}$ .",
"Remark 4.5 The proof of [35] also gives the contractions of the hyperbolic morphisms $H_{i,j}:{{\\mathbf {K}}}^Q_i\\rightarrow \\mathbf {GW}_i^j$ and forgetful morphisms $f_{i,j}:\\mathbf {GW}_i^j\\rightarrow {{\\mathbf {K}}}_i^Q$ .",
"We find $(H_{i,j})_{-1}=H_{i-1,j-1}$ and $(f_{i,j})_{-1}=f_{i-1,j-1}$ .",
"Proposition REF implies that, for any pair of integers $i,j$ , the sheaf $\\mathbf {GW}^j_i$ is a contraction.",
"As a consequence, the sheaf $\\mathbf {GW}^j_i$ is equipped with a natural action of ${{\\mathbf {G}}_{m}}$ that factors through an action of ${{\\mathbf {K}}}^{MW}_0 \\cong \\mathbf {GW}^0_0$ ; see the beginning of Section for a more discussion of this action.",
"On the other hand, the multiplicative structure on Grothendieck-Witt groups [40] determines a morphism $\\mathbf {GW}^0_0 \\times \\mathbf {GW}^j_i \\longrightarrow \\mathbf {GW}^j_i.$ The next lemma shows that, under the canonical isomorphism between ${{\\mathbf {K}}}^{MW}_0$ and $\\mathbf {GW}^0_0$ , these two actions coincide.",
"Lemma 4.6 The action of ${{\\mathbf {K}}}^{MW}_0$ on $\\mathbf {GW}^j_i$ coming from [31] coincides with the multiplicative action of $\\mathbf {GW}^0_0$ on $\\mathbf {GW}^j_i$ under the canonical isomorphism ${{\\mathbf {K}}}^{MW}_0 {\\stackrel{{\\scriptscriptstyle {\\sim }}}{\\;\\rightarrow \\;}}\\mathbf {GW}^0_0$ .",
"We unwind the definition of the action of ${{\\mathbf {K}}}^{MW}_0$ on $\\mathbf {GW}^0_0$ .",
"For any smooth scheme $X$ , we have an exact sequence of the form $0 \\longrightarrow \\mathbf {GW}^j_i(X) \\longrightarrow \\mathbf {GW}^j_{i}(X\\times {{\\mathbf {G}}_{m}}) \\stackrel{\\pi }{\\longrightarrow } \\mathbf {GW}^{j-1}_{i-1}(X) \\longrightarrow 0.$ One can construct a splitting of $\\pi $ as follows.",
"The projection map $p: X \\times {{\\mathbf {G}}_{m}}\\rightarrow X$ induces a pullback map $\\mathbf {GW}^{j-1}_{i-1}(X) \\rightarrow \\mathbf {GW}^{j-1}_{i-1}(X \\times {{\\mathbf {G}}_{m}})$ .",
"On the other hand, the canonical homomorphism ${{\\mathbf {K}}}^{MW}_1({{\\mathbf {G}}_{m}}) \\rightarrow \\mathbf {GW}^1_1({{\\mathbf {G}}_{m}})$ is an isomorphism (this follows from [7] after unwinding the definition of $\\mathbf {GW}^1_1$ and using the description of ${{\\mathbf {K}}}^{MW}_1$ as a fiber product).",
"Picking a coordinate $t$ on ${{\\mathbf {G}}_{m}}$ , there is, under this identification, a canonical class $[t] \\in \\mathbf {GW}^1_1$ .",
"The element $[t]$ then determines a class in $\\mathbf {GW}^{1}_{1}(X \\times {{\\mathbf {G}}_{m}})$ by pullback, and multiplication by this element determines a homomorphism $\\mathbf {GW}^{j-1}_{i-1}(X \\times {{\\mathbf {G}}_{m}}) \\rightarrow \\mathbf {GW}^j_i(X \\times {{\\mathbf {G}}_{m}})$ that provides the required splitting.",
"Now, given any $a \\in {\\mathcal {O}}_X(X)^{\\times }$ , the action on $\\mathbf {GW}^{j-1}_{i-1}(X)$ is given by the formula $a \\cdot \\alpha = \\pi (p^*\\alpha \\cdot [at])$ .",
"Since ${{\\mathbf {K}}}_1^{MW}(X\\times {{\\mathbf {G}}_{m}})=\\mathbf {GW}_1^1(X\\times {{\\mathbf {G}}_{m}})$ , we have $[at]=[a]+\\langle a\\rangle [t]$ where $\\langle a\\rangle $ is the class defined by $a$ in $\\mathbf {GW}_0^0(X)$ .",
"Since $\\pi $ is $\\mathbf {GW}_0^0$ -linear, we get $\\pi (p^*\\alpha \\cdot [at])=\\pi (p^*\\alpha \\cdot [a])+\\pi (p^*\\alpha \\cdot \\langle a\\rangle [t])=\\langle a\\rangle \\cdot \\pi (p^*\\alpha \\cdot [t]),$ which provides the required identification.",
"Using Proposition REF , it follows from [31] that the degree $n$ piece of the Rost-Schmid complex for $\\mathbf {A} = \\mathbf {GW}^j_i$ has terms of the form $\\bigoplus _{x\\in X^{(n)}} GW_{i-n}^{j-n}(k(x),\\omega _x^{{\\mathcal {L}}}),$ where $\\omega _x^{{\\mathcal {L}}}=\\mathrm {Ext}^n_{\\mathcal {O}_{X,x}}(k(x),{\\mathcal {L}}\\otimes \\mathcal {O}_{X,x})$ .",
"A priori, there is another action of ${{\\mathbf {G}}_{m}}$ on $\\mathbf {GW}_i^2$ for any $i\\in \\mathbb {N}$ defined as follows.",
"If $R$ is a smooth $k$ -algebra and $t\\in R^\\times $ is an invertible element, then let $\\sigma _{2,t}$ be the invertible matrix $\\begin{pmatrix} t & 0 \\\\ 0 & 1\\end{pmatrix}$ .",
"Define $2n \\times 2n$ -matrices inductively by means of the formula $\\sigma _{2n,t}:=\\sigma _{2n-2,t}\\perp \\sigma _{2,t}$ .",
"For any natural number $n$ , the function assigning to $t \\in R^{\\times }$ the operation of conjugating a matrix $X \\in Sp_{2n}(R)$ by $\\sigma _{2n,t}$ defines an action of $R^{\\times }$ on $Sp_{2n}(R)$ .",
"This construction defines an action of ${{\\mathbf {G}}_{m}}$ on $Sp_{2n}$ and consequently an action of ${{\\mathbf {G}}_{m}}$ on $BSp_{2n}$ .",
"The stabilization morphisms $BSp_{2n} \\rightarrow BSp_{2n+2}$ are, by construction, equivariant for the ${{\\mathbf {G}}_{m}}$ -actions so defined.",
"As a consequence, there is an induced action ${{\\mathbf {G}}_{m}}\\times \\mathbf {GW}_i^{2} \\longrightarrow \\mathbf {GW}_i^{2}.$ We can compare this action of ${{\\mathbf {G}}_{m}}$ to the one discussed above.",
"Lemma 4.7 For any smooth $k$ -algebra $R$ , and any $t \\in R^{\\times }$ , the action of $t$ on $\\mathbf {GW}^2_i(R)$ described in REF is given by multiplication by $\\langle t \\rangle $ .",
"Let $R$ be a smooth algebra as above, and let $t\\in R^\\times $ .",
"Let $\\mathcal {S}$ be the symmetric monoidal category with objects given by pairs $(P,\\varphi )$ where $P$ is a (finitely generated) projective $R$ -module equipped with a non-degenerate skew-symmetric form $\\varphi $ and with morphisms given by isometries.",
"An element $t \\in R^{\\times }$ determines a (strict) symmetric monoidal functor $\\cdot \\otimes \\langle t \\rangle : \\mathcal {S} \\rightarrow \\mathcal {S}$ as follows: on objects $(P,\\varphi ) \\otimes \\langle t \\rangle := (P,t\\varphi )$ , and act by the identity on morphisms; this specifies a categorical action of $R^{\\times }$ on $\\mathcal {S}$ .",
"The groups $GW^2_i(R)$ can be identified [40] as homotopy groups of $B \\mathcal {S}^{-1}\\mathcal {S}$ (here $\\mathcal {S}^{-1}\\mathcal {S}$ is Quillen's construction; see, e.g., [49]).",
"The categorical action of $R^{\\times }$ described above defines an action of $R^{\\times }$ on the space $B \\mathcal {S}^{-1}\\mathcal {S}$ .",
"The action of $R^{\\times }$ on the connected component of the base-point $(\\mathcal {S}^{-1}\\mathcal {S})_0$ of $B \\mathcal {S}^{-1}\\mathcal {S}$ is given as follows: if $h_{2n}$ is the usual hyperbolic space of rank $2n$ , then $t \\cdot (h_{2n},h_{2n}) = (h_{2n} \\otimes \\langle t \\rangle ,h_{2n} \\otimes \\langle t \\rangle )$ and $(f,g) \\mapsto (f,g)$ .",
"Now, the map $c:BSp\\rightarrow (\\mathcal {S}^{-1}\\mathcal {S})_0$ is the map $\\mathrm {hocolim}_{n\\in \\mathbb {N}}\\mathrm {Aut}(h_{2n})\\rightarrow (\\mathcal {S}^{-1}\\mathcal {S})_0$ defined on objects by $n\\mapsto (h_{2n},h_{2n})$ and on morphisms by $f\\mapsto (1,f)$ (where $f\\in \\mathrm {Aut}(h_{2n}$ )).",
"Now, consider the diagram ${\\mathrm {hocolim}_{n\\in \\mathbb {N}}\\mathrm {Aut}(h_{2n})[r][d]_-c & \\mathrm {hocolim}_{n\\in \\mathbb {N}}\\mathrm {Aut}(h_{2n})[d]^-c \\\\(\\mathcal {S}^{-1}\\mathcal {S})_0[r] & (\\mathcal {S}^{-1}\\mathcal {S})_0}$ where the top map is induced by the conjugation by $\\sigma _{2n,t}$ and the bottom map is the action of $t$ on $(\\mathcal {S}^{-1}\\mathcal {S})_0$ described above.",
"This diagram is not strictly commutative.",
"Indeed, one composite maps an object $n$ to $(h_{2n} \\otimes \\langle t \\rangle ,h_{2n} \\otimes \\langle t \\rangle )$ and a morphism $f$ to $(1,f)$ , while the other composite maps $n$ to $(h_{2n},h_{2n})$ and $f$ to $(1,\\sigma _{2n,t}^{-1}f\\sigma _{2n,t})$ .",
"However, there is a natural isomorphism given by $(\\sigma _{2n,t},\\sigma _{2n,t})$ as shown by the following diagram $@C=5.5em{(h_{2n},h_{2n})[r]^-{(1,\\sigma _{2n,t}^{-1}f\\sigma _{2n,t})}[d]_-{(\\sigma _{2n,t},\\sigma _{2n,t})} & (h_{2n},h_{2n})[d]^-{(\\sigma _{2n,t},\\sigma _{2n,t})} \\\\(t\\cdot h_{2n},t\\cdot h_{2n})[r]_-{(1,f)} & (t\\cdot h_{2n},t\\cdot h_{2n}).",
"}$ It follows that the diagram (REF ) commutes up to homotopy and the result stands proved.",
"As explained before the lemma, the action of ${{\\mathbf {G}}_{m}}$ on $Sp_{2n}$ stabilizes.",
"As a consequence, the homogeneous spaces $Sp_{2n+2}/Sp_{2n}$ inherit an action of ${{\\mathbf {G}}_{m}}$ such that the quotient map $Sp_{2n+2}\\rightarrow Sp_{2n+2}/Sp_{2n}$ is equivariant.",
"The projection onto the first row map $Sp_{2n+2} \\rightarrow {\\mathbb {A}}^{2n+2}\\setminus 0$ factors through a projection map $Sp_{2n+2}/Sp_{2n}\\rightarrow \\mathbb {A}^{2n+2}\\setminus 0$ that is an ${\\mathbb {A}}^1$ -weak equivalence.",
"If we equip ${\\mathbb {A}}^{2n+2}\\setminus 0$ with the ${{\\mathbf {G}}_{m}}$ -action given (functorially) by $t \\cdot (a_1,\\ldots ,a_{2n+2})) = (a_1,t^{-1}a_2,a_3,t^{-1}a_4,\\ldots ,a_{2n+1},t^{-1}a_{2n+2})$ the map $Sp_{2n+2}/Sp_{2n} \\rightarrow \\mathbb {A}^{2n+2}\\setminus 0$ becomes ${{\\mathbf {G}}_{m}}$ -equivariant.",
"Lemma 4.8 If $n \\in {\\mathbb {N}}$ is even, then the ${{\\mathbf {G}}_{m}}$ -action on ${\\mathbb {A}}^{2n}\\setminus 0$ described above is ${\\mathbb {A}}^1$ -homotopically trivial.",
"For any (finitely generated) field extension $F/k$ , we know from [31] that there is an isomorphism $[\\mathbb {A}^{2n}_F\\setminus 0,\\mathbb {A}^{2n}_F\\setminus 0]_{{\\mathbb {A}}^1}\\cong \\mathbf {GW}(F)$ called the motivic Brouwer degree.",
"By [16] (and the fact that the matrix $\\mathrm {diag}(t^{-1},t)$ is ${\\mathbb {A}}^1$ -homotopic to the identity), the motivic Brouwer degree of the map $(a_1,\\ldots ,a_{2n}) \\longmapsto (a_1,t^{-1}a_2,a_3,t^{-1}a_4,\\ldots ,a_{2n-1},t^{-1}a_{2n})$ is precisely $\\langle 1\\rangle $ .",
"Theorem REF shows that, for any $n\\in \\mathbb {N}$ , there is an exact sequence of sheaves of the form $0 \\longrightarrow {\\mathbf {T}}^{\\prime }_{2n+2} \\longrightarrow \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\longrightarrow \\mathbf {GW}_{2n+1}^2 \\longrightarrow 0.$ If we equip $Sp_{2n}$ , $Sp_{2n+2}$ and $\\mathbb {A}^{2n}\\setminus 0$ with the ${{\\mathbf {G}}_{m}}$ -actions specified above, there is an induced action of ${{\\mathbf {G}}_{m}}$ on $\\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n})$ such that the above sequence is an exact sequence of sheaves with ${{\\mathbf {G}}_{m}}$ -action.",
"The next result is then a consequence of Lemmas REF and REF .",
"Corollary 4.9 The induced ${{\\mathbf {G}}_{m}}$ -action on ${\\mathbf {T}}^{\\prime }_{2n+2}$ is trivial if $n$ is odd, and restricts to the action given by multiplication on $\\mathbf {GW}^2_{2n+1}$ under the map $\\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\rightarrow \\mathbf {GW}_{2n+1}^2$ .",
"If $X$ is a smooth $k$ -scheme (not necessarily a threefold), and ${\\mathcal {L}}$ is a line bundle on $X$ , then the ${{\\mathbf {G}}_{m}}$ -action on $\\mathbf {GW}^2_3$ described in the previous section can be used to define the sheaf $\\mathbf {GW}^2_3({\\mathcal {L}})$ .",
"In this section, we analyze the Gersten (or Rost-Schmid) resolution of $\\mathbf {GW}^2_3({\\mathcal {L}})$ in more detail in order to compute $H^3_{\\operatorname{Nis}}(X, \\mathbf {GW}^2_3({\\mathcal {L}}))$ .",
"Remark 4.10 A priori, there is another definition of the sheaf $\\mathbf {GW}^2_3({\\mathcal {L}})$ that appears in the literature: for any étale morphism $u: U \\rightarrow X$ , one can sheafify $U \\mapsto GW^2_3(U,u^*{\\mathcal {L}})$ for the Nisnevich topology.",
"The resulting sheaf is an unramified sheaf and the description of the Gersten resolution shows that this definition coincides with the one we gave.",
"Using Karoubi periodicity, we get an exact sequence of sheaves on $X$ ${{{\\mathbf {K}}}_3^Q[r]^-{H_{3,2}} & \\mathbf {GW}_3^2({\\mathcal {L}})[r]^-{\\eta } & \\mathbf {GW}_2^1({\\mathcal {L}})[r]^-{f_{2,1}} & {{\\mathbf {K}}}_2^Q};$ we denote by $\\mathbf {A}({\\mathcal {L}})$ the image of $H_{3,2}$ and by $\\mathbf {B}({\\mathcal {L}})$ the image of $\\eta $ .",
"We thus get an exact sequence ${0[r] & \\mathbf {A}({\\mathcal {L}})[r] & \\mathbf {GW}_3^2({\\mathcal {L}})[r]^-{\\eta } & \\mathbf {B}({\\mathcal {L}})[r] & 0.",
"}$ Lemma 4.11 The epimorphism $H_{3,2}:{{\\mathbf {K}}}_3^Q\\rightarrow \\mathbf {A}({\\mathcal {L}})$ induces an isomorphism ${{\\mathbf {K}}}^Q_1/2\\cong \\mathbf {A}_{-2}({\\mathcal {L}})$ .",
"Moreover, $\\mathbf {B}_{-2}({\\mathcal {L}})\\cong {\\mathbb {Z}}/2$ and the exact sequence ${0[r] & \\mathbf {A}({\\mathcal {L}})_{-2}[r] & \\mathbf {GW}_1^0({\\mathcal {L}})[r] & \\mathbf {B}({\\mathcal {L}})_{-2}[r] & 0}$ splits if ${\\mathcal {L}}$ is trivial.",
"Contracting (REF ) two times, we get an exact sequence of sheaves ${{{\\mathbf {K}}}_1^Q[r]^-{H_{1,0}} & \\mathbf {GW}_1^0({\\mathcal {L}})[r]^-{\\eta } & \\mathbf {GW}_0^3({\\mathcal {L}})[r]^-{f_{0,3}} & {\\mathbb {Z}}.",
"}$ By [19], the hyperbolic functor $H_{0,3}:{\\mathbb {Z}}\\rightarrow \\mathbf {GW}_0^3({\\mathcal {L}})$ induces an isomorphism ${\\mathbb {Z}}/2\\cong GW_0^3(k(X),{\\mathcal {L}})$ .",
"On the other hand, $(\\mathbf {GW}_0^3({\\mathcal {L}}))_{-1}=\\mathbf {W}_0^3({\\mathcal {L}})$ which vanishes on any field by [12].",
"It follows that $\\mathbf {GW}_0^3({\\mathcal {L}})={\\mathbb {Z}}/2$ and that $f_{0,3}=0$ .",
"Thus $\\mathbf {B}({\\mathcal {L}})_{-2}=\\mathbf {GW}_0^3({\\mathcal {L}})\\cong {\\mathbb {Z}}/2$ .",
"If ${\\mathcal {L}}$ is trivial, it is easy to check that the map $\\eta :\\mathbf {GW}_1^0\\rightarrow {\\mathbb {Z}}/2$ coincides with the determinant morphism, which is split.",
"Using [6], we see that the hyperbolic map $H_{1,0}:{{\\mathbf {K}}}_1^Q\\rightarrow \\mathbf {GW}_1^0({\\mathcal {L}})$ induces an injective map $k(X)^\\times /(k(X)^\\times )^2\\rightarrow GW_1^0(k(X),{\\mathcal {L}})$ .",
"It follows that $H_{1,0}$ induces an injective morphism ${{\\mathbf {K}}}_1^Q/2\\rightarrow \\mathbf {GW}_1^0({\\mathcal {L}})$ with, by definition, image equal to $\\mathbf {A}({\\mathcal {L}})_{-2}$ .",
"The exact sequence ${0[r] & \\mathbf {A}({\\mathcal {L}})[r] & \\mathbf {GW}_3^2({\\mathcal {L}})[r]^-\\eta & \\mathbf {B}({\\mathcal {L}})[r] & 0}$ yields an exact sequence of cohomology groups ${H^2_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))[r] & H^3_{\\operatorname{Nis}}(X,\\mathbf {A}({\\mathcal {L}}))[r] & H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}}))[r] & H_{\\operatorname{Nis}}^3(X,\\mathbf {B}({\\mathcal {L}}))}$ and we now turn to the task of understanding the cohomology groups of $\\mathbf {A}$ and $\\mathbf {B}$ .",
"We first introduce some notation.",
"Notation 4.12 We will denote by $Ch^n(X)$ the group $CH^n(X)/2$ , where $CH^n(X)$ is the Chow groups of codimension $n$ cycles in $X$ .",
"Lemma 4.13 We have $H^3_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))=0$ .",
"Moreover, the hyperbolic morphism $H_{3,2}:{{\\mathbf {K}}}_3^Q\\rightarrow \\mathbf {A}({\\mathcal {L}})$ induces an isomorphism $Ch^3(X)\\cong H^3_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}_3^Q/2)\\cong H^3_{\\operatorname{Nis}}(X,\\mathbf {A}({\\mathcal {L}}))$ .",
"Since $\\mathbf {B}({\\mathcal {L}})_{-2}\\cong {\\mathbb {Z}}/2$ , it follows that $\\mathbf {B}({\\mathcal {L}})_{-3}=0$ and therefore $H^3_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))=0$ .",
"Now the hyperbolic morphism induces an isomorphism ${{\\mathbf {K}}}_1^Q/2\\cong \\mathbf {A}({\\mathcal {L}})_{-2}$ and therefore an isomorphism ${\\mathbb {Z}}/2\\cong \\mathbf {A}({\\mathcal {L}})_{-3}$ .",
"It follows that $Ch^3(X)\\cong H^3_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}_3^Q/2)\\cong H^3_{\\operatorname{Nis}}(X,\\mathbf {A}({\\mathcal {L}}))$ .",
"We now compute the group $H^2_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))$ .",
"Lemma 4.14 The hyperbolic morphism $H_{2,1}:{{\\mathbf {K}}}_2^Q\\rightarrow \\mathbf {GW}_2^1({\\mathcal {L}})$ induces a morphism $H^\\prime _{2,1}:{{\\mathbf {K}}}_2^Q\\rightarrow \\mathbf {B}({\\mathcal {L}})$ .",
"By definition of $\\mathbf {B}({\\mathcal {L}})$ , it suffices to show that the composite ${{{\\mathbf {K}}}_2^Q[r]^-{H_{2,1}} & \\mathbf {GW}_2^1({\\mathcal {L}})[r]^-{f_{2,1}} & {{\\mathbf {K}}}_2^Q}$ is trivial.",
"By Matsumoto's theorem, the morphism of sheaves $\\mu _2:{{\\mathbf {K}}}_2^M\\rightarrow {{\\mathbf {K}}}_2^Q$ is an isomorphism, and it follows therefore from Lemma REF that $f_{2,1}H_{2,1}=0$ after evaluating at $k(X)$ .",
"Thus $f_{2,1}H_{2,1}=0$ .",
"Lemma 4.15 The morphism $H^\\prime _{2,1}:{{\\mathbf {K}}}_2^Q\\rightarrow \\mathbf {B}({\\mathcal {L}})$ yields an isomorphism $Ch^2(X)\\cong H^2_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}_2^Q/2)\\rightarrow H^2_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}})).$ By Lemma REF (or rather its proof), we know that $(H^\\prime _{2,1})_{-2}$ induces an isomorphism ${\\mathbb {Z}}/2\\cong \\mathbf {B}({\\mathcal {L}})_{-2}$ .",
"The same lemma shows that $H_{1,0}$ yields a monomorphism ${{\\mathbf {K}}}_1^Q/2\\rightarrow \\mathbf {GW}_{1}^0({\\mathcal {L}})$ whose cokernel is isomorphic to ${\\mathbb {Z}}/2$ .",
"This cokernel is identified with the image of $f_{1,0}:\\mathbf {GW}_1^0({\\mathcal {L}})\\rightarrow {{\\mathbf {K}}}_1^Q$ .",
"It follows that $(H^\\prime _{2,1})_{-1}$ yields an isomorphism ${{\\mathbf {K}}}_1^Q/2\\cong \\mathbf {B}({\\mathcal {L}})_{-1}$ .",
"Combining Lemmas REF and REF , we obtain the following result, which is a key tool in the proof of Theorem REF .",
"Proposition 4.16 If $k$ is a field having characteristic unequal to 2, $X$ is a smooth $k$ -scheme and ${\\mathcal {L}}$ is a line bundle on $X$ , then there is an exact sequence of the form ${Ch^2(X)[r]^-\\partial & Ch^3(X)[r] & H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}}))[r] & 0.",
"}$ It is possible to compute the map $\\partial :Ch^2(X)\\rightarrow Ch^3(X)$ of Proposition REF explicitly.",
"The discussion of the remainder of this section is devoted to providing an explicit description of this map, which is not strictly necessary in the sequel.",
"To state our result, recall first from [9] that one can define Steenrod squaring operations $Sq^2:Ch^n(X)\\rightarrow Ch^{n+1}(X)$ for any $n\\in \\mathbb {N}$ satisfying a number of useful properties.",
"In particular, if $f:Y\\rightarrow X$ is a proper morphism of smooth connected schemes, the formula [9]: $Sq^2(f_*[Y])=c_1(\\omega _{X/k})f_*([Y])-f_*(c_1(\\omega _{Y/k}))$ holds, where $\\omega _ {X/k}$ (resp.",
"$\\omega _ {Y/k}$ ) is the canonical sheaf of $X$ over $\\operatorname{Spec}k$ (resp.",
"$Y$ over $\\operatorname{Spec}k$ ).",
"In fact, as Totaro explains [46], this property characterizes $Sq^2$ .",
"Given a line bundle ${\\mathcal {L}}$ on $X$ , we denote by $c_1({\\mathcal {L}})$ its class in $Ch^1(X)$ .",
"We can then twist the Steenrod operations to obtain a new operation $Sq^2_{{\\mathcal {L}}}:Ch^n(X)\\rightarrow Ch^{n+1}(X)$ defined by $Sq^2_{{\\mathcal {L}}}(\\alpha )=Sq^2(\\alpha )+ c_1({\\mathcal {L}}) \\cdot \\alpha $ .",
"Theorem 4.17 If $X$ is a smooth scheme over a field $k$ having characteristic different from 2, then there is an exact sequence of the form ${Ch^{2}(X)[r]^-{Sq^2_{{\\mathcal {L}}}} & Ch^3(X)[r] & H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}}))[r] & 0.",
"}$ To prove the theorem, we first observe that $Ch^2(X)$ is generated by the classes of integral subvarieties of codimension 2.",
"Let then $Y$ be such a subvariety, and let $C$ be its normalization.",
"The morphism $i:C\\rightarrow Y\\subset X$ is finite.",
"The singular locus of $C$ is of codimension $\\ge 2$ and its image $T$ under $i$ (which is closed) is of codimension $\\ge 4$ .",
"Considering $X\\setminus T$ and $C\\setminus i^{-1}(T)$ instead of $X$ and $C$ , we see that we can suppose that the normalization $C$ of $Y$ is smooth.",
"Since $i$ is finite, [31] yields homomorphisms $i_*:H^n_{\\operatorname{Nis}}(C,\\mathbf {F}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\rightarrow H^{n+2}_{\\operatorname{Nis}}(X,\\mathbf {F}({\\mathcal {L}}))$ for any strictly ${\\mathbb {A}}^1$ -invariant sheaf $\\mathbf {F}$ (that is already a contraction) and any $n\\in \\mathbb {N}$ , where ${\\mathcal {N}}=\\omega _{C/k}\\otimes i^*\\omega _{X/k}^\\vee $ and $\\omega _{C/k}$ (resp.",
"$\\omega _{X/k}$ ) is the canonical sheaf of $C$ (resp.",
"$X$ ).",
"The exact sequence (REF ) then yields a commutative diagram ${H^0_{\\operatorname{Nis}}(C,\\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})[r]^-{\\partial ^\\prime }[d] & H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})[r][d] & H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}}))[d] \\\\H^2_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))[r]@{=}[d] & H^3_{\\operatorname{Nis}}(X,\\mathbf {A}({\\mathcal {L}}))[r]@{=}[d] & H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}})) \\\\Ch^2(X)[r]_-{\\partial } & Ch^3(X) & }$ Now, Lemma REF shows that $\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2}\\cong {{\\mathbf {K}}}_1^Q/2$ and $\\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2}\\cong {\\mathbb {Z}}/2$ .",
"There are therefore canonical identifications $H^0_{\\operatorname{Nis}}(C,\\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\cong {\\mathbb {Z}}/2$ and $H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\cong H^1_{\\operatorname{Nis}}(C,{{\\mathbf {K}}}_1^Q/2)\\cong Pic(C)/2$ .",
"The map $i_*:H^0_{\\operatorname{Nis}}(C,\\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\rightarrow H^2_{\\operatorname{Nis}}(X,\\mathbf {B}({\\mathcal {L}}))$ satisfies then $i_*(1)=[Y]$ .",
"It follows that it is sufficient to compute $\\partial ^\\prime $ in order to identify $\\partial $ .",
"Proposition 4.18 Under the previous identifications, the homomorphism $\\partial ^\\prime :{\\mathbb {Z}}/2\\cong H^0_{\\operatorname{Nis}}(C,\\mathbf {B}_{-2}({\\mathcal {N}}\\otimes {\\mathcal {L}}))\\rightarrow H^1_{\\operatorname{Nis}}(C,\\mathbf {A}_{-2}({\\mathcal {N}}\\otimes {\\mathcal {L}}))\\cong Pic(C)/2$ satisfies $\\partial ^\\prime (1)=c_1({\\mathcal {N}}\\otimes {\\mathcal {L}})$ .",
"We first suppose that $c_1({\\mathcal {N}}\\otimes {\\mathcal {L}})= 0$ in $Pic(C)/2$ .",
"There exists then a line bundle $\\mathcal {M}$ and an isomorphism ${\\mathcal {N}}\\otimes {\\mathcal {L}}\\cong \\mathcal {M}^{\\otimes 2}$ .",
"In this case, we have an isomorphism $\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}})\\cong \\mathbf {GW}_1^0$ by, e.g., [35].",
"The sequence ${0[r] & \\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2}[r] & \\mathbf {GW}_1^0[r] & \\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2}[r] & 0}$ is then split by Lemma REF and therefore $\\partial ^\\prime (1)=0$ .",
"We can therefore assume that $c_1({\\mathcal {N}}\\otimes {\\mathcal {L}}) \\ne 0$ in $Pic(C)/2$ .",
"Observe first that the sequence ${{\\mathbb {Z}}/2\\cong H^0_{\\operatorname{Nis}}(C,\\mathbf {B}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})[r]^-{\\partial ^\\prime } & H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})[r] & H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}}))}$ is exact.",
"It suffices thus to prove that $c_1({\\mathcal {L}}\\otimes {\\mathcal {N}})$ belongs to the kernel of the map $H^1_{\\operatorname{Nis}}(C,{{\\mathbf {K}}}_1^Q/2)\\cong H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\rightarrow H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}}))$ in order to conclude.",
"The Bloch-Ogus spectral sequence [8] (see also [20] in the context of Grothendieck-Witt groups and [36] in the context of $K$ -theory) yields homomorphisms $F^1GW_0^0(C,{\\mathcal {N}}\\otimes {\\mathcal {L}})\\rightarrow H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}}))$ and $\\tilde{K}_0(C)\\rightarrow Pic(C)$ where $F^1GW_0^0(C,{\\mathcal {N}}\\otimes {\\mathcal {L}})$ is the kernel of the localization map $GW_0^0(C,{\\mathcal {N}}\\otimes {\\mathcal {L}})\\rightarrow GW_0^0(k(C),{\\mathcal {N}}\\otimes {\\mathcal {L}})$ and $\\tilde{K}_0(C)$ is the kernel of the rank homomorphism.",
"Observe that $[{\\mathcal {N}}\\otimes {\\mathcal {L}}]$ in $Pic(C)$ is the image of $[{\\mathcal {N}}\\otimes {\\mathcal {L}}]-[{\\mathcal {O}}_C]$ under the second map.",
"The hyperbolic functor gives a commutative diagram of the form ${\\tilde{K}_0(C)[r][d]_{H_{0,0}} & Pic(C)[d] \\\\F^1GW_0^0(C,{\\mathcal {N}}\\otimes {\\mathcal {L}})[r] & H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}}))},$ and the right-hand vertical map is the composite $Pic(C)\\rightarrow Pic(C)/2\\cong H^1_{\\operatorname{Nis}}(C,{{\\mathbf {K}}}_1^Q/2)\\cong H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {N}}\\otimes {\\mathcal {L}})_{-2})\\rightarrow H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {N}}\\otimes {\\mathcal {L}})).$ If $\\mathcal {M}$ is a line bundle over $C$ , then the image of $[\\mathcal {M}]-[{\\mathcal {O}}_C]$ under the left-hand vertical map is $[\\mathcal {M}\\oplus (\\mathcal {M}^\\vee \\otimes {\\mathcal {N}}\\otimes {\\mathcal {L}}),h_{\\mathcal {M}}]-[{\\mathcal {O}}_C\\oplus ({\\mathcal {N}}\\otimes {\\mathcal {L}}),h_{{\\mathcal {O}}_C}],$ where $h_{\\mathcal {M}}$ and $h_{\\mathcal {O}_C}$ are the usual hyperbolic forms.",
"It is straightforward to check that $[{\\mathcal {N}}\\otimes {\\mathcal {L}}]-[{\\mathcal {O}}_C]$ vanishes under this map.",
"It follows that $c_1({\\mathcal {L}}\\otimes {\\mathcal {N}})$ belongs to the kernel of $H^1_{\\operatorname{Nis}}(C,{{\\mathbf {K}}}_1^Q/2)\\cong H^1_{\\operatorname{Nis}}(C,\\mathbf {A}({\\mathcal {L}})_{-2})\\rightarrow H^1_{\\operatorname{Nis}}(C,\\mathbf {GW}_1^0({\\mathcal {L}}))$ , giving $\\partial ^\\prime (1)= c_1({\\mathcal {L}}\\otimes {\\mathcal {N}})$ .",
"To conclude the proof of Theorem REF , it suffices now to observe that the diagram ${{\\mathbb {Z}}/2[r]^-{\\partial ^\\prime }[d] & Pic(C)/2[d] \\\\Ch^2(X)[r]_-{Sq^2_{\\mathcal {L}}} & Ch^3(X)}$ commutes by the formula (REF ) defining Steenrod operations and the projection formula."
],
[
"Vanishing theorems",
"In this section, we prove a number of cohomological vanishing results that will be used in Section to provide explicit descriptions of sets of isomorphism classes of vector bundles."
],
[
"Cohomology of ${{\\mathbf {K}}}^{\\mathrm {MW}}_j$ and line bundle twists",
"Recall some notation from the beginning of Section : $\\mathbf {I}^n$ is the unramified sheaf (in the Nisnevich or Zariski topology) of the $n$ -th power of the fundamental ideal, ${{\\mathbf {K}}}^{MW}_n$ is the unramified Milnor-Witt K-theory sheaf.",
"If ${\\mathcal {L}}$ is a line bundle over a smooth $k$ -scheme $X$ , $\\mathbf {I}^n({\\mathcal {L}})$ and ${{\\mathbf {K}}}^{MW}_n({\\mathcal {L}})$ are the corresponding ${\\mathcal {L}}$ -twisted sheaves (on the small Nisnevich site of $X$ ).",
"Furthermore, recall that if $\\ell $ is a prime number, then $cd_{\\ell }(k)$ is the smallest integer $n$ (or $\\infty $ ) such that $H^i_{\\text{ét}}(\\operatorname{Spec}k,{\\mathcal {F}}) = 0$ for $i > n$ and all $\\ell $ -torsion étale sheaves ${\\mathcal {F}}$ over $\\operatorname{Spec}k$ .",
"Proposition 5.1 Let $X$ be a smooth scheme of dimension $d$ over a field $k$ with $\\mathrm {cd}_2(k)=r<\\infty $ and let ${\\mathcal {L}}$ be a line bundle on $X$ .",
"Then the Zariski sheaf $\\mathbf {I}^{n}({\\mathcal {L}})=0$ for any $n\\ge r+d+1$ .",
"By definition of $\\mathbf {I}^{n}({\\mathcal {L}})$ it is sufficient to prove that $I^n(k(X),{\\mathcal {L}}\\otimes k(X))=0$ .",
"Choosing a generator of ${\\mathcal {L}}\\otimes k(X)$ yields an isomorphism $I^n(k(X))\\simeq I^n(k(X),{\\mathcal {L}}\\otimes k(X))$ and we can thus suppose that ${\\mathcal {L}}$ is trivial.",
"Consider the quotient group $\\overline{I}^n(k(X)):=I^n(k(X))/I^{n+1}(k(X))$ .",
"The affirmation of the Milnor conjecture yields an isomorphism $\\overline{I}^n(k(X))\\simeq H^n_{\\mathrm {Gal}}(k(X),\\mu _2^{\\otimes n})$ .",
"The latter is trivial since $\\mathrm {cd}_2(k(X))\\le r+d$ by [41].",
"It follows then from [3] that $I^n(k(X))=0$ .",
"We now prove a yet stronger vanishing statement for ${\\mathbf {I}}^{d+r}({\\mathcal {L}})$ .",
"Proposition 5.2 Let $X$ be a smooth affine scheme of dimension $d$ over a field $k$ with $\\mathrm {cd}_2(k)=r<\\infty $ and let ${\\mathcal {L}}$ be a line bundle on $X$ .",
"If $d\\ge 1$ , then $H^d_{\\operatorname{Nis}}(X,{\\mathbf {I}}^{j}({\\mathcal {L}}))=0$ for any $j\\ge d+r$ .",
"If $d\\ge 2$ then we have $H^{d-1}_{\\operatorname{Nis}}(X,{\\mathbf {I}}^{j}({\\mathcal {L}}))=0$ for any $j\\ge d+r$ .",
"We prove the vanishing result for cohomology computed in the Zariski topology since the Gersten resolution implies that Zariski and Nisnevich cohomology coincide.",
"By Proposition REF , we are reduced to the case $j=d+r$ .",
"The exact sequence of sheaves on the small Nisnevich site of $X$ [17] ${0[r] & {\\mathbf {I}}^{d+r+1}({\\mathcal {L}})[r] & {\\mathbf {I}}^{d+r}({\\mathcal {L}})[r] & \\overline{\\mathbf {I}}^{d+r}[r] & 0}$ and Proposition REF give an isomorphism of sheaves $\\mathbf {I}^{d+r}({\\mathcal {L}}) \\cong \\overline{\\mathbf {I}}^{d+r}$ .",
"Using the affirmation of Milnor's conjecture on quadratic forms [34] and Voevodsky's affirmation of the Milnor conjecture on the mod 2 norm residue homomorphism [47], there are isomorphisms $\\overline{\\mathbf {I}}^{d+r} \\cong {{\\mathbf {K}}}^{M}_{d+r}/2 \\cong \\mathcal {H}^{d+r}_{\\text{ét}}(\\mu _2^{d+r})$ , where $\\mathcal {H}^{i}_{\\text{ét}}(\\mu _2^{j})$ is the Nisnevich sheaf associated with the presheaf $U \\mapsto H^i_{\\text{ét}}(U,\\mu _2^{\\otimes j})$ .",
"To establish the result in the statement, we will show that $H^{i}(X,\\mathcal {H}^{d+r}_{\\text{ét}}(\\mu _2^{d+r})) = 0$ in the stated range.",
"To this end, consider the Bloch-Ogus spectral sequence [8] abutting to $H^*_{\\text{ét}}(X,\\mu _2^{\\otimes d+r})$ .",
"By the assumption on cohomological dimension, the lines resolving $\\mathcal {H}^{j}_{\\text{ét}}(\\mu _2^{\\otimes d+r})$ vanish for $j \\ge d+r+1$ (by the assumption on cohomological dimension).",
"The analysis of the spectral sequence therefore yields an isomorphism $H^{2d+r}_{\\text{ét}}(X,\\mu _2^{\\otimes d+r}) \\cong H^{d}_{\\operatorname{Nis}}(X,\\mathcal {H}^{d+r}_{\\text{ét}}(\\mu _2^{\\otimes d+r}))$ and a surjection $H^{2d+r-1}_{\\text{ét}}(X,\\mu _2^{\\otimes d+r}) \\twoheadrightarrow H^{d-1}_{\\operatorname{Nis}}(X,\\mathcal {H}^{d+r}_{\\text{ét}}(\\mu _2^{\\otimes d+r})).$ Now, since $X$ is affine, $H^i_{\\text{ét}}(X,\\mu _2^{\\otimes q})=0$ for $i\\ge d+r+1$ ([30] and ([30]; see also [30])).",
"Corollary 5.3 If $k$ is a quadratically closed field, $X$ is a smooth affine $k$ -scheme of dimension $d\\ge 2$ , and ${\\mathcal {L}}$ is a line bundle on $X$ , for any pair of integers $i,j\\ge d-1$ , there are isomorphisms $H^{i}_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^{\\mathrm {MW}}_j({\\mathcal {L}})) {\\stackrel{\\sim }{\\;\\longrightarrow \\;}}H^i_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_j).$ Use the long exact sequence in cohomology associated with the short exact sequence of sheaves $0 \\longrightarrow {\\mathbf {I}}^{j+1}({\\mathcal {L}}) \\longrightarrow {{\\mathbf {K}}}^{\\mathrm {MW}}_j({\\mathcal {L}}) \\longrightarrow {{\\mathbf {K}}}^M_j \\longrightarrow 0$ and Proposition REF .",
"Proposition 5.4 Let $X$ be a smooth variety of dimension $d$ over a field $k$ .",
"If there exists an integer $m > 0$ such that for any closed point $x\\in X$ the group $k(x)^\\times $ is $m$ -divisible, then $H^{d}_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_{d+1}/m) = 0$ .",
"The Gersten resolution for the sheaf ${{\\mathbf {K}}}^M_{d+1}/m$ gives an exact sequence of the form $\\bigoplus _{x \\in X^{(d)}}({{\\mathbf {K}}}^M_{d+1}/m)_{-d}(k(x)) \\longrightarrow H^d_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_{d+1}/m) \\longrightarrow 0.$ Furthermore $({{\\mathbf {K}}}^M_{d+1}/m)_{-d} = {{\\mathbf {K}}}^M_1/m$ , and ${{\\mathbf {K}}}^M_1/m(k(x)) = k(x)^\\times /(k(x)^\\times )^m = 0$ by assumption."
],
[
"Obstruction theory and classification results for vector bundles",
"In this section, we begin by reviewing aspects of obstruction theory involving the Postnikov tower in ${\\mathbb {A}}^1$ -homotopy theory.",
"We combine the results of the previous sections with obstruction theory for the Postnikov tower of $BGL_n$ to obtain information about vector bundles.",
"The main results of this section are Theorems REF and Corollary REF ."
],
[
"The Postnikov tower in ${\\mathbb {A}}^1$ -homotopy theory",
"If ${\\mathscr G}$ is a (Nisnevich) sheaf of groups, and ${\\mathbf {A}}$ is a (Nisnevich) sheaf of abelian groups on which $\\mathscr {G}$ acts, there is an induced action of ${\\mathscr G}$ on the Eilenberg-Mac Lane space $K({\\mathbf {A}},n)$ that fixes the base-point.",
"In that case, we set $K^{\\mathscr G}(\\mathbf {A},n) := E{\\mathscr G} \\times ^{\\mathscr G} K({\\mathbf {A}},n)$ .",
"The projection onto the first factor defines a morphism $K^{\\mathscr G}({\\mathbf {A}},n) \\rightarrow B{\\mathscr G}$ that is split by the inclusion of the base-point.",
"Just as simplicial homotopy classes of maps $[{\\mathscr X},K({\\mathbf {A}},n)]_{s}$ are in bijection with elements of $H^n_{\\operatorname{Nis}}({\\mathscr X},{\\mathbf {A}})$ , there is a corresponding classification theorem in this “twisted\" setting.",
"A map ${\\mathscr X} \\rightarrow K^{{\\mathscr G}}({\\mathbf {A}},n)$ gives, by composition, a morphism $\\mathscr {X} \\rightarrow B{\\mathscr G}$ , which yields a $\\mathscr {G}$ -torsor ${\\mathscr P} \\rightarrow \\mathscr {X}$ by pullback.",
"Then, the morphism of the previous sentence can be interpreted as a $\\mathscr {G}$ -equivariant map ${\\mathscr P} \\rightarrow K({\\mathbf {A}},n)$ , i.e., a $\\mathscr {G}$ -equivariant degree $n$ cohomology class on ${\\mathscr X}$ with coefficients in $\\mathbf {A}$ .",
"The following result summarizes the form of the Postnikov tower we will use; this result is collated from a collection of sources including [21], [33] and [31].",
"Theorem 6.1 If $({\\mathscr Y},y)$ is any pointed ${\\mathbb {A}}^1$ -connected space, then there are a sequence of pointed ${\\mathbb {A}}^1$ -connected spaces $({\\mathscr Y}^{(i)},y)$ , morphisms $p_i: {\\mathscr Y} \\rightarrow {\\mathscr Y}^{(i)}$ , and morphisms $f_i: {\\mathscr Y}^{(i+1)} \\rightarrow {\\mathscr Y}^{(i)}$ such that i) ${\\mathscr Y}^{(i)}$ has the property that $\\pi _j^{{\\mathbb {A}}^1}({\\mathscr Y}^{(i)}) = 0 $ for $j > i$ , ii) the morphism $p_i$ induces an isomorphism on ${\\mathbb {A}}^1$ -homotopy sheaves in degree $\\le i$ , iii) the morphism $f_i$ is an ${\\mathbb {A}}^1$ -fibration, and the ${\\mathbb {A}}^1$ -homotopy fiber of $f_i$ is a $K(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+1)$ , iv) the induced morphism ${\\mathscr Y} \\rightarrow \\operatorname{holim}_i {\\mathscr Y}^{(i)}$ is an ${\\mathbb {A}}^1$ -weak equivalence.",
"Furthermore, $f_i$ is a twisted ${\\mathbb {A}}^1$ -principal fibration, i.e., there is a unique (up to ${\\mathbb {A}}^1$ -homotopy) morphism $k_{i+1}: \\mathscr {Y}^{(i)} \\longrightarrow K^{\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y}),i+2)$ called a $k$ -invariant sitting in an ${\\mathbb {A}}^1$ -homotopy pullback square of the form ${\\mathscr {Y}^{(i+1)} [r][d] & B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y}) [d] \\\\\\mathscr {Y}^{(i)} [r]^-{k_{i+1}} & K^{\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y}),i+2),}$ where the action of $\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})$ on the higher ${\\mathbb {A}}^1$ -homotopy sheaves is the usual conjugation action induced by change of base-points.",
"Let $(\\mathscr {X},x)$ be a pointed space.",
"With notation as in Theorem REF , if $g^{(i)}: \\mathscr {X} \\rightarrow \\mathscr {Y}^{(i)}$ is a map, then $g^{(i)}$ lifts to a map $g^{(i+1)}:\\mathscr {X} \\rightarrow \\mathscr {Y}^{(i+1)}$ if and only if the composite $\\mathscr {X} \\stackrel{g^{(i)}}{\\longrightarrow } \\mathscr Y^{(i)} \\longrightarrow K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)$ lifts to a map $\\mathscr {X}\\rightarrow B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ .",
"Since each stage of the Postnikov tower is a twisted ${\\mathbb {A}}^1$ -principal fibration with fiber an Eilenberg-Mac Lane space, the set of lifts of a given $g^{(i)}$ admits a cohomological description.",
"The space $\\mathscr {Y}^{(0)}$ is ${\\mathbb {A}}^1$ -contractible by assumption, so any pointed map $\\mathscr {X} \\rightarrow \\mathscr {Y}^{(0)}$ is ${\\mathbb {A}}^1$ -homotopy equivalent to the constant map $\\mathscr {X} \\rightarrow y \\hookrightarrow \\mathscr {Y}^{(0)}$ .",
"There is no obstruction to lifting the constant map to the first stage of the Postnikov tower to obtain a map $\\xi : \\mathscr {X} \\rightarrow \\mathscr {Y}^{(1)} = B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ , and such a map yields a $\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})$ -torsor $\\mathscr {P}$ on $\\mathscr {X}$ .",
"We will describe lifts to higher stages of the Postnikov tower with $\\xi $ fixed.",
"Given a map $g^{(i)} : \\mathscr {X} \\rightarrow \\mathscr {Y}^{(i)}$ for which the obstruction to lifting vanishes, using the homotopy cartesian square that appears in Theorem REF , we can provide a description of the set of lifts of $g^{(i)}$ .",
"Indeed, Theorem REF guarantees that $\\mathscr {Y}^{(i+1)} \\rightarrow \\mathscr {Y}^{(i)}$ is an ${\\mathbb {A}}^1$ -fibration so any lift of $g^{(i)}$ can, after choosing appropriate fibrant models of the $\\mathscr {Y}^{(i)}$ , be assumed to be given by an actual morphism $\\mathscr {X} \\rightarrow \\mathscr {Y}^{(i+1)}$ .",
"Specifying a lift of the morphism $g^{(i)}$ is then equivalent to specifying a section of the map $\\mathscr {X} \\times _{g^{(i)},\\mathscr {Y}^{(i)},f_i} \\mathscr {Y}^{(i+1)} \\longrightarrow \\mathscr {X}.$ Since $\\mathscr {Y}^{(i+1)}$ is the homotopy fiber product of $\\mathscr {Y}^{(i)}$ and $B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ over $K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)$ , there is an ${\\mathbb {A}}^1$ -weak equivalence $\\mathscr {X} \\times _{g^{(i)},\\mathscr {Y}^{(i)},f_i} \\mathscr {Y}^{(i+1)} \\cong \\mathscr {X} \\times _{K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)} B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ (on the right, we mean homotopy fiber product).",
"Now, existence of a lift of $g^{(i)}$ is, as observed above, equivalent to the composite map $\\mathscr {X} \\rightarrow \\mathscr {Y}^{(i)} \\rightarrow K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)$ factoring through $B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ .",
"In fact, using the choices of the previous paragraph, the map $\\mathscr {X} \\rightarrow K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)$ appearing in the fiber product above factors through the map $\\xi : \\mathscr {X} \\rightarrow B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ and there is an induced ${\\mathbb {A}}^1$ -weak equivalence $\\mathscr {X} \\times _{K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)} B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y}) \\cong \\mathscr {X} \\times _{\\xi ,B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})} (B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y}) \\times _{K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)} B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})),$ where, again, both sides are homotopy fiber products.",
"The homotopy fiber product $B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y}) \\times _{K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)} B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ is precisely the space of fiberwise (simplicial) loops in $K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+2)$ , which is identified with $K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+1)$ (see, e.g., [37] for a related discussion).",
"Combining these equivalences, we see that the space of lifts of a given $g^{(i)}$ is a quotient of the space of ${\\mathbb {A}}^1$ -homotopy classes of sections of the projection map $\\mathscr {X} \\times _{\\xi ,B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})} K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}({\\mathscr Y}),i+1) \\longrightarrow \\mathscr {X},$ and this space of sections is precisely $H^{i+1}(\\mathscr {X},\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y})(\\xi ))$ .",
"Proposition 6.2 If $X$ is a smooth algebraic variety over a field $k$ of dimension $\\le d$ , $(\\mathcal {Y},y)$ is any pointed ${\\mathbb {A}}^1$ -connected space and $i\\ge 1$ , then the morphism $\\mathscr {Y} \\rightarrow \\mathscr {Y}^{(i)}$ induces a map $[X,\\mathscr {Y}]_{{\\mathbb {A}}^1} \\longrightarrow [X,\\mathscr {Y}^{(i)}]_{{\\mathbb {A}}^1}$ that is surjective if $i \\ge d-1$ , and bijective if $i \\ge d$ .",
"If moreover $H^d_{\\operatorname{Nis}}(X,\\pi _d^{{\\mathbb {A}}^1}(\\mathscr {Y})(\\xi )) = 0$ for any $\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})$ -torsor $\\xi $ obtained as the composite $[X,\\mathscr {Y}^{(i)}]_{{\\mathbb {A}}^1}\\rightarrow [X,\\mathscr {Y}^{(1)}]_{{\\mathbb {A}}^1}=[X,B\\pi _1^{{\\mathbb {A}}^1}(\\mathscr {Y})]_{{\\mathbb {A}}^1}$ , then the above map is bijective also for $i = d-1$ .",
"We use the Postnikov tower, which actually requires us to consider pointed maps.",
"However, by adjoining a disjoint base-point to $X$ and observing that there is a canonical bijection $[X_+,(\\mathscr {Z},z)]_{{\\mathbb {A}}^1} \\cong [X,\\mathscr {Z}]_{{\\mathbb {A}}^1}$ for any pointed space $\\mathscr {Z}$ , we can make statements about free homotopy classes.",
"Given an element of $[X,\\mathscr {Y}^{(i)}]_{{\\mathbb {A}}^1}$ , the obstruction to lifting it to $[X,\\mathscr {Y}^{(i+1)}]$ is given by the composite map $X \\longrightarrow \\mathscr {Y}^{(i)} \\longrightarrow K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y}),i+2).$ This composite is an element of $[X,K^{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y}),i+2)]_{{\\mathbb {A}}^1} \\cong H^{i+2}_{\\pi _1^{{\\mathbb {A}}^1}({\\mathscr Y})}(X,\\pi _{i+1}^{{\\mathbb {A}}^1}(\\mathscr {Y})),$ which vanishes for $i \\ge d-1$ because the cohomological dimension of $X$ is $\\le d$ .",
"Assuming this vanishing, we know that the set of lifts is a quotient of $H^d_{\\operatorname{Nis}}(X,\\pi _d^{{\\mathbb {A}}^1}(\\mathscr {Y})(\\xi ))$ .",
"The latter group vanishes if $i \\ge d$ , and the results follow immediately.",
"We now apply the discussion above in the case where $\\mathscr {Y} = BGL_n$ with its canonical basepoint.",
"We saw that $\\pi _1^{{\\mathbb {A}}^1}(BGL_n) \\cong {{\\mathbf {G}}_{m}}$ , independent of $n$ .",
"The space $B{{\\mathbf {G}}_{m}}$ has exactly one non-zero homotopy sheaf in degree 1, and this homotopy sheaf is isomorphic to ${{\\mathbf {G}}_{m}}$ .",
"The determinant map $GL_n \\rightarrow {{\\mathbf {G}}_{m}}$ yields a morphism $BGL_n \\rightarrow B{{\\mathbf {G}}_{m}}$ that allows us to identify $B{{\\mathbf {G}}_{m}}$ as the first stage of the ${\\mathbb {A}}^1$ -Postnikov tower of $BGL_n$ (see, e.g., [4] and the preceding discussion).",
"To describe the second stage of the ${\\mathbb {A}}^1$ -Postnikov tower of $BGL_n$ , we need to understand the action of $\\pi _1^{{\\mathbb {A}}^1}(BGL_n) = {{\\mathbf {G}}_{m}}$ on $\\pi _2^{{\\mathbb {A}}^1}(BGL_n)$ .",
"The homomorphism $SL_n \\rightarrow GL_n$ induces a map $BSL_n \\rightarrow BGL_n$ .",
"Since $BSL_n$ is ${\\mathbb {A}}^1$ -1-connected (this follows [31], but see, e.g., [2] for this statement), it follows that the ${\\mathbb {A}}^1$ -universal cover of $BGL_n$ has the ${\\mathbb {A}}^1$ -homotopy type of $BSL_n$ .",
"If $G$ is a sheaf of groups, and we write $EG$ for the Cech simplicial scheme associated with the structure map $G \\rightarrow \\operatorname{Spec}k$ , then $EG$ is simplicially contractible.",
"The inclusion $SL_n \\hookrightarrow GL_n$ determines maps $BSL_n = ESL_n/SL_n \\rightarrow EGL_n/SL_n$ , and $EGL_n/SL_n \\rightarrow EGL_n/GL_n=BGL_n$ .",
"The map $BSL_n \\rightarrow EGL_n/SL_n$ is a simplicial weak equivalence, while the map $EGL_n/SL_n \\rightarrow BGL_n$ is a ${{\\mathbf {G}}_{m}}$ -torsor and thus an ${\\mathbb {A}}^1$ -covering space by means of the identification $GL_n/SL_n {\\stackrel{{\\scriptscriptstyle {\\sim }}}{\\;\\rightarrow \\;}}{{\\mathbf {G}}_{m}}$ given by the determinant and [31].",
"In particular, $EGL_n/SL_n \\rightarrow BGL_n$ is a model of the ${\\mathbb {A}}^1$ -universal cover of $BGL_n$ by [31].",
"It follows that the action of ${{\\mathbf {G}}_{m}}$ on the higher ${\\mathbb {A}}^1$ -homotopy sheaves of $BGL_n$ is induced by “deck transformations\" from the standard ${{\\mathbf {G}}_{m}}$ -action on $EGL_n/SL_n$ .",
"There is a conjugation action of ${{\\mathbf {G}}_{m}}$ on $SL_n$ induced by the splitting $t \\mapsto diag(t,1,\\ldots ,1)$ of the determinant homomorphism.",
"This action determines ${{\\mathbf {G}}_{m}}$ -actions on $ESL_n$ and $EGL_n$ such that the map $ESL_n \\rightarrow EGL_n$ is ${{\\mathbf {G}}_{m}}$ -equivariant.",
"Thus, there is an ${{\\mathbf {G}}_{m}}$ -equivariant map $BSL_n \\rightarrow EGL_n/SL_n$ .",
"It is straightforward to check that the ${{\\mathbf {G}}_{m}}$ -action on $EGL_n/SL_n$ coming from the identification of $EGL_n/SL_n \\rightarrow BGL_n$ as a ${{\\mathbf {G}}_{m}}$ -torsor coincides with the action described in the previous paragraph.",
"In other words, the action of ${{\\mathbf {G}}_{m}}$ by “deck transformations\" on the higher ${\\mathbb {A}}^1$ -homotopy sheaves of $BGL_n$ comes from the conjugation action of ${{\\mathbf {G}}_{m}}$ on $SL_n$ .",
"When $n = 2$ , we can identify $SL_2 = Sp_2$ .",
"Combining Theorem REF and [31] we see that ${{\\mathbf {K}}}^{MW}_2 = \\mathbf {GW}^2_2$ .",
"Under these identifications, the conjugation action of ${{\\mathbf {G}}_{m}}$ on $SL_2$ induces, by Lemmas REF and REF , the multiplication action of ${{\\mathbf {K}}}^{MW}_0$ on ${{\\mathbf {K}}}^{MW}_2$ .",
"Given this description of the action, the next result provides an explicit description of maps to the second stage of the Postnikov tower of $BGL_2$ .",
"Proposition 6.3 For any smooth scheme $X$ , we have $[X,BGL_2^{(2)}]_{{\\mathbb {A}}^1} = [X,K^{{{\\mathbf {G}}_{m}}}({{\\mathbf {K}}}^{MW}_2,2)]_{{\\mathbb {A}}^1}$ , where the ${{\\mathbf {G}}_{m}}$ action on ${{\\mathbf {K}}}^{MW}_2$ is that described above, and an element of $[X,K^{{{\\mathbf {G}}_{m}}}({{\\mathbf {K}}}^{MW}_2,2)]_{{\\mathbb {A}}^1}$ corresponds to a pair $(\\xi ,\\alpha )$ , where $\\xi : X \\rightarrow B{{\\mathbf {G}}_{m}}$ corresponds to a line bundle ${\\mathcal {L}}$ on $X$ , and $\\alpha \\in H^2_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^{MW}_2({\\mathcal {L}}))$ .",
"The composite map $X \\rightarrow K^{{{\\mathbf {G}}_{m}}}({{\\mathbf {K}}}^{MW}_2,2) \\rightarrow B{{\\mathbf {G}}_{m}}$ yields a line bundle $\\xi : {\\mathcal {L}}\\rightarrow X$ on $X$ .",
"Write ${\\mathcal {L}}^{\\circ }$ for the complement of the zero section of ${\\mathcal {L}}$ .",
"The pullback of the universal ${{\\mathbf {G}}_{m}}$ -torsor along the map classifying $\\xi $ is ${\\mathcal {L}}^{\\circ }$ .",
"As explained above, the map $X \\rightarrow K^{{{\\mathbf {G}}_{m}}}({{\\mathbf {K}}}^{MW}_2,2)$ can then be identified as a morphism ${\\mathcal {L}}^{\\circ } \\rightarrow K({{\\mathbf {K}}}^{MW}_2,2)$ that is ${{\\mathbf {G}}_{m}}$ -equivariant.",
"Taking the product with the identity, one obtains a ${{\\mathbf {G}}_{m}}$ -equivariant map ${\\mathcal {L}}^{\\circ } \\rightarrow K({{\\mathbf {K}}}^{MW}_2,2) \\times {\\mathcal {L}}^{\\circ }$ that, after forming the quotient by the ${{\\mathbf {G}}_{m}}$ -action, determines a map $X \\rightarrow K({{\\mathbf {K}}}^{MW}_2,2) \\times ^{{{\\mathbf {G}}_{m}}} {\\mathcal {L}}^{\\circ }$ .",
"Now [31] provides a canonical isomorphism $K({{\\mathbf {K}}}^{MW}_2,2) \\times ^{{{\\mathbf {G}}_{m}}} {\\mathcal {L}}^{\\circ }\\cong K({{\\mathbf {K}}}^{MW}_2 \\times ^{{{\\mathbf {G}}_{m}}} {\\mathcal {L}}^{\\circ })$ .",
"As described just prior to the statement of the proposition, the action of ${{\\mathbf {G}}_{m}}$ on ${{\\mathbf {K}}}_2^{MW}$ is induced by the multiplication action of ${{\\mathbf {K}}}^{MW}_0$ .",
"It follows that the sheaf ${{\\mathbf {K}}}^{MW}_2 \\times ^{{{\\mathbf {G}}_{m}}} {\\mathcal {L}}^{\\circ }$ is precisely the sheaf ${{\\mathbf {K}}}^{MW}_2({\\mathcal {L}})$ described at the beginning of Section .",
"Remark 6.4 For $n \\ge 3$ , the induced action of ${{\\mathbf {G}}_{m}}$ on $\\pi _2^{{\\mathbb {A}}^1}(BGL_n) = {{\\mathbf {K}}}^M_2$ can be seen to be trivial by a more straightforward argument.",
"In that case, the action of ${{\\mathbf {G}}_{m}}$ factors through $\\operatorname{Hom}({{\\mathbf {K}}}^M_2,{{\\mathbf {K}}}^M_2)$ , which can be identified with the constant sheaf ${\\mathbb {Z}}$ .",
"Over any algebraically closed extension $L/k$ , this action is necessarily trivial because ${{\\mathbf {G}}_{m}}(L)$ divisible.",
"From the discussion of the previous section, the ${{\\mathbf {G}}_{m}}$ -action on $\\pi _3^{{\\mathbb {A}}^1}(BGL_2)$ is that described in Corollary REF .",
"In particular, recall that the ${{\\mathbf {G}}_{m}}$ -action on $\\mathbf {T}^{\\prime }_4$ is trivial, while the ${{\\mathbf {G}}_{m}}$ -action on $\\mathbf {GW}^2_3$ is the standard action induced by multiplication by $\\mathbf {GW}^0_0$ .",
"Repeating the argument of Proposition REF , we deduce the following result.",
"Proposition 6.5 If ${\\mathcal {L}}\\rightarrow X$ is a line bundle whose associated ${{\\mathbf {G}}_{m}}$ -torsor is classified by an element $\\xi \\in [X,B{{\\mathbf {G}}_{m}}]_{{\\mathbb {A}}^1}$ , then there are canonical isomorphisms $\\begin{split}H^3_{\\operatorname{Nis}}(X,\\mathbf {T}^{\\prime }_4(\\xi )) &\\cong H^3_{{{\\mathbf {G}}_{m}}}({\\mathcal {L}}^{\\circ },\\mathbf {T}^{\\prime }_4) \\cong H^3_{\\operatorname{Nis}}(X,\\mathbf {T}^{\\prime }_4), \\text{ and } \\\\ H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}^2_3(\\xi )) &\\cong H^3_{{{\\mathbf {G}}_{m}}}({\\mathcal {L}}^{\\circ },\\mathbf {GW}^2_3) \\cong H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}^2_3({\\mathcal {L}})).\\end{split}$ Theorem 6.6 If $k$ is an algebraically closed field having characteristic unequal to 2, and $X$ is a smooth affine 3-fold, the map sending a vector bundle of rank 2 to its Chern classes determines a bijection between the pointed set of isomorphism classes of rank 2 vector bundles on $X$ and $CH^1(X) \\times CH^2(X)$ .",
"By Proposition REF , we know that $[X,BGL_2^{(2)}]$ consists of pairs $({\\mathcal {L}},\\alpha )$ consisting of a line bundle ${\\mathcal {L}}\\in Pic(X)$ and an element $\\alpha \\in H^2_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^{MW}_2({\\mathcal {L}}))$ .",
"By Corollary REF , the canonical morphism $H^2_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^{MW}_2({\\mathcal {L}})) \\longrightarrow H^2_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_2)$ is an isomorphism.",
"The map $[X,BGL_2]_{{\\mathbb {A}}^1} \\rightarrow [X,B{{\\mathbf {G}}_{m}}]_{{\\mathbb {A}}^1}$ induced by the first stage of the Postnikov tower for $BGL_2$ sends the class of a vector bundle to its determinant, i.e., its first Chern class.",
"To identify the class $\\alpha $ , observe that, by [31], the composite map $BGL_2 \\rightarrow K^{{{\\mathbf {G}}_{m}}}({{\\mathbf {K}}}^{MW}_2,2) \\rightarrow K({{\\mathbf {K}}}^M_2,2)$ is given by the composition $BGL_2 \\rightarrow BGL_{\\infty } \\stackrel{c_2}{\\rightarrow }K({{\\mathbf {K}}}^M_2,2)$ , where $c_2$ is the second Chern class, considered as a morphism in $\\mathscr {H}_{\\bullet }({k})$ .",
"Therefore, the elements described at the end of the previous paragraph are precisely the first and second Chern classes.",
"Applying Proposition REF to $\\mathscr {Y} = BGL_2$ , we see that the map $[X,BGL_2]_{{\\mathbb {A}}^1} \\longrightarrow [X,BGL_2^{(2)}]_{{\\mathbb {A}}^1}$ is surjective.",
"To conclude the proof, it suffices to establish that the stronger hypothesis of Proposition REF is satisfied, i.e., that $H^3_{\\operatorname{Nis}}(X,\\pi _3^{{\\mathbb {A}}^1}(BGL_2)(\\xi )) = 0$ for $\\xi $ the ${{\\mathbf {G}}_{m}}$ -torsor corresponding to the line bundle ${\\mathcal {L}}$ above.",
"Since $\\pi _3^{{\\mathbb {A}}^1}(BGL_2) \\cong \\pi _2^{{\\mathbb {A}}^1}(SL_2)$ was computed in Theorem REF , we know that $\\pi _2^{{\\mathbb {A}}^1}(SL_2)$ is an extension of $\\mathbf {GW}_3^2$ by $\\mathbf {T}^{\\prime }_4$ which is itself an extension of $\\mathbf {S}^{\\prime }_4$ by a quotient of $\\mathbf {I}^5$ .",
"Furthermore, $\\mathbf {S}^{\\prime }_4$ admits an epimorphism from ${{\\mathbf {K}}}^M_4/12$ .",
"From this description and Proposition REF , the group $H^3_{\\operatorname{Nis}}(X,\\pi _3^{{\\mathbb {A}}^1}(BGL_2)(\\xi ))$ fits into an exact sequence of the form $H^3_{\\operatorname{Nis}}(X,\\mathbf {T}^{\\prime }_4) \\longrightarrow H^3_{\\operatorname{Nis}}(X,\\pi _3^{{\\mathbb {A}}^1}(BGL_2)(\\xi )) \\longrightarrow H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}^2_3({\\mathcal {L}})).$ It suffices then to prove that the groups $H^3_{\\operatorname{Nis}}(X,{\\mathbf {T}}^\\prime _4)$ and $H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}}))$ vanish for any ${\\mathcal {L}}\\in Pic(X)$ .",
"We now use the assumption that $k$ is algebraically closed.",
"By Proposition REF , we know that the sheaf $\\mathbf {I}^5$ restricted to $X$ is just the trivial sheaf so the sheaves $\\mathbf {T}^{\\prime }_4$ and $\\mathbf {S}^{\\prime }_4$ restricted to $X$ are isomorphic.",
"For reasons of cohomological dimension, the epimorphism ${{\\mathbf {K}}}^M_4/12 \\rightarrow \\mathbf {S}^{\\prime }_4$ induces a surjective map $H^3_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_4/12) \\rightarrow H^3_{\\operatorname{Nis}}(X,{\\mathbf {S}}^\\prime _4)$ , and $H^3_{\\operatorname{Nis}}(X,{{\\mathbf {K}}}^M_4/12)$ vanishes by Proposition REF .",
"Thus, $H^3_{\\operatorname{Nis}}(X,{\\mathbf {T}}^\\prime _4)=0$ .",
"Now, by Proposition REF , $H^3_{\\operatorname{Nis}}(X,\\mathbf {GW}_3^2({\\mathcal {L}}))$ is a quotient of $Ch^3(X)$ .",
"Because $X$ is affine of dimension 3, $CH^3(X)$ is divisible, and thus $Ch^3(X)$ is trivial (in fact, $CH^3(X)$ is uniquely divisible by [42]).",
"Remark 6.7 With more work, it is possible to reprove the result [29] of Mohan Kumar and Murthy for fields having characteristic unequal to 2.",
"More precisely, given a smooth affine threefold over an algebraically closed field having characteristic unequal 2, and given arbitrary classes $(c_1,c_2,c_3) \\in \\prod _{i=1}^3 CH^i(X)$ there exists a unique rank 3 vector bundle on $X$ with these Chern classes.",
"One uses obstruction theory in the same way as above.",
"In this case, the stronger hypothesis of Proposition REF for $\\mathscr {Y} = BGL_3$ is not satisfied for $d = 3$ .",
"Nevertheless, it is still possible to prove that lifts from the second stage of the Postnikov tower to the third stage of the Postnikov tower are parameterized as a set by $CH^3(X)$ ; this involves showing that the action of $\\pi _1^{{\\mathbb {A}}^1}(BGL_3)$ on $\\pi _i^{{\\mathbb {A}}^1}(BGL_3)$ is trivial for $i = 2,3$ (the case $i = 2$ is Remark REF ).",
"Since stably isomorphic vector bundles have equal Chern classes, the following statement, which is a strengthening of [15], is an immediate consequence of Theorem REF .",
"Corollary 6.8 If $X$ is a smooth affine 3-fold over an algebraically closed field $k$ having characteristic unequal to 2, and if $E$ and $E^\\prime $ are a pair of stably isomorphic rank 2 vector bundles over $X$ , then $E$ and $E^\\prime $ are isomorphic."
],
[
"Complex realization and ${\\mathbb {A}}^1$ -homotopy sheaves",
"Recall that if $k$ is a field that admits a complex embedding, then the assignment $X \\mapsto X({\\mathbb {C}})$ sending a smooth $k$ -scheme $X$ to $X({\\mathbb {C}})$ equipped with its usual structure of a complex manifold can be extended to a functor $\\mathscr {H}_{\\bullet }({k}) \\rightarrow \\mathscr {H}_{\\bullet }$ , where $\\mathscr {H}_{\\bullet }$ is the usual homotopy category of topological spaces [33].",
"In particular, this functor induces homomorphisms $\\pi _{i,j}^{{\\mathbb {A}}^1}(\\mathcal {X},x)({\\mathbb {C}}) \\longrightarrow \\pi _{i+j}(\\mathcal {X}({\\mathbb {C}}),x).$ By a result of Morel [31] (see [2] for a statement in the form we use), the sheaf appearing on the left hand side can be computed in terms of contractions of $\\pi _i^{{\\mathbb {A}}^1}(\\mathcal {X},x)$ .",
"We refer the reader to [2] for a convenient summary of other facts about contractions we will use here.",
"The goal of this section is to study this homomorphism for $\\mathcal {X} = Sp_{2n}$ ; the main results are Theorems REF and REF .",
"Along the way, we prove Lemmas REF and REF , which, in particular, establish non-triviality of the sheaf $\\mathbf {T}^{\\prime }_{2n}$ that appears in Theorem REF .",
"The results of this section are independent of those of Section ."
],
[
"Further computations of contracted homotopy sheaves",
"More precise statements regarding the structure of the sheaf ${\\mathbf {T}}^{\\prime }_{2n+2}$ of Theorem REF can be made after repeated contraction.",
"The following results show that the structure of the sheaf ${\\mathbf {T}}^{\\prime }_{2n+2}$ depends on the parity of $n$ .",
"We refer the reader to [2] for a convenient summary of other facts about contractions we will use here.",
"Lemma 7.1 If $n$ is an even natural number, then the morphism of sheaves ${{\\mathbf {K}}}^M_{2n+2}/(2(2n+1)!)",
"\\longrightarrow {\\mathbf {S}}^{\\prime }_{2n+2}$ induces an isomorphism ${{\\mathbf {K}}}^M_{2}/(2n+1)!",
"\\rightarrow (\\mathbf {S}^{\\prime }_{2n+2})_{-2n}$ .",
"Recall from Section that there is a commutative diagram of the form ${{{\\mathbf {K}}}_{2n+2}^{Sp}[r]^-{\\varphi _{2n+2}}[d]_-{f_{2n+2,2}} & {{\\mathbf {K}}}_{2n+2}^{MW}[d] \\\\{{\\mathbf {K}}}_{2n+2}^Q[r]_-{\\psi _{2n+2}} & {{\\mathbf {K}}}_{2n+2}^M.",
"}$ It follows that the sheaf ${\\mathbf {S}}^{\\prime }_{2n+2}$ is the cokernel of the composite map ${{{\\mathbf {K}}}^{Sp}_{2n+2}=\\mathbf {GW}_{2n+2}^2 [r]^-{f_{2n+2,2}} & {{\\mathbf {K}}}^Q_{2n+2}[r] & {{\\mathbf {K}}}^M_{2n+2}.",
"}$ Contracting $2n$ times and using Proposition REF , we obtain a composite $@C=3em{\\mathbf {GW}_2^{2-2n} [r]^-{f_{2,2-2n}} & {{\\mathbf {K}}}^Q_2[r] & {{\\mathbf {K}}}^M_2}$ whose cokernel is $({\\mathbf {S}}^{\\prime }_{2n+2})_{-2n}$ .",
"We know from [2] that the cokernel of the morphism ${{\\mathbf {K}}}^Q_2\\rightarrow {{\\mathbf {K}}}^M_2$ is precisely ${{\\mathbf {K}}}^M_2/((2n+1)!",
")$ and it suffices to show that $f_{2,2-2n}$ is onto to conclude.",
"Since $n$ is even, we can identify $\\mathbf {GW}_2^{2-2n}=\\mathbf {GW}_2^2$ and the forgetful map $f_{2,2}$ is the natural epimorphism $\\mathbf {GW}^2_2 \\cong {{\\mathbf {K}}}^{MW}_2\\rightarrow {{\\mathbf {K}}}^M_2$ .",
"Lemma 7.2 If $n$ is an odd natural number, then the morphism of sheaves ${{\\mathbf {K}}}^M_{2n+2}/(2(2n+1)!",
")\\rightarrow {\\mathbf {S}}^{\\prime }_{2n+2}$ induces an isomorphism after $j$ -fold contraction for any $j \\ge 2n+1$ .",
"Arguing as in the previous lemma, we obtain a composite morphism $@C=3em{\\mathbf {GW}_1^{1-2n} [r]^-{f_{1,1-2n}} & {{\\mathbf {K}}}^Q_1[r] & {{\\mathbf {K}}}^M_1}$ whose cokernel is $({\\mathbf {S}}^{\\prime }_{2n+2})_{-2n-1}$ .",
"The cokernel of ${{\\mathbf {K}}}^Q_1\\rightarrow {{\\mathbf {K}}}^M_1$ is ${{\\mathbf {K}}}_1^M/((2n+1)!",
")$ by [2] and it remains to show that the image of $\\mathbf {GW}_1^{1-2n}\\rightarrow {{\\mathbf {K}}}^Q_1$ is $2{{\\mathbf {K}}}^Q_1$ .",
"Since $n$ is odd, we can identify $\\mathbf {GW}_1^{1-2n}=\\mathbf {GW}_1^3$ by Proposition REF .",
"Combining [18] and Lemma REF yields the required statement regarding the image.",
"Lemma 7.3 If $n$ is an odd natural number, there is an isomorphism $\\mathbf {D}_{2n+3}\\cong \\mathbf {W}$ .",
"In particular, the sheaf $\\mathbf {D}_{2n+3}$ is non-trivial.",
"Consider the stabilization sequence $\\cdots \\longrightarrow {{\\mathbf {K}}}^{Sp}_{2n+2} \\longrightarrow {{\\mathbf {K}}}^{MW}_{2n+2} \\longrightarrow \\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}) \\longrightarrow {{\\mathbf {K}}}^{Sp}_{2n+1} \\longrightarrow 0.$ Since $n$ is odd, $({{\\mathbf {K}}}^{Sp}_{2n+2})_{-(2n+3)} = (\\mathbf {GW}^{2}_{2n+2})_{-(2n+3)} \\cong (\\mathbf {GW}^2_0)_{-1} \\cong (\\mathbb {Z})_{-1} = 0$ by Proposition REF .",
"Similarly, using [19], $({{\\mathbf {K}}}^{Sp}_{2n+1})_{-(2n+3)} = (\\mathbf {GW}^2_{2n+1})_{-(2n+3)} \\cong (\\mathbf {GW}^3_0)_{-2} \\cong ({\\mathbb {Z}}/2)_{-2} = 0$ .",
"Furthermore, $({{\\mathbf {K}}}^{MW}_{2n+2})_{-(2n+3)} = {{\\mathbf {K}}}^{MW}_{-1} \\cong \\mathbf {W}$ by [2].",
"Thus, one concludes that $(\\pi _{2n}^{{\\mathbb {A}}^1}(Sp_{2n}))_{-(2n+3)} \\cong \\mathbf {W}$ .",
"By Theorem REF , and the discussion of the previous paragraph, it follows that $(\\mathbf {T}^{\\prime }_{2n+2})_{-(2n+3)}\\cong \\mathbf {W}$ .",
"Now, since $(\\mathbf {S}^{\\prime }_{2n+2})_{-(2n+3)} \\cong ({{\\mathbf {K}}}^M_{2n+2}/(2(2n+1)!",
"))_{-(2n+3)}$ by Lemma REF , it follows from [2] that $(\\mathbf {S}^{\\prime }_{2n+2})_{-(2n+3)} = 0$ .",
"Therefore, $(\\mathbf {D}_{2n+3})_{-(2n+3)} \\cong \\mathbf {W}$ as well.",
"Theorem 7.4 The homomorphisms $\\begin{split}\\pi _{2,3}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}}) &\\longrightarrow \\pi _5(SL_2({\\mathbb {C}})) \\cong \\pi _5(SU(2)) = {\\mathbb {Z}}/2, \\text{ and } \\\\\\pi _{2,4}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}}) &\\longrightarrow \\pi _6(SL_2({\\mathbb {C}})) \\cong \\pi _6(SU(2)) = {\\mathbb {Z}}/12\\end{split}$ are isomorphisms.",
"Recall the description of $\\pi _2(SL_2)$ from Theorem REF : there is an exact sequence of the form: $\\mathbf {I}^5 \\longrightarrow \\mathbf {T}_4^{\\prime } \\longrightarrow \\mathbf {S}_4^{\\prime } \\longrightarrow 0.$ Contracting this sequence 4 times, evaluating on ${\\mathbb {C}}$ (using the fact that $\\mathbf {I}({\\mathbb {C}}) = 0$ ), and using Lemma REF yields an isomorphism ${\\mathbb {Z}}/12 {\\stackrel{{\\scriptscriptstyle {\\sim }}}{\\;\\rightarrow \\;}}(\\mathbf {S}_4^{\\prime })_{-4}({\\mathbb {C}})$ .",
"We observed that $({{\\mathbf {K}}}^{Sp}_3)_{-4} = 0$ in the proof of Lemma REF .",
"Thus, there is an isomorphism ${\\mathbb {Z}}/12 {\\stackrel{{\\scriptscriptstyle {\\sim }}}{\\;\\rightarrow \\;}}\\pi _{2,4}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}})$ .",
"By [10] one knows that the classifying map of the $Sp_2$ -bundle $Sp_{4}/Sp_2 \\rightarrow BSp_2$ provides a generator of $\\pi _6(S^3) \\cong \\pi _7(BSp_2({\\mathbb {C}}))$ .",
"The computation of $\\pi _{2}^{{\\mathbb {A}}^1}(SL_2)$ was achieved using the ${\\mathbb {A}}^1$ -fiber sequence $Sp_2 \\rightarrow Sp_4 \\rightarrow Sp_4/Sp_2$ , and the isomorphism ${\\mathbb {Z}}/12 \\rightarrow \\pi _{2,4}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}})$ is induced by the connecting homomorphism of the associated long exact sequence in ${\\mathbb {A}}^1$ -homotopy sheaves.",
"Since the fiber sequence $Sp_2 \\rightarrow Sp_4 \\rightarrow Sp_4/Sp_2$ is mapped under complex realization to one homotopy equivalent to that considered by Borel-Serre, the result follows.",
"For the first isomorphism, recall first that ${{\\mathbf {K}}}^M_1/12({\\mathbb {C}}) = 0$ .",
"Then, using the fact that ${{\\mathbf {K}}}^{Sp}_3 = \\mathbf {GW}^2_3$ observe that $(\\mathbf {GW}^2_3)_{-3} = (\\mathbf {GW}^{0}_1)_{-1} = {\\mathbb {Z}}/2$ (again, use Lemma [19] and Proposition REF ).",
"Now, by Lemma REF , the fact that ${{\\mathbf {K}}}^M_1/12({\\mathbb {C}}) = 0$ , the fact that $\\mathbf {I}^2({\\mathbb {C}}) = 0$ , and the fact that $({{\\mathbf {K}}}^{Sp}_{3})_{-3} = {\\mathbb {Z}}/2$ , we see that $\\pi _{2,3}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}}) \\cong {\\mathbb {Z}}/2$ .",
"Thus, complex realization gives a map $\\pi _{2,3}^{{\\mathbb {A}}^1}(SL_2)({\\mathbb {C}}) = {\\mathbb {Z}}/2 \\rightarrow {\\mathbb {Z}}/2$ .",
"The computation of [52] shows that (see the proof of [26] for more details) the generator of $\\pi _5(S^3)$ is obtained as follows: start with the Hopf map $\\eta _{{\\mathbb {C}}}: S^3 \\rightarrow S^2$ and consider the composition $\\Sigma \\eta _{{\\mathbb {C}}} \\circ \\Sigma ^2 \\eta _{{\\mathbb {C}}}$ .",
"Now, there is the algebro-geometric Hopf map $\\eta : {\\mathbb {A}}^2 \\setminus 0 \\rightarrow {\\mathbb {P}}^1$ (see [31]), and taking the ${{\\mathbf {G}}_{m}}$ and ${\\mathbb {P}}^1$ -suspensions of this map we obtain: $\\begin{split}\\Sigma _{{{\\mathbf {G}}_{m}}} \\eta &: {{\\mathbb {P}}^1}^{\\wedge 2} \\longrightarrow {\\mathbb {A}}^2 \\setminus 0 \\\\\\Sigma _{{\\mathbb {P}}^1} \\eta &: {\\mathbb {A}}^3 \\setminus 0 \\longrightarrow {{\\mathbb {P}}^1}^{\\wedge 2}.\\end{split}$ The composite of these two maps has complex realization the generator of the $\\pi _5(S^3)$ since the complex realization of $\\eta $ is the usual Hopf map $\\eta _{{\\mathbb {C}}}$ .",
"Theorem 7.5 The homomorphism $\\pi _{2n,2n+2}^{{\\mathbb {A}}^1}(Sp_{2n})({\\mathbb {C}}) \\longrightarrow \\pi _{4n+2}(Sp_{2n}({\\mathbb {C}}))$ induced by complex realization is an isomorphism.",
"The complex realization of $Sp_{2n}$ is the group $Sp_{2n}({\\mathbb {C}})$ , which is homotopy equivalent to its maximal compact subgroup (denoted $Sp_n$ in [22]).",
"It follows from the table on [22] that $\\pi _{4n+2}(Sp_{2n}({\\mathbb {C}}))$ is ${\\mathbb {Z}}/(2n+1)!$ if $n$ is even and ${\\mathbb {Z}}/(2(2n+1)!",
")$ if $n$ is odd.",
"There is a canonical morphism $\\pi _{2n,2n+2}^{{\\mathbb {A}}^1}(Sp_{2n})({\\mathbb {C}}) \\rightarrow \\pi _{4n+2}(Sp_{2n}({\\mathbb {C}}))$ .",
"Since $\\mathbf {W}({\\mathbb {C}}) = {\\mathbb {Z}}/2$ , in view of Lemmas REF and REF , it suffices to prove that in either case we can lift a generator.",
"In each case, the generator is the image of a generator of $\\pi _{2n+1,2n+2}^{{\\mathbb {A}}^1}({\\mathbb {A}}^{2n+2} \\setminus 0)({\\mathbb {C}}) \\cong ({{\\mathbf {K}}}^{MW}_{2n+2})_{-2n-2}({\\mathbb {C}}) \\cong \\mathbf {GW}({\\mathbb {C}}) = {\\mathbb {Z}}$ .",
"Since the generator of this group is mapped to the generator of $\\pi _{4n+3}(S^{4n+3})$ under complex realization, the result follows by comparison with the proofs in [22]."
]
] | 1204.0770 |
[
[
"Heavy quarkonium 2S states in light-front quark model"
],
[
"Abstract We study the charmonium 2S states $\\psi'$ and $\\eta_c'$, and the bottomonium 2S states $\\Upsilon'$ and $\\eta_b'$, using the light-front quark model and the 2S state wave function of harmonic oscillator as the approximation of the 2S quarkonium wave function.",
"The decay constants, transition form factors and masses of these mesons are calculated and compared with experimental data.",
"Predictions of quantities such as Br$(\\psi' \\to \\gamma \\eta_c')$ are made.",
"The 2S wave function may help us learn more about the structure of these heavy quarkonia."
],
[
"Introduction",
"Charmonium physics has long been an interesting issue as it is related with both the perturbative and non-perturbative QCD.",
"Charmonia not only provide us with the opportunity to investigate the interactions between the constituent quarks and the structure of quarkonia, but also the chance to learn and understand the QCD dynamics better.",
"As exited states of charmonia, $\\psi ^{\\prime }$ and $\\eta _c^{\\prime }$ have been studied by many authors.",
"The decay widths of $\\psi ^{\\prime } \\rightarrow e^+ e^-$ and $\\eta _c^{\\prime } \\rightarrow 2\\gamma $ were calculated with both relativistic and QCD radiative corrections [1] and the result $\\psi ^{\\prime } \\rightarrow e^+ e^-$ is in agreement with experimental data.",
"The nonrelativistic potential model [2] and the Godfrey and Isgur (GI) model [3], [4] have achieved much success, but their predictions of the decay widths of $\\psi ^{\\prime } \\rightarrow \\gamma \\eta _c~(\\gamma \\eta _c^{\\prime })$ are larger than experimental data.",
"The lattice QCD result [5], [6] of $J/\\psi \\rightarrow \\gamma \\eta _c$ is consistent with experimental data, but the result of $\\psi ^{\\prime } \\rightarrow \\gamma \\eta _c^{\\prime }$ has too large uncertainties.",
"The intermediate meson loop contribution to the decays $\\psi ^{\\prime } \\rightarrow \\gamma \\eta _c~(\\gamma \\eta _c^{\\prime })$ was investigated recently [7], [8], and the results are closer to experimental data.",
"We also investigate these decays, using light-front formalism and the harmonic oscillator wave functions as the approximate wave functions of the 1S and 2S quarkonia.",
"In fact, there are still some puzzles concerning $\\psi ^{\\prime }$ , such as the well-known “$\\rho \\pi $ puzzle\" [9], [10], [11], [12], and the recent unanticipated small experimental value of Br$(\\psi ^{\\prime }\\rightarrow \\gamma \\eta ) $ /Br$(\\psi ^{\\prime } \\rightarrow \\gamma \\eta ^{\\prime })$ [13], [14].",
"For the “$\\rho \\pi $ puzzle\", Ref.",
"[15] suggested the explanation that the $\\psi ^{\\prime } \\rho \\pi $ coupling is suppressed due to the mismatch between the nodeless wave function of the $\\bar{c}c$ in the $|u\\bar{d}\\bar{c}c\\rangle $ Fock state of $\\rho $ and the one-node 2S $\\bar{c}c$ wave function of $\\psi ^{\\prime }$ , and our postulation of the 2S wave function of $\\psi ^{\\prime }$ may be able to offer a numerical realization for this explanation.",
"The BES and CLEO collaborations have conducted many experimental measurements on 1S and 2S charmonia, and the decay mode $\\psi ^{\\prime } \\rightarrow \\gamma \\eta _c^{\\prime }$ is being studied by BES-III.",
"It is then important to learn carefully about the structures and decay mechanisms of the 2S charmonia.",
"The 2S bottomonia have been studied by some experiments, but many data about them are still not available, such as the mass and decay data of $\\eta _b^{\\prime }$ [16].",
"In Ref.",
"[1], the decay modes $\\eta _b^{\\prime } \\rightarrow 2\\gamma $ and $\\Upsilon ^{\\prime } \\rightarrow e^+ e^-$ were studied by considering both relativistic and QCD radiative corrections, and predictions were made.",
"With the same method of studying the 2S charmonia in the light-front quark model, we can study more decay modes, and calculate the masses of these bottomonia in this paper.",
"From an experimental viewpoint, a large amount of bottomonia and their excited states could be produced at the forthcoming LHC or by the Belle experiment in the near future, and they could provide important tests of different predictions.",
"Moreover, heavy quarkonia, especially charmonia and bottomonia, act as improtant diagnostic tools to probe the properties of the background QCD matter, such as of the formation of the quark-gluon plasma (QGP), in heavy ion collisions at RHIC and LHC [17].",
"As has been pointed in Ref.",
"[17], the study of heavy-quarkonium suppression at RHIC energy, which might be a signature for the QGP formation, calls for the knowledge of the light-front wave functions of the quarkonia: $f(x_\\perp ,x_\\perp ^\\prime ,\\tau _0)=\\varphi (x_\\perp )\\varphi ^*(x_\\perp ^\\prime )$ , where $\\varphi (x_\\perp )$ can be taken as the Fourier transform of our light-front momentum space wave functions for the 1S or 2S quarkonia (Eqs.",
"(REF ) and ()), and $f(x_\\perp ,x_\\perp ^\\prime ,\\tau _0)$ are the essential quantities to calculate the transverse momentum distribution of quarks.",
"Thus our wave functions can not only help us understand the structure of heavy quarkonia themselves, but also be used as inputs for other physical studies.",
"This paper is organized as follows.",
"In Sec.",
", we describe the light-front quark model and the 2S state wave function for the quarkonia.",
"In Sec.",
", we present our numerical results of the decay constants, form factors and masses of these charmonia and bottomonia, and compare them with experimental data.",
"A brief summary is given in Sec.",
"."
],
[
" Model description",
"Heavy quarkonia have been studied by non-relativistic treatments [18], [19], [20], but in some occasions related to non-perturbative scales, they have to use model dependent methods.",
"And as the virtual photon momentum $Q^2$ increases, the relativistic effects become important.",
"So it is useful to study quarkonia in a relativistic treatment.",
"Several powerful non-perturbative tools have been developed to study the structure and decays of mesons, such as the QCD sum-rule technique and the lattice gauge theory.",
"The light-front quark model is also an important model to do such studies [21], [22], [23], and it has a number of salient features.",
"Light-front quark model includes some important relativistic effects that are neglected in the traditional constituent quark model, and the vacuum in the light-cone coordinate is simple because the Fock vacuum is the exact eigenstate of the full Hamiltoian.",
"Light-front quark model has been successfully applied in many investigation of hadron structures [31], [32], [33], [34], [35], [36], [37], [38], [24], [25], [26], [27], [28], [29], [30].",
"In the light-front quark model, the states of quarkonia can be described by the Fock state expansion $|M\\rangle &=& \\sum |q\\bar{q}\\rangle \\psi _{q\\bar{q}}+ \\sum |q\\bar{q}g\\rangle \\psi _{q\\bar{q}g} + \\cdots ,$ and to simplify the problem, we adopt the lowest order of the above expansions and take only the quark-antiquark valence states of the mesons into consideration.",
"The quarkonium wave function in light-front formalism is [21], [22], [39] $|M (P^+, \\mathbf {P}_\\perp , S_z) \\rangle &=& \\int \\frac{\\mathrm {d} x \\mathrm {d}^2\\mathbf {k}_{\\perp }}{\\sqrt{x(1-x)}16\\pi ^3}\\nonumber \\\\&&\\cdot \\phi (x,\\mathbf {k}_{\\perp })\\chi _M^{S_z}(x,\\mathbf {k}_{\\perp },\\lambda _1,\\lambda _2),$ with the momentum of the struck quark being $(xP^+, [m^2+(x\\mathbf {P}_\\perp +\\mathbf {k}_{\\perp })^2]/(xP^+), x\\mathbf {P}_\\perp +\\mathbf {k}_{\\perp })$ , and $\\lambda _i$ being the helicity of the $i$ -th constituent quark.",
"$\\phi (x,\\mathbf {k}_{\\perp })$ is the radial wave function, and $\\chi _M^{S_z}(x,\\mathbf {k}_{\\perp }, \\lambda _1, \\lambda _2)$ is the light-front spin wave function, which is related to the instant-form spin wave function by the Melosh-Wigner rotation [40], [41], [42], [43], [24], [25], [26], [27], [28], [29], [30] $\\left\\lbrace \\begin{array}{lll}\\chi _i^\\uparrow (T) &=& w_i[(k_i^+ +m_i)\\chi _i^\\uparrow (F)-k_i^R\\chi _i^\\downarrow (F)],\\\\\\chi _i^\\downarrow (T)&=& w_i[(k_i^+ +m_i)\\chi _i^\\downarrow (F)+k_i^L\\chi _i^\\uparrow (F)],\\end{array}\\right.$ where $w_i=1/\\sqrt{2k_i^+ (k^0+m_i)}$ , $k^{R,L}=k^1\\pm k^2$ , $k^+=k^0+k^3=x \\mathcal {M}$ , $m_i$ is the mass of the constituent quark, and the invariant mass of the composite system $\\mathcal {M}\\equiv \\sqrt{(\\mathbf {k_\\perp }^2+m_1^2)/x+(\\mathbf {k_\\perp }^2+m_2^2)/(1-x)}$ .",
"The Melosh-Wigner rotation is an important ingredient of light-front quark model and plays an essential role in explaining the “proton spin puzzle\" [24], [25], [26], [27], [28], [29], [30].",
"In the above formalism, the Drell-Yan-West ($q^+=0$ ) frame [44], [45] is used because only valence contributions are needed in this frame when studying the decay of quarkonia.",
"For the radial wave function $\\phi $ , the harmonic oscillator wave function has been adopted to describe the 1S state mesons [39], [36], [37], [38], and it can well explain experimental data.",
"So we try to go further to use the 2S state harmonic oscillator wave function as the approximate wave function of the 2S quarkonia.",
"The wave functions of the 1S and 2S states of the non-relativistic 3-dimensional isotropical harmonic oscillator in momentum space are $\\varphi ^{1S}(\\mathbf {p})&=&\\frac{1}{\\pi ^{3/4}(\\alpha \\hbar )^{3/2}}\\exp (-\\frac{p^2}{2\\alpha ^2 \\hbar ^2}),\\\\\\varphi ^{2S}(\\mathbf {p})&=&\\frac{\\sqrt{6}}{3\\pi ^{3/4}(\\alpha \\hbar )^{7/2}}(p^2-\\frac{3}{2}\\alpha ^2 \\hbar ^2) \\exp (-\\frac{p^2}{2\\alpha ^2\\hbar ^2}), $ where $\\alpha =\\sqrt{\\mu \\omega /\\hbar }$ , $\\mu $ and $\\omega $ are the mass of the oscillating particle and the frequency of the corresponding classical oscillator respectively.",
"We use the connection between the equal-time wave function in the rest frame and the light-front wave function suggested by Brodsky-Huang-Lepage [21], [22], [39], for the quarkonia with $m_1=m_2\\equiv m_q$ , $p^2 \\longleftrightarrow \\frac{\\mathbf {k}_{\\perp }^2+m_q^2}{4x(1-x)}-m_q^2 $ and we use the prescription in Ref.",
"[46] to extend the non-relativistic form wave function into a relativistic one [36].",
"Then we have the corresponding relativistic wave functions in light-front formalism $\\phi ^{1S}(x_i,\\mathbf {k}_{i \\perp })&=&\\frac{4\\pi ^{3/4}}{\\beta ^{3/2}}\\sqrt{\\frac{\\partial k_z}{\\partial x}}\\exp (-\\frac{\\mathbf {k}^2}{2\\beta ^2}),\\\\\\phi ^{2S}(x_i,\\mathbf {k}_{i \\perp })&=&\\frac{4\\sqrt{6}\\pi ^{3/4}}{3\\beta ^{7/2}}\\sqrt{\\frac{\\partial k_z}{\\partial x}}(\\mathbf {k}^2-\\frac{3}{2}\\beta ^2)\\exp (-\\frac{\\mathbf {k}^2}{2\\beta ^2}),$ where $\\beta $ is the parameter equivalent with $\\alpha $ in Eqs.",
"(REF ) and (), and its value can be chosen to fit experimental data.",
"The longitudinal momentum $k_z=(x-1/2)\\mathcal {M}+(m^2_2-m^2_1)/2\\mathcal {M}$ , one can easily check that this is equivalent to Eq.",
"(REF ).",
"The factor $\\sqrt{\\partial k_z/\\partial x}$ in the above two equations comes from the Jacobian of the transformation $(x, \\mathbf {k}) \\rightarrow (\\mathbf {k}, k_z)$ , and the normalization factors are from the requirement of the normalization of the total wave function [47].",
"Using the above formalism and wave functions, we can calculate the decay constants and transition form factors of the quarkonia [48], [49].",
"In the $V\\rightarrow e^+e^-$ process, the decay constant of the vector meson $V$ is defined by $\\langle 0| j_\\mu |V(p,S_z)\\rangle = M_V f_V \\epsilon _\\mu (S_z),\\\\$ and with the same method as Ref.",
"[50], we have, for the vector quarkonium, $f_V &=&2\\sqrt{6}~e_q\\int \\frac{\\mathrm {d}x\\mathrm {d}^2\\mathbf {k}_\\perp }{16\\pi ^3} \\frac{1}{\\sqrt{x(1-x)}}~\\phi _V(x,\\mathbf {k}_\\perp )\\nonumber \\\\&&\\cdot \\frac{2\\mathbf {k}_\\perp ^2+m_q(\\mathcal {M}+2m_q)}{\\sqrt{\\mathbf {k}_\\perp ^2+m_q^2}(\\mathcal {M}+2m_q)},$ where $m_q$ and $e_q$ is the mass and electric charge of the constituent quark of the quarkonium respectively ($e_q=2/3$ for charmonia, and $-1/3$ for bottomonia).",
"In the $P\\rightarrow \\gamma \\gamma ^*$ process, the transition form factor of the pseudoscalar meson $P$ is defined by $\\langle \\gamma (p-q)|J^\\mu |P(p,\\lambda )\\rangle = ie^2F_{P\\rightarrow \\gamma \\gamma ^*}(Q^2)\\varepsilon ^{\\mu \\nu \\rho \\sigma }p_\\nu \\epsilon _\\rho (p-q,\\lambda )q_\\sigma ,$ and we have the formula for the pseudoscalar quarkonium $F_{P\\rightarrow \\gamma \\gamma ^*}(Q^2)&=&4\\sqrt{6}~e^2_q \\int \\frac{\\mathrm {d}x \\mathrm {d}^2\\mathbf {k}_\\perp }{16\\pi ^3}~\\phi _P(x,\\mathbf {k}_\\perp )\\nonumber \\\\&&\\cdot \\frac{m_q}{x\\sqrt{\\mathbf {k}_\\perp ^2+m^2}}\\frac{x(1-x)}{m_q^2+\\mathbf {k}_\\perp ^{^{\\prime }2}},$ where $\\mathbf {k}^{\\prime }_\\perp =\\mathbf {k}_\\perp -(1-x)\\mathbf {q}_\\perp $ , and $Q^2=-q^2=\\mathbf {q}_\\perp ^2$ , $q$ is the momentum of the virtual photon.",
"The radiative transition form factor between a vector meson $V$ and a pseudoscalar meson $P$ is defined by $\\langle P(p^{\\prime })|J^\\mu |V(p,\\lambda )\\rangle = ieF_{V\\rightarrow \\gamma P}(Q^2)\\varepsilon ^{\\mu \\nu \\rho \\sigma }\\epsilon _\\nu (p,\\lambda )p^{\\prime }_\\rho p_\\sigma ,$ and we have $F_{V\\rightarrow \\gamma P}(Q^2)=4~e_q\\int \\frac{\\mathrm {d}x\\mathrm {d}^2\\mathbf {k}_\\perp }{16\\pi ^3}~\\phi _P(x,\\mathbf {k}_\\perp ^{\\prime })\\phi _V(x,\\mathbf {k}_\\perp )\\nonumber \\\\\\cdot \\frac{m_q(\\mathcal {M}+2m_q)(1-x)+ 2(1-x)\\mathbf {k}_\\perp ^2\\sin ^2\\theta }{(\\mathcal {M}+2m_q)\\sqrt{\\mathbf {k}_\\perp ^2+m_q^2}\\sqrt{\\mathbf {k}_\\perp ^{^{\\prime }2}+m_q^2}},$ where $\\theta $ is the angle between $\\mathbf {k}_\\perp $ and $\\mathbf {q}_\\perp $ .",
"The above quantities are related to the decay width of the quarkonium by [48] $\\Gamma (V\\rightarrow e^+ e^-) &=& \\frac{4\\pi \\alpha ^2f_V^2}{3M_V},\\\\\\Gamma (P\\rightarrow \\gamma \\gamma ) &=& \\frac{1}{4}\\pi \\alpha ^2 M_P^3|F_{P\\rightarrow \\gamma \\gamma ^*}(0)|^2, \\\\\\Gamma _{V\\rightarrow \\gamma P}&=&\\frac{\\alpha }{3}\\left|F_{V\\rightarrow \\gamma ^*P}(0)\\right|^2\\left(\\frac{M_V^2-M_P^2}{2 M_V}\\right)^3.$ We can also calculate the mass of the quarkonium, using the QCD-motivated Hamiltonian for mesons [51] $H_{q\\bar{q}}=\\sqrt{m^2_q+\\mathbf {k}^2}+\\sqrt{m^2_{\\bar{q}}+\\mathbf {k}^2}+V_{q\\bar{q}},$ where $\\mathbf {k}$ is the momentum of the constituent quark, and $V_{q\\bar{q}}=a+br^2-\\frac{4\\alpha _s}{3r}+\\frac{2\\mathbf {S}_{q}\\cdot \\mathbf {S}_{\\bar{q}}}{3m_q m_{\\bar{q}}}\\bigtriangledown ^2V_{\\mathrm {coul}},$ with the last term being the hyperfine interaction that causes the mass splitting between vector and pseudoscalar mesons.",
"Here we choose the confining potential (the second term) to be the harmonic oscillator potential rather than the linear potential in order to keep consistency with our harmonic oscillator wave function for the quarkonium.",
"The values of parameters $a$ , $b$ and $\\alpha _s$ were given in Ref [51].",
"The mass of the meson is obtained as $M_{q\\bar{q}}= \\langle \\phi |H_{q\\bar{q}}| \\phi \\rangle $ [51].",
"For the 2S quarkonium, we have $M_{q\\bar{q}}&=&\\frac{16}{3\\sqrt{\\pi }\\beta ^7}\\int _0^\\infty (\\sqrt{m^2_q+p^2})p^2(p^2-\\frac{3}{2}\\beta ^2)^2e^{-p^2/\\beta ^2}\\mathrm {d}p\\nonumber \\\\&&+a+\\frac{7b}{2\\beta ^2}-\\frac{20\\alpha _s\\beta }{9\\sqrt{\\pi }}\\nonumber \\\\&&+\\left\\lbrace \\begin{array}{l} \\frac{4\\alpha _s\\beta ^3}{3\\sqrt{\\pi }m_q^2}~~\\mathrm {(vector~~quarkonia)},\\\\-\\frac{4\\alpha _s\\beta ^3}{\\sqrt{\\pi }m_q^2}~~\\mathrm {(pseudoscalar~~quarkonia)},\\end{array}\\right.$ and such formula of the mass of the 1S quarkonium can be found in Ref [52]."
],
[
" Numerical results",
"In our numerical calculation, the parameter $\\beta $ in the wave function and the mass of the constituent quark $m_q$ were chosen to fit experimental data.",
"Since the only difference between the vector and pseudoscalar quarkonia that share the same energy quantum number ($n$ ) is the hyperfine interaction term in this model, we choose the same $\\beta $ for them.",
"For charmonia, $m_c$ and $\\beta _{J/\\psi }~(\\beta _{\\eta _c})$ were fixed by Refs.",
"[47], [52], and their results are in good agreement with experimental data, so we use their values of $m_c$ and $\\beta _{J/\\psi }~(\\beta _{\\eta _c})$ , and we only have to fix the parameter $\\beta _{\\psi ^{\\prime }}(\\beta _{\\eta _c^{\\prime }})$ .",
"For the bottomoina, we fix all the parameters $m_b$ , $\\beta _{\\Upsilon }(\\beta _{\\eta _b})$ and $\\beta _{\\Upsilon ^{\\prime }}(\\beta _{\\eta _b^{\\prime }})$ .",
"The parameters of the charmonia are fixed as $&&m_c=1.8~\\mathrm {GeV},~~\\beta _{J/\\psi }(\\beta _{\\eta _c})=0.6998~\\mathrm {GeV},\\nonumber \\\\&&\\beta _{\\psi ^{\\prime }}(\\beta _{\\eta _c^{\\prime }})=0.630~\\mathrm {GeV},$ and our numerical results of 2S charmonia are listed in Table REF .",
"Table: Numerical results (GeV) of 2S charmonia.The numerical results of 1S charmonia can be found in Refs.",
"[47], [52].",
"The transition form factors $F_{\\psi ^{\\prime }\\rightarrow \\eta _c^{\\prime } \\gamma ^* }(0)$ and $F_{\\eta _c^{\\prime }\\rightarrow \\gamma \\gamma ^*}(0)$ are our predictions, and we see from the table that they are well below experimental upper limits.",
"We can also obtain the branching ratios of the two decay modes using Eqs.",
"(REF ) and () and the total widths of $\\psi ^{\\prime }$ and $\\eta _c^{\\prime }$ [16]: $\\mathrm {Br}(\\psi ^{\\prime } \\rightarrow \\gamma \\eta _c^{\\prime })&=&3.9012\\times 10^{-4},\\nonumber \\\\\\mathrm {Br}(\\eta _c^{\\prime } \\rightarrow 2\\gamma )&=&1.0555\\times 10^{-4}.$ BES-III collaboration reported very recently the first measurement of the branching ratio Br$(\\psi ^{\\prime }\\rightarrow \\gamma \\eta _c^{\\prime })= (4.7\\pm 0.9_{\\mathrm {stat}}\\pm 3.0_{\\mathrm {sys}})\\times 10^{-4}$ [53], and our prediction in Eq.",
"(REF ) is in agreement with the preliminary data within error bars.",
"The very recent theoretical study in Ref.",
"[8] gives $\\Gamma (\\psi ^{\\prime }\\rightarrow \\gamma \\eta _c^{\\prime })=0.08^{+0.03} _{-0.03}~\\mathrm {keV}$ , and converted into the branching ratio using the total width of $\\psi ^{\\prime }$ [16], it is Br$(\\psi ^{\\prime }\\rightarrow \\gamma \\eta _c^{\\prime })=(2.7972\\pm 1.1)\\times 10^{-4}$ .",
"Other theoretical predictions for Br$(\\psi ^{\\prime }\\rightarrow \\gamma \\eta _c^{\\prime })$ fall in a range of $(0.1-6.2)\\times 10^{-4}$ [54].",
"Assuming that $\\eta _c$ and $\\eta _c^{\\prime }$ have equal branching fractions to $K_SK\\pi $ , Ref.",
"[55] obtained the experimental data $\\Gamma _{\\gamma \\gamma }(\\eta _c^{\\prime })=1.3\\pm 0.6~\\mathrm {keV}$ , using the total width of $\\eta _c^{\\prime }$ [16], the branching ratio is $\\mathrm {Br} (\\eta _c^{\\prime } \\rightarrow 2\\gamma )=(0.9286\\pm 0.63)\\times 10^{-4}$ .",
"The theoretical prediction in Ref.",
"[1], converted into branching ratio, gives $\\mathrm {Br}(\\eta _c^{\\prime }\\rightarrow 2\\gamma )=1.4286\\times 10^{-4}$ .",
"We see that our prediction in Eq.",
"(REF ) is close to these data.",
"However, the accurate experimental data for this decay mode is still not available, and only the upper limit $\\mathrm {Br} (\\eta _c^{\\prime }\\rightarrow 2\\gamma )<5\\times 10^{-4}$ is given [16].",
"Future experimental measurements at BES and CLEO may provide tests for these predictions of Br$(\\psi ^{\\prime }\\rightarrow \\gamma \\eta _c^{\\prime })$ and $\\mathrm {Br}(\\eta _c^{\\prime } \\rightarrow 2\\gamma )$ .",
"For bottomonia, the parameters are fixed as $&&m_b=5.1~\\mathrm {GeV},~~\\beta _{\\Upsilon }(\\beta _{\\eta _b})=1.1656~\\mathrm {GeV},\\nonumber \\\\&&\\beta _{\\Upsilon ^{\\prime }}(\\beta _{\\eta _b^{\\prime }})=1.1050~\\mathrm {GeV},$ and our numerical results of the 1S and 2S bottomonia are listed in Table REF .",
"Table: Numerical results (in units of GeV) ofthe 1S and 2S bottomonia.Our results give the prediction $\\Gamma (\\eta _b^{\\prime }\\rightarrow 2\\gamma )=0.1494~\\mathrm {keV}$ , compared with the prediction given in Ref.",
"[1] $\\Gamma (\\eta _b^{\\prime }\\rightarrow 2 \\gamma )=0.21~\\mathrm {keV}$ .",
"Future experiments at LHC or by the Belle experiment on $\\eta _b^{\\prime }$ can not only test these predictions, but also help us learn more about this meson by providing more experimental information about it.",
"The small values of the experimental data for the branching ratio of the mode $V(2S)\\rightarrow \\gamma P(1S)$ in Table REF and Table REF can be easily understood with our wave functions, as the $2S$ and $1S$ wave functions are orthogonal to each other and their overlap in Eq.",
"(REF ) is suppressed.",
"Although nonrelativistic models have provided efficient and powerful theoretical tools to handle various problems related to hadron structure, the relativistic models have been successful in many investigation of hadron structures [31], [32], [33], [34], [35], [36], [37], [38], [24], [25], [26], [27], [28], [29], [30].",
"It is thus necessary to make an estimate of the effect due to nonrelativistic to relativistic treatments [56].",
"For simplicity, we assess the non-relativistic to relativistic effects by letting $\\mathbf {k}_\\perp ^2$ of $\\sqrt{\\mathbf {k}_\\perp ^2+m_q^2}$ in the expressions of the decay constants and form factors to be zero.",
"For examples, after this procedure, we have $F_{\\psi ^{\\prime }\\rightarrow \\eta _c \\gamma ^* }(0)=0.1167~\\mathrm {GeV}$ , compared with $F_{\\psi ^{\\prime }\\rightarrow \\eta _c\\gamma ^*}(0)=0.0402~\\mathrm {GeV}/0.0392~\\mathrm {GeV}$ from the relativistic treatment / experimental data, and $f_{\\Upsilon ^{\\prime }}(\\Upsilon ^{\\prime } \\rightarrow e^+ e^-)=0.2159~\\mathrm {GeV}$ compared with $f_{\\Upsilon ^{\\prime }}(\\Upsilon ^{\\prime } \\rightarrow e^+e^-)=0.1944~\\mathrm {GeV}/0.1657~\\mathrm {GeV}$ from the relativistic treatment / experimental data.",
"We see that the relativistic treatment is needed to describe the experimental data well."
],
[
" summary",
"In this work, we studied the 2S quarkonia $\\psi ^{\\prime }$ , $\\eta _c^{\\prime }$ , $\\Upsilon ^{\\prime }$ and $\\eta _b^{\\prime }$ in light-front quark model.",
"Similar with the 1S harmonic oscillator wave function that was commonly used as the 1S quarkonium wave function in light-front quark model studies, we tried to use the 2S harmonic oscillator wave function as the 2S quarkonium wave function.",
"The decay constants and transition form factors of these quarkonia are calculated.",
"Using the QCD-motivated Hamiltonian for mesons, we also calculated masses of these quarkonia.",
"Our numerical results of these quantities are in agreement with experimental data.",
"Predictions of transition form factors and masses of these quarkonia are made, and these predictions can be tested by future experiments.",
"The 1S and 2S wave functions could also be used as inputs to study other problems such as the “$\\rho \\pi $ puzzle\" and the suppression of heavy-quarkonia at RHIC energy.",
"This work is partially supported by National Natural Science Foundation of China (Grants No.",
"11021092, No.",
"10975003, No.",
"11035003, and No.",
"11120101004) and by the Research Fund for the Doctoral Program of Higher Education (China)."
]
] | 1204.0863 |
[
[
"Theoretical Spectra of Terrestrial Exoplanet Surfaces"
],
[
"Abstract We investigate spectra of airless rocky exoplanets with a theoretical framework that self-consistently treats reflection and thermal emission.",
"We find that a silicate surface on an exoplanet is spectroscopically detectable via prominent Si-O features in the thermal emission bands of 7 - 13 \\mu m and 15 - 25 \\mu m. The variation of brightness temperature due to the silicate features can be up to 20 K for an airless Earth analog, and the silicate features are wide enough to be distinguished from atmospheric features with relatively high-resolution spectra.",
"The surface characterization thus provides a method to unambiguously identify a rocky exoplanet.",
"Furthermore, identification of specific rocky surface types is possible with the planet's reflectance spectrum in near-infrared broad bands.",
"A key parameter to observe is the difference between K band and J band geometric albedos (A_g (K)-A_g (J)): A_g (K)-A_g (J) > 0.2 indicates that more than half of the planet's surface has abundant mafic minerals, such as olivine and pyroxene, in other words primary crust from a magma ocean or high-temperature lavas; A_g (K)-A_g (J) < -0.09 indicates that more than half of the planet's surface is covered or partially covered by water ice or hydrated silicates, implying extant or past water on its surface.",
"Also, surface water ice can be specifically distinguished by an H-band geometric albedo lower than the J-band geometric albedo.",
"The surface features can be distinguished from possible atmospheric features with molecule identification of atmospheric species by transmission spectroscopy.",
"We therefore propose that mid-infrared spectroscopy of exoplanets may detect rocky surfaces, and near-infrared spectrophotometry may identify ultramafic surfaces, hydrated surfaces and water ice."
],
[
"Introduction",
"Rocky exoplanets have been discovered by transit surveys.",
"A primary transit happens when a planet passes in front of its host star and blocks a part of the star, so that the depth of the transit corresponds to the ratio between the planetary radius and the stellar radius.",
"The planet's radius together with its mass provides the mean density and clues on its interior structure (See Rogers & Seager 2010 and references therein).",
"A few exoplanets have been suggested to have rocky surfaces, because the mass and radius constraints indicate that the planets are predominantly rocky with no extensive atmosphere envelope, i.e., likely rocky surfaces.",
"Corot-7b opens up the possibility of close-in airless rocky exoplanets (Léger et al.",
"2009; Queloz et al.",
"2009).",
"Due to the small star-planet distance, Corot-7b may have molten or even vaporized metals on its sub-stellar surface (Léger et al.",
"2011).",
"Recently observations of transits of 55 Cnc e, a 8-$M_{}$ planet around a G8V star, determine the planetary radius to be 2.1 $R_{}$ , which suggests that it can be a rocky planet (Winn et al., 2011; Demory et al., 2011).",
"The Kepler mission, with unprecedented photometric precision, is very powerful in discovering small-size transiting exoplanets.",
"Kepler-10b, a 4.5-$M_{}$ planet, is the first rocky exoplanet discovered by Kepler (Batalha et al.",
"2011).",
"Kepler recently discovered several planets with size in the“super earth\" regime, including Kepler-11b (Lissauer et al.",
"2011), Kepler-18b (Cochran et al.",
"2012), Kepler-20b (Gautier et al.",
"2012) and notably Kepler-22b in its host star's habitable zone (Borucki et al.",
"2012).",
"Due to uncertainties of the planets' radii and masses, however, the composition of these Kepler super-Earths is ambiguous.",
"They can be predominantly rocky like Earth, or have significant gas envelope like Neptune.",
"Also, Kepler has detected Earth-sized transiting planets, Kepler-20e and Kepler-20f, with no constraints on their masses due to difficulties of followup radial-velocity observations (Fressin et al.",
"2012).",
"With the progress of the Kepler mission, other transit surveys and followup observations, more and more exoplanets that potentially have rocky surfaces will be discovered and confirmed.",
"The purpose of this paper is to identify mineral-specific spectral features that would allow characterization of the surface composition of airless rocky exoplanets.",
"The analogs of airless or nearly airless rocky exoplanets in the Solar System are the Moon, Mars, Mercury and asteroids, whose surface compositions have been studied extensively by spectroscopy of reflected solar radiation and planetary thermal emission (see Pieters & Englert 1993 and de Pater & Lissauer 2001 and extensive references therein).",
"For investigations of the Solar System in 1970s and 1980s, the rocky bodies were still spatially unresolved or poorly resolved with resolution of a few thousand kilometers, and spectra of reflected solar radiation in the near-infrared (NIR) and spectra of planetary emission in the mid-infrared (MIR) were the only information to infer their surface composition.",
"Thanks to prominent absorption features of specific minerals, notably olivine, pyroxene and ices, infrared spectroscopy provided significant constraints on the surface composition of rocky planets of the Solar System (e.g., Pieters & Englert 1993).",
"We propose that using infrared spectroscopy to characterize mineral composition is applicable to study solid surfaces of exoplanets.",
"Both reflected stellar radiation and planetary thermal emission from an exoplanet are potentially observable by secondary eclipses if the planet is transiting, as well as by direct imaging (e.g., Seager & Sasselov 2000; Seager 2010).",
"Still, characterization of exoplanetary surfaces is very different from the investigation of the Solar System analog surfaces, due to the fact that an exoplanet cannot be spatially resolved and the spectroscopy of exoplanets is limited to very low spectral resolutions, probably broad-band photometry in the near term.",
"As a result, it is essential to focus on the most prominent spectral features in the disk-averaged radiation from an exoplanet.",
"We develop a theoretical framework to compute disk-integrated spectra of airless rocky exoplanets that self-consistently treats reflection and thermal emission and investigate the spectral features that can be used to interpret the mineral composition on exoplanetary surfaces.",
"So far, the study of spectral features due to exoplanets' surfaces has been limited to features in the reflected light by vegetation (see Seager et al.",
"2005 for the “red-edge\") and a liquid water ocean (e.g., Ford et al.",
"2001; Cowan et al.",
"2009).",
"In this paper we employ a generalized approach to investigate spectral features of solid materials on an exoplanet's surface, with a consistent treatment of reflected stellar irradiation and planetary thermal emission.",
"We focus on airless rocky exoplanets with solid surfaces, whose surface temperatures are lower than the melting temperature of silicates and other common minerals ($\\sim 1000$ - 2000 K).",
"This surface temperature requirement ensures that the planetary thermal emission and the reflected stellar radiation can be separated in wavelengths.",
"As shown in Figure REF , Kepler-22b, Kepler-20f and Kepler-11b can have an unmelted silicate surface.",
"Moreover, a large number of Kepler planetary candidates have the orbital distances that permit unmelted silicate surfaces.",
"In this paper we do not consider close-in rocky exoplanets that have molten lava surfaces, such as the case of Corot-7b.",
"Once melted, crystal-field features, such as Fe$^{2+}$ electronic transition at 1 $\\mu $ m, will no longer persist in the same manner as in crystalline minerals.",
"In situ measurements of active lava flow in Hawaii confirm that NIR spectra of molten lava are dominated by black-body emission (e.g., Flynn & Mouginis-Mark 1992).",
"We will address the spectral features of molten lava on close-in rocky exoplanet in a seperate paper.",
"The paper is organized as follows.",
"In § 2 we describe the background of geological surface remote sensing and the application in the study of the reflectance spectra of Solar System rocky planets.",
"In § 3 we define several types of planetary solid surfaces and describe a theoretical framework to compute the disk-average reflection and thermal emission spectra of an airless rocky exoplanet.",
"§ 4 contains the results of the paper in which we present reflection and thermal emission spectra of exoplanets with various types of crust, and summarize the most prominent and diagnostic features.",
"In § 5 we discuss the effect of atmospheres, the effect of space weathering, detection potential of the proposed spectra, and the connection between the surface characterization and the planetary formation and evolution.",
"We summarize the paper in § 6.",
"The spectral reflectances of minerals have features specific to their chemical compositions and crystal structures.",
"From the visible to the NIR wavelengths (VNIR; 0.3 - 3 $\\mu $ m), minerals such as pyroxene, olivine and hematite create prominent absorption features in reflection data, mostly due to electronic transitions of transition element ions (i.e., Fe$^{2+}$ , Fe$^{3+}$ , Ti$^{3+}$ , etc.)",
"in the the crystal structures of minerals (Burns 1993).",
"For example, Fe$^{2+}$ in the crystal field of olivine absorbs strongly at 1 $\\mu $ m. Position, strength, and number of features of the 1-$\\mu $ m absorption band are diagnostic of the relative proportions of the Mg and Fe cations in olivines (e.g., Sunshine & Pieters 1998).",
"In the ultraviolet (UV) and visible wavelengths, many minerals, such as hematite (Fe$_2$ O$_3$ ), have strong charge-transfer absorption bands, making them very dark (e.g., Clark 1999).",
"For secondary minerals, typically formed by interactions with water, their volatile components, such as hydroxyl (OH), water (H$_2$ O), and carbonates (CO$_3^{2+}$ ), absorb strongly in the NIR due to overtone and combination absorptions from vibrations, i.e.",
"bands and stretches of components in the crystal structure (e.g., Farmer 1974).",
"See Table REF for a list of common minerals on terrestrial planet surfaces and their key features in the NIR reflectance spectroscopy.",
"Table: Common minerals on terrestrial planet surfaces and their key features in the NIR reflectance spectroscopy.",
"Spectra used are from representative samples of average grain diameters less than 200 μ\\mu m, comparable to lunar soil (Duke et al., 1970).Similar to reflected stellar radiation, planetary thermal emission is encoded with information about the planet's surface composition.",
"Absorption in the mid-infrared (MIR; 3 - 25 $\\mu $ m) is due to vibrational motions in crystal lattices, so that their wavelengths are related to the crystal structure and elemental composition (i.e., mineralogy) (Farmer 1974).",
"Silicates have the most intense (i.e., the greatest absorption coefficient) spectral features between 8 and 12 $\\mu $ m due to the Si-O stretching, and the second most intense features between 15 and 25 $\\mu $ m due to the Si-O bending or deformation (Salisbury 1993).",
"These vibration bands are so strong that they can manifest as mirror-like reflectance peaks and therefore emittance troughs.",
"The shapes of these bands are complicated by the so-called “transparency feature\", features associated with a change from surface to volume scattering, and therefore are sensitive to particle size (Salisbury & Walter 1989; Salisbury 1993).",
"At the short-wavelength edge of the Si-O stretching band ($7.5\\sim 9.0$ $\\mu $ m), there is always a reflectance minimum (emittance maximum) as the refractive index of the mineral approaches that of the medium.",
"This emittance maximum is termed “Christiansen Feature\" (CF), unique and ubiquitous for silicates.",
"The wavelength of the CF is indicative of silica content of the material, i.e., more mafic silicates have the CF at longer wavelengths (Salisbury & Walter 1989; Walter & Salisbury 1989).",
"For example, Glotch et al.",
"(2010) inferred highly silica-rich compositions on the Moon by determining the CF wavelengths.",
"Iron oxides also show spectral features due to Fe-O fundamentals, but at longer wavelengths than Si-O fundamentals, because iron is more massive than silicon (Clark 1999).",
"For example, hematite, Fe$_2$ O$_3$ , has 3 strong stretching modes between 16 and 30 $\\mu $ m. Spectral features of pure particulate minerals discussed above can be wide and deep, and could stand out in low-resolution and low signal-to-noise ratio spectra.",
"In electronic processes, the electron participating in transitions may be shared between individual atoms and energy levels of shared electrons become smeared over wide energy bands, which results in wide spectral features (Burns 1993; Clark 1999).",
"In vibrational processes, absorption bands are typically narrower than electronic features but can be broadened if the crystal is poorly ordered or if bands overlap (Farmer 1974; Clark 1999).",
"In reality, the contrast of spectral features may be significantly reduced, especially for thermal emission in MIR.",
"Typical thermal emission band contrasts on planetary surfaces are less than $\\sim $ 0.1 in the Solar System (Sprague 2000; Christensen et al.",
"2000; Clark et al.",
"2007).",
"The reduction of spectral contrast in the thermal emission is mostly due to multiple scattering between regolith particles, with contributions of surface roughness and volume absorption (Hapke 1993; Kirkland et al.",
"2003).",
"Multiple reflection, as a result of surface roughness, increases the emergent emissivity ($\\epsilon _e$ ) towards unity, as illustrated by $\\epsilon _e = 1 - (1-\\epsilon )^{(n+1)} \\ , $ where $\\epsilon $ is the true emissivity of material and $n$ is the number of reflections (Kirkland et al.",
"2003).",
"As a result, unique identification of mineral compositions using planetary thermal emission usually requires high signal-to-noise ratio and high resolution spectroscopy.",
"Finally, mixing of minerals and space weathering may also reduce contrast of the spectral features.",
"There are two levels of mixing, microscopic and macroscopic.",
"Microscopic mixing (or intimate mixing) concerns particulates of different minerals that may be intimately mixed, such that photons are multiply-scattered by interactions with materials of different compositions.",
"This is especially relevant for reflectance in VNIR.",
"Macroscopic mixing (or areal mixing) concerns the planetary surface viewed as a disk average, which may contain discrete patches of different minerals or mineral assemblages.",
"The average of macroscopic mixing can be modeled as a linear combination of reflected flux, whereas the microscopic mixing has to be treated as a multi-component radiative transfer problem.",
"For spatially unresolved exoplanets, both microscopic mixing and macroscopic mixing need to be considered.",
"Space weathering does not change the bulk mineralogic composition of the rocks and soils, but leads to formation of nanophase iron throughout the whole of the mature portion of the regolith, occurring in both vapor-deposited coatings on grain surfaces and in agglutinate particles, which alters the spectral properties of the surface significantly (e.g., Pieters et al.",
"2000; Hapke 2001)."
],
[
"Solar System Airless Body Surface Spectra",
"Infrared spectral features have been used to study surface compositions of Solar System rocky planets.",
"In the early-stage investigations, ground-based or balloon-based telescopic observations have been used to characterize the surfaces, for example the Moon (e.g., Murcray et al.",
"1970; Pieters 1978; McCord et al.",
"1981; Pieters 1986; Tyler et al.",
"1988), Mars (e.g., Singer et al.",
"1979; McCord et al.",
"1982) and Mercury (e.g., Vilas 1985; Tyler et al.1988; Blewett et al.",
"1997; Sprague et al.",
"2002).",
"As a benchmark for the exoplanet investigations, we focus here on a description of the most prominent features in the low-spatial-resolution reflectance spectra and their implications.",
"Several representative ground-based spectra of the Moon, Mars, and Mercury and the prominent features therein are shown in Figure REF .",
"Reflectance spectra at NIR have been used to characterize the lunar mare and the lunar highlands.",
"The lunar mare are generally dark and absorb strongly at 1 $\\mu $ m and 2 $\\mu $ m, which indicate they are of basaltic composition (e.g., Pieters 1978; see Figure REF ).",
"The 1-$\\mu $ m absorption is due to iron-bearing glass, pyroxene, and olivine, and the 2-$\\mu $ m absorption is exclusively due to pyroxene (Pieters 1978).",
"Moreover, the relative strength of the 1-$\\mu $ m absorption and the 2-$\\mu $ m absorption, as well as the band center positions of these absorptions, are used to infer the amount of olivine versus pyroxene and pyroxene composition of the lunar mare (McCord et al.",
"1981).",
"The basaltic nature of the lunar mare indicates that they formed by volcanic eruptions.",
"In contrast to the lunar mare, the lunar highlands are bright and have nearly flat spectra at NIR with weak 1-$\\mu $ m and 2-$\\mu $ m absorptions, exhibiting a plagioclase composition with minor amounts of pyroxene (e.g., Pieters 1986; see Figure REF ).",
"The nearly pure plagioclase composition of the lunar highlands indicates that they are primary crust formed from solidification of a magma ocean (e.g., Warren 1985).",
"Spatially unresolved spectroscopy of Mars has provided essential information to determine the surface composition of the red planet.",
"Mars spectra feature strong Fe$^{3+}$ charge-transfer and crystal field absorptions from the near-UV to about 0.75 $\\mu $ m (McCord & Westphal 1971; Singer et al.",
"1979; see Figure REF ).",
"These spectral features, together with the visual red color and the polarization properties of Mars, established that the major component of the Martian surface is ferric oxides (e.g., hematite).",
"Also, the spectra from dark areas on Mars show features of mixtures of pyroxene and olivine, probably covered by a layer of ferric oxides (McCord et al.",
"1982; see Figure REF ).",
"The mineral composition of the Martian surface is interpreted as a result of secondary basaltic volcanism, affected by later oxidative weathering, perhaps in presence of liquid water to form Fe$^{3+}$ oxides (Mustard et al.",
"2005).",
"Telescopic spectra of Mercury suggest a surface similar to the lunar highlands.",
"The NIR spectra of Mercury are flat without the signature Fe$^{+2}$ absorptions at 1 and 2 $\\mu $ m (Vilas 1985; see Figure REF ).",
"Also, emission features in the mid-infrared (7 – 13 $\\mu $ m) characteristic of silicate materials have been reported for Mercury (e.g., Tyler 1988).",
"The spectroscopic evidence suggested a low-iron plagioclase surface similar to the lunar highlands (Blewett et al.",
"1997), consistent with the outcome of solidification of magma ocean without mantle overturn (Brown & Elkins-Tanton 2009).",
"Recent observations from a spacecraft orbiting Mercury have challenged this view by measuring Mg-rich, Fe-poor and Al-poor chemical compositions (ultramafic) and morphologies consistent with flood volcanism (Nittler et al.",
"2011; Head et al., 2011; Blewett et al.",
"2011).",
"Research is on-going to explain Mercury data; a potential explanation reconciling all observations is atypically low iron content ($<\\sim 1$ %) in common minerals forming ultramafic crusts (olivine, pyroxene), which usually exhibit Fe$^{+2}$ absorptions."
],
[
"Types of Planetary Solid Surface",
"The assemblage of minerals provides valuable information on the geological history and even the interior structure of the planet.",
"For example, in the Solar System, feldspathic surfaces, such as the lunar highlands, are primitive products of crystallization from a magma ocean, since plagioclase is light in density and floats on top of the magma ocean (e.g., Warren 1985).",
"For a relatively large planet with mass similar to that of Earth and Mars, the predicted crust composition after the overturn of the mantle in order to form a stable density stratification is dominated by Mg-rich olivines and pyroxenes, i.e., an ultramafic surface (Elkins-Tanton et al., 2005).",
"Subsequent partial melting of mantle (i.e., volcanism) leads to production of distinctive igneous rocks such as basalts, i.e.",
"the lunar mare.",
"Finally, re-processing, heating and partial melting of these materials leads to the generation of granites, a regime on Earth driven by plate tectonics and incorporation of water in subducted crustal materials to lower the melting point (e.g., Taylor 1989).",
"We consider multiple geologically plausible planetary surface types as several assemblages of minerals, tabulated in Table REF .",
"The types include primary crust, i.e., the crust that forms from solidification of magma ocean; secondary crust, i.e., the crust that forms from volcanic eruptions; and tertiary crust, i.e., the crust that forms from tectonic re-processing.",
"Each type of igneous crust differs from another by assemblages of minerals, which are in turn governed by thermodynamics, planetary composition, and planetary history (e.g.",
"Best 2002; Hazen et al.",
"2008).",
"Moreover, various modification processes in the planetary evolution may alter the surface spectral properties significantly.",
"The modification processes include aqueous alteration and oxidative weathering Space weathering is discussed in section REF.",
"Additionally we simulate an ice-rich planet and one lacking a silicate crust or mantle.",
"The list in Table REF encompasses the most common surface types; abiotic mechanisms of surface formation and modification are considered, and the list largely covers the diversified solid surfaces on Solar System rocky planets.",
"Table: Potential crustal compositions of rocky exoplanets.",
"NIR spectra were defined for eight notional exoplanet surface types from laboratory measurement of rock powders (sample) or radiative transfer modeling combining endmember mineral samples measured in the laboratory (modeled).As is evident from Table REF , a planet's surface is defined as the assemblage of several endmember minerals.",
"Moreover, a planet's surface may be composed of bulk patches of different crusts, as is the case for the lunar mare and highlands.",
"It is therefore essential to consider the macroscopic mixture of different crusts as well as the microscopic (intimate) mixture of different minerals for each crust.",
"We model the intimate mixture of minerals as follows.",
"For each endmember mineral, we use the measured bidirectional reflectance in the USGS Digital Spectral Library (Clark et al.",
"2007) or the RELAB Spectral Database (2010).",
"We retrieve the wavelength-dependent single scattering albedo ($\\omega $ ) of each type of mineral from the experimental data based on an analytical radiative transfer model of Hapke (1981, 2002), which will be detailed in section 3.2.",
"We average the mineral composition utilizing single scattering albedo spectra of endmembers, weighted by their mixing ratios.",
"The bidirectional, directional-hemispherical reflectance and directional emissivity of the mixture can thus be computed using the Hapke radiative transfer model in the forward sense (Hapke 1981, 2002).",
"This method of computing reflectance spectra of mineral mixtures has been proven within 10% error by experiments for binary and ternary mixing among components with moderate albedo contrast (Mustard & Pieters 1989).",
"There are some fundamental limitations of this approach to synthesize reflectance spectra of mineral mixtures.",
"First, the Hapke radiative-transfer method is designed for mixture of particulate materials whose grain sizes are small and whose phase dependent scattering behaviors are similar.",
"Although regoliths are widespread on the surfaces of rocky bodies in the Solar System as a result of extensive meteoritic bombardments, it is uncertain whether an exoplanet's surface is made of regolith or bulk rocks We focus on regolith surfaces in this paper.",
"Surfaces of an airless planetary body are most likely comprised of particulate regoliths, because impact gardening effectively converts surface rocks to a regolith layer.. Second, the Hapke radiative-transfer method may induce an error up to 25% for for dark materials (basalts) in mixtures with clay (Ehlmann & Mustard, in preparation).",
"For these reasons, we choose well-characterized actual materials rather than Hapke modeled spectra as representative of feldspathic and basaltic surface types (see Table REF ), and results from intimate mixing studies should be considered indicative rather than exact."
],
[
"Bidirectional Reflectance Spectra of Planetary Surface Material",
"For computation of planetary surface spectra the most important parameter is bidirectional reflectance of surface solid material, which is usually characterized by the radiance coefficient ($r_c$ ).",
"The radiance coefficient is defined as the brightness of a surface relative to the brightness of a Lambert surface identically illuminated, which depends on the direction of both incident and scattered light.",
"The radiance coefficient of a solid material not only depends on its chemical composition, but also depends on its crystal structures.",
"Reflectance spectra of these minerals are shown in Figure REF .",
"Hapke (1981, 2002) presents a straightforward method to compute the approximated radiance coefficient of any particulate material in terms its single scattering albedo.",
"In essence, Hapke (1981, 2002) treats the problem as the radiative transfer of a planar, semi-infinite, particulate medium illuminated by collimated light.",
"The key assumption to achieve a convenient analytical expression is to assume isotropic scatterers for multiple scattering and non-isotropic scatterers for single scattering (Hapke 1981).",
"The resulting expression has been proved to be correct and handy in the interpretation of surface composition of the Moon and Mars (e.g., Hapke 2002).",
"Experimental data of radiance coefficient ($r_c$ ) of common minerals and mixtures are typically presented for certain combinations of incidence angle ($\\mu _0$ ), scattering angle ($\\mu $ ) and phase angle ($g$ ).",
"For application to exoplanets, we extend the experimentally measured $r_c$ to any combination of $(\\mu _0,\\mu ,g)$ using the analytical expression of Hapke (2002).",
"Also, we compute the directional-hemispherical reflectance ($r_{\\rm h}$ ) for different minerals based on experimental bidirectional reflection using an analytical expression given by Hapke (2002).",
"The analytical form of Hapke (2002) employs a parameter $h$ to describe the opposition effect and a phase function to describe the single particle scattering.",
"For fine grains, $h\\sim 0.1-0.4$ .",
"We assume $h=0.2$ and the phase function to be the first order Legendre expansion as $p(g) = 1+b\\cos (g) \\ ,$ where $b$ is the anisotropy parameter in $[-1,1]$ .",
"We typically assume $b=0$ , i.e.",
"isotropic scattering.",
"Numerical experiments show that the retrieval of single scattering albedo and the computation of radiance coefficient do not sensitively depend on the assumptions of $h$ and $b$ .",
"For consistent treatment of reflection and thermal emission, we use experimentally measured reflectance and derive emissivities of mineral assemblages.",
"In the Hapke framework, the directional emissivity and the directional-hemispherical reflectance obey Kirchhoff's Law (Hapke 1993).",
"We performed cross-checks between our derived emissivity and experimentally measured emissivity of pure minerals (Christensen et al.",
"2000) and demonstrate that they are roughly consistent."
],
[
"Disk-Averaged Spectral Model",
"Where we differ from the solar-system models are that spectra of exoplanets are always disk integrated.",
"The observable quantity of exoplanet reflection is the occultation depth of the secondary eclipse.",
"Reflected light from an exoplanet is an average over the entire sub-stellar hemisphere.",
"Moreover, the planetary radiation flux combines the reflected stellar radiation and the thermal emission, as these two components may overlap in wavelength ranges.",
"Here we present a detailed formulation of disk-averaged reflection and thermal emission spectra from an airless rocky exoplanet with contribution from thermal emission, in terms of the radiance coefficient ($r_c$ ) and the directional-hemispherical reflectance ($r_{\\rm h}$ ) of its surface materials.",
"Radiative flux per unit area per wavelength from an exoplanet for an Earth-based observer $F_{\\rm p}$ (erg s$^{-1}$ cm$^{-2}$ nm$^{-1}$ ) is a hemisphere-integral as $F_{\\rm p} = \\bigg (\\frac{R_{\\rm p}}{D}\\bigg )^2\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}I_{\\rm p}(\\theta ,\\phi )\\cos ^2\\theta \\cos \\phi \\ d\\theta d\\phi \\ ,$ in which $R_{\\rm p}$ is the radius of the planet, $D$ is the distance to the observer, and $I_{\\rm p}(\\theta ,\\phi )$ is the intensity from a location on the hemisphere towards the observer specified by latitude-longitude coordinates $(\\theta ,\\phi )$ .",
"The coordinate system is chosen such that the observer is at the direction of $(\\theta =0,\\phi =0)$ .",
"The planetary radiance is composed of the reflection of stellar light and the thermal emission from the planet itself, $I_{\\rm p}(\\theta , \\phi ) = I_{\\rm s}(\\theta , \\phi ) + I_{\\rm t}(\\theta , \\phi ) \\ .$ The reflected intensity is related to the incident stellar irradiance ($F_{\\rm inc}$ ) as $I_{\\rm s}(\\theta , \\phi ) = F_{\\rm inc}\\frac{\\mu _0}{\\pi } r_c(\\mu _0,\\mu ,g) \\ ,$ where $r_c$ is the radiance coefficient as a function of the incidence angle $i$ , the scattering angle $e$ and the phase angle of scattering $g$ .",
"By definition, a Lambertian sphere has $r_c=1$ .",
"Let $\\alpha $ be the phase angle of the exoplanet with respect to the Earth, so that the stellar coplanar light comes from the direction of $(\\theta =0,\\phi =\\alpha )$ .",
"For each surface element, we have the following geometric relations : $&& \\mu _0\\equiv \\cos i = \\cos \\theta \\cos (\\alpha -\\phi ) \\ ,\\\\&& \\mu \\equiv \\cos e = \\cos \\theta \\cos \\phi \\ ,\\\\&& g = \\alpha \\ .$ The thermal emission of the planet depends on its surface temperature, which is controlled by both stellar radiation and planetary surface properties.",
"The irradiance incident on the planet is $F_{\\rm inc} = \\pi B_{\\lambda }[T_*]\\bigg (\\frac{R_*}{D_{\\rm p}}\\bigg )^2\\ ,$ and the stellar irradiance for the observer is $F_{*} = \\pi B_{\\lambda }[T_*]\\bigg (\\frac{R_*}{D}\\bigg )^2\\ ,$ in which $T_*$ and $R_*$ are the temperature and the radius of the star, $D_{\\rm p}$ is the semi-major axis of the planet's orbit, and $B_{\\lambda }$ is the Planck function for blackbody radiance.",
"The thermal emission intensity is $I_{\\rm t} (\\theta , \\phi ) = \\epsilon _{\\lambda }(\\mu )B_{\\lambda }[T(\\theta , \\phi )]\\ ,$ where $\\epsilon _{\\lambda }$ is the directional emissivity, and $T(\\theta , \\phi )$ is the temperature of the planet's surface.",
"According to the Kirchhoff's law of thermal radiation and the framework of Hapke (1993), $\\epsilon _{\\lambda }$ is tied to the directional-hemispherical reflectance $r_{\\rm dh} $ via $\\epsilon _{\\lambda } (\\mu ) = 1 - r_{\\rm dh} (\\mu ) \\ .$ An effectively airless rocky exoplanet, without efficient heat transport mechanisms, is likely to have local thermal equilibrium (e.g.",
"Léger et al.",
"2011 for CoRot-7b).",
"The energy balance equation can therefore be written as $\\mu _0 \\int \\epsilon _{\\lambda } (\\mu _0) F_{\\rm inc} d\\lambda = \\pi \\int \\epsilon _{\\lambda }^{h} B_{\\lambda }[T] d\\lambda \\ , $ where $\\epsilon _{\\lambda }^{h}$ is the hemispheric emissivity, i.e., the hemispherical average of the directional emissivity.",
"By solving this equation we determine the local surface temperature and then compute the directional thermal emission of each surface element.",
"Finally, the occultation depth of the secondary eclipse is $\\frac{F_{\\rm p}}{F_*} = \\frac{1}{F_{\\rm inc}} \\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}I_{\\rm p}(\\theta ,\\phi )\\cos ^2\\theta \\cos \\phi \\ d\\theta d\\phi \\ \\times \\bigg (\\frac{R_{\\rm p}}{D_{\\rm p}}\\bigg )^2 \\equiv A_{\\rm g} \\bigg (\\frac{R_{\\rm p}}{D_{\\rm p}}\\bigg )^2 \\ , $ where $A_{\\rm g}$ is the apparent geometric albedo of the planet, and $I_{\\rm p}(\\theta ,\\phi )$ should be evaluated from Equation (REF ) at $\\alpha =0$ .",
"Here we include thermal emission into the definition of geometric albedo, because for close-in exoplanets the reflected stellar light and the thermal emission may not be separated in spectra.",
"In case of negligible thermal emission, for example in the visible and NIR wavelengths for Earth-like planets, the geometric albedo in Equation (REF ) can be simplified to be the conventional definition as $A_{\\rm g} = \\frac{1}{\\pi }\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}\\int _{-\\frac{\\pi }{2}}^{\\frac{\\pi }{2}}r_c(\\mu _0,\\mu _0,0)\\cos ^3\\theta \\cos ^2\\phi \\ d\\theta d\\phi \\ \\ ,$ consistent with Sobolev (1972) and Seager (2010).",
"In addition, we use the apparent brightness temperature ($T_{\\rm b}$ ) to describe the thermal emissivity of the planet, namely $\\pi B_{\\lambda }[T_{\\rm b}] = F_{\\rm p} \\ .$ The brightness temperature defined as such can be compared with observations directly, and takes into account the disk average of directional emissivities and surface temperatures."
],
[
"Results",
"Our main findings are: rocky silicate surfaces lead to unique features in planetary thermal emission at the mid-infrared due to strong Si-O vibrational bands (7 - 13 $\\mu $ m and 15 - 25 $\\mu $ m); the location of the emissivity maxima at the short-wavelength edge of the silicate feature (7 - 9 $\\mu $ m) is indicative of the silica content in the surface silicates; ultramafic surfaces can be uniquely identified in the reflectance spectra via a prominent absorption feature at 1 $\\mu $ m (i.e., the J band); hydrous surfaces induce strong absorption at 2 $\\mu $ m (i.e., the K band); and surface water ice has a unique absorption feature in the reflected stellar light at 1.5 $\\mu $ m (i.e., the H band).",
"In the following we present disk-average spectra of airless rocky exoplanets and describe the main results.",
"We compute the disk average using Equation (REF ) based on measured or modeled bidirectional reflectance of mineral assemblages as tabulated in Table REF .",
"Figure REF shows the VNIR geometric albedos and MIR brightness temperatures of airless exoplanet fully covered by the 8 types of crust; Table REF lists the main spectral features due to surface minerals in the planetary thermal emission; Table REF lists the geometric albedos of the 8 cases averaged in the NIR J (1.1 - 1.4 $\\mu $ m), H (1.5 - 1.8 $\\mu $ m), and K (2 - 2.4 $\\mu $ m) bands; Figure REF is a scatter plot showing the relation between broad-band true geometric albedos for the 8 cases; Figure REF explores the effect of macroscopic mixture of two types of crust; and Figure REF shows the modeled planetary spectra of Kepler-20 f if the planet has a particulate solid surface."
],
[
"Silicate Features in the Thermal Emission of Rocky Exoplanets",
"Silicate surfaces possess prominent minima in the thermal emission spectra from 7 - 13 $\\mu $ m and 15 - 25 $\\mu $ m (see Figure REF ).",
"These Si-O stretching and bending vibrations manifest in the thermal emission spectra as troughs of complicated shapes, due to a strong reststrahlen reflection at the band center and volume scattering near the band edges.",
"Prominent Si-O features, e.g.",
"in the ultramafic and granitoid surfaces, have equivalent width (EW) larger than 1 $\\mu $ m and $\\Delta T_{\\rm b}$ larger than 20 K (see Table REF ).",
"For comparison, the atmospheric O$_3$ absorption line at 9.6 $\\mu $ m has $\\Delta T_{\\rm b}$ of about 30 K for the Earth (Des Marais et al.",
"2002; Belu et al.",
"2011).",
"For close-in rocky exoplanets, $\\Delta T_{\\rm b}$ of the silicate features can be as large as of 200 K (see Figure REF ), which corresponds to a variation of secondary transit depth of 2 part-per-million (ppm) for the case of Kepler-20f (see Figure REF ).",
"In contrast to silicate surfaces, iron-oxidized surfaces do not have thermal emission troughs in 7 - 13 $\\mu $ m, but usually have a clear double-peak Fe-O feature in the 15 - 25 $\\mu $ m band, as shown in Figure REF for the Fe-oxidized surface.",
"Silicates and iron-oxides can therefore be distinguished based on thermal emission spectra.",
"In summary, wide troughs of brightness temperature in both 7 - 13 $\\mu $ m and 15 - 25 $\\mu $ m constitute a unique signature of silicate surfaces of exoplanets.",
"Furthermore, high-resolution spectra of the silicate bands can allow identification of different kinds of silicate surfaces.",
"As shown in Table REF , as silica content in the mineral assemblage increases, the Christiansen feature (CF), defined as the emissivity maxima at the edge of the main Si-O band, shifts to shorter wavelengths.",
"Note that the ultramafic surface is the most silica-poor and the granitoid surface is the most silica-rich.",
"This well-known effect in mineralogy is applicable to spectral analysis of disk-integrated planetary thermal emission for characterizing rocky surfaces.",
"Determination of the CF wavelength, however, involves tracing the curvature of brightness temperature spectra and therefore requires high-resolution spectra with a high signal-to-noise ratio.",
"At a minimum, the determination of CF location and thus silica content requires three narrow bands in the 7 - 9 $\\mu $ m region (see Greenhagen et al.",
"2010 and Glotch et al.",
"2000 for an example applicable to the Moon).",
"Not only the spectral features, but also the thermal emission continuum provide valuable information on the planetary surfaces.",
"Thermal emission of an effectively airless rocky exoplanet probes the planet's surface temperature, which in turn depends on its surface composition.",
"A highly reflective surface at VNIR, for example the feldspathic surface, has much lower equilibrium temperature than a VNIR absorptive surface, for example the basaltic surface (e.g.",
"Figure REF ).",
"For an airless Earth analog, the brightness temperature continuum from 5 to 25 $\\mu $ m is about 340 K if the surface is basaltic and 290 K if the surface is feldspathic (see Figure REF ).",
"Solely due to different VNIR reflectivities, the apparent temperature difference of the planet, $\\Delta T_{\\rm b}$ , can be as large as 50 K. As a result, equilibrium temperature of planetary solid surfaces, derived from the planetary thermal emission, may coarsely constrain the planetary surface composition, for example to the level of feldspathic versus ultramafic.",
"Note in Figure REF that more than one types of crust have overall low VNIR albedos, including metal-rich, basaltic and Fe-oxidized crusts; several crusts have overall high VNIR albedos, including feldsphathic, granitoid and clay crusts; and several crusts have intermediate VNIR albedos, including ultramafic and ice-rich silicate crusts.",
"Moreover, the overall surface reflectivity sensitively depends on the surface roughness and particle size.",
"Intrinsic degeneracy between surface composition and roughness or particle size exists if thermal emission continuum, or equilibrium temperature, is the only piece of information.",
"This highlights the utility of multiple spectral channels in characterizing exoplanet surfaces.",
"Last but not the least, we comment that thermal emission does not cause the reduction of spectral feature contrasts at NIR wavelengths, unless the exoplanet is very hot when in a close-in orbit (i.e., semi-major axis less than $\\sim $ 0.2 AU).",
"For an analog of Earth or Mars, the thermal emission is negligible at NIR.",
"For close-in exoplanets, the sub-stellar temperature may be very high and their thermal emission may extend to the NIR if the planet's orbit is at 0.15 AU (e.g.",
"Figure REF upper left).",
"In this case, the thermal emission may severely reduce the spectral features at $>2$ $\\mu $ m, because the thermal emission compensates for the absorption of stellar radiation.",
"According to Kirchhoff's law, a surface has high emissivity at wavelengths where it absorbs strongly.",
"For example, the strong absorption at 2.3 $\\mu $ m of hydrated silicates is largely compensated by the strong thermal emission at the same wavelengths (see Figure REF ).",
"For even shorter orbital periods, the sub-solar point may reach the melting temperature of minerals on the surface, which creates molten lava on the surface (see Figure REF ).",
"For solar-type stars, the transition of solid surface and molten lava happens at about 0.1 AU.",
"To date exoplanets that could be rocky mostly lie in the regime of molten lava.",
"Notably, Kepler-20f is marginally at the melting point for a granitoid surface, and is certainly solid for more refractory surfaces such as ultramafic.",
"Since it is unlikely that Kepler-20f possesses a significant gas envelope (Fressin et al.",
"2012), Kepler-20f may be a good exoplanetary candidate for solid surface characterization via prominent silicate and iron-oxide features (see Figure REF ).",
"Table: Spectral features in planetary thermal emission that are characteristic of surface compositions."
],
[
"General Reflection Spectra of Exoplanets with Solid Surfaces",
"Planetary surfaces of different compositions can have very different overall reflectivity.",
"As shown in Figure REF , a pure feldspathic surface (e.g., the lunar highlands) is very bright, with a geometric albedo $\\sim $ 0.6 and a relatively flat NIR spectrum.",
"In contrast, basaltic, Fe-oxidized or metal-rich surfaces are very dark, with a geometric albedo less than 0.3.",
"As shown in Figure REF , when these spectra are downsampled to the resolution of the J, H, and K bands, there is a strong correlation between reflectivity in the three bands for most surface types.",
"The absorption band near 1 $\\mu $ m is deep and wide for the ultramafic surface.",
"The constituents of this surface are olivine and pyroxene, both of which have Fe$^{2+}$ and absorb strongly near 1.0 $\\mu $ m. Weaker absorptions in this band can also be found in the feldspathic, basaltic, Fe-oxidized, and ice-rich silicate cases because they all contain some olivine or pyroxene in their composition.",
"The absorptions are caused by ferrous iron, commonly present at $>1$ % levels in the igneous minerals comprising these surfaces (although see section 2.2 discussion on Mercury for the possibility of non-ferrous components).",
"For the feldspathic crust, the absorption peak is shifted to $\\sim $ 1.3 $\\mu $ m, which is distinctive for Fe-anorthite, a feldspar that has been positively identified in the lunar highlands (e.g., Pieters 1986).",
"The Fe-oxidzed surface, which contains substantial hematite, shows absorptions at wavelengths shorter than 1.0 $\\mu $ m due to charge transfer absorptions for Fe$^{3+}$ .",
"Signature narrow absorption features at NIR are characteristic of water ice as well as hydrated minerals.",
"As shown in Figures REF and REF , water ice and hydrated minerals have high reflectivities at 1 $\\mu $ m and absorb strongly at longer wavelengths.",
"The drop in reflectance at wavelengths longer than $\\sim $ 2 $\\mu $ m is due to the presence of both the very strong OH stretch fundamental and overtone of the H$_2$ O bend near 3 $\\mu $ m. We note that the amount of equivalent water in hydrated minerals is small ($<\\sim $ 15%), but even minor amounts of water (or OH) lead to prominent absorption features in the infrared.",
"Sharp features due to bends and stretches of OH and H$_2$ O occur at 1.4 and 1.9 $\\mu $ m in hydrated minerals and 1.5 and 2.0 $\\mu $ m in ice.",
"The non-linear effect of intimate mixing of different minerals can serve to subdue the absorptions of present phases.",
"Darker phases are especially effective in hiding other constituents.",
"Even though bright plagioclase is often the most abundant constituent in basalt, basalt is typically dark due to nonlinear mixing between plagioclase and dark constituents present at the few percent level.",
"These nonlinear affects require careful analysis of detectability thresholds.",
"Nevertheless, as discussed further in 4.2.1 and 4.2.2 below, the absorptions of ultramafic and hydrous surface types can be distinguished in broad-band telescopic data.",
"Table: Average geometric albedo and color of an airless exoplanet fully covered by 8 types of crust listed in Table in NIR J, H and K bands.",
"The two planet scenarios correspond to one case where reflection dominates the NIR planetary flux and the other case of close-in exoplanets where thermal emission extends to the NIR.",
"Note that in the reflection-dominated case, the ultramafic surface is the only type of crust considered that has significantly higher albedo in the K band than in the J band, hydrous crusts are the only ones with lower albedos in the K band than J band, and ice-rich silicate surface is the only type of crust that has lower albedo in the H band than in the J band."
],
[
"Broad-Band Spectral Feature of the Ultramafic Surface",
"The NIR J band (1.1-1.4 $\\mu $ m) is sensitive to the detection of ultramafic crusts that contain ferrous igneous minerals on an exoplanet's surface.",
"In the J band, surface absorbers include olivine and pyroxene (see Figure REF ).",
"Made of these two types of minerals, the ultramafic crust leads to a distinctive J-band geometric albedo significantly lower than the H band and the K band (see Figure REF and Table REF ), unique in our set of representative surfaces.",
"We define a key parameter for characterization of surface composition as the difference between the K-band geometric albedo ($A_{\\rm g} ({\\rm K})$ ) and the J-band geometric albedo ($A_{\\rm g} ({\\rm J})$ ).",
"This parameter basically describes the “color\" of exoplanetary surfaces in the NIR wavelengths.",
"Note that thermal emission may contribute to the apparent geometric albedo, as shown by Figure REF and Table REF .",
"By saying “color\" we refer to the true geometric albedo with component of thermal emission properly removed.",
"For an exoplanet completely covered by ultramafic crust, $A_{\\rm g} ({\\rm K}) - A_{\\rm g} ({\\rm J}) = 0.29$ , whereas all other crustal types give value less than 0.1 (see Table REF ).",
"It is therefore very likely that an ulframafic surface on exoplanets will stand out in the reflectance spectra.",
"Macroscopic mixtures between the ultramafic surface and other types of surfaces linearly lowers the spectral contrast of the J-band absorption feature.",
"When mixed with other surfaces that have relatively flat spectra, such as basalt and granite, the J band feature of an ultramafic surface will appear to be shallower.",
"As shown in Figure REF , the key parameter $A_{\\rm g} ({\\rm K}) - A_{\\rm g} ({\\rm J})$ depends linearly on the percentage of ultramafic crust on the planet's surface.",
"A detection of $A_{\\rm g} ({\\rm K}) - A_{\\rm g} ({\\rm J})>0.2$ indicates that more than a half of the planetary surface need to be covered by ultramatic materials."
],
[
"Broad-Band Spectral Feature of the Water and Hydrated Mineral Surfaces",
"The NIR H band (1.5-1.8 $\\mu $ m) is suitable for the detection of water ice on an exoplanet's surface.",
"At the H band, the most prominent absorber is water ice.",
"In fact, water ice absorption in the H band is so strong that small amount (10%) of water ice on a very reflective surface can significantly reduce the planetary geometric albedo.",
"An ice-rich surface is the only type of surface in our sample that has lower albedo in H band than in J band (Table REF ).",
"As a result, if one observes a high J-band albedo and a low H-band albedo for an airless exoplanet, it is very likely that water ice exists on the planet's surface.",
"If $A_{\\rm g} ({\\rm H}) - A_{\\rm g} ({\\rm J}) < -0.06$ , more than half of the planetary surface is likely to be ice-rich (see Figure REF ).",
"The NIR K band (2-2.4 $\\mu $ m) is sensitive to water ice and hydrated minerals on an exoplanet's surface.",
"We find that ice-rich surfaces and aqueously altered surfaces produce strong K-band absorption, which leads to a K-band albedo smaller than the J-band albedo (see Figure REF ).",
"For example, for a generally reflective surface in K band, e.g., plagioclase- or olivine-rich, a small amount of water ice (10%) can reduce the planetary geometric albedo significantly.",
"As shown in Figure REF , the key parameter $A_{\\rm g} ({\\rm K}) - A_{\\rm g} ({\\rm J})$ depends linearly on the percentage of clay crust on the planet's surface.",
"A detection of $A_{\\rm g} ({\\rm K}) - A_{\\rm g} ({\\rm J})<-0.09$ indicates that more than a half of the planetary surface need to be covered by hydrated materials."
],
[
"Effects of Atmospheres",
"If the exoplanet has a thin atmosphere, molecular NIR absorptions will introduce additional features in its reflection and thermal emission spectra (e.g.",
"Mars reflectance spectra in Figure REF ).",
"Common molecules that actively absorb at NIR in an rocky exoplanet's atmosphere include H$_2$ O, CO$_2$ , CO, CH$_4$ , NH$_3$ , SO$_2$ , etc (e.g., Seager & Deming 2010).",
"Based on the HITRAN molecular absorption line database, for an atmosphere of Earth-like temperature and pressure, 18 parts-per-million (ppm) of H$_2$ O or 173 ppm of CH$_4$ can produce an integrated optical depth larger than unity in the J band (Hu & Seager, in preparation).",
"Similarly, 32 ppm of CH$_4$ can produce an integrated optical depth larger than unity in the H band; and 104 ppm of CO$_2$ , or 4 ppm of CH$_4$ , or 5 ppm of NH$_3$ can produce an integrated optical depth larger than unity in the K band (Hu & Seager, in preparation).",
"For the broad-band infrared photometry, the effect of atmospheres on characterization of surface composition can be very serious.",
"For example, we have demonstrated that water ice on the planet's surface produce H-band absorption, or negative $A_{\\rm g} ({\\rm H}) - A_{\\rm g} ({\\rm J})$ .",
"However, water vapor in the planet's atmosphere absorbs in J band, which will reduce the apparent contrast of the surface-related H-band feature.",
"For another example, a J-band absorption feature in reflectance spectra can be interpreted as absorption of olivine and pyroxene on the surface, or as the absorption of water vapor in the atmosphere.",
"Without any prior knowledge on the planet's atmosphere, it could be hard to draw conclusive surface compositions from broad-band spectrophotometry observations.",
"There are two possible ways to break the surface-atmosphere degeneracy in the exoplanet reflection spectra.",
"First, observing the primary transit can determine the atmospheric composition of the exoplanet via transmission spectroscopy (e.g., Seager & Deming 2010).",
"The surface minerals do not induce any spectral features in the transmission of stellar radiation observable in primary transit.",
"The difficulty of this approach is that the magnitude of atmospheric signal from the primary transit is proportional to the atmospheric scale height (e.g., Miller-Ricci et al.",
"2009; Seager 2010), therefore for a terrestrial-like atmosphere the signal may be too low to be detected.",
"Second, high-resolution spectroscopy, rather than spectrophotometry, can distinguish a surface absorption from an atmospheric absorption.",
"For reflection, the iron-related absorption in silicates are typically very wide and smooth in wavelength (see Figure REF ), but the atmospheric molecular features are typically narrow and only occur at specific wavelengths; for thermal emission, the silicate vibrational features have particular shapes which can hardly be produced by any plausible temperature structures in the planetary atmosphere (see Figure REF and Figure REF for an example of Kepler-20f).",
"As a result, one can break the surface-atmosphere degeneracy in the exoplanet reflection spectra by achieving high spectral resolution.",
"In the extreme case, the refractory rocks may even be vaporized on the dayside and form a metal vapor atmosphere, as suggested for CoRot-7b by Schaefer & Fegley (2010) and Léger et al.",
"(2011).",
"It has been shown that metal vapor atmospheres create narrow and strong metal transition lines in the visible wavelengths (e.g., Seager & Sasselov 2000).",
"It is therefore unlikely that metal vapor atmospheres impede spectral characterization of the surface beneath."
],
[
"Effects of Space Weathering",
"The surface of an airless planet is subject to space weathering.",
"The processes of space weathering include collision of galactic cosmic rays, sputtering from solar wind particles, bombardment of micrometeorites, etc.",
"Space weathering will produce a thin layer of nanophase iron on mineral grain surfaces and agglutinates in the regolith, altering surface spectral properties significantly.",
"In general, as a surface matures due to space weathering, it becomes darker, redder, and the depth of its diagnostic absorption bands is reduced (Pieters et al.",
"1993; Hapke 2001).",
"The broad-band diagnostic features considered in this paper will be subdue for an exoplanet covered by space-weathered (mature) surfaces.",
"We present several reflectance spectra of selected mature lunar and martian surfaces in Figure REF .",
"Fe$^{2+}$ absorption features at 1 $\\mu $ m and 2 $\\mu $ m become very shallow for the mature lunar mare (e.g., Figure REF vs.",
"Figure REF ).",
"We see that space weathering is very effective in reducing the diagnostic features.",
"Nonetheless, various mechanisms can refresh a planet's surface in the course of evolution, which include impact cratering, volcanism, plate tectonics, etc.",
"As a result, the fact that space weathering reduces the 1-$\\mu $ m and 2-$\\mu $ m spectral feature makes this detection indicative of active or recent re-surfacing on the planet.",
"Finally, for weathered surfaces the slope of reflectance spectra could be broadly diagnostic of the composition of weathering products and the nature of weathering.",
"As shown in Figure REF , after space weathering, the reflectance spectrum from the basaltic lunar mare have a steep upward slope, as expected.",
"In contrast, the basaltic martian lowlands become bluer after weathering.",
"The downward slope in the spectra of mature martian lowlands likely indicates silica and nanophase iron oxides coatings caused by chemical reactions in thin films of water on basaltic rocks (Mustard et al., 2005).",
"Although weathering reduces the contrast of spectral features, it causes slopes in reflectance spectra relevant to understanding planetary surface processes."
],
[
"Detection Potential",
"The characterization of airless rocky exoplanets' solid surfaces is not inherently more difficult than the characterization of their atmospheres.",
"Via the thermal emission, the silicate, iron-oxide and metal-rich surfaces manifest themselves differently in the planetary spectra, with a spectral contrast comparable to that of atmospheric CO$_2$ or O$_3$ features.",
"The secondary transit signal of an exoplanet is larger if the planet is closer to its host star.",
"As shown in Figure REF , however, an airless rocky planet cannot be too close to the star in order to keep its surface solid, which limits the potential occultation depth of the planet.",
"For an Earth-sized planet around a G type star, the occultation depth can be larger than 10 ppm in MIR, and the surface characterization by transits requires a photometric precision of 2 ppm in MIR (see Figure REF for an example of Kepler-20f).",
"Such precision and corresponding signal-to-noise ratio may be attainable by the James Webb Space Telescope (JWST) if the telescope would observe all transits in its 5-year nominal mission time (Belu et al.",
"2011).",
"Much higher secondary transit depths may be possible if the rocky exoplanets are discovered to orbit around M dwarfs, because M dwarfs have smaller radius and surface temperature compared to solar-type stars.",
"M dwarfs are the most feasible targets for the characterization of rocky exoplanets' atmospheres (e.g., Seager & Deming 2010), the same for the characterization of rocky exoplanet's surfaces.",
"For instance, an Earth-sized planet at a 0.02-AU orbit of an M dwarf with effective temperature of 3000 K and size of 0.5 $R_{}$ can have solid silicate surfaces (Figure REF ), and the secondary transit depth ranges from 50 to 100 ppm in the mid-infrared.",
"Such planets, if discovered, may be proven to have certain type of silicate surfaces (e.g., ultramafic versus granitoid) or iron-oxide surfaces by MIR spectroscopy.",
"The possibility of rocky super-Earths should not be neglected.",
"A number of transiting exoplanets with size of about 2 $R_{}$ have been detected, and the mass and radius constraints cannot exclude the possibility of a silicate-dominant composition (e.g., Lissauer et al.",
"2011; Gautier et al.",
"2012; Borucki et al.",
"2012).",
"Notably, Kepler-11b and Kepler-22b may have solid silicate surfaces according to their orbital semi-major axis (Figure REF ).",
"The secondary transit depths are 20 - 40 ppm for Kepler-11b and 5 - 10 ppm for Kepler-22b in the mid-infrared.",
"As an application of our theoretical model, the contrast of the silicate features is computed to be 10 ppm in the 7 - 13 $\\mu $ m band for Kepler-11b and 2 ppm in the 15 - 25 $\\mu $ m band for Kepler-22b.",
"Furthermore, large planets around M dwarfs, although not yet discovered, have the best potential for surface characterization.",
"For example, a 2-$R_{}$ planet orbiting at 0.02 AU around a 0.5-$R_{}$ M star having effective temperature of 3000 K, the secondary transit depth is up to $\\sim $ 10 parts-per-million (ppm) at VNIR ($\\lambda <3$ $\\mu $ m), and up to 300 ppm in MIR.",
"The planet's thermal emission flux at $\\sim 10$ $\\mu $ m can vary between 200 and 300 ppm for different types of crust.",
"As a comparison, the warm SPITZER has achieved the photometric precision as high as 65 ppm for bright stars such as 55 Cnc (e.g.",
"Demory et al.",
"2011).",
"Characterization of rocky exoplanets' surfaces via the thermal emission is therefore possible with current spacebased facilities if suitable targets are discovered.",
"Surface characterization by VNIR reflectance of rocky exoplanets is beyond the reach of current observation technology.",
"The occultation depth of secondary eclipse at VNIR is fundamentally limited by the melting temperature of silicate rocks.",
"To maintain a solid surface, the exoplanet's surface temperature should be lower than the melting temperature.",
"Averaging the emissivity over the stellar-radiation wavelengths and the thermal-emission wavelengths, the local energy balance equation for surface temperature (Eq.",
"REF ) can be solved to give the planet's sub-stellar temperature ($T_{\\rm p}$ ) as $T_{\\rm p} = T_{*} \\bigg (\\frac{\\epsilon _{VNIR}}{\\epsilon _{MIR}}\\bigg )^{1/4} \\bigg (\\frac{R_*}{D_{\\rm p}}\\bigg )^{1/2} \\ ,$ where $\\epsilon _{VNIR}$ and $\\epsilon _{MIR}$ are the Planck mean emissivity in the wavelength ranges of stellar and planetary emission, respectively.",
"Algebra from equation (REF ) and equation(REF ) gives the following expression of the occultation depth: $\\frac{F_{\\rm p}}{F_*} = A_{\\rm g} \\frac{\\epsilon _{VNIR}}{\\epsilon _{MIR}} \\bigg (\\frac{R_{\\rm p}}{R_{\\rm *}}\\bigg )^{2} \\bigg (\\frac{T_{\\rm p}}{T_*}\\bigg )^4\\ .",
"$ Without considering the thermal emission, $A_{\\rm g}$ is the true geometric albedo of the planet and is in the order of unity.",
"We see that the occultation depth increases rapidly with the surface temperature and the radius of the planet.",
"For an Earth-like exoplanet around a Sun-like star, having surface temperature of 1000 K, the secondary occultation depth is about 0.1 ppm.",
"Even for close-in rocky planets around M dwarfs (see Figure REF ), the VNIR reflection only leads to a transit depth in the order of 10 ppm.",
"To characterize surface compositions by measuring secondary transits, the required photometric precision is in the order of 1 ppm.",
"Photometric precision is fundamentally limited by the photon noise and the stellar variability.",
"For Kepler observing a star of V mag of 13, the photometric precision per 6.5-hour transit is about 20 ppm (Koch et al.",
"2010).",
"Therefore, the required photometric precision for surface characterization by VNIR reflectance is currently not possible.",
"The photometric precision might be significantly improved by accumulating a large number of transits, with spacebased broad-band photometry instruments observing certain key objects for long periods, which requires a comprehensive understanding of noises, systematics, and stellar variability.",
"In the future, direct imaging can be suitable for characterizing solid surfaces of exoplanets.",
"If direct imaging could spatially resolve the exoplanetary system, long-cadence observations can be carried out to obtain low-resolution spectra of the planet's reflection and thermal emission.",
"Specific absorption features of mafic minerals, water ice and hydrated minerals in the reflection, as well as silicate and iron-oxide features in the thermal emission presented in this paper could stand out with broad-band photometry, and might be detectable by direct imaging."
],
[
"Connection of Surface Composition to the Planetary Interior and Evolution",
"Planetary radius and mass inferred from primary transits and radial velocity measurements constrain the density of transiting exoplanets.",
"The assemblage of minerals comprising the surface of a rocky planetary body provides additional valuable constraints on understanding exoplanet interior structure and geologic evolution.",
"Planetary surface composition depends on the bulk composition of the exoplanet, the history and nature of magmatic and thermal processes, and the subsequent interaction of produced solids with atmospheric volatiles and/or the space environment.",
"We list as follows, in the order of detection likelihood, several type of planetary surfaces and their implications on the planet's interior structure and geological history.",
"A silicate surface, detectable via the prominent silicate features in the MIR thermal emission, will resolve the ambiguity of whether or not the planet has a significant envelope of volatiles.",
"Due to the uncertainties in the measurements of mass and radius, the constraints of planetary interior are always ambiguous with various interpretation acceptable by data (e.g., Rogers & Seager 2010).",
"Theoretical studies of the volatile evolution may provide additional but indirect constraints (see Fressin et al.",
"2012 for an example of Kepler-20e and Kepler-20f).",
"With the detection of planetary thermal emission in MIR and the identification of surface silicate features described in this paper, the possibility of significant volatile envelope can be readily excluded, and the planet can be confirmed to have silicate surface and mantle.",
"The surface characterization enabled by MIR spectroscopy will therefore provide an essential dimension of constraints on the interpretation of mass-radius relationship of the planet.",
"A surface bright in VNIR, inferred from low equilibrium temperature, is likely to have felsic composition.",
"Although the overall VNIR reflectivity is largely controlled by unknown factors such as surface roughness and weathering processes, no known surface process can increase the surface reflectivity.",
"In other words, the detection of a bright surface indicates that the surface is intrinsically bright when fresh.",
"As shown in Figure REF , a bright surface can be feldspathic, granitoid, and clay; all are silica-rich (termed “felsic\").",
"A felsic surface on a planet of size larger than Mars is probably produced by slow intrusion of molten lava, which indicates plate-tectonics and potential geological setting for the origins of life (Best & Christiansen 2001; Southam et al.",
"2007).",
"A caveat here is that high VNIR reflectivity can also be attributed to atmospheric effects, such as bright clouds and hazes.",
"A careful study, probably with transmission spectra, needs to be carried out to distinguish a bright solid surface against a bright cloud deck.",
"An ultramafic surface indicated by the 1-$\\mu $ m absorption feature in reflectance spectra implies either mantle overturn of the planet or very high temperature lavas.",
"For rocky planets such as Mars and Earth, mantle pressures lead to the retention of Al in garnet, making it unavailable for feldspar formation.",
"Consequently, the predicted primary crust composition, following mantle overturn to form a stable density stratrification, is dominated by Mg-rich olivines and pyroxenes (Elkins-Tanton et al., 2005).",
"As the surface matures, the 1-$\\mu $ m absorption feature diminishes.",
"As a result, if strong 1-$\\mu $ m feature is identified from the J-band absorption, one may infer that the lava eruption or mantle overturn was geologically recent.",
"The key indicator mineral is olivine and its strong contrast in the J-band due to the 1-$\\mu $ m absorptions makes detection possible.",
"A surface with hydrous materials, indicated by signature absorption features of ice or OH, implies substantial volatile inventory and constrains the planetary temperature to less than $\\sim $ 700 K over geologic time to retain these materials on the surface.",
"Hydrous materials, such as clays, would indicate liquid water having interacted with the crust, a parameter relevant to the habitability of the planet."
],
[
"Summary",
"We have developed a theoretical framework to investigate reflection and thermal emission spectra of airless rocky exoplanets.",
"We have modeled representative planetary surface types as fully covered by particulate mineral assemblages, whose mineral compositions depend on formation and evolutionary history of the planet.",
"The most prominent spectral features and their geological implications are listed in the order of detectability.",
"The silicate surface leads to Si-O vibrational features in both the 7 - 13 $\\mu $ m band and the 15 - 25 $\\mu $ m band.",
"The silicate features are universal for all rocky surfaces, the magnitude of which can be as large as 20 K in terms of brightness temperature for an airless Earth analog.",
"The silicate features allow unambiguous detection of rocky exoplanets via mid-infrared spectroscopy.",
"Iron-oxidation leads to Fe-O vibrational features in the 15 - 25 $\\mu $ m band, which indicates an oxidized surface geochemical environment.",
"The location of emissivity maxima at the short-wavelength edge (7 - 9 $\\mu $ m) of the main silicate feature uniquely indicates the silica content, which can be used to determine whether the surface is ultramafic, mafic or felsic.",
"The location of the emissivity maxima may be found via photometry using 3 or more narrow-bands in the 7 - 9 $\\mu $ m region.",
"The iron crystal-field electron band at 1 $\\mu $ m is indicative of olivine and pyroxene minerals.",
"In terms of NIR broad-band photometry, the difference between the J-band geometric albedo and the K-band geometric albedo ($A_{\\rm g}({\\rm K})-A_{\\rm g}({\\rm J})$ ) may distinguish ultramafic surfaces with these minerals, implying either mantle overturn of the planet or very high temperature lavas.",
"The OH vibrational bands beyond 2 $\\mu $ m are indicative of either surface water ice or hydrated minerals, which indicates extant or past water on the planet's surface.",
"A broad vibrational absorption band at 1.5 $\\mu $ m is diagnostic of water ice on the surface.",
"We propose that observations of rocky exoplanet reflection and thermal emission will provide valuable information on the planet's surface mineral composition.",
"Broad-band photometry of secondary eclipses in MIR may be able to identify a planetary surface of silicates or iron oxides (i.e., a rocky surface) via prominent spectral features.",
"The required photometry precision is 2 ppm for planets around G stars and 20 ppm for planets around M stars.",
"Next-generation space infrared facilities, such as the JWST, will likely be able to identify silicate surfaces around Sun-like stars.",
"Reflected stellar light can also be used to uniquely determine the mineral composition of an exoplanet's surface, and is particularly useful for identifying ferrous or hydrated surface compositions.",
"The occultation depth of the secondary eclipse in VNIR, however, is very small and beyond the reach of current technology.",
"The required photometric precision for surface characterization via reflection is less than 0.1 part per million for an Earth-sized exoplanet around a Sun-like star, and 1 part per million for an Earth-sized exoplanet around an M dwarf.",
"Although reflection of Earth-sized rocky planets are difficult to observe, planets as large as 55 Cnc e, if orbiting around M dwarfs, are best targets for long-period spacebased photometric monitoring.",
"Eventually, the unique identification of minerals on exoplanets' surfaces may rely upon direct imaging.",
"There are degeneracies among surface-related absorption features and atmospheric features, which may be broken by achieving high spectral resolution or by observing the primary transit and obtaining the transmission spectra.",
"The surface mineral composition provides important constraints on the composition and the geological history of the exoplanet, which will constitute a new dimension of exoplanet characterization.",
"Thanks to P. Isaacson for providing M$^3$ lunar spectra and W. Calvin for providing modeled water ice spectra.",
"RH is supported by NASA Earth and Space Science Fellowship (NESSF/NNX11AP47H)."
]
] | 1204.1544 |
[
[
"ALICE soft physics summary"
],
[
"Abstract Within the first two years of the LHC operation ALICE addressed the major soft physics observables in Pb-Pb and pp collisions.",
"In this contribution we present a selection of these results, with the emphasis on the bulk particle production and on particle correlations.",
"The latter subject is discussed in detail in several dedicated ALICE talks in the same workshop; the reader is referred to the corresponding contributions."
],
[
"Introduction",
"ALICE (A Large Ion Collider Experiment) is an experiment at the Large Hadron Collider dedicated to studies of QCD matter created in energetic collisions between lead nuclei [1].",
"QCD predicts that at energy densities above 1 GeV/fm$^3$ a state of deconfined quarks and gluons occurs, possibly accompanied by chiral symmetry restoration in which quarks assume their current masses.",
"Assessing the properties of the created matter requires sound understanding of the underlying collision dynamics which can be best studied via the bulk particle production observables.",
"ALICE has a high granularity, a low transverse momentum threshold $p_T ^\\mathrm {min} \\approx 0.15$ GeV/$c$ , and a good hadron identification up to several GeV/$c$ and is thus perfectly suitable for addressing soft physics observables in heavy-ion collisions.",
"The setup is shown in Fig.",
"REF .",
"The central-barrel detectors, the Inner Tracking System (ITS), the Time Projection Chamber (TPC), the Transition Radiation Detector (TRD), and the Time Of Flight (TOF), cover the full azimuthal angle range at midrapidity $|\\eta |<0.9$ .",
"Figure: The ALICE experiment at the CERN LHC.",
"The central-barrel detectors(ITS, TPC, TRD, TOF), with a pseudorapidity coverage |η|<0.9|\\eta |<0.9, addressthe particle production at midrapidity.",
"The centrality is determined fromthe charged-particle multiplicity at 1.7<|η|<5.11.7<|\\eta |<5.1.The calorimeters EMCal and PHOS and the particle identification detector HMPID have partial coverage at midrapidity.",
"The V0 detector measures charged-particle multiplicity at $-3.7<\\eta <-1.7$ and $2.8<\\eta <5.1$ and is mainly used for triggering and centrality determination.",
"Several other detector systems exist but are not relevant to the results discussed in these proceedings.",
"The collision systems and energies measured by ALICE are summarized in Table REF .",
"Table: Data sets collected by ALICE in 2009–2012 (excluding rare triggers)."
],
[
"Charged-particle production",
"The bulk particle production is a basic indicator of entropy during a nuclear collision.",
"The charged-particle pseudorapidity density in central (0-5%) Pb–Pb collisions at $\\sqrt{s_\\mathrm {NN}}$ =2.76 TeV, normalized to the number of participant pairs, is $\\mathrm {d}N_\\mathrm {ch}/\\mathrm {d}\\eta /(0.5\\langle N_\\mathrm {part} \\rangle )=8.3\\pm 0.4$ , about two times higher than in pp collisions at the same energy, Figure: Charged-particle yields at midrapidity from pp and heavy-ion collisions .and also about twice as high as in the gold–gold collisions at RHIC (Fig.",
"REF ).",
"Both in pp and Pb–Pb the charged-particle pseudorapidity density is a power-law function of $\\sqrt{s}$ .",
"The exponent is higher in nuclear collisions.",
"The observed charged-particle pseudorapidity density exceeds the predictions of most models with the initial state gluon saturation.",
"The bulk particle production was subject of the first ALICE publication from the lead campaign at the LHC [2].",
"More information is contained in the centrality dependence of the charged-particle pseudorapidity density.",
"The centrality of the collision events, in terms of the fraction of the geometric cross section, was derived from the charged-particle multiplicity seen in the V0 detector at $1.7<|\\eta |<5.1$ .",
"For this, the multiplicity distribution (Fig.",
"REF ) was fitted by a simple model assuming $fN_\\mathrm {part} +(1-f)N_\\mathrm {coll} $ particle sources, each source producing particles following a negative binomial distribution (red line).",
"Figure: Determination of the centrality using the charged-particle multiplicityat 1.7<|η|<5.11.7<|\\eta |<5.1 .The number of participants and the number of binary collisions, $N_\\mathrm {part}$ and $N_\\mathrm {coll}$ , were calculated for each impact parameter using Glauber model.",
"The centrality resolution was better than 1%.",
"For the details of this analysis see Ref. [3].",
"The centrality dependence of the normalized charged-particle pseudorapidity density turns out to coincide in shape with the one measured at RHIC (Fig.",
"REF ).",
"Figure: Charged-particle production as a function of centrality .The LHC measurement coincides with the scaled RHIC data.This is against the expectation that an increased contribution of hard processes should lead to a steeper centrality dependence at the LHC.",
"The weak centrality dependence observed is presumably related to the nuclear shadowing and is, in fact, reasonably well reproduced by models which take this phenomenon into account [4]."
],
[
"Identified hadrons",
"A better handle on the reaction dynamics can be obtained using identified hadrons.",
"Transverse momentum spectra of pions, kaons, and protons from Pb–Pb collisions are shown in Fig.",
"REF .",
"The spectra at the LHC are harder than at RHIC [5].",
"A blast wave fit allows one to associate this fact with a 10% increase of the average transverse flow velocity.",
"Figure: Transverse momentum spectra of identified hadrons from ALICE.Hydrodynamic predictions [6] underestimate the high-$p_T$ part (Fig.",
"REF ).",
"Figure: Transverse momentum spectra of identified hadrons compared tohydro predictions.They also overestimate proton yield which might indicate a too high chemical freeze-out temperature $T_{\\rm ch}$ .",
"A similar discrepancy is present in the thermal model.",
"There, however, a lower $T_{\\rm ch}$ is excluded by the $\\Xi $ and $\\Omega $ yields [7].",
"The details of the identified hadron analysis are given in Ref. [5].",
"Another exciting prospect is to look for production of light nuclei and hypernuclei.",
"Combining the specific energy loss in the TPC with the time of flight from TOF, ALICE was able to identify four anti-alpha particles in the 2010 Pb–Pb data set.",
"An order of magnitude more abundant was the antihypertriton $^3_{\\overline{\\Lambda }}\\overline{\\rm H}$ detected via its decay into $^3\\overline{\\rm He}$ and $\\pi ^+$ .",
"For antihelium $^3\\overline{\\rm He}$ , a transverse momentum spectrum was measured in the range of 1-8 GeV/$c$ .",
"The search for such composed objects with ALICE is discussed in Refs.",
"[8], [9], [10]."
],
[
"Femtoscopy",
"An increase of the spatial extension of the particle source from RHIC to LHC energies was declared a decisive test of whether the concept of “matter” is at all applicable to the system created in energetic nuclear collisions [11].",
"Spatial extension of the source of pions with a fixed momentum vector (homogeneity length, or Hanbury Brown–Twiss (HBT) radius) is accessible via the Bose-Einstein correlations aka HBT analysis.",
"The pion homogeneity volume measured by ALICE in most central 5% Pb–Pb collisions at $\\sqrt{s_\\mathrm {NN}}$ =2.76 TeV [12] is shown in Fig.",
"REF .",
"The volume is about two times larger than the analogous volume at RHIC and scales linearly with multiplicity.",
"Figure: Homogeneity volume in central gold and lead collisions at differentenergies .While the homogeneity volume of central heavy-ion collisions observed at various collision energies nicely scales with the charged-particle pseudorapidity density, including other centralities and, in particular, other collision systems leads to a violation of this scaling.",
"This is particularly clear when comparing the volumes measured at the LHC in pp and Pb–Pb at the same charged-particle pseudorapidity density (Fig.",
"REF ) [13], [14].",
"The three HBT radii always scale linearly with the cube root of $\\langle \\mathrm {d}N_\\mathrm {ch}/\\mathrm {d}\\eta \\rangle $ but the slope of the scaling is different for protons and heavy-ions.",
"This indicates that the HBT radii are not driven exclusively by the final multiplicity but are also sensitive to the initial geometry of the collision.",
"Figure: Scaling of HBT radii with the charged-particle pseudorapidity density in pp and AA collisions.The collective outward motion of particles at freeze-out is the commonly accepted explanation for the fact that the HBT radii in heavy-ion collisions decrease with increasing transverse momentum.",
"The observation of a similar $p_T$ dependence in pp collisions at RHIC suggested that either this explanation has to be revised, or collective flow exists also in pp collisions.",
"Some clarification on the subject and support to the latter statement comes from the observation that the $p_T$ dependence in pp collisions develops with increasing multiplicity [15], [13].",
"This is shown in Fig.",
"REF and discussed in detail in Refs.",
"[13], [14], [16].",
"Figure: Evolution of the transverse momentum dependence of HBT radii with multiplicity inpp collisions ."
],
[
"Elliptic flow",
"After the RHIC experiments found that the QCD matter created in heavy-ion collisions behaved like a low-viscosity fluid, a natural question arose whether this behavior would continue at higher collision energies, or whether the system would get closer to a non-interacting gas of quarks and gluons.",
"This question was answered by the ALICE measurement of the elliptic flow coefficient $v_2$ which quantifies the azimuthal anisotropy of the particle emission in non-central collisions and which, therefore, is sensitive to the early stage of the collision [17].",
"As is shown in Fig.",
"REF , Figure: Collision energy dependence of the elliptic flow .The ALICE measurement matches the trend established at lower energies.the elliptic flow at the LHC turned out to be higher than at RHIC and to follow the trend observed at lower energies.",
"The $v_2$ increase at the LHC mostly comes from the increase in $\\langle p_T \\rangle $ , as the shape of the $v_2(p_T)$ dependence remains unchanged within the $\\sqrt{s_\\mathrm {NN}}$ range from 40 to 2760 GeV.",
"The elliptic flow coefficient of identified hadrons shows a splitting characteristic to the presence of transverse radial flow (Fig.",
"REF ) [18], [19].",
"Figure: Transverse momentum dependence of the elliptic flow of pions, kaons,and protons.",
"The splitting is presumably caused by transverse radial flow.The shape of the $p_T$ dependence and the presence of the splitting are fairly well reproduced by hydrodynamics [20].",
"The calculation, however, underpredicts the elliptic flow of protons.",
"This can be cured by adding hadronic rescattering [20].",
"An important role at the LHC is played by the triangular flow.",
"Its coefficient $v_3$ only weakly depends on centrality and its direction is not related to the orientation of the event plane which points to the initial energy density fluctuations as its origin.",
"There are indications that the “Mach cone”-like azimuthal emission pattern is actually just a superposition of the elliptic and triangular collective flow [21], [22], [19]."
],
[
"Fluctuations",
"The magnitude of charge fluctuations should reflect the number of charge carriers, and thus be different for QGP and hadron gas.",
"It is puzzling that the experimental results so far were closer to the hadron gas calculation than to QGP (although the distinction is somewhat obscured by resonances that introduce anticorrelations between positive and negative charges).",
"In an ALICE analysis of Pb–Pb at $\\sqrt{s_\\mathrm {NN}}$ =2.76 TeV the net-charge fluctuations, expressed in terms of $\\nu _{(+,-,{\\rm dyn})} = \\left\\langle \\left(\\frac{N_+}{\\langle N_+ \\rangle }-\\frac{N_-}{\\langle N_- \\rangle }\\right)^2\\right\\rangle - \\frac{1}{\\langle N_+ \\rangle } - \\frac{1}{\\langle N_- \\rangle } $ and corrected for the finite acceptance, approach the QGP value in most central collisions as shown in the left panel of Fig.",
"REF (see Ref.",
"[23] for details).",
"Enhanced transverse momentum fluctuations may signal vicinity to the critical point of the QCD phase transition.",
"The ALICE measurement of the $p_T$ fluctuations, expressed via the mean covariance between transverse momenta of track pairs $C_m$ , is shown for Pb–Pb and pp collisions in the right hand panel of Fig.",
"REF .",
"Unlike the HBT radii discussed above, the relative $p_T$ fluctuations fall on a universal curve when plotted against multiplicity.",
"The dependence is of the power-law type, except for deviations at very soft pp and at central Pb–Pb collisions.",
"The latter is not reproduced by the HIJING event generator.",
"The details of this analysis are given in Ref. [24].",
"Figure: Multiplicity dependence of charge (left) and p T p_T (right) fluctuationsmeasured by ALICE.The transient magnetic field of two ions colliding at a finite impact parameter may lead to charge-dependent angular correlations.",
"The STAR experiment at RHIC reported such correlations developing when going from central to peripheral collisions of gold nuclei [25].",
"The analogous measurement in Pb–Pb performed by ALICE agrees within errors with the STAR result [26]."
],
[
"Summary",
"During the first campaign of Pb–Pb collisions at the LHC, ALICE addressed the most important soft physics observables.",
"The lead collision studies were augmented by pp measurements at several energies.",
"New insights into the reaction dynamics include an alternative interpretation of the angular emission pattern (flow rather than the “Mach cone”), the $p_T$ dependence of HBT radii developing with multiplicity in pp collisions, and the proton puzzle (lower than expected yield and elliptic flow).",
"As the c.m.s.",
"energy increase from RHIC to LHC is unprecedentedly large it is interesting to verify whether the energy dependence trends found at lower energies are still valid at the LHC.",
"The results discussed in this paper fall in three categories.",
"First, about two times larger than at RHIC are the particle yield and the homogeneity volume.",
"Second, an increase by 10-30% is observed in the transverse flow, mean transverse momentum, integrated elliptic flow, and the mass splitting of $v_2$ .",
"(The latter three may be a consequence of the first.)",
"Finally, unchanged with respect to RHIC remained the centrality dependencies of the particle yield and of $v_2$ , the multiplicity dependencies of pion HBT radii and of particle ratios, the $p_T$ dependence of $v_2$ , and the charge and $p_T$ fluctuations.",
"The author thanks the organizers for the pleasant and inspiring atmosphere during the meeting."
]
] | 1204.1224 |
[
[
"Probing nuclear symmetry energy with the sub-threshold pion production"
],
[
"Abstract Within the framework of semiclassical Boltzmann-Uehling-Uhlenbeck (BUU) transport model, we investigated the effects of symmetry energy on the sub-threshold pion using the isospin MDI interaction with the stiff and soft symmetry energies in the central collision of $^{48}$Ca + $^{48}$Ca at the incident beam energies of 100, 150, 200, 250 and 300 MeV/nucleon, respectively.",
"We find that the ratio of $\\pi^{-}/\\pi^{+}$ of sub-threshold charged pion production is greatly sensitive to the symmetry energy, particularly around 100 MeV/nucleon energies.",
"Large sensitivity of sub-threshold charged pion production to nuclear symmetry energy may reduce uncertainties of probing nuclear symmetry energy via heavy-ion collision."
],
[
"GBKsong Probing nuclear symmetry energy with the sub-threshold pion production Zhang Fang School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Liu Yang, Yong Gao-Chan and Zuo Wei Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Within the framework of semiclassical Boltzmann-Uehling-Uhlenbeck (BUU) transport model, we investigated the effects of symmetry energy on the sub-threshold pion using the isospin MDI interaction with the stiff and soft symmetry energies in the central collision of $^{48}$ Ca + $^{48}$ Ca at the incident beam energies of 100, 150, 200, 250 and 300 MeV/nucleon, respectively.",
"We find that the ratio of $\\pi ^{-}/\\pi ^{+}$ of sub-threshold charged pion production is greatly sensitive to the symmetry energy, particularly around 100 MeV/nucleon energies.",
"Large sensitivity of sub-threshold charged pion production to nuclear symmetry energy may reduce uncertainties of probing nuclear symmetry energy via heavy-ion collision.",
"25.70.-z, 21.65.Ef The density dependence of the nuclear symmetry energy is not only important for nuclear physics, but also crucial for many astrophysical processes, such as the structure of neutron stars and the dynamical evolution of proto-neutron stars [1].",
"Though considerable progress has been made recently in determining the density dependence of the nuclear symmetry energy around the normal nuclear matter density from studying the isospin diffusion in heavy-ion reactions at intermediate energies [2], [3], [4], much more work is still needed to probe the high-density behavior of the nuclear symmetry energy.",
"Fortunately, heavy-ion reactions, especially those induced by radioactive beams, provide a unique opportunity to constrain the EOS of asymmetric nuclear matter, and a number of such observables have been already identified in heavy-ion collisions induced by neutron-rich nuclei, such as the free neutron/proton ratio [5], the isospin fractionation [6], [7], the neutron-proton transverse differential flow [8], the neutron-proton correlation function [9], t/3He [10], the isospin diffusion [11], the proton differential elliptic flow [12].",
"Currently, to pin down the symmetry energy, the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University, the Geselschaft fuer Schwerionenforschung (GSI) at Darmstadt, the Rikagaku Kenkyusho (RIKEN, The Institute of Physical and Chemical Research) of Japan, as well as the Cooler Storage Ring (CSR) in Lanzhou, are planning to do related experiments.",
"Some of related information can be find via Ref. [13].",
"Recently, pion production in heavy-ion collisions has attracted much attention in the nuclear physics community [14], [15], [19], [21].",
"One important reason for this is that pion production is connected with the high-density behavior of nuclear symmetry energy, especially around pion production threshold.",
"The latter is crucial for understanding many interesting issues in both nuclear physics and astrophysics.",
"And several hadronic transport models have quantitatively shown that $\\pi ^{-}/\\pi ^{+}$ ratio is indeed sensitive to the symmetry energy, especially around pion production threshold [19] and above pion production threshold [24].",
"In the previous studies, pion production mainly from energetic heavy-ion collisions, in which pions are mainly produced above their threshold energy.",
"What is the case of effects of nuclear symmetry energy on charged pion ratio via sub-threshold production?",
"To answer this question, we studies the effects of nuclear symmetry energy on sub-threshold pion production.",
"We select $^{48}$ Ca+$^{48}$ Ca as the reaction system due to it large asymmetry.",
"Based on the isospin-dependent Boltzmann-Uehling- Uhlenbeck (IBUU) transport model, we studied the effects of the symmetry energy in the central reaction $^{48}$ Ca+$^{48}$ Ca and find that the ratio of $\\pi ^{-}/\\pi ^{+}$ of sub-threshold charged pion production, compared with above threshold case, is particularly sensitive to the symmetry energy.",
"Our present work is based on the semi-classical transport model IBUU04, in which nucleons, $\\Delta $ and $N^{\\ast }$ resonances as well as pions and their isospin-dependent dynamics are included.",
"We use the isospin-dependent in-medium nucleon-nucleon ($NN$ ) elastic cross sections from the scaling model according to nucleon effective masses.",
"For the inelastic cross sections we use the experimental data from free space $NN$ collisions since at higher incident beam energies the $NN$ cross sections have no evident effects on the slope of neutron-proton differential flow.",
"The total and differential cross sections for all other particles are taken either from experimental data or obtained by using the detailed balance formula.",
"The isospin dependent phase-space distribution functions of the particles involved are solved by using the test-particle method numerically.",
"The isospin-dependence of Pauli blockings for fermions is also considered.",
"The isospin-dependence of Pauli blockings for fermions is also considered.",
"More details can be found in Refs.",
"[4], [5], [17], [16], [18], [19].",
"In the present studies, the momentum-dependent single nucleon potential (MDI) adopted here is: $U(\\rho ,\\delta ,\\mathbf {p},\\tau ) &=&A_{u}(x)\\frac{\\rho _{\\tau ^{\\prime }}}{\\rho _{0}}+A_{l}(x)\\frac{\\rho _{\\tau }}{\\rho _{0}} \\nonumber \\\\&&+B(\\frac{\\rho }{\\rho _{0}})^{\\sigma }(1-x\\delta ^{2})-8x\\tau \\frac{B}{\\sigma +1}\\frac{\\rho ^{\\sigma -1}}{\\rho _{0}^{\\sigma }}\\delta \\rho _{\\tau ^{\\prime }} \\nonumber \\\\&&+\\frac{2C_{\\tau ,\\tau }}{\\rho _{0}}\\int d^{3}\\mathbf {p}^{\\prime }\\frac{f_{\\tau }(\\mathbf {r},\\mathbf {p}^{\\prime })}{1+(\\mathbf {p}-\\mathbf {p}^{\\prime })^{2}/\\Lambda ^{2}} \\nonumber \\\\&&+\\frac{2C_{\\tau ,\\tau ^{\\prime }}}{\\rho _{0}}\\int d^{3}\\mathbf {p}^{\\prime }\\frac{f_{\\tau ^{\\prime }}(\\mathbf {r},\\mathbf {p}^{\\prime })}{1+(\\mathbf {p}-\\mathbf {p}^{\\prime })^{2}/\\Lambda ^{2}}.",
"$ In the above equation, $\\delta =(\\rho _{n}-\\rho _{p})/(\\rho _{n}+\\rho _{p})$ is the isospin asymmetry parameter, $\\rho =\\rho _{n}+\\rho _{p}$ is the baryon density and $\\rho _{n},\\rho _{p}$ are the neutron and proton densities, respectively.",
"$\\tau =1/2(-1/2)$ for neutron (proton) and $\\tau \\ne \\tau ^{\\prime }$ , $\\sigma =4/3$ , $f_{\\tau }(\\mathbf {r},\\mathbf {p})$ is the phase-space distribution function at coordinate $\\mathbf {r}$ and momentum $\\mathbf {p}$ .",
"The parameters $A_{u}(x),A_{l}(x),B,C_{\\tau ,\\tau }$ , $C_{\\tau ,\\tau ^{\\prime }}$ and $\\Lambda $ were set by reproducing the momentum-dependent potential $U(\\rho ,\\delta ,\\mathbf {p},\\tau )$ predicted by the Gogny Hartree-Fock and/or the Brueckner-Hartree-Fock calculations.",
"The momentum-dependence of the symmetry potential steams from the different interaction strength parameters $C_{\\tau ,\\tau ^{\\prime }}$ and $C_{\\tau ,\\tau }$ for a nucleon of isospin $\\tau $ interacting, respectively, with unlike and like nucleons in the background fields, more specifically, $C_{unlike}=-103.4$ $MeV$ while $C_{like}=-11.7$ $MeV$ .",
"The parameters $A_{u}(x)$ and $A_{l}(x)$ depend on the $x$ parameter according to $Au(x)=-95.98-x\\frac{2B}{\\sigma +1}$ and $A_{l}(x) =-120.57+x\\frac{2B}{\\sigma +1}$ .",
"The saturation properties of symmetric nuclear matter and the symmetry energy of about 32 MeV at normal nuclear matter density $\\rho _{0}=0.16$ fm$^{-3}$ .",
"The incompressibility of symmetric nuclear matter at normal density is set to be 211 MeV.",
"According to essentially all microscopic model calculations, the EOS for isospin asymmetric nuclear matter can be expressed as $E(\\rho ,\\delta )=E(\\rho ,0)+E_{\\text{sym}}(\\rho )\\delta ^{2}+\\mathcal {O}(\\delta ^{4}),$ where $E(\\rho ,0)$ and $E_{\\text{sym}}(\\rho )$ are the energy per nucleon of symmetric nuclear matter and nuclear symmetry energy, respectively.",
"For a given value $x$ , with the single particle potential $U(\\rho ,\\delta ,\\mathbf {p},\\tau )$ , one can readily calculate the symmetry energy $E_{\\text{sym}}(\\rho )$ as a function of density.",
"The main reaction channels related to pion production and absorption are $&& NN \\longrightarrow NN \\nonumber \\\\&& NR \\longrightarrow NR \\nonumber \\\\&& NN \\longleftrightarrow NR \\nonumber \\\\&& R \\longleftrightarrow N\\pi ,$ where $R$ denotes $\\Delta $ or $N^{\\ast }$ resonances.",
"In the present work, we use the isospin-dependent in-medium reduced $NN$ elastic scattering cross section from the scaling model according to nucleon effective mass [19] to study the effect of symmetry energy on pion production.",
"Assuming in-medium $NN$ scattering transition matrix is the same as that in vacuum, the elastic $NN$ scattering cross section in medium $\\sigma _{NN}^{medium}$ is reduced compared with their free-space value $\\sigma _{NN}^{free}$ by a factor of [20] $R_{medium}(\\rho ,\\delta ,\\textbf {p})&\\equiv & \\sigma _{NN_{elastic}}^{medium}/\\sigma _{NN_{elastic}}^{free}\\nonumber \\\\&=&(\\mu _{NN}^{\\ast }/\\mu _{NN})^{2}.$ where $\\mu _{NN}$ and $\\mu _{NN}^{\\ast }$ are the reduced masses of the colliding nucleon pair in free space and medium, respectively.",
"For in-medium $NN$ inelastic scattering cross section, even assuming in-medium $NN \\rightarrow NR$ scattering transition matrix is the same as that in vacuum, the density of final states $D_{f}^{^{\\prime }}$ of $NR$ is very hard to calculate due to the fact that the resonance's potential in matter is presently unknown.",
"The in-medium $NN$ inelastic scattering cross section is thus quite controversial.",
"Because the purpose of present work is just study the effect of symmetry energy on pion production and charged pion ratio, to simplify the question, for the $NN$ inelastic scattering cross section we use the free $NN$ inelastic scattering cross section.",
"The effective mass of nucleon in isospin asymmetric nuclear matter is $\\frac{m_{\\tau }^{\\ast }}{m_{\\tau }}=\\left\\lbrace 1+\\frac{m_{\\tau }}{p}\\frac{dU_{\\tau }}{dp}\\right\\rbrace ^{-1}.$ From the definition and Eq.",
"(REF ), we can see that the effective mass depends not only on density and asymmetry of medium but also the momentum of nucleon.",
"Figure: (Color online) Density dependence ofnuclear symmetry energy with parameters x=1,-1x= 1, -1,respectively.Figure: (Color online) Maximal baryon densityreached in the central reaction 48 ^{48}Ca+ 48 ^{48}Ca with differentincident beam energies.Fig.",
"REF shows the density dependence of nuclear symmetry energy with parameter $x=1, -1$ , respectively.",
"As discussed in the previous part, the single particle used has an $x$ parameter, different specific $x$ parameter denotes different density dependent symmetry energy.",
"For the central reaction $^{48}$ Ca+$^{48}$ Ca, the maximal density reached is about 1.5 $\\sim $ 2 times saturation density as shown in Fig.",
"REF .",
"From Fig.",
"REF , we can also see that the low density behaviors of nuclear symmetry energy separate from each other with different $x$ parameters.",
"At the saturation point there is a cross and then they separate from each other again.",
"At lower densities, the value of symmetry energy of $x=0$ is lower than that of $x=1$ .",
"But at high densities, the value of symmetry energy of $x=0$ is higher than that of $x=1$ .",
"From Fig.",
"REF , we can see that the maximal baryon density reached in the central reaction $^{48}$ Ca+$^{48}$ Ca increases with the incident beam energy.",
"At 100 MeV/nucleon, the maximal baryon density reaches about 1.5 times saturation density and at 300 MeV/nucleon, the maximal baryon density reaches about 2 times saturation density.",
"Therefore the ratio of $\\pi ^{-}/\\pi ^{+}$ of sub-threshold charged pion production still mainly reflects the high density behavior of nuclear symmetry energy.",
"Figure: (Color online) Kinetic energydistribution of the π - /π + \\pi ^{-}/\\pi ^{+} ratio using the MDIinteraction with x=1x=1 and -1 for the central collision of 48 ^{48}Ca + 48 ^{48}Ca at the incident beam energies of 150MeV/nucleon.Figure: (Color online) Kinetic energydistribution of the π - /π + \\pi ^{-}/\\pi ^{+} ratio using the MDIinteraction with x=1x=1 and -1 for the central collision of 48 ^{48}Ca + 48 ^{48}Ca at the incident beam energies of 300MeV/nucleon.Fig.",
"REF shows the kinetic energy distribution of the $\\pi ^{-}/\\pi ^{+}$ ratio using the MDI interaction with $x=1$ and -1 for the central collision of $^{48}$ Ca + $^{48}$ Ca at the incident beam energies of 150 MeV/nucleon.",
"It is seen that at pion kinetic energies of 30 $\\sim $ 120 MeV, the $\\pi ^{-}/\\pi ^{+}$ ratio is very sensitive to the symmetry energy.",
"The soft symmetry energy (x= 1) gives larger $\\pi ^{-}/\\pi ^{+}$ ratio whereas the stiff symmetry energy ($x=-1$ ) gives smaller $\\pi ^{-}/\\pi ^{+}$ ratio.",
"This is consistent with the previous studies [14], [19], [21] for the neutron-rich reactions.",
"And we can see that the sensitivity of $\\pi ^{-}/\\pi ^{+}$ ratio to the high density behavior of symmetry energy is quite large, about 80%, is quite sensitive to cases of pion production at higher incident beam energies [22].",
"The sub-threshold charged pion production is thus a very sensitive probe of the symmetry energy.",
"Fig.",
"REF is the case at 300 MeV/nucleon incident beam energy.",
"Again we clearly see that at pion kinetic energies of 30 $\\sim $ 120 MeV, the $\\pi ^{-}/\\pi ^{+}$ ratio is sensitive to the symmetry energy.",
"The soft symmetry energy ($x=1$ ) gives larger $\\pi ^{-}/\\pi ^{+}$ ratio whereas the stiff symmetry energy ($x=-1$ ) gives smaller $\\pi ^{-}/\\pi ^{+}$ ratio.",
"Sensitivity of charged pion ratio to the symmetry energy at higher incident beam energy 300 MeV/nucleon is smaller than the case at lower incident beam energy 150 MeV/nucleon.",
"In the heavy-ion collisions at intermediate energies, generally speaking, mean-field effect and collision effect compete each other.",
"The effect of mean-field increases with the decreasement of incident beam energy.",
"Thus at lower incident beam energies effects of the symmetry energy (which is the isovector part of nuclear mean-field potential) are larger than that with higher incident beam energies.",
"From Fig.",
"REF , we can also see that there is an evident Coulomb peak [19], and around the Coulomb peak $\\pi ^{-}/\\pi ^{+}$ ratio is more sensitive to the symmetry energy.",
"At lower incident beam energy 150 MeV/nucleon, however, there is no Coulomb peak at at pion kinetic energies of 30 $\\sim $ 120 MeV.",
"Figure: (Color online) Excitation function of theπ - /π + \\pi ^{-}/\\pi ^{+} ratio using the MDI interaction with x=1x=1 and-1 for the central collision of 48 ^{48}Ca + 48 ^{48}Ca at theincident beam energies of 100, 150, 200, 250 and 300 MeV/nucleon,respectively.Fig.",
"REF shows the excitation function of the $\\pi ^{-}/\\pi ^{+}$ ratio using the MDI interaction with $x=1$ and -1 for the central collision of $^{48}$ Ca + $^{48}$ Ca at the incident beam energies of 100, 150, 200, 250 and 300 MeV/nucleon, respectively.",
"From Fig.",
"REF we can clearly see that sensitivity of charged pion ratio $\\pi ^{-}/\\pi ^{+}$ reaches the maximum at the lower incident beam energy 100 MeV/nucleon, about 100% sensitive to the symmetry energy.",
"The trend of the sensitivity of the charged pion ratio $\\pi ^{-}/\\pi ^{+}$ to symmetry energy decreases with the incident beam energy is consistent with the previous studies [14].",
"Note here that at the incident beam energy region studied here, the soft symmetry energy always corresponds large value of $\\pi ^{-}/\\pi ^{+}$ ratio and the stiff symmetry energy corresponds relative small value of $\\pi ^{-}/\\pi ^{+}$ ratio [23].",
"However, in Ref.",
"[24], Feng et al.",
"systematically investigated the pion production in heavy-ion collisions in the region of below 1 AGeV energies by using the ImIQMD model.",
"They found that the excitation functions of the $\\pi ^{-}/\\pi ^{+}$ ratio increases with the stiffness of the symmetry energy.",
"This is inconsistent with our studies qualitatively and the reasons are needed to be clarified.",
"In summary, based on the semiclassical Boltzmann-Uehling-Uhlenbeck (BUU) transport model, we studied the effects of symmetry energy on the sub-threshold pion production in the central reaction $^{48}$ Ca+$^{48}$ Ca.",
"We find that the ratio of $\\pi ^{-}/\\pi ^{+}$ of sub-threshold charged pion production is particularly sensitive to the symmetry energy with decrease of incident beam energy of heavy-ion collision.",
"The highly sensitive charged sub-threshold pion ratio to the symmetry energy may reduce the uncertainties of probing nuclear symmetry energy via heavy-ion collision.",
"The work is supported by the National Natural Science Foundation of China (10875151, 10947109, 11175219, 10740420550), the Knowledge Innovation Project (KJCX2-EW-N01) of Chinese Academy of Sciences, the Major State Basic Research Developing Program of China under No.2007CB815004, the CAS/SAFEA International Partnership Program for Creative Research Teams (CXTD-J2005-1), the Fundamental Research Fund for Physics and Mathematic of Lanzhou University LZULL200908, the Fundamental Research Funds for the Central Universities lzujbky-2010-160."
]
] | 1204.0853 |
[
[
"Matched pairs of Courant algebroids"
],
[
"Abstract We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids.",
"We consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair."
],
[
"Introduction",
"Matched pairs of algebraic structures occur naturally in several contexts of mathematics.",
"For instance, a matched pair of groupoids, introduced by Mackenzie in [10] while studying double (Lie) groupoids, are two groupoids $G\\rightrightarrows M$ and $H\\rightrightarrows M$ over the same base $M$ together with a representation of $G$ on $H$ and a representation of $H$ on $G$ compatible such that their product $G\\bowtie H$ is again a groupoid.",
"The infinitesimal version is a matched pair of Lie algebroids, which were introduced by Lu in [9] and studied by Mokri in [12].",
"They consist of two Lie algebroids $A_1$ and $A_2$ over the same base manifold $M$ , together with an $A_1$ -module structure on $A_2$ and an $A_2$ -module structure on $A_1$ , such that their direct sum $A_1\\bowtie A_2$ is again a Lie algebroid.",
"Further examples arise from the study of holomorphic Poisson structures and holomorphic Lie algebroids [7].",
"The main goal of this note is to study matched pairs of Courant algebroids.",
"More precisely, we investigate the question under which conditions the direct sum of two Courant algebroids over the same base manifold is still a Courant algebroid.",
"We derive conditions which are similar to those of Lu and Mokri [9], [12].",
"However, there is a significant difference between matched pairs of Courant algebroids and matched pairs of Lie algebroids.",
"It turns out, unlike Lie algebroids, that each component of the direct sum Courant algebroid of a matched pair is no longer a Courant subalgebroid.",
"Examples of matched pairs of Courant algebroids have appeared in literature.",
"In connection with the study of port-Hamiltonian systems, Merker considered the Courant algebroid $TM\\oplus T^*M\\oplus E\\oplus E^*$ , where $E\\rightarrow M$ be a vector bundle endowed with a flat connection $\\nabla $ [11].",
"This is indeed a very simple example of matched pairs of Courant algebroids.",
"Another class of matched pairs of Courant algebroids arise when studying holomorphic Courant algebroids along a similar line as in the study of holomorphic Lie algebroids [7].",
"In particular, we prove that a holomorphic Courant algebroid over a complex manifold $X$ is equivalent to a matched pair of (smooth) Courant algebroids satisfying certain special properties, one of which is the standard Courant algebroid $T_X^{0, 1}\\oplus (T_X^{0, 1})^*$ .",
"A third class of examples arise from the construction of regular Courant algebroids, which were recently classified by one of the authors in a joint work [1].",
"The paper is organized as follows.",
"In Section we review the notion of Courant algebroids and matched pairs of Lie algebroids.",
"In Section we introduce the definition of matched pairs of Courant algebroids.",
"In Section we give four classes of examples: Courant algebroids with flat connections, complex manifolds, holomorphic Courant algebroids, and flat regular Courant algebroids.",
"In Section , we give a definition of matched pairs of Dirac structures and show that a matched pair of Dirac structures is a matched pair of Lie algebroids.",
"A supergeometric description of matched pairs of Courant algebroids will be discussed elsewhere.",
"We would like to thank Thomas Strobl and Ping Xu for enlightening discussions."
],
[
"Preliminaries",
"We recall the definition of Courant algebroids based on the Dorfman bracket originally introduced in [4], [3].",
"For a comparison to the bracket introduced by Courant [8], see [13], [2].",
"A (real) Courant algebroid is a real vector bundle $E\\rightarrow M$ endowed with a symmetric non-degenerate $$ -bilinear form $\\langle ~,~\\rangle $ on $E$ with values in $$ , an ${\\mathbb {R}}$ -bilinear product $\\diamond $ on the space of sections ${E}$ called Dorfman bracket and a bundle map $\\rho :E\\rightarrow TM$ (over the identity) called anchor map satisfying $\\phi \\diamond (\\phi _1\\diamond \\phi _2) = \\phi _1\\diamond (\\phi \\diamond \\phi _2) +(\\phi \\diamond \\phi _1)\\diamond \\phi _2 \\\\\\phi \\diamond (f\\phi ^{\\prime }) = \\big (\\rho (\\phi )f\\big )\\phi ^{\\prime } +f(\\phi \\diamond \\phi ^{\\prime }) \\\\\\phi \\diamond \\phi = {\\mathcal {D}}\\langle \\phi ,\\phi \\rangle \\\\\\rho (\\phi )\\langle \\phi ^{\\prime },\\phi ^{\\prime }\\rangle = 2\\langle \\phi \\diamond \\phi ^{\\prime },\\phi ^{\\prime }\\rangle , $ where $\\phi ,\\phi _1,\\phi _2,\\phi ^{\\prime }\\in {E}$ , $f\\in {C^\\infty }(M)$ and ${\\mathcal {D}}:{C^\\infty }(M)\\rightarrow {E}$ is defined by the relation $\\langle {\\mathcal {D}}f,\\phi \\rangle =\\rho ^*({\\mathrm {d}}f)\\phi $ .",
"For any $\\phi ,\\psi \\in {E}$ , we have [14] rhomor()=[(),()] .Moreover $({\\mathcal {D}}f)\\diamond \\phi =0$ .",
"Complex Courant algebroids are defined similarly except that the pairing is $$ -valued, the anchor is $TM\\otimes $ -valued, and $$ -linearity replaces $$ -linearity.",
"In one of his letters to Alan Weinstein, Pavol Ševera described the following example: To each closed 3-form $H$ on $M$ is associated a Courant algebroid structure on $TM\\oplus T^*M$ with inner product inner X,Y = (X)+(Y) ,anchor map anchor (X) = X ,and Dorfman bracket bracket (X)(Y) = [X,Y]( LX-iYd+ iXY H) ,where $X,Y\\in {\\mathfrak {X}}(M)$ and $\\alpha ,\\beta \\in ^1 (M)$ .",
"This Dorfman bracket is called standard if $H=0$ and $H$ -twisted if $H\\ne 0$ .",
"Given an anchored vector bundle $E\\xrightarrow{}TM$ and a vector bundle $V$ over a smooth manifold $M$ , an $E$ -connection on $V$ is a bilinear operator $\\nabla :{E}\\otimes {V}\\rightarrow {V}$ fulfilling $\\nabla _{\\!",
"f\\psi }\\,v = f\\nabla _{\\!",
"\\psi }\\,v , \\\\\\nabla _{\\!",
"\\psi }\\,(fv) = \\big (\\rho (\\psi )f\\big ) v + f \\nabla _{\\!",
"\\psi }\\,v $ for all $f\\in {C^\\infty }(M)$ , $\\psi \\in {E}$ , and $v\\in {V}$ .",
"[[12]] Two Lie algebroids $A$ and $A^{\\prime }$ form a matched pair when a flat $A$ -connection $$ on $A^{\\prime }$ and a flat $A^{\\prime }$ -connection $$ on $A$ satisfying $_{\\!\\!\\alpha }\\,[b,c] = [_{\\!\\!\\alpha }\\,b,c] +[b, _{\\!\\!\\alpha }\\,c]+_{\\!\\!_{\\!\\!c}\\,\\alpha }\\,b -_{\\!\\!_{\\!\\!b}\\,\\alpha }\\,c \\\\_{\\!\\!a}\\,[\\beta ,\\gamma ] = [_{\\!\\!a}\\, \\beta ,\\gamma ] +[\\beta , _{\\!\\!a}\\, \\gamma ]+_{\\!\\!_{\\!\\!\\gamma }\\,a}\\,\\beta -_{\\!\\!_{\\!\\!\\beta }\\,a}\\,\\gamma $ (for all $\\alpha ,\\beta ,\\gamma \\in {A^{\\prime }}$ and $a,b,c\\in {A}$ ) are specified.",
"The following theorem is due to Mokri.",
"[[12]] Let $A$ and $A^{\\prime }$ be a pair of Lie algebroids (with anchors $\\rho _A$ and $\\rho _{A^{\\prime }}$ , and Lie brackets $[,]_A$ and $[,]_{A^{\\prime }}$ , resp.).",
"If a pair of connections $$ and $$ makes $(A,A^{\\prime })$ into a matched pair of Lie algebroids, then the vector bundle $A\\oplus A^{\\prime }$ is a Lie algebroid when endowed with the anchor map $\\rho _A + \\rho _{A^{\\prime }}$ and the bracket $ [a+\\alpha ,b+\\beta ]= ( [a,b]_A + _{\\!\\!\\alpha }\\, b - _{\\!\\!\\beta }\\, a )+ ( [\\alpha ,\\beta ]_{A^{\\prime }} + _{\\!\\!a}\\, \\beta - _{\\!\\!b}\\, \\alpha ) .$ Conversely, given a Lie algebroid $L$ and two Lie subalgebroid $A$ and $A^{\\prime }$ such that $L=A\\oplus A^{\\prime }$ as vector bundles, then $(A,A^{\\prime })$ is a matched pair of Lie algebroids whose pair of connections is determined by the following relation: $ [a,\\beta ]=-_{\\!\\!\\beta }\\,a+_{\\!\\!a}\\,\\beta .$"
],
[
"Matched pairs",
"Let $(E,{}{},\\rho ,{}{})$ be a Courant algebroid.",
"Assume we are given two subbundles $E_1,E_2$ of $E$ such that $E=E_1\\oplus E_2$ and $E_1^\\perp =E_2$ .",
"Let $_1$ (resp.",
"$_2$ ) denote the projection of $E$ onto $E_1$ (resp.",
"$E_2$ ) and $_1$ (resp.",
"$_2$ ) denote the inclusion of $E_1$ (resp.",
"$E_2$ ) into $E$ , respectively.",
"Assume that $E_k$ is itself a Courant algebroid with anchor $\\rho _k=\\rho \\circ _k$ , inner product ${a}{b}_k={_k a}{_k b}$ , and Dorfman bracket ${a}{b}{k}=_k({_k a}{_k b})$ .",
"A natural question is how to recover the Courant algebroid structure on $E$ from $E_1$ and $E_2$ .",
"The inner product and the anchor map of $E$ are uniquely determined by their restrictions to the subbundles $E_1$ and $E_2$ .",
"Indeed, for $a,b\\in {E_1}$ and $\\alpha ,\\beta \\in {E_2}$ , we have $\\langle a\\oplus \\alpha ,b\\oplus \\beta \\rangle = \\langle a,b\\rangle _1 +\\langle \\alpha ,\\beta \\rangle _2 , \\\\\\rho (a\\oplus \\alpha ) = \\rho _1(a)+\\rho _2(\\alpha ) , $ where $a\\oplus \\alpha $ is shorthand for $_1(a)+_2(\\alpha )$ .",
"The bracket on $E$ induces an $E_1$ -connection on $E_2$ : bird a =2((1 a)(2 )) and an $E_2$ -connection on $E_1$ : fish a=1((2 )(1 a)) .These connections preserve the inner products on $E_1$ and $E_2$ : $\\rho (\\alpha ){a}{b}_1= {_{\\!\\!\\alpha }\\,a}{b}_1 +{a}{_{\\!\\!\\alpha }\\,b}_1 \\\\\\rho (a){\\alpha }{\\beta }_2= {_{\\!\\!a}\\, \\alpha }{\\beta }_2 +{\\alpha }{_{\\!\\!a}\\,\\beta }_2 $ Moreover, we have parrot (a0)(0) = - a a for $a\\in {E_1}$ and $\\beta \\in {E_2}$ .",
"The first two equations follow from Leibniz rule ().",
"The next two equations follow from ad-invariance ().",
"The last equation uses in addition axiom ().",
"For all $a,b\\in {E_1}$ , we have pig (a0)(b0) = (ab1) ( 12 2 ab1 + (a,b) ) ,where $\\Omega \\colon \\wedge ^2{E_1}\\rightarrow \\Gamma (E_2)$ is defined by the relation monkey (a,b)=2 (1 a1 b-1 b1 a) .In fact $\\Omega $ is entirely determined by the connection $\\colon {E_2}\\otimes {E_1}\\rightarrow {E_1}$ through the relation donkey (a,b)2 = ( ab1 - a b1) For all $\\alpha ,\\beta \\in {E_2}$ , we have snake (0)(0) = ( 12 1 2 + (,)) (2) ,where $\\mho \\colon \\wedge ^2{E_2}\\rightarrow {E_1}$ is defined by the relation elephant (,)=1 (2 2 - 2 2 ) .In fact, $\\mho $ is entirely determined by the connection $:{E_1}\\otimes {E_2}\\rightarrow {E_2}$ through the relation shark c(,)1 = (c 2 - c 2) .",
"From axiom () we conclude that the symmetric part of the bracket is given by $ (a\\oplus 0)\\diamond (a\\oplus 0)=\\tfrac{1}{2}{\\mathcal {D}}\\langle a,a\\rangle _1=\\tfrac{1}{2}{\\mathcal {D}}_1\\langle a,a\\rangle _1\\oplus \\tfrac{1}{2}{\\mathcal {D}}_2 \\langle a,a\\rangle _1 .$ Moreover using () we get $\\langle 0\\oplus \\gamma ,(a\\oplus 0)\\diamond (b\\oplus 0)\\rangle = \\rho (a\\oplus 0)\\langle 0\\oplus \\gamma ,b\\oplus 0\\rangle -\\langle (a\\oplus 0)\\diamond (0\\oplus \\gamma ),(b\\oplus 0)\\rangle =\\langle _{\\!\\!\\gamma }\\,a,b\\rangle _1$ The formula for $\\mho $ occurs under analog considerations for $0\\oplus \\alpha $ and $0\\oplus \\beta $ .",
"As a consequence, we obtain the formula ${(a\\oplus \\alpha )}{(b\\oplus \\beta )}= \\big ( {a}{b}{1} +_{\\!\\!\\alpha }\\, b-_{\\!\\!\\beta }\\, a+\\mho (\\alpha ,\\beta )+_1{\\alpha }{\\beta }_2 \\big ) \\\\\\oplus \\big ( {\\alpha }{\\beta }{2} +_{\\!\\!a}\\, \\beta -_{\\!\\!b}\\, \\alpha +\\Omega (a,b)+_2{a}{b}_1 \\big ) ,$ which shows that the Dorfman bracket on ${E}$ can be recovered from the Courant algebroid structures on $E_1$ and $E_2$ together with the connections $$ and $$ .",
"For any $f\\in {C^\\infty }(M)$ , $b\\in {E_1}$ , and $\\beta \\in {E_2}$ , we have $_{\\!\\!_1 f}\\, \\beta =0$ and $_{\\!\\!_2 f}\\, b =0$ .",
"We have $ _{\\!\\!",
"{\\mathcal {D}}_1 f}\\,\\beta = _2(({\\mathcal {D}}_1 f\\oplus 0)\\diamond (0\\oplus \\beta )) = _2({\\mathcal {D}}f\\diamond (0\\oplus \\beta )) = 0 $ .",
"Set $ (a,b)\\alpha &:= _{\\!\\!a}\\,_{\\!\\!b}\\, \\alpha -_{\\!\\!b}\\,_{\\!\\!a}\\, \\alpha -_{\\!\\!",
"{a}{b}{1}}\\,\\alpha ,\\\\ (\\alpha ,\\beta )a &:= _{\\!\\!\\alpha }\\,_{\\!\\!\\beta }\\,a-_{\\!\\!\\beta }\\,_{\\!\\!\\alpha }\\,a -_{\\!\\!",
"{\\alpha }{\\beta }{2}}\\,a \\;.$ The curvatures $$ and $$ are sections of $\\wedge ^2 E_1^*\\otimes \\mathfrak {o}(E_2)\\cong \\wedge ^2 E_1^*\\otimes \\wedge ^2 E_2^*$ , where $\\mathfrak {o}(E_2)$ is the bundle of skew-symmetric endomorphisms of $(E_2,\\langle ~,~\\rangle _2)$ .",
"The ${C^\\infty }(M)$ -linearity of $(a,b)\\alpha $ in $\\alpha $ follows from ().",
"The curvature $$ is also ${C^\\infty }(M)$ -linear in $b$ .",
"In view of Lemma , it is also skew-symmetric with respect to $a$ and $b$ .",
"Assume we are given two Courant algebroids $E_1$ and $E_2$ over the same manifold $M$ and two connections $\\colon {E_1}\\otimes {E_2}\\rightarrow {E_2}$ and $\\colon {E_2}\\otimes {E_1}\\rightarrow {E_1}$ preserving the fiberwise metrics and satisfying $_{\\!\\!_1 f}\\, \\beta =0$ and $_{\\!\\!_2 f}\\, b =0$ for all $f\\in {C^\\infty }(M)$ , $b\\in {E_1}$ , and $\\beta \\in {E_2}$ .",
"Then the inner product (REF ), the anchor map (), and the bracket (REF ) on the direct sum $E=E_1\\oplus E_2$ satisfy (), (), and ().",
"Moreover, the Jacobi identity (REF ) is fully equivalent to the following group of properties: $\\begin{split}&_{\\!\\!\\alpha }\\,(a_1\\diamond _1 a_2)-(_{\\!\\!\\alpha }\\,a_1)\\diamond _1 a_2-a_1\\diamond _1(_{\\!\\!\\alpha }\\,a_2)-_{\\!\\!_{\\!\\!a_2}\\,\\alpha }\\,a_1 +_{\\!\\!_{\\!\\!a_1}\\,\\alpha }\\,a_2 \\\\&\\qquad =-\\mho (\\alpha ,\\Omega (a_1,a_2)+{\\mathcal {D}}_2\\langle a_1,a_2\\rangle )-{\\mathcal {D}}_1\\big \\langle \\alpha ,\\Omega (a_1,a_2)+{\\mathcal {D}}_2\\langle a_1,a_2\\rangle \\,\\big \\rangle \\end{split}\\\\\\begin{split}&_{\\!\\!a}\\,(\\alpha _1\\diamond _2\\alpha _2)-(_{\\!\\!a}\\, \\alpha _1)\\diamond _2\\alpha _2-\\alpha _1\\diamond _2(_{\\!\\!a}\\, \\alpha _2)-_{\\!\\!_{\\!\\!\\alpha _2}\\,a}\\,\\alpha _1 +_{\\!\\!_{\\!\\!\\alpha _1}\\,a}\\,\\alpha _2 \\\\&\\qquad =-\\Omega (a,\\mho (\\alpha _1,\\alpha _2)+{\\mathcal {D}}_1\\langle \\alpha _1,\\alpha _2\\rangle )-{\\mathcal {D}}_2\\big \\langle a,\\mho (\\alpha _1,\\alpha _2)+{\\mathcal {D}}_1\\langle \\alpha _1,\\alpha _2\\rangle \\,\\big \\rangle \\end{split}\\\\+= 0\\\\_{\\!\\!\\Omega (a_1,a_2)}\\, a_3 +\\text{c.p.",
"}=0 \\\\_{\\!\\!\\mho (\\alpha _1,\\alpha _2)}\\, \\alpha _3 +\\text{c.p.",
"}=0 $ Axioms (), (), and () are straightforward computations from the definition (REF ) using () and ().",
"To check the Jacobi identity, it is helpful to compute for triples of sections of $E_1$ and $E_2$ , respectively, which gives 8 cases.",
"Equations (), (), and () arise when pairing the Jacobiator with an arbitrary section of $E_1\\oplus E_2$ .",
"We are now ready to introduce the main notion of the paper.",
"Two Courant algebroids $E_1$ and $E_2$ over the same manifold $M$ together with a pair of connections satisfying the properties listed in Theorem are said to form a matched pair.",
"The induced Courant algebroid structure on the direct sum vector bundle $E_1\\oplus E_2$ is called the matched sum of the pair $(E_1,E_2)$ ."
],
[
"Courant algebroids with a flat connection",
"The first example goes back to Merker in [11].",
"Start with a Courant algebroid $(E_1,\\diamond _1,\\rho _1)$ and assume that $\\nabla $ is a metric connection on a pseudo Euclidean vector bundle $(V,\\langle ~,~\\rangle )$ over the same manifold.",
"If, in addition, $\\nabla _{\\!",
"{\\mathcal {D}}f}\\,s=0$ for all smooth functions $f\\in {C^\\infty }(M)$ , then the curvature $R$ , defined as $ R(\\psi _1,\\psi _2)v = \\nabla _{\\!",
"\\psi _1}\\,\\nabla _{\\!",
"\\psi _2}\\,v -\\nabla _{\\!",
"\\psi _2}\\,\\nabla _{\\!",
"\\psi _1}\\,v - \\nabla _{\\!",
"\\psi _1\\diamond \\psi _2}\\,v$ is an element of $\\Gamma (\\wedge ^2E_1\\otimes \\mathfrak {o}(V))$ .",
"We require this to vanish.",
"We can endow $E_2=V$ with the trivial Courant bracket and trivial anchor.",
"Furthermore, we assume that the $E_2$ -connection on $E_1$ is trivial.",
"Then, $(\\psi \\oplus v)\\diamond (\\psi ^{\\prime }\\oplus v^{\\prime }) = \\big ((\\psi \\diamond _1\\psi ^{\\prime }) +\\tfrac{1}{2}{\\mathcal {D}}\\langle v,v^{\\prime }\\rangle +\\Omega (v,v^{\\prime })\\big )\\oplus \\big (\\nabla _{\\!",
"\\psi }\\,v^{\\prime }-\\nabla _{\\!",
"\\psi ^{\\prime }}\\, v\\big ).$ This Courant algebroid plays an important role in the study of port-Hamiltonian systems [11].",
"Let $E\\rightarrow M$ be a vector bundle endowed with a flat connection $\\nabla $ .",
"The port-Hamiltonian system can be described by a Dirac structure $D\\subset CM\\oplus V$ , where $CM:=TM\\oplus T^*M$ is the standard Courant algebroid, $V:=E\\oplus E^*$ the trivial Courant algebroid with vanishing bracket and anchor, and $CM\\oplus V$ their matched pair as explained above.",
"An interesting family of Dirac structures that does not-necessarily project to Dirac structures on $CM$ or $V$ arises from a 2-form $\\omega \\in \\Omega ^2(M)$ and a so-called port map, which is a bundle map $A\\colon T^*M\\rightarrow E$ .",
"The Dirac structure is now the graph of the bundle map $ TM\\oplus E^*{\\begin{pmatrix}\\omega ^\\#& -(A\\circ \\omega ^\\#)^* \\\\ A\\circ \\omega ^\\#&0\\end{pmatrix}}T^*M\\oplus E .$ The integrability conditions are ${\\mathrm {d}}\\omega =0$ and ${\\mathrm {d}}_\\oplus (A\\circ \\omega ^\\#)=0$ , where ${\\mathrm {d}}_\\oplus $ is the Lie algebroid differential of the sum Lie algebroid $TM\\oplus E$ of the matched pair of Lie algebroids $(TM, E)$ .",
"Conversely, given a bivector field $\\pi \\in {\\wedge ^2TM}$ and a port map $A: T^*M\\rightarrow E$ , we can consider the graph of the bundle map $\\begin{pmatrix}\\pi & -A^*\\\\ A & 0\\end{pmatrix}: T^*M\\oplus E\\rightarrow T^*M\\oplus E$ .",
"This is always isotropic.",
"It is integrable iff $[\\pi ,\\pi ]=0$ , $[\\pi ,A]_\\oplus =0$ , and $[A,A]_\\oplus =0$ , where $[~,~]_\\oplus $ is the Schouten bracket of the sum Lie algebroid of the matched pair $(TM, E)$ ."
],
[
"Complex manifolds",
"Let $X$ be a complex manifold.",
"Its tangent bundle $T_X$ is a holomorphic vector bundle.",
"Consider the almost complex structure $j\\colon T_X\\rightarrow T_X$ .",
"Since $j^2=-$ , the complexified tangent bundle $T_X\\otimes {\\mathbb {C}}$ decomposes as the direct sum of $T^{0,1}_X$ and $T^{1,0}_X$ .",
"Let $\\colon T_X\\otimes {\\mathbb {C}}\\rightarrow T_X$ and $\\colon T_X\\otimes {\\mathbb {C}}\\rightarrow T_X$ denote the canonical projections.",
"The complex vector bundles $T_X$ and $T_X$ are canonically identified with one another by the bundle map $(-\\sqrt{-1} j):T_X\\rightarrow T_X$ .",
"A complex Courant algebroid is a complex vector bundle $E\\rightarrow M$ endowed with a symmetric nondegenerate $$ -bilinear form ${.}{.",
"}$ on the fibers of $E$ with values in $$ , a $$ -bilinear Dorfman bracket ${}{}$ on the space of sections ${E}$ and an anchor map $\\rho :E\\rightarrow TM\\otimes $ satisfying relations (REF ), (), (), and (), where ${\\mathcal {D}}:{C^\\infty }(M;)\\rightarrow {E}$ is defined by $\\langle {\\mathcal {D}}f,\\phi \\rangle =\\rho ^*({\\mathrm {d}}f) \\phi $ , $\\forall \\phi \\in {E}, f\\in {C^\\infty }(M;)$ The complexified tangent bundle $T_X\\otimes $ of a complex manifold $X$ is a smooth complex Lie algebroid.",
"As a vector bundle, it is the direct sum of $T_X$ and $T_X$ , which are both Lie subalgebroids.",
"The Lie bracket of (complex) vector fields induces a $T_X$ -module structure on $T_X$ : horse X Y=XY (XX, YX) ,and a $T_X$ -module structure on $T_X$ : deer Y X=YX (XX, YX) .The flatness of these connections is a byproduct of the integrability of $j$ .",
"We use the same symbol $$ to denote the induced connections on the dual spaces: $ _X \\beta =(\\mathcal {L}_X \\beta ) \\qquad (X\\in {\\mathfrak {X}}, \\beta \\in ) , \\\\_Y \\alpha =(\\mathcal {L}_Y \\alpha ) \\qquad (Y\\in {\\mathfrak {X}}, \\alpha \\in ) .$ As in Example , associated to each $H^{3,0}\\in \\Omega ^{3,0}(X)$ such that $\\partial H^{3,0}=0$ , there is a complex twisted Courant algebroid structure on $C_X=T_X\\oplus (T_X)^*$ .",
"Moreover, if two $(3,0)$ -forms $H^{3,0}$ and $H^{\\prime 3,0}$ are $\\partial $ -cohomologous, then the associated twisted Courant algebroid structures on $C_X$ are isomorphic.",
"Similarly, associated to each $H^{0, 3}\\in \\Omega ^{0, 3}(X)$ such that $\\bar{\\partial } H^{0, 3}=0$ , there is a complex twisted Courant algebroid structure on $C_X=T_X\\oplus (T_X)^*$ .",
"Let $H=H^{3,0}+H^{2, 1}+H^{1, 2}+H^{0, 3}\\in \\Omega ^3(X)\\otimes $ , where $H^{i, j}\\in \\Omega ^{i, j}(X)$ , be a closed 3-form.",
"Let $(C^{1,0}_X)_{H^{3,0}}$ be the complex Courant algebroid structure on $C_X=T_X\\oplus (T_X)^*$ twisted by $H^{3,0}$ , and $(C^{0, 1}_X)_{H^{0, 3}}$ be the complex Courant algebroid structure on $C_X=T_X\\oplus (T_X)^*$ twisted by $H^{0, 3}$ .",
"Then $(C^{1,0}_X)_{H^{3,0}}$ and $(C^{0, 1}_X)_{H^{0, 3}}$ form a matched pair of Courant algebroids, with connections given by $_{\\!\\!X\\oplus \\alpha }\\,(Y\\oplus \\beta )=&\\, \\nabla ^o_X Y \\oplus \\nabla ^o_X \\beta +H^{1, 2}(X,Y, \\cdot ) \\\\_{\\!\\!Y\\oplus \\beta }\\,(X\\oplus \\alpha )=&\\, \\nabla ^o_Y X \\oplus \\nabla ^o_Y \\alpha +H^{2, 1}(Y,X, \\cdot ) $ for all $X\\in {\\mathfrak {X}}^{1,0}$ , $\\alpha \\in ^{1,0}$ , $Y\\in {\\mathfrak {X}}^{0,1}$ .",
"The resulting matched sum Courant algebroid is isomorphic to the standard complex Courant algebroid $(T_X\\oplus T_X^*)\\otimes $ twisted by $H$ .",
"The Courant algebroid $(C_X)_{H^{3,0}}$ is the twisted complex Courant algebroid $(T_X \\oplus {T_X}^*)_{H^{3,0}}$ .",
"Therefore the 3-form $H^{3,0}$ must be closed under $\\partial $ .",
"This is true since the $\\Omega ^{(4,0)}$ -component of ${\\mathrm {d}}H$ vanishes.",
"The analog considerations are true for $(C_X)_{H^{0,3}}$ .",
"It is clear that $(T_X\\oplus T^*_X)\\otimes {\\mathbb {C}}$ can be twisted by $H$ .",
"Straightforward computations show that this induces the twisted standard brackets on $C_X$ and $C_X$ respectively.",
"Also the connections are induced by this Courant bracket."
],
[
"Holomorphic Courant algebroids",
"Let $X$ be a complex manifold.",
"We denote by $C_X=T_X\\oplus (T_X)^*$ and $C_X=T_X\\oplus (T_X)^*$ the standard Courant algebroid.",
"A holomorphic Courant algebroid consists of a holomorphic vector bundle $E$ over a complex manifold $X$ , with sheaf of holomorphic sections ${E}$ , endowed with a fiberwise $$ -valued inner product ${~}{~}$ inducing a homomorphism of sheaves of $$ -modules ${E}\\otimes _{}{E}{{~}{~}}$ , a bundle map $E\\rightarrow T_X$ inducing a homomorphism of sheaves of $$ -modules ${E}{\\rho }$ , where $$ denotes the sheaf of holomorphic sections of $T_X$ , and a homomorphism of sheaves of $$ -modules ${E}\\otimes _{}{E}{{}{}}{E}$ satisfying relations (REF ), (), (), and (), where ${\\mathcal {D}}=\\rho ^*\\circ \\partial :\\rightarrow {E}$ and $\\phi ,\\phi _1,\\phi _2,\\phi ^{\\prime }\\in {E}$ , $f\\in $ .",
"Holomorphic Courant algebroids were also studied by Gualtieri, and we refer to [5] for more details.",
"[Theorem 2.6.26 in [6]] Let $E$ be a complex vector bundle over a complex manifold $X$ , and let ${E}$ be a sheaf of $$ -modules of sections of $E\\rightarrow X$ such that, for each $x\\in X$ , there exists an open neighborhood $U\\in X$ with ${U;E}={U,}\\cdot {E}(U)$ .",
"Then the following assertions are equivalent.",
"The vector bundle $E$ is holomorphic with sheaf of holomorphic sections ${E}$ .",
"There exists a (unique) flat $T_X$ -connection $\\nabla $ on $E\\rightarrow X$ such that $ {E}(U)=\\lbrace \\sigma \\in {U;E} \\text{ s.t. }",
"\\nabla _{\\!",
"Y}\\,\\sigma =0,\\;\\forall Y\\in {\\mathfrak {X}}(X) \\rbrace .$ Let $E\\rightarrow X$ be a holomorphic vector bundle with ${E}$ as sheaf of holomorphic sections, and $\\nabla $ the corresponding flat $T_X$ -connection on $E$ .",
"Let ${~}{~}$ be a smoothly varying $$ -valued fiberwise symmetric nondegenerate $$ -bilinear form on $E$ .",
"The following assertions are equivalent: The inner product ${.}{.",
"}$ induces a homomorphism of sheaves of $$ -modules ${E}\\otimes _{}{E}\\rightarrow $ .",
"For all $\\phi ,\\psi \\in {E}$ and $Y\\in {\\mathfrak {X}}(X)$ , we have $ Y{\\phi }{\\psi }={\\nabla _{\\!",
"Y}\\,\\phi }{\\psi }+{\\phi }{\\nabla _{\\!",
"Y}\\,\\psi } .$ Let $E\\rightarrow X$ be a holomorphic vector bundle with ${E}$ as the sheaf of holomorphic sections, and $\\nabla $ the corresponding flat $T_X$ -connection on $E$ .",
"Let $\\rho $ be a homomorphism of (complex) vector bundles from $E$ to $T_X$ .",
"The following assertions are equivalent: The homomorphism $\\rho $ induces a homomorphism of sheaves of $$ -modules ${E}\\rightarrow $ .",
"The homomorphism $\\rho :E\\rightarrow T_X$ obtained from $\\rho $ by identifying $T_X$ to $T_X$ satisfies the relation $ \\rho (\\nabla _{\\!",
"Y}\\, \\phi )={Y}{\\rho \\phi } ,\\quad \\forall Y\\in {T_X},\\;\\phi \\in {E} .$ Let $(E,{.}{.",
"},\\rho ,{.}{.",
"})$ be a holomorphic Courant algebroid over a complex manifold $X$ .",
"Denote the sheaf of holomorphic sections of the underlying holomorphic vector bundle by ${E}$ and the corresponding $T_X$ -connection on $E$ by $\\nabla $ .",
"Then, there exists a unique complex Courant algebroid structure on $E\\rightarrow X$ with inner product ${~}{~}$ and anchor map $\\rho =(-i j)\\circ \\rho \\colon E\\rightarrow T_X\\subset T_X\\otimes $ , the restriction of whose Dorfman bracket to ${E}$ coincides with ${}{}$ .",
"Such a complex Courant algebroid is denoted by $E^{1,0}$ .",
"To prove the uniqueness, assume that there are two complex Courant algebroid structures on the smooth vector bundle $E\\rightarrow X$ whose anchor map and inner products coincide.",
"Moreover the Dorfman brackets coincide on ${E}$ , the sheaf of holomorphic sections.",
"Then the Dorfman brackets coincide on all of $\\Gamma (E)$ , because the holomorphic sections over a coordinate neighborhood of $X$ are dense in the set of smooth sections.",
"To argue for the existence, note that we want the Leibniz rules $\\phi \\diamond (g\\cdot \\psi ) &= \\rho (\\phi )[g]\\cdot \\psi +g\\cdot (\\phi \\diamond \\psi ) \\\\(f\\cdot \\phi )\\diamond \\psi &= -\\rho (\\psi )[f]\\cdot \\phi +f(\\phi \\diamond \\psi ) +\\langle \\phi ,\\psi \\rangle \\cdot \\rho ^{(1,0)*}{\\mathrm {d}}f$ for $\\phi ,\\psi \\in \\Gamma (E)$ and $f,g\\in {C^\\infty }(X)$ .",
"Therefore we define the Dorfman bracket of the smooth sections $f\\cdot \\phi $ and $g\\cdot \\psi $ , $\\forall \\phi ,\\psi \\in {E}$ , as $(f\\cdot \\phi )\\diamond (g\\cdot \\psi ) = f\\rho (\\phi )[g]\\cdot \\psi +fg\\cdot (\\phi \\diamond \\psi )-g\\rho (\\psi )[f]\\cdot \\phi +\\langle \\phi ,\\psi \\rangle g\\cdot \\rho ^{(1,0)*}{\\mathrm {d}}f \\;.$ First we should argue that this extension is consistent with the bracket defined for holomorphic sections.",
"Assume therefore that $f\\cdot \\phi $ is again holomorphic, where $f\\in {C^\\infty }(X)$ and $\\phi \\in {E}$ .",
"But this implies that $f$ is holomorphic on the open set where $\\phi $ is not zero.",
"Therefore on this open set the extension says $f\\rho (\\phi )[g]\\cdot \\psi &+fg\\cdot (\\phi \\diamond \\psi ) -g\\rho (\\psi )[f]\\cdot \\phi +\\langle \\phi ,\\psi \\rangle g\\cdot \\rho ^{(1,0)*}{\\mathrm {d}}f \\\\&= f\\rho (\\phi )[g]\\cdot \\psi +fg\\cdot (\\phi \\diamond \\psi ) -g\\rho (\\psi )[f]\\cdot \\phi +\\langle \\phi ,\\psi \\rangle g\\cdot \\rho ^*\\partial f$ where we have used the fact that the two anchor maps coincide for holomorphic sections and $\\rho ^{(1,0)*}{\\mathrm {d}}f=\\rho ^*\\partial f$ .",
"Therefore the term is just $(f\\cdot \\phi )\\diamond (g\\cdot \\psi )$ .",
"Since the first derivatives of $f$ and $g$ are continuous, the above relation also holds on the closure of the open domain where both $\\phi $ and $\\psi $ do not vanish.",
"But on the complement, which is open, this term is just 0, because at least one of the terms $\\phi $ or $\\psi $ as well as the bracket $\\phi \\diamond \\psi $ is also 0.",
"It follows from a straightforward but tedious computation that this bracket fulfills the four axioms (REF )–().",
"As an example, we show the proof of the axiom (): $(f\\cdot \\phi )\\diamond (f\\cdot \\phi ) &= f^2\\cdot (\\phi \\diamond \\phi ) +\\langle \\phi ,\\phi \\rangle f\\cdot \\rho ^{(1,0)*}{\\mathrm {d}}f \\\\&= \\tfrac{1}{2}\\rho ^*\\partial \\langle \\phi ,\\phi \\rangle +\\langle \\phi ,\\phi \\rangle f\\cdot \\rho ^{(1,0)*}{\\mathrm {d}}f \\\\&=\\tfrac{1}{2}\\rho ^{(1,0)*}{\\mathrm {d}}\\langle f\\cdot \\phi ,f\\cdot \\phi \\rangle $ where, in the last step, we used again the fact that $\\rho ^*\\partial =\\rho ^{(1,0)*}{\\mathrm {d}}$ when applied to holomorphic functions.",
"Define a flat $C_X$ -connection $$ on $E$ by squalor Y e= Y e ,and a flat $E$ -connection $$ on $C_X$ by filth e (Y)=o(e)(Y)=(L(e)(Y)) =((e)YL(e)) ,where $$ denotes the projection of $(T_X\\oplus T_X^*)\\otimes $ onto $T_X\\oplus (T_X)^*$ .",
"We can write the identity (REF ) as $ \\rho (_{\\!\\!Y}\\,\\phi )-_{\\!\\!\\phi }\\,Y = [Y,\\rho (\\phi )] \\;.$ Thus we have the following Let $(E,{.}{.",
"},\\rho ,{}{})$ be a holomorphic Courant algebroid over a complex manifold $X$ .",
"Denote the sheaf of holomorphic sections by ${E}$ and the corresponding flat $T_X$ -connection on $E$ by $\\nabla $ .",
"Then the two complex Courant algebroids $C_X$ and $E$ , together with the flat connections $$ and $$ given by () and (), constitute a matched pair of complex Courant algebroids, which we call the companion matched pair of the holomorphic Courant algebroid $E$ .",
"We check that $C_X\\oplus E^{1,0}$ is a Courant algebroid.",
"The axioms ()–() are straightforward computations by using the fact that the connections preserve the inner product on each Courant algebroid.",
"It remains to check that the Jacobi identity holds.",
"By Proposition , it suffices to check that the five properties (REF )–() hold.",
"All equations are trivially satisfied when $a_i\\in {E}$ and $\\alpha _i\\in \\bar{\\Theta }\\oplus \\bar{}_X\\subset \\Gamma (C_X)$ .",
"So we are left to check that multiplying the $a_i$ (or the $\\alpha _i$ respectively) with smooth functions we get the same additional terms on each side of the equations.",
"This will be demonstrated for (REF ).",
"The left hand side of (REF ) is $(a_2) := _{\\!\\!\\alpha }\\,(a_1\\diamond a_2) - (_{\\!\\!\\alpha }\\,a_1)\\diamond a_2-a_1\\diamond (_{\\!\\!\\alpha }\\,a_2) -_{\\!\\!_{\\!\\!a_2}\\,\\alpha }\\, a_1 +_{\\!\\!_{\\!\\!a_1}\\,\\alpha }\\, a_2.$ Then $ (f\\cdot a_2)=f\\cdot (a_2)+\\left(\\left[\\rho (\\alpha ),\\rho (a_1)\\right]+\\rho (_{\\!\\!a_1}\\,\\alpha )-\\rho (_{\\!\\!\\alpha }\\,a_1)\\right)[f]\\cdot a_2 $ and the second term vanishes due to (REF ).",
"For the right hand side $ (a_2) := -\\mho (\\alpha ,\\Omega (a_1,a_2)+{\\mathrm {d}}\\langle a_1,a_2\\rangle ) -{\\mathcal {D}}\\langle \\alpha , \\Omega (a_1,a_2)+{\\mathrm {d}}\\langle a_1,a_2\\rangle \\rangle ,$ we have $ (f\\cdot a_2) = f\\cdot (a_2) .$ Thus the right hand side is also ${C^\\infty }(X)$ -linear in $a_2$ .",
"Analog considerations result in coinciding terms for both sides of (REF ) when multiplying $a_1$ or $\\alpha $ by a smooth function.",
"In fact, the converse is also true.",
"Let $X$ be a complex manifold.",
"Assume that $(C_X,B)$ is a matched pair of complex Courant algebroids such that the anchor of $B$ takes values in $T_X$ , both connections $$ and $$ are flat with the $B$ -connection $$ on $C_X$ being given by $_{\\!\\!e}\\, (Y\\oplus \\eta )=\\nabla ^o_{\\rho (e)}(Y\\oplus \\eta )$ .",
"Then there is a unique holomorphic Courant algebroid $E$ such that $B=E^{1, 0}$ .",
"The flat $C$ -connection induces a flat $T$ -connection on $B$ .",
"Hence there is a holomorphic vector bundle $E\\rightarrow X$ such that $E=B$ , according to Lemma REF .",
"Since the connection $$ preserves the inner product on $B$ and is compatible with the anchor map (REF ), $E$ inherits a holomorphic inner product and a holomorphic anchor map.",
"It remains to check that the induced Dorfman bracket is holomorphic as well.",
"If $a_1,a_2\\in {E}$ are two holomorphic sections of $E$ and $\\alpha \\in \\bar{\\Theta }_X\\oplus \\bar{\\Omega }_X$ is an anti-holomorphic section of $C_X$ , then all terms of Equation (REF ), except the first one, vanish.",
"But then the first one $_{\\!\\!\\alpha }\\,(a_1\\diamond a_2)$ also has to vanish.",
"This shows that the Dorfman bracket of two holomorphic sections is itself holomorphic."
],
[
"Flat regular Courant algebroid",
"A Courant algebroid $E$ is said to be regular if $F:=\\rho (E)$ has constant rank, in which case $\\rho (E)$ is an integrable distribution on the base manifold $M$ and $\\ker \\rho /(\\ker \\rho )$ is a bundle of quadratic Lie algebras over $M$ .",
"It was proved by Chen et al.",
"[1] that the vector bundle underlying a regular Courant algebroid $E$ is isomorphic to $\\oplus \\oplus F$ , where $F$ is the integrable subbundle $\\rho (E)$ of $TM$ and $$ the bundle $\\ker \\rho /(\\ker \\rho )^\\perp $ of quadratic Lie algebras over $M$ .",
"Thus we can confine ourselves to those Courant algebroid structures on $$ whose anchor map is $ \\rho (\\xi _1+_1+x_1)=x_1 ,$ whose pseudo-metric is $ {\\xi _1+_1+_1}{\\xi _2+_2+_2} = {\\xi _1}{_2}+{\\xi _2}{_1}+{_1}{_2},$ and whose Dorfman bracket satisfies $ {} ({_1}{_2})={_1}{_2} ,$ where $\\xi _1,\\xi _2\\in $ , $_1,_2\\in $ , and $_1,_2\\in F$ .",
"We call them standard Courant algebroid structures on $$ .",
"[[1]] A Courant algebroid structure on $$ , with pseudo-metric (REF ) and anchor map (REF ), and satisfying (REF ), is completely determined by an $F$ -connection $\\nabla $ on $$ , a bundle map $:\\wedge ^2 F\\rightarrow $ , and a 3-form $\\in {\\wedge ^3 }$ satisfying the compatibility conditions $ { x}{}{}={\\nabla _{\\!",
"x}\\,{}}{}+{}{\\nabla _{\\!",
"x}\\,{}},\\\\ \\nabla _{\\!",
"x}\\,{}{}={\\nabla _{\\!",
"x}\\,}{}+{}{\\nabla _{\\!",
"x}\\,},\\\\\\big (\\nabla _{\\!",
"x}\\,(y,z)-({x}{y},z)\\big ) +c.p.=0,\\\\ \\nabla _{\\!",
"x}\\,\\nabla _{\\!",
"y}\\,-\\nabla _{\\!",
"y}\\,\\nabla _{\\!",
"x}\\, -\\nabla _{\\!",
"{x}{y}}\\,={(x,y)}{}, \\\\ =$ for all $x,y,z\\in {F}$ and $,\\in {}$ .",
"Here $$ denotes the 4-form on $F$ given by $ (x_1,x_2,x_3,x_4) = \\tfrac{1}{4} \\sum _{\\sigma \\in S_4} (\\sigma ){(x_{\\sigma (1)},x_{\\sigma (2)})}{(x_{\\sigma (3)},x_{\\sigma (4)})}, $ where $x_1,x_2,x_3,x_4\\in F$ .",
"The Dorfman bracket on $E=$ is then given by ${_1}{_2}&=(_1,_2,\\_) + (_1,_2)+{_1}{_2}, \\\\{_1}{_2}&=(_1,_2)+{_1}{_2},\\\\{\\xi _1}{_2}&={_1}{\\xi _2}={\\xi _1}{\\xi _2}=0,\\\\{_1}{\\xi _2}&=_{_1}\\xi _2, \\\\{\\xi _1}{_2}&=-_{_2}\\xi _1+{\\xi _1}{_2},\\\\{_1}{_2}&=-{_2}{_1}=-2\\mathcal {Q}(_1,_2)+\\nabla _{\\!",
"_1}\\,_2,$ for all $\\xi _1,\\xi _2\\in {}$ , $_1,_2\\in {}$ , $_1,_2\\in {F}$ .",
"Here $:{M}\\rightarrow {F^*}$ denotes the leafwise de Rham differential.",
"The maps $:{}\\otimes {}\\rightarrow {}$ and $\\mathcal {Q}: {F}\\otimes {}\\rightarrow {F^*}$ are defined, respectively, by the relation $ {(_1,_2)}{y} = 2{_2}{\\nabla _{\\!",
"y}\\,_1}$ and $ {\\mathcal {Q}(x,)}{y} = {}{(x,y)}.$ A standard Courant algebroid $E=$ is said to be flat if $$ vanishes.",
"Let $E=$ be a flat standard Courant algebroid.",
"Then $(F\\oplus F^*)_{H}$ and $$ form a matched pair of Courant algebroids, and $E$ is isomorphic to their matched sum, where $(F\\oplus F^*)_{H}$ denotes the twisted Courant algebroid on $F\\oplus F^*$ by the 3-form $H$ .",
"Here the $(F\\oplus F^*)_{H}$ connection on $$ is given by rayx+ = x ,while the $$ -connection on $(F\\oplus F^*)_{H}$ is given by manta r (X) = 2Q(X,r)0 .",
"It follows from the flatness of the Courant algebroid that the 3-form $H\\in ^3_M(F)$ is $d_F$ closed.",
"Therefore we can construct the twisted standard Courant algebroid $(F\\oplus F^*)_H$ .",
"By comparing each component of the Dorfman bracket of the sum Courant algebroid $\\oplus (F\\oplus F^*)_H$ with that of the bracket (REF )–(), we see that, for our choice () and (), both coincide."
],
[
"Matched pairs of Dirac structures",
"Recall that a Dirac structure $D$ in a Courant algebroid $(E,\\langle .,.\\rangle ,\\diamond ,\\rho )$ with split signature is a maximal isotropic and integrable subbundle.",
"Given a matched pair $(E_1, E_2)$ of Courant algebroids of split signature and Dirac structures $D_1\\subset E_1$ and $D_2\\subset E_2$ , the direct sum $D_1\\oplus D_2$ is a Dirac structure in the Courant algebroid $E_1\\oplus E_2$ iff $_{\\!\\!\\alpha }\\,a\\in {D_1}$ and $_{\\!\\!a}\\, \\alpha \\in {D_2}$ for all $\\alpha \\in {D_2}$ and $a\\in {D_1}$ .",
"It is obvious that $D_1\\oplus D_2$ is maximal isotropic.",
"It remains to check that the $E_2$ ($E_1$ )-component of the bracket of any two sections of $D_1$ is automatically in $D_2$ ($D_1$ ).",
"Indeed, we have $ \\langle 0\\oplus \\alpha ,(a\\oplus 0)\\diamond (b\\oplus 0)\\rangle = \\langle _{\\!\\!\\alpha }\\,a,b\\rangle _1 \\;.",
"$ Since $D_1$ is isotropic, the RHS vanishes.",
"It thus follows from the maximal isotropy of $D_2$ that $(a\\oplus 0)\\diamond (b\\oplus 0)$ is in $D_2$ .",
"Let $(E_1,E_2)$ be a matched pair of Courant algebroids.",
"A Dirac structure $D_1$ in $E_1$ and a Dirac structure $D_2$ in $E_2$ are said to form a matched pair of Dirac structures if their direct sum $D_1\\oplus D_2$ is a Dirac structure in the matched sum $E_1\\oplus E_2$ .",
"Let $D_1$ (resp.",
"$D_2$ ) be a Dirac structure in a Courant algebroid $E_1$ (resp.",
"$E_2$ ).",
"If $(E_1,E_2)$ is a matched pair of Courant algebroids and $(D_1,D_2)$ is a matched pair of Dirac structures, then $(D_1,D_2)$ is a matched pair of Lie algebroids.",
"Let $CM:=TM\\oplus T^*M$ be the standard Courant algebroid, $V\\rightarrow M$ a vector bundle with a flat connection $\\nabla $ .",
"Endowing $V^*$ with the dual connection, $CM$ and $V\\oplus V^*$ are matched pair of Courant algebroids.",
"Let $\\omega \\in \\Omega ^2(M)$ and $L\\in {\\wedge ^2V^*}$ .",
"$\\operatorname{Graph}\\omega \\subset CM$ is a Dirac structure in $CM$ iff ${\\mathrm {d}}\\omega =0$ .",
"On the other hand, $\\operatorname{Graph}L^\\#\\subset V\\oplus V^*$ is automatically a Dirac structure.",
"Then $(\\operatorname{Graph}\\omega , \\operatorname{Graph}L^\\#)$ is a matched pair of Dirac structure iff $[\\nabla _{\\!",
"X}\\,,L^\\#]=0$ for all $X\\in {TM}$ .",
"In this case the direct sum Dirac structure is the graph of bundle map $ TM\\oplus V {\\begin{pmatrix}\\omega ^\\#&0\\\\ 0&L^\\#\\end{pmatrix}} T^*M\\oplus V^* .$ On the other hand, we can consider the Dirac structure on $CM$ given by the graph of a Poisson bivector $\\pi $ on $M$ , and the Dirac structure on $V\\oplus V^*$ given by the graph of $\\Lambda \\in {\\wedge ^2 V}$ .",
"They form a matched pair of Dirac structures if and only if $[\\pi ,\\Lambda ]_\\oplus =0$"
]
] | 1204.1311 |
[
[
"In-medium vector mesons and low mass lepton pairs from heavy ion\n collisions"
],
[
"Abstract The rho and omega meson self-energy at finite temperature and baryon density have been analysed for an exhaustive set of mesonic and baryonic loops in the real time formulation of thermal field theory.",
"The large enhancement of spectral strength below the nominal rho mass is seen to cause a substantial enhancement in dilepton pair yield in this mass region.",
"The integrated yield after space-time evolution using relativistic hydrodynamics with quark gluon plasma in the initial state leads to a very good agreement with the experimental data from In-In collisions obtained by the NA60 collaboration."
],
[
"Introduction",
"Colliding heavy ions at ultra-relativistic energies is the only way to produce and study bulk properties of strongly interacting matter.",
"Systematic efforts, both theoretical and experimental over the last few decades [1], [2] have addressed various facets of the thermodynamics of the underlying theory $-$ QCD.",
"The spectra of hadrons emitted after freeze-out of the fireball produced in heavy ion collisions have provided us with a wealth of information.",
"This includes the recent discovery made by studying the elliptic flow of hadrons that the quark gluon plasma (QGP) produced in Au+Au collisions at RHIC actually behaves as a strongly interacting fluid [3] as opposed to a asymptotically free gas of quarks and gluons.",
"However, by virtue of the fact that electromagnetic probes (real photons and dileptons) are emitted all through the lifetime of the fireball coupled with their low rescattering probability make them penetrating probes capable of mapping the space-time history of the collision [4].",
"Lepton pairs with both invariant mass and transverse momentum information are in fact preferable to real photons.",
"Large mass pairs produced by Drell-Yan process and from the decays of heavy quarkonia are emitted early whereas pairs with low invariant mass radiated from thermal hadronic matter and Dalitz decays of hadrons are produced late in the collision.",
"The invariant mass spectra of dileptons thus carry time information as displayed explicitly in [5] using the invariant mass dependence of the elliptic flow of lepton pairs.",
"The rate of production of thermal dileptons is proportional to the two-point correlator of vector currents [6].",
"In the low invariant mass region which is dominated by lepton pairs produced during the later stages of the collision, dilepton emission takes place due to the decay of vector mesons.",
"Consequently, the spectral properties of vector mesons, the $\\rho $ meson in particular has been a subject of intense discussion [7], [4], [2], [8].",
"We find that it is only for the $\\pi -\\pi $ loop that one calculates the thermal self-energy loop for the vector mesons directly.",
"In the case of other loops typically involving one heavy and one light particle or both heavy particles one uses in general either the virial formula or the Lindhard function.",
"The sources modifying the free propagation of a particle find a unified description in terms of contributions from the branch cuts of the self energy function.",
"In addition to the unitary cut present already in vacuum, the thermal amplitude generates a new cut, the so called the Landau cut which provides the effect of collisions with the surrounding particles in the medium.",
"This formalism was applied to obtain the $\\rho $ self-energy in hot mesonic [9] and baryonic [10] matter considering an exhaustive set of one-loop diagrams.",
"The framework of real time thermal field theory that we use, enables us to evaluate the imaginary part of the self-energy from the branch cuts for real and positive values of energy and momentum without having to resort to analytic continuation as in the imaginary time approach.",
"To evaluate the baryonic loops we work with the full relativistic baryon propagator in which baryons and anti-baryons manifestly appear on an equal footing.",
"Thus the contributions from all the singularities in the self-energy function including the distant ones coming from the unitary cut of the loops involving heavy baryons are also included.",
"These are not considered in the Lindhard function approach but can contribute appreciably to the real part of the $\\rho $ meson self-energy as shown [11] in the case of a $N\\Delta $ loop.",
"The broadening of the vector meson spectral functions leads to an enhancement of lepton pair production in the invariant mass region below the $\\rho $ peak.",
"The effect of the evolving matter is handled by relativistic hydrodynamics.",
"The integrated yield is seen to agree very well with the NA60 data [12] from In-In collisions at 17.3 AGeV.",
"The article is organised as follows.",
"We will begin with a short derivation of the the dilepton emission rate in terms of the current correlation function in section 2.",
"The relation to the spectral function of vector mesons is specified in section 3.",
"This will be followed by a discussion on $\\rho $ and $\\omega $ self-energies in hot and dense matter in section 4.",
"In section 5 a brief account of the space-time evolution and initial conditions will be provided followed by the dilepton invariant mass spectra.",
"We will end with a summary in section 6."
],
[
"Dilepton emission rate and the current correlation function",
"Let us consider an initial state $|I\\rangle $ which goes to a final state $|F\\rangle $ producing a lepton pair $l^+l^-$ with momenta $p_1$ and $p_2$ respectively.",
"The dilepton multiplicity thermally averaged over initial states is given by [13] $N=\\sum _{I}\\sum _{F}|\\langle F,l^+l^-|e^{i\\int {\\cal L}_{int}d^4x}| I\\rangle |^2 \\frac{e^{-\\beta E_I}}{Z}\\frac{d^3p_1}{(2\\pi )^32E_1}\\frac{d^3p_2}{(2\\pi )^32E_2}$ where $Z=Tr[e^{-\\beta H}]$ and ${\\cal L}_{int}=e\\overline{\\psi }_l(x)\\gamma _{\\mu }\\psi _l(x)A^{\\mu }(x)+eJ_\\mu ^{h}(x)A^{\\mu }(x)$ in which $\\psi _l(x)$ is the lepton field operator and $J_\\mu ^{h}(x)$ is the electromagnetic current of hadrons.",
"Following [6], [13], [4] this expression can be put in the form $\\frac{dN}{d^4xd^4q}=\\frac{e^4L(q^2)}{3(2\\pi )^5q^4}e^{-\\beta q_0} W^{>}_{\\mu \\nu }(q)(q^{\\mu }q^{\\nu }-q^2g^{\\mu \\nu })$ where, $W^{>}_{\\mu \\nu }=\\int d^4x\\ e^{iq\\cdot x}\\langle J^h_{\\mu }(x)J^h_{\\nu }(0)\\rangle _{\\beta }$ is the Fourier transform of the thermal expectation value of the two-point correlator of the hadronic currents and $L(q^2)=(1+\\frac{2m^2_l}{q^2})\\sqrt{1-\\frac{4m^2_l}{q^2}}$ .",
"We now define $W^{<}_{\\mu \\nu }$ by interchanging the order of the currents getting $W^{<}_{\\mu \\nu }=e^{-\\beta q_0}W^{>}_{\\mu \\nu }$ .",
"Using these to define the commutator $W_{\\mu \\nu }=W^{>}_{\\mu \\nu }-W^{<}_{\\mu \\nu }=\\int d^4x\\ e^{iq\\cdot x}\\langle [J^h_{\\mu }(x),J^h_{\\nu }(0)]\\rangle _{\\beta }$ we finally have $\\frac{dN}{d^4xd^4q}=-\\frac{\\alpha ^2}{6\\pi ^3}\\frac{g^{\\mu \\nu }}{q^2}L(q^2)f_{BE}(q_0) W_{\\mu \\nu }(q_0,\\vec{q})$ for conserved hadronic currents.",
"This expression appears in different forms in the literature.",
"Replacing $f_{BE}(q_0)W_{\\mu \\nu }$ in (REF ) by (i)$W^{M}_{\\mu \\nu }=\\int d^4x\\ e^{iq\\cdot x}\\langle J^h_{\\mu }(x)J^h_{\\nu }(0)\\rangle _{\\beta }$ yields the expression in [6], by (ii) $2{\\rm {Im}} W^{R}_{\\mu \\nu }f_{BE}(q_0)$ where $W^{R}_{\\mu \\nu }=i\\int d^4x\\ e^{iq\\cdot x}\\theta (t-t^{\\prime })\\langle [J^h_{\\mu }(x),J^h_{\\nu }(0)]\\rangle _{\\beta }$ yields the rate in [14] and by (iii) $2{\\rm {Im}} W^{T}_{\\mu \\nu }/(1+e^{\\beta q_0})$ where $W^{T}_{\\mu \\nu }=i\\int d^4x\\ e^{iq\\cdot x}\\langle TJ^h_{\\mu }(x)J^h_{\\nu }(0)\\rangle _{\\beta }$ gives the rate in [15].",
"The rate given by eq.",
"(REF ) is to leading order in electromagnetic interactions but exact to all orders in the strong coupling encoded in the current correlator $W_{\\mu \\nu }$ .",
"The $q^2$ in the denominator indicates the exchange of a single virtual photon and the Bose distribution implies the thermal weight of the source."
],
[
"Current correlator and the spectral function of vector mesons",
"Thermal field theory is the appropriate framework to carry out perturbative calculations in the medium.",
"In the real time version of this formalism two point functions assume a $2\\times 2$ matrix structure on account of the shape of the contour in the complex time plane [16], [17].",
"It is convenient to begin with the quantity, $T^{ab}_{\\mu \\nu }=i\\int d^4x e^{iq\\cdot x}\\langle T_cJ^h_{\\mu }(x)J^h_{\\nu }(0)\\rangle _{\\beta }$ where $T_c$ denotes ordering along the contour and $a, b$ are the thermal indices which take values 1 and 2.",
"This quantity can be diagonalised by means of a matrix $U$ so that $T^{ab}_{{\\mu \\nu }}=U\\left(\\begin{array}{cc}\\overline{T}_{{\\mu \\nu }} & 0\\\\0 & -\\overline{T}_{{\\mu \\nu }}^*\\end{array}\\right)U~;~~~~U=\\left(\\begin{array}{cc}\\sqrt{1+n} & \\sqrt{n}\\\\\\sqrt{n} & \\sqrt{1+n}\\end{array}\\right)~,~~~n=\\frac{1}{e^{\\beta |q_0|}-1}$ where $\\overline{T}_{\\mu \\nu }$ is an analytic function.",
"Equating both sides it follows that this function is obtainable from any one (thermal) component of $T^{ab}_{{\\mu \\nu }}$ .",
"It is related e.g.",
"to the 11-component as, ${\\rm Re} \\overline{T}_{{\\mu \\nu }}(q_0,\\vec{q})={\\rm Re} T^{11}_{{\\mu \\nu }}(q_0,\\vec{q})~;~~{\\rm Im} \\overline{T}_{{\\mu \\nu }}(q_0,\\vec{q})=\\coth (\\frac{\\beta |q_0|}{2}){\\rm Im} T^{11}_{{\\mu \\nu }}(q_0,\\vec{q})~.$ Furthermore, $\\overline{T}_{{\\mu \\nu }}$ has the spectral representation [18], [16] $\\overline{T}_{{\\mu \\nu }}(q_0,\\vec{q})=\\int \\frac{dq^{\\prime }_0}{2\\pi }\\frac{W_{\\mu \\nu }(q_0^{\\prime },\\vec{q})}{q^{\\prime }_0-q_0-i\\eta \\epsilon (q_0)}$ which immediately leads to $W_{\\mu \\nu }(q_0,\\vec{q})=2\\epsilon (q_0) {\\rm Im} \\overline{T}_{{\\mu \\nu }}(q_0,\\vec{q})~.$ In the QGP where quarks and gluons are the relevant degrees of freedom, the time ordered correlation function $T^{11}_{{\\mu \\nu }}$ can be directly evaluated by writing the hadron current in terms of quarks of flavour $f$ i.e.",
"$J_\\mu ^{h}=\\sum _{f}e_f \\overline{\\psi }_f\\gamma _{\\mu }\\psi _f$ .",
"To leading order we obtain using relations (REF ) and (REF ), $g^{{\\mu \\nu }}W_{\\mu \\nu }=-\\frac{3q^2}{2\\pi }\\sum _{f}e^2_f(1-\\frac{4m^2_q}{q^2})~.$ The rate in this case corresponds to dilepton production due to process $q\\overline{q}\\rightarrow \\gamma ^*\\rightarrow l^+l^-$ .",
"To obtain the rate of dilepton production from hadronic interactions it is convenient to break up the quark current $J_\\mu ^{h}$ into parts with definite isospin $J^{h}_\\mu &=&\\frac{1}{2}(\\bar{u}\\gamma _\\mu u -\\bar{d}\\gamma _\\mu d)+\\frac{1}{6}(\\bar{u}\\gamma _\\mu u +\\bar{d}\\gamma _\\mu d) + \\cdots \\nonumber \\\\&=&J^{V}_\\mu +J^{S}_\\mu + \\cdots \\nonumber \\\\&=&J^{\\rho }_\\mu +J^{\\omega }_\\mu /3 + \\cdots $ where $V$ and $S$ denote iso-vector and iso-scalar currents and the dots denote currents comprising of quarks with strangeness and heavier flavours.",
"These currents couple to individual hadrons as well as multiparticle states with the same quantum numbers and are usually labelled by the lightest meson in the corresponding channel [19].",
"We thus identify the isovector and isoscalar currents with the $\\rho $ and $\\omega $ mesons respectively.",
"Using eq.",
"(REF ) in eq.",
"(REF ) and neglecting possible mixing between the isospin states, we write $T^{ab}_{{\\mu \\nu }}=T^{(\\rho )ab}_{{\\mu \\nu }} + T^{(\\omega )ab}_{{\\mu \\nu }}/9 + \\cdots $ where $T^{(\\rho )ab}_{{\\mu \\nu }}=i\\int d^4x e^{iq\\cdot x}\\langle T_cJ^\\rho _{\\mu }(x)J^\\rho _{\\nu }(0)\\rangle _{\\beta }$ and similarly for the scalar current.",
"The current commutator in the isospin basis follows as $W_{{\\mu \\nu }}=W^\\rho _{{\\mu \\nu }} + W^\\omega _{{\\mu \\nu }}/9 + ...$ The correlator of vector-isovector currents $W^\\rho _{{\\mu \\nu }}$ have in fact been measured [20] in vacuum along with the axial-vector correlator by studying $\\tau $ decays into even and odd number of pions.",
"The former is found to be dominated at lower energies by the prominent peak of the $\\rho $ meson followed by a continuum at high energies.",
"The axial correlator, on the other hand, is characterised by the broad hump of the $a_1$ .",
"The distinctly different shape in the two spectral densities is an experimental signature of the fact that chiral symmetry of QCD is dynamically broken by the ground state [21].",
"It is expected that this symmetry may be restored at high temperature and/or density and will be signalled by a complete overlap of the vector and axial-vector correlators [22].",
"In the medium, both the pole and the continuum structure of the correlation function gets modified [4], [23].",
"We will first evaluate the modification of the pole part due to the self-energy of vector mesons in the following.",
"Using Vector Meson Dominance the isovector and scalar currents are written in terms of dynamical field operators for the mesons allowing us to express the correlation function in terms of the exact(full) propagators of the vector mesons in the medium.",
"Writing $J_\\mu ^{\\rho }(x)=F_\\rho m_\\rho \\rho _\\mu (x)$ and $J_\\mu ^{\\omega }(x)=3F_\\omega m_\\omega \\omega _\\mu (x)$ in eq.",
"(REF ) and using eq.",
"(REF ) the current commutator becomes $W_{\\mu \\nu }=2\\epsilon (q_0)F^2_\\rho m^2_\\rho {\\rm Im} \\overline{D}^{\\rho }_{{\\mu \\nu }}+2\\epsilon (q_0)F^2_\\omega m^2_\\omega {\\rm Im} \\overline{D}^{\\omega }_{{\\mu \\nu }}+ \\cdots $ where $\\overline{D}_{{\\mu \\nu }}$ is the diagonal element of the thermal propagator matrix which is a two point function of the fields of vector mesons and is diagonalisable as in (REF ).",
"The exact propagator is obtained in terms of the in-medium self-energies using the Dyson equation.",
"Following [24] this is given by $\\overline{D}_{{\\mu \\nu }}(q)=-\\frac{P_{{\\mu \\nu }}}{q^2-m_\\rho ^2-\\overline{\\Pi }_t(q)}-\\frac{Q_{{\\mu \\nu }}/q^2}{q^2-m_\\rho ^2-q^2\\overline{\\Pi }_l(q)}-\\frac{q_\\mu q_\\nu }{q^2m_\\rho ^2}$ where $P_{{\\mu \\nu }}$ and $Q_{{\\mu \\nu }}$ are the transverse and longitudinal projections.",
"The imaginary part is then put in eqs.",
"(REF ) and then in eq.",
"(REF ) to arrive at the dilepton emission rate [24] $\\frac{dN}{d^4qd^4x}=\\frac{\\alpha ^2}{\\pi ^3q^2}L(q^2)f_{BE}(q_0) \\left[{F^2_\\rho m^2_\\rho } A_\\rho (q_0,\\vec{q})+{F^2_\\omega m^2_\\omega } A_\\omega (q_0,\\vec{q})+\\cdots \\right]$ where e.g.",
"$A_\\rho (=-g^{{\\mu \\nu }}{\\rm Im}\\overline{D}^{\\rho }_{{\\mu \\nu }}/3)$ is given by $A_\\rho =-\\frac{1}{3}\\left[\\frac{2\\sum {\\rm Im}\\Pi ^R_t}{(q^2-m_\\rho ^2-\\sum \\mathrm {Re}\\Pi ^R_t)^2+(\\sum {\\rm Im}\\Pi ^{R}_t)^2}+\\frac{q^2\\sum {\\rm Im}\\Pi ^R_l}{(q^2-m_\\rho ^2-q^2\\sum \\mathrm {Re}\\Pi ^R_l)^2+q^4(\\sum {\\rm Im}\\Pi ^{R}_l)^2}\\right]$ the sum running over all mesonic and baryonic loops.",
"Thus, the dilepton emission rate in the present scenario actually boils down to the evaluation of the self energy graphs (shown in Fig.",
"REF ).",
"The self-energy is also a $2\\times 2$ matrix and is diagonalisable by the matrix $U^{-1}$ .",
"The real and imaginary parts of the self energy function can then be obtained from the 11-component as [17], [10] $&&{\\rm Re}\\Pi ^R_{{\\mu \\nu }}(q_0,\\vec{q})={\\rm Re}\\overline{\\Pi }_{{\\mu \\nu }}(q_0,\\vec{q})={\\rm Re}\\Pi ^{11}_{{\\mu \\nu }}(q_0,\\vec{q})\\nonumber \\\\&&{\\rm Im}\\Pi ^R_{{\\mu \\nu }}(q_0,\\vec{q})=\\epsilon (q_0){\\rm Im}\\overline{\\Pi }_{\\mu \\nu }(q_0,\\vec{q})=\\tanh (\\beta q_0/2){\\rm Im}\\Pi ^{11}_{{\\mu \\nu }}(q_0,\\vec{q})$ where $\\Pi ^R$ denotes the retarded self-energy.",
"Figure: One-loop Feynman diagrams for ρ\\rho or ω\\omega self-energy involvingmesons (first figure) andbaryons (second and third figures).",
"VV stands for theρ\\rho or ω\\omega in the external line.In the internal lines, hh stands for π\\pi , ω\\omega , a 1 a_1 and h 1 h_1 mesons.For the baryonic loops, NN and BB indicate respectively nucleon andbaryonic internal lines.As indicated earlier, coupling of the hadronic current to multiparticle states gives rise to a continuum structure in the current correlation function $W^{\\mu \\nu }$ .",
"Following Shuryak [19] we take a parametrised form for this contribution and augment the dilepton emission rate with $\\frac{dN}{d^4qd^4x}=\\frac{\\alpha ^2}{\\pi ^3}L(q^2)f_{BE}(q_0) \\sum _{V=\\rho ,\\omega }A^{\\rm cont}_V.$ where $A_\\rho ^{\\rm cont}=\\frac{1}{8\\pi }\\left(1+\\frac{\\alpha _s}{\\pi }\\right)\\frac{1}{1+\\exp (\\omega _0-q_0)/\\delta }$ with $\\omega _0 = 1.3, 1.1$ GeV for $\\rho , \\omega $ and $\\delta = 0.2$ for both $\\rho $ and $\\omega $ .",
"The continuum contribution for the $\\omega $ contains an additional factor of $\\frac{1}{9}$ ."
],
[
"Spectral function and self-energy of vector mesons",
"Obtaining the in-medium spectral functions for the $\\rho $ and $\\omega $ mesons essentially involves the evaluation of the self-energies.",
"For the $\\rho $ meson these have been evaluated recently for mesonic [9] and baryonic [10] loops.",
"In the following we summarise these results for the $\\rho $ followed by those for the $\\omega $ meson."
],
[
"$\\rho $ meson",
"The 11-component of the self-energy of vector mesons for loops containing mesons or baryons can be generically expressed as $\\Pi ^{11}_{{\\mu \\nu }}(q)=i\\int \\frac{d^4k}{(2\\pi )^4}L_{{\\mu \\nu }}(k,q)\\Delta ^{11}(k,m_k)\\Delta ^{11}(p,m_p) ~.$ where the term $L_{{\\mu \\nu }}(k,q)$ contains factors from the numerators of the two propagators in the loop as well as from the two vertices and $\\Delta ^{11}(k,m_k)=\\frac{-1}{k^2-m_k^2+i\\epsilon }+a2\\pi i\\delta (k^2-m_k^2)N_1~.$ For bosons, $a=1$ and $N_1=n$ with $n(\\omega _k)=\\frac{1}{e^{\\beta \\omega _k}-1}$ whereas for fermions, $a=-1$ and $N_1=n_+\\theta (k_0)+n_-\\theta (-k_0)$ with $n_{\\pm }(\\omega _k)=\\frac{1}{e^{\\beta (\\omega _k\\mp \\mu )}+1}$ .",
"The loop momentum $p$ in (REF ) for the various cases are as shown in Fig.",
"(REF ).",
"We begin with mesonic loops.",
"The $\\rho $ self-energy for four possible $\\pi $ -$h$ loops, where $h=\\pi ,\\omega ,h_1,a_1$ have been evaluated.",
"The imaginary part of the retarded self-energy (REF ) is given by, $&&{\\rm Im}\\Pi ^{{\\mu \\nu }}_R (q_0,\\vec{q})=-\\pi \\int \\frac{d^3\\vec{k}}{(2\\pi )^3 4\\omega _\\pi \\omega _h}\\times \\nonumber \\\\&&[L^{{\\mu \\nu }}_1\\lbrace (1+n_\\pi +n_h)\\delta (q_0-\\omega _\\pi -\\omega _h)-(n_\\pi -n_h)\\delta (q_0-\\omega _\\pi +\\omega _h)\\rbrace \\nonumber \\\\&& +L^{{\\mu \\nu }}_2 \\lbrace (n_\\pi -n_h)\\delta (q_0+\\omega _\\pi -\\omega _h)-(1+n_\\pi +n_h)\\delta (q_0+\\omega _\\pi +\\omega _h)\\rbrace ]~.$ where the Bose distribution functions $n_\\pi \\equiv n(\\omega _\\pi )$ with $\\omega _\\pi =\\sqrt{\\vec{k}^2+m_\\pi ^2}$ and $n_h\\equiv n(\\omega _h)$ with $\\omega _h=\\sqrt{(\\vec{q}-\\vec{k})^2+m_h^2}$ .",
"$L^{{\\mu \\nu }}_i(i=1,2)$ are the values of $L^{{\\mu \\nu }}(k_0)$ for $k_0=\\omega _\\pi ,-\\omega _\\pi $ respectively and the vertices used in $L^{{\\mu \\nu }}(k_0)$ have been obtained from the chiral Lagrangians which are specified in [9].",
"Figure: Branch cuts of self-energy function in q 0 q_0 plane for fixed q →\\vec{q}given by πh\\pi h loop.",
"The quantities q 1,2,3 q_{1,2,3} denote the end points of cutsdiscussed in the text : q 1 =(m h +m π ) 2 +|q →| 2 q_1=\\sqrt{(m_h+m_\\pi )^2+|\\vec{q}|^2}, q 2 =(m h -m π ) 2 +|q →| 2 q_2=\\sqrt{(m_h-m_\\pi )^2+|\\vec{q}|^2} and q 3 =|q →|q_3=|\\vec{q}|.The regions, in which the four terms of eq.",
"(REF ) are non-vanishing, give rise to cuts in the self-energy function (Fig.",
"REF ).",
"These regions are controlled by the respective $\\delta $ -functions.",
"Thus, the first and the fourth terms are non-vanishing for $q^2 \\ge (m_h +m_\\pi )^2$ , giving the unitary cut, while the second and the third are non-vanishing for $q^2 \\le (m_h -m_\\pi )^2$ , giving the so-called Landau cut.",
"The unitary cut arises from the states, which can communicate with the $\\rho $ .",
"These states are, of course, the same as in vacuum, but, as we see above, the probabilities of their occurrence in the medium are modified by the distribution functions.",
"On the other hand, the Landau cut appears only in medium and arises from scattering of $\\rho $ with particles present there.",
"We note that this contribution appears as the first term in the virial expansion of the self-energy function.",
"Integrating over the angle and restricting to the kinematic region $q_0,q^2>0$ the contribution to the imaginary part coming from the unitary cut is given by [9] ${\\rm Im}\\Pi ^{{\\mu \\nu }}_{R}=-\\frac{1}{16\\pi |\\vec{q}|}\\int _{\\omega ^-_{\\pi }}^{\\omega ^+_{\\pi }}d\\omega _\\pi L^{{\\mu \\nu }}_1\\lbrace 1+n(\\omega _{\\pi })+n(q_0-\\omega _{\\pi })\\rbrace $ and that from the Landau cut is given by ${\\rm Im}\\Pi ^{{\\mu \\nu }}_{R}=-\\frac{1}{16\\pi |\\vec{q}|}\\int _{{\\omega ^-_{\\pi }}^{\\prime }}^{{\\omega ^+_{\\pi }}^{\\prime }}d\\omega _{\\pi } L^{{\\mu \\nu }}_2\\lbrace n(\\omega _{\\pi })-n(q_0+\\omega _{\\pi })\\rbrace $ where $\\omega _{\\pi \\pm }=\\frac{S^2_{\\pi }}{2q^2}(q_0\\pm |\\vec{q}|W_{\\pi })$ , ${\\omega _{\\pi \\pm }}^{\\prime }=\\frac{S^2_{\\pi }}{2q^2}(-q_0\\mp |\\vec{q}|W_{\\pi })$ with $W_{\\pi }=\\sqrt{1 -\\frac{4q^2 m_\\pi ^2}{S^4_{\\pi }}}$ and $S^2_{\\pi }=q^2-m^2_{h}+m^2_{\\pi }$ .",
"The real part can be obtained from the imaginary part by a dispersion relation or can be evaluated directly from the graphs.",
"Let us now turn to the baryonic loops in the $\\rho $ meson self-energy.",
"Using $p=k-q$ for the third diagram of Fig.",
"REF in eq.",
"(REF ) the retarded self-energy is evaluated for $NB$ loops including all spin one-half and three-half $4-$ star resonances listed by the Particle Data Group so that $B$ stands for the $N^*(1520)$ , $N^*(1650)$ , $N^*(1700)$ , $N^*(1720)$ $\\Delta (1230)$ , $\\Delta ^*(1620),$ as well as the $N(940)$ itself.",
"As before, the imaginary part can be evaluated from the discontinuities of the self-energy.",
"However, the threshold for the unitary cut for the baryon loops being far away from the $\\rho $ pole we only consider the Landau part.",
"The expression of the self-energy corresponding to the second and third diagrams of Fig.",
"REF can be obtained from one another by inverting the sign of $q$ .",
"Adding the contributions coming from the two diagrams, the imaginary part of the self energy is given by ${\\rm Im}\\Pi ^{{\\mu \\nu }}_R=-\\frac{1}{16\\pi |\\vec{q}|}\\int _{\\omega ^+_{N}}^{\\omega ^-_{N}}d\\omega _N[L^{{\\mu \\nu }}_1(-q)\\lbrace n_+(q_0+\\omega _N)-n_+(\\omega _N)\\rbrace +L^{{\\mu \\nu }}_2(q)\\lbrace n_-(q_0+\\omega _N)-n_-(\\omega _N)\\rbrace ]$ where $\\omega ^{\\pm }_{N}=\\frac{S^2_{N}}{2q^2}(-q_0 \\pm |\\vec{q}| W_{N})$ with $W_{N}=\\sqrt{1-\\frac{4q^2m_N^2}{S^4_N}}$ , $S^2_{N}=q^2-m_B^2+m_N^2$ .",
"The factors $L^{{\\mu \\nu }}_i(i=1,2)$ in eq.",
"(REF ) are the values of $L^{{\\mu \\nu }}(k_0)$ for $k_0=\\omega _N,-\\omega _N$ respectively and is obtained using gauge invariant interactions details of which are provided in [10].",
"The calculations described above treats the heavy mesons and baryon resonances in the narrow width approximation.",
"We have included their widths by folding with their vacuum spectral functions as done in [25].",
"Figure: Left panel shows the imaginary part from baryonic loops and the rightpanel shows the total contribution from meson and baryon loops.Figure: The spectral function of the ρ\\rho mesonfor (left) different values of the temperature TT and(right) different values of the baryonic chemical potential (μ\\mu ).We now present the results of numerical evaluation.",
"We plot in the left panel of Fig.",
"REF the imaginary part of the $\\rho $ self-energy for baryon loops as a function of the invariant mass $\\sqrt{q^2}\\equiv M$ for $\\vec{q}=300$ .",
"The transverse (solid line) and longitudinal (dashed line) components ${\\rm Im}\\Pi _t$ and $q^2{\\rm Im}\\Pi _l$ have been shown separately.",
"The $NN^*(1520)$ loop makes the most significant contribution followed by the $N^*(1720)$ and $\\Delta (1700)$ .",
"On the right panel is plotted the spin-averaged $\\rho $ self-energy defined by $\\Pi =\\frac{1}{3}(2\\Pi _t+q^2\\Pi _l)$ showing contributions from the baryon and meson loops for two values of the baryonic chemical potential.",
"The small positive contribution from the baryon loops to the real part is partly compensated by the negative contributions from the meson loops.",
"The substantial baryon contribution at vanishing baryonic chemical potential reflects the importance of anti-baryons.",
"We now turn to the spin averaged spectral function defined in eq.",
"(REF ).",
"First, in the left panel of Fig.REF we plot the spectral function at fixed values of the baryonic chemical potential and three-momentum for various representative values of the temperature.",
"We observe an increase of spectral strength at lower invariant masses resulting in broadening of the spectral function with increase in temperature.",
"This is purely a Landau cut contribution from the baryonic loop arising from the scattering of the $\\rho $ from baryons in the medium.",
"We then plot in the right panel of Fig.",
"REF , the spectral function for various values of the baryonic chemical potential for a fixed temperature.",
"For high values of $\\mu $ we observe an almost flattened spectral density of the $\\rho $ ."
],
[
"$\\omega $ meson",
"The $\\omega $ self-energy is evaluated along similar lines [26].",
"The $\\omega $ meson decays mostly into three pions.",
"Assuming this decay to proceed via an intermediate $\\rho $ meson i.e.",
"$\\omega \\rightarrow \\rho \\pi \\rightarrow 3\\pi $ , the dominant contribution to the $\\omega $ self-energy in meson matter can be expressed as [27] $\\Pi ^{{\\mu \\nu }}_{R(3\\pi )}(q)=\\frac{1}{N_{\\rho }}\\int ^{(q-m_{\\pi })^2}_{4m^2_{\\pi }} dM^2[\\Pi ^{{\\mu \\nu }}_{R(\\pi \\rho )}(q,M)]A_{\\rho }(M)$ where $N_{\\rho }=\\int ^{(q-m_{\\pi })^2}_{4m^2_{\\pi }} dM^2 A_{\\rho }(M^2)$ , $A_{\\rho }$ being the vacuum spectral function of the $\\rho $ .",
"Here $\\Pi ^{{\\mu \\nu }}_{R(\\pi \\rho )}(q,M)$ can be obtained by evaluating the first diagram of Fig.",
"(REF ) with $h=\\rho $ .",
"For $\\omega $ self-energy due to baryons we have evaluated $NB$ loops where $B = N^*(1440), N^*(1520), N^*(1535), N^*(1650), N^*(1720), N(940)$ .",
"The calculation proceeds similarly as in case of the $\\rho $ .",
"Figure: Same as Fig.",
"() for the ω\\omega .Figure: The spectral function of the ω\\omega for differentvalues of TT (left panel) and μ\\mu (right panel).Left panel of Fig.",
"(REF ) shows the transverse and longitudinal components of the imaginary part coming from the baryonic loops and the right panel shows the individual mesonic and baryonic loop contributions to the $\\omega $ self-energy at two different chemical potentials.",
"We see that the $N^*(1535)$ plays the dominating role mainly due to the strong coupling compared to the other baryonic resonances [27].",
"The spin averaged spectral function at different temperatures and chemical potentials are plotted in the left and right panels of Fig.",
"(REF ) respectively.",
"We observe a slight positive shift in the peak position.",
"Having obtained the in-medium spectral functions of the two most important low-lying vector mesons, we are now in a position to evaluate the static rate of dilepton production using eq.",
"(REF ).",
"Integrating over the transverse momentum $q_T$ and rapidity $y$ of the electron pairs we plot $dR/dM^2$ vs $M$ in Fig.",
"REF for $T$ =175 MeV.",
"Because of the kinematical factors multiplying the $\\rho $ spectral function the broadening appears magnified in the dilepton emission rate.",
"A significant enhancement is seen in the low mass lepton production rate due to baryonic loops over and above the mesonic ones shown by the dot-dashed line.",
"The substantial contribution from baryonic loops even for vanishing chemical potential points to the important role played by antibaryons in thermal equilibrium in systems created at RHIC and LHC energies.",
"Figure: The lepton pair emission rate at T=175T=175 MeV with and withoutbaryon (B) loops in addition to the meson (M) loops."
],
[
"Space time evolution and dilepton spectra in In-In collisions",
"Since dileptons are produced at all stages of the collision it is necessary to integrate the emission rates over the space-time volume from creation to freeze-out.",
"We assume that quark gluon plasma having a temperature $T_i$ is produced at an initial time $\\tau _i$ .",
"Hydrodynamic expansion and cooling follows up to a temperature $T_c$ where QGP undergoes a transition to hadronic matter.",
"Subsequent cooling leads to freeze-out of the fluid element into observable hadrons.",
"In the present work the fireball is taken to undergo an azimuthally symmetric transverse expansion along with a boost invariant longitudinal expansion [28].",
"The local temperature of the fluid element and the associated flow velocity as a function of the radial coordinate and proper time is obtained by solving the the energy momentum conservation equation $\\partial _\\mu \\,T^{\\mu \\nu }=0$ where $T^{\\mu \\nu }=(\\epsilon +P)u^{\\mu }u^{\\nu }\\,+\\,g^{\\mu \\nu }P$ is the energy momentum tensor for ideal fluid.",
"This set of equations are closed with the Equation of State (EoS); typically a functional relation between the pressure $P$ and the energy density $\\epsilon $ .",
"It is a crucial input which essentially controls the profile of expansion of the fireball.",
"The initial temperature is constrained by the experimentally measured hadron multiplicity through entropy conservation [29], $T_i^3(b_m)\\tau _i=\\frac{2\\pi ^4}{45\\zeta (3)\\pi \\,R_{\\perp }^2 4a_k}\\langle \\frac{dN}{dy}(b_m)\\rangle $ where $\\langle dN/dy(b_m)\\rangle $ is the hadron (predominantly pions) multiplicity for a given centrality class with maximum impact parameter $b_m$ , $R_{\\perp }$ is the transverse dimension of the system and $a_k$ is the degeneracy of the system created.",
"The initial radial velocity, $v_r(\\tau _i,r)$ and energy density, $\\epsilon (\\tau _i,r$ ) profiles are taken as [30], $v_r(\\tau _i,r)=0~~{\\rm and}~~\\epsilon (\\tau _i,r)={\\epsilon _0}/({e^{\\frac{r-R_A}{\\delta }}+1})$ where the surface thickness, $\\delta =0.5$ fm.",
"In the present work we assume $T_c = 175$ MeV [31].",
"In a quark gluon plasma to hadronic matter transition scenario, we use the bag model EoS for the QGP phase and all resonances with mass $\\le 2.5$ GeV for the hadronic gas.",
"The transition region is parametrized as [32] $s=f(T)s_q + (1-f(T))s_h~~{\\rm with}~~f(T)=\\frac{1}{2}(1+\\mathrm {tanh}(\\frac{T-T_c}{\\Gamma }))~.$ where $s_q$ ($s_h$ ) is the entropy density of the quark (hadronic) phase at $T_c$ .",
"The value of the parameter $\\Gamma $ can be varied to make the transition strong first order or continuous.",
"We take $\\Gamma =20$ MeV in this work.",
"The ratios of various hadrons measured experimentally at different $\\sqrt{s_{\\mathrm {NN}}}$ indicate that the system formed in heavy ion collisions chemically decouple at a temperature ($T_{\\mathrm {ch}}$ ) which is higher than the temperature for kinetic freeze-out ($T_f$ )determined by the transverse spectra of hadrons [33].",
"Therefore, the system remains out of chemical equilibrium from $T_{\\mathrm {ch}}$ to $T_f$ .",
"The chemical non-equilibration affects the dilepton yields through (a) the emission rate through the phase space factor and (b) the space-time evolution of the matter via the equation of state.",
"The value of the chemical potential and its inclusion in the EoS has been taken into account following [34].",
"Finally, we have obtained the dimuon yield ($dN/dM$ ) in In-In collisions at SPS at a center of mass energy of 17.3 AGeV.",
"The initial energy density is taken as 4.5 GeV/fm$^3$ corresponding to a thermalisation time $\\tau _i=0.7$ fm.",
"We take the QGP to hadronic matter transition temperature $T_c=$ 175 MeV and the freeze-out temperature $T_f=$ 120 MeV which can reproduce the slope of the hadronic spectra measured by the NA60 Collaboration.",
"In Fig.",
"(REF ) we have shown the invariant mass spectra for different transverse momentum ($p_T$ ) windows.",
"The theoretical curves agree quite well with the experimental data [12] for all the $p_T$ ranges.",
"The strong enhancement in the low $M$ domain is clearly due to the large broadening of the $\\rho $ (and $\\omega $ ) in the thermal medium which comes entirely from the Landau cut in the self-energy diagrams.",
"In the last panel we also plot for comparison the spectra calculated in [35] where the self-energy due to baryons has been evaluated following the approach of [36].",
"It is seen that this approach depicted by the dashed curve does not produce the required enhancement to explain the data in the range $0.35\\le M\\le 0.65$ GeV.",
"Figure: Dilepton invariant mass spectra for different p T p_T-bins compared with the NA60data."
],
[
"Summary",
"The self-energy of $\\rho $ and $\\omega $ mesons have been computed in nuclear matter at finite temperature and baryon density.",
"Loop graphs involving mesons, nucleons and 4-star $N^*$ and $\\Delta $ resonances up to spin 3/2 were calculated using gauge invariant interactions in the framework of real time thermal field theory to obtain the correct relativistic expressions for the self-energy.",
"The singularities in the complex energy plane were analysed and the imaginary part obtained from the Landau cut contribution.",
"Results for the real and imaginary parts at non-zero three-momenta for various values of temperature and baryonic chemical potential were shown for the individual loop graphs.",
"The spectral function of the $\\rho $ was observed to undergo a significant modification at and below the nominal rho mass which was seen to bring about a large enhancement of lepton pair yield in this region.",
"After a space-time evolution using relativistic hydrodynamics, the invariant mass spectra for various $p_T$ windows was found to be in very good agreement with the experimental data obtained in In-In collisions at 17.3 AGeV."
]
] | 1204.0893 |
[
[
"The origin of the late rebrightening in GRB 080503"
],
[
"Abstract GRB 080503, detected by Swift, belongs to the class of bursts whose prompt phase consists of an initial short spike followed by a longer soft tail.",
"It did not show any transition to a regular afterglow at the end of the prompt emission but exhibited a surprising rebrightening after one day.",
"We aim to explain this rebrightening with two different scenarios - refreshed shocks or a density clump in the circumburst medium - and two models for the origin of the afterglow, the standard one where it comes from the forward shock, and an alternative one where it results from a long-lived reverse shock.",
"We computed afterglow light curves either using a single-zone approximation for the shocked region or a detailed multizone method that more accurately accounts for the compression of the material.",
"We find that in several of the considered cases the detailed model must be used to obtain a reliable description of the shock dynamics.",
"The density clump scenario is not favored.",
"We confirm previous results that the presence of the clump has little effect on the forward shock emission, except if the microphysics parameters evolve when the shock enters the clump.",
"Moreover, we find that the rebrightening from the reverse shock is also too weak when it is calculated with the multi-zone method.",
"On the other hand, in the refreshed-shock scenario both the forward and reverse shock models provide satisfactory fits of the data under some additional conditions on the distribution of the Lorentz factor in the ejecta and the beaming angle of the relativistic outflow."
],
[
"Introduction",
"Short bursts with a duration of less than 2 s represent about 25% of the BATSE sample [24] but had to wait until 2005 (i.e.",
"eight years after long bursts) to enter the afterglow era [15], [13], [22], [7].",
"This is due to two reasons: (i) short bursts tend to emit less photons because of harder spectra and lower fluences [24], which makes their localization more difficult; (ii) they have fainter afterglows, which are harder to detect.",
"Following the discovery of the first afterglows, it appeared that the nature of the host galaxy, the location of the afterglow, and the absence of a supernova imprint in the visible light curve (even when the host is located at a redshift below 0.5) were indicative of progenitors that were different from those of long bursts [15], [13], [49].",
"Several short burts are clearly associated to elliptical galaxies [8], [6] while others with accurate positions appear to have no coincident hosts, which clearly excludes progenitors belonging to the young population and favors merger scenarios involving compact objects [35], [32], [44], [3].",
"About 40% of the short bursts have no detectable afterglows after about 1000 s while the other 60% [45] have long-lasting afterglows comparable to those of long bursts (see the review on short bursts by [33] and references therein).",
"If short bursts indeed result from the merging of two compact objects, the kick received when the black hole or neutron star components formed in past supernova explosions [23], [3] can allow the system to reach the low-density outskirts of the host galaxy (or even to leave the galaxy) before coalescence occurs.",
"This can naturally explains why some afterglows are so dim or have no coincident host (the observational data presented in [52] show that the galactocentric offset of short bursts is on average much larger than for long bursts).",
"The direct and simple connection between duration and progenitor class became fuzzier when it was found that in some bursts an initial short duration spike is followed by a soft tail lasting several tens of seconds [2], [55], [36].",
"It was then suggested [56], [14] to introduce a new terminology that would distinguish type-I bursts resulting from mergers and type-II events coming from collapsars.",
"In the absence of a detected afterglow that can help to relate the burst to either the old or young stellar population, a vanishing spectral lag (for both the spike and the extended emission) has been proposed as an indicator for a type-I identification [14].",
"GRB 080503 belongs to the class of short bursts with extended emission.",
"The extended emission ended with a steep decay that was not immediately followed by a standard afterglow component.",
"A peculiar feature in GRB 080503 is that after remaining undetected for about one day, it showed a spectacular rebrightening (both in X-rays and the visible), which could be followed for five days in the visible.",
"[38] described in great detail the multi-wavelength data they collected for this event and discussed different possibilities that could account for the late rebrightening: (i) a delayed rise of the afterglow due to an extremely low density of the surrounding medium; (ii) the presence of a density clump in the burst environment; (iii) an off-axis jet that becomes visible when relativistic beaming has been reduced by deceleration (see e.g.",
"[20]); (iv) a refreshed shock, when a slower part of the ejecta catches up with the shock, again as a result of deceleration (see e.g.",
"[46]); and finally (v) a “mini-supernova” from a small amount of ejected material powered by the decay of $^{56}$ Ni [25].",
"Case (i) imposes an external density below $10^{-6}$ cm$^{-3}$ , which seems unreasonably low; case (iii) implies a double-jet structure [18] with one on-axis component producing the prompt emission but no visible afterglow (which can be possible only if the prompt phase has a very high efficiency) and the other one (off-axis) producing the delayed afterglow; case (v) can account for the rebrightening in the visible, but not in X-rays.",
"We therefore reconsider in this work the two most promising cases (ii) and (iv) in the context of the standard model where the afterglow is produced by the forward shock [30], [47] but also in the alternative one where it comes from a long-lived reverse shock [16], [53].",
"The paper is organized as follows.",
"We briefly summarize the observational data in Sect.",
"and list in Sect.",
"possible sources for the initial spike and extended emission.",
"We constrain in Sect.",
"the energy released in these two components and discuss in Sect.",
"different ways to explain the rebrightening with a special emphasis on cases (ii) and (iv) above.",
"Finally Sect.",
"is our conclusion.",
"The Swift-BAT light-curve of GRB 080503 consists of a short bright initial spike followed by a soft extended emission of respective durations $t_{90,{\\rm spike}} =0.32 \\pm 0.07$ s and $t_{90,{\\rm ee}} =170 \\pm 40$ s [27], [38].",
"The fluence of the extended emission from 5 to 140 s and between 15 and 150 keV was $S_{\\rm ee}^{15 - 150}=(1.86\\pm 0.14)\\,10^{-6}$ erg.cm$^{-2}$ while the fluence of the spike $S_{\\rm spike}^{15 - 150}$ was 30 times lower.",
"The spectra of both the spike and the extended emission were fitted by single power-laws with respective photon indices $1.59\\pm 0.28$ and $1.91\\pm 0.12$ .",
"The position of the initial spike in the duration-hardness diagram and the absence of any significant spectral lag (together with the absence of a candidate host galaxy directly at the burst location) make it consistent with a short (type I) burst classification, resulting from the merging of two compact objects.",
"No spectral lag analysis could be performed on the extended emission, which was weaker and softer than the spike."
],
[
"Afterglow emission",
"The afterglow of GRB 080503 was very peculiar.",
"The prompt extended emission ended in X-rays with a steep decay phase of temporal index $\\alpha =2 - 4$ ($F(t)\\propto t^{-\\alpha }$ ), which is common to most long and short bursts.",
"This decay did not show any transition to a “regular afterglow” and went below the detection limit in less than one hour.",
"This behavior has been observed in about 40% of the short burst population [45] but in GRB 080503 it covered nearly six orders of magnitude.",
"In the visible, except for a single Gemini $\\it {g}$ band detection at 0.05 day, the afterglow remained undetected until it exhibited a surprising late rebrightening (both in X-rays and the visible) starting at about one day after trigger.",
"Following the peak of the rebrightening, the available optical data points (extending up to five days) and subsequent upper limits show a steep decay of temporal index $\\alpha \\sim 2$ [38]."
],
[
"Origin of the different emission components",
"The different temporal and spectral properties of the prompt initial spike and extended emission indicate that they are produced by distinct parts of the outflow, possibly even with different dissipation or radiative mechanisms.",
"The temporal structure of the extended emission, showing a short time-scale variability (with $t_{\\rm var}\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$$ 1 s), excludes the possibility of anyconventional afterglow origin.",
"Models of the central engine have been proposed, which are able toproduce a relativistic outflow made of two distinct componentswith kinetic powersand temporal properties similar to what is seen in short GRBs with extended emission.For example, in compact binary progenitors, the extendedemission could be caused by the fallback of material, following coalescence \\cite {rosswog:2007, troja:2008}.For a magnetar progenitor, \\cite {metzger:2008} suggested that the initial spike is produced by accretiononto the protomagnetar from a small disk, while the extended emission comes from rotational energy extractedon a longer time scale.",
"Finally, \\cite {barkov:2011} recently described a two-component jet modelthat could explain short GRBs both with and without extended emission, where a wide, short-lived jet is powered by$$ annihilation and a narrow, long-lived one by the Blandford-Znajek mechanism.$ For the rest of this study it will be assumed that the outflow in GRB 080503 consisted of two main sub-components, responsible for the initial spike and the extended emission, respectively, and that the afterglow emission is associated to the interaction of this structured outflow with the circumburst medium.",
"The energy content of each component can be estimated from the observed fluences (Sect. ).",
"For the refreshed-shock scenario (see Sect.",
"REF below) their typical Lorentz factors are somewhat constrained by the time of the rebrightening for a given value of the external density."
],
[
"Kinetic energy of the outflow",
"To obtain the kinetic energy carried by the different parts of the outflow, one should start estimating the correction factor between the 15 - 150 keV and bolometric fluences $C^{\\rm bol} = S^{\\rm bol} / S^{15-150}$ for both components.",
"Unfortunately, the shape of the spectrum is poorly constrained so that we will simply assume that $ 2 < C^{\\rm bol} < 4$ .",
"This is the range obtained with the simplifying assumption that the spectrum can be represented by a broken power-law of low and high-energy photon indices $\\alpha =-1.5$ , $\\beta =-2.5$ , and peak energy between 20 and 300 keV (with $C^{\\rm {bol}}\\sim 2$ - 2.5 for $E_{\\rm p}$ between 20 and 100 keV and rising to 4 at $E_{\\rm p}=300$ keV).",
"From the fluence, we can express the total isotropic energy release in gamma-rays as a function of redshift ${\\cal E}_{\\gamma , {\\rm iso}}={4\\pi D_{\\rm L}^2(z)S^{\\rm bol}\\over 1+z} \\, ,$ where $D_{\\rm L}(z)$ is the luminosity distance.",
"To finally obtain the kinetic energy, one has to assume a radiative efficiency $f_{\\rm rad}$ , defined as the fraction of the initial kinetic energy of the flow eventually converted to gamma-rays.",
"The remaining energy at the end of the prompt phase is then given by ${\\cal E}_{{\\rm K}, {\\rm iso}}={1-f_{\\rm rad}\\over f_{\\rm rad}}\\,{\\cal E}_{\\gamma , {\\rm iso}}\\ .$ We adopted $f_{\\rm rad}\\approx 0.1$ as a typical value.",
"It could be lower for internal shocks [39], [10] or higher for magnetic reconnection [50], [12], [17], [28] or modified photospheric emission [41], [5].",
"We did not consider scenarii where the radiative efficiency would be very different for the spike and extended emission even if this possibility cannot be excluded a priori.",
"Because the redshift of GRB 080503 is not known, we adopted $z=0.5$ as a “typical” value for a type-I burst for the different examples considered in Sect. .",
"This yields ${\\cal E}_{{\\rm K}, {\\rm iso}} \\simeq C^{\\rm bol}_{\\rm ee} \\times 1.1\\,10^{52} \\ \\mathrm {erg}$ and $C^{\\rm bol}_{\\rm spike} \\times 3\\,10^{50} \\ \\mathrm {erg}$ for the extended emission and spike.",
"The dominant uncertainties on these energies clearly come from the unknown radiative efficiency and distance of the burst.",
"We briefly discuss below how our results are affected when assuming a different redshift or a different radiative efficiency."
],
[
"Forward and long-lived reverse shocks",
"We considered two different mechanisms that can explain GRB afterglows.",
"The first one corresponds to the standard picture where the afterglow results from the forward shock propagating in the external medium, following the initial energy deposition by the central engine [47].",
"The second one was proposed by [16] and [53] to account for some of the unexpected features revealed by Swift observations of the early afterglow.",
"It considers that the forward shock is still present but radiatively inefficient and that the emission comes from the reverse shock that sweeps back into the ejecta as it is decelerated.",
"The reverse shock is long-lived because it is supposed that the ejecta contains a tail of material with low Lorentz factor (possibly going down to $\\Gamma =1$ ).",
"We performed the afterglow simulations using two different methods to model the shocked material.",
"In the first one it is represented by one single zone as in [47]: the physical conditions just behind the shock are applied to the whole shocked material.",
"At any given time, all shocked electrons are considered as a single population, injected at the shock with a power-law energy distribution.",
"Then the corresponding synchrotron spectrum can be calculated, taking into account the effect of electron cooling over a dynamical timescale.",
"The second method is more accurate, considering separately the evolution of each elementary shocked shell [4] except for the pressure, which is uniform throughout the whole shocked ejecta.",
"The electron population (power-law distribution) and magnetic field of each newly shocked shell are computed taking into account the corresponding shock physical conditions and microphysics parameters.",
"Then each electron population is followed individually during the whole evolution, starting from the moment of injection, and taking into account radiative and adiabatic cooling.",
"The evolution of the magnetic field – assuming that the toroidal component is dominant – is estimated using the flux conservation condition.",
"Furthermore, it was checked that the magnetic energy density never exceeds equipartition.",
"Finally, we made a few more assumptions to somewhat restrict the parameter space of the study.",
"We adopted a uniform external medium of low density because GRB 080503 was probably a type-I burst, resulting from the coalescence of two compact objects in a binary system at the periphery of its host galaxy.",
"We also assumed that the redistribution microphysics parameters $\\epsilon _e$ and $\\epsilon _B$ – respectively the fraction of the shock dissipated energy that is injected in the population of accelerated relativistic electrons (power-law distribution with a slope $-p$ ) and in the amplified magnetic field – follow the prescription $\\epsilon _e=\\epsilon _B^{1/2}$ , which results from the acceleration process of electrons moving toward current filaments in the shocked material [29].",
"This assumption simplifies the discussion but is not critical for the general conclusions of our study.",
"We did not try to fit the initial steep decay in X-rays because it is generally interpreted as the high-latitude emission ending the prompt phase and not as a true afterglow component.",
"In that respect, it is not clear if the optical data point at $\\sim 0.05$ day should be associated to the high-latitude emission or already belongs to the afterglow.",
"We assumed that it is of afterglow origin (the most constraining option) and imposed that the simulated light curve goes through it.",
"This leads to some specific consequences, mainly for the reverse shock model (see discussion in Sect.",
"REF )."
],
[
"Refreshed shocks",
"One way to explain the late rebrightening is to consider that the forward or reverse shocks have been refreshed by a late supply of energy [40], [46].",
"This is possible if the initial short duration spike in the burst profile was produced by a “fast” relativistic outflow (of Lorentz factor $\\Gamma _{\\rm spike}$ ) while the extended emission came from “slower” material with $\\Gamma _{\\rm ee}<\\Gamma _{\\rm spike}$ .",
"Then, at early times, only the fast part of the flow is decelerated and contributes to the afterglow.",
"When the slower part is finally able to catch up, energy is added to the shocks and the emission is rebrightened."
],
[
"Forward shock model",
"In the standard forward shock model the lack of any detectable afterglow component before one day imposes severe constraints on either the density of the external medium or the values of the microphysics parameters.",
"Fixing $\\epsilon _e$ and $\\epsilon _B$ to the commonly used values 0.1 and 0.01 implies to take $n\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 10-6$ cm$ -3$\\cite {perley:2009}.",
"This very low density would likely correspond to the intergalactic medium, which mightbe consistent with the absence of a candidate host galaxy down to a visual magnitude of$ 28.5$.",
"We preferred to adopt a lessextreme value $ n=10-3$ cm$ -3$, more typical of the interstellar medium at the outskirts of a galaxy (see e.g.",
"\\cite {steidel:2010}).Then, decreasing the microphysics parameters to $ e=B1/2=0.05$ becomes necessary toremain consistent with the data.$ To obtain a rebrightning at one day we adopted $\\Gamma _{\\rm ee}=20$ and $\\Gamma _{\\rm spike}=300$ .",
"The outflow lasts for a total duration of 100 s (1s for the the spike and 99 s for the tail).",
"We injected a kinetic energy $E_{\\rm kin}=7\\,10^{50}$ erg in the spike and 50 times more in the tail.",
"It can be seen that the results, shown in Fig.",
"REF , are consistent with the available data and upper limits except possibly after the peak of the rebrightening where the decline of the synthetic light curve is not steep enough.",
"This can be corrected if a jet break occurs close to the peak, which is possible if the jet opening angle $\\theta _{\\rm jet}$ is on the order of $1/\\Gamma _{\\rm ee} \\simeq 0.05$ rad.",
"This beaming angle is somewhat smaller than the values usually inferred from observations of short burst afterglows (see e.g.",
"[9], [21]) or suggested by simulations of compact binary mergers (see e.g.",
"[43]).",
"An example of a light curve with a jet break is shown in Fig.",
"REF , assuming that the jet has an opening angle of 3.4$^\\circ $ (0.06 rad) and is seen on-axis.",
"A detailed study of the jet-break properties is beyond the scope of this paper and we therefore did not consider the case of an off-axis observer and neglected the lateral spreading of the jet, expected to become important when $\\Gamma \\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 1/jet$.",
"Detailed hydrodynamical studies (see e.g.",
"\\cite {granot:2007,zhang:2009, vaneerten:2011, lyutikov:2011}) tend to show, however, that as long asthe outflow remains relativistic, the jet-break is more caused by the ``missing^{\\prime \\prime } sideways emitting materialthan by jet angular spreading.$ Figure: Refreshed shocks: forward shock model.Left panel:initial distribution of the Lorentz factor (lower part) and kinetic power (upper part) in the flow as a functionof injection time t inj t_{\\rm inj}.Right panel:synthetic light curves at 2 eV (black, dotted line) and 10 keV(gray, dotted line) compared to the data from .The kinetic energies injected in the spikeand extended emissioncomponents are E kin spike =710 50 E_{\\rm kin}^{\\rm spike} = 7 \\ 10^{50} erg andE kin ee =50E kin spike E_{\\rm kin}^{\\rm ee} = 50 \\ E_{\\rm kin}^{\\rm spike}.We adopt ϵ e =ϵ B 1/2 =510 -2 \\epsilon _e = \\epsilon _B^{1/2} = 5 \\ 10^{-2}, p=2.5p = 2.5 in the shocked external medium together withn=10 -3 n=10^{-3} cm -1 ^{-1} and z=0.5z=0.5.The steeper thin lines at latetimes correspond to a conical jet (seen on-axis) of opening angle θ jet =0.06\\theta _{\\rm jet}=0.06 rad.We finally checked how our results are affected if different model parameters are adopted.",
"If the density $n$ of the external medium is increased or decreased, similar light curves can be obtained by changing the Lorentz factors (to still achieve the rebrightning at one day) and the microphysics parameters (to recover the observed flux).",
"For example, increasing the density to $n=0.1$ cm$^{-3}$ requires $\\Gamma _{\\rm ee}\\simeq 10$ (keeping $\\Gamma _{\\rm spike}=300$ ) and $\\epsilon _e=\\epsilon _B^{1/2}=0.02$ .",
"Conversely, with $n=10^{-5}$ cm$^{-3}$ , $\\Gamma _{\\rm ee}\\simeq 35$ and $\\epsilon _e=\\epsilon _B^{1/2}=0.08$ are needed.",
"If the kinetic energy of the outflow is increased (resp.",
"decreased) because the radiative efficiency $f_{\\rm rad}$ is lower (resp.",
"higher) or the redshift higher (resp.",
"lower), light curves agreeing with the data can again be obtained by increasing (resp.",
"decreasing) $\\Gamma _{\\rm ee}$ and decreasing (resp.",
"increasing) $\\epsilon _e=\\epsilon _B^{1/2}$ .",
"Also note that the spread of the Lorentz factor $\\delta \\Gamma _{\\rm ee}$ around $\\Gamma _{\\rm ee}$ at the end of the prompt phase has to be limited to ensure that the slower material is able to catch up in a sufficiently short time to produce an effective rebrightening.",
"In the case shown in Fig.",
"REF we have $\\delta \\Gamma _{\\rm ee}/\\Gamma _{\\rm ee}=0$ but we have checked from the numerical simulation that acceptable solutions can be obtained as long as $\\delta \\Gamma _{\\rm ee}/\\Gamma _{\\rm ee}\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 0.2$.",
"This configuration is for example naturallyexpected after an internal shock phase where fast and slow parts of the flow collide, resulting in a shocked region witha nearly uniform Lorentz factor distribution (see e.g.",
"\\cite {daigne:2000}).$"
],
[
"Long-lived reverse shock model",
"If the afterglow is produced by the reverse shock, similar good fits of the data can be obtained.",
"Fig.",
"REF shows an example of synthetic light curves for $E_{\\rm kin}^{\\rm spike} = 7 \\ 10^{50}$ erg and $E_{\\rm kin}^{\\rm ee} = 30 \\ E_{\\rm kin}^{\\rm spike}$ , $\\epsilon _e = \\epsilon _B^{1/2} = 0.16$ , $p = 2.5$ in the shocked ejecta and $n=10^{-3}$ cm$^{-3}$ .",
"The microphysics parameters have to be higher than in the forward shock case because the reverse shock is dynamically less efficient, which requires a higher radiative efficiency to obtain the same observed fluxes.",
"The Lorentz factor distribution in the ejecta is also slightly different to guarantee that the light curve (i) goes through the optical point at 0.05 day and (ii) decays steeply after the peak.",
"No jet break has to be invoked here because the decay rate (in contrast to what happens in the forward shock model) depends on the distribution of energy as a function of the Lorentz factor in the ejecta.",
"Again, if the external density and kinetic energy of the flow are varied, satisfactory fits of the data can be recovered by slightly adjusting the Lorentz factor and microphysics parameters."
],
[
"Density clump in the external medium",
"We now investigate the possibility that the rebrightening is caused by the encounter of the decelerating ejecta with a density clump in the external medium.",
"For illustration, we adopted a simple distribution of the Lorentz factor that linearly decreases with injection time from 300 to 2 so that, in the absence of the density clump, afterglow light curves from either the forward or reverse shocks would be smooth and regular (we have checked that the exact shape of the low Lorentz factor tail is not crucial in this scenario).",
"To model the clump, we assumed that the circumburst medium is uniform (with $n=10^{-3}$ cm$^{-3}$ ) up to 1.7 $10^{18}$ cm (0.55 pc) and that the density then rises linearly to $n=1$ cm$^{-3}$ over a distance of $10^{18}$ cm (0.32 pc).",
"The ejecta is strongly decelerated after entering the high-density region and we find that the forward shock is still inside the clump at the end of the calculation (at $t_{\\rm obs}=8$ days)."
],
[
"Forward shock model",
"As [34] showed by coupling their hydrodynamical calculation to a detailed radiative code, a density clump in the external medium has little effect on the forward shock emission.",
"Therefore a clump cannot produce the rebrightening in GRB 080503.",
"In the simple case where the shocked medium is represented by a single zone, the effect of the clump is barely visible.",
"With the detailed multi-zone model a stronger rebrightening is found, because the effects of the compression resulting from the deceleration of the flow are better described, but even in this case the calculated flux remains nearly one order of magnitude below the data.",
"Fig.",
"REF illustrates these results and confirms that the forward shock emission does not strongly react to the density clump.",
"Even if, from an hydrodynamical point of view, the forward shock is sensitive to the clump, the observed synchrotron emission is only moderately affected because the increase in upstream density is nearly counterbalanced by the decrease of Lorentz factor in the shocked material.",
"Of course, spectral effects complicate the picture, but the essence of the result remains the same (see [34] for details)."
],
[
"Possible evolution of the microphysics parameters",
"In view of the many uncertainties in the physics of collisionless shocks it is often assumed for simplicity, as we did so far, that the microphysics redistribution parameters $\\epsilon _e$ and $\\epsilon _B$ stay constant during the whole afterglow evolution.",
"However, particle-in-cell simulations of acceleration in collisionless shocks (see e.g.",
"[48]) do not show any evidence of universal values of the parameters.",
"If $\\epsilon _e$ or/and $\\epsilon _B$ are allowed to change during afterglow evolution, the problem of the forward shock encountering a density clump can be reconsidered, now with the possibility of a sudden increase of radiative efficiency triggered by the jump in external density.",
"Fig.",
"REF shows the resulting light curves when the microphysics parameters $\\epsilon _e$ and $\\epsilon _B$ of the forward shock are increased at the density clump.",
"If the prescription $\\epsilon _e= \\epsilon _B^{1/2}$ is maintained, no satisfactory solution can be found in the simple model where the shocked medium is represented by one single zone.",
"In this case, the optical frequency lies between the injection and cooling frequencies ($\\nu _i<\\nu _{\\rm opt}<\\nu _c$ ) while the X-ray frequency satisfies $\\nu _{\\rm X}>\\nu _c$ so that the visible and X-ray flux densities depend on the microphysics parameters in the following way [37] $f_{\\nu ,{\\rm opt}}\\propto \\epsilon _e^{p-1}\\epsilon _B^{p+1\\over 4} \\ {\\rm and} \\ \\ f_{\\nu ,{\\rm X}}\\propto \\epsilon _e^{p-1}\\epsilon _B^{p-2\\over 4} .$ With the prescription $\\epsilon _e=\\epsilon _B^{1/2}$ we obtain $f_{\\nu ,{\\rm opt}}\\propto \\epsilon _e^{3p-1\\over 2}$ and $f_{\\nu ,{\\rm X}}\\propto \\epsilon _e^{3 p-4 \\over 2}$ The optical flux is therefore much more sensitive than the X-ray flux to a change of the microphysics parameters and a simultaneous fit of the data in both energy bands is not possible.",
"A simple solution to this problem is to change $\\epsilon _e$ alone, keeping $\\epsilon _B$ constant.",
"In this case, increasing $\\epsilon _e$ by a factor of 25 (from 0.01 to 0.25) is required to reproduce the rebrightening in both the X-ray and visible ranges.",
"In the more detailed model with a multi-zone shocked region, the situation is different.",
"In the shells that contribute most to the emission, we find that both $\\nu _{\\rm opt}$ and $\\nu _{\\rm X}$ are larger than $\\nu _c$ and therefore $f_{\\nu ,{\\rm opt}}$ and $f_{\\nu ,{\\rm X}}$ depend in the same way on the microphysics parameters.",
"It is then possible to achieve a satisfactory solution (dashed lines in Fig.",
"REF ) that keeps the prescription $\\epsilon _e=\\epsilon _B^{1/2}$ (with $\\epsilon _e$ increased by a factor 5).",
"As in Sect.",
"REF we introduce a jet break (now assuming $\\theta _{\\rm jet} = 0.08$ rd) to account for the decay of the optical flux following the peak of the rebrightening.",
"Notice that the decay is steeper here (compare Fig.",
"REF and Fig.",
"REF ) owing to the rapid decrease of the Lorentz factor inside the clump."
],
[
"Long-lived reverse shock model",
"With the simple one-zone model, the reverse shock emission is found to be much more sensitive to the density clump than the forward shock emission.",
"Indeed, when the ejecta starts to be decelerated, its bulk Lorentz factor suddenly decreases and slow shells from the tail material pile up at a high temporal rate and with a strong contrast in Lorentz factor.",
"These two combined effects lead to a sharp rise of the flux from the reverse shock.",
"Synthetic light curves showing a satisfactory agreement with the data are shown in Fig.",
"REF .",
"However, the detailed multi-zone model gives different results, where the rebrightening is dimmer and cannot fit the data.",
"The main reason is that the higher contrast in Lorentz factor, leading to a higher specific dissipated energy, now only concerns the freshly shocked shells, while in the single zone model it is applied to the whole shocked region.",
"This example (as well as the one already discussed in Sect.",
"REF ) shows that using a detailed description of the shocked material can be crucial when dealing with complex scenarios (i.e.",
"not the standard picture where the blast-wave propagates in a smooth external medium, with constant microphysics parameters)In the refreshed-shock scenario (Sect.",
"REF ) where the dynamics is simpler, the single and multi-zone models give comparable results..",
"It is still possible to fit the data by increasing the microphysics parameters during the propagation in the clump.",
"However, this seems less natural than for the forward shock (Sect.",
"REF ) because the upstream density of the reverse shock does not change.",
"On the other hand, a modification of the microphysics parameters could still be due to the sudden increase in the reverse shock Lorentz factor triggered by the clump encounter."
],
[
"Conclusion",
"GRB 080503 belongs to the special group of short bursts where an initial bright spike is followed by an extended soft emission of much longer duration.",
"It did not show a transition to a standard afterglow after the steep decay observed in X-rays at the end of the extended emission.",
"This behavior has been observed previously in short bursts, but GRB 080503 was peculiar because it exhibited a spectacular rebrightening after one day, both in X-rays and the visible.",
"The presence of the extended emission prevents one from classifying GRB 080503 on the basis of duration only, but the lack of any candidate host galaxy at the location of the burst and the vanishing spectral lag of the spike component are consistent with its identification as a type-I event resulting from the coalescence of a binary system consisting of two compact objects.",
"From its formation to the coalescence, the system can migrate to the external regions of the host galaxy allowing the burst to occur in a very low density environment, accounting for the initial lack of a detectable afterglow.",
"To explain the late rebrightening, we considered two possible scenarios – refreshed shocks from a late supply of energy or a density clump in the circumburst medium – and two models for the origin of the afterglow, the standard one where it comes from the forward shock and the alternative one where it is made by a long-lived reverse shock.",
"In the refreshed-shock scenario we supposed that the initial spike was produced by fast moving material (we adopted $\\Gamma =300$ ) while the one making the soft tail was slower ($\\Gamma \\sim 20$ ).",
"Initially, only the spike material is decelerated and contributes to the afterglow until the tail material is eventually able to catch up, which produces the rebrightening.",
"Both the forward and reverse shock models provide satisfactory fits of the data under the condition that the material making the tail has a limited spread in Lorentz factor $\\delta \\Gamma /\\Gamma \\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 0.2$.",
"This allows the rise time of the rebrightening to be sufficiently short.This condition might be satisfied from the beginning but can also result from a previoussequence of internal shocks that has smoothed most of the fluctuations of the Lorentz factor initiallypresent in the flow.",
"In addition, a jet break is required in the forward shock caseto reproduce the steep decline that follows the rebrightening.",
"This implies that the jet should be beamed withinan opening angle of 3 - 5$$, which appears somewhat smaller than the values usually preferred for type-I bursts.In the long-lived reverse shock model a jet break is not necessarybecause the shape of the light curve now depends on the energy distribution in the ejecta, which can beadjusted to fit the data.$ In the scenario where a density clump is present in the burst environment, the rebrightening resulting from the forward shock is weak, in agreement with the previous work of [34].",
"We performed the calculation in two ways: first with a simple method where the shocked material was represented by one single zone, then using a more detailed, multi-zone approach.",
"The impact of the clump was barely visible in the first case.",
"The rebrightening was larger in the second one but still remained nearly one order of magnitude below the data.",
"We then considered the possibility that the shock microphysics might change inside the clump.",
"We found that by increasing $\\epsilon _e$ by a factor of five (and with the prescription that $\\epsilon _e=\\epsilon _B^{1/2}$ ) it was possible to fit the data with the multi-zone model under the additional condition to have a jet break at about 2 - 3 days (corresponding to a jet opening angle $\\mathrel {\\hbox{t}o 0pt{}\\hss }$$$ 5$).With the simplifiedmodel the results were more extreme, imposing to increase $ e$ alone by a very large factor of25.$ If the afterglow is made by the reverse shock, the effect of the clump is strong with the simple model.",
"It is however much reduced with the detailed model and the observed rebrightening cannot be reproduced, the synthetic light curve lying nearly one order of magnitude below the observed one.",
"It appears that only the multi-zone approach provides a proper description of the compression resulting from the encounter with the density barrier.",
"Conversely, in the refreshed-shock scenario the simple and detailed models give comparable results.",
"From the different possibilities we considered, which could explain the late rebrightening in GRB 080503, several appear compatible with the data, but none is clearly favored.",
"The refreshed-shock scenario may seem more natural because the initial spike and extended emission probably correspond to different phases of central engine activity.",
"It is not unreasonable to suppose that the material responsible for the extended emission had a lower Lorentz factor, as required by the refreshed-shock scenario.",
"Then, both the forward and reverse shock models lead to satisfactory fits of the X-ray and visible light curves, if two conditions on the Lorentz factor distribution and jet opening angle (see above) are satisfied.",
"The density clump scenario does not seem able to account for the rebrightening if the afterglow is made by the reverse shock.",
"The conclusion is the same with the forward shock, except if the microphysics parameters are allowed to change when the shock enters the clump.",
"The authors acknowledge the French Space Agency (CNES) for financial support.",
"R.H.'s PhD work is funded by a Fondation CFM-JP Aguilar grant."
]
] | 1204.0941 |
[
[
"Quantum spin liquids and the metal-insulator transition in doped\n semiconductors"
],
[
"Abstract We describe a new possible route to the metal-insulator transition in doped semiconductors such as Si:P or Si:B.",
"We explore the possibility that the loss of metallic transport occurs through Mott localization of electrons into a quantum spin liquid state with diffusive charge neutral \"spinon\" excitations.",
"Such a quantum spin liquid state can appear as an intermediate phase between the metal and the Anderson-Mott insulator.",
"An immediate testable consequence is the presence of metallic thermal conductivity at low temperature in the electrical insulator near the metal-insulator transition.",
"Further we show that though the transition is second order the zero temperature residual electrical conductivity will jump as the transition is approached from the metallic side.",
"However the electrical conductivity will have a non-monotonic temperature dependence that may complicate the extrapolation to zero temperature.",
"Signatures in other experiments and some comparisons with existing data are made."
],
[
"Appendix – Disorder Averaged Properties of the Low–Energy Effective Slave–Rotor Theory",
"In this appendix, we develop further, the properties of the low-energy effective field theory for the metal-insulator transition and spin-liquid phase within the slave-rotor theory.",
"In principle, a full treatment of disorder would need to account for correlations in the effects of local disorder fluctuations among the boson, fermion and the gauge field sectors.",
"We assume that the principal effect of such correlations is to produce local patches where the rotor and fermion are locally confined, i.e.",
"to produce the dilute fraction of local-moments that are observed in each phase, but appear to decouple from the remainder of the system.",
"We expect that the formation of local-moments accounts for the effects of the rare-long tails of the disorder distribution, and that the remaining connected component of the system is reasonably characterized by treating disorder separately in each sector.",
"In the boson-sector, the superfluid-insulator transition will occur in the presence of a random potential, and the resulting insulating phase will be a mixture of glassy puddles of superfluid which do not percolate, coexisting with a Mott-localized bulk.",
"In the spin-liquid scenario, the thermally conducting fluid of spinons (not including local moments) form a diffusive metal.",
"As shown below, the phase transition in the boson sector is not affected by the presence of gapless fermions or gauge fluctuations, and is identical to that of the ordinary dirty Bose-Hubbard model with random chemical potential and long-range Coulomb interactions[41]."
],
[
"A. Irrelevance of Spinon and Gauge Fluctuations on the Slave–Particle Boson–Mott Transition",
"Coupling to Fermions – Including gauge fluctuations generically gives rise to a spinon-rotor density-density coupling of the form $\\lambda _0 \\delta n_f n_b$ , where $\\delta n_f$ is the deviation of $n_f$ from its average value.",
"Here we give a simple scaling argument that such a coupling does not alter the critical behavior of the boson–sector near the Mott transition.",
"Consider integrating out the spinons.",
"The leading order term in the effective action for the bosons will be of the form: $\\lambda \\sum _{\\omega ,\\mathbf {q}}\\langle \\delta n_f(\\omega ,\\mathbf {q})\\delta n_f(-\\omega ,-\\mathbf {q})\\rangle |n_b(\\omega ,\\mathbf {q})|^2$ .",
"In the transition to the spin-liquid, the spinon density-density correlator is diffusive and evolves smoothly across the transition: $\\langle \\delta n_f(\\omega ,\\mathbf {q})\\delta n_f(-\\omega ,-\\mathbf {q})\\rangle \\sim \\frac{Dq^2}{|\\omega |+Dq^2}$ .",
"After a momentum-shell renormalization–group (RG) step, integrating out modes with $q\\in [\\Lambda /s,\\Lambda ]$ , with $s\\gtrsim 1$ , one rescales $q\\rightarrow sq$ and $\\omega \\rightarrow s^z\\omega $ to compare the new effective action to the original.",
"Due to the random chemical potential provided by the spinon sector, the Boson Mott transition is in the same universality class whether one tunes through the transition either by changing chemical potential or hopping strength (this would not be true without the presence of a random chemical potential, where the Bose–Mott transition takes place at fixed density per site, with the random potential however, the density per site is only fixed on average).",
"For the chemical potential driven transition, under an RG step, the scaling part of the boson density rescales as $n_b(r,t)\\rightarrow s^{d+z-1/\\nu }n_b(r,t)$ where $\\nu $ is the correlation length exponent.",
"Equivalently, the fourier component rescales as $n_b(q,\\omega )\\rightarrow s^{-1/\\nu }n_b(q,\\omega )$ .",
"For $z<2$ , the denominator of the diffusive fermion correlator is dominated by the $|\\omega |$ term, and the density–coupling term scales as: $\\lambda g\\int d^dqd\\omega q^{d-z}|n_b(q,\\omega )|^2\\rightarrow \\lambda ^{\\prime } s^{2(d-1/\\nu )}\\int d^dqd\\omega q^{d-z}|n_b(q,\\omega )|^2$ , indicating that the coupling constant $\\lambda $ rescales as $\\lambda \\rightarrow s^{2(1/\\nu -d)}\\lambda $ .",
"Since $\\nu d\\ge 2$ , $\\lambda $ is irrelevant.",
"Similarly if $z\\ge 2$ , the diffusive fermion correlator scales like a constant under RG, and the coupling constant $\\lambda \\rightarrow s^{2\\nu -d-z}\\lambda $ is again irrelevant.",
"Coupling to Gauge Fluctuations – The rotor–gauge field coupling generically takes the form $\\int d^dr d\\tau a_\\mu j_b^\\mu $ , where $j_b$ is the boson current.",
"Integrating out the gauge-field at the RPA level generates a term of the form $g \\int d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^d qd \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega G_a|j_b(\\omega ,\\mathbf {q})|^2$ , where $G_a = \\langle |\\mathbf {a}(\\omega ,q)|^2\\rangle $ .",
"Current–continuity requires that $j_b(\\omega ,q)\\sim \\frac{\\omega }{q}n_b(\\omega ,q)$ , where $n_b$ is the total boson density which scales like $n_b\\sim 1/L^d$ , indicating that $j_b(\\omega ,q)\\rightarrow s^{-1}j_b(\\omega ,q)$ in each RG step.",
"As shown below, the gauge-field propagator scales like $G_a\\sim (\\omega +q^2)^{-1}$ .",
"For $z\\le 2$ , under RG the gauge-fluctuation term rescales as $g \\int d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^d qd \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega G_a|j_b(\\omega ,\\mathbf {q})|^2 \\rightarrow g^{\\prime } \\int s^{d+z}d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^d qd \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega s^{-z}G_a s^{-2}|j_b(\\omega ,\\mathbf {q})|^2$ .",
"Consequently $g\\rightarrow gs^{2-d}$ flows to zero under RG and is irrelevant.",
"Similarly, for $z\\ge 2$ , the term rescales as $g \\int d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^d qd \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega G_a|j_b(\\omega ,\\mathbf {q})|^2 \\rightarrow g^{\\prime } \\int s^{d+z}d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^d qd \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega s^{-2}G_a s^{-2}|j_b(\\omega ,\\mathbf {q})|^2$ , indicating $g\\rightarrow s^{4-(d+z)}g$ , and the term is again irrelevant."
],
[
"B. RPA Effective Action for the Emergent Gauge Field in the Diffusive Spinon Metal",
"Since the spinons and bosons have opposite charge under the emergent gauge field $a$ , the gauge field couples to the currents as $S_{a-j} = \\int d\\tau d^3r \\left(j_\\mu ^b-j_\\mu ^f\\right)a^\\mu $ (here and throughout, we work in imaginary time).",
"Therefore integrating out the spinon and boson fields within the RPA approximation, gives the following disorder-averaged effective action for the gauge field: $ S_{\\text{eff}}^{\\text{(RPA)}} = \\sum _{\\omega ,q} a_\\mu (\\omega ,q)\\left[K_b^{\\mu \\nu }(\\omega ,q)+K_f^{\\mu \\nu }(\\omega ,q)\\right]a_\\nu (\\omega ,q) $ where $K_{f/b}^{\\mu \\nu }$ are the disorder–averaged current–current correlators (equivalently linear-response kernels) for the fermions and bosons respectively.",
"The temporal fluctuations of the gauge field are screened by the compressible fermions and become massive.",
"At the critical point and in the insulating phase the Boson conductivity vanishes, and consequently the gauge field dynamics are determined by the fermion response.",
"The disorder-averaged density-density part of the fermion electric response kernel is diffusive: $ K^f_{00} = \\frac{2N(0)Dq^2}{|\\omega |+Dq^2} $ where, we have expanded the diffusion “constant\" $D(\\omega ,q)$ (which generally has some $\\omega $ and $q$ dependence, but does not vanish for $q,\\omega \\rightarrow 0$ outside of the disorder–localized phase) near $\\omega =0=q$ , and dropped the irrelevant higher order terms.",
"The other components of electric field response-function are related to $K^f_{00}$ by gauge invariance and charge conservation: $K^f_{0i} = K^f_{i0} = \\frac{-\\omega q_i}{q^2}K^f_{00}$ , and $K^f_{ij} = \\delta _{ij} \\frac{-\\omega ^2}{q^2}K^f_{00}$ .",
"Furthermore, we can identify $2N(0)D$ as the static uniform spinon–conductivity $\\sigma _f$ .",
"In addition, there is the usual diamagnetic response to fluctuating magnetic fields for $\\omega \\ll q$ : $\\chi _dq^2$ , where $\\chi _d$ is the Landau diamagnetic susceptibility of the spinons (whose average value is not altered by disorder).",
"Combining these considerations, and working in the Coulomb gauge ($\\nabla \\cdot \\mathbf {a}=0$ so that only the transverse gauge field $\\mathbf {a}_\\perp $ remains) gives the following RPA action for the gauge-field $\\mathbf {a}$ : $ S_a^\\text{(RPA)} = \\sum _{\\omega ,q,\\mu }\\left[\\chi _d q^2+\\frac{\\sigma _f\\omega ^2}{|\\omega |+Dq^2}\\right]|\\mathbf {a}_\\perp (\\omega ,q)|^2$ where $D = \\frac{v_F^2\\tau }{d}$ is the diffusion constant ($\\tau $ is the disorder scattering time), and $\\sigma _f$ is the spinon-conductivity."
],
[
"C. Inelastic Scattering of Spinons from Gauge Fluctuations",
"Using the RPA expression for the gauge field propagator, one can find the leading (one-loop) self-energy for low-energy spinons near the Fermi-energy: $ \\Sigma (i\\omega ,k) &= \\int d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}\\omega d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}^2q_\\perp d \\hspace{-3.40001pt}\\rule [1.2ex]{0.8ex}{.1ex}q_\\parallel D_a(i\\Omega ,q)\\left(v_F \\frac{q_\\perp }{q}\\right)^2G_f\\left(i(\\omega -\\Omega ),k+q\\right)\\nonumber \\\\G_f(i\\omega ,\\mathbf {k}) &= \\frac{1}{i\\omega -\\xi _{\\mathbf {k}}+\\frac{i}{2\\tau }\\text{sgn}\\omega }\\nonumber \\\\D_a(\\Omega ,q) &= \\frac{1}{\\chi _D q^2+\\frac{\\sigma _f\\Omega ^2}{|\\Omega |+Dq^2}}$ In the fermion Green's function $\\xi _{k+q}\\sim v_Fq_\\parallel +\\mathcal {O}(q^2)$ , indicating that $q_\\parallel \\sim \\Omega $ .",
"Consequently one may approximate $q_\\parallel ^2\\ll q_\\perp ^2$ in the gauge field propagator, making $D_a$ a function of $q_\\perp $ only.",
"Furthermore, in this approximation, the current vertex $\\left(v_F \\frac{q_\\perp }{q}\\right)^2\\approx v_F^2$ .",
"Performing the $q_\\parallel $ integral then gives $N(0)\\text{sgn}(\\omega -\\Omega )$ independent of the disorder scattering time $\\tau $ .",
"Considering the case of $\\omega >0$ for definiteness, this limits the range of $\\Omega $ integration from 0 to $\\omega $ .",
"Since the dominant contributions come from $\\Omega \\sim q_\\perp ^2$ , one finds that the spinon self-energy due to diffusively–screened gauge fluctuations scales like: $\\Sigma (i\\omega ) \\sim \\omega \\log (1/\\omega )$ Continuing to real-time one finds the inelastic gauge field scattering rate scales like $\\text{Im}\\left[\\Sigma ^R(\\omega )\\right]\\sim \\omega $ , or equivalently $\\tau _{\\text{inelastic}}^{-1}\\sim T$ .",
"At low-temperature, this inelastic scattering is clearly sub-dominant compared to the elastic impurity scattering."
]
] | 1204.1342 |
[
[
"Lorentz violation bounds on Bhabha scattering"
],
[
"Abstract We investigate the effect of Lorentz-violating terms on Bhabha scattering in two distinct cases correspondent to vectorial and axial nonminimal couplings in QED.",
"In both cases, we find significant modifications with respect to the usual relativistic result.",
"Our results reveal an anisotropy of the differential cross section which imply new constraints on the possible Lorentz violating terms."
],
[
"Introduction",
"Since the Carroll-Field-Jackiw seminal paper [1] and after the construction of the extended Standard Model (SME) by Colladay and Kostelecky [2], [3] (see also [4] and references therein), the possibility of Lorentz covariance breakdown in the context of Quantum Field Theory has been extensively studied.",
"The interest in this issue appears in different contexts, such as supersymmetric models [5], [6], noncommutative geometry [7], gravity and cosmology [9], [10], [11], [8], high derivative models [12], [13], [14], renormalization [15], [16], [17], [18] and scattering processes [19], [20] in quantum electrodynamics (QED), condensed matter systems [21], [22], [23], and so on.",
"Following these theoretical developments, many experimental tests on Lorentz-violating (LV) corrections have also been carried out and several constraints on LV parameters were established [24].",
"One of the most precise experiments, the clock anisotropy, which is a spectroscopic experiment, determines bounds of $10^{-33}$ GeV [25] when LV parameters are introduced as in the SME [2], [3].",
"However, for scattering processes, there are few studies about possible effects of LV on cross sections aimed to determination of upper bounds on the breaking parameters [19], [20], [26].",
"In the usual aproach to LV theories, the breaking term is implemented on the kinetic sector and implies in modifications on the energy-momentum relations, the free propagators and scattering states as have been stressed in Refs.",
"[19], [20].",
"An alternative procedure, is to modify just the interactions part via a nonminimal coupling with terms like $\\epsilon _{\\mu \\nu \\alpha \\beta }v^{\\nu }F^{\\alpha \\beta }$ and $\\epsilon _{\\mu \\nu \\alpha \\beta }\\gamma _{5}b^{\\nu }F^{\\alpha \\beta }$ .",
"In Ref.",
"[21] this possibility was used used to evaluate the induction of topological phases on fermion systems.",
"Later on, its implication on the spectrum of the hydrogen atom providing the determination of bounds on the magnitude of the LV coefficients were reported in Ref.",
"[22].",
"However, possible effects on scattering processes in the framework of QED by these nonminimal couplings have not been investigated.",
"That is the main objective of this paper, i.e.",
"to obtain a bound to Lorentz violation from a scattering process involving a nonminimal coupling.",
"Bounds obtained from noncolliders experiments [22] usually depend on the study of the hyperfine structure what is outside of the scope of this work.",
"Collision experiments in high energy physics provide a suitable environment where Lorentz symmetry breaking can be tested.",
"Moreover, Bhabha scattering is one of the most fundamental reactions in QED processes and has been extensively studied in colliders [27], [28], [29].",
"It is particularly important since it is used to determine the luminosity of the $\\mbox{e}^{+}\\mbox{e}^{-}$ collisions [30], [31].",
"This fact motivated us to evaluate and analise the behavior of the differential cross section for Bhabha scattering in the presence of nonminimal couplings and to directly obtain upper bounds on LV coefficients.",
"As we will show, our calculations can be done similarly to those in standard QED.",
"We found that the breaking of Lorentz symmetry leads to an unusual dependence of the cross section on the orientation of the scattering plane in the center of mass reference frame.",
"This paper is organized as follows.",
"In Sec.",
", the differential cross section for Bhabha scattering on the presence of the vectorial nonminimal coupling is calculated.",
"The results obtained are analyzed and a bound to the magnitude of the Lorentz violation is established.",
"In Sec.",
", the axial-like nonminimal coupling is considered.",
"In Sec.",
", some final remarks are made."
],
[
"Bhabha scattering: vectorial nonminimal coupling ",
"In this section we calculate the unpolarized differential cross section for Bhabha scattering $e^{+}e^{-}\\rightarrow e^{+}e^{-}$ , in an extended version of QED characterized by a nonminimal covariante derivative [21], [22]: $D_{\\mu }=\\partial _{\\mu }+ieA_{\\mu }+igv^{\\nu }F_{\\mu \\nu }^{\\ast },$ where $F_{\\mu \\nu }^{\\ast }=\\frac{1}{2}\\varepsilon _{\\mu \\nu \\alpha \\beta }F^{\\alpha \\beta }$ is the dual electromagnetic tensor with $\\epsilon ^{0123}=1$ ; $e$ , $g$ , $v^{\\mu }$ are the electron charge, a coupling constant and a constant four vector, respectively.",
"With such modification the QED Lagrangian is $\\mathcal {L}&=&-\\frac{1}{4}F^{\\mu \\nu }F_{\\mu \\nu }+\\bar{\\psi }(i\\gamma ^{\\mu }\\partial _{\\mu }-m)\\psi -\\frac{1}{2\\alpha }(\\partial _{\\mu }A^{\\mu })^{2}\\nonumber \\\\&-&e\\bar{\\psi }\\gamma ^{\\mu }\\psi A_{\\mu }-gv^{\\nu }\\bar{\\psi }\\gamma ^{\\mu }\\psi \\partial ^{\\alpha }A^{\\beta }\\epsilon _{\\mu \\nu \\alpha \\beta }.$ The additional vertex is gauge invariant, but explicitly violates Lorentz symmetry, since $v^{\\mu }$ defines a privileged direction in the space-time.",
"Furthermore, it is not perturbatively renormalizable, since their coupling constant has mass dimension $\\left[gv^{\\mu }\\right]=-1$ .",
"As in standard QED, the Feynman rules can be read directly from Eq.",
"(REF ), telling us how to write down the tree-level diagrams related in the process $e^{-}(p_{1})e^{+}(q_{1})\\rightarrow e^{-}(p_{2})e^{+}(q_{2})$ .",
"In this work we will assume the Feynman gauge $(\\alpha =1)$ and the result, to lowest order, for the $S$ -matrix element is therefore $i\\mathcal {M}_{\\mbox{total}}=i\\mathcal {M}_{0}+i\\mathcal {M}_{1}+i\\mathcal {M}_{2},$ where $i\\mathcal {M}_{0}$ is just the matrix element in conventional QED: $i\\mathcal {M}_{0}&=&ie^{2}\\left[\\frac{\\overline{u}(p_{2})\\gamma ^{\\alpha }u(p_{1})\\overline{v}(q_{1})\\gamma _{\\alpha }v(q_{2})}{(p_{1}-p_{2})^{2}}\\right.\\nonumber \\\\&&\\left.-\\frac{\\overline{u}(p_{2})\\gamma ^{\\alpha }v(q_{2})\\overline{v}(q_{1})\\gamma _{\\alpha }u(p_{1})}{(p_{1}+q_{1})^{2}}\\right].$ The matrix element $i\\mathcal {M}_{1}$ is linear in $(gv^{\\mu })$ being formed by an usual vertex and another with the Lorentz-violating term: $i\\mathcal {M}_{1}&=&2egv^{\\nu }\\epsilon _{\\mu \\nu \\sigma \\rho }\\left[\\frac{(p_{1}-p_{2})^{\\sigma }\\overline{u}(p_{2})\\gamma ^{\\rho }u(p_{1})\\overline{v}(q_{1})\\gamma ^{\\mu }v(q_{2})}{(p_{1}-p_{2})^{2}}\\right.\\nonumber \\\\&&\\left.+\\frac{(p_{1}+q_{1})^{\\sigma }\\overline{u}(p_{2})\\gamma ^{\\text{$\\rho $}}v(q_{2})\\overline{v}(q_{1})\\gamma ^{\\text{$\\mu $}}u(p_{1})}{(p_{1}+q_{1})^{2}}\\right].$ Finally, $i\\mathcal {M}_{2}$ is quadratic in $(gv^{\\mu })$ as it results purely from the Lorentz-violating vertex: $i\\mathcal {M}_{2}&=& ig^{2}v^{\\gamma }v^{\\delta }g^{\\kappa \\lambda }\\epsilon _{\\epsilon \\delta \\tau \\lambda }\\epsilon _{\\omega \\gamma \\sigma \\kappa }\\nonumber \\\\&&\\left[\\frac{(p_{1}-p_{2})^{\\sigma }(p_{1}-p_{2})^{\\tau }\\overline{u}(p_{2})\\gamma ^{\\omega }u(p_{1})\\overline{v}(q_{1})\\gamma ^{\\epsilon }v(q_{2})}{(p_{1}-p_{2})^{2}}\\right.\\nonumber \\\\&&-\\left.\\frac{(p_{1}+q_{1})^{\\sigma }(p_{1}+q_{1})^{\\tau }\\overline{u}(p_{2})\\gamma ^{\\text{$\\epsilon $}}v(q_{2})\\overline{v}(q_{1})\\gamma ^{\\text{$\\omega $}}u(p_{1})}{(p_{1}+q_{1})^{2}}\\right].\\nonumber \\\\$ To evaluate the cross section, we now compute $\\left|i\\mathcal {M}_{\\mbox{total}}\\right|^{2}$ , taking an average over the spin of the incoming particles and summing over the outgoing particles.",
"This can be accomplished using the completeness relations: $\\sum u^{s}(p)\\overline{u}^{s}(p)={p}+m$ and $\\sum v^{r}(p)\\overline{v}^{r}(p)={p}-m$ , leading to traces of Dirac matrices products.",
"We performed these trace calculations, which involves the product of up to eight gamma matrices and the Levi-Civita symbol using the FeynCalc package [32].",
"Furthermore, as our main goal is to consider the behavior of the scattering process in the high energy limit, we set $p_{1,2}^{2}=q_{1,2}^{2}=m^{2}=0$ .",
"This is possible because the $(gv^{\\mu })$ factors are overall on all terms.",
"In this way, we arrive at the following expression: $\\frac{1}{4}\\sum _{\\mbox{spins}}|\\mathcal {M}_{total}|^{2}&=&e^{4}\\left(\\frac{2(s^{2}+u^{2})}{t^{2}}+\\frac{4u^{2}}{st}+\\frac{2(t^{2}+u^{2})}{s^{2}}\\right)\\nonumber \\\\&+&\\mbox{{\\bf A}}(v,p_{1,2},q_{1,2})+\\mbox{{\\bf B}}(v,p_{1,2},q_{1,2}),$ with $s$ , $t$ , and $u$ being the Mandelstam variables.",
"The first term in (REF ) consists of the usual squared amplitude of Bhabha scattering and the second and third terms are the corrections of second and fourth order in $(gv^{\\mu })$ , represented by $\\mbox{{\\bf A}}(v,p_{1,2},q_{1,2})$ and $\\mbox{{\\bf B}}(v,p_{1,2},q_{1,2})$ respectively.",
"The exact form of these corrections are lengthy and will not be displayed in detail.",
"However, we notice that the interference terms of odd order cancel each other.",
"In order to complete the cross section calculation, we must adopt a frame of reference to express the kinematic variables.",
"Bhabha scattering is conventionally analyzed in the center of mass frame, where the 4-momenta take the form $p_{1}&=&(E,\\ {\\bf p}),\\ q_{1}=(E,-{\\bf p}),\\ p_{2}=(E,{\\bf \\ q}),\\nonumber \\\\q_{2}&=&(E,-{\\bf q}),\\ s=(2E)^{2}=E_{cm}^{2},$ with ${\\bf p}=E{\\bf \\hat{z}}$ , ${\\bf q}\\cdot \\hat{{\\bf z}}=E\\cos \\theta $ and the expression of the differential cross section becomes $\\frac{d\\sigma }{d\\Omega }_{\\mbox{cm}}=\\frac{1}{64\\pi ^{2}E_{cm}^{2}}.\\frac{1}{4}\\sum _{\\mbox{spins}}|\\mathcal {M}_{total}|^{2}.$ We will consider two possibilities according $v^{\\mu }$ being time-like or space-like.",
"For the first case where $(v^{\\mu }=v_0,0)$ is time-like , we can simplify (REF ) and make use of (REF ), to obtain $\\frac{d\\sigma }{d\\Omega }_{cm} & = & \\frac{e^{4}(\\cos 2\\theta +7)^{2}}{256\\pi ^{2}E_{cm}^{2}(\\cos \\theta -1)^{2}}+\\frac{v_{0}^{2}g^{2}e^{2}\\sin ^{2}\\frac{\\theta }{2}(-65\\cos \\theta +6\\cos 2\\theta +\\cos 3\\theta +122)}{256\\pi ^{2}(\\cos \\theta -1)^{2}}\\nonumber \\\\& + & \\frac{v_{0}^{4}g^{4}E_{cm}^{2}\\sin ^{4}\\frac{\\theta }{2}(-4\\cos \\theta +\\cos 2\\theta +11)}{128\\pi ^{2}(\\cos \\theta -1)^{2}},$ where the first term is the usual QED differential cross section at lowest order and the second and third terms contain the contributions of the LV background.",
"This result shows that the differential cross section remains symmetrical with respect to the colliding beams and its assymptotic angular dependence is qualitatively the same as the usual, as can be seen in Fig.REF .",
"For the second case of interest, we consider $(v^{\\mu }=0,\\mbox{\\textbf {v}})$ space-like and assuming an arbitrary direction.",
"In this way, we can write the scalar product of vectors as follows: ${\\bf p}\\cdot {\\bf v}&=&E\\mbox{v}\\cos (\\theta _{v}), \\nonumber \\\\{\\bf q}\\cdot {\\bf v}&=&E\\mbox{v}(\\sin \\theta \\sin \\theta _{v}\\cos (\\varphi -\\varphi _{v})+\\cos \\theta \\cos \\theta _{v})\\nonumber \\\\&\\equiv &E\\mbox{v}\\cos (\\Psi ),$ Thus, after some algebraic simplifications, we get $\\frac{d\\sigma }{d\\Omega }_{cm} & = & \\frac{e^{4}(\\cos 2\\theta +7)^{2}}{256\\pi ^{2}E_{cm}^{2}(\\cos \\theta -1)^{2}}\\nonumber \\\\& + & \\frac{e^{2}g^{2}\\text{v}^{2}}{256\\pi ^{2}(\\cos \\theta -1)^{2}}\\left[2(32\\cos \\theta +\\cos 2\\theta -49)\\cos ^{2}\\frac{\\theta }{2}\\left(\\cos ^{2}\\theta _{v}+\\cos ^{2}\\Psi \\right)\\right.\\nonumber \\\\& + & 2(61\\cos \\theta -22\\cos 2\\theta +3\\cos 3\\theta -10)\\cos \\Psi \\cos \\theta _{v}\\nonumber \\\\& + & \\left.\\sin ^{2}\\theta (-64\\cos \\theta +3\\cos 2\\theta +85)\\right]\\nonumber \\\\& + & \\frac{g^{4}\\text{v}^{4}E_{cm}^{2}}{1024\\pi ^{2}(\\cos \\theta -1)^{2}}\\left[-4(8\\cos \\theta +\\cos 2\\theta +7)\\cos ^{3}\\Psi \\cos \\theta _{v}+\\right.\\nonumber \\\\& + & 2\\cos ^{2}\\Psi \\left((12\\cos \\theta +7\\cos 2\\theta +29)\\cos ^{2}\\theta _{v}-2\\sin ^{2}\\theta (3\\cos \\theta +7)\\right)\\nonumber \\\\& - & 4\\cos \\Psi \\cos \\theta _{v}\\left((8\\cos \\theta +\\cos 2\\theta +7)\\cos ^{2}\\theta _{v}-4\\sin ^{2}\\theta (3\\cos \\theta +2)\\right)\\nonumber \\\\& + & (4\\cos \\theta +\\cos 2\\theta +11)\\cos ^{4}\\theta _{v}-4\\sin ^{2}\\theta (3\\cos \\theta +7)\\cos ^{2}\\theta _{v}\\nonumber \\\\& + & \\left.",
"(4\\cos \\theta +\\cos 2\\theta +11)\\cos ^{4}\\Psi +8\\sin ^{4}\\frac{\\theta }{2}(24\\cos \\theta +7\\cos 2\\theta +25)\\right].$ In the above result, we note the dependence of the cross section with respect to the azimuthal angle $\\varphi $ .",
"For the fixed background ${\\bf v}$ perpendicular to the beam collision $(\\theta _v=\\pi /2)$ , this effect is maximal and it is characterized by a set of periodic sharp peaks, as illustrated in Fig.REF .",
"For the Compton scattering with the LV term in the kinetic sector a similar result was reported [26].",
"To conclude this section, we will determine upper bounds for the products of the parameters $(gv^{\\mu })$ in the cases evaluated above.",
"Our choice to study Bhabha scattering was motivated, in addition to the questions outlined in the introduction, by practical reasons, i.e, the experimental data on precision tests for this kind of scattering in QED are readily available in Ref.",
"[30].",
"In the experiment reported in that paper, the measurements of the differential cross sections for $e^{+}e^{-}\\rightarrow e^{+}e^{-}$ and $e^{+}e^{-}\\rightarrow \\gamma \\gamma $ scatterings were evaluated at a center-of-mass energy of 29 GeV and in the polar-angular region $\\left|\\cos \\theta \\right|<0.55$ .",
"For Bhabha scattering, small deviations on the magnitude of the QED tree results may be expressed in the form: $\\Bigg {|}\\left.\\left(\\frac{d\\sigma }{d\\Omega }\\right)\\right\\bad.\\left(\\frac{d\\sigma }{d\\Omega }\\right)_{QED}-1\\Bigg {|}\\approx \\left(\\frac{3s}{\\Lambda ^{2}}\\right),$ where $s=E_{cm}^{2}$ and $\\Lambda $ is a small parameter representing possible experimental departures from the theoretical predictions (see Table XIV of Ref.",
"[30]).",
"Considering the leading corrections for small ($gv^{\\mu }$ ) in (REF ) and (REF ), we can show that the magnitude of these corrections are of order $g^{2}v^{2}s/e^{2}$ , and therefore when compared with (REF ) may not be larger than $3s/\\Lambda ^{2}$ .",
"Thus, we obtain the upper bound $(gv^{\\mu })\\le 10^{-12}(\\mbox{eV})^{-1},$ for $\\Lambda =200$ GeV.",
"In the above calculations we provided a way to obtain bounds to LV from the analyses of the Bhabha scattering experiment using only QED interactions.",
"The inclusion of QCD effects would improve the value of $\\Lambda $ (consequently the bound) and should allow a better comparison with the results encountered for atomic clocks or torsion balances."
],
[
"Bhabha scattering: axial-like nonminimal coupling",
"We turn our attention now to the nonminimal coupling of chiral character, defined as $D_{\\mu }=\\partial _{\\mu }+ieA_{\\mu }+ig_{5}\\gamma ^{5}b^{\\nu }F_{\\mu \\nu }^{\\ast },$ which was also examined in Refs.",
"[21], [22].",
"The calculation of the unpolarized cross section may be worked out similarly as in the previous section.",
"Note that the expression for $i\\mathcal {M}_{2}$ differ from (REF ) just for the insertion of the $\\gamma _5$ matrix in each matrix element, whereas for $i\\mathcal {M}_{1}$ we have the mixture of the vertices: $i\\mathcal {M}_{1}&=&\\frac{eg_{5}b^{\\nu }(p_{1}-p_{2})^{\\theta }\\epsilon _{\\mu \\nu \\theta \\rho }}{(p_{1}-p_{2})^{2}}\\left[u(p_{2})\\gamma ^{\\rho }u(p_{1})\\overline{v}(q_{1})\\gamma ^{\\mu }\\gamma ^{5}v(q_{2})\\right.\\nonumber \\\\&&\\left.-\\overline{u}(p_{2})\\gamma ^{\\text{$\\mu $}}\\gamma ^{5}u(p_{1})\\overline{v}(q_{1})\\gamma ^{\\text{$\\rho $}}v(q_{2})\\right]\\nonumber \\\\&&+\\frac{eg_{5}b^{\\nu }(p_{1}+q_{1})^{\\theta }\\epsilon _{\\mu \\nu \\theta \\rho }}{(p_{1}+q_{1})^{2}}\\left[\\overline{u}(p_{2})\\gamma ^{\\text{$\\rho $}}v(q_{2})\\overline{v}(q_{1})\\gamma ^{\\text{$\\mu $}}\\gamma ^{5}u(p_{1})\\right.\\nonumber \\\\&&\\left.-\\overline{u}(p_{2})\\gamma ^{\\text{$\\mu $}}\\gamma ^{5}v(q_{2})\\overline{v}(q_{1})\\gamma ^{\\text{$\\rho $}}u(p_{1})\\right].$ In the high energy limit and the center of mass frame the differential cross section for the case $b^{\\mu }=(b_{0},0)$ is given by $\\frac{d\\sigma }{d\\Omega }_{cm}&=&\\frac{e^{4}(\\cos 2\\theta +7)^{2}}{256\\pi ^{2}E_{cm}^{2}(\\cos \\theta -1)^{2}}\\nonumber \\\\&-&\\frac{b_{0}^{2}g_{5}^{2}e^{2}\\sin ^{4}\\frac{\\theta }{2}(8\\cos \\theta +\\cos 2\\theta +23)}{64\\pi ^{2}(\\cos \\theta -1)^{2}}\\nonumber \\\\&+&\\frac{b_{0}^{4}g_{5}^{4}E_{cm}^{2}\\sin ^{4}\\frac{\\theta }{2}(-4\\cos \\theta +\\cos 2\\theta +11)}{128\\pi ^{2}(\\cos \\theta -1)^{2}}.$ Similarly to the time-like case evaluated in the previous section, the above result contains only even terms in $(g_{5}b_{0})$ .",
"However, the asymptotic behavior is quite different and the leading-order contribution is finite in the limit $\\theta \\rightarrow 0$ , as shown by the dotted line in Fig.REF .",
"For the case $b^{\\mu }=(0,{\\bf b})$ , the differential cross section becomes $\\frac{d\\sigma }{d\\Omega }_{cm} & = & \\frac{e^{4}(\\cos 2\\theta +7)^{2}}{256\\pi ^{2}E_{cm}^{2}(\\cos \\theta -1)^{2}}\\nonumber \\\\& + & \\frac{e^{3}g_{5}\\text{b}\\sin ^{2}\\theta (\\cos \\theta +1)(\\cos \\theta _{5}+\\cos \\Psi _{5})}{16\\pi ^{2}E_{cm}(\\cos \\theta -1)^{2}}\\nonumber \\\\& + & \\frac{e^{2}g_{5}^{2}\\text{b}^{2}}{512\\pi ^{2}(\\cos \\theta -1)^{2}}\\left[8(7\\cos \\theta -6\\cos 2\\theta +\\cos 3\\theta +14)\\cos \\Psi _{5}\\cos \\theta _{5}\\right.\\nonumber \\\\& + & (-\\cos \\theta +26\\cos 2\\theta +\\cos 3\\theta -90)(\\cos ^{2}\\theta _{5}+\\cos ^{2}\\Psi _{5})\\nonumber \\\\& + & \\left.6\\sin ^{2}\\theta (-24\\cos \\theta +\\cos 2\\theta +31)\\right]\\nonumber \\\\& + & \\frac{eg_{5}^{3}\\text{b}^{3}E_{cm}\\sin ^{2}\\theta }{32\\pi ^{2}(\\cos \\theta -1)^{2}}\\left[3(\\cos \\theta -1)+(\\cos \\Psi _{5}-\\cos \\theta _{5}){}^{2}\\right](\\cos \\theta _{5}+\\cos \\Psi _{5})\\nonumber \\\\& + & \\frac{g_{5}^{4}\\text{b}^{4}E_{cm}^{2}}{1024\\pi ^{2}(\\cos \\theta -1)^{2}}\\left[-4(8\\cos \\theta +\\cos 2\\theta +7)\\cos ^{3}\\Psi _{5}\\cos \\theta _{5}\\right.\\nonumber \\\\& + & 2(12\\cos \\theta +7\\cos 2\\theta +29)\\cos ^{2}\\Psi _{5}\\cos ^{2}\\theta _{5}\\nonumber \\\\& + & 4\\cos \\Psi _{5}\\cos \\theta _{5}\\left(4\\sin ^{2}\\theta (3\\cos \\theta +2)-(8\\cos \\theta +\\cos 2\\theta +7)\\cos ^{2}\\theta _{5}\\right)\\nonumber \\\\& - & 2\\sin ^{2}\\theta (3\\cos \\theta +7)\\left(\\cos 2\\theta _{5}+\\cos 2\\Psi _{5}+2\\right)\\nonumber \\\\& + & (4\\cos \\theta +\\cos 2\\theta +11)\\cos ^{4}\\theta _{5}+(4\\cos \\theta +\\cos 2\\theta +11)\\cos ^{4}\\Psi _{5}\\nonumber \\\\& + & \\left.8\\sin ^{4}\\frac{\\theta }{2}(24\\cos \\theta +7\\cos 2\\theta +25)\\right],$ where the presence of odd-order corrections in $(g_{5}\\mbox{b})$ , absent in the previous vectorial cases is to be noticed.",
"Furthermore, the effect of anisotropy in the cross-section is highlighted, as indicated in Fig.",
"REF .",
"Now, an analyses similar to the previous section allows to set up an upper bound to the breaking parameter $(g_{5}b^{\\mu })$ .",
"Taking into account the magnitude of the leading-order corrections for the time-like and space-like cases given respectively by $4g_{5}^{2}b_{0}^{2}s/e^{2}$ and $16g_{5}\\mbox{b}\\sqrt{s}/e$ and assuming that $s=29$ GeV and $\\Lambda _{\\pm }=200$ GeV, we find $g_{5}b_{0}\\le 10^{-12}(\\mbox{eV})^{-1}\\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ g_{5}\\mbox{b}\\le 10^{-14}\\mbox{(eV)}^{-1}.$"
],
[
"Conclusion",
"In this paper, the implications of Lorentz symmetry breaking on Bhabha scattering have been studied.",
"The LV background terms were introduced by nonminimal couplings between the fermion and gauge fields.",
"We calculated the differential cross sections for the vector and axial couplings and determined upper bounds on the magnitude of the corresponding LV coefficients, by making use of accurate experimental data, available in the literature.",
"In particular, when we consider the vector backgrounds, $v^{\\mu }, b^{\\mu }$ , as being purely spatial, the cross section acquires a nontrivial dependence on the direction of these vectors.Finally, we hope that these results may be usefull as a guide in the investigation of the Lorentz violation phenomena in high energy scattering processes.",
"This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP).",
"Figure: Angular dependence of the differentialcross section for Bhabha scattering to the time-likeLorentz violation: QED prediction (solid line),vectorial (dashed line) and axial-like (dotted line) nonminimal couplings.Figure: Low order correction to the vectorial cross section (space-like case) for different directions of the background vector: (θ v =0,ϕ v =0)(\\theta _{v}=0,\\varphi _{v}=0) and (θ v =π/2,ϕ v =0)(\\theta _{v}=\\pi /2,\\varphi _{v}=0), respectively.Figure: Low order correction to the axial-like cross section (space-like case) for different directions of the background vector: (θ 5 =0,ϕ 5 =0)(\\theta _{5}=0,\\varphi _{5}=0), (θ 5 =π/2,ϕ 5 =0)(\\theta _{5}=\\pi /2,\\varphi _{5}=0) and (θ 5 =π,ϕ 5 =0)(\\theta _{5}=\\pi ,\\varphi _{5}=0), respectively."
]
] | 1204.0755 |
[
[
"Actions for an Hierarchy of Attractive Nonlinear Oscillators Including\n the Quartic Oscillator in 1+1 Dimensions"
],
[
"Abstract In this paper, we present an explicit form in terms of end-point data for the classical action $S_{2n}$ evaluated on extremals satisfying the Hamilton-Jacobi equation for each member of a hierarchy of classical non-relativistic oscillators characterized by even power potentials (i.e., attractive potentials $V_{2n}(y_{2n})={\\frac{1}{2n}}k_{2n}y_{2n}^{2n}(t)|_{n{\\geq}1}$).",
"The nonlinear quartic oscillator corresponds to $n=2$ while the harmonic oscillator corresponds to $n=1$."
],
[
"INTRODUCTION",
"The linearization map in [1] gives the solution for all members of the hierarchy $V_{2n}(y_{2n})={\\frac{1}{2n}}k_{2n}y_{2n}^{2n}(t)|_{n{\\ge }1}$ in terms of the linear (harmonic) oscillator ($n=1$ ).",
"It consists of an explicit nonlinear deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature: $x({\\hat{t}})&=(k_{2n}/nk_{2})^{1/2}y_{2n}(t)(y_{2n}^{2}(t))^{(n-1)/2},\\\\y_{2n}(t)&=(nk_{2}/k_{2n})^{1/2n}x({\\hat{t}})(x^{2}({\\hat{t}}))^{(1/2)(1-n)/n}, $ ${\\frac{dt}{d{\\hat{t}}}}=n^{-(2n-1)/2n}(k_{2}/k_{2n})^{1/2n}(x^{2}({\\hat{t}}))^{-(n-1)/2n},\\\\{\\frac{d{\\hat{t}}}{dt}}={\\sqrt{n}}(k_{2n}/k_{2})^{1/2}(y_{2n}^{2}(t))^{(n-1)/2},$ So, it important to keep in mind that all the quantities in this paper are known in principle as a consequence of (1.1) and (1.2)!",
"However, it is a non-canonical map.",
"Therefore, to find a form for the action evaluated on an extremal, we can only use the linearization map as a guide, albeit an extremely useful one.",
"Critical to the form stated below is the fact that $y_{{2n}_{\\max }}$ and $t_{\\max }$ are constants of the motion and they uniquely characterize every extremal of the periodic systems studied in this paper.",
"This allows us to find a form in terms of end-point data for the classical action $S_{2n}$ evaluated on extremals for each member of a hierarchy of classical non-relativistic oscillators characterized by even power potentials (i.e.,attractive potentials $V_{2n}(y_{2n})={\\frac{1}{2n}}k_{2n}y_{2n}^{2n}(t)|_{n{\\ge }1}$ ).",
"(The quartic oscillator corresponds to $n=2$ while the harmonic oscillator corresponds to $n=1$ .",
"The form is new for the harmonic linear oscillator and using the material in Part II one can readily check that it is equal in value to the one given in [2]-[3].)",
"(See Appendix B.)",
"In particular, we arrive at a form for $S_{(2n)}(t_a,y_{{2n}_{a}};t_b,y_{{2n}_{b}})={\\overset{t_b}{\\underset{t_{\\max }}{\\int }}}L_{2n}(y_{2n}(t),{\\frac{d}{dt}}y_{2n}(t))dt|_{extremal}{\\hspace{6.0pt}},\\\\+{\\overset{t_{\\max }}{\\underset{t_{a}}{\\int }}}L_{2n}(y_{2n}(t),{\\frac{d}{dt}}y_{2n}(t))dt|_{extremal}{\\hspace{6.0pt}},$ where $L_{2n}$ equals the Lagrangian for the $2n$ oscillator, which satisfies ${\\frac{{\\partial }S_{2n}}{{\\partial }y_{{2n}_b}}}&=p_{{2n}_b}{\\hspace{6.0pt}},\\\\{\\frac{{\\partial }S_{2n}}{{\\partial }t_b}}&=-E=-H_{2n}({\\frac{{\\partial }S_{2n}}{{\\partial }y_{{2n}_b}}},V_{2n}(y_{{2n}_b})){\\hspace{6.0pt}} ,$ and the time-reversed motion ${\\frac{{\\partial }S_{2n}}{{\\partial }y_{{2n}_a}}}&=-p_{{2n}_a}{\\hspace{6.0pt}},\\\\{\\frac{{\\partial }S_{2n}}{{\\partial }t_a}}&=+E=H_{2n}({\\frac{{\\partial }S_{2n}}{{\\partial }y_{{2n}_a}}},V_{2n}(y_{{2n}_a})){\\hspace{6.0pt}} ,$ where ${H_{2n}}$ = Hamiltonian for the $2n$ oscillator.",
"We shall use the notation $y=y_{2n}$ where it does not cause confusion.",
"$&S_{2n}(t_a,y_a;t_b,y_b) ={\\frac{m{\\omega }}{(n+1)}}({\\frac{k_{2n}}{nk_2}})^{1/2}{\\hspace{411.0pt}}\\\\&{\\lbrace }[(y^2_b)^{(n+1)/2}{\\cos }{\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt-(n+1)y_b{\\hspace{2.0pt}}y_{\\max }(y_{{\\max }^2})^{(n-1)/2}+n(y^2_{{\\max }})^{(n+1)/2} ({\\cos }^2{\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt)^{(n+1)/2n}/{\\cos }({\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt)]\\\\&/{\\sin }({\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt)\\\\&+{\\hspace{540.0pt}}\\\\&[(y^2_a)^{(n+1)/2}{\\cos }{\\omega }{\\int }^{t_{\\max }}_{t{_{a}}}{\\gamma }(t)dt-(n+1)y_ay_{\\max }(y_{{\\max }^2})^{(n-1)/2}+n(y^2_{{\\max }})^{(n+1)/2}({\\cos }^2{\\omega }{\\int }^{t_{\\max }}_{t{_{a}}}{\\gamma }(t)dt)^{(n+1)/2n}/{\\cos }({\\omega }{\\int }^{t_{\\max }}_{t{_{a}}}{\\gamma }(t)dt)]\\\\&/{\\sin }({\\omega }{\\int }^{t_{\\max }}_{t{_a}}{\\gamma }(t)dt){\\rbrace }\\\\&+{\\hspace{540.0pt}}\\\\&{\\frac{(n-1)}{(n+1)}}{\\frac{k_{2n}}{2n}}y^{2n}(t_b-t_a).$ where ${\\gamma }(t)$ = $({\\frac{nk_{2n}}{k_2}})^{1/2}$ $(y^2(t))^{n-1/2}$ .",
"In Part II, we discuss the description of the hierarchy extremals using the known results for the harmonic (linear) oscillator ($n=$ 1) that are implied by the linearization map presented in [1].",
"We start with the quartic oscillator and then present the general case.",
"In Part III, we set $n=2$ in (1.6) and show that the classical action evaluated on an extremal $S_{qo}$ satisfies the Hamilton-Jacobi equation equations (1.4)-(1.5)$|_{n=2}$ for the quartic oscillator.",
"In Part IV, we take $n$ arbitrary in (1.6) and show that the classical action evaluated on an extremal $S_{2n}$ satisfies the Hamilton-Jacobi equations (1.4)-(1.5) for each member of the hierarchy.",
"This reproduces the quartic oscillator result in Part III and as well as yielding a new form for the harmonic oscillator."
],
[
"Extremals",
"The set of extremals for the harmonic (linear) oscillator is described by the endpoint solution of Newton's equation of motion $x({\\hat{t}})=(x_b{\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_a)+x_a{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}})/{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a), $ where the spring constant $k_2$ = $m{\\omega }^2$ , $m=mass$ , ${\\hat{t}}$ denotes the harmonic oscillator time and $x$ denotes the space coordinate of the linear oscillator (ref [3] and eq.",
"(1.1) in [1]).",
"It is important to note that each extremal is uniquely characterized by $x_{\\max }$ and a ${\\hat{t}}_{{\\max }}$ .",
"Specifically, we take $0{\\hspace{2.0pt}}{\\le }{\\hspace{3.0pt}}{\\hat{t}}_{{\\max }}<{\\frac{2{\\pi }}{{\\omega }}}$ and of course $x_{\\max }$ is fixed by the energy.",
"In practice the evaluation of quantities here we can take ${\\hat{t}}_{{\\max }}$ up to a multiple of the period.",
"Now the extremals are also described by the equation $x({\\hat{t}})=x_{\\max }{\\hspace{2.0pt}}{\\cos }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }) $ [(see Appendix A for a demonstation)].",
"The set of extremals for a quartic oscillator with mass $m$ is described by the equivalent endpoint solution of Newton's equation of motion $&(y^2(t))^{1/2}{\\hspace{2.0pt}}y(t)=\\\\&{\\Biggl {[}}{\\frac{y_b(y^2_b)^{(1/2)}{\\sin }{\\omega }{\\overset{t}{\\underset{t_{a}}{\\int }}}({\\frac{2k_{4}}{k_2}})^{1/2}(y^2(t^{\\prime }))^{1/2}dt^{\\prime }{\\hspace{4.0pt}}+{\\hspace{4.0pt}}y_a(y{_a^2})^{1/2}{\\sin }{\\omega }{\\overset{t_{b}}{\\underset{t}{\\int }}}({\\frac{2k_{4}}{k_2}})^{1/2}(y^2(t^{\\prime }))^{1/2}dt^{\\prime }}{{\\sin }{\\omega }{\\overset{b}{\\underset{a}{\\int }}}({\\frac{2k_{4}}{k_2}})^{1/2}(y^2(t^{\\prime }))^{1/2}dt^{\\prime }}}{\\Biggr {]}} $ and the equivalent integral equation $(y(t)(y^2(t))^{1/2})-(y_{\\max }(y^2_{\\max })^{1/2}){\\cos }({\\omega }{\\int }^{t}_{t{_{max}}}{\\gamma }(t)dt^{\\prime })=0$ where $\\gamma $ = (2$k_4/k_2)^{1/2}(y^2)^{1/2},{\\hspace{4.0pt}}y_{max}$ = $(4E/k_4)^{1/4}$ , $k_4$ = denotes the quartic spring constant, $t$ denotes quartic oscillator time and $y=y_{qo}=y_4$ denotes the space coordinate of the quartic oscillator.",
"(We remind the reader that because of the linearization map given in [1], the integral equation (2.4) is solved.",
"However, as an aside, we would like to point out that ${\\int }^{t}_{t{_{max}}}{\\gamma }(t^{\\prime })dt^{\\prime }$ can also be determined from (2.4) since we know the pairs $(t,y(t))$ and $(t_{\\max }, y_{\\max })$ .)",
"The argument in Appendix A generalizes to this case for the equivalence of (2.3) and (2.4).",
"Further, that the integral equation (2.4) satisfies Newton’s equation of motion ${\\frac{d^2}{dt^2}}my(t)$ = $-k_4y^3(t)$ follows by direct differentiation twice.",
"The pairs (2.1)-(2.2) and (2.3)-(2.4) are connected by the linearization map (1.1)-1.2) given in [1] between the linear oscillator and the quartic oscillator.",
"Using the preceding arguments, (2.3)-(2.4) generalize to $&(y^2_{2n}(t))^{(n-1)/2}{\\hspace{2.0pt}}y_{2n}(t)=\\\\&{\\Biggl {[}}{\\frac{y_{2n_b}(y_{2n_b}^2)^{(n-1)/2}{\\sin }{\\omega }{\\overset{t}{\\underset{t_{a}}{\\int }}}{\\gamma }(t^{\\prime })dt^{\\prime }{\\hspace{4.0pt}}+{\\hspace{4.0pt}}y_{2n_a}(y_{2n_a}^2)^{(n-1)/2}{\\sin }{\\omega }{\\overset{t_b}{\\underset{t}{\\int }}}{\\gamma }(t^{\\prime })dt^{\\prime }}{{\\sin }{\\omega }{\\overset{t_b}{\\underset{t_a}{\\int }}}{\\gamma }(t^{\\prime })dt^{\\prime }}}{\\Biggr {]}} $ which is equivalent to the integral equation $(y_{2n}(t)(y^{2}_{2n}(t))^{(n-1)/2})-(y_{{2n}_{\\max }}(y^2_{2n_{\\max }})^{(n-1)/2}){\\cos }({\\omega }{\\int }^{t}_{t{_{\\max }}}{\\gamma }(t^{\\prime })dt^{\\prime })=0,$ where ${\\gamma }(t^{\\prime })$ = $({\\frac{nk_{2n}}{k_2}})^{1/2}(y_{2n}(t^{\\prime }))^{(n-1)/2}$ .",
"The above imply the following momenta since all systems have the same mass $m$ : $p_{{2n}_b}={\\frac{m{\\omega }}{ }}({\\frac{k_{2n}}{nk_2}})^{1/2}{\\bigl {[}}(y^2_{{2n}_b})^{(n-1)/2}y_{{2n}_b}{\\cos }({\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt)-(y^2_{{2n}_{\\max }})^{(n-1)/2}y_{{2n}_{\\max }}{\\bigr {]}}/{\\sin }({\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt),$ $p_{{2n}_a}={\\frac{m{\\omega }}{ }}({\\frac{k_{2n}}{nk_2}})^{1/2}{\\bigl {[}}(y^2_{{2n}_a})^{(n-1)/2}y_{{2n}_a}{\\cos }({\\omega }{\\int }^{t_{\\max }}_{t{_a}}{\\gamma }(t)dt)-(y^2_{{2n}_{\\max }})^{(n-1)/2}y_{{2n}_{\\max }}{\\bigr {]}}/{\\sin }({\\omega }{\\int }^{t_{\\max }}_{t{_a}}{\\gamma }(t)dt),$ These are the equations we have to reproduce with our ${S_{2n}}$ .",
"Note (2.6) implies $(y^2_{2n}(t_b))^{(n-1)/2})=(y^{2}_{{2n}_{\\max }})^{(n-1)/2})({\\cos }^2({\\omega }{\\int }^{t_b}_{t{_{\\max }}}{\\gamma }(t)dt))^{(n-1)/2n},$ and $(y^2_{2n}(t_a))^{(n-1)/2})=(y^{2}_{{2n}_{\\max }})^{(n-1)/2})({\\cos }^2({\\omega }{\\int }^{t_{\\max }}_{t{_a}}{\\gamma }(t)dt))^{(n-1)/2n}.$ These latter relations are needed to evaluate ${\\gamma }$ when we differentiate w.r.t time for $n=2$ in Part III and arbitrary $n$ in Part IV."
],
[
"Nonlinear action for the quartic oscillator",
"As mentioned above, because the linearization map given in [1] is a non-canonical one, we do not have a derivation of the actions for the nonlinear quartic oscillator in terms of $y_b,y_a,t_b$ and $t_a$ , rather we have constructed it, namely (1.6) with $n=2$ , using the linearization map as a guide.",
"$&S_{qo}(t_a,y_a;t_b,y_b)=S_4(t_a,y_a;t_b,y_b,)\\\\&={\\frac{m{\\omega }}{3}}({\\frac{k_4}{2k_2}})^{1/2}\\\\&{\\lbrace }{\\Biggl {[}}(y^2_b)^{3/2}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)-3y_b{\\hspace{4.0pt}}y_{\\max }(y^2_{\\max })^{1/2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}2(y^2_{\\max })^{3/2}({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))^{3/4}{\\hspace{4.0pt}}/{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)){\\Biggr {]}}\\\\&/{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\hspace{360.0pt}}\\\\&+\\\\&{\\Biggl {[}}(y^2_a)^{3/2}{\\cos }({\\omega }{\\int }^{t_{\\max }}_{t_a}{\\gamma }(t)dt)-3y_a{\\hspace{4.0pt}}y_{\\max }(y^2_{\\max })^{1/2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}2(y^2_{\\max })^{3/2}({\\cos }^2({\\omega }{\\int }^{t_{\\max }}_{t_a}{\\gamma }(t)dt))^{3/4}{\\hspace{4.0pt}}/{\\cos }({\\omega }{\\int }^{t_{\\max }}_{t_a}{\\gamma }(t)dt)){\\Biggr {]}}\\\\&/{\\sin }({\\omega }{\\int }^{t_{\\max }}_{t_a}{\\gamma }(t)dt){\\rbrace }\\\\&+{\\frac{1}{12}}{\\hspace{2.0pt}}k_4{\\hspace{2.0pt}}y^4_{\\max }(t_b-t_a)$ This implies $&{\\frac{\\partial }{{\\partial }y_b}}S_{qo}(t_a,y_a;t_b,y_b)\\\\&=m{\\omega }({\\frac{k_4}{2k_2}})^{1/2}\\\\&{\\Biggl {[}}(y^2_b)^{1/2}{\\hspace{4.0pt}}y_b{\\hspace{4.0pt}}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)-(y^2_{\\max })^{1/2}y_{max}{\\Biggr {]}}/{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)=p_b \\\\$ and $&{\\frac{\\partial }{{\\partial }t_b}}S_{qo}(t_a,y_a;t_b,y_b)\\\\&={\\frac{m{\\omega }^2}{3}}({\\frac{k_4}{2k_2}})^{1/2}{\\gamma }_b{\\lbrace }\\\\&{\\Biggl {[}}(y^2_b)^{3/2}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }dt)-3y_b{\\hspace{4.0pt}}y_{\\max }(y^2_{\\max })^{1/2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}2(y^2_{\\max })^{3/2}({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)^{3/4}{\\hspace{4.0pt}}/{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)){\\Biggr {]}}\\\\&(-{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)/{\\sin }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))\\\\&+[(y^2_b)^{3/2}(-{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)\\\\&+2(y^2_{\\max })^{3/2}(3/4){\\langle }-2{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\rangle }/(({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))^{1/4}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))\\\\&+2(y^2_{\\max })^{3/2}({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))^{3/4}{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)/{\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)]\\\\&/{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\rbrace }\\\\&+{\\frac{1}{12}}{\\hspace{2.0pt}}k_4{\\hspace{2.0pt}}y^4_{\\max }\\\\&=-{\\frac{k_4}{3}}y^4_{\\max }+{\\frac{1}{12}}{\\hspace{2.0pt}}k_4{\\hspace{2.0pt}}y^4_{\\max }=-{\\frac{k_{4}{\\hspace{2.0pt}}y^4_{\\max }}{4}}$ (We have used (2.4) $|_{t=t_b}$ and (2.8a) $|_{n=2}$ after differentiating wrt $t_b$ to obtain (3.3).)",
"The first four terms sum to zero.",
"${\\frac{{\\partial }S_{qo}}{{\\partial }y_{_b}}}&=p_{{qo}_b}{\\hspace{6.0pt}},\\\\{\\frac{{\\partial }S_{qo}}{{\\partial }t_b}}&=-E=H_{qo}({\\frac{{\\partial }S_{qo}}{{\\partial }y_{_b}}},V_{qo}(y_b)){\\hspace{6.0pt}}.$ This agrees with (2.7a) and (1.4) for $n=2$ .",
"The $a$ -differentiations parallel the $b$ -differentiations and yield ${\\frac{{\\partial }S_{qo}}{{\\partial }y_{_a}}}&=-p_{{qo}_a}{\\hspace{6.0pt}},\\\\{\\frac{{\\partial }S_{qo}}{{\\partial }t_a}}&=+E=H_{qo}({\\frac{{\\partial }S_{qo}}{{\\partial }y_{_a}}},V_{qo}(y_a)){\\hspace{6.0pt}}.$ This agrees with (2.7b) and (1.5) for $n=2$ .",
"It is important to note that the value of $S_{qo}$ is not changed if the substitution of (2.4)$|_{t=t_b}$ is made in (3.1).",
"However, this substitution cannot be made before all differentiations are made because the choice of the form in terms of space-time endpoints for $S_{qo}$ is critical."
],
[
"Actions for the nonlinear hierarchy $V_2(y_{2n})={\\frac{1}{2_n}}k_{2n}y_{2n}^{2n}(t){\\hspace{4.0pt}}|_{n{\\ge }1^.}$",
"Here we generalize the approach from Part III.",
"The action on an extremal $S_{2n}(t_a,y_a;t_b,y_b)$ is given (1.4) (Here, we shall use the notation $y=y_{2n}$ .",
").",
"We now proceed to verify that (1.6) satisfies the Hamilton-Jacobi equations (1.4)-(1.5) by verifying that the following equations are satisfied (We shall use the notation $y=y_{2n}$ .",
"): $&{\\frac{\\partial }{{\\partial }y_b}}S_{2n}(t_a,y_a;t_b,y_b)\\\\&={\\hspace{4.0pt}}m{\\omega }({\\frac{k_{2n}}{nk_2}})^{1/2}\\\\&{\\Biggl {[}}(y^2_b)^{(n-1)/2}{\\hspace{4.0pt}}y_b{\\hspace{4.0pt}}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)-y_{\\max }(y^2_{\\max })^{(n-1)/2}{\\Biggr {]}}\\\\&/{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)=p_{{2n}_b},$ and $&{\\frac{\\partial }{{\\partial }t_b}}S_{2n}(t_a,y_a;t_b,y_b)={\\frac{m{\\omega }^2}{(n+1)}}({\\frac{k_{2n}}{nk_2}})^{1/2}{\\gamma }_b\\\\&{\\lbrace }{\\Biggl {[}}(y^2_b)^{(n+1)/2}{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt-(n+1)y_b{\\hspace{4.0pt}}y_{\\max }(y^2_{\\max })^{(n-1)/2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}n(y^2_{\\max })^{(n+1)/2}({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))^{(n+1)/2n}/{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\Biggr {]}}\\\\&(-{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)/{\\sin }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)\\\\&+{\\Biggl {[}}(y^2_a)^{(n+1)/2}-{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt))+n(y^2_{\\max })^{(n+1)/2n}((n+1)/2){\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)^{((n+1)/2n-1}{\\langle }-2{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)\\\\&{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt){\\rangle }/{\\cos }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)+n(y^2_{\\max })^{(n+1)/2}{\\cos }^2{\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)^{(n+1)/2n}{\\hspace{4.0pt}}{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt)/({\\cos }^2({\\omega }{\\int }^{t_b}_{t_{\\max }}{\\gamma }(t)dt.",
"){\\Biggr {]}}\\\\&/{\\sin }({\\omega }{\\int }^{t_b}_{t_{\\max }}(t)dt){\\rbrace }\\\\&+{\\frac{n-1}{n+1}}{\\hspace{2.0pt}}{\\frac{k_{2n}}{2n}}{\\hspace{2.0pt}}y^{2n}_{\\max }\\\\&=-{\\frac{k_{2n}}{n+1}}y^{2n}_{\\max }+{\\frac{(n-1)}{(n+1)}}{\\frac{k_{2n}}{2n}}y^{2n}_{\\max },$ where ${\\gamma }_b$ = $({\\frac{nk_{2n}}{k_2}})^{1/2}(y^{2}_{b})^{(n-1)/2}$ .",
"(We have used (2.4) $|_{t=t_b}$ and (2.8a) after differentiating $t_b$ to obtain (4.2).)",
"The first four terms sum to zero.",
"Thus, ${\\frac{{\\partial }S_{2n}}{{\\partial }y_b}}&=p_{{2n}_b}{\\hspace{3.0pt}},\\\\{\\frac{{\\partial }S_{2n}}{{\\partial }t_b}}&=-E,{\\hspace{3.0pt}}$ which agrees with (2.7a) and (1.4).",
"The $a$ -differentiations parallel the $b$ -differentiations and yield.",
"${\\frac{{\\partial }S_{2n}}{{\\partial }y_a}}&=p_{{2n}_a}{\\hspace{3.0pt}},\\\\{\\frac{{\\partial }S_{2n}}{{\\partial }t_a}}&=E,{\\hspace{3.0pt}}$ which agrees with (2.7b) and (1.5)."
],
[
"Concluding Remarks",
"To the best of the author's knowledge of the existing literature on classical mechanics, he can not find any literature on general transformation theory devoted to transforming one system at a given time to another at a distinctly different time, hence the absence of references to the classical literature on this point in this paper."
],
[
"Acknowledgements",
"The author wishes to acknowledge those who participated in a seminar organized by David Edwards and Robert Varley in AY 2006-2007 to study Feynman Path Integrals and especially two students Emily Pritchett and Justin Manning.",
"The seminar provided the original motivation for exploring the extent of the connection between the linear oscillator and the Feynman’s Path Integral Method of which this is a part.",
"Further, Robert Varley has continued to this date to provide insight and encouragement in this effort.",
"The author wishes to thank Professor Howard Lee for his insightful discussions and his constant encouragement."
],
[
"References",
"[1] Robert L. Anderson, “An Invertible linearization map for the quartic oscillator”, JMP 51 , 122904 (2010).",
"[2] Feynman’s Thesis-A New Approach To Quantum Theory, Laurie M. Brown-editor.",
"Singapore: World Scientific, 2005.",
"See equation (27), p 18 and set .",
"We are grateful to Justin Manning for bringing this result to our attention.",
"[3] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.",
"As stated in Part II, the set of extremals for the harmonic (linear) oscillator is given by $x({\\hat{t}})=(x_b{\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_a)+x_a{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}})/{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)$ , ${\\hspace{45.0pt}}$ (2.1) where the spring constant $k_2$ = $m{\\omega }^2$ , $m$ = $mass$ , ${\\hat{t}}$ denotes the harmonic oscillator time and $x$ denotes the space coordinate of the linear oscillator.",
"We now demonstrate that equation (A.1) is equivalent to the relationship $x({\\hat{t}})=x_{\\max }{\\hspace{2.0pt}}{\\cos }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max })$ .${\\hspace{150.0pt}}$ (2.2) Rewriting (2.1) as $x({\\hat{t}})=(x_b{\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}_a)+x_a{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}))/{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}_a), $ cross-multiplying and expanding, we have $x({\\hat{t}}){\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}_a)=(x_b{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}_a)+x_a{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }+{\\hat{t}}_{\\max }-{\\hat{t}}))$ ${\\hspace{50.0pt}}$ or $x({\\hat{t}}){\\lbrace }{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)+{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a){\\rbrace }\\\\=x_b{\\lbrace }{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }){\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)+{\\cos }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }){\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a){\\rbrace }\\\\+x_a{\\lbrace }{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}})+{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}){\\rbrace }\\\\ $ Substituting (2.2), we obtain the identity $x({\\hat{t}}){\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\frac{x_a}{x_{\\max }}}+x({\\hat{t}}){\\frac{x_b}{x_{\\max }}}{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)\\\\=x_b{\\sin }{\\omega }({\\hat{t}}-{\\hat{t}}_{\\max }){\\frac{x_a}{x_{\\max }}}+x_b{\\frac{x({\\hat{t}})}{x_{\\max }}}{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)\\\\+x_a{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max }){\\frac{x({\\hat{t}})}{x_{\\max }}}+x_a{\\frac{x_b}{x_{\\max }}}{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}), $ where the 1st and the 4th terms on the r.h.s.",
"of (A.3) cancel.",
"You can now run the above argument backwards.",
"Thus we have shown that (2.1) and (2.2) are equivalent.",
"It follows from differentiating (2.1) and (2.2) and setting ${\\hat{t}}$ = ${\\hat{t}}_b$ that $-x_{\\max }{\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })={\\frac{x_b{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)-x_a}{{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)}}.$ Similarly, it follows that $x_{\\max }{\\hspace{2.0pt}}{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)={\\frac{x_b-x_a{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)}{{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)}}.$ Hence $&m{\\omega }[{\\frac{(x_b^2+x_a^2){\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)-2x_bx_a}{2{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_a)}}]\\\\=&{\\frac{m{\\omega }}{2}}[-x_{\\max }x_b{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })-x_{\\max }x_a{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)].$ where the l.h.s.",
"is the $S_{ho}$ of [2] - [3].",
"The action $S_{2n}|_{n=1}=S_{ho}$ with $x=y_2$ given by (1.6) is $&S_{ho}={\\frac{m{\\omega }}{2}}{\\lbrace }[x^2_b{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })-2x_bx_{\\max }+{\\frac{x^2_{\\max }{\\cos }^2{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })}{{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })}}]/{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })\\\\&+[x^2_a{\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)-2x_ax_{\\max }+{\\frac{x^2_{\\max }{\\cos }^2{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)}{{\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)}}]/{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)$ Now (B.4)$|_{(2.2)}$ in value is given by $&{\\frac{m{\\omega }}{2}}{\\lbrace }x^2_b[{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })-{\\frac{1}{{\\cos }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })}}]/{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })\\\\&+x^2_a[{\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)-{\\frac{1}{{\\cos }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)}}]/{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a){\\rbrace }\\\\&={\\frac{m{\\omega }}{2}}[-x_bx_{\\max }{\\sin }{\\omega }({\\hat{t}}_b-{\\hat{t}}_{\\max })-x_ax_{\\max }{\\sin }{\\omega }({\\hat{t}}_{\\max }-{\\hat{t}}_a)].$ The r.h.s.",
"of (B.5) equals the l.h.s.",
"of (B.3)."
]
] | 1204.0768 |
[
[
"Space efficient streaming algorithms for the distance to monotonicity\n and asymmetric edit distance"
],
[
"Abstract Approximating the length of the longest increasing sequence (LIS) of an array is a well-studied problem.",
"We study this problem in the data stream model, where the algorithm is allowed to make a single left-to-right pass through the array and the key resource to be minimized is the amount of additional memory used.",
"We present an algorithm which, for any $\\delta > 0$, given streaming access to an array of length $n$ provides a $(1+\\delta)$-multiplicative approximation to the \\emph{distance to monotonicity} ($n$ minus the length of the LIS), and uses only $O((\\log^2 n)/\\delta)$ space.",
"The previous best known approximation using polylogarithmic space was a multiplicative 2-factor.",
"Our algorithm can be used to estimate the length of the LIS to within an additive $\\delta n$ for any $\\delta >0$ while previous algorithms could only achieve additive error $n(1/2-o(1))$.",
"Our algorithm is very simple, being just 3 lines of pseudocode, and has a small update time.",
"It is essentially a polylogarithmic space approximate implementation of a classic dynamic program that computes the LIS.",
"We also give a streaming algorithm for approximating $LCS(x,y)$, the length of the longest common subsequence between strings $x$ and $y$, each of length $n$.",
"Our algorithm works in the asymmetric setting (inspired by \\cite{AKO10}), in which we have random access to $y$ and streaming access to $x$, and runs in small space provided that no single symbol appears very often in $y$.",
"More precisely, it gives an additive-$\\delta n$ approximation to $LCS(x,y)$ (and hence also to $E(x,y) = n-LCS(x,y)$, the edit distance between $x$ and $y$ when insertions and deletions, but not substitutions, are allowed), with space complexity $O(k(\\log^2 n)/\\delta)$, where $k$ is the maximum number of times any one symbol appears in $y$."
],
[
"Introduction",
"Two classic optimization problems concerning subsequences (substrings) of arrays (strings) are the longest increasing subsequence (LIS) and longest common subsequence (LCS) problems.",
"A string of length $n$ over alphabet $\\Sigma $ is represented as a function $x:[n] \\rightarrow \\Sigma $ .",
"A subsequence of length $k$ is a string $x(i_1)x(i_2)\\ldots x(i_k)$ , where $1 \\le i_1 < i_2 < \\cdots < i_k \\le n$ .",
"In the LIS problem, the alphabet $\\Sigma $ comes equipped with a (total or partial) order $\\triangleleft $ , and we look for the longest subsequence whose terms are in increasing order.",
"In the LCS problem we are given two strings $x$ and $y$ and look for the longest string which is a subsequence of each of them.",
"Note that the LIS of $x$ is the LCS of $x$ and its sorted version.",
"Both of these problems can be solved by dynamic programs.",
"The LIS can be found on $O(n\\log n)$ time [28], [17], [2].",
"This is known to be optimal, even for (comparison based) algorithms that only determine the length of the LIS [27].",
"The LCS problem has a fairly direct $O(n^2)$ algorithm [11], which can be improved to $O(n^2/\\log ^2n)$ [24], [9].",
"It is a notoriously difficult open problem to improve this bound, or prove some matching lower bounds.",
"It is often natural to focus on the complements of the LIS and LCS lengths, which are related to some notion of distances between strings.",
"The distance to monotonicity of (the length $n$ string) $x$ , denoted $DM(x)$ is defined to be $n-LIS(x)$ , and is the the minimum number of values that need to be changed to make $x$ .",
"The (insertion-deletion) edit distance of (two length $n$ strings) $x,y$ , denoted $E(x,y)$ is defined to be $n-LCS(x,y)$ and is the minimum number of insertions and deletions needed to change one string into the other.",
"(Note that $E(x,y)$ is bounded between $L(x,y)$ and $2L(x,y)$ where $L(x,y)$ is the Levenshtein distance, where insertions, deletions, and substitutions are allowed.)",
"Of course the algorithmic problems of exactly computing $LIS(x)$ and $DM(x)$ are equivalent, but approximating them can be very different.",
"In recent years, there has been a lot of attention on giving approximate solutions for LIS and LCS that are much more efficient that the basic dynamic programming solutions.",
"Any improved results for LCS would be very interesting, since the best known quadratic time solution is infeasible for very large strings.",
"These problems can be studied in a variety of settings - sampling, streaming, and communication.",
"The streaming setting has been the focus of many results [20], [30], [18], [13].",
"The model for the LIS is that we are allowed one (or constant) passes over the input string $x$ , and only have access to sublinear storage.",
"The usual formulation of LCS in the streaming model postulates that we have only one-way access to both strings $x$ and $y$ .",
"We consider an alternative asymmetric model in which we have one-way access to string $x$ (called the input string) but random accesss to string $y$ (called the fixed string).",
"This model is more powerful that the standard one, but it is still far from clear how to obtain space efficient approximations to $E(x,y)$ in this model.",
"(This model was inspired by recent work of [6] concerning the time complexity of approximating edit distance in the random access model.",
"One part of their work introduced an asymmetric version of the random access model in which one pays only for accesses to one of the strings, and established time lower bounds for good approximations that hold even in this more powerful model.)"
],
[
"Results",
"Our first result is a streaming algorithm for approximating the distance to monotonicity.",
"There is a randomized one-pass streaming algorithm that for any $\\delta >0$ , takes as input an array (of length $n$ ), makes one-pass through the array, uses space $O(\\delta ^{-1}\\log ^2 n)$ and with error probability $n^{-\\Omega (1)}$ outputs an estimate to $DM(x)$ that is between $DM(x)$ and $(1+\\delta )DM(x)$ .",
"Previously there was a polylogarithmic time algorithm that gave a factor 2-approximation [13], and an algorithm that gave arbitrarily good multiplicative approximations to $LIS(x)$ (which is harder than approximating $DM(x)$ ) but required $\\Omega (\\sqrt{n})$ space [20].",
"The improvement in the approximation ratio from 2 to $1+\\delta $ (for polylogarithmic space algorithms) is not just “chipping away” at a constant, but provides a significant qualitative difference: previous polylogarithmic space algorithms might return an estimate of 0 when the LIS length is $n/2$ , while our algorithm can detect increasing subsequences of length a small fraction of $n$ .",
"More precisely, it is easy to see that if $V$ is an estimate of $DM(x)$ that is between $DM(x)$ and $(1+\\delta )DM$ then $n-V$ is within an additive $\\frac{\\delta }{1+\\delta } n$ of $LIS(x)$ , and so our algorithm can provide an estimation interval for $LIS(x)/n$ of arbitrarily small width.",
"The previous polylogarithmic time streaming algorithm only gave such an algorithm for $\\delta \\ge 1$ , which only guarantees an estimation interval for $LIS(x)/n$ of width 1/2.",
"The algorithm promised by Theorem REF is derived as a special case of a more general algorithm (Theorem REF ) that finds increasing sequences in partial orders.",
"This algorithm will also be applied to give a good (additive) approximation algorithm for edit distance in the asymmetric setting, whose space is polylogarithmic in the case that no symbol appears many times in the fixed string.",
"Let $\\delta \\in (0,1]$ .",
"Suppose $y$ is a fixed string of length $n$ and $x$ an input string of length $n$ to which we have streaming access.",
"There is a randomized algorithm that makes one pass through $x$ and, with error probability $n^{-\\Omega (1)}$ , outputs an additive $\\delta n$ -approximation to $E(x,y)$ and uses space $O(k\\log ^2 n/\\delta )$ where $k$ is the maximum number of times any symbol appears in $y$ .",
"There is a deterministic algorithm that runs in space $O(\\sqrt{(n\\log n)/\\delta })$ -space and outputs a $(1+\\delta )$ -multiplicative (which is also a $\\delta n$ -additive) approximation to $E(x,y)$ ."
],
[
"Techniques",
"A notable feature of our algorithm is its conceptual simplicity.",
"The pseudocode for the LIS approximation is just a few lines.",
"The algorithm has parameters $\\alpha (i,t)$ for $1 \\le i < t$ ) whose exact formula is a bit cumbersome to state at this point.",
"We set $\\alpha (i,t)$ to be 0 for $i \\ge t-O(\\log (n))$ and approximately $1/(t-i)$ otherwise.",
"The algorithm maintains a set of indices $R$ , and for each $i \\in R$ , we store $x(i)$ and an estimate $r(i)$ of $DM(x[1,i])$ , where $x[1,i]$ is the length $i$ prefix of $x$ .",
"For convenience, we add dummy elements $x(0) = x(n+1) = -\\infty $ and begin with $R = \\lbrace 0\\rbrace $ .",
"For each time $t \\ge 1$ , we perform the following update: Define $R^{\\prime } = \\lbrace i \\in R | x(i) \\le x(t)\\rbrace $ .",
"Set $r(t)=\\min _{i \\in R^{\\prime }}(r(i)+t-1-i)$ .",
"$R \\longleftarrow R \\cup \\lbrace t\\rbrace $ .",
"Remove each $i \\in R$ independently with probability $\\alpha (i,t)$ .",
"The final output is $r(n+1)$ .",
"The space used by the algorithm is (essentially) the maximum size of $|R|$ .",
"The update time is determined by step 1, which runs in time $O(|R|)$ .",
"Without the third step, the algorithm is a simple quadratic time exact algorithm for $DM(x)$ using linear space.",
"(The $O(n\\log n)$ time algorithms [17], [2] also work in a streaming fashion, but store data much more cleverly.)",
"More precisely, at step $t$ , $R=\\lbrace 0,\\ldots ,t\\rbrace $ and for each $i \\le t$ , $r(i)=DM(x[1,i])$ .",
"The third step reduces the set $R$ , thereby reducing the space of the algorithm.",
"The space used by the algorithm is (essentially) the maximum size of $R$ .",
"Intuitively, the algorithm “forgets” $(x(i),r(i))$ for those $i$ removed from $R$ .",
"The set $R$ of remembered indices is a subset of $[1,t]$ whose density decays as one goes back in time from the present time $t$ .",
"When we compute $r(t)$ , it may no longer be equal to the distance to monotonicity of the prefix of $x$ of length $i$ , but it will be at least this value.",
"This forgetting strategy is tailored to ensure that $r(t)$ is also at most a $(1+\\delta )$ -factor away from the distance to monotonicity.",
"We also ensure that that (with high probability) the set $R$ does not exceed size $O(\\delta ^{-1}\\log ^2 n)$ Just to give an indication of the difficulty, consider an algorithm that forgets uniformly at random.",
"At some time $t$ , the set of remembered indices $R$ is a uniform random set of size $O(\\delta ^{-1}\\log ^2 n)$ up to index $t$ .",
"These are used to compute $r(t)$ and include $t$ in $R$ .",
"The algorithm then forgets a uniform random index in $R$ to maintain the space bound.",
"Since we want to get a $(1+\\delta )$ -factor approximation, the algorithm must be able to detect an LIS of length $\\Omega (\\delta n)$ .",
"Of the indices in $R$ (up to time $t$ ), it is possible that around a $O(\\delta )$ -fraction of them are in the LIS.",
"Suppose we reach a small stretch of indices not on the LIS.",
"If this has size even $\\textrm {poly}(\\delta ^{-1}\\log n)$ , it is likely that all LIS indices in $R$ are forgotten.",
"But how do we selectively remember the LIS indices without knowing the LIS in advance?",
"That is the challenge of the forgetting strategy.",
"All past polylogarithmic space algorithms [20], [13] for LIS use combinatorial characterizations of increasing sequences based on inversion counting [14], [12], [26], [1].",
"While this is a very powerful technique, it does not lead to accurate approximations for the LIS, and (apparently) do not yield any generalizations to LCS.",
"The idea of remembering selected information about the sequence that becomes sparser as one goes back in time was first used by [20] for the inversion counting approach.",
"Our work seems to be the first to use this to directly mimic the dynamic program, though the idea is quite natural and has almost certainly been considered before.",
"The main contribution here is to analyze this algorithm, and determine the values of parameters that allow it to be both space efficient and a good approximation.",
"This line of thinking can be exploited to deal with asymmetric streaming LCS.",
"We construct a simple reduction of LCS to finding the longest chain in a specific partial order.",
"This reduction has a streaming implementation, so the input stream can be directly seen as just elements of this resulting partial order.",
"This reduction blows up the size of the input, and the size of the largest chain can become extremely small.",
"If each symbol occurs $k$ times in $x$ and $y$ , then the resulting partial order has $nk$ elements.",
"Nonetheless, the longest chain still has length at most $n$ .",
"We require very accurate estimates for the length of the longest chain.",
"This is where the power of the $(1+\\delta )$ -approximation comes in.",
"We can choose $\\delta $ to be much smaller to account for the input blow up, and still get a good approximation.",
"Note that if we only had a $1.01$ -approximation for the longest chain problem, this reduction would not be useful.",
"Our $\\widetilde{O}(\\sqrt{n})$ -space algorithm also works according to the basic principle of following a dynamic program, although it uses one different from the previous algorithms.",
"This can be thought of as generalization of the $\\widetilde{O}(\\sqrt{n})$ -space algorithm for LIS[20].",
"We maintain a $\\widetilde{O}(\\sqrt{n})$ -space deterministic sketch of the data structure maintained by the exact algorithm.",
"By breaking the stream up into the right number of chunks, we can update this sketch using $\\widetilde{O}(\\sqrt{n})$ -space.",
"Previous work The study of LIS and LCS in the streaming setting was initiated by Liben-Nowell et al [23], although their focus was mostly on exactly computing the LIS.",
"Sun and Woodruff [30] improved upon these algorithms and lower bounds and also proved bounds for the approximate version.",
"Most relevant for our work, they prove that randomized protocols that compute a $(1+\\varepsilon )$ -approximation of the LIS length essentially require $\\Omega (\\varepsilon ^{-1}\\log n)$ .",
"Gopalan et al [20] provide the first polylogarithmic space algorithm that approximates the distance to monotonicity.",
"This was based on inversion counting ideas in [26], [1].",
"Ergun and Jowhari [13] give a 2-approximation using the basic technique of inversion counting, but develop a different algorithm.",
"Gál and Gopalan [18] and independently Ergun and Jowhari [13] proved an $\\Omega (\\sqrt{n})$ lower bound for deterministic protocols that approximate that LIS length up to a multiplicative constant factor.",
"For randomized protocols, the Sun and Woodruff bound of $\\Omega (\\log n)$ is the best known.",
"One of the major open problems is to get a $o(\\sqrt{n})$ space randomized protocol (or an $\\Omega (\\sqrt{n})$ lower bound) for constant factor approximations for the LIS length.",
"Note that our work does not imply anything non-trivial for this problem.",
"We are unaware of any lower bounds for estimating the distance to monotonicity in the streaming setting.",
"A significant amount of work has been done in studying the LIS (or rather, the distance to monotonicity) in the context of property testing [14], [12], [15], [26], [1].",
"The property of monotonicity has been studied over a variety of domains, of which the boolean hypercube and the set $[n]$ (which is the LIS setting) have usually been of special interest [19], [12], [16], [22], [1], [26], [10].",
"In previous work, the authors of this paper found a $(1+\\delta )$ -multiplicative approximation algorithm for the distance to monotonicity (in the random access model) that runs in time $O(\\textrm {poly}\\log (n))$ [29].",
"As the present result does for the streaming model, that result also improved on the previous best factor 2 approximation for that model.",
"Despite the superficial similarity between the statement of results, the models considered in these two papers are quite different, and the algorithm we give here in the streaming model is completely different from the complicated algorithm we gave in the sublinear time model.",
"The LCS and edit distance have an extemely long and rich history, especially in the applied domain.",
"We point the interesting reader out to [21], [25] for more details.",
"Andoni et al [6] achieved a breakthrough by giving a near-linear time algorithm (in the random access model) that gives polylogarithmic time approximations for the edit distance.",
"This followed a long line of results well documented in [6].",
"They initiate the study of the asymmetric edit distance, where one string is known and we are only charged for accesses to the other string.",
"For the case of non-repetitive strings, there has been a body of work on studying the Ulam distance between permutations [4], [5], [3], [7].",
"Paths in posets We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).",
"We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.",
"Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.",
"This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP.",
"Weighted $P$ -sequences and the approximate minimum-defect path problem We use $P$ to denote a fixed set endowed with a partial order $\\triangleleft $ .",
"The partial order relation is given by an oracle which, given $u,v \\in P$ outputs $u \\triangleleft v$ or $\\lnot (u \\triangleleft v)$ .",
"For a natural number $n$ we write $[n]$ for the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"A sequence ${\\bf \\sigma }=(\\sigma (1),\\ldots ,\\sigma (n)) \\in P$ is called a $P$ -sequence.",
"The number of terms ${\\bf \\sigma }$ is called the length of ${\\bf \\sigma }$ and is denoted $|{\\bf \\sigma }|$ ; we normally use $n$ to denote $|{\\bf \\sigma }|$ .",
"A weighted $P$ -sequence consists of a $P$ -sequence ${\\bf \\sigma }$ together with a sequence $(w(1),\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .",
"In all our final applications $w(i)$ will always be 1.",
"Nonetheless, we solve this slightly more general weighted version.",
"We have the following additional definitions: For $t \\in [n]$ , ${\\bf \\sigma }_{\\le t}$ denotes the sequence $(\\sigma _1,\\ldots ,\\sigma _t)$ .",
"Also for $J \\subseteq [n]$ , $J_{\\le t}$ denotes the set $J \\cap \\lbrace 1,\\ldots ,t\\rbrace $ .",
"For $J \\subseteq [n]$ , $w(J)=\\sum _{j \\in J} w(j)$ .",
"The digraph $D=D({\\bf \\sigma })$ associated to the $P$ -sequence ${\\bf \\sigma }$ has vertex set $[n]$ (where $n=|{\\bf \\sigma }|$ ) and arc set $\\lbrace i \\rightarrow j: i<j \\text{ and } \\sigma (i) \\triangleleft \\sigma (j)\\rbrace $ .",
"A path $\\pi $ in $D({\\bf \\sigma })$ is called a ${\\bf \\sigma }$ -path.",
"Such a path is a sequence $1 \\le \\pi _1 < \\ldots <\\pi _k \\le n$ of indices with $\\pi _1 \\longrightarrow \\cdots \\longrightarrow \\pi _k$ .",
"We say that $\\pi $ ends at $\\pi _k$ .",
"The defect of path $\\pi $, $\\mbox{\\rm defect}(\\pi )$ is defined to be $w([n]-\\pi )$ .",
"$\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is defined to be the minimum of $\\mbox{\\rm defect}(\\pi )$ over all ${\\bf \\sigma }$ -paths $\\pi $ .",
"We now define the Approximate Minimum-defect path problem (AMDP).",
"The input is a weighted $P$ -sequence $({\\bf \\sigma },w)$ , an approximation parameter $\\delta \\in (0,1]$ , and an error parameter $\\gamma >0$ .",
"The output is a number $A$ such that: ${\\rm Prob}[A \\in [\\mbox{\\rm min-defect}({\\bf \\sigma },w),(1+\\delta ) \\mbox{\\rm min-defect}({\\bf \\sigma },w)]] \\ge 1-\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is said to be a one-sided error algorithm.",
"Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.",
"For the AMDP, the input consists of the parameters $\\delta $ and $\\gamma $ together with a sequence of $n$ pairs $((\\sigma (t),w(t)):t \\in [n])$ .",
"We think of the input as arriving in a sequence of discrete time steps, where $\\delta ,\\gamma $ arrive at time step 0 and for $t \\in [n]$ , $(\\sigma (t),w(t))$ arrives at time step $t$ .",
"The main complexity parameter of interest is the auxiliary memory needed.",
"For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.",
"Associated to a weighted $P$ -sequence $({\\bf \\sigma },w)$ we define the parameter: $\\rho =\\rho (w)=\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\rho =n^{O(1)}$ .",
"The main technical theorem about AMDP is the following.",
"There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\frac{\\ln (n/\\gamma )\\ln (\\rho )}{\\delta })$ .",
"In particular, if $\\rho =n^{O(1)}$ and $\\gamma =1/n^{O(1)}$ then the space is $O(\\frac{(\\ln (n))^2}{\\delta })$ .",
"The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"We first review this dynamic program.",
"Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .",
"Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.",
"Set $w(n+1)=0$ .",
"We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.",
"We initialize $s(0) = 0$ and $W(0) = 0$ .",
"For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .",
"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .",
"In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .",
"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).",
"The polylog space streaming algorithm We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .",
"Our approximation algorithm is a natural variant of the exact algorithm.",
"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .",
"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.",
"The amount of space used by the algorithm is proportional to the maximum size of $R$ .",
"We first define the probabilities $p(i,t)$ .",
"Similar quantities were defined in [20].",
"$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .",
"We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .",
"The following update is performed for each time step $t \\in [n+1]$ .",
"The final output is just $r(n+1)$ .",
"$W(t)=W(t-1)+w(t)$ .",
"$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .",
"The index $t$ is inserted in $R$ .",
"Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .",
"On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .",
"The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .",
"The above theorem does not exactly give what was promised in Theorem REF .",
"For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.",
"To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.",
"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .",
"We now prove Theorem REF .",
"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"The second part will be proved in the two subsection.",
"The final assertion of Theorem REF showing the space bound is deferred to Appendix .",
"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.",
"Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .",
"Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .",
"We need some additional definitions.",
"For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .",
"Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .",
"We assume without loss of generality that $n+1 \\in C$ .",
"We write $R^t$ for the subset $R$ at the end of step $t$ .",
"Note that $R^t \\subseteq [t]$ .",
"We define $F^t=[t]-R^t$ .",
"An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.",
"Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"We write $U^t$ for the set of indices that are unsafe at time $t$ .",
"An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.",
"We denote the set of unsafe indices by $U$ .",
"On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .",
"On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .",
"We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .",
"Assume $t \\ge 1$ and that the result holds for $j<t$ .",
"We consider two cases.",
"Case i.",
"$U^{t-1} = C_{\\le t-1}$ .",
"Then $w(\\bar{C}_{\\le t-1} \\cup U^{t-1})=W(t-1)$ .",
"By definition $r(t) \\le r(0)+W(t-1)-W(0)=W(t-1)$ , as required.",
"Case ii.",
"$U^{t-1} \\ne C_{\\le t-1}$ .",
"Let $j$ be the maximum index in $C_{\\le t-1}-U^{t-1}$ .",
"Since $j,t \\in C$ we must have $\\sigma (j) \\longrightarrow \\sigma (t)$ .",
"Therefore by the definition of $r(t)$ we have: $r(t) \\le r(j)+W(t-1)-W(j)$ .",
"By the induction hypothesis we have $r(j) \\le w(\\bar{C}_{\\le j-1} \\cup U^{j-1})$ .",
"Since $j$ is the largest element of $C_{\\le t-1}-U^{t-1}$ we have: $\\bar{C}_{\\le t-1}\\cup U^{t-1}=\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]$ , and so: $r(t) & \\le & r(j)+W(t-1)-W(j) \\\\& \\le & w(\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]) \\\\& \\le & w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ $\\Box $ By Lemma REF the output of the algorithm is at most $w(\\bar{C})+w(U))=\\mbox{\\rm min-defect}({\\bf \\sigma },w)+w(U)$ .",
"It now suffices to prove: ${\\rm Prob}[w(U) \\ge \\delta w(\\bar{C})] \\le \\gamma /2.$ Call an interval $[i,j]$ dangerous if $w(C \\cap [i,j]) \\le w([i,j])(\\delta /(1+\\delta ))$ .",
"In particular $[i,i]$ is dangerous iff $i \\notin C$ .",
"Call an index $i$ dangerous if it is the left endpoint of some dangerous interval.",
"Let $D$ be the set of all dangerous indices.",
"We define a sequence $I_1,I_2,\\ldots ,I_\\ell $ of disjoint dangerous intervals as follows.",
"If there is no dangerous interval then the sequence is empty.",
"Otherwise: Let $i_1$ be the smallest index in $D$ and let $I_1$ be the largest interval with left endpoint $i_1$ .",
"Having chosen $I_1,...,I_j$ , if $D$ contains no index to the right of all of the chosen intervals then stop.",
"Otherwise, let $i_{j+1}$ be the least index in $D$ to the right of all chosen intervals and let $I_{j+1}$ be the largest dangerous interval with left endpoint $i_{j+1}$ .",
"It is obvious from the definition that each successive interval lies entirely to the right of the previously chosen intervals.",
"Let $B=I_1 \\cup \\cdots \\cup I_\\ell $ and let $\\bar{B}=[n]-B$ .",
"We now make a series of observations: $\\bar{C} \\subseteq D \\subseteq B$ .",
"$w(B) \\le w(\\bar{C})(1+\\delta )$ .",
"${\\rm Prob}[U \\subseteq B] \\ge 1-\\gamma /2$ .",
"By Claims REF and REF , we have $U \\cup \\bar{C} \\subseteq B$ with probability at least $1-\\gamma /2$ , and so by Claim REF , $w(U \\cup \\bar{C}) \\le w(\\bar{C})(1+\\delta )$ with probability at least $1-\\gamma /2$ , establishing (REF ).",
"Thus it remains to prove the claims.",
"Proof of Claim REF: If $i \\in \\bar{C}$ then, as noted earlier, $i$ is dangerous so $i \\in D$ .",
"Now suppose $i \\in D$ .",
"By the construction of the sequence of intervals, there is at least one interval $I_1$ and the left endpoint $i_1$ is at most $i$ .",
"If $i \\in I_1 \\subseteq B$ , we're done.",
"So assume $i \\notin I_1$ and so $i$ is to the right of $I_1$ .",
"Let $j$ be the largest index for which $i$ is to the right of $I_j$ .",
"Then $I_{j+1}$ exists and $i_{j+1} \\le i$ .",
"Since $I_{j+1}$ is not entirely to the right of $i$ we must have $i \\in I_{j+1} \\subset B$ .",
"Proof of Claim REF : For each $I_j$ we have $w(I_j \\cap C) \\le w(I_j)\\delta /(1+\\delta )$ .",
"Therefore $w(I_j \\cap \\bar{C}) \\ge w(I_j)/(1+\\delta )$ and so $(1+\\delta )w(I_j \\cap \\bar{C}) \\ge w(I_j)$ .",
"Summing over $I_j$ we get $(1+\\delta ) w(\\bar{C}) \\ge w(B)$ .",
"Proof of Claim REF : We fix $t \\in [n]$ and $i \\in \\bar{B} \\cap [t]$ and show ${\\rm Prob}[i \\in U^t] \\le \\frac{\\gamma }{4t^3}$ .",
"This is enough to prove the claim since we will then have: ${\\rm Prob}[U \\subseteq B] & = & 1-{\\rm Prob}[\\bar{B} \\cap U \\ne \\emptyset ] \\\\& \\ge & 1-\\sum _{t=1}^n {\\rm Prob}[\\bar{B} \\cap U^t \\ne \\emptyset ]\\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} {\\rm Prob}[i \\in U^t] \\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} \\frac{\\gamma }{4t^3}\\\\&\\ge & 1 - \\frac{\\gamma }{4} \\sum _{t=1}^n \\frac{1}{t^2}\\ge 1- \\gamma /2.$ So fix $t$ and $i \\in \\bar{B} \\cap [t]$ .",
"Since $i \\notin B$ , the interval $[i,t]$ is not dangerous, and so $w(C \\cap [i,t]) \\ge w([i,t])\\delta /(1+\\delta )$ , and so $ w([i,t]) \\le \\frac{1+\\delta }{\\delta } w(C \\cap [i,t]).$ We have $i \\in U^t$ only if every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"For $j \\le t$ , the probability that index $j \\in t$ has been forgotten by time $t$ is $1-q(j,t)$ so ${\\rm Prob}[i \\in U^t] = \\prod _{j \\in C \\cap [i,t]}(1- q(j,t))$ .",
"If $q(j,t)=1$ for any of the multiplicands then the product is 0.",
"Otherwise for each $j \\in C \\cap [i,t]$ : $q(j,t) & = & \\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma ) \\frac{w(j)}{(W(t)-W(j-1)} \\\\& \\ge & \\ln (4t^3/\\gamma ) \\frac{1+\\delta }{\\delta }\\frac{w(j)}{w([i,t])}\\ge \\ln (4t^3/\\gamma ) \\frac{w(j)}{w(C \\cap [i,t])},`$ where the final inequality uses (REF ).",
"Therefore: ${\\rm Prob}[i \\in U(t)] & \\le & \\prod _{j \\in C \\cap [i,t]}(1- q(j,t)) \\\\& \\le & \\exp (-\\sum _{j \\in C \\cap [i,t]} q(j,t)) \\le \\gamma /4t^3,$ as required to complete the proof of Claim REF , and of the second assertion of Theorem REF .",
"Applying AMDP to LIS and LCS We now show how to apply Theorem REF to LIS and LCS.",
"The application to LIS is quite obvious.",
"We first set some notation about points in the two-dimensional plane.",
"We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.",
"$z(2)$ ) refers to the first (resp.",
"second) coordinate of $z$ .",
"We use the standard coordinate-wise partial order on $z$ .",
"So $z \\triangleleft z^{\\prime }$ iff $z(1) < z^{\\prime }(1)$ and $z(2) < z^{\\prime }(2)$ .",
"(of Theorem REF ) The input is a stream $x(1), x(2), \\ldots , x(n)$ .",
"Think of the $i$ th element of the stream as the point $(i,x(i))$ .",
"So the input is thought of as a sequence of points.",
"Note that the points arrive in increasing order of first coordinate.",
"Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).",
"We set $\\gamma = n^{O(1)}$ and $\\rho = n$ in Theorem REF .",
"$\\Box $ The application to LCS is somewhat more subtle.",
"Again, we think of the input as a set of points in the two-dimensional plane.",
"But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\delta $ .",
"Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .",
"Then there is a $O(\\delta ^{-1}k\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\delta n$ -approximation of $E(x,y)$ .",
"We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.",
"Let $P$ be the set of pairs $\\lbrace (i,j)|x(i)=y(j)\\rbrace $ under the partial order $(i,j) < (i^{\\prime },j^{\\prime })$ if $i < i^{\\prime }$ and $j < j^{\\prime }$ .",
"It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.",
"Now we associate to the pair of strings $x,y$ the sequence $\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\prime },j^{\\prime })$ is $i < i^{\\prime }$ or if $i=i^{\\prime }$ and $j < j^{\\prime }$ .)",
"Note that $\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.",
"Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\sigma $ -paths as defined in the AMDP.",
"Thus the length of the LCS is equal to the size of the largest $\\sigma $ -path.",
"It is not true that $E(x,y)$ is equal to $min-defect(\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\sigma $ is in general longer than $n$ .",
"Given full access to $y$ , and a streamed $x$ .",
"We have a bound on $|\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .",
"We now argue that an additive $\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\delta /k)$ -approximation for AMDP of $P$ .",
"Let the length of the longest chain in $P$ be $\\ell $ and the min-defect be $m$ .",
"Let $d$ be a shorthand for $E(x,y)$ .",
"We have $\\ell + m = |P|$ and $\\ell + d = n$ .",
"The output of AMDP is an estimate $est$ such that $m \\le est \\le (1+\\delta /k)m$ .",
"We estimate $d$ by $est_d = est + n - |P|$ .",
"We show that $est_d \\in [d,d + \\delta n]$ .",
"We have $est_d = est + n - |P| \\ge m + n - |P| = n-\\ell = d$ .",
"We can also get an upper bound.",
"$est_d & = & est + n - |P| \\\\& \\le & m + n - |P| + \\delta m/k \\\\& = & d + \\delta m/k \\ \\textrm {(since |P| - m = \\ell and d = n - \\ell )} \\\\& \\le & d + \\delta n \\ \\textrm {(since m \\le |P| \\le nk)}$ Hence, we use the parameters $\\delta /k, \\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.",
"An application of Theorem REF completes the proof.",
"$\\Box $ Deterministic streaming algorithm for LCS We now discuss a deterministic $\\sqrt{n}$ -space algorithm for LCS.",
"This can be used for large alphabets to beat the bound given in Theorem .",
"For any consistent sequence (CS), the size of the complement is called the defect.",
"For indices $i,j \\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.",
"The main theorem is: Let $\\delta > 0$ .",
"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .",
"There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.",
"The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.",
"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].",
"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .",
"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .",
"Our aim is to implement (an approximation of) this algorithm in sublinear space.",
"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.",
"In some sense, we only maintain a small subset of the partial solutions.",
"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .",
"We convert an LCS into finding the longest chain in a set of points $P$ .",
"We construct a set of anchor points in the plane, which may not be in $P$ .",
"Our aim is to just maintain the longest chain between pairs of anchor points.",
"Let $\\delta > 0$ be some fixed parameter.",
"We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .",
"For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.",
"$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.",
"Our analysis remains identical..",
"So we can drop the floors used in the definition of $S_i$ .",
"Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .",
"We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .",
"This is the set of anchor points that we discussed earlier.",
"Note that they are placed according to a geometric grid.",
"A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.",
"There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .",
"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).",
"This gives us a simple bound on the defect that we shall exploit.",
"Lemma makes an important argument.",
"It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.",
"Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.",
"Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .",
"The defect $n-r$ is at least $|j_a - i|$ .",
"There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .",
"We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.",
"Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .",
"Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .",
"Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .",
"We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .",
"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .",
"If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .",
"Otherwise, it is not added.",
"Note that $j_b \\ge \\ell _p$ , simply by construction.",
"The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.",
"It now remains to bound the defect of $L^{\\prime }$ .",
"Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .",
"Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .",
"This means that ${j_b} > \\ell _{p}$ .",
"Let $i_c$ be the smallest index larger than $p\\bar{n}$ .",
"So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .",
"Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .",
"We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .",
"Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .",
"By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .",
"The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .",
"The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .",
"$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.",
"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .",
"Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .",
"Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.",
"The total running time is $O(n\\bar{n}^2)$ .",
"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.",
"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.",
"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .",
"A union bound over all $t$ proves the third assertion of Theorem REF .",
"Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .",
"Fix $t \\in [n]$ .",
"Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .",
"For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.",
"Then $|R_t|=\\sum _{i \\le t} Z_i$ .",
"Let $\\mu =\\mathbb {E}[|R_t|]$ .",
"Below we show that $\\mu \\le M$ .",
"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .",
"Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .",
"Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).",
"It remains to show that $\\mu \\le M$ .",
"We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.",
"For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .",
"We prove by backwards induction on $r$ .",
"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.",
"Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .",
"We start with a technical statement.",
"$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .",
"This completes the proof.",
"$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.",
"Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .",
"(of Claim ) Assume wlog that $i_k \\ge j_k$ .",
"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .",
"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .",
"Hence the defect is at least $i_k - j_k$ .",
"Repeating over all $k$ , we complete the proof.",
"$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).",
"If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .",
"Suppose neither of these are true.",
"Then $j_a > i_a \\ge i$ .",
"Let us focus on the characters of $x$ that are not matched.",
"No character of $x$ with index in $[i,i_a)$ is matched.",
"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).",
"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .",
"So the number of unmatched characters in $x$ is at least $j_a - i$ .",
"$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .",
"We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .",
"This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .",
"The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .",
"Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"This is obtained by just maximizing over all possible $j^{\\prime }$ .",
"We now apply Claim .",
"We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .",
"Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.",
"This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .",
"This can be done for all $y(j)$ , $j \\in S_i$ .",
"The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .",
"$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.",
"Consider the index $i\\bar{n}$ .",
"Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"This requires space $O(|S_i|) = O(\\bar{n}))$ .",
"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.",
"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.",
"Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .",
"The space bound is $O(\\bar{n})$ .",
"The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .",
"$\\Box $"
],
[
"Paths in posets",
"We begin by defining a streaming problem called the Approximate Minimum-Defect Path problem (AMDP).",
"We define it formally below, but intuitively, we look at the stream as a sequence of elements from some poset.",
"Our aim is to estimate the size of the complement of the longest chain, consistent with the stream ordering.",
"This is more general than LIS, and we will show how streaming algorithms for LIS and LCS can be obtained from reductions to AMDP."
],
[
"Weighted $P$ -sequences and the approximate minimum-defect path problem",
"We use $P$ to denote a fixed set endowed with a partial order $\\triangleleft $ .",
"The partial order relation is given by an oracle which, given $u,v \\in P$ outputs $u \\triangleleft v$ or $\\lnot (u \\triangleleft v)$ .",
"For a natural number $n$ we write $[n]$ for the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"A sequence ${\\bf \\sigma }=(\\sigma (1),\\ldots ,\\sigma (n)) \\in P$ is called a $P$ -sequence.",
"The number of terms ${\\bf \\sigma }$ is called the length of ${\\bf \\sigma }$ and is denoted $|{\\bf \\sigma }|$ ; we normally use $n$ to denote $|{\\bf \\sigma }|$ .",
"A weighted $P$ -sequence consists of a $P$ -sequence ${\\bf \\sigma }$ together with a sequence $(w(1),\\ldots ,w(n))$ of nonnegative integers; $w(i)$ is called the weight of index $i$ .",
"In all our final applications $w(i)$ will always be 1.",
"Nonetheless, we solve this slightly more general weighted version.",
"We have the following additional definitions: For $t \\in [n]$ , ${\\bf \\sigma }_{\\le t}$ denotes the sequence $(\\sigma _1,\\ldots ,\\sigma _t)$ .",
"Also for $J \\subseteq [n]$ , $J_{\\le t}$ denotes the set $J \\cap \\lbrace 1,\\ldots ,t\\rbrace $ .",
"For $J \\subseteq [n]$ , $w(J)=\\sum _{j \\in J} w(j)$ .",
"The digraph $D=D({\\bf \\sigma })$ associated to the $P$ -sequence ${\\bf \\sigma }$ has vertex set $[n]$ (where $n=|{\\bf \\sigma }|$ ) and arc set $\\lbrace i \\rightarrow j: i<j \\text{ and } \\sigma (i) \\triangleleft \\sigma (j)\\rbrace $ .",
"A path $\\pi $ in $D({\\bf \\sigma })$ is called a ${\\bf \\sigma }$ -path.",
"Such a path is a sequence $1 \\le \\pi _1 < \\ldots <\\pi _k \\le n$ of indices with $\\pi _1 \\longrightarrow \\cdots \\longrightarrow \\pi _k$ .",
"We say that $\\pi $ ends at $\\pi _k$ .",
"The defect of path $\\pi $, $\\mbox{\\rm defect}(\\pi )$ is defined to be $w([n]-\\pi )$ .",
"$\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is defined to be the minimum of $\\mbox{\\rm defect}(\\pi )$ over all ${\\bf \\sigma }$ -paths $\\pi $ .",
"We now define the Approximate Minimum-defect path problem (AMDP).",
"The input is a weighted $P$ -sequence $({\\bf \\sigma },w)$ , an approximation parameter $\\delta \\in (0,1]$ , and an error parameter $\\gamma >0$ .",
"The output is a number $A$ such that: ${\\rm Prob}[A \\in [\\mbox{\\rm min-defect}({\\bf \\sigma },w),(1+\\delta ) \\mbox{\\rm min-defect}({\\bf \\sigma },w)]] \\ge 1-\\gamma .$ An algorithm for AMDP that has the further guarantee that $A \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is said to be a one-sided error algorithm.",
"Streaming algorithms and the main result In a one-pass streaming algorithm, the algorithm has one-way access to the input.",
"For the AMDP, the input consists of the parameters $\\delta $ and $\\gamma $ together with a sequence of $n$ pairs $((\\sigma (t),w(t)):t \\in [n])$ .",
"We think of the input as arriving in a sequence of discrete time steps, where $\\delta ,\\gamma $ arrive at time step 0 and for $t \\in [n]$ , $(\\sigma (t),w(t))$ arrives at time step $t$ .",
"The main complexity parameter of interest is the auxiliary memory needed.",
"For simplicity, we assume that each memory cell can store any one of the following: a single element of $P$ , an index in $[n]$ , or an arbitrary sum $w(J)$ of distinct weights.",
"Associated to a weighted $P$ -sequence $({\\bf \\sigma },w)$ we define the parameter: $\\rho =\\rho (w)=\\sum _i w_i.$ Typically one should think of the weights as bounded by a polynomial in $n$ and so $\\rho =n^{O(1)}$ .",
"The main technical theorem about AMDP is the following.",
"There is a randomized one-pass streaming algorithm for AMDP that operates with one-sided error and uses space $O(\\frac{\\ln (n/\\gamma )\\ln (\\rho )}{\\delta })$ .",
"In particular, if $\\rho =n^{O(1)}$ and $\\gamma =1/n^{O(1)}$ then the space is $O(\\frac{(\\ln (n))^2}{\\delta })$ .",
"The algorithm Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"We first review this dynamic program.",
"Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .",
"Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.",
"Set $w(n+1)=0$ .",
"We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.",
"We initialize $s(0) = 0$ and $W(0) = 0$ .",
"For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .",
"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .",
"In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .",
"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ ).",
"The polylog space streaming algorithm We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .",
"Our approximation algorithm is a natural variant of the exact algorithm.",
"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .",
"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.",
"The amount of space used by the algorithm is proportional to the maximum size of $R$ .",
"We first define the probabilities $p(i,t)$ .",
"Similar quantities were defined in [20].",
"$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .",
"We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .",
"The following update is performed for each time step $t \\in [n+1]$ .",
"The final output is just $r(n+1)$ .",
"$W(t)=W(t-1)+w(t)$ .",
"$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .",
"The index $t$ is inserted in $R$ .",
"Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .",
"On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .",
"The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .",
"The above theorem does not exactly give what was promised in Theorem REF .",
"For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.",
"To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.",
"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .",
"We now prove Theorem REF .",
"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"The second part will be proved in the two subsection.",
"The final assertion of Theorem REF showing the space bound is deferred to Appendix .",
"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.",
"Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .",
"Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .",
"We need some additional definitions.",
"For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .",
"Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .",
"We assume without loss of generality that $n+1 \\in C$ .",
"We write $R^t$ for the subset $R$ at the end of step $t$ .",
"Note that $R^t \\subseteq [t]$ .",
"We define $F^t=[t]-R^t$ .",
"An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.",
"Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"We write $U^t$ for the set of indices that are unsafe at time $t$ .",
"An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.",
"We denote the set of unsafe indices by $U$ .",
"On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .",
"On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .",
"We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .",
"Assume $t \\ge 1$ and that the result holds for $j<t$ .",
"We consider two cases.",
"Case i.",
"$U^{t-1} = C_{\\le t-1}$ .",
"Then $w(\\bar{C}_{\\le t-1} \\cup U^{t-1})=W(t-1)$ .",
"By definition $r(t) \\le r(0)+W(t-1)-W(0)=W(t-1)$ , as required.",
"Case ii.",
"$U^{t-1} \\ne C_{\\le t-1}$ .",
"Let $j$ be the maximum index in $C_{\\le t-1}-U^{t-1}$ .",
"Since $j,t \\in C$ we must have $\\sigma (j) \\longrightarrow \\sigma (t)$ .",
"Therefore by the definition of $r(t)$ we have: $r(t) \\le r(j)+W(t-1)-W(j)$ .",
"By the induction hypothesis we have $r(j) \\le w(\\bar{C}_{\\le j-1} \\cup U^{j-1})$ .",
"Since $j$ is the largest element of $C_{\\le t-1}-U^{t-1}$ we have: $\\bar{C}_{\\le t-1}\\cup U^{t-1}=\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]$ , and so: $r(t) & \\le & r(j)+W(t-1)-W(j) \\\\& \\le & w(\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]) \\\\& \\le & w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ $\\Box $ By Lemma REF the output of the algorithm is at most $w(\\bar{C})+w(U))=\\mbox{\\rm min-defect}({\\bf \\sigma },w)+w(U)$ .",
"It now suffices to prove: ${\\rm Prob}[w(U) \\ge \\delta w(\\bar{C})] \\le \\gamma /2.$ Call an interval $[i,j]$ dangerous if $w(C \\cap [i,j]) \\le w([i,j])(\\delta /(1+\\delta ))$ .",
"In particular $[i,i]$ is dangerous iff $i \\notin C$ .",
"Call an index $i$ dangerous if it is the left endpoint of some dangerous interval.",
"Let $D$ be the set of all dangerous indices.",
"We define a sequence $I_1,I_2,\\ldots ,I_\\ell $ of disjoint dangerous intervals as follows.",
"If there is no dangerous interval then the sequence is empty.",
"Otherwise: Let $i_1$ be the smallest index in $D$ and let $I_1$ be the largest interval with left endpoint $i_1$ .",
"Having chosen $I_1,...,I_j$ , if $D$ contains no index to the right of all of the chosen intervals then stop.",
"Otherwise, let $i_{j+1}$ be the least index in $D$ to the right of all chosen intervals and let $I_{j+1}$ be the largest dangerous interval with left endpoint $i_{j+1}$ .",
"It is obvious from the definition that each successive interval lies entirely to the right of the previously chosen intervals.",
"Let $B=I_1 \\cup \\cdots \\cup I_\\ell $ and let $\\bar{B}=[n]-B$ .",
"We now make a series of observations: $\\bar{C} \\subseteq D \\subseteq B$ .",
"$w(B) \\le w(\\bar{C})(1+\\delta )$ .",
"${\\rm Prob}[U \\subseteq B] \\ge 1-\\gamma /2$ .",
"By Claims REF and REF , we have $U \\cup \\bar{C} \\subseteq B$ with probability at least $1-\\gamma /2$ , and so by Claim REF , $w(U \\cup \\bar{C}) \\le w(\\bar{C})(1+\\delta )$ with probability at least $1-\\gamma /2$ , establishing (REF ).",
"Thus it remains to prove the claims.",
"Proof of Claim REF: If $i \\in \\bar{C}$ then, as noted earlier, $i$ is dangerous so $i \\in D$ .",
"Now suppose $i \\in D$ .",
"By the construction of the sequence of intervals, there is at least one interval $I_1$ and the left endpoint $i_1$ is at most $i$ .",
"If $i \\in I_1 \\subseteq B$ , we're done.",
"So assume $i \\notin I_1$ and so $i$ is to the right of $I_1$ .",
"Let $j$ be the largest index for which $i$ is to the right of $I_j$ .",
"Then $I_{j+1}$ exists and $i_{j+1} \\le i$ .",
"Since $I_{j+1}$ is not entirely to the right of $i$ we must have $i \\in I_{j+1} \\subset B$ .",
"Proof of Claim REF : For each $I_j$ we have $w(I_j \\cap C) \\le w(I_j)\\delta /(1+\\delta )$ .",
"Therefore $w(I_j \\cap \\bar{C}) \\ge w(I_j)/(1+\\delta )$ and so $(1+\\delta )w(I_j \\cap \\bar{C}) \\ge w(I_j)$ .",
"Summing over $I_j$ we get $(1+\\delta ) w(\\bar{C}) \\ge w(B)$ .",
"Proof of Claim REF : We fix $t \\in [n]$ and $i \\in \\bar{B} \\cap [t]$ and show ${\\rm Prob}[i \\in U^t] \\le \\frac{\\gamma }{4t^3}$ .",
"This is enough to prove the claim since we will then have: ${\\rm Prob}[U \\subseteq B] & = & 1-{\\rm Prob}[\\bar{B} \\cap U \\ne \\emptyset ] \\\\& \\ge & 1-\\sum _{t=1}^n {\\rm Prob}[\\bar{B} \\cap U^t \\ne \\emptyset ]\\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} {\\rm Prob}[i \\in U^t] \\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} \\frac{\\gamma }{4t^3}\\\\&\\ge & 1 - \\frac{\\gamma }{4} \\sum _{t=1}^n \\frac{1}{t^2}\\ge 1- \\gamma /2.$ So fix $t$ and $i \\in \\bar{B} \\cap [t]$ .",
"Since $i \\notin B$ , the interval $[i,t]$ is not dangerous, and so $w(C \\cap [i,t]) \\ge w([i,t])\\delta /(1+\\delta )$ , and so $ w([i,t]) \\le \\frac{1+\\delta }{\\delta } w(C \\cap [i,t]).$ We have $i \\in U^t$ only if every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"For $j \\le t$ , the probability that index $j \\in t$ has been forgotten by time $t$ is $1-q(j,t)$ so ${\\rm Prob}[i \\in U^t] = \\prod _{j \\in C \\cap [i,t]}(1- q(j,t))$ .",
"If $q(j,t)=1$ for any of the multiplicands then the product is 0.",
"Otherwise for each $j \\in C \\cap [i,t]$ : $q(j,t) & = & \\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma ) \\frac{w(j)}{(W(t)-W(j-1)} \\\\& \\ge & \\ln (4t^3/\\gamma ) \\frac{1+\\delta }{\\delta }\\frac{w(j)}{w([i,t])}\\ge \\ln (4t^3/\\gamma ) \\frac{w(j)}{w(C \\cap [i,t])},`$ where the final inequality uses (REF ).",
"Therefore: ${\\rm Prob}[i \\in U(t)] & \\le & \\prod _{j \\in C \\cap [i,t]}(1- q(j,t)) \\\\& \\le & \\exp (-\\sum _{j \\in C \\cap [i,t]} q(j,t)) \\le \\gamma /4t^3,$ as required to complete the proof of Claim REF , and of the second assertion of Theorem REF .",
"Applying AMDP to LIS and LCS We now show how to apply Theorem REF to LIS and LCS.",
"The application to LIS is quite obvious.",
"We first set some notation about points in the two-dimensional plane.",
"We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.",
"$z(2)$ ) refers to the first (resp.",
"second) coordinate of $z$ .",
"We use the standard coordinate-wise partial order on $z$ .",
"So $z \\triangleleft z^{\\prime }$ iff $z(1) < z^{\\prime }(1)$ and $z(2) < z^{\\prime }(2)$ .",
"(of Theorem REF ) The input is a stream $x(1), x(2), \\ldots , x(n)$ .",
"Think of the $i$ th element of the stream as the point $(i,x(i))$ .",
"So the input is thought of as a sequence of points.",
"Note that the points arrive in increasing order of first coordinate.",
"Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).",
"We set $\\gamma = n^{O(1)}$ and $\\rho = n$ in Theorem REF .",
"$\\Box $ The application to LCS is somewhat more subtle.",
"Again, we think of the input as a set of points in the two-dimensional plane.",
"But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\delta $ .",
"Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .",
"Then there is a $O(\\delta ^{-1}k\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\delta n$ -approximation of $E(x,y)$ .",
"We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.",
"Let $P$ be the set of pairs $\\lbrace (i,j)|x(i)=y(j)\\rbrace $ under the partial order $(i,j) < (i^{\\prime },j^{\\prime })$ if $i < i^{\\prime }$ and $j < j^{\\prime }$ .",
"It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.",
"Now we associate to the pair of strings $x,y$ the sequence $\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\prime },j^{\\prime })$ is $i < i^{\\prime }$ or if $i=i^{\\prime }$ and $j < j^{\\prime }$ .)",
"Note that $\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.",
"Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\sigma $ -paths as defined in the AMDP.",
"Thus the length of the LCS is equal to the size of the largest $\\sigma $ -path.",
"It is not true that $E(x,y)$ is equal to $min-defect(\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\sigma $ is in general longer than $n$ .",
"Given full access to $y$ , and a streamed $x$ .",
"We have a bound on $|\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .",
"We now argue that an additive $\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\delta /k)$ -approximation for AMDP of $P$ .",
"Let the length of the longest chain in $P$ be $\\ell $ and the min-defect be $m$ .",
"Let $d$ be a shorthand for $E(x,y)$ .",
"We have $\\ell + m = |P|$ and $\\ell + d = n$ .",
"The output of AMDP is an estimate $est$ such that $m \\le est \\le (1+\\delta /k)m$ .",
"We estimate $d$ by $est_d = est + n - |P|$ .",
"We show that $est_d \\in [d,d + \\delta n]$ .",
"We have $est_d = est + n - |P| \\ge m + n - |P| = n-\\ell = d$ .",
"We can also get an upper bound.",
"$est_d & = & est + n - |P| \\\\& \\le & m + n - |P| + \\delta m/k \\\\& = & d + \\delta m/k \\ \\textrm {(since |P| - m = \\ell and d = n - \\ell )} \\\\& \\le & d + \\delta n \\ \\textrm {(since m \\le |P| \\le nk)}$ Hence, we use the parameters $\\delta /k, \\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.",
"An application of Theorem REF completes the proof.",
"$\\Box $ Deterministic streaming algorithm for LCS We now discuss a deterministic $\\sqrt{n}$ -space algorithm for LCS.",
"This can be used for large alphabets to beat the bound given in Theorem .",
"For any consistent sequence (CS), the size of the complement is called the defect.",
"For indices $i,j \\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.",
"The main theorem is: Let $\\delta > 0$ .",
"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .",
"There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.",
"The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.",
"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].",
"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .",
"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .",
"Our aim is to implement (an approximation of) this algorithm in sublinear space.",
"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.",
"In some sense, we only maintain a small subset of the partial solutions.",
"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .",
"We convert an LCS into finding the longest chain in a set of points $P$ .",
"We construct a set of anchor points in the plane, which may not be in $P$ .",
"Our aim is to just maintain the longest chain between pairs of anchor points.",
"Let $\\delta > 0$ be some fixed parameter.",
"We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .",
"For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.",
"$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.",
"Our analysis remains identical..",
"So we can drop the floors used in the definition of $S_i$ .",
"Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .",
"We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .",
"This is the set of anchor points that we discussed earlier.",
"Note that they are placed according to a geometric grid.",
"A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.",
"There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .",
"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).",
"This gives us a simple bound on the defect that we shall exploit.",
"Lemma makes an important argument.",
"It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.",
"Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.",
"Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .",
"The defect $n-r$ is at least $|j_a - i|$ .",
"There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .",
"We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.",
"Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .",
"Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .",
"Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .",
"We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .",
"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .",
"If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .",
"Otherwise, it is not added.",
"Note that $j_b \\ge \\ell _p$ , simply by construction.",
"The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.",
"It now remains to bound the defect of $L^{\\prime }$ .",
"Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .",
"Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .",
"This means that ${j_b} > \\ell _{p}$ .",
"Let $i_c$ be the smallest index larger than $p\\bar{n}$ .",
"So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .",
"Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .",
"We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .",
"Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .",
"By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .",
"The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .",
"The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .",
"$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.",
"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .",
"Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .",
"Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.",
"The total running time is $O(n\\bar{n}^2)$ .",
"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.",
"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.",
"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .",
"A union bound over all $t$ proves the third assertion of Theorem REF .",
"Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .",
"Fix $t \\in [n]$ .",
"Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .",
"For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.",
"Then $|R_t|=\\sum _{i \\le t} Z_i$ .",
"Let $\\mu =\\mathbb {E}[|R_t|]$ .",
"Below we show that $\\mu \\le M$ .",
"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .",
"Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .",
"Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).",
"It remains to show that $\\mu \\le M$ .",
"We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.",
"For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .",
"We prove by backwards induction on $r$ .",
"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.",
"Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .",
"We start with a technical statement.",
"$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .",
"This completes the proof.",
"$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.",
"Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .",
"(of Claim ) Assume wlog that $i_k \\ge j_k$ .",
"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .",
"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .",
"Hence the defect is at least $i_k - j_k$ .",
"Repeating over all $k$ , we complete the proof.",
"$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).",
"If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .",
"Suppose neither of these are true.",
"Then $j_a > i_a \\ge i$ .",
"Let us focus on the characters of $x$ that are not matched.",
"No character of $x$ with index in $[i,i_a)$ is matched.",
"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).",
"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .",
"So the number of unmatched characters in $x$ is at least $j_a - i$ .",
"$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .",
"We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .",
"This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .",
"The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .",
"Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"This is obtained by just maximizing over all possible $j^{\\prime }$ .",
"We now apply Claim .",
"We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .",
"Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.",
"This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .",
"This can be done for all $y(j)$ , $j \\in S_i$ .",
"The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .",
"$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.",
"Consider the index $i\\bar{n}$ .",
"Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"This requires space $O(|S_i|) = O(\\bar{n}))$ .",
"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.",
"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.",
"Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .",
"The space bound is $O(\\bar{n})$ .",
"The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .",
"$\\Box $"
],
[
"The algorithm",
"Our streaming algorithm can be viewed as a modification of a standard dynamic programming algorithm for exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"We first review this dynamic program."
],
[
"Exact computation of $\\mbox{\\rm min-defect}({\\bf \\sigma },w)$",
"It will be convenient to extend the $P$ -sequence by an element $\\sigma (n+1)$ that is greater than all other elements of $P$ .",
"Thus all arcs $j \\longrightarrow n+1$ for $j \\in [n]$ are present.",
"Set $w(n+1)=0$ .",
"We define sequences $s(0),\\ldots ,s(n+1)$ and $W(0),\\ldots ,W(n+1)$ as follows.",
"We initialize $s(0) = 0$ and $W(0) = 0$ .",
"For $t \\in [n+1]$ : $W(t) & = & W(t-1)+w(t)\\\\s(t) & = & \\min (s(i)+W(t-1)-W(i): i < t \\\\& & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)).$ Thus $W(t)=w([t])$ .",
"It is easy to prove by induction that $s(t)$ is equal to the minimum of $W(t)-w(\\pi )$ over all paths $\\pi $ whose maximum element is $\\sigma (t)$ .",
"In particular, $\\mbox{\\rm min-defect}({\\bf \\sigma },w)=s(n+1)$ .",
"The above recurrence can be implemented by a one-pass streaming algorithm that uses linear space (to store the values of $s(t)$ and $W(t)$ )."
],
[
"The polylog space streaming algorithm",
"We denote our streaming algorithm by $\\Gamma =\\Gamma ({\\bf \\sigma },w,\\delta ,\\gamma )$ .",
"Our approximation algorithm is a natural variant of the exact algorithm.",
"At step $t$ the algorithm computes an approximation $r(t)$ to $s(t)$ .",
"The difference is that rather than storing $r(i)$ and $W(i)$ for all $i$ , we store them only for an evolving subset $R$ of indices, called the active set of indices.",
"The amount of space used by the algorithm is proportional to the maximum size of $R$ .",
"We first define the probabilities $p(i,t)$ .",
"Similar quantities were defined in [20].",
"$& & q(i,t) \\\\& = & \\min \\left\\lbrace 1,\\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma )\\frac{w(i)}{W(t)-W(i-1)}\\right\\rbrace \\\\& & p(i,i) = 1 \\ \\ \\ \\ p(i,t) = \\frac{q(i,t)}{q(i,t-1)} \\text{ for $t>i$},$ Note that in the typical case that $\\delta = \\theta (1)$ and $\\gamma = \\log (n)^{-\\Theta (1)}$ , we have $q(i,t)$ is $\\Theta (\\ln (n)/(t-i))$ .",
"We initialize $R=\\lbrace 0\\rbrace $ , $r(0)=0$ and $W(0)=0$ .",
"The following update is performed for each time step $t \\in [n+1]$ .",
"The final output is just $r(n+1)$ .",
"$W(t)=W(t-1)+w(t)$ .",
"$r(t)=\\min (r(i)+W(t-1)-W(i): i \\in R\\mbox{ such that } \\sigma _i \\longrightarrow \\sigma _t)$ .",
"The index $t$ is inserted in $R$ .",
"Each element $i \\in R$ is (independently) discarded with probability $1-p(i,t)$ .",
"On input $({\\bf \\sigma },w,\\delta ,\\gamma )$ , the algorithm $\\Gamma $ satisfies: $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"${\\rm Prob}[r(n+1) > (1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)] \\le \\gamma /2$ .",
"The probability that $|R|$ ever exceeds $ \\frac{2e^2}{\\delta }\\ln (2\\rho )\\ln (4n^3/\\gamma )$ is at most $\\gamma /2$ .",
"The above theorem does not exactly give what was promised in Theorem REF .",
"For the algorithm $\\Gamma $ , there is a small probability that the set $R$ exceeds the desired space bound while Theorem REF promises an upper bound on the space used.",
"To achieve the guarantee of Theorem REF we modify $\\Gamma $ to an algorithm $\\Gamma ^{\\prime }$ which checks whether $R$ ever exceeds the desired space bound, and if so, switches to a trivial algorithm which only computes the sum of all weights and outputs that.",
"This guarantees that we stay within the space bound, and since the probability of switching to the trivial algorithm is at most $\\gamma /2$ , the probability that the output of $\\Gamma ^{\\prime }$ exceeds $(1+\\delta )\\mbox{\\rm min-defect}({\\bf \\sigma },w)$ is at most $\\gamma $ .",
"We now prove Theorem REF .",
"The first assertion is a direct consequence of the following proposition whose easy proof (by induction on $t$ ) is omitted: For all $j \\le n+1$ we have $r(j) \\ge s(j)$ and thus $r(n+1) \\ge \\mbox{\\rm min-defect}({\\bf \\sigma },w)$ .",
"The second part will be proved in the two subsection.",
"The final assertion of Theorem REF showing the space bound is deferred to Appendix .",
"Quality of estimate bound of Theorem REF We prove the second assertion of Theorem REF , which is the main technical part of the proof.",
"Let $R_t$ denote the set $R$ after processing $\\sigma (t),w(t)$ .",
"Observe that the definition of $p(i,j)$ implies: For each $i \\le t \\le n$ , ${\\rm Prob}[i \\in R_t]=\\prod _{j \\in [i,t]}p(i,j)=q(i,t)$ .",
"We need some additional definitions.",
"For $I \\subseteq [n+1]$ , we denote $[n+1]-I$ by $\\bar{I}$ .",
"Let $C$ be the index set of some fixed chain having minimum $\\mbox{\\rm defect}$ , so that the minimum defect is equal to $w(\\bar{C})$ .",
"We assume without loss of generality that $n+1 \\in C$ .",
"We write $R^t$ for the subset $R$ at the end of step $t$ .",
"Note that $R^t \\subseteq [t]$ .",
"We define $F^t=[t]-R^t$ .",
"An index $i \\in R^t$ is said to be remembered at time $t$ and $i \\in F^t$ is said to be forgotten by time $t$.",
"Index $i \\in C$ is said to be unsafe at time $t$ if every index in $C \\cap [i,t] \\subseteq F^t$ , i.e., every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"We write $U^t$ for the set of indices that are unsafe at time $t$ .",
"An index $i \\in C$ is said to be unsafe if it is unsafe for some time $t>i$ and is safe otherwise.",
"We denote the set of unsafe indices by $U$ .",
"On any execution, the set $U$ is determined by the sequence $R^1,\\ldots ,R^n$ .",
"On any execution of the algorithm, $r(n+1) \\le w(\\bar{C} \\cup U)$ .",
"We prove by induction on $t$ that if $t \\in C$ then $r(t) \\le w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ .",
"Assume $t \\ge 1$ and that the result holds for $j<t$ .",
"We consider two cases.",
"Case i.",
"$U^{t-1} = C_{\\le t-1}$ .",
"Then $w(\\bar{C}_{\\le t-1} \\cup U^{t-1})=W(t-1)$ .",
"By definition $r(t) \\le r(0)+W(t-1)-W(0)=W(t-1)$ , as required.",
"Case ii.",
"$U^{t-1} \\ne C_{\\le t-1}$ .",
"Let $j$ be the maximum index in $C_{\\le t-1}-U^{t-1}$ .",
"Since $j,t \\in C$ we must have $\\sigma (j) \\longrightarrow \\sigma (t)$ .",
"Therefore by the definition of $r(t)$ we have: $r(t) \\le r(j)+W(t-1)-W(j)$ .",
"By the induction hypothesis we have $r(j) \\le w(\\bar{C}_{\\le j-1} \\cup U^{j-1})$ .",
"Since $j$ is the largest element of $C_{\\le t-1}-U^{t-1}$ we have: $\\bar{C}_{\\le t-1}\\cup U^{t-1}=\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]$ , and so: $r(t) & \\le & r(j)+W(t-1)-W(j) \\\\& \\le & w(\\bar{C}_{\\le j-1} \\cup U^{j-1} \\cup [j+1,t-1]) \\\\& \\le & w(\\bar{C}_{\\le t-1} \\cup U^{t-1})$ $\\Box $ By Lemma REF the output of the algorithm is at most $w(\\bar{C})+w(U))=\\mbox{\\rm min-defect}({\\bf \\sigma },w)+w(U)$ .",
"It now suffices to prove: ${\\rm Prob}[w(U) \\ge \\delta w(\\bar{C})] \\le \\gamma /2.$ Call an interval $[i,j]$ dangerous if $w(C \\cap [i,j]) \\le w([i,j])(\\delta /(1+\\delta ))$ .",
"In particular $[i,i]$ is dangerous iff $i \\notin C$ .",
"Call an index $i$ dangerous if it is the left endpoint of some dangerous interval.",
"Let $D$ be the set of all dangerous indices.",
"We define a sequence $I_1,I_2,\\ldots ,I_\\ell $ of disjoint dangerous intervals as follows.",
"If there is no dangerous interval then the sequence is empty.",
"Otherwise: Let $i_1$ be the smallest index in $D$ and let $I_1$ be the largest interval with left endpoint $i_1$ .",
"Having chosen $I_1,...,I_j$ , if $D$ contains no index to the right of all of the chosen intervals then stop.",
"Otherwise, let $i_{j+1}$ be the least index in $D$ to the right of all chosen intervals and let $I_{j+1}$ be the largest dangerous interval with left endpoint $i_{j+1}$ .",
"It is obvious from the definition that each successive interval lies entirely to the right of the previously chosen intervals.",
"Let $B=I_1 \\cup \\cdots \\cup I_\\ell $ and let $\\bar{B}=[n]-B$ .",
"We now make a series of observations: $\\bar{C} \\subseteq D \\subseteq B$ .",
"$w(B) \\le w(\\bar{C})(1+\\delta )$ .",
"${\\rm Prob}[U \\subseteq B] \\ge 1-\\gamma /2$ .",
"By Claims REF and REF , we have $U \\cup \\bar{C} \\subseteq B$ with probability at least $1-\\gamma /2$ , and so by Claim REF , $w(U \\cup \\bar{C}) \\le w(\\bar{C})(1+\\delta )$ with probability at least $1-\\gamma /2$ , establishing (REF ).",
"Thus it remains to prove the claims.",
"Proof of Claim REF: If $i \\in \\bar{C}$ then, as noted earlier, $i$ is dangerous so $i \\in D$ .",
"Now suppose $i \\in D$ .",
"By the construction of the sequence of intervals, there is at least one interval $I_1$ and the left endpoint $i_1$ is at most $i$ .",
"If $i \\in I_1 \\subseteq B$ , we're done.",
"So assume $i \\notin I_1$ and so $i$ is to the right of $I_1$ .",
"Let $j$ be the largest index for which $i$ is to the right of $I_j$ .",
"Then $I_{j+1}$ exists and $i_{j+1} \\le i$ .",
"Since $I_{j+1}$ is not entirely to the right of $i$ we must have $i \\in I_{j+1} \\subset B$ .",
"Proof of Claim REF : For each $I_j$ we have $w(I_j \\cap C) \\le w(I_j)\\delta /(1+\\delta )$ .",
"Therefore $w(I_j \\cap \\bar{C}) \\ge w(I_j)/(1+\\delta )$ and so $(1+\\delta )w(I_j \\cap \\bar{C}) \\ge w(I_j)$ .",
"Summing over $I_j$ we get $(1+\\delta ) w(\\bar{C}) \\ge w(B)$ .",
"Proof of Claim REF : We fix $t \\in [n]$ and $i \\in \\bar{B} \\cap [t]$ and show ${\\rm Prob}[i \\in U^t] \\le \\frac{\\gamma }{4t^3}$ .",
"This is enough to prove the claim since we will then have: ${\\rm Prob}[U \\subseteq B] & = & 1-{\\rm Prob}[\\bar{B} \\cap U \\ne \\emptyset ] \\\\& \\ge & 1-\\sum _{t=1}^n {\\rm Prob}[\\bar{B} \\cap U^t \\ne \\emptyset ]\\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} {\\rm Prob}[i \\in U^t] \\\\& \\ge & 1-\\sum _{t=1}^n \\sum _{i \\in \\bar{B} \\cap [t]} \\frac{\\gamma }{4t^3}\\\\&\\ge & 1 - \\frac{\\gamma }{4} \\sum _{t=1}^n \\frac{1}{t^2}\\ge 1- \\gamma /2.$ So fix $t$ and $i \\in \\bar{B} \\cap [t]$ .",
"Since $i \\notin B$ , the interval $[i,t]$ is not dangerous, and so $w(C \\cap [i,t]) \\ge w([i,t])\\delta /(1+\\delta )$ , and so $ w([i,t]) \\le \\frac{1+\\delta }{\\delta } w(C \\cap [i,t]).$ We have $i \\in U^t$ only if every index of $C \\cap [i,t]$ is forgotten by time $t$ .",
"For $j \\le t$ , the probability that index $j \\in t$ has been forgotten by time $t$ is $1-q(j,t)$ so ${\\rm Prob}[i \\in U^t] = \\prod _{j \\in C \\cap [i,t]}(1- q(j,t))$ .",
"If $q(j,t)=1$ for any of the multiplicands then the product is 0.",
"Otherwise for each $j \\in C \\cap [i,t]$ : $q(j,t) & = & \\frac{1+\\delta }{\\delta }\\ln (4t^3/\\gamma ) \\frac{w(j)}{(W(t)-W(j-1)} \\\\& \\ge & \\ln (4t^3/\\gamma ) \\frac{1+\\delta }{\\delta }\\frac{w(j)}{w([i,t])}\\ge \\ln (4t^3/\\gamma ) \\frac{w(j)}{w(C \\cap [i,t])},`$ where the final inequality uses (REF ).",
"Therefore: ${\\rm Prob}[i \\in U(t)] & \\le & \\prod _{j \\in C \\cap [i,t]}(1- q(j,t)) \\\\& \\le & \\exp (-\\sum _{j \\in C \\cap [i,t]} q(j,t)) \\le \\gamma /4t^3,$ as required to complete the proof of Claim REF , and of the second assertion of Theorem REF .",
"Applying AMDP to LIS and LCS We now show how to apply Theorem REF to LIS and LCS.",
"The application to LIS is quite obvious.",
"We first set some notation about points in the two-dimensional plane.",
"We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.",
"$z(2)$ ) refers to the first (resp.",
"second) coordinate of $z$ .",
"We use the standard coordinate-wise partial order on $z$ .",
"So $z \\triangleleft z^{\\prime }$ iff $z(1) < z^{\\prime }(1)$ and $z(2) < z^{\\prime }(2)$ .",
"(of Theorem REF ) The input is a stream $x(1), x(2), \\ldots , x(n)$ .",
"Think of the $i$ th element of the stream as the point $(i,x(i))$ .",
"So the input is thought of as a sequence of points.",
"Note that the points arrive in increasing order of first coordinate.",
"Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).",
"We set $\\gamma = n^{O(1)}$ and $\\rho = n$ in Theorem REF .",
"$\\Box $ The application to LCS is somewhat more subtle.",
"Again, we think of the input as a set of points in the two-dimensional plane.",
"But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\delta $ .",
"Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .",
"Then there is a $O(\\delta ^{-1}k\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\delta n$ -approximation of $E(x,y)$ .",
"We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.",
"Let $P$ be the set of pairs $\\lbrace (i,j)|x(i)=y(j)\\rbrace $ under the partial order $(i,j) < (i^{\\prime },j^{\\prime })$ if $i < i^{\\prime }$ and $j < j^{\\prime }$ .",
"It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.",
"Now we associate to the pair of strings $x,y$ the sequence $\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\prime },j^{\\prime })$ is $i < i^{\\prime }$ or if $i=i^{\\prime }$ and $j < j^{\\prime }$ .)",
"Note that $\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.",
"Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\sigma $ -paths as defined in the AMDP.",
"Thus the length of the LCS is equal to the size of the largest $\\sigma $ -path.",
"It is not true that $E(x,y)$ is equal to $min-defect(\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\sigma $ is in general longer than $n$ .",
"Given full access to $y$ , and a streamed $x$ .",
"We have a bound on $|\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .",
"We now argue that an additive $\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\delta /k)$ -approximation for AMDP of $P$ .",
"Let the length of the longest chain in $P$ be $\\ell $ and the min-defect be $m$ .",
"Let $d$ be a shorthand for $E(x,y)$ .",
"We have $\\ell + m = |P|$ and $\\ell + d = n$ .",
"The output of AMDP is an estimate $est$ such that $m \\le est \\le (1+\\delta /k)m$ .",
"We estimate $d$ by $est_d = est + n - |P|$ .",
"We show that $est_d \\in [d,d + \\delta n]$ .",
"We have $est_d = est + n - |P| \\ge m + n - |P| = n-\\ell = d$ .",
"We can also get an upper bound.",
"$est_d & = & est + n - |P| \\\\& \\le & m + n - |P| + \\delta m/k \\\\& = & d + \\delta m/k \\ \\textrm {(since |P| - m = \\ell and d = n - \\ell )} \\\\& \\le & d + \\delta n \\ \\textrm {(since m \\le |P| \\le nk)}$ Hence, we use the parameters $\\delta /k, \\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.",
"An application of Theorem REF completes the proof.",
"$\\Box $ Deterministic streaming algorithm for LCS We now discuss a deterministic $\\sqrt{n}$ -space algorithm for LCS.",
"This can be used for large alphabets to beat the bound given in Theorem .",
"For any consistent sequence (CS), the size of the complement is called the defect.",
"For indices $i,j \\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.",
"The main theorem is: Let $\\delta > 0$ .",
"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .",
"There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.",
"The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.",
"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].",
"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .",
"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .",
"Our aim is to implement (an approximation of) this algorithm in sublinear space.",
"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.",
"In some sense, we only maintain a small subset of the partial solutions.",
"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .",
"We convert an LCS into finding the longest chain in a set of points $P$ .",
"We construct a set of anchor points in the plane, which may not be in $P$ .",
"Our aim is to just maintain the longest chain between pairs of anchor points.",
"Let $\\delta > 0$ be some fixed parameter.",
"We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .",
"For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.",
"$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.",
"Our analysis remains identical..",
"So we can drop the floors used in the definition of $S_i$ .",
"Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .",
"We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .",
"This is the set of anchor points that we discussed earlier.",
"Note that they are placed according to a geometric grid.",
"A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.",
"There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .",
"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).",
"This gives us a simple bound on the defect that we shall exploit.",
"Lemma makes an important argument.",
"It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.",
"Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.",
"Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .",
"The defect $n-r$ is at least $|j_a - i|$ .",
"There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .",
"We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.",
"Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .",
"Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .",
"Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .",
"We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .",
"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .",
"If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .",
"Otherwise, it is not added.",
"Note that $j_b \\ge \\ell _p$ , simply by construction.",
"The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.",
"It now remains to bound the defect of $L^{\\prime }$ .",
"Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .",
"Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .",
"This means that ${j_b} > \\ell _{p}$ .",
"Let $i_c$ be the smallest index larger than $p\\bar{n}$ .",
"So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .",
"Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .",
"We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .",
"Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .",
"By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .",
"The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .",
"The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .",
"$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.",
"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .",
"Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .",
"Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.",
"The total running time is $O(n\\bar{n}^2)$ .",
"Acknowledgements The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.",
"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions.",
"The space bound of Theorem REF The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .",
"A union bound over all $t$ proves the third assertion of Theorem REF .",
"Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .",
"Fix $t \\in [n]$ .",
"Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .",
"For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.",
"Then $|R_t|=\\sum _{i \\le t} Z_i$ .",
"Let $\\mu =\\mathbb {E}[|R_t|]$ .",
"Below we show that $\\mu \\le M$ .",
"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .",
"Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .",
"Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).",
"It remains to show that $\\mu \\le M$ .",
"We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.",
"For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .",
"We prove by backwards induction on $r$ .",
"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.",
"Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .",
"We start with a technical statement.",
"$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .",
"This completes the proof.",
"$\\Box $ Proofs from Section We first prove another claim from which the proof of Claim follows.",
"Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .",
"(of Claim ) Assume wlog that $i_k \\ge j_k$ .",
"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .",
"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .",
"Hence the defect is at least $i_k - j_k$ .",
"Repeating over all $k$ , we complete the proof.",
"$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).",
"If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .",
"Suppose neither of these are true.",
"Then $j_a > i_a \\ge i$ .",
"Let us focus on the characters of $x$ that are not matched.",
"No character of $x$ with index in $[i,i_a)$ is matched.",
"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).",
"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .",
"So the number of unmatched characters in $x$ is at least $j_a - i$ .",
"$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .",
"We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .",
"This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .",
"The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .",
"Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"This is obtained by just maximizing over all possible $j^{\\prime }$ .",
"We now apply Claim .",
"We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .",
"Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.",
"This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .",
"This can be done for all $y(j)$ , $j \\in S_i$ .",
"The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .",
"$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.",
"Consider the index $i\\bar{n}$ .",
"Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"This requires space $O(|S_i|) = O(\\bar{n}))$ .",
"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.",
"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.",
"Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .",
"The space bound is $O(\\bar{n})$ .",
"The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .",
"$\\Box $"
],
[
"Applying AMDP to LIS and LCS",
"We now show how to apply Theorem REF to LIS and LCS.",
"The application to LIS is quite obvious.",
"We first set some notation about points in the two-dimensional plane.",
"We will label the axes as 1 and 2, and for a point $z$ , $z(1)$ (resp.",
"$z(2)$ ) refers to the first (resp.",
"second) coordinate of $z$ .",
"We use the standard coordinate-wise partial order on $z$ .",
"So $z \\triangleleft z^{\\prime }$ iff $z(1) < z^{\\prime }(1)$ and $z(2) < z^{\\prime }(2)$ .",
"(of Theorem REF ) The input is a stream $x(1), x(2), \\ldots , x(n)$ .",
"Think of the $i$ th element of the stream as the point $(i,x(i))$ .",
"So the input is thought of as a sequence of points.",
"Note that the points arrive in increasing order of first coordinate.",
"Hence, a chain in this poset corresponds exactly to an increasing sequence (and vice versa).",
"We set $\\gamma = n^{O(1)}$ and $\\rho = n$ in Theorem REF .",
"$\\Box $ The application to LCS is somewhat more subtle.",
"Again, we think of the input as a set of points in the two-dimensional plane.",
"But this transformation will lead to a blow up in size, which we counteract by choosing a small value of $\\delta $ .",
"Let $x$ and $y$ be two strings of length where each character occurs at most $k$ times in $y$ .",
"Then there is a $O(\\delta ^{-1}k\\log ^2n)$ -space algorithm for the asymmetric setting that outputs an additive $\\delta n$ -approximation of $E(x,y)$ .",
"We show how to convert an instance of approximating $E(x,y)$ in the asymmetric model to an instance of AMDP.",
"Let $P$ be the set of pairs $\\lbrace (i,j)|x(i)=y(j)\\rbrace $ under the partial order $(i,j) < (i^{\\prime },j^{\\prime })$ if $i < i^{\\prime }$ and $j < j^{\\prime }$ .",
"It is easy to see that common subsequences of $x$ and $y$ correspond to chains in this poset.",
"Now we associate to the pair of strings $x,y$ the sequence $\\sigma $ consisting of points in $P$ listed lexicographically ($(i,j)$ precedes $(i^{\\prime },j^{\\prime })$ is $i < i^{\\prime }$ or if $i=i^{\\prime }$ and $j < j^{\\prime }$ .)",
"Note that $\\sigma $ can be constructed online given streaming access to $x$ : when $x(i)$ arrives we generate all pairs with first coordinate $i$ in order by second coordinate.",
"Again it is easy to check that common subsequences of $x$ and $y$ correspond to $\\sigma $ -paths as defined in the AMDP.",
"Thus the length of the LCS is equal to the size of the largest $\\sigma $ -path.",
"It is not true that $E(x,y)$ is equal to $min-defect(\\sigma )$ (here we omit the weight function, which we take to be identically 1), because the length of $\\sigma $ is in general longer than $n$ .",
"Given full access to $y$ , and a streamed $x$ .",
"We have a bound on $|\\sigma |$ of $nk$ since each symbol appears at most $k$ times in $x$ .",
"We now argue that an additive $\\delta n$ -approximation for $E(x,y)$ can be obtained from a $(1+\\delta /k)$ -approximation for AMDP of $P$ .",
"Let the length of the longest chain in $P$ be $\\ell $ and the min-defect be $m$ .",
"Let $d$ be a shorthand for $E(x,y)$ .",
"We have $\\ell + m = |P|$ and $\\ell + d = n$ .",
"The output of AMDP is an estimate $est$ such that $m \\le est \\le (1+\\delta /k)m$ .",
"We estimate $d$ by $est_d = est + n - |P|$ .",
"We show that $est_d \\in [d,d + \\delta n]$ .",
"We have $est_d = est + n - |P| \\ge m + n - |P| = n-\\ell = d$ .",
"We can also get an upper bound.",
"$est_d & = & est + n - |P| \\\\& \\le & m + n - |P| + \\delta m/k \\\\& = & d + \\delta m/k \\ \\textrm {(since |P| - m = \\ell and d = n - \\ell )} \\\\& \\le & d + \\delta n \\ \\textrm {(since m \\le |P| \\le nk)}$ Hence, we use the parameters $\\delta /k, \\gamma = n^{O(1)}$ for the AMDP instance created by our reduction.",
"An application of Theorem REF completes the proof.",
"$\\Box $"
],
[
"Deterministic streaming algorithm for LCS",
"We now discuss a deterministic $\\sqrt{n}$ -space algorithm for LCS.",
"This can be used for large alphabets to beat the bound given in Theorem .",
"For any consistent sequence (CS), the size of the complement is called the defect.",
"For indices $i,j \\in [n]$ , $x(i,j)$ refers to the substring of $x$ from the $i$ th character up to the $j$ th character.",
"The main theorem is: Let $\\delta > 0$ .",
"We have strings $x$ and $y$ with full access to $y$ and streaming access to $x$ .",
"There is a deterministic one-pass streaming algorithm that computes a $(1+\\delta )$ -approximation to $E(x,y)$ that uses $O(\\sqrt{(n \\ln n)/\\delta })$ space.",
"The algorithm performs $O(\\sqrt{(\\delta n)/\\ln n})$ updates, each taking $O(n^2\\ln n/\\delta )$ time.",
"The following claim is a direct consequence of the standard dynamic programming algorithm for LCS [11].",
"Suppose we are given two strings $x$ and $y$ , with complete access to $y$ and a one-pass stream through $x$ .",
"There is an $O(n)$ -space algorithm that guarantees the following: when we have seen $x(1,i)$ , we have the lengths of the LCS between $x(1,i)$ and $y(1,j)$ , for all $j \\in [n]$ .",
"Our aim is to implement (an approximation of) this algorithm in sublinear space.",
"As before, we maintain a carefully chosen portion of the $O(n)$ -space used by the algorithm.",
"In some sense, we only maintain a small subset of the partial solutions.",
"Although we do not explicitly present it in this fashion, it may be useful to think of the reduction of Theorem .",
"We convert an LCS into finding the longest chain in a set of points $P$ .",
"We construct a set of anchor points in the plane, which may not be in $P$ .",
"Our aim is to just maintain the longest chain between pairs of anchor points.",
"Let $\\delta > 0$ be some fixed parameter.",
"We set $\\bar{n} = \\sqrt{(n\\ln n)/\\delta }$ and $\\mu = (\\ln n)/\\bar{n} = \\sqrt{(\\delta \\ln n)/n}$ .",
"For each $i \\in [n/\\bar{n}]$ , the set $S_i$ of indices is defined as follows.",
"$ S_i = \\lbrace \\lfloor i\\bar{n} + b(1+\\mu )^r \\rfloor \\big | r \\ge 0, b \\in \\lbrace -1, +1\\rbrace \\rbrace $ For convenience, we treat $\\bar{n}$ , $n/\\bar{n}$ , and $(1+\\mu )^r$ as integersFormally, we need to take floors of these quantities.",
"Our analysis remains identical..",
"So we can drop the floors used in the definition of $S_i$ .",
"Note that the $|S_i| = O(\\mu ^{-1}\\ln n) = O(\\bar{n})$ .",
"We refer to the family of sets $\\lbrace S_1, S_2, \\ldots \\rbrace $ by ${\\cal S}$ .",
"This is the set of anchor points that we discussed earlier.",
"Note that they are placed according to a geometric grid.",
"A common subsequence of $x$ and $y$ is consistent with ${\\cal S}$ if the following happens.",
"There exists a sequence of indices $\\ell _1 \\le \\ell _2 \\le \\ldots \\ell _m$ such that $\\ell _i \\in S_i$ and if character $x(k)$ ($k \\in [i\\bar{n}, (i+1)\\bar{n}]$ ) in the common subsequence is matched to $y(k^{\\prime })$ , then $k^{\\prime } \\in [\\ell _i, \\ell _{i+1}]$ .",
"We have a basic claim about the LCS of two strings (proof deferred to Appendix ).",
"This gives us a simple bound on the defect that we shall exploit.",
"Lemma makes an important argument.",
"It argues that the the anchor points ${\\cal S}$ were chosen such that an ${\\cal S}$ -consistent sequence is “almost\" the LCS.",
"Suppose that $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ are identical subsequences of $x$ and $y$ , respectively.",
"Let $i \\in [n]$ be arbitrary and let $i_a$ be the smallest index of the $x$ subsequence such that $i_a \\ge i$ .",
"The defect $n-r$ is at least $|j_a - i|$ .",
"There exists an ${\\cal S}$ -consistent common subsequence of $x$ and $y$ whose defect is at most $(1+\\delta )E(x,y)$ .",
"We start with an LCS $L$ of $x$ and $y$ and “round\" it to be ${\\cal S}$ -consistent.",
"Let $L$ be $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ .",
"Consider some $p \\in [n/\\bar{n}]$ , and let $i_a$ be the smallest index larger than $p\\bar{n}$ .",
"Set $\\ell _p$ to be the largest index in $S_p$ smaller than $j_a$ .",
"We construct a new common sequence $L^{\\prime }$ by removing certain matches from $L$ .",
"Consider a matched pair $(x(i_b), y(j_b))$ in $L$ .",
"If $i_b \\in [p\\bar{n},(p+1)\\bar{n}]$ and $j_b \\le \\ell _{p+1}$ , then we add this pair to $L^{\\prime }$ .",
"Otherwise, it is not added.",
"Note that $j_b \\ge \\ell _p$ , simply by construction.",
"The new common sequence $L^{\\prime }$ is ${\\cal S}$ -consistent.",
"It now remains to bound the defect of $L^{\\prime }$ .",
"Consider a matched pair $(x(i_b), y(j_b)) \\in L$ that is not present in $L^{\\prime }$ .",
"Let $i_b \\in [(p-1)\\bar{n},p\\bar{n}]$ .",
"This means that ${j_b} > \\ell _{p}$ .",
"Let $i_c$ be the smallest index larger than $p\\bar{n}$ .",
"So $\\ell _{p}$ is the largest index in $S_{p}$ smaller than $j_c$ .",
"Let $\\ell _{p} = p\\bar{n} + (1+\\mu )^r $ .",
"We have $j_c - p\\bar{n} = [(1+\\mu )^r, (1+\\mu )^{r+1}]$ .",
"Since $j_b \\in [\\ell _p, j_c]$ , the total possible values for $j_b$ is at most $(1+\\mu )^{r+1} - (1+\\mu )^r$ $= \\mu (1+\\mu )^r$ .",
"By Claim , $E(x,y) \\ge j_c - p\\bar{n} \\ge (1+\\mu )^r$ .",
"The number of characters of $x$ with indices in $[(p-1)\\bar{n},p\\bar{n}]$ that are not in $L^{\\prime }$ is at most $\\mu E(x,y)$ .",
"The total number of characters of $L^{\\prime }$ not in $L$ is at most $\\mu (n/\\bar{n}) E(x,y)$ $\\le \\delta E(x,y)$ .",
"$\\Box $ The final claim shows how we to update the set of partial LCS solutions consistent with the anchor points.",
"The proof of this claim and the final proof of the main theorem (that puts everything together) is given in Appendix .",
"Suppose we are given the lengths of the largest ${\\cal S}$ -consistent common subsequences between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"Also, suppose we have access to $x(i\\bar{n},(i+1)\\bar{n})$ and $y$ .",
"Then, we can compute the lengths of the largest ${\\cal S}$ -consistent common sequences between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ (for all $j \\in S_{i+1}$ ) using $\\bar{n}$ space.",
"The total running time is $O(n\\bar{n}^2)$ ."
],
[
"Acknowledgements",
"The second author would like to thank Robi Krauthgamer and David Woodruff for useful discussions.",
"He is especially grateful to Ely Porat with whom he discussed LCS to LIS reductions."
],
[
"The space bound of Theorem ",
"The following claim shows that the probability that $|R_t|$ exceeds the space bound is at most $\\gamma /2n$ .",
"A union bound over all $t$ proves the third assertion of Theorem REF .",
"Let $M=\\frac{2}{\\delta } \\ln (4n^3/\\gamma )\\ln (e\\rho )$ .",
"Fix $t \\in [n]$ .",
"Then ${\\rm Prob}[|R_t| \\ge e^2M] \\le \\gamma /2n$ .",
"For $i \\in [t]$ let $Z_i=1$ if $i \\in R_t$ and 0 otherwise.",
"Then $|R_t|=\\sum _{i \\le t} Z_i$ .",
"Let $\\mu =\\mathbb {E}[|R_t|]$ .",
"Below we show that $\\mu \\le M$ .",
"We need the following tail bound (which is equivalent to the bound of [8], Theorem A.12): Let $Z_1,\\ldots ,Z_m$ be independent 0/1-valued random variables, let $Z=\\sum _i Z_i$ , and let $\\mu =\\mathbb {E}[Z]$ .",
"Then for any $C \\ge 0$ , ${\\rm Prob}[Z \\ge C] \\le (e\\mu /C)^{C}$ .",
"Applying this proposition with $C=e^2M$ gives ${\\rm Prob}[|R_t| \\ge e^2M] \\le e^{-C}$ which is at most $\\gamma /2n$ (with a lot of room to spare).",
"It remains to show that $\\mu \\le M$ .",
"We have: $ \\mu & = & \\sum _{i=1}^t \\mathbb {E}[Z_i] = \\sum _{i=1}^t q(i,t) \\\\& & \\le \\frac{2}{\\delta }\\ln (4n^3/\\gamma ) \\sum _{i=1}^t w(i)/(W(t)-W(i-1).$ We note the following fact.",
"For $r \\ge 1$ , $\\sum _{i=r}^t w(i)/(W(t)-W(i-1)) \\le \\ln (\\frac{e(W(t)-W(r-1))}{w(t)})$ .",
"We prove by backwards induction on $r$ .",
"For $r=t$ , the left side is $w(t)/(W(t)-W(t-1)) = 1$ , the same as the right side.",
"Assume up to $r \\ge 2$ , and we shall prove the statement for $r-1$ .",
"We start with a technical statement.",
"$& & \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& = & \\ln (\\frac{W(t) - W(r-2)}{(W(t) - W(r-2)) - w(r-1)}) \\\\& = & - \\ln (1 - w(r-1)/(W(t) - W(r-2))) \\\\& \\ge & w(r-1)/(W(t) - W(r-2))$ Combining the induction hypothesis with this inequality, $& & \\sum _{i=r-1}^t \\frac{w(i)}{W(t)-W(i-1)} \\\\& = & \\sum _{i=r}^t \\frac{w(i)}{W(t)-W(i-1)} + \\frac{w(r-1)}{W(t) - W(r-2)} \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-1))}{w(t)}) + \\ln (\\frac{W(t) - W(r-2)}{W(t) - W(r-1)}) \\\\& \\le & \\ln (\\frac{e(W(t)-W(r-2))}{w(t)})$ $\\Box $ Thus $\\sum _{i=1}^t w(i)/(W(t)-W(i-1) \\le \\ln (eW(t)/w(t)) \\le \\ln (e\\rho )$ , and so $\\mu \\le M$ .",
"This completes the proof.",
"$\\Box $"
],
[
"Proofs from Section ",
"We first prove another claim from which the proof of Claim follows.",
"Given a common subsequence $x(i_1), x(i_2), \\ldots , x(i_r)$ and $y(j_1), y(j_2), \\ldots , y(j_r)$ , the defect is at least $\\max _{k \\le r} (|i_k - j_k|)$ .",
"(of Claim ) Assume wlog that $i_k \\ge j_k$ .",
"Since the $i_k$ th character of $x$ is matched to $j_k$ th character of $y$ , the length of this common subsequence is at most $LCS(x(1,i_k), y(1,j_k)) + LCS(x(i_k+1,n), y(j_k+1,n))$ .",
"This can be bounded above trivially by $j_k + (n-i_k) = n - (i_k - j_k)$ .",
"Hence the defect is at least $i_k - j_k$ .",
"Repeating over all $k$ , we complete the proof.",
"$\\Box $ (of Claim ) The defect is at least $|j_a - i_a|$ (by Claim ) and is also at least $|i_a - i|$ (by definition of $i_a$ ).",
"If either $j_a \\in [i,i_a]$ or $i \\in [j_a,i_a]$ , then the defect is certainly at least $|j_a - i|$ .",
"Suppose neither of these are true.",
"Then $j_a > i_a \\ge i$ .",
"Let us focus on the characters of $x$ that are not matched.",
"No character of $x$ with index in $[i,i_a)$ is matched.",
"The characters in $(i_a,n]$ can only be matched to characters of $y$ in $(j_a,n]$ (since $(x(i_a),y(j_a))$ is a match).",
"So the number of characters in $(i_a,n]$ that are not matched is at least $(n-i_a) - (n-j_a)$ $=(j_a - i_a)$ .",
"So the number of unmatched characters in $x$ is at least $j_a - i$ .",
"$\\Box $ (of Claim ) Consider some $j \\in S_{i+1}$ , and set $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ .",
"We wish to compute the largest ${\\cal S}$ -consistent CS between in $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"Suppose we look at the portion of this CS in $x(1,i\\bar{n})$ .",
"This forms a ${\\cal S}$ -consistent sequence between $\\bar{x}$ and $y(1,j^{\\prime })$ , for some $j^{\\prime } \\in S_i$ .",
"The remaining portion of the CS is just the LCS between $\\bar{x} = x(i\\bar{n},(i+1)\\bar{n})$ and $y(j^{\\prime },j)$ .",
"Hence, given the LCS length of $\\bar{x}$ and $y(j^{\\prime },j)$ , for all $j^{\\prime } \\in S_i$ , we can compute the length of the largest ${\\cal S}$ -consistent CS between $x(1,(i+1)\\bar{n})$ and $y(1,j)$ .",
"This is obtained by just maximizing over all possible $j^{\\prime }$ .",
"We now apply Claim .",
"We have $\\bar{x}$ in hand, and stream in reverse order through $y(1,j)$ .",
"Using $O(\\bar{n})$ space, we can compute all the LCS lengths desired.",
"This gives the length of the largest ${\\cal S}$ -consistent CS that ends at $y(j)$ .",
"This can be done for all $y(j)$ , $j \\in S_i$ .",
"The total running time is $O(|S_{i+1}|n\\bar{n}) = O(n\\bar{n}^2)$ .",
"$\\Box $ (of Theorem ) Our streaming algorithm will compute the length of the longest ${\\cal S}$ -consistent CS.",
"Consider the index $i\\bar{n}$ .",
"Suppose we have currently stored the lengths of the largest ${\\cal S}$ -consistent CS between $x(1,i\\bar{n})$ and $y(1,j)$ , for all $j \\in S_i$ .",
"This requires space $O(|S_i|) = O(\\bar{n}))$ .",
"By Claim , we can compute the corresponding lengths for $S_{i+1}$ using an additional $O(\\bar{n})$ space.",
"Hence, at the end of the stream, we will have the length (and defect) of the longest ${\\cal S}$ -consistent CS.",
"Lemma tells us that this defect is a $(1+\\delta )$ - approximation to $E(x,y)$ .",
"The space bound is $O(\\bar{n})$ .",
"The number of updates is $O(n/\\bar{n}) = O(\\sqrt{(\\delta n)/\\ln n})$ , and the time for each update is $O(n{\\bar{n}}^2) = O((n^2\\ln n)/\\delta )$ .",
"$\\Box $"
]
] | 1204.1098 |
[
[
"Transport in coupled graphene-nanotube quantum devices"
],
[
"Abstract We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.",
"We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.",
"The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.",
"We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20% of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.",
"Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.",
"This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads."
],
[
"Fabrication of coupled graphene-nanotube quantum devices S. Engels$^{1,2,3}$ , P. Weber$^{1,2,3}$ , B. Terrés$^{1,2,3}$ , J. Dauber$^{1,2,3}$ , C. Meyer$^{2,3}$ , C. Volk$^{1,2,3}$ , S. Trellenkamp$^{2}$ , U. Wichmann$^{1}$ and C. Stampfer$^{1,2,3}$ [email protected] $^1$ II.",
"Institute of Physics B, RWTH Aachen University, 52074 Aachen, Germany, EU $^2$ Peter Grünberg Institute (PGI-6/8/9), Forschungszentrum Jülich, 52425 Jülich, Germany, EU $^3$ JARA – Fundamentals of Future Information Technologies We report on the fabrication and characterization of all-carbon hybrid quantum devices based on graphene and single-walled carbon nanotubes.",
"We discuss both, carbon nanotube quantum dot devices with graphene charge detectors and nanotube quantum dots with graphene leads.",
"The devices are fabricated by chemical vapor deposition growth of carbon nanotubes and subsequent structuring of mechanically exfoliated graphene.",
"We study the detection of individual charging events in the carbon nanotube quantum dot by a nearby graphene nanoribbon and show that they lead to changes of up to 20$\\%$ of the conductance maxima in the graphene nanoribbon acting as a good performing charge detector.",
"Moreover, we discuss an electrically coupled graphene-nanotube junction, which exhibits a tunneling barrier with tunneling rates in the low GHz regime.",
"This allows to observe Coulomb blockade on a carbon nanotube quantum dot with graphene source and drain leads.",
"graphene, carbon nanotube, charge detector, quantum dot 71.10.Pm, 73.21.-b, 81.07.Ta, 81.05.ue Carbon nanomaterials, such as graphene and carbon nanotubes (CNTs) attract increasing interest mainly due to their promises for flexible electronics, high-frequency devices and spin-based quantum circuits [1], [2], [3].",
"Both materials consist of sp$^2$ -bound carbon and exhibit unique electronic properties resulting in the suppression of direct backscattering, high carrier mobilities and low intrinsic spin noise.",
"In particular the weak hyperfine interaction makes graphene and CNTs interesting host materials for quantum dots which promise the implementation of long-living spin qubits [4].",
"Up to the present state quantum dots (QDs) and double quantum dots have been demonstrated successfully in both carbon nanomaterials.",
"In particular, the fabrication of ultra-clean few-carrier QDs with well-defined spin states in quasi one-dimensional (1-D) carbon nanotubes attracted great interest [5], [6], [7].",
"A comparable quality in graphene is not yet reached, which is mainly due to its gap less band structure making it hard to controllably confine electrons and holes.",
"State-of-the-art graphene QDs are therefore either based on nanoribbons [8], [9] or etched islands [10], [11], [12], [13], [14] and in both cases edge roughness and disorder are dominating their properties.",
"However, in contrast to quasi 1-D nanotubes, the 2-D nature of graphene makes it easy to integrate lateral graphene gates and in-plane charge sensors [15], which are both important for the control and readout of QD states.",
"Here, we present the fabrication and characterization of quantum devices based on both graphene and carbon nanotubes, which combine the advantages of the two carbon allotropes and open the route to unprecedented quantum devices.",
"Figure: (color online) (a),(b) and (c) Schematic illustrations of the three main fabrication steps.",
"(d) Scanning force microscopy (SFM) image of a SiO 2 \\rm {SiO}_2 substrate with CVD grown CNTs (white arrows) and subsequently deposited graphene (black arrows).",
"(e) SFM image of a similar sample after the dry etching process.",
"White dashed lines indicate areas exposed to an Ar /O 2 \\rm {Ar/O}_{2} plasma which results in etching both carbon materials as indicated by the black points.",
"(f) Raman spectrum of the graphene flake at the position indicated by the green dot in panel (e).",
"The spectrum shows clear characteristics of single-layer graphene.In particular, we first discuss a carbon nanotube QD with a capacitively coupled graphene nanoribbon acting as electrostatic gate and charge detector.",
"Charge read-out schemes have proven to be difficult for carbon nanotubes QDs, the main challenge lying in the random orientation and position of the nanotubes on the substrate.",
"Two possible strategies have been put forward so far to overcome this difficulty.",
"The first one consists in placing a metallic single electron transistor (SET) close the the CNT QD [16], [17].",
"The second relies in capacitively coupling the QD to an SET realized in the same [6] or in a neighboring CNT [18] via a deposited metallic gate.",
"Our work adds a third detection scheme based on a graphene charge detector which is easy to fabricate and has a more than sufficient charge sensitivity to detect single charging events.",
"In addition to the charge sensing, we show a graphene-carbon nanotube hybrid device where a nanotube QD is contacted with both source and drain graphene leads.",
"Thus, a quantum dot device exclusively built of the two different carbon allotropes.",
"This configuration is especially interesting since it opens the route to quantum dot devices with a tunable density of states in the source and drain leads.",
"In particular, it may allow to experimentally investigate the pseudo-gap Kondo model which predicts that graphene undergoes a quantum phase transition [19], [20] from a phase with a screened to an unscreened impurity moment at low charge carrier densities [21].",
"The predicted quantum phase transition might manifest itself in the characteristic scaling of observable quantities (e.g.",
"the Kondo resonance) in a graphene-contacted quantum dot.",
"Figure: (color online) Scanning force micrograph ofa carbon nanotube (CNT) quantum dot device with a nearby structuredgraphene nanoribbon (GNR) for charge sensing.",
"The schematic illustrationshows the used measurement circuit and the applied voltages.",
"For detailedinformation see text.The fabrication process is based on chemical vapor deposition (CVD) growth of carbon nanotubes and subsequent deposition of mechanically exfoliated natural graphite.",
"In Figs.",
"REF (a) to REF (c) we show the three main fabrication steps for making all-carbon graphene-nanotube devices.",
"As a first step single-walled carbon nanotubes are grown on 290 nm $\\rm {SiO}_{2}$ on highly p-doped Si substrates by CVD using a Ferritin-based iron catalyst method [22].",
"The single-walled carbon nanotubes have a diameter of around 1.5-2 nm and are up to several micrometers in length.",
"In a next step, graphene is deposited on these pretreated substrates by mechanical exfoliation of natural graphite [23].",
"Finally, a graphene nanoribbon (GNR) or graphene leads are patterned from the deposited graphene by an electron beam lithography (EBL) step followed dry etching.",
"Figs.",
"REF (d) and REF (e) show scanning force microscopy (SFM) images of two examples of the deposition process.",
"Due to the stochastic nature of the method a great variety of different configurations is obtained.",
"For example, the graphene flake shown in Fig.",
"REF (d) (black arrow) lies on top or nearby of a number of individual CNTs (white arrows) whereas in Fig.",
"REF (e) we show an example where a nanotube (white arrow) is contacted by two graphene flakes (black arrows).",
"A crucial parameter for the success of this stochastic fabrication process is the density of grown CNTs which can be well controlled by the Ferritin-based CVD process [22].",
"In this study we used a density of approx.",
"1-2 CNTs per $\\rm {\\mu m}^2$ .",
"In Fig.",
"REF (f) we show a Raman spectrum taken at the position indicated by the green dot in Fig.",
"REF (e).",
"From the full width at half maximum of the 2D peak of 34 $\\rm {cm}^{-1}$ and the relative intensity of the 2D and G peak of I(G/2D)$\\approx $ 0.65, we conclude that the graphene is of single-layer nature [24], [25] (laser excitation of 532 nm).",
"A Raman spectrum of the bottom graphene flake at the position of the intersecting CNT reveals a bilayer flake (not shown).",
"The contacted CNT structure consists of two segments.",
"Most likely the upper segment is a small bundle of single-walled CNTs whereas the lower segment (white arrow in Fig.",
"REF (e)) is an individual single-walled CNT.",
"This conclusion is supported by a comparison of the SFM profiles of the single-layer graphene and the lower carbon nanotube segment, as well as the high accuracy of the Ferritin-based CVD growth process [22].",
"The third fabrication step consists of EBL followed by an $\\rm {Ar/O}_{2}$ based dry etching step for (i) structuring graphene and (ii) removing unwanted CNTs.",
"In Fig.",
"REF (e) we highlight areas (dashed lines) where graphene and CNTs have been successfully removed (see black points).",
"Finally, we used EBL, metal evaporation (5 nm Cr/ 50 nm Au) and lift-off for contacting the devices.",
"Figure: (color online) (a),(b) Back gate characteristics of the carbon nanotube QD (a) and the graphene nanoribbon (b).",
"Both measurements were recorded at a source-drain bias voltage of V CNT \\rm {V}_{CNT}=V GNR \\rm {V}_{GNR}=0.5 mV and V REF \\rm {V}_{REF}=0 V. The upper inset in panel (a) shows Coulomb blockade resonances of the CNT QD as a function of V BG \\rm {V}_{BG}.",
"The inset in panel (b) highlights the conductance resonances in the GNR in the same range of V BG \\rm {V}_{BG}.",
"(c) Differential conductance on the CNT QD.",
"The white dotted line indicates an excited state with an energy of Δ\\Delta =0.9 meV.",
"(d) Conductance of the CNT as function of ΔV BG \\Delta \\rm {V}_{BG}=V BG \\rm {V}_{BG}-13.63 V and V REF * \\rm {V}_{REF^{*}} at V CNT \\rm {V}_{CNT}=20 mV.",
"The black dashed line highlights the relative lever arm α BG , GNR \\rm {\\alpha }_{BG,GNR}.Figure: (color online) (a) QD conductance and (b) GNR conductance as a function of V REF \\rm {V}_{REF} with a simultaneous change in BG voltage following V BG \\rm {V}_{BG}=a-b·\\cdot V REF \\rm {V}_{REF} with a=48.166 V and b=-0.335.",
"A bias voltage of V CNT \\rm {V}_{CNT}=V GNR \\rm {V}_{GNR}=0.50.5 mV was applied to both structures.",
"The inset in (b) shows a schematic of the charge detecting mechanism.",
"(c) Derivative of (b) showing well pronounced peaks at every charging event.",
"(d) Dependence of the CNT QD conductance on V BG \\rm {V}_{BG} and V REF \\rm {V}_{REF} measured at V CNT \\rm {V}_{CNT}=0.5 mV on a logarithmic scale.",
"Periodic Coulomb blockade resonances with a positive slope of α REF , BG =\\rm {\\alpha }_{REF,BG}=1.25 are observed.",
"The resulting electron occupation numbers of the QD are given in white letters.",
"(e) Conductance of the GNR depending on V BG \\rm {V}_{BG} and V REF \\rm {V}_{REF} with V GNR \\rm {V}_{GNR}=0.5 mV.",
"(f) Absolute value of the GNR transconductance.",
"Elevated conductance traces with a positive slope (dotted line) match perfectly with the Coulomb resonances in (d).",
"Detection lines are even visible where the current is too small to be measured directly (see white arrows).",
"Measurements shown in (a)-(c) were measured parallel to line (2).",
"All measurements were recorded simultaneously.Fig.",
"REF shows a SFM image of an all-carbon device consisting of a carbon nanotube lying in the close vicinity to an etched graphene nanoribbon (GNR) which acts as a charge detector (CD).",
"By Raman spectroscopy we identify the bright colored area to be a graphene nanoribbon of bilayer nature (not shown) and following the above argument we conclude that the nanotube is a single-walled carbon nanotube.",
"Both carbon nanostructures are separated by roughly 150 nm, the nanoribbon has a width of around 100 nm and the CNT quantum dot is defined by two metal electrodes (indicated in blue) which are separated by 350 nm.",
"As illustrated in Fig.",
"REF , we apply a symmetric bias voltage ($\\rm {V}_{CNT}$ and $\\rm {V}_{GNR}$ respectively) to both structures.",
"The overall Fermi level can be tuned by the back gate voltage $\\rm {V}_{BG}$ applied to the highly doped Si substrate.",
"Additionally, we can use the CNT (GNR) as a lateral gate for the GNR (CNT) by applying a reference potential $\\rm {V}_{REF}$ ($\\rm {V}_{REF^{*}}$ ).",
"All presented measurements were performed in a pumped $^4\\rm {He}$ -cryostat at a base temperature of T$\\approx $ 1.5 K using low-frequency lock-in techniques.",
"Figs.",
"REF (a) and REF (b) show the back gate characteristics i.e.",
"the conductance as function of $\\rm {V}_{BG}$ of both, the CNT (Fig.",
"REF (a)) and the graphene nanoribbon (Fig.",
"REF (b)) for a constant bias voltage $\\rm {V}_{CNT}$ =$\\rm {V}_{GNR}$ =0.5 mV.",
"The CNT reveals a semiconducting behavior with a large band gap resulting in an extended BG region of suppressed current (-60 V$< \\rm {V}_{BG} < $ 40 V).",
"A high resolution measurement performed at the edge of this gap (see inset in Fig.",
"REF (a)) exhibits reproducible, well resolved and sharp Coulomb peaks.",
"The peak width of the sharpest resonances is given by the electron temperature [26] which can be extracted to be 2 K. In Fig.",
"REF (c) we show so-called Coulomb diamond measurements i.e.",
"the differential conductance d$\\rm {I}_{CNT}$ /d$\\rm {V}_{CNT}$ plotted as function of $\\rm {V}_{BG}$ and $\\rm {V}_{CNT}$ .",
"From the extent of suppressed current of the diamonds in bias ($\\rm {V}_{CNT}$ ) direction we can extract a charging energy of $\\rm {E}_C \\approx $ 6-9 meV of the CNT quantum dot and a BG lever arm, $\\rm {\\alpha }_{BG}$ =0.22.",
"Following Ref.",
"[27] we can relate the charging energy with the length of the nanotube segment forming the quantum dot ($\\rm {L}_{QD}$ ) by roughly $\\rm {E}_C$ =1.4 eV/$\\rm {L}_{QD}$ (nm).",
"This provides an order of magnitude estimate of $\\rm {L}_{QD}\\approx $ 150-235nm which is in reasonable agreement with the device geometry.",
"The change in differential conductance parallel to the diamond edges is attributed to excited states providing additional transport channels.",
"The observed excited states exhibit an energy of $\\Delta $$\\approx $ 0.9 meV (see e.g.",
"white dotted line in Fig.",
"REF (c)).",
"The independently measured back gate characteristic of the bilayer graphene nanoribbon is shown in Fig.",
"REF (b).",
"This low bias ($\\rm {V}_{GNR}$ =0.5 mV) measurement highlights that transport can be tuned from the hole regime (left inset) into the so-called transport gap [28], [29], [30], [31], [32], [33], [34], [35] starting at around $\\rm {V}_{BG}$$\\approx $ 40 V. Within the gap region transport is governed by localized states resulting in sharp resonances of the conductance as shown in the right inset of Fig.",
"REF (c).",
"Interestingly, both carbon nanostructures have different doping levels.",
"In contrast to the n-doped CNT we observe a significant p-doping of the GNR which is most likely due to atmospheric $\\rm {O}_{2}$ binding on the graphene edges [36].",
"Fig.",
"REF (d) shows the conductance of the CNT in dependence of the BG voltage $\\rm {V}_{BG}$ =13.63 V + $\\Delta \\rm {V}_{BG}$ and the reference voltage on the nanoribbon $\\rm {V}_{REF^{*}}$ at $\\rm {V}_{CNT}$ =20 mV, which demonstrates the gating effect of the GNR on the CNT QD.",
"Lines of higher conductance can be attributed to Coulomb resonances of the CNT and from their slopes we extract a relative lever arm of $\\rm {\\alpha }_{BG,GNR}$ =0.23.",
"If both quantum devices are now operated simultaneously, the nanoribbon device can be used to detect individual charging events on the nanotube QD device.",
"This phenomenon is shown in Fig.",
"4.",
"The BG voltage is put to an offset ($\\rm {V}_{BG}$ =48.166 V) such that (i) the CNT is in the Coulomb blockade regime and (ii) the conductance of the graphene nanoribbon exhibits sharp and well-reproducible resonances.",
"Figs.",
"REF (a) and REF (b) show the simultaneously measured low-bias ($\\rm {V}_{CNT}$ =0.5 meV) conductance through the nanotube $\\rm {G}_{CNT}$ and the nanoribbon $\\rm {G}_{GNR}$ as function of $\\rm {V}_{REF}$ where the BG voltage is simultaneously adjusted according to $\\rm {V}_{BG}$ =a-b$\\cdot $$\\rm {V}_{REF}$ with a=48.166 V and b=-0.335.",
"Similar to the inset of Fig.",
"REF (a), $\\rm {G}_{CNT}$ exhibits Coulomb peaks, which are indicating single charging events in the CNT QD.",
"In the simultaneously measured trace of the GNR, we observe distinct steps in the conductance at the exact positions of the CNT QD charging events.",
"The steps in conductance can measure up to 20 $\\%$ of the total resonance amplitude and are due to the capacitive coupling of both nanostructures.",
"Increasing the reference potential $\\rm {V}_{REF}$ and decreasing $\\rm {V}_{BG}$ both leads to a higher chemical potential in the QD and subsequently to a lower occupation number at every event.",
"Consequently, the GNR resonance shifts to lower values of $\\rm {V}_{REF}$ giving rise to the shape of the charge detecting resonance in Fig.",
"REF (b).",
"The unconventional shape of the detection signal is explained by the inset of Fig.",
"REF (b).",
"In this schematic, each resonance curve (in gray) would represent the conductance resonance of the GNR at a fixed charge state of the CNT QD.",
"However, the latter is itself a (discontinuous) function of $\\rm {V}_{REF}$ , so that the measured conductance through the GNR is not a smooth curve but show a step whenever the charge state of the CNT QD (and thus the conductance resonance) changes.",
"By relating the step height to the noise level of our measurement system we achieve an estimate for the charge sensitivity with an upper limit of $~10^{-3} \\rm {e}/\\sqrt{\\rm {Hz}}$ which is in agreement with previous experiments on GNR charge detectors [15].",
"In order to highlight the charge detection we further plot the transconductance i.e.",
"the derivative of $\\rm {G}_{GNR}$ with respect to $\\rm {V}_{REF}$ as shown in Fig.",
"REF (c).",
"Each individual transconductance peak (dip) is very well aligned with the directly measured Coulomb peaks.",
"For proving this more rigerously we performed measurements as function of both $\\rm {V}_{REF}$ and $\\rm {V}_{BG}$ and extracted relative lever arms.",
"Corresponding charge stability diagrams are shown in Figs.",
"REF (d) and REF (e).",
"As anticipated, the evolution of Coulomb resonances of the CNT QD (shown in Fig.",
"REF (d)) follows a constant positive slope with a relative lever arm of $\\rm {\\alpha }_{REF,BG}\\approx $ 1.25 and the occupation numbers (denoted in white letters) decrease with increasing $\\rm {V}_{REF}$ and decreasing $\\rm {V}_{BG}$ .",
"For the charge detector the observed patterns in Fig.",
"REF (e) show similarities to those of a multi-dot system.",
"Further analysis of the transconductance $\\rm {dG}_{CD}/\\rm {dV}_{REF}$ reveals at least three lines with negative slopes.",
"These lines show that the transport through the GNR is governed by a series of quantum dots where each of the lines can be attributed to single QDs located at different positions along the nanoribbon.",
"The extracted lever arms of the visible conductance resonances are $\\rm {\\alpha }_{REF,BG}^{(1)}$ =$-0.43$ , $\\rm {\\alpha }_{REF,BG}^{(2)}$ =$-0.31$ and $\\rm {\\alpha }_{ REF,BG}^{(3)}$ =$-0.12$ as indicated in Fig.",
"REF (f).",
"Additionally the plot exhibits clearly visible features (see white dotted line) with in this case positive slopes with their positions matching perfectly to those in Fig.",
"REF (d).",
"Consequently they can be associated with charging events in the CNT QD which are detected by the quantum dots situated in the graphene nanoribbon.",
"Detection lines are even visible where the current through the dot is too low to be measured directly (see white arrows in Figs.",
"REF (d) and (f)).",
"In order to further improve the sensitivity, the detector could potentially be included in a RF-readout setup.",
"The charge sensitivities reported for this type of setups are in the order of $~10^{-4} \\rm {e}/\\sqrt{\\rm {Hz}}$ to $~10^{-5} \\rm {e}/\\sqrt{\\rm {Hz}}$ [16], [17].",
"While a quantitative comparison of the presented dc-readout technique with other dc-charge sensors on carbon nanotube QDs [6], [18] can not be provided (no sensitivities are given), a qualitative comparison of the data shows that the graphene charge detector yields at least equal visibility of the charge states.",
"Finally, we discuss a carbon nanotube device where both metal leads (source and drain) are substituted by two graphene flakes.",
"The device structure has been discussed earlier (see fabrication section and Fig.",
"REF (e)) and a close up of the final device is shown as inset in Fig.",
"REF (a).",
"For measuring the conductance of the nanotube, Cr/Au metal contacts are deposited on each graphene flake by an EBL step followed by metalization and lift-off.",
"The metal contacts are designed in a way that they do not touch the carbon nanotube.",
"As a first important observation we see that current flows from one graphene flake to the other through the carbon nanotube.",
"Moreover, we do not observe a $\\rm {V}_{BG}$ -regime where the current is fully suppressed (not shown) which suggests the presence of a metallic nanotube.",
"In Fig.",
"REF (b) we show the current $\\rm {I}_{CNT}$ in dependence of the voltage $\\rm {V}_{SG}$ applied to a metallic side gate which was deposited next to the nanotube as indicated by the blue area in Fig.",
"REF (a) and allows to locally gate the central segment of the carbon nanotube.",
"The current $\\rm {I}_{CNT}$ exhibits periodic Coulomb oscillations.",
"In Fig.",
"REF (c) we plot the current through the nanotube $\\rm {I}_{CNT}$ on a logarithmic scale as function of the bias voltage $\\rm {V}_{BIAS}$ and $\\rm {V}_{SG}$ for a fixed value of $\\rm {V}_{BG}$ =35.15 V. The measurement exhibits clear diamond shaped patterns which can be attributed to Coulomb diamonds.",
"From the addition energies of 1-1.5 $\\rm {meV}$ we can estimate the quantum dot length of 0.9-1.5 $\\rm {\\mu m}$ [27], which is in good agreement with the 1.2 $\\rm {\\mu m}$ spacing between the two graphene leads.",
"This leads to the conclusion that the quantum dot in the nanotube extends over the full length set by the distance of the graphene leads.",
"Moreover, the observation of Coulomb blockade oscillations and peaks manifests the presence of tunneling barriers at the interface of both carbon allotropes.",
"From the maxima of the Coulomb peaks and assuming similar rates for tunneling in and out of the dot we extract tunneling rates in the low GHz regime.",
"The origin of the observed tunneling barriers is not fully understood at the present state.",
"Since graphene is a semi-metal and the investigated carbon nanotube is a metallic carbon nanotube, it is believed that the tunneling barriers form similarly to those metallic-CNT devices with metal contacts.",
"Here, tunneling barriers may arise from imperfect electrical contacts to the nanotube [37].",
"Figure: (color online)(a) SFM image of the investigated device (see also Fig. (e)).",
"(b) Current I CNT \\rm {I}_{CNT} through the nanotube as function of the voltage V SG \\rm {V}_{SG} applied to a metal electrode placed roughly 50 nm from the nanotube (see blue area in (a)) for a fixed back gate voltage V BG \\rm {V}_{BG}=35 V and V BIAS \\rm {V}_{BIAS}=0.3 mV.",
"(c) Current I CNT \\rm {I}_{CNT} through the nanotube on a logarithmic scale as function of V BIAS \\rm {V}_{BIAS} and V SG \\rm {V}_{SG}.",
"V BG \\rm {V}_{BG} is constant at 35.15 V.In summary, we present the fabrication and characterization of carbon nanotube-graphene hybrid devices.",
"We show the example of a structure where a nanotube quantum dot is capacitively coupled to a graphene nanoribbon.",
"Sharp resonances in the graphene nanoribbon conductance give rise to clear charge detection signals with an upper estimate for the charge sensitivity of $~10^{-3} \\rm {e}/\\sqrt{\\rm {Hz}}$ .",
"This kind of graphene charge detector represents a third charge detection scheme for CNT QDs in addition to the existing strategies based on metal single electron transistors [16], [17] and capacitively coupled carbon nanotube segments [6], [18].",
"The presented charge detector is rather easy to fabricate.",
"Its sensitivity can be further improved by including it into a RF-circuit.",
"In addition, we study a device where two carbon allotropes (graphene and CNT) are electrically coupled.",
"We find that the interfaces between the graphene and the carbon nanotube resembles tunneling barriers.",
"This device represents an important preliminary step towards experiments investigating the quantum phase transition from screened to unscreened impurity moments.",
"Both results open the road to more sophisticated devices which are entirely fabricated out of carbon nanostructures and exploit the different advantages of these promising materials.",
"Acknowledgment — The authors wish to thank L. Durrer and C. Hierold for their support and help on the growth of the CNT samples.",
"We thank A. Steffen, R. Lehmann and J. Mohr for the help on the sample fabrication.",
"Discussions with F. Haupt, J. Splettstösser, D. Schuricht and M. Wegewijs and support by the JARA Seed Fund and the DFG (SPP-1459 and FOR-912) are gratefully acknowledged."
]
] | 1204.1474 |
[
[
"Phase space reduction of the one-dimensional Fokker-Planck (Kramers)\n equation"
],
[
"Abstract A pointlike particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered.",
"We present a rigorous mapping of the Fokker-Planck equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x.",
"The result is the Smoluchowski equation, valid in the overdamped limit, m->0, with a series of corrections expanded in powers of m. They are determined unambiguously within the recurrence mapping procedure.",
"The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator."
],
[
"I. Introduction",
"The Brownian motion of a particle in a confined system represents an essential model used in description of stochastic transport through quasi one-dimensional (1D) systems, e.g.",
"channels in nanomaterials, pores or fibers in biological structures.",
"In a 1D system, the trajectory $x(t)$ of a Brownian particle is described by the Langevin equation $m\\ddot{x}+\\gamma \\dot{x}+\\partial _xU(x)=f(t)\\ .$ Here, $m$ denotes mass of the particle, $U(x)$ represents the driving potential, $\\gamma $ is an effective friction coefficient and $f(t)$ is the stochastic force, satisfying the usual conditions on averaged values, $\\left< f(t)\\right>=0$ and $\\left<f(t)f(t^{\\prime })\\right>=2\\gamma k_BT\\delta (t-t^{\\prime })$ ; $T$ is the temperature and $k_B$ the Boltzmann constant.",
"The corresponding phase space density $\\rho (x,v,t)$ of the particle satisfies the Fokker-Planck (FP), or Kramers (kinetic) equation, $\\left(\\partial _t+v\\partial _x-\\frac{U^{\\prime }(x)}{m}\\partial _v\\right)\\rho (x,v,t)&& \\cr && \\hspace{-108.405pt}=\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\rho (x,v,t),\\hspace{21.68121pt}$ where $x$ is the spatial coordinate, $v$ denotes its velocity and $\\beta =1/k_BT$ is the inverse temperature.",
"Solutions of the Langevin equation, as well as the corresponding kinetic equation, are studied over almost a century [1], [2], [3].",
"Still, motion of a particle in confined geometries represents usually a complicated problem, requiring next reductions of the used description.",
"Due to simplicity, and often for relevance, mainly the overdamped limit is studied.",
"The mass dependent term in Eq.",
"(REF ), $m\\ddot{x}$ , is then considered negligible and the particle's spatial density $p(x,t)$ is governed by the Smoluchowski equation, $\\partial _tp(x,t)=D_0\\partial _xe^{-\\beta U(x)}\\partial _xe^{\\beta U(x)}p(x,t)\\ ,$ containing no information about the mass of the particle; $D_0$ denotes the diffusion constant.",
"Instead of the full phase space, one works with only the spatial coordinate $x$ .",
"Of course, the question is the price for this simplification, as well as possibility of also including properly inertia of the massive particles in the reduced (real space only) description of the Brownian dynamics.",
"Recent studies showed its importance in understanding rectification of the transport in ratchets [4], or its influence on the effective diffusion coefficient [5], [6], [7] in a narrow channel.",
"For our purpose to demonstrate the phase space reduction, we will deal here with only the 1D FP (Kramers) equation (REF ).",
"The eq.",
"(REF ) is reducible to the Smoluchowski eq.",
"(REF ) in the limit $m\\rightarrow 0$ .",
"The reduction procedure [8], [9], [10] is based on an instant thermalization of the particle's velocity after any move in the $x$ direction in the case of an infinitely small mass.",
"The situation resembles derivation of the Fick-Jacobs (FJ) equation [11], [12], $\\partial _tp(x,t)=\\partial _x A(x)\\partial _x\\frac{p(x,t)}{A(x)},$ appearing as the result of the dimensional reduction of the diffusion equation in a 2D channel with varying cross section $A(x)$ , onto the longitudinal coordinate $x$ , if equilibration in the transverse ($y$ ) direction is instant.",
"The function $p(x,t)$ denotes the linear (1D) density of the particle.",
"Of course, as in reduction of diffusion to the FJ equation, that of (REF ) to (REF ) as $m\\rightarrow 0$ is a singular limit, and must be handled with care, but from the viewpoint of the reduction of (REF ), no such caveat is needed.",
"Recently, an exact mapping procedure was proposed [13], [14], enabling us to also derive the corrections to the FJ equation (REF ), which are necessary, if the transverse equilibration is not instant.",
"The key was to introduce anisotropy of the diffusion constant in the diffusion equation, $\\partial _t\\rho (x,y,t)=\\left(\\partial _x^2+\\frac{1}{\\epsilon }\\partial _y^2\\right)\\rho (x,y,t),$ governing the 2D (spatial) density $\\rho (x,y,t)$ .",
"For $\\epsilon \\rightarrow 0$ , the infinitely fast transverse diffusion immediately flattens the $y$ profile of $\\rho (x,y,t)\\rightarrow \\rho (x,t)=p(x,t)/A(x)$ .",
"Then integration of Eq.",
"(REF ) over the cross section, together with the reflecting boundary conditions satisfied at the hard walls, results in the FJ eq.",
"(REF ).",
"In the case of a slower transverse diffusion, the mapping procedure generates a series of corrections to the FJ equation controlled by $\\epsilon >0$ , developing the $y$ profile of $\\rho (x,y,t)$ , which is already curved depending on the flux and geometry of the channel.",
"The procedure was extended to mapping of diffusion in an external field $U(x,y)$ , e.g.",
"for diffusion in a channel with soft walls [15], where the particle is kept near the $x$ axis by the parabolic potential, $\\beta U(x,y)=\\alpha (x)y^2$ ; $\\alpha (x)$ represents the varying stiffness of the walls.",
"The equation to be mapped onto the $x$ coordinate is the 2D Smoluchowski equation, $\\left(\\partial _t-\\partial _xe^{-\\alpha (x)y^2}\\partial _xe^{\\alpha (x)y^2}\\right)\\rho (x,y,t)=\\hspace{57.81621pt}\\cr =\\frac{1}{\\epsilon }\\partial _ye^{-\\alpha (x)y^2}\\partial _ye^{\\alpha (x)y^2}\\rho (x,y,t).\\hspace{14.45377pt}$ Integrating over $y$ and applying the mapping scheme gives the mapped 1D equation in an extended Smoluchowski form, $\\partial _tp(x,t)=\\partial _xe^{-V(x)}\\left[1+\\epsilon \\hat{Z}(x,\\partial _x)\\right]\\partial _xe^{V(x)}p(x,t),$ governing the 1D density $p(x,t)$ , where $V(x)$ stands for an effective potential and $\\hat{Z}$ is the correction operator sought as an expansion in the small parameter $\\epsilon $ .",
"The potential $V(x)$ and the coefficients of $\\hat{Z}$ depend on $\\alpha (x)$ and both are fixed unambiguously within the recurrence mapping procedure.",
"The central idea of this paper is a conjecture that the FP eq.",
"(REF ) can be reduced to an extended Smoluchowski-like 1D form, governing the spatial density $p(x,t)$ , after integration over $v$ and applying a similar mapping scheme.",
"The velocity $v$ thus represents a \"transverse\" coordinate instead of $y$ , with the mass $m$ playing the role of the small parameter $\\epsilon $ .",
"Then the Smoluchowski equation (REF ) should be obtained in the limit $m\\rightarrow 0$ [8], [9].",
"Performing the recurrence procedure, a series of corrections to this equation in powers of $m$ would be derived.",
"Then the final mapped equation would also respect inertia of the Brownian particles, although working only in real space.",
"Use of the mapping procedure developed for diffusion [13], [14], [15], [16] is not straightforward; the left-hand side operator of the FP equation (REF ) has a different structure than that of the diffusion (REF ) or Smoluchowski equation (REF ).",
"Still, there is a way to apply the general scheme of the mapping in this case and to perform reduction of the phase space onto the real space in the way described above.",
"Presentation of this algorithm is the primary aim of this study.",
"The result of the mapping of the FP equation (REF ) is again an equation of the form (REF ) with $V(x)=\\beta U(x)$ and $\\epsilon $ replaced by $m$ .",
"In the limit of stationary flow, this equation can be simplified by subsequent reduction of the operator $1-m\\hat{Z}$ to a function $D(x)$ , a spatially dependent effective diffusion coefficient [12], [17], $\\partial _tp(x,t)=\\partial _xe^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x,t)\\ .$ The leading term of $D(x)$ is proportional to $mU^{\\prime \\prime }(x)$ and the whole series of corrections to the Smoluchowski equation can be summed up, giving $D(x)=D_0\\frac{1-\\sqrt{1-4mU^{\\prime \\prime }(x)/\\gamma ^2}}{2mU^{\\prime \\prime }(x)/\\gamma ^2},$ with $D_0=1/\\gamma \\beta $ , if the higher derivatives of $U(x)$ are neglected.",
"In the following section, we analyze how to apply the mapping scheme for reduction of the phase space in the FP equation (REF ).",
"In Sect.",
"III, our considerations are verified on an exactly solvable model, the FP equation with no field, $U(x)=0$ .",
"Analysis of this example helps us to construct the recurrence scheme for calculation of the series of corrections to the zeroth order approximation, the Smoluchowski eq.",
"(REF ), in the small parameter $m$ .",
"Finally, the complete mapping procedure for an arbitrary potential $U(x)$ is presented in Sect.",
"IV.",
"The mapped equation of type (REF ), as well as the formula (REF ) is derived, and checked on the damped linear harmonic oscillator."
],
[
"II. Preliminary considerations",
"The key points of the mapping procedure, as formulated for diffusion [13], [14], are recalled in this Section.",
"Based on physical considerations, we adjust the general scheme of the mapping for dimensional reduction of the FP equation (REF ) to this situation.",
"The mapping procedure represents a consistent transition from a fine grain to a coarse grain description of some evolution process.",
"The process is described in details by some partial differential equation (PDE), governing the density of particles $\\rho ({\\bf r},t)$ in the full space, defined by the coordinates ${\\bf r}$ .",
"The dimensional reduction projects this equation onto another PDE, which governs the density $p(x,t)$ in the reduced space of the coordinate $x$ .",
"The coordinate $x$ is one of the coordinates of the full space, ${\\bf r}=(x,{\\bf y})$ , and the mapping accomplishes integration over the transverse coordinates ${\\bf y}$ .",
"In the case of the FP equation (REF ), phase space $(x,v)$ represents the full space and the dimensional reduction integrates over the \"transverse\" coordinate, the velocity $v$ .",
"Hence we have the defining relation between the densities $\\rho $ and $p$ : $p(x,t)=\\int _{-\\infty }^{\\infty }\\rho (x,v,t)dv\\ .$ The phase-space density $\\rho (x,v,t)$ is expected to be near the thermodynamic equilibrium and so approximately proportional to $\\exp (-\\beta mv^2/2)$ , which provides convergence of the integral in Eq.",
"(REF ).",
"Then the first step of the mapping is also integration of Eq.",
"(REF ) over $v$ .",
"If completed, we get $\\int _{-\\infty }^{\\infty }\\left(\\partial _t+v\\partial _x\\right)\\rho (x,v,t)dv=0;$ the other terms are zero due to $\\rho (x,v,t)\\rightarrow 0$ in the limit $v\\rightarrow \\pm \\infty $ .",
"This equation represents nothing but mass conservation in the reduced space, $\\partial _tp(x,t)+\\partial _xj(x,t)=0,$ where the 1D flux $j$ is defined by the relation $j(x,t)=\\int _{-\\infty }^{\\infty }v\\rho (x,v,t)dv\\ .$ In contrast to diffusion, where $j=-D_0\\partial _xp$ is fixed, here the flux $j$ is a function formally independent of the density $p$ , so we also need the evolution equation for this quantity.",
"After integration of Eq.",
"(REF ) multiplied by $v$ , we obtain $\\partial _tj(x,t)+\\partial _x\\int _{-\\infty }^{\\infty }v^2\\rho (x,v,t)dv+\\frac{U^{\\prime }(x)}{m}p(x,t)=\\ \\ \\ \\cr =-\\frac{\\gamma }{\\beta m^2}\\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\rho (x,v,t)dv\\ \\ $ [after some algebra and applying the definitions (REF ) and (REF ).",
"This step recalls the Grad's method of moments [18], [19].",
"For the 1D Kramers equation (REF ), taking only a couple of the zeroth ($p$ ) and the first ($j$ ) order moment of the phase space density $\\rho $ is satisfactory for generating the selfconsistent system of the mapped (real space) equations (REF ) and (REF ).]",
"The next key point of the mapping algorithm is that of expressing the full-space density $\\rho (x,v,t)$ using the 1D density $p(x,t)$ and also the flux $j(x,t)$ in this case.",
"This relation enables us to complete integrations in Eq.",
"(REF ) and get the evolution equation for $j(x,t)$ , together with the 1D mass conservation, in closed form.",
"The initial task is to find the zeroth order approximation, valid in the limit $m\\rightarrow 0$ .",
"Our first proposal for such a relation between $\\rho (x,v,t)$ and the reduced space quantities $p$ and $j$ is based on the following physical construction: For an infinitely small mass of the particle, the stochastic force thermalizes its velocity $v$ almost immediately after any move along the spatial coordinate $x$ .",
"Similar to the transverse equilibration of the 2D density of a particle diffusing in a narrow channel with biasing transverse force [15], [20], [21], [22], one could try the formula with separated Boltzmann factor in the fast relaxing \"direction\" $v$ , $\\rho (x,v,t)\\simeq \\sqrt{\\beta m/2\\pi }p(x,t)\\exp (-\\beta mv^2/2)$ .",
"It is easy to check that it does not work here; the flux $j$ becomes zero according to Eq.",
"(REF ).",
"To prevent this failure, let us suppose that the distribution in $v$ is shifted by the local mean (macroscopic) velocity $v_0$ , depending on the local flux, $j(x,t)=v_0(x,t)p(x,t)$ .",
"Then we have $\\nonumber \\rho (x,v,t)&\\simeq & \\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta m(v-v_0)^2/2}p(x,t)\\cr &\\simeq &\\sqrt{\\frac{\\beta m}{2\\pi }}\\left[1+\\beta mvv_0+ ...\\right]e^{-\\beta mv^2/2}p(x,t).$ Retaining only these two terms in the square brackets and replacing $v_0p$ by the flux $j$ , one gets $\\rho _0(x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}\\left[p(x,t)+\\beta mvj(x,t)\\right],$ which will be taken as the sought zeroth order relation between $\\rho $ and the reduced space quantities, $p$ and $j$ .",
"This heuristic formula will be verified later by the exact mapping algorithm.",
"Still, one can check immediately that the relation (REF ) represents correctly a kind of backward mapping of the 1D (spatial) functions $p$ and $j$ onto the phase space densities $\\rho $ ; if substituted for $\\rho (x,v,t)$ in the defining relations (REF ) and (REF ), we obtain identities.",
"Applying Eq.",
"(REF ) to Eq.",
"(REF ), the integrals over $v$ can be completed and the result, $\\partial _tj(x,t)+\\partial _x\\frac{p(x,t)}{\\beta m}+\\frac{U^{\\prime }(x)}{m}p(x,t)=-\\frac{\\gamma }{m}j(x,t),$ together with the mass conservation, Eq.",
"(REF ), forms a closed couple of PDE, governing the mapped quantities $p$ and $j$ .",
"In the limit $m\\rightarrow 0$ , the first term in Eq.",
"(REF ), $\\partial _tj$ , becomes negligible and the equation expresses the zeroth order relation between the flux $j$ and the density $p$ , $j(x,t)=-\\frac{1}{\\beta \\gamma }e^{-\\beta U(x)}\\partial _xe^{\\beta U(x)}p(x,t)\\ .$ If combined with the mass conservation, Eq.",
"(REF ), we get the Smoluchowski equation (REF ); $1/\\beta \\gamma =D_0$ represents the diffusion constant.",
"The calculation presented shows that the Smoluchowski equation (REF ) is related to the FP equation (REF ) in the same way as the Fick-Jacobs equation [11] to the diffusion equation valid in a narrow 2D channel.",
"Both mapped equations describe an asymptotic behavior of the full-space density infinitely rapidly equilibrating in the transverse direction; the velocity $v$ plays the role of the transverse coordinate for the FP equation.",
"Our considerations indicate that the mass of the particle, $m$ , becomes the small parameter, controlling the series of corrections to the Smoluchowski equation in the case when the transverse equilibration is not infinitely fast.",
"Following the scheme of the mapping procedure [13], [14], the next point is that of searching for the true relation between the phase space density $\\rho (x,v,t)$ and the 1D quantities $p(x,t)$ and $j(x,t)$ , replacing the heuristic formula (REF ), valid for nonzero $m$ .",
"Without losing generality, it can be written in the form $\\rho (x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}\\Big [\\hat{\\omega }(x,v)p(x,t)\\hspace{28.90755pt}\\cr \\hspace{-28.90755pt}+\\beta mv\\hat{\\eta }(x,v)j(x,t)\\Big ].$ If the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ (with $\\partial _x$ implicit) are expandable in $m$ , one can substitute for $\\rho (x,v,t)$ in the FP equation (REF ) and fix the coefficients of these operators to satisfy this equation in each order of $m$ , similar to the mapping of diffusion.",
"Then, using the relation of backward mapping (REF ) in Eq.",
"(REF ) gives the expansion of the evolution equation for $j$ and finally, in combination with mass conservation (REF ), the sought series of corrections to the Smoluchowski equation (REF ) in terms of the finite mass $m$ .",
"To verify whether this scheme is viable, we analyze the exactly solvable case with $U(x)=0$ in the next Section."
],
[
"III. Exactly solvable model",
"The exact solution of the FP equation (REF ) with no potential, $U(x)=0$ , is presented in this Section.",
"We demonstrate the mapping on the example of the phase space density $\\rho (x,v,t)$ evolving from the initial density $\\rho (x,v,0)$ with thermalized velocity $v$ .",
"The mapped equation, as well as the form of the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ in Eq.",
"(REF ) can be found explicitly in this case.",
"This analysis will direct us in construction of the recurrence mapping scheme in Sect.",
"IV.",
"The case $U(x)=0$ is exactly solvable [23], the Green's function $G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })$ of the FP equation (REF ), $&&\\hspace{-14.45377pt}\\left[\\partial _t+v\\partial _x-\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\right]G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })\\cr &&\\hspace{72.26999pt}=\\delta (x-x^{\\prime })\\delta (v-v^{\\prime })\\delta (t-t^{\\prime })\\ ,$ can be derived explicitly (see Appendix A), $G&=&\\frac{\\gamma \\beta }{4\\pi }\\frac{\\Theta (\\tau -\\tau ^{\\prime })}{\\sqrt{\\tau -\\tau ^{\\prime }-\\tanh (\\tau -\\tau ^{\\prime })}\\sqrt{1-q^2}}\\cr &&\\times \\exp \\bigg (-\\frac{\\left[2(\\xi -\\xi ^{\\prime })-(u+u^{\\prime })\\tanh (\\tau -\\tau ^{\\prime })\\right]^2}{4[\\tau -\\tau ^{\\prime }-\\tanh (\\tau -\\tau ^{\\prime })]}\\ \\ \\cr &&\\hspace{28.90755pt}-\\frac{q}{1-q^2}\\left[q(u^2+u^{\\prime 2})-2uu^{\\prime }\\right]-u^2\\bigg )\\ ,$ if expressed in the scaled coordinates, $\\tau &=&\\gamma t/2m,\\cr \\xi &=&(\\beta m/2)^{3/2}\\frac{\\gamma x}{\\beta m^2},\\cr u&=&\\sqrt{\\beta m/2}\\ v,$ and $q=\\exp [-2(\\tau -\\tau ^{\\prime })]$ .",
"If the thermalized particle is inserted at time $t=0$ with a spatial distribution $p_0(x)$ , $\\rho (x,v,0)=\\sqrt{\\frac{\\beta m}{2\\pi }}p_0(x)e^{-\\beta mv^2/2},$ evolution of the density $\\rho $ is given by the formula $\\rho (x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}\\int _{-\\infty }^{\\infty }dv^{\\prime }\\int dx^{\\prime } G(x,v,t;x^{\\prime },v^{\\prime },0)\\ \\ \\cr &&\\hspace{72.26999pt}\\times p_0(x^{\\prime })e^{-\\beta mv^{\\prime 2}/2}\\cr &=&\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{4\\pi D_0\\sqrt{Z}}\\exp \\bigg (-\\frac{[\\xi -\\xi ^{\\prime }]^2}{Q}\\cr &&\\hspace{14.45377pt}-\\frac{Q}{Z}\\left[u-\\frac{(1-e^{-2\\tau })}{2Q}(\\xi -\\xi ^{\\prime })\\right]^2\\bigg );$ $D_0=1/\\gamma \\beta $ , the abbreviations $Q&=&\\tau -\\frac{1}{2}\\left(1-e^{-2\\tau }\\right),\\cr Z&=&Q-\\frac{1}{4}\\left(1-e^{-2\\tau }\\right)^2$ are used and the integration over $x^{\\prime }$ runs over the whole (unspecified) 1D spatial domain.",
"Then the spatial (1D) density $p$ and the flux $j$ are integrated directly according to Eqs.",
"(REF ) and (REF ), $p(x,t)&=&\\sqrt{\\frac{2}{\\beta m}}\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{4D_0\\sqrt{\\pi Q}}e^{-(\\xi -\\xi ^{\\prime })^2/Q},\\hspace{50.58878pt}\\cr j(x,t)&=&\\frac{2}{\\beta m}\\int \\frac{p_0(x^{\\prime })dx^{\\prime }}{8D_0\\sqrt{\\pi Q^3}}\\left(1-e^{-2\\tau }\\right)\\cr &&\\hspace{65.04256pt}\\times \\left(\\xi -\\xi ^{\\prime }\\right) e^{-(\\xi -\\xi ^{\\prime })^2/Q}.$ It is easy to check that the mass conservation (REF ), $\\partial _{\\tau }p+\\sqrt{\\beta m/2}\\partial _{\\xi }j=0$ in the scaled coordinates, is satisfied.",
"The quantity $Q$ plays the role of a \"stretched\" time [24].",
"For short times, $t\\ll 2m/\\gamma $ , $Q\\simeq \\tau ^2$ and $Z\\simeq 4\\tau ^3/3$ .",
"The formulas (REF ) and (REF ) describe correctly behavior of the Newtonian particles in this limit.",
"The mapping procedure, as outlined in the previous Section, requires us to study asymptotic behavior in the opposite limit, $t\\gg 2m/\\gamma $ .",
"For large times, $\\tau \\rightarrow \\infty $ , the stretched time $Q$ becomes $\\tau $ and the formulas (REF ) represent the general solution of the diffusion equation, as expected according to the previous Section.",
"Now it is necessary to verify that the mass $m$ can serve as a small parameter controlling the series of corrections to the diffusion equation and its solution.",
"It may look problematic at first glance, because $Q$ contains $\\exp (-2\\tau )=\\exp (-\\gamma t/m)$ , representing essential singularity of the variable $m\\rightarrow 0$ .",
"Then the formulas (REF ) [and similarly Eq.",
"(REF )] are not expandable in $m$ .",
"Nevertheless, this property is still consistent with the general scheme of the mapping, as analyzed in Ref.",
"[14].",
"The dimensional reduction, as demonstrated on anisotropic diffusion in a narrow channel [14], also reduces the Hilbert space of the full-space problem.",
"Let us denote $\\hat{M}(\\epsilon )$ the spatial operator of the evolution equation, i.e.",
"$\\hat{M}(\\epsilon )=\\partial _x^2+(1/\\epsilon )\\partial _y^2$ for anisotropic 2D diffusion; the eigenvalues $\\lambda _i$ and the eigenfunctions $\\psi _i(x,y)$ are given by the equation $-\\hat{M}(\\epsilon )\\psi _i(\\epsilon ;x,y)=\\lambda _i(\\epsilon )\\psi _i(\\epsilon ;x,y),$ supplemented by proper boundary conditions at the walls of the channel.",
"The parameter of anisotropy $\\epsilon <1$ splits the spectrum into two parts, the low-lying states, whose eigenvalues $\\lambda _l(\\epsilon )$ remain finite for $\\epsilon \\rightarrow 0$ and the transients with $\\lambda _r(\\epsilon )$ diverging $\\sim 1/\\epsilon $ .",
"Then the exact 2D density $\\rho $ evolves as $\\rho (\\epsilon ;x,y,t)=\\sum _{i}c_i\\psi _i(\\epsilon ;x,y)e^{-\\lambda _i(\\epsilon ) t},$ the constants $c_i$ are given by the initial condition and the summation runs over the whole spectrum.",
"The transients contribute to the sum by the terms proportional to $\\exp (-\\bar{\\lambda }_r t/\\epsilon )$ , where $\\bar{\\lambda }_r=\\epsilon \\lambda _r(\\epsilon )$ are finite in the limit $\\epsilon \\rightarrow 0$ .",
"The result is a formula containing the essential singularity in the parameter $\\epsilon $ near zero, similar to Eq.",
"(REF ) with singular $\\exp (-\\gamma t/m)$ for $m\\rightarrow 0$ .",
"The mapping procedure reduces the full Hilbert space of all $\\psi _i$ onto the space defined only by the low-lying states $\\psi _l$ .",
"If the 1D density $p(x,t)$ is integrated from Eq.",
"(REF ) and mapped backward onto the full Hilbert space (by some operator $\\hat{\\omega }$ ), the transients will be canceled; the sum (REF ) after the mapping there and back runs only over the low-lying states $\\psi _l$ .",
"The terms retained involve no essential singularity in $\\epsilon $ ; the formula for $\\rho $ considered in the mapping procedure represents the regular part of the exact 2D density $\\rho $ with respect to the parameter $\\epsilon $ near zero.",
"This reduction is natural for the zero-th order (Fick-Jacobs) approximation, as the transients decay infinitely fast due to their infinite eigenvalues $\\lambda _r(\\epsilon \\rightarrow 0)$ .",
"Nevertheless, the mapping based on fixing the series of corrections expanded in $\\epsilon $ can work only with the regular part of the 2D density.",
"Correspondingly, the formulas (REF ) and (REF ) are exact, including the contributions of the transients, which are represented by the singular terms $\\sim \\exp (-\\gamma t/m)$ .",
"The mapping requires us to analyze only the regular parts, $p_{reg}(x,t)&=&\\int \\frac{p(x^{\\prime })dx^{\\prime }}{2\\sqrt{\\pi D_0(t-D_0\\beta m)}}\\cr &&\\times \\exp \\left[-(x-x^{\\prime })^2)/4D_0(t-D_0\\beta m)\\right],\\hspace{14.45377pt}\\cr j_{reg}(x,t)&=&\\int \\frac{(x-x^{\\prime })p(x^{\\prime })dx^{\\prime }}{4\\sqrt{\\pi D_0(t-D_0\\beta m)^3}}\\cr &&\\times \\exp \\left[-(x-x^{\\prime })^2/4D_0(t-D_0\\beta m)\\right]$ and $\\rho _{reg}(x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}\\int \\frac{p(x^{\\prime })dx^{\\prime }}{2\\sqrt{\\pi D_0(t-3D_0\\beta m/2)}}\\hspace{28.90755pt}\\cr &&\\hspace{-36.135pt}\\times \\exp \\left[-\\frac{(x-x^{\\prime }-D_0\\beta mv)^2}{4D_0(t-3D_0\\beta m/2)}-\\frac{1}{2}\\beta mv^2\\right],$ written in the unscaled coordinates, obtained after taking only the regular parts of $Q$ and $Z$ (REF ), $Q_{reg}=\\tau -1/2$ and $Z_{reg}=\\tau -3/4$ , in Eqs.",
"(REF ) and (REF ).",
"Of course, the formulas are applied for $t\\gg D_0\\beta m$ , when the transients vanish.",
"We omit writing the subscript \"reg\" in the following text.",
"In comparison with the overdamped limit $m\\rightarrow 0$ , evolution of the spatial (1D) density $p$ and the flux $j$ is only corrected by a time shift, $t\\rightarrow t-D_0\\beta m$ in the formulas (REF ).",
"The Gaussian distribution is retarded by the time $t_0=D_0\\beta m=m/\\gamma $ , corresponding to the mean time necessary for losing information about the original velocity.",
"The value of the shift $t_0$ is constant in $x$ and $t$ , so the density $p$ (REF ) still obeys the diffusion equation, $\\partial _tp(x,t)=D_0\\partial _x^2p(x,t),$ and $j(x,t)=-D_0\\partial _xp(x,t)$ .",
"There is no correction of the zero-th order mapped equation due to the finite mass of the particle.",
"Let us stress that the case of $U(x)=0$ is extremely simple, similar to the mapping of the 2D diffusion in a flat narrow channel, also giving no corrections to the Fick-Jacobs approximation.",
"Nevertheless, the relation of the backward mapping, generating the phase-space density $\\rho $ from the mapped quantities $p$ and $j$ , is not quite trivial.",
"In the formula for $\\rho $ , Eq.",
"(REF ), the time is shifted by $3t_0/2$ and the displacement $x-x^{\\prime }$ by $vt_0$ with respect to the distribution of a massless particle.",
"The shortest way to construct the relation of backward mapping is by applying the shift operators in $t$ and $x$ on $p(x,t)$ , compensating the different shifts of time and displacement in Eqs.",
"(REF ) and (REF ), $\\rho (x,v,t)=\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2}e^{-(t_0/2)\\partial _t-vt_0\\partial _x}p(x,t).$ Due to Eq.",
"(REF ), the operator $\\partial _t$ in the exponent can be replaced by $D_0\\partial _x^2$ .",
"After expanding the shift operator $\\exp (-vt_0\\partial _x)$ and using $j=-D_0\\partial _xp$ , we arrive at the relation of the form (REF ), $\\rho (x,v,t)&=&\\sqrt{\\frac{\\beta m}{2\\pi }}e^{-\\beta mv^2/2-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }(\\beta mv^2)^k\\hspace{21.68121pt}\\cr &&\\hspace{-36.135pt}\\times (D_0t_0)^k\\partial _x^{2k}\\left[\\frac{1}{(2k)!",
"}p(x,t)+\\frac{\\beta mv}{(2k+1)!",
"}j(x,t)\\right];$ the explicit formulas for $\\hat{\\omega }$ and $\\hat{\\eta }$ to be substituted in Eq.",
"(REF ) in the case $U(x)=0$ are $\\hat{\\omega }(x,v)&=&e^{-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }\\frac{(\\beta mv^2)^k}{(2k)!",
"}(D_0t_0)^k\\partial _x^{2k},\\hspace{21.68121pt}\\cr \\hat{\\eta }(x,v)&=&e^{-(D_0t_0/2)\\partial _x^2}\\sum _{k=0}^{\\infty }\\frac{(\\beta mv^2)^k}{(2k+1)!",
"}(D_0t_0)^k\\partial _x^{2k}.$ Both operators are expandable in $m$ , ($t_0=m/\\gamma $ ); their zero-th order coefficients equal unity, consistent with the heuristic formula (REF ).",
"Also applying the relation (REF ) in the definitions (REF ) and (REF ) gives identity.",
"The final step is that of verifying the evolution equation for $j$ , Eq.",
"(REF ).",
"Two integrals are to be completed with $\\rho $ expressed by the backward mapping, Eq.",
"(REF ), $\\int _{-\\infty }^{\\infty }v^2\\rho (x,y,t)dv=\\left(1+D_0t_0\\partial _x^2\\right)p(x,t)/\\beta m,\\cr \\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _ve^{\\beta mv^2/2}\\rho (x,v,t)dv=\\beta m j(x,t),\\ \\ $ which are now valid exactly.",
"If substituted in Eq.",
"(REF ), and the equation (REF ) is applied, we get $(1+t_0\\partial _t)[j(x,t)+D_0\\partial _xp(x,t)]=0$ after simple algebra.",
"This equation validates the relation $j=-D_0\\partial _xp$ for nonzero $m$ as well and thus, if combined with the mass conservation (REF ), also the diffusion equation (REF ) without any corrections.",
"If compared with the calculation of Eq.",
"(REF ), the 1-st order term $\\sim \\partial _tj$ , neglected in the previous Section, is compensated here by other 1-st order term coming from the exact relation (REF ), appearing in the integrals (REF ).",
"The FP equation (REF ) with zero potential is too simple to give nonzero corrections to the diffusion equation (REF ).",
"Nevertheless, it helped us to understand the structure of the mapping.",
"It shows that the scheme suggested in the previous section is viable.",
"The relation of the backward mapping has the form of Eq.",
"(REF ), at least for $U(x)=0$ , and the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ can be expanded in $m$ , or $t_0=m/\\gamma $ , $\\hat{\\omega }=\\sum _{k=0}^{\\infty }t_0^k\\hat{\\omega }_k(x,u),\\hspace{14.45377pt}\\hat{\\eta }=\\sum _{k=0}^{\\infty }t_0^k\\hat{\\eta }_k(x,u).$ Integration in Eqs.",
"(REF ) indicates that the coefficients $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ should be sought dependent up on the scaled velocity $u$ rather than $v$ (compare to Ref.",
"[25]); otherwise each term in Eq.",
"(REF ) would contribute in several succeeding orders in the integrals (REF ).",
"The mixing of orders would hinder us in constructing the recurrence scheme generating corrections to Eq.",
"(REF ).",
"In the notation of Eq.",
"(REF ), $\\hat{\\omega }_0=\\hat{\\eta }_0=1$ and $\\hat{\\omega }_1(x,u)&=&(u^2-1/2)D_0\\partial _x^2,\\hspace{21.68121pt}\\cr \\hat{\\eta }_1(x,u)&=&(u^2/3-1/2)D_0\\partial _x^2,\\cr \\hat{\\omega }_2(x,u)&=&(u^4/6-u^2/2+1/8)D_0^2\\partial _x^4,\\cr .&.&.$ valid for $U(x)=0$ according to Eqs.",
"(REF ), will be used for testing the results of the recurrence procedure in the next Section."
],
[
"IV. Mapping procedure",
"We now finish the construction of the mapping procedure, outlined in the Section II, for an arbitrary analytic potential $U(x)$ .",
"Supposing the backward mapping of the form (REF ) with the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ expanded in $t_0$ ($m$ ) according to Eqs.",
"(REF ), we find recurrence relations fixing the coefficients $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ .",
"Completing the integrals in Eq.",
"(REF ), we obtain a series of corrections to the zero-th order relation $j=-D_0\\partial _xp$ .",
"Combined with mass conservation, Eq.",
"(REF ), it gives the Smoluchowski equation corrected due to nonzero mass of the particle.",
"The essential relation determining the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ is the FP equation (REF ), which has to be satisfied for any solution of the reduced problem, the density $p(x,t)$ and the flux $j(x,t)$ , after their backward mapping (REF ) onto the full-dimensional Hilbert space.",
"If the expansion in $t_0=m/\\gamma $ of both operators, Eq.",
"(REF ), is supposed, we have $&&\\hspace{-21.68121pt}\\left[\\partial _t+\\sqrt{\\frac{2}{\\beta m}}u\\partial _x-\\frac{\\beta U^{\\prime }(x)}{\\sqrt{2\\beta m}}\\partial _u-\\frac{1}{2t_0}\\partial _u e^{-u^2}\\partial _u e^{u^2}\\right]\\sum _{k=0}^{\\infty }t_0^k\\cr &&\\hspace{-21.68121pt}\\times e^{-u^2}\\left[\\hat{\\omega }_k(x,u)p(x,t)+\\sqrt{2\\beta m}u\\hat{\\eta }_k(x,u)j(x,t)\\right]=0$ after introducing the scaled velocity $u=\\sqrt{\\beta m/2}v$ in Eq.",
"(REF ).",
"The factor $\\beta m$ is replaced by $t_0/D_0$ in the following calculations.",
"Thus half-integer powers of $t_0$ appear in Eq.",
"(REF ) [25].",
"As this equation has to be satisfied for any $t_0$ , we can split it for clarity into two relations: the first one, including only the integer powers of $t_0$ , $\\sum _{k=0}^{\\infty }t_0^k\\bigg [\\partial _t\\hat{\\omega }_kp+2u^2\\partial _x\\hat{\\eta }_kj-\\beta U^{\\prime }(x)e^{u^2}\\partial _uue^{-u^2}\\hat{\\eta }_kj\\cr -\\frac{1}{2t_0}e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_kp\\bigg ]=0,\\ \\ \\ $ and the second one, collecting the half-integer powers, $\\sum _{k=0}^{\\infty }t_0^{k-1/2}\\bigg [2t_0u\\partial _t\\hat{\\eta }_kj+D_0\\Big (2u\\partial _x-\\beta U^{\\prime }(x)e^{u^2}\\partial _ue^{-u^2}\\Big )\\cr \\times \\hat{\\omega }_kp-e^{u^2}\\partial _u e^{-u^2}\\partial _u u\\hat{\\eta }_k j\\bigg ]=0.\\hspace{21.68121pt}$ Notice that Eqs.",
"(REF ) and (REF ) do not violate parity of $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ in $u$ .",
"If used for construction of the recurrence relations between the coefficients, all they have to have the same parity as $\\hat{\\omega }_0=\\hat{\\eta }_0=1$ ; hence $\\hat{\\omega }_k(x,u)=\\hat{\\omega }_k(x,-u)$ and $\\hat{\\eta }_k(x,u)=\\hat{\\eta }_k(x,-u)$ .",
"This symmetry enables us to find the normalization (or identity) conditions for $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ .",
"The backward mapped $\\rho $ , Eq.",
"(REF ), with the operators $\\hat{\\omega }$ , $\\hat{\\eta }$ expanded in $t_0$ , Eq.",
"(REF ), substituted in the definitions (REF ) and (REF ) has to give identities for any $t_0$ , $p(x,t)$ and $j(x,t)$ .",
"Thus we obtain $\\frac{1}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }du e^{-u^2}\\hat{\\omega }_k(x,u)=\\delta _{0,k},$ $\\frac{1}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }2u^2du e^{-u^2}\\hat{\\eta }_k(x,u)=\\delta _{0,k}.$ The operators $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ are supposed not to depend on time, so the time derivative commutes with them and acts directly on $p(x,t)$ or $j(x,t)$ in Eqs.",
"(REF ), (REF ).",
"To derive the operators unambiguously, using only spatial derivatives, we express $\\partial _tp=-\\partial _xj$ from the mass conservation, Eq.",
"(REF ).",
"However, the time derivative of $j$ cannot be expressed in a similar way from Eq.",
"(REF ), because $\\partial _tj$ is not the leading term there.",
"If the backward mapping, Eqs.",
"(REF ) and (REF ), is applied to the integrals of Eq.",
"(REF ), we get $\\nonumber \\int _{-\\infty }^{\\infty }v^2\\rho (x,v,t) dv=\\frac{1}{\\beta m}\\left(1+\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k(x)\\right)p(x,t),$ where the operators $\\hat{I}_k$ are given by $\\hat{I}_k(x)=\\frac{2}{\\sqrt{\\pi }}\\int _{-\\infty }^{\\infty }u^2du e^{-u^2}\\hat{\\omega }_k(x,u);$ the right-hand side integral of Eq.",
"(REF ) results in $\\nonumber \\frac{\\gamma }{\\beta m^2}\\int _{-\\infty }^{\\infty }e^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\rho (x,v,t)dv&=&\\cr \\frac{\\gamma }{\\sqrt{\\pi }m}\\sum _{k=0}^{\\infty }t_0^k\\int _{-\\infty }^{\\infty }due^{-u^2}\\partial _uu\\hat{\\eta }_k(x,u)j(x,t)&=&\\frac{j(x,t)}{t_0}$ after integrating by parts and using the normalization relation (REF ).",
"Then, instead of the evolution equation for $j$ , Eq.",
"(REF ) has to be understood as an expression relating $j$ and $p$ , $(1+t_0\\partial _t)j(x,t)&=&-D_0\\bigg [e^{-\\beta U(x)}\\partial _x e^{\\beta U(x)}\\hspace{50.58878pt}\\cr &&\\hspace{21.68121pt}+\\partial _x\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k(x)\\bigg ]p(x,t),$ and the flux $j$ , as well as its time derivative, is expressed using $p$ according to this equation.",
"The term $t_0\\partial _t$ acts now on $p$ as the 1-st order correction in $(1+t_0\\partial _t)^{-1}p$ after completing inversion and commutation with the spatial operators in the square brackets of Eq.",
"(REF ).",
"Then $\\partial _tp$ is systematically replaced by $-\\partial _x j$ , contributing to the next corrections in the higher orders of $t_0$ .",
"The result is a formula for $j$ expressed by some purely spatial operator acting on $p$ , $j(x,t)=-D_0 e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]\\partial _x e^{\\beta U(x)}p(x,t),$ where the operators $\\hat{Z}_k$ are related to $\\hat{I}_k$ , Eq.",
"(REF ).",
"Using this form, we are able to write the final mapped equation explicitly after combination with Eq.",
"(REF ), $\\partial _tp&=&D_0\\partial _x e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k\\right]\\partial _x e^{\\beta U(x)}p\\cr &=&\\sum _{k=0}^{\\infty }t_0^k\\hat{Q}_kp=\\hat{Q}p,$ which is the Smoluchowski equation (REF ) in the zero-th order, extended by a series of mass dependent corrections in $t_0$ .",
"The operators $\\hat{Q}_k$ are introduced to simplify notation in the following calculations.",
"Now the operators $\\hat{Z}_k$ can be expressed explicitly using $\\hat{I}_k$ .",
"Expanding $(1+t_0\\partial _t)^{-1}p$ in $t_0$ and applying Eq.",
"(REF ), we have $\\nonumber (1+t_0\\partial _t)^{-1}p=(1-t_0\\hat{Q}+...)p=\\Big (1+\\sum _{k=0}^{\\infty }t_0^{k+1}\\hat{Q}_k\\Big )^{-1}\\hspace{-7.22743pt}p,$ which is to be used in the operator equation, $&&\\hspace{-14.45377pt}e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k\\right]\\partial _xe^{\\beta U(x)}=\\bigg [e^{-\\beta U(x)}\\partial _x e^{\\beta U(x)}\\cr &&\\hspace{43.36243pt}+\\partial _x\\sum _{k=1}^{\\infty }t_0^k\\hat{I}_k\\bigg ]\\Big (1+\\sum _{k=0}^{\\infty }t_0^{k+1}\\hat{Q}_k\\Big )^{-1},$ obtained after comparison of Eqs.",
"(REF ) and (REF ).",
"Expanding in $t_0$ and comparing the coefficients of the same powers of $t_0$ , we get a sequence of relations fixing $\\hat{Z}_k$ , $e^{-\\beta U}\\hat{Z}_1\\partial _x e^{\\beta U}&=&\\partial _x \\hat{I}_1-e^{-\\beta U}\\partial _x e^{\\beta U}\\hat{Q}_0,\\cr e^{-\\beta U}\\hat{Z}_2\\partial _x e^{\\beta U}&=&\\partial _x \\hat{I}_2-\\partial _x \\hat{I}_1\\hat{Q}_0\\cr &&+e^{-\\beta U}\\partial _x e^{\\beta U}(\\hat{Q}_0^2-\\hat{Q}_1),\\cr .&.&.$ Finally, we derive the recurrence scheme, generating the operators $\\hat{\\omega }_k$ , determining $\\hat{I}_k$ , Eq.",
"(REF ), and thus also $\\hat{Z}_k$ or $\\hat{Q}_k$ , Eqs.",
"(REF ), (REF ).",
"Notice that the $\\hat{\\eta }_k$ do not directly enter the mapped equation (REF ), but they are necessary in the recurrence formulas for $\\hat{\\omega }_k$ .",
"The recurrence scheme for $\\hat{\\omega }_k$ is defined by Eq.",
"(REF ).",
"This equation has to be satisfied for any $p$ and $j$ , solving the mapped problem, but these quantities, although considered before as formally independent, are related by Eq.",
"(REF ).",
"To obtain an equation for operators, $j$ has to be expressed by $p$ .",
"Applying the relation (REF ) and Eq.",
"(REF ) for $\\partial _tp$ , we get $&&\\hspace{-14.45377pt}\\sum _{n=0}^{\\infty }t_0^n\\bigg [D_0\\Big (\\hat{\\omega }_n\\partial _x-2u^2\\partial _x\\hat{\\eta }_n+\\beta U^{\\prime }e^{u^2}\\partial _uue^{-u^2}\\hat{\\eta }_n\\Big )e^{-\\beta U(x)}\\cr &&\\hspace{-7.22743pt}\\times \\sum _{l=0}^{\\infty }t_0^l\\hat{Z}_l\\partial _xe^{\\beta U(x)}-\\frac{1}{2t_0}e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_n\\bigg ]p=0,$ valid for any function $p(x,t)$ ; we have $\\hat{Z}_0=1$ .",
"To lowest order, $t_0^{-1}$ , only the term $e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_0=0$ .",
"It is satisfied by $\\hat{\\omega }_0=1$ , the only solution nondiverging at $u\\rightarrow \\pm \\infty $ and also satisfying the normalization, Eq.",
"(REF ).",
"In the higher orders, we derive the recurrence relation, $e^{u^2}\\partial _ue^{-u^2}\\partial _u\\hat{\\omega }_{n+1}&=&2D_0\\sum _{k=0}^n\\Big (\\hat{\\omega }_k\\partial _x-2u^2\\partial _x\\hat{\\eta }_k+\\beta U^{\\prime }(x)\\cr &&\\hspace{-57.81621pt}\\times e^{u^2}\\partial _uu e^{-u^2}\\hat{\\eta }_k\\Big )e^{-\\beta U(x)}\\hat{Z}_{n-k}\\partial _xe^{\\beta U(x)}.$ Calculation of the $\\hat{\\omega }_{n+1}$ requires us to know the $\\hat{\\eta }_k$ up to $k=n$ .",
"They are generated from Eq.",
"(REF ).",
"Again, $j$ , as well as $\\partial _t j$ , have to be expressed by $p$ using the relations (REF ) and (REF ).",
"Then Eq.",
"(REF ) becomes $&&\\sum _{n=0}^{\\infty }t_0^{n-1/2}\\bigg [e^{u^2}\\partial _u e^{-u^2}\\partial _u u\\hat{\\eta }_n e^{-\\beta U}\\sum _{k=0}^{\\infty }t_0^k\\hat{Z}_k\\partial _xe^{\\beta U}\\cr &&-2D_0t_0u\\hat{\\eta }_ne^{-\\beta U}\\sum _{k=0}^{\\infty }t_0^k\\hat{Z}_k\\partial _xe^{\\beta U}\\partial _xe^{-\\beta U}\\sum _{l=0}^{\\infty }t_0^l\\hat{Z}_l\\partial _xe^{\\beta U}\\cr &&+\\big (2u\\partial _x-\\beta U^{\\prime }e^{u^2}\\partial _u e^{-u^2}\\big )\\hat{\\omega }_n\\bigg ]p=0$ valid for any function $p(x,t)$ .",
"In lowest order, $t_0^{-1/2}$ , $e^{u^2}\\partial _ue^{-u^2}\\partial _u u\\hat{\\eta }_0e^{-\\beta U}\\partial _xe^{\\beta U}&=&-2u\\partial _x+\\beta U^{\\prime }e^{u^2}\\partial _ue^{-u^2}\\cr =-2u(\\partial _x+\\beta U^{\\prime })&=&-2ue^{-\\beta U}\\partial _x e^{\\beta U},$ where we have used $\\hat{\\omega }_0=1$ and $\\hat{Z}_0=1$ .",
"After the first integration, we have $\\partial _uu\\hat{\\eta }_0=-e^{u^2}\\int 2udue^{-u^2}=1+\\hat{C}_1e^{u^2};$ the integration constant $\\hat{C}_1=0$ provides convergence as $u\\rightarrow \\pm \\infty $ .",
"The next integration gives $\\hat{\\eta }_0=1+(1/u)\\hat{C}_0$ ; $\\hat{C}_0=0$ .",
"This calculation validates our heuristic formula (REF ) in the zero-th order approximation.",
"In the higher orders, Eq.",
"(REF ) generates the relations $&&\\hspace{-14.45377pt}e^{u^2}\\partial _ue^{-u^2}\\partial _u u\\hat{\\eta }_ne^{-\\beta U}\\partial _xe^{\\beta U}=(\\beta U^{\\prime }e^{u^2}\\partial _ue^{-u^2}-2u\\partial _x)\\hat{\\omega }_n\\cr &&+2D_0\\sum _{k,l=0}^{k+l<n} u\\hat{\\eta }_{n-k-l-1}e^{-\\beta U}\\hat{Z}_k\\partial _x e^{\\beta U}\\partial _x e^{-\\beta U}\\hat{Z}_l\\partial _x e^{\\beta U}\\cr &&-\\sum _{k=0}^{n-1}e^{u^2}\\partial _ue^{-u^2}\\partial _uu\\hat{\\eta }_k e^{-\\beta U}\\hat{Z}_{n-k}\\partial _xe^{\\beta U},$ forming the recurrence scheme for $\\hat{\\eta }_n$ .",
"Completing the operations in Eqs.",
"(REF ) and (REF ) one has to keep in mind that the equation acts on an arbitrary function $p(x,t)$ , not depending on $u$ .",
"On the other hand, the operators $\\hat{\\omega }_k$ and $\\hat{\\eta }_k$ for $k>0$ depend on $u$ .",
"The recurrence procedure starts from $\\hat{\\omega }_0=1$ and $\\hat{Z}_0=1$ .",
"Calculation of the next order correction requires first expressing the $\\hat{\\eta }_n$ according to Eq.",
"(REF ), or (REF ) for $n=0$ , as shown above.",
"Then $\\hat{\\omega }_{n+1}$ is derived from Eq.",
"(REF ), $\\hat{I}_{n+1}$ integrated according to Eq.",
"(REF ) and finally $\\hat{Z}_{n+1}$ expressed from Eq.",
"(REF ).",
"To demonstrate the procedure, we derive the first order correction, $\\hat{Z}_1$ .",
"We use already calculated $\\hat{\\eta }_0=1$ .",
"For $n=0$ , Eq.",
"(REF ) becomes $e^{u^2}\\partial _u e^{-u^2}\\partial _u\\hat{\\omega }_1=2D_0(1-2u^2)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}.$ After the first integration, $\\partial _u\\hat{\\omega }_1=2D_0e^{u^2}\\left(ue^{-u^2}+C_1\\right)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)},$ the integration constant $C_1=0$ , to provide convergence for $u\\rightarrow \\pm \\infty $ .",
"The integration constant $C_0$ after the next integration is fixed to satisfy the normalization, Eq.",
"(REF ), $\\int _{-\\infty }^{\\infty }due^{-u^2}D_0(u^2+C_0)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}=0,$ hence $C_0=-1/2$ and $\\hat{\\omega }_1=D_0(u^2-1/2)e^{-\\beta U(x)}\\partial _x^2 e^{\\beta U(x)}.$ For $U(x)=0$ , we recover the corresponding formula in Eq.",
"(REF ).",
"Integration over $u$ in Eq.",
"(REF ) results in $\\hat{I}_1=D_0\\exp [-\\beta U(x)]\\partial _x^2\\exp [\\beta U(x)]$ , giving finally $\\hat{Z}_1=D_0\\left(e^{\\beta U}\\partial _x e^{-\\beta U}\\partial _x-\\partial _x e^{\\beta U}\\partial _xe^{-\\beta U}\\right)=D_0\\beta U^{\\prime \\prime }(x)$ from Eq.",
"(REF ).",
"In the higher orders, $\\hat{\\eta }_n$ , $\\hat{\\omega }_{n+1}$ are calculated according to Eqs.",
"(REF ) and (REF ).",
"The integration constants after double integration have to provide convergence for $u\\rightarrow \\pm \\infty $ , requiring the operators to be even in $u$ , and the normalization, Eq.",
"(REF ).",
"The condition (REF ) for $\\hat{\\eta }_n$ is satisfied automatically; it serves as a check on the computation.",
"The derivation is tedious, and we present only the results in second order, $\\hat{\\eta }_1&=&D_0(u^2/3-1/2)e^{-\\beta U(x)}\\partial _x^2e^{\\beta U(x)},\\cr \\hat{\\omega }_2&=&\\frac{D_0^2}{2}e^{-\\beta U(x)}\\Big [\\Big (\\frac{u^4}{3}-u^2+\\frac{1}{4}\\Big )\\partial _x^3\\cr &&+\\Big (u^2-\\frac{1}{2}\\Big )\\big (4\\beta U^{\\prime \\prime }(x)\\partial _x+3\\beta U^{(3)}(x)\\big )\\Big ]\\partial _x e^{\\beta U(x)},\\cr \\hat{Z}_2&=&\\frac{D_0^2}{2}\\Big [4\\big (\\beta U^{\\prime \\prime }(x)\\big )^2-\\beta ^2U^{\\prime }(x)U^{(3)}(x)+\\beta U^{(4)}(x)\\cr &&\\hspace{21.68121pt}+3\\beta U^{(3)}(x)\\partial _x\\Big ].$ Again, the formulas (REF ) for $U(x)=0$ are recovered.",
"There are no contributions to $\\hat{Z}_n$ in this case, too, as expected according to the analysis in the previous Section.",
"Also, linear potentials, $U(x)=-Fx$ , have no effect on validity of the uncorrected Smoluchowski equation (REF ).",
"The particle driven by a constant force $F$ move asymptotically with constant mean velocity $v_0=F/\\gamma $ and the distribution $p(y,t)$ in coordinate $y$ , shifted by the drift, $y=x-v_0t$ , is again Gaussian as in the case of no potential.",
"The situation becomes different if the driving force $F(x)$ is not constant.",
"If the mass $m$ or the time of the thermalization $t_0=m/\\gamma $ is small, but nonzero, the particle appearing at a new position $x$ has to accommodate to the new local mean velocity.",
"On the other hand, it carries some mean momentum from its previous position and needs some time to change it.",
"Meanwhile it slips to some other position than predicted by purely stochastic dynamics due to its inertia, or non-zero mass.",
"The effects of such slipping are indicated by the corrections $\\hat{Z}_n$ of the Smoluchowski equation and they are nonzero for potentials with nonzero $U^{\\prime \\prime }(x)$ , or higher derivatives.",
"As seen from Eq.",
"(REF ), the $\\hat{Z}_n$ are not only functions, but operators, containing $\\partial _x$ in the higher orders.",
"So the mapped equation (REF ) has exactly the same structure as the mapped equations for diffusion [16], or biased diffusion [15], [22], [26].",
"Being inspired by these works, Eq.",
"(REF ) can be simplified by replacing the correction operators $D_0[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k]$ by a function $D(x)$ , a spatially dependent effective diffusion coefficient, $\\partial _tp(x,t)=\\partial _xe^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x,t),$ which becomes valid in the limit of stationary flow, i.e.",
"when the spatial density $p$ and the flux $j$ change very slowly, $p(x,t)\\rightarrow p(x)$ .",
"Due to mass conservation, Eq.",
"(REF ), the flux $j(x,t)=j$ is constant (but nonzero) in $x$ as well.",
"If expressed from Eq.",
"(REF ), $j=-e^{-\\beta U(x)}D(x)\\partial _xe^{\\beta U(x)}p(x),$ the function $\\partial _x(\\exp [\\beta U(x)]p(x))=-j\\exp [\\beta U(x)]/D(x)$ is dependent only on the system; $m$ , $\\gamma $ and the potential $U(x)$ , for any stationary solution $p(x)$ .",
"So we can substitute for it in Eq.",
"(REF ), $j=D_0e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]e^{\\beta U(x)}\\frac{j}{D(x)},$ valid for stationary flow, and calculate $D(x)$ unambiguously from the expansion of the corrections $\\hat{Z}_k$ , $\\frac{D_0}{D(x)}=e^{-\\beta U(x)}\\left[1+\\sum _{k=1}^{\\infty }t_0^k\\hat{Z}_k(x)\\right]^{-1}e^{\\beta U(x)},$ as a series in $t_0$ , $D(x)/D_0&=&1+D_0t_0\\beta U^{\\prime \\prime }+(D_0t_0)^2\\Big [2(\\beta U^{\\prime \\prime })^2\\cr &&\\hspace{14.45377pt}+\\beta ^2U^{\\prime }U^{(3)}+\\beta U^{(4)}/2\\Big ]+...\\ .$ The next simplification is that of neglecting all the derivatives but $U^{\\prime \\prime }(x)$ , i.e.",
"approximating the real $U(x)$ locally by a quadratic potential.",
"In this case, the expansion (REF ) can be summed up to infinity, $D(x)/D_0&\\simeq &\\sum _{n=0}^{\\infty }\\frac{(2n)!}{n!(n+1)!",
"}\\left[D_0t_0\\beta U^{\\prime \\prime }(x)\\right]^n\\cr &&=\\frac{1-\\sqrt{1-4D_0t_0\\beta U^{\\prime \\prime }(x)}}{2D_0t_0\\beta U^{\\prime \\prime }(x)},$ the proof is given in the Appendix B.",
"The formula works for $D_0t_0\\beta U^{\\prime \\prime }(x)=mU^{\\prime \\prime }(x)/\\gamma ^2<1/4$ , which is the condition for non-oscillatory movement of a particle in a quadratic well with friction, the damped harmonic oscillator.",
"If $U^{\\prime \\prime }(x)=\\kappa $ is constant, the trajectory of a particle averaged over the stochastic force is governed by $m\\langle \\ddot{x}\\rangle +\\gamma \\langle \\dot{x}\\rangle +\\kappa \\langle x\\rangle =0$ from Eq.",
"(REF ).",
"The particular solutions are $\\langle x(t)\\rangle =\\exp (\\alpha t)$ with $\\alpha =-(\\gamma \\pm \\sqrt{\\gamma ^2-4m\\kappa })/2m$ .",
"Requiring $\\alpha $ to be a real number gives the same condition.",
"This simple example demonstrates restriction of the theory presented to nonoscillatory movement of the particle in potential wells on its way along a 1D channel.",
"A small mass $m$ is expected, to enable the friction quickly to damp the momentum of a particle; i.e.",
"to have the relaxation in the velocity faster than in the real space coordinate $x$ .",
"Figure: Plot of the effective diffusion coefficient D(x)D(x) dependenton the mass of the particle mm and friction γ\\gamma according toEq.",
"(), valid if the derivatives of the potential higherthan U '' (x)U^{\\prime \\prime }(x) are neglected.A more detailed insight to the restrictions of the dimensional reduction of the phase space controlled by the mass $m$ can be obtained by comparison of the Green's function (GF) of the mapped equation (REF ) with $D(x)$ given by Eq.",
"(REF ) and GF of the Kramers equation (REF ) for the damped harmonic oscillator, $U(x)=\\kappa x^2/2=m\\omega _0^2x^2/2$ , which is exactly solvable.",
"The solution $G=G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })$ of the equation $\\Big (\\partial _t+v\\partial _x-\\omega _0^2x\\partial _v-\\frac{\\gamma }{\\beta m^2}\\partial _ve^{-\\beta mv^2/2}\\partial _v e^{\\beta mv^2/2}\\Big ) G\\cr =\\delta (x-x^{\\prime })\\delta (v-v^{\\prime })\\delta (t-t^{\\prime })\\hspace{28.45274pt}$ reads [1] $G=\\frac{(s_1-s_2)e^{\\gamma t/m}}{2\\pi \\sqrt{ab-h^2}}\\exp \\Big [-\\Big (a(\\xi -\\xi _0)^2+b(\\eta -\\eta _0)^2\\cr +2h(\\xi -\\xi _0)(\\eta -\\eta _0)\\Big )\\big /2(ab-h^2)\\Big ],\\hspace{28.45274pt}$ where $s_{1,2}=-\\frac{\\gamma }{2m}\\pm \\sqrt{\\frac{\\gamma ^2}{4m^2}-\\omega _0^2},$ $\\xi =(s_1x-v)e^{-s_2t},\\hspace{28.45274pt}\\xi _0=(s_1x^{\\prime }-v^{\\prime }),\\hspace{36.98866pt}\\cr \\eta =(s_2x-v)e^{-s_1t},\\hspace{28.45274pt}\\eta _0=(s_2x^{\\prime }-v^{\\prime }),\\hspace{36.98866pt}\\cr a=\\frac{\\gamma }{\\beta m^2s_1}\\Big (1-e^{-2s_1t}\\Big ),\\ \\ b=\\frac{\\gamma }{\\beta m^2s_2}\\Big (1-e^{-2s_2t}\\Big ),\\cr h=\\frac{2}{\\beta m}\\Big (1-e^{\\gamma t/m}\\Big ).\\hspace{56.9055pt}$ Similar to the case of no potential in the Section III, let us suppose that a thermalized particle (equilibrated in velocity) was inserted at a position $x_0$ at time $t^{\\prime }=0$ , $\\rho _0(x^{\\prime },v^{\\prime })=\\sqrt{\\frac{\\beta m}{2\\pi }}\\delta (x^{\\prime }-x_0)e^{-\\beta mv^{\\prime 2}/2}.$ After integration over $v^{\\prime }$ and $v$ , we get the corresponding spatial density $p(x,t)=\\int _{-\\infty }^{\\infty }G(x,v,t;x^{\\prime }v^{\\prime },0)\\rho _0(x^{\\prime },v^{\\prime })dx^{\\prime }dv^{\\prime }dv=\\cr \\sqrt{\\frac{\\beta m}{2\\pi Z}}\\omega _0(s_1-s_2)\\exp \\Big [-\\frac{\\beta m\\omega _0^2}{2 Z}\\hspace{42.67912pt}\\cr \\times \\Big (s_1(x-x_0e^{s_2t}) -s_2(x-x_0e^{s_1t})\\Big )^2\\Big ];\\hspace{14.22636pt}$ $\\nonumber Z=\\Big [s_1(1+e^{s_2t})-s_2(1+e^{s_1t})\\Big ]\\Big [s_1(1-e^{s_2t})-s_2(1-e^{s_1t})\\Big ].$ On the other hand, the coefficient $D(x)$ , Eq.",
"(REF ), becomes constant for the quadratic potential, $D(x)=\\frac{\\gamma /2m-\\sqrt{\\gamma ^2/4m^2-\\omega _0^2}}{\\beta m\\omega _0^2}=\\frac{-s_1}{\\beta m\\omega _0^2},$ and GF of the corresponding mapped equation (REF ), $\\Big (\\partial _t-\\partial _xD(x)e^{-\\beta m\\omega _0^2x^2/2}\\partial _xe^{\\beta m\\omega _0^2x^2/2}\\Big )g(x,t;x_0,t_0)\\cr =\\delta (x-x_0)\\delta (t-t_0),\\hspace{56.9055pt}$ can be easily found by a calculation similar to the derivation of Eq.",
"(REF ), Appendix A.",
"The result, $g(x,t;x_0,0)=\\sqrt{\\frac{\\beta m}{2\\pi (1-e^{2s_1t})}}\\omega _0\\exp \\Big [-\\frac{1}{2}\\beta m\\omega _0^2\\cr \\times \\big (x-x_0e^{s_1t}\\big )^2\\big /\\big (1-e^{2s_1t}\\big )\\Big ],\\hspace{14.22636pt}$ describes evolution of the real space density of a thermalized particle inserted at $x_0$ , too, and can be directly compared with the formula (REF ).",
"First, let us notice that in comparison with Eq.",
"(REF ), the exponential $e^{s_2t}$ disappeared from the formula (REF ).",
"The root $s_2\\simeq -\\gamma /m$ for $m\\rightarrow 0$ makes $e^{s_2t}\\simeq e^{-\\gamma t/m}$ the term essentially singular in $m$ and so invisible for the recurrence procedure, which works with the operators $\\hat{\\omega }$ and $\\hat{\\eta }$ expanded in $m$ .",
"Using our argumentation from the Section III, $e^{s_2t}$ represents the \"transients\" neglected by the mapping.",
"On the other hand, $e^{s_1t}\\simeq e^{-m\\omega _0^2t/\\gamma }$ is regular in $m$ small, representing the contribution of the low-lying states, retained by the method.",
"Next, let us stress that the equation (REF ) with $D(x)$ expressed by the expansion (REF ) was derived in the limit of the stationary flow; for the net flux almost constant, which is not the case of the process described by the Eqs.",
"(REF ) and (REF ).",
"Nevertheless, using the approximations $\\nonumber s_1-s_2\\big (1\\pm e^{s_1t}\\big )&\\simeq & (s_1-s_2)\\big (1\\pm e^{s_1t}\\big ),\\cr s_1x-s_2\\big (x-x_0e^{s_1t}\\big )&\\simeq & (s_1-s_2)\\big (x-x_0e^{s_1t}\\big ),$ applicable for $e^{s_1t}\\ll 1$ , the regularized formula (REF ) (with $e^{s_2t}$ neglected) becomes finally Eq.",
"(REF ); i.e.",
"it represents correctly the asymptotic behavior of the spatial density $p(x,t)$ for large time $t$ .",
"If $m$ approaches $\\gamma /2\\omega _0$ , the transients contributing by $e^{s_2t}$ , neglected by the mapping, become important.",
"In the oscillating regime, $s_{1,2}$ are complex numbers and both are necessary for expressing the real density $p(x,t)$ in Eq.",
"(REF ).",
"The mapping which splits the Hilbert space to the retained low-lying states and the neglected transients, controlled by $m$ small, loses its justification and the method stops working.",
"Mapping in this region requires a different method to be applied.",
"It will be an object of our study in the future."
],
[
"V. Conclusion",
"Although modeling of transport in confined systems is often based on study of the Langevin equation, Eq.",
"(REF ) in the simplest 1D case, the solutions necessary in practical applications are often accessible only in two limits: either the friction $\\gamma \\rightarrow 0$ , when the particles obey Newtonian dynamics, or the mass $m\\rightarrow 0$ , which corresponds to stochastic dynamics.",
"Any solution in the region between these limits requires working in phase space, which makes the problem much more complicated.",
"The present paper shows how to describe the region of finite $m/\\gamma $ while still working in real space, as in the case of stochastic dynamics.",
"The equation governing evolution of the spatial density $p(x,t)$ is the Smoluchowski equation, corresponding to the limit of a massless particle, extended by a series of corrections in powers of $t_0=m/\\gamma $ ; $t_0$ can be interpreted as the typical time of thermalizing of the particle's initial velocity by the stochastic force.",
"In general, the extended Smoluchowski equation has the form of Eq.",
"(REF ), $D_0=1/\\gamma \\beta $ denotes the diffusion constant and the operators $\\hat{Z}_k$ are systematically derived within the recurrence procedure presented in the Sect.",
"IV.",
"In the limit of stationary flow, i.e.",
"when the flux is almost constant but nonzero in time and space, this equation can be simplified to Eq.",
"(REF ), where the effective diffusion coefficient $D(x)$ is calculated unambiguously from the operators $\\hat{Z}_k$ , Eq.",
"(REF ).",
"In the simplest approximation, when all the derivatives of the potential higher than $U^{\\prime \\prime }(x)$ are neglected, the series of corrections can be summed up to infinity, giving the formula for $D(x)$ in a closed form, Eq.",
"(REF ), described in Fig.",
"1.",
"Then the equation describes stationary flow in a quadratic potential.",
"The theory works while $4mU^{\\prime \\prime }(x)<\\gamma ^2$ , until the averaged trajectory of a single particle is not oscillatory in the potential wells along the 1D channel.",
"The mapping in the oscillatory regime requires the next analysis, which will be done in the future.",
"Technically, the paper demonstrates that the projection technique developed for mapping of diffusion in 2D (3D) channels with varying cross section [13], [14], [16], can be adapted for the dimensional reduction of a process described by an evolution equation of a different type than the diffusion or Smoluchowski equation.",
"The method has been modified significantly; $m/\\gamma $ had to be confirmed as the small parameter controlling the expansion of the corrections $\\hat{Z}_k$ , as well as the operators of the backward mapping, $\\hat{\\omega }$ and $\\hat{\\eta }$ .",
"In contrast to diffusion, the flux $j(x,t)$ is handled here as a quantity independent of the density $p(x,t)$ .",
"Thus the recurrence procedure, calculating expansions of the correction operators $\\hat{Z}_k$ and the operators $\\hat{\\omega }$ , $\\hat{\\eta }$ , is in principle the result of combination of three equations, Eqs.",
"(REF ), (REF ) and also (REF ), with the relation of the backward mapping, Eq.",
"(REF ).",
"It is worthwhile to notice that including the mass dependent corrections to the Smoluchowski equation results in the equations (REF ) or (REF ), which are of the same form as the comparable equations obtained from the mapping of diffusion in channels with varying cross section.",
"On the other hand, the effective coefficient $D(x)$ (REF ) has a different symmetry than the similar formulas extending the Fick-Jacobs equation [16], [17] for confined diffusion.",
"Also $D(x)$ can be greater than 1 here (see Fig.",
"1); i.e.",
"the quasi stationary flux is accelerated when passing through a shallow potential well, depending on the nonzero mass of the particles.",
"These interesting properties could be observed in simulations similar to that verifying $D(x)$ in the extended Fick-Jacobs equation [27], [28].",
"The effects of slipping of the particles diffusing under a nonconstant force $F(x)$ due to their inertia, as described in Sect.",
"IV, might also influence the interesting phenomena in the micro and nano world, such as Brownian pumps [29], [30], rectification of the flux in quasi 1D structures [31], stochastic resonance [20], [32], [33], [34], or the negative mobility [35].",
"Study of such applications of the theory presented is expected in the future."
],
[
"Acknowledgments",
"Support from VEGA grant No.",
"2/0049/12 and CE SAS QUTE project is gratefully acknowledged.",
"P.K.",
"also thanks CIMS, New York University for kind hospitality."
],
[
"Appendix A: Exact solution",
"The Green's function (REF ) solving the FP equation with zero potential, Eq.",
"(REF ), is calculated here.",
"First we introduce the scaled coordinates $\\xi $ , $u$ , $\\tau $ according to Eqs.",
"(REF ) and define the function $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })$ , $G(x,v,t;x^{\\prime },v^{\\prime },t^{\\prime })=e^{-u^2/2}\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })e^{u^{\\prime 2}/2},$ satisfying the transformed equation (REF ), $\\left[\\partial _{\\tau }+u\\partial _{\\xi }-\\partial _u^2+u^2-1\\right]\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })\\hspace{36.135pt}\\cr =\\frac{\\gamma \\beta }{4}e^{(u^2-u^{\\prime 2})/2}\\delta (\\xi -\\xi ^{\\prime })\\delta (u-u^{\\prime })\\delta (\\tau -\\tau ^{\\prime }),\\ $ the exponential factor becomes unity due to $\\delta (u-u^{\\prime })$ .",
"After the Fourier transform in $\\xi $ and $\\tau $ , $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })=\\int \\frac{dkd\\nu }{4\\pi ^2}e^{ik(\\xi -\\xi ^{\\prime })-i\\nu (\\tau -\\tau ^{\\prime })}\\Gamma _{k,\\nu }(u;u^{\\prime }),$ and shifting the velocities by $ik/2$ , $w=u+ik/2$ and $w^{\\prime }=u^{\\prime }+ik/2$ , the equation $\\left[-i\\nu -\\partial _w^2+w^2-1+k^2/4\\right]\\Gamma _{k,\\nu }(w;w^{\\prime })=\\frac{\\gamma \\beta }{4}\\delta (u-u^{\\prime })$ becomes solvable if $\\Gamma _{k,\\nu }(w;w^{\\prime })$ is expressed in the basis set of the linear harmonic oscillator $\\psi _n(w)$ , $\\Gamma _{k,\\nu }(w;w^{\\prime })=\\sum _{n=0}^{\\infty }\\Gamma _n(k,\\nu )\\psi _n(w)\\psi _n^*(w^{\\prime }).$ The eigenfunctions $\\psi _n(w)$ satisfy $\\left(-\\partial _w^2+w^2\\right)\\psi _n(w)=\\lambda _n\\psi _n(w)=(2n+1)\\psi _n(w)$ and we use the integral representation of the Hermite polynomials $H_n(w)$ [36] $\\psi _n(w)&=&\\frac{1}{\\@root 4 \\of {\\pi }\\sqrt{2^n n!",
"}}H_n(w)e^{-w^2/2}\\cr &=&\\sqrt{\\frac{2^n}{n!\\pi ^{3/2}}}e^{-w^2/2}\\int _{-\\infty }^{\\infty }(w+ir)^ne^{-r^2}dr\\hspace{14.45377pt}$ in the next calculation.",
"Using the transformations above, we find $\\Gamma _n(k,\\nu )=\\frac{\\gamma \\beta /4}{-i\\nu +\\lambda _n-1+k^2/4}.$ Applying it in the formulas (REF ) and (REF ), we integrate the last one over $\\nu $ in the complex plane, $\\Gamma (\\xi ,u,\\tau ;\\xi ^{\\prime },u^{\\prime },\\tau ^{\\prime })&&\\hspace{-14.45377pt}=\\frac{\\Theta (\\tau -\\tau ^{\\prime })}{8\\pi D_0}\\int _{-\\infty }^{\\infty }dke^{ik(\\xi -\\xi ^{\\prime })-k^2(\\tau -\\tau ^{\\prime })/4}\\cr &&\\times \\sum _{n=0}^{\\infty }e^{-2n(\\tau -\\tau ^{\\prime })}\\psi _n(w)\\psi _n^*(w^{\\prime }),$ $\\Theta (x)$ denotes the Heaviside unit step function and $D_0=1/\\gamma \\beta $ is the diffusion constant.",
"Now the integral relation (REF ) is used for $\\psi _n(w)$ and $\\psi _n^*(w^{\\prime })$ and the summation over $n$ can be readily completed.",
"Finally, the straightforward triple integration over $k$ , $r$ , $r^{\\prime }$ is performed and using the transformation (REF ) results in the formula (REF )."
],
[
"Appendix B: Quadratic approximation",
"Derivation of the formula (REF ) for the effective diffusion coefficient $D(x)$ with all the derivatives higher than $U\"(x)$ neglected is presented here.",
"This approximation corresponds to local replacing of the potential by a parabola, $U(x)\\simeq \\kappa (x-x_0)^2/2+U_0$ , where $\\kappa ,\\ x_0$ and $U_0$ are fitting parameters.",
"First we simplify Eq.",
"(REF ).",
"For quadratic potential, the right hand side can be rewritten as $\\nonumber e^{-\\beta U(x)}\\left(1+t_0\\hat{Z}\\right)^{-1}e^{\\beta U(x)}=\\left(1+t_0 e^{-\\beta U}\\hat{Z}(x)e^{\\beta U}\\right)^{-1},$ $t_0\\hat{Z}=\\sum _{n=1}^{\\infty }t_0\\hat{Z}_n$ ; the difference contains only the higher derivatives of $U(x)$ , which are zero.",
"Hence $D(x)/D_0=1+e^{-\\beta U(x)}\\sum _{n=1}^{\\infty }t_0^n\\hat{Z}_ne^{\\beta U(x)}.$ The formulas for $D(x)$ have been derived considering stationary flow; $j(x,t)=j$ is constant.",
"It simplifies the relation (REF ); $\\partial _tj=0$ .",
"Thus the right hand side represents stationary flux, which can be directly compared with Eq.",
"(REF ), giving a much simpler relation between $\\hat{Z}_n$ and $\\hat{I}_n$ than Eq.",
"(REF ), $e^{-\\beta U(x)}\\hat{Z}_n(x)\\partial _xe^{\\beta U(x)}p(x)=\\partial _x\\hat{I}_n(x)p(x)$ for any stationary solution $p(x)$ .",
"Calculation of the coefficients of $D(x)$ according to Eq.",
"(REF ) requires us to take $\\partial _x\\exp [\\beta U(x)]p(x)=\\exp [\\beta U(x)]$ , hence finally $e^{-\\beta U(x)}\\hat{Z}_ne^{\\beta U(x)}&=&\\frac{2}{\\sqrt{\\pi }}\\partial _x\\int _{-\\infty }^{\\infty }u^2due^{-u^2}\\hspace{57.81621pt}\\cr &&\\times \\hat{\\omega }_n(x,u)e^{-\\beta U(x)}\\int dx e^{\\beta U(x)}$ after application of Eq.",
"(REF ).",
"Before writing the explicit formulas for $\\hat{\\omega }_n$ for the quadratic potential, we define the polynomials $P_n(u)&=&\\sum _{k=0}^{n}\\frac{(-1)^{n-k}2^{2k-n}}{(2k)!(n-k)!",
"}u^{2k},\\cr Q_n(u)&=&\\sum _{k=0}^{n}\\frac{(-1)^{n-k}2^{2k-n}}{(2k+1)!(n-k)!",
"}u^{2k},$ $n=1,2, ...$ , coming from the expansions of $\\hat{\\omega }_n$ and $\\hat{\\eta }_n$ for zero potential in $t_0$ , Eqs.",
"(REF ).",
"The first few polynomials are visible in the round brackets of Eq.",
"(REF ).",
"One can check by direct integration that $\\int _{-\\infty }^{\\infty }Q_n(u)u^2e^{-u^2}du&=&\\frac{\\sqrt{\\pi }}{2^{n+1}}\\sum _{k=0}^n\\frac{(-1)^{k-n}}{k!(n-k)!",
"}=0,\\cr \\int _{-\\infty }^{\\infty }P_n(u)u^2e^{-u^2}du&=&\\frac{\\sqrt{\\pi }}{2}\\delta _{n,1},$ corresponding to the normalization of $\\hat{\\eta }_n$ , Eq.",
"(REF ), and the relations (REF ), (REF ), proving no correction to the Smoluchowski equation in the case $U(x)=0$ .",
"The operators $\\hat{\\omega }_n$ and $\\hat{\\eta }_n$ for the quadratic potential have the form $\\hat{\\omega }_n&=&e^{-\\beta U(x)}\\sum _{k=1}^nc_{n,k}P_k(u)\\left(\\beta U^{\\prime \\prime }\\right)^{n-k}\\partial _x^{2k}e^{\\beta U(x)},\\hspace{21.68121pt}\\cr \\hat{\\eta }_n&=&e^{-\\beta U(x)}\\sum _{k=1}^nc_{n,k}Q_k(u)\\left(\\beta U^{\\prime \\prime }\\right)^{n-k}\\partial _x^{2k}e^{\\beta U(x)},$ with the coefficients $c_{n,k}=D_0^n\\frac{2k\\ (2n-1)!}{(n-k)!(n+k)!",
"}.$ Due to the integrals, Eq.",
"(REF ), only the first terms with $P_1(u)$ in Eq.",
"(REF ) contribute to the expansion of $D(x)$ , Eq.",
"(REF ).",
"Then the functions become $e^{-\\beta U(x)}\\hat{Z}_ne^{\\beta U(x)}&=&c_{n,1}\\partial _xe^{-\\beta U(x)}\\left(\\beta U^{\\prime \\prime }\\right)^{n-1}\\partial _xe^{\\beta U(x)}\\cr &=&\\frac{2\\ (2n-1)!}{(n-1)!(n+1)!",
"}\\left(D_0\\beta U^{\\prime \\prime }\\right)^n,$ taking $U^{(3)}(x)=0$ into account.",
"Applied in Eq.",
"(REF ) it results in the expansion of $D(x)$ , Eq.",
"(REF ).",
"Finally, one has to verify that the formulas (REF ) satisfy the recurrence relations (REF ) and (REF ), acting on the function $p(x)=\\exp [-\\beta U(x)]\\int dx\\exp [\\beta U(x)]$ .",
"Although the equations simplify notably due to neglecting the derivatives higher than $U^{\\prime \\prime }(x)$ , we omit the details of this tedious but straightforward calculation."
]
] | 1204.1145 |
[
[
"The paradox of soft singularity crossing and its resolution by\n distributional cosmological quantitities"
],
[
"Abstract A cosmological model of a flat Friedmann universe filled with a mixture of anti-Chaplygin gas and dust-like matter exhibits a future soft singularity, where the pressure of the anti-Chaplygin gas diverges (while its energy density is finite).",
"Despite infinite tidal forces the geodesics pass through the singularity.",
"Due to the dust component, the Hubble parameter has a non-zero value at the encounter with the singularity, therefore the dust implies further expansion.",
"With continued expansion however, the energy density and the pressure of the anti-Chaplygin gas would become ill-defined, hence from the point of view of the anti-Chaplygin gas only a contraction is allowed.",
"Paradoxically, the universe in this cosmological model would have to expand and contract simultaneously.",
"This obviosly could not happen.",
"We solve the paradox by redefining the anti-Chaplygin gas in a distributional sense.",
"Then a contraction could follow the expansion phase at the singularity at the price of a jump in the Hubble parameter.",
"Although such an abrupt change is not common in any cosmological evolution, we explicitly show that the set of Friedmann, Raychaudhuri and continuity equations are all obeyed both at the singularity and in its vicinity.",
"We also prove that the Israel junction conditions are obeyed through the singular spatial hypersurface.",
"In particular we enounce and prove a more general form of the Lanczos equation."
],
[
"Introduction",
"The problem of cosmological singularities has been attracting the attention of theoreticians working in gravity and cosmology since the early fifties [1], [2], [3].",
"In the sixties general theorems about the conditions for the appearance of singularities were proven [4], [5] and the oscillatory regime of approaching the singularity [6], the Mixmaster universe [7] was discovered.",
"Basically, until the end of nineties almost all discussions about singularities were devoted to the Big Bang and Big Crunch singularities, which are characterized by a vanishing cosmological radius.",
"However, kinematical investigations of Friedmann cosmologies have raised the possibility of sudden future singularity occurrence [8]-[18], characterized by a diverging $\\ddot{a}$ whereas both the scale factor $a$ and $\\dot{a}$ are finite.",
"Then, the Hubble parameter $H=\\dot{a}/a~$ and the energy density $\\rho $ are also finite, while the first derivative of the Hubble parameter and the pressure $p$ diverge.",
"Until recent years, however, sudden future singularities attracted only a limited interest among researchers.",
"The interest grew due to two reasons.",
"The recent discovery of the cosmic acceleration [19] has stimulated the elaboration of dark energy models, responsible for such a phenomenon (see e.g.",
"for review [20]).",
"Remarkably in some of these models the sudden singularities arise quite naturally.",
"Another source of the interest to sudden singularities is the development of brane models [10], [11], [18], where singularities of this kind could arise naturally (sometimes these singularities, arising in brane-world models, are called “quiescent” [10]).",
"In the investigations devoted to sudden singularities one can distinguish three main topics.",
"The first of them deals with the question of the compatibility of the models possessing soft singularities with observational data [15], [21], [22].",
"The second direction is connected with the study of quantum effects [11], [17], [23], [24], [25].",
"Here one can see two subdirections: the study of quantum corrections to the effective Friedmann equation, which can eliminate classical singularities or at least, change their form [10], [17], [23]; and the study of solutions of the Wheeler-DeWitt equation for the quantum state of the universe in the presence of sudden singularities [24], [25].",
"The third direction is connected with the possibility of the sudden singularity crossing in classical cosmology [26], [27], [28], [29], [25].",
"The present paper is devoted exactly to this topic.",
"A particular feature of the sudden future singularities is their softness [26].",
"As the Christoffel symbols depend only on the first derivative of the scale factor, they are regular at these singularities.",
"Hence, the geodesics are well behaved and they can cross the singularity [26].",
"One can argue that the particles crossing the singularity will generate the geometry of the spacetime, providing in such a way a “soft rebirth” of the universe after the singularity crossing [29].",
"Note that the possibility of crossing of some kind of cosmological singularities was noticed already in the early paper by Tipler [30].",
"A close idea of integrable singularities in black holes, which can give origin to a cosmogenesis was recently put forward in [31].",
"As a starting point we consider an interesting example of a sudden future singularity - the Big Brake which was discovered in Ref.",
"[32] while studying a particular tachyon cosmological model.",
"The particularity of the Big Brake singularity consists in the fact that the time derivative of the scale factor is not only finite, but exactly equal to zero.",
"That makes the analysis of the behavior in the vicinity of singularity especially convenient.",
"In particular, in Ref.",
"[22] it was shown that the predictions of the future of the universe in this model [32] are compatible with the supernovae type Ia data, while in Refs.",
"[24], [25] some quantum cosmological questions were studied in the presence of the Big Brake singularity.",
"The simplest cosmological model allowing a Big Brake singularity was also introduced in Ref.",
"[32].",
"This model is based on the perfect fluid, dubbed “anti-Chaplygin gas”.",
"This fluid is characterized by the equation of state $p=\\frac{A}{\\rho }, $ where $A$ is a positive constant.",
"Such and equation of state arises, for example, in the theory of wiggly strings [33].",
"In paper [32] a fluid obeying the equation of state (REF ) was called \"anti-Chaplygin gas\" in analogy with the Chaplygin gas [34] which has the equation of state $p=-A/\\rho $ and has acquired some popularity as a candidate for a unified theory of dark energy and dark matter [35].",
"An explicit example of the crossing of the Big Brake singularity was described in detail in paper [29], were the tachyon model [32], [22] was investigated.",
"In this model the tachyon field passes through the singularity, continuing its evolution with a recollapse towards a Big Crunch.",
"In a simpler model, based on the anti-Chaplygin gas, such a crossing is even easier to understand.",
"The next natural step in the analysis of the soft singularities seems to be obvious.",
"One can consider a soft singularity of more general type than the big brake by adding to the tachyon matter or to the anti-Chaplygin gas some dustlike matter.",
"However, in this case the traversability of the singularity seems to be obstructed.",
"The main reason for this is that while the energy density of the tachyonic field (or of the anti-Chaplygin gas) vanishes at the singularity, the energy density of the matter component does not, leaving the Hubble parameter at the singularity with a finite value.",
"Then some kind of the paradox arises: if the universe continues its expansion, and if the equation of state of the component of matter, responsible for the appearance of the soft singularity (in the simplest case, the anti-Chaplygin gas) is unchanged, then the expression for the energy density of this component becomes imaginary, which is unacceptable.",
"The situation looks rather strange: indeed, the model, including dust should be in some sense more regular, than a single exotic fluid, the anti-Chaplygin gas.",
"Thus, if the model based on the pure anti-Chaplygin gas has a traversable Big Brake singularity, than the more general singularity arising in the model based on the mixture of the anti-Chaplygin gas and dust should also be traversable.",
"Related to that, it was recently shown that general soft singularities arising in the Friedmann model, filled with the scalar field with a negative potential, inversely proportional to this field are traversable.",
"So, what could be wrong with the simple two-fluid model?",
"One can see that what we face is some sort of a clash between the equation of state of one of these fluids and the dynamics (the Friedmann and Raychaudhuri equations) and energy conservation equations.",
"In this paper we shall try to resolve this paradox, insisting on the preservation of the equation of state of the anti-Chaplygin gas.",
"The price which one has to pay for it is the obligatory use of the generalized functions for some cosmological quantities.",
"Namely, the anti-Chaplygin gas remains physical if rather a recollapse follows, but then the Hubble parameter would have a sharp jump, obstructing the validity of the Raychaudhuri equation (the second Friedmann equation) in the usual sense of functions.",
"Thus, apparently, the evolution cannot be continued through the soft singularity, unless treating the cosmological quantities as distributions.",
"We claim that such a generalization is mathematically rigorous, moreover, the introduction of distributions is not so drastic, as it looks at the first sight, as the pressure of the anti-Chaplygin gas diverges anyhow at the soft singularity (as it so does for the tachyon field).",
"Then in the Conclusion we shall dwell on the possible physical sense of the proposed constructions and its possible alternatives.",
"The plan of the paper is as follows.",
"In Section we discuss generic Friedmann space-times, which admit $\\dot{H}=-\\infty $ type singularities, while the Hubble parameter $H$ remains finite.",
"Such singularities are related to corresponding divergencies in the pressure of the perfect fluid filling the Friedmann universe (while its energy density stays finite).",
"We investigate the kinematics, the geodesic equations, the geodesic deviation equations in the vicinity of these singularities and also prove that these singularities are weak.",
"In Section we discuss a mixture of the anti-Chaplygin gas and dust in a flat Friedmann universe and explain the essence of the paradox.",
"We explicitly derive the behavior of the energy density and pressure in the vicinity of the soft singularity and we solve the geodesic equations in this region.",
"While the singularity turns to be traversable by the geodesics, the explicit solution also shows that the Raychaudhuri equation is violated at the singularity.",
"In Section we add generalized distributional contributions to both the pressure and energy density, such that (a) the equation of state of the anti-Chaplygin gas still holds and (b) the singularity becomes traversable.",
"We also perform checks of the Friedmann, Raychaudhuri and continuity equations, which all hold valid across the singularity in a distributional sense.",
"In the process we employ a number of Propositions on distributions presented and proved in Appendix A.",
"For the convenience of the reader we present a related semi-heuristic discussion of two known distributional identities in Appendix B.",
"We stress that the distributional modifications of the energy density and pressure do not modify the cosmological evolution, but they make possible the soft singularity crossing.",
"In Section we revisit the junction conditions along a spacelike hypersurface in a flat Friedmann universe.",
"The future soft singularity represents such a spatial hypersurface, along which the energy-momentum tensor diverges.",
"Extending the space-time through this hypersurface is possible by obeying both Israel junction conditions.",
"While the first condition, requiring the continuity of the induced metric is easy to satisfy (the metric stays regular at the soft singularity), the second condition relates the jump in the extrinsic curvature to the distributional part of the energy-momentum tensor through the Lanczos equation.",
"We will show that in flat Friedmann space-times the Lanczos equation holds for a more general class of distributional energy-momentum tensors.",
"With this we give a second proof that the generalized distributional energy-momentum tensor assures the traversability of the soft singularity.",
"In the process we employ a simple form of the Lanczos equation valid in flat Friedmann universes, derived in Appendix .",
"We summarize our results and give some further outlook in the Concluding Remarks.",
"We chose $c=1$ and $8\\pi G/3=1$ .",
"A subscript $S$ denotes the value of the respective quantity at the soft singularity."
],
[
"Pressure singularities in flat Friedmann universes",
"The line element squared of a flat Friedmann universe can be written as $ds^{2}=-dt^{2}+a^{2}\\left( t\\right) \\sum _{\\alpha }\\left( dx^{\\alpha }\\right)^{2}\\ ,$ where $x^{\\alpha }$ ($\\alpha =1,2,3$ ) are Cartesian coordinates.",
"The evolution of the Friedmann universe is governed by the Raychaudhuri (second Friedmann) equation $\\dot{H}=-\\frac{3}{2}\\left( \\rho +p\\right) \\ , $ and by the continuity equation for the fluid $\\dot{\\rho }+3H\\left( \\rho +p\\right) =0\\ .",
"$ Here the dot denotes the derivative with respect to cosmological time $t$ .",
"A first integral of this system is given by the first Friedmann equation $H^{2}=\\rho \\ .",
"$ It is easy to see that the Raychaudhuri equation can be obtained from the first Friedmann and the continuity equations."
],
[
"Kinematics in the vicinity of sudden singularities",
"Sudden singularities are characterized by finite $H$ and $\\dot{H}\\rightarrow -\\infty $ (finite $\\dot{a}$ and $\\ddot{a}\\rightarrow -\\infty $ ) at some finite scale factor $a$ .",
"The energy density of the fluid is finite but its pressure diverges at this type of singularity, therefore the term “pressure singularity” is also in use.",
"Then, we would like to emphasize the fact that there is an essential difference between the sudden singularities with $H=0$ and with $H>0$ .",
"As has been already mentioned in the Introduction, in the first case, which is called Big Brake, the universe begin contracting and running towards the Big Crunch singularity.",
"Exactly this occurs in models based on tachyon field with a particular potential [32], [22], [29] or in the anti-Chaplygin gas models.",
"In the case of the model based on the mixture of one of this fluids and dust, we encounter the second situation when the value of the Hubble constant is positive at the moment of encounter with the sudden singularity.",
"That means that after crossing the singularity the universe should continue its expansion, but the anti-Chaplygin gas becomes ill-defined, as it will be shown in detail in Section , devoted to the model based on mixture of the anti-Chaplygin gas and dust.",
"One possible way of overcoming this obstacle is to allow the jump in the sign of the Hubble parameter, which as was mentioned in the Introduction leaves valid the first Friedmann equation, the continuity equation and the equation of state, while making invalid the Raychaudhuri equation.",
"This last obstacle can be cured by the acceptance the distributional Dirac $\\delta $ -function type contributions into the pressure and the energy density, which is described in detail in the section IV."
],
[
"Geodesics in the vicinity of sudden singularities",
"The geodesic equations in flat Friedmann space-time are $\\frac{d^{2}x^{\\alpha }}{d\\lambda ^{2}}+2\\frac{\\dot{a}}{a}\\frac{dt}{d\\lambda }\\frac{dx^{\\alpha }}{d\\lambda }=0\\ ,$ $\\frac{d^{2}t}{d\\lambda ^{2}}+a\\dot{a}\\sum _{\\alpha }\\left( \\frac{dx^{\\alpha }}{d\\lambda }\\right) ^{2}=0\\ ,$ where $\\lambda $ is an affine parameter.",
"Integrating these equations yields $\\frac{dx^{\\alpha }}{d\\lambda }=\\frac{P^{\\alpha }}{a^{2}}\\ ,$ $\\left( \\frac{dt}{d\\lambda }\\right) ^{2}=\\epsilon +\\frac{P^{2}}{a^{2}}\\ ,$ with $P^{\\alpha }$ , $\\epsilon $ integration constants and $P^{2}=\\sum _{\\alpha }\\left( P^{\\alpha }\\right) ^{2}$ .",
"The quantity $\\epsilon $ is fixed by the length of the tangent vector $u^{a}$ of the geodesic as $\\epsilon =-u_{a}u^{a}$ ; i.e.",
"one for timelike and zero for lightlike orbits.",
"In a comoving system $P^{\\alpha }=0$ and $t=\\lambda $ is affine parameter.",
"Eqs.",
"(REF ) and (REF ) are singular only for vanishing scale factor (see also Ref.",
"[26]).",
"Therefore, the existence of a solution $t\\left( \\lambda \\right) $ , $x^{\\alpha }\\left(\\lambda \\right) $ of Eqs.",
"(REF ) and (REF ) is assured by the Cauchy-Peano theorem for any nonzero $a$ (including the soft singularity).",
"Thus the functions $t\\left( \\lambda \\right) $ and $x^{\\alpha }\\left( \\lambda \\right) $ , i.e.",
"the geodesics can be continued through the singularity occurring at finite scale factor.",
"Only derivatives of higher order than two of $t\\left( \\lambda \\right) $ and $x^{\\alpha }\\left( \\lambda \\right) $ are singular (as they contain $\\ddot{a}$ ), however these do not appear in the geodesic equations.",
"Pointlike particles moving on geodesics do not experience any singularity.",
"Thus, as we argued in the preceding paper [29] one is not obliged to consider such a singularity as a final state of the universe.",
"Indeed, passing through this singularity the matter recreates also the spacetime in a unique way, at least for such simple models, as those based on Friedmann metrics."
],
[
"Deviation equation in the vicinity of sudden singularities",
"The 3-spaces with $t=$ const have vanishing Riemann curvature.",
"However, the 4-dimensional Riemann curvature tensor has the nonvanishing components: $R_{\\,\\,\\,\\,\\,t\\beta t}^{\\alpha } &=&-\\frac{\\ddot{a}}{a}\\delta _{\\beta }^{\\alpha }=\\left( -\\dot{H}+H^{2}\\right) \\delta _{\\beta }^{\\alpha }\\ ,\\\\R_{\\,\\,\\,\\,\\,212}^{1} &=&R_{\\,\\,\\,\\,\\,313}^{1}=R_{\\,\\,\\,\\,\\,323}^{2}=\\dot{a}^{2} $ and the corresponding components arising from symmetry.",
"Here $\\alpha ,\\beta =1,2,3$ .",
"Remarkably, all components which diverge at the singularity are of the type $R_{tata}$ [29].",
"Therefore, the singularity arises in the mixed spatio-temporal components.",
"The geodesic deviation equation along the integral curves of $u=\\partial /\\partial t$ (which are geodesics with affine parameter $t$ ) is $\\dot{u}^{a}=-R_{\\ \\ cbd}^{a}\\eta ^{b}u^{c}u^{d}\\ ,$ where $\\eta ^{b}$ is the deviation vector separating neighboring geodesics, chosen to satisfy $\\eta ^{b}u_{b}=0$ .",
"For a Friedmann universe it becomes $\\dot{u}^{a}=-R_{\\ \\ tbt}^{a}\\eta ^{b}\\propto \\ddot{a}\\ ,$ which at the singularity diverges as $-\\infty $ .",
"Therefore, when approaching the singularity, the tidal forces manifest themselves as an infinite braking force stopping the further increase of the separation of geodesics, but not the evolution along the geodesics.",
"With $\\ddot{a}<0$ in the vicinity of the singularity, once the geodesics have passed through, they will approach each other.",
"Therefore a contraction phase will follow: everything that has reached the singularity will bounce back."
],
[
"The type of the singularity",
"In this subsection we shall present the classification of singularities, based on the point of view of finite size objects, which approach these singularities.",
"In principle, finite size objects could be destroyed while passing through the singularity due to the occurring infinite tidal forces.",
"A strong curvature singularity is defined by the requirement that an extended finite object is crushed to zero volume by tidal forces.",
"We give below Tipler's [30] and Królak's [36] definitions of strong curvature singularities together with the relative necessary and sufficient conditions.",
"An alternative definition of the softness of a singularity, based on a Raychaudhuri averaging, was developed by Dabrowski [37].",
"According to Tipler's definition if every volume element defined by three linearly independent, vorticity-free, geodesic deviation vectors along every causal geodesic through a point $p$ vanishes, a strong curvature singularity is encountered at the respective point $p$ [30], [26].",
"The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\lambda _{s}$ ($\\lambda $ is affine parameter of the curve) [38] is that the double integral $\\int _{0}^{\\lambda }d\\lambda ^{\\prime }\\int _{0}^{\\lambda ^{\\prime }}d\\lambda ^{\\prime \\prime }\\left|R_{\\,\\,ajb}^{i}u^{a}u^{b}\\right|$ diverges as $\\lambda \\rightarrow \\lambda _{s}$ .",
"A similar condition is valid for lightlike geodesics, with $R_{\\,\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\,ab}u^{a}u^{b}$ in the double integral.",
"Królak's definition is less restrictive.",
"A future-endless, future-incomplete null (timelike) geodesic $\\gamma $ is said to terminate in the future at a strong curvature singularity if, for each point $p\\in \\gamma $ , the expansion of every future-directed congruence of null (timelike) geodesics emanating from $p$ and containing $\\gamma $ becomes negative somewhere on $\\gamma $ [36], [39].",
"The necessary and sufficient condition for a causal geodesic to run into a strong singularity at $\\lambda _{s}$ [38] is that the integral $\\int _{0}^{\\lambda }d\\lambda ^{\\prime }\\left|R_{\\,\\,ajb}^{i}u^{a}u^{b}\\right|$ diverges as $\\lambda \\rightarrow \\lambda _{s}$ .",
"Again, a similar condition is valid for lightlike geodesics, with $R_{\\,\\,ajb}^{i}u^{a}u^{b}$ replacing $R_{\\,ab}u^{a}u^{b}$ in the integral.",
"In flat Friedmann space-time the comoving observers move on geodesics having four velocity $u=\\partial /\\partial t$ , where $t$ is affine parameter.",
"The nonvanishing components of Riemann tensor are given by Eq.",
"(REF ).",
"Since $H$ is finite along the geodesics, neither of the integrals (REF ) and (REF ) diverge at the singularity.",
"The singularity is weak (soft) according to both Tipler's and Królak's definitions.",
"That means although the tidal forces become infinite, the finite objects are not necessarily crushed when reaching the singularity (see also [26])."
],
[
"The paradox of the soft singularity crossing in the cosmological\nmodel based on the anti-Chaplygin gas and dust universe",
"We discuss an universe filled with two components.",
"One is the anti-Chaplygin gas with the equation of state (REF ) and other is the pressureless dust.",
"The solution of the continuity equation for the anti-Chaplygin gas gives the following dependence of its energy density on the scale factor: $\\rho _{ACh}=\\sqrt{\\frac{B}{a^{6}}-A}\\ , $ where $B$ is a positive constant, determining the initial condition.",
"The energy density of the dust-like matter is as usual $\\rho _{m}=\\frac{\\rho _{m,0}}{a^{3}}\\ ,~ $ where $\\rho _{m,0}$ is a constant.",
"It is clear that when during the expansion of the universe, its scale factor approaches the value $a_{S}=\\left( \\frac{B}{A}\\right) ^{\\frac{1}{6}}\\ , $ the energy density of the anti-Chaplygin gas vanishes, and its pressure grows to infinity.",
"That means that the deceleration also becomes infinite.",
"However, the energy density of dust remains finite, hence the same is true also for the Hubble parameter.",
"It is here that the paradox arises: if the universe continues to expand, the expression under the sign of the square root in Eq.",
"(REF ) becomes negative and the energy density of the anti-Chaplygin gas becomes ill-defined.",
"A way out of this situation is only by assuming that at this moment the Hubble parameter changes its sign, while keeping its absolute value (such that the energy density will not have a jump, as implied by the Friedmann equation).",
"This possibility will be studied in detail in the following subsections."
],
[
"Evolutions in the vicinity of the singularity",
"Let us substitute the expressions (REF ) and (REF ) into the first Friedmann equation.",
"We shall find its solution for the universe approaching to the soft singularity point at the moment $t_{S}$ (the latter cannot be found analytically, but its value is not important for our analysis): $a(t)=a_{S}-\\sqrt{\\frac{\\rho _{m,0}}{a_{S}}}(t_{S}-t)-\\sqrt{\\frac{2Aa_{S}^{2}}{3H_{S}}}(t_{S}-t)^{3/2}\\ , $ where $H_{S}=\\sqrt{\\frac{\\rho _{m,0}}{a_{S}^{3}}} $ is the value of the Hubble parameter at $t=t_{S}$ .",
"Correspondingly the leading terms of the energy densities of the anti-Chaplygin gas and dust, also of the pressure of the anti-Chaplygin gas are $\\rho _{m}=H_{S}^{2}+3H_{S}^{3}(t_{S}-t)\\ , $ $\\rho _{ACh}=\\sqrt{6AH_{S}(t_{S}-t)}\\ , $ $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}(t_{S}-t)}}\\ .",
"$ One can see that the expressions (REF ), (REF ) and (REF ) cannot be continued for $t>t_{S}$ due to the emerging negative quantities under the square roots.",
"The assumption of a sharp transition from expansion to contraction implies the following changes in Eqs.",
"(REF ) – (REF ): $a(t)=a_{S}-\\sqrt{\\frac{\\rho _{m,0}}{a_{S}}}|t_{S}-t|-\\sqrt{\\frac{2Aa_{S}^{2}}{3H_{S}}}|t_{S}-t|^{3/2}\\ , $ $\\rho _{m}=H_{S}^{2}+3H_{S}^{3}|t_{S}-t|\\ , $ $\\rho _{ACh}=\\sqrt{6AH_{S}|t_{S}-t|}\\ , $ $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}\\ .",
"$ The quantities (REF )–(REF ) are well-defined and continuous at the moment of the singularity crossing.",
"The expression for the pressure (REF ) is divergent, but this divergence is integrable and this is sufficient for our purposes.",
"These new expressions satisfy the Friedmann equation, the continuity equations and the equation of state for the anti-Chaplygin gas.",
"However, the time derivatives of these quantities are not continuous and it is the reason of the failure of the Raychaudhuri equation.",
"We shall analyze this problem in the following section, but before we discuss the geodesics in the vicinity of the singularity."
],
[
"Singularity crossing geodesics",
"We can integrate explicitly the geodesics equations (REF ) and (REF ) in the vicinity of singularity, using the expression (REF ) for the cosmological factor, also taken in the vicinity of singularity.",
"Choosing the affine parameter in such a way that the point $\\lambda =0$ corresponds to the singularity crossing we obtain up to the second order in $\\lambda $ terms $t=t_{S}+\\sqrt{\\epsilon +\\frac{P^{2}}{a_{S}^{2}}}\\lambda +\\frac{P^{2}H_{S}}{2a_{S}^{2}}sgn(\\lambda )\\lambda ^{2}\\ , $ $x^{\\alpha }=x_{S}^{\\alpha }+\\frac{P^{\\alpha }}{a_{S}^{2}}\\lambda +\\sqrt{\\epsilon +\\frac{P^{2}}{a_{S}^{2}}}\\frac{P^{\\alpha }H_{S}}{a_{S}^{2}}sgn(\\lambda )\\lambda ^{2}\\ .",
"$ One can see from Eqs.",
"(REF ) and (REF ) that not only the time and spatial coordinates of the geodesics are continuous at the soft singularity crossing, but also their first derivatives with respect to the affine parameter $\\lambda $ ."
],
[
"Singularity crossing, the Raychaudhuri equation and distributions\n",
"Let us discuss the expressions for the Hubble parameter and its time derivative in the vicinity of the singularity.",
"Starting from the expression (REF ) we obtain $H(t) &=&H_{S}sgn(t_{S}-t) \\\\&&+\\sqrt{\\frac{3A}{2H_{S}a_{S}^{4}}}sgn(t_{S}-t)\\sqrt{|t_{S}-t|}~,$ $\\dot{H}=-2H_{S}\\delta (t_{S}-t)-\\sqrt{\\frac{3A}{8H_{S}a_{S}^{4}}}\\frac{sgn(t_{S}-t)}{\\sqrt{|t_{S}-t|}}~.",
"$ Naturally, the $\\delta $ -term in $\\dot{H}$ arises because of the jump in $H$ , as the expansion of the universe is followed by a contraction.",
"To restore the validity of the Raychaudhuri equation we shall add a singular $\\delta $ -term to the pressure of the anti-Chaplygin gas, which will acquire the form $p_{ACh}=\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}+\\frac{4}{3}H_{S}\\delta (t_{S}-t)~.$ The equation of state (REF ) of the anti-Chaplygin gas is preserved, if we also modify the expression for its energy density: $\\rho _{ACh}=\\frac{A}{\\sqrt{\\frac{A}{6H_{S}|t_{S}-t|}}+\\frac{4}{3}H_{S}\\delta (t_{S}-t)}~.",
"$ The last expression should be understood in the sense of the composition of distributions (see Appendix A and the references therein).",
"In order to prove that $p_{ACh}$ and $\\rho _{ACh}$ represent a self-consistent solution of the system of cosmological equations, we shall use the following distributional identities: $\\left[ sgn\\left( \\tau \\right) g\\left( \\left|\\tau \\right|\\right) \\right] \\delta \\left( \\tau \\right) &=&0\\ , \\\\\\left[ f\\left( \\tau \\right) +C\\delta \\left( \\tau \\right) \\right] ^{-1}&=&f^{-1}\\left( \\tau \\right) \\ , \\\\\\frac{d}{d\\tau }\\left[ f\\left( \\tau \\right) +C\\delta \\left( \\tau \\right) \\right] ^{-1} &=&\\frac{d}{d\\tau }f^{-1}\\left( \\tau \\right) \\ .$ Here $g\\left( \\left|\\tau \\right|\\right) $ is bounded on every finite interval, $f\\left( \\tau \\right) >0$ and $C>0$ is a constant.",
"These identities follow from the Propositions 1, 2 and the Corollary enounced and proved in the Appendix A.",
"The parameter $\\tau $ stays instead of the difference $t_{S}-t$ .",
"Because of Eq.",
"(), the energy density (REF ) behaves as a continuous function which vanishes at the singularity.",
"The first term in the expression for the pressure (REF ) diverges at the singularity.",
"Therefore the addition of a Dirac delta term, which is not changing the value of $p_{ACh}$ at any $\\tau \\ne 0$ (i.e.",
"$t\\ne t_{S}$ ) does not look too drastic and might be considered as a some kind of renormalization.",
"To prove that Friedmann, Raychaudhuri and continuity equations are satisfied we must only investigate those terms, appearing in the field equations, which contain Dirac $\\delta $ -functions, since without them, these equations can be reduced to those we have found in the previous section.",
"First, we check the continuity equation for the anti-Chaplygin gas.",
"Due to the identities ()-(), the $\\delta \\left(\\tau \\right) $ -terms occurring in $\\rho _{ACh}$ and $\\dot{\\rho }_{ACh}$ could be dropped.",
"We keep them however in order to have the equation of state explicitly satisfied.",
"Then the $\\delta \\left( \\tau \\right) $ -term appearing in $3Hp_{ACh}$ vanishes, because the Hubble parameter changes sign at the singularity [see Eq.",
"(REF )].",
"The $\\delta \\left( \\tau \\right) $ -term appearing in $\\rho _{ACh}$ does not affect the Friedmann equation due to the identity ().",
"Finally, the $\\delta $ -term arising in the time derivative of the Hubble parameter in the left-hand side of the Raychaudhuri equation is compensated by the conveniently chosen $\\delta $ -term in the right-hand side of Eq.",
"(REF )."
],
[
"The junction conditions across the singularity",
"In this section we discuss the singularity crossing in a slightly different way, by analyzing the junction conditions.",
"We have to match two space-time regions across the space-like hypersurface $\\tau =0$ .",
"The junction of two space-time regions has to obey the Israel matching conditions [40], namely, the induced metric should be continuous and the extrinsic curvature of the junction hypersurface could possibly have a jump, which is related to the distributional energy-momentum tensor on the hypersurface by the Lanczos equation.",
"The scale factor being continuous across the singularity the first Israel condition is obeyed.",
"We will next prove that the second Israel junction condition (the Lanczos equation [41], [40]) is also satisfied.",
"For this we have to check whether Eq.",
"$\\frac{\\partial }{\\partial t}\\left( Ha^{2}\\right) =\\left\\lbrace -H^{2}+\\frac{3}{2}\\left[ \\widetilde{\\rho }-\\widetilde{p}-\\overline{p}\\delta \\left( \\tau \\right) \\right] \\right\\rbrace a^{2}\\ , $ (see Appendix ), still implies the Lanczos equation $\\Delta H|_{t_{s}}=-\\frac{3}{2}\\overline{p}\\ , $ derived in Appendix , when $\\widetilde{p}+\\overline{p}\\delta \\left( \\tau \\right) =p_{ACh}$ , $\\widetilde{\\rho }=\\rho _{m}+\\rho _{ACh}$ and $\\rho _{ACh}$ is generalized to a distribution $\\rho _{ACh}a^{2}=\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}~.$ Here $\\omega >0$ , $R\\left( \\tau \\right) >0$ , $Q\\left( \\tau \\right) >0$ and $P\\left( \\tau \\right) $ is bounded.",
"When Eq.",
"(REF ) is applied to a test function $\\varphi \\left( \\tau \\right) $ , the terms containing $H^{2}$ and $\\rho _{m}$ give regular contributions and the limits of the respective integrals vanish, similarly as discussed in Appendix .",
"Also, due to Proposition 2 given in the Appendix A, the integral of the distributional term containing $\\rho _{ACh}$ becomes $\\int _{-\\varepsilon }^{\\varepsilon }\\frac{P\\left( \\tau \\right) \\varphi \\left(\\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }d\\tau ~,$ which also vanishes for $\\varepsilon \\rightarrow 0$ .",
"We still have to consider the contributions $\\int _{-\\varepsilon }^{\\varepsilon }\\left[ \\widetilde{p}+\\overline{p}\\delta \\left( \\tau \\right) \\right] \\varphi \\left( \\tau \\right) a^{2}d\\tau \\ .$ Although the contribution $\\widetilde{p}\\varphi \\left( \\tau \\right) a^{2}$ to the integrand is singular at $\\tau =0$ , its integral can be conveniently evaluated by the Residue Theorem.",
"For this we remark, that the integrand is an analytically extendible function into the complex plane in the vicinity of $\\tau =0$ and its residue is zero, therefore the integral vanishes.",
"Finally, the contribution containing $\\overline{p}$ leads by integration and the limiting process to the right hand side of the Lanczos equation (REF ).",
"Therefore we have proven that the space-time regions separated by the singular spatial hypersurface, representing the pressure singularity, can be joined.",
"In other words, the singularity becomes traversable."
],
[
"Concluding Remarks",
"It is known that certain models of cosmological fluids in Friedmann universes, like the anti-Chaplygin gas or the tachyon field with a special potential [32], evolve into a sudden future singularity, which in spite of a diverging pressure, is weak.",
"It was argued that singularities of this kind could be traversable despite infinite tidal forces emerging at the singularity for an infinitesimally short time [26].",
"In Ref.",
"[29] the process of crossing of the Big Brake singularity was described in some detail for the tachyon model [32].",
"(The particularity of the Big Brake singularity, consists in the fact that at the crossing of such a singularity the Hubble variable is not only finite, but vanishes.)",
"We also note recent discussions [42] on crossing the “traditional” Big Bang and Big Crunch singularities.",
"In the present paper we considered a simple cosmological model containing a mixture of anti-Chaplygin gas and dust.",
"We have shown that the geodesics equations and their solutions are still well-defined in this case, however the inclusion of dust generates a nonzero value of the Hubble parameter at the singularity encounter, generating the following paradox.",
"The dust would require a continued expansion, which would make the energy density and pressure of the anti-Chaplygin gas ill-defined.",
"A contraction in turn, would be compatible with the anti-Chaplygin gas, nevertheless implying an abrupt change of the Hubble parameter from expansion to contraction.",
"The jump in the Hubble parameter implies the appearance of the $\\delta $ function in the Raychaudhuri equation (which contains $\\dot{H}$ ).",
"We have cured this situation by redefining the pressure and energy density of the anti-Chaplygin gas as distributions.",
"As an equivalent interpretation, the pressure can be generalized by the addition of a distributional contribution, while the energy density left unchanged, at the price of redefining the equation of state of the anti-Chaplygin gas in a distributional sense.",
"Then all cosmological equations are satisfied in the same distributional sense.",
"We have also shown, that the Israel junction conditions are obeyed through the singular spatial hypersurface, in particular we have enounced and proved a more general form of the Lanczos equation.",
"The results rely on two Propositions and a Corollary proven in Appendix A.",
"The resolution of the paradox at the soft singularity crossing by the introduction of distributional quantities and equations may look unusual, however distributional quantities, localized on hypersurfaces are quite commonly used in general relativity and other gravitational theories.",
"Spacetime regions are frequently matched by the inclusion of distributional layers; also shock-waves can be modeled by Dirac $\\delta $ -functions.",
"Braneworld models [44], [43] arise due to the orbifold boundary conditions, the non-smoothness of the 5-dimensional metric at the brane (the jump in its extrinsic curvature) being directly related to the distributional 3+1 standard model fields embedded in the 5-dimensional spacetime.",
"Besides, metrics allowing distributional curvature were considered earlier for studying strings and other distributional sources in general relativity [45].",
"The applications of the distributional quantities to the study of Schwarzschild geometry and point massive particles in general relativity were used in Refs.",
"[46] and [47] respectively.",
"More generically the connection between singularities and the distributional treatment of the physical quantities is well-known in quantum field theory.",
"Indeed, the appearance of the ultraviolet divergences can be understood as the result of the indefiniteness of the product of distributions and the renormalization procedure could be interpreted as a definition of such a product [48].",
"We hope that the investigations presented here may turn useful in deriving similar results in connection with the traversability of other types of sudden singularities.",
"While mathematically self-consistent, the scenario presented in this paper may look somewhat counter-intuitive from the physical point of view.",
"This is because its essential ingredient is the abrupt change of the expansion into a contraction.",
"However, such a behavior is not more counter-intuitive that the absolutely elastic bounce of the ball from a rigid wall, as known in classical mechanics.",
"Indeed, in the latter case the velocity and the momentum of the ball change their direction abruptly.",
"That means that an infinite force acts from the wall onto the ball during an infinitely small interval of time.",
"The result of this action is however integrable and results in a finite change of the momentum of the ball.",
"In fact, the absolutely elastic bounce is an idealization of a process of finite time-span during which inelastic deformations of both the ball and the wall are likely.",
"It is reasonable to think that something similar occurs also in the two-fluid universe model presented in this paper, which undergoes a transition from an expanding to a contracting phase.",
"The smoothing of this process should involve some (temporary) geometrically induced change of the equation of state of matter.",
"Note, that such changes are not uncommon in cosmology.",
"In the tachyon model [32] which was starting point of our studies of the Big Brake singularities, there was the tachyon -pseudotachyon transformation driven by the continuity of the cosmological evolution.",
"In a cosmological model with the phantom field with the cusped potential [49], the transformations between phantom and standard scalar field were considered.",
"Thus, one can imagine that the real process of the transition from the expansion to contraction induced by passing through a soft singularity can imply some temporary change of the equation of state which makes the above processes smoother.",
"We hope to explore such a scenario in the future."
],
[
"ACKNOWLEDGMENTS",
"We are grateful to V. Gorini, M. O. Katanaev, V. N. Lukash, U. Moschella, D. Polarski and A.",
"A. Starobinsky for useful discussions.",
"The work of ZK was supported by OTKA grant no.",
"100216 and AK was partially supported by the RFBR grant no.",
"11-02-00643."
],
[
"Propositions on the product and the composition of distributions\n",
"To investigate how the Friedmann universe crosses a soft singularity, we must solve the field equation in distributional sense.",
"For this purpose we give the definitions of the product and of the composition of distributions and prove two propositions.",
"Fisher derived the following result: $\\left[sgn\\left( \\tau \\right) \\left|\\tau \\right|^{\\lambda }\\right] \\delta \\left( \\tau \\right) =0$ for $\\lambda >-1$ [50].",
"Our first proposition generalizes this equation for $\\lambda \\ge 0$ .",
"The second proposition generalizes Antosik's result: $\\left[ 1+\\delta \\left( \\tau \\right) \\right] ^{-1}=1$ [51].",
"Finally, we show a corollary.",
"Let $\\rho \\left( \\tau \\right) $ be any infinitely differentiable function having the following properties: $i)$ $\\rho \\left( \\tau \\right) =0$ for $\\left|\\tau \\right|\\ge 1$ ; $ii)$ $\\rho \\left( \\tau \\right) \\ge 0$ ; $iii)$ $\\rho \\left( \\tau \\right) =\\rho \\left( -\\tau \\right) $ ; $iv)$ $\\int _{-1}^{1}\\rho \\left( \\tau \\right) d\\tau =1$ .",
"Then $\\delta _{n}\\left(\\tau \\right) =n\\rho \\left( n\\tau \\right) $ (with $n=1$ , 2, ...) is a regular sequence of infinitely differentiable functions converging to Dirac delta function: $\\lim _{n\\rightarrow \\infty }\\left\\langle \\delta _{n},\\varphi \\right\\rangle =\\left\\langle \\delta ,\\varphi \\right\\rangle $ for any $\\varphi \\in \\mathcal {D}$ [52].",
"Here $\\mathcal {D}$ denotes the space of test functions having continuous derivatives of all orders and compact support.",
"The action of an $f\\in \\mathcal {D}^{\\prime }$ distribution on test functions $\\varphi $ is given by $\\left\\langle f,\\varphi \\right\\rangle $ , which in the case when $f$ is an ordinary locally summable function is nothing but $\\int _{-\\infty }^{\\infty }f\\left( \\tau \\right) \\varphi \\left(\\tau \\right) d\\tau $ .",
"We note that $\\delta _{n}\\left( \\tau \\right) $ has the compact support: $\\left( -1/n,1/n\\right) $ .",
"We will also use the $n$ -th derivative of $f\\in \\mathcal {D}^{\\prime }$ acts as$\\ \\left\\langle df\\left(\\tau \\right) /d\\tau ^{n},\\varphi \\right\\rangle =\\left( -1\\right)^{n}\\left\\langle f\\left( \\tau \\right) ,d\\varphi /d\\tau ^{n}\\right\\rangle $ .",
"For an arbitrary distribution $f$ , the function $f_{n}\\left( \\tau \\right)=f\\ast \\delta _{n}\\equiv \\left\\langle f\\left( \\tau -x\\right) ,\\delta _{n}\\left( x\\right) \\right\\rangle $ gives a sequence of infinitely differentiable functions converging to $f$ .",
"Definition 1 : The commutative product of $f$ and $g$ exists and is equal to $h$ on the open interval $\\left( a,b\\right) $ ($-\\infty \\le a<b\\le \\infty $ ) if $\\lim _{n\\rightarrow \\infty }\\left\\langle f_{n}g_{n},\\varphi \\right\\rangle =\\left\\langle h,\\varphi \\right\\rangle $ for any $\\varphi \\in \\mathcal {D}$ with support contained in the interval $\\left( a,b\\right) $ [52] This definition can be generalized for the cases when the usual limit does not exist by taking the so-called neutrix limit [52], [53].",
"However, we do not need for this more general definition here..",
"Proposition 1 : The commutative product of $sgn\\left( \\tau \\right) g\\left( \\left|\\tau \\right|\\right) $ and $\\delta \\left( \\tau \\right) $ exists and $\\left[ sgn\\left( \\tau \\right) g\\left( \\left|\\tau \\right|\\right) \\right] \\delta \\left( \\tau \\right) =0$ for arbitrary $g\\left( \\left|\\tau \\right|\\right) $ bounded on every finite interval.",
"We would like to show that $\\left\\langle \\left[ sgn\\left( \\tau \\right)g\\left( \\left|\\tau \\right|\\right) \\right] \\delta \\left( \\tau \\right) ,\\varphi \\right\\rangle =0$ .",
"Using the mean value theorem $\\varphi \\left( \\tau \\right) =\\varphi \\left( 0\\right) +\\tau d\\varphi \\left( \\xi \\tau \\right) /d\\tau $ with $0\\le \\xi \\le 1$ , we have $&&\\left|\\left\\langle \\left[ sgn\\left( \\tau \\right) g\\left( \\left|\\tau \\right|\\right) \\right] _{n}\\delta _{n}\\left( \\tau \\right) ,\\varphi \\right\\rangle \\right|\\\\&\\le &\\left|\\varphi \\left( 0\\right) \\int _{-1/n}^{1/n}\\left[ sgn\\left(\\tau \\right) g\\left( \\left|\\tau \\right|\\right) \\right] _{n}\\delta _{n}\\left( \\tau \\right) d\\tau \\right|\\\\&&+\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|\\frac{d\\varphi \\left( \\tau \\right) }{d\\tau }\\right|\\\\&&\\times \\int _{-1/n}^{1/n}\\left|\\tau \\left[ sgn\\left( \\tau \\right)g\\left( \\left|\\tau \\right|\\right) \\right] _{n}\\delta _{n}\\left(\\tau \\right) \\right|d\\tau \\ .$ The first integral on the right side of the above equation vanishes because the integrand is an odd function.",
"For the second integrand, we have $&&\\int _{-1/n}^{1/n}\\left|\\tau \\left[ sgn\\left( \\tau \\right) g\\left(\\left|\\tau \\right|\\right) \\right] _{n}\\delta _{n}\\left( \\tau \\right) \\right|d\\tau \\\\&=&\\int _{-1/n}^{1/n}\\left|\\tau \\delta _{n}\\left( \\tau \\right)\\right|\\int _{-1/n}^{1/n}\\left|g\\left( \\left|\\tau -x\\right|\\right) \\right|\\delta _{n}\\left( x\\right) dxd\\tau \\\\&\\le &n\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|\\rho \\left(\\tau \\right) \\right|\\int _{-1/n}^{1/n}\\left|\\tau \\delta _{n}\\left(\\tau \\right) \\right|\\int _{-1/n}^{1/n}\\left|g\\left( \\left|\\tau -x\\right|\\right) \\right|dxd\\tau \\\\&\\le &2\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|\\rho \\left(\\tau \\right) \\right|\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|g\\left( \\left|\\tau \\right|\\right) \\right|\\int _{-1/n}^{1/n}\\left|\\tau \\delta _{n}\\left( \\tau \\right) \\right|d\\tau \\\\&=&\\frac{2}{n}\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|\\rho \\left( \\tau \\right) \\right|\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|g\\left( \\left|\\tau \\right|\\right) \\right|\\int _{-1}^{1}\\left|y\\rho \\left( y\\right) \\right|dy \\\\&\\le &\\frac{2}{n}\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|\\rho \\left( \\tau \\right) \\right|\\sup _{\\left|\\tau \\right|\\le 1/n}\\left|g\\left( \\left|\\tau \\right|\\right) \\right|\\ ,$ that vanishes for $n\\rightarrow \\infty $ .",
"Definition 2 : The composition $F\\left( f\\right) $ of distributions $F$ and $f$ exists and is equal to $h\\in \\mathcal {D}^{\\prime }$ on the interval $\\left(a,b\\right) $ if $\\lim _{n\\rightarrow \\infty }\\left[ \\lim _{m\\rightarrow \\infty }\\int _{a}^{b}F_{n}\\left( f_{m}\\left( \\tau \\right) \\right) \\ \\varphi \\left(\\tau \\right) d\\tau \\right] =\\left\\langle h,\\varphi \\right\\rangle $ for all $\\varphi \\in \\mathcal {D}$ with support contained in the interval $\\left( a,b\\right) $ This definition can be generalized for the cases when the usual limit does not exist by taking double neutrix limit [54], [55], [56]..",
"Proposition 2 : The composition of distribution $P\\left( \\tau \\right) \\left[ R\\left( \\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{-\\omega }$ (where $\\omega >0$ , $P\\left( \\tau \\right) $ is bounded, $R\\left( \\tau \\right) \\ne 0$ , and in some range close $\\tau =0$ the signs of $R\\left(\\tau \\right) $ and $Q\\left( \\tau \\right) $ are the same if $Q\\left( \\tau \\right) \\ne 0$ ) exists if $P\\left( \\tau \\right) /R^{\\omega }\\left( \\tau \\right) $ existsWe note that this proposition can be held even if $P\\left( \\tau \\right) =1$ and $R\\left( \\tau \\right) =\\delta \\left( \\tau \\right) $ with $\\omega =1,$ $2,$ $...$ .",
"Indeed, $\\delta ^{-\\omega }\\left( \\tau \\right) $ exists in neutrix limit and $\\delta ^{-\\omega }\\left( \\tau \\right) =0$ [55].",
"Thus in the definition 2, the usual limit must be changed for neutrix limit for this case.",
"and $\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}=\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }\\ .$ By the definition of composition of distributions, we should calculate $&&\\left\\langle \\frac{P_{n}\\left( \\tau \\right) }{\\left[ R_{m}\\left( \\tau \\right) +Q_{m}\\left( \\tau \\right) \\delta _{m}\\left( \\tau \\right) \\right]_{n}^{\\omega }},\\varphi \\left( \\tau \\right) \\right\\rangle \\\\&=&\\int _{-\\infty }^{\\infty }\\int _{-1/n}^{1/n}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( x\\right) dxd\\tau }{\\left[R_{m}\\left( \\tau -x\\right) +Q_{m}\\left( \\tau \\right) \\delta _{m}\\left( \\tau -x\\right) \\right] ^{\\omega }}\\ .$ Performing a change of the variables as $\\tau =\\tau $ , $y=m\\left( \\tau -x\\right) $ , we have $&=&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{-\\infty }^{\\infty }\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\\\&=&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{1}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{R_{m}^{\\omega }\\left( y/m\\right) }dyd\\tau \\\\&&-\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{2}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\ ,$ where $\\Omega _{2}=\\left\\lbrace y:\\left|y\\right|<1\\text{ and }\\rho \\left( y\\right) \\ne 0\\right\\rbrace $ and $\\Omega _{1}=\\mathbb {R}-\\Omega _{2}$ .",
"The double limit of the first term is $&&\\lim _{n\\rightarrow \\infty }\\lim _{m\\rightarrow \\rightarrow \\infty }-\\frac{1}{m}\\int _{-\\infty }^{\\infty }d\\tau \\varphi \\left( \\tau \\right) P_{n}\\left(\\tau \\right) \\\\&&\\times \\int _{\\Omega _{1}}\\frac{\\delta _{n}\\left( \\tau -y/m\\right) }{R_{m}^{\\omega }\\left( y/m\\right) }dy \\\\&=&\\lim _{n\\rightarrow \\infty }\\lim _{m\\rightarrow \\rightarrow \\infty }\\int _{-\\infty }^{\\infty }d\\tau \\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\\\&&\\times \\int _{\\begin{array}{c} \\left|x\\right|<1/n, \\\\ m\\left|\\tau -x\\right|\\in \\Omega _{1}\\end{array}}\\frac{\\delta _{n}\\left( x\\right) }{R_{m}^{\\omega }\\left( \\tau -x\\right) }dx \\\\&=&\\left\\langle \\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) },\\varphi \\left( \\tau \\right) \\right\\rangle \\ .$ We investigate the absolute value of the second integral.",
"According to our assumptions for $R$ and $Q$ , and since we are interested in $m\\rightarrow \\infty $ , we can choose $m$ large enough to let the signs of $R$ and $Q$ be the same, then for $\\omega >0$ : $&&\\left|\\frac{1}{m}\\int _{-\\infty }^{\\infty }\\int _{\\Omega _{2}}\\frac{\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\delta _{n}\\left( \\tau -y/m\\right) }{\\left[ R_{m}\\left( y/m\\right) +mQ_{m}\\left( y/m\\right) \\rho \\left( y\\right) \\right] ^{\\omega }}dyd\\tau \\right|\\\\&\\le &\\left|\\frac{1}{m^{1+\\omega }}\\int _{-\\infty }^{\\infty }\\varphi \\left( \\tau \\right) P_{n}\\left( \\tau \\right) \\int _{\\Omega _{2}}\\frac{\\delta _{n}\\left( \\tau -y/m\\right) }{Q_{m}\\left( y/m\\right) \\rho ^{\\omega }\\left(y\\right) }dyd\\tau \\right|\\ .$ Performing a change of the variables as $z=n\\left( \\tau -y/m\\right) $ , $y=y$ , we have $&\\le &\\frac{1}{m^{1+\\omega }}\\int _{-1}^{1}\\int _{\\Omega _{2}}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\frac{\\rho \\left( z\\right) }{\\rho ^{\\omega }\\left( y\\right) }\\right|dydz \\\\&\\le &\\frac{1}{m^{1+\\omega }}\\sup _{\\Omega _{2},\\left|z\\right|\\le 1}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\rho ^{-\\omega }\\left( y\\right) \\right|\\\\&&\\times \\int _{-1}^{1}\\rho \\left( z\\right) dz\\int _{-1}^{1}dy \\\\&=&\\frac{2}{m^{1+\\omega }}\\sup _{\\Omega _{2},\\left|z\\right|\\le 1}\\left|\\varphi \\left( \\frac{z}{n}+\\frac{y}{m}\\right) P_{n}\\left( \\frac{z}{n}+\\frac{y}{m}\\right) \\rho ^{-\\omega }\\left( y\\right) \\right|\\ ,$ that vanishes for $m\\rightarrow \\infty $ if $P$ is bounded.",
"Corollary 1 : The distribution $d\\left\\lbrace P\\left( \\tau \\right) \\left[ R\\left( \\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{-\\omega }\\right\\rbrace /d\\tau $ (with the same properties for $P$ , $R$ , $Q$ and $\\omega $ as in proposition 2) exists if $P\\left( \\tau \\right) /R^{\\omega }\\left( \\tau \\right) $ and its derivative exist, and $\\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }}=\\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) }\\ .$ Applying the derivative of a distribution at tests functions, and using the fact that $d\\varphi /d\\tau \\in \\mathcal {D}$ for any $\\varphi \\in \\mathcal {D}$ , and by the proposition 2, we have $&&\\left\\langle \\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{\\left[ R\\left(\\tau \\right) +Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right]^{\\omega }},\\varphi \\right\\rangle \\\\&=&-\\left\\langle \\frac{P\\left( \\tau \\right) }{\\left[ R\\left( \\tau \\right)+Q\\left( \\tau \\right) \\delta \\left( \\tau \\right) \\right] ^{\\omega }},\\frac{d}{d\\tau }\\varphi \\right\\rangle \\\\&=&-\\left\\langle \\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right)},\\frac{d}{d\\tau }\\varphi \\right\\rangle =\\left\\langle \\frac{d}{d\\tau }\\frac{P\\left( \\tau \\right) }{R^{\\omega }\\left( \\tau \\right) },\\varphi \\right\\rangle \\ .$"
],
[
"Two simple examples of the product and of the decomposition of\ndistributions",
"The definition of the product and the composition of distributions, used in this paper and presented in the Appendix A are not often encountered in physics.",
"Thus, to give the reader some flavor of the corresponding considerations, using simpler means we decided to give two semi-heuristic examples of such products and compositions.",
"We consider first a remarkable formula $\\mathcal {P}\\left( \\frac{1}{x}\\right) \\delta (x)=-\\frac{1}{2}\\delta ^{\\prime }(x)~, $ which was first proven in [57].",
"Here $\\mathcal {P}$ means the principal value of the corresponding function.",
"We shall prove here that the regularizing succession of functions with compact support $\\rho $ , employed in the Appendix A and the references therein, can be chosen alternatively as the family of the Cauchy-Lorentz functions $f_{\\epsilon }(x)=\\frac{1}{\\pi }\\frac{\\epsilon }{x^{2}+\\epsilon ^{2}}~.$ It is well known that when the small parameter $\\epsilon \\rightarrow 0$ , the functions of this family tend in the distributional sense to the Dirac $\\delta $ function.",
"Obviously, the convolution of the function (REF with the Dirac $\\delta $ function gives again the same function (REF ): $f_{\\epsilon }\\ast \\delta (x)=f_{\\epsilon }(x)~.",
"$ The calculation of the convolution of the principal value $\\mathcal {P}\\left(\\frac{1}{x}\\right) $ with the function $f_{\\epsilon }(x)$ is slightly more complicated: $&&\\mathcal {P}\\left( \\frac{1}{x}\\right) \\ast f_{\\epsilon }(x)=\\lim _{\\varepsilon \\rightarrow 0}\\left( \\int _{-\\infty }^{x-\\varepsilon }dy\\frac{1}{x-y}\\frac{\\epsilon }{\\pi (y^{2}+\\epsilon ^{2})}\\right.",
"\\\\&&\\left.",
"+\\int _{x+\\varepsilon }^{\\infty }dy\\frac{1}{x-y}\\frac{\\epsilon }{\\pi (y^{2}+\\epsilon ^{2})}\\right) =\\frac{x}{x^{2}+\\epsilon ^{2}}~.$ The product of the expressions (REF ) and (REF ) is $\\mathcal {P}\\left( \\frac{1}{x}\\right) _{\\epsilon }\\ast \\delta _{\\epsilon }(x)=\\frac{\\epsilon x}{\\pi (x^{2}+\\epsilon ^{2})^{2}}~.",
"$ Let us now consider a family of functions $\\frac{df_{\\epsilon }(x)}{dx}=-\\frac{2x\\epsilon }{\\pi (x^{2}+\\epsilon ^{2})^{2}}~.",
"$ One can easily prove that if the family of functions (REF ) converges in the distributional sense to the Dirac $\\delta $ function, the family of their derivatives (REF ) converges to the derivative of the delta function.",
"Now, comparing the right-hand sides of Eqs.",
"(REF ) and (REF ) we see that when $\\epsilon \\rightarrow 0$ the product in the left-hand side of Eq.",
"(REF ) converges in the distributional sense to $-\\frac{1}{2}\\delta ^{\\prime }(x)$ and thus the correctness of the equality (REF ) is checked.",
"Now let us discuss the Antosik identity [51] $\\frac{1}{1+\\delta (x)}=1.",
"$ Here we have the composition of the distributions $F(g)$ , where $F=\\frac{1}{g}$ and $g(x)=1+\\delta (x)$ .",
"Calculating the convolutions of the distributions $F$ and $g$ with the Cauchy-Lorentz functions (REF ) we obtain $F_{\\sigma }(g)=F\\ast f_{\\sigma }(g)=\\frac{g}{g^{2}+\\sigma ^{2}},$ $g_{\\epsilon }(x)=1+\\frac{\\epsilon }{x^{2}+\\epsilon ^{2}}.",
"$ Correspondingly the composition of these functions is $F_{\\sigma }(g_{\\epsilon })=\\frac{1+\\frac{\\epsilon }{\\pi (x^{2}+\\epsilon ^{2}}}{\\sigma ^{2}+\\left( 1+\\frac{\\epsilon }{\\pi (x^{2}+\\epsilon ^{2}}\\right) ^{2}} $ and it is easy to check that $\\lim _{\\sigma \\rightarrow 0}\\lim _{\\epsilon \\rightarrow 0}F_{\\sigma }(g_{\\epsilon })=1,$ confirming the identity (REF )."
],
[
"The Lanczos equation",
"For a generic junction surface the Lanczos equation emerges from the Gauss-Codazzi relations [58], [43].",
"The projected Lie derivative of the extrinsic curvature $K_{ab}$ in the normal direction $n$ to the surface is $h_{a}^{i}h_{b}^{j}\\mathcal {L}_{\\mathbf {n}}K_{ij}=-3\\epsilon \\left(h_{a}^{i}h_{b}^{k}T_{ik}-\\frac{h_{ab}}{2}g^{ik}T_{ik}\\right) +\\mathcal {Z}_{ab}\\ $ (Eq.",
"(21) of [43] in the units $8\\pi G/3=1$ ), with $\\mathcal {Z}_{ab} &=&-\\epsilon \\mathcal {R}_{ab}+2K_{ac}K_{b}^{c}-g^{ik}K_{ik}K_{ab} \\\\&&+D_{b}\\alpha _{a}-\\epsilon \\alpha _{b}\\alpha _{a}\\ .$ Here $g_{ab}$ is the space-time metric, $h_{ab}=g_{ab}-\\epsilon n_{a}n_{b}$ ($\\epsilon =n^{a}n_{a}=\\left\\lbrace -1,1\\right\\rbrace $ ) is the induced metric on the junction surface, and $T_{ab}$ is the energy-momentum tensor.",
"The tensor $\\mathcal {Z}_{ab}$ depends only on geometrical quantities: $\\mathcal {R}_{ab}$ and $D$ are the Ricci tensor and covariant derivative induced on the hypersurface, and $\\alpha _{a}=n^{c}\\nabla _{c}n_{a}$ , with $\\nabla $ the 4-dimensional covariant derivative.",
"When the energy-momentum tensor is a sum $T_{ik}=\\Pi _{ik}+\\Upsilon _{ik}\\delta \\left( \\tau \\right) $ (where $\\tau $ is the coordinate adapted to $n$ , i.e.",
"$n=t_{S}^{-1}\\partial /\\partial \\tau $ , and $n^{a}\\Upsilon _{ab}=0$ ), with $\\Pi _{ik}$ the regular 4-dimensional part and $\\Upsilon _{ik}$ the distributional part on the hypersurface, integration of Eq.",
"(REF ) across $\\tau $ through an infinitesimal range containing the hypersurface keeps only the distributional part, leading to the Lanczos equation [41], [40].",
"$\\Delta K_{ab}=-3\\epsilon \\left( \\Upsilon _{ab}-\\frac{\\Upsilon }{2}h_{ab}\\right) \\ ,$ or equivalently $-3\\epsilon \\Upsilon _{ab}=\\Delta K_{ab}-h_{ab}\\Delta K\\ .$ Here $\\Upsilon $ is the trace of $\\Upsilon _{ab}$ .",
"As $\\mathcal {Z}_{ab}$ is finite, its contribution to the integral across the infinitesimal range also vanishes.",
"Without a distributional energy-momentum part, the extrinsic curvature should be continuous.",
"Let us now specialize this for a junction along a maximally symmetric $\\tau =0$ spacelike hypersurface (a hyperplane with $\\mathcal {R}_{ab}=0$ ) embedded in a flat Friedmann space-time.",
"The normal vector $n$ has zero acceleration $\\alpha _{a}=0$ and the extrinsic curvature becomes $K_{ab}=\\dot{a}a\\widetilde{h}_{ab}$ , with $\\widetilde{h}_{ab}$ the 3-dimensional Euclidean metric.",
"The curvature term is $\\mathcal {Z}_{ab}=-H^{2}a^{2}\\widetilde{h}_{ab} $ and the energy momentum tensors are $\\Pi _{ab}=\\widetilde{\\rho }n_{a}n_{b}+\\widetilde{p}a^{2}\\widetilde{h}_{ab}$ and $\\Upsilon _{ab}=\\overline{p}a^{2}\\widetilde{h}_{ab}$ .",
"Since the projected Lie-derivative in Eq.",
"(REF ) becomes a time derivative, the equation reads $\\frac{\\partial }{\\partial t}\\left( Ha^{2}\\right) =\\left\\lbrace -H^{2}+\\frac{3}{2}\\left[ \\widetilde{\\rho }-\\widetilde{p}-\\overline{p}\\delta \\left( \\tau \\right) \\right] \\right\\rbrace a^{2}\\ , $ which is a combination of the Raychaudhuri and Friedmann equations.",
"For finite $H$ , $\\widetilde{\\rho }$ and $\\widetilde{p}$ as before the integration of Eq.",
"(REF ) across an infinitesimal time range $\\tau $ leads to the Lanczos equation $\\Delta H|_{t_{s}}=-\\frac{3}{2}\\overline{p}\\ .",
"$"
]
] | 1204.1199 |
[
[
"Ulta-slow relaxation in discontinuous-film based electron glasses"
],
[
"Abstract We present field effect measurements on discontinuous 2D thin films which are composed of a sub monolayer of nano-grains of Au, Ni, Ag or Al.",
"Like other electron glasses these systems exhibit slow conductance relaxation and memory effects.",
"However, unlike other systems, the discontinuous films exhibit a dramatic slowing down of the dynamics below a characteristic temperature $T^*$.",
"$T^*$ is typically between 10-50K and is sample dependent.",
"For $T<T^*$ the sample exhibits a few other peculiar features such as repeatable conductance fluctuations in millimeter size samples.",
"We suggest that the enhanced system sluggishness is related to the current carrying network becoming very dilute in discontinuous films so that the system contains many parts which are electrically very weakly connected and the transport is dominated by very few weak links.",
"This enables studying the glassy properties of the sample as it transitions from a macroscopic sample to a mesocopic sample, hence, the results provide new insight on the underlying physics of electron glasses."
],
[
"Ulta-slow relaxation in discontinuous-film based electron glasses T. Havdala A. Eisenbach A. Frydman The Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel We present field effect measurements on discontinuous 2D thin films which are composed of a sub monolayer of nano-grains of Au, Ni, Ag or Al.",
"Like other electron glasses these systems exhibit slow conductance relaxation and memory effects.",
"However, unlike other systems, the discontinuous films exhibit a dramatic slowing down of the dynamics below a characteristic temperature $T^*$ .",
"$T^*$ is typically between 10-50K and is sample dependent.",
"For $T<T^*$ the sample exhibits a few other peculiar features such as repeatable conductance fluctuations in millimeter size samples.",
"We suggest that the enhanced system sluggishness is related to the current carrying network becoming very dilute in discontinuous films so that the system contains many parts which are electrically very weakly connected and the transport is dominated by very few weak links.",
"This enables studying the glassy properties of the sample as it transitions from a macroscopic sample to a mesocopic sample, hence, the results provide new insight on the underlying physics of electron glasses.",
"75.75.Lf; 72.80.Ng; 72.20.Ee; 73.40.Rw Glassy behavior of the conductivity, $\\sigma $ , in strongly disordered systems that are characterized by strong electronic interactions were predicted by several groups [1], [2], [3], [4], [5].",
"Exciting such a system out of equilibrium leads to an increase in conductivity, $\\sigma $ , after which the relaxation towards equilibrium is characterized by extremely long times, memory phenomena and aging.",
"Since the slow dynamics are related to their electronic properties these systems were termed electron glasses [4].",
"Experimentally, glassy features were observed in a verity of systems including granular Au [6], amorphous and poly-crystalline indium oxide films [7], [8], [9], [10], [11], ultrathin Pb films [12], granular aluminum [13], [14] and thin beryllium films [15].",
"A standard way of excitation in these experiments is by applying a gate voltage, $V_{g}$ , in a MOSFET setup.",
"Conductivity increases for both orientations of $V_{g}$ followed by very slow relaxation of $\\sigma $ which is found to follow an approximate logarithmic dependence on time and may be measured over time-scales of days.",
"A typical feature which has been suggested as the hallmark of intrinsic electron glasses [16] is a \"memory dip\" (MD) which shows up as a minimum in the $\\sigma (V_g)$ curve when $V_{g}$ is scanned fast compared to the characteristic relaxation time.",
"The dip is centered around the gate voltage at which the sample was allowed to equilibrate.",
"The origin of the extremely slow relaxation and the memory dip as well as their dependence on parameters such as temperature, bias voltage, carrier concentration etc.",
"are still under debate and more experimental information may help shedding light on the physics of electron glasses.",
"In this letter we present results on the glassy properties of two dimensional discontinuous films.",
"We find that these systems exhibit a dramatic slowing down of the dynamics below a characteristic temperature $T^*$ .",
"For $T<T^*$ the conductance of the sample exhibits reproducible fluctuations with exponentially growing amplitude as the temperature is lowered indicating that the effective electronic size of the sample has become very small.",
"We discuss the influence of the sample geometry and its effective size on the conductance relaxation properties.",
"All conductance results presented in this work were obtained by standard 2-wire lock-in techniques performed on thin discontinuous Au, Al, Ag or Ni films prepared by the quench condensation technique, i.e.",
"thermal evaporation on a cryocooled substrate [17], [18], [19].",
"dc techniques were used for comparison in a few cases yielding similar results.",
"For achieving field effect geometry we used a doped Si substrate (that was utilized as a gate electrode) coated by a 0.5 $\\mu m$ insulating SiO layer.",
"Gold pads were pre-prepared on the substrate so that, together with a shadow mask, they defined a sample area of 0.6mm by 0.6mm.",
"The substrate was then placed on a sample holder within a vacuum chamber.",
"After the chamber was pumped out, the substrate was cooled to cryogenic temperatures and thin layers of Au, Ni, Al or Ag were deposited while monitoring the film thickness and resistance.",
"For thin enough layers this technique yields a film that is discontinuous, consisting of a sub monolayer of metallic grains, 10-20nm in diameter, separated by vacuum as seen in fig.",
"1.",
"A major advantage of this method is that throughout the entire process of sample growth and measurement the samples are kept in ultra-high vacuum and not exposed to air, thus protecting the grains from oxidation or contamination.",
"This is especially important for nano-grains in which the surface area to volume ratio is very high.",
"Figure: Resistance versus temperature for a 7 nm thick discontinuous gold film (black full squares)measured using dc methods and a 10nm Ni Film (yellow empty circles) measured with ac methods.",
"The lines are fits to Efros Shklovskii like behavior.",
"Insert: An SEM image of the Au film after heating it to room temperature (color online).Granular metals which are on the insulating side of the metal-insulator-transition are known to be hopping systems that exhibit resistance versus temperature (R(T)) curves that follow: $ R\\propto exp[\\frac{T_{0}}{T}]^{\\alpha }$ Experimentally, $\\alpha s$ between 0.5 and 1 have been reported.",
"In our films eq.",
"REF is fulfilled only for relatively high T. Fig.",
"1. depicts the R(T) curves of a gold film and a nickel film.",
"It is seen that for temperatures higher than a characteristic temperature, $T^*$ (in fig 1 $T^* \\simeq 20K$ for the gold sample and $\\simeq 10K$ for the Ni sample), we observe a usual Efros Shklovskii [20] like dependence ($\\alpha =0.5$ ).",
"However, for $T<T*$ the conductance depends much weaker on temperature and seems to be approaching saturation at low temperatures.",
"It turns out that T* manifests itself in other transport properties of the sample.",
"As in other electron glasses, after the sample is allowed to equilibrate at a certain gate voltage, $V_{g0}$ for a long time, a MD is observed in conductance versus gate voltage curve.",
"In all our films the MD is accompanied by reproducible conductance fluctuations (see fig.",
"2).",
"These fluctuations are random, however they are reproducible for sequential experiments performed under similar conditions.",
"Such fluctuations have been observed in the past in indium oxide [21] and granular Al [22] electron glasses with dimensions smaller than $100\\mu m$ .",
"We observe these fluctuations for much larger samples.",
"Fig.",
"2 shows that the rms amplitude of the fluctuations, $\\frac{\\delta G}{<G>}$ , of a 0.6*0.6 mm film becomes measurable for $T<T^*$ and grows exponentially with decreasing temperature.",
"The increase of $\\frac{\\delta G}{<G>}$ with lowering T was observed in other systems [22], however, in those cases it was found that $\\frac{\\delta G}{<G>} \\propto T^{ - \\beta } $ with $\\beta \\sim 2$ .",
"The power law dependence was attributed to the fact that the magnitude of the fluctuations are proportional to $L_{0}^{2}$ where $L_{0}$ is the spatial scale of an independent microscopic fluctuations which was ascribed to the correlation length of the percolation network.",
"This length is predicted to have a power law dependence on temperature [23], [24], [25] $ L_0 \\propto [r_h]^\\nu \\propto T^{-\\alpha \\nu }$ where $r_h$ is the characteristic hopping length, $\\alpha $ is the power of the temperature dependence of the resistance given by eq.",
"REF and $\\nu $ is between 1 and 2.",
"An exponential dependence of $\\frac{\\delta G}{<G>}$ as seen in fig.",
"2 implies an unconventional change of the percolation network for $T<T^*$ .",
"Figure: a: Conductance as a function of gate voltage for one of our0.6*0.6 mm Au films at different temperatures.",
"The thickness was 7nm.",
"V g0 =0VV_{g0}=0V.",
"The inset shows the rms amplitude of the conductance fluctuations as a function of temperature.b: δG <G>\\frac{\\delta G}{<G>} (blue circles) and the dip amplitude 20 minutes after the quench-coolling of the sample (black triangles) versus T.(color online)A similar conclusion can be derived from the temperature dependence of the current-voltage characteristics.",
"It is useful to study conductivity versus voltage curves, $\\sigma (V)$ , such as that shown in fig.",
"3 for an Au film.",
"These curves are characterized by a voltage $V_{0}$ above which the conductivity increases with voltage thus deviating from ohmic behavior.",
"Several theoretical works [23], [24], [25] predict that non-Ohmic conductivity should occur for: $ k_{B}T<eFL_{0}$ where F is the field applied across $L_0$ .",
"Hence $V_0$ is expected to follow [26]: $ V_0 \\propto \\frac{T}{L_0} \\propto T^m$ where m is between 1 and 2.",
"The dependence of $V_0$ on T for a discontinuous Au film is shown in fig.",
"3.",
"It is seen that for $T>T^*$ the I-V characteristic yields $V_0 \\propto T^2$ as expected.",
"For $T<T^*$ there is a sharp drop in $V_0$ and a clear deviation from eq.",
"REF .",
"Such behavior could be interpreted as an unusually rapid increase in $L_{0}$ as the temperature is lowered below $T^*$ .",
"Figure: V 0 V_0 as a function of T for a discontinuous Au film.",
"The determination of V 0 V_0 is illustrated in the inset which shows a conductance versus voltagecurve at T=6K.",
"V 0 V_0 is defined as the voltage at which the conductance increases by 5 % from its ohmic value.Perhaps the most remarkable phenomenon that occurs at low temperatures is a dramatic slowing down of the system's electronic dynamics.",
"Since the relaxation after any excitation follows an approximate logarithmic dependence it does not have a natural characteristic decay time.",
"Nevertheless a number of methods have been proposed to experimentally define the \"slowness\" of the relaxation in electron glasses [27], all giving equivalent characteristic times, $\\tau $ .",
"A popular method employs the \"two dip experiment\" (TDE) [11].",
"In this experiment the sample is allowed to equilibrate for a long time (of the order of a day) at a certain gate voltage $V_{g1}$ .",
"At this stage a fast $\\sigma (Vg)$ scan yields a conductivity memory dip centered around $V_{g1}$ .",
"At time t=0 the gate voltage is abruptly changed to $V_{g2}$ and fast $\\sigma (V_g)$ scans are performed at selected time intervals.",
"As a function of time the dip at $V_{g1}$ is slowly suppressed while a new dip develops around $V_{g2}$ .",
"The characteristic relaxation time, $\\tau $ , is defined as the time at which the amplitude of the two dips is equal.",
"A TDE for a discontinuous Au film sample at T=55K is shown in fig.",
"4e.",
"It is seen that $\\tau _{55k} \\approx 1000 s$ , which is a typical value for disordered samples having high carrier concentration [16].",
"Upon lowering the temperature, all our films exhibits a huge increase in the characteristic relaxation time.",
"For example, $\\tau _{15k}$ is found to be $\\sim 10^6 s$ .",
"This makes the systematic study of $\\tau _{TDE}$ at different temperatures unpractical.",
"Therefore we measure the fraction of relaxation that the system undergoes over a time of 1 hour.",
"This is done in the following way: We allow the sample to equilibrate at relatively high T ($T>T^*$ ) while applying $V_{g}$ =0 for a time $t_{1}$ after which a dip is well developed.",
"We then cool the sample to a different temperature, perform a gate voltage sweep to determine the size of the dip and change the gate voltage to $V_{g}=-13V$ .",
"We then wait for one hour and measure the amplitude of the dip at $V_{g}=-13$ .",
"We define the amplitude ratio between the new dip and the old dip (at $V_{g}=0$ ) as $A_{1h}$ .",
"This is taken as a measure for the \"slowness\" of the relaxation.",
"The dependance of $A_{1h}$ on temperature is shown in fig.",
"4.",
"It is seen that there is a dramatic decrease of $A_{1h}$ at $T \\sim T^*$ .",
"Similar behavior was obtained for all studied samples (over 12).",
"The relaxation times and $T^*$ did not seem to depend on resistance but the amplitude of the memory dip decreased as the sample approached the metal-insulator transition.",
"Figure: a-d: Conductance versus gate voltage at t=0 (black lightline) and at t=1 hour (blue heavy line) for differenttemperatures.",
"e: More detailed two dip experiment results for T=55K.",
"f: Second dip amplitude of the Au sample of panels a-e (red full squares) and a Ni sample (blue empty circles)as a function of temperature (color online).Slowing down of relaxation processes with decreasing temperature may seem natural, however it is contrary to the situation in other electron glass systems.",
"In amorphous and crystalline indium oxide [30], and granular aluminum [14] the dynamics were found either to be independent on temperature or to slow down upon increasing T. The latter has been suggested as evidence that disordered electronic systems are quantum glasses [30].",
"In contrast, our Au films exhibit a dramatic slowing of the dynamics upon cooling.",
"Furthermore, the temperature dependence of $\\tau $ is very peculiar.",
"Fig.",
"4f depicts $A_{1h}$ (which inversely depends on $\\tau $ ) as a function of T. It is seen that there is a sharp increase of relaxation times over a small temperature range.",
"The fact that the slowing down of the dynamics occurs at $T \\sim T^*$ , where mesoscopic conductance fluctuations become significant, leads us to postulate that it is related to a significant dilution of the current carrying network (CCN).",
"Note that, unlike in other electron glasses, the conductivity in the discontinuous films is strictly two dimensional.",
"In addition, the SEM image of figure 1 shows that the grains are not closely packed, but rather the film morphology is composed of fractal shaped clusters connected by thin (single grain) bottlenecks [31].",
"Hence this film is characterized by geometrical and not only electronic percolation which is typical of other hopping systems.",
"At low temperatures many of these bottlenecks may disconnect from the CCN because of energy mismatches thus leaving most of the sample electrically cut-off.",
"This gives rise to enhanced conductance fluctuations since the effective electronic system size is small.",
"This also accounts for the apparent saturation of the R(T) curve at low T as shown in fig 1.",
"Upon lowering T the electric current network becomes progressively dilute and the variable range hopping mechanism becomes less relevant as fewer sections of the sample dominate the conductivity.",
"Eventually, at the extreme case where the conductivity governed by a single weak link it is expected to be temperature independent.",
"Under these conditions, at low T there is a very small probability for an electron to tunnel into and out of the current carrying network and most of the charge carriers are trapped in isolated regions of the sample.",
"This causes a significant slowing down of the relaxation processes to equilibrium because these rely on many body hopping processes taking place in various parts of the sample.",
"Since most of the sample is very weakly connected, the relaxation to equilibrium becomes extremely slow, thus hindering the development of a new MD at low temperatures.",
"This can be realized from fig.",
"2b that shows the amplitude of the memory dip 20 minutes after the cool down, $A_{20m}$ , as a function of temperature.",
"It is seen that for $T<T^*$ this amplitude reduces with decreasing temperature.",
"The amplitude of the memory dip in other electron glasses has shown to increase rapidly with decreasing temperature [8], [14].",
"The decrease of $A_{20m}$ with lowering temperature reflects the fact that the dynamics of the disconnected sections (most of the sample) have become extremely slow that the CCN is so dilute so that glassy properties are no longer relevant.",
"Indeed some of the mesoscopic electron glasses studied in the past also did not exhibit a measurable MD [21], [22].",
"In conclusion we have shown that electron glasses based on 2D discontinuous films exhibit a sharp increase of relaxation times at low temperatures.",
"These systems are characterized by a tenuous morphology which was not studied in the context of electron glasses so far.",
"Our results demonstrate that relaxation to equilibrium hinges upon electronic transition in a wide region of the sample and when these become unavailable the relaxation processes are considerably impeded.",
"The many-body and many-electron-hopping nature of the electron glass thus becomes strikingly apparent in these discontinuous films.",
"We are grateful for useful discussions with A. Amir, Z. Ovadyahu and M. Pollak.",
"This research was supported by the Israeli academy of science (grant number 399/09) M. Grunewald, B. Pohlman, L. Schweitzer, and D. Wurtz, J. Phys.",
"C 15, L1153 (1982).",
"M. Pollak and M. Ortuno, Sol.",
"Energy Mater.",
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]
] | 1204.1142 |
[
[
"Eigenvalue Distributions of Reduced Density Matrices"
],
[
"Abstract Given a random quantum state of multiple distinguishable or indistinguishable particles, we provide an effective method, rooted in symplectic geometry, to compute the joint probability distribution of the eigenvalues of its one-body reduced density matrices.",
"As a corollary, by taking the distribution's support, which is a convex moment polytope, we recover a complete solution to the one-body quantum marginal problem.",
"We obtain the probability distribution by reducing to the corresponding distribution of diagonal entries (i.e., to the quantitative version of a classical marginal problem), which is then determined algorithmically.",
"This reduction applies more generally to symplectic geometry, relating invariant measures for the coadjoint action of a compact Lie group to their projections onto a Cartan subalgebra, and can also be quantized to provide an efficient algorithm for computing bounded height Kronecker and plethysm coefficients."
],
[
"=1 matrix,arrows satzthm satz lemthm lem corthm cor prpthm prp cnjthm cnj quethm que fctthm fct obsthm obs algthm alg asmthm asm dfnthm dfn ntnthm ntn remthm rem ntethm nte exlthm exl =1200 same [4] pdfauthor=Matthias Christandl and Brent Doran and Stavros Kousidis and Michael Walter,pdftitle=Eigenvalue Distributions of Reduced Density Matrices"
]
] | 1204.0741 |
[
[
"On torsion anomalous intersections"
],
[
"Abstract A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$.",
"We prove some cases of this conjecture.",
"We show that the $V$-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety $V$ embedded in a product of elliptic curves with CM.",
"We give explicit uniform bounds in the dependence on $V$.",
"As an immediate consequence we prove the conjecture for $V$ of codimension two in a product of CM elliptic curves.",
"We also point out some implications on the effective Mordell-Lang Conjecture."
],
[
"Introduction",
"In this article, by variety we mean an algebraic variety defined over the algebraic numbers.",
"We denote by $G$ a semi-abelian variety defined over a number field $k$ and by $k_{\\mathrm {tor}}$ the field of definition of the torsion points of $G$ .",
"Let $V$ be a subvariety of $G$ .",
"The variety $V$ is a translate, respectively a torsion variety, if it is a finite union of translates of proper algebraic subgroups by points, respectively by torsion points.",
"An irreducible variety $V$ is transverse, respectively weak-transverse, if it is not contained in any translate, respectively in any torsion variety.",
"Of course a torsion variety is in particular a translate, and a transverse variety is weak-transverse.",
"In addition transverse implies non-translate, and weak-transverse implies non-torsion.",
"Notice that we are considering only proper subvarieties.",
"It is a natural problem to investigate when a geometric assumption on $V$ is equivalent to the non-density of some special subsets of $V$ .",
"Several classical statements, such as, for instance, the Manin-Mumford, the Mordell-Lang and the Bogomolov Conjectures—nowadays theorems, are all of this nature.",
"More recently, new questions of similar type have been raised.",
"The Zilber-Pink Conjecture asserts that, for a transverse variety $V$ , the intersection of $V$ with the union of all algebraic subgroups of codimension at least $\\dim V+1$ translated by points in a subgroup of finite rank, is non-dense in $V$ .",
"This conjecture has been tackled from several points of view, but it has only been answered partially.",
"For instance it is known for curves in some semi-abelian varieties.",
"E. Bombieri, D. Masser and U. Zannier in [11] give a new approach for general dimensions.",
"They introduce the notions of anomalous and torsion anomalous subvarieties.",
"In their definitions they always avoid points.",
"For us points can be torsion anomalous, but not anomalous.",
"This gives a perfect match with the CIT conjecture, as clarified below.",
"An irreducible subvariety $Y$ of $V$ is a $V$ -torsion anomalous variety if - $Y$ is an irreducible component of $V\\cap (B+\\zeta )$ with $B+\\zeta $ an irreducible torsion variety; - the dimension of $Y$ is larger than expected, i.e.",
"${\\mathrm {codim}\\,} Y < {\\mathrm {codim}\\,} V + {\\mathrm {codim}\\,} B.$ The variety $B+\\zeta $ is minimal for $Y$ if it satisfies the above conditions and has minimal dimension.",
"The relative codimension of $Y$ is the codimension of $Y$ in its minimal $B+\\zeta $ .",
"We say that $Y$ is a maximal $V$ -torsion anomalous variety if it is $V$ -torsion anomalous and it is not contained in any $V$ -torsion anomalous variety of strictly larger dimension.",
"The complement in $V$ of the union of all $V$ -torsion anomalous varieties is denoted by $V^{ta}$ .",
"Clearly $V^{ta}$ is obtained removing from $V$ all maximal $V$ -torsion anomalous varieties.",
"Again, for Bombieri, Masser and Zannier $V^{ta}$ is the complement of the union of all $V$ -torsion anomalous varieties of positive dimension.",
"Furthermore, an irreducible variety $Y$ of positive dimension is $V$ -anomalous if it is a component of $V\\cap (B+p)$ with $B+p$ an irreducible translate and in addition $Y$ has dimension larger than expected.",
"The complement in $V$ of the union of all $V$ -anomalous varieties is denoted by $V^{oa}.$ Clearly points should be excluded from the definition of $V$ -anomalous varieties because otherwise all points would be anomalous, making the notion uninteresting.",
"On the other hand we allow points to be torsion anomalous varieties: they are exactly the torsion anomalous varieties which are not anomalous.",
"It may be possible that only some components of $V\\cap (B+\\zeta )$ are anomalous, so each component has to be treated separately.",
"This justifies the assumption of $Y$ being irreducible.",
"In the Torsion Openness Conjecture [11], Bombieri, Masser and Zannier conjectured that the complement of the set of the torsion anomalous varieties of positive dimension is open.",
"In addition, in the Torsion Finiteness Conjecture, they claim that there are only finitely many maximal torsion anomalous points.",
"Here we state a slightly stronger conjecture, which includes both their conjectures.",
"In addition, it specifies that $V^{ta}$ is empty exactly when $V$ is not weak-transverse.",
"In other words we say that there are only finitely many maximal torsion anomalous varieties of any dimension.",
"Conjecture 1 (Bombieri-Masser-Zannier) Let $V$ be a weak-transverse variety in a (semi-)abelian variety.",
"Then $V^{ta}$ is a dense open subset of $V$ .",
"For a hypersurface the conjecture is clearly true.",
"Indeed the intersection of an irreducible torsion variety $B+\\zeta $ with a hypersurface is either the variety $B+\\zeta $ itself or it has the right dimension $\\dim B-1$ .",
"So the only $V$ -torsion anomalous varieties are torsion varieties contained in $V$ ; but we know by the Manin-Mumford Conjecture that the maximal torsion varieties contained in $V$ are finitely many.",
"Among other results, Bombieri, Masser and Zannier in [11], Theorem 1.7, prove the openness of $V^{ta}$ for an irreducible variety $V$ of codimension 2 in $\\mathbb {G}^n_m$ .",
"Among the main ingredients in their proof is a result of Ax consisting in the analogue of Schanuel's Conjecture in fields of complex power series in several variables.",
"In this paper, we first prove Conjecture REF for weak-transverse translates in a general abelian variety.",
"Then we give a totally effective method which shows that in any weak-transverse variety $V$ in a product of elliptic curves with CM, the $V$ -torsion anomalous varieties of relative codimension one are non-dense.",
"An immediate consequence is Conjecture REF for weak-transverse varieties of codimension 2, proving the analogue of Theorem 1.7 of [11] in a product of CM elliptic curves.",
"Our method differs from theirs; in particular it is completely effective, and we also avoid the use of Ax's theorem.",
"An intrinsic consequence of our effective bounds on the height is the effective CIT (stated below) for weak-transverse varieties of codimension 2 in a product of elliptic curves with CM.",
"Finally we point out some implications on the effective Mordell-Lang Conjecture.",
"We also show how the method for elliptic curves generalises to abelian varieties.",
"The following first result, based only on some geometric considerations, is proved in Section REF .",
"Theorem 1.1 Let $H+p$ be a weak-transverse translate in an abelian variety.",
"Then the set of $(H+p)$ -torsion anomalous varieties is empty.",
"This clarifies the situation and brings some simplifications in the proof of our main result: Theorem 1.2 Let $V$ be a weak-transverse variety in a product of elliptic curves with CM defined over a number field $k$ .",
"Then the maximal $V$ -torsion anomalous varieties $Y$ of relative codimension one are finitely many; in addition their degree and normalised height are effectively bounded as $ h(Y) &\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-1}{N-1-\\dim V}+\\eta }[k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]^{\\frac{\\dim V}{N-1-\\dim V}+\\eta },\\\\\\deg Y &\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-2}{N-1-\\dim V}+\\eta }[k_\\mathrm {tor}(V):k_\\mathrm {tor}]^{\\frac{\\dim V-1}{N-1-\\dim V}+\\eta }.$ These are only some of the bounds we obtain.",
"The method is totally effective, and we give explicit dependence on $V$ .",
"The notations will be made precise in the next sections.",
"In addition we bound the degree of the torsion varieties $B+\\zeta $ minimal for the maximal $V$ -torsion anomalous that we consider; these bounds provide, in principle, an algorithm to find all such anomalous varieties.",
"The effective statements require some further notation and they are given in Theorem REF for maximal torsion anomalous varieties which are not translates, in Theorems REF and REF for maximal torsion anomalous points and in Theorems REF and REF for maximal torsion anomalous translates of positive dimension.",
"The proof of our main theorem has four main ingredients: the deep Zhang inequality, the strong explicit Arithmetic Bézout theorem by Philippon, a sharper variant of an effective result by Galateau on the Bogomolov Conjecture, and the relative Lehmer bound in CM abelian varieties by Carrizosa.",
"In the proofs we need to distinguish whether a $V$ -torsion anomalous variety $Y$ is or not itself a translate.",
"If $Y$ is not a translate then the Zhang inequality, the explicit Arithmetic Bézout theorem, and a functorial version of the effective Bogomolov Conjecture are sufficient.",
"This case is proved in a product of elliptic curves, as the Bogomolov type bounds do not require any assumption on complex multiplication.",
"If $Y$ is a point we use the Zhang inequality, the explicit Arithmetic Bézout theorem and the relative Lehmer bound; here we need to assume that the elliptic curve has CM, as a sharp Lehmer type bound is known only under this hypothesis.",
"Finally, we reduce the case of translates of positive dimension to the case of points.",
"To this aim we also use a result recalled in the appendix by Patrice Philippon, which relates the essential minimum of a translate to the height of the point of translation.",
"Theorem REF can be easily generalised to abelian varieties, in the following sense.",
"Theorem 1.3 Let $V$ be a weak-transverse variety in an abelian variety $A$ with CM.",
"Let $g$ be the maximal dimension of a simple factor of $A$ .",
"If ${\\rm codim\\,}V\\ge g+1$ , then the maximal $V$ -torsion anomalous varieties of relative codimension 1 are finitely many and they have degree and normalised height effectively bounded.",
"For clarity, we first present the proof of Theorem REF , where the technicalities are simpler to follow.",
"We then explain how this generalises to abelian varieties in Section .",
"We notice that the most interesting case remains the case of elliptic curves, in the sense that there is the largest number of subgroups.",
"A breaking-through result would be the proof of Theorem REF for any relative codimension with an effective method.",
"Such a general result would imply the effective Mordell-Lang Conjecture.",
"We now give several consequences and applications of our main theorem.",
"By definition all $V$ -torsion anomalous varieties satisfy a dimensional inequality.",
"As a straightforward consequence, we see that in a variety $V$ of codimension 2 the relative codimension of a $V$ -torsion anomalous variety is either one or zero.",
"Then, from Theorem REF , it follows immediately the following: Theorem 1.4 Let $V$ be a weak-transverse variety of codimension 2 in a product of elliptic curves with CM.",
"Then $V^{ta}$ is a dense open subset of $V$ .",
"It is sufficient to consider maximal $V$ -torsion anomalous varieties.",
"Let $Y$ be a maximal $V$ -torsion anomalous component of $V\\cap (B+\\zeta )$ .",
"Then, by definition of $V$ -torsion anomalous variety, $ {\\rm codim\\,}Y< {\\rm codim\\,}V+ {\\rm codim\\,}B.$ Equivalently $\\dim B-\\dim Y<{\\rm codim\\,}V=2.$ If $Y$ has relative codimension zero, then $Y= B+\\zeta $ and $Y$ is a component of the closure of the torsion contained in $V$ , which is a proper closed by the Manin-Mumford Conjecture.",
"If $Y$ has relative codimension one, we apply our main theorem.",
"Conjecture REF is well known to be related to the following conjecture, which in turn is equivalent to the Zilber-Pink Conjecture.",
"For a natural number $r$ , define $S_{r}(V)=V \\cap \\bigcup _{{\\rm codim\\,}\\, H\\ge r}H,$ where $H$ runs over all algebraic subgroups of codimension at least $r$ .",
"Conjecture 2 (CIT, Conjecture on the Intersection with Torsion varieties) Let $V$ be a weak-transverse variety in an abelian variety.",
"Then $S_{\\dim V+1}(V)$ is non-dense in $V$ .",
"By definition, for any torsion variety $B+\\zeta $ , the intersection $V^{ta}\\cap (B+\\zeta )$ has the right dimension.",
"In particular, if a point of $V$ lies in some algebraic subgroup of codimension $\\ge \\dim V +1$ , then that point is contained in a $V$ -torsion anomalous variety (we would expect empty intersection), and so it does not belong to $V^{ta}$ .",
"Then, as a consequence of the effective version of Theorems REF and REF , we obtain a completely effective version of the following theorem.",
"Theorem 1.5 Let $V$ be a weak-transverse variety of codimension 2 or a weak-transverse translate in a product of elliptic curves with CM.",
"Then $S_{\\dim V+1}(V)$ is non-dense in $V$ and its closure is $V \\setminus V^{ta}$ .",
"If ${\\rm codim\\,}(B+\\zeta ) \\ge \\dim V+1$ , then all components of $V\\cap (B+\\zeta )$ are torsion anomalous so they do not intersect $V^{ta}$ .",
"Therefore $V^{ta }\\cap \\bigcup _{{\\rm codim\\,}\\, H\\ge \\dim V+1}H=\\varnothing $ and $S_{\\dim V+1}(V) \\subseteq V\\setminus V^{ta}.$ By Theorems REF and REF , $V^{ta}$ is an open dense set in $V$ .",
"That the closure of $S_{\\dim V+1}(V)$ is $V \\setminus V^{ta}$ is proven exactly as in [11], page 25, for tori.",
"Recall that for them points are not torsion anomalous varieties.",
"As an immediate corollary we obtain: Corollary 1.6 Let $C$ be a weak-transverse curve in $E^3$ where $E$ is an elliptic curve with CM defined over a number field $k$ .",
"Then $S_2(C)=C\\setminus C^{ta}$ is a finite set of cardinality and Néron-Tate height effectively bounded.",
"In particular every non-torsion point $Y_0\\in S_2(C)$ satisfies $\\hat{h}(Y_0) &\\ll _\\eta (h(C)+\\deg C)^{2+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{1+\\eta },\\\\[k (Y_0):\\mathbb {Q}] &\\ll _\\eta (\\deg C (h(C)+\\deg C)[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}][k(C):k])^{2+\\eta }.$ That $S_2(C)$ is a finite set is known for weak-transverse curves in tori ([22], Theorem 1.2) and in any product of elliptic curves (see [33]).",
"That the height is bounded is proved using the Vojta inequality in a non-effective way.",
"Effective bounds for the height of $S_2(C)$ are given for weak-transverse curves in tori in [9], using an effective Mordell-Lang Theorem, and in abelian varieties only for transverse curves in a product of elliptic curves in [32].",
"So this corollary is a first example of an effective bound for the height for a weak-transverse curve in abelian varieties.",
"In higher dimensions, the Bounded Height Conjecture proved by Habegger ([20], theorem at page 407), together with the Effective Bogomolov Bound by Galateau ([18], Theorem 1.1) and the non-density Theorem by Viada ([34], Theorem 1.6) imply the CIT for varieties with $V^{ta}$ non empty embedded in certain abelian varieties which include all CM abelian varieties.",
"The original formulation of the Bounded Height Theorem in [19] did not contain an explicit height bound and the author did not discuss the effectivity of the result (however, in a forthcoming paper, Habegger provides an explicit version of the Bounded Height Theorem for tori).",
"Even if the method is made effective, the assumption $V^{ta}\\ne \\varnothing $ is stronger than transversality.",
"In this respect, ours is a new effective method in the context of the CIT for weak-transverse varieties.",
"Many are the contributions on the CIT of several authors.",
"For a more extensive list, we refer to the references given in the papers mentioned above.",
"In the last part of the paper we generalise our method, obtaining an effective weak result for curves, related to the CIT.",
"Theorem REF gives a complete effective version of the following result.",
"Theorem 1.7 Let $C$ be a weak-transverse curve in $E^N$ , where $E$ is an elliptic curve with CM defined over a number field $k$ .",
"Then, $C\\cap \\cup _{{\\rm codim\\,}H>\\dim H}H$ is a finite set of cardinality and Néron-Tate height effectively bounded.",
"Here $H$ ranges over all algebraic subgroups of codimension larger than their dimension.",
"In particular every non-torsion point $Y_0\\in C\\cap \\cup _{{\\rm codim\\,}H>\\dim H}H$ satisfies $\\hat{h}(Y_0) &\\ll _\\eta (h(C)+\\deg C)^{\\frac{N+1}{2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{N-1}{2}+\\eta },\\\\[k(Y_0):\\mathbb {Q}] &\\ll _\\eta (\\deg C [k(C):k])^{\\frac{N+1}{N-1}+\\eta }((h(C)+\\deg C)[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}])^{\\frac{N+1}{2}+\\eta }.$ In the following section we emphasise the implications of these theorems on the effective and quantitative Mordell-Lang Conjecture."
],
[
"Applications to the effective Mordell-Lang Conjecture",
"The CIT is well known to have implications on the Mordell-Lang Conjecture.",
"The toric case of this conjecture has been extensively studied, also in its effective form, by many authors.",
"A completely effective version in the toric case can be found in [7], Theorem 5.4.5, and generalisations in [8].",
"However an effective general result in abelian varieties is not known.",
"Since we prove an effective version of Theorem REF , we obtain the following effective cases of the Mordell-Lang Conjecture for curves in products of elliptic curves with CM, where effective means that we give a bound for the height of the set of points in $C$ and in a group of finite rank $\\Gamma $ ; the dependence on $C$ and $\\Gamma $ is completely explicit.",
"The assumption on the relative codimension gives a condition on the rank of $\\Gamma $ .",
"This method is completely different from the two classical effective methods known in abelian varieties.",
"The method by Chabauty and Coleman, surveyed, for example, in [27], gives a Mordell-Lang statement for a curve in its Jacobian and $\\Gamma $ of rank less than the genus; it involves a Selmer group calculation and smart computations with the Jacobian of the curve, and it can be made effective in some particular cases (see [23]).",
"The method by Manin and Demjanenko, described in Chapter 5.2 of [31], gives an effective Mordell theorem for curves with many independent morphisms to an abelian variety.",
"As remarked by Serre, the method remains of difficult application.",
"Let $E$ be an elliptic curve defined over the algebraic numbers.",
"We let $\\hat{h}$ be the standard Néron-Tate height on $E^N$ ; if $V$ is a subvariety of $E^N$ , we shall denote by $h(V)$ the normalized height of $V$ , as defined in [25].",
"The height of a set is as usual the supremum of the heights of its points.",
"If $E$ is defined over a field $k$ , we denote by $k_\\mathrm {tor}$ the field of definition of all torsion points of $E$ .",
"All the constants in the following theorems become explicit if the constant for the Lehmer type bound of Carrizosa in [13] is made explicit.",
"Theorem 2.1 Let $C$ be a weak-transverse curve in $E^N$ , with $E$ a CM elliptic curve and $N>2$ .",
"Let $k$ be a field of definition for $E$ .",
"Let $\\Gamma $ be a subgroup of $E^N$ such that the group generated by its coordinates is an ${\\mathrm {End}}(E)$ -module of rank one.",
"Then, for any positive $\\eta $ , there exists a constant $c_1$ , depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_1 (h(C)+\\deg C)^{\\frac{N-1}{N-2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{1}{N-2}+\\eta }.$ Let $g$ be a generator of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .",
"If a point $x=(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $0\\ne a_i,b_i \\in {\\mathrm {End}}(E)$ and torsion points $\\zeta _i\\in E$ such that $a_ix_i=b_i g+\\zeta _i, \\qquad i=1,\\cdots ,N.$ If all $b_i=0$ then $x$ is a torsion point, thus it has height zero.",
"We can suppose, for instance, that $b_1\\ne 0$ .",
"So $g$ depends on $x_1$ .",
"Substituting this dependence in the last $N-1$ equations, we obtain a system of $N-1$ linearly independent equations in the variables $x_1, \\dots ,x_N$ given by $ b_1 a_j x_j=a_1 b_j x_1+\\zeta ^{\\prime }_j, \\qquad j=2,\\cdots ,N$ for $\\zeta ^{\\prime }_j$ torsion points.",
"These equations define a torsion variety $H$ of codimension $N-1$ in $E^N$ .",
"Thus, $ (C\\cap \\Gamma )\\subseteq (C\\cap \\cup _{\\dim H=1}H) \\cup (C\\cap {\\mathrm {Tor}}_{E^N})=S_{N-1}(C)$ for $H$ ranging over all algebraic subgroup of dimension one.",
"However, if $N>2$ , any point $x$ in the intersection $C\\cap H$ is a $C$ -torsion anomalous point.",
"In addition, as $C$ is weak-transverse, each such $C$ -torsion anomalous point is maximal.",
"If $x$ is not a torsion point then $H$ is minimal for $x$ , because $H$ has dimension one.",
"Thus the relative codimension of $x$ in $H$ is one.",
"Applying Theorem REF we deduce the bound.",
"If $N=2$ , the intersection $C \\cap H$ is not torsion anomalous, so we must follow another line.",
"For this reason, we need the assumption of $C$ being transverse.",
"Theorem 2.2 Let $C$ be a transverse curve in $E^2$ with $E$ a CM elliptic curve defined over a number field $k$ .",
"Let $\\Gamma $ be a subgroup of $E^2$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank one, generated by $g$ .",
"Then, for any positive $\\eta $ there exists a constant $c_2$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_2[k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}]^{1+\\eta }(h(C)+(\\hat{h}(g)+1)\\deg C)^{2+\\eta }.$ Let $g$ be a generator of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .",
"Consider the curve $C^{\\prime }=C\\times g$ in $E^3$ .",
"Since $C$ is transverse then $C^{\\prime }$ is weak-transverse.",
"If a point $(x_1, x_2)$ is in $\\Gamma $ then there exist $0\\ne a_i,b_i \\in {\\mathrm {End}}(E)$ and torsion points $\\zeta _i$ such that $a_1x_1=b_1 g+\\zeta _1 , \\,\\,\\, \\,\\,\\, a_2x_2=b_2 g+\\zeta _2.$ Thus the point $(x_1,x_2,g) $ belongs to the intersection $C^{\\prime }\\cap H$ with $H$ the torsion variety of codimension 2 in $E^3$ defined by the equations $a_1x_1=b_1 x_3+\\zeta _1, \\quad a_2x_2=b_2 x_3+\\zeta _2.$ Thus $(x_1,x_2,g)\\in (C^{\\prime }\\cap \\cup _{\\dim H=1}H)$ for $H$ ranging over all algebraic subgroups of dimension one in $E^3$ .",
"Therefore $C\\cap \\Gamma $ is embedded in $C^{\\prime }\\cap \\cup _{\\dim H=1}H$ and $\\hat{h}(C\\cap \\Gamma )\\le \\hat{h}(C^{\\prime }\\cap \\cup _{\\dim H=1}H)$ .",
"However any point $x$ in the intersection $C^{\\prime }\\cap H$ is a maximal $C^{\\prime }$ -torsion anomalous point.",
"If $x$ is not a torsion point, then it has relative codimension one in $H$ .",
"We apply Theorem REF to $C^{\\prime }\\subseteq E^3$ , with $\\deg C^{\\prime }=\\deg C$ and $h(C^{\\prime })\\le 2( h(C)+\\hat{h}(g)\\deg C)$ by Zhang's inequality, recalled in Section REF below.",
"This gives the bound for the height of a point in $C\\cap \\Gamma $ .",
"The assumptions on the curve are necessary in both theorems.",
"Indeed for a weak-transverse curve $C$ in $E^2$ the above theorem is not true.",
"Consider the weak-transverse curve $C=E\\times g$ ; for any positive integer $m$ , let $\\Gamma _m$ be generated by the point $\\gamma _m=(mg,g)$ .",
"For any $m$ , the point $\\gamma _m$ belongs to $C$ and, for $m$ which goes to infinity, the height of $\\gamma _m$ tends to infinity.",
"Thus there cannot be general bounds independent of $\\Gamma $ .",
"This does not happen in higher codimension.",
"The analogue would be $C=E\\times g\\times g^{\\prime }$ where $g$ and $g^{\\prime }$ are linearly independent to ensure the weak-transversality.",
"No point of $C$ can be in $\\Gamma $ , as no point of the type $(x,g,g^{\\prime })$ has coordinates in $\\overline{\\Gamma }$ , which has rank one.",
"We also remark that for $C$ weak-transverse, $C\\times g$ is not necessarily weak-transverse: let $C=E\\times g$ , then $E\\times g\\times g$ is contained in the abelian subvariety $x_2=x_3$ .",
"With this method, the bound in the case $N=2$ depends on $g$ .",
"It is possible to remove such a dependence.",
"However one has to prove that for a transverse curve $C$ , the set $C\\cap \\cup _{\\dim H=1}H$ has bounded height.",
"This is proved in [32] Theorem 1.",
"The proof is effective, but not explicit in the dependence on $C$ and on $E$ .",
"The dependence on $C$ is given comparing the height function relative to a point with the height function relative to the intersections of $C\\cap \\Delta ^-$ where $ \\Delta ^-$ is the divisor defined by $x_1=-x_2$ .",
"We use [32] to deduce another effective case of the Mordell-Lang Conjecture.",
"Theorem 2.3 Let $C$ be a transverse curve in $E^N$ with $E$ an elliptic curve defined over a number field.",
"Let $\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank at most $N-1$ .",
"Then the set $C\\cap \\Gamma $ is finite of Néron-Tate height bounded by a constant $c(C,E^N)$ depending on $C$ and $E^N$ i.e.",
"$\\hat{h}(C\\cap \\Gamma ) \\le c(C,E^N).$ The proof follows the same idea as the proof of Theorem REF .",
"Let $g_1, \\dots , g_{N-1}$ be generators of $\\overline{\\Gamma }$ , the group generated by all the coordinates of any element in $\\Gamma $ .",
"If a point $(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $a_i\\in {\\mathrm {End}}(E)$ , an $N\\times (N-1)$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1, \\ldots , a_Nx_N)^{t}=B\\cdot (g_1,\\ldots ,g_{N-1})^{t}+\\zeta .$ Explicitating $(g_1, \\dots , g_{N-1})$ in terms of $x_1, \\dots , x_{N-1}$ from the first $N-1$ equations and substituting in the last equation, we obtain one equation in the variables $x_1, \\dots ,x_N$ $ a^{\\prime }_1x_1+ \\dots +a^{\\prime }_Nx_N=\\zeta ^{\\prime },$ where $\\zeta ^{\\prime }\\in E$ is a torsion point and $a^{\\prime }_i\\in {\\mathrm {End}}(E)$ .",
"This equation defines a torsion variety $H$ of codimension one in $E^N$ .",
"Thus, $ (C\\cap \\Gamma )\\subseteq (C\\cap \\cup _{{\\rm codim\\,}H\\ge 1}H)=S_1(C)$ for $H$ ranging over all algebraic subgroups of codimension at least one.",
"By [32] Theorem 1, this set has height bounded by an effective constant depending only on $E^N$ and $C$ .",
"At this point we want to extend our main theorem, at least in the case of curves, to see if with our method we could prove other cases of the Mordell-Lang Conjecture.",
"Theorem REF enables us to obtain a more general version of Theorem REF for $C$ weak-transverse and larger rank of $\\overline{\\Gamma }$ .",
"We also obtain a theorem similar to Theorem REF , with explicit dependence on $C$ .",
"Theorem 2.4 Let $C$ be a weak-transverse curve in $E^N$ with $E$ an elliptic curve with CM.",
"Let $k$ be a field of definition for $E$ .",
"Let $\\Gamma $ be a subgroup of $E^N$ such that the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank $ t< N/2$ .",
"Then, for any positive $\\eta $ , there exists a constant $c_3$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_3(h(C)+\\deg C)^{\\frac{N-t}{N-2t}+\\eta }[k_\\mathrm {tor}(C):k_\\mathrm {tor}]^{\\frac{t}{N-2t}+\\eta }.$ Let $g_1, \\dots , g_t$ be generators of the free part of $\\overline{\\Gamma }$ , the group generated by all coordinates of any element in $\\Gamma $ .",
"If a point $(x_1,\\dots , x_N)$ is in $\\Gamma $ then there exist $0\\ne a_i\\in {\\mathrm {End}}(E)$ , an $N\\times t$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (g_1,\\ldots ,g_t)^{t}+\\zeta .$ If the rank of $B$ is zero, then $x$ is a torsion point and so it has height zero.",
"If $B$ has positive rank $m$ , we can choose $m$ equations of the system corresponding to $m$ linearly independent rows of $B$ .",
"We use these equations to write the $g_j$ in terms of the $x_i$ and we substitute these expressions in the remaining equations.",
"We obtain a system of maximal rank with $N-m\\ge N-t$ linearly independent equations in the variables $x_1, \\dots ,x_N$ : ${\\left\\lbrace \\begin{array}{ll}a^{\\prime }_{11}x_1+ \\dots +a^{\\prime }_{1N}x_N=\\zeta ^{\\prime }_1\\\\\\vdots \\\\a^{\\prime }_{N-m,1}x_1+ \\dots +a^{\\prime }_{N-m,N}x_N=\\zeta ^{\\prime }_{N-m}\\end{array}\\right.",
"}$ where $\\zeta ^{\\prime }_i\\in E$ are torsion points and $a^{\\prime }_{ij}\\in {\\mathrm {End}}(E)$ .",
"These equations define a torsion variety $H$ of codimension $N-m$ in $E^N$ .",
"Thus $ (C\\cap \\Gamma )\\subseteq S_{ N-t}(C)$ for $H$ ranging over all algebraic subgroups of codimension at least $N-t$ .",
"However $\\dim H=N-{\\rm codim\\,}H=m\\le t$ .",
"Thus, if $N>2t$ $ S_{ N-t}(C)\\subseteq (C\\cap \\cup _{{\\rm codim\\,}H>\\dim H}H).$ Applying Corollary REF with $r=N-t$ gives the wished bound for the height of $C\\cap \\Gamma $ .",
"We notice that Theorems REF and REF are proved for weak-transverse curves, while previous effective results assumed transversality.",
"If we assume the transversality of $C$ we can relax the hypothesis on the rank of $\\overline{\\Gamma }$ .",
"Theorem 2.5 Let $C$ be a transverse curve in $E^N$ with $E$ a CM elliptic curve defined over $k$ .",
"Let $\\Gamma $ be a subgroup of $E^N$ such that the free part of the group of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank $t\\le N-1$ , generated by $g_1, \\dots , g_t$ .",
"Then, for any positive $\\eta $ there exists a constant $c_4$ depending only on $E^N$ and $\\eta $ , such that the set $C\\cap \\Gamma $ has Néron-Tate height bounded as $\\hat{h}(C\\cap \\Gamma ) \\le c_4 [k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}]^{\\frac{t}{N-t}+\\eta }(h(C)+ (\\hat{h}(g)+1)\\deg C)^{\\frac{N}{N-t}+\\eta }$ where $g=(g_1, \\dots , g_t)$ .",
"Consider the curve $C^{\\prime }=C\\times g$ in $E^{N+t}$ .",
"Since $C$ is transverse then $C^{\\prime }$ is weak-transverse.",
"If a point $(x_1, \\dots , x_N)$ is in $\\Gamma $ , then there exist $0\\ne a_i\\in {\\mathrm {End}}(E)$ , an $N\\times t$ matrix $B$ with coefficients in ${\\mathrm {End}}(E)$ and a torsion point $\\zeta \\in E^N$ such that $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (g_1,\\ldots ,g_t)^{t}+\\zeta .$ Thus the point $(x_1,\\dots x_N,g_1, \\dots , g_t) $ belongs to the intersection $C^{\\prime }\\cap H$ with $H$ the torsion variety of codimension $N$ and dimension $t$ in $E^{N+t}$ defined by the equations $(a_1x_1,\\ldots ,a_Nx_N)^{t}=B (y_{N+1},\\cdots , y_{N+t})^{t}+\\zeta .$ Thus, $C\\cap \\Gamma $ is embedded in $C^{\\prime }\\cap \\cup _{\\dim H=t}H$ , and $\\hat{h}(C\\cap \\Gamma )\\le \\hat{h}(C^{\\prime }\\cap \\cup _{\\dim H=t}H)$ for $H$ ranging over all algebraic subgroups of dimension $t$ .",
"If $N> t$ then ${\\rm codim\\,}H>\\dim H$ .",
"The bound for the height is then given by Corollary REF applied to $C^{\\prime }\\subseteq E^{N+t}$ , where $\\deg C=\\deg C^{\\prime }$ , $h(C^{\\prime })\\le 2( h(C)+\\hat{h}(g)\\deg C)$ by Zhang's inequality, recalled in Section REF , below."
],
[
"Applications to the quantitative Mordell-Lang Conjecture",
"In [29], Theorem 1.2 Rémond gives a bound on the cardinality of the intersection $C\\cap \\Gamma $ for a transverse curve in $E^ N$ and $\\Gamma $ a $\\mathbb {Z}$ -module of rank $r$ .",
"He obtains the following bound $\\#(C\\cap \\Gamma )\\le {c(E^N, \\mathcal {L})}^{r +1}(\\deg C)^{(r +1) N^{20}},$ where $c(E^N, \\mathcal {L})$ is a positive effective constant depending on $E^N$ and on the choice of an invertible, symmetric and ample sheaf $\\mathcal {L}$ on $E^N$ .",
"The Manin-Mumford Conjecture is a special case of the Mordell-Lang conjecture.",
"Explicit bounds on the number of torsion points in $C$ are given, for instance, by David and Philippon in [17] and by Hrushovski in [21].",
"David and Philippon in [17], Proposition 1.12 show that the number of torsion points on a non-torsion curve $C$ is at most $\\#(C\\cap \\mathrm {Tor}_{E^N})\\le (10^{2N+53}\\deg C)^{33},$ where $\\mathrm {Tor}_{E^N}$ is the set of all torsion points of $E^N$ .",
"Our method enables us to give a sharp bound for the number of non torsion points in $C\\cap \\Gamma $ for $C$ weak-transverse in $E^N$ , which, together with the just mentioned bounds for the torsion, improves in some cases the bounds of Rémond.",
"Notice that below we use the rank $t$ of $\\overline{\\Gamma }$ , the $\\mathrm {End}(E)$ -module of the coordinates of $\\Gamma $ .",
"To compare with Rémond's result, we can use the trivial relation $r<2Nt$ and $t<Nr$ .",
"The following theorem is obtained combining the results from Section REF with bounds from Theorems REF and Corollary REF .",
"Theorem 2.6 Let $C$ be a curve in $E^N$ , where $E$ has CM and is defined over a number field $k$ .",
"Let $\\Gamma $ be a subgroup of $E^N$ such that the group $\\overline{\\Gamma }$ of its coordinates is an ${\\mathrm {End}}(E)$ -module of rank $t$ .",
"Let $S$ be the number of non-torsion points in the intersection $C\\cap \\Gamma $ .",
"Then, for every positive real $\\eta $ there exist constants $d_1, d_2,d_3, d_4$ depending only on $E^N$ and $\\eta $ , such that: if $C$ is weak-transverse in $E^N$ , $N>2$ and $t=1$ , we have $S\\le d_1 & {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{(N-1)(4N^2-N-4)}{2(N-2)^2}+\\eta }(\\deg C)^{\\frac{2N^3-N^2+N-4}{2(N-2)}+\\eta }\\\\&\\cdot [k(C):k]^{\\frac{N(N-1)(2N+1)}{2(N-2)}+\\eta };$ if $C$ is transverse in $E^2$ and $t=1$ , we have $S\\le d_2 &{([k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}](h(C)+(\\hat{h}(g)+1)\\deg C))}^{29+\\eta }(\\deg C)^{22+\\eta }\\\\&\\cdot [k(C\\times g):k]^{21+\\eta }$ where $g$ is a generator of $\\overline{\\Gamma }$ ; if $C$ is weak-transverse in $E^N$ and $t<N/2$ , we have $S\\le d_3& {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{t(N-t)(4N^2-2Nt+N-2t-2)}{2(N-2t)(N-t-1)}+\\eta }(\\deg C)^{1+\\frac{N(2N+1)(N-t)}{2(N-t-1)}+\\eta }\\\\&\\cdot [k(C):k]^{\\frac{N(2N+1)(N-t)}{2(N-t-1)}+\\eta };$ if $C$ is transverse in $E^N$ and $t\\le N-1$ , we have $ S\\le d_4 & {([k_\\mathrm {tor}(C\\times g):k_\\mathrm {tor}](h(C)+(\\hat{h}(g)+1)\\deg C))}^{\\frac{Nt(4N^2+2t^2+6Nt+N-t-2)}{2(N-t)(N-1)}+\\eta }\\\\&\\cdot (\\deg C)^{1+\\frac{(N+t)N(2N+2t+1)}{2(N-1)}+\\eta }[k(C\\times g):k]^{\\frac{(N+t)N(2N+2t+1)}{2(N-1)}+\\eta },$ where $\\overline{\\Gamma }$ is generated by $g_1,\\ldots ,g_t$ and $g=(g_1,\\ldots ,g_t)$ .",
"Cases REF and REF follow from the proofs of Theorem REF and Theorem REF , respectively, together with the bounds from Theorem REF .",
"Cases REF and REF follow from the proofs of Theorem REF and Theorem REF , respectively, together with the bounds from Corollary REF ."
],
[
"Torsion anomalous varieties: preliminary results ",
"In this section we denote by $G$ an abelian variety or a torus and by $\\mathrm {Tor}_G$ the set of all torsion points in $G$ .",
"In the next statements we consider subvarieties $V$ of $G$ ."
],
[
"Maximality and minimality",
"To show that the set of all $V$ -torsion anomalous varieties is non-dense, we only need to consider maximal ones.",
"We recall from the introduction the following definition.",
"W̥e say that a $V$ -torsion anomalous variety $Y$ is maximal if it is not contained in any $V$ -torsion anomalous variety of strictly larger dimension.",
"On the other hand, a variety $Y$ can be a component of the intersection of $V$ with different torsion varieties.",
"We want to choose the minimal torsion variety which makes $Y$ anomalous.",
"L̥et $Y$ be a $V$ -torsion anomalous variety.",
"We say that the irreducible torsion variety $B+\\zeta $ is minimal for $Y$ if $Y$ is an irreducible component of $V\\cap (B+\\zeta )$ , ${\\rm codim\\,}Y<{\\rm codim\\,}V+{\\rm codim\\,}B,$ and $B+\\zeta $ has minimal dimension among the irreducible torsion varieties with such properties.",
"Note that the minimal torsion variety for $Y$ is unique.",
"Indeed if $B+\\zeta $ and $B^{\\prime }+\\zeta ^{\\prime }$ are two torsion varieties which are minimal for $Y$ , then also $ (B+\\zeta )\\cap (B^{\\prime }+\\zeta ^{\\prime })$ is minimal for $Y$ , and by minimality $\\dim ((B+\\zeta )\\cap (B^{\\prime }+\\zeta ^{\\prime }))=\\dim (B+\\zeta )$ .",
"So $(B^{\\prime }+\\zeta ^{\\prime })\\subseteq (B+\\zeta )$ .",
"But $(B+\\zeta )$ and $(B^{\\prime }+\\zeta ^{\\prime })$ are irreducible and reduced, thus $(B+\\zeta )=(B^{\\prime }+\\zeta ^{\\prime })$ ."
],
[
"Relative position of the torsion anomalous varieties",
"Without loss of generality we can work with a maximal $V$ -torsion anomalous $Y$ and its minimal torsion variety $B+\\zeta $ .",
"The maximality for $Y$ avoids redundancy and the minimality assures the weak-transversality of $Y$ in $B+\\zeta $ , as defined below.",
"The relative position of a $V$ -torsion anomalous variety $Y$ in $B+\\zeta $ is determinant and leads to the following natural definition.",
"Ḁn irreducible variety $Y$ is weak-transverse in a torsion variety $B+\\zeta $ if $Y\\subseteq (B+\\zeta )$ and $Y$ is not contained in any proper torsion subvariety of $B+\\zeta $ .",
"Similarly $Y$ is transverse in a translate $B+p$ if $Y\\subseteq (B+p)$ is not contained in any translate strictly contained in $B+p$ .",
"The codimension of $Y$ in $B+\\zeta $ is called the relative codimension of $Y$ in $B+\\zeta $; we simply say the relative codimension of $Y$ if $Y$ is $V$ -torsion anomalous and $B+\\zeta $ is minimal for $Y$ .",
"Then, we have the following lemma.",
"Lemma 3.1 Let $Y$ be a maximal $V$ -torsion anomalous variety and let $B+\\zeta $ be minimal for $Y$ .",
"Then $Y$ is weak-transverse in $B+\\zeta $ .",
"Assume that $Y$ is not weak-transverse in $B+\\zeta $ , then it is contained in an irreducible torsion subvariety $B^{\\prime }+\\zeta ^{\\prime }$ of $B+\\zeta $ with ${\\rm codim\\,}(B^{\\prime }+\\zeta ^{\\prime })>{\\rm codim\\,}(B+\\zeta )$ .",
"So $Y$ is a component of $ V \\cap (B^{\\prime }+\\zeta ^{\\prime })$ .",
"In addition ${\\rm codim\\,}Y <{\\rm codim\\,}V+{\\rm codim\\,}(B+\\zeta )<{\\rm codim\\,}V+{\\rm codim\\,}(B^{\\prime }+\\zeta ^{\\prime }),$ which contradicts the minimality of $ (B+\\zeta )$ ."
],
[
"Torsion anomalous varieties as components of different intersections",
"In the next lemma we prove that every $V$ -torsion anomalous variety which is a component of $V\\cap (B+\\zeta )$ , is also a component of any intersection $V\\cap (A+\\zeta ^{\\prime })$ with $B+\\zeta \\subseteq A+\\zeta ^{\\prime }$ .",
"Moreover, we can choose $\\zeta =\\zeta ^{\\prime }$ , in fact if a translate $B+p$ is contained in another $ A+p^{\\prime }$ , then $A+p=A+p^{\\prime }$ .",
"Indeed $B$ is a subgroup and it contains 0, so $p-p^{\\prime }\\in A$ and $A+p^{\\prime }=A+(p-p^{\\prime })+p^{\\prime }=A+p$ .",
"Lemma 3.2 Let $Y$ be a maximal $V$ -torsion anomalous variety, and let $B+\\zeta $ be minimal for $Y$ .",
"Then $Y$ is a component of $V\\cap (A+\\zeta )$ for every algebraic subgroup $A\\supseteq B$ , with ${\\rm codim\\,}A\\ge \\dim V-\\dim Y$ .",
"Clearly $Y\\subseteq V\\cap (A+\\zeta )$ .",
"Let $X$ be an irreducible component of $V\\cap (A+\\zeta )$ which contains $Y$ , then ${\\rm codim\\,}X \\le {\\rm codim\\,}V+{\\rm codim\\,}A.$ If the inequality is strict then $X$ is anomalous and by the maximality of $Y$ we get $Y=X$ .",
"Otherwise, since by assumption ${\\rm codim\\,}A\\ge \\dim V-\\dim Y$ , we have ${\\rm codim\\,}X = {\\rm codim\\,}V+{\\rm codim\\,}A\\ge {\\rm codim\\,}V +\\dim V-\\dim Y={\\rm codim\\,}Y,$ but $Y\\subseteq X$ , so the opposite inequality also holds.",
"This implies ${\\rm codim\\,}X={\\rm codim\\,}Y$ .",
"Therefore both varieties are irreducible, so $Y=X$ and $Y$ is a component of $V\\cap (A+\\zeta )$ ."
],
[
"No torsion anomalous varieties on a weak-transverse translate",
"The simple choice of maximal and minimal varieties and the group structure allow us to prove Conjecture REF for weak-transverse translates.",
"Let $Y$ be an $(H+p)$ -torsion anomalous variety and let $B+\\zeta $ be minimal for $Y$ .",
"Then $Y$ is a component of $(H+p)\\cap (B+\\zeta )$ and $\\dim H+\\dim B-\\dim Y<N.$ Remark that whenever $(H+p)\\cap (B+\\zeta )\\ne \\varnothing $ , then $p=h+b+\\zeta $ for some $h\\in H$ and $b\\in B$ .",
"Thus $(H+p)\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+b+\\zeta )=(H\\cap B)+b+\\zeta $ and $\\dim (H\\cap B)=\\dim ((H+p) \\cap (B+\\zeta )).$ In our case $Y\\subseteq (H+p)\\cap (B+\\zeta )$ , then $\\dim (H\\cap B)\\ge \\dim Y.$ Thus $\\dim (H+B)=\\dim H +\\dim B -\\dim (H\\cap B)\\le \\dim H +\\dim B -\\dim Y <N,$ where the last inequality is obtained from the fact that $Y$ is $V$ -torsion anomalous.",
"Therefore $H+B$ is a proper algebraic subgroup of the ambient variety.",
"Since $p\\in H+B+\\zeta $ , then $H+p\\subseteq H+B+\\zeta $ , against the weak-transversality of $H+p$ ."
],
[
"Finitely many maximal $V$ -torsion anomalous varieties in {{formula:7fcfe8cd-fb0f-41fd-93eb-571843c691f8}}",
"Let $V$ be a weak-transverse variety in $G$ .",
"Let us fix an irreducible torsion subvariety $B$ of $G$ ; we end this section by showing the finiteness of the maximal $V$ -torsion anomalous varieties in $ V\\cap (B+{\\mathrm {Tor}}_{G})$ for which $B+\\zeta $ is minimal, for some torsion point $\\zeta $ .",
"This result implies the finiteness of the maximal $V$ -torsion anomalous, if one can uniformly bound the degree of the corresponding minimal torsion variety $B+\\zeta $ .",
"We fist prove that the maximal $V$ -torsion anomalous components in $B+{\\mathrm {Tor}}_{G}$ are non-dense.",
"Lemma 3.3 Let $V$ be a weak-transverse variety in $G$ .",
"Let $B$ be an abelian subvariety of $G$ .",
"Then the set of $V$ -torsion anomalous varieties in $V\\cap (B+{\\mathrm {Tor}_{G}})$ is non-dense in $V$ .",
"Suppose that there exists a dense subset of maximal $V$ -torsion anomalous varieties which are components of $V\\cap (B+\\zeta )$ , for $\\zeta $ ranging over all torsion points of $G$ .",
"By the box principle, there exists a dense subset of $d$ -dimensional $V$ -torsion anomalous varieties $Y_i\\subseteq V\\cap (B+\\zeta _i)$ , for $\\zeta _i$ torsion points, where $d$ is the maximal integer having this property.",
"Consider the natural projection $\\pi _B: G\\rightarrow G/B$ .",
"As the $Y_i$ are dense and project to torsion points, the dimensional formula tells us that $\\dim V = \\dim Y_i + \\dim \\pi _B(V).$ Note that $V$ is not $V$ -torsion anomalous, because $V$ is weak-transverse, thus $\\dim Y_i<\\dim V$ and $\\dim \\pi _B(V)$ is at least one.",
"However the $Y_i$ are anomalous therefore ${\\rm codim\\,}Y_i <{\\rm codim\\,}V + {\\rm codim\\,}B.$ We deduce that $\\dim \\pi _B(V) < {\\rm codim\\,}B=\\dim G/B.$ This shows that ${\\pi _B}_{|V}$ is not surjective on $G/B$ .",
"Since $V$ is weak-transverse, also $\\pi _B(V)$ is weak-transverse in $A/B$ .",
"Notice that the image via $\\pi $ of any $V$ -torsion anomalous component in $B+{\\mathrm {Tor}}_{G}$ is a torsion point.",
"By the Manin-Mumford Conjecture the closure of the torsion of $\\pi _B(V)$ is non-dense, thus also its preimage is non-dense in $V$ .",
"This contradicts the density of the $Y_i$ .",
"An application of this same result actually shows the finiteness of the maximal $V$ -torsion anomalous varieties in $ V\\cap (B+{\\mathrm {Tor}}_{G})$ for which $B+\\zeta $ is minimal, for some torsion point $\\zeta $ .",
"Lemma 3.4 Let $V$ be weak-transverse in $G$ .",
"Let $B$ be an abelian subvariety of $G$ .",
"Then there exist only finitely many torsion points $\\zeta $ such that $V\\cap (B+\\zeta )$ has a maximal $V$ -torsion anomalous component for which $B+\\zeta $ is minimal.",
"Let $\\dim G=N$ .",
"From Lemma REF the closure of all maximal $V$ -torsion anomalous $Y_i$ which are components of $V\\cap (B+\\mathrm {Tor}_{G})$ is a proper closed subset of $V$ .",
"In particular, if we consider only those for which $B+\\zeta $ is minimal, their closure is still a proper closed subset.",
"We write it as the finite union of its irreducible components $X_1 \\cup \\dots \\cup X_R$ .",
"We now show that the $X_j$ are exactly all the maximal $V$ -torsion anomalous components such that $B+\\zeta $ is minimal, for some $\\zeta \\in \\mathrm {Tor}_{G}$ .",
"Suppose that $X_1$ is not a maximal $V$ -torsion anomalous component for which $B+\\zeta $ is minimal, with $\\zeta \\in \\mathrm {Tor}_{G}$ .",
"Then some of the equidimensional $Y_i$ are dense in $X_1$ , $\\dim X_1> \\dim Y_i$ and $X_1$ is not $V$ -torsion anomalous due to the maximality of the $Y_i$ .",
"By assumption, $B+\\zeta _i$ is minimal for $Y_i$ for some $\\zeta _i\\in {\\mathrm {Tor}_{G}}$ .",
"Then $X_1$ is not contained in any $B+\\zeta $ , otherwise $Y_i\\subseteq X_1\\subseteq V\\cap (B+\\zeta )$ and $X_1$ would be a $V$ -torsion anomalous variety; this gives a contradiction.",
"Let $H+\\zeta $ be the torsion variety (not necessarily proper) of smallest dimension containing $X_1$ .",
"Then $X_1$ is weak-transverse in $H+\\zeta $ , and since $X_1$ is not $V$ -torsion anomalous $N-\\dim X_1=N-\\dim V+N-\\dim H.$ Recall that, the $Y_i$ are $V$ -torsion anomalous varieties and use (REF ) to obtain $N-\\dim Y_i<N-\\dim V+N-\\dim B=\\dim H-\\dim X_1+N-\\dim B.$ whence $ \\dim H-\\dim Y_i<\\dim H-\\dim X_1+\\dim H-\\dim B.$ Since $B+\\zeta _i$ is minimal for $Y_i$ , we see that $(B+\\zeta _i)\\subseteq (H+\\zeta )$ and thus $(B+\\zeta _i-\\zeta )\\subseteq H$ .",
"Translating every variety by $\\zeta $ , we obtain that $X_1-\\zeta $ is weak-transverse in $H$ and the $Y_i-\\zeta $ are dense in $X_1-\\zeta $ and $(X_1-\\zeta )$ -torsion anomalous.",
"This contradicts Lemma REF applied to $X_1-\\zeta $ in $H$ ."
],
[
"Notation",
"Recall that all varieties are assumed to be defined over the field of algebraic numbers.",
"Let $A$ be an abelian variety; to a symmetric ample line bundle ${\\mathcal {L}}$ on $A$ we attach an embedding $i_{\\mathcal {L}}: A\\hookrightarrow \\mathbb {P}^m$ defined by the minimal power of ${\\mathcal {L}}$ which is very ample.",
"Heights and degrees corresponding to ${\\mathcal {L}}$ are computed via such an embedding.",
"More precisely, the degree of a subvariety of $A$ is the degree of its image under $i_{\\mathcal {L}}$ ; $\\hat{h}=\\hat{h}_{\\mathcal {L}}$ is the ${\\mathcal {L}}$ -canonical Néron-Tate height of a point in $A$ , and $h$ is the normalized height of a subvariety of $A$ as defined, for instance, in [25].",
"Most often we consider products of an elliptic curve $E$ .",
"Then, we denote by ${\\mathcal {O}}_1$ the line bundle on $E$ defined by the neutral element, and by ${\\mathcal {O}_{N}}$ the bundle on $E^N$ obtained as the tensor product of the pull-backs of ${\\mathcal {O}}_1$ through the $N$ natural projections.",
"Unless otherwise specified, we compute degrees and heights on $E^N$ with respect to ${\\mathcal {O}_{N}}$ .",
"We note that by a result of Masser and Wüstholz in [24] Lemma 2.2, every abelian subvariety of $E^N$ is defined over a finite extension of $k$ of degree bounded by $3^{16 N^4}$ .",
"For this reason we always assume that all abelian subvarieties are defined over $k$ .",
"Up to a field extension of degree two, we also assume that every endomorphism of $E$ is defined over $k$ .",
"By $\\ll $ we always denote an inequality up to a multiplicative constant depending only on $E$ and $N$ ."
],
[
"Subgroups and torsion varieties",
"Let $B+\\zeta $ be an irreducible torsion variety of $E^N$ with ${\\rm codim\\,}B=r$ .",
"We associate $B$ with a morphism $\\varphi _B:E^N \\rightarrow E^{r}$ such that $\\ker \\varphi _B=B+\\tau $ with $\\tau $ a torsion set of absolutely bounded cardinality (by [24] Lemma 1.3).",
"In turn $\\varphi _B$ is identified with a matrix in $\\mathrm {Mat}_{r\\times N}(\\mathrm {End}(E))$ of rank $r$ such that the degree of $B$ is essentially (up to constants depending only on $N$ ) the sum of the squares of the determinants of the minors of $\\varphi _B$ .",
"By Minkosky's theorem, such a sum is essentially the product of the squares $d_i$ of the norms of the rows of the matrix representing $ \\varphi _B$ (see for instance [14] for more details).",
"In short $B+\\zeta $ is a component of the torsion variety given as the zero set of forms $h_1, \\dots , h_r$ , which are the rows of $\\varphi _B$ , of degree $d_i$ .",
"In addition $d_1 \\cdots d_r \\ll \\deg (B+\\zeta )\\ll d_1 \\cdots d_r .$ We assume to have ordered the $h_i$ by increasing degree.",
"We also recall that, as is well known, we can use Siegel's lemma to complete the matrix defining $B$ to a square invertible matrix; this gives a construction for the orthogonal complement $B^\\perp $ and shows that $\\#(B\\cap B^\\perp )\\ll (\\deg B)^2$ .",
"As remarked in [33], in a product of different elliptic curves an algebraic subgroup is associated with a matrix where the entries corresponding to the non-isogenous factors are all zero.",
"For this reason our theorems, which we prove in $E^N$ for simplicity, hold in products of different elliptic curves as well."
],
[
"The Zhang Estimate",
"We recall the following definition.",
"defiessmin For a variety $V\\subseteq A$ , the essential minimum $\\mu (V)$ is the supremum of the reals $\\theta $ such that the set $ \\lbrace x \\in V(\\overline{\\mathbb {Q}}) \\mid \\hat{h} (x)\\le \\theta \\rbrace $ is non-dense in $V$ .",
"The Bogomolov Conjecture, proved by Ullmo and Zhang in 1998, asserts that the essential minimum $\\mu (Y)$ is strictly positive if and only if $Y$ is non-torsion.",
"From the crucial result in Zhang's proof of the Bogomolov Conjecture (see [35]) and from the definition of normalized height, we have that for an irreducible subvariety $X$ of an abelian variety: $\\mu (X) \\le \\frac{h(X)}{\\deg X} \\le (1+\\dim X) \\mu (X).$"
],
[
"The Arithmetic Bézout theorem",
"The following version of the Arithmetic Bézout theorem is due to Philippon [26].",
"Theorem 4.1 (Philippon) Let $X$ and $Y$ be irreducible subvarieties of the projective space $\\mathbb {P}^n$ defined over $\\overline{\\mathbb {Q}}$ .",
"Let $Z_1, \\dots , Z_g$ be the irreducible components of $X\\cap Y$ .",
"Then $ \\sum _{i=1}^g h(Z_i)\\le \\deg X h(Y) +\\deg Y h(X) +c(n) \\deg X \\deg Y,$ where $c(n)$ is a constant depending only on $n$ ."
],
[
"An effective Bogomolov Estimate for relative transverse varieties",
"The following theorem is a sharp effective version of the Bogomolov Conjecture for weak-transverse varieties.",
"It is an elliptic analogue, up to a lower order term, of a toric conjecture of Amoroso and David in [1].",
"Theorem 4.2 (Checcoli-Veneziano-Viada) Let $E^N$ be a product of elliptic curves, and let $Y$ be an irreducible subvariety of $E^N$ transverse in a translate $B+p$ .",
"Then, for any $\\eta >0$ , there exists a positive constant $c_1$ depending on $E^N$ and $\\eta $ , such that $\\mu (Y) \\ge c_1 \\frac{(\\deg B)^{\\frac{1}{\\dim B-\\dim Y}-\\eta }}{(\\deg Y)^{{\\frac{1}{\\dim B-\\dim Y}}+ \\eta }}.$ This theorem is a special case of the main theorem of [14].",
"For our applications, the lower bound must take into account the degree of $B$ .",
"To do so one needs to consider several different line bundles on $E^N$ , and the proof in [14] is based on an equivalence between two line bundles, and on the lower bound for the essential minimum of a transverse variety with respect to the standard bundle ${\\mathcal {O}_{N}}$ .",
"Such a bound is given by Galateau in [18].",
"The result of Galateau is inspired by the toric version of Amoroso and David [1] Theorem 1.4.",
"The constant $c_1$ is effective and becomes explicit if the constants in [18] are made explicitIn a personal communication Galateau provided us explicit computations.."
],
[
"A relative Lehmer Estimate for points",
"In [13], Theorem 1.15, Carrizosa proves the so called relative Lehmer problem for CM abelian varieties.",
"Theorem 4.3 (Carrizosa) Let $A$ be an abelian defined over a number field $k$ and having CM.",
"Let $\\mathcal {L}_0$ be an ample symmetric bundle on $A$ , and let $\\mathcal {L}=\\mathcal {L}_0^{4}$ .",
"Let $H$ be an abelian subvariety of $A$ of dimension $g_0>0$ and $P$ be a point in $H$ , which is not a torsion point modulo all proper abelian subvarieties of $H$ .",
"Then there exists a positive constant $c(A, k, \\mathcal {L})$ depending only on $A$ , $k$ and $\\mathcal {L}$ such that $\\hat{h}_\\mathcal {L}(P)\\ge \\frac{c(A,k,\\mathcal {L})}{\\omega _{k_\\mathrm {tor}}(P,H)}\\left( \\frac{ \\log \\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}} H)^2)}{\\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}} H)^2)}\\right)^{\\kappa (g_0)}$ where $\\kappa (g_0)=2^{2g_0+1}g_0^{4g_0}(g_0+1)!^{2g_0}$ .",
"Here $k_\\mathrm {tor}=k(A_\\mathrm {tor})$ is the field of definition of all the torsion in $A$ and ${\\omega _{k_\\mathrm {tor}}(P,H)}$ is defined in the following way (see Definition 1.12 of [13]): $\\omega _{k_\\mathrm {tor}}(P,H)=\\min _V\\left( \\frac{\\deg _\\mathcal {L} V}{\\deg _\\mathcal {L} H} \\right)^{1/{\\rm codim\\,}_H V}$ where $V$ ranges over all proper subvarieties of $H$ defined over $k_\\mathrm {tor}$ , and containing $P$ .",
"This theorem generalises to abelian varieties a result of Ratazzi in [28] for one elliptic curve.",
"The proof of Ratazzi is inspired to the theorem of Amoroso and Zannier in [4] for algebraic numbers.",
"The effectivity in the relative Lehmer is not explicitly stated in the theorem of Carrizosa.In a short personal communication she claims that her constants are effective.",
"Using also the effectivity of other results, as for instance the result of Amoroso and Zannier in [4]Though in [4] the authors are not concerned with effectivity, their result can be made effective (see [2]).",
"and of David and Hindry [16], one may check that Carrizosa's constant can be made effective.",
"In addition, the complicated descent in her article can be replaced by the simple induction argument presented for tori in [3].",
"An analogous effective method for the relative Lehmer in tori is given by Delsinne, see [15].",
"As a straightforward corollary of Theorem REF we have the following: Theorem 4.4 Let $E$ be an elliptic curve with CM defined over a field $k$ .",
"Let $P$ be a point of infinite order in $E^N$ , and let $B+\\zeta $ be the torsion variety of minimal dimension containing $P$ , with $B$ an abelian subvariety and $\\zeta $ a torsion point.",
"Then for every $\\eta >0$ there exists a positive constant $c_2$ depending on $E^N$ and $\\eta $ , such that $\\hat{h}(P)\\ge c_2 \\frac{(\\deg B)^{\\frac{1}{\\dim B}-\\eta }}{[k_\\mathrm {tor}(P):k_\\mathrm {tor}]^{\\frac{1}{\\dim B}+ \\eta }}.$ We clarify how to obtain this theorem from Theorem REF .",
"We recall that we are assuming all abelian subvarieties to be defined over $k$ , We remark that $P-\\zeta \\in B$ .",
"In addition $\\hat{h}(P)=\\hat{h}(P-\\zeta )$ and $[k_\\mathrm {tor}(P):k_\\mathrm {tor}]=[k_\\mathrm {tor}(P-\\zeta ):k_\\mathrm {tor}]$ so we can apply Theorem REF to $P-\\zeta $ in $B$ .",
"Let $V$ be the set of all conjugates of $P-\\zeta $ over $k_\\mathrm {tor}$ .",
"Clearly the zero-dimensional variety $V$ is defined over $k_\\mathrm {tor}$ , and it is properly contained in $B$ .",
"Therefore we can say that $\\omega _{k_\\mathrm {tor}}(P-\\zeta ,B)\\le \\left( \\frac{[k_\\mathrm {tor}(P):k_\\mathrm {tor}]}{\\deg _{\\mathcal {L}}B} \\right)^{1/\\dim B}.$ Notice also that, if we take as $\\mathcal {L}$ the fourth power ${\\mathcal {O}_{N}}^{ 4}$ of the bundle corresponding to the canonical embedding, this only introduces a constant in the heights and the degrees.",
"Finally, it is clear that in the statement of Theorem REF the corrective factor can be replaced by $\\left( \\frac{ \\log \\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H)^2)}{\\log (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H)^2)}\\right)^{\\kappa (g_0)}\\gg _\\eta (\\omega _{k_\\mathrm {tor}}(P,H)(\\deg _{\\mathcal {L}}H))^{-\\eta }.$ Notice at last that if $B+\\zeta $ is the torsion variety of minimal dimension containing $P$ , then $P-\\zeta $ is not a torsion point modulo all proper abelian subvarieties of $B$ ."
],
[
"Torsion anomalous varieties which are not translates",
"We let $V$ be a weak-transverse variety in a power of elliptic curves.",
"In this section we prove the finiteness of the maximal $V$ -torsion anomalous varieties which are not translates and have relative codimension one.",
"Note that this theorem holds in any power of elliptic curves, independently if it has or not CM.",
"Theorem 5.1 Let $V\\subseteq E^N$ be a weak-transverse variety.",
"Then the maximal $V$ -torsion anomalous varieties of relative codimension one which are not translates are finitely many.",
"More precisely, let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.",
"Assume that $Y$ has relative codimension one in its minimal $B+\\zeta $ .",
"Then for any $\\eta >0$ there exist constants depending only on $E^N$ and $\\eta $ such that: $ \\deg B \\ll _\\eta (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V -1}+\\eta }, $ $h(Y) \\ll _\\eta (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1}+\\eta } $ and $ \\deg Y \\ll _\\eta \\deg V (h(V)+ \\deg V)^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1} -1+\\eta }.$ In addition the torsion points $\\zeta $ belong to a finite set.",
"Let $Y$ be a maximal $V$ -torsion anomalous variety which is not a translate.",
"Let $B+\\zeta $ be minimal for $Y$ .",
"Then $Y$ is a component of $V\\cap (B+\\zeta )$ with $B$ an abelian variety and $\\zeta $ a torsion point.",
"In addition ${\\rm codim\\,}Y<{\\rm codim\\,}V+{\\rm codim\\,}B$ .",
"We prove that $\\deg B$ is bounded only in terms of $V$ and $E^N$ ; we then deduce the bounds for ${h}(Y)$ and $\\deg Y$ .",
"By Lemma REF $Y$ is weak-transverse in $B+\\zeta $ , and by assumption ${\\rm codim\\,}_{B+\\zeta } Y=1$ ; therefore $Y$ is transverse in $B+\\zeta $ .",
"As points are translates and $V$ is not contained in a torsion variety, we have $1\\le \\dim Y<\\dim V.$ Applying the Bogomolov estimate Theorem REF to $Y$ in $B+\\zeta $ we get $\\frac{(\\deg B)^{1-\\eta }}{(\\deg Y)^{{1}+\\eta }}\\ll _\\eta \\mu (Y).$ We set $r={\\rm codim\\,}B$ .",
"Let $h_1, \\dots , h_r$ be the forms of increasing degrees $d_i$ such that $B+\\zeta $ is a component of their zero set, as recalled in Section REF .",
"Then $d_1 \\cdots d_r \\ll \\deg (B+\\zeta )=\\deg B \\ll d_1 \\cdots d_r.$ We denote $r_1=\\dim V-\\dim Y.$ Note that $r_1< r$ , because $Y$ is $V$ -torsion anomalous.",
"Let $A$ be the algebraic subgroup given by the first $h_1\\cdots h_{r_1}$ forms.",
"Then $\\deg A \\ll d_1 \\cdots d_{r_1} .$ Let $A_0$ be an irreducible component of $A$ containing $B+\\zeta $ .",
"Then by (REF ) we have $\\deg A_0 \\ll d_1 \\cdots d_{r_1}\\ll (\\deg B)^\\frac{r_1}{r} , $ and ${\\rm codim\\,}A_0=r_1=\\dim V-\\dim Y$ .",
"By Lemma REF , $Y$ is a component of $V\\cap A_0$ .",
"We apply the Arithmetic Bézout theorem to $V\\cap A_0$ and recall that $h(A_0)=0$ , because $A_0$ is a torsion variety.",
"Then $h(Y) \\ll (h(V)+ \\deg V) \\deg A_0\\ll (h(V)+\\deg V)(\\deg B)^{\\frac{r_1}{r}}.$ For the irreducible variety $Y$ of positive dimension, Zhang's inequality (REF ) says $\\mu (Y)\\le \\frac{h(Y)}{\\deg Y}.$ Combining this with (REF ) and (REF ) we obtain $\\frac{{(\\deg B)}^{1-\\eta }}{(\\deg Y)^{1+\\eta }}\\ll _\\eta \\mu (Y)\\ll (h(V)+ \\deg V)\\frac{ (\\deg B)^\\frac{r_1}{r}}{\\deg Y}.$ Recall that $Y$ is a component of $ V\\cap (B+\\zeta )$ .",
"By Bézout's theorem $\\deg Y \\le \\deg B\\deg V$ .",
"Thus ${(\\deg B)}^{1-\\eta }\\ll _\\eta (h(V)+\\deg V)(\\deg B)^\\frac{r_1}{r}{(\\deg B\\deg V)}^{\\eta }$ and therefore ${(\\deg B)^{\\frac{r-r_1}{r}-2\\eta }}\\ll _\\eta (h(V)+\\deg V){ (\\deg V)}^{\\eta }.$ Since $r-r_1={\\rm codim\\,}V -1$ , for $\\eta $ small enough we get $\\deg B \\ll _\\eta (h(V)+ \\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}+\\eta }(\\deg V)^{\\eta } .$ So we have proved that the degree of $B$ is bounded only in terms of $V$ and $E^N$ .",
"Since the abelian subvarieties of bounded degree are finitely many, applying Lemma REF we conclude that $\\zeta $ belongs to a finite set.",
"Finally, the bound on the height of $Y$ is given by (REF ) and (REF ) $h(Y)\\ll _\\eta (h(V)+\\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}+\\eta }(\\deg V)^{\\eta }.$ The bound on the degree is given by Bézout's theorem for the component $Y$ of $V\\cap A_0$ and (REF ) $\\deg Y\\ll _\\eta (h(V)+\\deg V)^{\\frac{r}{{\\rm codim\\,}V -1}-1+\\eta }(\\deg V)^{1+\\eta }.$"
],
[
"Torsion anomalous points",
"In this and the following section we prove that, if $V$ is a weak-transverse variety in a power of elliptic curves with CM, then the $V$ -torsion anomalous varieties which are translates are non-dense in $V$ .",
"We now prove that the $V$ -torsion anomalous varieties of dimension zero are non-dense.",
"The proof relies on the Arithmetic Bézout theorem, the Zhang's inequality and on the relative Lehmer, Theorem REF .",
"As the last bound is proved only for CM elliptic curves we need this assumption.",
"Theorem 6.1 Let $V\\subseteq E^N$ be a weak-transverse variety, where $E$ has CM.",
"Then, the set of maximal $V$ -torsion anomalous points of relative codimension one is a finite set of explicitly bounded height and relative degree.",
"More precisely, let $k$ be a field of definition for $E$ and let $k_\\mathrm {tor}$ be the field of definition of all torsion points of $E^N$ .",
"Let $d$ be the dimension of $V$ .",
"Let $Y_0$ be a maximal $V$ -torsion anomalous point and let $B+\\zeta $ be minimal for $Y_0$ , with $\\dim B=1$ .",
"Then $\\deg B &\\ll _\\eta ((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{\\frac{N-1}{N-1-d}+\\eta } ,\\\\\\hat{h}(Y_0) &\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-1}{N-1-d}+\\eta }[k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]^{\\frac{d}{N-1-d}+\\eta },\\\\[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}] &\\ll _\\eta \\deg V[k_\\mathrm {tor}(V):k_\\mathrm {tor}]^{\\frac{N-1}{N-1-d}+\\eta }{(h(V)+\\deg V)}^{\\frac{d}{N-1-d}+\\eta }.$ In addition the torsion points $\\zeta $ have order effectively bounded in Theorem REF .",
"Let $Y_0$ be a maximal $V$ -torsion anomalous point, with $B+\\zeta $ minimal for $Y_0$ .",
"By assumption $\\dim B={\\rm codim\\,}_{B+\\zeta }Y_0=1$ .",
"We proceed to bound $\\deg B$ and, in turn, the height of $Y_0$ and its degree over $k_\\mathrm {tor}$ .",
"To this aim we shall use Theorem REF and the Arithmetic Bézout theorem.",
"By Section REF , the variety $B+\\zeta $ is a component of the torsion variety defined as the zero set of forms $h_1,\\dots ,h_{N-1}$ of increasing degrees $d_i$ and $d_1 \\cdots d_{N-1}\\ll \\deg B=\\deg (B+\\zeta )\\ll d_1 \\cdots d_{N-1}.$ Consider the torsion variety defined as the zero set of the first $d$ forms $h_1, \\dots , h_d$ , and take a connected component $A_0$ containing $B+\\zeta $ .",
"Then $\\deg A_0\\ll d_1 \\cdots d_d\\ll {(\\deg B)}^{\\frac{d}{N-1}}$ and ${\\rm codim\\,}A_0=d=\\dim V-\\dim Y_0.$ By Lemma REF , each component of $V\\cap (B+\\zeta )$ is a component of $V\\cap A_0$ .",
"All conjugates of $Y_0$ over $k_\\mathrm {tor}(V)$ are in $V\\cap (B+\\zeta )$ , so the number of components of $V\\cap A_0$ is at least $[k_\\mathrm {tor}(V,Y_0):k_\\mathrm {tor}(V)]\\ge \\frac{[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]}{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}.$ We then apply the Arithmetic Bézout theorem to $Y_0\\subseteq V\\cap A_0$ obtaining $[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]\\hat{h}(Y_0)\\ll {(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}(\\deg B)^{\\frac{d}{N-1}}.$ Applying Theorem REF to $Y_0$ in $B+\\zeta $ , we obtain that for every positive real $\\eta $ $\\hat{h}(Y_0)\\gg _\\eta \\frac{(\\deg B)^{1-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{1+\\eta }}.$ Combining (REF ) and (REF ) we have $\\frac{(\\deg B)^{1-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\eta }}&\\ll _\\eta [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\hat{h}(Y_0)\\ll \\\\&\\ll {(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}(\\deg B)^{\\frac{d}{N-1}}.$ For $\\eta $ small enough we obtain $\\deg B\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{N-1}{N-1-d}+\\eta }[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^\\eta .$ Apply now Bézout's theorem to $V\\cap A_0$ .",
"All the conjugates of $Y_0$ over $k_\\mathrm {tor}(V)$ are components of the intersection, so $\\frac{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]}{[k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]}\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d}{N-1-d}+\\eta } (\\deg V)^{1+\\eta },$ which gives the last bound in the statement.",
"Substituting (REF ) back into (REF ) we have the bound on $\\deg B$ .",
"Finally apply the Arithmetic Bézout theorem to $V\\cap A_0$ to get $\\hat{h}(Y_0)\\ll (h(V)+\\deg V)(\\deg B)^{\\frac{d}{N-1}}\\ll _\\eta (h(V)+\\deg V)^{\\frac{N-1}{N-1-d}}+\\eta [k_{\\mathrm {tor}}(V):k_{\\mathrm {tor}}]^{\\frac{d}{N-1-d}+\\eta }.$ Having bounded $\\deg B$ , in view of Lemma REF , the points $\\zeta $ belong to a finite set.",
"Notice that in Theorem REF we have effectively bounded the degree of the abelian variety $B$ , and we applied Lemma REF to prove the finiteness of the points $Y_0$ in a non effective way.",
"In the following theorem we give a completely effective result.",
"We explicitly bound the degree $[k(Y_0):\\mathbb {Q}]$ for $Y_0$ not a torsion point; this, together with the bound for $\\hat{h}(Y_0)$ in Theorem REF , allows to effectively find all $V$ -torsion anomalous points of relative codimension one which are not torsion points.",
"The effectivity of this result is relevant for the applications to the Mordell-Lang Conjecture shown in Section .",
"Theorem 6.2 Let $V$ be a weak-transverse variety in $E^N$ , where $E$ has CM.",
"Let $k$ be a field of definition for $E$ .",
"Let $d$ be the dimension of $V$ .",
"Let $Y_0$ be a maximal $V$ -torsion anomalous point which is not a torsion point, and let $B+\\zeta $ be minimal for $Y_0$ with $\\dim B=1$ .",
"Then $[k(Y_0):\\mathbb {Q}]\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d(N-1)}{(N-1-d)^2}+\\eta }{(\\deg V[k(V):k])}^{\\frac{N-1}{N-1-d}+\\eta }.$ In addition the torsion points $\\zeta $ can be chosen with $[k(\\zeta ):\\mathbb {Q}]\\ll [k(Y_0):\\mathbb {Q}]$ and order bounded by $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }.$ Finally let $S$ be the number of maximal $V$ -torsion anomalous points of relative codimension one.",
"Then $S\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{A_1+\\eta }{(\\deg V)}^{A_2+1+\\eta }[k(V):k]^{A_2+\\eta }$ where $A_1&= \\frac{(N-1)(2(N+1)(N-d-1)+dN(2N+1))}{2(N-d-1)^2} \\le (N+1)^4,\\\\A_2&= \\frac{N(N-1)(2N+1)}{2(N-d-1)} \\le N^3.$ In view of Theorem REF , we know that $\\deg B$ and $\\hat{h}(Y_0)$ are bounded.",
"We now proceed to bound $[k(Y_0):k]$ .",
"To this aim we need to construct an algebraic subgroup $G$ of codimension $d$ defined over $k$ , containing $Y_0$ and of controlled degree.",
"In order to do this we use Siegel's lemma in a similar way as in [32], Proposition 3, which in turn follows the work [10] of Bombieri, Masser and Zannier in tori.",
"Here we use Siegel's lemma directly on equations with coefficients in the endomorphism ring of $E$ .",
"We notice that $\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\mathcal {O}$ .",
"The coordinates of $Y_0=(x_1,\\ldots ,x_N)$ generate an $\\mathcal {O}$ -module $\\Gamma $ of rank one.",
"The torsion submodule of $\\Gamma $ is well known; for instance in [32], Proposition 2 we find a description for it.",
"Such a torsion module is clearly $\\mathcal {O}$ invariant.",
"As a $\\mathbb {Z}$ -module it is generated by two points $T,\\tau T$ of exact orders $R,R^{\\prime }$ respectively, where $R=c(\\tau )R^{\\prime }$ and $c(\\tau )$ is essentially the real part of $\\tau ^2$ , so a constant of the problem.",
"Therefore we can write $x_i=\\alpha _i g +\\beta _i T$ for a fixed point of infinite order $g$ in $E$ , with coefficients $\\alpha _i,\\beta _i\\in \\mathcal {O}$ such that $\\hat{h}(x_i)=|N_L(\\alpha _i)|\\hat{h}(g)$ and $N_L(\\beta _i)\\ll R^2.$ We want to find coefficients $a_i\\in \\mathcal {O}$ such that $\\sum _i^N a_i x_i=0$ .",
"This gives a linear system of 2 equations, obtained equating to zero the coefficients of $g$ and $T$ .",
"The system has coefficients in $\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.",
"We use a version of Siegel's lemma over $\\mathcal {O}$ as stated in [7], Section 2.9, to get $d$ equations with coefficients in $\\mathcal {O}$ ; multiplying them by a constant depending only on $E$ , we may assume that they have coefficients in $\\mathrm {End}(E)$ .",
"Thus they define the sought-for algebraic subgroup $G$ of degree $\\deg G\\ll \\left((\\max _i N_L(\\alpha _i))(\\max _i N_L(\\beta _i)) \\right)^{\\frac{d}{N-1}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .",
"Then $\\deg G_0\\ll \\left((\\max _i N_L(\\alpha _i)) R^2 \\right)^\\frac{d}{N-1}.$ By the maximality of $Y_0$ , the point $Y_0$ is a component of $V\\cap G_0$ and by Bézout's theorem we get $\\frac{[k(Y_0):k]}{[k(V):k]}\\le \\deg V\\deg G_0.$ Using also (REF ), we obtain $\\frac{[k(Y_0):k]}{[k(V):k]}&\\ll \\deg V \\left((\\max _i N_L(\\alpha _i)) R^2 \\right)^\\frac{d}{N-1}\\le \\\\&\\le \\deg V \\left(R^2\\frac{ \\max _i \\hat{h}(x_i)}{\\hat{h}({g})}\\right)^\\frac{d}{N-1}.$ Notice that $\\hat{h}(x_i)\\le \\hat{h}(Y_0)$ for all $i$ .",
"We can now apply Theorem REF to $g$ in $E$ , obtaining for every $\\eta >0$ $\\frac{1}{\\hat{h}(g)}\\ll _\\eta {[k_\\mathrm {tor}(g):k_\\mathrm {tor}]^{1+\\eta }}\\le [k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]^{1+\\eta }$ because $g$ is defined over $k(Y_0)$ .",
"The product $[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]\\hat{h}(Y_0)$ was bounded in (REF ), so using also the bounds in Theorem REF we obtain $[k(Y_0):k]\\ll _\\eta \\deg V [k(V):k]((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{\\frac{d}{N-1-d}+\\eta }R^{2\\frac{d}{N-1}}.$ By a result of Serre in [30], recalled also in [32], Corollary 3, for $R$ larger than a constant and $\\varphi $ the Euler function, we have $\\varphi (R)\\varphi (R^{\\prime })\\ll [k(Y_0):k] .$ In addition $R^{2-\\eta }\\ll _\\eta \\varphi (R)\\varphi (R^{\\prime })$ since in general $\\varphi (x)\\gg _\\eta x^{1-\\eta }$ and $R$ and $R^{\\prime }$ are related by a constant.",
"From this and (REF ), for $\\eta $ small enough we obtain $[k(Y_0):k]\\ll _\\eta \\deg V [k(V):k]{((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d}{N-d-1}+\\eta } [k(Y_0):k]^{\\frac{d}{N-1}+\\eta }.$ Since $Y_0$ is $V$ -torsion anomalous of relative dimension one, we have $d<N-1$ and we deduce $&[k(Y_0):\\mathbb {Q}]\\ll [k(Y_0):k] \\ll _\\eta \\\\&\\ll _\\eta ({\\deg V[k(V):k]})^{\\frac{N-1}{N-d-1}+\\eta }{((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{d(N-1)}{(N-d-1)^2}+\\eta }.$ We now want to bound the degree of $\\zeta $ over $k$ and the order of $\\zeta $ .",
"Let $K$ be the field of definition of $B+\\zeta $ ; we are going to prove that $[k(\\zeta ):K]$ is absolutely bounded and that $[K:k]\\le [k(Y_0):k]$ .",
"From [6], we can choose $\\zeta $ in a complement $B^{\\prime }$ of $B$ such that $B\\cap B^{\\prime }$ has cardinality bounded only in terms of $N$ .",
"Moreover, we notice that $K\\subseteq k(\\zeta )$ : in fact if $\\sigma \\in \\mathrm {Gal}(\\overline{k}/k(\\zeta ))$ , then $\\sigma (B+\\zeta )=B+\\zeta $ .",
"Now let $\\sigma \\in \\mathrm {Gal}(\\overline{k}/K)$ and suppose that $\\sigma (\\zeta )\\ne \\zeta $ .",
"Then $\\sigma (B+\\zeta )=B+\\sigma (\\zeta )=B+\\zeta $ , because $B$ is defined over $k$ .",
"Since $\\zeta , \\sigma (\\zeta )\\in B^{\\prime }$ we have $\\sigma (\\zeta )-\\zeta \\in B\\cap B^{\\prime }$ .",
"So $[k(\\zeta ):K]\\ll 1$ .",
"We also notice that $K\\subseteq k(Y_0)$ , otherwise we would have a $\\sigma \\in \\mathrm {Gal}(\\overline{k}/k)$ such that $\\sigma (Y_0)=Y_0$ , but $\\sigma (B+\\zeta )\\ne B+\\zeta $ .",
"If this were the case, $Y_0$ would be a component of $V\\cap (B+\\zeta )\\cap \\sigma (B+\\zeta )$ , against the minimality of $B+\\zeta $ .",
"Thus $[k(\\zeta ):k]=[k(\\zeta ):K][K:k]\\ll [K:k]\\le [k(Y_0):k].$ In view of (REF ), $\\zeta $ generates an extension of $k$ of bounded degree.",
"By Serre's result mentioned above $\\textrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }.$ We are left to give an explicit bound for the number of maximal $V$ -torsion anomalous points $Y_0$ of relative codimension one.",
"This is obtained in the following way: we first bound the number of possible subgroups $B$ and possible torsion points $\\zeta $ such that $B+\\zeta $ is minimal for some $Y_0$ .",
"Then we apply Bézout's theorem to every intersection $V\\cap (B+\\zeta )$ .",
"We already proved in Theorem REF and (REF ) that if $B+\\zeta $ is minimal for $Y_0$ , then $\\deg B$ and $\\textrm {ord}(\\zeta )$ are bounded.",
"By Section REF , the number of abelian subvarieties $B$ in $E^N$ of dimension one and degree at most $\\deg B$ is $\\ll _\\eta ({\\deg B})^{N+\\eta }$ , for every $\\eta >0$ .",
"In fact, if $B$ is such a abelian subvariety, consider its associated matrix, as recalled in Section REF ; it is an $(N-1)\\times N$ matrix and we call $d_i$ the square of the norm of its $i$ -th row.",
"We let $\\deg B={p_1}^{e_1}\\cdots {p_r}^{e_r}$ be the factorization of $\\deg B$ into distinct prime factors.",
"Then the number of possible choices for the elements $d_i$ is bounded by $\\delta (\\deg B)^{N-1}$ , where, for a positive integer $n$ , $\\delta (n)$ counts the number of divisors of $n$ .",
"We notice that, for every $\\eta >0$ we have $ \\delta (\\deg B)\\ll _\\eta (\\deg B)^{\\eta }.$ Now, for every choice of the $d_i$ , the number $D$ of $(N-1)\\times N$ matrices in which the square of the norm of the $i$ -th row is at most $d_i$ is bounded in the following way $D\\ll \\left(\\prod _{i=1}^{N-1} d_i\\right)^N\\ll ({\\deg B})^N.$ So for every $\\eta >0$ , the number of possible subgroups $B$ is $\\ll _\\eta (\\deg B)^{N+\\eta }.$ As for the point $\\zeta $ , it is well known that the number of torsion points in $E^N$ of order bounded by a constant $M$ is at most $M^{2N+1}$ .",
"In fact the number of points of order dividing a positive integer $i$ is $i^{2N}$ ; so a bound for the number of torsion points of order at most $M$ is given by $\\sum _{i=1}^M i^{2N}\\ll M^{2N+1}.$ Applying Bézout's theorem to every intersection $V\\cap (B+\\zeta )$ , we obtain that for every $\\eta >0$ the number $S$ of $V$ -torsion anomalous points of relative codimension one is bounded by $S\\ll _\\eta \\deg V (\\deg B)^{N+1+\\eta } {\\textrm {ord}(\\zeta )}^{2N+1}.$ This, combined with Theorem REF , (REF ) and (REF ), gives the required explicit bounds."
],
[
"Torsion anomalous translates of positive dimension",
"We let $V$ be a weak-transverse variety in a power of elliptic curves with CM.",
"In this section we study $V$ -torsion anomalous varieties which are translates of positive dimension.",
"We reduce this case to the zero dimensional case.",
"First we compare the $V$ -torsion anomalous translates with translates contained in $V$ .",
"Lemma 7.1 Let $V$ be a weak-transverse subvariety of an abelian variety of dimension $N$ .",
"Let $Y$ be a maximal $V$ -torsion anomalous translate, then $Y$ is a maximal translate contained in $V$ (i.e.",
"$Y$ is not strictly contained in any translate contained in $V$ ).",
"Let $Y$ be a maximal $V$ -torsion anomalous translate and suppose that $Y$ is contained in a maximal translate $(H+p)\\subseteq V$ with $\\dim (H+p)> \\dim Y $ .",
"Let $B+\\zeta $ be minimal for $Y$ .",
"Then $Y$ is a component of $V\\cap (B+\\zeta )$ and ${\\rm codim\\,}Y <{\\rm codim\\,}V +{\\rm codim\\,}B .$ Since $(H+p)\\cap (B+\\zeta )\\supset Y$ then $p=h+b+\\zeta $ for some $h\\in H$ and $b\\in B$ .",
"Therefore $(H+p)\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+\\zeta )=(H+b+\\zeta )\\cap (B+b+\\zeta )=(H\\cap B)+b+\\zeta $ and $\\dim (H\\cap B)=\\dim ((H+p) \\cap (B+\\zeta ))\\ge \\dim Y.$ By (REF ), we deduce $\\begin{split}\\dim (H+B)&=\\dim H +\\dim B -\\dim (H\\cap B)\\\\& \\le \\dim H +\\dim B -\\dim Y <N.\\end{split}$ So $H+B+ \\zeta $ is a proper torsion subvariety of the ambient variety.",
"Moreover $H+p=H+h+b+\\zeta \\subseteq H+B+\\zeta .$ Thus $H+p\\subseteq V\\cap (H+B+\\zeta ).$ By (REF ) and (REF ), we deduce $\\begin{split}N-\\dim H &< N-\\dim V + N -\\dim H- \\dim B+\\dim Y\\\\&\\le N-\\dim V + N -\\dim H- \\dim B+\\dim (B\\cap H)\\\\&=N-\\dim V + N -\\dim (H+B).\\end{split}$ This implies that $H+p$ is a $V$ -torsion anomalous translate, against the maximality of $Y$ .",
"The following lemma is a straightforward corollary of the result proved by Patrice Philippon in the Appendix .",
"It relates the essential minimum of a translate to the height of the point of translation.",
"Lemma 7.2 Let $H+Y_0$ be a weak-transverse translate in $E^N$ , with $Y_0$ a point in the orthogonal complement $H^\\perp $ of $H$ .",
"Then $ \\mu (Y_0)= \\mu (H+Y_0).$ The points $Y_0+\\zeta $ , for $\\zeta \\in {\\mathrm {Tor}}_{H}$ , are dense in $H+Y_0$ and they have height equal to $\\hat{h} (Y_0)$ .",
"So we get $\\mu (H+Y_0)\\le \\mu (Y_0).$ To obtain the other inequality, consider a set of points of the form ${x_i+Y_0}$ with $x_i \\in H$ , which is dense in $H+Y_0$ .",
"By the Lemma of Philippon in the Appendix , we get $\\hat{h} (x_i+Y_0)= \\hat{h}(x_i) +\\hat{h}(Y_0) \\le \\hat{h}(Y_0) $ .",
"We now state our main theorem for $V$ -torsion anomalous translates.",
"Let $H+Y_0$ be a maximal $V$ -torsion anomalous translate of relative codimension one.",
"The idea is to apply the functorial Lehmer-type bound by Carrizosa to the point $Y_0$ in the complement $H^\\perp $ of $H$ , so that the problem becomes zero dimensional.",
"We then apply the Arithmetic Bézout theorem to $H+Y_0$ in the usual way.",
"The link between $\\mu (Y_0)$ and $\\mu (H+Y_0)$ is then given by Lemma REF .",
"Theorem 7.3 Let $V$ be a weak-transverse subvariety of $E^N$ , where $E$ has CM.",
"Then the set of $V$ -torsion anomalous translates of relative codimension one is a finite set of explicitly bounded normalized height and degree.",
"More precisely, let $k$ be a field of definition for $E$ and let $k_\\mathrm {tor}$ be the field of definition of all torsion points of $E^N$ .",
"Let $H+p$ be a maximal $V$ -torsion anomalous translate of relative codimension one.",
"Let $B+\\zeta $ be minimal for $H+p$ .",
"Then, for every real positive $\\eta $ there exist constants depending only on $E^N$ and $\\eta $ , such that $\\deg B&\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1}+\\eta },\\\\h(H+p)&\\ll _\\eta {(h(V)+\\deg V)}^{\\frac{{\\rm codim\\,}B}{{\\rm codim\\,}V-1}+\\eta }{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}^{\\frac{\\dim V-\\dim B +1}{{\\rm codim\\,}V-1}+\\eta },\\\\\\deg (H+p)&\\ll _\\eta (\\deg V){((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{\\dim V-\\dim B +1}{{\\rm codim\\,}V-1}+\\eta }.$ In addition the points $\\zeta $ belong to a finite set (of cardinality absolutely bounded in Theorem REF ).",
"As we remarked in Section REF , we can assume all abelian subvarieties of $E^N$ to be defined over $k$ .",
"Let $Y_0$ be a point in the orthogonal complement $H^\\perp $ of $H$ , such that $H+p=H+Y_0$ .",
"By Lemma REF , $\\mu (Y_0)=\\mu (H+Y_0).$ We are going to use the Arithmetic Bézout theorem to find an upper bound for $\\mu (H+Y_0)$ and Theorem REF to find a lower bound.",
"Let $B+\\zeta $ be minimal for $H+Y_0$ , with $\\zeta $ in the orthogonal complement of $B$ .",
"Then $H+Y_0$ is a component of $V\\cap (B+\\zeta )$ and by assumption ${\\rm codim\\,}_{B+\\zeta } (H+Y_0) =1$ .",
"By Lemma REF $H+Y_0$ is weak-transverse in $B+\\zeta $ , thus $Y_0$ is not a torsion point and the coordinates of $Y_0$ generate a module of rank ${\\rm codim\\,}_{B+\\zeta } (H+Y_0) =1$ .",
"Let $r$ be the codimension of $B$ .",
"The variety $B+\\zeta $ is a component of the torsion variety given by the zero set of the forms $h_1,\\dots ,h_{r}$ of increasing degrees $d_i$ and $d_1 \\cdots d_{r}\\ll \\deg B=\\deg (B+\\zeta )\\ll d_1 \\cdots d_{r}.$ Note that $H+Y_0$ is contained in $V$ and, by assumption, it has relative codimension one.",
"Thus ${\\rm codim\\,}V<{\\rm codim\\,}H=r+1$ .",
"Denote $r_1= r+1-{\\rm codim\\,}V= \\dim V-\\dim H.$ Consider the torsion variety defined as the zero set of the first $r_1$ forms $h_1,\\dots ,h_{r_1}$ , and take one of its irreducible component $A_0$ passing through $H+Y_0$ .",
"Then $\\deg A_0\\ll {(\\deg B)}^{\\frac{r_1}{r}}$ and ${\\rm codim\\,}A_0=r_1.$ Recall that $k_\\mathrm {tor}$ is the field of definition of all torsion points in $E^N$ .",
"By Lemma REF , $H+Y_0$ is a component of $V\\cap A_0$ .",
"Since the intersection $V\\cap A_0$ is defined over $k_\\mathrm {tor}$ , every conjugate of $H+Y_0$ over $k_\\mathrm {tor}$ is a component of $V\\cap A_0$ and all such components have the same normalized height.",
"All conjugates of $H+Y_0$ over $k_\\mathrm {tor}(V)$ are in $V\\cap (B+\\zeta )$ .",
"So the number of components of $V\\cap A_0$ is at least $ \\frac{[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]}{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}.$ We apply the Arithmetic Bézout theorem to $H+Y_0$ in $V\\cap A_0$ and we obtain ${h}(H+Y_0)\\ll (h(V)+\\deg V) {(\\deg B)}^{\\frac{r_1}{r}}$ and estimating the components we have ${h}(H+Y_0)\\frac{[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]}{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}\\ll (h(V)+\\deg V) {(\\deg B)}^{\\frac{r_1}{r}}.$ By the Zhang's inequality, (REF ) and (REF ), we deduce $\\mu (Y_0)\\ll \\frac{(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]{(\\deg B)}^{\\frac{r_1}{r}}}{[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]\\deg H}.$ Consider the smallest abelian subvariety $H^{\\prime }$ of $B$ containing $Y_0-\\zeta $ , so $Y_0 $ is not contained in any torsion subvariety of $H^{\\prime }+\\zeta $ .",
"The relative codimension of $H+Y_0$ in $B+\\zeta $ is one therefore the dimension of $H^{\\prime }$ is one.",
"Moreover $ H^\\perp \\cap (B+\\zeta )$ has dimension one and it contains $Y_0$ .",
"Consider the irreducible component $ H^\\perp _0$ of the intersection containing $Y_0$ : since $Y_0$ is not torsion, then $H^\\perp _0$ has dimension one.",
"So $H^{\\prime }+\\zeta = H^\\perp _0$ , because both varieties are irreducible, contain $Y_0$ and are one dimensional.",
"Notice in particular that $\\zeta \\in H_0^\\perp \\subseteq H^\\perp $ , and therefore $H^{\\prime }\\subseteq H^\\perp $ .",
"We also notice that $[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]\\le [k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]\\cdot \\#(H\\cap H^{\\prime }).$ In fact if $\\sigma \\in \\mathrm {Gal}(\\overline{k_\\mathrm {tor}}/k_\\mathrm {tor}(H+Y_0))$ then $\\sigma (Y_0)-Y_0\\in H$ .",
"Since $H^{\\prime }+\\zeta =H^{\\prime }+Y_0$ and it is defined over $k_\\mathrm {tor}$ , we have that $\\sigma (Y_0)-Y_0\\in H^{\\prime }$ as well.",
"Hence the number of conjugates of $Y_0$ over $k_\\mathrm {tor}$ is at most $\\#(H\\cap H^{\\prime })$ .",
"For the lower bound for $\\mu (Y_0)$ , the proof follows the case of dimension zero.",
"In particular applying Theorem REF to $Y_0$ in $H^{\\prime }+\\zeta $ we get that, for every positive real $\\eta $ $\\mu (Y_0)=\\hat{h}(Y_0)\\gg _\\eta \\frac{(\\deg H^{\\prime })^{1-\\eta }}{[k_\\mathrm {tor}(Y_0):k_\\mathrm {tor}]^{1+\\eta }}.$ Combining (REF ), (REF ) and (REF ) we get $\\begin{split}(\\deg H^{\\prime })^{1-\\eta }\\ll _\\eta {(h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}]} \\frac{(\\deg B)^{\\frac{r_1}{r}}}{\\deg H}\\cdot \\\\ \\cdot \\#(H\\cap H^{\\prime })^{1+\\eta }[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]^{\\eta }.\\end{split}$ Notice that, by the definition of $H^{\\prime }$ , $B=H+H^{\\prime }$ and $\\#(H\\cap H^{\\prime })\\deg B = \\deg H \\deg H^{\\prime }.$ Thus, possibly changing $\\eta $ , we have $\\deg B\\ll _\\eta ((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{\\frac{r}{{\\rm codim\\,}V-1}+\\eta } \\left( \\#(H\\cap H^{\\prime })[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]\\right)^{\\eta }.$ We now want to remove the dependence on $\\#(H\\cap H^{\\prime })$ and $[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]$ and bound the degree of the translate $H+Y_0$ .",
"We first apply Bézout's theorem to the intersection $V\\cap A_0$ , obtaining: $\\deg H\\le \\deg V\\deg A_0\\ll \\deg V(\\deg B)^\\frac{r_1}{r}$ and estimating the components we obtain $\\deg H\\frac{[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]}{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}\\le \\deg V\\deg A_0\\ll \\deg V(\\deg B)^\\frac{r_1}{r}.$ We remarked that $H^{\\prime }\\subseteq H^\\perp $ , therefore $H\\cap H^{\\prime } \\subseteq H\\cap H^\\perp $ and, $\\#(H\\cap H^{\\prime })\\le \\#(H\\cap H^\\perp )\\ll (\\deg H)^2.$ This and (REF ) give $\\#(H\\cap H^{\\prime })[k_\\mathrm {tor}(H+Y_0):k_\\mathrm {tor}]\\ll ([k_\\mathrm {tor}(V):k_\\mathrm {tor}]\\deg V)^2 (\\deg B)^{\\frac{2r_1}{r}}.$ Substituting in (REF ) we get $ \\deg B\\ll _\\eta {((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{r}{{\\rm codim\\,}V-1}+\\eta }.$ Then, using (REF ) and replacing (REF ) we have $\\deg (H+Y_0) &\\ll (\\deg V)(\\deg B)^\\frac{r_1}{r}\\ll _\\eta \\\\&\\ll _\\eta (\\deg V){((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])}^{\\frac{r_1}{{\\rm codim\\,}V-1}+\\eta }.$ Finally, from (REF ) and (REF ) we obtain $h(H+Y_0)\\ll _\\eta {(h(V)+\\deg V)}^{\\frac{r}{{\\rm codim\\,}V-1}+\\eta }{[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}^{\\frac{r_1}{{\\rm codim\\,}V-1}+\\eta }.$ Since we have bounded $\\deg B$ , we can conclude from Lemma REF that the points $\\zeta $ belong to a finite set.",
"In the proof we bounded the degree of $H+Y_0$ using Bézout's theorem, and we obtained a bound depending on $\\deg V$ and $h(V)$ .",
"The dependence on $h(V)$ may be removed with a different argument.",
"Let $H+p\\subseteq V$ be a translate which is maximal with respect to the inclusion among all such translates.",
"Bombieri and Zannier in [12], Lemma 2, proved that only finitely many such abelian subvarieties $H$ can occur.",
"More precisely the maximal translates contained in $V$ have degree bounded only in terms of the degree and the dimension of $V$ .",
"As a corollary of their proof we obtain the following lemma.",
"Lemma 7.4 (Bombieri-Zannier) Let $V$ be a weak-transverse variety.",
"Then the maximal translates contained in $V$ are of the form $H+p$ for finitely many abelian subvarieties $H$ with $\\deg H \\le (\\deg V)^{2^{\\dim V }}.$ Using this lemma we obtain a bound which is more uniform, though the dependence on $\\deg V$ shows a bigger exponent.",
"As remarked for the zero dimensional case, in Theorem REF we proved finiteness using the non effective Lemma REF .",
"We now give a completely effective result which is the analogous of Theorem REF in positive dimension.",
"We bound the degrees of the fields of definition of the translates $H+p$ and of the torsion points $\\zeta $ .",
"Theorem 7.5 Let $V$ be a weak-transverse subvariety of $E^N$ , where $E$ has CM.",
"Let $k$ be a field of definition for $E$ .",
"Let $H+p$ be a maximal $V$ -torsion anomalous translate of relative codimension one, and let $B+\\zeta $ be minimal for $H+p$ .",
"Set $r={\\rm codim\\,}B$ ; then the field $k(H+p)$ of definition of $H+p$ has degree bounded by $[k(H+p):\\mathbb {Q}]\\ll _\\eta [k(V):k]^{r+\\eta } ({\\deg V})^{3r-1}\\cdot \\\\\\cdot (h(V)+\\deg V)^{\\frac{(2r-1)(r-{\\rm codim\\,}V+1)+r(r-1)}{{\\rm codim\\,}V-1}+\\eta } {[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}^{\\frac{(3r-2)(r-{\\rm codim\\,}V +1)}{{\\rm codim\\,}V-1}}.$ Moreover the torsion points $\\zeta $ can be chosen so that $[k(\\zeta ):\\mathbb {Q}]\\ll _\\eta [k(H+p):\\mathbb {Q}]$ and $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(H+p):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }.$ We keep all the notations and definitions used in Theorem REF .",
"Of course $[k(H+p):k]=[k(H+Y_0):k]\\le [k(Y_0):k],$ because we are assuming all abelian subvarieties of $E^N$ to be defined over $k$ .",
"The bound on the degree $[k(Y_0):k]$ is obtained following the same idea of Theorem REF : since $H^{\\prime }$ has dimension one, the group generated by the coordinates of $Y_0$ is an $\\mathrm {End}(E)$ -module of rank one.",
"We can apply Siegel's lemma in a similar way to the zero-dimensional case, where we use the estimate (REF ) for the height of $Y_0$ .",
"In this case, however, we want to find an algebraic subgroup $G$ of dimension 2, containing $Y_0$ , and contained in $H^\\perp $ .",
"We know that $\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ .",
"Let $\\lambda _1,\\cdots , \\lambda _{N-r-1}$ be linear forms which give equations for $H^\\perp $ , and define $\\delta _i=\\max _j |N_L(l_{ij})|,$ where $(l_{ij})_{j}$ is the vector of coefficients of the form $\\lambda _i$ .",
"We now follow the steps of the proof of Theorem REF and apply Siegel's lemma to obtain $r-1$ independent solutions which are also orthogonal to the vectors of coefficients of $\\lambda _1,\\cdots , \\lambda _{N-r-1}$ ; this time, in addition to the two equations of Theorem REF , we have also one equation for each of the $\\lambda _i$ , for a total of $N-r+1$ equations in $N+1$ unknowns ($N$ coefficients and one for the torsion point).",
"These $r-1$ vectors give $r-1$ linear forms which, together with the $\\lambda _1,\\cdots , \\lambda _{N-r-1}$ , provide the $N-2$ equations needed to define $G$ .",
"The bounds provided by Siegel's lemma give $\\deg G&\\ll _\\eta \\left(\\prod _{i=1}^{N-r-1}\\delta _i \\right)\\left(\\hat{h}(Y_0)[k(Y_0):k]^{1+\\eta } \\prod _{i=1}^{N-r-1}\\delta _i\\right)^\\frac{r-1}{r}\\ll _\\eta \\\\&\\ll _\\eta (\\deg H)^{2-\\frac{1}{r}}\\hat{h}(Y_0)^\\frac{r-1}{r}[k(Y_0):k]^{\\frac{r-1}{r}+\\eta }.$ Now that we have found $G$ , we first show that $H+Y_0$ is a component of $V\\cap (H+G)$ : indeed $\\dim (H+G)=\\dim B +1$ , because $G\\subseteq H^\\perp $ has dimension 2; therefore any component of the intersection $V\\cap (H+G)$ with dimension greater than $\\dim H$ is anomalous.",
"But $H+Y_0$ is clearly contained in $V\\cap (H+G)$ and, by the maximality of $H+Y_0$ , it is not contained in any $V$ -anomalous subvariety of greater dimension; hence it is itself a component of $V\\cap (H+G)$ and so are all its conjugates over $k(V)$ .",
"Applying Bézout's theorem to $V\\cap (H+G)$ we get $\\frac{[k(Y_0):k]}{[k(V):k]}\\deg H &\\le \\deg V\\deg (H+G)=\\deg V\\deg H\\deg G,\\\\ [k(Y_0):k] &\\le [k(V):k]\\deg V\\deg G,$ and this gives the desired bound.",
"Substituting (REF ) in (REF ), and using all the bounds from Theorem REF we obtain $\\begin{split}[k(Y_0):\\mathbb {Q}]\\ll _\\eta [k(V):k]^{r+\\eta } ({\\deg V})^{3r-1} (h(V)+\\deg V)^{\\frac{(2r-1)r_1+r(r-1)}{{\\rm codim\\,}V-1}+\\eta }\\cdot \\\\\\cdot {[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}^{\\frac{(3r-2)r_1}{{\\rm codim\\,}V-1}}.\\end{split}$ Finally, as in the proof of Theorem REF , $\\zeta $ can be chosen so that $[k(\\zeta ):\\mathbb {Q}] \\ll [k(Y_0): \\mathbb {Q}]$ ; again by Serre's theorem, the points $\\zeta $ have order bounded by $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }$ and therefore belong to an explicit finite set.",
"Finally, the set of all possible translates $H+p$ has cardinality $S$ bounded by $S\\ll _\\eta [k(V):k]^{D_1} ({\\deg V})^{D_2}(h(V)+\\deg V)^{D_3} {[k_\\mathrm {tor}(V):k_\\mathrm {tor}]}^{D_4}$ where $D_1&=\\frac{rN(2N+1)}{2}\\\\D_2&=\\frac{(3r-1)N(2N+1)}{2}+1\\\\D_3&=\\frac{(N+1)r}{{\\rm codim\\,}V -1} + \\frac{N(2N+1)}{2}\\frac{(2r-1)(r-{\\rm codim\\,}V +1)+r(r-1)}{{\\rm codim\\,}V -1}\\\\D_4&=\\frac{(N+1)r}{{\\rm codim\\,}V -1} +\\frac{N(2N+1)}{2}\\frac{(3r-2)(r-{\\rm codim\\,}V +1)}{{\\rm codim\\,}V-1}.$"
],
[
"Proof of Theorem ",
"We now give the proof of Theorem REF , restated here for clarity.",
"Theorem 8.1 Let $V$ be a weak-transverse variety in an abelian variety $A$ with CM.",
"Let $g$ be the maximal dimension of a simple factor of $A$ .",
"If ${\\rm codim\\,}V\\ge g+1$ , then the maximal $V$ -torsion anomalous varieties of relative codimension 1 are finitely many and they have degree and normalised height effectively bounded.",
"Let $X$ be a maximal $V$ -torsion anomalous variety of relative codimension 1 and let $B+\\zeta $ be minimal for $X$ .",
"Then $X$ is a component of the intersection $V\\cap (B+\\zeta )$ and ${\\rm codim\\,}V>\\dim B-\\dim X=1$ .",
"Since ${\\rm codim\\,}V\\ge g+1$ by the hypothesis and $\\dim V\\ge \\dim X+1=\\dim B$ , then ${\\rm codim\\,}B\\ge g+1$ .",
"From Section REF , $B+\\zeta $ is a component of the torsion variety given by the zero set of $r\\ge 2$ forms $h_1,\\ldots ,h_r$ of increasing degrees $d_i$ and $d_1 \\cdots d_r \\ll \\deg (B+\\zeta )\\ll d_1 \\cdots d_r .$ Notice that one single equation gives ${\\rm codim\\,}B\\le g$ , since $g$ is the maximal dimension of a simple factor of $A$ .",
"This implies that $r\\ge 2$ .",
"Denote by $A_0$ the algebraic subgroup given by $h_1,\\ldots ,h_{r-1}$ .",
"Since $r\\ge 2$ , we have $A_0\\subsetneq A$ .",
"In addition $\\deg A_0\\ll (\\deg B)^{\\frac{r-1}{r}}$ .",
"We now show that $X$ is a component of $V\\cap A_0$ .",
"Suppose, to the contrary, that $X\\subsetneq Y\\subseteq V\\cap A_0$ where $Y$ is a component of $V\\cap A_0$ strictly containing $X$ .",
"By maximality of $X$ , $Y$ cannot be $V$ -torsion anomalous, hence ${\\rm codim\\,}Y={\\rm codim\\,}V+{\\rm codim\\,}A_0.$ Moreover by construction $\\dim A_0\\le \\dim B+g$ .",
"This implies $\\dim B-1=\\dim X<\\dim Y=\\dim A_0-{\\rm codim\\,}V\\le \\dim B+g-{\\rm codim\\,}V$ and so ${\\rm codim\\,}V<g+1$ , which contradicts the hypothesis on $V$ .",
"So $X$ is a component of $V\\cap A_0$ .",
"The bounds for the normalised height and for the degree of $X$ are now obtained following the proof in the case of elliptic curves.",
"Namely we combine the Zhang inequality, the Arithmetic Bézout theorem, the Bogomolov estimate in Theorem REF and the relative Lehmer estimate in Theorem REF .",
"The appendix is here replaced by the general result of Bertrand in [5].",
"If $X$ is not a translate, the same argument of Theorem REF applies to show that for any positive real $\\eta $ , there exist constants depending only on $A$ and $\\eta $ such that $h(X)\\ll _\\eta (h(V)+\\deg V)^{{\\rm codim\\,}B+\\eta }(\\deg V)^\\eta $ and $\\deg X\\ll _\\eta {(h(V)+\\deg V)}^{{\\rm codim\\,}B -1+\\eta }(\\deg V)^{1+\\eta }.$ If $X$ is a translate, we proceed as in the proofs of Theorem REF if $\\dim X=0$ and of Theorem REF if $\\dim X\\ge 1$ , obtaining that for any positive real $\\eta $ , there exist constants depending only on $A$ and $\\eta $ such that $h(X)\\ll _\\eta (h(V)+\\deg V)^{{\\rm codim\\,}B+\\eta }[k_\\mathrm {tor}(V):k_\\mathrm {tor}]^{{\\rm codim\\,}B-1+\\eta }$ and $\\deg X\\ll _\\eta \\deg V ((h(V)+\\deg V)[k_\\mathrm {tor}(V):k_\\mathrm {tor}])^{{\\rm codim\\,}B-1+\\eta }.$ Here $k$ is the field of definition for $A$ , $k_\\mathrm {tor}$ is the field of definiton for the torsion points of $A$ and $\\deg X=[k_\\mathrm {tor}(X):k_\\mathrm {tor}]$ when $X$ is a point."
],
[
"The case of a curve",
"In this section we extend our method in order to get some further results for curves.",
"Theorem 9.1 Let $C$ be a weak-transverse curve in $E^N$ , where $E$ has CM.",
"Let $k$ be a field of definition for $E$ and denote by $k_{\\mathrm {tor}}$ the field of definition of all torsion points of $E^N$ .",
"Then the set $\\mathcal {S}(C)= C \\cap \\left(\\cup _{{\\rm codim\\,}H>\\dim H} H\\right)$ is a finite set of effectively bounded Néron-Tate height.",
"Here $H$ ranges over all algebraic subgroups of codimension larger than the dimension.",
"More precisely the union can be taken over finitely many algebraic subgroups $\\mathcal {S}(C)=C\\cap \\bigcup _{i=1}^M H_i;$ the $H_i$ are algebraic subgroups with $\\dim H <{\\rm codim\\,}H$ , and for any real $\\eta >0$ , there exist constants depending only on $E^N$ and $\\eta $ such that $ \\deg H_i \\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)(N+2r-2)}{2(r-1)(2r-N)}+\\eta } ([k(C):k]\\deg C)^{\\frac{Nr}{2(r-1)}+\\eta },$ where $r={\\rm codim\\,}H_i$ .",
"Moreover $M$ can be effectively bounded.",
"Furthermore, if $Y_0\\in \\mathcal {S}(C)$ we have $\\hat{h}(Y_0) \\ll _\\eta (h(C)+\\deg C)^{\\frac{N+1}{2}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{N-1}{2}+\\eta };$ if $Y_0$ is not a torsion point we also have $[k(Y_0):\\mathbb {Q}]\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{N+1}{N-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{N+1}{2}+\\eta },$ and the cardinality of the points in $\\mathcal {S}(C)$ which are not torsion points is bounded as $S\\ll _\\eta [k(C):k]^{B_1}(\\deg C)^{B_1+1+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{B_2+\\eta },$ where $B_1=\\frac{N(N+1)(2N+1)}{2(N-1)}, \\qquad B_2=\\frac{(3N^2+N-1)(N+1)}{4}.$ We notice that the set of all torsion points in $C$ is finite and has cardinality bounded by the effective Manin-Mumford Conjecture.",
"From now on, we will be concerned with points in $\\mathcal {S}(C)$ that are not torsion.",
"Clearly all the points in the intersection $C\\cap (\\cup _{{\\rm codim\\,}H>\\dim H}H) $ are $C$ -torsion anomalous.",
"In addition since $C$ is a weak-transverse curve each torsion anomalous point is maximal.",
"Let $Y_0\\in \\mathcal {S}(C)$ be a non-torsion point; then $Y_0\\in C\\cap H$ , with $H$ the minimal subgroup containing $Y_0$ with respect to the inclusion, and let $r={\\rm codim\\,}H$ .",
"Let also $B+\\zeta $ be a component of $H$ containing $Y_0$ .",
"Clearly $\\dim B=\\dim H$ and $Y_0\\in C\\cap (B+\\zeta )$ with $B+\\zeta $ minimal for $Y_0$ ; as done several times, we can also choose the torsion point $\\zeta $ in the orthogonal complement of $B$ .",
"Notice that $\\deg H\\le (\\deg B) \\mathrm {ord}(\\zeta ).$ In fact $B+\\langle \\zeta \\rangle $ is an algebraic subgroup of dimension equal to $\\dim H$ , it contains $Y_0$ and it is contained in $H$ ; thus $B+\\langle \\zeta \\rangle =H$ , by the minimality of $H$ .",
"The proof now follows the lines of the proofs of Theorems REF and REF .",
"We proceed to bound $\\deg B$ and, in turn, the height of $Y_0$ , using Theorem REF and the Arithmetic Bézout theorem.",
"Then, using Siegel's lemma, we get a bound for $[k(Y_0):k]$ and for the order and the number of torsion points $\\zeta $ , providing also a bound for $\\deg H$ .",
"Recall that we have $r={\\rm codim\\,}B={\\rm codim\\,}H$ .",
"We first exclude the case $r=1$ and show that the case $N-r=1$ is covered by Theorems REF and REF .",
"If $r=1$ then $\\dim B<{\\rm codim\\,}B$ implies $\\dim B=0$ and $N=\\dim B+{\\rm codim\\,}B=1$ contradicting the weak-transversality of $C$ .",
"The case $N-r=1$ corresponds to $\\dim B=1$ and $Y_0$ of relative codimension one, that can be treated with Theorems REF and REF .",
"Thus we can assume $r>N-r\\ge 2$ .",
"Moreover $2r>N$ , since by assumption ${\\rm codim\\,}B> \\dim B$ .",
"As done several times, by Section REF , the variety $B+\\zeta $ is a component of the zero set of forms $h_1,\\dots ,h_{r}$ of increasing degrees $d_i$ and $d_1 \\cdots d_{r}\\ll \\deg B=\\deg (B+\\zeta )\\ll d_1 \\cdots d_{r}.$ Consider the torsion variety defined as the zero set of $h_1$ , and let $A_0$ be one of its connected components containing $B+\\zeta $ .",
"Then $\\deg A_0\\ll d_1\\ll (\\deg B)^{\\frac{1}{r}}.$ From Theorem REF applied to $Y_0$ in $B+\\zeta $ , for every positive real $\\eta $ we get $\\hat{h}(Y_0)\\gg _\\eta \\frac{(\\deg B)^{\\frac{1}{N-r}-\\eta }}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\frac{1}{N-r}+\\eta }}.$ Notice that all conjugates of $Y_0$ over $k_\\mathrm {tor}(C)$ are components of $C\\cap A_0$ .",
"Applying the Arithmetic Bézout theorem to $Y_0$ in $C\\cap A_0$ and arguing as in the proof of Theorem REF , we have $\\frac{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]}{[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]}\\hat{h}(Y_0)\\ll (h(C)+\\deg C) (\\deg B)^{\\frac{1}{r}}.$ From (REF ) and (REF ) we get $(\\deg B)^{\\frac{2r-N}{r(N-r)}-\\eta }\\ll _\\eta (h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}][k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{\\frac{1}{N-r}-1+\\eta }.$ Since $2r>N$ , $N-r>1$ and $ [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\ge 1$ , for $\\eta $ small enough we obtain $\\deg B\\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{2r-N}+\\eta }.$ So, from (REF ) we have $ [k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]\\hat{h}(Y_0)\\ll _\\eta {((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r}{2r-N}+\\eta };$ while, using the right-hand side of (REF ) as a bound for $\\hat{h}(Y_0)$ and (REF ) we get $\\hat{h}(Y_0)\\ll _\\eta (h(C)+\\deg C)^{\\frac{r}{2r-N}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{(N-r)}{2r-N}+\\eta }$ as required.",
"Having bounded $\\deg B$ and $\\hat{h}(Y_0)$ , we now proceed to bound $[k(Y_0):k]$ only in terms of $C$ and $E^N$ .",
"We use Siegel's lemma to construct an algebraic subgroup $G$ of codimension $1=\\dim C$ defined over $k$ , containing $Y_0$ and of controlled degree.",
"The construction is exactly as in [32] Propositions 3 and 4.",
"We present the steps of the proof.",
"We know that $\\mathrm {End}(E)$ is an order in an imaginary quadratic field $L$ with ring of integers $\\mathcal {O}$ .",
"By minimality of $B+\\zeta $ , the coordinates of $Y_0=(x_1,\\ldots ,x_N)$ generate an $\\mathcal {O}$ -module $\\Gamma $ of rank equal to the dimension of $B+\\zeta $ which is $N-r$ .",
"Let $g_1, \\dots , g_{N-r}$ be generators of the free part of $\\Gamma $ which give the successive minima, and are chosen as in [32], Proposition 2.",
"Then $\\hat{h}(\\sum _j \\alpha _j g_j)\\gg \\sum _i |N_L(\\alpha _j)|\\hat{h}(g_j)$ for coefficients $\\alpha _j$ in $\\mathcal {O}$ .",
"In addition, like in the case of relative codimension one, the torsion part is generated by a torsion point $T$ of exact order $R$ .",
"Therefore we can write $x_i=\\sum _j\\alpha _{i,j} g_j +\\beta _i T$ with coefficients $\\alpha _i,\\beta _i\\in \\mathcal {O}$ and $N_L(\\beta _i)\\ll R^2.$ As in Proposition 2 of [32], we have $\\hat{h}(x_i)\\gg \\sum _j \\left|N_L(\\alpha _{i,j})\\right|\\hat{h}(g_j).$ We define $\\nu _j=(\\alpha _{1,j}, \\dots , \\alpha _{N,j}) \\quad {\\mathrm {and}} \\quad |\\nu _j|=\\max _i|N_L(\\alpha _{i,j})|.$ Then $|\\nu _j|\\ll \\frac{\\hat{h}(Y_0)}{\\hat{h}(g_j)}.$ We want to find coefficients $a_i\\in \\mathcal {O}$ such that $\\sum _i^N a_i x_i=0$ .",
"This gives a linear system of $N-r+1$ equations, obtained equating to zero the coefficients of $g_j$ and of $T$ .",
"The system has coefficients in $\\mathcal {O}$ and $N+1$ unknowns: the $a_i$ 's and one more unknown for the congruence relation arising from the torsion point.",
"We use the Siegel's lemma over $\\mathcal {O}$ as stated in [7], Section 2.9.",
"We get one equation with coefficients in $\\mathcal {O}$ ; multiplying it by a constant depending only on $E$ we may assume that it has coefficients in $\\mathrm {End}(E)$ .",
"Thus it defines the sought-for algebraic subgroup $G$ of degree $\\deg G\\ll \\left((\\max _i N_L(\\beta _i))\\left(\\prod ^{N-r}_j|\\nu _j|\\right) \\right)^{\\frac{1}{r}}.$ Let $G_0$ be a $k$ -irreducible component of $G$ passing through $Y_0$ .",
"Then $\\deg G_0\\ll \\left(R^2\\prod ^{N-r}_j|\\nu _j| \\right)^\\frac{1}{r}.$ Since $C$ is weak-transverse, the point $Y_0$ is a component of $C\\cap G_0$ .",
"In addition $C$ and $G_0$ are defined over $k$ and Bézout's theorem gives $[k(Y_0):k]\\le [k(C):k]\\deg C\\deg G_0.$ Hence $[k(Y_0):k]\\ll [k(C):k]\\deg C \\left(R^2\\prod ^{N-r}_j|\\nu _j| \\right)^\\frac{1}{r}.$ Using (REF ) we get $[k(Y_0):k]\\ll [k(C):k]\\deg C \\left(R^2\\frac{\\hat{h}(Y_0)^{N-r}}{\\prod ^{N-r}_j\\hat{h}({g_j})}\\right)^\\frac{1}{r}.$ Following step by step the proof of Proposition 4 in [32], using Theorem REF in place of Theorem 1.3 of [16], we get $\\prod _{i=1}^{N-r}\\hat{h}(g_i)\\gg _\\eta \\frac{1}{[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]^{1+\\eta }}.$ By a result of Serre, recalled also in [32], Corollary 3, we know $[k(Y_0):k]\\gg _\\eta R^{2-\\eta }$ .",
"Moreover from (REF ), the product $\\hat{h}(Y_0)[k_{\\mathrm {tor}}(Y_0):k_{\\mathrm {tor}}]$ is bounded.",
"Substituting these bounds in (REF ) and recalling that $r>2$ , for $\\eta $ small enough we obtain $[k(Y_0):k]\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{r}{r-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{(2r-N)(r-1)}+\\eta }.$ Moreover, as in the proof of Theorem REF , we can choose $\\zeta $ so that $[k(\\zeta ):k]\\ll [k(Y_0):\\mathbb {Q}], $ and $\\mathrm {ord}(\\zeta )\\ll _\\eta [k(Y_0):\\mathbb {Q}]^{\\frac{N}{2}+\\eta }$ .",
"Substituting this and (REF ) in (REF ), we get the bound for $\\deg H$ .",
"Now, as in the last part of the proof of Theorem REF , for every $\\eta >0$ a bound for the number $S$ of non-torsion points $Y_0$ is given by $S\\ll _\\eta {\\deg C} \\phantom{.",
"}({\\deg B})^{N+1+\\eta } {\\mathrm {ord}(\\zeta )}^{2N+1}$ and combining this with the previous results we obtain the bounds.",
"Finally we notice that we have bounded the degree $\\deg H$ for $H$ of fixed codimension $r$ ; letting $r$ vary this proves that the subgroups $H$ can be taken in a finite set $\\lbrace H_1,\\cdots , H_M\\rbrace $ .",
"A bound for $M$ can be given effectively, as done in the proof of Theorem REF .",
"In Theorem REF we have stated a simplified version of the bounds for $\\hat{h}(Y_0),[k(Y_0):\\mathbb {Q}]$ and $S$ .",
"This has been done for clarity, but a closer inspection of the proof shows that we have proved the following, sharper bounds.",
"Corollary 9.2 With the same setting and notation of Theorem REF , take $Y_0\\in \\mathcal {S}(C)$ is not a torsion point, and $Y_0\\in C\\cap H$ with $H$ of minimal dimension $N-r$ .",
"Then for any real $\\eta >0$ , there exist constants depending only on $E^N$ and $\\eta $ such that $\\hat{h}(Y_0) &\\ll _\\eta (h(C)+\\deg C)^{\\frac{r}{2r-N}+\\eta }[k_{\\mathrm {tor}}(C):k_{\\mathrm {tor}}]^{\\frac{(N-r)}{2r-N}+\\eta },\\\\[k(Y_0):\\mathbb {Q}]&\\ll _\\eta ([k(C):k]\\deg C)^{\\frac{r}{r-1}+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{\\frac{r(N-r)}{(2r-N)(r-1)}+\\eta }.$ Moreover $Y_0$ belongs to a finite set of cardinality $S$ bounded as $S\\ll _\\eta [k(C):k]^{B_1}(\\deg C)^{B_1+1+\\eta }{((h(C)+\\deg C)[k_\\mathrm {tor}(C):k_\\mathrm {tor}])}^{B_2+\\eta },$ where $ B_1=\\frac{rN(2N+1)}{2(r-1)},$ $B_2=\\frac{r(N-r)(2rN+2r-2+2N^2-N)}{2(2r-N)(r-1)}.$"
],
[
"Sur une question d'orthogonalité dans les puissances de courbes elliptiques",
"We are indebted to Patrice Philippon for writing this appendix which is essentially the proof of Lemma REF .",
"Let us mention that deeper properties of orthogonality in Mordell-Weil groups have been studied, see [5].",
"Soit $E$ une courbe elliptique définie sur le corps des nombres algébriques $\\overline{\\mathbb {Q}}$ , considéré plongé dans le corps des nombres complexes $\\mathbb {C}$ .",
"Soit $L={\\rm Frac}({\\rm End}(E))$ le corps des multiplications de $E$ , qu'on considérera comme un sous-corps de $\\overline{\\mathbb {Q}}\\subset {\\mathbb {C}}$ .",
"On a $L=\\mathbb {Q}$ ou $L=\\mathbb {Q}(\\sqrt{-D})$ pour un entier $D$ positif, sans facteur carré.",
"On suppose $E$ plongée dans un espace projectif par un diviseur ample et symétrique défini sur $\\overline{\\mathbb {Q}}$ et on note $\\hat{h}:E\\left(\\overline{\\mathbb {Q}}\\right)\\rightarrow {\\mathbb {R}}$ la hauteur normalisée (i.e.",
"de Néron-Tate) correspondante, qui satisfait $\\hat{h}(\\tau p)=N_{K/{\\mathbb {Q}}}(\\tau )\\hat{h}(p)$ pour $\\tau \\in {\\rm End}(E)$ , $p\\in E(\\overline{\\mathbb {Q}})$ , et qui ne s'annule qu'aux points de torsion de $E$ .",
"L'accouplement de Néron-Tate associé s'écrit $\\langle p,q\\rangle _{NT}=\\frac{1}{2}\\left( \\hat{h}(p+q)-\\hat{h}(p)-\\hat{h}(q)\\right)\\in \\mathbb {R}$ pour $p,q\\in E(\\overline{\\mathbb {Q}})$ , il induit naturellement une forme $\\mathbb {Q}$ -bilinéaire symétrique sur le $L$ -espace vectoriel $E(\\overline{\\mathbb {Q}})\\otimes _\\mathbb {Z}\\mathbb {Q}$ .",
"On introduit alors le produit scalaire (à valeurs dans $L\\otimes _\\mathbb {Q}\\mathbb {R}$ ) $\\langle p,q \\rangle = {\\left\\lbrace \\begin{array}{ll}\\langle p,q\\rangle _{NT} &\\text{si $L=\\mathbb {Q}$}\\\\\\langle p,q\\rangle _{NT}-\\frac{1}{\\sqrt{-D}}\\langle p,\\sqrt{-D} q\\rangle _{NT} &\\text{si $L=\\mathbb {Q}(\\sqrt{-D})$,}\\end{array}\\right.",
"}$ dont on vérifie qu'il est $L$ -sesqui-linéaire (relativement à la conjugaison sur $L$ ) et hermitien.",
"On étend ces produits scalaires à $E(\\overline{\\mathbb {Q}})^n\\otimes _\\mathbb {Z}\\mathbb {Q}$ par les formules $\\langle p,q\\rangle _{NT}&=\\sum _{i=1}^n \\langle p_i,q_i \\rangle _{NT}\\\\\\langle p,q\\rangle &=\\sum _{i=1}^n \\langle p_i,q_i \\rangle $ pour $p=(p_1,\\cdots ,p_n)$ et $q=(q_1,\\cdots ,q_n)$ dans $E(\\overline{\\mathbb {Q}})^n$ .",
"Et on munit $\\mathbb {C}^n$ , identifié à l'espace tangent en l'origine de $E(\\overline{\\mathbb {C}})^n$ , de son produit hermitien standard.",
"Lemma 1.1 Soient $H$ et $H^{\\prime }$ deux sous-groupes algébriques, connexes, de $E^n$ , alors leurs espaces tangents à l'origine $\\mathrm {T}H(\\mathbb {C})$ et $\\mathrm {T}H^{\\prime }(\\mathbb {C})$ sont orthogonaux dans $\\mathbb {C}^n$ si et seulement si $H(\\overline{\\mathbb {Q}})$ et $H^{\\prime }(\\overline{\\mathbb {Q}})$ sont orthogonaux dans $E^n$ (pour le produit scalaire $\\langle \\cdot ,\\cdot \\rangle $ ou de façon équivalente pour l'accouplement de Néron-Tate).",
"Soient $d$ et $d^{\\prime }$ les dimensions respectives de $H$ et $H^{\\prime }$ , il existe des homomorphismes $\\varphi :E^d\\rightarrow E^n$ et $\\varphi ^{\\prime }:E^{d^{\\prime }}\\rightarrow E^n$ dont les images sont $H$ et $H^{\\prime }$ respectivement.",
"On peut décrire ces homomorphismes sous la forme $(p_1,\\ldots ,p_d)&\\mapsto p=\\left(\\sum _{j=1}^d a_{1,j}p_j,\\ldots ,\\sum _{j=1}^d a_{n,j}p_j\\right)\\\\(q_1,\\ldots ,q_{d^{\\prime }})&\\mapsto q=\\left(\\sum _{k=1}^{d^{\\prime }} b_{1,k}q_k,\\ldots ,\\sum _{k=1}^{d^{\\prime }} b_{n,k}q_k\\right)$ et leurs applications tangentes $\\mathrm {T}_\\varphi :\\mathbb {C}^{d}\\rightarrow \\mathrm {T} H(\\mathbb {C})\\subseteq \\mathbb {C}^n$ et $\\mathrm {T}_\\varphi ^{\\prime }:\\mathbb {C}^{d^{\\prime }}\\rightarrow \\mathrm {T} H^{\\prime }(\\mathbb {C})\\subseteq \\mathbb {C}^n$ par $(u_1,\\ldots ,u_d)&\\mapsto u=\\left(\\sum _{j=1}^d a_{1,j}u_j,\\ldots ,\\sum _{j=1}^d a_{n,j}u_j\\right)\\\\(v_1,\\ldots ,v_{d^{\\prime }})&\\mapsto v=\\left(\\sum _{k=1}^{d^{\\prime }} b_{1,k}v_k,\\ldots ,\\sum _{k=1}^{d^{\\prime }} b_{n,k}v_k\\right)$ où $a_{i,j}, b_{i,k}\\in \\mathrm {End}(E)$ pour tous $i,j,k$ .",
"On calcule alors les produits scalaires en développant par sesquilinéarité $\\langle u,v \\rangle =\\sum _{i=1}^n\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }} a_{i,j}u_j \\overline{b_{i,k}v_k}=\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }}\\left(\\sum _{i=1}^n a_{i,j}\\overline{b_{i,k}}\\right)u_j\\overline{v_k}$ et $\\langle p,q \\rangle =\\sum _{i=1}^n\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }} a_{i,j} \\overline{b_{i,k}}\\langle p_j,q_k \\rangle =\\sum _{j=1}^d\\sum _{k=1}^{d^{\\prime }}\\left(\\sum _{i=1}^n a_{i,j}\\overline{b_{i,k}}\\right)\\langle p_j,q_k \\rangle .$ Si les espaces tangents en l'origine à $H$ et $H^{\\prime }$ sont orthogonaux (resp.",
"si $H$ et $H^{\\prime }$ sont orthogonaux) on a $\\langle u,v\\rangle =0$ pour tous $u_1,\\dots ,u_d,v_1,\\dots ,v_{d^{\\prime }}\\in {\\mathbb {C}}$ (resp.",
"$\\langle p,q\\rangle =\\langle p,q\\rangle _{\\rm NT}=0$ pour tous $p_1,\\dots ,p_d,q_1,\\dots ,q_{d^{\\prime }}\\in E(\\overline{\\mathbb {Q}}$ ).",
"Appliqué avec $u_{j^{\\prime }}=v_{k^{\\prime }}=0$ pour $j^{\\prime }\\ne j$ , $k^{\\prime }\\ne k$ et $u_j=v_k=u\\ne 0$ (resp.",
"$p_{j^{\\prime }}=q_{k^{\\prime }}=0$ pour $j^{\\prime }\\ne j$ , $k^{\\prime }\\ne k$ et $p_j=q_k=p$ non de torsion), ceci entraîne les égalités $\\sum _{i=1}^na_{i,j}\\overline{b_{i,k}}=0$ , pour tous $j,k$ .",
"Réciproquement, ces dernières entraînent $\\langle u,v\\rangle =0$ et $\\langle p,q\\rangle =\\langle p,q\\rangle _{\\rm NT}=0$ pour tous $u\\in {\\rm T}H({\\mathbb {C}})$ , $v\\in {\\rm T}H^{\\prime }({\\mathbb {C}})$ , $p\\in H(\\overline{\\mathbb {Q}})$ et $q\\in H^{\\prime }(\\overline{\\mathbb {Q}})$ , ce qui établit l'équivalence énoncée."
],
[
"Acknowledgements",
"We wish to thank P. Philippon for useful discussions and for his kind communication of the appendix.",
"We kindly thank A. Galateau for his communication of the constants in [18].",
"We thank D. Bertrand, P. Habegger, M. Hindry and D. Masser for providing us useful references.",
"We warmly thank the FNS for the great support we receive in doing research in Mathematics.",
"E. Viada also likes to thank the AIM and the organizers J. Pila and U. Zannier for supporting her participation to a conference on unlikely intersections in Pisa, where this work was inspired.",
"S. Checcoli and F. Veneziano would like to address a special thank to their advisor U. Zannier for introducing them to mathematical research."
]
] | 1204.1435 |
[
[
"How Many Vote Operations Are Needed to Manipulate A Voting System?"
],
[
"Abstract In this paper, we propose a framework to study a general class of strategic behavior in voting, which we call vote operations.",
"We prove the following theorem: if we fix the number of alternatives, generate $n$ votes i.i.d.",
"according to a distribution $\\pi$, and let $n$ go to infinity, then for any $\\epsilon >0$, with probability at least $1-\\epsilon$, the minimum number of operations that are needed for the strategic individual to achieve her goal falls into one of the following four categories: (1) 0, (2) $\\Theta(\\sqrt n)$, (3) $\\Theta(n)$, and (4) $\\infty$.",
"This theorem holds for any set of vote operations, any individual vote distribution $\\pi$, and any integer generalized scoring rule, which includes (but is not limited to) almost all commonly studied voting rules, e.g., approval voting, all positional scoring rules (including Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.",
"We also show that many well-studied types of strategic behavior fall under our framework, including (but not limited to) constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.",
"Therefore, our main theorem naturally applies to these problems."
],
[
"Introduction",
"Voting is a popular method used to aggregate voters' preferences to make a joint decision.",
"Recently, voting has been used in many fields of artificial intelligence, for example in multi-agent systems [15], recommender systems [21], [32], and web-search engines [12].",
"One of the most desired properties for voting rules is strategy-proofness, that is, no voter has incentive to misreport her preferences to obtain a better outcome of the election.",
"Unfortunately, strategy-proofness is not compatible with some other desired properties, due to the celebrated Gibbard-Satterthwaite theorem [22], [36], which states that when there are at least three alternatives, no strategy-proof voting rule satisfies the following two natural properties: non-imposition (every alternative can win) and non-dictatorship (no voter is a dictator, whose top ranked alternative is always selected to be the winner).",
"Even though manipulation is inevitable, researchers have set out to investigate whether computational complexity can serve as a barrier against various types of strategic behavior, including manipulation.",
"The idea is, if we can prove that it is computationally too costly for a strategic individual to find a beneficial operation, she may give up doing so.",
"Initiated by Bartholdi, Tovey, and Trick [2], a fair amount of work has been done to characterize the computational complexity of various types of strategic behaviorSee [19], [17], [35] for recent surveys., including the following.",
"$\\bullet $ Manipulation: a voter or a coalition of voters cast false vote(s) to change the winner (and the new winner is more preferred).",
"$\\bullet $ Bribery: a strategic individual changes some votes by bribing the voters to make the winner preferable to her [16].",
"The bribery problem is closely related to the problem of computing the margin of victory [5], [25], [47].",
"$\\bullet $ Control: a strategic individual adds or deletes votes to make the winner more preferable to her [3].",
"Most previous results studying “using computational complexity as a barrier against strategic behavior” conduct worst-case analyses of computational complexity.",
"Recently, an increasing number of results show that manipulation, as a particular type of strategic behavior, is typically not hard to compute.",
"One direction, mainly pursued in the theoretical computer science community, is to obtain a quantitative version of the Gibbard-Satterthwaite theorem, showing that for any given voting rule that is “far” enough from any dictatorships, an instance of manipulation can be found easily with high probability.",
"This line of research was initiated by Friedgut, Kalai, and Nisan [20], where they proved the theorem for 3 alternatives and neutral voting rules.",
"The theorem was extended to an arbitrary number of alternatives by Isaksson, Kindler, and Mossel [23], and finally, the neutrality constraint was removed by Mossel and Racz [27].",
"Other extensions include Dobzinski and Procaccia [11] and Xia and Conitzer [49].",
"Another line of research is to characterize the “frequency of manipulability”, defined as the probability for a randomly generated preference-profile to be manipulable by a group of manipulators, where the non-manipulators' votes are generated i.i.d.",
"according to some distribution (for example, the uniform distribution over all possible types of preferences).",
"Peleg [31], Baharad and Neeman [1], and Slinko [37], [38] studied the asymptotic frequency of manipulability for positional scoring rules when the non-manipulators' votes are drawn i.i.d.",
"uniformly at random.",
"Procaccia and Rosenschein [34] showed that for positional scoring rules, when the non-manipulators’ votes are drawn i.i.d.",
"according to some distribution that satisfies some natural conditions, if the number of manipulators is $o(\\sqrt{n})$ , where $n$ is the number of non-manipulators, then the probability that the manipulators can succeed goes to 0 as $n$ goes to infinity; if the number of manipulator is $\\omega (n)$ , then the probability that the manipulators can succeed goes to 1.",
"This dichotomy theorem was generalized to a class of voting rules called generalized scoring rules (GSRs) by Xia and Conitzer [48].",
"A GSR is defined by two functions $f,g$ , where $f$ maps each vote to a vector in multidimensional space, called a generalized scoring vector (the dimensionality of the space is not necessarily the same as the number of alternatives).",
"Given a profile $P$ , let total generalized scoring vector be the sum of $f(V)$ for all votes $V$ in $P$ .",
"Then, $g$ selects the winner based on the total preorder of the components of the total generalized scoring vector.",
"We call a GSR an integer GSR, if the components of all generalized scoring vectors are integers.",
"(Integer) GSRs are a general class of voting rules.",
"One evidence is that many commonly studied voting rules are integer GSRs, including (but not limited to) approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.The definition of these commonly studied voting rules can be found in, e.g., [48].",
"In this paper, we define GSRs as voting rules where the inputs are profiles of linear orders.",
"GSRs can be easily generalized to include other types of voting rules where the inputs are not necessarily linear orders, for example, approval voting.",
"As another evidence, GSRs admit a natural axiomatic characterization [50], which also suggests that GSRs are equivalent to hyperplane rules [26].",
"The knife-edge case of $\\Theta (\\sqrt{n})$ was studied experimentally for STV and veto in [45], showing that the probability for the manipulators to succeed has a smooth phase transition.",
"More recently, [26] extends the dichotomy theorem to all anonymous voting rules for distributions that satisfy some mild conditions, and theoretically proved that for all generalized scoring rules, for the knife-edge case, the probability that the manipulators can achieve their goal is continuously differentiable, which suggests a smooth phase transition.",
"While most of the aforementioned results are about manipulation, in this paper, we focus the optimization variants of various types of strategic behavior, including manipulation, bribery, and control.",
"Despite being natural, to the best of our knowledge, such optimization variants have been investigated for only three types of strategic behavior.",
"The first is the unweighted coalitional optimization (UCO) problem, where we are asked to compute the minimum number of manipulators who can make a given alternative win [53].",
"Approximation algorithms have been proposed for UCO for specific voting systems, including positional scoring rules and maximin [53], [51], [52].",
"The second is the margin of victory problem, where we are asked to compute the smallest number of voters who can change their votes to change the winner [25], [5], [47].",
"The third is the minimum manipulation coalition size problem, which is similar to the margin of victory, except that all voters who change their votes must prefer the new winner to the old winner [33].",
"In this paper, we introduce a unified framework to study a class of strategic behavior for generalized scoring rules, which we call vote operations.",
"In our framework, a strategic individual seeks to change the winner by applying some operations, which are modeled as vectors in a multidimensional space.",
"We study three goals of the strategic individual: (1) making a favored alternative win, called constructive vote operation ($\\text{\\sc CVO}$), (2) making a disfavored alternative lose, called destructive vote operation ($\\text{\\sc DVO}$ ), and (3) change the winner of the election, called change-winner vote operation ($\\text{\\sc CWVO}$ ).",
"The framework will be formally defined in Section .",
"This is our main conceptual contribution.",
"Our main technical contribution is the following asymptotical characterization of the minimum number of operations that are needed for the strategic individual to achieve her goal.",
"Theorem REF (informally put) Fix the number of alternatives and the set of vote operations.",
"For any integer generalized scoring rule and any distribution $\\pi $ over votes, we generate $n$ votes i.i.d.",
"according to $\\pi $ and let $n$ go to infinity.",
"Then, for any $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\rbrace $ and any $\\epsilon >0$ , with probability at least $1-\\epsilon $ , the minimum number of operations that are necessary for the strategic individual to achieve $\\text{\\sc VO}$ falls into one of the following four categories: (1) 0, (2) $\\Theta (\\sqrt{n})$ , (3) $\\Theta (n)$ , and (4) $\\infty $ .",
"More informally, Theorem REF states that in large elections, to achieve a specific goal (one of the three goals described above), with probability that can be infinitely close to 1 the strategic individual needs to either do nothing (the goal is already achieved), apply $\\Theta (\\sqrt{n})$ vote operations, apply $\\Theta (n)$ vote operations, or the goal cannot be achieve no matter how many vote operations are applied.",
"This characterization holds for any integer generalized scoring rule, any set of vote operations, and any distribution $\\pi $ for individual votes.",
"The proof of Theorem REF is based on the Central Limit Theorem and on sensitivity analyses for the integer linear programmings (ILPs).",
"It works as follows.",
"We will formulate each of the strategic individual's three goals as a set of ILPs in Section .",
"By applying Central Limit Theorem, we show that with probability that goes to 1 the random generated preference-profile satisfies a desired property.",
"Then, for each such preference-profile we apply the sensitivity analyses in [8] to show that with high probability the number of operations that are necessary is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .",
"While Theorem REF may look quite abstract, we show later in the paper that many well-studied types of strategic behavior fall under our vote operation framework, including constructive/destructive manipulation, bribery, and control by adding/deleting votes, margin of victory, and minimum manipulation coalition size.We defer the definition of these types of strategic behavior to Section .",
"Therefore, we naturally obtain corollaries of Theorem REF for these types of strategic behavior.",
"The theorem also applies to other types of strategic behavior, for example the mixture of any types mentioned above, which is known as multimode control attacks [18]."
],
[
"Related Work and Discussion",
"Our main theorem applies to any integer generalized scoring rule for destructive manipulation, constructive and destructive bribery and control by adding/deleting votes.",
"To the best of our knowledge, no similar results were obtained even for specific voting rules for these types of strategic behavior.",
"Three previous papers obtained similar results for manipulation, margin of victory, and minimum manipulation coalition size.",
"The applications of our main theorem to these types of strategic behavior are slightly weaker, but we stress that our main theorem is significantly more general.",
"Three related papers.",
"First, the dichotomy theorem in [48] implies that, (informally) when the votes are drawn i.i.d.",
"from some distribution, with probability that goes to 1 the solution to constructive and destructive UCO is either 0 or approximately $\\sqrt{n}$ for some favored alternatives.",
"However, this result only works for the UCO problem and some distributions over the votes.",
"Second, it was proved in [47] that for any non-redundant generalized scoring rules that satisfy a continuity condition, when the votes are drawn i.i.d.",
"and we let the number of voters $n$ go to infinity, either with probability that can be arbitrarily close to 1 the margin of victory is $\\Theta (\\sqrt{n})$ , or with probability that can be arbitrarily close to 1 the margin of victory is $\\Theta (n)$ .",
"It is easy to show that for non-redundant voting rules, the margin of victory is never 0 or $\\infty $ .",
"Though it was shown in [47] that many commonly studied voting rules are GSRs that satisfy such continuity condition, in general it is not clear how restrictive the continuity condition is.",
"More importantly, the result only works for the margin of victory problem.",
"Third, in [33], the authors investigated the distribution over the minimum manipulation coalition size for positional scoring rules when the votes are drawn i.i.d.",
"from the uniform distribution.",
"However, it is not clear how their techniques can be extended beyond the uniform distributions and positional scoring rules, which are a very special case of generalized scoring rules.",
"Moreover, the paper only focused on the minimum manipulation coalition size problem.",
"Our results has both negative and positive implications.",
"On the negative side, our results provide yet another evidence that computational complexity is not a strong barrier against strategic behavior, because the strategic individual now has some information about the number of operations that are needed, without spending any computational cost or even without looking at the input instance.",
"Although the estimation of our theorem may not be very precise (because we do not know which of the four cases a given instance belongs to), such estimation may be explored to designing effective algorithms that facilitate strategic behavior.",
"On the positive side, this easiness of computation is not always a bad thing: sometimes we want to do such computation in order to test how robust a given preference-profile is.",
"For example, computing the margin of victory is an important component in designing novel risk-limiting audit methods [25], [5], [47], [39], [40], [42], [41], [43].",
"While being quite general, our results have two main limitations.",
"First, they are asymptotical results, where we fix the number of alternatives and let the number of voters go to infinity.",
"We do not know the convergence rate, or equivalently, how many voters are needed for the observation to hold.",
"In fact, this is a standard setting in previous work, especially in the studies of “frequency of manipulability”.",
"We feel that our results work well in settings where there are small number of alternatives and large number of voters, e.g., political elections.",
"Second, our results show that with high probability one of the four cases holds (0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , $\\infty $ ), but we do not know which case holds more often.",
"We will briefly discuss this issue in .",
"It is possible to refine our study for specific voting rules and specific types of strategic behavior that fall under our framework, which we leave as future work."
],
[
"Structure of the Paper",
"After recalling basic definitions of voting and generalized scoring rules, we present the framework in Section , where we define vote operations as well as the strategic individual's objectives.",
"Then, in Section we formulate the optimization problem for the strategic individual as a set of integer linear programmings (ILPs).",
"The main theorem will be presented in Section .",
"To show the wide application of the framework and the main theorem, we show that many commonly studied types of strategic behavior can be modeled as vote operations for generalized scoring rules in Section , which means that our main theorem naturally applies to these cases.",
"We add some discussions and point out some future directions in Section and .",
"Some proofs are relegated to ."
],
[
"Preliminaries",
"Let $\\mathcal {C}$ denote the set of alternatives (or candidates), $|\\mathcal {C}|=m$ .",
"We assume strict preference orders.",
"That is, a vote is a linear order over $\\mathcal {C}$ .",
"The set of all linear orders over $\\mathcal {C}$ is denoted by $L(\\mathcal {C})$ .",
"A preference-profile $P$ is a collection of $n$ votes for some $n\\in \\mathbb {N}$ , that is, $P\\in L(\\mathcal {C})^n$ .",
"Let $L(\\mathcal {C})^*=\\bigcup _{n=1}^\\infty L(\\mathcal {C})^n$ .",
"A voting rule $r$ is a mapping that assigns to each preference-profile a single winner.",
"That is, $r: L(\\mathcal {C})^*\\rightarrow \\mathcal {C}$ .",
"Throughout the paper, we let $n$ denote the number of votes and let $m$ denote the number of alternatives.",
"We now recall the definition of generalized scoring rules (GSRs) [48].",
"For any $K\\in \\mathbb {N}$ , let ${\\mathcal {O}}_K=\\lbrace o_1,\\ldots ,o_K\\rbrace $ .",
"A total preorder (preorder for short) is a reflexive, transitive, and total relation.",
"Let $\\text{Pre}({\\mathcal {O}}_K)$ denote the set of all preorders over ${\\mathcal {O}}_K$ .",
"For any $\\vec{p}\\in {\\mathbb {R}}^K$ , we let $\\text{Ord}(\\vec{p})$ denote the preorder $\\trianglerighteq $ over ${\\mathcal {O}}_K$ where $o_{k_1}\\trianglerighteq o_{k_2}$ if and only if $p_{k_1}\\ge p_{k_2}$ .",
"That is, the $k_1$ -th component of $\\vec{p}$ is as large as the $k_2$ -th component of $\\vec{p}$ .",
"For any preorder $\\trianglerighteq $ , if $o\\trianglerighteq o^{\\prime }$ and $o^{\\prime }\\trianglerighteq o$ , then we write $o=_\\trianglerighteq o^{\\prime }$ .",
"Each preorder $\\trianglerighteq $ naturally induces a (partial) strict order $\\vartriangleright $ , where $o\\vartriangleright o^{\\prime }$ if and only if $o\\trianglerighteq o^{\\prime }$ and $o^{\\prime }\\ntrianglerighteq o$ .",
"Definition 1 Let $K\\in \\mathbb {N}$ , $f:L(\\mathcal {C})\\rightarrow {\\mathbb {R}}^K$ and $g:\\text{Pre}({\\mathcal {O}}_K)\\rightarrow \\mathcal {C}$ .",
"$f$ and $g$ determine a generalized scoring rule (GSR) $GS(f,g)$ as follows.",
"For any preference-profile $P=(V_1,\\ldots , V_n)\\in L(\\mathcal {C})^n$ , abusing the notation we let $f(P)=\\sum _{i=1}^nf(V_i)$ , and let $GS(f,g)(P)=g(\\text{Ord}(f(P)))$ .",
"We say that $GS(f,g)$ is of order $K$.",
"If $f(V)\\in {\\mathbb {Z}}^K$ holds for all $V\\in L(\\mathcal {C})$ , then we call $\\text{GS}(f,g)$ an integer GSR.",
"For any $V\\in L(\\mathcal {C})$ , $f(V)$ is called a generalized scoring vector, $f(P)$ is called a total generalized scoring vector, and $\\text{Ord}(f(P))$ is called the induced preorder of $P$ .",
"The class of integer GSRs is equivalent to the class of rational GSRs, where the components of each generalized scoring vector is in $\\mathbb {Q}$ , because for any $l>0$ , $\\text{GS}(f,g)=\\text{GS}(l\\cdot f,g)$ .",
"Almost all commonly studied voting rules are generalized scoring rules, including (but not limited to) approval voting, Bucklin, Copeland, maximin, plurality with runoff, ranked pairs, and multi-stage voting rules that use GSRs in each stage to eliminate alternatives (including Nanson's and Baldwin's rule).",
"As an example, we recall the proof from [48] that the single transferable vote (STV) rule (a.k.a.",
"instant-runoff voting or alternative vote for single-winner elections) is an integer generalized scoring rule.",
"Example 1 STV selects the winner in $m$ rounds.",
"In each round, the alternative that gets the lowest plurality score (the number of times that the alternative is ranked in the top position) drops out, and is removed from all of the votes (so that votes for this alternative transfer to another alternative in the next round).",
"Ties are broken alphabetically.",
"The last-remaining alternative is the winner.",
"To see that STV is an integer GSR, we will use generalized scoring vectors with many components.",
"For every proper subset $S$ of alternatives, for every alternative $c$ outside of $S$ , there is a component in the vector that contains the number of times that $c$ is ranked first if all alternatives in $S$ are removed.",
"Let $\\bullet $ $K_{STV}=\\sum _{i=0}^{m-1}{m \\atopwithdelims ()i} (m-i)$ ; the components are indexed by $(S,j)$ , where $S$ is a proper subset of $\\mathcal {C}$ and $j\\le m,c_j \\notin S$ .",
"$\\bullet $ $(f_{STV}(V))_{(S,j)}=1$ , if after removing $S$ from $V$ , $c_j$ is at the top; otherwise, let $(f_{STV}(V))_{(S,j)}=0$ .",
"$\\bullet $ $g_{STV}$ selects the winners based on $\\trianglerighteq $ as follows.",
"In the first round, let $j_1$ be the index such that $o_{(\\emptyset ,j_1)}$ is ranked the lowest in $\\trianglerighteq $ among all $o_{(\\emptyset ,j)}$ (if there are multiple such $j$ 's, then we break ties alphabetically to select the least-preferred one).",
"Let $S_1=\\lbrace c_{j_1}\\rbrace $ .",
"Then, for any $2\\le i\\le m-1$ , define $S_i$ recursively as follows: $S_i=S_{i-1}\\cup \\lbrace c_{j_i}\\rbrace $ , where $j_i$ is the index such that $o_{(S_{i-1},j_i)}$ is ranked the lowest in $\\trianglerighteq $ among all $o_{(S_{i-1},j)}$ ; finally, the winner is the unique alternative in $(\\mathcal {C}\\setminus S_{m-1})$ .",
"GSRs admit a natural axiomatic characterization [50].",
"That is, GSRs are the class of voting rules that satisfy anonymity, homogeneity, and finite local consistency.",
"Anonymity says that the winner does not depend on the name of the voters, homogeneity says that if we duplicate the preference-profile multiple times, then the winner does not change, and finite local consistency is an approximation to the well-studied consistency axiom.",
"Not all voting rules are GSRs, for example, Dodgson's rule is not a GSR because it violates homogeneity [4], and the following skewed majority rule is also not a GSR because it also violates homogeneity.",
"Example 2 For any $\\frac{1}{2}<\\gamma <1$ , the $\\gamma $ -majority rule is defined for two alternatives $\\lbrace a,b\\rbrace $ as follows: $b$ is the winner if and only if the number of voters who prefer $b$ is more than the number of voters who prefer $a$ by at least $n^{\\gamma }$ .",
"Admittedly, these $\\gamma $ -majority rules are quite artificial.",
"Later in this paper we will see that the observation made for GSRs in our main theorem (Theorem REF ) does not hold for $\\gamma $ -majority rules for any $\\frac{1}{2}<\\gamma <1$ .",
"Notice that these rules satisfy anonymity, which means that the observation made in Theorem REF cannot be extended to all anonymous voting rules."
],
[
"Vote Operations",
"All types of strategic behavior mentioned in the introduction share the following common characteristics.",
"The strategic individual (who can be a group of manipulators, a briber, or a controller, etc.)",
"changes the winner by changing the votes in the preference-profile.",
"Therefore, for generalized scoring rules, any such an operation can be uniquely represented by changes in the total generalized scoring vector.",
"This is in contrast to some other types of strategic behavior where the strategic individual changes the set of alternatives or the voting rule [3], [44].",
"In this section, we first define the set of operations the strategic individual can apply, then define her goals.",
"Given a generalized scoring rule of order $K$ , we model the strategic behavior, called vote operations, as a set of vectors, each of which has $K$ elements, representing the changes made to the total generalized scoring vector if the strategic individual applies this operation.",
"We focus on integer vectors in this paper.",
"Definition 2 Given a GSR $\\text{GS}(f,g)$ of order $K$ , let $\\Delta =[\\vec{\\delta }_1 \\cdots \\vec{\\delta }_T]$ denote the vote operations, where for each $i\\le T$ , $\\vec{\\delta }_i\\in {\\mathbb {Z}}^K$ is a column vector that represents the changes made to the generalized scoring vector by applying the $i$ -th vote operation.",
"For each $l\\le K$ , let $\\Delta _l$ denote the $l$ -th row of $\\Delta $ .",
"We will show examples of these vote operations for some well-studied types of strategic behavior in Section .",
"Given the set of available operations $\\Delta $ , the strategic individual's behavior is characterized by a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ , where $\\vec{v}$ is a column vector and for each $i\\le T$ , $v_i$ represents the number of $i$ -th operation (corresponding to $\\vec{\\delta }_i$ ) that she applies.",
"Let $\\Vert \\vec{v}\\Vert _1=\\sum _{i=1}^Tv_i$ denote the total number of operations in $\\vec{v}$ , which is the L$_1$ -norm of $\\vec{v}$ .",
"It follows that $\\Delta \\cdot \\vec{v}$ is the change in the total generalized scoring vector introduced by the strategic individual, where for any $l\\le K$ , $\\Delta _l\\cdot \\vec{v}$ is the change in the $l$ -th component.",
"Next, we give definitions of the strategic individual's three goals and the corresponding computational problems studied in this paper.",
"Definition 3 In the constructive vote operation (CVO) problem, we are given a generalized scoring rule $\\text{GS}(f,g)$ , a preference-profile $P$ , a favored alternative $c$ , and a set of vote operations $\\Delta =[\\vec{\\delta }_1 \\cdots \\vec{\\delta }_T]$ , and we are asked to compute the smallest number $k$ , denoted by $\\text{\\sc CVO}(P,c)$ , such that there exists a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ with $\\Vert \\vec{v}\\Vert _1= k$ and $g\\left(\\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})\\right)=c$ .",
"If such $\\vec{v}$ does not exist, then we denote $\\text{\\sc CVO}(P,c)=\\infty $ .",
"The destructive vote operation (DVO) problem is defined similarly, where $c$ is the disfavored alternative, and we are asked to compute the smallest number $k$ , denoted by $\\text{\\sc DVO}(P,c)$ , such that there exists a vector $\\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^{T}$ with $\\Vert \\vec{v}\\Vert _1= k$ and $g\\left(\\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})\\right)\\ne c$ .",
"In the change-winner vote operation (CWVO) problem, we are not given $c$ and we are asked to compute $\\text{\\sc DVO}(P,\\text{GS}(f,g)(P))$ , denoted by $\\text{\\sc CWVO}(P)$ .",
"In CVO, the strategic individual seeks to make $c$ win; in DVO, the strategic individual seeks to make $c$ lose; and in CWVO, the strategic individual seeks to change the current winner.",
"For a given instance $(P,r)$ , $\\text{\\sc CWVO}$ is a special case of $\\text{\\sc DVO}$ , where $c=\\text{GS}(f,g)(P)$ .",
"We distinguish these two problems because in this paper, the input preference-profiles are generated randomly, so the winners of these preference-profiles might be different.",
"Therefore, when the preference-profiles are randomly generated, the distribution for the solution to $\\text{\\sc DVO}$ does not immediately give us a distribution for the solution to $\\text{\\sc CWVO}$ ."
],
[
"The ILP Formulation",
"Let us first put aside the strategic individual's goal for the moment (i.e., making a favored alternative win, making a disfavored alternative lose, or changing the winner) and focus on the following question: given a preference-profile $P$ and a preorder $\\trianglerighteq $ over the $K$ components of the generalized scoring vector, that is, $\\trianglerighteq \\in \\text{Pre}({\\mathcal {O}}_K)$ , how many vote manipulations are needed to change the order of the total generalized scoring vector to $\\trianglerighteq $ ?",
"Formally, given a $\\text{GS}(f,g)$ , a preference-profile $P$ and $\\trianglerighteq \\in \\text{Pre}({\\mathcal {O}}_K)$ , we are interested in $\\min \\lbrace \\Vert \\vec{v}\\Vert _1: \\vec{v}\\in {\\mathbb {N}}_{\\ge 0}^K, \\text{Ord}(f(P)+\\Delta \\cdot \\vec{v})=\\trianglerighteq \\rbrace $ .",
"This can be computed by the following integer linear programming ILP$_\\trianglerighteq $ , where $v_i$ represents the $i$ th component in $\\vec{v}$ , which must be a nonnegative integer.",
"We recall that $\\Delta _l$ denotes the $l$ -th row vector of $\\Delta $ .",
"$\\begin{tabular}{r@{\\hspace{28.45274pt}}rl}& \\min &\\Vert \\vec{v}\\Vert _1\\\\s.t.",
"&\\forall o_i=_\\trianglerighteq o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}= [f(P)]_j-[f(P)]_i\\\\&\\forall o_i\\vartriangleright o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}\\ge [f(P)]_j-[f(P)]_i+1\\\\&\\forall i:& v_i\\ge 0\\end{tabular}&&(\\text{LP}_{\\trianglerighteq })$ Now, we take the strategic individual's goal into account.",
"We immediately have the following lemma as a warmup, whose proofs are straightforward and are thus omitted.",
"Lemma 1 Given a GSR $\\text{GS}(f,g)$ , an alternative $c$ , and a preference-profile $P$ , $\\bullet $ $\\text{\\sc CVO}(P,c)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_\\trianglerighteq $ has an integer solution; $\\bullet $ $\\text{\\sc DVO}(P,c)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )\\ne c$ and LP$_\\trianglerighteq $ has an integer solution; $\\bullet $ $\\text{\\sc CWVO}(P)<\\infty $ if and only if there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )\\ne \\text{GS}(f,g)(P)$ and LP$_\\trianglerighteq $ has an integer solution.",
"(We do not need the input $c$ for this problem.)",
"Moreover, the solution to each of the three problems is the minimum objective value in all LPs corresponding to the problem.",
"For example, if $\\text{\\sc CVO}(P,c)<\\infty $ , then $\\hfill \\text{\\sc CVO}(P,c)=\\min _{\\Vert \\vec{v}\\Vert _1}\\lbrace \\vec{v}\\text{ is the solution to some LP}_\\trianglerighteq \\text{ where }g(\\trianglerighteq )=c\\rbrace \\hfill $"
],
[
"The Main Theorem",
"In this section we prove the main theorem, which states that for any fixed $m$ , any generalized scoring rules, and any set of vote operations $\\Delta $ , if $n$ votes are generated i.i.d., then for CVO (respectively, DVO, CWVO), with probability that can be infinitely close to 1, the solution is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .",
"We first present a simple example for the majority rule for two alternatives $\\lbrace a,b\\rbrace $ to show the taste of the proof for a very special case, and then comment on why this idea cannot be extended to GSRs.",
"After the proof of Theorem REF we will add more comments on the non-triviality of proof.",
"Example 3 Suppose there are $n$ voters, whose votes are drawn i.i.d.",
"from a distribution $\\pi $ over all possible votes (i.e., voting for $a$ with probability $\\pi (a)$ and voting for $b$ with probability $\\pi (b)$ , w.l.o.g.",
"$\\pi (a)\\ge \\pi (b)$ ).",
"Let $Y_a$ (respectively, $Y_b$ ) denote random variable that represents the total number of voters for $a$ (respectively, for $b$ ).",
"The number of manipulators that are needed to make $b$ to win (i.e., the solution to the UCO problem, see Section REF for formal definition) is thus a random variable $Y_a-Y_b$ .If $Y_a-Y_b<0$ then no manipulator is needed.",
"Let $X$ denote the random variable that takes 1 with probability $\\pi (a)$ and takes $-1$ with probability $\\pi (b)$ .",
"It follows that $Y_a-Y_b=\\underbrace{X+\\cdots +X}_{n}$ .",
"By the Central Limit Theorem, $Y_a-Y_b$ converges to a normal distribution with mean $n\\cdot E(X)$ and variance ${n\\cdot \\text{Var}(X)}$ .",
"We are interested in usually how large is $Y_a-Y_b$ .",
"Not surprisingly, the answer depends on the distribution $\\pi $ .",
"If $\\pi (a)=\\pi (b)=1/2$ , then the mean of $Y_a-Y_b$ is zero, and the probability that it is a few standard deviations away from the mean is small.",
"For example, the probability that its absolute value is larger than $4\\sqrt{n\\cdot \\text{Var}(X)}$ is less than $0.01$ , which means that with $99\\%$ probability the solution of UCO is no more than $4\\sqrt{n\\cdot \\text{Var}(X)}$ .",
"On the other hand, if $\\pi (a)> \\pi (b)$ , then the mean of $Y_a-Y_b$ is $n(\\pi (a)-\\pi (b))$ , which means that with high probability the solution of UCO is very close to $n(\\pi (a)-\\pi (b))=\\Theta (n)$ .",
"The idea behind the argument in Example REF can be easily extended to positional scoring rules [34].",
"However, we do not believe that it can be extended to generalized scoring rules, even for the case of manipulation, for the following two reasons.",
"First, for generalized scoring rules, the components of the generalized scoring vector do not correspond to the “scores” of alternatives.",
"Therefore, having two components tied in the total generalized scoring vector does not mean that two alternatives are tied.",
"Second, the conditions for an alternative to win can be much more complicated than the condition for positional scoring rules, which amounts to requiring that a corresponding component of the total generalized scoring vector is the largest.",
"Therefore, it is not easy to figure out whether the manipulators can achieve their goal by just knowing the asymptotic relationship between the components of the total generalized scoring vector.",
"Theorem 1 Let $\\text{GS}(f,g)$ be an integer generalized scoring rule, let $\\pi $ be a distribution over all linear orders, and let $\\Delta $ be a set of vote operations.",
"Suppose we fix the number of alternatives, generate $n$ votes i.i.d.",
"according to $\\pi $ , and let $P_n$ denote the preference-profile.",
"Then, for any alternative $c$ , $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\footnote {When \\text{\\sc VO}=\\text{\\sc CWVO}, we let \\text{\\sc VO}(P_n,c) denote \\text{\\sc CWVO}(P_n).",
"}\\rbrace $ , and any $\\epsilon >0$ , there exists $\\beta ^*>1$ such that as $n$ goes to infinity, the total probability for the following four events sum up to more than $1-\\epsilon $ : (1) $\\text{\\sc VO}(P_n,c)=0$ , (2) $\\frac{1}{\\beta ^*}\\sqrt{n}<\\text{\\sc VO}(P_n,c)<\\beta ^* \\sqrt{n}$ , (3) $\\frac{1}{\\beta ^*} n<\\text{\\sc VO}(P_n,c)<\\beta ^* n$ , and (4) $\\text{\\sc VO}(P_n,c)=\\infty $ .",
"Proof of Theorem REF : Let $f(P_\\pi )=\\sum _{V\\in L(\\mathcal {C})}\\pi (V)\\cdot f(V)$ , and $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ .",
"We first prove the theorem for $\\text{\\sc CVO}$ , and then show how to adjust the proof for $\\text{\\sc DVO}$ and $\\text{\\sc CWVO}$ .",
"The theorem is proved in the following two steps.",
"Step 1: we show that as $n$ goes to infinity, with probability that goes to one we have the following: in a randomly generated $P_n$ , the difference between any pair of components in $f(P_n)$ is either $\\Theta (\\sqrt{n})$ or $\\Theta (n)$ .",
"Step 2: we apply sensitivity analyses to ILPs that are similar to the ILP given in Section to prove that for any such preference-profile and any $\\text{\\sc VO}\\in \\lbrace \\text{\\sc CVO},\\text{\\sc DVO},\\text{\\sc CWVO}\\rbrace $ , $\\text{\\sc VO}(P_n,c)$ is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ .",
"The idea behind Step 2 is, for any preference-profile $P_n$ , if the difference between a pair of components in $f(P_n)$ is $\\Theta (\\sqrt{n})$ , then we consider this pair of components (not alternatives) to be “almost tied”; if the difference is $\\Theta (n)$ , then we consider them to be “far away”.",
"Take $\\text{\\sc CVO}$ as an example, we can easily identify the cases where $\\text{\\sc CVO}(P_n,c)$ is either 0 (when $\\text{GS}(f,g)=c$ ) or $\\infty $ (by Lemma REF ).",
"Then, we will first try to break these “almost tied” pairs by using LPs that are similar to LP$_\\trianglerighteq $ introduced in Section , and show that if there exists an integer solution $\\vec{v}$ , then the objective value $\\Vert \\vec{v}\\Vert _1$ is $\\Theta (\\sqrt{n})$ .",
"Otherwise, we have to change the orders between some “far away” pairs by using LP$_\\trianglerighteq $ 's, and show that if there exists an integer solution to some LP$_\\trianglerighteq $ with $g(\\trianglerighteq )=c$ , then the objective value is $\\Theta (n)$ .",
"Formally, given $n\\in \\mathbb {N}$ and $\\beta >1$ , let ${\\cal P}_\\beta $ denote the set of all $n$ -vote preference-profiles $P$ that satisfy the following two conditions (we recall that $f(P_\\pi )=\\sum _{V\\in L(\\mathcal {C})}\\pi (V)\\cdot f(V)$ ): for any pair $i,j\\le K$ , 1. if $[f(P_\\pi )]_i=[f(P_\\pi )]_j$ then $\\frac{1}{\\beta }\\sqrt{n}<|[f(P)]_i-[f(P)]_j|<\\beta \\sqrt{n}$ ; 2. if $[f(P_\\pi )]_i\\ne [f(P_\\pi )]_j$ then $\\frac{1}{\\beta } n<|[f(P)]_i-[f(P)]_j|<\\beta n$ .",
"The following lemma was proved in [47], which follows after the Central Limit Theorem.",
"Lemma 2 For any $\\epsilon >0$ , there exists $\\beta $ such that $\\lim _{n\\rightarrow \\infty } \\text{Pr}\\left(P_n\\in {\\cal P}_\\beta \\right)>1-\\epsilon $ .",
"For any given $\\epsilon $ , in the rest of the proof we fix $\\beta $ to be a constant guaranteed by Lemma REF .",
"The next lemma (whose proof is deferred to the appendix) will be frequently used in the rest of the proof.",
"Lemma 3 Fix an integer matrix $\\bf A$ .",
"There exists a constant $\\beta _{\\bf A}$ that only depends on $\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .",
"$\\min \\Vert \\vec{x}\\Vert _1, \\text{ s.t.",
"}{\\bf A}\\cdot \\vec{x}\\ge \\vec{b}$ To prove that with high probability $\\text{\\sc CVO}(P_n,c)$ is either 0, $\\Theta (\\sqrt{n})$ , $\\Theta (n)$ , or $\\infty $ , we introduce the following notation.",
"A preorder $\\trianglerighteq ^{\\prime }$ is a refinement of another preorder $\\trianglerighteq $ , if $\\vartriangleright ^{\\prime }$ extends $\\vartriangleright $ .",
"That is, $\\vartriangleright \\subseteq \\vartriangleright ^{\\prime }$ .",
"We note that $\\trianglerighteq $ is a refinement of itself.",
"Let $\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq $ denote the strict orders that are in $\\vartriangleright ^{\\prime }$ but not in $\\vartriangleright $ .",
"That is, $(o_i,o_j)\\in (\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq )$ if and only if $o_i\\vartriangleright ^{\\prime } o_j$ and $o_i=_{\\trianglerighteq }o_j$ .",
"We define the following LP that is similar to LP$_\\trianglerighteq $ defined in Section , which will be used to check whether there is a way to break “almost tied” pairs of components to make $c$ win.",
"For any preorder $\\trianglerighteq $ and any of its refinement $\\trianglerighteq ^{\\prime }$ , we define LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ as follows.",
"$\\begin{tabular}{r@{\\hspace{28.45274pt}}rl}& \\min &\\Vert \\vec{v}\\Vert _1\\\\s.t.",
"&\\forall o_i=_{\\trianglerighteq ^{\\prime }} o_j: & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}= [f(P)]_j-[f(P)]_i\\\\&\\forall (o_i, o_j)\\in (\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq ): & (\\Delta _i-\\Delta _j)\\cdot \\vec{v}\\ge [f(P)]_j-[f(P)]_i+1\\\\&\\forall i:& v_i\\ge 0\\end{tabular}&&(\\text{LP}_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq })$ LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ is defined with a little abuse of notation because some of its constraints depend on $\\trianglerighteq $ (not only the pairwise comparisons in $(\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq )$ ).",
"This will not cause confusion because we will always indicate $\\trianglerighteq $ in the subscript.",
"We note that there is a constraint in LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ for each pair of components $o_i,o_j$ with $o_i=_{\\trianglerighteq }o_j$ .",
"Therefore, LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ is used to find a solution that breaks ties in $\\trianglerighteq $ .",
"It follows that LP$_{\\trianglerighteq ^{\\prime }\\ominus \\trianglerighteq }$ has an integer solution $\\vec{v}$ if and only if the strategic individual can make the order between any pairs of $o_i,o_j$ with $o_i=_{\\trianglerighteq }o_j$ to be the one in $\\trianglerighteq ^{\\prime }$ by applying the $i$ -th operation $v_i$ times, and the total number of vote operations is $\\Vert \\vec{v}\\Vert _1$ .",
"The following two claims identify the preference-profiles in ${\\cal P}_\\beta $ for which $\\text{\\sc CVO}$ is $\\Theta (\\sqrt{n})$ and $\\Theta (n)$ , respectively, whose proofs are deferred to the appendix.",
"Claim 1 There exists $N\\in \\mathbb {N}$ and $\\beta ^{\\prime }>1$ such that for any $n\\ge N$ , any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that $g(\\trianglerighteq ^*)=c$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .",
"Claim 2 There exists $\\beta ^{\\prime }>1$ such that for any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, and (3) there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_{\\trianglerighteq }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .",
"Lastly, for any $P\\in {\\cal P}_\\beta $ such that $\\text{GS}(f,g)(P)\\ne c$ , the only case not covered by Claim REF and Claim REF is that there does not exist $\\trianglerighteq $ with $\\text{GS}(f,g)(\\trianglerighteq )=c$ such that LP$_\\trianglerighteq $ has an integer solution.",
"It follows from Lemma REF that in this case $\\text{\\sc CVO}(P,c)=\\infty $ .",
"We note that $\\beta ^{\\prime }$ in Claim REF and Claim REF does not depend on $n$ .",
"Let $\\beta ^*$ be an arbitrary number that is larger than the two $\\beta ^{\\prime }$ s. This proves the theorem for $\\text{\\sc CVO}$ .",
"For $\\text{\\sc DVO}$ , we only need to change $g(\\trianglerighteq ^*)=c$ to $g(\\trianglerighteq ^*)\\ne c$ in Claim REF , and change $g(\\trianglerighteq )=c$ to $g(\\trianglerighteq )\\ne c$ in Claim REF .",
"For $\\text{\\sc CWVO}$ , $\\text{\\sc CWVO}(P)$ is never 0 and we only need to change $g(\\trianglerighteq ^*)=c$ to $g(\\trianglerighteq ^*)\\ne \\text{GS}(f,g)(P)$ in Claim REF , and change $g(\\trianglerighteq )=c$ to $g(\\trianglerighteq ) \\ne \\text{GS}(f,g)(P)$ in Claim REF .",
"$\\blacksquare $ More comments on the non-triviality of the proof.",
"Lemma REF has been proved in [47], whose intuition is quite straightforward and naturally corresponds to a random walk in multidimensional space.",
"However, we did not find an obvious connection between random walk theory and the observation made in Theorem REF .",
"We believe that it is unlikely that an obvious connection exists.",
"One evidence is that the observation made in Theorem REF does not hold for some voting rules.",
"For example, consider the $\\gamma $ -majority rule defined in Example REF .",
"It is not hard to see that as $n$ goes to infinity, with probability that goes to 1 we have $\\text{\\sc CVO}(P_n,b)=\\text{\\sc DVO}(P_n,a)=\\text{\\sc CWVO}(P_n)=n^{\\gamma }/2$ , which is not any of the four cases described in Theorem REF if $\\frac{1}{2}<\\gamma <1$ .",
"(This implies that for any $\\frac{1}{2}<\\gamma <1$ , $\\gamma $ -majority is not a generalized scoring rule, which we already know because they do not satisfy homogeneity.)",
"Therefore, the proof of Theorem REF should involve analyses on the specific structure of GSRs.",
"The main difficulty in proving Theorem REF is, for generalized scoring rules we have to handle the cases where some components of the total generalized scoring vector are equivalent.",
"This only happens with negligible probability for the randomly generated preference-profile $P_n$ , but it is not clear how often the strategic individual can make some components equivalent in order to achieve her goal.",
"This is the main reason for us to convert the vote manipulation problem to multiple ILPs and apply Lemma REF to analyze them."
],
[
"Applications of the Main Theorem",
"In this section we show how to apply Theorem REF to some well-studied types of strategic behavior, including constructive and destructive unweighted coalitional optimization, bribery and control, and margin of victory and minimum manipulation coalition size.",
"In the sequel, we will use each subsection to define these problems and describe how they fit in our vote operation framework, and how Theorem REF applies.",
"In the end of the section we present a unified corollary for all these types of strategic behavior."
],
[
"Unweighted Coalitional Optimization",
"Definition 4 In a constructive (respectively, destructive) unweighted coalitional optimization (UCO) problem, we are given a voting rule $r$ , a preference-profile $P^{NM}$ of the non-manipulators, and a (dis)favored alternative $c\\in \\mathcal {C}$ .",
"We are asked to compute the smallest number of manipulators who can cast votes $P^{M}$ such that $c=r(P^{NM}\\cup P^{M})$ (respectively, $c\\ne r(P^{NM}\\cup P^{M})$ ).",
"To see how UCO fits in the vote operation model, we view the group of manipulators as the strategic individual, and each vote cast by a manipulator is a vote operation.",
"Therefore, the set of operations is exactly the set of all generalized scoring vectors $\\lbrace f(V): V\\in L(\\mathcal {C})\\rbrace $ .",
"To apply Theorem REF , for constructive UCO we let $\\text{\\sc VO}=\\text{\\sc CVO}$ and for destructive UCO we let $\\text{\\sc VO}=\\text{\\sc DVO}$ ."
],
[
"Bribery",
"In this paper we are interested in the optimization variant of the bribery problem [16].",
"Definition 5 In a constructive (respectively, destructive) opt-bribery problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\in \\mathcal {C}$ .",
"We are asked to compute the smallest number $k$ such that the strategic individual can change no more than $k$ votes such that $c$ is the winner (respectively, $c$ is not the winner).",
"To see how opt-bribery falls under the vote operation framework, we view each action of “changing a vote” as a vote operation.",
"Since the strategic individual can only change existing votes in the preference-profile, we define the set of operations to be the difference between the generalized scoring vectors of all votes and the generalized scoring vectors of votes in the support of $\\pi $ , that is, $\\lbrace f(V)-f(W): V,W\\in L(\\mathcal {C})\\text{ s.t.",
"}\\pi (W)>0\\rbrace $ .",
"Then, similarly the constructive variant corresponds to $\\text{\\sc CVO}$ and the destructive variant corresponds to $\\text{\\sc DVO}$ .",
"In both cases Theorem REF cannot be directly applied, because in the ILPs we did not limit the total number of each type of vote operations that can be used by the strategic individual.",
"Nevertheless, we can still prove a similar proposition by taking a closer look at the relationship between $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) and opt-bribery as follows: For any preference-profile, the solution to $\\text{\\sc CVO}$ (respectively, $\\text{\\sc DVO}$ ) is a lower bound on the solution to constructive (respectively, destructive) opt-bribery, because in $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ there are no constraints on the number of each type of vote operations.",
"We have the following four cases.",
"1.",
"If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is 0, then the solution to constructive (destructive) opt-bribery is also 0.",
"2.",
"If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\Theta (\\sqrt{n})$ , as $n$ become large enough, with probability that goes to 1 each type of votes in the support of $\\pi $ will appear $\\Theta (n)$ , which is $>\\Theta (\\sqrt{n})$ , times in the randomly generated preference-profile, which means that there are enough votes of each type for the strategic individual to change.",
"3.",
"If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\Theta (n)$ , then the solution to constructive (destructive) opt-bribery is either $\\Theta (n)$ (when the strategic individual can change all votes to achieve her goal), or $\\infty $ .",
"4.",
"If the solution to $\\text{\\sc CVO}$ ($\\text{\\sc DVO}$ ) is $\\infty $ , then the solution to constructive (destructive) opt-bribery is also $\\infty $ .",
"It follows that the observation made in Theorem REF holds for opt-bribery."
],
[
"Margin of Victory (MoV)",
"Definition 6 Given a voting rule $r$ and a preference-profile $P$ , the margin of victory (MoV) of $P$ is the smallest number $k$ such that the winner can be changed by changing $k$ votes in $P$ .",
"In the mov problem, we are given $r$ and $P$ , and are asked to compute the margin of victory.",
"For a given instance $(P,r)$ , mov is equivalent to destructive opt-bribery, where $c=r(P)$ .",
"However, when the input preference-profiles are generated randomly, the winners in these profiles might be different.",
"Therefore, the corollary of Theorem REF for opt-bribery does not directly imply a similar corollary for mov.",
"This relationship is similar to the relationship between $\\text{\\sc DVO}$ and $\\text{\\sc CWVO}$ .",
"Despite this difference, the formulation of mov in the vote operation framework is very similar to that of opt-bribery: The set of all operations and the argument to apply Theorem REF are the same.",
"The only difference is that for mov, we obtain the corollary from the $\\text{\\sc CWVO}$ part of Theorem REF , while the corollary for opt-bribery is obtained from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF ."
],
[
"Minimum Manipulation Coalition Size (MMCS)",
"The minimum manipulation coalition size (MMCS) problem is similar to mov, except that in MMCS the winner must be improved for all voters who change their votes [33].",
"Definition 7 In an MMCS problem, we are given a voting rule $r$ and a preference-profile $P$ .",
"We are asked to compute the smallest number $k$ such that a coalition of $k$ voters can change their votes to change the winner, and all of them prefer the new winner to $r(P)$ .",
"Unlike mov, MMCS falls under the vote operation framework in the following dynamic way.",
"For each preference-profile, suppose $c$ is the current winner.",
"For each adversarial $d\\ne c$ , we use $\\lbrace f(V)-f(W):V,W\\in L(\\mathcal {C})\\text{ s.t.",
"}d\\succ _W c\\text{ and } \\pi (W)>0\\rbrace $ as the set of operations.",
"That is, we only allow voters who prefer $d$ to $c$ to participate in the manipulative coalition.",
"We also replace each of LP$_{\\trianglerighteq }$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ by multiple LPs, each of which is indexed by a pair of alternatives $(d,c)$ and the constraints are generated by using the corresponding set of operations.",
"Then, the corollary for MMCS follows after a similar argument to that of $\\text{\\sc CVO}$ in Theorem REF ."
],
[
"Control by Adding/Deleting Votes (CAV/CDV)",
"Definition 8 In a constructive (respectively, destructive) optimal control by adding votes (opt-CAV) problem, we are given a preference-profile $P$ , a (dis)favored alternative $c\\in \\mathcal {C}$ , and a set $N^{\\prime }$ of additional votes.",
"We are asked to compute the smallest number $k$ such that the strategic individual can add $k$ votes in $N^{\\prime }$ such that $c$ is the winner (respectively, $c$ is not the winner).",
"For simplicity, we assume that $|N^{\\prime }|=n$ and the votes in $N^{\\prime }$ are drawn i.i.d.",
"from a distribution $\\pi ^{\\prime }$ .",
"To show how opt-CAV falls under the vote operation model, we let the set of operations to be the generalized scoring vectors of all votes that are in the support of $\\pi ^{\\prime }$ , that is, $\\lbrace f(V): V\\in L(\\mathcal {C})\\text{ and }\\pi ^{\\prime }(V)>0\\rbrace $ .",
"Then, the corollary follows from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery.",
"Definition 9 In a constructive (respectively, destructive) optimal control by deleting votes (opt-CDV) problem, we are given a preference-profile $P$ and a (dis)favored alternative $c\\in \\mathcal {C}$ .",
"We are asked to compute the smallest number $k$ such that the strategic individual can delete $k$ votes in $P$ such that $c$ is the winner (respectively, $c$ is not the winner).",
"To show how opt-CDV falls under the vote operation framework, we let the set of operations to be the negation of generalized scoring vectors of votes in the support of $\\pi ^{\\prime }$ , that is, $\\lbrace -f(V): V\\in L(\\mathcal {C})\\text{ and }\\pi ^{\\prime }(V)>0\\rbrace $ .",
"Then, the corollary follows from the $\\text{\\sc CVO}$ and $\\text{\\sc DVO}$ parts of Theorem REF via a similar argument to the argument for opt-bribery."
],
[
"A Unified Corollary",
"The next corollary of Theorem REF summarizes the results obtained for all types of strategic behavior studied in this section.",
"Corollary 1 For any integer generalized scoring rule, any distribution $\\pi $ over votes, and any $X\\in $ $\\big ($ {constructive, destructive}$\\times $ {UCO, opt-bribery, opt-CAV, opt-CDV}$\\big )$$\\cup $ {MoV, MMCS}, suppose the input preference-profiles are generated i.i.d.",
"from $\\pi $ .For CAV, the distribution over the new votes can be generated i.i.d.",
"from a different distribution $\\pi ^{\\prime }$ .",
"Then, for any alternative $c$ and any $\\epsilon >0$ , there exists $\\beta ^*>1$ such that the total probability for the solution to $X$ to be one of the following four cases is more than $1-\\epsilon $ as $n$ goes to infinity: (1) 0, (2) between $\\frac{1}{\\beta ^*}\\sqrt{n}$ and $\\beta ^* \\sqrt{n}$ , (3) between $\\frac{1}{\\beta ^*} n$ and $\\beta ^* n$ , and (4) $\\infty $ ."
],
[
"Discussions and Future Work",
"In this paper, we proposed a general framework to study vote operations for generalized scoring rules.",
"Our main theorem is a characterization for the number of vote operations that are needed to achieve the strategic individual's goal.",
"We showed that the main theorem can be applied to many types of strategic behavior and many commonly used voting rules, for most of which no similar results were previously known.",
"We further discuss the generality of our framework in the next two paragraphs.",
"GSRs vs. integer GSRs.",
"Though integer GSRs are a subclass of GSRs, we feel that from a computational point of view, focusing integer generalized scoring rules does not sacrifice much generality.",
"Because the $g$ function only depends on the preorder among components in the total generalized scoring vector, if the $f$ function is scaled up by a constant, then the $g$ function will select the same winner.",
"Therefore, integer GSRs are equivalent to GSRs where components in the generalized scoring vectors are rational numbers.",
"When the components are irrational numbers, two computational problems arise.",
"First, it is not clear how these irrational numbers are represented, and second, it is hard to compare two irrational numbers computationally, thus even harder to compute the preorder of the components of the total generalized scoring vector.",
"On the other hand, integer GSRs do not have such computational constraints.",
"In fact, all commonly studied voting rules that are known to be GSRs are integer GSRs.",
"Therefore, we believe that our main theorem has a wide application (at least can be applied to many commonly studied voting rules).",
"On the generality of vote operations.",
"While the framework we proposed covers many types of strategic behavior, some other types of strategic behavior that have been widely studied are not covered by our framework.",
"These types of strategic behavior can be roughly categorized as follows: (1) controls that changes the set of alternatives, for example, control by adding/deleting alternatives [3] and control by introducing clones of alternatives [44], [13]; and (2) controls that change the procedure of voting, for example control by (runoff) partition of alternatives and control by partition of voters [3], and control by changing the agenda of voting [24].",
"Building a more general framework that covers more types of strategic behavior and studying their properties are interesting directions for future research.",
"As we discussed in the introduction, on the positive side, our main theorem suggests that computing the margin of victory is usually not hard, which helps implementing efficient post-election auditing methods.",
"One promising future direction is to design practical computational techniques for computing the margin of victory for generalized scoring rules, based on the ILP proposed in this paper.",
"On the negative side, our main theorem suggests that computational complexity might merely be a weak barrier against many types of strategic behavior.",
"Therefore, we should look for new ways to protect voting, for example introducing randomization [6], [14], [46], [29], [30], using multiple rounds [9], [28], [10], or limiting the strategic individuals' information about other voters [7].",
"Another interesting research direction is to investigate the phase transition of the probability for a coalition of strategic individuals to achieve their goal by using vote operations, as it was done for manipulation [45], [26]."
],
[
"Proofs",
"Lemma REF Fix an integer matrix $\\bf A$ .",
"There exists a constant $\\beta _{\\bf A}$ that only depends on $\\bf A$ , such that if the following LP has an integer solution, then the solution is no more than $\\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .",
"$\\min \\Vert \\vec{x}\\Vert _1, \\text{ s.t.",
"}{\\bf A}\\cdot \\vec{x}\\ge \\vec{b}$ Proof of Lemma REF : Let $\\bf A$ be a $m^*\\times n^*$ integer matrix, which includes the constraints $\\vec{x}\\ge \\vec{0}$ .",
"Suppose LP (REF ) has a (non-negative) integer solution.",
"We note that $\\vec{0}$ is an optimal integer solution to $\\min \\vec{1}\\cdot (\\vec{x})^{\\prime } \\text{ s.t.",
"}{\\bf A}\\cdot \\vec{x}\\ge \\vec{0}$ .",
"Then, it follows from Theorem 5 (ii) in [8] that LP (REF ) has a (non-negative) integer solution $\\vec{z}$ such that $\\Vert \\vec{z}-\\vec{0}\\Vert _\\infty \\le n^*\\cdot M({\\bf A})\\cdot (\\Vert \\vec{b}-\\vec{0}\\Vert _\\infty +2),$ where $M({\\bf A})$ is the maximum of the absolute values of the determinants of the square sub-matrices of $\\bf A$ .",
"Since $\\bf A$ is fixed, the right hand side becomes a constant, that is, $\\Vert \\vec{z}\\Vert _\\infty =O(\\Vert \\vec{b}\\Vert _\\infty )$ .",
"Therefore, there exists $\\beta _{\\bf A}$ such that the optimal value in the ILP (REF ) is no more than $\\vec{1}\\cdot (\\vec{z})^{\\prime }\\le n^*\\Vert \\vec{z}\\Vert _\\infty \\le \\beta _{\\bf A}\\cdot \\Vert \\vec{b}\\Vert _\\infty $ .",
"$\\blacksquare $ Claim REF There exists $N\\in \\mathbb {N}$ and $\\beta ^{\\prime }>1$ such that for any $n\\ge N$ , any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , and (2) there exists a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that $g(\\trianglerighteq ^*)=c$ and LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .",
"Proof of Claim REF : Let $\\trianglerighteq =\\text{Ord}(f(P))$ .",
"Because $\\text{GS}(f,g)(P)\\ne c$ , $g(\\trianglerighteq )\\ne c$ .",
"Therefore, the strategic individual has to change the order of some components in the generalized scoring vector to make $c$ win.",
"We note that $P\\in {\\cal P}_\\beta $ , which means that the difference between any pair of components of $f(P)$ is more than $\\frac{1}{\\beta } \\sqrt{n}$ .",
"Let $d_{max}$ denote the maximum difference between any pair of components in generalized score vectors.",
"That is, $d_{max}=\\max _{t,t^{\\prime },L\\in L(\\mathcal {C})}\\lbrace (f(L))_t-(f(L))_{t^{\\prime }}\\rbrace $ .",
"In order for $c$ to win, the number of vote operations must be at least $\\frac{1}{\\beta } \\sqrt{n}/d_{max}$ .",
"Therefore, $\\text{\\sc CVO}(P,c)>\\frac{1}{\\beta d_{max}} \\sqrt{n}$ .",
"We next show the upper bound.",
"Because $P\\in {\\cal P}_\\beta $ , for any pair $o_i,o_j$ with $o_i=_{\\trianglerighteq _\\pi }o_j$ , $|[f(P)]_i-[f(P)]_j|<\\beta \\sqrt{n}$ .",
"Therefore, the right hand side of each (in)equality in LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ is no more than $\\beta \\sqrt{n}$ .",
"Applying Lemma REF to LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ , we have that there exists a constant $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ that only depends on $\\trianglerighteq ^*$ and $\\trianglerighteq _\\pi $ , and an integer solution $\\vec{v}$ with $\\Vert \\vec{v}\\Vert _1\\le \\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }\\sqrt{n}$ (the $\\bf A$ matrix in Lemma REF is fixed because we fix the number of alternatives $m$ , and the left hand side of each (in)equality in LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ does not depend on $n$ ).",
"Let $\\beta ^{\\prime }$ be the maximum of $d_{max}\\beta $ and all $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ (since we fix the number of alternatives, there are finite many $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }$ 's).",
"Since $\\beta _{\\trianglerighteq ^*,\\trianglerighteq _\\pi }>\\frac{1}{d_{max}\\beta }$ , $\\beta ^{\\prime }>1$ .",
"It follows that $\\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ .",
"We next show that for a sufficiently large $n$ , if the strategic individual applies $\\vec{v}$ , then the order over components of the total scoring vector will become $\\trianglerighteq ^*$ .",
"That is, $c$ can be made win.",
"The idea is, LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ ensures that by applying $\\vec{v}$ , ties between the “almost tied” components are broken as in $\\trianglerighteq ^*$ .",
"Since $\\Vert \\vec{v}\\Vert _1=O(\\sqrt{n})$ , when $n$ is large enough the order between any pair of “far away” components will not be affected.",
"Formally, let $\\vec{x}=f(P)+\\Delta \\cdot \\vec{v}$ .",
"That is, $\\vec{x}$ is the total generalized scoring vector after the strategic individual applied $\\vec{v}$ .",
"Because $\\vec{v}$ is a solution to LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ , for any pair $o_i,o_j$ with $o_i=_{\\trianglerighteq _\\pi }o_j$ , the order between $o_i$ and $o_j$ in $\\trianglerighteq ^*$ is the same as the order between $o_i$ and $o_j$ in $\\text{Ord}(\\vec{x})$ .",
"Since $\\trianglerighteq ^*$ is an extension of $\\trianglerighteq _\\pi $ , if $o_i\\vartriangleright _\\pi o_j$ , then we must have $o_i\\vartriangleright ^* o_j$ .",
"Therefore, we only need to check that for any $o_i\\vartriangleright _\\pi o_j$ , we have $[\\vec{x}]_i>[\\vec{x}]_j$ .",
"Because $P\\in {\\cal P}_\\beta $ , $|[f(P)]_i-[f(P)]_j|>\\frac{1}{\\beta }n$ .",
"We note that $\\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ , which means that by applying $\\vec{v}$ , the strategic individual can only change the difference between any pair of components by no more than $d_{max} \\beta ^{\\prime }\\sqrt{n}$ .",
"Let $N=(d_{max} \\beta ^{\\prime }\\beta )^2+1$ .",
"When $n\\ge N$ , $d_{max} \\beta ^{\\prime }\\sqrt{n}<\\frac{1}{\\beta }n$ , which means that for any $o_i\\vartriangleright _\\pi o_j$ , applying $\\vec{v}$ will not change the order between $o_i$ and $o_j$ in the total generalized scoring vector.",
"This means that by applying $\\vec{v}$ , the strategic individual can make $c$ win.",
"Therefore, $ \\text{\\sc CVO}(P,c)\\le \\Vert \\vec{v}\\Vert _1<\\beta ^{\\prime }\\sqrt{n}$ .",
"It follows that for any $n\\ge N$ , $\\frac{1}{\\beta ^{\\prime }}\\sqrt{n}< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime }\\sqrt{n}$ .",
"$\\blacksquare $ Claim REF There exists $\\beta ^{\\prime }>1$ such that for any $P\\in {\\cal P}_\\beta $ , if (1) $c$ is not the winner for $P$ , (2) there does not exist a refinement $\\trianglerighteq ^*$ of $\\trianglerighteq _\\pi =\\text{Ord}(f(P_\\pi ))$ such that LP$_{\\trianglerighteq ^*\\ominus \\trianglerighteq _\\pi }$ has an integer solution, and (3) there exists $\\trianglerighteq $ such that $g(\\trianglerighteq )=c$ and LP$_{\\trianglerighteq }$ has an integer solution, then $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .",
"Proof of Claim REF : Let $\\trianglerighteq =\\text{Ord}(f(P))$ .",
"Because the premises of Claim REF do not hold, the strategic individual has to change the order of some pair of components that are “far away” (that is, the difference between them is $\\Theta (n)$ before the strategic individual applies vote operations) to make $c$ win.",
"We note that one operation can only change the difference between a pair of components by at most $d_{max}$ .",
"Therefore, $\\text{\\sc CVO}(P,c)\\ge \\frac{1}{\\beta }n/d_{max}$ .",
"On the other hand, it follows from Lemma REF and condition (3) in the statement of the claim that $\\text{\\sc CVO}(P,c)<\\infty $ .",
"The only thing left to show is that there exists $\\beta ^{\\prime }>1$ such that $\\text{\\sc CVO}(P,c)<\\beta ^{\\prime }n$ for all $P\\in {\\cal P}_\\beta $ and for all $n$ .",
"Because $P\\in {\\cal P}_\\beta $ , for any pair $o_i,o_j$ , $|[f(P)]_i-[f(P)]_j|<\\beta n$ .",
"Applying Lemma REF to LP$_{\\trianglerighteq }$ and condition (3) in the statement of the claim, we have that for any $\\trianglerighteq $ , there exits $\\beta _{\\trianglerighteq }$ that only depends on $\\trianglerighteq $ , and an optimal integer solution $\\vec{v}$ such that $\\Vert \\vec{v}\\Vert _1\\le \\beta _\\trianglerighteq n$ .",
"Let $\\beta ^{\\prime }$ be the maximum of $d_{max} \\beta $ and all $\\beta _{\\trianglerighteq }$ (again, there are finite number of $\\beta _{\\trianglerighteq }$ 's).",
"It follows that $\\frac{1}{\\beta ^{\\prime }} n< \\text{\\sc CVO}(P,c)<\\beta ^{\\prime } n$ .",
"$\\blacksquare $"
],
[
"Discussion: How often the solution is 0 or $\\infty $ ?",
"One important question is: how large is the probability that the solution to problems studied in this section is 0 or $\\infty $ ?",
"Not surprisingly the answer depends on both the voting rule and the type of vote operations.",
"The probability can be large for some voting rules.",
"For example, for any voting rule that always selects a given alternative $d$ as the winner, the solution to $\\text{\\sc CVO}$ (respectively, $\\text{\\sc DVO}$ ) is always 0 (respectively $\\infty $ ) for $c=d$ and is always $\\infty $ (respectively 0) for $c\\ne d$ .",
"However, for common voting rules the alternatives are treated almost equally (except for cases with ties).",
"Therefore, we may expect that for a preference-profile whose votes are generated i.i.d., each alternative has almost the same probability of being selected as the winner.",
"This is indeed the case in all commonly used voting rules, including approval voting, all positional scoring rules (which include Borda, plurality, and veto), plurality with runoff, Bucklin, Copeland, maximin, STV, and ranked pairs.",
"Therefore, for these voting rules, when the votes are drawn i.i.d.",
"uniformly at random, the probability for $\\text{\\sc CVO}$ is approximately $\\frac{1}{m}$ and the probability for $\\text{\\sc DVO}$ is approximately $\\frac{m-1}{m}$ .",
"For $\\text{\\sc CWVO}$ , the answer is never 0 because changing 0 votes cannot change the winner.",
"We would also expect for common voting rules, for some types of strategic behavior studied in this section, with low probability the solution is $\\infty $ .",
"For UCO, the strategic individual can introduce many (but finitely many) votes such that the non-manipulators' votes are negligible.",
"For opt-bribery and mov, the strategic individual can change all votes to achieve her goal.",
"For MMCS, CAV, and CDV, it is not clear how large such probability is.",
"The following table summarizes folklore results for common voting rules when votes are drawn i.i.d.",
"uniformly at random.",
"Table: Probability for solutions to some problems to be 0 or ∞\\infty for common voting rules."
]
] | 1204.1231 |
[
[
"Assessment of density functional approximations for the hemibonded\n structure of water dimer radical cation"
],
[
"Abstract Due to the severe self-interaction errors associated with some density functional approximations, conventional density functionals often fail to dissociate the hemibonded structure of water dimer radical cation (H2O)2+ into the correct fragments: H2O and H2O+.",
"Consequently, the binding energy of the hemibonded structure (H2O)2+ is not well-defined.",
"For a comprehensive comparison of different functionals for this system, we propose three criteria: (i) The binding energies, (ii) the relative energies between the conformers of the water dimer radical cation, and (iii) the dissociation curves predicted by different functionals.",
"The long-range corrected (LC) double-hybrid functional, omegaB97X-2(LP) [J.-D. Chai and M. Head-Gordon, J. Chem.",
"Phys., 2009, 131, 174105.",
"], is shown to perform reasonably well based on these three criteria.",
"Reasons that LC hybrid functionals generally work better than conventional density functionals for hemibonded systems are also explained in this work."
],
[
"Introduction",
"Water can be decomposed when it is exposed to high-energy flux.",
"The products of water radiolysis may contain various radical species, e.g.",
"hydrogen atoms (H), hydroxide radicals (OH), oxygen anions (O$^-$ ), and water cations (H$_2$ O$^+$ ), depending on the radiation infrastructure setup.",
"For example the overall decomposition scheme activated by $\\beta $ particles has been outlined by Garrett et al.",
"in 2005 where three main channels of decomposition were listed.",
"The cationic channel leads to the formation of ionized water living in about several tens femtoseconds and hydrated electron, followed by the generation of hydronium (H$_3$ O$^+$ ) and OH radicals through proton transfer process , .",
"The energized-neutral and anionic channels could result in the cleavage the oxygen-hydrogen chemical bonds to produce hydrogen and oxygen derivatives, i.e.",
"H, H$^-$ , H$_2$ , O, O$^-$ , OH$^-$ etc.",
"Subsequent chemical reactions can progress further up to the desorption of stable gas molecules H$_2$ and O$_2$ being driven by those reactive radical species .",
"The cationic channel is therefore particularly interesting due to its dominant products — OH radicals and solvated electrons.",
"The smallest system to understand the chemical dynamics of ionized water is the water dimer radical cation (H$_2$ O)$_2$$^+$ , and it has been approached by several experimental studies in the past.",
"Angel and Stace reported the predominant H$_3$ O$^+$ –OH central core by the collision-induced fragmentation experiment against the earlier theoretical assignment of charge-resonance hydrazine structure .",
"Dong et al.",
"observed a weak signal corresponding the formation of (H$_2$ O)$_2$$^+$ near the low-mass side of (H$_2$ O)$_2$ H$^+$ using 26.5 eV soft X-ray laser .",
"Gardenier, Johnson, and McCoy reported the argon-tagged predissociation infrared spectra of (H$_2$ O)$_2$$^+$ and assigned its structural pattern as a charge-localized H$_3$ O$^+$ –OH complex .",
"Recently, Fujii's group reported the infrared spectroscopic observations of larger (H$_2$ O)$_n$$^+$ clusters, $n = 3-11$ , where the OH radical vibrational signal was clearly identified for $n\\leqq 5$ clusters, but vibrational signature of OH radical become inseparable due to the overlap with H-bonded OH stretch in $n>6$ .",
"As being evidenced in the earlier studies , , theoretical investigations such as ab initio electronic structure theory and density functional theory (DFT) play an important role in understanding the infrared spectroscopic features of the ionized water clusters.",
"Because high-level ab initio calculations are computationally prohibited for larger ionized water clusters, e.g.",
"fully solvated cationic moiety, a reliable DFT method is necessary.",
"In earlier theoretical reports, two minimum structures of the water dimer radical cation have been identified: the proton transferred structure and the hemibonded structure, as shown in Fig.",
"REF , , , .",
"The previous DFT calculations have shown that many exchange-correlation (XC) functionals fail to predict reasonable results , , given rise to the presence of hemibonding interaction.",
"The hemibonding interaction, could be theoretically located in (H$_2$ O)$_n$$^+$ systems, is notorious for the serious self-interaction errors (SIEs) associated with some density functional approximations.",
"Both local density approximation (LDA) and generalized gradient approximations (GGAs) were reported to contain non-negligible amount of SIEs for describing the hemibonded structure , .",
"It has been suggested to adopt hybrid functionals with larger fractions of the exact Hartree-Fock (HF) exchange for more accurate results in the hemibonded structure , .",
"However, as the SIEs of functionals become larger at the dissociation limit, these suggested functionals can yield spurious barriers on their dissociation curves , which can lead to unphysical results in molecular dynamics simulations.",
"Clearly, more stringent criteria for choosing suitable functionals are needed.",
"In this work, we propose three different criteria for a comprehensive comparison of different functionals for this system.",
"Figure: (a) The proton transferred structure, (b) the hemibonded structure, and (c) the transition state between the structures of (a) and (b)."
],
[
"Computational Methods",
"Calculations are performed on the optimized geometries of the two structures of the water dimer cation and the transition state between them, optimized with the ab initio MP2 theory and various XC functionals involving BLYP , , PBE , and M06L , which are pure density functionals (i.e.",
"the fraction of HF exchange $\\alpha _{\\mathrm {HF}}=0.00$ ), B97 with $\\alpha _{\\mathrm {HF}}=0.19$ , B3LYP , , with $\\alpha _{\\mathrm {HF}}=0.20$ , PBE0 with $\\alpha _{\\mathrm {HF}}=0.25$ , M06 with $\\alpha _{\\mathrm {HF}}=0.27$ , M05 with $\\alpha _{\\mathrm {HF}}=0.28$ , BH&HLYP with $\\alpha _{\\mathrm {HF}}=0.50$ , M06-2X with $\\alpha _{\\mathrm {HF}}=0.54$ , M05-2X with $\\alpha _{\\mathrm {HF}}=0.56$ , M06HF with $\\alpha _{\\mathrm {HF}}=1.00$ , the $\\omega \\mbox{B97}$ series ($\\omega \\mbox{B97}$ , $\\omega \\mbox{B97}$ X , $\\omega \\mbox{B97}$ X-D , and $\\omega \\mbox{B97}$ X-2(LP) ), which are long-range corrected (LC) hybrid functionals (i.e.",
"the fraction of HF exchange depends on the interelectronic distance ), and the double-hybrid functional B2PLYP with $\\alpha _{\\mathrm {HF}}=0.53$ .",
"The DFT and MP2 calculations are performed with the 6-311++G(3df,3pd) basis set, where the reference values of binding energy are obtained from Ref.",
".",
"For efficiency, the resolution-of-identity (RI) approximation is used for calculations with the MP2 correlation (using sufficiently large auxiliary basis sets).",
"The CCSD(T) dissociation curves are calculated on the fixed monomer geometries (using the CCSD(T) optimized geometry of Ref.",
"), with the aug-cc-pVTZ basis set.",
"Note that although the ZPE corrected energy of the proton transferred structure (or, referred to as the Ion structure in Ref.",
".)",
"is inconsistent with Ref.",
".",
"However, adopting the geometries obtained from Ref.",
"yields results that are consistent with Ref.",
".",
"All of the calculations are performed with the development version of Q-Chem 3.2 .",
"As the basis set superposition error (BSSE) for the ionized water dimer has been shown to be insignificant (if the diffuse basis functions are adopted) , , we do not perform BSSE correction throughout this paper."
],
[
"Results and Discussion",
"The ZPE corrected binding energies and relative energies of the water dimer cation calculated by various XC functionals are shown in Table REF and Table REF , respectively.",
"The calculated dissociation curves for the hemibonded structure are shown in Fig.",
"REF .",
"A summary of the results based on these three different criteria is shown in Table REF .",
"The notation used for characterizing statistical errors is as follows: Mean signed errors (MSEs), mean absolute errors (MAEs) and root-mean-square (RMS) errors."
],
[
"Criterion I: Binding Energies",
"Table REF shows the binding energies of the two structures of water dimer radical cation and the transition state between them.",
"The reference data obtained from Ref.",
"are based on the CCSD(T) calculations with the aug-cc-pVQZ basis set.",
"Since the errors of XC functionals for the hemibonded structure are much larger than those for the proton transferred structure, we focus our discussion on the hemibonded structure.",
"From Table REF , the functionals with MAE less than 2.5 kcal/mol are $\\omega \\mbox{B97X-2(LP)}$ , M06HF, and BH&HLYP.",
"Table REF also confirms the trend that has been mentioned previously: Functionals with larger fractions of HF exchange give more accurate results for the hemibonded structure.",
"To give reasonable results for the hemibonded structure, the $\\alpha _{\\mathrm {HF}}$ of a global hybrid functional should be at least larger than 0.43, as observed in Ref.",
"for the MPW1K functional.",
"Although M06HF, containing a full HF exchange, gives a small error of the hemibonded structure, it yields a large error for the proton transferred structure (due to the incomplete cancelation of errors between the exact exchange and semilocal correlation), as shown in Table REF .",
"Although functionals with $\\alpha _{\\mathrm {HF}}$ larger than 0.43 have been suggested, functionals with $\\alpha _{\\mathrm {HF}}\\ge 0.5$ are not always reliable, which can be observed from the errors of the hemibonded structure calculated by B2PLYP, M05-2X and M06-2X.",
"But this trend still holds: the results of the M05-2X and M06-2X are much better than those of the M05 and M06.",
"This means that although the energy of the hemibonded structure is sensitive to the $\\alpha _{\\mathrm {HF}}$ values in XC functionals, it may also be affected by the associated density functional approximations (DFAs).",
"Also note that some GGA functionals, such as BLYP and PBE, cannot predict that the transition state between the two structures of the water dimer radical cation.",
"The HF exchange included in the $\\omega \\mbox{B97}$ series is given by $E^{\\omega \\mathrm {B97~series} }_{\\mathrm {HF~exchange} }=E^{\\mathrm {LR-HF} }_x+C_xE^{\\mathrm {SR-HF} }_x,$ where $E^{\\mathrm {LR-HF}}_x=&-\\frac{1}{2}\\sum _{\\sigma }\\sum ^{occu.",
"}_{i,j}\\int \\int \\psi ^*_{i\\sigma }(\\textbf {r}_1)\\psi ^*_{j\\sigma }(\\textbf {r}_2)\\nonumber \\\\&\\times \\frac{\\mbox{erf}(\\omega r_{12}) }{r_{12}}\\psi _{j\\sigma }(\\textbf {r}_1)\\psi _{i\\sigma }(\\textbf {r}_2)d\\textbf {r}_1d\\textbf {r}_2,$ and $E^{\\mathrm {SR-HF} }_x=&-\\frac{1}{2}\\sum _{\\sigma }\\sum ^{occu.",
"}_{i,j}\\int \\int \\psi ^*_{i\\sigma }(\\textbf {r}_1)\\psi ^*_{j\\sigma }(\\textbf {r}_2)\\nonumber \\\\&\\times \\frac{\\mbox{erfc}(\\omega r_{12}) }{r_{12}}\\psi _{j\\sigma }(\\textbf {r}_1)\\psi _{i\\sigma }(\\textbf {r}_2)d\\textbf {r}_1d\\textbf {r}_2.$ Here $r_{12}\\equiv \\left|{\\bf r}_{12}\\right|=\\left|{\\bf r}_{1}-{\\bf r}_{2}\\right|$ (atomic units are used throughout this paper).",
"The parameter $\\omega $ defines the range of the splitting operators.",
"The coefficients for the $\\omega \\mbox{B97}$ series are listed in Table REF .",
"Since the fraction of HF exchange in $\\omega \\mbox{B97}$ series depends on the interelectronic distance $r_{12}$ , the trend mentioned previously is not as obvious as the global hybrid functionals.",
"But it is clear that the $\\omega \\mbox{B97X-2(LP)}$ , a LC double-hybrid functional, gives the most accurate results compare to the other functionals in the $\\omega \\mbox{B97}$ series.",
"Table: Appendix"
]
] | 1204.1522 |
[
[
"Some limit theorems for flows of branching processes"
],
[
"Abstract We construct two kinds of stochastic flows of discrete Galton-Watson branching processes.",
"Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line."
],
[
"Introduction",
"Continuous-state branching processes (CB-processes) arose as weak limits of rescaled discrete Galton-Watson branching processes; see, e.g., Jiřina (1958) and Lamperti (1967).",
"Continuous-state branching processes with immigration (CBI-processes) are generalizations of them describing the situation where immigrants may come from other sources of particles.",
"Those processes can be obtained as the scaling limits of discrete branching processes with immigration; see, e.g., Kawazu and Watanabe (1971) and Li (2006).",
"A CBI-process was constructed in Dawson and Li (2006) as the strong solution of a stochastic equation driven by Brownian motions and Poisson random measures; see also Fu and Li (2010).",
"A similar construction was given in Li and Ma (2008) using a stochastic equation driven by time-space Gaussian white noises and Poisson random measures.",
"In the study of scaling limits of coalescent processes with multiple collisions, Bertoin and Le Gall (2006) constructed a flow of jump-type CB-processes as the weak solution flow of a system of stochastic equations driven by Poisson random measures; see also Bertoin and Le Gall (2003, 2005).",
"A more general flow of CBI-processes was constructed in Dawson and Li (2012) as strong solutions of stochastic equations driven by Gaussian white noises and Poisson random measures.",
"The flows in Bertoin and Le Gall (2006) and Dawson and Li (2012) were also treated as path-valued processes with independent increments.",
"Motivated by the works of Aldous and Pitman (1998) and Abraham and Delmas (2010) on tree-valued Markov processes, another flow of CBI-processes was introduced in Li (2012), which was identified as a path-valued branching process.",
"From the flows in Bertoin and Le Gall (2006), Dawson and Li (2012) and Li (2012), one can define some superprocesses or immigration superprocesses over the positive half line with local and nonlocal branching mechanisms.",
"To study the genealogy trees for critical branching processes conditioned on non-extinction, Bakhtin (2011) considered a flow of continuous CBI-processes driven by a time-space Gaussian white noise.",
"He obtained the flow as a rescaling limit of systems of discrete Galton-Watson processes and also pointed out the connection of the model with a superprocess conditioned on non-extinction.",
"In this paper, we consider two flows of discrete Galton-Watson branching processes and show suitable rescaled sequences of the flows converge to the flows of Dawson and Li (2012) and Li (2012), respectively.",
"The main motivation of the work is to understand the connection between discrete and continuum tree-valued processes.",
"Our results generalize those of Bakhtin (2011) to flows of discontinuous CB-processes.",
"To simplify the presentation, we only treat models without immigration, but the arguments given here carry over to those with immigration.",
"We shall first prove limit theorems for the induced superprocesses, from which we derive the convergence of the finite-dimensional distributions of the path-valued branching processes.",
"In Section 2, we give a brief review of the flows of Dawson and Li (2012) and Li (2012).",
"In Section 3, we consider flows consisting of independent branching processes and show their scaling limit gives a flow of the type of Dawson and Li (2012).",
"The formulation and convergence of interactive flows were discussed in Section 4, which lead to a flow in the class studied in Li (2012).",
"Let $\\mathbb {N}$ = $\\lbrace 0,1,2,\\cdots \\rbrace $ and $\\mathbb {N}_+$ = $\\lbrace 1,2,\\cdots \\rbrace $ .",
"For any $a\\ge 0$ let $M[0,a]$ be the set of finite Borel measures on $[0,a]$ endowed with the topology of weak convergence.",
"We identify $M[0,a]$ with the set $F[0,a]$ of positive right continuous increasing functions on $[0,a]$ .",
"Let $B[0,a]$ be the Banach space of bounded Borel functions on $[0,a]$ endowed with the supremum norm $\\Vert \\cdot \\Vert $ .",
"Let $C[0,a]$ denote its subspace of continuous functions.",
"We use $B[0,a]^+$ and $C[0,a]^+$ to denote the subclasses of positive elements and $C[0,a]^{++}$ to denote the subset of $C[0,a]^+$ of functions bounded away from zero.",
"For $\\mu \\in M[0,a]$ and $f\\in B[0,a]$ write $\\langle \\mu , f\\rangle $ = $\\int f d\\mu $ if the integral exists."
],
[
"Local and nonlocal branching flows",
"In this section, we recall some results on constructions and characterizations of the flow of CB-processes and the associated superprocess.",
"It is well-known that the law of a CB-process is determined by its branching mechanism $\\phi $ , which is a function on $[0,\\infty )$ and has the representation $\\phi (z)=bz+\\frac{1}{2}\\sigma ^2z^2 +\\int _0^\\infty (e^{-zu}-1+zu)m(du),$ where $\\sigma \\ge 0$ and $b$ are constants and $(u\\wedge {u}^2)m(du)$ is a finite measure on $(0,\\infty )$ .",
"Let $W(ds,du)$ be a white noise on $(0,\\infty )^2$ based on $dsdu$ and $\\tilde{N}(ds,dz,du)$ a compensated Poisson random measure on $(0,\\infty )^3$ with intensity $dsm(dz)du$ .",
"By Theorem 3.1 of Dawson and Li (2012), a CB-process with branching mechanism $\\phi $ can be constructed as the pathwise unique strong solution $\\lbrace Y_t:t\\ge 0\\rbrace $ to the stochastic equation: $Y_t \\!\\!&=\\!\\!&Y_0 + \\sigma \\int _0^t\\int _0^{Y_{s-}}W(ds,du) - \\int _0^tbY_{s-}ds \\cr \\!\\!&\\!\\!&+ \\int _0^t\\int _0^{\\infty }\\int _0^{Y_{s-}}z \\tilde{N}(ds,dz,du).$ Let us fix a constant $a\\ge 0$ and a function $\\mu \\in F[0,a]$ .",
"Let $\\lbrace Y_t(q): t\\ge 0\\rbrace $ denote the solution to (REF ) with $Y_0(q) =\\mu (q)$ .",
"We can consider the solution flow $\\lbrace Y_t(q): t\\ge 0, q\\in [0,a]\\rbrace $ of (REF ).",
"As observed in Dawson and Li (2012), there is a version of the flow which is increasing in $q\\in [0,a]$ .",
"Moreover, we can regard $\\lbrace (Y_t(q))_{t\\ge 0}: q\\in [0,a]\\rbrace $ as a path-valued stochastic process with independent increments.",
"Let $\\lbrace Y_t: t\\ge 0\\rbrace $ denote the $M[0,a]$ -valued process so that $Y_t[0,q] = Y_t(q)$ for every $t\\ge 0$ and $q\\in [0,a]$ .",
"Then $\\lbrace Y_t: t\\ge 0\\rbrace $ is a càdlàag superprocess with branching mechanism $\\phi $ and trivial spatial motion; see Theorems 3.9 and 3.11 in Dawson and Li (2012).",
"For $\\lambda \\ge 0$ let $t\\mapsto v(t,\\lambda )$ be the unique locally bounded positive solution of $v(t,\\lambda )= \\lambda -\\int _0^t\\phi (v(s,\\lambda ))ds, \\qquad t\\ge 0.$ For any $f\\in B[0,a]^+$ define $x\\mapsto v(t,f)(x)$ by $v(t,f)(x) =v(t,f(x))$ .",
"Then the superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ has transition semigroup $(Q_t)_{t\\ge 0}$ on $M[0,a]$ defined by $\\int _{M[0,a]} e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,\\nu )=\\exp \\left\\lbrace -\\langle \\mu ,v(t,f)\\rangle \\right\\rbrace , \\qquad f\\in B[0,a]^+.$ By Proposition 3.1 in Li (2011) one can see that $v(t,f)\\in C[0,a]^{++}$ for every $f\\in C[0,a]^{++}$ .",
"Then it is easy to verify that $(Q_t)_{t\\ge 0}$ is a Feller semigroup.",
"We can define another branching flow.",
"For this purpose, let us consider an admissible family of branching mechanisms $\\lbrace \\phi _q: q\\in [0,a]\\rbrace $ , where $\\phi _q$ is given by (REF ) with parameters $(b,m)=(b_q,m_q)$ depending on $q\\in [0,a]$ .",
"Here by an admissible family we mean for each $z\\ge 0$ , the function $q\\mapsto \\phi _q(z)$ is decreasing and continuously differentiable with the derivative $\\psi _\\theta (z) =-(\\partial /\\partial \\theta )\\phi _{\\theta }(z)$ of the form $\\psi _\\theta (z)=h_{\\theta }z+\\int _0^{\\infty }(1-e^{-zu})n_{\\theta }(du),$ where $h_{\\theta }\\ge 0$ and $n_{\\theta }(du)$ is a $\\sigma $ -finite kernel from $[0,a]$ to $(0,\\infty )$ satisfying $\\sup _{0\\le \\theta \\le a}\\Big [h_{\\theta }+\\int _0^{\\infty }un_{\\theta }(du)\\Big ]<\\infty .$ Then we have $\\phi _q(z)=\\phi _0(z)-\\int _0^q \\psi _{\\theta }(z)d\\theta , \\qquad z\\ge 0.$ Let $m(dz,d\\theta )$ be the measure on $(0,\\infty )\\times [0,a]$ defined by $m([c,d]\\times [0,q])=m_q[c,d],\\quad q\\in [0,a], d>c>0.$ Suppose that $W(ds,du)$ is a white noise on $(0,\\infty )^2$ based on $dsdu$ and $\\tilde{N}(ds,dz,d\\theta ,du)$ is a compensated Poisson random measure on $(0,\\infty )^2\\times [0,a]\\times (0,\\infty )$ with intensity $dsm(dz,d\\theta )du$ .",
"By the results in Li (2012), for any $\\mu \\in F[0,a]$ the stochastic equation $Y_t(q) \\!\\!&=\\!\\!&\\mu (q)-b_q\\int _0^t Y_{s-}(q)ds+\\sigma \\int _0^t\\int _0^{Y_{s-}(q)}W(ds,du)\\cr \\!\\!&\\!\\!&+\\int _0^t\\int _0^{\\infty }\\int _{[0,q]}\\int _0^{Y_{s-}(q)}z \\tilde{N}(ds,dz,d\\theta ,du)$ has a unique solution flow $\\lbrace Y_t(q): t\\ge 0,q\\in [0,a]\\rbrace $ .",
"For each $q\\in [0,a]$ , the one-dimensional process $\\lbrace Y_t(q): t\\ge 0\\rbrace $ is a CB-process with branching mechanism $\\phi _q$ .",
"It was proved in Li (2011) that there is a version of the flow which is increasing in $q\\in [0,a]$ .",
"Moreover, we can also regard $\\lbrace (Y_t(q))_{t\\ge 0}: q\\in [0,a]\\rbrace $ as a path-valued branching process.",
"The solution flow of (REF ) also induces a càdlàag superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ with state space $M[0,a]$ .",
"Let $f\\mapsto \\Psi (\\cdot ,f)$ be the operator on $C^+[0,a]$ defined by $\\Psi (x,f)=\\int _{[0,a]} f(x\\vee \\theta )h_{\\theta }d \\theta +\\int _{[0,a]}d\\theta \\int _0^{\\infty }(1-e^{-zf(x\\vee \\theta )})n_{\\theta }(dz).$ The superprocess $\\lbrace Y_t: t\\ge 0\\rbrace $ has local branching mechanism $\\phi _0$ and nonlocal branching mechanism given by (REF ); see Theorem 6.2 in Li (2012).",
"Then the transition semigroup $(Q_t)_{t\\ge 0}$ of $\\lbrace Y_t: t\\ge 0\\rbrace $ is defined by $\\int _{M[0,a]}e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,d\\nu )=\\exp \\Big \\lbrace -\\langle \\mu ,V_tf\\rangle \\Big \\rbrace ,\\qquad f\\in C^+[0,a],$ where $t\\mapsto V_tf$ is the unique locally bounded positive solution of $V_tf(x)=f(x)-\\int _0^t [\\phi _0(V_sf(x))-\\Psi (x,V_sf)]ds,\\qquad t\\ge 0,x\\in [0,a].$ To study the scaling limit theorems of the discrete branching flows, we need to introduce a metric on $M[0,a]$ .",
"Let $\\lbrace h_0,h_1,h_2,\\cdots \\rbrace $ be a countable dense subset of $\\lbrace h\\in C[0,a]^+:\\Vert h\\Vert \\le 1\\rbrace $ with $h_0\\equiv 1$ .",
"For convenience we assume each $h_i$ is bounded away from zero.",
"Then $\\lbrace h_0,h_1,h_2,\\cdots \\rbrace \\subset C[0,a]^{++}$ .",
"Now we define a metric $\\rho $ on $M[0,a]$ by $\\rho (\\mu ,\\nu ) = \\sum _{i=0}^{\\infty }\\frac{1}{2^i}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |), \\qquad \\mu , \\nu \\in M[0,a].$ It is easy to see that the metric is compatible with the weak convergence topology of $M[0,a]$ .",
"In other words, we have $\\mu _n\\rightarrow \\mu $ in $M[0,a]$ if and only if $\\rho (\\mu _n,\\mu )\\rightarrow 0$ .",
"For $\\nu \\in M[0,a]$ , set $\\mbox{\\rm e}_{h_i}(\\nu )= \\mbox{\\rm e}^{-\\langle \\nu ,h_i\\rangle }$ .",
"Theorem 2.1 The metric space $(M[0,a], \\rho )$ is a locally compact Polish (complete and separable) space, and $\\lbrace \\mbox{\\rm e}_{h_{i}}:i=0,1,2,\\cdots \\rbrace $ is strongly separating points of $M[0,a]$ , that is, for every $\\nu \\in M[0,a]$ and $\\delta >0$ , there exists a finite set $\\lbrace \\mbox{\\rm e}_{h_{i_1}},\\mbox{\\rm e}_{h_{i_2}},\\cdots ,\\mbox{\\rm e}_{h_{i_k}}\\rbrace \\subset \\lbrace \\mbox{\\rm e}_{h_i}:i=0,1,2,\\cdots \\rbrace $ such that $\\inf _{\\mu :\\rho (\\mu ,\\nu )\\ge \\delta }\\max _{1\\le j\\le k}|\\mbox{\\rm e}_{h_{i_j}}(\\mu )-\\mbox{\\rm e}_{h_{i_j}}(\\nu )|>0.$ Proof.",
"By Li (2011, p.4 and p.7) we know $M[0,a]$ is separable and locally compact, so there is a complete metric on $M[0,a]$ compatible with the weak convergence topology.",
"The following argument shows the metric $\\rho $ defined above is complete.",
"Suppose $\\lbrace \\mu _n\\rbrace _{n\\ge 1}\\subset M[0,a]$ is a Cauchy sequence under $\\rho $ .",
"Then for every $m\\ge 1$ , $\\lbrace \\langle \\mu _n,h_m\\rangle \\rbrace _{n\\ge 1}$ is also a Cauchy sequence.",
"We denote the limit by $\\Phi (h_m)$ .",
"For $f\\in C[0,a]^+$ satisfying $\\Vert f\\Vert \\le 1$ , let $\\lbrace h_{i_k}\\rbrace _{k\\ge 1}\\subset \\lbrace h_0,h_1,h_2,\\cdots \\rbrace $ be a sequence so that $\\Vert h_{i_k}-f\\Vert \\rightarrow 0$ as $k\\rightarrow \\infty $ .",
"For $n\\ge m\\ge 1$ we have $\\limsup _{m,n\\rightarrow \\infty }|\\langle \\nu _n,f\\rangle -\\langle \\nu _m,f\\rangle |\\!\\!&\\le \\!\\!&\\limsup _{m,n\\rightarrow \\infty }\\Big [|\\langle \\nu _n,f\\rangle -\\langle \\nu _n,h_{i_k}\\rangle |\\cr \\!\\!&\\!\\!&+|\\langle \\nu _n,h_{i_k}\\rangle -\\langle \\nu _m,h_{i_k}\\rangle |+|\\langle \\nu _m,h_{i_k}\\rangle -\\langle \\nu _m,f\\rangle |\\Big ]\\cr \\!\\!&\\le \\!\\!&2\\Phi (1)\\Vert f-h_{i_k}\\Vert .$ Then letting $k\\rightarrow \\infty $ we have $\\limsup _{m,n\\rightarrow \\infty }|\\langle \\nu _n,f\\rangle -\\langle \\nu _m,f\\rangle |=0.$ By linearity the above relation holds for all $f\\in C[0,a]$ , so the limit $\\Phi (f)=\\lim \\limits _{n\\rightarrow \\infty }\\langle \\mu _n,f\\rangle $ exists for each $f\\in C[0,a]$ .",
"Clearly, $f\\rightarrow \\Phi (f)$ is a positive linear functional on $C[0,a]$ .",
"By the Riesz representation theorem there exists $\\mu \\in M[0,a]$ so that $\\langle \\mu ,f\\rangle =\\Phi (f)$ for every $f\\in C[0,a]$ .",
"By the construction of $\\Phi $ we have $\\mu _n\\rightarrow \\mu $ , therefore, $\\rho (\\mu _n,\\mu )\\rightarrow 0$ .",
"That proves the first assertion of the theorem.",
"For any $\\nu \\in M[0,a]$ and $\\delta \\ge 0$ , there exists an $N_0\\in \\mathbb {N}_+$ such that $\\sum _{i=N_0+1}^{\\infty }1/{2^i}<{\\delta }/{2}$ .",
"Consider $\\lbrace h_0,h_1,\\cdots ,h_{N_0}\\rbrace $ , for any $\\mu \\in M[0,a]$ satisfying $\\rho (\\mu ,\\nu )\\ge \\delta $ , we have $\\sum _{i=0}^{N_0}\\frac{1}{2^i}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |)\\ge \\frac{\\delta }{2},$ and thus, $\\sum _{i=0}^{N_0}(1\\wedge |\\langle \\mu ,h_i\\rangle -\\langle \\nu ,h_i\\rangle |)\\ge \\frac{\\delta }{2}.$ It follows that $|\\langle \\mu ,h_j\\rangle -\\langle \\nu ,h_j\\rangle |\\ge \\frac{\\delta }{2N_0}~~\\mbox{for some}~~0\\le j\\le N_0.$ Since $|\\mbox{\\rm e}^{-x}-\\mbox{\\rm e}^{-y}|=\\mbox{\\rm e}^{-y}|\\mbox{\\rm e}^{y-x}-1|\\ge \\mbox{\\rm e}^{-y}\\Big [(\\mbox{\\rm e}^{|y-x|}-1)\\wedge (1-\\mbox{\\rm e}^{-|y-x|})\\Big ],\\qquad x,y\\in \\mathbb {R},$ we have $\\!\\!&\\!\\!&\\inf _{\\mu :\\rho (\\mu ,\\nu )\\ge \\delta }\\max _{0\\le i\\le N_0}|\\mbox{\\rm e}_{h_{i}}(\\mu )-\\mbox{\\rm e}_{h_{i}}(\\nu )|\\cr \\!\\!&\\!\\!&\\qquad \\quad \\ge \\mbox{\\rm e}^{-\\max _{0\\le i\\le N_0}\\langle \\nu ,h_i\\rangle }\\Big [(\\mbox{\\rm e}^{\\frac{\\delta }{2N_0}}-1)\\wedge (1-\\mbox{\\rm e}^{-\\frac{\\delta }{2N_0}})\\Big ]\\cr \\!\\!&\\!\\!&\\qquad \\quad > 0.$ That proves the second assertion.",
"$\\Box $"
],
[
"Flows of independent branching processes",
"In this section, we consider some flows of independent Galton-Watson branching processes.",
"We shall study the scaling limit in the setting of superprocesses.",
"Then we derive the convergence of the finite-dimensional distributions of the path-valued processes.",
"Let $\\lbrace g_i: i=0,1,2,\\cdots \\rbrace $ be a family of probability generating functions.",
"Given a family of $\\mathbb {N}$ -valued independent random variables $\\lbrace X_0(i): i=0,1,2,\\cdots \\rbrace $ , for each $i\\in \\mathbb {N}$ suppose that there are $X_0(i)$ independent Galton-Watson trees originating at time 0 and at place $i$ with offspring distribution given by $g_i$ .",
"Let us denote by $X_n(i)$ the numbers of vertices in the $n$ -th generation of the trees with root at $i$ .",
"In addition, we assume $(X_n(i))_{n\\ge 0}$ , $i=1,2,\\cdots $ are mutually independent.",
"It is well-known that for each $i\\in \\mathbb {N}$ , $(X_n(i))_{n\\ge 0}$ is a Galton-Watson branching process (GW-process) with parameter $g_i$ ; i.e., a discrete-time $\\mathbb {N}$ -valued Markov chain with $n$ -step transition matrix $P^n(j,k)$ defined by $\\sum _{k=0}^{\\infty }P^n(j,k)z^k=(g_i^n(z))^j,\\qquad |z|\\le 1,$ where $g_i^n(z)$ is defined by $g_i^n(z)=g_i(g_i^{n-1}(z))$ successively with $g_i^0(z)=z$ .",
"Suppose that for each integer $k\\ge 1$ we have a sequence of GW-processes $\\lbrace (X_n^{(k)}(i))_{n\\ge 0}:i\\ge 0\\rbrace $ with parameter $g_i^{(k)}$ .",
"Let $\\gamma _k$ be a positive real sequence so that $\\gamma _k\\rightarrow \\infty $ increasingly as $k\\rightarrow \\infty $ .",
"For $m,n\\in \\mathbb {N}$ , define $\\bar{X}_n^{(k)}(m)=\\sum _{i=0}^m X_n^{(k)}(i),$ and $\\qquad \\qquad \\qquad \\qquad Y_t^{(k)}(x)=\\displaystyle \\frac{1}{k}\\bar{X}_{[\\gamma _kt]}^{(k)}([kx]),\\qquad k=1,2,\\cdots ,$ where $[\\cdot ]$ denotes the integer part.",
"Then the increasing function $x\\mapsto Y_t^{(k)}(x)$ induces a random measure $Y_t^{(k)}(dx)$ on $[0,\\infty )$ so that $Y_t^{(k)}([0,x])=Y_t^{(k)}(x)$ for $x\\ge 0$ .",
"For convenience we fix a constant $a\\ge 0$ and consider the restriction of $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ to $[0,a]$ without changing the notation.",
"Clearly, $Y_0^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_0^{(k)}(i)\\delta _{\\frac{i}{k}}$ and $Y_t^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_{[\\gamma _kt]}^{(k)}(i)\\delta _{\\frac{i}{k}}.$ In view of (REF ), for each $i\\ge 0$ , given $X_0^{(k)}(i)=x_i\\in \\mathbb {N}$ , the conditional distribution $Q_{i,k}^{[\\gamma _kt]}(x_i/k,\\cdot )$ of $\\lbrace k^{-1}X_{[\\gamma _kt]}^{(k)}(i):t\\ge 0\\rbrace $ on $E_k=\\lbrace 0,1/k,2/k,\\cdots \\rbrace $ is determined by $\\int _{E_k} e^{-\\lambda y}Q_{i,k}^{[\\gamma _kt]}({x_i}/{k},dy) =\\exp \\left\\lbrace -\\frac{x_i}{k}v_i^{(k)}(t,\\lambda )\\right\\rbrace ,$ where $v_i^{(k)}(t,\\lambda )=-k\\log (g_i^{(k)})^{[\\gamma _kt]}(e^{-\\lambda /k})$ .",
"Let $Q_{\\mu _k}^{(k)}$ denote the conditional law given $Y_0^{(k)}=\\mu _k=k^{-1}\\sum _{i=0}^{[ka]}x_i\\delta _{i/{k}}\\in M_k[0,a]$ , where $M_k[0,a]:=\\lbrace k^{-1}\\sum _{i=0}^{[ka]}x_i\\delta _{i/{k}}: ~x_i\\in {\\mathbb {N}}, ~{k}^{-1}\\sum _{i=0}^{[ka]}{x_i}<\\infty \\rbrace $ .",
"For $f\\in B[0,a]^+$ , from (REF ) we have $Q_{\\mu _k}^{(k)}\\exp \\Big \\lbrace -\\langle Y_t^{(k)},f\\rangle \\Big \\rbrace \\!\\!&=\\!\\!&Q_{\\mu _k}^{(k)}\\exp \\bigg \\lbrace -\\sum _{i=0}^{[ka]}\\frac{1}{k}X_{[\\gamma _kt]}^{(k)}(i)f{({i}/{k})}\\bigg \\rbrace \\cr \\!\\!&=\\!\\!&\\prod _{i=1}^{[ka]}\\int _{E_k}e^{-f{({i}/{k})}y}Q_{i,k}^{[\\gamma _kt]}({x_i}/{k},dy)\\cr \\!\\!&=\\!\\!&\\exp \\bigg \\lbrace -\\sum _{i=0}^{[ka]}\\frac{x_i}{k}v_i^{(k)}(t,f({i}/{k}))\\bigg \\rbrace \\cr \\!\\!&=\\!\\!&\\exp \\Big \\lbrace -\\langle \\mu _k,v^{(k)}(t,f)\\rangle \\Big \\rbrace ,$ where $x\\mapsto v^{(k)}(t,f)(x)$ is defined by $v^{(k)}(t,f)(x)=v_{[kx]}^{(k)}(t,f(x))$ .",
"For any $x, z\\ge 0$ define $\\phi _k(x,z)=k\\gamma _k[g_{[kx]}^{(k)}(e^{-z/k})-e^{-z/k}].$ For convenience of statement of the results, we formulate the following condition: Condition (3.A) For each $a\\ge 0$ the sequence $\\lbrace \\phi _k(x,z)\\rbrace $ is Lipschitz with respect to $z$ uniformly on $[0,\\infty )\\times [0,a]$ and there is a continuous function $(x,z)\\mapsto \\phi (x, z)$ such that $\\phi _k(x, z) \\rightarrow \\phi (x, z)$ uniformly on $[0,\\infty )\\times [0,a]$ as $k\\rightarrow \\infty $ .",
"Before giving the limit theorem for the sequence of the rescaled processes, we first introduce the limit process.",
"By Proposition 4.3 in Li (2011), if Condition (3.A) is satisfied, the limit function $\\phi $ has the representation $\\phi (x,z)=b(x)z+\\frac{1}{2}c(x)z^2 +\\int _0^\\infty (e^{-zu}-1+zu)m(x,du),\\qquad x,z\\ge 0.$ where $b$ is a bounded function on $[0,\\infty )$ and $c$ is a positive bounded function on $[0,\\infty )$ .",
"$(u\\wedge {u}^2)m(x,du)$ is a bounded kernel from $[0,\\infty )$ to $(0,\\infty )$ .",
"Conversely, for any continuous function $(x,z)\\mapsto \\phi (x, z)$ given by (REF ), we can construct a family of probability generating functions $\\lbrace g_i^{(k)}:i=0,1,2,\\cdots \\rbrace $ so that the sequence (REF ) satisfies Condition (3.A); see, e.g., Li (2011, p.93).",
"For any $l\\ge 0$ , let $B_l[0,\\infty )^+$ be the set of positive bounded functions on $[0,\\infty )$ satisfying $\\Vert f\\Vert \\le l$ .",
"By a modification of the proof of Theorem 3.42 in Li (2011), it is not hard to show that for each $T\\ge 0$ and $l\\ge 0$ , $v^{(k)}(t,f)(x)$ converges uniformly on the set $[0,T]\\times [0,\\infty )\\times B_l[0,\\infty )^+$ of $(t,x,f)$ to the unique locally bounded positive solution $(t,x)\\mapsto v(t,f)(x)$ of the evolution equation $v(t,f)(x)=f(x)-\\int _0^t\\phi (x,v(s,f)(x))ds.",
"$ Let $\\lbrace Y_t: t\\ge 0\\rbrace $ be the superprocess with state space $M[0,a]$ and transition semigroup $(Q_t)_{t\\ge 0}$ defined by $\\int _{M[0,a]} e^{-\\langle \\nu ,f\\rangle }Q_t(\\mu ,\\nu )=\\exp \\left\\lbrace -\\langle \\mu ,v(t,f)\\rangle \\right\\rbrace , \\qquad f\\in B[0,a]^+.$ Using (REF ) and Gronwall's inequality one can see $x\\mapsto v(t,f)(x)$ is continuous on $[0,a]$ for every $f\\in C[0,a]^+$ .",
"Then by Proposition 3.1 in Li (2011) it is easy to see that $ v(t,f)\\in C[0,a]^{++}$ for every $f\\in C[0,a]^{++}$ .",
"From this and (REF ) it follows that $(Q_t)_{t\\ge 0}$ is a Feller semigroup.",
"Note that if $\\phi (x, z)= \\phi (z)$ independent of $x\\ge 0$ , then $(Q_t)_{t\\ge 0}$ is the same transition semigroup as that defined by (REF ) and (REF ).",
"In this case, the corresponding superprocess can be defined by the stochastic integral equation (REF ).",
"Let $D([0,\\infty ),M[0,a])$ denote the space of càdlàg paths from $[0,\\infty )$ to $M[0,a]$ furnished with the Skorokhod topology.",
"The proof of the next theorem is a modification of that of Theorem 3.43 in Li (2011).",
"Theorem 3.1 Suppose that Condition (3.A) is satisfied.",
"Let $\\lbrace Y_t:t\\ge 0\\rbrace $ be a càdlàg superprocess with transition semigroup $(Q_t)_{t\\ge 0}$ defined by (REF ) and (REF ).",
"If $Y_0^{(k)}$ converges to $Y_0$ in distribution on $M[0,a]$ , then $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ converges to $\\lbrace Y_t:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .",
"Proof.",
"For $f\\in C[0,a]^{++}$ and $\\nu \\in M[0,a]$ set $e_f(\\nu )=e^{-\\langle \\nu ,f\\rangle }$ .",
"Clearly, the function $\\nu \\mapsto e_f(\\nu )$ is continuous in $\\rho $ .",
"We denote by $D_1$ the linear span of $\\lbrace \\mbox{\\rm e}_f:f\\in C[0,a]^{++}\\rbrace $ .",
"By Theorem REF we have $D_1$ is an algebra strong separating the points of $M[0,a]$ .",
"Let $C_0(M[0,a])$ be the space of continuous functions on $M[0,a]$ vanishing at infinity.",
"Then $D_1$ is uniformly dense in $C_0(M[0,a])$ by the Stone-Weierstrass theorem; see, e.g., Hewitt and Stromberg (1975, pp.98-99).",
"On the other hand, for any $f\\in C[0,a]^{++}$ , since $v(t,f)$ is bounded away from zero and $v_k(t,f)(x)\\rightarrow v(t,f)(x)$ uniformly on $[0,\\infty )$ for every $t\\ge 0$ , we have $v_k(t,f)$ is also bounded away from zero for $k$ sufficiently large.",
"Without loss of generality we may assume $v_k(t,f)\\ge c$ and $v(t,f)\\ge c$ for some $c>0$ .",
"Let $Q_t^{(k)}$ denote the transition semigroup of $Y_t^{(k)}$ .",
"We get from (REF ) and (REF ) that, for any $M\\ge 0$ , $\\!\\!&\\!\\!&\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} e_f(\\nu )-Q_te_f(\\nu )\\right|\\cr \\!\\!&\\!\\!&\\qquad \\quad =\\sup _{\\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le \\sup _{\\langle \\nu ,1\\rangle \\le M\\atop \\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\qquad +\\sup _{\\langle \\nu ,1\\rangle > M\\atop \\nu \\in M_k[0,a]}\\Big |\\exp \\Big \\lbrace -\\langle \\nu ,v_k(t,f)\\rangle \\Big \\rbrace -\\exp \\Big \\lbrace -\\langle \\nu ,v(t,f)\\rangle \\Big \\rbrace \\Big |\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le \\sup _{\\langle \\nu ,1\\rangle \\le M\\atop \\nu \\in M_k[0,a]}|\\langle \\nu ,v_k(t,f)\\rangle -\\langle \\nu ,v(t,f)\\rangle | + \\sup _{\\langle \\nu ,1\\rangle > M\\atop \\nu \\in M_k[0,a]}2e^{-\\langle \\nu ,c\\rangle }\\cr \\!\\!&\\!\\!&\\qquad \\quad \\le M\\Vert v_k(t,f)-v(t,f)\\Vert +2e^{-Mc}.$ Since $M\\ge 0$ was arbitrary, we have $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} e_f(\\nu )-Q_te_f(\\nu )\\right|=0$ for every $t\\ge 0$ .",
"Thus $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}\\left|Q_t^{(k)} F(\\nu )-Q_tF(\\nu )\\right|=0$ for every $t\\ge 0$ and $F\\in C_0(M[0,a])$ .",
"By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ converges to $\\lbrace Y_t:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .",
"$\\Box $ Let $\\lbrace 0\\le a_1<a_2<\\cdots <a_n=a\\rbrace $ be an ordered set of constants.",
"Denote by $\\lbrace Y_{t,a_i}:t\\ge 0\\rbrace $ and $\\lbrace Y_{t,a_i}^{(k)}:t\\ge 0\\rbrace $ the restriction of $\\lbrace Y_t:t\\ge 0\\rbrace $ and $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ to $[0,a_i]$ , $i=1,2,\\cdots ,n$ , respectively.",
"The following theorem is an extension of Theorem REF .",
"Theorem 3.2 Suppose that Condition (3.A) is satisfied.",
"If $Y_{0,a}^{(k)}$ converges to $Y_{0,a}$ in distribution on $M[0,a]$ , then $\\lbrace (Y_{t,a_1}^{(k)}, \\cdots ,Y_{t,a_n}^{(k)}):t\\ge 0\\rbrace $ converges to $\\lbrace (Y_{t,a_1}, \\cdots ,Y_{t,a_n}):t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a_1]\\times \\cdots \\times M[0,a_n])$ .",
"Proof.",
"Let $f_i\\in C[0,a_i]$ for $i=1, \\cdots , n$ .",
"By Theorem REF we see that for every $1\\le i\\le n$ , $\\lbrace \\langle Y_{t,a_i}^{(k)},f_i\\rangle : t\\ge 0\\rbrace $ is tight in $D([0,\\infty ), \\mathbb {R})$ .",
"Thus $\\lbrace \\sum _{i=1}^n\\langle Y_{t,a_i}^{(k)},f_i\\rangle : t\\ge 0\\rbrace $ is tight in $D([0,\\infty ), \\mathbb {R})$ .",
"Then the tightness criterion of Roelly (1986) implies $\\lbrace (Y_{t,a_1}^{(k)}, \\cdots , Y_{t,a_n}^{(k)}):t\\ge 0\\rbrace $ is tight in $D([0,\\infty ),M[0,a_1]\\times \\cdots \\times M[0,a_n])$ .",
"Let $\\lbrace (Z_{t,a_1}, \\cdots , Z_{t,a_n}):t\\ge 0\\rbrace $ be a weak limit point of $\\lbrace (Y_{t,a_1}^{(k)}, \\cdots , Y_{t,a_n}^{(k)}):t\\ge 0\\rbrace $ .",
"By an argument similar to the proof of Theorem 5.8 in Dawson and Li (2012) one can show that $\\lbrace (Z_{t,a_1}, \\cdots , Z_{t,a_n}):t\\ge 0\\rbrace $ and $\\lbrace (Y_{t,a_1},\\cdots , Y_{t,a_n}):t\\ge 0\\rbrace $ have the same distributions on $D([0,\\infty ),M[0,a_1]\\times \\cdots \\times M[0,a_n])$ .",
"That gives the desired result.$\\Box $ Corollary 3.3 Suppose that Condition (3.A) is satisfied.",
"Let $\\lbrace 0\\le a_1<a_2<\\cdots <a_n=a\\rbrace $ be an ordered set of constants.",
"Let $Y_{t}(a_i):=Y_t[0,a_i]$ and $Y_{t}^{(k)}(a_i):=Y_t^{(k)}[0,a_i] $ for every $t\\ge 0$ , $i=1,2,\\cdots ,n$ .",
"If $(Y_{0}^{(k)}(a_1), \\cdots , Y_{0}^{(k)}(a_n))$ converges to $(Y_{0}(a_1),\\cdots , Y_{0}(a_n))$ in distribution on $\\mathbb {R}_+^{n}$ , then $\\lbrace (Y_{t}^{(k)}(a_1), \\cdots , Y_{t}^{(k)}(a_n)):t\\ge 0\\rbrace $ converges to $\\lbrace (Y_{t}(a_1), \\cdots , Y_{t}(a_n)):t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ), \\mathbb {R}_+^{n})$ ."
],
[
"Flows of interactive branching processes",
"In this section, we prove some limit theorems for a sequence of flows of interactive branching processes, which leads to a superprocesses with local branching and nonlocal branching.",
"From those limit theorems we derive the convergence of the finite-dimensional distributions of the path-valued branching processes.",
"Let $g_0$ be a probability generating function and $\\lbrace h_i:i=1,2,\\cdots \\rbrace $ a family of probability generating functions.",
"For each $i\\ge 1$ define $g_i:=g_0h_1\\cdots h_i$ and suppose that $\\lbrace \\xi _{n,j}(i):n=0,1,2,\\cdots ;j=1,2,\\cdots \\rbrace $ and $\\lbrace \\eta _{n,j}(i):n=0,1,2,\\cdots ;j=1,2,\\cdots \\rbrace $ are two independent families of positive integer-valued i.i.d.",
"random variables with distributions given by $g_i$ and $h_i$ , respectively.",
"Given another family of positive integer-valued random variables $\\lbrace z_i:i=1,2,\\cdots \\rbrace $ independent of $\\lbrace \\xi _{n,j}(i):i=1,2,\\cdots \\rbrace $ and $\\lbrace \\eta _{n,j}(i):i=1,2,\\cdots \\rbrace $ , we define inductively $X_0(0)=z_0$ and $X_{n+1}(0)=\\sum _{j=1}^{X_{n}(0)}\\xi _{n,j}(0),\\qquad n=0,1,2,\\cdots .$ Suppose that $\\lbrace X_{n}(i):n=0,1,2,\\cdots \\rbrace $ has been constructed for $i=0,1,\\cdots ,m-1$ , we define $\\lbrace X_{n}(m):n=0,1,2,\\cdots \\rbrace $ by $X_0(m)=z_{m}$ and $X_{n+1}(m)=\\sum _{j=1}^{X_{n}(m)}\\xi _{n,j}(m)+\\sum _{j=1}^{\\bar{X}_{n}(m-1)}\\eta _{n,j}(m), \\qquad n=0,1,2,\\cdots ,$ where $\\bar{X}_n(m-1)=\\sum _{i=0}^{m-1} X_{n}(i),~n=0,1,2,\\cdots $ .",
"It is easy to show that for any $m\\in {\\mathbb {N}}$ , $\\lbrace (X_{n}(0),X_{n}(1),\\cdots ,X_{n}(m)): n=0,1,2,\\cdots \\rbrace $ is a discrete-time $\\mathbb {N}^{m+1}$ -valued Markov chain with one-step transition probability $Q(x,dy)$ determined by, for $\\lambda , x\\in \\mathbb {N}^{m+1}$ , $\\int _{\\mathbb {N}^{m+1}}e^{-\\langle \\lambda , y\\rangle }Q(x,dy) =\\prod _{i=0}^m[g_i(e^{-\\lambda _i})]^{x_i}[h_i(e^{-\\lambda _i})]^{\\sum _{j=0}^{i-1}x_j},$ where $x_i$ and $\\lambda _i$ denote the $i$ -th component of $x$ and $\\lambda $ , respectively.",
"Suppose that for each integer $k\\ge 1$ we have two sequence of processes $\\lbrace (X_n^{(k)}(i))_{n\\ge 0}: {i\\ge 0}\\rbrace $ and $\\lbrace (\\bar{X}_n^{(k)}(i))_{n\\ge 0}: i\\ge 0\\rbrace $ with parameters $g_0^{(k)}$ and $\\lbrace h_i^{(k)}: i=1,2,\\cdots \\rbrace $ .",
"Suppose that $\\gamma _k$ is a positive real sequence so that $\\gamma _k\\rightarrow \\infty $ increasingly as $k\\rightarrow \\infty $ .",
"Let $[\\gamma _kt]$ denote the integer part of $\\gamma _kt\\ge 0$ .",
"Define $Y_t^{(k)}(x) :=\\frac{1}{k}\\bar{X}_{[\\gamma _kt]}^{(k)}([kx])=\\frac{1}{k}\\sum _{i=0}^{[kx]}{X}_{[\\gamma _kt]}^{(k)}(i), \\qquad k=1,2,\\cdots .$ Let $Y_t^{(k)}(dx)$ denote the random measure on $[0,\\infty )$ induced by the random function $Y_t^{(k)}(x)$ .",
"We are interested in the asymptotic behavior of the continuous-time process $\\lbrace Y_{t}^{(k)}(dx):t\\ge 0\\rbrace $ as $k\\rightarrow \\infty $ .",
"Let $h_0^{(k)}\\equiv 1$ .",
"For any $z\\ge 0$ and $\\theta \\ge 0$ set $\\phi _{\\theta }^{(k)}(z) = k\\gamma _k\\Big [g_{[k\\theta ]}^{(k)}(e^{-z/k})-e^{-z/k}\\Big ]$ and $\\psi _{\\theta }^{(k)}(z) = k^2\\gamma _k\\Big [1-h_{[k\\theta ]}^{(k)}(e^{-z/k})\\Big ].$ Let us consider the following set of conditions: Condition (4.A) For every $l\\ge 0$ , the sequence $\\lbrace \\phi _0^{(k)}\\rbrace $ is uniformly Lipschitz on $[0,l]$ and there is a function $\\phi _0$ on $[0,\\infty )$ such that $\\phi _0^{(k)}(z)\\rightarrow \\phi _0 (z)$ uniformly on $[0,l]$ as $k\\rightarrow \\infty $ .",
"Condition (4.B) There is a function $\\psi $ on $[0,\\infty )^2$ such that, for every $l\\ge 0$ , $\\psi _{\\theta }^{(k)}(z)\\rightarrow \\psi _{\\theta }(z)$ uniformly on $[0,l]^2$ as $k\\rightarrow \\infty $ and $\\sup _{\\theta \\in [0,a]}\\frac{d}{dz}\\psi _{\\theta }(z)|_{z=0^+}< \\infty .$ Proposition 4.1 If Conditions (4.A) and (4.B) hold, then for every $q\\ge 0$ there is a branching mechanism $\\phi _q$ such that $\\phi _q^{(k)}(z)\\rightarrow \\phi _q (z)$ uniformly on $[0,l]$ for every $l\\ge 0$ as $k\\rightarrow \\infty $ .",
"Moreover, the family of branching mechanisms $\\lbrace \\phi _q:q\\ge 0\\rbrace $ is admissible with $(\\partial /\\partial \\theta ) \\phi _{\\theta }(z)= - \\psi _{\\theta }(z)$ .",
"Proof.",
"If Conditions (4.A) and (4.B) hold, then the limit function $\\phi _0$ has the representation (REF ) with $(b,m)=(b_0,m_0)$ and $\\psi _{\\theta }$ has the representation (REF ); see, e.g., Li (2011, p.76).",
"By the definition of $g_i^{(k)}$ it is simple to check that, for every $q\\ge 0$ , $\\phi _q^{(k)}(z)\\!\\!&=\\!\\!&k\\gamma _k[g_0^{(k)}(e^{-z/k})-e^{-z/k}]\\prod _{i=1}^{[kq]}h_i^{(k)}(e^{-z/k})\\cr \\!\\!&\\!\\!&-\\sum _{i=1}^{[kq]}k\\gamma _k[1-h_i^{(k)}(e^{-z/k})]e^{-z/k}\\prod _{j=i+1}^{[kq]}h_j^{(k)}(e^{-z/k}).$ By elementary calculations, $\\prod _{i=1}^{[kq]}h_i^{(k)}(e^{-z/k})=\\exp \\bigg \\lbrace -\\sum _{i=1}^{[kq]}\\frac{1}{k^{2}\\gamma _k\\zeta _i^{(k)}}\\psi _{\\frac{i}{k}}^{(k)}(z)\\bigg \\rbrace ,$ where $\\zeta _i^{(k)}\\in [h_i^{(k)}(e^{-z/k}),1]$ .",
"It is easy to show that $\\prod _{i=1}^{[kq]}h_i^{(k)}(e^{-z/k})$ converges to 1 uniformly on $[0,l]$ for every $l\\ge 0$ if Condition (4.B) holds, and hence for each $1\\le i\\le [kq]$ , $\\prod _{j=i+1}^{[kq]}h_j^{(k)}(e^{-z/k})$ converges to 1 uniformly on $[0,l]$ for every $l\\ge 0$ .",
"By letting $k\\rightarrow \\infty $ in (REF ) we see $\\phi _q^{(k)}(z)$ uniformly converge to a function $\\phi _q (z)$ on $[0,l]$ for every $l\\ge 0$ and (REF ) holds.",
"Then the desired result follows readily.$\\Box $ Proposition 4.2 To each admissible family of branching mechanisms $\\lbrace \\phi _q: q\\ge 0\\rbrace $ with $(\\partial /\\partial \\theta )\\phi _{\\theta }(z)=-\\psi _{\\theta }(z)$ , there correspond two sequences $\\lbrace \\phi _{0}^{(k)}\\rbrace $ and $\\lbrace \\psi _{\\theta }^{(k)}\\rbrace $ in form of (REF ) and (REF ), respectively, so that Conditions (4.A) and (4.B) are satisfied.",
"Proof.",
"By Li (2011, p.93) there is a sequence $\\lbrace \\phi _{0}^{(k)}\\rbrace $ in form of (REF ) satisfying Condition (4.A).",
"By Li (2011, p.102), there is a family of probability generating functions $\\lbrace \\bar{h}_{\\theta }^{(k)}\\rbrace $ such that $k[1-\\bar{h}_{\\theta }^{(k)}(e^{-z/k})]\\rightarrow \\psi _{\\theta }(z)$ uniformly on $[0,l]^2$ for every $a\\ge 0$ as $k\\rightarrow \\infty $ .",
"Let $\\tilde{h}_{\\theta }^{(k)}(z)=1+\\frac{1}{k\\gamma _k}[\\bar{h}_{\\theta }^{(k)}(z)-1],\\quad \\theta \\ge 0, \\quad |z|\\le 1.$ Clearly, $\\lbrace \\tilde{h}_{\\theta }^{(k)}:\\theta \\ge 0\\rbrace $ is a family of probability generating functions and $k^2\\gamma _k [1-\\tilde{h}_{\\theta }^{(k)}(e^{-z/k})]\\rightarrow \\psi _{\\theta }(z)$ uniformly on $[0,l]^2$ for every $l\\ge 0$ as $k\\rightarrow \\infty $ .",
"For each $k\\ge 1$ , define $h_{i}^{(k)}=\\tilde{h}_{i/k}^{(k)}$ , $i=1,2,\\cdots $ .",
"Then by the continuity of $(\\theta ,z)\\mapsto \\psi _{\\theta }(z)$ we get the result.$\\Box $ Given a constant $a\\ge 0$ , denote by $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ the restriction of $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ to $[0,a]$ .",
"Then it is easy to see $Y_{0,a}^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_0^{(k)}(i)\\delta _{\\frac{i}{k}}\\quad \\text{and}\\quad Y_{t,a}^{(k)}=\\frac{1}{k}\\sum _{i=0}^{[ka]}X_{[\\gamma _kt]}^{(k)}(i)\\delta _{\\frac{i}{k}}.$ Then $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ is a measure-valued Markov process with state space $M_k[0,a]$ .",
"From (REF ) one can see the (discrete) generator $L_k$ of $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ is given by, for $\\nu =k^{-1}\\sum _{i=0}^{[ka]}x_i^{(k)}\\delta _{i/k}\\in M_k[0,a]$ and $f\\in C[0,a]^{++}$ , $L_ke^{-\\langle \\nu ,f\\rangle } \\!\\!&=\\!\\!&{\\gamma _k}\\Big [\\prod _{i=0}^{[ka]}g_i^{(k)}(e^{-f(\\frac{i}{k})/k})^{x_i}h_i^{(k)}(e^{-f(\\frac{i}{k})/k})^{\\sum _{j=0}^{i-1}x_j}-e^{-\\langle \\nu ,f\\rangle }\\Big ]\\cr \\!\\!&=\\!\\!&e^{-\\langle \\nu ,f\\rangle }{\\gamma _k}\\Big [\\exp \\Big \\lbrace \\sum _{i=0}^{[ka]}\\log \\Big (g_i^{(k)}(e^{-f(\\frac{i}{k})/k})^{x_i}h_i^{(k)}(e^{-f(\\frac{i}{k})/k})^{\\sum _{j=0}^{i-1}x_j}\\Big )\\cr \\!\\!&\\!\\!&\\qquad \\qquad \\qquad \\qquad + \\langle \\nu ,f\\rangle \\Big \\rbrace -1\\Big ]\\cr \\!\\!&=\\!\\!&e^{-\\langle \\nu ,f\\rangle }{\\gamma _k}\\Big [\\exp \\lbrace \\alpha _k+\\beta _k\\rbrace -1\\Big ],$ where $\\alpha _k =\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_i^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ], \\quad \\beta _k=\\sum _{i=0}^{[ka]}\\sum _{j=0}^{i-1}x_j\\log h_i^{(k)}(e^{-f(\\frac{i}{k})/k}).$ By the definition of $g_i^{(k)}$ we have $\\alpha _k \\!\\!&=\\!\\!&\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_0^{(k)}(e^{-f(\\frac{i}{k})/k}) +\\sum _{j=0}^{i}\\log h_j^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ]\\cr \\!\\!&=\\!\\!&\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_0^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ]+\\sum _{i=0}^{[ka]}\\sum _{j=0}^{i}x_i\\log h_j^{(k)}(e^{-f(\\frac{i}{k})/k}).$ It follows that $\\alpha _k+\\beta _k \\!\\!&=\\!\\!&\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_0^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ]+\\sum _{i=0}^{[ka]}\\sum _{j=0}^{i}x_i\\log h_j^{(k)}(e^{-f(\\frac{i}{k})/k})\\cr \\!\\!&\\!\\!&+\\sum _{i=0}^{[ka]}\\sum _{j=0}^{i-1}x_j\\log h_i^{(k)}(e^{-f(\\frac{i}{k})/k})\\cr \\!\\!&=\\!\\!&\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_0^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ]+\\sum _{i=0}^{[ka]}\\sum _{j=0}^{i}x_i\\log h_j^{(k)}(e^{-f(\\frac{i}{k})/k})\\cr \\!\\!&\\!\\!&+\\sum _{i=0}^{[ka]}\\sum _{j=i+1}^{[ka]-1}x_i\\log h_j^{(k)}(e^{-f(\\frac{j}{k})/k})\\cr \\!\\!&=\\!\\!&\\sum _{i=0}^{[ka]}x_i\\Big [\\log g_0^{(k)}(e^{-f(\\frac{i}{k})/k})+f(\\frac{i}{k})/k\\Big ]+\\sum _{i=0}^{[ka]}\\sum _{j=0}^{[ka]}x_i\\log h_j^{(k)}(e^{-f(\\frac{i\\vee j}{k})/k})\\cr \\!\\!&=\\!\\!&\\frac{1}{\\gamma _k}\\bigg [\\sum _{i=0}^{[ka]}\\frac{x_i}{k\\zeta _i^{(k)}}\\phi _0^{(k)}(f(\\frac{i}{k}))-\\sum _{i=0}^{[ka]}\\frac{x_i}{k}\\Big (\\sum _{j=0}^{[ka]}\\frac{1}{k\\zeta _{i,j}^{(k)}}\\psi _{\\frac{j}{k}}^{(k)}({f(\\frac{i\\vee j}{k}}))\\Big )\\bigg ],$ where $\\zeta _i^{(k)}$ is between $e^{-f(\\frac{i}{k})/k}$ and $g_0^{(k)}(e^{-f(\\frac{i}{k})/k})$ , $\\zeta _{i,j}^{(k)}\\in [h_j^{(k)}(e^{-f(\\frac{i\\vee j}{k})/k}),1]$ .",
"Clearly, both $\\zeta _i^{(k)}$ and $\\zeta _{i,j}^{(k)}$ converge to 1 uniformly as $k\\rightarrow \\infty $ if Conditions (4.A) and (4.B) hold.",
"Then the above equality implies $\\alpha _k+\\beta _k=\\frac{1}{\\gamma _k}\\Big [\\langle \\nu ,\\phi _0^{(k)}(f(\\cdot ))\\rangle -\\langle \\nu ,\\Psi ^{(k)}(\\cdot ,f)\\rangle +o(1)\\Big ],$ where $\\Psi ^{(k)}(\\cdot ,f)=\\sum _{j=0}^{[ka]}\\frac{1}{k}\\psi _{\\frac{j}{k}}^{(k)}({f(\\cdot \\vee \\frac{j}{k}})).$ Let $\\lbrace Y_{t,a}:t\\ge 0\\rbrace $ be the càdlàg superprocess with transition semigroup $(Q_t)_{t\\ge 0}$ defined by (REF ) and (REF ).",
"Theorem 4.3 Suppose that Conditions (4.A) and (4.B) are satisfied.",
"If $Y_{0,a}^{(k)}$ converges to $Y_{0,a}$ in distribution on $M[0,a]$ , then $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ converges to $\\lbrace Y_{t,a}:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .",
"Proof.",
"As in the proof of Theorem 2.1 in Li (2006), we shall prove the convergence of the generators.",
"Let $D_1$ be the algebra as defined in Theorem REF .",
"For $f\\in C[0,a]^{++}$ , let $Le^{-\\langle \\nu ,f\\rangle }=e^{-\\langle \\nu ,f\\rangle }\\Big [\\langle \\nu , \\phi _0(f(\\cdot ))\\rangle -\\langle \\nu ,\\Psi (\\cdot ,f)\\rangle \\Big ], \\qquad \\nu \\in M[0,a],$ and extend the definition of $L$ to $D_1$ by linearity.",
"By (REF ) one can check that $L$ is a restriction of strong generator of $(Q_t)_{t\\ge 0}$ .",
"Note also that $L:=\\lbrace (f,Lf): f\\in D_1\\rbrace $ is a linear space of $C_0(M[0,a])\\times C_0(M[0,a])$ .",
"On the other hand, let $f(x)=\\lambda $ in (REF ) and (REF ), we have the function $\\lambda \\mapsto V_t(\\lambda )$ is strictly increasing on $[0,\\infty )$ for every $t\\ge 0$ ; see, e.g., Li (2011, p.58).",
"Therefore, $V_t(\\lambda )>0$ for every $\\lambda >0$ and $t\\ge 0$ .",
"In view of (REF ), for any $f\\in C[0,a]^{++}$ we have $V_tf\\in C[0,a]^{++}$ for every $t\\ge 0$ .",
"Then $D_1$ is invariant under $(Q_t)_{t\\ge 0}$ , which is a core of the strong generator of $(Q_t)_{t\\ge 0}$ ; see, e.g., Ethier and Kurtz (1986, p.17).",
"In other words, the closure of $L$ generates $(Q_t)_{t\\ge 0}$ uniquely; see, e.g., Ethier and Kurtz (1986, p.15 and p.17).",
"Based on (REF ) and (REF ) one can see $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}|L_ke^{-\\langle \\nu ,f\\rangle }-Le^{-\\langle \\nu ,f\\rangle }|=0$ for every $f\\in C[0,a]^{++}$ , which implies $\\lim _{k\\rightarrow \\infty }\\sup _{\\nu \\in M_k[0,a]}|L_kF(\\nu )-LF(\\nu )|=0$ for every $F\\in D_1$ .",
"By Ethier and Kurtz (1986, p.226 and pp.233-234) we conclude that $\\lbrace Y_{t,a}^{(k)}:t\\ge 0\\rbrace $ converges to the immigration superprocess $\\lbrace Y_{t,a}:t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a])$ .",
"$\\Box $ Let $\\lbrace 0\\le a_1<a_2<\\cdots <a_n=a\\rbrace $ be an ordered set of constants.",
"Denote by $\\lbrace Y_{t,a_i}:t\\ge 0\\rbrace $ and $\\lbrace Y_{t,a_i}^{(k)}:t\\ge 0\\rbrace $ the restriction of $\\lbrace Y_t:t\\ge 0\\rbrace $ and $\\lbrace Y_t^{(k)}:t\\ge 0\\rbrace $ to $[0,a_i]$ , respectively.",
"Let $Y_{t}(a_i):=Y_t[0,a_i]$ and $Y_{t}^{(k)}(a_i):=Y_t^{(k)}[0,a_i] $ for every $t\\ge 0$ , $i=1,2,\\cdots ,n$ .",
"By arguments similar to those in Section 3 we have: Theorem 4.4 Suppose that Conditions (4.A) and (4.B) are satisfied.",
"If $Y_{0,a}^{(k)}$ converges to $Y_{0,a}$ in distribution on $M[0,a]$ , then $\\lbrace (Y_{t,a_1}^{(k)}, \\cdots , Y_{t,a_n}^{(k)}):t\\ge 0\\rbrace $ converges to $\\lbrace (Y_{t,a_1}, \\cdots , Y_{t,a_n}):t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ),M[0,a_1]\\times \\cdots \\times M[0,a_n])$ .",
"Corollary 4.5 Suppose that Conditions (4.A) and (4.B) are satisfied.",
"If $(Y_{0}^{(k)}(a_1), \\cdots , Y_{0}^{(k)}(a_n))$ converges to $(Y_{0}(a_1), \\cdots , Y_{0}(a_n))$ in distribution on $\\mathbb {R}_+^{n}$ , then $\\lbrace (Y_{t}^{(k)}(a_1), \\cdots ,Y_{t}^{(k)}(a_n)):t\\ge 0\\rbrace $ converges to $\\lbrace (Y_{t}(a_1), \\cdots ,Y_{t}(a_n)):t\\ge 0\\rbrace $ in distribution on $D([0,\\infty ), \\mathbb {R}_+^{n})$ .",
"Acknowledgements.",
"This paper was written under the supervision of Professor Zenghu Li, to whom gratitude is expressed.",
"We also would like to acknowledge the Laboratory of Mathematics and Complex Systems (Ministry of Education, China) for providing us the research facilities.",
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]
] | 1204.1248 |
[
[
"The CMB and the measure of the multiverse"
],
[
"Abstract In the context of eternal inflation, cosmological predictions depend on the choice of measure to regulate the diverging spacetime volume.",
"The spectrum of inflationary perturbations is no exception, as we demonstrate by comparing the predictions of the fat geodesic and causal patch measures.",
"To highlight the effect of the measure---as opposed to any effects related to a possible landscape of vacua---we take the cosmological model, including the model of inflation, to be fixed.",
"We also condition on the average CMB temperature accompanying the measurement.",
"Both measures predict a 1-point expectation value for the gauge-invariant Newtonian potential, which takes the form of a (scale-dependent) monopole, in addition to a related contribution to the 3-point correlation function, with the detailed form of these quantities differing between the measures.",
"However, for both measures both effects are well within cosmic variance.",
"Our results make clear the theoretical relevance of the measure, and at the same time validate the standard inflationary predictions in the context of eternal inflation."
],
[
"Introduction",
"Spacetime might feature eternal inflation [1], [2].",
"Theoretically, eternal inflation occurs about any metastable state with a decay rate smaller than its Hubble rate [3], and such states are expected to exist in string theory [4], [5], [6].",
"Moreover, the spectrum of temperature fluctuations in the cosmic microwave background (CMB) is broadly consistent with expectations from an early period of slow-roll inflation [7], and the simplest models of slow-roll inflation also exhibit eternal inflation (see for example [8]).",
"Observationally, supernova and CMB data indicate that the energy budget of our universe is dominated by some `dark energy' with equation of state $w\\simeq -1$ [9], [10], [7], and the simplest explanation of this involves a cosmological constant (vacuum energy density), which would also imply eternal inflation.",
"Eternal inflation generates diverging spacetime volume.",
"Consequently, the pre-conditions to any type of measurement are established a diverging number of times, and to predict the relative probability of different possible outcomes of a measurement requires a procedure to regulate these divergences.",
"Such a regulation procedure is called a measure—the measure in effect constructs the probability space for physical observables in an infinite spacetime—and the `measure problem' refers to the fact that different seemingly sensible measure proposals sometimes give dramatically different physical predictions (for recent reviews of the measure problem, see for example [11], [12]).",
"The choice of measure in principle affects all probabilistic predictions.",
"This includes, in particular, any predictions related to the spectrum of inflationary perturbations.",
"Of course, insofar as the standard formulation of inflationary predictions (see for example [13]) has met with empirical success, the choice of measure should make predictions at least approximately in agreement.",
"Nevertheless, the standard formulation cannot stand by itself, as an arbitrarily precise and self-contained framework, as it relies on assumptions that require justification in the context of eternal inflation.",
"It is worthwhile to elaborate on this point.",
"Given suitable initial conditions, inflation generates a spacetime region with approximate Friedmann-Robertson-Walker (FRW) symmetries.",
"These symmetries are broken by quantum fluctuations, which decohere to represent an ensemble of classical perturbations, with branching ratios given by projection onto the Bunch-Davies vacuum [14].",
"The branching ratios are related to probabilities via the usual Born rule, but these are probabilities for perturbations, not the outcomes of measurements.",
"The standard formulation makes predictions by assuming that measurements are equally likely to be performed at any comoving coordinate in the FRW geometry.",
"(Put another way, it assumes that there are no correlations between the locations of measurements and the perturbations that they measure.)",
"Actually, this assumption is not exactly valid even in a finite spacetime with global FRW symmetries.",
"For example, a subset of the ensemble of inflationary perturbations is, by chance, homogeneous at a level far below the variance characterizing the entire ensemble.",
"Elements of this subset feature stunted structure formation and therefore relatively few observers to measure them.",
"Therefore, perturbations in this subset should contribute less to expectation values than the above assumption would indicate.",
"This `anthropic' effect is very small, and ignoring it can be viewed as a convenient approximation.",
"Anthropic effects notwithstanding, the above assumption would be well motivated if the global spacetime were finite (and nearly all measurements occurred in the FRW region) or if the FRW symmetries described the global spacetime.",
"However, neither of these conditions holds in the context of eternal inflation.",
"To illustrate how the choice of measure could invalidate the above assumption, consider as an analogy the selection of random points via throwing darts at a wall.",
"If the wall has a hemispherical bulge, an observer on the bulge who is small compared to the bulge might recognize the local O(3) symmetry and suppose that her random position is selected according to that.",
"This observer would be incorrect.",
"This situation is not unlike that of the worldline-based measures studied in this paper, with the darts representing the future histories of the worldline, the wall representing the reference frame of the worldline, and the bulge representing a frame with local FRW symmetries.",
"Here the effect of the `measure' is large in the sense that random points are much less likely to lie where the tangent to the bulge is parallel to the trajectories of the darts.",
"On the other hand, insofar as the local O(3) symmetry implies that a local environment is independent of its coordinates on the bulge, the observer cannot discern her (likely) special location.",
"Nevertheless, the same effect applies with respect to any small perturbation on the bulge, though now the size of the effect is suppressed in terms of the smallness of the perturbation.",
"This foreshadows the conclusions of our analysis.",
"We study the spectrum of inflationary perturbations in the context of two measures, the fat geodesic measure [15] (see also [16], [17]) and the causal patch measure [18].",
"We focus on these measures in part because we find their formulations to be particularly amenable to the analysis of inflationary perturbations, and in part because these measures are among a small set known to pass three important phenomenological tests.",
"In particular, they do not possess a youngness problem [19], [8], [20],These measures still feature what might be called a youngness pressure: all else being equal, they assign greater weight to sequences of events that span less time.",
"However, this effect is comparatively small—roughly speaking, it is only significant for sequences of events that span a Hubble time, and therefore it does not pose any obvious problems for phenomenology—though its theoretical implications are a matter of debate [21], [22].",
"nor do they feature the $Q$ or $G$ catastrophes [23], [24], [25], and they can avoid Boltzmann-brain domination [26], [27], [28], [29], [30], given plausible assumptions about decay rates in the landscape [29], [15].",
"(It should be noted that the causal patch measure assigns an overwhelming probability to observe a negative vacuum energy [31], [32].",
"This is a serious issue, but one might speculate that it can be resolved without affecting predictions for positive-energy vacua such as ours.",
"The causal patch measure might also predict too small of a positive vacuum energy, depending on the details of the landscape [32].)",
"The causal patch measure is closely related to the lightcone-time cutoff measure [33], [34], and the assumptions of our analysis establish a condition by which these measures make identical predictions [35].",
"The fat geodesic measure is closely related to the scale-factor cutoff measure [36], [37], but the correspondence is not precise [15], and we expect these measures to give different predictions at our level of analysis.",
"The CAH+ measure [38], [39] has some similarities with the lightcone-time cutoff measure, but again the relationship is not precise, and we expect these measures to make different predictions at our level of analysis.",
"The `equilibrium' measure [40] has some similarities with the fat geodesic measure, however the two also have important differences and so we cannot assert that they make the same predictions here.",
"To highlight the effects of the measure—as opposed to any effects related to performing cosmological measurements in the context of a landscape of vacua—we take the cosmological model, including the model of inflation, to be specified.",
"We also condition on a given average CMB temperature at the point of measurement.",
"(To keep the analysis simple, we only set this condition in an approximate manner, however we do not expect this to affect the qualitative form of the results.)",
"We find that both measures predict a `1-point' expectation value for the Fourier components of the gauge-invariant Newtonian potential $\\Psi $ , this taking the form of a scale-dependent monopole.",
"Accordingly, they both predict a contribution to the expectation value of the 3-point correlation function, when one of the three insertions of $\\Psi $ is a monopole.",
"The predictions of both measures have the same order of magnitude, but the precise size and form of the predictions differ between the measures.",
"With the fat geodesic measure, the effect comes in part from the tendency of the worldline that defines the measure to gravitate toward over-densities, these over-densities correlating with $\\Psi $ .",
"With the causal patch measure, there is a similar effect, and in addition there is an effect coming from the correlation between the size of the causal patch—which affects the number of measurements that are performed in a given causal patch—and $\\Psi $ .",
"While the monopole is just a constant background with respect to the primary CMB perturbations, it appears in secondary effects, including the angular size of acoustic peaks (see for example [41]).",
"Nevertheless, we find the sizes these measure effects to be well within cosmic variance.",
"While our work demonstrates the measure dependence of the inflationary spectrum, the smallness of the effects can be seen as validation of the standard formulation of inflationary predictions.",
"Yet, it is important to keep in mind that while the fat geodesic and causal patch measures have achieved some phenomenological successes, no measure has yet to acquire a broadly compelling theoretical motivation.",
"Hence, the validation of the standard formulation should be seen more as a proof of principle.",
"Nevertheless, familiarity with the phenomenology of measures lends one to suspect that any measure that survives the three major phenomenological tests listed above (for instance the scale-factor cutoff measure and CAH+ measure) will permit similar conclusions with respect to inflationary observables.",
"Before proceeding, we note that an effect on the inflationary spectrum coming from the proper-time cutoff measure [42], [43] was previously discussed in [44].",
"Compared to our analysis, that work is less explicit in the size of the effect (which also includes a scale-dependent 1-point expectation value in the form of a monopole).",
"Also, the proper-time cutoff measure is known to exhibit the youngness problem as well as the $Q$ and $G$ catastrophes mentioned above, and is therefore no longer considered as a viable measure on eternal inflation.",
"The remainder of this paper is organized as follows.",
"In Section , we briefly review the fat geodesic and causal patch measures, and then set up the general framework for predicting inflationary observables in each case.",
"For concreteness we introduce a number of conditional assumptions; these are sufficient but not necessary to demonstrate the measure dependence.",
"Section describes our model assumptions, which are intended to provide a simple approximation of the standard cosmological model.",
"In Section , we compute the expectation values of correlation functions of $\\Psi $ in the context of the fat geodesic measure, while in Section we perform the same calculations but for the causal patch measure.",
"Finally, we summarize our results and draw our conclusions in Section .",
"We first review the fat geodesic and causal patch measures.",
"Both of these measures focus on the future histories surrounding a worldline emanating from some initial conditions, taking for granted a probability space defined over the set of possible initial conditions.",
"It is assumed that for practical purposes, these future histories can be represented by an ensemble of semi-classical spacetimes.",
"Different elements of this ensemble correspond to different Coleman–De Luccia (CDL) bubbles [45], [46] nucleating in different places, different configurations of decohered inflaton fluctuations, etc.",
"Eventually, the worldline encounters a CDL bubble with negative vacuum-energy density and is inevitably drawn into a big crunch singularity, at which point the worldline is terminated.",
"(Some worldlines never encounter such a singularity, but these form a set of measure zero in the fat geodesic and causal patch measures.)",
"The fat geodesic measure only counts events that occur within some small, fixed physical distance orthogonal to the worldline (the `fat geodesic'), whereas the causal patch measure only counts events that occur within the past lightcone of the end of the worldline (the `causal patch').",
"We denote the ensemble of causal patches $\\Sigma $ , and take this to define an ensemble of fat geodesics as well, in the latter case simply restricting attention to the spacetime within the fat geodesic.",
"Each element $i$ of $\\Sigma $ is assigned a weight ${\\cal I}_i$ corresponding to the quantum-mechanical branching ratio to the semi-classical spacetime that it represents.",
"The fat geodesic and causal patch measures use their corresponding rules to assign relative probabilities to classes of local events in the semi-classical spacetime.",
"Specifically, the relative probability assigned to some event of type $E$ in the fat geodesic measure is $P(E) \\propto \\sum _{i\\in \\Sigma }\\,\\, {\\cal I}_i\\,{\\cal N}_i(E) \\,,$ where ${\\cal N}_i(E)$ is the number of events of type $E$ within the fat geodesic of $i$ .",
"In the causal patch measure, this relative probability is $P(E) \\propto \\sum _{i\\in \\Sigma }\\,\\,{\\cal I}_i\\,\\overline{\\cal N}_{\\!i}(E) \\,,$ where $\\overline{\\cal N}_{\\!i}(E)$ is the number of events $E$ within the causal patch $i$ .",
"One is usually interested in predicting conditional probabilities, in which case one only counts events when they satisfy the prescribed conditions, and one can likewise restrictthe ensemble of causal patches $\\Sigma $ to the subset containing events satisfying those conditions."
],
[
"Assumptions and approximations",
"These probabilities can be made more concrete with some plausible assumptions about the landscape.",
"First of all, when making predictions for observers like us, we can condition on our extremely small (in Planck units) and therefore extremely rare (in the landscape) vacuum-energy density.",
"Thus it can be assumed that no vacua with smaller-magnitude vacuum-energy densities are nearby in the landscape configuration space.",
"Since transitions to higher-energy vacua are exponentially suppressed next to transitions to lower-energy vacua, this means that among the set of worldlines that enter our vacuum, the overwhelming majority transition next to a negative-energy vacuum, after which the worldline quickly terminates.",
"Therefore, to a good approximation we can restrict attention to the sub-ensemble of $\\Sigma $ in which the worldline encounters our vacuum only once, terminating soon after our vacuum decays.",
"Since the decay of our vacuum is itself likely to be strongly exponentially suppressed (in Hubble units), the causal patch of the worldline can be approximated as the past lightcone of future infinity, for this purpose treating our vacuum as if it does not decay.",
"Moreover, to demonstrate the measure dependence of inflationary observables, it is sufficient to condition on a specific model of the local cosmology.",
"In particular, we assume that the observable universe is contained in a CDL bubble, which nucleated many Hubble times after the nucleation of its progenitor bubble, and for which the effects of bubble-wall perturbations and collisions with other bubbles are negligible.",
"We assume there is a finite period of slow-roll inflation within the bubble to erase its initial spatial curvature.",
"(This period of inflation generates the inflationary observables.)",
"The particular model of inflation, as well as the model describing the subsequent cosmological evolution, is taken to be completely fixed.",
"We even condition on a given value of the average CMB temperature, which we denote $T_{\\rm obs}$ , at the point of measurement.We could forego this condition, but this would invite us to condition more explicitly on anthropic criteria, which at this stage we prefer to avoid (we discuss this and related issues in Section REF ).",
"Figure: Left: toy conformal diagram of a comoving (with respect to the spatially flat de Sitter chart) worldline γ\\gamma (thick curve) entering a CDL bubble (thick dashed line), in which it is boosted with respect to the comoving open-FRW chart (solid and dotted curves), but quickly becomes comoving during inflation.",
"The diagram indicates the times η 0 \\eta _0 and η obs \\eta _{\\rm obs} referred to in the text.",
"One should imagine a small negative-vacuum-energy CDL bubble at the tip of the worldline.",
"Right: FRW symmetry in the bubble has been exploited to make γ\\gamma asymptote toward the center of the bubble.",
"The causal patch of the worldline is indicated (thick dotted curve), as are the three-volume of the causal patch at η obs \\eta _{\\rm obs} (thick curve) and the curvature perturbation at η 0 \\eta _0 (dashed curve).Figure REF provides two equivalent conformal diagrams of the geometry.",
"Our assumptions allow us to focus on a single bubble embedded in the spatially flat de Sitter chart.",
"The interior of the bubble is an infinite, open-FRW universe.",
"The worldline defining either measure can enter the bubble at any point along the bubble wall, and for entry points displaced from the center of the bubble the worldline will be boosted with respect to the open-FRW chart.",
"Nevertheless, this boost is redshifted as the scale factor within the bubble expands, and the worldline quickly becomes comoving in the open-FRW frame.",
"To simplify visualization, one can then exploit the FRW symmetries in the bubble to boost the worldline so that it appears to have come from the origin of coordinates in the bubble."
],
[
"Formal application to inflationary observables",
"Consider the decohered curvature perturbation $\\zeta $ on some fixed-FRW-time hypersurface $\\eta =\\eta _0$ near the end of inflation in some bubble like ours (a gauge-invariant specification of $\\zeta $ is given in Section REF ).",
"Given our assumptions, the (relevant) future evolution of the bubble depends only on the specific configuration of this (classical) perturbation.",
"Therefore, we can label fat geodesics / causal patches according to their particular perturbation $\\zeta (\\eta _0,{\\mathbf {x}})$ , taking the worldline defining each measure to pass through ${\\mathbf {x}}=0$ .",
"The branching ratio ${\\cal I}_{\\zeta }$ is then just the quantum-mechanical branching ratio to create the specified perturbation $\\zeta $ .",
"To make this precise, note that our assumptions allow us to work in the effective Fock space $\\lbrace |\\zeta \\rangle \\rbrace $ of curvature perturbations on an open–de Sitter chart (the usual `Fock space' of cosmological perturbation theory in the single-bubble background [47], [48], [49]).",
"Then we can write the branching ratio ${\\cal I}_{\\zeta } = |\\langle \\zeta |0\\rangle |^2 \\,,$ where $|0\\rangle $ denotes the Bunch-Davies vacuum.Distinct configurations of $\\zeta $ will sometimes correspond to the same fat geodesic / causal patch, in part because the positive cosmological constant in our bubble implies that the defining worldline is in causal contact with only a finite portion of the infinite $\\eta =\\eta _0$ hypersurface.",
"This degeneracy would be relevant to computing the branching ratios to distinct fat geodesics / causal patches if the field configuration within the causal patch were not statistically independent of the field configuration outside of the causal patch.",
"Our adoption of the Bunch-Davies vacuum guarantees the requisite statistical independence, however this is ultimately an assumption about the probability space over initial conditions in these measures.",
"To better understand the counting of events with ${\\cal N}_{\\zeta }$ and $\\overline{\\cal N}_{\\!\\zeta }$ , it will help to refine our notation.",
"The spectrum of inflationary perturbations seen by a given observer contains a lot of information, and one may wish to characterize it with some statistic $z$ .",
"This statistic is a functional of the particular curvature perturbation $\\zeta $ within the past lightcone of the observer, who resides at some point $\\lbrace \\eta ,{\\mathbf {x}}\\rbrace $ .",
"Thus we should write $z=z[\\eta ,{\\mathbf {x}},\\zeta ]$ .",
"(We emphasize that $z$ is a statistic characterizing the data set of a single observer at a specific location in a particular semi-classical spacetime.)",
"Let $E_{z\\rightarrow \\tilde{z}}$ denote the event of measuring $z$ to attain the value $\\tilde{z}$ , regardless of where the measurement is performed or what is the specific perturbation $\\zeta $ .",
"The number of such events in a given fat geodesic is ${\\cal N}_{\\zeta } (E_{z\\rightarrow \\tilde{z}}) =\\int d\\eta \\int _{{\\rm FG}[\\zeta ]}\\!",
"\\sqrt{-g[\\eta ,{\\mathbf {x}},\\zeta ]}\\,d{\\mathbf {x}}\\,\\delta \\big [\\eta -\\eta _{\\rm obs}({\\mathbf {x}})\\big ]\\,\\delta \\big \\lbrace z\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ]-\\tilde{z}\\big \\rbrace \\,{\\cal A}\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ] \\,,$ where the inner integral is understood to cover the fixed physical three-volume within the fat geodesic at time $\\eta $ , the first delta function selects for when the average CMB temperature is $T_{\\rm obs}$ —with $\\eta _{\\rm obs}({\\mathbf {x}})$ being the time at which this happens according to a comoving geodesic through ${\\mathbf {x}}$ —the second delta function selects for when the statistic $z$ gives $\\tilde{z}$ , and ${\\cal A}[\\eta ,{\\mathbf {x}},\\zeta ]$ gives the probability for there to be a measurement at $\\lbrace \\eta ,{\\mathbf {x}}\\rbrace $ , which depends on $\\zeta $ .",
"We have set up the calculation so that the worldline defining the fat geodesic measure is comoving and passes through the coordinate ${\\mathbf {x}}=0$ at some time $\\eta _0$ near the end of inflation.",
"Nevertheless, this worldline will subsequently be drawn toward over-densities.",
"Therefore, the inner integral in (REF ) is not centered at ${\\mathbf {x}}=0$ , but instead follows some trajectory that is correlated with the particular perturbation $\\zeta $ through which the worldline evolves.",
"Insofar as an initially uniform distribution of comoving worldlines is eventually distributed according to the cold dark matter density, we can account for this effect by including a factor of the matter density $\\rho _{\\rm m}[\\eta ,{\\mathbf {x}},\\zeta ]$ in ${\\cal A}$ , after which we can again take the defining worldline to be centered at ${\\mathbf {x}}=0$ .",
"Note that $\\rho _{\\rm m}$ is a gauge-dependent quantity.",
"Since we are conditioning on a given average CMB temperature $T_{\\rm obs}$ , we take $\\rho _{\\rm m}$ to be specified with respect to a foliation on which hypersurfaces of fixed $\\eta $ see fixed average CMB temperature.",
"Meanwhile, we take the probability for there to be a measurement to be proportional to the matter density, so that ${\\cal A}$ contains an additional factor of $\\rho _{\\rm m}$ .",
"For simplicity we do not distinguish between cold dark matter and baryonic matter.",
"Meanwhile, the physical three-volume within the fat geodesic is intended to be very small—in particular much smaller than the scales of non-linear structure formation—which means the integrals and delta functions of (REF ) essentially select for fat geodesics for which $z[\\eta _{\\rm obs}(0),0,\\zeta ]=\\tilde{z}$ .",
"Putting all of this together, we write $P(E_{z\\rightarrow \\tilde{z}}) \\propto \\int [d\\zeta ]\\,|\\langle \\zeta |0\\rangle |^2\\,\\rho _{\\rm m}^2\\big [\\eta _{\\rm obs}(0),0,\\zeta \\big ]\\,\\delta \\big \\lbrace z\\big [\\eta _{\\rm obs}(0),0,\\zeta \\big ]-\\tilde{z}\\big \\rbrace \\,.$ We interpret $P(E_{z\\rightarrow \\tilde{z}})$ as the probability distribution for measured values of the observable $z$ .",
"Accordingly, we write the (semi-classical) expectation value $\\langle z\\rangle _{\\rm FG} = \\frac{1}{N}\\int d\\tilde{z}\\,\\tilde{z}\\,P(E_{z\\rightarrow \\tilde{z}}) \\,,$ where $N\\equiv \\int d\\tilde{z}\\,P(E_{z\\rightarrow \\tilde{z}}) \\,.$ While the above analysis is semi-classical, it is illuminating to express the result in a more quantum-mechanical language.",
"Consider the promotion of the classical observables $z$ and $\\rho _{\\rm m}$ to quantum operators $\\hat{z}$ and $\\hat{\\rho }_{\\rm m}$ (on the effective Fock space $\\lbrace |\\zeta \\rangle \\rbrace $ ), according to replacing their dependence on $\\zeta $ with a dependence on $\\hat{\\zeta }\\equiv |\\zeta \\rangle \\langle \\zeta |\\,\\zeta $ .",
"Then we can write $\\langle z\\rangle ^{\\phantom{^{\\prime }}}_{\\rm FG} &=& \\frac{1}{N} \\int d\\tilde{z}\\int [d\\zeta ]\\,\\delta \\big \\lbrace z\\big [\\eta _{\\rm obs}(0),0,\\zeta \\big ]-\\tilde{z}\\big \\rbrace \\,\\langle \\zeta |0\\rangle \\langle 0|\\hat{\\rho }_{\\rm m}^2\\hat{z}|\\zeta \\rangle \\quad \\,\\,\\\\&=& \\frac{1}{N}\\,\\langle 0|\\hat{\\rho }_{\\rm m}^2\\hat{z}|0\\rangle \\,,$ where now $N=\\langle 0|\\hat{\\rho }_{\\rm m}^2|0\\rangle $ .",
"We emphasize that we have not actually solved a quantum-mechanical problem; we have merely found an effective Fock space $\\lbrace |\\zeta \\rangle \\rbrace $ on which certain operators reproduce the predictions of the fat geodesic measure in a semi-classical picture of the multiverse.",
"The prediction is apparently different than that of the standard approach.",
"In the standard formulation, the expectation value for $z$ is $\\langle z\\rangle ^{\\phantom{^{\\prime }}}_{\\rm SM}=\\langle 0|\\hat{z}|0\\rangle $ .",
"The result (REF ) differs because $\\rho _{\\rm m}$ depends perturbatively on $\\zeta $ , whose statistics are being assessed by $z$ .",
"Essentially, the fat geodesic measure enhances the probability to observe curvature perturbations $\\zeta $ that increase the matter density at the origin (the location of the defining worldline), as this increases the likelihood that the fat geodesic includes a measurement of $\\zeta $ .",
"There is both an anthropic effect, as we assume measurements are performed in proportion to the matter density, and what might be called a measure selection effect, due to the tendency of over-densities to attract the fat geodesic.",
"Of course, the correlation between $\\rho _{\\rm m}[\\eta _{\\rm obs}(0),0,\\zeta ]$ and $z[\\eta _{\\rm obs}(0),0,\\zeta ]$ is very small.",
"Our goal is merely to identify the main effect of the measure.",
"The density of events $E_{z\\rightarrow \\tilde{z}}$ in the causal patch measure is $\\overline{\\cal N}_{\\!\\zeta }(E_{z\\rightarrow \\tilde{z}})\\propto \\int d\\eta \\int _{{\\rm CP}[\\zeta ]}\\!\\sqrt{-g[\\eta ,{\\mathbf {x}},\\zeta ]}\\,d{\\mathbf {x}}\\,\\delta \\big [\\eta -\\eta _{\\rm obs}({\\mathbf {x}})\\big ]\\,\\delta \\big \\lbrace z\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ]-\\tilde{z}\\big \\rbrace \\,{\\cal A}\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ] \\,,$ where the inner integral is understood to run over the three-volume in the causal patch at time $\\eta $ (which we emphasize is a functional of the curvature perturbation $\\zeta $ ), and the terms in the integrand play the same roles as before.",
"The quantity $\\overline{\\cal N}_{\\!\\zeta }$ is superficially very similar to ${\\cal N}_\\zeta $ , but the integration over the causal patch—which is comparable to the Hubble volume in size—as opposed to an integration over the fat geodesic—which is much smaller than the scales of non-linear structures—creates a crucial difference.",
"The argument concerning ${\\cal A}$ for the fat geodesic measure applies here, except while the defining worldline is boosted to ${\\mathbf {x}}=0$ , any given observer is not.",
"Therefore ${\\cal A}[\\eta ,{\\mathbf {x}},\\zeta ] \\propto \\rho _{\\rm m}\\big [\\eta ,0,\\zeta \\big ]\\,\\rho _{\\rm m}\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ] \\,.$ As before, it is illuminating to promote the various quantities to quantum operators, but here we should be mindful of the dependence on ${\\mathbf {x}}$ .",
"For the moment we simply write $\\hat{Z} \\equiv \\int d\\eta \\int _{{\\rm CP}[\\zeta ]}\\!\\sqrt{-g[\\eta ,{\\mathbf {x}},\\zeta ]}\\, d{\\mathbf {x}}\\,\\delta \\big [\\eta -\\eta _{\\rm obs}({\\mathbf {x}})\\big ]\\,{\\cal A}\\big [\\eta ,{\\mathbf {x}},\\zeta \\big ]\\,z\\big [\\eta _{\\rm obs}({\\mathbf {x}}),{\\mathbf {x}},\\zeta \\big ]\\Big |_{\\zeta \\rightarrow \\hat{\\zeta }}\\,\\,.",
"$ Then, following the steps outlined above for the fat geodesic measure, we obtain $\\langle z\\rangle ^{\\phantom{^{\\prime }}}_{\\rm CP} = \\frac{1}{\\overline{N}}\\int [d\\zeta ]\\, \\langle \\zeta |0\\rangle \\langle 0|\\hat{Z}|\\zeta \\rangle = \\frac{1}{\\overline{N}}\\,\\langle 0|\\hat{Z}|0\\rangle \\,,$ where the normalization factor is now $\\overline{N}=\\langle 0|\\hat{Z}|_{z\\rightarrow 1}|0\\rangle $ .",
"We reiterate that we have not actually solved a quantum-mechanical problem; we have merely identified certain operators that, when applied to the effective Fock space, reproduce the predictions of the causal patch measure in a semi-classical picture of the multiverse.",
"Note that the expectation value (REF ) for the causal patch measure is different than the expectation value (REF ) for the fat geodesic measure.",
"In particular, the integrations over the causal patches in $\\hat{Z}$ and $\\hat{Z}|_{z\\rightarrow 1}$ do not factor out and cancel in the ratio of (REF ), in part because the volume of the causal patch depends on $\\zeta $ and in part because ${\\cal A}$ and $z$ depend on the position in the causal patch.",
"The fat geodesic measure assigns a higher weight to curvature perturbations $\\zeta $ that increase the matter density at the origin, as this increases the likelihood for the fat geodesic to enclose a measurement of $\\zeta $ , while the causal patch measure in addition assigns a higher weight to curvature perturbations that generate a larger causal patch, as this encloses more measurements elsewhere in the causal patch.",
"Granted, in each case the measure dependence on $\\zeta $ is perturbative in $\\zeta $ , so to a good approximation the two measures make the same inflationary predictions.",
"Our goal is merely to identify the leading order measure dependence."
],
[
"FRW cosmology",
"We continue to adopt the assumptions and approximations of Section REF .",
"Correspondingly, we take the observable universe to be contained in a CDL bubble.",
"On small comoving scales the line element has negligible spatial curvature, and at leading order can be written $ds^2 = a^2(\\eta )\\left( -d\\eta ^2 + d{\\mathbf {x}}^2 \\right) ,$ where $d{\\mathbf {x}}^2$ is the three-dimensional Cartesian line element.",
"In this section we describe the FRW cosmology.",
"Most of the details are unimportant, and our primary interest is to establish some notation and understand the relative sizes of important comoving scales.",
"For the purpose of understanding the background geometry, we model inflation as essentially vacuum-energy domination, and denote the (constant) Hubble rate $H_{\\rm inf}$ .",
"After several $e$ -folds of inflation, the scale factor can be written $a(\\eta ) = -\\frac{1}{H_{\\rm inf}\\,(\\eta +\\eta _\\infty )} \\,,$ where $\\eta _\\infty $ is an integration constant, which we set so that $\\eta $ asymptotically approaches zero (from below) during the late-time cosmological-constant dominated phase in our bubble.",
"We adopt this convention so that the comoving radius of the causal patch is given simply by $|\\eta |$ .",
"Note that during the early-time period of inflation, $\\eta $ asymptotes toward $-\\eta _\\infty $ .",
"We assume that inflation gives way directly to a period of radiation domination, followed by matter domination, followed by cosmological-constant domination.",
"In each case we approximate the transition as instantaneous, solving the FRW field equations on each side of the transition by assuming the energy density behaves as a perfect fluid with the appropriate equation of state: $w\\equiv p/\\rho =1/3$ during radiation domination, $w=0$ during matter domination, and $w=-1$ during cosmological-constant domination (here $p$ is the pressure and $\\rho $ is the energy density).",
"The integration constants are set so as to make the scale factor and its first time derivative continuous across the transition.",
"The details are exactly the same as in [39], and below we simply summarize the relevant results.",
"Figure: Zoom-in on a conformal diagram illustrating the local cosmology in our bubble (to scale).",
"The horizontal dotted line indicates the time of last scattering; the periods of slow-roll inflation and cosmological-constant domination are shaded gray.",
"The light dashed lines indicate the comoving sizes of the boundary of the causal patch and the past lightcone of an observer at time η obs =-(1/3)η ∞ \\eta _{\\rm obs}=-(1/3)\\,\\eta _\\infty .",
"The thick dashed line is the comoving size of the apparent horizon.Figure REF illustrates the important cosmological transitions as a function of $\\eta $ .",
"At the end of inflation / onset of radiation domination, $\\eta $ is exponentially close to $-\\eta _\\infty $ .",
"The fractional change in $\\eta $ during radiation domination is negligible compared to the change in $\\eta $ during the subsequent matter domination, and by the end of matter domination we have $\\eta =-(1/3)\\,\\eta _\\infty $ .",
"(In writing this and subsequent results, we have solved for the conformal-time evolution all the way to future infinity, so as to determine $\\eta _\\infty $ in terms of the other model parameters, and then have used that result to express the quantities of interest in terms of $\\eta _\\infty $ , as opposed to in terms of the other model parameters.)",
"Surfaces of constant average CMB temperature correspond to surfaces of constant $\\eta $ , and to avoid a proliferation of scales we assume that measurements are performed at matter / cosmological-constant equality, $\\eta _{\\rm obs} = -(1/3)\\,\\eta _\\infty \\,.$ As for the scale-factor evolution of this toy model, it is only important to note the behavior after cosmological-constant domination, $a(\\eta ) = -\\frac{1}{H_\\Lambda \\eta } \\,, \\qquad $ where $H_\\Lambda $ is the (constant) Hubble rate during cosmological-constant domination.",
"The `integration constant' $\\eta _\\infty $ sets the scale for all of the important comoving distances in this cosmology.",
"While our results do not depend on the actual size of $\\eta _\\infty $ , we note that it can be written $\\eta _\\infty = 3\\sqrt{\\Omega _{\\rm c}(\\eta _{\\rm obs})}$ , where $\\Omega _{\\rm c}$ is the usual curvature parameter.",
"The comoving radius of the causal patch at the time of observation, $r_{\\!\\rm cp}$ , is simply given by the magnitude of the conformal time at the time of observation, that is $r_{\\!\\rm cp}= |\\eta _{\\rm obs}|=(1/3)\\,\\eta _\\infty \\,.$ We also take interest in the comoving size of the surface of last scattering $r_{\\!\\rm cmb}$ , which is given by the comoving size of the observer's past lightcone at the time of last scattering.",
"Since the change in conformal time between the end of inflation and the time of last scattering is very small compared to $\\eta _\\infty $ , this can be approximated $r_{\\!\\rm cmb}= (2/3)\\,\\eta _\\infty \\,.$ Note that $r_{\\!\\rm cmb}$ is of the same order as $r_{\\!\\rm cp}$ .",
"Finally, we also refer to the comoving size of the apparent horizon at matter / radiation equality, which we denote $r_{\\!\\rm eq}$ .",
"For our purposes, it is only important that $r_{\\!\\rm eq}$ is very small compared to $r_{\\!\\rm cp}$ , but to be more precise we write $r_{\\!\\rm eq}= \\big [(2/27)\\,H_\\Lambda \\tau _{\\rm eq}\\big ]^{1/3}\\,\\eta _\\infty \\,,$ where $\\tau _{\\rm eq}$ denotes the proper time (defined with respect to the time of reheating) of matter / radiation equality.",
"The relative size of $r_{\\!\\rm eq}$ and $r_{\\!\\rm cp}$ is at the percent level."
],
[
"Cosmological perturbation theory",
"We assume that metric perturbations are sourced entirely by a single inflaton field, we ignore tensor perturbations, and we focus on scales that are small compared to the curvature radius of our CDL bubble.",
"The most general line element can then be written [13] $ds^2 &=& a^2(\\eta ) \\Big \\lbrace -\\!\\big [1+2\\phi (\\eta ,{\\mathbf {x}})\\big ] d\\eta ^2+2\\,\\partial _i B(\\eta ,{\\mathbf {x}})\\,d\\eta \\,dx^i+ \\big [1-2\\psi (\\eta ,{\\mathbf {x}})\\,\\delta _{ij} \\big .",
"\\Big .\\nonumber \\\\& & + \\Big .",
"\\big .",
"\\,2\\,\\partial _i\\partial _j E(\\eta ,{\\mathbf {x}}) \\big ] dx^i\\,dx^j \\Big \\rbrace \\,.$ Among $\\phi $ , $B$ , $\\psi $ , $E$ , and the inflaton perturbation $\\delta \\varphi $ , there is actually only one scalar degree of freedom; the others are related to diffeomorphism (gauge) invariance.",
"In particular, under the generic `scalar' infinitesimal coordinate transformation $\\eta \\rightarrow \\eta +\\sigma (\\eta ,{\\mathbf {x}}) \\,, \\quad {\\mathbf {x}}\\rightarrow {\\mathbf {x}}+\\mathbf {\\partial }\\xi (\\eta ,{\\mathbf {x}}) \\,,$ the metric perturbations transform according to $\\phi \\rightarrow \\phi -\\dot{\\sigma }-{\\cal H}\\sigma \\,, \\quad \\,B\\rightarrow B+\\sigma -\\dot{\\xi }\\,, \\quad \\,\\psi \\rightarrow \\psi +{\\cal H}\\sigma \\,, \\quad \\,E\\rightarrow E-\\xi \\,,$ where ${\\cal H}\\equiv \\dot{a}/a$ , and dots denote derivatives with respect to $\\eta $ .",
"To help transform from one gauge to another, one can refer to the gauge-invariant variables $\\begin{array}{ll}\\Phi \\equiv \\phi + {\\cal H}\\big (B-\\dot{E}\\big )+\\dot{B}-\\ddot{E}\\phantom{\\Big [\\Big ]} \\quad \\,\\, &\\Psi \\equiv \\psi - {\\cal H}\\big (B-\\dot{E}\\big )\\phantom{\\Big [\\Big ]} \\\\\\Upsilon \\equiv \\delta \\varphi + \\dot{\\varphi }_0 \\big (B-\\dot{E}\\big )\\phantom{\\Big [\\Big ]} &\\Delta \\equiv \\delta \\rho + \\dot{\\rho }_0 \\big (B-\\dot{E}\\big )\\,, \\phantom{\\Big [\\Big ]}\\end{array}$ where $\\varphi _0$ is the homogeneous component of the inflaton field and $\\delta \\rho $ represents the perturbation in a more generic energy density, $\\rho _0$ being the homogeneous component.",
"We henceforth use the gauge freedom associated with ${\\mathbf {x}}$ translation to set $E=0$ .",
"The Einstein field equations then give the constraint $\\Phi =\\Psi \\,,$ and, on scales larger than the apparent horizon, $3{\\cal H}^2\\Psi + 3{\\cal H}\\dot{\\Psi } &=& -4\\pi G\\Big [\\dot{\\varphi }_0\\Upsilon +a^2(dV/d\\varphi )\\Upsilon -\\dot{\\varphi }_0^2\\Psi \\Big ] \\\\3{\\cal H}^2\\Psi + 3{\\cal H}\\dot{\\Psi } &=& -4\\pi G\\,a^2\\Delta \\phantom{\\Big [\\Big ]} \\,.$ As we have remarked, among the (scalar) perturbations there is one degree of freedom.",
"A careful exposition of the self-interactions of this degree of freedom is given in [50], which works in a gauge defined by $\\delta \\varphi =E=0$ .",
"The scalar degree of freedom can be written in terms of gauge-invariant variables according to $\\zeta = -\\Psi + \\frac{{\\cal H}^2}{{\\cal H}^2-\\dot{{\\cal H}}}\\big (\\Phi -\\dot{\\Psi }) \\,.$ The leading-order action for $\\zeta $ is that of a free field in de Sitter space, in particular $S = \\frac{1}{2}\\int d\\eta \\,d{\\mathbf {x}}\\,\\frac{a\\,\\dot{\\varphi }^2_0}{{\\cal H}^2}\\Big [ \\dot{\\zeta }^2 -(\\mathbf {\\partial }\\zeta )\\!\\cdot \\!\\mathbf {\\partial }\\zeta \\Big ] \\,.$ We use $\\zeta $ to establish the quantum theory of the effective Fock space of Section .",
"To begin, we perform a mode expansion of $\\zeta $ .",
"We refer to both Cartesian mode functions, $\\zeta _{\\mathbf {k}}&\\equiv & \\int d{\\mathbf {x}}\\, e^{-i{\\mathbf {k}}\\cdot {\\mathbf {x}}}\\,\\zeta (\\eta _0,{\\mathbf {x}})= v_k(\\eta _0)\\,a_{\\mathbf {k}}+v^*_k(\\eta _0)\\,a^*_{-{\\mathbf {k}}} \\,,$ and spherically symmetric mode functions $\\zeta _{k\\ell m} &\\equiv & \\int d\\Omega _2 \\int r^2dr\\,\\sqrt{2/\\pi }\\,k\\,j_\\ell (kr)\\, Y^m_\\ell (\\Omega _2)\\,\\zeta (\\eta _0,{\\mathbf {x}}) \\\\&=& v_k(\\eta _0)\\,a_{k\\ell m} + v^*_k(\\eta _0)\\,a^*_{k\\ell m} \\,.\\phantom{T^{T^{T^{T^T}}}} $ In each case the modes are evaluated at some reference time $\\eta =\\eta _0$ near the end of inflation.",
"The $v_k(\\eta )$ are taken to satisfy the equation of motion (with $\\mathbf {\\partial }\\!\\cdot \\!\\mathbf {\\partial }\\rightarrow -k^2$ ), given Bunch-Davies boundary conditions, with their normalization set so that the Fourier coefficients $a_{\\mathbf {k}}$ and $a^*_{-{\\mathbf {k}}}$ ($a_{k\\ell m}$ and $a^*_{k\\ell m}$ ) satisfy the usual ladder-operator commutation relations when promoted to quantum operators.",
"We use the real basis of the spherical harmonics $Y^m_\\ell $ ($\\Omega _2$ represents the angular coordinates on the 2-sphere), and the $j_\\ell $ are spherical Bessel functions.",
"The classical perturbation $\\zeta $ is promoted to a quantum operator $\\hat{\\zeta }$ by replacing the classical Fourier coefficients $a^{\\phantom{*}}_{\\mathbf {k}}$ and $a^*_{-{\\mathbf {k}}}$ ($a^{\\phantom{*}}_{k\\ell m}$ and $a^*_{k\\ell m}$ ) with the ladder operators $\\hat{a}^{\\phantom{*}}_{\\mathbf {k}}$ and $\\hat{a}^\\dagger _{-{\\mathbf {k}}}$ ($\\hat{a}^{\\phantom{*}}_{k\\ell m}$ and $\\hat{a}^\\dagger _{k\\ell m}$ ).",
"We promote the corresponding mode functions $\\zeta _{\\mathbf {k}}$ and $\\zeta _{k\\ell m}$ similarly.",
"As a matter of shorthand, we also write other perturbative quantities as quantum operators, for instance the gauge-invariant perturbation $\\hat{\\Psi }$ or its Fourier mode $\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}$ .",
"It should be understood that these quantities are defined in relation to $\\hat{\\zeta }$ and $\\hat{\\zeta }_{\\mathbf {k}}$ , via multiplicative factors deduced from the gauge transformations and constraint equations given above.",
"Although we compute sub-leading effects coming from the fat geodesic and causal patch measures, our calculus does not require using the sub-leading interaction terms in the action to compute them.",
"Therefore we only need refer to leading-order description outlined above.",
"Of course, the effects of sub-leading interaction terms compete with the corrections we compute; however these can be found elsewhere in the literature (see for example [50]).",
"We now briefly describe the evolution of these perturbations.",
"It is convenient to do this in longitudinal gauge—defined by setting $B=E=0$ —both because in this gauge the metric perturbations are equal to gauge-invariant quantities, $\\phi =\\Phi =\\psi =\\Psi $ , and because it allows us to draw upon textbook results (see for example [51]).",
"On scales larger than the apparent horizon, $\\Psi $ is roughly constant with respect to time during radiation and matter domination.",
"Indeed, during matter domination $\\Psi $ is approximately constant with respect to time even on scales within the apparent horizon.",
"During radiation domination, $\\Psi $ decays on scales within the apparent horizon.",
"The net effect is usually modeled by working in Fourier space and by introducing a transfer function $T(k)$ , whereby $\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _{\\rm obs}) \\approx \\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _{\\rm eq})= T(k)\\,\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _0) \\,,$ with $T(k) \\approx \\left\\lbrace \\begin{array}{ll}\\displaystyle \\,\\, 1 \\phantom{T_{T_{T_{T}}}} \\qquad & {\\rm for}\\,\\, k\\lesssim 1/r_{\\!\\rm eq}\\\\\\displaystyle \\,\\, \\frac{\\ln (kr_{\\!\\rm eq})}{k^2r_{\\!\\rm eq}^2} \\qquad & {\\rm for}\\,\\, k\\gg 1/r_{\\!\\rm eq}\\,\\,, \\\\\\end{array}\\right.$ where $\\eta _{\\rm eq}$ denotes the time of matter / radiation equality and we ignore factors of order unity.",
"That is, until cosmological-constant domination $\\Psi $ is roughly unchanged from its primordial value all the way down to the comoving scale $r_{\\!\\rm eq}$ , below which it decreases with decreasing scale.",
"During cosmological-constant domination, $\\Psi $ decays with time on all scales obeying a linearized analysis.",
"Solving (REF ) using results from the toy model of Section REF , we find $\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta ) = \\frac{\\eta }{\\eta _{\\rm obs}}\\,\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _{\\rm obs}) \\qquad \\,\\,{\\rm for}\\,\\, \\eta \\ge \\eta _{\\rm obs} \\,.$ Our cosmological model takes the density perturbation to be adiabatic.",
"Thus, on scales larger than the apparent horizon, the total density contrast $\\delta \\rho /\\rho $ is equal to the radiation density contrast $\\delta \\rho _{\\rm rad}/\\rho _{\\rm rad}$ which is equal to the matter density contrast $\\delta \\equiv \\delta \\rho _{\\rm m}/\\rho _{\\rm m}$ .",
"These quantities are constrained by (REF ).",
"Accordingly, on scales larger than the apparent horizon and at times below the onset of cosmological constant domination, we have $\\delta ^{\\mbox{\\tiny L}}= -2\\Psi \\qquad \\,\\, \\mbox{(beyond the apparent horizon)}\\,,$ and likewise for $\\delta \\rho _{\\rm rad}/\\rho _{\\rm rad}$ .",
"We have added the superscript to $\\delta ^{\\mbox{\\tiny L}}$ to avoid confusion with the density contrast computed in another gauge, given below.",
"It can be shown, by studying the divergencelessness of the stress-energy tensor, that these results continue to hold after cosmological-constant domination, at least on scales larger than the apparent horizon.",
"On scales within the apparent horizon, $\\delta ^{\\mbox{\\tiny L}}$ grows with time.",
"During radiation domination this growth is logarithmic with respect to the growth of the scale factor, while during matter domination the growth is linear with respect to the growth of the scale factor.",
"The net effect can be modeled in Fourier space, giving $\\delta ^{\\mbox{\\tiny L}}_{\\mathbf {k}}(\\eta _{\\rm obs}) =-\\left(2+\\gamma k^2\\eta _{\\rm obs}^2\\right)T(k)\\,\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _0) \\,.$ On scales larger than the apparent horizon—scales corresponding to $k\\gtrsim r_{\\!\\rm cp}=|\\eta _{\\rm obs}|$ at the time $\\eta _{\\rm obs}$ —the matter contrast is given by its primordial value, reflected by the first term in (REF ).",
"On scales within the apparent horizon, $\\delta ^{\\mbox{\\tiny L}}$ is enhanced by the growth of modes after they enter the horizon, the net effect being modeled by the factor $\\gamma k^2\\eta ^2_{\\rm obs}$ .",
"We have included an order-unity factor $\\gamma $ to balance the different levels of precision entering the calculations of these two terms.",
"The sub-horizon evolution of $\\delta ^{\\mbox{\\tiny L}}$ after cosmological-constant domination does not enter into our calculations.",
"Finally, we return to the radiation contrast $\\delta \\rho _{\\rm rad}/\\rho _{\\rm rad}$ .",
"Its sub-horizon evolution is more complicated than that of $\\Psi $ and $\\delta ^{\\mbox{\\tiny L}}$ , however for our analysis is only important to note that it does not grow—aside from an order-unity enhancement on scales corresponding to the first few acoustic peaks in the CMB—and indeed on sufficiently small scales (scales well below $r_{\\!\\rm eq}$ at the time of last scattering), it decays, due to Silk damping."
],
[
"Conditioning on the average CMB temperature",
"In Section , we conditioned predictions using a given average CMB temperature $T_{\\rm obs}$ .",
"This condition is automatically satisfied on a constant-time hypersurface in a gauge that sets the radiation perturbation to zero, $\\delta \\rho _{\\rm rad}=0$ .",
"To keep the analysis simple, we set the gauge so as to satisfy this condition on scales larger than the apparent horizon, and then take this as an approximation for satisfying the condition on all scales.",
"The accuracy of this approximation is enhanced by the fact that the average CMB temperature is dominated by scales larger than the apparent horizon at last scattering—this because last scattering occurs soon after matter / radiation equality, and $r_{\\!\\rm cmb}\\gg r_{\\!\\rm eq}$ —while the sum variance on smaller scales is suppressed, due to statistical regression to the mean.",
"To find the appropriate gauge, first note that on scales larger than the apparent horizon, the radiation perturbation is set to zero when the total density perturbation $\\delta \\rho $ is set to zero.",
"Since the quantity $\\Delta $ in (REF ) is gauge-invariant, $\\delta \\rho $ is set to zero by shifting $B$ by an amount $\\delta \\rho /\\dot{\\rho }_0$ .",
"The shift in $B$ is accomplished by an appropriate redefinition of the time parameter, according to (REF ) and (REF ).",
"For example, starting in longitudinal gauge, we use $\\sigma = \\frac{\\delta \\rho }{\\dot{\\rho }_0} =-\\frac{\\delta ^{\\mbox{\\tiny L}}}{3{\\cal H}} = \\frac{2}{3}\\frac{\\Psi }{{\\cal H}}\\,,$ where the the last two expressions incorporate our interest in setting the condition on scales larger than the apparent horizon, during and after matter domination.",
"This gives $\\phi = \\bigg (\\frac{1}{3}+\\frac{2}{3}\\frac{\\dot{{\\cal H}}}{{\\cal H}^2}\\bigg )\\Psi -\\frac{2}{3}\\frac{\\dot{\\Psi }}{{\\cal H}} \\,,\\qquad B = \\frac{2}{3}\\frac{\\Psi }{{\\cal H}} \\,,\\qquad \\psi = \\frac{5}{3}\\,\\Psi \\,,\\qquad E = 0 \\,.",
"\\,\\,$ While the gauge choice (REF ) sets the matter perturbation to zero on scales larger than the apparent horizon, within the apparent horizon the matter perturbation grows relative to its primordial value and is therefore non-zero.",
"Given the matter contrast in longitudinal gauge, $\\delta ^{\\mbox{\\tiny L}}$ , we solve for the density contrast in the $\\delta \\rho _{\\rm rad}=0$ gauge by using the gauge invariance of $\\Delta $ .",
"This gives $\\delta _{\\mathbf {k}}(\\eta _{\\rm obs}) = -\\gamma k^2\\eta _{\\rm obs}^2\\,T(k)\\,\\Psi _{\\!",
"{\\mathbf {k}}}(\\eta _0) \\,.$ Note that while our calculation of $\\delta _{\\mathbf {k}}(\\eta _{\\rm obs})$ is simplified by the approximate manner in which we impose the gauge condition $\\delta \\rho _{\\rm rad}=0$ , it is not non-zero simply as a consequence of this approximation.",
"In particular, the radiation perturbation does not significantly increase on scales within the apparent horizon, while the matter perturbation does.",
"Clearly, (REF ) arises entirely as a consequence of the relative growth of the matter contrast.",
"We emphasize that we use the gauge choice (REF ) because in this gauge, hypersurfaces of constant time correspond to surfaces of constant-average-CMB temperature (at our level of approximation).",
"Insofar as our results depend on gauge-dependent quantities—such as the matter contrast or the three-volume on a fixed-time slice in the causal patch—it is because our calculations depend on the conditional data—the average CMB temperature $T_{\\rm obs}$ —and not because of residual gauge dependence.",
"That is, we could perform the calculations in any gauge and get the same results, it is just much simpler to work in the gauge (REF ), which makes our conditioning data manifest on constant-time slices.",
"It is not unexpected that predictions depend on the conditioning data, but this raises the question of what is the correct data to condition on.",
"To make the most accurate predictions, one should condition on all data of which one is aware.",
"Yet, as a matter of expediency, or to explain data that one already possesses, one may wish to condition on less [52].",
"This is the context of our work.",
"When conditioning on less, it is important to remember that predictions of probabilistic outcomes imply a notion of typicality—that is, it is only meaningful to compare a measured distribution to the predicted distribution if the hypothesis asserts that the measurements are drawn randomly from the predicted distribution—and so the conditional data should be sufficient to select an ensemble from which our measurements can be considered randomly drawn.",
"While it only increases the accuracy of our prediction to condition on a given average CMB temperature $T_{\\rm obs}$ , we do not condition on our residing in a galaxy much like the Milky Way (we simply take the number of observers to be proportional to the matter density), which is another important datum.",
"One could in principle condition on both, but this would be technically challenging and is beyond the scope of this paper.",
"Finally, it is worth noting that while these considerations are important for the physical interpretation of our calculations, the actual numerical order of magnitude of our results is insensitive to the precise details of the conditioning data that is employed."
],
[
"Correlation functions in the fat geodesic measure",
"Section demonstrates that the fat geodesic and the causal patch measures in general predict different values for inflationary observables.",
"Furthermore, these predictions differ from those of the standard formulation of inflationary predictions.",
"The differences are apparently small.",
"To better appreciate the size and form of the measure selection effects, we here compute expectation values for correlation functions of the Fourier components of the gauge-invariant Newtonian potential $\\Psi $ .",
"While these are not actually physical observables—in the context of CMB measurements, the physical observables relate to temperature variations with respect to position in the sky of the temperature of photons, which arrive at a local detector after free-streaming from the surface of last scattering—there is nothing new in converting the correlation functions of $\\Psi $ into a spectrum of temperature fluctuations on a local sky (see for example [41]).",
"Therefore, we set these complications aside.",
"The analysis of Section is straightforward to apply to the fat geodesic measure.",
"To begin, consider the expectation value of the 2-point correlation function of the Fourier modes $\\Psi _{\\!",
"{\\mathbf {k}}}$ .",
"According to the result (REF ), this can be written $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\rangle _{\\rm obs} &=&\\frac{\\langle 0|\\overline{\\rho }_{\\rm m}^2(\\eta _{\\rm obs})\\big [1+\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]^2\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}\\,\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}^{\\prime }}|0\\rangle }{\\langle 0|\\overline{\\rho }_{\\rm m}^2(\\eta _{\\rm obs})\\big [1+\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]^2|0\\rangle } \\\\&=& \\frac{\\langle 0|\\big [1+2\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}\\,\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}^{\\prime }}|0\\rangle }{\\langle 0|\\big [1+2\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]|0\\rangle } \\,,$ where $\\overline{\\rho }_{\\rm m}$ is the homogeneous component of the matter density, which cancels between the numerator and denominator, and we have expanded perturbatively in the matter contrast $\\delta $ .",
"The expression still looks complicated but, given the ladder-operator algebra, these quantities are easy to compute.",
"Only even factors of field insertions survive the expectation value in the Bunch-Davies vacuum, so the normalization factor can always be ignored (after accounting for the factor of $\\overline{\\rho }^2_{\\rm m}$ ).",
"In the case of the 2-point function, the term involving $\\hat{\\delta }$ can also be ignored, and we recover the standard prediction: $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\rangle _{\\rm obs} = P(k)\\,(2\\pi )^3\\delta ({\\mathbf {k}}+{\\mathbf {k}}^{\\prime })\\,,$ where $P(k)\\propto T^2(k)\\,|v_k|^2$ is the power spectrum of $\\Psi $ .",
"Note that, to keep expressions simple, we include in $P(k)$ the effects of the transfer function on scales $k\\ll 1/r_{\\!\\rm eq}$ .",
"Aside from that, $P(k)$ is proportional to $1/k^3$ times a weak function of $k$ , which depends on the details of the inflationary cosmology (but not on the measure, given our conditioning assumptions).",
"Although the fat geodesic measure does not modify the inflationary prediction for the 2-point correlator, it is now clear that we should look at the 1-point and 3-point correlation functions.",
"Indeed, we find the 1-point expectation value $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\rangle _{\\rm obs} &=&\\langle 0|\\big [1+2\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}|0\\rangle \\phantom{\\Big [\\Big ]} \\\\&=& -2\\,\\langle 0|\\int \\frac{d{\\mathbf {q}}}{(2\\pi )^3}\\,\\gamma q^2\\eta _{\\rm obs}^2\\hat{\\Psi }_{\\!",
"{\\mathbf {q}}}\\,\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}|0\\rangle \\\\&=& -2\\gamma k^2\\eta _{\\rm obs}^2\\, P(k) \\,,\\phantom{\\Big [\\Big ]}$ where we have used (REF ) to express the matter contrast $\\delta $ in terms of $\\Psi $ , evaluating $\\delta $ in the gauge (REF ) to automatically incorporate the fixed-average-CMB-temperature condition.",
"The 1-point expectation value is usually zero by construction.",
"However, in the present case each perturbation $\\Psi $ is defined against its `mean' value (on the $\\eta =\\eta _0$ hypersurface in a given bubble) before the fat geodesic of the bubble is weighted according to the measure.",
"When we subsequently weight the fat geodesic of the perturbation $\\Psi $ according to the measure, we find the expectation value of $\\Psi $ is shifted relative to this `prior' mean.",
"If we were computing statistics of $\\Psi $ in position space, we might first subtract off the mean value of (the observed portion of) the perturbation $\\Psi $ .",
"Since we work in Fourier space, the spatially-averaged mean affects only the ${\\mathbf {k}}=0$ mode, which has already been discarded.",
"We are left with the scale-dependence of the shift, given by (REF ).",
"The 1-point expectation value (REF ) takes the form of a monopole.",
"This could have been guessed based on the (statistical) symmetries of the physical scenario, and is evinced by the fact that (REF ) depends only on the magnitude of ${\\mathbf {k}}$ .",
"It is manifest when we express the 1-point function in terms of the spherically symmetric mode functions, $\\langle \\Psi _{\\!k\\ell m}\\rangle _{\\rm obs} &=&-2\\,\\langle 0|\\int dq\\sum _{\\ell ^{\\prime },m^{\\prime }}\\sqrt{2/\\pi }\\,q\\,j_{\\ell ^{\\prime }}(0)\\,Y^{m^{\\prime }}_{\\ell ^{\\prime }}\\!",
"(0)\\,\\gamma q^2\\eta _{\\rm obs}^2\\hat{\\Psi }_{\\!q00}\\,\\hat{\\Psi }_{\\!k\\ell m}|0\\rangle \\\\&=& -\\big (\\sqrt{2}/\\pi \\big )\\,\\gamma k^3\\eta _{\\rm obs}^2 P(k)\\,\\delta _{\\ell 0}\\,\\delta _{m0} \\,,\\phantom{\\Big [\\Big ]}$ where we have input $j_\\ell (0)=\\delta _{\\ell 0}$ and $Y^0_0=(4\\pi )^{-1/2}$ .",
"The extra power of $k$ relative to (REF ) expresses the different measure on radial modes between Cartesian and spherically symmetric coordinates.",
"While the scale-dependence of a monopole mode is in principle detectable in the CMB—it affects the apparent scale of the acoustic peaks, for example—its imprints are subdominant.",
"As a qualitative guide, one can liken the observability of the monopole to the observability of spatial curvature in an open (FRW) cosmology.",
"To appreciate the size of the measure selection effect, first note that the scale of observable wavenumbers is set by the comoving size of the surface of last scattering, $r_{\\!\\rm cmb}$ , in particular observable scales correspond to $k\\gtrsim 1/r_{\\!\\rm cmb}$ .",
"Meanwhile, $|\\eta _{\\rm obs}|=(1/2)\\,r_{\\!\\rm cmb}$ .",
"Therefore, the factors of $k$ provides some enhancement of the monopole on scales of interest.",
"On the other hand, the size of the 1-point function is always below cosmic variance.",
"This can be seen by simply noting the size and scale dependence of a typical $\\Psi $ perturbation, $\\Psi _{\\!",
"{\\mathbf {k}}}\\sim \\big [P(k)\\big ]^{1/2}\\sim k^{-3/2}\\, \\big [k^3P(k)\\big ]^{1/2} \\,.$ Up to slow-roll corrections and a loss of power on comoving scales below $r_{\\!\\rm eq}$ , the factor in brackets in the second expression is dimensionless, independent of $k$ , and of order $10^{-5}$ .",
"Our result (REF ) differs by a factor of order $k^{1/2}\\big [k^3P(k)\\big ]^{1/2}$ .",
"The term in brackets provides an order $10^{-5}$ suppression.",
"To compensate for this suppression with the factor of $k^{1/2}$ would require examining extremely small scales, $kr_{\\!\\rm cmb}\\sim 10^{10}$ , well beyond the validity of our analysis and beyond any foreseeable observation.",
"Finally, we turn to the 3-point correlation function.",
"It is given by $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime \\prime }}\\rangle _{\\rm obs} =-2\\gamma k^2\\eta _{\\rm obs}^2P(k)P(k^{\\prime })\\,(2\\pi )^3\\delta ({\\mathbf {k}}^{\\prime }+{\\mathbf {k}}^{\\prime \\prime }) + {\\rm perms}\\,,$ or, in terms of spherical harmonics, $\\langle \\Psi _{\\!k\\ell m}\\,\\Psi _{\\!k^{\\prime }\\ell ^{\\prime }m^{\\prime }}\\,\\Psi _{\\!k^{\\prime \\prime }\\ell ^{\\prime \\prime }m^{\\prime \\prime }}\\rangle _{\\rm obs}&=& -\\big (\\sqrt{2}/\\pi \\big )\\,\\gamma k^3\\eta _{\\rm obs}^2 P(k)P(k^{\\prime })\\,\\delta (k^{\\prime }-k^{\\prime \\prime })\\,\\delta _{\\ell ^{\\prime }\\ell ^{\\prime \\prime }}\\,\\delta _{m^{\\prime }m^{\\prime \\prime }}\\,\\delta _{\\ell 0}\\,\\delta _{m0} \\qquad \\, \\nonumber \\\\& & +\\, {\\rm perms} \\,.\\phantom{T^{T^{T^T}}}$ Notice that in each term of the sum, one of the wavenumbers is unconstrained.",
"Usually, the 3-point function is proportional to a total-wavenumber-conserving delta function; for example in Cartesian coordinates there is a factor of $\\delta ({\\mathbf {k}}+{\\mathbf {k}}^{\\prime }+{\\mathbf {k}}^{\\prime \\prime })$ .",
"This factor arises because the usual 3-point correlator involves field insertions coming from an expansion in the local self-interaction terms of the sub-leading action, and the local interactions conserve momentum.",
"The 3-point correlator (REF ) comes from a non-local correlation between the matter contrast $\\delta $ and the perturbation mode $\\Psi _{\\!",
"{\\mathbf {k}}}$ —a consequence of the measure—therefore there is no reason to expect total wavenumber conservation.",
"The 3-point function is associated with non-Gaussianity.",
"This is usually parametrized in terms of a quantity $f_{\\rm NL}$ [53], which in terms of $\\Psi $ would roughly correspond to $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime \\prime }}\\rangle \\sim f_{\\rm NL}\\,\\big [ P(k)P(k^{\\prime }) + {\\rm perms.}",
"\\big ] \\,\\delta ({\\mathbf {k}}+{\\mathbf {k}}^{\\prime }+{\\mathbf {k}}^{\\prime \\prime })\\,.$ Since the source of non-Gaussianity from the measure selection effect is non-local, we cannot draw a direct connection to the parameter $f_{\\rm NL}$ .",
"Nevertheless, setting aside the delta functions, it appears as if the fat-geodesic measure predicts non-Gaussianity of a magnitude in rough correspondence with $f_{\\rm NL}\\sim \\gamma k^2r_{\\!\\rm cmb}^2$ .",
"This is actually larger than the standard slow-roll result [50], which does not feature the rising $k$ dependence and is also suppressed by a slow-roll parameter.",
"However, expanding our attention to include the delta functions, we see that the present non-Gaussianity arises from correlations involving the monopole; specifically one of the modes must have $\\ell =m=0$ .",
"This alone would seem to make it indiscernible in the CMB, where the effects of the monopole are indirect.",
"Yet, even in a map of perturbations with more three-dimensional resolution, we expect the 3-point correlator to fall within cosmic variance, since restricting to the single $\\ell =m=0$ mode does not allow one to reduce the uncertainty as is usually done by averaging over many $m$ at large $\\ell $ ."
],
[
"Observables and operators",
"The analysis of Section describes how to make predictions using the causal patch measure, however the main result (REF ) is not expressed in a form that is amenable to predicting the expectation values of correlation functions.",
"We here construct a more tractable expression for the operator $\\hat{Z}$ that appears in (REF ).",
"The operator $\\hat{Z}$ depends on the physical observable $z$ that one wishes to predict; we are interested in expectation values of correlation functions involving products of the Cartesian Fourier modes $\\Psi _{\\!",
"{\\mathbf {k}}}$ .",
"Note that these quantities are independent of the location in the causal patch at which they are measured.The Fourier modes $\\Psi _{\\!",
"{\\mathbf {k}}}$ are not directly observable, while the actual observables—the spherical harmonic coefficients $a_{\\ell m}$ of temperature variations in CMB photons—depend on the location at which they are observed.",
"However, the expectation values of products of $a_{\\ell m}$ can be written as functionals of the expectation values of products of $\\Psi _{\\!",
"{\\mathbf {k}}}$ , where the latter are computed as if the $\\Psi _{\\!",
"{\\mathbf {k}}}$ were observable.",
"The same applies here, except the expectation values of products of $\\Psi _{\\!",
"{\\mathbf {k}}}$ are computed with the measure, as opposed to as straight correlation functions.",
"In the end these are the quantities that we compute, and they are independent of the location at which they are measured.",
"This means that they factor out of the volume integral of $\\hat{Z}$ , which makes it worthwhile to study the operator $\\hat{Z}$ but with the various insertions of $\\Psi _{\\!",
"{\\mathbf {k}}}$ factored out—an operator that we denote $\\hat{Z}_0$ —inserting the factors of $\\Psi _{\\!",
"{\\mathbf {k}}}$ back into $\\hat{Z}$ nearer to the end of the calculation.",
"The volume integral of $\\hat{Z}_0$ integrates over both space and time, but with a delta function selecting for the time at which the average CMB temperature attains the given value $T_{\\rm obs}$ .",
"This delta function is automatically satisfied (at our level of approximation) on an appropriate fixed-time hypersurface in the gauge (REF ).",
"Therefore, in this gauge we can write $\\hat{Z}_0 \\propto \\int _{\\rm CP[\\Psi ]} \\sqrt{-g[\\eta _{\\rm obs},\\Psi ]}\\,d{\\mathbf {x}}\\,\\,\\overline{\\rho }_{\\rm m}^2(\\eta _{\\rm obs})\\big [1+\\hat{\\delta }(\\eta _{\\rm obs},0)\\big ]\\big [1+\\hat{\\delta }(\\eta _{\\rm obs},{\\mathbf {x}})\\big ] \\,.$ The rotational symmetry of the causal patch makes the spherically symmetric mode functions most convenient.",
"Expanding to linear order in perturbative quantities, we write $\\hat{Z}_0 &\\propto & \\int d\\Omega _2 \\int _0^{\\tilde{r}(\\Omega _2)}r^2dr\\,\\bigg [ 1 + \\int dq\\sum _{\\ell ,m}\\sqrt{2/\\pi }\\,q\\,j_\\ell (qr)\\,Y^m_\\ell (\\Omega _2)\\!\\left( \\hat{\\delta }_{q\\ell m} - 5\\hat{\\Psi }_{\\!q\\ell m}\\right)\\bigg .",
"\\nonumber \\\\& & +\\, \\bigg .",
"\\int dq\\sum _{\\ell ,m}\\sqrt{2/\\pi }\\,q\\,j_\\ell (0)\\,Y^m_\\ell (0)\\,\\hat{\\delta }_{q\\ell m} \\bigg ] \\,,$ where $\\tilde{r}$ is the comoving radius of the causal patch, which in the presence of metric perturbations depends on the angular coordinates.",
"We have dropped the factor of $a^3(\\eta _{\\rm obs})\\,\\overline{\\rho }_{\\rm m}^2(\\eta _{\\rm obs})$ and have expressed the metric perturbation $\\psi $ in terms of $\\Psi $ using (REF ), delaying the corresponding substitution with $\\delta $ to keep the expressions simpler.",
"The radial integration can be performed for all terms, but the expressions are rather complicated for some of them.",
"In the end, the details are not important, so we simply define $I_{q\\ell }(\\tilde{r}) \\equiv \\int _0^{\\tilde{r}} r^2dr\\,q^3\\,j_\\ell (qr) \\,.$ Performing the radial integration, we obtain $\\hat{Z}_0 &\\propto & \\int d\\Omega _2\\, \\bigg [ \\frac{1}{3}\\,\\tilde{r}^3(\\Omega _2) +\\int \\frac{dq}{q^2}\\sum _{\\ell ,m}\\sqrt{2/\\pi }\\,I_{q\\ell }\\big [\\tilde{r}(\\Omega _2)\\big ]\\,Y^m_\\ell (\\Omega _2)\\!\\left( \\hat{\\delta }_{q\\ell m} - 5\\hat{\\Psi }_{\\!q\\ell m}\\right)\\bigg .",
"\\nonumber \\\\*& & +\\,\\bigg .",
"\\frac{1}{3}\\,\\tilde{r}^3(\\Omega _2)\\int \\frac{q\\,dq}{\\sqrt{2}\\pi }\\,\\hat{\\delta }_{q00} \\bigg ] \\,.$ Note that while our analysis does not describe the perturbations $\\delta $ and $\\Psi $ down to arbitrarily small scales, we have nevertheless integrated over all scales in the causal patch.",
"Since the integrations were performed in Fourier space, the analysis is accurate so long as we do not re-sum the Fourier mode expansion and so long as we restrict attention to scales on which the perturbations $\\delta $ and $\\Psi $ are accurately described by the mode expansion.",
"Since in the end the observables we are interested in are statistics of Fourier components of $\\Psi $ , we only re-sum the modes on the scales corresponding to those Fourier components.",
"Meanwhile, although the description of the evolution of modes within the apparent horizon in Section REF is not precise, it is accurate at an order-of-magnitude level over scales well below $r_{\\!\\rm eq}$ , which is very small next to $r_{\\!\\rm cmb}$ , which is of order $r_{\\!\\rm cp}$ .",
"Therefore, there are plenty of modes on scales for which our integration over the causal patch is accurate.",
"The comoving distance to the boundary of the causal patch $\\tilde{r}$ is obtained by solving for the trajectory of a radial null ray backwards from future infinity.",
"The null condition is $-(1+2\\phi )\\,d\\eta ^2 + 2\\partial _rB\\,d\\eta \\,dr + (1-2\\psi )\\, dr^2 =0\\,.$ Since we integrate from future infinity to $\\eta _{\\rm obs}$ , we are interested in the behavior of the metric perturbations during cosmological-constant domination.",
"In the gauge (REF ), this gives $\\phi = \\frac{5}{3}\\frac{\\eta }{\\eta _{\\rm obs}}\\,\\Psi \\,,\\qquad B = -\\frac{2}{3}\\frac{\\eta ^2}{\\eta _{\\rm obs}}\\,\\Psi \\,, \\qquad \\psi = \\frac{5}{3}\\frac{\\eta }{\\eta _{\\rm obs}}\\,\\Psi \\,, \\quad $ where we have used (REF ), along with ${\\cal H}=-1/\\eta $ during cosmological-constant domination in our toy cosmological model, and where $\\Psi $ is understood to be evaluated at $\\eta _{\\rm obs}$ .",
"Using the quadratic formula and expanding in perturbations, the null condition (REF ) becomes $-d\\eta = \\left( 1 + \\frac{2}{3}\\frac{\\eta ^2}{\\eta _{\\rm obs}}\\,\\partial _r\\Psi - \\frac{10}{3}\\frac{\\eta }{\\eta _{\\rm obs}}\\,\\Psi \\right) dr \\,.$ Simple as it looks, this equation does not permit a separation of variables.",
"Nevertheless, it is apparent that the solution takes the form $r = -\\eta + {\\cal O}(\\Psi )$ .",
"We can use this to convert the terms in parentheses into functions of $\\eta $ , then expand in $\\Psi $ , then perform a mode expansion of $\\Psi |_{r\\rightarrow -\\eta }$ , then integrate over $\\eta $ .",
"This gives (after integration by parts) $\\tilde{r}(\\Omega _2) = r_{\\!\\rm cp}+ \\int dq\\sum _{\\ell ,m}\\sqrt{2/\\pi }\\,\\Psi _{\\!q\\ell m}\\,\\bigg [\\frac{2}{3}qr_{\\!\\rm cp}\\,j_\\ell (qr_{\\!\\rm cp})+\\frac{2J_{q\\ell }(r_{\\!\\rm cp})}{qr_{\\!\\rm cp}}\\bigg ]\\, Y^m_\\ell (\\Omega _2) \\,,$ where $r_{\\!\\rm cp}= -\\eta _{\\rm obs}$ is the previously-defined average comoving distance to the boundary of the causal patch (at the time $\\eta =\\eta _{\\rm obs}$ ), and we have defined $J_{q\\ell }(r_{\\!\\rm cp}) \\equiv \\int _0^{r_{\\!\\rm cp}} rdr\\,q^2\\,j_\\ell (qr) \\,.$ We proceed by inserting the expression for $\\tilde{r}$ (REF ) into expression for $\\hat{Z}_0$ (REF ).",
"Expanding in the perturbations, the angular integration can be performed, and we obtain $\\hat{Z}_0 &\\propto & \\frac{4\\pi }{3}r_{\\!\\rm cp}^3+\\sqrt{8}\\,r_{\\!\\rm cp}^3\\int q\\,dq\\,\\bigg \\lbrace \\bigg [\\frac{2}{3}\\,j_0(qr_{\\!\\rm cp})+2\\,\\frac{J_{q0}(r_{\\!\\rm cp})}{q^2r_{\\!\\rm cp}^2}-5\\,\\frac{I_{q0}(r_{\\!\\rm cp})}{q^3r_{\\!\\rm cp}^3}\\bigg ]\\hat{\\Psi }_{\\!q00} \\bigg .",
"\\nonumber \\\\*& & +\\,\\bigg .",
"\\bigg [1+\\frac{I_{q0}(r_{\\!\\rm cp})}{q^3r_{\\!\\rm cp}^3}\\bigg ]\\hat{\\delta }_{q00}\\,\\bigg \\rbrace \\,.$ Note that the angular integrations kill all of the modes except for the monopole, $\\ell =m=0$ .",
"So far it has been convenient to work in terms of spherically symmetric mode functions, but we are interested in predicting expectation values for correlation functions of Cartesian Fourier modes.",
"(It is important to use Cartesian Fourier modes because, unlike the spherically symmetric mode functions, they do not depend on the origin of coordinates.)",
"To express $\\hat{Z}_0$ in terms of Fourier mode functions, we first note that $\\Psi _{\\!q00} &=& \\int d\\Omega _2 \\int r^2dr\\,\\sqrt{2/\\pi }\\,q\\,j_0(qr)\\,Y^0_0(\\Omega _2)\\,\\Psi (\\eta _{\\rm eq},{\\mathbf {x}}) \\nonumber \\\\&=& \\int d\\Omega _2 \\int \\frac{rdr}{\\sqrt{2}\\pi }\\,\\sin (qr) \\int \\frac{d{\\mathbf {p}}}{(2\\pi )^3}\\,e^{i{\\mathbf {p}}\\cdot {\\mathbf {x}}}\\,\\Psi _{\\!",
"{\\mathbf {p}}} \\,,$ where in the second line we have substituted for the particular spherically symmetric mode functions and have inserted the Fourier mode expansion for the perturbation $\\Psi $ .",
"To simplify this expression, we write ${\\mathbf {p}}=p\\,\\hat{{\\mathbf {p}}}$ , where $p$ is the magnitude of ${\\mathbf {p}}$ , and we align the coordinate systems so that $\\hat{{\\mathbf {x}}}\\cdot \\hat{{\\mathbf {p}}}=\\cos (\\theta )$ , where $d\\Omega _2=\\sin (\\theta )\\,d\\theta \\,d\\phi $ (and ${\\mathbf {x}}=r\\,\\hat{{\\mathbf {x}}}$ ).",
"We can then perform the first angular integral, giving $\\Psi _{\\!q00} = \\sqrt{2} \\int dr\\,\\sin (qr)\\int \\frac{d{\\mathbf {p}}}{(2\\pi )^3}\\frac{1}{p}\\,\\sin (pr)\\,\\Psi _{\\!",
"{\\mathbf {p}}} \\,.$ We now recognize the integral over $r$ in the context of the orthonormality condition of spherical Bessel functions: $\\int r^2dr\\,j_0(qr)\\,j_0(pr) = \\int dr \\sin (qr)\\sin (pr) = \\frac{\\pi }{2q^2}\\,\\delta (p-q) \\,.$ Plugging into (REF ), we obtain our final expression for $\\hat{Z}_0$ : $\\hat{Z}_0 \\propto \\frac{4\\pi }{3}r_{\\!\\rm cp}^3+r_{\\!\\rm cp}^3\\int \\frac{d{\\mathbf {q}}}{(2\\pi )^2}\\bigg \\lbrace \\bigg [\\frac{2}{3}\\,j_0(qr_{\\!\\rm cp})+2\\,\\frac{J_{q0}(r_{\\!\\rm cp})}{q^2r_{\\!\\rm cp}^2}-5\\,\\frac{I_{q0}(r_{\\!\\rm cp})}{q^3r_{\\!\\rm cp}^3}\\bigg ]\\hat{\\Psi }_{\\!",
"{\\mathbf {q}}} +\\bigg [1+\\frac{I_{q0}(r_{\\!\\rm cp})}{q^3r_{\\!\\rm cp}^3}\\bigg ]\\hat{\\delta }_{\\mathbf {q}}\\bigg \\rbrace \\,.$"
],
[
"Correlation functions",
"With the expression (REF ) in hand, the calculation of expectation values of correlation functions is straightforward, if sometimes a bit messy.",
"Following the treatment of the fat geodesic measure in Section , we first consider the 2-point correlation function.",
"It is $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\rangle _{\\rm obs} =\\frac{\\langle 0|\\hat{Z}_0\\,\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}}\\,\\hat{\\Psi }_{\\!",
"{\\mathbf {k}}^{\\prime }}|0\\rangle }{\\langle 0|\\hat{Z}_0|0\\rangle } = P(k)\\,(2\\pi )^3\\delta ({\\mathbf {k}}+{\\mathbf {k}}^{\\prime }) \\,.$ The first expression restates (REF ); the second expression follows from inserting (REF ), noting that only even-numbered products of ladder operators give non-zero expectation values in the Bunch-Davies vacuum (and thus the non-trivial terms in $\\hat{Z}_0$ do not contribute).",
"We see that, as in the case of the fat geodesic measure, the causal patch measure does not modify the standard inflationary prediction at leading or first sub-leading order.",
"Meanwhile, for the 1-point function, we find $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\rangle _{\\rm obs} = -\\frac{3}{2}\\,\\bigg \\lbrace \\!\\gamma k^2r_{\\!\\rm cp}^2 \\bigg [1+\\frac{I_{k0}(r_{\\!\\rm cp})}{k^3r_{\\!\\rm cp}^3}\\bigg ]-\\frac{2}{3}\\,j_0(kr_{\\!\\rm cp}) - 2\\,\\frac{J_{k0}(r_{\\!\\rm cp})}{k^2r_{\\!\\rm cp}^2}+5\\,\\frac{I_{k0}(r_{\\!\\rm cp})}{k^3r_{\\!\\rm cp}^3} \\bigg \\rbrace \\, P(k) \\,,$ where we have used (REF ) with $|\\eta _{\\rm obs}|=r_{\\!\\rm cp}$ .",
"As with the fat geodesic measure, the 1-point expectation value takes the form of a monopole.",
"(Again, this could have been guessed based on the symmetries of the physical scenario.)",
"The functions $I_{k0}(r_{\\!\\rm cp})$ , $J_{k0}(r_{\\!\\rm cp})$ , and $j_0(kr_{\\!\\rm cp})$ are at most of order unity for all values of $kr_{\\!\\rm cp}$ .",
"Note also that $r_{\\!\\rm cp}= (1/2)\\,r_{\\!\\rm cmb}$ , with observable scales corresponding to $k\\gtrsim 1/r_{\\!\\rm cmb}$ .",
"Therefore, the size of the 1-point expectation value is of the same order as the 1-point expectation value for the fat geodesic measure.",
"In particular, it is well within cosmic variance, despite the enhancement of the first term for large $k$ .",
"While the computations leading the various terms in (REF ) are somewhat complicated, it is evident that several effects contribute to the final result.",
"As with the fat geodesic measure, there is an effect coming from the tendency of the worldline defining the measure to gravitate toward over-densities.",
"Also as with the fat geodesic measure, the calculation incorporates an anthropic effect, whereby the probability of a measurement is taken to be proportional to the matter density; however unlike in the fat geodesic measure in the causal patch measure this factor is integrated over the entire causal patch.",
"And finally, this integration over the causal patch depends on the total volume in the causal patch, which depends on the metric and its perturbations.",
"Finally, we turn to the 3-point correlation function.",
"It can be written $\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime }}\\,\\Psi _{\\!",
"{\\mathbf {k}}^{\\prime \\prime }}\\rangle _{\\rm obs} =\\langle \\Psi _{\\!",
"{\\mathbf {k}}}\\rangle _{\\rm obs}\\,P(k^{\\prime })\\,(2\\pi )^3\\delta ({\\mathbf {k}}^{\\prime }+{\\mathbf {k}}^{\\prime \\prime }) + {\\rm perms}\\,,$ where we refer to the result (REF ) to avoid repeating the long expression.",
"Each term depends only on the magnitude of one of the wavenumbers $\\lbrace {\\mathbf {k}},{\\mathbf {k}}^{\\prime },{\\mathbf {k}}^{\\prime \\prime }\\rbrace $ and so, as with the fat geodesic measure, the 3-point function is only non-zero when one of the field insertions is a monopole.",
"The overall magnitude is also the same as in the expectation value of the 3-point function in the fat geodesic measure, so all of the comments there apply here."
],
[
"Conclusions",
"We have computed the spectrum of inflationary perturbations in the context of the fat geodesic and causal patch measures.",
"Both measures predict a 1-point expectation value for the gauge-invariant Newtonian potential $\\Psi $ .",
"This takes the form of a scale-dependent monopole, $\\langle \\Psi _{\\!k\\ell m}\\rangle _{\\rm obs} =\\overline{\\Psi }_{\\!k00}\\,k\\,\\delta _{\\ell 0}\\,\\delta _{m0}$ , where the precise form of $\\overline{\\Psi }_{\\!k00}$ differs between the two measures, but in both cases the dominant behavior is $\\overline{\\Psi }_{\\!k00}\\sim k^2r_{\\!\\rm cmb}^2\\,P(k)$ , where $r_{\\!\\rm cmb}$ is the comoving size of the surface of last scattering and $P(k)$ is the power spectrum of $\\Psi $ .",
"Each measure also predicts a contribution to the expectation value of the 3-point correlation function, when one of the three field insertions is the monopole $\\Psi _{\\!k00}$ , this effect evidently due to correlations with the background $\\overline{\\Psi }_{\\!k00}$ .",
"In each case the prediction is well within cosmic variance.",
"These predictions take the local cosmological model, including the model of inflation, to be completely specified.",
"A more general approach would survey the landscape of vacua in the theory and take into account all of the models of inflation (and of the subsequent cosmology) that are consistent with our knowledge of the environment.",
"Consequently, the more general predictions would include distributions for the amplitude of the perturbations, the tilt, etc., and the profiles of these distributions would depend on the choice of measure.",
"While this suggests interesting possibilities for testing phenomenology, we stress that our analysis is orthogonal to these considerations.",
"Indeed, we never actually specified the model of inflation.",
"The inflaton model parameters determine properties of the power spectrum $P(k)$ , however our analysis is independent of the precise form of this function.",
"We can now rephrase the analogy of the introduction, involving darts being thrown at a wall, in terms of these worldline-based measures.",
"Consider the worldline in the bubble-nucleation geometry illustrated in Figure REF .",
"As it enters the bubble, the worldline is not comoving with respect to the open-FRW frame in the bubble.",
"Although the worldline quickly becomes comoving, the initial misalignment of comoving frames gives what might be seen as a large effect: surveying all future histories of the worldline—over which the bubble nucleates, with uniform frequency, at all points in the local spacetime—the worldline overwhelmingly passes through points in the bubble within a few curvature radii of the center of the bubble.",
"(Here, the center of the bubble is defined by the comoving worldline that passes through the center of the bubble-nucleation event.)",
"This effect might be seen as large because the three-volume on constant-time hypersurfaces in any given bubble diverges.",
"That is, based on the symmetries, one might naively conclude that there is zero probability to lie within a few curvature radii of the center.",
"On the other hand, the FRW symmetries in the bubble makes it difficult for an observer to discern his distance from the center.",
"Nevertheless, given some small perturbation to this background, we recover the effect.",
"Specifically, the worldline, which becomes comoving with respect to surfaces of uniform energy density, is not comoving with respect to surfaces of uniform radiation density (after structure formation), and meanwhile an observer can discern his location in relation to the perturbation.",
"This is the source of one measure selection effect on the observed inflationary spectrum.",
"The size of the effect is, roughly speaking, suppressed by the smallness of the perturbation.",
"With the causal patch measure, there is also an effect coming from the size of the causal patch.",
"In the introduction we mentioned a related analysis with respect to the proper-time cutoff measure.",
"In that case the scale-dependent monopole is expected to be much more significant than with the fat geodesic and causal patch measures.",
"We believe this is due to the youngness problem of the proper-time cutoff measure, by which the measure gives overwhelming weight to regions featuring perturbations that would be on the extreme tail of the (approximately) Gaussian distribution predicted by assuming global FRW symmetries.",
"Thus, while the size of the effect is `suppressed' by the size of the perturbations, the measure selects for unusually large perturbations and so the size of the effect is significant.",
"One can view selection effects associated with the choice of measure as a consequence of broken FRW symmetries on the largest scales.",
"In light of this, one might be tempted to view the choice of measure as an infrared effect, providing corrections that can be parametrized in terms of the usual tools of effective field theory.",
"While this view might have some (limited) validity, we have yet to find a crisp formulation of it.",
"The immediate problem with the usual approach is we do not know our location in the global spacetime—meaning we must consider that we could be at any point consistent with our knowledge of the local environment—which forces us to consider the global properties of spacetime even as we focus on local observations.",
"And indeed, while the measure effects studied in this paper are small, they are suppressed in terms of the size of the inflationary perturbations, not in terms of any infrared cosmological scale.",
"This failure of effective-field-theoretic intuition is more dramatic in light of the three phenomenological tests mentioned in Section , which demonstrate how the choice of measure can have very large effects.",
"Therefore, we consider it an important proof of principle that the fat geodesic and causal patch measures validate the standard inflationary predictions, up to effects within cosmic variance.",
"The author thanks Dionysios Anninos, Raphael Bousso, Daniel Harlow, Steve Shenker, and Douglas Stanford for many invaluable discussions.",
"The author is also grateful for the support of the Stanford Institute for Theoretical Physics."
]
] | 1204.1569 |
[
[
"Non-Gaussianity from extragalactic point-sources"
],
[
"Abstract The population of compact extragalactic sources contribute to the non-Gaussianity at Cosmic Microwave Background frequencies.",
"We study their non-Gaussianity using publicly available full-sky simulations.",
"We introduce a parametrisation to visualise efficiently the bispectrum and we describe the scale and frequency dependences of the bispectrum of radio and IR point-sources.",
"We show that the bispectrum is well fitted by an analytical prescription.",
"We find that the clustering of IR sources enhances their non-Gaussianity by several orders of magnitude, and that their bispectrum peaks in the squeezed triangles.",
"Examining the impact of these sources on primordial non-Gaussianity estimation, we find that radio sources yield an important positive bias to local fNL at low frequencies but this bias is efficiently reduced by masking detectable sources.",
"IR sources produce a negative bias at high frequencies, which is not dimmed by the masking, as their clustering is dominated by faint sources."
],
[
"Introduction",
"The CMB, the dominant signal on the sky around 100 GHz, is a powerful probe of the early universe which significantly contributes to the establishment of the standard model of cosmology.",
"In the 30-350 GHz frequency range, other signals contribute at small angular scales and modify the statistical distribution (initially close to Gaussian) of the measured CMB anisotropies.",
"Three of these signals are associated with extragalactic sources: - galaxy clusters, in which the electrons of the hot ionised gas scatter off the CMB photons leaving a distinct spectral signature (Sunyaev-Zeldovich effect) - radio loud galaxies with Active Galactic Nuclei emitting through synchrotron and free-free processes - dusty star-forming galaxies, where the UV emission from stars heats the dust which consequently reemits in the infrared domain We will focus on the characterisation of the distribution of these two latter populations.",
"Radio sources can be considered randomly distributed on the sky and are hence modeled as a white-noise entirely described by the sources number counts.",
"Their harmonic “N-point\" correlation functions (power spectrum, bispectrum, etc) are then flat and related to the corresponding moment of the 1-point distribution.",
"By contrast, IR sources are highly clustered therefore their harmonic correlation functions require the inclusion of the spatial distribution of the sources in addition to their number counts.",
"The primary NG in the CMB can break the degeneracy between the primordial processes generating the cosmological perturbations e.g.",
"the inflations models.",
"While most models predict similar power spectra, they may be distinguished at the 3-point or higher level.",
"As an example, standard single-field models predict undetectably small NG while multifield models, non-canonical kinetic terms or vacuum initial condition etc produce potentially detectable NG.",
"As of today, no definite deviation from Gaussianity has been found in the CMB data, even if there is a hint of `local'-type NG in WMAP7 data [1].",
"The local-type NG, produced generically by several inflation models, takes the form : $\\Phi (\\mathbf {x}) = \\Phi _G(\\mathbf {x}) + f_\\mathrm {NL}\\left(\\Phi _G(\\mathbf {x})^2- \\langle \\Phi _G(\\mathbf {x})^2 \\rangle \\right)$ where $\\Phi $ is the Bardeen potential and the subscript G denotes the Gaussian linear part.",
"To study the distribution of radio and IR galaxies we use publicly available full-sky simulated maps of these sources [2] between 30 and 350 GHz at high angular resolution.",
"The results presented in this communication are presented in greater details in [3]"
],
[
"Bispectrum",
"A gaussian field on the sphere is entirely characterised by its mean and 2-point correlation function (or its power spectrum).",
"The 3-point correlation function is thus the lowest order, and most prominent, indicator of NG.",
"The bispectrum is this 3-point correlation function in harmonic space : $\\langle a_{\\ell _1 m_1} a_{\\ell _2 m_2} a_{\\ell _3 m_3}\\rangle = G_{\\ell _1 \\ell _2 \\ell _3}^{m_1 m_2 m_3} \\times b_{\\ell _1 \\ell _2 \\ell _3} \\quad \\mathrm {with} \\quad G_{\\ell _1 \\ell _2 \\ell _3}^{m_1 m_2 m_3} = \\int \\mathrm {d}^2 n \\, Y_{\\ell _1 m_1}(n) Y_{\\ell _2 m_2}(n) Y_{\\ell _3 m_3}(n)$ A full calculation of the bispectrum at WMAP or Planck resolution ($\\ell _\\mathrm {max} \\sim 700$ and $\\ell _\\mathrm {max} \\sim 2000$ respectively) is computationally very intensive so one usually resort to binning multipoles.",
"The bispectrum is invariant under permutation of $(\\ell _1,\\ell _2,\\ell _3)$ , it is a function of the triangle shape only.",
"Three triangles shapes are of particular importance : squeezed (one side much smaller than the two others), equilateral and folded (flat isosceles).",
"To plot a bispectrum in a efficient way, e.g.",
"without redundancy of information, we introduced a parametrisation accounting for this permutation invariance, and we defined three parameters (P,F,S) based on the elementary symmetric polynomials $\\sigma _1=\\ell _1+\\ell _2+\\ell _3$ , $\\sigma _2=\\ell _1\\ell _2+\\ell _1\\ell _3+\\ell _2\\ell _3$ and $\\sigma _3=\\ell _1\\ell _2\\ell _3$ .",
"The locations of the different triangles of constant perimeter P in the (F,S) plan is shown in Fig.REF (left panel).",
"One can then plot a bispectrum by making slices of perimeters and colour-coding the value of the bispectrum.",
"Figure: Left panel: Positions of the triangle of a given perimeter in the (F,S) plan of the parametrisation.Right panel: IR bispectrum at 148 GHz in our parametrisationThe local-type primordial NG defined in Eq.REF leaves a characteristic imprint on the CMB, which can be detected at the 3-point level.",
"The bispectrum thus generated peaks in the squeezed triangles, and because it has a separable form [4] it has been shown that a fast estimator for its amplitude $f_\\mathrm {NL}$ can be built, which does not need the –prohibitive– computation of the full bispectrum.",
"This, commonly-called KSW, estimator is optimal in the sense that it minimizes the $\\chi ^2$ of the fit of the observed bispectrum to the local bispectrum : $\\hat{f}_\\mathrm {NL} = \\sigma ^2(f_\\mathrm {NL}) \\sum _{\\ell _1\\le \\ell _2\\le \\ell _3} N_{\\ell _1\\ell _2\\ell _3} \\frac{b_{\\ell _1\\ell _2\\ell _3}^\\mathrm {obs}\\, b_{\\ell _1\\ell _2\\ell _3}^\\mathrm {loc}}{C_{\\ell _1} C_{\\ell _2} C_{\\ell _3}} \\\\\\mathrm {with} \\quad \\sigma ^2(f_\\mathrm {NL}) = \\sum _{\\ell _1\\le \\ell _2\\le \\ell _3} N_{\\ell _1\\ell _2\\ell _3} \\frac{\\left(b_{\\ell _1\\ell _2\\ell _3}^\\mathrm {loc}\\right)^2}{C_{\\ell _1} C_{\\ell _2} C_{\\ell _3}}$ where $N_{\\ell _1\\ell _2\\ell _3} = \\frac{(2\\ell _1+1)(2\\ell _2+1)(2\\ell _3+1)}{4\\pi } \\begin{pmatrix} \\ell _1 & \\ell _2 & \\ell _3 \\\\ 0 & 0 & 0 \\end{pmatrix}^2$ is the number of such configurations on the sky"
],
[
"Point-sources",
"Radio sources are randomly distributed on the sky, so they produce a constant bispectrum $b_\\mathrm {ps} \\propto \\int S^3 \\frac{\\mathrm {d}N}{\\mathrm {d}S} \\mathrm {d}S$ , but as higher frequencies are surveyed e.g.",
"with Planck, the IR population becomes of growing importance and one must account for the superposition of these two signals.",
"From a method proposed in [5], we have developped a full-sky analytical prescription to compute the bispectrum and higher order moments based on number counts and power spectrum.",
"The prescription can handle individual populations and their superposition [3].",
"Analytically, at the bispectrum level, it yields : $b^\\mathrm {IR}_{\\ell _1\\ell _2\\ell _3} = \\alpha \\sqrt{C^\\mathrm {IR}_{\\ell _1} C^\\mathrm {IR}_{\\ell _2} C^\\mathrm {IR}_{\\ell _3}} \\quad \\mathrm {with} \\quad \\alpha = \\frac{\\int S^3 \\frac{\\mathrm {d}N}{\\mathrm {d}S} \\mathrm {d}S}{\\left(\\int S^2 \\frac{\\mathrm {d}N}{\\mathrm {d}S} \\mathrm {d}S\\right)^{3/2}}$ We used publicly available simulations of these two populations between 30 and 350 GHz by [2] to compare the measured bispectrum with our prescription.",
"Fig.REF shows the measured and predicted bispectra at 350 GHz in the case of superposition of IR and radio sources.",
"Figure: IR+radio bispectrum at 350 GHz in some configurations.",
"Black line is the measured bispectrum from the simulations, red line is the result of prescription, blue dashed line is the case where radio and IR populations are assumed correlated with each other.We found a good agreement, within cosmic variance error bars, between the prescription and the measured bispectrum at all frequencies and in all cases: radio sources alone, IR alone, radio and IR together.",
"It is also noticeable that the IR clustering increases NG by several order of magnitudes at large angular scales – as compared to unclustered case.",
"While the radio bispectrum is flat, the IR one exhibits a configuration dependence due to clustering (see at 148 GHz visualisation in Fig.REF , right panel).",
"The IR bispectrum peaks in the squeezed triangles –as predicted by our prescription– with a slight flattening at high multipoles due to shot-noise, especially at the higher frequencies."
],
[
"Estimating the contamination to primordial NG",
"We used the KSW estimator described previously (Eq.REF , [4]) on the simulated maps, to estimate the bias of the primordial local NG parameter $f_\\mathrm {NL}$ due to foreground signals.",
"We computed this bias for two angular resolution (WMAP-like, Planck-like), with and without masking sources above the Planck ERCSC catalog [6] flux cuts.",
"Results are presented in Table REF &REF for radio and IR sources respectively.",
"Table: Bias from IR sources to local NG parameter f NL f_\\mathrm {NL}The relative error bars of these biases are of 1.5-2.5% for radio sources and 3 to 7% for IR sources.",
"We find that for both populations the bias increases at higher maximum multipoles, as the CMB signal plummets due to Silk damping.",
"The radio bias is positive and maximal at low frequencies, while the IR bias is negative and peaks at the higher frequencies.",
"At a WMAP-like resolution, except at 30 GHz, these biases can be neglected compared tho the expected error bars on $f_\\mathrm {NL}$ due to cosmic variance.",
"However at Planck-like resolution, these biases become non-negligible ; while the radio bias is very efficiently reduced by masking detectable sources, it is not the case for IR sources where the bias is unaffected.",
"Indeed the IR population, and especially the clustered galaxies, is dominated by faint sources much below the detection limit."
]
] | 1204.1480 |
[
[
"A finite oscillator model related to sl(2|1)"
],
[
"Abstract We investigate a new model for the finite one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1).",
"In this setting, it is natural to present the position and momentum operators of the oscillator as odd elements of the Lie superalgebra.",
"The model involves a parameter p (0<p<1) and an integer representation label j.",
"In the (2j+1)-dimensional representations W_j of sl(2|1), the Hamiltonian has the usual equidistant spectrum.",
"The spectrum of the position operator is discrete and turns out to be of the form $\\pm\\sqrt{k}$, where k=0,1,...,j.",
"We construct the discrete position wave functions, which are given in terms of certain Krawtchouk polynomials.",
"These wave functions have appealing properties, as can already be seen from their plots.",
"The model is sufficiently simple, in the sense that the corresponding discrete Fourier transform (relating position wave functions to momentum wave functions) can be constructed explicitly."
],
[
"Introduction",
"Finite quantum harmonic oscillator models (simply referred to as finite oscillator models) are of importance in optical image processing [7], or in models where only a finite number of eigenmodes can exist such as in signal processing [2], [3], [4].",
"Quantum kinematics of finite oscillators has also been used to remove infinities or divergences in quantum theory or quantum field theory [22].",
"The main idea underlying these finite oscillator models is to replace the Heisenberg algebra of the standard quantum oscillator, which allows only infinite-dimensional representations, by a “closely related” algebra with the same dynamics but which allows finite-dimensional representations.",
"More concretely, for a one-dimensional finite oscillator one considers three (essentially self-adjoint) operators: a position operator $\\hat{q}$ , its corresponding momentum operator $\\hat{p}$ and a Hamiltonian $\\hat{H}$ which is the generator of time evolution.",
"These operators should satisfy the Hamilton-Lie equations (or the compatibility of Hamilton's equations with the Heisenberg equations): $[\\hat{H}, \\hat{q}] = -\\mathrm {i}\\hat{p}, \\qquad [\\hat{H},\\hat{p}] = \\mathrm {i}\\hat{q},$ in units with mass and frequency both equal to 1, and $\\hbar =1$ .",
"The third relation of the canonical oscillator, $[\\hat{q}, \\hat{p}] = \\mathrm {i}$ , is dropped since otherwise the corresponding algebra (the Heisenberg algebra) has infinite-dimensional representations only.",
"Instead, one requires [5]: all operators $\\hat{q}$ , $\\hat{p}$ , $\\hat{H}$ belong to some (Lie) algebra (or superalgebra) $\\cal A$ ; the spectrum of $\\hat{H}$ in (unitary) representations of $\\cal A$ is equidistant.",
"The most popular model is with ${\\cal A}= \\mathfrak {su}(2)$ (or its enveloping algebra), see [5], [6], [7].",
"This model is also underlying the theory in [22].",
"The relevant finite-dimensional representations are the common $\\mathfrak {su}(2)$ representations labelled by an integer or half-integer $j$ .",
"In such a representation, the Hamiltonian is taken as $\\hat{H}=J_0+j+\\frac{1}{2}$ , where $J_0=J_z$ is the diagonal $\\mathfrak {su}(2)$ operator, and thus the spectrum of $\\hat{H}$ is $n+\\frac{1}{2}$ ($n=0,1,\\ldots ,2j$ ).",
"The operators $\\hat{q}$ and $\\hat{p}$ are linear combinations of the other $\\mathfrak {su}(2)$ operators $J_+$ and $J_-$ , such that the relations (REF ) are satisfied.",
"They turn out to have a finite spectrum given by $\\lbrace -j,-j+1,\\ldots ,+j\\rbrace $ [5], see Figure 1(a).",
"In this context, one can also construct the discrete position and momentum wave functions.",
"For the $\\mathfrak {su}(2)$ case, these are given by Krawtchouk functions (normalized symmetric Krawtchouk polynomials), i.e.",
"Krawtchouk polynomials with parameter $p=1/2$ .",
"These discrete wave functions have many interesting properties, and their shape is reminiscent of those of the canonical oscillator [5].",
"In [13], [14], deformations of $\\mathfrak {su}(2)$ were considered by introducing a deformation parameter $\\alpha >-1$ .",
"For the even-dimensional representations [13] ($j$ half-integer), this led to an alternative model of the finite oscillator with the spectrum of $\\hat{H}$ again given by $n+\\frac{1}{2}$ ($n=0,1,\\ldots ,2j$ ), and with the spectrum of the position operator $\\hat{q}$ given by $\\pm (\\alpha +k) \\qquad (k=1,2,\\ldots ,j+\\frac{1}{2}),$ i.e.",
"a finite and mostly equidistant spectrum apart from a gap of size $2\\alpha +2$ in the middle, see Figure 1(b).",
"For the odd-dimensional representations [14] ($j$ integer), this led to a similar model with the same spectrum of $\\hat{H}$ but with the spectrum of $\\hat{q}$ given by (Figure 1(c)) $0, \\quad \\pm \\sqrt{k(2\\alpha +k+1)}, \\qquad (k=1,\\ldots ,j).$ In both deformations, the position wave functions could be constructed explicitly and turned out to be related to normalized Hahn (or dual Hahn) polynomials [13], [14].",
"It was also shown how these wave functions could be interpreted as the finite version of a paraboson oscillator.",
"Recall however that Wigner [25] was among the first to drop the canonical commutation relation $[\\hat{q}, \\hat{p}] = \\mathrm {i}$ , proposing a model with relations (REF ) with the extra condition $\\hat{H} = \\frac{1}{2}(\\hat{p}^2 + \\hat{q}^2).$ This is known as the Wigner quantum oscillator (or paraboson oscillator) [19], [20].",
"This paraboson oscillator still has an infinite energy spectrum of the form $n+a$ (with a positive representation parameter $a$ ), and the spectrum of the position operator is the real axis.",
"The canonical oscillator is recovered from the paraboson oscillator in the representation with $a=1/2$ , i.e.",
"one of the representations of the paraboson oscillator coincides with the canonical oscillator [17], [18], [12].",
"The algebraic structure equivalent with (REF ) and (REF ) is the Lie superalgebra $\\mathfrak {osp}(1|2)$ [18], [12].",
"In this context, the position and momentum operators $\\hat{q}$ and $\\hat{p}$ are odd elements of the Lie superalgebra, whereas $\\hat{H}$ is an even element.",
"This observation, and the fact that the canonical oscillator fits in one of the $\\mathfrak {osp}(1|2)$ representations, leads to the idea that it is perhaps more natural to consider the position and momentum operators of alternative oscillator models as odd elements of a Lie superalgebra rather than just (even) elements of a Lie algebra.",
"Following this last idea, we propose and investigate in this paper a finite oscillator model based on the Lie superalgebra $\\mathfrak {sl}(2|1)$ .",
"Indeed, apart from $\\mathfrak {osp}(1|2)$ , $\\mathfrak {sl}(2|1)$ is the simplest Lie superalgebra that can be considered as a superversion of the Lie algebra $\\mathfrak {su}(2)$ [21].",
"The Lie superalgebra $\\mathfrak {sl}(2|1)$ has moreover a class of representations $W_j$ of dimension $2j+1$ ($j$ integer) which are similar to the common $\\mathfrak {su}(2)$ representations, see Section 2.",
"In the finite oscillator model studied here in Section 3, the Hamiltonian $\\hat{H}$ is a diagonal operator with spectrum $n+\\frac{1}{2}$ ($n=0,1,\\ldots ,2j$ ) in $W_j$ .",
"The position operator $\\hat{q}$ is an arbitrary (self-adjoint) odd element from $\\mathfrak {sl}(2|1)$ , and the form of the momentum operator $\\hat{p}$ follows from (REF ); the model introduces in a natural way a real parameter $p$ with $0<p<1$ .",
"Our investigation shows that the spectrum of $\\hat{q}$ in the representation $W_j$ is still very simple, given by $\\pm \\sqrt{k} \\qquad (k=0,1,\\ldots ,j),$ see Figure 1(d).",
"In order to prove this statement, we construct the eigenvectors of $\\hat{q}$ in $W_j$ explicitly.",
"For this construction, we need some properties of Krawtchouk polynomials (but now with general parameter $p$ ), a well known set of discrete orthogonal polynomials.",
"The properties of the position and momentum wave functions for the new $\\mathfrak {sl}(2|1)$ oscillator model are investigated in Section 4.",
"In particular, we study first some plots of the wave functions for different values of the parameter $p$ .",
"These discrete wave function plots are rather different from the $\\mathfrak {su}(2)$ plots in [5]; only for small $p$ -values these plots show some similarity with the plots of the continuous canonical oscillator wave functions.",
"We explore also the behaviour of the discrete wave functions when the representation parameter $j$ tends to infinity, and discover a relation with the paraboson oscillator.",
"When both position wave functions and momentum wave functions are known explicitly, one can examine the transformation that relates the two.",
"In the canonical case this is just the Fourier transform.",
"In a finite oscillator model, this gives a discrete version of the Fourier transform, determined by a Fourier matrix $F$ .",
"For the $\\mathfrak {sl}(2|1)$ model, we manage to give an explicit expression for an arbitrary matrix element $F_{kl}$ of $F$ , see Section 5.",
"The properties of the matrix $F$ are studied and prove to be similar to those of the standard Discrete Fourier Transform used in spectral analysis.",
"Our paper closes with some concluding remarks in Section 6."
],
[
"The Lie superalgebra $\\mathfrak {sl}(2|1)$ and a class of representations",
"The Lie superalgebra $\\mathfrak {gl}(2|1)$ is well known: it can be defined as the set of all $3\\times 3$ matrices $x=(x_{ij})$ with rows and columns labelled by indices $i,j=1,2,3$ .",
"As a basis in $\\mathfrak {gl}(2|1)$ we choose the Weyl matrices $e_{ij}, \\; i,j=1,2,3$ , where the odd elements are $\\lbrace e_{1i}, e_{i1} | i=2,3\\rbrace $ , and the remaining elements are even.",
"The Lie superalgebra bracket is determined by $[\\!",
"[ e_{ij}, e_{kl} ]\\!]",
"\\equiv e_{ij} e_{kl}-(-1)^{{\\deg (e_{ij})\\deg (e_{kl})}}e_{kl}e_{ij} = \\delta _{jk} e_{il} - (-1)^{\\deg (e_{ij}) \\deg (e_{kl})}\\delta _{il} e_{kj}.",
"$ Note that the even part $\\mathfrak {gl}(2|1)_0 = \\mathfrak {gl}(2) \\oplus \\mathfrak {gl}(1)$ , where $\\mathfrak {gl}(2)=\\hbox{span}\\lbrace e_{ij}|i,j=1,2\\rbrace $ and $\\mathfrak {gl}(1) =\\hbox{span}\\lbrace e_{33}\\rbrace $ .",
"For elements $x$ of $\\mathfrak {gl}(2|1)$ , one defines the supertrace as $\\mathop {\\rm str}\\nolimits (x)=x_{11}+x_{22}-x_{33}$ .",
"The Lie superalgebra $\\mathfrak {gl}(2|1)$ is not simple, and one can define the simple superalgebra $\\mathfrak {sl}(2|1)$ as the subalgebra consisting of elements with supertrace 0.",
"As a basis for $\\mathfrak {sl}(2|1)$ , it is convenient to follow the choice of [10], where one can find the actual matrices of the basis [10]: $& F^+=e_{32},\\ G^+=e_{13},\\ F^-=e_{31},\\ G^-=e_{23}, \\\\& H=\\frac{1}{2}(e_{11}-e_{22}),\\ E^+=e_{12},\\ E^-=e_{21},\\ Z=\\frac{1}{2}(e_{11}+e_{22})+e_{33}.",
"$ So $\\mathfrak {sl}(2|1)$ has four odd (or `fermionic') basis elements $F^+, F^-, G^+, G^-$ and four even (or `bosonic') basis elements $H, E^+, E^-, Z$ .",
"The basis for the $\\mathfrak {gl}(2)$ subalgebra is $\\lbrace H,E^+,E^-\\rbrace $ and the $\\mathfrak {gl}(1)\\cong U(1)$ subalgebra is spanned by $Z$ .",
"The basic Lie superalgebra brackets can be found in [10] or, in a different notation, in [21].",
"For the odd elements, the anti-commutators are given by $& \\lbrace F^\\pm ,G^\\pm \\rbrace =E^\\pm , \\quad \\lbrace F^\\pm ,G^\\mp \\rbrace =Z\\mp H; \\nonumber \\\\& \\lbrace F^\\pm ,F^\\pm \\rbrace =\\lbrace G^\\pm ,G^\\pm \\rbrace =\\lbrace F^\\pm ,F^\\mp \\rbrace =\\lbrace G^\\pm ,G^\\mp \\rbrace =0.",
"$ For the even elements, the commutators are given by $[H,E^\\pm ]=\\pm E^\\pm ,\\quad [E^+,E^-]=2H,\\quad [Z,H]=[Z,E^\\pm ]=0.",
"$ The mixed commutation relations read: $&[H,F^\\pm ]=\\pm \\frac{1}{2} F^\\pm , \\quad [Z,F^\\pm ]=\\frac{1}{2} F^\\pm , \\quad [E^\\pm ,F^\\pm ]=0,\\quad [E^\\mp ,F^\\pm ]=-F^\\mp ; \\nonumber \\\\&[H,G^\\pm ]=\\pm \\frac{1}{2} G^\\pm , \\quad [Z,G^\\pm ]=-\\frac{1}{2} G^\\pm , \\quad [E^\\pm ,G^\\pm ]=0,\\quad [E^\\mp ,G^\\pm ]=G^\\mp .",
"$ The irreducible representations of $\\mathfrak {sl}(2|1)$ have been studied by Scheunert et al [21] and Marcu [16]; for a summary, see [10].",
"The superalgebra $\\mathfrak {sl}(2|1)$ has typical and atypical irreducible representations.",
"Here, we shall consider a class of atypical irreducible representations, labelled by a non-negative integer $j$ (these are denoted by $\\pi _-(j/2)$ in [10]).",
"In order to describe these representations explicitly, let us choose a basis for the representation space $W_j$ of the form $|j,m\\rangle , \\qquad m=-j,-j+1,\\ldots ,+j.$ So $\\dim (W_j)=2j+1$ .",
"For the actions of the $\\mathfrak {sl}(2|1)$ basis elements on these vectors, it is handy to use the following “even” and “odd” functions, defined on integers $n$ : $& {\\cal E}(n)=1 \\hbox{ if } n \\hbox{ is even and 0 otherwise},\\nonumber \\\\& {\\cal O}(n)=1 \\hbox{ if } n \\hbox{ is odd and 0 otherwise}.",
"$ Note that ${\\cal O}(n)=1-{\\cal E}(n)$ , but it is convenient to use both notations.",
"The actions of the odd generators are now given by: $& F^\\pm |j,m\\rangle = \\pm {\\cal O}(j-m) \\sqrt{\\frac{j\\pm m+1}{2}}\\; |j,m\\pm 1\\rangle , \\nonumber \\\\& G^\\pm |j,m\\rangle = \\pm {\\cal E}(j-m) \\sqrt{\\frac{j\\mp m}{2}} \\;|j,m \\pm 1\\rangle .$ The actions of the even generators can in principle be computed from (REF ), and are $& Z |j,m\\rangle = -{\\cal E}(j-m) \\frac{j}{2}\\; |j,m\\rangle -{\\cal O}(j-m) (\\frac{j+1}{2})\\; |j,m\\rangle , \\nonumber \\\\& H |j,m\\rangle = \\frac{m}{2}\\; |j,m\\rangle , \\nonumber \\\\& E^\\pm |j,m\\rangle = \\frac{1}{2}{\\cal E}(j- m) \\sqrt{(j\\mp m)(j\\pm m+2)}\\; |j,m\\pm 2\\rangle \\nonumber \\\\& \\qquad +\\frac{1}{2}{\\cal O}(j-m) \\sqrt{(j \\mp m-1)(j\\pm m+1)}\\; |j,m\\pm 2\\rangle .$ It is easy to verify that the relations (REF )-(REF ) are satisfied for these actions.",
"One can also see that with respect to the even subalgebra $\\mathfrak {gl}(2)\\oplus \\mathfrak {gl}(1) \\cong \\mathfrak {su}(2) \\oplus U(1)$ , $W_j$ decomposes as $(\\frac{j}{2}; -\\frac{j}{2}) \\oplus (\\frac{j-1}{2}; -\\frac{j+1}{2})$ , where $(l;b)$ denotes the $\\mathfrak {su}(2) \\oplus U(1)$ representation “with isospin $l$ and hypercharge $b$ ” [10].",
"The above representation is a star representation (or unitary representation) for the adjoint operation $Z^\\dagger =Z,\\quad H^\\dagger =H,\\quad (E^\\pm )^\\dagger = E^\\mp ,\\quad (F^\\pm )^\\dagger = -G^\\mp ,\\quad (G^\\pm )^\\dagger = -F^\\mp ,$ compatible with the positive definite inner product on the representation space $W_j$ : $\\langle j,m | j, m^{\\prime }\\rangle = \\delta _{m,m^{\\prime }}.$"
],
[
"The $\\mathfrak {sl}(2|1)$ model for a finite one-dimensional oscillator",
"We wish to investigate how $\\mathfrak {sl}(2|1)$ can be used for a finite oscillator model.",
"Inspired by the seminal paper [7] on the $\\mathfrak {su}(2)$ model for a finite oscillator, and by the requirements for the spectrum of the Hamiltonian $\\hat{H}$ , one should take $\\hat{H} = 2H +j+\\frac{1}{2}$ as operator for $\\hat{H}$ in the representation space $W_j$ .",
"This operator is diagonal, self-adjoint, and has indeed the equidistant spectrum: $n+\\frac{1}{2}$ ($n=0,1,2,\\ldots , 2j$ ).",
"Next comes the choice for the position operator $\\hat{q}$ , which should be a self-adjoint element of $\\mathfrak {sl}(2|1)$ .",
"There are two reasons to choose an odd element of $\\mathfrak {sl}(2|1)$ .",
"First, choosing an even element would yield a model that is essentially the same as the $\\mathfrak {su}(2)$ model of [7].",
"Second, as explained in the introduction, in Wigner's paraboson oscillator the position and momentum operator are elements of the odd part of the Lie superalgebra $\\mathfrak {osp}(1|2)$ [25], [18], [12].",
"And since the canonical oscillator is a special case of the paraboson oscillator (corresponding to one particular $\\mathfrak {osp}(1|2)$ representation [18], [12]), the position and momentum operator can also be considered as odd elements of a superalgebra in a particular representation.",
"The most general real self-adjoint odd element of $\\mathfrak {sl}(2|1)$ is given by $A F^+ +B G^+-B F^- -AG^-,$ with $A$ and $B$ real constants.",
"An overall constant does not play a crucial role, so let us assume that there is some normalization like $A^2+B^2=1$ .",
"Consider now first the case that $A$ and $B$ have the same sign, say both positive (the case $A$ positive and $B$ negative will be very similar, and is described at the end of this section).",
"In that case, one can write $\\hat{q} = \\sqrt{p}\\; F^+ + \\sqrt{1-p}\\; G^+-\\sqrt{1-p}\\; F^- -\\sqrt{p}\\;G^-, \\qquad (0 \\le p \\le 1).$ We shall in fact consider $0<p<1$ and later view the values $p=0$ and $p=1$ as a limit.",
"Once $\\hat{q}$ is fixed, the form of $\\hat{p}$ follows from the first equation of (REF ), and thus $\\hat{p} = \\mathrm {i}(\\sqrt{p}\\; F^+ + \\sqrt{1-p}\\; G^+ +\\sqrt{1-p}\\; F^- +\\sqrt{p}\\;G^-), \\qquad (0 \\le p \\le 1).$ With these operators, (REF ) and the conditions described in Section 1 are satisfied, and we can truly speak of an $\\mathfrak {sl}(2|1)$ model for the oscillator.",
"Now it remains to study the operators $\\hat{q}$ and $\\hat{p}$ , in particular their spectrum and eigenvectors in the representation $W_j$ .",
"Note that, due to the actions (REF ), in the (ordered) basis $\\lbrace |j,j\\rangle , |j,j-1\\rangle , \\ldots , |j,-j+1\\rangle , |j,-j\\rangle \\rbrace $ of $W_j$ , the operators $\\hat{q}$ and $\\hat{p}$ are tridiagonal matrices.",
"In particular, for $\\hat{q}$ , one has: $\\hat{q}=\\left(\\begin{array}{cccccccc}0 & R_1& 0 & \\cdots & & & & 0 \\\\R_1 & 0 & S_1 & \\cdots & & & & 0\\\\0 & S_1 & 0 & R_2 & & & & \\vdots \\\\\\vdots & & R_2 & 0 & S_2 & & & \\\\& & & S_2 & 0 & \\ddots & &\\vdots \\\\& & & & \\ddots & \\ddots & R_j & 0 \\\\& & & & & R_j & 0 & S_j \\\\0 & \\cdots & & & \\cdots & 0 & S_j & 0\\end{array}\\right) \\equiv M_q,$ where $R_k=\\sqrt{p}\\sqrt{j+1-k}, \\qquad S_k=\\sqrt{1-p}\\sqrt{k} \\qquad (k=1,2,\\ldots ,j).$ The matrix form $M_p$ of $\\hat{p}$ is similar.",
"For these matrices, we need to study the spectrum and the eigenvectors.",
"It is at this point that Krawtchouk polynomials play a role.",
"Krawtchouk polynomials $K_n(x;p, N)$ of degree $n$ in the variable $x$ , with parameter $p$ are defined by [15], [11], [1]: $K_n(x;p,N) = {\\;}_2F_1 \\left( \\genfrac{}{}{0.0pt}{}{-n,-x}{-N} ; \\frac{1}{p} \\right), \\qquad (n=0,1,\\ldots ,N)$ in terms of the hypergeometric series $_2F_1$ [1], [8], [23] (which is terminating here because of the appearance of $-n$ as numerator parameter).",
"Their (discrete) orthogonality relation reads [15], [11], [1]: $\\sum _{x=0}^N w(x;p,N) K_l(x;p, N) K_n(x;p,N) = h(n;p,N)\\, \\delta _{ln}, \\qquad (0<p<1)$ where $w(x;p,N) = \\binom{N}{x} p^x(1-p)^{N-x} \\quad (x=0,1,\\ldots ,N); \\qquad h(n;p,N)= \\frac{n!(N-n)!}{N!}",
"\\left( \\frac{1-p}{p}\\right)^n.$ For the orthonormal Krawtchouk functions we use the notation: $\\tilde{K}_n(x;p,N) \\equiv \\frac{\\sqrt{w(x;p,N)}}{\\sqrt{h(n;p,N)}}\\, K_n(x;p,N).$ Now we are able to describe the eigenvalues and (orthonormal) eigenvectors of $M_q$ .",
"In this context, two sets of Krawtchouk polynomials play a role: $K_n(x;p,j)$ (with $N=j$ ) and $K_n(x;p,j-1)$ (with $N=j-1$ ).",
"Proposition 1 Let $M_q$ (i.e.",
"the matrix representation of $\\hat{q}$ ) be the tridiagonal $(2j+1)\\times (2j+1)$ -matrix (REF ) and let $U=(U_{kl})_{0\\le k,l\\le 2j}$ be the $(2j+1)\\times (2j+1)$ -matrix with matrix elements: $& U_{2n,j} =(-1)^n \\tilde{K}_0(n;p,j), \\;n\\in \\lbrace 0,1,\\ldots ,j\\rbrace ;\\;U_{2n+1,j} = 0, \\; n\\in \\lbrace 0,\\ldots ,j-1\\rbrace ; \\\\& U_{2n,j-k} = U_{2n,j+k} = \\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_k(n;p,j), \\;n\\in \\lbrace 0,1,\\ldots ,j\\rbrace , \\; k\\in \\lbrace 1,\\ldots ,j\\rbrace ; \\\\& U_{2n+1,j-k} = -U_{2n+1,j+k} = -\\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_{k-1}(n;p,j-1), \\;n\\in \\lbrace 0,1,\\ldots ,j-1\\rbrace , \\quad k \\in \\lbrace 1,\\ldots ,j\\rbrace .$ Then $U$ is an orthogonal matrix: $U U^T = U^TU=I.$ The columns of $U$ are the eigenvectors of $M_q$ , i.e.",
"$M_q U = U D,$ where $D$ is a diagonal matrix containing the eigenvalues of $M_q$ : $D=\\mathop {\\rm diag}\\nolimits (-\\sqrt{j},-\\sqrt{j-1},\\ldots ,-\\sqrt{2}, -1,0,1,\\sqrt{2}, \\ldots ,\\sqrt{j-1},\\sqrt{j}).",
"$ Proof.",
"Using the orthogonality of the Krawtchouk polynomials, and the explicit expressions (REF )-(), a simple computation shows that $(U^TU)_{kl}=\\delta _{kl}$ .",
"Thus $U^TU=I$ , the identity matrix, and hence $UU^T=I$ holds as well.",
"It remains to verify (REF ) and that the eigenvalues are indeed (REF ).",
"Due to the tridiagonal form (REF ) of $M_q$ , one has: $&\\big (M_qU\\big )_{2n,k}= S_{n}U_{2n-1,k}+R_{n+1}U_{2n+1,k}, \\\\&\\big (M_qU\\big )_{2n+1,k}= R_{n+1}U_{2n,k}+S_{n+1}U_{2n+2,k}.$ For the first case (REF ), we need to consider three distinct subcases, according to $k$ belonging to $\\lbrace 0,1,\\ldots ,j-1\\rbrace $ , to $\\lbrace j+1,j+2,\\ldots ,2j\\rbrace $ or $k=j$ .",
"For $k\\in \\lbrace 0,1,\\ldots ,j-1\\rbrace $ , this gives: $&(M_qU)_{2n,j-k}=S_{n}U_{2n-1,j-k}+R_{n+1}U_{2n+1,j-k} \\\\& =\\frac{(-1)^{n}}{\\sqrt{2}}\\sqrt{1-p}\\sqrt{n}\\tilde{K}_{k-1}(n-1;p,j-1) +\\frac{(-1)^{n+1}}{\\sqrt{2}}\\sqrt{p}\\sqrt{j-n}\\tilde{K}_{k-1}(n;p,j-1) \\\\&=\\frac{(-1)^{n+1}}{\\sqrt{2}}\\frac{(\\sqrt{p})^{n+1} (\\sqrt{1-p})^{j-n-1} \\sqrt{(j-1)!}}{\\sqrt{n!",
"(j-n)!h(k-1;p,j-1)}} \\\\& \\qquad \\times [ (j-n) K_{k-1}(n;p,j-1)- n(\\frac{1-p}{p})K_{k-1}(n-1;p,j-1)].$ For the last linear combination between squared brackets, the backward shift operator formula for Krawtchouk polynomials [15] can be applied, and yields $(M_qU)_{2n,j-k}& =\\frac{(-1)^{n+1}}{\\sqrt{2}}\\frac{(\\sqrt{p})^{n+1} (\\sqrt{1-p})^{j-n-1} \\sqrt{(j-1)!}}{\\sqrt{n!",
"(j-n)!h(k-1;p,j-1)}}j K_k(n;p,j) \\\\&= -\\sqrt{k}\\; U_{2n,j-k} = \\big (UD\\big )_{2n,j-k}.$ For the other two subcases with first index $2n$ , the computations are similar.",
"For the case (), we need again to consider three subcases.",
"Now one finds for $k\\in \\lbrace 0,1,\\ldots ,j-1\\rbrace $ : $&(M_qU)_{2n+1,j-k}=R_{n+1}U_{2n,j-k}+S_{n+1}U_{2n+2,j-k} \\\\& =\\frac{(-1)^{n}}{\\sqrt{2}}\\sqrt{p}\\sqrt{j-n}\\tilde{K}_{k}(n;p,j) +\\frac{(-1)^{n+1}}{\\sqrt{2}}\\sqrt{1-p}\\sqrt{n+1}\\tilde{K}_{k}(n+1;p,j) \\\\&=\\frac{(-1)^{n+1}}{\\sqrt{2}}\\frac{(\\sqrt{p})^{n+1} (\\sqrt{1-p})^{j-n} \\sqrt{j!}}{\\sqrt{n!",
"(j-n-1)!h(k;p,j)}}[ K_{k}(n+1;p,j)- K_{k}(n;p,j)].$ For the last linear combination, the forward shift operator formula for Krawtchouk polynomials [15] can be applied, and yields $(M_qU)_{2n+1,j-k}& =- \\frac{(-1)^{n+1}}{\\sqrt{2}}\\frac{(\\sqrt{p})^{n+1} (\\sqrt{1-p})^{j-n} \\sqrt{j!}}{\\sqrt{n!",
"(j-n-1)!h(k;p,j)}} \\frac{k}{pj} K_{k-1}(n;p,j-1) \\\\&= -\\sqrt{k}\\; U_{2n+1,j-k} = \\big (UD\\big )_{2n+1,j-k}.$ The other two subcases are similar.",
"This completes the proof.",
"$\\Box $ The above proposition gives, besides the eigenvectors, also the spectrum of the position operator $\\hat{q}$ in the representation $W_j$ .",
"It will be appropriate to denote these $\\hat{q}$ -eigenvalues by $q_k$ , where $k=-j,-j+1,\\ldots ,+j$ , so $q_{\\pm k} = \\pm \\sqrt{k}, \\qquad k=0, 1, \\ldots , j.$ To our knowledge, this is the first time we come across a tridiagonal operator with such a spectrum.",
"Note also that the spectrum of $\\hat{q}$ is independent of the parameter $p$ in (REF ), but the eigenvectors themselves do depend on $p$ .",
"As far as the eigenvectors of $\\hat{q}$ are concerned, these are the columns of the matrix $U$ .",
"It will be useful to introduce a notation for these eigenvectors: the orthonormal eigenvector of the position operator $\\hat{q}$ in $W_j$ for the eigenvalue $q_k$ , denoted by $|j,q_k)$ , is given in terms of the standard basis by $|j,q_k) = \\sum _{m=-j}^j U_{j+m,j+k} |j,-m\\rangle .$ For the matrix representation $M_p$ of $\\hat{p}$ , the analysis is essentially the same.",
"Without going into the details, we give the final result: Proposition 2 Let $M_p$ be the tridiagonal $(2j+1)\\times (2j+1)$ -matrix representing $\\hat{p}$ in $W_j$ , and let $V=(V_{kl})_{0\\le k,l\\le 2j}$ be the $(2j+1)\\times (2j+1)$ -matrix with matrix elements $V_{2k,l}=-\\mathrm {i}(-1)^k U_{2k,l},\\qquad V_{2k+1,l}= (-1)^k U_{2k+1,l},$ where $U$ is the matrix determined by (REF )-().",
"Then $V$ is a unitary matrix, $V V^\\dagger = V^\\dagger V=I$ .",
"The columns of $V$ are the eigenvectors of $M_p$ , i.e.",
"$M_p V = V D,$ where $D$ is the same diagonal matrix as in Proposition 1.",
"In other words, the eigenvalues of $\\hat{p}$ are also given by: $-\\sqrt{j},-\\sqrt{j-1},\\ldots ,-\\sqrt{2}, -1,0,1,\\sqrt{2}, \\ldots ,\\sqrt{j-1},\\sqrt{j}.$ The matrix $V$ of eigenvectors satisfies: $V^T V = \\left( \\begin{array}{cccc}0 & \\cdots & 0 & -1 \\\\ 0 & \\cdots & -1 & 0 \\\\ \\vdots & \\mathinner {\\hspace{0.55542pt}\\hspace{7.0pt}\\hbox{.",
"}}\\hspace{1.111pt}.\\end{array}\\hspace{1.111pt}.\\right.\\hspace{0.55542pt}$ $\\vdots $$\\vdots $ -1 0 0 ), and $V={\\cal J}U \\qquad \\hbox{where}\\qquad {\\cal J}=-\\mathrm {i}\\mathop {\\rm diag}\\nolimits (\\mathrm {i}^0,\\mathrm {i}^1,\\mathrm {i}^2,\\mathrm {i}^3,\\ldots , \\mathrm {i}^{2j}) = \\mathop {\\rm diag}\\nolimits (-\\mathrm {i},1,\\mathrm {i},-1,\\ldots ).$ The last assertions follow from the explicit expressions (REF ), (REF )-() and the orthogonality properties of Krawtchouk polynomials.",
"Also here, it will be useful to denote the $\\hat{p}$ -eigenvalues by $p_k$ , where $k=-j,-j+1,\\ldots ,+j$ (so $p_{\\pm k} = \\pm \\sqrt{k}$ , $k=0, 1, \\ldots , j$ ), and to write the normalized eigenvectors as: $|j,p_k) = \\sum _{m=-j}^j V_{j+m,j+k} |j,-m\\rangle .$ We end this section with two remarks.",
"First, let us briefly return to the remaining case (REF ) with $A$ positive and $B$ negative; or, without losing generality, $A=\\sqrt{p}$ and $B=-\\sqrt{1-p}$ .",
"The corresponding matrix $M_q^{\\prime }$ is then the same as in (REF )-(REF ), but with $S_k$ replaced by $-\\sqrt{1-p}\\sqrt{k}$ .",
"In other words, one can write $M_q^{\\prime }=D_1 M_q D_1$ where $D_1=\\mathop {\\rm diag}\\nolimits (1,1,-1,-1,1,1,-1,-1,1,1,\\ldots )$ .",
"This implies that $M_q^{\\prime } U^{\\prime }= U^{\\prime } D$ , where $D$ is the same matrix as in Proposition 1 and $U^{\\prime }=D_1U$ .",
"To conclude for this second case: the eigenvalues remain the same, and the matrix $U^{\\prime }$ of eigenvectors is the same as that of the first case, up to sign changes in rows.",
"For this reason, we shall not return to this second case, and just continue with the first case (REF ) for our analysis.",
"Secondly, we have so far considered (REF ) with $0<p<1$ , but what about the cases $p=0$ and $p=1$ ?",
"In these limiting cases, the form of $M_q$ in (REF ) remains valid, and since the eigenvalues of $M_q$ in (REF ) are independent of $p$ , the eigenvalues are again given by (REF ).",
"For the matrix of eigenvectors $U$ , one can simply take the right limit $p\\rightarrow 0$ or the left limit $p\\rightarrow 1$ in the matrix $U$ given in Proposition 1.",
"For example, it is easy to compute: $U_0=\\lim _{ \\genfrac{}{}{0.0pt}{}{\\scriptstyle p\\rightarrow 0}{\\scriptstyle p>0}} U, \\qquad U_0 = \\frac{1}{\\sqrt{2}}\\left( \\begin{array}{ccccccc}\\cdots & 0 & 0 & \\sqrt{2} & 0 & 0 & \\cdots \\\\& 0 & -1 & 0 & 1 & 0 & \\\\& 0 & 1 & 0 & 1 & 0 & \\\\& -1 & 0 & 0 & 0 & 1 & \\\\\\cdots & 1 & 0 & 0 & 0 & 1 & \\cdots \\\\& \\vdots &&&& \\vdots &\\end{array}\\right),$ where the general form of the orthogonal matrix $U_0$ (with only two nonzero elements per row, starting from the second row onwards) is clear from the above.",
"The left limit $p\\rightarrow 1$ yields a similar matrix form for $U$ ."
],
[
"Position and momentum wave functions and their properties",
"The position (resp.",
"momentum) wave functions of the $\\mathfrak {sl}(2|1)$ finite oscillator are the overlaps between the $\\hat{q}$ -eigenvectors (resp.",
"$\\hat{p}$ -eigenvectors) and the $\\hat{H}$ -eigenvectors (or equivalently, the $J_0$ -eigenvectors $|j,m\\rangle $ ).",
"Let us denote them by $\\phi ^{(p)}_{j+m}(q)$ (resp.",
"$\\psi ^{(p)}_{j+m}(\\bar{p})$ ), where $m=j,j-1,\\ldots ,-j$ , and where $q$ (resp.",
"$\\bar{p}$ ) assumes one of the discrete values $q_k$ (resp.",
"$p_k$ ) $(k=-j,-j+1,\\ldots ,+j)$ .",
"Observe that we have denoted the momentum variable of the wave function by $\\bar{p}$ in order not to confuse with the parameter $p$ ($0<p<1$ ) which appears in expression (REF ) and which is also the parameter of the Krawtchouk polynomials occurring here.",
"In the notation of the previous section (and where we want to emphasize the dependence on the parameter $p$ ), we have $& \\phi ^{(p)}_{j+m}(q_k)= \\langle j,-m | j,q_k ) = U_{j+m,j+k}, \\\\& \\psi ^{(p)}_{j+m}(p_k)= \\langle j,-m | j,p_k ) = V_{j+m,j+k}.",
"$ Let us consider the explicit form of these wave functions, first for the position variable.",
"For $j+m$ even, $j+m=2n$ , and for positive $q$ -values one has $\\phi ^{(p)}_{2n} (q_{k}) = \\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_{k}(n;p,j), \\qquad n=0, 1, \\ldots , j,\\qquad k=1, \\ldots , j,$ or equivalently, with $q_k=\\sqrt{k}$ ($k=1,2,\\ldots ,j$ ): $\\phi ^{(p)}_{2n} (q_{k}) = \\frac{(-1)^n}{\\sqrt{2}} j!\\sqrt{\\frac{p^{n+k}(1-p)^{j-n-k}}{n!(j-n)!k!(j-k)!",
"}}{\\ }_2F_1 \\left( \\genfrac{}{}{0.0pt}{}{-k,-n}{-j} ; \\frac{1}{p} \\right).$ The expression for $\\phi ^{(p)}_{2n} (q_{-k})$ , with $q_{-k}=-\\sqrt{k}$ ($k=1,2,\\ldots ,j$ ) is also given by the right hand side of (REF ).",
"So $\\phi ^{(p)}_{2n}$ is an even function.",
"For the argument 0, one simply has $\\phi ^{(p)}_{2n} (0) = (-1)^n \\sqrt{\\binom{j}{n}p^{n}(1-p)^{j-n}} .$ When $j+m$ is odd, $j+m=2n+1$ , one finds for positive $q$ -values $\\phi ^{(p)}_{2n+1} (q_{k}) = \\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_{k-1}(n;p,j-1), \\qquad n=0, 1, \\ldots , j-1,\\qquad k=1, \\ldots , j,$ or, with $q_k=\\sqrt{k}$ ($k=1,2,\\ldots ,j$ ): $\\phi ^{(p)}_{2n+1} (q_{k}) = \\frac{(-1)^n}{\\sqrt{2}} (j-1)!\\sqrt{\\frac{p^{n+k-1}(1-p)^{j-n-k}}{n!(j-1-n)!(k-1)!(j-k)!",
"}}{\\ }_2F_1 \\left( \\genfrac{}{}{0.0pt}{}{-k+1,-n}{-j+1} ; \\frac{1}{p} \\right).$ The expression for $\\phi ^{(p)}_{2n+1} (q_{-k})$ , with $q_{-k}=-\\sqrt{k}$ ($k=1,2,\\ldots ,j$ ) is given by minus the right hand side of (REF ), and $\\phi ^{(p)}_{2n+1} (0)=0$ .",
"So $\\phi ^{(p)}_{2n+1}$ is an odd function.",
"Let us now consider some plots of these discrete wave functions.",
"In Figure 2 we have plotted these functions for the representation $j=10$ (so discrete plots with $2j+1=21$ points).",
"We have considered three values for the parameter $p$ : $p=0.1$ , $p=1/2$ and $p=0.9$ .",
"The spectrum of $\\hat{q}$ is independent of $p$ , so for each of these three cases we plot points corresponding to the values $\\pm \\sqrt{k}$ ($k=0,1,\\ldots ,10$ ) on the horizontal axis.",
"For each of the considered $p$ -values, we have plotted $\\phi ^{(p)}_0$ (the ground state), $\\phi ^{(p)}_1$ (the first excited state), $\\phi ^{(p)}_2$ and $\\phi ^{(p)}_3$ .",
"The behaviour of these discrete wave functions is reminiscent of that of the corresponding continuous wave functions of the canonical oscillator, especially when $p$ is small.",
"As $p$ increases, the wave function values for positions near the origin tend to decrease.",
"In fact, for increasing $p$ -values, the behaviour of the discrete wave functions rather tends to the corresponding wave functions of the paraboson oscillator (see e.g.",
"Figure 3 of [13]).",
"Secondly, it is interesting to investigate what happens when the representation parameter $j$ increases, i.e.",
"when the dimension of the representation $W_j$ increases.",
"For this purpose, we have plotted the ground state and the first excited state, for a fixed $p$ -value, and for the values $j=10$ , $j=30$ and $j=60$ in Figure 3.",
"These plots remind of the shape of (continuous) paraboson wave functions $\\Psi _n^{(a)}(q)$ for increasing values of $a$ .",
"In Figure 4 we have illustrated some of these functions, and the similarity is indeed striking.",
"This can also be confirmed by a limit calculation.",
"The limit is not simply $\\lim _{j\\rightarrow \\infty } \\phi ^{(p)}_{n} (q_{k});$ that would just yield zero, as the non-zero contributions are shifted further away from the origin (see Figure 3).",
"Instead, we need to involve another set of orthogonal polynomials, the dual Hahn polynomials $R_n(\\lambda (x);\\gamma ,\\delta ,N)$ defined by [15] $R_n(\\lambda (x);\\gamma ,\\delta ,N) = {\\;}_3F_2 \\left( \\genfrac{}{}{0.0pt}{}{-n,-x,x+\\gamma +\\delta +1}{-N,\\gamma +1} ; 1 \\right), \\qquad (n=0,1,\\ldots ,N),$ where $\\lambda (x)=x(x+\\gamma +\\delta +1)$ .",
"These polynomials satisfy a discrete orthogonality relation ($\\gamma >-1$ , $\\delta >-1$ ): $\\sum _{x=0}^N \\bar{w}(x;\\gamma ,\\delta ,N) R_m(\\lambda (x);\\gamma ,\\delta ,N)R_n(\\lambda (x);\\gamma ,\\delta ,N)=\\bar{h}(n;\\gamma ,\\delta ,N)\\delta _{mn},$ with $\\bar{w}(x;\\gamma ,\\delta ,N)$ and $\\bar{h}(n;\\gamma ,\\delta ,N)$ given by [15].",
"Let us also fix a notation for the orthonormal functions: $\\tilde{R}_n(\\lambda (x);\\gamma ,\\delta ,N)= \\sqrt{ \\frac{\\bar{w}(x;\\gamma ,\\delta ,N)}{\\bar{h}(n;\\gamma ,\\delta ,N)} }R_n(\\lambda (x);\\gamma ,\\delta ,N).$ Consider now some positive parameter $\\alpha >0$ .",
"Let $j$ be the representation parameter, $n=0,1,\\ldots ,j$ , and $0<p<1$ .",
"The following limit is obvious: $\\lim _{\\alpha \\rightarrow \\infty } {\\;}_3F_2 \\left( \\genfrac{}{}{0.0pt}{}{-n,-k,k+2\\alpha +1}{-j,2p\\alpha +1} ; 1 \\right) ={\\;}_2F_1 \\left( \\genfrac{}{}{0.0pt}{}{-n,-k}{-j} ; \\frac{1}{p} \\right).$ In other words, $\\lim _{\\alpha \\rightarrow \\infty } R_n(\\lambda (k);2p\\alpha ,2(1-p)\\alpha ,j) = K_n(k;p,j).$ Using some elementary limits for the expressions in (REF ), one can in fact show: $\\lim _{\\alpha \\rightarrow \\infty } \\tilde{R}_n(\\lambda (k);2p\\alpha ,2(1-p)\\alpha ,j) = \\tilde{K}_n(k;p,j).$ For the wave functions (REF ) under consideration, this means $\\phi ^{(p)}_{2n} (q_{k}) = \\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_k(n;p,j) = \\frac{(-1)^n}{\\sqrt{2}} \\tilde{K}_n(k;p,j) =\\frac{(-1)^n}{\\sqrt{2}} \\lim _{\\alpha \\rightarrow \\infty }\\tilde{R}_n(\\lambda (k);2p\\alpha ,2(1-p)\\alpha ,j),$ for values $q_k=\\sqrt{k}$ , $k=1,2,\\ldots ,j$ .",
"On the other hand, one finds with [13] that for $jx^2=\\lambda (k)=k(k+2\\alpha +1)$ , $\\lim _{j\\rightarrow \\infty } {\\;}_3F_2 \\left( \\genfrac{}{}{0.0pt}{}{-n,-k,k+2\\alpha +1}{-j,2p\\alpha +1} ; 1 \\right) ={\\;}_1F_1 \\left( \\genfrac{}{}{0.0pt}{}{-n}{2p\\alpha +1} ; x^2 \\right) = \\frac{n!",
"}{(2p\\alpha +1)_n} L_n^{(2p\\alpha )}(x^2),$ where $L_n^{(a)}$ is a (generalized) Laguerre polynomial.",
"Following the limit computations in [13], one obtains $& \\lim _{j\\rightarrow \\infty } \\frac{(-1)^n}{\\sqrt{2}} j^{1/4} \\tilde{R}_n(j x^2;2p\\alpha ,2(1-p)\\alpha ,j) \\nonumber \\\\& = (-1)^n \\sqrt{ \\frac{n!",
"}{\\Gamma (n+2p\\alpha +1)} }|x|^{2p\\alpha +1/2} \\mathrm {e}^{-x^2/2} L_n^{(2p\\alpha )}(x^2)= \\Psi _{2n}^{(2p\\alpha -1)}(x),$ where $\\Psi _n^{(a)}(x)$ is the paraboson wave function with parameter $a$ (see [13]).",
"Combining (REF ) and (REF ), it follows indeed that for large $j$ -values the behaviour of the wave functions $\\phi _{2n}^{(p)}(q_k)$ is the same as the behaviour of the paraboson wave functions $\\Psi _{2n}^{(2p\\alpha -1)}(x)$ for large values of $\\alpha $ .",
"For the explanation above, we have used even wave functions; clearly for wave functions of degree $2n+1$ the computation is similar and the conclusion is the same."
],
[
"The corresponding discrete Fourier transform",
"In canonical quantum mechanics, the momentum wave function (in $L^2({\\mathbb {R}})$ ) is given by the Fourier transform of the position wave function (and vice versa): $\\psi (\\bar{p})= \\frac{1}{\\sqrt{2\\pi }} \\int \\mathrm {e}^{-i\\bar{p} q}\\phi (q)dq.$ In the present situation we are dealing with discrete wave functions, and an analogue of this should be viewed as follows.",
"Let $\\phi ^{(p)} (q_k)=\\left(\\begin{array}{c}\\phi _0^{(p)}(q_k) \\\\\\phi _1^{(p)} (q_k) \\\\\\vdots \\\\\\phi _{2j}^{(p)}(q_k)\\end{array}\\right), \\qquad \\psi ^{(p)} (p_k)=\\left(\\begin{array}{c}\\psi _0^{(p)}(p_k) \\\\\\psi _1^{(p)} (p_k) \\\\\\vdots \\\\\\psi _{2j}^{(p)}(p_k)\\end{array}\\right)\\qquad (k=-j,\\ldots ,+j).$ In this case, the corresponding discrete Fourier transform is defined as the matrix $F=(F_{kl})_{-j\\le k,l\\le +j}$ relating these two wave functions: $\\psi ^{(p)}(p_l)=\\sum _{k=-j}^j F_{kl}\\;\\phi ^{(p)}(q_k).$ By (REF )-(), the columns of $U$ consist of the column vectors $\\phi ^{(p)}(p_k)$ ($k=-j..,+j$ ) and similarly for the matrix $V$ .",
"So (REF ) means that $V = U \\cdot F$ , or: $F=U^T \\cdot V = U^T {\\cal J} U,$ with $J$ given by (REF ).",
"Using the explicit matrix elements for $U$ and ${\\cal J}$ , this leads to the following form of the matrix elements of $F$ : $&F_{j-k,j\\mp l}=F_{j+k,j\\pm l}=-\\frac{\\mathrm {i}}{2} S(k,l;p,j) \\pm \\frac{1}{2} S(k-1,l-1;p,j-1),\\quad (k,l=1,\\ldots ,j); \\\\&F_{j\\mp k,j}=F_{j,j\\mp k}=-\\frac{\\mathrm {i}}{\\sqrt{2}} S(k,0;p,j), \\quad (k=1,\\ldots ,j);\\\\& F_{jj}=-\\mathrm {i}S(0,0;p,j), $ where $S(k,l;p,j) = \\sum _{n=0}^j(-1)^n \\tilde{K}_k(n;p,j) \\tilde{K}_l(n;p,j).$ This last expression is easy to simplify, using [24] or [9].",
"One finds: $S(k,l;p,j)= \\sqrt{\\binom{j}{k}\\binom{j}{l} } 2^{k+l} (p(1-p))^{(k+l)/2} (1-2p)^{j-k-l}{\\;}_2F_1 \\left( \\genfrac{}{}{0.0pt}{}{-k,-l}{-j} ; \\frac{1}{4p(1-p)} \\right).$ So from (REF ) and (REF )-() we have an explicit form for the elements $F_{kl}$ of the matrix $F$ .",
"Just as in [14], one can state the following properties of the discrete Fourier transform matrix $F$ : Proposition 3 The $(2j+1)\\times (2j+1)$ -matrix $F$ is symmetric, $F^T=F$ , and unitary, $F^\\dagger F=F F^\\dagger =I$ .",
"Furthermore, it satisfies $F^4=I$ , so its eigenvalues are $\\pm 1, \\pm \\mathrm {i}$ .",
"A set of orthonormal eigenvectors of $F$ is given by the rows of $U$ , determined in Proposition REF .",
"The multiplicity of the eigenvalues depends on the parity of $j$ .",
"When $j=2n$ is even, then the multiplicity of $-\\mathrm {i},1,\\mathrm {i},-1$ is $n+1,n,n,n$ respectively.",
"When $j=2n+1$ is odd, then the multiplicity of $-\\mathrm {i},1,\\mathrm {i},-1$ is $n+1,n+1,n+1,n$ respectively.",
"Proof.",
"The symmetry of $F$ is easily seen from the expressions (REF )-().",
"The unitarity of $F$ follows from (REF ), the orthogonality of the real matrix $U$ and ${\\cal J}^\\dagger {\\cal J}=I$ .",
"Again using (REF ) and the orthogonality of $U$ , one finds $F^2=FF^T= V^TUU^TV=V^TV$ .",
"But the explicit form of $V^TV$ is known, see (REF ).",
"Since $(V^TV)^2=I$ , the result $F^4=I$ follows.",
"So the eigenvalues can only be $\\pm 1, \\pm \\mathrm {i}$ .",
"From the last part of (REF ), one has $F U^T = U^T {\\cal J}$ .",
"So the columns of $U^T$ (or the rows of $U$ ) form a set of orthonormal eigenvectors of $F$ , and the eigenvalues of $F$ are found in the diagonal matrix ${\\cal J}$ , see (REF ).",
"$\\Box $ We are dealing here with a simple discrete Fourier transform matrix $F$ , with parameter $0<p<1$ : the matrix elements of $F$ are given by ${}_2F_1$ expressions (REF ), the eigenvectors of $F$ are given by the rows of $U$ in Proposition 1, and the eigenvalues of $F$ (with their multiplicities) are given above.",
"$F$ still has the property that it transforms position wave functions into momentum wave functions.",
"Note that due to expression (REF ), the matrix $F$ is particularly simple when $p=1/2$ , as most of its matrix elements are zero.",
"For $j=3$ , the form of $F$ for $p=1/2$ reads $2F = \\left(\\begin{array}{ccccccc}0&0&1&-\\mathrm {i}\\sqrt{2}&-1&0&0\\\\0&1&-\\mathrm {i}&0&-\\mathrm {i}&-1&0\\\\1&-\\mathrm {i}&0&0&0&-\\mathrm {i}&-1\\\\-\\mathrm {i}\\sqrt{2}&0&0&0&0&0&-\\mathrm {i}\\sqrt{2}\\\\-1&-\\mathrm {i}&0&0&0&-\\mathrm {i}&1\\\\0&-1&-\\mathrm {i}&0&-\\mathrm {i}&1&0\\\\0&0&-1&-\\mathrm {i}\\sqrt{2}&1&0&0\\end{array} \\right),$ and it is clear how this generalizes for arbitrary $j$ ."
],
[
"Conclusions",
"We have, for the first time, explored the possibility of using a Lie superalgebra as the basic structure underlying a finite oscillator model.",
"This was inspired by the idea that the position and momentum operators of an oscillator model could most naturally be represented by odd (rather than even) elements of a superalgebra.",
"In the case presented here, we have taken the Lie superalgebra $\\mathfrak {sl}(2|1)$ for this purpose, being a simple generalization of the Lie algebra $\\mathfrak {su}(2)$ .",
"The $\\mathfrak {sl}(2|1)$ representations most suitable to use in the model are the $(2j+1)$ -dimensional representations $W_j$ .",
"Indeed, in the standard basis for these representations the Hamiltonian is diagonal and the position and momentum operators are self-adjoint tridiagonal matrices.",
"The most general form for the position operator $\\hat{q}$ involves a parameter $p$ ($0<p<1$ ), see (REF ).",
"The first main result of the paper is the determination of the eigenvalues and eigenvectors of $\\hat{q}$ in explicit form (Proposition 1).",
"The spectrum is very simple, see (REF ); the eigenvectors are in terms of Krawtchouk polynomials $K_n(x;p,j)$ or $K_n(x;p,j-1)$ .",
"The eigenvalues and eigenvectors of the momentum operator $\\hat{p}$ are similar, see Proposition 2.",
"The matrix $U$ of eigenvectors of $\\hat{q}$ is interesting from a second point of view.",
"Indeed, its rows correspond to the discrete position wave functions of the finite oscillator.",
"These wave functions have been examined in Section 4, by means of plots and by investigating a limit.",
"Although the discrete wave functions are also given in terms of Krawtchouk polynomials, as in the $\\mathfrak {su}(2)$ case, the behaviour is quite different.",
"This is because in the $\\mathfrak {su}(2)$ case [5] the Krawtchouk polynomials appearing in the wave functions are $K_n(x;\\frac{1}{2},2j+1)$ , whereas here we have a combination of $K_n(x;p,j)$ and $K_n(x;p,j-1)$ .",
"For large values of $j$ , the discrete wave functions tend to certain paraboson wave functions.",
"Since the position and momentum wave functions have fairly simple forms, we are able to construct the matrix $F$ that transforms them into each other.",
"This matrix is a discrete analogue of the Fourier transform.",
"The matrix elements of $F$ are terminating ${\\,}_2F_1$ series.",
"The discrete version of the Fourier transform $F$ has many properties in common with the standard Discrete Fourier Transform, see Proposition 3.",
"Apart from the finite-dimensional representations $W_j$ considered here, the Lie superalgebra $\\mathfrak {sl}(2|1)$ also has an interesting class of infinite-dimensional representations in $\\ell ^2({\\mathbb {Z}}_+)$ .",
"It would be tempting to study $\\mathfrak {sl}(2|1)$ oscillator models in these representations, and we hope to tackle this problem in a future paper."
],
[
"Acknowledgments",
"E.I.",
"Jafarov was supported by a postdoc fellowship from the Azerbaijan National Academy of Sciences."
]
] | 1204.1135 |
[
[
"Quantum transport of two-dimensional Dirac fermions in SrMnBi2"
],
[
"Abstract We report two-dimensional quantum transport in SrMnBi$_2$ single crystals.",
"The linear energy dispersion leads to the unusual nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.",
"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.",
"With increase in the temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.",
"The effective magnetoresistant mobility $\\mu_{MR}\\sim 3400$ cm$^2$/Vs is derived.",
"Angular dependent magnetoresistance and quantum oscillations suggest dominant two-dimensional (2D) Fermi surfaces.",
"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ and imply that bulk crystals with Bi square nets can be used to study low dimensional electronic transport commonly found in 2D materials like graphene."
],
[
"Quantum transport of two-dimensional Dirac fermions in SrMnBi$_2$ Kefeng Wang Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA D. Graf National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA Hechang Lei Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA S. W. Tozer National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4005, USA C. Petrovic Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton New York 11973 USA We report two-dimensional quantum transport in SrMnBi$_2$ single crystals.",
"The linear energy dispersion leads to the unusual nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.",
"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.",
"With increase in the temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.",
"The effective magnetoresistant mobility $\\mu _{MR}\\sim 3400$ cm$^2$ /Vs is derived.",
"Angular dependent magnetoresistance and quantum oscillations suggest dominant two-dimensional (2D) Fermi surfaces.",
"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ and imply that bulk crystals with Bi square nets can be used to study low dimensional electronic transport commonly found in 2D materials like graphene.",
"72.20.My,72.80.-r,75.47.Np Dirac fermions have raised great interest in condensed matter physics, as seen on the example of materials such as graphene [1] and topological insulators (TIs).",
"[2] The linear dispersion between momentum and energy of Dirac fermions brings forth some spectacular properties, such as zero effective mass and large transport mobility.",
"[1], [2] In addition to the surface/interface states in TIs and graphene, Dirac states in bulk materials were discussed in organic conductors [3] and iron-based superconductors such as BaFe$_2$ As$_2$ .",
"[4], [5] Recently, highly anisotropic Dirac states were observed in SrMnBi$_2$ ,[6], [7] where linear energy dispersion originates from the crossing of two Bi $6p_{x,y}$ bands in the double-sized Bi square nets.",
"SrMnBi$_2$ has a crystal structure similar to that of the superconducting Fe pnictides and is a bad metal.",
"[7], [8] The Fermi velocity along $\\Gamma -M$ symmetry line is $\\nu _{F}^{\\parallel }\\approx 1.51\\times 10^{6}$ m/s, whereas the Fermi velocity in the orthogonal direction experiences nearly one order of magnitude decrease.",
"[7], [8] One of the interesting properties of Dirac materials is the quantum transport phenomena.",
"[9], [10] Unlike the conventional electron gas with parabolic energy dispersion, where Landau levels (LLs) are equidistant,[11] the distance between the lowest and 1st LLs of Dirac fermions in magnetic field is very large and the quantum limit where all of the carriers occupy only the lowest LL is easily realized under moderate fields.",
"[12], [13] Consequently some quantum transport phenomena such as quantum Hall effect and large linear magnetoresistance (MR) could be observed by conventional experimental methods in Dirac fermion system.",
"[14], [15], [16], [17] Here we show two-dimensional (2D) quantum transport in bulk SrMnBi$_2$ single crystals.",
"The linear energy dispersion leads to the unusual nonsaturated linear MR since all Dirac fermions occupy the lowest LL in the quantum limit.",
"The transverse MR exhibits a crossover at a critical field $B^*$ from semiclassical weak-field MR $\\sim $ $B^2$ to the high-field MR $\\sim $ B dependence.",
"The critical field $B^*$ increases with the increase in temperature and its temperature dependence satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.",
"Angular dependent MR and oscillation indicates the quasi-2D Fermi surfaces (FS).",
"We derive the effective magnetoresistant mobility $\\mu _{MR}\\sim 3400$ cm$^2$ /Vs.",
"Our results illustrate the dominant 2D Dirac fermion states in SrMnBi$_2$ .",
"Single crystals of SrMnBi$_2$ were grown using a self-flux method.",
"[18] Stoichiometric mixtures of Sr (99.99$\\%$ ), Mn (99.9$\\%$ ) and excess Bi (99.99$\\%$ ) with ratio Sr:Mn:Bi=1:1:9 were sealed in a quartz tube, heated to 1050 $^{\\circ }C$ and cooled to 450 $^{\\circ }C$ where the crystals were decanted.",
"X-ray diffraction (XRD) data were taken with Cu K$_{\\alpha }$ ($\\lambda =0.15418$ nm) radiation of Rigaku Miniflex powder diffractometer.",
"Transport measurements were conducted in a Quantum Design PPMS-9 with conventional four-wire method.",
"The crystal was cleaved to a rectangular shape with dimension 4$\\times $ 1 mm$^{2}$ in the ab-plane and 0.2 mm thickness along the c-axis.",
"For in-plane resistivity $\\rho _{ab}(T)$ , the current path was in the ab-plane, whereas magnetic field was perpendicular to the current and parallel to the c-axis except in the rotator experiments.",
"The $c$ -axis resistivity $\\rho _c(T)$ was measured by attaching current and voltage wires to the opposite sides of the plate-like crystals.",
"[19] High field MR oscillation were performed at National High Magnetic Field Laboratory in the same configuration to the in-plane MR.",
"Figure: (a) Powder XRD patterns and structural refinement results.",
"The data were shown by (++) , and the fit is given by the red solid line.",
"The difference curve (the green solid line) is offset.",
"The inset shows the single crystal XRD pattern indicating the c-axis orientation of crystal.",
"(b) Temperature dependence of the in-plane resistivity ρ ab (T)\\protect \\rho _{ab}(T) (open symbols) and cc-axis resistivity ρ c (T)\\rho _c(T) (filled symbols) in B=0B=0 T (squares) and B=9B=9 T (circles) magnetic field respectively.All powder and single crystal XRD reflections can be indexed in the I4/mmm space group by RIETICA software (Fig.",
"1(a)).",
"[20] The determined lattice parameters are $a=b=0.4561(8)$ nm and $c=2.309(6)$ nm in agreement with the published data.",
"[6] The in-plane resistivity $\\rho _{ab}(T)$ shown in Fig.",
"1(b) exhibits a metallic behavior.",
"An external magnetic field enhances the resistivity.",
"As the temperature is increased, MR is gradually suppressed and is rather small above $\\sim 60$ K. Resistivity along the $c$ -axis ($\\rho _c(T)$ ) is nearly two orders of magnitude larger than $\\rho _{ab}(T)$ and exhibits a weak crossover at high temperature.",
"In what follows we will discuss in-plane MR. Angular dependent MR $\\rho (B,\\theta )$ at $T\\sim 2$ K is shown in Fig.",
"2(a)-(b).",
"The crystal was mounted on a rotating stage such that the tilt angle $\\theta $ between sample surface ($ab$ -plane) and the magnetic field can be continuously changed, with currents flowing in the $ab$ -plane perpendicular to magnetic field (inset in Fig.",
"2(a)).",
"The magnetoresistance of SrMnBi$_2$ exhibits significant angular dependence (Fig.",
"2(a,b)).",
"When $B$ is parallel to the $c$ -axis ($\\theta =0^{o}$ ), the MR is maximized and is linear above a characteristic field ($\\sim 1$ T).",
"With increase in the tilt angle $\\theta $ , MR gradually decreases and becomes nearly negligible for $B$ in the $ab$ -plane ($\\theta =90^o$ ).",
"Figure: (a) In-plane resistivity ρ\\rho vs. the tilt angle θ\\protect \\theta from 0o to 360o at BB = 3, 6 and 9 T and TT = 2 K. The red line is the fitting curve using |cos(θ)||\\cos (\\protect \\theta )| (see text).",
"Inset shows the configuration of the measurement.",
"(b) In-plane Resistivity ρ\\protect \\rho vs. magnetic field BB with different tilt angles θ\\protect \\theta at 2 K. (c) MR SdH oscillations ΔR=R xx -<R>\\Delta R=R_{xx}-<R> as a function of field BB below 35 T with tilt angles θ\\theta from 0o to 35o at 2 K. The dashed lines indicate the SdH dips at Landau filling factor v=5v=5 and 6.",
"The different curves are offset for clarification.",
"(d) Position of dips with v=5v=5 and 6, as well as the position of peaks with v=5v=5, plotted against the tilt angle θ\\theta .",
"The data are consistent with the 1/|cos(θ)|1/|\\cos (\\theta )| dependence (red lines).",
"The inset shows the Fourier transform of the SdH oscillation which gives a single frequency of F=138(7)F=138(7) T.The response of the carriers to the applied magnetic field and the magnitude of MR is determined by the mobility in the plane perpendicular to the magnetic field.",
"[11] For nearly isotropic three-dimensional (3D) FS, there should be no significant angle-dependent MR (AMR).",
"In (quasi-)2D systems, 2D states will only respond to perpendicular component of the magnetic field $B|\\cos (\\theta )|$ and consequently longitudinal AMR and AMR-oscillation were observed in some quasi-2D conductors (such as Sr$_2$ RuO$_4$ and $\\beta -$ (BEDT-TTF)$_2$ I$_3$ ) and the surface states of TIs.",
"[21], [22], [14], [15] For example, the MR of the bulk state in topological insulator has only $\\sim 10\\%$ angular dependence while the angular dependence of MR in the surface state is about ten times larger.",
"[15] Significant AMR was also observed in some materials with highly anisotropic 3D FS such as Bi and Cu, but the period of AMR is determined by the shape of the Fermi surface and is very different from the one in 2D systems.",
"In Bi, electrons exhibit a threefold valley degeneracy and in-plane mass anisotropy, so the AMR peaks each time when the magnetic field is oriented along the bisectrix axis and has a 60o period.",
"[23] In Cu the AMR is more complex due to the complex FSs and peaks about every 25o.",
"[24] The electronic structure calculations in SrMnBi$_2$ show that the states near the Fermi energy $E_F$ are dominated by the Bi states in the Bi square nets.",
"Consequently the dominant FS should be quasi-2D.",
"Angular dependent resistivity in $B=9$ T and $T=2$ K shows wide maximum when the field is parallel to the $c$ -axis ($\\theta =0^o, 180^o$ ), and sharper minimum around $\\theta =90^o, 270^o$ (Fig.",
"2(a)).",
"The whole curve of AMR in SrMnBi$_2$ follows the function of $|\\cos (\\theta )|$ very well with a 180o period (red line in Fig.",
"2(a)).",
"Moreover, the larger $\\rho _{c}$ than $\\rho _{ab}$ in Fig.",
"1(b) implies that the transfer integral and the coupling between layers along the $c$ -axis is very small.",
"All this implies that the FSs in SrMnBi$_2$ should be highly anisotropic and that the mobility of carriers along $k_z$ is much smaller than the value in $k_xk_y$ plane.",
"Angular dependent MR quantum oscillations are directly related to the cross section of FS.",
"In Fig.",
"2(c), the in-plane $\\Delta R=R-<R>$ measured using same configuration as shown in the inset of Fig.",
"2(a) exhibits clear Shubnikov-de Hass (SdH) oscillations with tilt angles $\\theta $ from 0o to 35o.",
"In metals, SdH oscillations correspond to successive emptying of LLs by the magnetic field and the LLs index $n$ is related to the cross section of FS $S_F$ by $2\\pi (n+\\gamma )=S_F\\frac{\\hbar }{eB}$ .",
"[25], [14], [15] For a 2D FS (a cylinder), the cross section has $S_F(\\theta )=S_0/|\\cos (\\theta )|$ angular dependence and the LLs positions should be inversely proportional to $|cos(\\theta )|$ .",
"[14], [25] The peak (dip) positions in SrMnBi$_2$ rapidly shift toward higher field direction with increase in $\\theta $ (as indicated by the dashed lines in Fig.",
"2(c)).",
"In Fig.",
"2(d), the dip positions corresponding to LLs $n=5, 6$ and the peak position with $n=5$ were plotted against the tilt angle $\\theta $ and can be described very well by $1/|\\cos (\\theta )|$ (the red lines in Fig.",
"2(d)).",
"Similar behavior was observed in the surface states of TIs [14], [15] and some other layered structures.",
"[25], [26] The Fourier transform of the SdH oscillation (inset of Fig.",
"2(d)) revealed that the oscillation component shows a periodic behavior in $1/B$ with a single frequency $F=138(7)$ T. The small value of frequency is consistent with previous value in Ref.",
"[7] and demonstrates that the dominant FSs are very small since the Onsager relation is $F=(\\Phi _0/2\\pi ^2)A_k$ where $\\Phi _0$ is the flux quantum and $A_k$ is the cross sectional area of FS.",
"[25] This clearly shows that the dominant two-dimensional FSs found in Fig.",
"2(b) are indeed the small FSs between $\\Gamma $ and $M$ points, rather than the large FS at $\\Gamma $ point in the Brillouin zone.",
"Above SdH oscillation combined with the angular MR clearly suggests that the dominant FSs of SrMnBi$_2$ are small quasi-2D cylinders along $k_z$ originating from Bi square nets.",
"In addition, there are still conventional parabolic bands with three dimensional characteristic close to the Fermi level,[7] causing the small deviation from quasi-2D transport.",
"Figure: (a) The magnetic field (BB) dependence of the in-plane magnetoresistance (MR =(ρ(B)-ρ(0))/ρ(0)=(\\protect \\rho (B)-\\protect \\rho (0))/\\protect \\rho (0)) at differenttemperatures.",
"(b) The field derivative of in-plane MR at different temperaturerespectively.",
"The lines in high field regions were fitting results using MR =A 1 B+O(B 2 )=A_1B+O(B^2) and the lines in low field regions using MR =A 2 B 2 =A_2B^2.Figure: (a) Temperature dependence of the critical field B * B^* (blacksquares) and the high field MR linear coefficient A 1 A_1 (blue circles) up to50 K. The red solid line is the fitting results using B * =1 2eℏv F 2 (E F +k B T) 2 B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2.",
"(b) Temperature dependence of theeffective MR mobility μ MR \\protect \\mu _{MR} extracted from the weak-field MR.Now we turn to the linear nonsaturated in-plane magnetoresistance in SrMnBi$_2$ (Fig.",
"3(a)).",
"The MR is linear over a wide field range up to 50 K. This behavior extends to very low fields where the MR naturally reduces to a weak-field semiclassical quadratic dependence.",
"The cross-over from the weak-field $B^2$ dependence to the high-field linear dependence can be best seen by considering the field derivative of the MR, $d$ MR$/dB$ (Fig.",
"3(b)).",
"In the low field range ($B<$ 1 T at 2 K), $d$ MR$/dB$ is proportional to $B$ (as shown by lines in low-field regions), indicating the semiclassical MR $\\sim A_2B^2$ .",
"But above a characteristic field $B^*$ , $d$ MR$/dB$ deviates from the semiclassical behavior and saturates to a much reduced slope (as shown by lines in the high-field region).",
"This indicates that the MR for $B>B^*$ is dominated by a linear field dependence plus a very small quadratic term (MR$=A_1B+O(B^2)$ ).",
"The linear MR deviates from the semiclassical $B^2$ dependence of MR in the low field region and a saturating MR in high fields.",
"[11] The unusual nonsaturating linear magnetoresistance has been reported in gapless semiconductor Ag$_{2-\\delta }$ (Te/Se) [27], [28] with the linear energy spectrum in the quantum limit.",
"[10], [28] Recent first principle calculations confirmed that these materials have a gapless Dirac-type surface state.",
"[29] Linear magnetoresistance is also observed in topological insulators [14], [15] and BaFe$_2$ As$_2$ [16] with Dirac fermion states.",
"Another possible origin of the large linear magnetoresistance is the mobility fluctuations in a strongly inhomogeneous system.",
"[30] This does not apply in SrMnBi$_2$ since our sample is stoichiometric crystal without doping/disorder.",
"Below we show that the nonsaturating linear magnetoresistance and the deviation from the semiclassical transport in SrMnBi$_2$ is due to the linear energy dispersion.",
"The application of a strong perpendicular external magnetic field ($B$ ) would lead to a complete quantization of the orbital motion of carriers with linear energy dispersion, resulting in quantized LLs $E_n=sgn(n)v_F\\sqrt{2e\\hbar B|n|}$ where $n=0,\\pm 1,\\pm 2,\\cdots $ is the LL index and $v_F$ is the Fermi velocity.",
"[12], [13] Then the energy splitting between the lowest and 1st LLs is described by $\\triangle _{LL}=\\pm v_F\\sqrt{2e\\hbar B}$ .",
"[12], [13] In the quantum limit at specific temperature and field, $\\triangle _{LL}$ becomes larger than both the Fermi energy $E_F$ and the thermal fluctuations $k_BT$ at a finite temperature.",
"Consequently all carriers occupy the lowest Landau level and eventually the quantum transport with linear magnetoresistance shows up.",
"The critical field $B^*$ above which the quantum limit is satisfied at specific temperature $T$ is $B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2$ .",
"[16] The splitting of LLs in conventional parabolic bands is $\\triangle _{LL}=\\frac{e\\hbar B}{m^*}$ .",
"Hence the evolution of $\\triangle _{LL}$ with field for parabolic bands is much slower than that for Dirac fermion states, and it is difficult to observe quantum limit behavior in the moderate field range.",
"The temperature dependence of critical field $B^*$ in SrMnBi$_2$ clearly deviates from the linear relationship and can be well fitted by $B^*=\\frac{1}{2e\\hbar v_F^2}(E_F+k_BT)^2$ , as shown in Fig.",
"4(a).",
"The fitting gives the Fermi velocity $v_F\\sim 5.13\\times 10^5$ ms$^{-1}$ and $\\Delta _1\\sim 4.97$ meV.",
"This confirms the existence of Dirac fermion states in SrMnBi$_2$ .",
"In a multiband system with both Dirac and conventional parabolic-band carriers (including electrons and holes) the magnetoresistance in the semiclassical transport can be described as $MR=\\frac{\\sigma _e\\sigma _h(\\mu _e+\\mu _h)^2}{(\\sigma _e+\\sigma _h)^2}B^2$ where $\\sigma _e, \\sigma _h, \\mu _e, \\mu _h$ are the effective electron and hole conductivity and mobility in zero field respectively, when the Dirac carriers are dominant in transport.",
"[16], [17] Then the coefficient of the low-field $B^2$ quadratic term, $A_2$ , is related to the effective MR mobility $\\sqrt{A_2}=\\frac{\\sqrt{\\sigma _e\\sigma _h}}{\\sigma _e+\\sigma _h}(\\mu _e+\\mu _h)=\\mu _{MR}$ , which is smaller than the average mobility of carriers $\\mu _{ave}=\\frac{\\mu _e+\\mu _h}{2}$ and gives an estimate of the lower bound.",
"Fig.",
"4(b) shows the dependence of $\\mu _{MR}$ on the temperature.",
"At 2 K, the value of $\\mu _{MR}$ is about 3400 cm$^2$ /Vs.",
"The large effective MR mobility also implies that Dirac fermions dominate the transport behavior.",
"With increase in temperature, the value of $\\mu _{MR}$ and the coefficient of high-field linear term $A_1$ (Fig.",
"4(a)) decrease sharply.",
"This is due to thermal fluctuation smearing out the LL splitting.",
"In summary, we demonstrate quantum transport of 2D Dirac fermion states in bulk SrMnBi$_2$ single crystals.",
"The bands with linear energy dispersion lead to the large nonsaturated linear magnetoresistance since all Dirac fermions occupy the lowest Landau level in the quantum limit.",
"The transverse magnetoresistance exhibits a crossover at a critical field $B^*$ from semiclassical weak-field $B^2$ dependence to the high-field linear-field dependence.",
"With increase in temperature, the critical field $B^*$ increases and the temperature dependence of $B^*$ satisfies quadratic behavior which is attributed to the Landau level splitting of the linear energy dispersion.",
"The effective magnetoresistant mobility $\\mu _{MR}\\sim 3400$ cm$^2$ /Vs comparable to values observed in graphene is observed.",
"The angle-dependence of magnetoresistance shows a $|\\cos (\\theta )|$ dependence while the LL positions in SdH oscillations are inversely proportional to $|\\cos (\\theta )|$ , indicating the dominant quasi-2D Fermi surfaces.",
"Our results show that the crystals with Bi square nets can host phenomena commonly observed so far in 2D structures and materials like graphene.",
"We than John Warren for help with SEM measurements.",
"Work at Brookhaven is supported by the U.S. DOE under contract No.",
"DE-AC02-98CH10886.",
"Work at the National High Magnetic Field Laboratory is supported by the DOE NNSA DEFG52-10NA29659 (S. W. T and D. G.), by the NSF Cooperative Agreement No.",
"DMR-0654118 and by the state of Florida."
]
] | 1204.1049 |
[
[
"Disordered quantum wires: microscopic origins of the DMPK theory and\n Ohm's law"
],
[
"Abstract We study the electronic transport properties of the Anderson model on a strip, modeling a quasi one-dimensional disordered quantum wire.",
"In the literature, the standard description of such wires is via random matrix theory (RMT).",
"Our objective is to firmly relate this theory to a microscopic model.",
"We correct and extend previous work (arXiv:0912.1574) on the same topic.",
"In particular, we obtain through a physically motivated scaling limit an ensemble of random matrices that is close to, but not identical to the standard transfer matrix ensembles (sometimes called TOE, TUE), corresponding to the Dyson symmetry classes \\beta=1,2.",
"In the \\beta=2 class, the resulting conductance is the same as the one from the ideal ensemble, i.e.\\ from TUE.",
"In the \\beta=1 class, we find a deviation from TOE.",
"It remains to be seen whether or not this deviation vanishes in a thick-wire limit, which is the experimentally relevant regime.",
"For the ideal ensembles, we also prove Ohm's law for all symmetry classes, making mathematically precise a moment expansion by Mello and Stone.",
"This proof bypasses the explicit but intricate solution methods that underlie most previous results."
],
[
"Introduction",
"We start below with a brief introduction to the physics of quasi one-dimensional quantum wires.",
"In Section REF , we sketch the scope of this paper and its relation to previous works.",
"The appropriate random matrix theory is discussed in Section .",
"Our microscopic model, convergence results and proofs are presented in Sections and ."
],
[
"Phenomenology",
"Without yet introducing a concrete mathematical framework, we present the basic physics setup of quantum wires and try to elucidate the questions of charge transport and conductance fluctuations.",
"We rely heavily on the excellent review [2].",
"Disordered quantum wires are quasi one-dimensional pieces of dirty (disordered) conductor.",
"The wire has a physical length $L$ , which is most conveniently expressed in units of the mean free path $\\ell $ so that we shall use $s:= L/\\ell $ .",
"In a microscopic model where the parameter $\\lambda \\ge 0$ measures the strength of the disorder $\\ell \\sim \\lambda ^{-2}$ .",
"The width $W$ of the wire is expressed by an integer $N$ that corresponds to the number of different modes that `fit' in the wire.",
"Physically, $N \\sim W/\\lambda _F$ with $\\lambda _F$ the Fermi wavelength of the electrons sent through the wire, which is in turn determined by the energy of those incoming electrons.",
"For a purely one-dimensional wire, $N=1$ , it is well-known that an electron travelling through the wire gets localized with localization length of the order of the mean free path $\\ell $ , hence $s \\sim 1$ .",
"However, the localization length increases with $N$ (roughly as $s \\sim N$ , at least in the weak disorder limit $\\lambda \\rightarrow 0$ ) and we can ask how the system behaves for $s \\ll N$ , before localization sets in.",
"There, one can distinguish the ballistic regime $s \\le 1$ , where incoming electrons did not yet get scattered by the impurities, and the most interesting diffusive regime characterized by $1 \\ll s, \\qquad s/ N \\ll 1\\,.$ One of the fascinating aspects of this regime is the phenomenon of universal conductance fluctuations (UCF) first discussed in [14].",
"Let $g= g(s,N, \\lambda )$ be the conductance of the wire, expressed in units of the conductance quantum $2e^2/\\hbar $ .",
"It is a random quantity due to the disorder.",
"In the thick wire limit, its disorder average ${\\mathbb {E}}(g)$ , is roughly given by ${\\mathbb {E}}(g) \\sim \\left\\lbrace \\begin{array}{lllll} N/s & \\qquad & 1 \\ll s, \\quad s/ N \\ll 1 & \\qquad & \\textrm {(Ohm^{\\prime }s law)} \\\\[2mm] \\exp {\\lbrace - s /N \\rbrace } & \\qquad & s > N & \\qquad & \\textrm {(localization)} \\end{array} \\right.",
"$ Furthermore, universal conductance fluctuations mean that, in the diffusive regime defined by (REF ), $\\mathrm {Var}(g) = 2/(15\\beta ),$ independently of the microscopic details of the wire, or its length and width.",
"The only parameter that remains in this regime is the symmetry index $\\beta $ that refers to Dyson's symmetry classes.",
"We emphasize that these phenomena should emerge in a large $N$ limit only.",
"On the other hand, $N$ cannot be too large because then we enter the regime of two-dimensional localization, at least if we assume that the wire has one transverse dimension.",
"However, even if the transverse dimension is higher, the reasoning breaks down as soon as $W > \\ell $ .",
"It is therefore important to take a weak-disorder limit first, $\\lambda \\rightarrow 0$ , which also means that the wire's microscopic length $L=\\lambda ^{-2}s$ diverges.",
"Below, we try to distill some precise conjectures that are generally accepted.",
"From the mathematical perspective, they can be partially proven if one accepts RMT as a starting point (see Section ), but open if one starts from a more realistic model, as the one treated in Section of this article.",
"Conjecture 1 (Ohm's law) $\\lim _{N \\rightarrow \\infty } \\quad \\lim _{\\lambda \\rightarrow 0} \\quad \\frac{1}{N}{\\mathbb {E}}(g) = \\frac{1}{s} + o(1/s), \\qquad s \\rightarrow \\infty .$ Conjecture 2 (Universal conductance fluctuations) $\\quad \\lim _{N \\rightarrow \\infty } \\quad \\lim _{\\lambda \\rightarrow 0} \\quad \\mathrm {Var}(g) = \\frac{2}{15 \\beta } +o(1), \\qquad s \\rightarrow \\infty .$ We stress here that these conjectures reflect the minimum of what should be true according to the literature, and that the underlying heuristics is quite involved.",
"The present paper does partially settle theses conjectures starting from a microscopic model but with an additional scaling limit, as will be explained in the next section."
],
[
"Setup, goals, and results",
"The standard approach to disordered quantum wires is to model the transfer matrix of such a wire by an appropriate ensemble of random matrices.",
"The matrices under consideration belong to a subgroup of pseudo-unitary matrices.",
"Following [8] we shall call their ensembles TOE, TUE and TSE in analogy to the better known ensembles of Hamiltonians, the hermitian GOE, GUE and GSE, or the circular ensembles of unitaries: COE, CUE and CSE.",
"In fact, ensembles of transfer matrices come with a real positive parameter, called $s$ above and physically corresponding to the length of the wire.",
"They are therefore more complicated, but also more interesting: in particular the parameter $s$ tunes a localization-delocalization transition.",
"This can be observed for example in the Fokker-Planck equation describing the $s$ -dependence of the conductance, which is the equation usually referred to as the DMPK equation.",
"The natural question arises whether the RMT ensembles allow for a verification of the conjectures mentioned at the end of the previous section, with the proviso that the $\\lambda \\rightarrow 0$ should be omitted as the RMT assumes weak coupling from the start.",
"In the physics literature, there is overwhelming evidence for an affirmative answer, and the conjectures have been verified in [15], [3], [7], [21].",
"In that perspective, we shall here give a rigorous proof of Ohm's law based on a moment argument of [17], thereby confirming Conjecture 1 for the TOE, TUE and TSE.",
"The ultimate goal of our work is a derivation of the conjectures from a more realistic model of the wire, i.e.",
"from a `reasonable' microscopic Hamiltonian, namely the Anderson model on a tube of width $N$ with a disordered region of length $L$ and disorder strength $\\lambda $ .",
"First, we need to be in the weak coupling regime $\\lambda \\rightarrow 0$ , and therefore $L=\\lambda ^{-2}s\\rightarrow \\infty $ .",
"This first scaling limit yields a random matrix ensemble ${\\mathcal {G}}(s)$ , see Proposition REF .",
"For the conjectures to hold, a second scaling is certainly necessary, namely that of a broad wire, $N\\rightarrow \\infty $ .",
"At the time of writing, the validity of the conjectures in this scaling regime remains an open question.",
"However, if we consider an additional scaling limit in which the transversal hopping in the wire is small compared to the longitudinal hopping, see Theorem REF , we obtain, instead of the ensemble ${\\mathcal {G}}(s)$ , a new transfer matrix ensemble ${\\mathcal {A}}(s)$ that is very close to the ideal ensemble.",
"In fact, for $\\beta =2$ , the conductance calculated from that ensemble is the same as that calculated from the TUE.",
"Since we proved Ohm's law for the $N\\rightarrow \\infty $ limit of the random matrix ensemble in the first place, this provides a proof of Conjecture 1 in a weaker sense for $\\beta =2$ .",
"In Section REF , we comment on the ensemble ${\\mathcal {A}}(s)$ , pointing out to how and why it fails to satisfy all the symmetry properties of the ideal ensembles.",
"This article is to a large extent based on a previous paper [1] by two of us, which appeared on the arXiv shortly after and independently of [20].",
"Despite their similarity these two articles stressed different aspects of the resulting transfer matrix evolutions.",
"However, [1] contained an error, as pointed out by the second author of the present paper, and the symmetry properties of the model were not consistently treated.",
"In this article, which supersedes [1], we first extend the setup by constructing models for both $\\beta =1$ and $\\beta =2$ symmetry classesThe physically most natural way to discuss $\\beta =4$ as well would be to consider electrons with spin, which we chose not to do for reasons of simplicity.",
"Moreover, we incorporate technical improvements (among other things borrowing some terminology from [20]), mostly concerning the statement of the joint scaling limit in Theorem REF .",
"Finally, we study the convergence as $N\\rightarrow \\infty $ of a hierarchy of equations for the moments of the conductance introduced by [17].",
"We prove that the limit satisfies Ohm's law, see Theorem REF ."
],
[
"Random matrix theory: the DMPK equation",
"Transport properties of a quasi one-dimensional system are most conveniently approached through its scattering matrix, or equivalently its transfer matrix.",
"In this section we shall consider these objects as the fundamental quantities of the theory, understand what symmetries imply on their general structure and derive a stochastic differential equation describing their behavior as a function of the length of the disordered wires, based on an isotropy assumption, also called `(local) maximal entropy' Ansatz.",
"In particular, we do not assume that the transfer matrices here arise from some sort of microscopic Hamiltonian dynamics."
],
[
"Transfer matrices and symmetries",
"Heuristically speaking the transfer matrix of a quasi one-dimensional wire maps free waves on the far right of the sample to free waves on the far left of it.",
"Although this picture is physically meaningful, we shall only refer to it explicitly in Section and keep an abstract point of view here.",
"We first fix a preferred basis in ${\\mathbb {C}}^{2N}$ and make the following definition.",
"Definition 1 A transfer matrix for a wire of width $N$ is a $2N\\times 2N$ pseudo-unitary matrix, ${\\mathcal {M}}^{*}\\Sigma _z {\\mathcal {M}}= \\Sigma _z\\,,\\qquad \\text{where}\\qquad \\Sigma _z = \\begin{pmatrix} 1 & 0 \\\\ 0 & -1\\end{pmatrix}\\,.$ Furthermore, a transfer matrix ${\\mathcal {M}}$ is time reversal invariant if $\\Sigma _x {\\mathcal {M}}\\Sigma _x = \\overline{{\\mathcal {M}}}\\,,\\qquad \\text{where}\\qquad \\Sigma _x = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}\\,,$ and $\\overline{{\\mathcal {M}}}$ is the complex conjugate of ${\\mathcal {M}}$ .",
"Transfer matrices have a simple multiplicative composition rule.",
"If ${\\mathcal {M}}_1$ and ${\\mathcal {M}}_2$ are transfer matrices for two wires, then ${\\mathcal {M}}_2{\\mathcal {M}}_1$ is the transfer matrix for the composite system obtained from gluing the two pieces in series.",
"In view of (REF , REF ), it is natural to write a transfer matrix in block form ${\\mathcal {M}}= \\begin{pmatrix} {\\mathcal {M}}_{++} & {\\mathcal {M}}_{+-} \\\\ {\\mathcal {M}}_{-+} & {\\mathcal {M}}_{--} \\end{pmatrix}\\,.$ Combining (REF ) and the singular value decompositions ${\\mathcal {M}}_{++} = U_+ S_+ V_+$ and ${\\mathcal {M}}_{--} = U_- S_- V_-$ of the diagonal blocks, we obtain the following factorization ${\\mathcal {M}}= \\begin{pmatrix} U_+ & 0 \\\\ 0 & U_- \\end{pmatrix} \\begin{pmatrix} S & (S^2-1)^{1/2} \\\\ (S^2-1)^{1/2} & S \\end{pmatrix} \\begin{pmatrix} V_+ & 0 \\\\ 0 & V_- \\end{pmatrix}\\,.$ where $S=S_+=S_-$ .",
"If, moreover, time reversal invariance is imposed, then $U_- = \\overline{U_+}$ and $V_- = \\overline{V_+}$ .",
"Let $r$ be the reflection and $t$ be the transmission matrices, defined through ${\\mathcal {M}}\\begin{pmatrix} 1 \\\\ r \\end{pmatrix} = \\begin{pmatrix} t \\\\ 0 \\end{pmatrix}$ In particular, $t = {\\mathcal {M}}_{++}-{\\mathcal {M}}_{+-}{\\mathcal {M}}_{--}^{-1}{\\mathcal {M}}_{-+}= \\left({\\mathcal {M}}_{++}^{*}\\right)^{-1}\\,,$ where we used (REF ) in the second equality.",
"The so-called transmission eigenvalues $(T_k)_{k=1}^N$ are defined as the eigenvalues of the matrix $t^{*}t$ , i.e.",
"of $({\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++})^{-1}$ .",
"Hence, the transmission eigenvalues are also the inverses of the squares of the singular values contained in $S$ .",
"Let $T = S^{-2}$ be the diagonal matrix of transmission eigenvalues.",
"Many transport properties of the disordered wire can be expressed as functions thereof.",
"In particular, the conductance $g$ is given in units of $2e^2 / \\hbar $ by the Landauer-Büttiker formula [5], $g:= \\sum _{i=1}^N T_i = \\mathrm {Tr}\\, T = \\mathrm {Tr}\\, t^{*}t\\,,$ a formula that we accept here as a definition of $g$ ."
],
[
"The DMPK Theory",
"The DMPK theory introduced by [9] and independently by [16] is an evolution equation for the transfer matrix ${\\mathcal {M}}(r,s)$ of a wire on $[r,s]$ .",
"By the composition rule, for any $s_1\\le s_2$ , ${\\mathcal {M}}(0,s_2) ={\\mathcal {M}}(s_1,s_2) {\\mathcal {M}}(0,s_1)\\,,$ with ${\\mathcal {M}}(s,s)=1$ .",
"The first crucial idea is to take $s_2-s_1$ infinitesimal and write ${\\mathcal {M}}(s,s+\\mathrm {d}s) \\sim 1 + \\mathrm {d}{\\mathcal {L}}(s)$ such that $\\mathrm {d}{\\mathcal {L}}(s)$ is independent of ${\\mathcal {M}}(s)$ and contains only diffusive terms but no drift.",
"Mathematically, this translates into the assumption that ${\\mathcal {M}}(s)$ satisfies an Itô stochastic differential equation (SDE) $\\begin{split}\\mathrm {d}{\\mathcal {M}}(s) &= \\mathrm {d}{\\mathcal {L}}(s) {\\mathcal {M}}(s)\\,,\\\\{\\mathcal {M}}(0)&=1\\,,\\end{split}$ where ${\\mathcal {L}}(s)$ is a matrix valued Brownian motion and ${\\mathcal {L}}(0) = 0$ .",
"Lemma 1 Let ${\\mathcal {M}}(s)$ be a solution of the SDE (REF ).",
"Assume that $\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z + \\Sigma _z \\mathrm {d}{\\mathcal {L}}&= 0\\,, \\\\\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z \\mathrm {d}{\\mathcal {L}}&= 0\\,.",
"$ Then ${\\mathcal {M}}(s)$ is pseudo unitary, eq.",
"(REF ).",
"If moreover $\\Sigma _x \\mathrm {d}{\\mathcal {L}}\\Sigma _x = \\overline{\\mathrm {d}{\\mathcal {L}}}\\,,$ then ${\\mathcal {M}}(s)$ is also time reversal invariant, eq.",
"(REF ).",
"For the first part, we take the differential of (REF ), use (REF ) and Itô calculus to obtain ${\\mathcal {M}}^{*}\\left(\\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z + \\Sigma _z \\mathrm {d}{\\mathcal {L}}+ \\mathrm {d}{\\mathcal {L}}^{*}\\Sigma _z \\mathrm {d}{\\mathcal {L}}\\right){\\mathcal {M}}= 0\\,,$ which holds if and only if both (REF ) and () hold as ${\\mathcal {M}}$ is nonsingular.",
"Similarly, the differential of (REF ) immediately yields (REF ).",
"Secondly, the DMPK theory prescribes a particular invariance of the distribution of $\\mathrm {d}{\\mathcal {L}}(s)$ .",
"The law of the increments $\\mathrm {d}{\\mathcal {L}}(s)$ shall be independent of $s$ and maximally isotropic in the sense that ${\\mathcal {W}}^{*}\\mathrm {d}{\\mathcal {L}}\\, {\\mathcal {W}}\\mathop {=}\\limits ^{d} \\mathrm {d}{\\mathcal {L}}\\qquad \\text{for any unitary}\\qquad {\\mathcal {W}}= \\begin{pmatrix} W_+ & 0 \\\\ 0 & W_- \\end{pmatrix}\\,.$ The unitary blocks $W_\\pm $ are independent of each other if ${\\mathcal {M}}$ does not exhibit any symmetry, whereas $W_- = \\overline{W_+}$ if time reversal symmetry is imposed.",
"For notational simplicity, we cast ${\\mathcal {L}}$ in block form, ${\\mathcal {L}}(s) = \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\ \\mathfrak {b}(s)^{*}& \\mathfrak {a}^{\\prime }(s) \\end{pmatrix}$ where $\\mathfrak {a}(s),\\mathfrak {a}^{\\prime }(s), \\mathfrak {b}(s)$ are independent local martingales, with $\\mathfrak {a}(s) = -\\mathfrak {a}(s)^{*}$ , similarly for $\\mathfrak {a}^{\\prime }(s)$ , and $\\mathrm {d}\\mathfrak {a}^{*}\\mathrm {d}\\mathfrak {a} = \\mathrm {d}\\mathfrak {b}\\mathrm {d}\\mathfrak {b}^{*}= \\mathrm {d}\\mathfrak {a}^{\\prime }{\\mathrm {d}\\mathfrak {a}^{\\prime }}^{*}\\,.$ The isotropy assumption (REF ) reduces to invariance conditions on the blocks.",
"First, $ \\mathfrak {a}_{ij}(s) = {\\left\\lbrace \\begin{array}{ll}1 / \\sqrt{2N} \\cdot (B^R_{ij}(s) + \\mathrm {i}B^I_{ij}(s)) & 1 \\le i < j \\le N \\\\\\mathrm {i}/ \\sqrt{N} \\cdot B^I_{ii}(s) & i=j \\\\-\\overline{\\mathfrak {a}_{ji}(s)} & \\text{otherwise}\\end{array}\\right.",
"}\\,,$ where $B^R$ and $B^I$ are independent real standard Brownian motions, and similarly but independently for $\\mathfrak {a}^{\\prime }(s)$ .",
"Secondly, $ \\mathfrak {b}_{ij}(s) = 1 / \\sqrt{2N} \\cdot (\\tilde{B}^R_{ij}(s) + \\mathrm {i}\\tilde{B}^I_{ij}(s))\\,,\\quad \\text{for all }i,j\\,.$ Note that the relative normalization of $\\mathfrak {a}(s)$ and $\\mathfrak {b}(s)$ are fixed by pseudounitarity, i.e.",
"(REF ).",
"In the time reversal invariant case, the matrix $\\mathfrak {a}(s)$ does not change, but $\\mathfrak {a}^{\\prime }(s) = \\overline{\\mathfrak {a}(s)}\\,,$ and $\\mathfrak {b}(s)$ becomes symmetric, $\\mathfrak {b}(s)^{*}= \\overline{\\mathfrak {b}(s)}$ with real and imaginary parts orthogonally invariant, namely $ \\mathfrak {b}_{ij}(s) = {\\left\\lbrace \\begin{array}{ll}1 / \\sqrt{2(N+1)} \\cdot (B^R_{ij}(s) + \\mathrm {i}B^I_{ij}(s)) & 1 \\le i < j \\le N \\\\1/\\sqrt{N+1} \\cdot (B^R_{ii}(s) + \\mathrm {i}B^I_{ii}(s)) & i=j \\\\\\mathfrak {b}_{ji}(s) & \\text{otherwise}\\end{array}\\right.",
"}\\,.$ Here again, the relative factor $\\sqrt{N/(N+1)}$ is imposed by (REF ).",
"From a physical point of view, the DMPK theory's interest lies in its predictions for the statistics of the transmission eigenvalues.",
"Indeed, the unitary invariance of the increments $\\mathrm {d}{\\mathcal {L}}$ implies that the set of $T_k$ satisfies an autonomous equation, which can be formally derived by Itô calculus from the matrix SDE (REF ): $\\begin{split}\\mathrm {d}T_k(s)&=v_k(T(s))\\mathrm {d}s+D_k(T(s))\\mathrm {d}B_k(s),\\\\T_k(0)&=1,\\end{split}$ for all $k=1,...,N$ .",
"The Brownian motions $B_k$ are independent, and the drift and diffusion coefficients are given explicitly by $\\begin{split}v_k&=-T_k+\\frac{2T_k}{\\beta N+2-\\beta }\\left(1-T_k+\\frac{\\beta }{2}\\sum _{j\\ne k}\\frac{T_k+T_j-2T_kT_j}{T_k-T_j}\\right)\\,,\\\\D_k&=\\sqrt{4\\frac{T_k^2(1-T_k)}{\\beta N+2-\\beta }}.\\end{split}$ The first term in the drift $v_k$ contracts all transmission eigenvalues towards 0 as the length $s$ of the wire increases.",
"However, and similarly to Dyson's Brownian motion, the drift also contains repulsion terms originating from second order perturbation theory.",
"As a consequence, the eigenvalues $T_k$ `try to avoid' degeneracy.",
"What makes a naive derivation formal is that Itô's formula is only applicable if the denominator $T_k-T_j$ never becomes singular, i.e.",
"${\\mathcal {M}}^*_{++}(s){\\mathcal {M}}_{++}(s)$ never has degenerate eigenvalues.",
"This is a nontrivial property for $s>0$ , and even more so as $s\\rightarrow 0^+$ since (REF ) starts with the completely degenerate ${\\mathcal {M}}(0)=1$ .",
"Both issues can however be tackled and the SDE (REF ) has a unique weak and strong solution, see [6].",
"Finally, let us comment on some deeper principles underlying the process ${\\mathcal {M}}$ and the resulting DMPK equation.",
"The maximal isotropy assumption that was used above, can be derived from a simple `maximal entropy assumption' on the set of infinitesimal transfer matrices $1+\\mathrm {d}{\\mathcal {L}}(s)$ that have a fixed 'scattering strength' $\\sum _k T_k$ .",
"Alternatively, as remarked by [12], one can also guess the DMPK equation from geometric considerations, since it is the radial part of the canonical Brownian motion on a certain symmetric space.",
"The reduction from Lie group to symmetric space is obtained by identifying certain transfer matrices that, in particular, have the same transmission eigenvalues.",
"This geometric approach was very fruitful.",
"For example, in [4] it was shown how it naturally explains the appearance of non-universal conductance properties in wires with off-diagonal disorder."
],
[
"Ohm's law",
"In the context of the DMPK theory, a treatment, or even proof, of the conjectures mentioned in the introduction is possible, as already indicated in Section REF .",
"The existing approaches rely on explicit calculations and are quite intricate.",
"Nevertheless, if one is solely after Ohm's law (and not the universal conductance fluctuations (UCF)), there is an appealing and compact approach by [17].",
"Below we present a rigorous version of this approach.",
"The following theorem shows that in the large $N$ limit, the rescaled moments of the conductance have an Ohmic behavior.",
"In particular, Conjecture 1 holds for the TOE, TUE and TSE.",
"We note that the symmetry index $\\beta $ drops out in that particular scaling.",
"Theorem 2 (Ohm's law) Let $\\left(T_k(s)\\right)_{k=1}^N$ be the solution of the DMPK process (REF ), and let $g_N(s) = \\sum _{k=1}^N T_k(s)\\,.$ Then for all $p\\ge 1$ and $T>0$ , $ \\lim _{N\\rightarrow \\infty }\\frac{{\\mathbb {E}}(g^p_N(s))}{N^p} = \\frac{1}{(1+s)^p}$ uniformly for $s\\in [0,T]$ .",
"The proof goes through a sequence of lemmas.",
"Lemma 3 Let $g_N^{(j)} = \\sum _k T_k^j$ for $j>1$ .",
"Then, for any $p\\ge 1$ , $ \\frac{\\mathrm {d}}{\\mathrm {d}s}{\\mathbb {E}}(g_N^p) = -p\\gamma _N(\\beta )\\left[{\\mathbb {E}}(g_N^{p+1}) - \\left(1-\\frac{2}{\\beta }\\right){\\mathbb {E}}(g_N^{p-1} g_N^{(2)}) -\\frac{2(p-1)}{\\beta }{\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)}))\\right].$ where $\\gamma _N(\\beta ) = \\frac{\\beta }{\\beta N +2 -\\beta }.$ Itô's formula yields $\\mathrm {d}(g_N^p) = \\sum _{i_1,\\ldots ,i_p = 1}^N \\left[\\sum _{j=1}^p T_{i_1}\\cdots dT_{i_j}\\cdots T_{i_p} + \\sum _{j\\ne k=1}^p T_{i_1}\\cdots dT_{i_j}\\cdots dT_{i_k}\\cdots T_{i_p}\\right]$ so that $ \\frac{\\mathrm {d}}{\\mathrm {d}s}{\\mathbb {E}}(g_N^p) = p{\\mathbb {E}}\\left(g_N^{p-1}\\sum _k v_k\\right) + \\frac{p(p-1)}{2}{\\mathbb {E}}\\left(g_N^{p-2}\\sum _k D_k^2\\right)\\,.$ where we used the DMPK equation (REF ).",
"In order to rewrite the right-hand side, note the simple identity $\\sum _k T_k \\sum _{j\\ne k}\\frac{T_k+T_j-2T_kT_j}{T_k-T_j} = \\sum _k\\sum _{j < k}(T_k+T_j-2T_kT_j) = (N-1) g_N - g_N^2 + \\sum _k T_k^2$ Therefore, $\\sum _k v_k &= -g_N + \\frac{2}{\\beta N +2 -\\beta }\\left[g_N - \\sum _k T_k^2 + (\\beta /2)\\left((N-1) g_N - g_N^2 + \\sum _k T_k^2\\right) \\right] \\nonumber \\\\&=-\\gamma _N(\\beta )\\left( g_N^2 - (1-2/\\beta )g_N^{(2)}\\right)\\,, $ and the lemma follows upon substituting this in (REF ).",
"Let us now consider $\\Psi _{N}(p,s) :=\\frac{{\\mathbb {E}}(g^p_N(s))}{N^p}\\,,$ The following properties are immediate from the definition and the differential equation (REF ) $\\Psi _{N}(p,0)=1$ .",
"$ |\\Psi _{N}(p,s) |\\le 1$ .",
"The function $s \\mapsto \\Psi _{N}(p,s)$ is continuously differentiable and $ |\\frac{\\partial \\Psi _{N}}{\\partial s}(p,s) |\\le c(p) < \\infty $ .",
"We consider the Banach space ${\\mathcal {L}}={\\mathcal {C}}([0,T], {\\mathbb {R}})$ for some $T>0$ , equipped with the supremum norm.",
"Let ${\\mathcal {L}}_p, p=1,2,\\ldots $ be copies of ${\\mathcal {L}}$ and define the cartesian product ${\\mathcal {K}}= \\mathop {\\times }\\limits _{p=1}^{\\infty } {\\mathcal {L}}_p$ , equipped with the product topology.",
"Lemma 4 There is an increasing sequence $N_n, n \\in {\\mathbb {N}}$ and an element $\\Psi \\in {\\mathcal {K}}$ such that for each $p$ , $\\Psi _{N_n}(p,\\cdot ) \\rightarrow \\Psi (p,\\cdot ) $ in ${\\mathcal {L}}$ , as $n \\rightarrow \\infty $ .",
"For $p=1,2,\\ldots $ , consider the sets $ {\\mathcal {S}}_p= \\left\\lbrace \\Psi _{N}(p,\\cdot ), N \\in {\\mathbb {N}}\\right\\rbrace \\subset {\\mathcal {L}}_p.$ By the properties $\\mathrm {i,ii,iii}$ above and Arzela-Ascoli's theorem, each of these sets is sequentially compact (s.c.).",
"Countable products of s.c. sets are s.c. in the product topology (sequential Tychonov's theorem), hence $\\times _p{\\mathcal {S}}_p \\subset {\\mathcal {K}}$ is s.c.",
"Therefore, the sequence $\\Psi _N \\subset \\times _p{\\mathcal {S}}_p$ has a convergent subsequence.",
"Since convergence in the product topology implies convergence for any $p$ , the lemma follows.",
"Lemma 5 Any limit point $\\Psi \\in {\\mathcal {K}}$ as in Lemma REF satisfies $\\Psi (p,s_2)-\\Psi (p,s_1) = - p \\int _{s_1}^{s_2} \\mathrm {d}s \\Psi (p+1,s), \\qquad \\Psi (p,0) = 1$ The equation (REF ) is rewritten as $ \\Psi _N(p,s_2)-\\Psi _N(p,s_1)= -p (1-r_1(N)) \\int _{s_1}^{s_2} \\mathrm {d}s \\left[\\Psi _N(p+1,s) - r_2(N,s,p) \\right]$ where $r_2$ is the sum of the second and third term between square brackets in (REF ) and $r_1(N) &= 1-N\\gamma _{N} = {\\mathcal {O}}(1/N), \\\\r_2(N,s,p) &= \\left(1-\\frac{2}{\\beta }\\right)\\frac{{\\mathbb {E}}(g_N^{p-1} g_N^{(2)})}{N^{p+1}} + \\frac{2(p-1)}{\\beta }\\frac{{\\mathbb {E}}(g_N^{p-2} (g_N^{(2)}-g_N^{(3)})}{N^{p+1}} = {\\mathcal {O}}(1/N)$ with the bounds ${\\mathcal {O}}(1/N)$ uniform in $s$ but not necessarily in $p$ .",
"The lemma follows by considering (REF ), for fixed $p$ , along the sequence ${N_n}$ .",
"Sloppily put, the above two lemmas show that $\\Psi _N(p,\\cdot )$ converges to a solution of the `limiting hierarchy of equations' (REF ).",
"It remains to prove that the limiting hierarchy has a unique solution, namely the right-hand side of (REF ).",
"Therefore, the proof of Theorem REF is completed by the next lemma.",
"Lemma 6 $\\Psi (p,s): = \\frac{1}{(1+s)^p}$ is the unique element in ${\\mathcal {K}}$ that satisfies (REF ) and $\\sup _p ||\\Psi (p,\\cdot ) ||_\\infty \\le 1$ .",
"We proceed by induction in the interval length $T$ .",
"Assume that the claim is proven for $T\\ge 0$ (for $T=0$ it is trivial).",
"Take then $0 \\le s_1 \\le T$ and $ s_1< s_2 <s_1+1$ .",
"We choose a $\\Psi (p,\\cdot )$ satisfying (REF ) and $p\\ge 1$ .",
"We iterate (REF ) $k$ times to obtain $\\Psi (p,s_2)- \\Psi (p,s_1) &= \\sum _{j=1}^k \\frac{(s_2-s_1)^j}{j!}",
"a(j,p) \\Psi (p+j,s_1) \\\\&\\quad + \\int _0^{s_2-s_1} \\mathrm {d}t_1\\cdots \\int _0^{t_{k}}\\mathrm {d}t_{k+1} a(k+1,p) \\Psi (p+k+1,s_1+t_{k+1}) \\,.$ where $a(k,p) = (-1)^k(p+k-1)!",
"/ (p-1)!$ .",
"By the induction hypothesis, $\\Psi (p+j,s_1)= (1+s_1)^{-(p+j)}$ and hence the sum on the right-hand side is the $k$ th order Taylor polynomial of the function $s \\mapsto (1+s)^{-p}$ at $s=s_1$ .",
"The series is absolutely convergent for $s_2-s_1 < 1+s_1$ .",
"Upon using $\\sup _p ||\\Psi (p,\\cdot ) ||_\\infty \\le 1$ , the second term is bounded as $\\int _0^{s_2-s_1} \\mathrm {d}t_1\\cdots \\int _0^{t_{k}}\\mathrm {d}t_{k+1} |a(k+1,p) |\\le \\frac{(s_2-s_1)^{k+1}}{(k+1)!}",
"\\frac{(p+k)!}{(p-1)!",
"}$ which converges to zero as $k\\rightarrow \\infty $ whenever $s_2-s_1<1$ .",
"Therefore $ \\Psi (p,s_2)=(1+s_2)^{-p}$ , completing the induction step."
],
[
"A microscopic model",
"The DMPK theory is a macroscopic theory based on few symmetry assumptions, but does not refer to any particular physically relevant microscopic (Hamiltonian) model.",
"We now introduce a concrete quantum lattice model with disorder, identify the physical symmetries, define the corresponding ensemble of transfer matrices and study its properties for long wires.",
"In the relevant weak coupling limit, we shall derive the stochastic differential equation to be compared with the DMPK evolution."
],
[
"The Hamiltonian; symmetries and spectrum",
"The system is an infinitely extended wire, modeled by the Hilbert space ${\\mathcal {H}}= l^2 ({\\mathbb {Z}}\\times {\\mathbb {Z}}_N ) = l^2({\\mathbb {Z}})\\otimes {\\mathbb {C}}^N\\,.$ A vector $\\Psi \\in {\\mathcal {H}}$ is a sequence $\\Psi (x,z)$ , with longitudinal coordinate $x$ and transverse coordinate $z$ , or rather $\\Psi _x(z)$ if we prefer to think of a ${\\mathbb {C}}^N$ -valued sequence.",
"The Hamiltonian $H = H_{\\mathrm {kin}} + \\lambda V$ has a deterministic, translation invariant kinetic term and a random on-site potential $(V\\Psi )(x,z) = V(x,z)\\Psi (x,z)\\,.$ The disorder is limited to a finite region, namely $V(x,z) = 0$ for $x\\notin \\lbrace 1,\\ldots ,L\\rbrace $ .",
"The non vanishing elements $V(x,z)$ are i.i.d.",
"real random variables, with ${\\mathbb {E}}(V(x,z)) = 0$ and normalized to have ${\\mathbb {E}}(V(x,z)^2) = 1$ so that the strength of the disorder is exclusively controlled by the parameter $\\lambda $ .",
"The specific form of the kinetic Hamiltonian will only play a role in determining the symmetry class to which the system belongs, and could therefore be left essentially open, up to these limited symmetry requirements.",
"For simplicity and definiteness, we shall however make here a particular choice that allows for an explicit tracking of the symmetries and their consequences.",
"Henceforth $H_\\mathrm {kin}$ will describe a nearest neighbor and diagonal hopping in the presence of a magnetic field in the longitudinal direction, namely $\\left(H_\\mathrm {kin}\\Psi \\right)(x,z) = \\Psi (x+1,z) + \\Psi (x-1,z) \\\\+ h_1 \\left[ \\mathrm {e}^{\\mathrm {i}\\gamma }\\Psi (x,z+1) + \\mathrm {e}^{-\\mathrm {i}\\gamma }\\Psi (x,z-1) \\right] + h_2 \\left[ \\mathrm {e}^{\\mathrm {i}\\gamma }\\Psi (x-1,z+1) + \\mathrm {e}^{-\\mathrm {i}\\gamma }\\Psi (x+1,z-1)) \\right] $ where $h_1,h_2>0$ and $0\\le \\gamma < 2\\pi / N$ .",
"Note that as indicated in the definition of $\\mathcal {H}$ , periodic boundary conditions in the transverse direction are imposed on all operators.",
"As we have briefly discussed in Section , the DMPK equation comes in various guises, depending on the abstract symmetry group of the transmission matrix.",
"At the microscopic level, time reversal $T$ is naturally implemented by complex conjugation in the `position basis', namely $(T\\Psi )(x,z) = \\overline{\\Psi (x,z)}\\,.$ It is immediate to check that $H_\\mathrm {kin}$ is invariant under $T$ iff $\\gamma =0$ : the magnetic field indeed breaks time reversal invariance."
],
[
"Eigenvalues, eigenvectors and chaoticity",
"We consider the eigenvalue problem for the kinetic Hamiltonian $H_{\\mathrm {kin}}\\Psi =E\\Psi $ at some fixed energy $E$ .",
"By translation invariance and the periodic boundary conditions, the (non-normalizable) solutions are given by $\\Psi _{k,\\nu }(x,z)=\\frac{1}{\\sqrt{N}}e^{ik x}e^{\\frac{2\\pi i}{N}\\nu z}$ for $\\nu =1,\\ldots ,N$ , with $E =E(k,\\nu )= 2\\cos (k) + 2h_1\\cos \\left(\\gamma + \\frac{2\\pi }{N}\\nu \\right) + 2 h_2\\cos \\left(k - \\gamma - \\frac{2\\pi }{N}\\nu \\right)$ We now look for the condition on $E$ so that (REF ) has plane wave solutions, rather than exponentially decaying ones, i.e.",
"such that $E=E(k,\\nu )$ for some $k,\\nu $ .",
"In physical terms, this means that we do not want to study evanescent modes, also called `elliptic channels'.",
"In fact, we first fix the energy $E$ and consider solutions $k = k_\\nu (E)$ of (REF ) for any `transversal mode' $\\nu $ .",
"We shall drop the $E$ dependence in the sequel.",
"There are two such wavevectors $k_\\nu ^\\pm $ corresponding to a right moving and a left moving wave.",
"The relations $ k_\\nu ^+ = -k_{-\\nu }^-$ holds in the time reversal invariant case, but is broken if $\\gamma >0$ .",
"If $h_2=0$ , the residual symmetry $\\Psi (x,z) \\mapsto \\Psi (x,-z)$ induces the additional degeneracy $k_\\nu ^+ = -k_\\nu ^-$ that needs to be avoided.",
"It is a straightforward exercise to check Lemma 7 For an energy $E\\ne 0$ , $|E|<2$ , and a kinetic Hamiltonian $H_{\\mathrm {kin}}$ with parameters $0\\le \\gamma <\\frac{\\pi }{N}$ , and $h_1,h_2>0$ sufficiently small, in particular $|E|+2h_1+2h_2<2\\,,$ the equation $H_{\\mathrm {kin}}\\Psi =E\\Psi $ has $2N$ plane wave solutions $\\Psi _{k,\\nu }(x,z)=\\frac{1}{\\sqrt{N}}e^{ik_\\nu ^{\\sigma } x}e^{\\frac{2\\pi i}{N}\\nu z}$ with $\\nu \\in {\\mathbb {Z}}_N$ , $\\sigma \\in \\lbrace {+,-\\rbrace }$ .",
"For $\\gamma \\ne 0$ , the longitudinal wave numbers $k_\\nu ^{\\sigma }$ are non degenerate in the sense that $\\varsigma _1 k_{\\nu (1)}^{\\sigma _1}+\\varsigma _2 k_{\\nu (2)}^{\\sigma _2}+\\varsigma _3 k_{\\nu (3)}^{\\sigma _3}+\\varsigma _4 k_{\\nu (4)}^{\\sigma _4}=0\\mbox{ }(\\mathrm {mod}\\mbox{ }2\\pi )$ for signs $\\varsigma _1,...,\\varsigma _4$ only in the trivial case $\\big (\\varsigma _1, \\sigma _1, \\nu (1)\\big ) = \\big (-\\varsigma _2, \\sigma _2, \\nu (2)\\big )\\quad \\text{and}\\quad \\big (\\varsigma _3, \\sigma _3, \\nu (3)\\big ) = \\big (-\\varsigma _4, \\sigma _4, \\nu (4)\\big )\\,,$ and if $\\gamma =0$ also in the $T$ -symmetric situation $\\big (\\varsigma _1, \\sigma _1, \\nu (1)\\big ) = \\big (\\varsigma _2, -\\sigma _2, -\\nu (2)\\big )\\quad \\text{and}\\quad \\big (\\varsigma _3, \\sigma _3, \\nu (3)\\big ) = \\big (\\varsigma _4, -\\sigma _4, -\\nu (4)\\big )\\,,$ all of this up to relabeling of the indices.",
"The above lemma shows that the Hamiltonian has no other symmetry than time reversal invariance, in accordance with the macroscopic theory of the previous section.",
"It is important to realize that a residual symmetry of the kinetic Hamiltonian could, and in fact would, leave a trace in the scaling limit, even if that symmetry got broken by the disorder.",
"The DMPK theory considers a single ensemble of transfer matrices, whereas there are really two microscopic models, one without and one with disorder.",
"For our derivation, it is essential that both have the correct symmetry properties.",
"For further considerations, it is useful to define the chaoticity of the kinetic Hamiltonian in analogy to [20] by $\\mathrm {cha}\\left(\\gamma ,h_1,h_2\\right)=\\min \\left\\lbrace \\left|\\sum _{i=1}^4 \\varsigma _i k_{\\nu (i)}^{\\sigma _i}\\right|, ({\\varsigma },{\\sigma },{\\nu })\\mbox{ not solving } (\\ref {exceptionTRI}) \\right\\rbrace \\,,$ for $\\gamma >0$ and $\\mathrm {cha}\\left(0,h_1,h_2\\right)=\\min \\left\\lbrace \\left|\\sum _{i=1}^4 \\varsigma _i k_{\\nu (i)}^{\\sigma _i}\\right|,({\\varsigma },{\\sigma },{\\nu })\\mbox{ not solving } (\\ref {exceptionTRI})\\mbox{ or } (\\ref {exception})\\right\\rbrace .$ if $\\gamma =0$ ."
],
[
"The transfer matrix",
"We decompose the kinetic Hamiltonian $(H_\\mathrm {kin}\\Psi )_x= H_\\perp \\Psi _x + P \\Psi _{x+1} + P^* \\Psi _{x-1}$ as a strictly transverse operator $(H_\\perp \\phi )(z) = h_1 \\left( \\mathrm {e}^{\\mathrm {i}\\gamma }\\phi (z+1) + \\mathrm {e}^{-\\mathrm {i}\\gamma }\\phi (z-1) \\right)\\,,$ and the components inducing hopping in the longitudinal direction $(P\\phi )(z) = \\phi (z) + h_2 \\mathrm {e}^{\\mathrm {i}\\gamma } \\phi (z+1)\\,,\\quad \\text{and}\\quad (P^{*}\\phi )(z) = \\phi (z) + h_2 \\mathrm {e}^{-\\mathrm {i}\\gamma } \\phi (z-1)\\,,$ where $\\phi (z)\\in {\\mathbb {C}}^N$ .",
"Similarly, the random potential can be seen as a sequence of $N\\times N$ diagonal matrices $V_x$ .",
"With these notations, the eigenvalue equation with disorder, $\\lambda >0$ , reads $\\begin{pmatrix}\\Psi _{x+1}\\\\\\Psi _{x}\\end{pmatrix}=\\begin{pmatrix}\\left(P^*\\right)^{-1}(E-H_\\perp -\\lambda V_x)&-\\left(P^*\\right)^{-1}P\\\\{1}&{0}\\end{pmatrix}\\begin{pmatrix}\\Psi _{x}\\\\\\Psi _{x-1}\\end{pmatrix}=:T_x^\\lambda \\begin{pmatrix}\\Psi _{x}\\\\\\Psi _{x-1}\\end{pmatrix},$ thereby defining the transfer matrix $T_x^\\lambda $ (of dimension $2N$ ) for the layer $x$ as it usually appears in the mathematical literature.",
"By the multiplicative property, the transfer matrix for $L$ layers is simply given by the product of the one-layer matrices, $\\begin{pmatrix}\\Psi _{L+1}\\\\\\Psi _{L}\\end{pmatrix}=T_L^\\lambda \\cdots T_1^\\lambda \\begin{pmatrix}\\Psi _{1}\\\\\\Psi _{0}\\end{pmatrix},$ In order to allow for a comparison with the DMPK theory, it is important to write the transfer matrix in the natural basis for ${\\mathbb {C}}^{2N}$ , i.e.",
"the basis in which the first $N$ components correspond to left moving waves, and the other $N$ components are right moving.",
"Morever, in the time-reversal invariant case we need the correct identification between left and right moving channels, given by (REF ).",
"In other words, the time reversal of the vector $(\\phi _1, \\phi _2)^t$ must correspond to $(\\bar{\\phi }_2, \\bar{\\phi }_1)^t$ .",
"Only in that basis do the definitions of the transfer matrix given in Section REF apply.",
"We shall refer to it as the channel basis.",
"Let $Q$ be the $N\\times N$ matrix whose columns are the transverse eigenvectors $\\phi _\\nu (z) = 1/\\sqrt{N}\\exp (2\\pi \\mathrm {i}\\nu z/N)$ .",
"Furthermore, let $\\Pi $ be the permutation matrix that interchanges the $\\nu $ and $-\\nu $ channels, and let $\\Upsilon :=\\begin{pmatrix}\\frac{\\mathrm {e}^{\\mathrm {i}k^+}}{\\sqrt{\\left|v^+\\right|}}&\\frac{\\mathrm {e}^{\\mathrm {i}k^-}}{\\sqrt{\\left|v^-\\right|}}\\\\\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^-\\right|}}\\end{pmatrix}\\begin{pmatrix}1&0\\\\0&\\Pi \\end{pmatrix}$ where $k^\\pm $ and $v^\\pm $ denote diagonal matrices with elements $k^\\pm _{\\nu }$ and $v^\\pm _\\nu $ , and $v^\\pm _\\nu $ are the velocities in each channel, $v^\\pm _\\nu =\\left.\\frac{\\partial }{\\partial k}E(k,\\nu )\\right|_{k=k_\\nu ^\\pm }=-2\\sin \\left(k^\\pm _\\nu \\right)-2h_2\\sin \\left(k^\\pm _\\nu - \\frac{2\\pi }{N}\\nu - \\gamma \\right).$ Note that all $v^\\pm _\\nu $ are nonzero by (REF ).",
"The action of $T$ (complex conjugation) on the eigenvectors yields $T:\\,(k_\\nu ^\\sigma ,\\nu )\\mapsto (-k_\\nu ^\\sigma ,-\\nu )$ which equals $(k_{-\\nu }^{-\\sigma },-\\nu )$ if time reversal invariance holds, so that the $(\\sigma = -)$ block needs to be reordered by $\\nu \\leftrightarrow -\\nu $ .",
"Hence the permutation $\\Pi $ .",
"With these notations, Lemma REF reads $ \\Upsilon ^{-1}(Q\\otimes 1)^{*}T_x^0(Q\\otimes 1)\\Upsilon = \\begin{pmatrix}\\mathrm {e}^{\\mathrm {i}k^+}&0\\\\0&\\Pi \\mathrm {e}^{\\mathrm {i}k^-} \\Pi \\end{pmatrix} =: M_x^0\\,.$ The matrix $M_x^0$ is the transfer matrix for the purely kinetic transport in a single layer, which is diagonal the channel basis.",
"As expected, it simply adds a phase to the traveling plane wave.",
"It is now easily checked that $M_x^0$ is a bonafide transfer matrix in the sense of Definition REF : Eq.",
"(REF ) is always satisfied and (REF ) holds at $\\gamma =0$ , as conjugation by $\\Sigma _x$ precisely maps $k^\\sigma _\\nu $ to $k^{-\\sigma }_{-\\nu }=-k^\\sigma _{\\nu }$ .",
"The transfer matrix for the total system is again obtained by multiplication $M^\\lambda (L) := M^\\lambda _L \\cdots M^\\lambda _1 = K^{-1} \\, T_L^\\lambda \\cdots T_1^\\lambda \\, K\\,,$ where $K = (Q\\otimes 1)\\Upsilon $ is the change of basis from the position basis to the channel basis.",
"This random matrix and its relation the DMPK theory is the central object of study in the following."
],
[
"A scaling limit",
"The DMPK theory suggests that the microscopic transfer matrix for a disordered wire of length $L$ should converge to a solution of the DMPK equation in the correct macroscopic limit.",
"As discussed in the introduction, the natural scaling between the microscopic length $L$ and the macroscopic length $s$ is through the mean free path, $L = \\lambda ^{-2}s$ .",
"A naive interpretation of the DMPK theory would then be the convergence of $M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor )$ to ${\\mathcal {M}}(s)$ .",
"This cannot possibly hold as $M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor )$ contains rapidly oscillating terms as a function of $s$ , as already exemplified at the level of the unperturbed system (REF ): $M^0(\\lfloor \\lambda ^{-2}s \\rfloor )=\\begin{pmatrix}\\exp \\left(\\mathrm {i}\\lfloor \\lambda ^{-2}s \\rfloor k^+\\right)&0\\\\0&\\Pi \\exp \\left(\\mathrm {i}\\lfloor \\lambda ^{-2}s \\rfloor k^-\\right) \\Pi \\end{pmatrix}\\,.$ To obtain a reasonable limit, we therefore consider $A^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor ) := (M^0(\\lfloor \\lambda ^{-2}s \\rfloor ))^{-1}M^\\lambda (\\lfloor \\lambda ^{-2}s \\rfloor ).$ As the set of matrices of Definition REF form a group, $A^\\lambda $ is a transfer matrix again.",
"Moreover, it is an easy check that in the polar decomposition (REF ), the matrix $S$ corresponding to $A^\\lambda $ is the same as that corresponding to $M^\\lambda $ , so that they have the same transmission eigenvalues.",
"In order to state the result of the scaling limit, $\\lambda \\rightarrow 0$ , we introduce the following processes.",
"For $\\gamma >0$ , let ${\\mathcal {Z}}_{\\gamma }(s) := \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\\\mathfrak {b}^*(s) &{ \\mathfrak {a}^{\\prime }}(s)\\end{pmatrix}$ with $\\mathfrak {a}_{\\mu \\mu }(s)=-\\mathfrak {a}^{\\prime }_{\\mu \\mu }(s)=\\frac{\\mathrm {i}}{\\sqrt{(4-E^2)N}}W(s)$ with the same standard real Brownian motion $W$ for all $\\mu =1,\\ldots ,N$ .",
"All diagonal elements of ${\\mathcal {Z}}$ are thus perfectly correlated.",
"For the off-diagonal elements, $\\mathfrak {a}_{\\mu \\nu }(s)=-\\overline{\\mathfrak {a}_{\\nu \\mu }(s)}&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{++}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu <\\nu \\le N\\\\\\mathfrak {a}^{\\prime }_{\\mu \\nu }(s)=-\\overline{\\mathfrak {a}^{\\prime }_{\\nu \\mu }(s)}&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{--}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu <\\nu \\le N\\\\\\mathfrak {b}_{\\mu \\nu }(s)&=\\frac{1}{\\sqrt{(4-E^2)N}}B^{+-}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu ,\\nu \\le N$ with all the elements of $B^{++},B^{--}$ and $B^{+-}$ standard complex Brownian motions, mutually independent and independent of $W$ .",
"For $\\gamma =0$ , ${\\mathcal {Z}}_0(s) := \\begin{pmatrix} \\mathfrak {a}(s) & \\mathfrak {b}(s) \\\\ \\overline{\\mathfrak {b}(s)} &\\overline{ \\mathfrak {a}(s)}\\end{pmatrix}$ with the definition of $\\mathfrak {a}$ unchanged, but $\\mathfrak {a}^{\\prime }=\\overline{\\mathfrak {a}}$ now, and a symmetric $\\mathfrak {b}$ : $\\mathfrak {b}_{\\mu \\nu }(s)=\\mathfrak {b}_{\\nu \\mu }&=\\frac{1}{\\sqrt{(4-E^2)N}}\\tilde{B}^{+-}_{\\mu \\nu }(s)&\\mbox{ for }&1\\le \\mu \\le \\nu \\le N,$ with the entries of the $\\tilde{B}^{+-}$ independent standard complex Brownian motions again, independent of the elements of $B^{++}$ and $W$ .",
"The result of the scaling limit, $\\lambda \\rightarrow 0$ is summarized in the main theorem.",
"Theorem 8 If $h_1$ and $h_2$ depend on $\\lambda $ so that $h_1(\\lambda ) \\longrightarrow 0\\,,\\qquad h_2(\\lambda ) \\longrightarrow 0, \\qquad \\text{and}\\qquad \\lambda ^{-2}\\mathrm {cha}(\\gamma ,h_1(\\lambda ),h_2(\\lambda ))\\longrightarrow \\infty \\,,$ as $\\lambda \\rightarrow 0$ , then the process $\\left(A^\\lambda \\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right)_{s\\ge 0}$ converges in distribution to the process $\\left({\\mathcal {A}}(s)\\right)_{s\\ge 0}$ on the path space of ${\\mathbb {C}}^{2N\\times 2N}$ -valued processes endowed with Skorhod topology.",
"For $\\gamma \\ge 0$ , $\\left({\\mathcal {A}}(s)\\right)_{s\\ge 0}$ is given as the unique solution for $s\\ge 0$ to $\\begin{split}\\mathrm {d}{\\mathcal {A}}(s)&=\\mathrm {d}{\\mathcal {Z}}_\\gamma (s){\\mathcal {A}}(s)\\\\{\\mathcal {A}}(0)&=1.\\end{split}$ Remark.",
"Existence and uniqueness of ${\\mathcal {A}}$ is a standard result, as all entries of ${\\mathcal {Z}}$ are Brownian motions.",
"Note that no additional moment condition is needed on the random variables $V(x,z)$ .",
"The convergence of the hopping parameters $h_i$ to zero merely brings the resulting process ${\\mathcal {A}}$ into a isotropic form close to the DMPK process.",
"It is however not essential for the scaling limit per se, and the same techniques used in the proof of Theorem REF yield a limiting process for fixed $h_1, h_2\\ne 0$ , but a less isotropic one.",
"For $\\gamma >0$ , ${\\mathcal {Y}}_{\\gamma }(s) := \\begin{pmatrix} \\alpha (s) & \\beta (s) \\\\ \\beta (s)^{*}&{ \\alpha ^{\\prime }}(s)\\end{pmatrix}$ where $\\alpha (s), \\alpha ^{\\prime }(s)$ and $\\beta (s)$ differ from $\\mathfrak {a}(s),\\mathfrak {a}^{\\prime }(s)$ and $\\mathfrak {b}(s)$ only through their covariances, namely by the replacement $\\frac{1}{\\sqrt{(4-E^2)N}} \\quad \\longrightarrow \\quad \\frac{1}{\\sqrt{N \\left|v^+_{\\sigma _1\\nu _1}v^+_{\\sigma _2\\nu _2}\\right|}}\\,.$ For $\\gamma =0$ , ${\\mathcal {Y}}_0(s) := \\begin{pmatrix} \\alpha (s) & \\beta (s) \\\\[1mm] \\overline{\\beta (s)} &\\overline{\\alpha (s)}\\end{pmatrix}$ with the same substitution.",
"Proposition 9 Let $(\\gamma , h_1, h_2)$ be fixed (in particular, not dependent on $\\lambda $ ) and such that Lemma REF holds.",
"As $\\lambda \\rightarrow 0$ , the process $\\left(A_\\lambda \\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right)_{s\\ge 0}$ converges in distribution to $\\left({\\mathcal {G}}(s)\\right)_{s\\ge 0}$ on the path space of ${\\mathbb {C}}^{2N\\times 2N}$ -valued processes endowed with Skorhod topology.",
"$\\left({\\mathcal {G}}(s)\\right)_{s\\ge 0}$ is given as the unique solution for $s\\ge 0$ to $\\begin{split}\\mathrm {d}{\\mathcal {G}}(s)&=\\mathrm {d}{\\mathcal {Y}}(s){\\mathcal {G}}(s) \\\\{\\mathcal {G}}(0)&=1.",
"\\\\\\end{split}$"
],
[
"Discussion",
"Let us comment on the limiting process of transfer matrices ${\\mathcal {A}}(s)$ , and compare it to the ideal ensemble ${\\mathcal {M}}$ of Section .",
"As already discussed in [1], the overall factor $\\sqrt{4-E^2}$ only corresponds to a redefinition of the mean free path and is irrelevant here.",
"The major difference lies in the diagonal of the processes $\\mathfrak {a}, \\mathfrak {a}^{\\prime }$ generating ${\\mathcal {A}}$ , which have perfectly correlated diagonal elements, whereas they are independent of each other in their cousins generating ${\\mathcal {M}}$ .",
"In the case $\\beta =1$ , an additional deviation can be found in the variance of the diagonal elements of $\\mathfrak {b}$ , which are smaller here than in the ideal case by a factor $\\sqrt{2}\\cdot \\sqrt{N/(N+1)}$ .",
"Despite these differences, we have: Corollary 10 If $\\beta =2$ , the law of the process of transmission eigenvalues $(T_k)_{k=1}^N$ induced by ${\\mathcal {A}}$ is the same as that induced by ${\\mathcal {M}}$ .",
"Mathematically, this can be observed in the fact that the SDE for ${\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++}$ , obtained in a straightforward way by Itô calculus, $\\mathrm {d}({\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++} )= {\\mathcal {M}}_{++}^{*}\\mathrm {d}\\mathfrak {b}{\\mathcal {M}}_{-+} + {\\mathcal {M}}_{-+}^{*}\\mathrm {d}\\mathfrak {b}^{*}{\\mathcal {M}}_{++} + {\\mathcal {M}}_{++}^{*}{\\mathcal {M}}_{++} \\mathrm {d}s + {\\mathcal {M}}_{-+}^{*}{\\mathcal {M}}_{-+} \\mathrm {d}s$ does not depend on $\\mathrm {d}\\mathfrak {a}^\\sharp $ , and by recalling that the transmission eigenvalues are directly related to the eigenvalues $\\lambda _k$ of this matrix.",
"For $\\beta =2$ , the limiting $\\mathrm {d}\\mathfrak {b}$ is exactly equal to the ideal one.",
"The underlying physical reason is that the $\\mathfrak {a}^\\sharp $ blocks in the infinitesimal transfer matrix merely change the basis of left-, respectively right-moving channels.",
"As such they do not change the magnitude of either the scattered or the reflected waves, hence they do not contribute to the transmission eigenvalues $T_{k}$ .",
"For the $\\beta =1$ case, the process of transmission eigenvalues induced by ${\\mathcal {A}}$ is different from that induced by ${\\mathcal {M}}$ .",
"Of course one may suspect that in the limit $N \\rightarrow \\infty $ , these processes have similar features, in particular, that the variance of the conductance in the diffusive regime is the same in both cases, and equal to its universal value.",
"In fact in the same limit, we even expect the non-isotropic processes of Proposition REF to share the same property.",
"This remains currently at the level of speculations, the reason being the relatively poor understanding of the properties of the DMPK equation itself in the large $N$ limit.",
"This should be contrasted with recent efforts in the study of hermitian random matrices, where the Gaussian ensembles are very well-known and the challenge is to show that other ensembles share some of their properties, see e.g.",
"[10] and references therein."
],
[
"Symmetry considerations",
"As already mentioned in Section , the driving process ${\\mathcal {L}}$ in the SDE for ${\\mathcal {M}}$ satisfies the invariance property ${\\mathcal {W}}{\\mathcal {L}}{\\mathcal {W}}^{-1} \\mathop {=}\\limits ^{d} {\\mathcal {L}}$ for ${\\mathcal {W}}= \\begin{pmatrix} {\\mathcal {W}}_+ &0 \\\\ 0 &{\\mathcal {W}}_- \\end{pmatrix}$ with ${\\mathcal {W}}_+, {\\mathcal {W}}_-$ arbitrary unitaries in the $\\beta =2$ case and satisfying ${\\mathcal {W}}_-=\\overline{{\\mathcal {W}}_+}$ in the $\\beta =1$ case.",
"In other words, all channel bases are assumed to be equivalent.",
"It is exactly this equivalence that is lost in our model.",
"This can be understood heuristically as follows.",
"Here, the size of the impurities is assumed to be much smaller than the wavelength of the scattered waves, a fact expressed by the $\\delta $ -correlation in space of the potential, i.e.",
"$V=\\sum _y V_y$ with $y=(x,z)$ and ${\\mathbb {E}}(V_yV_{y^{\\prime }})= \\delta _{y,y^{\\prime }}$ .",
"Therefore, the position basis (or its dual, the momentum basis) is naturally singled out.",
"There is no reason to expect another choice of basis to be equivalent and to allow for arbitrary unitaries ${\\mathcal {W}}_+,{\\mathcal {W}}_-$ in (REF ).",
"The symmetry that still survives (but in fact, only so because we performed the additional scaling limit $h_1, h_2 \\rightarrow 0$ ) is the relabeling of channels; indeed the scaling limit $h_1, h_2 \\rightarrow 0$ makes the group velocity of all channels $\\nu $ equal at a given energy $E$ .",
"Up to phases, this corresponds to restricting ${\\mathcal {W}}_+,{\\mathcal {W}}_-$ to be permutation matrices.",
"One easily checks that, with this restriction, the driving process ${\\mathcal {Z}}_\\gamma $ still satisfies (REF ).",
"Another consequence of the locality of the impurities is captured by the following heuristic argument.",
"Let us consider the kernel $S_{p,p^{\\prime }}$ of the scattering matrix in momentum basis, i.e.",
"$p=(k,\\nu )$ .",
"Since the scattering is weak (as already indicated, this is inherent to the setup and it is forced in our model by the $\\lambda \\rightarrow 0$ limit), multiple scattering can be neglected for short slabs of material, hence we can use the Born approximation for $S_{p,p^{\\prime }}$ : $S_{p,p^{\\prime }} = \\delta _{p,p^{\\prime }}-2 \\pi \\mathrm {i}\\lambda \\sum _y (\\tilde{V}_y)_{p,p^{\\prime }} \\delta (E(p)-E(p^{\\prime })) + {\\mathcal {O}}(\\lambda ^2)$ where $(\\tilde{V}_y)=V_ye^{\\mathrm {i}(p-p^{\\prime })y}$ .",
"For $p=p^{\\prime }$ , this expression depends on $p$ only through the factor $\\delta (E(p)-E(p^{\\prime }))$ which contributes a $p$ -dependent group velocity, a dependence which vanishes in the subsequent scaling limit $h_1,h_2 \\rightarrow 0$ .",
"This means that the diagonal elements of the transmission matrix all coincide and this is exactly what we find in the ensemble ${\\mathcal {A}}(s)$ , since in lowest order the $\\mathfrak {a}$ block corresponds to the transmission matrix.",
"Finally, we mention [18], [19] where geometric methods are used to analyze a similar weak coupling limit of the Anderson model on tubes, and contact is made with random matrix theory."
],
[
"Proof of transfer matrix limits",
"We prove Theorem REF .",
"The evolution of $\\left(A^\\lambda (x)\\right)_{x\\ge 0}$ is given by $A^\\lambda (0)=1$ and the stochastic difference equation $A^\\lambda (x)-A^\\lambda (x-1) &= \\left((M^0(x))^{-1} K^{-1}T^\\lambda _x K M^0(x-1)-1\\right) A^\\lambda (x-1) \\nonumber \\\\&=: \\lambda Z_x A^\\lambda (x-1),$ where $Z_x = (M^0(x))^{-1} R_x M^0(x)$ and we defined $\\lambda R_x := K^{-1}T^\\lambda _x K (M^0_x)^{-1} -1 = K^{-1}(T^\\lambda _x-T^0_x) K (M^0_x)^{-1}.$ Recall that $K$ , introduced in the previous section, stands for the change from the position basis to the channel basis.",
"It follows that $R_x = \\Upsilon ^{-1} \\begin{pmatrix}-Q^{*}(P^{*})^{-1}V_xQ & 0 \\\\ 0 & 0\\end{pmatrix}\\Upsilon (M^0_x)^{-1}$ Using the explicit forms of $\\Upsilon , Q$ and $P$ , this matrix reads $R_x=\\mathrm {i}\\begin{pmatrix}\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\Pi \\\\-\\Pi \\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&-\\Pi \\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\Pi \\end{pmatrix},$ which of course satisfies $(R_{++})^{*}= -R_{++}$ , $(R_{--})^{*}= -R_{--}$ and $(R_{-+})^{*}= R_{+-}$ .",
"In the time reversal invariant case, since $Q=Q^T,\\qquad Q\\Pi =Q^*,\\qquad \\Pi Q^*=Q$ this simplifies to $R_x=\\mathrm {i}\\begin{pmatrix}\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&\\frac{1}{\\sqrt{\\left|v^+\\right|}}Q^*V_xQ^*\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\\\-\\frac{1}{\\sqrt{\\left|v^+\\right|}}QV_xQ\\frac{1}{\\sqrt{\\left|v^+\\right|}}&-\\frac{1}{\\sqrt{\\left|v^+\\right|}}QV_xQ^*\\frac{1}{\\sqrt{\\left|v^+\\right|}}\\end{pmatrix}\\,.$ Since $Q^*=\\overline{Q}$ , we also have that $R_{--} =\\overline{ R_{++}}$ , and in a similar fashion $R_{-+} = \\overline{R_{+-}}$ .",
"Before we go further into the proof, let us explain the heuristics of the convergence to the DMPK equation.",
"The matrix $R_x$ contains the $N$ i.i.d random variables $V(x,z)$ , $z=1,\\ldots ,N$ , and $R_x,R_y$ are independent for $x\\ne y$ .",
"Under the appropriate technical conditions, we have convergence, as $ \\lambda \\rightarrow 0$ ; $\\lambda \\sum _{x=0}^{\\lfloor \\lambda ^{-2}s\\rfloor }R_x \\stackrel{\\mathrm {d}}{\\longrightarrow }{\\mathcal {R}}(s)\\,.$ where the $V(x,z)$ in $R_x$ are replaced by Brownian motions $B_z(s)$ in ${\\mathcal {R}}(s)$ .",
"Of course, this is nothing else than the convergence of a random walk to Brownian motion.",
"Now, the matrices $Z_x$ generating the discrete process $A(x)$ do contain the highly oscillating phases of $M^0(x)$ .",
"Let $Z^\\lambda (s) = \\lambda \\sum _{x=0}^{\\lfloor \\lambda ^{-2}s\\rfloor }Z_x\\,.$ The correlation of any two matrix elements reads ${\\mathbb {E}}\\left[\\left(Z^\\lambda (s)\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right] \\\\=\\lambda ^{2}\\sum _{x=1}^{\\lfloor \\lambda ^{-2}s\\rfloor }\\exp \\left(\\mathrm {i}x\\left(-k^{\\sigma _m}_{\\sigma _m\\nu _m}+k^{\\sigma _n}_{\\sigma _n\\nu _n}-k^{\\sigma _p}_{\\sigma _p\\nu _p}+k^{\\sigma _r}_{\\sigma _r\\nu _r}\\right)\\right){\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right].$ where the $\\sigma $ 's and $\\nu $ 's denote the block and position in the block of a certain element, the `physical' momentum, due to the permutation $\\Pi $ is not $\\nu $ , but $\\sigma \\nu $ .",
"The expectation in the r.h.s is independent of $x$ , so that this sum is highly oscillatory and formally converges to a $\\delta $ function.",
"Hence, the phases create a limiting process ${\\mathcal {Z}}(s)$ with almost completely uncorrelated entries, apart from the exceptional conditions of Lemma REF .",
"As a result, the number of independent random variables in ${\\mathcal {Z}}(s)$ is of order $N^2$ , whereas it was only $N$ in ${\\mathcal {R}}(s)$ , a phenomenon called `noise explosion' in [20].",
"Precisely, we prove: Lemma 11 Under the conditions of Theorem REF and in the same topology, the process $\\left(Z^\\lambda (s)\\right)_{s\\ge 0}$ converges in distribution to the process $\\left({\\mathcal {Z}}_\\gamma (s)\\right)_{s\\ge 0}$ , for $\\gamma \\ge 0$ .",
"We first recall that $Q$ is the matrix of transversal plane waves so that $(Q^{*}V_x Q)_{\\mu \\nu } = \\widehat{V_x}(\\mu -\\nu )\\,.$ In particular, all elements on the diagonal are perfectly correlated.",
"Now, we consider (REF ) for $s\\ge 0$ , and note that the wavevectors $k_\\nu ^\\sigma $ and the matrices $R_x$ all depend on $\\lambda $ implicitly through the dependence $h_i(\\lambda )$ .",
"However, ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right]$ is independent of $x$ and can be taken out of the sum.",
"For all choices $\\lbrace (\\nu _i,\\sigma _i): i=m,n,p,r\\rbrace $ for which the exponent does not vanish, we have that $\\left| \\lambda ^{2}\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\exp \\left(ix\\left(-k^{\\sigma _m}_{\\sigma _m\\nu _m}+k^{\\sigma _n}_{\\sigma _n\\nu _n}-k^{\\sigma _p}_{\\sigma _p\\nu _p}+k^{\\sigma _r}_{\\sigma _r\\nu _r}\\right)\\right)\\right|\\le \\frac{2\\lambda ^2}{\\mathrm {cha}(\\lambda )}\\longrightarrow 0,$ as $\\lambda \\rightarrow 0$ , by Assumption (REF ).",
"By Lemma REF , the the wavenumbers cancel out whenever $\\left(\\sigma _m,\\nu _m\\right)=\\left(\\sigma _n,\\nu _n\\right)\\mbox{ and }\\left(\\sigma _p,\\nu _p\\right)=\\left(\\sigma _r,\\nu _r\\right)\\,,$ corresponding to any two diagonal elements in $Z^\\lambda $ , or $\\left(\\sigma _m,\\nu _m\\right)=\\left(\\sigma _r,\\nu _r\\right)\\mbox{ and }\\left(\\sigma _p,\\nu _p\\right)=\\left(\\sigma _n,\\nu _n\\right)\\,,$ namely two mutually `transpose' entries of $Z^\\lambda $ .",
"In the case $\\gamma =0$ , a last possibility is given by $\\left(\\sigma _m,\\nu _m\\right)=\\left(-\\sigma _p,\\nu _p\\right)\\mbox{ and }\\left(\\sigma _n,\\nu _n\\right)=\\left(-\\sigma _r,\\nu _r\\right)\\,,$ corresponding to two elements in the same position, but within the opposite blocks.",
"In these three cases and for all $\\lambda $ , $\\lambda ^{2}\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }1=s.$ Furthermore, using the explicit form (REF ) of $R_x$ and the definition of $Q$ , ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right]=-\\frac{\\sigma _m\\sigma _p}{N^2\\sqrt{\\left|v^+_{\\sigma _m\\nu _m}v^+_{\\sigma _n\\nu _n}v^+_{\\sigma _p\\nu _p}v^+_{\\sigma _r\\nu _r}\\right|}} \\\\\\cdot \\sum _{z=0}^{N-1}\\exp \\left(\\frac{2\\pi i}{N}z\\left(-\\sigma _m\\nu _m+\\sigma _n\\nu _n-\\sigma _p\\nu _p+\\sigma _r\\nu _r\\right)\\right)$ which we only need to evaluate in the `stationary phase' situations (REF ), (REF ), and (REF ).",
"In all these cases, ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right] = -\\frac{\\sigma _m\\sigma _p}{N\\sqrt{\\left|v^+_{\\sigma _m\\nu _m}v^+_{\\sigma _n\\nu _n}v^+_{\\sigma _p\\nu _p}v^+_{\\sigma _r\\nu _r}\\right|}}.$ Note that the $v^+_{\\nu }$ still depend on $\\lambda $ and $\\nu $ , but by (REF ), they converge to the $\\nu $ -independent limit $|2\\sin (k)|$ with $2\\cos (k)=E$ .",
"Hence, in the limit $\\lambda \\rightarrow 0$ , we have ${\\mathbb {E}}\\left[(R_x)_{mn}(R_x)_{pr}\\right] \\longrightarrow -\\frac{\\sigma _m\\sigma _p}{(4-E^2)N}\\,.$ A very similar oscillatory sum appears in ${\\mathbb {E}}\\left[\\left(Z^\\lambda (s)^{*}\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right]$ , with parallel conclusions.",
"In summary, for all $s\\ge 0$ $\\begin{split}\\lim _{\\lambda \\rightarrow 0}{\\mathbb {E}}\\left[\\left(Z^\\lambda (s)\\right)_{mn}\\left(Z^\\lambda (s)\\right)_{pr}\\right]&=\\int _0^s\\mathrm {d}\\left\\langle \\left({\\mathcal {Z}}\\right)_{mn}, \\left({\\mathcal {Z}}\\right)_{pr}\\right\\rangle _t,\\\\\\lim _{\\lambda \\rightarrow 0}{\\mathbb {E}}\\left[\\left(Z^\\lambda (s)^{*}\\right)_{mn}, \\left(Z^\\lambda (s)\\right)_{pr}\\right]&=\\int _0^s\\mathrm {d}\\left\\langle \\left({\\mathcal {Z}}^*\\right)_{mn},\\left({\\mathcal {Z}}\\right)_{pr}\\right\\rangle _t.\\end{split}$ with ${\\mathcal {Z}}$ given in the previous section, and $\\langle M,N \\rangle _t$ the bracket process of two martingales $M,N$ .",
"In particular, the block structure of ${\\mathcal {Z}}$ arises from the corresponding relations noted above in $R_x$ .",
"Moreover, perfect correlation of the diagonal elements of $Q^{*}V_x Q$ and the exceptional case (REF ) imply the perfect correlation of the diagonal elements of $\\mathrm {d}\\mathfrak {a}$ and $\\mathrm {d}\\mathfrak {a}^{\\prime }$ .",
"All other exceptional cases impose the correlations $\\vert \\mathrm {d}\\mathfrak {b}_{\\mu \\nu }\\vert ^2$ and $\\vert \\mathrm {d}\\mathfrak {a}_{\\mu \\nu }\\vert ^2$ .",
"Now the lemma follows as a simple generalization of Donsker's invariance principle, for example by using Chapter VII, Theorem 3.7 in [13].",
"They check the convergence of three characteristics, of which, in their notation, $\\left[\\sup -\\beta _3^{\\prime }\\right]$ is trivially fulfilled as $Z^\\lambda $ and ${\\mathcal {Z}}$ are martingales, $\\left[\\gamma _3^{\\prime }-{\\mathbb {R}}^+\\right]$ is the convergence of brackets as shown in (REF ), and $\\left[\\delta _{3,1}-{\\mathbb {R}}^+\\right]$ is a simple estimate on the jumps of $Z^\\lambda $ , namely $\\lim _{\\lambda \\rightarrow 0}\\sum _{x=1}^{\\lfloor \\lambda ^{-2}s\\rfloor }\\mathbb {E}\\left(g_a\\left(\\lambda \\left\\Vert R_x\\right\\Vert \\right)\\right)=0$ for all $g_a(y)=y^21_{\\lbrace |y|>a\\rbrace }$ , $a>0$ , which is trivial by $\\mathbb {E}\\left(v^2\\right)=1$ .",
"Notice that the covariances of the less isotropic ensemble of Proposition REF can be read off from this proof.",
"The main theorem will now follow from this lemma and the difference equation (REF ).",
"We simplify the notation of (REF ) and (REF ) by writing the real and imaginary parts of the matrix entries as elements of vectors in ${\\mathbb {R}}^d$ , $d=8N^2$ .",
"In this notation (REF ) reads $X^{\\lambda }_j(y)-X^{\\lambda }_j(y-1)=\\lambda \\sum _{k=1}^d\\xi ^{\\lambda }_{jk}(y)X^{\\lambda }_k(y-1)$ for all $y$ in ${\\mathbb {N}}$ , with $\\xi ^{\\lambda }(y)\\in {\\mathbb {R}}^{d\\times d}$ independent of $X^{\\lambda }(z), \\xi ^{\\lambda }(z)$ $z\\in \\lbrace 0,...,y-1\\rbrace $ .",
"Because of the phase factors, the law of $\\xi ^{\\lambda }(y)$ is not independent of $y\\in {\\mathbb {N}}$ , but ${\\mathbb {E}}(\\xi ^{\\lambda }(y))=0$ and $ \\Vert \\xi ^{\\lambda }(y)\\Vert ^2 \\le c^{\\prime } (\\sum _{z=1}^N |v(y,z) |)^2 \\le c \\sum _{z=1}^N v^2(y,z)$ where $c,c^{\\prime }< \\infty $ are $\\lambda $ -independent for sufficiently small $\\lambda $ , but they depend on $N$ .",
"This follows from (REF ) or (REF ) by noting that the velocity matrix $v^+$ converges to a nonsingular limit as $\\lambda \\rightarrow 0$ ; we will henceforth assume without comment that $\\lambda $ is sufficiently small.",
"Furthermore, we know from the proof of Lemma REF $\\lim _{\\lambda \\rightarrow 0}\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)=C_{ikjl}\\cdot s$ uniformly for $s\\ge 0$ from bounded intervals.",
"If we define $\\left({\\mathcal {B}}(s)\\right)_{s\\ge 0}$ as the matrix-valued Brownian motion with bracket process $\\left\\langle {\\mathcal {B}}_{ik},{\\mathcal {B}}_{jl}\\right\\rangle _s= C_{ikjl}\\cdot s,$ the equation (REF ) transforms to $\\mathrm {d}{\\mathcal {X}}_j(s)=\\sum _{k=1}^d\\mathrm {d}{\\mathcal {B}}_{jk}{\\mathcal {X}}_k(s).$ The initial values for $X^{\\lambda }$ and ${\\mathcal {X}}$ are identical and deterministic, the ${\\mathbb {R}}^d$ vector corresponding to the unit matrix $1_{2N}$ .",
"For notational convenience, we have chosen $X^{\\lambda }$ to still live on the microscopic, discrete space, what we really want to investigate is the cadlag process $\\overline{X}^{\\lambda }(s)=X^{\\lambda }\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right).$ For cadlag processes we define $\\overline{X}^{\\lambda }(s-)$ as the leftside limit of $\\overline{X}^{\\lambda }$ at $s$ .",
"With the filtration ${\\mathcal {F}}^{\\lambda }_s=\\sigma \\left\\lbrace \\overline{X}^{\\lambda }(t):t\\le s\\right\\rbrace $ , $\\left(\\overline{X}^{\\lambda }(s)\\right)_{s\\ge 0}$ is a $\\left\\lbrace {\\mathcal {F}}^{\\lambda }_s \\right\\rbrace $ -martingale.",
"Furthermore defining $V^{\\lambda }_{ij}(y)=\\lambda ^2\\sum _{x=1}^y\\sum _{k,l=1}^d\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)X^{\\lambda }_k(x-1)X^{\\lambda }_l(x-1)$ and the corresponding macroscopic $\\overline{V}^{\\lambda }_{ij}(s)=V^{\\lambda }\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)$ for all $i,j=1,...,d$ , the process $\\overline{X}^{\\lambda }_i\\overline{X}^{\\lambda }_j-\\overline{V}^{\\lambda }_{ij}$ is a $\\left\\lbrace {\\mathcal {F}}^{\\lambda }_s \\right\\rbrace $ -martingale as well, and we have by Theorem 7.4.1, [11], Lemma 12 If for any $T>0$ , and any stopping time $T^{\\lambda }_r=\\inf \\left\\lbrace s:\\left|\\overline{X}^{\\lambda }(s)\\right|\\ge r \\mbox{ or } \\left|\\overline{X}^{\\lambda }(s-)\\right|\\ge r\\right\\rbrace ,$ $r>0$ , for all $i,j=1,...,d$ $&\\lim _{\\lambda \\rightarrow 0}\\mathbb {E}\\left(\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{X}^{\\lambda }(s)-\\overline{X}^{\\lambda }(s-)\\right|^2\\right)=0\\\\&\\lim _{\\lambda \\rightarrow 0}\\mathbb {E}\\left(\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\overline{V}^{\\lambda }_{ij}(s-)\\right|\\right)=0$ and $\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\int _0^s\\mathrm {d}t\\sum _{k,l=1}^dC_{ikjl}\\overline{X}^{\\lambda }_k(t)\\overline{X}^{\\lambda }_l(t)\\right|\\stackrel{{\\mathbb {P}}}{\\rightarrow }0,$ then $\\left(\\overline{X}^{\\lambda }(s)\\right)_{s\\ge 0}$ converges in distribution on $D_{{\\mathbb {R}}^d}[0,\\infty )$ to $\\left({\\mathcal {X}}(s)\\right)_{s\\ge 0}$ .",
"So to prove Theorem REF , we only have to verify the conditions of Lemma REF .",
"We start with the following observation, Lemma 13 Let $Z_k$ , $k\\in {\\mathbb {N}}$ be i.i.d.",
"distributed, positive random variables, with $\\mathbb {E}\\left(Z_1\\right)<\\infty .$ Then $\\frac{1}{n}\\mathbb {E}\\left(\\max _{1\\le k\\le n}Z_k\\right)\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"With $q(x)={\\mathbb {P}}\\left(Z_1\\ge x\\right)$ for $x\\ge 0$ , we have $\\int _0^\\infty q(x)\\mathrm {d}x=\\mathbb {E}\\left(Z_1\\right),$ while ${\\mathbb {P}}\\left(\\max _{1\\le k\\le n}Z_k\\ge x\\right)=1-(1-q(x))^n\\le n q(x).$ Thus, $\\frac{1}{n}\\mathbb {E}\\left(\\max _{1\\le k\\le n}Z_k\\right)=\\int _0^\\infty \\frac{1-(1-q(x))^n}{n}\\mathrm {d}x$ with the integrand on the right side converging to zero and dominated by the integrable $q(x)$ , an the claim follows by dominated convergence.",
"For any $T>0, r>0$ given, we have $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\overline{X}^{\\lambda }(s)-\\overline{X}^{\\lambda }(s-)\\right|^2\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|X^{\\lambda }(y)-X^{\\lambda }(y-1)\\right|^2\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\lambda ^2\\sum _{j,k,l=1}^d\\xi ^{\\lambda }_{jk}(y)\\xi ^{\\lambda }_{jl}(y)X^{\\lambda }_{k}(y-1)X^{\\lambda }_{l}(y-1)\\\\\\le &r^2\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor }\\left\\Vert \\xi ^{\\lambda }(y)\\right\\Vert ^2\\\\\\le &c(\\lambda )r^2\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor }\\sum _{z=0}^{N-1}v(y,z)^2,\\end{split}$ where we used (REF ) in the last line.",
"Now the last line of (REF ) vanishes in expectation by Lemma REF and the fact that ${\\mathbb {E}}(v(y,z)^2)=1$ .",
"This proves (REF ).",
"For the proof of (), note $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\overline{V}^{\\lambda }_{ij}(s)-\\overline{V}^{\\lambda }_{ij}(s-)\\right|\\\\=&\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|V^{\\lambda }_{ij}(y)-V^{\\lambda }_{ij}(y-1)\\right|\\\\=&\\lambda ^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} \\left(T\\wedge T^{\\lambda }_r\\right)\\right\\rfloor }\\left|\\sum _{k,l=1}^d\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right|\\\\\\le &\\lambda ^2r^2\\max _{1\\le y\\le \\left\\lfloor \\lambda ^{-2} T\\right\\rfloor }\\mathbb {E}\\left(\\left\\Vert \\xi ^{\\lambda }(y)\\right\\Vert ^2\\right),\\end{split}$ which obviously vanishes in expectation as $\\lambda \\rightarrow 0$ .",
"For (REF ), start with $\\begin{split}\\overline{V}^{\\lambda }_{ij}(s)&-\\int _0^s\\mathrm {d}t\\sum _{k,l=1}^dC_{i,k,j,l}\\overline{X}^{\\lambda }_k(t)\\overline{X}^{\\lambda }_l(t)\\\\=&\\sum _{k,l=1}^d\\left(\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right)\\\\&\\hspace{56.9055pt}-\\left(s-\\lambda ^2\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{k,l=1}^dC_{ikjl}X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right).\\end{split}$ For the last line, we have $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\left(s-\\lambda ^2\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{k,l=1}^dC_{ikjl}X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right|\\\\&\\le \\lambda ^2r^2d^2\\max _{k,l}\\left|C_{ikjl}\\right|\\rightarrow 0\\end{split}$ almost surely, and thus in probability, as $\\lambda \\rightarrow 0$ .",
"We omit the (finite) $k,l$ sum in (REF ) from our notation, and use partial summation with respect to $y$ to obtain $\\begin{split}\\lambda ^2&\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\\\&=\\lambda ^2X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)\\\\&\\hspace{5.69054pt}-\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\sum _{x=1}^{y}\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)-C_{ikjl}\\right)\\left(X^{\\lambda }_k(y)X^{\\lambda }_l(y)-X^{\\lambda }_k(y-1)X^{\\lambda }_l(y-1)\\right).\\end{split}$ We know from the convergences (REF ) and $\\lim _{\\lambda \\rightarrow 0}\\lambda ^2\\sum _{x=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }C_{ikjl}=C_{ikjl}\\cdot s,$ which are both uniform for $s$ from compact sets, that $a^{\\lambda }_{ikjl}(y):=\\lambda ^2\\sum _{x=1}^{y}\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(x)\\xi _{jl}^{\\lambda }(x)\\right)-C_{ikjl}\\right)\\rightarrow 0$ as $\\lambda \\rightarrow 0$ uniformly in $y$ as long as $1\\le y\\le \\left\\lfloor \\lambda ^{-2}T\\right\\rfloor $ for fixed positive $T$ .",
"Thus for the first term on the right-hand side in (REF ), $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}&\\left|\\lambda ^2X^{\\lambda }_k\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)X^{\\lambda }_l\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }\\left(\\mathbb {E}\\left(\\xi _{ik}^{\\lambda }(y)\\xi _{jl}^{\\lambda }(y)\\right)-C_{ikjl}\\right)\\right|\\\\&\\le r^2\\sup _{s\\le T}\\max _{i,k,j,l}\\left|a^{\\lambda }_{ikjl}\\left(\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor \\right)\\right|\\rightarrow 0\\end{split}$ as $\\lambda \\rightarrow 0$ .",
"After plugging (REF ) into the second term on the right-hand side of (REF ), we are left with the sum of $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\lambda \\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }a^{\\lambda }_{ikjl}(y)\\left(\\sum _{k^{\\prime }=1}^d \\xi ^{\\lambda }_{kk^{\\prime }}(y)X^{\\lambda }_{k^{\\prime }}(y-1)X^{\\lambda }_l(y-1)\\right.\\right.\\\\+\\left.\\left.\\sum _{l^{\\prime }=1}^d \\xi ^{\\lambda }_{ll^{\\prime }}(y)X^{\\lambda }_{k}(y-1)X^{\\lambda }_{l^{\\prime }}(y-1)\\right)\\vphantom{\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }}\\right|\\end{split}$ and $\\begin{split}\\sup _{s\\le T\\wedge T^{\\lambda }_r}\\left|\\lambda ^2\\sum _{y=1}^{\\left\\lfloor \\lambda ^{-2}s\\right\\rfloor }a^{\\lambda }_{ikjl}(y)\\left(\\sum _{k^{\\prime },l^{\\prime }=1}^d \\xi ^{\\lambda }_{kk^{\\prime }}(y)\\xi ^{\\lambda }_{ll^{\\prime }}(y)X^{\\lambda }_{k^{\\prime }}(y-1)X^{\\lambda }_{l^{\\prime }}(y-1)\\right)\\right|\\\\\\end{split}$ converging to zero in $L^2({\\mathbb {P}})$ and $L^1({\\mathbb {P}})$ , respectively."
],
[
"Acknowledgements",
"Maximilian Butz benefited a lot from discussions with members of Antti Kupiainen's group at Helsinki University, and is grateful for financial support by the Academy of Finland during his stay there.",
"Sven Bachmann gratefully acknowledges the support of the National Science Foundation under Grant #DMS-0757581"
]
] | 1204.0823 |
Subsets and Splits